2004:17

D O C T O R A L T H E S I S L

Anisotropy and Non-linear Effects in SMC Composites From Material Data to FE-simulation of Structures

Luleå University of Technology Department of Applied Physics and Mechanical Engineering, Division of Polymer Engineering

2004:17 I issn: 1402-15441 isrn: ltu-dt— 04/17 — se

L U L E A •

U N I V E R S I T Y

O F T E C H N O L O G Y

Anisotropy and non-linear effects i n SMC composites

From material data to FE-simulation of structures

M . Oldenbo

Volvo Car Corporation, Exterior Engineering Dept. 93610, PV3C2

SE-405 31 GÖTEBORG, SWEDEN E-mail: moldenbo@volvocars.com

And

Division of Polymer Engineering Department of Applied Physics and Mechanical Engineering

Luleå University of Technology SE-971 87 Luleå, Sweden

Maj 2004

Abstract

In the design of composite structures for the automotive industry it is vital to have good tools because development time is very critical. Normally simulation of mechanical properties of sheet moulding compound (SMC) composite structures is limited to stiffness based on an isotropic material model. Design guidelines and the designers experience are common tools to estimate that the strength of the part is sufficient. With better design tools an optimal design could be expected to be found more quickly, giving improved cost efficiency and lower weight.

The research question in the present thesis is therefore dealing with advanced simulation use in design of SMC composite structures. The approach has been to do (a) material analysis of two different types of SMC using mechanical tests and in-situ microscopy, and (b) based on these results, develop material models and implement them in a commercial FE-solver.

A typical effect in SMC structures normally not considered when simulating the stiffness is anisotropy due to production of the prepreg or during the moulding of the structure. In the present thesis (Paper B), it is shown how to include this effect with a material model based on micromechanics. The model determines the local stiffness as function of fiber orientation distribution and is validated for a SMC with 30 weight-% glass fibers and 45 weight-% CaCU3 filler. Further on, the model has been successfully implemented in the FE software ABAQUS through a subroutine. The fiber orientation distribution is described with a second order fiber orientation tensor.

In all SMCs studied, significant modulus reductions were observed with increasing strain due to extensive damage. A critical modulus reduction is suggested as a failure criterion rather than strength. Simulation not only of stiffness, but also to use a design prerequisite for allowed load is an interesting new approach to improve the accuracy of the design. In doing this i t is important to have an accurate material model. Viscoelastic effects has been studied in cyclic loading test and creep test. A material model that considers the SMC composite as linear-viscoelastic material with evolving damage has been suggested to explain the non-linear stress-strain behaviour and the observed damage accumulation with increasing stress levels. I t is seen that almost nothing has been done on short fiber composites with random fiber distribution considering models for both viscoelasticity and damage evolution. The simplifying assumption in the model presented here is that damage development may be considered as an elastic process and hence depends only on the maximum stress experienced by the material. This allows for damage quantification in terms of stiffness reduction in quasi-static tensile loading-unloading tests. Then the time- and damage-dependent viscoelastic functions of the composite are described as a product of damage- and time-dependent terms, where the time-dependence of the viscoelastic behaviour is described by

i

creep compliance functions of undamaged composite. Hence, the damage-dependent term serves as a scaling factor.

An incremental formulation of the non-linear model usable in FE-simulation is derived, presented and implemented in FE-programme ABAQUS. Application to constant strain rate tensile test comparing analytical and FE result prove accuracy of the formulation and the subroutine.

Finally a stiffness reduction model for SMC composites with evolving damage is suggested and validated for a standard SMC material.

Preface

The work presented in this thesis has been carried out at Volvo Car Corporation AB (VCC) in collaboration with Luleå University of Technology (LTU) during the period of July 1999 until Maj 2004. I t has been financed by VCC and VINNOVA in Sweden thru the Swedish Vehicle Research Program (PFF), Grants no 2001-02546 and 2002-01578. Furthermore, this research student project has been a part of the clusters Integral Vehicle Structure Research school (IVS) and VCC's post graduate program.

There are numerous people who deserve my gratitude for their valuable help in performing this thesis. First of all I like to thank my main supervisors Prof. Lars Berglund and Prof. Janis Varna for their contribution and commitment to this work. I would also like to thank my industrial advisor Dr. Per Bengtsson-Meuller as well as my colleagues at exterior engineering department at Volvo, Mr Staffan Ek and Mr Peter Porsgaard. I like to thank my manager at Volvo, Mr Stefan Johansson-Tingström who has granted my endless travelling to Luleå. Materials centre, VCC, is greatly acknowledged for allowing a person from engineering department to do tests in their facilities. A lot of interesting collaboration has been done with the Swedish institute for composites AB (SICOMP) and especially I like to thank Dr Anders Holmberg, Dr Magnus Svanberg and Dr Patrik Fernberg for f ru i t fu l discussions and a great helping hand. I like to thank my industrial colleagues that have helped me understand the production of composites as well as being an invaluable source of test material and information. I 'm thinking here of Mr Jim-Einar Karlsen at Reichhold, Mr Gunter Torn at Peguform, Mr Philippe Coudron at Inoplast, Mr Christer Lundemo at Scania and Mr Hans Bernlind at Polytec Composites. Finally I like to thank all my colleagues at LTU whom I have shared many good times with during these years.

Göteborg, Maj 2004

Magnus Oldenbo

i i i

List of publications

The thesis comprises a introduction and the following appended papers:

A. Oldenbo M, Fernberg SP and Berglund LA. Mechanical behavior of SMC composites with toughening and low density additives, Composites part A 2003; 34:875-885

B. Oldenbo M , Mattsson D, Varna J and Berglund LA. Global stiffness of a SMC panel considering process induced fibre orientation, Journal of Reinforced Plastics and Composites 2004; 23:37-49

C. Oldenbo M, Varna J. A constitutive model for non-linear behaviour of SMC accounting for linear viscoelasticity and micro damage. Submitted to Polymer Composites.

D. Oldenbo M, Varna J. A 2D constitutive model for FE-simulations of SMC composite accounting for linear viscoelasticity and damage development. Submitted to International Journal for Numerical Methods in Engineering

E. Oldenbo M, Lundmark P, Varna J. Stiffness reduction model for SMC composites with evolving damage. Submitted to Composites Science and Technology

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Other publications by the author of relevance to this thesis

M . Oldenbo, TPO-nanocompositesfor automotive exterior bodypanels -Potential and experiences from evaluations of commercial materials, In: Proceedings of Nanocomposites 2001, Renaissance Chicago Hotel, Chicago, IL , USA, June 25-27, 2001.

M . Oldenbo, Constitutive model for non-linear behaviour of SMC, In : Proceedings of 2003 Annual Technical Conference of the Society of Plastics Engineers (ANTEC'03), Nashville, Tennessee, USA, May 4-8, 2003.

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Contents

Abstract i Preface i i i List of publications v 1 Introduction to research fields and contribution of this thesis l

1.1 Background and Research Question 1 1.2 Approach and Contributions 2

2 SMC Composite material and industrial use 3 2.1 Origin and development of SMC material 3

2.2 Processing 4

2.3 Anisotropy 4

2.4 Composition of SMC 5

2.4.1 Standard SMC 5 2.4.2 Low Density SMC 6

2.5 Mechanical properties 7

2.6 Importance of SMC material for the automotive industry 8

3 Damage evolution with increasing strain 9

3.1 Damage development in SMC composite - micro level 9

3.2 Damage development in SMC composite - macro level 10

3.3 Coupling micro damage with macro damage 10

4 Material models for SMC 12

4.1 Elastic material model - dependent on fiber orientation 13

4.2 In-elastic material model - use of damageable and viscoelastic

material model 14 4.2.1 Elastic-damageable material model for SMC 14 4.2.2 Adding viscoelastic effects 15 4.2.3 Viscoelastic-damageable material model 15

5 Implementation of material models in FE-solver 17 6 Case studies 18

6.1 Elastic deformations in a hood structure 18

6.2 In-elastic deformation and damage development in bending test

specimen 19 7 Concluding remarks 19 8 Future work 20 References 21

Paper A Paper B Paper C Paper D Paper E

v i i

l Introduction to research fields and contribution of this thesis

l.i Background and Research Question

It is seen in Figure l that the use of polymers in the automotive industry has increased continuously since the introduction of these materials. Polymers used for cars today include thermoplastics for panels, consoles, pipes etc, and thermosets (polyurethane foam) in seats. Glass mat reinforced thermoplastics (GMT) is used in seat frames and similar applications. Recently Volvo Cars introduced a tailgate made mainly in sheet moulding compound composite (SMC) on a high volume car; the 1999 Volvo V70 (Annual production of 120.000 in 2003). I n 2002 the Volvo XC90 was introduced with SMC in the upper part of the divided tailgate. SMC is a material usable in semi-structural parts, such as doors, lids etc, and it is interesting to hear from the co-manager of BMW material department, Dr. Stauber, that the increase of polymers in the future is expected mainly in those parts [1].

Figure 1. Historic use of polymers (plastics + rubber, excl. tires) in Volvo models. Source: Volvo Car Corporation.

In the design of composite structures for the automotive industry i t is vital to have good tools because development time is very critical. Normally FE-simulation of mechanical properties of sheet moulding compound (SMC) composite structures is limited to stiffness based on an isotropic material model. Design guidelines and the designers experience are common tools to

1

estimate that the strength of the part is sufficient. With better design tools an optimal design could be expected to be found more quickly, giving improved cost efficiency and lower weight. Simulation not only of stiffness, but also strength is an interesting new approach to improve the accuracy of the design. In doing this it is important to have an accurate material model.

As a conclusion, the research question in the present thesis is advanced simulation in design of SMC composite structures.

1.2 Approach and Contributions

The focus of the present work has been in materials modelling and application of those models to SMC materials and structures. The approach has been to do a) material analysis of two different types of SMCs using mechanical tests and in-situ microscopy, and b) based on these results develop material models and implement them in a commercial FE-programme. The material models are based on micromechanics for stiffness calculation and continuum damage mechanics combined with linear viscoelasticity for non-linear simulations. Other research fields entered includes material science (analysing different SMC materials) and structural mechanics (case studies of developed and implemented models).

An interesting finding in Paper A in this thesis was the extent of flow induced anisotropy in SMC panels. A new methodology to calculate the influence of anisotropy on stiffness is proposed in Paper B. Existing micromechanical models is applied to SMC composite and averaging over the fiber orientation distribution is performed. The model is also implemented in a FE-programme and tested on a hood structure with known state of fiber orientation (data is taken from a mould filling simulation).

Another result presented in Paper A was an interesting possibility for a design prerequisite for strength of a SMC panel. I t was suggested to f ind an acceptable damage level as determined by stiffness reduction. Paper C includes developmental work of a model describing SMC material as linear viscoelastic with evolving damage. In Paper D an incremental formulation of this model is presented and implemention of the model in FE-programme ABAQUS is described. The model is validated in a bending test.

Paper E deals with the influence of evolving crack density on stiffness reduction in SMC composites. The crack density is described as function of fiber orientation to applied load and strain level using fabricated cross-ply laminates with bundle structure. A model based on micromechanics and laminate theory is applied to SMC through use of a theoretical quasi-isotropic laminate.

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2 SMC Composite material and industrial use

2.1 Origin and development of SMC material

An early application of SMC with high demands was the 1969 Chevrolet Corvette body panels including complete doors. SMC replaced a hand lay-up polyester glass fiber composite and the obvious advantage of the new material was the fast cycle times using combined compression moulding and curing in hot tools. The stamping process to make SMC panels out of SMC prepreg is for composites very efficient and the total cycle time including charging, forming, curing and removal of the part produced is just a few minutes. SMC is still used by Corvette today. Even though SMC material has been used from the 60s the research papers on this subject are rare until mid eighties. Early work is about material properties obtained from tests or stiffness comparisons between SMC and sheet metal panels. In 1979, at the centre for composite materials in university of Delaware, Taggart et al. did an analysis of three SMC materials with 25 weight-%, 30 weight-% and 65 weight-% fibers as well as two materials with continuous fibers [2].

The continuous improvement of the material and related process concerning panel aesthetics, processing and weight governs growth in the automotive sector and it is found, by example, on many tailgates where the increased design freedom compared to sheet metal is used. This is explained more in section 2.6. Today SMC stands for nearly 40% of the total volume of fiber reinforced thermoset composites [3] worldwide and over 25% in Europe which equals 260 kton [4]. Traditionally, composites have been used for low-volume production due to the low investment needed compared to sheet metal stamping. However as previously discussed, by annual production volume, SMC is competitive in between traditional composites, such as resin transfer moulding (RTM), and sheet metal. This is due to the investment part - part price relation of using this technology. Even though the use of SMC in the automotive industry is wide spread, the most SMC is still used in the electronics industry.

SMC is a type of short fiber composite (SFC). SFC differs from continuous fiber composites (CFC) used in well known processes such as Prepreg and RTM in the way that the fibers are chopped and spread initially randomly in the material. Other SFC include chopped strand mat (CSM) and injection moulded fiber composites. Related processes and materials are Dough moulding compound (DMC) and Bulk moulding compound (BMC) which means hot pressing of injected material with less amount of fibers [5]. BMC actually has longer history than SMC and dates back to 1954 [6]. Glass mat reinforced thermoplastic (GMT) and long fiber technology (LFT) is related in the sense that i t is a stamped material with relative similar and high modulus.

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2.2 Processing

SMC differs to other SFCs in the way that it is first produced as a prepreg in a SMC-machine. This machine produces a roll of SMC-mat containing a layer of chopped roving surrounded by two layers of paste. The paste contains filler and polyester resin that is to be cut in suitable pieces before moulding, see for example [5,7]. Before compression moulding, the material is left for a maturing process involving viscosity increase in the SMC-paste from 20-40 Pa up to 40-100 kPa [5]. The chemistry behind this is that temporary bindings are created between magnesium ions and unsaturated polyester chains [7].

9 f l G y 11

• % \ s r --*"- *

i> i

Figure 2. Production of SMC body panels at Polytec Composites Sweden AB, Ljungby, Sweden. Photo: M. Oldenbo 2000.

Moulding is a stamping process done in a hot tool and the resin includes inhibitor that prevents the material f rom cure until the forming is complete. Figure 2 show production of SMC composite body panels at the currently only Swedish producer, Polytec composites AB in Ljungby.

2.3 Anisotropy

Anisotropy in SMC composite panels is common and this is due to a fiber orientation distribution that is not equal for all directions in the plane. Sources for this include: a) orientation from the belt in SMC-machine [3] and: b) from moulding of the SMC prepreg [8,9,10]. Fiber orientation is described using a fiber orientation distribution function, . This function describes the probability of an individual fiber having an orientation within a specified angular interval between <p = 0 and 0 = 180 degrees. For more reading about the fiber orientation distribution function, see for example [11].

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The fiber orientation distribution function can be written in tensor form described by Advani and Tucker [12]. Whereas the fourth order tensor is a fairly exact representation of the fiber orientation distribution, the second order fiber orientation represents a simplified curve. The second order fiber orientation tensor is:

A = a,, av (1)

With:

au = j ^ t ^ c o s 2 ødip ii

au = J v ^ s i n ø c o s ø t / ø o n

a22 = jV(<2>)sin2

(

(2)

(3)

(4)

McGee and McCullough have suggested the use of fiber orientation parameters ( / andg ) for planar fiber orientation distributions, see [13]. The relations

between / and gp and the second order fiber orientation tensor is shown in

Paper B. Fiber orientation could be determined in mould fil l ing simulation [9,10] through the use of a transport equation of the orientation tensor [14,8,9]. Another alternative is to measure the fiber orientation from a physical sample, either by burning off the matrix and study the fiber orientation distribution or by careful mechanical testing. Fiber orientation distribution after the matrix has been removed can be studied by for example using image analysis [15]. Use of mechanical testing to measure fiber orientation distribution requires an accurate model that links the mechanical properties to the fiber orientation.

2.4 Composition of SMC

2.4.1 Standard S M C

Standard SMC (Std-SMC) is a ternary composite made from thermoset polyester, calcium carbonate particulate filler and 25 mm E-glass fibers [7]. A typical composition is seen in Table 1 together with composition of Flex-SMC, see section 2.4.2. The polyester resin used for SMC contains thermoplastic additive for curing shrinkage compensation, i.e. low profile additive (LPA) solved in a styrene solution [7]. The volume fraction of fibers is typically 20-22 % and the fibers are dispersed in bundles. The bundle contain 180-400 fibers [16,17] and the volume fraction of fibers in the bundle is in the area of 0.5 [18]. The effective aspect ratio is related to the bundle rather than the individual

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fibers [17,19] but is still to be find between 89 and 200 [17,19], giving a reinforcement comparable to unidirectional (UD) composites.

Table 1. Material composition of Std-SMC and Flex-SMC. From Paper A. Vrvolume fraction, W:weight fraction, lfibe r:fiber length and p:density.

Flex-SMC A Flex-SMC B Std-SMC

V r e s i n M 30 b 2 3 b 43 b

Vflex additive etci 7"-19 b 16b

3 b

v f i b > ] 18b 21 b 22 b

V f i l l e r M 18 b 20 b 32 b

Vspheres ^ /

18 b 18 b -

W r e s i J %

21 16 24

Wflex additive etcL ] 1 2 10 2

W f , b e r [ % -30 34 30

l f i b J 1 ™ 25 25 25

W f , l l e r [ % 32 35 44

Wsphere S [% ]

5 b 5 b -

Pcomposite

3 g/cm

1-57 d 1.58d 1-95 d

a Resin is unsaturated polyester and low-profile (LP) additive in styrene solution. b Calculated using density of composite and different parts. Densities are 0.4 g/cm3 for the hollow spheres (most used glass spheres have density 0.37 g/cm3 [1,2]), 2.7 g/cms for the filler, 2.6 g/cms for the fibre, 1.0 g/cm3 for others and 1.1 g/cms for the resin. c Flex additive as well as initiator and mould release agent. d Measured following ISO 1183:1987.

2.4.2 L o w Density SMC

SMC materials are tending to move towards using different new additives to improve their performance such as hollow glass spheres for lower density (this type of SMC is denoted LD-SMC) or toughening additive for improved damage resistance. In the late 90s 3M company published research about including their hollow glass spheres in SMC composite to obtain lower density [20,21,22] while reducing the stiffness slightly. Paper A includes mechanical analysis of two materials containing both hollow glass spheres and toughening additives (Flex-SMC). The compositions of these materials are seen in Table 1 and the microstructure is seen in Figure 3. The circles in Figure 3 are hollow glass spheres which are 40-80 um in diameter. In paper C, the viscoelasticity and damage evolution with increasing strain of this material is studied. The damage development of this material is modelled using continuum damage mechanics.

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Figure 3. SEM picture showing microstructure ofLD-SMC with toughening additive ("Flex-SMC"). From Paper A.

2.5 Mechanical properties

Std-SMC is a stiff material with a Young's modulus of 10-12 GPa for a standard formulation. Tensile test data for two SMC materials with nearly in-plane isotropic fiber orientation distribution are given in Table 2.

Table 2. Mechanical data from tests of two different SMC's. Data from Paper A. Standard deviation is given in parentheses.

Property Flex-SMC Std-SMC Tensile Modulus [GPa] 8.3 (1.2) 11.7(0.8) Strength in Tension [MPa] 79 (11) 79 (11) Strain at Break in Tension [%] 1.3 (0.2) 1.2 (0.2) Compression Strength 126 (11) 155 (10)

t%] 0.2-0.3 <0.2

Jicb [kJ /m 2 ] 56 (7) 26 (2) !1 Threshold strain for onset of damage development in tensile loading. b Fracture toughness measured with DCB test with pure bending moment. This method is described in [15,23].

It is important to consider that the mechanical properties are highly dependent on fiber orientation. In general, the often high spread of reported mechanical data for SMC composites is due to fiber orientation distribution effects. More mechanical data for SMC is given in for example [2,3,5,7,19].

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2.6 Importance of SMC material for the automotive industry

SMC is found on cars in semi-structural parts such as hoods, doors, fenders, deck lids and tailgates. Structural parts with high degree of integrated fixing points and non visible surfaces such as grill opening reinforcements and sun roof reinforcements, are other popular applications for these materials.

The use of composite material in automotive body panel applications are on a general perspective driven by lower tool investments compared with sheet metal, weight savings and an increase in design freedom, concerning items such as press radii and integration of parts in the panel. Other features include no corrosion problems and electromagnetic properties allowing for antennas to be placed behind a composite panel. In Figure 4 a CAD model of the composite panels in the Volvo V70 tailgate is seen. Note that due to the press depth utilised with composite, no extra handle have to be added. The two large panels are made out of SMC and the small, upper and outer, panel is made of BMC, which is considered to give slightly smoother surface.

Figure 4. Design of SMC/BMC panels for Volvo V70 (1999) tailgate. Source: Volvo Car Corporation.

Demand for Class A surface is in favour of thermoplastics because of porosity and waviness in the SMC panels. The wall thickness used vary f rom 2 mm up to 3.5-4 mm in SMC designs and normally ribs are not used on visible panels due to risk for sink-marks on the visible surface. Decrease of the wall thickness is driven by cost and weight reduction whereas increase is driven by need for flexural stiffness and for risk of waviness as well as poor tools to predict the mechanical properties. The total European market size for automotive compression moulded parts is believed to be 918 million euros [24]. The annual market growth is believed to be 8-10% in total, and less for the rather

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mature materials, such as SMC (2-3% annual growth is estimated for them), and more for new technologies such as LFT.

3 Damage evolution with increasing strain

3.1 Damage development in SMC composite - micro level

The development of damage in SMC is due to micro cracking in the highly heterogeneous material and it is the reason for its non-linear behaviour. I n general terms this damage development can be described as formation of multiple cracks and increased crack density with increasing stress [25]. Crack initiation in composites may occur by fibre/matrix debonding, matrix/filler debonding and at fibre ends [25,26,27]. Transversely oriented fibre bundles have been reported as initiating sites for cracks in SMC [17]. Final failure is considered to occur as various small damage entities distributed in the material join and form heterogeneous macroscopic cracks [26]. In in-plane isotropic SMC and short fiber composites in uniaxial tension, cracks develop mainly transverse to the loading direction, see for example [28,29,30]. For materials with highly oriented fibers and high fiber content, cracks on the other hand tend to follow the main fiber direction, see for example the studies by Wang et al. [26,31] on fatigue damage evolution of short fiber composites. Development of large matrix cracks due to difference in stiffness between bundles and matrix coupled with shear stress is discussed in [29].

Figure 5. SEM pictures of damage development initiated in a transverse bundle of Flex-SMC. From Paper A.

In Paper A it was concluded that for Flex-SMCs, damage initiates in transverse fibre bundles and extends to macro-scale transverse cracks. Further on microscopic crack extension could be observed in all SMCs at constant load. This suggests a significant time-dependence of damage development processes in SMC. A SEM picture showing a crack in a transverse bundle is seen in

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Figure 5. The left picture in Figure 5 shows a crack inside a transverse oriented bundle at 0.35% global strain and the right picture shows the same crack at 0.44% strain. It is seen in the left picture that the crack in the transverse oriented fibre bundle has started to grow out to the surrounding matrix.

Jiao et al. have concluded that the damage modes can be determined by the assessment of the acoustic emission (AE) wave frequency [32,33]. Cracking is more likely to appear in regions where the matrix dominates or fibers are aligned normal to the applied load. From AE analysis of std-SMC two basic mechanisms was determined: a) Interfacial debonding of CaC03 particles at moderate loading and (b) debonding between the fibers and resin, at higher loading.

3.2 Damage development in SMC composite - macro level

By studying phenomenological aspects of the composite such as material degradation it is possible to build up a basic understanding of the material to be studied. It is seen that damage evolution of short fiber composites (SFC) in tensile loading is linked to stiffness degradation [34,35,36,37]. The degree of damage in this case is determined only by the magnitude of maximum stress applied. In Figure 6 the stiffness reduction as function of maximum strain is seen.

1.1 1 1

0 7

0.6 -I 1 i 1 j

0 0 .25 0.5 0 .75 1

Strain [%]

Figure 6. Stiffness reduction as function of maximum strain experienced for Flex SMC. From Paper C.

3.3 Coupling micro damage with macro damage

The stiffness reduction has been modelled with UD-plies by Kabelka et al . [ i8] . In their approach the fiber bundles were modelled as unidirectional (UD) plies

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followed by orientation averaging of the fiber orientation distribution. I n this way the material is built up through the thickness of a plate. The model predicted failure in bundles as a function of orientation to the load using failure criteria for the UD-plies which contain transverse tensile strength and in-plane shear strength of the bundles as parameters.

In Paper E, a model for stiffness reduction of SMC composite is derived. The damage model is based on a master curve for crack density as function of maximum strain history of a [0,90,0] laminate and one crack density curve of a [o,±6o,o] laminate to determine the influence of shear stress. Two categories of cracks are identified; bundle cracks and transverse layer cracks. The damage properties of SMC is obtained by modelling the composite as a quasi isotropic laminate and making a crack density curve for four directions, 9, in the range from 45 to 90 degrees to the applied load, see Figure 7. Here, the two types of cracks is averaged to one effective crack density curve showing cracks in transverse layer. Damage evolution is considered as being independent of whether a layer is oriented +9 or -9 degrees to the applied load.

SMC k=0.35

1000 i

Q , , , , ;

0.5 0.6 0.7 0,8 0,9 1.0

ex (%)

Figure 7. Averaged (all crack types) crack density of laminates with bundle structure at different directions to applied load. From Paper E.

A micromechanics based model for determination of the reduction in stiffness due to damage derived by Lundmark and Varna [38] is applied to the data in Figure 7. This model is developed in the framework of the laminate theory and is valid for general laminates with cracks in arbitrary off-axis layers. Two types of cracks are considered in simulations: (a) bundle cracks (TT-cracks) covering the thickness of the bundle and growing parallel to fibers; (b) matrix cracks initiated by bundle cracks which are transverse to the load direction. The results of predictions are seen in Figure 8.

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0,5 -i 1 1 1 i

0,5 0,6 0,7 0,8 0,9 1,0

ex (%)

Figure 8. Prediction of stiffness reduction as function of applied strain in SMC composite. From Paper E.

Since continuum mechanics models are more convenient for implementation the micromechanics models wil l serve as a basis for identification of evolution laws of damage parameters.

4 Material models for SMC

From an industrial perspective i t is of interest to be able to simulate performance of SMC panels with FE-analysis which requires that an adequate material model is available. When simulating performance correctly, competition from alternative materials is met by an optimized SMC structure which is not over-dimensioned or has a lot of expensive metal reinforcement.

The SMC material can be described with an orthotropic material model, which has three planes of symmetry [19]. I f the fibers have random in-plane distribution the material becomes transversally isotropic and the stiffness matrix has only five independent material constants. In this work shell elements are used which reduces the stiffness matrix to a dimension of 3x3, since the constitutive assumptions of plane stress is applicable:

Qu Qu o

s y m Q 6 6

(5)

For the FE simulation it is needed to have the out-of-plane shear stiffness (G, 3 and G2i) as well, which wil l be explained in chapter 5.

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4-1 Elastic material model - dependent on fiber orientation

The use of elastic fiber orientation distribution dependent models for FE-simulation is found to be limited even though the material models have been developed and validated on specimens. I f combining these theories with fiber orientation data, f rom for example mould fil l ing simulation, global stiffness of a structure can be calculated. This is done in Paper B.

Elastic material models for SMC are based on micromechanics for SFC with fiber bundle morphology. A review of elastic properties modelling based on the micro structure of SMC is done by McCullough et al. in the early 90s [19,39]. The basic approaches are to model the fiber bundle as either a ellipsoidal- or a cylindrical inclusion. Properties of a hypothetical laminate with aligned inclusions are to be calculated and the in-plane stiffness of the composite is obtained through averaging over the fiber orientation distribution. The matrix containing resin and filler is considered isotropic.

Orientation averaging of a short fiber composite has been made by Piry and Michaeli [40]. They used Halpin-Tsai equations [41] to simulate stiffness of unidirectional fiber layer with fibers oriented in a certain angle interval. The thickness of the layer corresponds to the amount of fibers in the angle interval. It has been shown by McGee and McCullough that the stiffness matrix of a composite with planar distribution of fibers can be obtained through the aggregate averaging based on the stiffness matrix of aligned ellipsoidal inclusions and the fiber orientation parameters f p and g p [ i 3 ] . Eduljee and

McCullough have explained how to calculate the stiffness matrix for aligned ellipsoidal inclusions [42] based on fiber properties and properties of the matrix. In this model, the lower bound relationship for the stiffness matrix of aligned inclusions, written in tensor form as [ c ] * M , is used. I t has shown to

give reasonable estimates of the elastic properties of a composite with aligned ellipsoidal inclusions. The calculation routine is written in tensor form [42]:

and[fiXis a modified Eshelby [43] tensor, i.e. tensor for the influence of inclusion geometry on the stress field, which is calculated based on the isotropic matrix properties as well as the ellipsoids effective aspect ratio. The new indexes used in Eqs. (6)-(7), m and/, denote matrix respectively fiber.

[clb,alAc]m+Vf*([H\-Vm[El)

Where [h\ = ( [ c l - [ c ] m ,

- 1 (6)

(7)

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4-2 Inelastic material model - use of damageable and viscoelastic material model

4.2.1 Elastic-damageable material model for SMC Guo et al. [27] have modelled the damage development f rom a micromechanics approach describing fiber bundles with the theory of ellipsoids, originated from Eshelby's (1957) [43] description of elastic field of an ellipsoidal inclusion. A statistical failure criterion for the identified failure mechanism of interfacial shear failure in fiber bundles was introduced. Kabelka et al. [18] used an approach with fiber bundles modelled as unidirectional (UD) plies combined with orientation averaging of the fiber orientation distribution. The model predicted failure in bundles as function of orientation to the load using failure criteria for the UD-plies which contain transverse tensile strength and in-plane shear strength of the bundles as parameters. Unidirectional loading curves have been predicted using common plasticity models [44].

In short fiber (or fiber bundle) composites with random or preferred fiber orientation distribution the variety of damage modes is much larger and they are interacting much more. Each of the damage modes start to develop at a certain level of stress and evolution curves are result of interaction of increasing loading and statistical distribution of strength (or fracture toughness). Therefore, i t appears reasonable in the first approximation to use continuous descriptors of damage related to stiffness changes instead. Damage is characterized in elastic tests and the damage parameters are used to scale down the composite viscoelastic response comparing to the virgin laminate response. Using a continuum damage mechanics approach it is possible to describe damage in the material in terms of decreasing stiffness over strain, see for example [34,35,36,37], and to measure the decrease in low frequency cyclic tests with increasing maximum strain per cycle. Dano et al. [28,36] have showed that a general damage mechanics theory by Chow and Wang [45], derived from the theory of equivalent strain energy, could be used to model the in-plane damage development in short fiber composites. The stiffness is dependent on the loading history which, in this theory, is described with the maximum thermodynamic force, %, experienced. In a unidirectional load case the relations are:

£, = £ ( i - A )

(8)

where £>, is the damage parameter which describes damage state in i-direction.

The plot of £>, Vs Yx based on the data in Figure 6 is seen in Figure 9.

14

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

Result:

f']( K,) = afp; - 5 j )

a = 0.205 Mpa"1. Y„ = 0.0654 MPa valid for r < 0.5 MPa

0.1 0.4 0.5 0.6 0.7

Yl [Mpa]

Figure 9. Damage law based on Figure 6 and damage mechanics theory.

4.2.2 Adding viscoelastic effects 1874 first formulations for 3D theory of isotropic viscoelasticity are described by Maxwell, Kelvin and Voigt, Bolzmann. The development of viscoelastic formulations have escalated since mid 1900s because of the entering polymer materials which typically behave viscoelastic [46]. The viscoelastic viscoplastic description of the composite may be performed based on the theory developed by Schapery [47]. The experimental identification of the viscoplastic terms is rather time consuming and may be obtained in strain recovery tests. The viscoelastic part of the composite response may be calculated from the known matrix viscoelastic behaviour using the same models as calculating the elastic response, see section 4.1. Due to the correspondence principle which is valid for linear viscoelastic materials, the micromechanics expressions developed for composite stiffness calculation in the elastic case may be used in the Laplace domain also in the viscoelastic case. Using the quasi-elastic method of solution, expressions for composite relaxation functions may be obtained in the time domain [48]. In linear viscoelasticity Prony series are used to describe the relaxation modulus:

E(t) = C0 + YJCke-"T> (9)

where Ck are the Prone series coefficients to be determined experimentally and

tk denotes the relaxation times.

4.2.3 Viscoelastic-damageable material model SMC materials are tending to move towards using different new additives to improve their performance, such as hollow glass spheres for lower density or toughening additives for improved damage resistance. A material containing both those additives has been tested (see Paper A, C-D and [16]) and in a cyclic

15

tensile test the material was found to show hysteresis and residual strain, indicating viscoelasticity, damage and possibly viscoplasticity. Both LPA and toughening additives, generally, are of thermoplastic content and thermoplastics generally are considered as much more viscoelastic than thermosets. As a result, this composite is probably more viscoelastic than could be expected f rom a standard SMC material containing only low-profile additive. Description of a heterogeneous material stress-strain response i f constituents are viscoelastic and evolving damage is a very complex task. Generally speaking the elastic - viscoelastic correspondence principle is not valid i f the damage evolution is analyzed and an accurate approach involves incremental numerical routines. For this reason most of the research on the viscoelasticity of damaged composites has been performed considering fixed damage state. In this way time effects are assumed to be negligible to damage evolution. Another drawback is that most of the papers are either purely theoretical or purely experimental and a proper experimental validation has been very limited. Results are available mostly for laminated composites where the damage modes are easy to identify and to quantify (layers have intra laminar cracks parallel to fibers, delamination etc. A coupled viscoelasticity-damageable material model for composites has been described in the literature, but rarely for short fiber composites and not at all for SMC, to the author's best knowledge. Weitsman [49] has used a power law to describe viscoelasticity and a scaling factor, for damage to model swirl-mat composite in unidirectional creep. Kumar et al. [50] have introduced a pseudo strain energy defined in Laplace domain to study viscoelasticity of laminates with different degree of damage.

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Strain [%]

Figure 10. Cyclic stress strain curve from test and simulation using a viscoelastic-damageable material model. From Paper C.

In paper C a simple model is derived which applies the Chow and Wang continuum damage mechanics theory [45] in combination with linear

16

viscoelasticity to SMC. In general, the stress-strain relationship with this model is formulated

' de ai{t,D)=\Cij{t-T,D)—^dr (10)

0 " T

By adding the viscoelastic effects, the hysteresis in the stress-strain curve in a cyclic loading can be explained. This would not be the case using a pure elastic damageable material model. The approach is based on the assumption that damage and viscoelastic effects may be separated. Thus, damage development is considered in the elastic formulation. The relaxation functions of the damaged composite are presented as a product of two terms representing damage and viscoelasticity, respectively. The model is validated in unidirectional creep test of samples with different degrees of damage. The model is exemplified in a cyclic stress-strain curve in Figure 10.

5 Implementation of material models in FE-solver

For FE-implementation of the material model an incremental formulation is needed. In an incremental formulation of the constitutive model stress, strain and damage state is known at time t and has to be calculated at time? + At. In a strain controlled analysis e{t + A?) is set and <r(? + A?, D) is to be calculated. Eq. (10) for time instant t + At is

a\t + At,D)= \ CJt + At-r,D)—J-dT ( l l ) o d T

We define:

Ar = tkM - tk

• tø, (%) = £,• (tM)-£, (tk) (12)

A<7j (tk ) = { 7 , (tk+l) — Gj (tk)

Where ACT, is calculated from the strain increment ACj .The solution is of type

ACT, = Cl]A£j (13)

where Cy is the tangent stiffness, i.e. Jacobian. The incremental formulation of

the viscoelastic-damageable material model is derived in Paper D. The material model is implemented in the UMAT subroutine for user-defined material model. The subroutine updates stress and tangent stiffness for the next increment.

17

Transverse shear stiffness is given as a constant calculated f rom out-of-plane shear relaxation stiffness, G 1 3, for linear elastic orthotropic material [51]:

Ku=Kn=^Gnh (14) o

Here h is the thickness of the shell.

Implementation of non-linear models for composites have been done previously. Fitoussi, Guo and Baptiste have applied their elastic-damageable model to simple structures through implementation into the FE-program ABAQUS [52]. Zocher et al. [53] have made a general incremental formulation suitable for orthotropic viscoelasticity. This model has been implemented in ABAQUS by SICOMP in an earlier project [54] where creep of GMT structures was calculated. Hinterhoelzl [55] has implemented a linear viscoelastic-isotropic damage model for reinforced plastics in FE-program ABAQUS.

6 Case studies

6.1 Elastic deformations in a hood structure

In Paper B it is shown how the deformation in a hood structure for a typical load case is to be obtained. For this calculation the elastic material model that takes into account fiber orientation distribution is used. The model is generated in a mesh programme based on a CAD model and is built up of shell elements. The result of the simulation is seen in Figure 11.

[stept inet, v-i.ODe+aCl

Figure 11. Elastic deformations in a hood structure. From Paper B.

18

6.2 Inelastic deformation and damage development in bending test specimen

In Paper D the in-elastic material model is applied to a simulation of a bending test. The model is built using shell theory and the load case is modelled using locks in out of plane direction in the ends respectively nodal forces to simulate the moving load noses. Five integration points through the thickness is used to handle the different damage development in thickness direction. Damage development is considered to appear only in the parts of the specimen that is in tensile loading. Damage level, determined as a damage parameter value on the face in traction is seen in Figure 12.

Figure 12. Damage parameter D^on the face in traction in a simulated bending test. From Paper D.

7 Concluding remarks

Different SMC materials have been analysed regarding composition, microstructure and mechanical properties. Micromechanics have been used to model the stiffness and the model has been implemented in a commercial FE software to be able to predict stiffness properties of a geometry with known state of fiber orientation.

In all SMCs studied, significant modulus reductions were observed with increasing strain due to extensive damage. A critical modulus reduction is suggested as a failure criterion rather than strength. Simulation not only of stiffness, but also to use a design prerequisite for allowed load is an interesting new approach to improve the accuracy of the design. In doing this i t is important to have an accurate material model. Viscoelastic effects were studied in cyclic loading test and creep test. A material model that considers the SMC composite as linear-viscoelastic material with evolving damage was suggested

19

to explain the nonlinear stress-strain behaviour and the observed damage accumulation with increasing stress levels. I t was seen that almost nothing has been done on short fiber composites with random fiber distribution considering models for both viscoelasticity and damage evolution. The simplifying assumption in the model is that damage development may be considered as an elastic process and hence depends only on the maximum stress experienced by the material. This allows for damage quantification in terms of stiffness reduction in quasi-static tensile loading and unloading testing. Then the time- and damage-dependent viscoelastic functions of the composite are described as a product of damage- and time-dependent terms, where the time-dependence of the viscoelastic behaviour is described by creep compliance functions of undamaged composite. Hence, the damage-dependent term serves as a scaling factor.

An incremental formulation of the non-linear model usable in FE-simulation is derived, presented and implemented in FE-programme ABAQUS. Application to constant strain rate tensile test comparing analytical and FE result prove accuracy of the formulation and the subroutine.

Finally, a stiffness reduction model for SMC composites with evolving damage is suggested and validated for a standard SMC material.

8 Future work

The work presented here has been dealing with developmental work of models to simulate stiffness and strength of SFC composites and especially SMC composites and structures. Next step include introducing the models in industry including refinement and testing on complicated structures. Easy to use tools for the engineer is needed.

20

References

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3 Ehrenstein GW, Polymeric materials- structure, Properties, Applications 2001. Munich, Hanser publishers.

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15 Vahlund CF, Gebart R. Analysis of an image processing method for fiber orientation in polymer composites, Pol. Comp. 2001; 22:327-336.

16 Fernberg SP, Berglund LA. Bridging law and toughness characterisation of CSM and SMC composites. Comp. Sei. Tech. 2002; 61, 2445-2454.

21

17 Hull D. An introduction to composite materials 1981. Cambridge solid state science.

18 Kabelka J, Hoffman L, Ehrenstein GW. Damage Process Modelling on SMC. d.Appl. Pol. Sei. 1996; 62:181-198.

19 McCullough RL. In: Carlsson LA and Gillespie JW Eds. Delaware Composites Design Encyclopedia 1990; Volume 2, Technomic Publishing Co., Lancaster, PA, pp 93-142.

20 Gregl BV. Glass microspheres produce lower weight SMC. Reinforced Plastics 1997. October issue.

21 Larson LD, Robertson DL, Ingham TL, Shah VC, Botts T, Anderson RD. SMC glass microspheres as a low-density alternative to traditional fillers. SAE Tech. paper series 1998, no 980982.

22 Gregl BV, Larson LD, Sommer M and Lemkie JR. Formulation Advancements in Hollow-Glass Filled SMC. SAB Technical paper series 1999, No 1999-01-0980.

23 Lindhagen JE, Berglund LA. Application of bridging-law concepts to short-fibre composites. 1) DCB test procedures for bridging law and fracture energy. Comp. Sei. Tech. 2000; 60:871-883.

24 Sonnen A. Menzolit-Fibron, Company presentation held at VCC 2004.

25 Hour KY, Sehitoglu H. Damage development in a short fibre composite. J. Comp. Mat. 1993; 27:782-805

26 Wang SS, Chim ESM. Fatigue damage and degradation in random short-fiber SMC composite. J. Comp. Mat. 1983; 17:114-134.

27 Fitoussi J, Guo G, Baptiste D. A statistical micromechanical model of anisotropic damage for SMC composites. Comp. Sei. Tech. 1998; 58:759-763.

28 Dano ML, Gendron G, Mir H. Mechanics of damage and degradation in random short glass fiber reinforced composites. J. Thermopl. Comp. Mat. 2002; 15:169-177.

29 Von Bernstoff B, Ehrenstein GW. Failure mechanisms in SMC subjected to alternating stresses. J. Mat. Sei. 1990; 25:4087-4097.

30 Aymerich F, Priolo P. Characterization of a SMC material for bumper applications. Key Eng. Mat. 1998; 144:179-190.

31 Wang SS, Suemasu H, Chim ESM. Analysis of fatigue damage evolution and associated anisotropic elastic property degradation in random short-fiber composite. Eng. Fract. Mech. 1986; 25:829-844.

22

32 Jiao GQ, Zheng ST, Suzuki M, Iwamoto M. Damage evaluation of sheet moulding compound composite by acoustic emission. Theor. Appl. Fract. Mech. 1990; 14:135-140.

33 Suzuki M, Jiao GQ. A study on damage growth of short fiber SMC composites in the case of quasi-static and tensile loading. VI International Congress on Experimental Mechanics. Vol. I , Portland, Oregon; USA; 6-10 June 1988. pp 344-349-

34 Ladeveze P. In: Talreja R editor. Damage Mechanics of Composite materials 1994. pp 117-138, Elsevier Science B.V., Amsterdam.

35 Lemaitre J, Dufailly J. Damage Measurements. Eng. Fract. Mech. 1987; 28:643-661.

36 Dano ML, Maillette F, Gendron G, Bissonnette B. Damage modelling of random short glass fiber reinforced composites. In: Proc. Cancom 2001, Quebec, Canada, Aug 21-24, 2001, pp 9-17.

37 Ye L. On fatigue damage accumulation and material degradation in composite materials. Comp. Sei. Tech. 1989; 36:339-350.

38 Lundmark P, Varna J. Constitutive relationships for damaged laminates in in-plane loading. Submitted to Int. J. of Damage Mechanics.

39 Eduljee RF, McCullough RL. Elastic Properties of Composites, In: Structures and properties of composites 1993. pp 381-474, VCH, New York.

40 Piry M, Michaeli W. Stiffness and failure analysis of SMC components considering the anisotropic material properties, Macromol. Mater. Eng. 2000; 284/285:40-45.

41 Halpin JC, Kardos JL. The Halpin-Tsai Equations: A Review, Pol. Eng. Sei. 1976; 16.

42 Eduljee RF, McCullough RL, Gillespie JW. The influence of inclusion geometry on the elastic properties of discontinous fiber composites, Pol. Eng. Sei. 1994; 34:352-360.

43 Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. Soc. A 1957; 241:376-396.

44 Jekabsons N, Fernberg SP, Prediction of progressive fracture of SMC by application of bridging laws. Comp. Sei. Tech. 2003; 63: 2133-2142

45 Chow CL, Wang J. An anisotropic theory of elasticity for continuum damage mechanics. Int. J. Fract. 1987; 33:3-16

46 Christensen RM, Theory of viscoelasticity, second edition 1982. Academic Press, New York.

23

47 Schapery RA. Nonlinear Viscoelastic and Viscoplastic Constitutive Equations Based on Thermodynamics. Mech. Time-Dep. Mat. 1997; i(2):209-240.

48 Megnis M, Varna J. Micromechanics based modelling of nonlinear viscoplastic response of unidirectional composite. Comp. Sei. Tech. 2003; 63:19-31.

49 Weitsman YJ. Time-dependent behavior of randomly reinforced polymeric composites. Mech. Comp. Mater 2002; 38:381-385.

50 Kumar RS, Talreja R. A continuum damage model for linear viscoelastic composite materials. Mech. Mater. 2003; 35:463-480.

51 ABAQUS 6.3-1 Documentation, Standard User's manual, Hibbitt, Karlsson & Sorensen, Inc., 2002

52 Guo G, Fitoussi J, Baptiste D. Modelling of damage behaviour of a short-fiber reinforced composite structure by the finite element analysis using a micro-macro law. Int. d. Dam. Mech. 1997; 6:278-297.

53 Zocher MA, Groves SE, Allen DH. A three-dimensional finite element formulation for thermoviscoelastic orthotropic media. Int. J. Num. Meth. Eng. 1997; 40:2267-2288.

54 Allen DH, Holmberg JA, Ericson M, Lans L, Svensson N, Holmberg S. Modeling the viscoelastic response of GMT structural components. Comp. Sei. Tech. 2001; 61:503-515.

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24

Available online at www.sciencedirect.com

ELSEVIER Composites: Part A 34 (2003) 875-885

Part A: applied science and manufacturing

www.elsevier.com/iocale/'conipositesa

Mechanical behaviour of SMC composites with toughening

and low density additives

M . 01denbo a b'*, S.P. Fernberg0, L.A. Berglundd

aVolvo Car Corporation. Exterior Engineering. Department 93610 PV3C2. SE-405 31 Göteborg. Sv eden bDivision oj Polymer Engineering. Luleå University of Technology, SE-971 87 Luleå. Sweden

'SICOMP AB. P.O. Box 271. SE-94 1 76 Piteå. Sweden JDepartment of Fiber and Polymer Technology. Royal Institute of Technology. SE-100 44 Stockholm. Sweden

Received 9 April 2002; revised 26 March 2003; accepted 11 April 2003

Abstract

A new type of SMC material (Flex-SMC) developed for automotive exterior body panels has been investigated. Flex-SMC contains hollow glass micro-spheres and thermoplastic toughening additives. A conventional SMC (Std-SMC) was used as a reference material. Materials were tested in monotonic tension and compression. Sliffness degradation with strain as well as fracture toughness was determined. In situ SEM was used lo study failure mechanisms. Flex-SMC has a density almost 20% lower than Std-SMC and has higher impact resistance. The damage threshold strain of the Flex-SMCs is higher than for Std-SMC. Flex-SMCs have more than twice the fracture toughness of Std-SMC. The major reason identified is that Flex-SMCs shows extensive fibre pullout. © 2003 Elsevier Ltd. All rights reserved.

Keywords: A. Moulding compound; B , Mechanical properties; D. Mechanical testing

1. Introduction

Sheet moulding compounds (SMC) have been used for a long time in the automotive industry. The major advantages of SMC are:

• Up to 30% lower weight for a part made from SMC compared with the corresponding steel part.

• SMC is less expensive than sheet metal for low-volume production. The reason is lower tool costs.

• SMC is a non-corroding polymer composite material. • There is more design freedom with SMC-materials

compared to sheet metal since smaller press radii are possible. It is also possible to integrate functions in the SMC structure, i.e. a car door handle in the rear door.

• The non-isolating electromagnetic properties of SMC allow antennas to be placed behind an SMC panel.

There are also some disadvantages of using SMC in automotive body panels. For example, small impact events

* Corresponding author. Address: Volvo Car Corporation. Exterior

Engineering, Department 93610 PV3C2, SE-405 31 Göteborg, Sweden.

Tel.: +46-31-32-56-924; fax: +46-31-59-10-34.

E-mail address: moldenbo@volvocars.com (M. Oldenbo).

such as minor parking lot incidents may cause cracks in the panel. These events are usually simulated by falling bail impact tests. SMC panels have approximately five times higher resistance to such impacts compared to steel i f dents in the steel panel are considered. However, since cracks are not common in conventional body panels this may lead to negative market reactions. Today there is also an increasing competition from light-weight alloys. The weight of a 1 mm thick aluminium panel (the conventional thickness) is about 50% of the weight of a corresponding 0.75 mm (conven­tional steel panel thickness) steel sheet metal panel. Similar considerations show an aluminium panel to be much lighter than a conventional 2.5 mm thick SMC panel.

There is an interesting development of low-density SMCs based on hollow-glass micro-spheres. The mechan­ical properties as well as surface quality aspects have been studied [1,2]. Typically, Young's modulus is reduced by 20% (corresponding to a density reduction of 20-30%), and the surface requires in mould coating (IMC) to allow painting. This problem is addressed by charging the mould with multiple materials, using low density in the centre and Std-SMC in the outer layers [2,3]. In the present paper, two low-density and toughened SMCs (Flex-SMC) developed by two major European SMC-suppliers, have been studied

1359-835X703/$ - see front matter © 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S1359-835X(03)00155-6

K I f , U. Oldenbo el al. / Composites: Part A 34 {2003) 875-885

and compared with a conventional SMC (Std-SMC). The Flex-SMC materials were developed for application in vertical automotive exterior body panels. These new materials are of lower density than Std-SMC and have higher resistance to the type of impacts previously mentioned. The objective was to compare the mechanical properties of the materials and clarify the reasons for differences in behaviour. Conventional tensile and com­pression tests were employed. The state of damage in the material was characterised by microscopy and measure­ments of stiffness changes with increasing strain. Fracture toughness tests were performed on double cantilever beam (DCB) specimens.

2. Experimental

2.1. Materials

Three materials were used in this study; two high-flex and low density SMCs (Flex-SMC A and B) and as reference one conventional, low profile, SMC material (Std-SMC). A l l materials are commercially available for production of exterior automotive body panels. Plate of all the materials were pressed by the material suppliers (Polytec Composites Sweden AB. Reichhold AS in Norway and Mecelec Composites Et Recyclage in France) and delivered as 2 mm X 300 mm x 300 mm panels as well as 3 mm X 300 mm X 300 mm panels.

2.2. Tensile tests

The samples for tensile test were of dimensions 300 mm x 50 mm and machined from moulded Plate of 2 mm thickness. The specimens were conditioned according to DIN 50014 (normal climate) at 23 ± 2 °C and 50 ± 5% relative humidity for at least 24 h prior to testing. Tensile tests were performed using a Zwick tensile tester with optical extensiometer and the grip to grip separation was about 220 mm. The gauge length was set to 180 mm for the cyclic tensile tests and 100 mm for the monotonic tensile test. The switch to shorter gauge-length was in order to avoid non-uniform strain-field effects close to the grips. However, a long gauge length (100 mm) was still desirable since it provides better average data for heterogeneous materials. E-modulus was measured between 0.05 and 0.25% strain for the monotonic tensile test. The test speed was 1 mm/min before and during £-modulus measurement followed by 50 mm/min until break. For the cyclic tests loading and unloading was carried out at a crosshead speed of 2 mm/min. In the first cycle, the sample was loaded to a stress corresponding to a strain of 0.1%. Cyclic loading was then conducted with increments of 0.1 % of strain per cycle. The unloading within a cycle was stopped once a low but finite tensile stress level, corresponding to a tensile load of 100 N, was attained to avoid compressive loads. The initial

£-modulus ( £ 0 ) was measured at the unloading path of the first cycle and the ratio of E/E0 was then plotted versus the maximum strain of the cycle. Five samples were tested for each material for both monotonic and cyclic tests.

2.3. Compression tests

Compression tests were conducted on Std-SMC and Flex-SMC B specimen of dimensions 80 mm X 15 mm X 3 mm. The tests were performed at a constant crosshead speed of 1 mm/min in a Dartec servo hydraulic test machine with 250 kN capacity. A test fixture [ 4J , designed and manu­factured at the Institute of Polymer Mechanics (IPM) in Riga, Latvia, was used to transfer the compressive load to the specimen. In total, nine specimens of each material were tested. In order to control whether macrobuckling occurs, three specimens had two strain gauges (one on each face).

2.4. In situ microscopy

The specimens for In situ optical microscopy were 200 mm X 15 mm X 2 mm and the tensile tester was an Instron 4411 with a video camera and a lens with 400 times magnification capability. Test speed was 2 mm/min and the gauge length was 50 mm. The edge that was to be studied was polished. In situ SEM for off-plane studies utilised specimens 54 mm X 14 mm X 2 mm with holes in the ends, which were used to mount the specimen in the tensile stage inside the SEM. This method is described in detail by Hagstrand and Rychwalski [5], The material around the holes was reinforced with 0.5 mm sheet metal Plate, which were bonded with epoxy adhesive. The edges of the samples were polished using 400 and 600 grade paper, followed by 6. 3 and 1 u.m diamond pastes. Gold sputtering was done for 1 min before mounting the sample in a Raith tensile stage inside the SEM-microscope (DSM 940 h). The crosshead speed was 2 pm/s. Strain at the in situ SEM measurements was determined by monitoring a point at the specimen relative to a carbon fibre positioned along the longitudinal axis that was fixed with a droplet of soft adhesive in one end. The gauge length was set to be equal to the free fibre end plus half of the imbedded length, because the fibre is stationary in the middle of the droplet. The error of measurements was estimated to be within ± 0 . 7 u,m [5J. Since the gauge length was approximately 7 mm the error was negligible.

2.5. Double cantilever beam tests

In these tests, Plate approximately 300 mm X 300 mm X 3 mm were adhesively bonded together using an epoxy adhesive (Araldite 2011) so that the desired thickness B (Fig. 1) of a DCB test specimen was achieved. Specimens were machined to size from the stack of SMC Plate. They were tested in a rig (Fig. 2) that provides pure moment loading of the DCB specimen 16,7], The tests were

M. Oldenbo el al. / Composites: Part A 34 (2003) 875-885 877

I» 1

\ T

3 au

H: 43 - 50 mm L: 500 mm B: 12-17 mm b: 3 - 4 mm

Fig, I. Drawing of DCB-specimen. Position of exlensiometer for measurement of end opening displacements is indicated. Dimensions used in the test are given

in the figure.

performed using a servo-hydraulic test machine. Instron 8501, at a constant cross head speed of 5 mm/min. Strain energy release rate can in this test configuration be evaluated by the /-integral applied along the specimen boundaries to give [6,8,9]

12(1 - i r ) M1 B

B1HyE ~b (1)

where v is Poisson's ratio. E is elastic modulus in tension and H. B. b are specimen dimensions.

Parts of bridging laws (or cohesive zone laws) can be estimated in this test if applied moment and the crack opening displacement A"' at the end of the pre-cut crack tip

1 —1 1 , . / f i

-

TK

\

i i i 1

+ •

i j

j

\

R \ « . V —1 >>

V \

IV l<

/

is continuously recorded throughout the tests. A' was measured by an extensiometer mounted on pins attached to the specimen at the neutral axis of each beam. The relationship

d7

åA* cr„(zf): (2)

then gives the experimental bridging law. As pointed out in Ref. [10], Eq. (2) wil l not give the true bridging relation during the early part of DCB tests. The registered changes of displacements A* are solely due to beam deformations at that stage. At later stages (after an apparent peak stress in the bridging relation). Eq. (2) is likely to reflect the true trends of the bridging relations since bulk contributions to

are more or less constant at that stage.

3. Results and discussion

3.1. Material composition and microstructure

The composition of the different materials is presented in Table I . The fibre volume fractions are fairly similar and

Table 1

Material composition (based on supplier's data)

Flex-SMC A Flex-SMC B Std-SMC

v r e > 1 „ W 3()b 23 b 43 b

Vflex addiiive etc (%) 19" 16" jb

V„s , r (%) 18" 21" 22 b

Vni,« ( * ) I 8 b 2(f 32 b

I 8 h 18 b -W r e s i „ (%)" 21 16 24

W|ie.x adiiiiivc etc ( ^ ) 12 10 2

W n t e r ( * ) 30 34 30

W » > 25 25 25

w f , „ „ (%) 32 35 44

w s p h e „ s ( * ) 5 b 5 b -Pcomposite (g/cm ) 1.57J 1.58J l .95 d

Fig. 2. The test rig for pure moment loading of D C B specimen.

J Resin is unsaturated polyester and low-profile (LP) additive in styrene

solution. b Calculated using density of composite and different parts. Densities are

0.4 g/cnv for the hollow spheres (most used glass spheres have density

0.37 g/cm 3 (1,2]), 2.7 g/cm 3 for the filler, 2.6 g/cm 3 for the fibre, 1,0 g/cm 3

for others and 1.1 g/cm 3 for the resin. L Flex additive as well as initiator and mould release agent.

d Measured following I S O 1183:1987.

87X M. Oldenbo el al. / Composites: Part A 34 (2003) 875-885

Table 2

Properties of materials determined in monotonic tensile test

Flex-SMC A

(batch I)

F lex-SMC B Std-SMC

£-modu(us

(GPa)

Strength

(MPa)

Strain at

break (%)

10 ± 0.4"

7.5 + 0.5"

8.6 ± 1.4"

100 ± 8 longit.

38 ± 9 transv.

69 s 34 ave.

1.3 ± 0.1 longit.

0.8 ± 0.3 transv.

1.1 ± 0.3 ave.

9.1 ± 0.7 longit.

7.5 ± 0.9 transv.

8.3 ± 1.2 ave.

85 ± 13 longit.

73 ± 2 transv.

79 nr. 11 ave.

1.1 ± 0 . 2 longit.

1.5 ± 0.1 transv.

1.3 ± 0.2 ave.

No value" longit.

11.1 ± 0 . 8 transv.

11.7 ± 0.8" ave.

84 ± 13 longit,

73 ± 7 transv.

79 ± 11 ave.

1.4 ± 0.2 longit.

1.1 ± 0.2 transv.

1.2 ± 0.2 ave.

200 («vt

Fig. 3. S E M micrograph of typical cross-section in F lex -SMC material.

Note the carbon fibre in the centre that was used for measuring of the strain,

any differences in fibre strength distribution are expected to be negligible [7J. The Flex-SMCs have significantly lower densities due to lower inorganic content (filler and fibre) and the addition of 18% by volume of hollow glass spheres 11, 2]. The glass spheres have a density of approximately 0.4 g/ cm . Some of the glass spheres are expected to break during processing and therefore the measured Flex-SMC densities in Table 1 are slightly higher than predicted.

In the Flex-SMCs. the high content of thermoplastic flex and low profile additives (up to 40% by volume of the matrix) is approximately twice as much as in Std-SMC. The flex additive is added to the unsaturated polyester system in a styrene solution. The microstructure of Flex-SMC A is presented in Fig. 3. The fibres are distributed in bundles with the hollow glass spheres distributed in the matrix-rich regions. The largest spheres have a diameter around 80 pm, with average diameter around 40 pm. The typical sphere wall thickness is approximately a few microns. At higher magnification, the presence of crushed spheres is apparent. In a different batch of flex SMC. the surface regions showed an increased sphere content.

3.2. Young's modulus and strength in tension and compression

Results from tensile tests are presented in Table 2. Al l materials showed some degree of anisotropy. For industrial practice, this information is important since Plate provided by material suppliers are often assumed to be isotropic. If different materials have different degrees of anisotropy. comparison of properties will be misleading. For processing reasons, material flow is desirable during the pressing operation and this is likely to cause some anisotropy.

The Young's moduli of Flex-SMCs are slightly lower than for Std-SMC. Reasons include lower inorganic filler content and the softening effect from the hollow glass

a A few values are missing due to technical problems, values from cyclic

test presented if available.

sphere additives. Previously, a 20% Young's modulus reduction was observed with about 10 wt% hollow micro glass spheres [2J.

Disregarding the highly anisotropic Flex-SMC A, the maximum tensile strength is 85 MPa. This is slightly higher than data for another standard SMC [2J. One may note that the tensile strength of Flex-SMC A in its weakest direction was only 38 MPa. Std-SMC and Flex-SMC were observed to have similar tensile strengths. In Ref. [2], addition of hollow glass spheres was found to reduce not only stiffness, but also strength. Regarding (he strength of short fibre composites, the average glass fibre strength is an important factor. This may vary between different £-glass fibres, due to differences in diameter, surface flaw distribution etc.

The strains to failure vary in the range 0.8-1.5%. An interesting observation is that the Flex-SMCs do not show higher strain to failure than Std-SMC. The lowest strain to failure was observed for the Flex-SMC A specimens with a high proportion of transverse oriented fibres.

Stress-strain curves from a few complementary tensile tests at 1 mm/min displacement rate are presented in Fig. 4. The major difference in response is that the Std-SMC material shows a distinct knee in the stress-strain curve at about 0.3% strain. This is attributed to the onset of damage development processes in the material.

Results from compression tests are presented in Table 3 and Fig, 5. Strengths are in general much higher in compression compared with tension. Flex-SMC B is significantly weaker than Std-SMC. The reason is the softening and weakening effect from hollow glass spheres and lower CaCO, content. It is also interesting to note that no distinct knee in Std-SMC compression curve is apparent.

3.3. Stiffness degradation

Stiffness degradation tests are often used as an indirect measure of damage development in materials. The method

M. Oldenbo el al. / Composites: Part A 34 (2003) 875-885 879

Fig. 4. Stress-strain curves for Std-SMC and Flex-SMC B.

has previously been applied to short fibre composites [11, 12J. General aspects of stiffness degradation tests are

discussed by Ladeveze [13] as well as Lemaitre and

Dufailly [14J. In Table 4 the properties are summarised from the cyclic

tests for stiffness degradation. It is interesting to compare the strength data from monotonic and cyclic tensile tests (Tables 2 and 4). For both the Std-SMC and Flex-SMC B, the ultimate strength is significantly lower in the cyclic test. The strain to failure is also slightly lower. The main reason

is probably that the damage state after repeated cycles is slightly more severe than after monotonic loading. With Flex-SMC A this effect is not very pronounced. The

threshold strain for onset of damage, s t h, is defined as the threshold strain at which stiffness reduction is observed. £,„ is lowest for Std-SMC, which is in agreement with the monotonic stress-strain data in Fig. 4. This may be related

to the higher filler content in this material. One may also note that eth is more sensitive to composite matrix (polymer matrix and filler) characteristics than monotonic strain to

failure. Typical stress-strain curves from the cyclic tests are

presented in Fig. 6. For clarity, only three curves are shown. Considerable hysteresis was observed in all three materials. The extent of these energy losses increased with strain. Sources of energy dissipation include viscoelastic losses, crack formation and friction associated with crack surfaces. The irreversible strain also increases with strain. This strain

Table 3

Results from compression tests of S M C materials Std-SMC and Flex-

S M C B

Property Std-SMC Flex-SMC B

Strength (MPa) 155 ± 9.6 126 = 11.3

200

S [% ]

Fig. 5. Compressive stress-strain curves for Std-SMC and Flex-SMC B.

is the point at which unloading ends and the new load cycle starts. The same phenomenon was observed in a stress controlled fatigue test of SMC [15]. Cracking in the transverse direction as well as along fibre bundles was used as an explanation.

In Fig. 7, the reduction in Young's modulus E/E0 is plotted versus strain for Std-SMC and Flex-SMC B. The damage threshold strain, e,„, is between 0.2 and 0.3% for Flex-SMC B and less than 0.2% for the Std-SMC material. EIE0 decreases rapidly for Std-SMC in the early stages of damage development. Flex-SMC B shows a much slower reduction in E/E0 with strain. Later the slope is similar to the one for Std-SMC, although displaced in strain.

I f we arbitrarily compare the strain at which E/E0 is 0.96, the value is 0.2% for Std-SMC and 0.4% for Flex-SMC B. Another interesting observation is that close to failure, E/E0 is 0.79 for Std-SMC and 0.74 for Flex-SMC

Table 4

Mechanical properties determined in cyclic tensile test

Material £ -modulus of Failure Strength e l h (9C )

first cycle (GPa) strain (%) (MPa)

Flex-SMC A 1 10 ± 0.4 longit. 1.2 ± 0.2 91 13 longit. 0.3-0.4

(batch 1) longit.

7.5 ± 0.5 transv. 0.8 ± 0.2 40 ± 10 transv. <0.2

transv.

8.6 ± 1.4 ave. 1.0 ± 66 ± 29 ave.

0.3 ave.

F lex-SMC B" 8.5 ± 0.7 ave. 1.1 ± 62 ± 10 ave. 0.2-0.3

0.2 ave.

Std-SMC" 11.7 ± 1.0 ave. 0.9 ± 68 ± 15 ave. <0.2

0.2 ave.

a F lex-SMC A (batch 1) was anisotropic (very visible flow lines on the

surface of the plate) and therefore tested in the assumed longitudinal and

transverse directions. b F lex-SMC B and Std-SMC were assumed planar isotropic.

880 M. Oldenbo el al. I Composites: Part A 34 (2003) S75-ÄÄ5

Fig. 6. Illustration of a typical result from the cyclic tensile test and

visualisation of the method to measure the E-modulus for the loading

cycles.

B. The strain at failure was for Flex-SMC B in monotonic

tension was also larger. In Fig. 8, data for Flex-SMC A are presented with data

for Flex-SMC B. The Flex-SMC A material was highly anisotropic, therefore transverse as well as longitudinal data

is presented. Longitudinal data were measured parallel to the direction of the majority of the fibres. e I h in the transverse direction is as low as 0.2%, whereas e t h in the longitudinal direction is 0.3-0.4%. The longitudinal e t h for

Flex-SMC A is the highest measured in the present study. In

addition, the slope of E/Eu versus strain is much less steep for longitudinal Flex-SMC A than for the other two materials in Fig. 8. The behaviour of Std-SMC in Fig. 7 is very similar to Flex-SMC A transverse in Fig. 8. The average strain at failure (data at highest strain) is also the highest for longitudinal Flex-SMC A. but the lowest for transverse Flex-SMC A.

An important message in Fig. 8 is that despite strong effects from material composition, the fibre orientation distribution can be a dominating factor. Unfavourable fibre orientation may compromise the properties of any SMC material. Fig. 8 also shows that transverse fibre orientation is associated with a faster damage development rate.

Similar studies on polypropylene (PP) glass mat thermoplastics (GMT) [12] resulted in s t h between 0.2 and 0.5%. E/E0 only decreased by approximately 10% before ultimate failure at 1.5% strain. Cracking was not wide­spread. The low yield stress and ductility of PP and the development of toughening mechanisms such as plastic deformation and drawing are likely causes of these effects.

Hagstrand and Rychwalski studied melamine formal­dehyde composites in order to find composition effects on damage development 15]. e t h was very low, usually below 0.2%. The changes in E/E0 were often larger than in the present study. This implies widespread cracking although the fibres are able to keep the material together. The reduction in modulus is caused by the extra global deformation resulting from local crack opening displacements.

The methodology of stiffness reduction studies is of great interest in the context of microstructural optimisation. Although stiffness and strength from monotonic tensile tests were very similar, damage development processes can be

1 4

«, 0.8

0.6

0,4

I 1

0.1

Li . r i

1 Flex-SMC B meanvalue (E„=8.5 GPa) i Std-SMC meanvalue (E0=11.7 GPa)

0.3 1.1 1.3 0.5 0,7 0.9

strain (%)

Fig. 7. E/EQ versus strain for Flex-SMC B and Std-SMC. The error bars represent standard deviation and is for clarity only marked in one direction.

M. Oldenbo el al. / Composites: Part A 34 12003) 875-885 XXI

06

0.4

0.1

Bf Flex-SMC B meanvalue (E0=8.5 GPa) Flex-SMC A (longitudinal) meanvalue (E0=10.0 GPa)

• Flex-SMC A (transverse) meanvalue (E<>=7.5 GPa)

0.3 0.5 0.7

strain (%)

1.1 1.3

Fig. 8. Variation in modulus as a function of strain for Flex-SMC A (anisotropic), batch 1, F lex-SMC B (nearly isotropic). The error bars represent standard

deviation and is for clarity only marked in one direction.

significantly different. The present Std-SMC material shows large reduction in E/E0 with strain and a low e t h.

Another conclusion from Figs. 7 and 8 is that as these materials fail in a monotonic tensile test, there is widespread damage. For this reason one may debate whether a strength criterion is reasonable to use in engineering design of SMC components. A possible alternative for a given material is to use a critical value for E/E0. In this way, the design criterion is related to a specific damage state in the material. One advantage with such a criterion is that it can be subjected to physical interpretation. It may also be relevant in creep and dynamic fatigue.

3.4. In silu microscopy

The non-linear behaviour of SMC has been connected to the formation of multiple cracks and increased crack density with increasing stress, see Ref. [15],

By in situ optical microscopy, large matrix cracks were observed at 0.4% strain in transversely loaded Flex-SMC A. Cracks extended at strains over 0.4% and the specimen failed between 0.4 and 0.6% strain. Note that the ultimate strain is much higher in the longitudinal direction. Flex-SMC B showed a similar behaviour as Flex-SMC A.

In situ SEM was then used to study Flex-SMC B and Std-SMC. The first damage observations in Flex-SMC B were made at 0.35% strain and consisted of diffuse matrix damage within transversely oriented fibre bundles, see Fig. 9. Note also the variations in fibre diameter. The matrix cracks are short, appear to be shallow and are numerous within small local regions. This damage observation corresponds with s,h of 0.2-0.3% observed for Flex-SMC B. The decrease in E/E0 at 0.35% strain is only 2-3%. The limited and localised matrix damage within transverse

fibre bundles is consistent with the small stiffness change. No larger scale cracks were observed at this strain. The location of matrix damage within transverse bundles is also consistent with the large matrix strain magnification in regions with high fibre volume fraction (12J. The stiff glass fibres result in large local strains in the matrix between neighbouring transverse fibres. The absence of debonding was also noted. This may be interpreted as high interfacial adhesion although a more likely explanation is that the matrix is easily damaged. In SMC, the 'matrix' is a polymer-filler composite. The presence of inorganic filler particles wil l surely govern the matrix damage mechanisms.

In Fig. 10, the crack from Fig. 9 is magnified. The crack is moving in the matrix around the fibre in the top of the picture and the CaC0 3 filler particle in the middle.

Fig. 9. Crack in matrix, initiated inside a transverse oriented fibre bundle.

This is the first crack seen in Flex-SMC B and the strain is here 0.35%. The

crack has stopped as it reached the matrix outside the bundle.

The lighter area in the centre of the picture indicates plasticity around the crack since the thickness of the gold layer here is thinner. This is because the darkness of the

picture corresponds to the conductivity, which in turn is determined by the thickness of the gold layer. This is also the reason why cracks often appear very light in the pictures.

A filler particle is present just below the fibre and possibly another one further down. The appearance of matrix damage at this higher magnification is of interest. In addition to the diffuse, larger cracks, additional matrix damage is observable as submicron cracks. The larger cracks follow a path around fibres and filler particles. The lighter area in the centre of the picture has been subjected to large deformation. The granular appearance of the polymer matrix, in particular at crack surfaces, is probably related to the thermoset/thermoplastic blend morphology. The crack appearance is very different from typical well-defined matrix cracks in thermoset composites; see Ref. [16]. In Fig. 11, the crack is at 0.44% strain and has now grown into

Fig. 11. This is the crack from Figs. 9 and 10 but the strain is now 0.44%.

The crack in the transverse oriented fibre bundle has started to grow out to

tbe surrounding matrix.

the surrounding matrix. During observation, the crack

extended to a macroscopic crack at constant load. In the Std-SMC materials, in situ SEM observations

revealed the first damage mechanism to be filler debond cracks and micro-cracks in regions with filler/polymer

material only. These events started at 0,19% strain. The phenomenon is exemplified in Fig. 12 (i.e. s = 0.35%). The

submicron voids are due to the low-profile additives

discussed in Ref. [3]. During several observations, extensive crack growth was evident although the strain was held constant. This means that the time-scale of loading is critical and will govern the resulting damage state. At a later stage,

distinct micro-cracks transverse to the loading direction were observed at a strain of 0.47%, see Fig. 13.

Crack initiation in composites may occur by fibre/matrix

debonding, matrix/filler debonding and at fibre ends [15,17. 18]. Transversely oriented fibre bundles have been reported as initiating sites for cracks in SMC [16]. The non-linear

behaviour of SMC is due to multiple microcracks and increased crack density with strain. In cyclic tensile fatigue

Fig. 13. Crack that has moved through a substantial pan of this Std-SMC

specimen, e = 0.35%. Crack growth occurred during microscopy.

M. Oldenbo el al. / Composites: Part A 34 (2003) 875-885 883

[17], final failure is considered to occur as various small damage entities distributed in the material join and form heterogeneous macroscopic cracks.

It seems reasonable that a proper constitutive model for SMC should be derived based on the physical mechanisms observed in our experiments, see Figs. 4 and 5. Models based on damage coupled to elastic deformations are in that sense preferable compared to plasticity models very often used for metals and ductile thermoplastics. This does not mean that no yielding takes place in SMC. Local yielding is detected as one of the active mechanisms in a zone at a crack under formation in Fig. 10.

3.5. Fracture energies and bridging laws

With composite materials based on fibres longer than a few millimetres, the experimental conditions required by linear elastic fracture mechanics (LEFM) are often difficult to meet. This is because a comparably large fracture process zone develops in the vicinity of an advancing crack in these materials 16.19). The process zone is often too large compared with the crack length and other specimen dimensions. Critical strain energy release rates, were therefore determined using large double cantilever beam (DCB) specimens loaded by pure bending moments [6.8.91. J\c is used to denote the critical strain energy release rate since a 7-integral procedure was used to determine the material toughness.

Data for 7k., are presented in Table 5. Std-SMC exhibits by far the lowest value, 7 l c = 26kJ/m 2 , Flex-SMC B exhibits a comparably high / k . = 56 k j / n r , whereas the measured value for Flex-SMC A varies between 38 and 62 kJ/nr. Previous work [7] showed that the large

difference between Std-SMC and Flex-SMC B was due to longer pullout lengths of bridging fibres for the latter material. A similar but less obvious tendency for extensive pull out was observed with Flex-SMC A. Still, specimens from one batch (Batch 2) of this material exhibit Jic values of similar magnitude as Flex-SMC B.

Optical microscopy of polished cross sections showed substantial damage in the form of cracks (mainly along fibre bundles) within the fracture process zone for both Flex-SMCs. Estimations of the damage zone heights in the direction perpendicular to the crack plane were made from these observations. The process zone height extended about

Table 5

Results from D C B tests

Flex-SMC A Flex-SMC A Flex- Std-SMC

(batch 1) (batch 2) S M C B

No. of specimens 3 transv. 3 longit. 5 8

3 transv.

Jlc (Ulm-) 38 ± 5 62 ± 4 longit. 56 ± 7 26 ± 2

54 ± 2 transv.

10 mm in this direction for both Flex-SMCs. The same measure was between 1 and 3 mm for Std-SMC. Note that the fibre strength distribution is expected to be similar for the materials.

The high J\c observed for Flex-SMCs is because a large proportion of the supplied energy is dissipated in a larger material volume during fracture. Mechanisms such as matrix cracking, debonding and frictional slip between fibre and matrix during pullout are dissipating energy. Thermoplastic additives increase the toughness of the material. Since debonding and successive fibre pullout is known to be the most important mechanisms of energy dissipation, a likely mechanism for increased toughness is reduced interfacial strength [20.21] possibly through poor interfacial adhesion and reduced matrix yield strength.

The material bridging law can also be estimated from this experiment 16.22,23*. The bridging law, see Eq. (2). is a material property describing the relationship between bridging stress rrh (supplied by a number of inelastic mechanisms within the fracture process zone) and crack opening displacement 5. Important information regarding the coupling between microstructural parameters and fracture behaviour can be supplied by the bridging law.

Typical bridging laws obtained by applying Eq. (2) to experimental results are presented in Fig. 14. Following the findings in Ref. [10] it must be emphasised that the initial increase in (7 b registered for all materials does not correspond to the real bridging behaviour since no macrocrack is developed at that stage. The registered displacement that increases rrb initially is rather a consequence of beam deformation. This part should there­fore be discarded from interpretation of the true materials bridging response. Until further improvements of the data reduction procedure of DCB test curves are available, it is not possible to qualitatively study bridging stresses associated with very small S. When 5 becomes larger i.e. after the peaks in Fig. 14. the contribution to S from beam deformations are assumed to be constant, and are therefore not influencing the derivative in Eq. (2). Al l Flex-SMC curves are shifted towards larger crack openings compared to Std-SMC. For Flex-SMC B, this is attributed to the large extent of debonding and successive fibre pullout [7], The same mechanism shifts the bridging laws of Flex-SMC A towards large crack openings.

Two different curves for Flex-SMC A are presented in 14, one curve for the highly oriented material obtained with batch 1 and another for material from batch 2 (isotropic in-the-plane). Flex-SMC A (batch 1) was tested in the 'transverse' direction. It was observed that bridging stresses are generally lower in this case compared to batch 2 materials. J k , see Table 5, is also lower for Flex-SMC A (batch 1). The improvement in bridging behaviour and corresponding fracture toughness for the isotropic Flex-SMC A (batch 2) clearly stems from the larger number of fibres oriented perpendicular to the crack path.

884 M. Oldenbo el al. / Composites: Pari A 34 (2003) 875-885

4. Conclusions

Hollow glass spheres added to SMC materials (about 18% by volume) result in reduced density by about 20%. Since the softer glass spheres are replacing stiff CaCOj fillers. Young's modulus and compressive strength is reduced.

In addition, the modified Flex-SMCs contain 16-19% by volume of a thermoplastic additive. Damage mechanisms are therefore shifted to higher strains as compared with the

reference material, a conventional Std-SMC. Ultimate strain of Flex-SMCs in monotonic tensile tests is not influenced as compared with Std-SMC. In all SMCs, significant modulus reductions were observed with increasing strain due to extensive damage. A critical modulus reduction is suggested as a failure criterion rather than strength.

In Std-SMC, damage initiation is due to debonding at the CaCO} filler particle/matrix interface followed by develop­ment into microcracks. At slightly higher strains, larger matrix cracks are formed in transverse fibre bundles. In Flex-SMCs. damage initiates in transverse fibre bundles and extends to macro-scale transverse cracks. Microscopic crack extension was observed in all SMCs at constant load. This suggests a significant time-dependence of damage development processes in SMC.

J k . toughness values for Flex-SMCs were more than twice as high as Std-SMC (56 kJ/rrT as compared with 26 kJ/nr). The major reason for this is that longer fibre pullout lengths lead to high bridging stress at much higher crack-opening displacements. Since fibre tensile strength

distributions are expected to be similar it follows that the major reason is weaker interfacial adhesion in Flex-SMC.

In a fracture mechanics sense. SMCs are very tough materials as compared with, for instance, thermoplastics such as polypropylene (Jlc £ 15 kJ/m2). The reputation of brittleness is instead associated with the low threshold strain ( — 0.2%) for onset of damage mechanisms such as transverse matrix cracks. The somewhat improved perform­ance of Flex-SMCs in this respect is caused by reduced CaCO} filler content and the addition of 16-19% by volume of a thermoplastic additive. It is apparent from our data that a high proportion of transverse fibre orientation severely decreases the threshold strain for microcracking, also in Flex-SMCs.

Acknowledgements

The authors wish to thank Reichhold AS in Norway, MCR in France and Polytec Composites Sweden AB for supplying test material, test Plate and material data. This research is funded by Volvo Car Corporation. The vehicle research program (PFF) in Sweden and national program for Integrated Vehicle Structures (IVS) in Sweden.

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Global Stiffness of a SMC Panel Considering Process Induced Fiber Orientation

M . O L D E N B O , 1 , 2 ' * D. M A T T S S O N , 1 ' 2 J. V A R N A 2 A N D L. A . B E R G L U N D 2

1 Volvo Car Corporation Exterior Engineering

Dept 93610 PV3C SE-405 31 Göteborg

Sweden 2Luleå University of Technology Division of Polymer Engineering

SE-971 87 Luleå Sweden

ABSTRACT: A material model, that translates into a stiffness matrix, the second order fiber orientation tensor, described by Advani and Tucker, and the stiffness matrix of a composite with aligned ellipsoidal inclusions, has been implemented in a FE programme and validated. The stiffness of a SMC panel with known state of fiber orientation is calculated using FEM. The influence of process induced fiber orientation is analysed. The fiber orientation for a realistic charge pattern for the panel has been obtained through mould filling simulation in a separate project. It is found that the fiber orientation has a rather small impact on the global stiffness. Only 0.8% lower stiffness compared to isotropic material model is obtained taking into account the fiber orientation distribution. The main reason for the low impact of the process induced fiber orientation is that the charge is symmetrically placed in the mould leading to a symmetric fiber orientation distribution.

KEY WORDS: SMC, stiffness prediction, anisotropic stiffness, material modelling.

INTRODUCTION

TH E M A I N P O L Y M E R alternative to sheet metal in car body components are compression-moulded composites such as SMC, BMC, and GMT. The reasons are

high stiffness and low tooling costs combined with a time efficient process. In the use of these materials, there are still important improvements to be made. For instance, the anisotropy of the material is not usually taken into account in the materials selection or in the engineering design process. This paper includes calculation of the global stiffness for a hypothetical exterior hood structure considering the process induced fiber orientation distribution. The fiber orientation distribution of the hood has been simulated through

•Author to whom correspondence should be addressed. E-mail: moldenbo@volvocars.com

Journal of R E I N F O R C E D P L A S T I C S A N D C O M P O S I T E S , Vol. 23, No. 112004 37

0731-6844/04/01 0037-13 $10.00/0 DOI: 10.1177/0731684404028700 © 2004 Sage Publications

38 M . O L D E N B O E T A L .

mould filling analysis in a separate project f l ] . The use of mould filling simulation for determination of fiber orientation for a compression-moulded structure is well established. The Folgar and Tucker transport equation [2] for the orientation tensor (1984) marks the start of these calculations. Jackson, Advani, and Tucker have adapted this model for thin compression mouldings [3]. Reifschneider and Akay [4] showed that they could predict the fiber orientation of a truck hood structure through implementation of a transport equation for the orientation tensor.

An overview of a large variety of material models is presented by McCullough and Eduljee [5,6], The general idea is first to calculate the properties of a hypothetical layer with aligned inclusions followed by averaging over all fiber orientations. Ellipsoidal inclusions could be used to describe fiber bundles in SMC and a bounding type micro mechanical model based on the lower bound is developed and described by Eduljee, McCullough, and Gillespie [6,7]. Other researchers [8] model the case of aligned inclusions using models for aligned unidirectional fibers.

No work could be found where the stiffness of a geometry was calculated through implementation of one of these models into a FE programme.

The objective of the present study is (a) to implement a material modej for stiffness matrix determination in all finite elements as a function of the fiber orientation distribution. This is done through a subroutine for the FE-programme ABAQUS; (b) to study the effect of fiber orientation on the stiffness of the panel. For comparison a conventional calculation assuming isotropic material is performed. In such a material model only two independent elastic constants (E-modulus and Poisson's ratio) are used to describe the material. The material is SMC with 30 weight percent glass fibers and 45 weight percent calcium carbonate filler.

For stiffness calculation a material model based on micromechanics described by Eduljee, McCullough and Gillespie [6,7] is validated and implemented and the results are compared with the conventional calculation assuming isotropic material. The material model has been reported to give higher values than experimentally determined and Eduljee with co-workers has identified the reasons to be unconsidered structural features when establishing the elastic behaviour [7]. It is not mentioned what these features could be but possible contributors not taken into account in the model are fiber-matrix interface properties or polymer-filler interface properties of the matrix. They report that the lower bound relationship combined with an adjustable so called effective aspect ratio, when needed, gives good agreement with experimental data but reduces the model to semi-empirical status.

GEOMETRY, LOAD CASE AND FIBER ORIENTATION DATA

Meshed CAD model with the load case chosen to study is seen in Figure 1. The mesh is generated in the ANSA programme, based on the CAD model, and is constituted of quadratic and triangular shell elements. The mesh is then imported to the Hypermesh programme where load case is established. Hypermesh is also used for writing the input deck for the FE-programme ABAQUS and post processing of the calculation result. The analysed geometry is the outer structure of the hood for the Volvo S60/V70, which in production is made of aluminium. In this paper, a calculation of a hypothetical outer structure in SMC is presented. The mesh is made with quadratic and triangular shell elements with a set thickness (2.6mm), that is constant throughout the hood. The number

Global Stiffness of a SMC Panel 39

I mk j ,}2345fi

123456 / v :

I F O R C E - 2.50e+02

A Ü i

,i- v

y Figure 1. Meshed CAD model of hood (outer shell) with load case established.

of elements is 12427 quads and 542 triads giving a total of 12969 elements. The analysed loading is a global torsion case where the rear corners (where the hinges are located on the inner structure) are fixed in all three directions and one corner in the front fixed in the --direction. In the free front corner, a load (250 N) is applied and the maximum deflection on the hood, which is found near the load application point, is measured.

The fiber orientation induced during the compression moulding of this geometry has been calculated by Vahlund [l] for a realistic material charge in a hypothetical tool. The fiber orientation distribution function is given by ^(ø) that describes the probability of finding a fiber with orientation between <p and <f> + dqb relative to a reference axis [3], here chosen to be the global .v-axis of the geometry. The global .v-axis goes along the centre line of the structure, from front end to the rear end, see Figure I . I t is possible to compress *(ø) for a point of the geometry to a fiber orientation tensor, described by Advani and Tucker [9]. Whereas the fourth order tensor is a fairly exact representation of the fiber orientation distribution, the second order fiber orientation represents a more smooth curve. Here, the fiber orientation data is given in the form of the second order fiber orientation tensor.

^23458

& • / ' !

A = £7|| tl\2

« 2 1 Ö22 0 )

« I I = I Jo

*((/)) cos2 <pd<p (2)

40 M . O L D E N B O E T A L .

Ö | 2 = Ö 2 I = f sm<pco5(pcl<j), (3) Jo

«22 = / sin2 (4) JO

*(ø) is normalised so that f *fø>)d<p = \. (5)

Jo

The geometry is symmetric around the centre line, see Figure 1, therefore calculations of Fiber orientation could be reduced to calculation of one half. The fiber orientation distribution is obtained in 2500 calculation points and since only one half of the structure was used in the mould filling analysis, it means there is data for 5000 points on the complete outer hood structure. The 1-2 axes of the fiber orientation tensor coincide with the global coordinate system (x-y) of the mesh used in the mould filling simulation [1]. The local axes of the shell elements in the structural mechanics mesh are set to be projections of the global coordinate system for the structural mechanics mesh. Therefore, coordinate transformation is done to transform fiber orientation data to the global coordinate system of the structural mechanics mesh.

The principal moment A\, of the fiber orientation distribution function, is a vector in the 1-2 plane pointing at direction of highest fiber orientation distribution, i.e. the longitudinal direction. For calculation of the accurate stiffness matrix, using the fiber orientation distribution averaging scheme described below, it is necessary to have the principal material directions, 1-2, in the directions of the principal moment, Ai, and perpendicular to it. The stiffness matrix must be rotated by angle 6 which is the angle between the directions of A\ and the global .v-axis before calculating the global stiffness. The relations between the second order fiber orientation tensor, the principal moment Ax and the angle between the principal moment and the global .v-axis are [10]:

A\ = 0.5 + yjiau - 0 . 5 ) 2 + ö 2

2 (6)

tan(20) = — ^ — (o,, > 0.5) (7)

« 1 1 - «22 I t is seen that for an =«22 = 0-5 the angle 0 could be arbitrary chosen since this corresponds to perfectly random fiber orientation distribution.

Since tan (20) is periodic with the period 2B=-JI, Equation (7) is only valid for ci\\ > 0.5 and for an < 0.5 it becomes:

tan(2fl = 2 a n , 0 = ^ + ^ (an < 0.5) (8) «11—«22 2

IMPLEMENTATION OF THE MATERIAL MODEL

The SMC material could be described by an orthotropic material model, which has three planes of symmetry (for example there is no variation of the properties through the

Glohal Stiffness of a SMC Panel 41

thickness of the SMC-panel). The general stiffness matrix for an orthotropic material contains nine independent constants and is written:

Ci2 C13 0 0 0 -

C22 C23 0 0 0

C|3 C23 C33 0 0 0

0 0 0 C44 0 0

0 0 0 0 C55 0

. 0 0 0 0 0 Cf>6 -

(9)

If the fibers have random in-plane distribution, the material becomes transversally isotropic and the stiffness matrix has only five independent material constants. I f the 1-2 plane is the plane of the panel, the simplifications become: C, 1 = C22> Q23 = C| 3, C^-= C55, C« = ( l /2 )x ( C „ - C , a ) .

In this work, shell elements were used and this reduces the stiffness matrix to 3 x 3 dimension since the constitutive assumptions of plane stress is applicable:

"f in Ö12 0

I 612 Ö22 0

0 0 Ö33 .

It will be shown later that for the FE programme the transverse shear stiffness (C 4 4 and C55 in (9)) is also needed.

The microstructure of SMC consists of fiber bundles (25 mm long E-glass fibers) dispersed in a matrix. The matrix is a composite itself and is therefore often called surrogate matrix. It contains polyester resin and CaC0 3 filler particles with a diameter of around 5pm. The fiber bundles typically consist of 400 fibers, giving bundles of effective (cylindrical) aspect ratio of L/40/'ni,ei- [11]. The SMC used in this work had fiber radius of 7 urn, giving the aspect ratio 89. This relatively high aspect ratio means that the fiber bundles behave almost as continuous fibers. Since fibers are 25 mm long and the thickness of SMC-panels is 2-3 mm, it is a good estimate to assume that the system is planar with all fibers lying perpendicular to the normal of the plane. Planar systems are usually treated through orientation averaging of the elastic constant array derived from the consideration of composite with aligned inclusions [6]. Elastic constants of aligned short fiber composite can be calculated using approximate expressions suggested by Halpin-Tsai [11] or using more sophisticated micromechanics analysis of aligned ellipsoidal inclusions [7].

Orientation averaging of a short fiber composite has been made by Piry and Michaeli [8]. They used Halpin-Tsai equations to simulate stiffness of unidirectional fiber layer with fibers oriented in a certain angle interval. The thickness of the layer corresponds to the amount of fibers in the angle interval.

It has been shown by McGee and McCullough that the stiffness matrix of a composite with planar distribution of fibers could be obtained through the aggregate averaging based on the stiffness matrix of aligned ellipsoidal inclusions and the fiber orientation parameters f p and gp [12], see Table 1. Eduljee and McCullough have explained how to calculate the stiffness matrix for aligned ellipsoidal inclusions [7] based on fiber properties and properties of the matrix. The lower bound relationship has shown to give reasonable estimates of the elastic properties of a composite with aligned ellipsoidal

42 M . O L D E N B O E T A L .

Table 1. Algorithm for calculation of the stiffness matrix based on the stiffness matrix for composite with aligned ellipsoidal Inclusions and the fiber orientation parameters f p and gp.

CA I = C , , , f a n d o m - [dCi , +5 x dC 6 6 ] x ip + 5dC 6 8 x g p

C i 2 = C 1 2 , randorn + 4 x d C 1 2 X fp - 5c/C 1 2 x g p

Cia = Ci3, r a n d o m - dC« X fp

C 2 2 = C 2 Z , random - [dC2z + 5 x dC 6 6 ] x fp + 5dC 6 6 x g p

C 2 3 = C 2 3 , r a n d o m — d C 2 3 X fp

O33 = £ 3 3 . r a n d o m

C 4 4 = C 4 4^ r a n d o m ~ dC^ X fp

C 5 5 = Cgs, r andom — dCgs X fp

0BB = Ces, r a n d o m + 4 x dC 6s x /p - 5 d C 6 6 x g p

Where: dCy = C//( r a n d o m - C//, al igned

r a n d o m = C 2 2 _ r a n d o m = ' ( + u /f=(1/4) x [Cu, allgned + C 2 2 , al ignad+2 X C 1 2 i a l igned]

C 1 3 , r a n d o m — C 2 3 , r andom = ' U=(1/B) X [Cu, a l l g n e d + C22, aligned — 2 X C i 2 , a l laned + 4 X Cee, a l igned]

C 4 4 , r a n d o m = C 5 5 , random = g I = (1/2) X [Ci 3 i al igned + C 2 3 , a l igned]

C-I2, r a n d o m = ^ — U g = (1 /2) X [ C ^ al igned + C 5 5 , a l igned]

C 3 3 , r a n d o m = " n = C 3 3 , a l igned

Cee, r a n d o m — U

Reprinted from [12] with kind permission from S. H. Munson-McGee. © 1983 Plenum Press, New York.

inclusions, [Cflh g l :

= [CL + V*f{[H]j - Vm[EfmTX (11)

where

[H]f = {[C]f-{Qmr\ (12)

and [E]®tl is a modified Eshelby tensor which is calculated based on the isotropic matrix properties as well as the ellipsoids effective aspect ratio. The indexes used in (11) Jb, al denote lower bound properties for aligned ellipsoidal inclusions whereas in and / denote matrix respectively fiber.

The fp and gp are calculated using the second order fiber orientation tensor (1) obtained in the mould Filling analysis described before. The definitions of the fiber orientation parameters and relations with the fiber orientation tensor are:

f p = 2* [ cos 2<pd<p-\ (13) Jo

Through identification with the fiber orientation tensor described above it is seen that:

f p = 2*an-\ (a,2 = 0) (14)

Since the averaging scheme, see Table 1, is only accurate for principal material directions in the direction of A\, corresponding to A 1 2 = 0 and A \ =a\ \ the general relation between the second order fiber orientation tensor and the fiber orientation parameter, f p , to be implemented is:

fp = 2*AX - 1 (15)

Globol Sliffness of a SMC Panel 43

gp = (5/5)* (&* j f cos4 qbd<p-3^ (16)

An approximation for possible for planar orientations, is [12]:

2%(7 - 2//;) 5(4-2/; ;)

(17)

The expressions for composite stiffness expressed through/ y„ gp, and the stiffness matrix for composite with aligned ellipsoidal inclusions are given in Table 1.

RESULTS AND DISCUSSION

Calculation of Stiffness Matrix for Aligned Ellipsoidal Inclusions

Values for modulus and Poisson's ratio of E-glass fibers have been reported to be between 70-76 GPa [5,6,8,11,13] and 0.21-0.22 [6,13] with an average of around 72GPa respectively 0.22. The Poisson's ratio for polyester resin has been reported to be 0.37-0.39 [11] and for CaC0 3 a Poisson's ratio of 0.3 have been used [5], Calculation of the surrogate matrix's Poisson's ratio with a composite sphere model, such as the S-mixing rule [5] gives a result of 0.35. References of matrix modulus are not so easy to find. In [8], the modulus of a surrogate matrix consisting of polyester resin and filler was reported to be 4.2 GPa for a similar SMC as tested here. This is very low since the modulus of unfilled thermoset polyester is reported to be 2.1-4.1 Gpa [11] and in our case the volume fraction filler in the surrogate matrix is 41%. Matlab was used to determine the matrix stiffness by fitting the model to the experimental data in Table 2. In doing so it is necessary to have all other parameters, except the fiber orientation distribution set. As discussed before, the elastic behaviour of SMC is very close to the behaviour of continuous fiber composites meaning a set aspect ratio could not be expected to add a lot to the error of the calculations. Fitting Matlab calculations to test results we obtained surrogate matrix stiffness of 5.07 GPa.

The resulting stiffness matrix for the composite with aligned ellipsoidal inclusions under assumption of plane stress becomes:

C y , aligned

20.1624 2.5234 0

2.5234 7.9539 0

0 0 2.7816

(GPa) (18)

The resulting engineering constants are:

^Longitudinal = 19.36 GPa ^Transverse - 7.64 GPa G L T = 2.75 GPa

vTL = 0.317 vLT = 0.125

The two transverse shear stiffness components are very similar at moderate orientations [5]. The transverse shear stiffness was calculated, with Equations (11)-(17). For random fiber orientation distribution we obtain G ] 3 = C? 2 3 = 2.71 GPa and for the aligned

44 M . O L D E N B O E T A L .

ellipsoidal inclusions it is G I 3=2.78 and G 2 3 = 2.63. On the other hand, for the fiber orientation distribution identified in Table 2 it becomes G ) 3 = 2.73 and G 2 3 = 2.69. This illustrates that for a moderate fiber orientation it makes no sense to implement a fiber orientation distribution dependent transverse shear stiffness. The transverse shear stiffness for the case of random fiber orientation distribution was used in the calculations.

Go = Gn = 2.71. (19)

Validation of the Material Model and the Subroutine

In order to validate the material model as well as the subroutine for the FE-programme ABAQUS compression moulding of plates with high degree of fiber orientation has been performed. In an area with good margins to the edges, tensile test specimens were cut in two directions (in flow direction and perpendicular to flow direction), see Figure 2. Tensile tests were done, following ASTM-D3039, to measure the Young's modulus. The results are seen in Table 2. As described before, Matlab was used for testing of the micromechanical model and to determine the matrix stiffness by fitting to experimental data. Here there are two unknown parameters (matrix modulus and state of fiber orientation) and two test results (longitudinal and transverse E-modulus) meaning the material model can be arranged to give exact results, see Table 2. In order to make sure that the subroutine would

Table 2. Results of tensile tests. Validation of the subroutine and micromechanical model by fitting the matrix stiffness to experimental data. 0 corresponds to the angle between the

direction of the principal m o m e n t , A 1 t ot the fiber orientation and the reference axle.

Strain, e = AL/L, Fiber Measured with AL from Orientation, E-Modulus ABAQUS and E-modulus Identified with

[GPa (stdev)] Subroutine. Table 1 [an /a^]

Longitudinal 12,59 (0.38) 0.318% 12,59 0,640/0 (6 = 0°) Transverse 9.55 (0.39) 0,430% 9.55 0.360/0 (0 = 90°)

Figure 2. Picture showing how panels were pressed and test specimens were cut for mechanical model, A and B are tensile test specimens.

validation of the micro

Global Stiffness of a SMC Panel 45

also give the correct value, simulation was made in ABAQUS with a single element mesh with load case as in a tensile test. The behaviour of the specimen is described in the material model implemented in the subroutine. The resulting displacement, AL, is used for calculation of the strain in the virtual specimen, see Table 2. The E-modulus of the composite is verified using uniaxial applied stress, a. The fiber orientation distribution of the virtual specimens, which have identical modulus as specimens taken from the test plaques, could finally be identified, using Table 1, to fp=0.28.

FE Calculations

The FE solver used is ABAQUS and the shell elements used is S3R and S4R type which means reduced integration to one calculation point in the centre of the element. Since there are 12969 elements this means that there are also 12969 calculation points. In a conventional calculation the material is considered isotropic and described only with the E-modulus and the Poisson's ratio. This means that the stiffness matrix is identical at all calculation points. The E-modulus and Poisson's number determined in Table 1 for the case of random fiber orientation distribution (11.28 GPa respectively 0.32) is used. The resulting maximum deformation using the conventional, isotropic, material model is 84.4 mm giving a stiffness of 2.96 N/mm for the load case. The same maximum deformation and stress is found with the implemented material model if setting the transverse shear stiffness equal to the isotropic case and fiber orientation distribution state to perfectly random (an =0.5 and ct\2 = 0) at all calculation points (which verifies the subroutine).

In the next step a transversally isotropic material model is used. This calculation is performed using the subroutine, described below, and with perfectly random fiber orientation distribution in all calculation points. The only difference to the isotropic material model is that the transverse shear stiffness (C44 and C5s) differs from C 6 6 . The maximum deformation now becomes 84.9 mm whereas the maximum stress remains unchanged. This corresponds to a stiffness of 2.94N/mm which is 0.6% less than for isotropic material.

The micromechanical material model was implemented using a Fortran code in ABAQUS through the subroutine of user defined material (UMAT). As previously described, the stiffness matrix must be rotated by an angle 9 determined as described in Equations (7) and (8) to be valid. The equation becomes [14]:

^-corrected — ^ ' C B, (20)

A =

2 2 c s

2 2 S C z

2sc

—2sc 2 2

—CS es c — s

B =

c 2 s 2 se

s 2 c 2 —SC

-2cs 2cs c 1 — s 2

with c = cos 6 and s — sin(9

At each calculation point, the fiber orientation tensor is imported from a datafile using the subroutine SDVINI, This subroutine is used in ABAQUS for defining initial values of solution dependent state variables. The complete fiber orientation data file is processed at each calculation point and the fiber orientation tensor closest to it is chosen for calculation

46 M . O L D E N B O E T A L .

stepl Incl, Ul.OOe-tOO

Displacements

> 7.29e+01 < 7.29e+01 < 6.086401 •c 4.86e+01 < 3.65e+01 < 2.43e+01 < 1.22e+01 < 3.68e-34

> 7.29e+01 < 7.29e+01 < 6.086401 •c 4.86e+01 < 3.65e+01 < 2.43e+01 < 1.22e+01 < 3.68e-34

> 7.29e+01 < 7.29e+01 < 6.086401 •c 4.86e+01 < 3.65e+01 < 2.43e+01 < 1.22e+01 < 3.68e-34

> 7.29e+01 < 7.29e+01 < 6.086401 •c 4.86e+01 < 3.65e+01 < 2.43e+01 < 1.22e+01 < 3.68e-34 •

> 7.29e+01 < 7.29e+01 < 6.086401 •c 4.86e+01 < 3.65e+01 < 2.43e+01 < 1.22e+01 < 3.68e-34

Max . Min =

• &61e+01 3.68e-34

F/gure 3. ffesu/r of calculation (hood outer shell) using material model which take process induced fiber orientation into account. Maximum deflection for the load case is 85.1 mm.

of the stiffness matrix with the UMAT subroutine. This means that the local stiffness matrix is defined for all elements of the mesh. The transverse shear stiffness (19) is given as a set value for all calculation points. The calculation result for a material model that takes local fiber orientation into account is seen in Figure 3. The resulting maximum deformation now is 85.1 mm giving a stiffness of 2.94 N/mm for the load case. This is only 0.8% less than for isotropic material. The process induced fiber orientation from the used charge pattern was moderate. The principal moment of fiber orientation, A\, had a maximum of about 0.6 [1], This could be one reason for the small difference in results between isotropic (random) and fiber orientation dependent model. For comparison, the fiber orientation is increased by an arbitrary A/J, = 0.2 maintaining the directions of the principal moment A\. This corresponds to a centred charge but with smaller tool covering than previously. The maximum deformation is only slightly changed and is now 85.6 mm giving a stiffness of 2.92 N/mm, which is 1.4% less than for the isotropic material. The charge pattern is centred in the middle of the structure, and as the fibers rotate in the material flow that appears during moulding they tend to point in direction of the edges [1,4]. In this load case it means that weaker areas with unfavourable fiber orientation distribution are compensated with other areas with favourable fiber orientation. To verify this it would be interesting to see a calculation where a more obviously unfavourable fiber orientation distribution is set on the hood. In. Figure 4, the result of such a calculation is seen. In this calculation, the fibers are set to be 100% aligned in the direction perpendicular to the diagonal of the hood on which the load is applied in one end. This corresponds to 0-30° which translates to an =0.75 and o 1 2 = 0.433 in all calculation points. Now the deformation is 93.2 mm corresponding to a stiffness of 2.68 N/mm, which is 9.5% less than in the isotropic case.

One might argue whether Mises equality stress is applicable on anisotropic material but it is a well-known method to determine the stress state in a shell. The maximum (Mises) stress is 106MPa for the fiber orientation distribution obtained in the mould filling

Global Stiffness of a SMC Panel 47

stepl incl, t=»1.00e+QÖ Displacements

> 7.9964-01 <7.99e+01

6.66*1 Ql < 6.336+01 «;3.99e+01

BB <r2£Bo+01

Hi «a.33e+01 < 3.90e --34

M M - • 9.32e+01 Mia = 3.906-34

Figure 4. flesu/f of calculation (hood outer shell) with fibers set to be alligned along the diagonal, starting on the front, right side. Maximum deflection for the load case is 93.2 mm.

stopl incl, t*1.00e+00

Mises Stress

r

0M >0.1Ie+OX <9.lle+0l < 7,«>e+01 < 6.07e+01 < 4.55e+01

• 1 <3.04e+01 H i < J..52e+01

< 1.l8e-42

Max: Mia - l.Wc-42

F/gure 5. Result of stress distribution (Mises) calculation (hood outer shell) using a material model which take process induced fiber orientation into account. Maximum stress for the load case is seen to be 106 MPa.

analysis, see Figure 5, The std-SMC cannot be loaded over 20-30 MPa continuously, because severe damage, leading to failure, will develop quickly [15], 106 MPa is also higher than the strength of std-SMC which is reported to be 68-90 MPa [15,16]. I t could be seen, in Figure 5, that the maximum stress outside the areas where the constraints are located is less than 30.4 MPa. Normally, reinforcement is used at these points, reducing the stress

48 M . O L D E N B O E T A L .

concentration. For a real case it is necessary to include the reinforcements when calculating if the design is strong enough.

The calculation time using the anisotropic model is 4700 s with the used computer (112 CPU IBM SP with 375 MHz per CPU). The major time consuming operation is as expected running through the complete list of 5000 points fiber orientation distribution data at each calculation point. I f a list of only one point of fiber orientation distribution data is used, the calculation time is reduced to 210 s.

CONCLUSIONS

1. A material model based on micromechanics, for determination of local stiffness as function of fiber orientation distribution, has been validated for a SMC with 30 w% glass fibers and 45 w% CaC03 filler and successfully implemented in the FE-programme ABAQUS through the user defined material option of the UMAT subroutine. The subroutine performs orientation averaging based on theory by McCullough, Jarzebski and McGee of fiber orientation distribution described with the Advani and Tucker second order fiber orientation tensor.

2. Stiffness of the outer structure for a hypothetical SMC hood has been successfully calculated based on a meshed CAD model and a global torsion loadcase. Shell elements are used for these calculations. The local stiffness has been defined in two ways: (a) using the micromechanical model; (b) an isotropic material model. For the micromechanical model the local fiber orientation distribution determined with mould filling simulation has been used.

3. In the analysed case, the difference between results using an isotropic material model and using a material model that takes fiber orientation into account was only 0.8%. The process induced fiber orientation from the used charge pattern is moderate, due to a fairly large charge area (about 50% mould coverage). However in this case, where global stiffness is calculated, the main reason for the small difference is that the weaker areas are compensated for by stiffer areas. This happens because the hood is symmetric around the centre line and the charge is placed in the centre of the mould. Under these circumstances, the influence of fiber orientation on global torsion stiffness of a SMC panel seems to be very moderate, since increasing the fiber orientation by A/^ = 0.2 increases the gap to the isotropic case only to 1.4%. The fiber orientation has, however, a large influence if it is very unfavourable. I f the orientation is mainly perpendicular to the loading direction the difference to a isotropic calculation will be almost 10%.

ACKNOWLEDGEMENTS

The authors wish to thank calculation engineer Johan Klingberg at "Volvo Car Corporation (VCC) and mechanics engineer Magnus Svanberg at the Swedish Institute for Composites (SICOMP) for support on the FE programmes used. This research is funded by VCC and the Vehicle Research Programme (PFF) in Sweden.

Global Stiffness of a SMC Panel 49

R E F E R E N C E S

1. Vahlund, C F . (2003). Using a Finite Volume Approach to Simulate the Mouldfilling in Compression Moulding, Journal of Reinforced Plastics and Composites, 22(6): 499-515.

2. Folgar, F . and Tucker, C L . (1984). Orientation Behaviour of Fibres in Concentrated Suspensions, Jownal of Reinforced Plastics and Composites, 3: 98-119.

3. Jackson, W . C , Advani, S .G. and Tucker, C L . (1986). Predicting the Orientation or Short Fibres in Thin Compression Moldings. Journal of Composite Materials, 20: 539-557.

4. Reifschneider, L . G . and Akay, H.U. (1994). Applications of a Fibre Orientation Prediction Algorithm for Compression Molded Parts with Multiple Charges, Polymer Composites, 15(4): 261-269.

5. McCullough, R . L . (1990), Micro-Models for Composite Materials - Particulate and Discontinous Fibre Composites. In: Carlsson, L .A. and Gillespie Jr., J.W. (Eds.), Delaware Composites Design Encyclopedia, Vol. 2, pp. 93-142, Technomic Publishing Co., Inc., Lancaster, Pa.

6. Eduljee, R . F . and McCullough, R . L . (1993). Elastic Properties of Composites, Structures and Properties of Composites, pp. 381-474, V C H , New York.

7. Eduljee, R . F . , McCullough, R . L . and Gillespie, J.W. (1994). The Influence of Inclusion Geometry on the Elastic Properties of Discontinous Fibre Composites, Polymer Engineering and Science, 34(4): 352-360.

R. Piry, M. and Michaeli, W. (2000). Stiffness and Failure Analysis of S M C Components Considering the Anisotropic Material Properties, Macromol. Mater. Eng., 284/285: 40-45.

9. Advani, S.G. and Tucker III , C L . (1987). The use of Tensors to Describe and Predict Fibre Orientation in Short Fibre Composites, Journal of /theology, 31(8): 761-784.

10. Lovrish, M.L. and Tucker, C L . (1985), Automated Measurements of the Fibre Orientation in Short Fibre Composites, ANTEC'85, pp. 1119-22.

11. Hull, D. (1981). An Introduction to Composite Materials, Cambridge Solid State Science Series.

12. McCullough, R . L . , Jnrzebski, G . J . and McGee, S.H. (1981). Constitutive Relationships for Sheet Molding Materials. In: Proceedings of a Joint U.S. - Italy Symposium on Composite Materials, 15-19 June.

13. Kia, H . G . (1993). Sheet Molding Compounds - Science and Technology, Hanser Verlag, Munich.

14. Wysocki, M . (2001). In: H . Damberg, (Ed.) , Beräkning och simulering, Komposithandboken, Industrilitteratur AB.

15. Oldenbo, M., Fernberg, S.P. and Berglund, L . A . (2002). Mechanical Behaviour of S M C Composites with Toughening and Low Density Additives, Composites Part A (submitted).

16. Taggart, D . G . et al. (1979). Properties of S M C Composites, Center for Composite Materials, University of Delaware, Newark, Delaware, Report No C C M - 7 9 - I .

A constitutive model for non-linear behavior of SMC accounting for linear viscoelasticity and

micro-damage

M. Oldenbo1'2' * and J. Varna2

1) Volvo Car Corporation, Exterior Engineering, dept 93610, PV3C2 SE-405 31 Göteborg, Sweden

2) Luleå University of Technology, Division of Polymer Engineering SE-971 87Luleå, Sweden

Submitted to Polymer Composites

Fundamentals of the present paper have been presented at The 2003 Annual Technical Conference of the Society of Plastics Engineers (ANTEC'03), Nashville, Tennessee,

USA, May 4-8 (2003), Paper no 345

A B S T R A C T

An approach for modeling SMC composites as viscoelastic damageable material is presented. Continuum damage mechanics theory by Chow and Wang is used in combination with linear viscoelasticity. The model is applied to a modern SMC composite material containing both hollow glass spheres for low density and toughening additive for improved impact resistance. Tensile tests and uniaxial creep test are employed to build the constitutive model. Validation is done by comparing test data with simulations of uniaxial creep on material with different degrees of damage. The model has good accuracy at moderate damage levels under controlled time-dependent crack propagation. Tensile testing at two different fixed strain rates is simulated using quasi-elastic method to calculate relaxation modulus. The model predicts the stress-strain curve with good accuracy until the region close to failure, where new mechanisms not accounted for are taking place. Finally a simulation of a cyclic tensile test with increasing maximum strain per cycle is performed, and since both damage and viscoelasticity are included in the model, the slope change, accumulation of residual strain and hysteresis in the stress-strain, loading-unloading curve are predicted.

* Corresponding author. Address: Volvo Car Corp., Dept. 93610, PV3C2, SE-405 31 GÖTEBORG, SWEDEN. Tel.: +46-31-32-56-924; Fax.: +46-31-59-10-34; E-mail: moldenbo@volvocars.com

1

NOMENCLATURE

er , e, Stress and strain.

E, v , C/j Young's modulus, Poisson's ratio and stiffness matrix for the

undamaged planar isotropic composite. M,j Damage effect tensor.

/_) Damage parameters.

Yi Thermodynamic forces for damage evolution.

aj, ej, Yj Bar above parameter denotes max value of the parameter.

Ej, v„, CJJ , Mechanical properties of the damaged composite.

o i (/),£,(/) Relaxation stress and creep strain.

E(t), C, (t), 5- (t) Time-dependent mechanical properties of the undamaged

composite.

r,. Relaxation times in Prony series.

AK,BK,CK Coefficients in Prony series.

2

1 INTRODUCTION

Sheet Molding Compound (SMC) is a well known ternary composite made from thermoset polyester, calcium carbonate particulate filler and 25 mm E-glass fibers dispersed in bundles [1]. The bundles contain 180-400 fibers [2,3] and the effective aspect ratio is related to the bundle rather than the individual fibers [3,4]. The polyester resin contains thermoplastic additive for curing shrinkage compensation, i.e. low profile additive (LPA) which is dissolved in a styrene solution [1]. The stamping process to make panels of SMC out of SMC prepreg is a very efficient process for composites, with cycle times down to a few minutes. During the stamping process, fibers are oriented because of the material flow. In a previous work we showed how to take the fiber orientation into account when calculating the stiffness [5]. Because of its fast cycle time, SMC is suitable for relatively high production volume and it is used, for example, for exterior panels on cars. It is noteworthy that SMC has the highest production volume of the thermoset fiber reinforced composites, accounting for nearly 40% [6]. From an industrial perspective, it is of interest to be able to simulate performance of such SMC panels with finite element (FE) analysis, which requires that an adequate material model is available. When simulating perfonnance correctly, competition from alternative materials is met by an optimized SMC structure which is not over-dimensioned and does not have a lot of expensive metal reinforcement.

The development of damage in SMC is due to micro-cracking in a highly heterogeneous material. Fitoussi et al. [7] have modeled the damage development from a micromechanics approach; describing fibers by the theory of ellipsoids, originated from Eshelby's (1957) [8] description of the elastic field of an ellipsoidal inclusion. A statistical failure criterion for the identified failure mechanism of interfacial shear failure of the fibers was introduced. Guo et al. [9] have applied the model to three simple structures through implementation into the FE-program ABAQUS. Kabelka et al. [10] used a different approach. The fiber bundles were modeled as unidirectional (UD) plies followed by orientation averaging of the fiber orientation distribution. In this way the material is built up through the thickness of a plate. The model predicted failure in bundles as a function of orientation to the load using failure criteria for the UD-plies which contain transverse tensile strength and in-plane shear strength of the bundles as parameters. Using a continuum damage mechanics approach, it is possible to describe damage in the material in terms of decreasing stiffness over strain, see for example [11,12,13,14], and to measure the decrease in low-frequency cyclic tests with increasing maximum strain per cycle. Dano et al. [13,15] have shown that a general damage mechanics theory by Chow and Wang [16], derived from the theory of equivalent strain energy, could be used to model the in-plane damage development in short fiber composites. The analyzed composite was fabricated using a spray-up open mould technology with 30 % by weight of 30 mm long fiber bundles. The model was validated for a unidirectional load case and, with certain assumptions, it can be used to describe 2D damage development.

3

SMC materials are tending to move towards using different new additives to improve their performance, such as hollow glass spheres for lower density or toughening additives for improved damage resistance. Hollow glass spheres in SMC have been studied, see for example the papers by Ingham et al. and Gregl et al. [17,18]. A material containing both those additives has been tested [2,19], and in a cyclic tensile test the material was found to show hysteresis and residual strain, indicating viscoelasticity, damage and possibly viscoplasticity. Both LPA and toughening additives, generally, are of thermoplastic content and thermoplastics generally are considered as much more viscoelastic than thermosets. As a result, this composite is probably more viscoelastic than could be expected from a standard SMC material containing only low-profile additive. A coupled viscoelasticity-damageable material model for composites has been described in the literature, but rarely for short fiber composites and not at all for SMC, to our best knowledge. Weitsman [20] has used a power law to describe viscoelasticity and a scaling factor, for damage to model swirl-mat composite in unidirectional creep. Kumar et al. [21] have introduced a pseudo strain energy defined in Laplace domain to study viscoelasticity of laminates with different degree of damage. The objective of the present paper is to develop a simple model which applies the Chow and Wang continuum damage mechanics theory [16] to SMC, and include also linear viscoelastic effects. By adding the viscoelastic effects, we can, for example, explain the hysteresis in the stress-strain curve in a cyclic loading. This would not be the case using a pure elastic damageable material model. The approach is based on the assumption that damage and viscoelastic effects may be separated. Thus, damage development is considered in the elastic formulation. The relaxation functions of the damaged composite are presented as a product of two terms representing damage and viscoelasticity, respectively. The model is validated in unidirectional creep test of samples with different degrees of damage.

2 M A T E R I A L M O D E L O F T H E D A M A G E D V I S C O E L A S T I C C O M P O S I T E

The material model is developed using the following simplifying assumptions, which separate damage and the viscoelastic response: A. The damage evolution is assumed time-independent and the damage state depends

only on the maximum stress state experienced in the loading history. This allows the damage evolution law to be determined in high strain rate loading tests where viscoelastic effects do not have time to develop.

B. The viscoelastic response of the damaged composite is analyzed considering creep compliance in the form of a product of two terms: damage related and material viscoelasticity related. The latter is determined in creep tests using undamaged specimens.

C. Validation is performed considering viscoelasticity in fixed damage conditions. This may be tested by first loading the composite to a high load level to reach a given damage state and then performing viscoelastic characterization.

4

2.1 Elastic analysis of damage accumulation and related material response

2.1.1 Continuum damage mechanics model The composite, which in its virgin state may be considered as a macroscopically in-plane isotropic material, becomes anisotropic due to damage accumulation in uniaxial tensile loading. We assume that the damage state depends only on the maximum load to which the specimen was subjected in the previous loading history. Further, we assume in this section that elastic properties' degradation is the only measure of damage, and all viscoelastic effects on damage development may be neglected. It is reasonable to assume that the composite in the damaged state is macroscopically orthotropic with symmetry axes coinciding with the loading axes. It follows that, in general in-plane static loading the orthotropy axes of the damaged material, that was originally in-plane isotropic, will coincide with the principal strain directions. This is because damage will develop mainly in the directions of principal strain. Further on, we assume that damage develops homogenously through the cross section.

Under these assumptions, the anisotropic elastic damage theory by C L Chow and J. Wang [16] may be used. For completeness we repeat here the basics of this theory. In this theory an effective stress tensorcf, (written in vectorial form) is introduced to represent the stress magnification due to a decrease of the effective cross section as a result of damage development. This effective stress is related to the "usual" stress tensor cr, through the damage effect tensor M0{Dk):

ai=Mi]{Dk)ai ( 1 )

Here damage parameters Dk is represented in MH by three damage variables in the three

principal directions. The damage variables are implicitly related to micromechanical damage mechanisms. In the principal coordinate system, which was assumed to coincide with the global system, the formulation forA/v ^ ) is [16]:

I M „ N =

-D, o

0

' " A

0 d .

<>• !) • 1)

o

o

(2)

The elastic stress-strain constitutive relationships of the damaged material were obtained using the elastic energy equivalence concept. It states that the complementary energy of a damaged material has the usual form i f the effective stress <r, is used.

5

Simple derivation using this postulate leads to the following expression for the effective elastic stiffness matrix of the damaged composite:

[SY =[Mf[cr[M]

In an explicit form we have:

1 - V -V

(\-DxJ ( ' -D .Xi -Dj (I-TJ.XI-D,)

1 -V

(\-D2l v

20 + '-)

( i -D jb -D , )

0

0

2(1 +v)

d-D,*-/?,;

(3)

2(1 +v)

(4)

Here £ is the elastic modulus and v is the Poisson's ratio of the undamaged composite. The random composite material analyzed in this paper, is not isotropic because its out-of plane properties are different from the properties in the plane of the plate. When focusing as usual on thin composite plates, the above relationships are applicable for in-plane properties only and the elastic stress-strain response simplifies as follows:

\£'\ - V

( • - A X i - A ) (\-D2j 0 0 y)

(5)

I f the loading is uniaxial with onlyu, * 0, equation (5) becomes:

1

E(\-Dx) e7

-v E(\-Dx)(\-D2)

-CT, (6)

2 . 1 . 2 Dependence of damage variables on applied strain Elastic stress-strain relationships of damaged laminate in uniaxial tension may be formally expressed using elastic constants of damaged composite E] and vu:

6

Comparing (7) to (6) one may obtain:

/? =E(\-D,y (8)

.1Z£L (9) 1 - A

'2

Since £, and v p , after loading to a certain strain (or stress) level, may be determined using the er, -e, and e2 - f , curves, Eqs. (8) and (9) may be used to determine damage variables as a function of the maximum load CT , applied during the loading history:

- r M y E (io)

V 1 2(<J,)V

2.1.3 Dependence of damage variables on associated thermodynamic forces

In a thermodynamics approach, the damage evolution is governed by associated thermodynamic forces. The thermodynamic forced associated to a certain damage variable Dj may be found using the expression for strain energy of the damaged composite, *P, and taking derivative with respect toZX:

ye _ jgjgj i g2 , + f l 2 )

2£Y1-Z),)2 £ ( l - A X l - ß j ) 2£ (1 -D 2 )2 £(1-D,)(1-/J>.)

^ = - V - (13) ' 3D,

The expressions of the thermodynamic forces for isotropic material with anisotropic damage introduced as described above, stand as follows [13]:

r x - a - 3 + 4 (14) £(1-D,) £ - ( l - D , ) 2 ( l - A ) 2G(1-D,)-(1-Z)2)

F = _ ^ v ° £ l + < (15) 2 E(l-D2Y £ (1-Z) , ) 2 ( l - / J l ) 2G(1-/J>,)2(1-Z)1)

E where G is the shear modulus of the virgin material (G = ).

& 2(1 + v)

7

I f the loading is uniaxial and applied in direction 1:

7,= !

r Y2 = 0 (16) £(1-Z),) 3 - v '

The evolution law is formulated i f the dependence of damage variables on thermodynamic forces is defined. In a general loading case both damage variables are functions of both associated forces, and identification of these functions using test data is not simple. However, when it is done

A = f ^ X ) (17)

D2=f2{Yx,Y2)

Eqs. (14), (15) and (17) may be used to determine damage variables for an arbitrary combination of stress components. Using the given stress state and the calculated damage variables, one may calculate all strain components using Eq. (5). In case of uniaxial loading Y2 = 0, and this dependence is given by the two unknown functions:

A = / ; W D2=f2(Y}) (18)

Here yjis the maximum value of ^during the loading ramp; corresponding to maximum stress state c?, during the loading ramp. It should be emphasized that Eq.

(18) alone can not be used to determine the function/, because there is a third

unknown parameter which is the maximum stress history, CT , . Experimental data relating stress and the damage variable must be involved in the identification procedure. The procedure is the following (see [13]): we determine for each specimen D},D2 and 7, from Eqs. (10), (11) and (16) where er, is used as a measured parameter. A plot of D, versus Yj is made and used to determine the functional relationship Eq. (18).

Simulation of the loading curves of the composite in elastic material formulation with growing damage may be performed as follows. For example, the stress-strain curve in uniaxial loading is calculated point by point using Eq. (6). The damage variables corresponding to a certain stress state in Eq. (6) may be obtained solving the system of Eqs. (16) and (18). However, in the simple uniaxial case, the damage parameter dependence on stress is obtained directly from experiment (see Eqs. (10) and (11)) and the application of the described methodology is just for illustrative purposes to understand the damage variable and associated force relationships.

8

2.1.4 Viscoelastic material model of damaged composite The constitutive relationships of a linear viscoelastic composite at a fixed state of damage may be obtained from elastic relationships, using the principle of correspondence in linear viscoelasticity:

V~ da ei(t)=\SiJ(t-t,Dk)—^dT (19)

0 d T

1 fa

ai(i)=\Cv(t-T,D1<)—EdT (20) 0 " T

Here,S \t,Dk) is the viscoelastic creep compliance matrix corresponding to the elastic

compliance matrix in elasticity given by Eq. (5). Eq. (20) is obtained in solving Eq.

(19) with respect to stress. In general \s\* [c]"' in viscoelasticity, but since the validity of the quasi-elastic method of solution of stress relaxation and creep problems has been proven experimentally as well as theoretically, the difference may be considered as negligible. Additionally, we assume in this preliminary study that/J^ depends only on the maximum stress during the loading history and does not depend on time at a given stress state. In uniaxial loading the stress component in Eq. (20) in the loading direction may be written as:

' ~ dp oft)=\Eft-T,Dk)^dr (21)

Considering a creep test with a f t ) =a0 H(t), where H(t) is the Heavyside unit step

function defined by:

fO, t<tB

^ - ^ i i , t > , ( 2 2 )

Eq. (19) turns to:

£,{t)=Sn{t,Dk)aQ, U = l,2 (23)

Hence, creep strain measurements using composites with fixed damage states render creep compliance functions of damaged composite. From the elastic damage model presented above (see Eq. (6)) and the correspondence principle follows the explicit

form of S„:

9

E{t\\-Dxf

(24)

Ut,Dt) = ^ ± 1

E{t) ( l - A X i - A )

According to this model, damage variables play the role of scaling factors in viscoelastic creep response. The accuracy of these assumptions must be verified in tests. I f the assumption is valid, then creep tests for S,y (;,£>,.) determination are

necessary only on virgin specimens. The damage dependence on stress is obtained in elastic tests.

The results may be validated also in a relaxation test. In this test £,(1) = e0H(t) is

applied to the specimen and the loading is uniaxial. Therefore, the stress - strain relationship for this loading may be written as:

CT, {t) = \EX (t-t,Di)^dT = Ej (/, D, K (25)

where the expression for £,(?,£>,) follows from Eq. (8):

El{t,Ds) = E(t\l-D1f (26)

Eq. (25) and relaxation tests may be used for experimental determination of £, (t, Dx).

The results are used to validate Eq. (26). The transverse strain in the relaxation test is calculated as:

V de £i (0 = - K (t - r), Dl)-±dT = -vu (t, TJ, )e0 (27)

This expression and test results may be used to determine the viscoelastic Poisson's ratio for fixed damage states.

3 EXPERIMENTAL

The material analyzed is SMC, modified with hollow glass spheres and toughening additive (SMC A), see Table 1.

10

Table 1. Material composition of SMC A (based on supplier's data). From [19].

V:volume fraction, W:weight fraction, l f i b e r:fiber length and p:density.

v a

r e s i n M 23 b

Vadditiveetc ^ 16 b

V f l b e r M 21 b

20 b

Vspherest"7^ 18 b

W r e S m [ % ] 16

Wadditiveetc[ 0 / °] 10

w f i b e r M 34

W f i l l J % ] 35

W S p h e r e S [% ] 5 b

l f iber t™ 1 1 ] 25

P composite g ^ c m 1,58

a The resin is unsaturated polyester and low-profile additive ( L P A ) in styrene solution. b Calculated using density of composite and densities of different constituents. Densities are presumably 0.4 g/cm3 for the hollow spheres (glass spheres mostly used have a density of 0.37

g/cm3), 2.7 g/cm3 for the filler, 2.6 g/cm3 for the fiber, 1.0 g/cm3 for additives etc. and 1.05 g/cm3 for the hardened resin. 1 Toughening additive as well as initiator and mould release agent. 1 Measured following ISO 1183

Typical damage: transverse cracks originating from a transverse oriented fiber bundle, is seen in Figure 1. The damage development in this material typically starts in transverse oriented fiber bundles (see also refs[ 19,22]). When reaching longitudinally oriented bundles the crack is either temporarily stopped or shear failure of the matrix in the interface of the longitudinal bundle starts, ie pull-out of bundle is initiated. At high loading level, fiber breaks are also possible.

11

Loading direction

Location^ of cracks

Hollow glass sphefi

Figure 1. SEM image of microstructure and typical damage development in SMC A under unidirectional tensile loading (e = 0.44%).

3.1 Determination of damage parameters dependence on thermodynamic forces

The relationships D, = /i(Yj)and£>2 = / 2(F,)for SMC A composite are determined; based on uniaxial tensile loading tests. An initial test sequence has been conducted as follows, see Figure 2: Three SMC A specimens (numbers 1-3), were progressively pre­loaded to three longitudinal tensile strain levels: £, = 0.25%, £,= 0.5% and £,=0.75%. In previous work [19] it was found that the degree of damage in the material is negligible b e l o w = 0.25%. Consequently the elastic properties of the material, Young's modulus, E, and Poisson's ratio, v , are calculated from the measurement of material pre-loaded toe, = 0.25%. Upon obtaining initial results, we added specimens to the test sample with the aim to pre-load them to higher load levels than in the initial tests. This was done to gain knowledge of how the model will work in the stress region when the hypothesis that there is no crack growth at constant load could not be expected to be accurate. At first, one specimen (number 4), was pre-loaded toe, =0.875% including E-modulus measurement, followed by pre-loading to £,=1.0%. Another specimen (number 5) was loaded directly to £,=1.0% after measuring its elastic properties. Finally, two specimens (numbers 6-7) were progressively loaded (increments of 0.25% strain as in the initial tests) to£,=1.0%. It proved not to work because the specimens failed upon loading the last increment. In summary, a total of seven specimens have been tested.

12

Measurement of undamaged material's Young's modulus, Poisson's ratio and creep compliance

I Loading to achieve specified damage level

i Young's modulus and Poisson's ratio

measurement in tensile test

i Creep compliance measurement

£i<0.75% j — — 1

No

END

Figure 2. Schematic sequence of the testing on specimens preloaded progressively up toex =0.75%.

The pre-loaded specimens were unloaded immediately after loading. The following tensile tests to measure Young's modulus and Poisson's ratio of the preloaded specimens were performed a minimum of 15 minutes after pre-loading. This time was considered long enough to allow total strain recovery and fading of the previous loading history. Young's modulus and Poisson's ratio were always measured on the loading curve in the strain range £,=0.05% to £,=0.25%. Usually, the unloading curve is preferred [11,12,15], but we used the loading curve for more instant response with less time for viscoelastic effects to develop. No further damage development was observed during stiffness measurements. An Instron 4411 tensile test machine with an extensometer of 50 mm gauge length together with a strain gauge for measuring the transverse strain were used for these tests. Tensile test cross-head speed was 2 mm/min. The specimens were of size approximately 200 x 20 x 2 mm with tabs glued to the ends.

The damage parameters D\ and D2 are calculated from tensile test data with Eqs. (10)

and (11). Damage parameters as functions of the thermodynamic force % (see Eq. (18)) have to be obtained experimentally, as described in the previous Section. This was done through a plot of Damage variables D, and D2 against 7, followed by curve fitting to the test data.

13

3.2 Determination of creep compliance in creep tests of undamaged and damaged composite

Stiffness measurement of pre-loaded samples was followed by a creep test for a minimum of 1 hour to measure the creep compliances of undamaged and damaged material (see steps 1 and 4 in Figure 2). Specimens number 1 thru 5 were used in this analysis, but only specimens number 1 thru 3 were used to obtain time dependency of undamaged material. An extensometer was used to measure £, and a strain gauge was used to measure e2. The applied <jo in creep tests on undamaged and damaged specimens corresponded to an initial elastic response £,= 0.2%, to make sure that the

finale, would stay below 0.25%. After the creep test, the samples were left for at least

2 hours to allow for strain recovery before conducting any new tests, e.g. next loop in Figure 2. The viscoelastic behavior is assumed linear and modeled with Prony series as follows:

Sll(t,Dl) = A0+YJAte-"T'

(28)

Sn(t,Dk) = B0+JjBke-'r>

Five terms were kept in the series with:

T, =20, r 2 =10 2, r 3 =10% r 4 =104 seconds (29)

The Prony series exponentsA0,...,A4, respectivelyB0,...,B4were obtained through curve

fitting, using a least square method. The Prony series coefficients in Eq. (28) for three samples were averaged to obtain the composite creep functions. Averaging in this case leads to calculation of the average value of the corresponding Prony series coefficients of the specimens.

4 R E S U L T S AND DISCUSSION

4.1 Material model for S M C A composite

4.1.1 Initial elastic properties and the damage evolution law The stress-strain curves obtained at modulus measurements of all seven samples are seen in Figure 3. The initial three specimens are seen above, and the additional four specimens are seen below. In the initial analysis three curves for each specimen, corresponding to a loading history of £,=0.25%, £,=0.5% and £,=0.75%, were obtained. Average E-modulus of the undamaged material in these tests, obtained from the £,=0.25% curve, was 8.87 GPa. Average elastic E-modulus for SMC A composite with standard deviation from all seven measurements is:

14

£•=8440 (490) MPa v= 0.30 (0.02) (30)

0 0.1 0.2 0.3 0.4 0.5 0.6

Strain [%]

Figure 3. Stress-strain curves used for measuring of EandE^ of specimens 1

through 3 (above) and specimens 4 through 7 (below).

The normalized measured modulus and Poisson's ratio as function of preloading strain for SMC A are shown in Figure 4. It is seen that the standard deviation of the normalized modulus is low up to the pre-loading level corresponding to e, =0.875%. The values at £,=1.0% have higher spread, which could be explained by the material being close to failure with varying and high macro-crack growth rate. Poisson's ratio has much higher variations at all strain levels, even though the trend shows a clear

15

decrease. One value, at£, = 1.0%, is missing due to strain gage failure. Variations are due to the strain gauge, the gauge length of which being much smaller compared to the extensometer gauge length. The trend line for normalized E-modulus is obtained from average values at each strain level combined with the prerequisite to have undamaged material ate, =0.25%. Trend lines for normalized Poisson's ratio are obtained using a method of least squares (in which the last two data points are not included due to their spread) and prerequisite of undamaged state at £,=0.25%. The rather large reduction of both E-modulus and Poisson's ratio means that cracks mainly develop transverse to the loading direction. This is in good agreement with our previous observations of this material [19] as well as with reported behavior of in-plane isotropic SMC and short fiber composites in uniaxial tension, see for example [15,23,24]. For materials with highly oriented fibers, cracks on the other hand tend to follow the main fiber direction, see for example the studies by Wang et al. [25,26] on fatigue damage evolution of short fiber composites. Another conclusion is that the damaged composite will be anisotropic after loading even though it was considered isotropic in the beginning.

For the four damage states (corresponding to pre-loading to £,=0.5, 0.75, 0.875 and

1.0%) it is possible to plot the damage parameters Z), and D2 against the

thermodynamic force Yj, see Figure 5, using the previously explained methodology.

The relation Z), = / j (Y x ) from Eq. (18) can be modeled with a linear function:

D\ - a(Y\ ~Y0 ), (31) with a = 0.205 M P a a n d Y0 = 0.0654MPa, Yj < 0.5 MPa.

The accuracy of using a linear function is seen to be good for Y{ < 0.5 MPa. Over 0.5 MPa the relation becomes uncertain because the material is in a stress region where macro-cracks grow at constant load. These values are therefore not included in Eq. (31). The few measured values indicate a nonlinear damage increase as a function of F, curve at Yj>0.5 MPa. The relation/J, = / 2(F,)in Figure 4 is unexpected because a

negative/), is not physically correct. Trend lines for vn I v and£, / E in Figure 3 were used in order to reduce the variations within the same specimen and between specimens. The flat solid line obtained indicates a very small change of damage parameter, but the values are still negative. The reason for a negative value is too large change in Poisson's ratio (most likely due to initially not perfect unidirectional stress state). Based on these unrealistic values, we conclude that D2 should be very small in uniaxial tension in direction 1, and thus this test is not suitable to determine the D2 values. As a result we get:

A = 0 forY 2=0 and Y^<0.5MPa (32)

We used measurement of transverse strain for calculation of/_), = f2(Yl). However more exact Poisson's value measurements must be done to reduce errors due to scatter and non-uniaxial effects. For example dog-bone specimens and multiple strain gages

16

could be used. The change of Poisson's ratio by 0.01 leads to change of£>, by more than 0.03, see Eq. (11). This illustrates the necessity to have very accurate Poisson's value data in order to obtainD2 = f2(FJ). Another approach [13] to obtainD2 = f2(Yx) is to use measurement of E2 but this requires that very wide specimens must be used to introduce damage.

1.1 i

0.7 -i

0.6 J 1

0 0.25 0.5 0.75 1

Strain [%]

l . l i

* ° - 9 t

— Estimated trend

• Test data

•

0 0.25 0.5 0.75 1

Strain [%]

Figure 4. Normalized Young's modulus (above) and Poisson's ratio (below) as function of strain for SMC A.

17

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

Result: f i ( ? ; ) = a(y, - F 0 ) _

a = 0.205 Mpa"1, F 0 = 0.0654 M P a

valid for V < 0.5 M P a

0.1 0.2 0.3 0.4

K [Mpa]

Test

Linear fit (10 points)

0.5 0.6 0.7

0.00

-0.01

-0.02

-0.03

-0.04

-0.05

Result

Valid for K < 0.5 MPa

• From trendlines in figure 3

— Linear fit

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

% [Mpa]

Figure 5. Identification of the damage law relating the thermodynamic force Yx

the damage parameters Dj (above) respectively D2 (below) of SMC A composite.

18

4.1.2 Viscoelasticity of the undamaged and damaged composite As previously described, creep tests of undamaged material (specimens number 1 thru 3) are the base for modeling the time dependency. With the creep test performed on undamaged material, we obtain creep compliances Sn(t) and Sl2(t) of the undamaged SMC A composite, using the known constant stress <r0 in the creep test. The creep test curves of undamaged material (specimens 1 through 3) are seen in Figure 6.

cd O h

o (D O I a. E o

U

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

-i.- v.-X x ****

s „ x *

* 4 $ + ++++++

+ Specimen 1, E=9170 MPa A Specimen 2, E=8930 MPa X Specimen 3, E=8500 MPa

as* s s a a M süss * f f T ^ T T W ^ H -

10 too

Time [seconds]

1000 10000

Figure 6. Creep test data (3 specimens) of undamaged material.

The relaxation modulus E(t) of the undamaged composite is obtained using the quasi-elastic method and taking the average value of creep data for the three specimens in undamaged state. Prony series are fitted also to the resulting curve describing Ex (t).

The Prony series coefficients, A0 A}, £„,..., 5., (for creep compliances), respectively

C0,...,C, (for relaxation modulus) describing the undamaged material, are given in

Table 2.

Table 2. Prony sseries coefficients.

Prony series

exponents

S„(0 [MPa"

' ] 4 *io 6

Sn(t) [MPa"1]

B, *106

E(t) [MPa]

ck

k=0 129.13 41.96 7751 k=l -1.41 -0.29 462 k=2 -5.25 -0.77 314 k=3 -3.41 -1.88 267 k=4 -7.49 -5.47 457

19

The first coefficient of the Prony series describing E(t), C0, is equal to the stiffness at infinite time and is found to be 7.75 GPa. The initial modulus, calculated as the sum of Prony series coefficients, is found to be higher than measured in tensile test (9.25 GPa compared to 8.44 GPa), see Eq. (30). However Eq. (30) is based on all seven specimens whereas only specimens 1 through 3 were used to calculate the Prony series coefficients. The average of initial stiffness of those three specimens was 8.87 GPa. Still this is lower than the sum of Prony series coefficients and the reason is that the material has time to relax in the tensile test for E-modulus measurement, whereas the creep test response is instantaneous.

4.1.3 Validation

It follows from Eqs. (24) that S, ,(/,£),) and Su(t,Dk) may be simulated using the

measured Su(t) and Sn(t) for the undamaged composite, i f the damage parameters

D, and D, are known. Using compliance instead of E-modulus in Eq. (24) it is seen

how this should be done:

Sn(t,D.) = d - A ) 2

Sl2(t,Dk

Sr_0) ( l - a x i - A )

(33)

As seen in Figure 2, the three specimens of undamaged SMC A composite used in initial stiffness and creep compliance measurements were also followed by

measurements of the creep compliances of damaged material, 5, ,(?,£),) and

Sl2[t,Dk). It is now possible to compare simulations with measurements. Functions

Su(t,Dj) and S]2 (t, Dk) simulated with Eq. (33) using the average damage state of the

specimens are presented in Figures 7-9. The damage states correspond to £,=0.5% in Figure 7, £,=0.75% in Figure 8 and £, = 1.0% in Figure 9. The creep response from preloaded specimens 1 thru 3, respectively specimens 4 and 5 (in figure 9) is also seen in the figures. In a strain controlled test program as conducted here, there will be differences in<7, history of the samples, due to small variations in the initial modulus E. This will correspond to differences in damage state, i.e. differences in parameters/), and£>2 of the samples. In the validation of the model in creep of damaged material the approach is to use average damage parameters from tests in the model and compare the resulting curve with all specimens. Eventually, differences of prediction and simulation could be explained using damage state and initial E-modulus of the individual specimens. It is seen that the model has good accuracy for Sn(t) at a maximum load history corresponding to £,=0.5%, see Figure 7. For Sn(t) the result is good for specimens loaded to both £,=0.5% and £,=0.75%. At higher damage level, the scatter is higher. The model seems to slightly underestimate the value for Su(t) for

20

specimens with higher damage level corresponding to £, =0.75%. One could speculate that for highly pre-damaged material the assumption of no crack growth in creep is not valid even at the low load of the creep test corresponding to £,=0.2%. However, this was not confirmed when testing the material with the highest damage level corresponding to £,=1.0%, because the trend in Figure 9, in that case, is the opposite. This relation is even more unexpected considering the low initial modulus of specimens 4 and 5. It appears that the reason is a high scatter in measured values at higher damage level.

o c3

O

U

0.16

0.14

0.1

0.08

0.06

0.04

0.02

0

+ Specimen 1, D,=0.018, E=9170 MPa

A Specimen 2, Di=0.022, E=8930 MPa

% Specimen 3, Di=0.034,E=8500 MPa

— Model, D|=0.025

+ Specimen 1, D,=0.018, E=9170 MPa

A Specimen 2, Di=0.022, E=8930 MPa

% Specimen 3, Di=0.034,E=8500 MPa

— Model, D|=0.025

- S , 2

- t t l Ä ^ Ä - ! * m i m å * Mk å å å å m å * å $ i * * x x x >

10 100

Time [seconds]

1000 10000

Figure 7. Creep test data (3 specimens) and simulation of damaged material (AverageD i=0.025).

21

0.16

0.14

£ 0-12

^ 0.1

C 0.08

.2 "H. 0.06

E

(2 0.04

0.02

0

+ Specimen 1, D,=0.061, E=9170 MPa

A Specimen 2, D,=0.066, E=8930 MPa

% Specimen 3, Di=0.079,E=8500 MPa

— Model, Di=0.069

-Sn ü 6.A,-VV, A / V A . A A A ^ i l W ^ £ £ Å Ü

10 100

Time [seconds]

1000 10000

Figure 8. Creep test data (3 specimens) and simulation of damaged material (Average D,=0.069).

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

4. Specimen 4, Di=0.158, E=8500 MPa

A Specimen 5, D,=0.116, E=7920 MPa

— Model, D|=0.137

+ ++++H- ++H- +++++++++ +H- + ++-H-H+++ + + +

S i 4 4 A i U A i A AA", A A A A A A A a A AAA A A A A i V W \ A A A A A i P

10 100

Time [seconds]

1000 10000

Figure 9. Creep test data (2 specimens) and simulation of damaged material (Average D,=0.137).

22

4.2 Simulation of the uniaxial tensile test with constant strain rate

Consider a loading ramp in which the applied strain is increasing linearly with time:

£, = ä (34)

The relaxation modulus is described by a Prony series and obtained from compliance data using the quasi-elastic method (coefficients are given in Table 2):

£(O = C 0 + X C / - " r " (35)

The stress-strain relationship, Eq. (21), in uniaxial loading for the constant strain rate is:

<T,(0= \É{t-x,Dl)^-dT = \E(t-T){\-D,)2édT--

( l -D 1 )2

£ ' f (C 0 +XC t e - , ' - r > ' r ' ) C /T o t

After integration:

(36)

C0t + ^CkTk(l-e-'^) (37)

Eqs. (16), (31) and (37) can now be solved for any time instant, t. Since Eq. (31) is linear, the solution may be obtained in an analytical form. The resulting expression for CT, as a function of time is found to be:

f(t)(l + aY0)2

i + ^ / ( 0 2

E

(38)

with:

f(t) = £ C0/ + X Q r , ( l - e - " r ' ) (39)

Matlab was used to simulate the tensile test at several strain rates. Tensile testing was performed at constant cross-head speeds of: a) 2 mm/min (following ASTM D3039) corresponding to a standard test strain rate and: b) 0.01 mm/min corresponding to a very slow strain rate. The latter was done to look for differences due to loading rate

23

and to check the ability of the viscoelastic damageable material model to predict them. The strain rate calculated using the constant cross-head speed is usually not equal to the strain rate in the specimen. Therefore the strain rate, é, was calculated from time and strain data measured in experiments and the average value was taken for simulation. Test data from [19] for the standard test strain rate were compared with low strain rate data obtained for two specimens. The strain rates for the standard test and the slow rate test were found to be 2.0xl0~4 s~ , respectively 9.8xl0"7 s"1, and i f rounding off the latter value to l.OxlO"6 s"1 the difference is of a factor 200.

It is possible to simulate a standard tensile test with good accuracy up to £, ~ 1.0%, see Figure 10, but above £, = 1.0% the model underestimates the stress level. The measured curve is more linear at high load level and the failure is brittle. This discrepancy is unexpected considering the relation of parameter £>, to the thennodynamic force Yj in Figure 5. The few data at high load level in Figure 5 indicate that the damage growth is increasing, and higher damage growth than in the linear model should lead to even lower stress in testing than predicted. On the contrary, in a slow strain rate test the model predicts higher stress at large strains than observed experimentally, see Figure 11. An explanation is found in the way measurement and simulation differ for the two strain rates, i.e. there is an underestimation at standard test strain rate and an overestimation at a slow strain rate. The model is based on damage data that may include a small amount of time-dependent crack growth due to the time needed for unloading of pre-loaded specimens. This could explain the underestimation in a

24

standard test rate simulation and the overestimation in a low strain rate simulation. This means that the time-dependent damage evolution during the slow strain rate test is the reason for the overestimation in the simulation of it. The propagation of cracks at constant stress level of this material has been confirmed in earlier research[19]. There is no time-dependent damage growth incorporated in the model at this point. Time-dependent damage growth is also the reason why the material fails at lower strain and stress in a slow strain rate test than in a standard strain rate test.

The damage model developed can not predict the brittle failure seen in SMC composite, but it predicts a yield point (maximum in the stress-strain curve) which is used here as point of failure. The yield point, rather than brittle failure, is also found by Hinterhoelzl [27] using an isotropic viscoelasticity-damage mechanical model for reinforced plastics. The final failure of the composite involves different mechanisms (fiber fracture, bundle and fiber pull-out) that are not present in the initial stage of damage accumulation, and what is described by the continuum damage model developed here.

4.3 Simulation of a cyclic loading test with increasing maximum strain per cycle

A progressive low-frequency cyclic loading test is often employed to study the damage evolution with increasing strain on a short fiber composite, see for example [19,24,28]. In these tests, it is vital to control phenomena such as viscoelasticity or crack growth.

25

Here we focus on the viscoelasticity. In [19] we applied cyclic loading and could see hysteresis in the loops. The experimental set-up for this test is found in [19]. Such hysteresis is usually considered as due to viscous deformation, crack formation and internal friction on new surfaces created with cracks [24,28]. However, i f the viscoelastic-damageable material model is employed to simulate these curves more realistically, the role of viscoelasticity and damage could be understood. In other words it is possible to see how much of the hysteresis is due to time-dependency of the material and what is due to crack formation and internal friction. Further on, the true damage state - as determined by stiffness degradation measured usually in this test -may be analyzed i f accounting for the time-dependency of the material. Here we refer to the fact that the slope of the unloading stress-strain curves IS NOT equal to the composite stiffness i f the material is viscoelastic.

100 200 300 400 500 600

Time [seconds]

700

Figure 12. Strain ramp in cyclic loading with strain rate, é = 2x10 4 sec . First seven direction changes in Eq. (41) are annotated.

The test used is strain controlled and the strain ramp applied is seen in Figure 12. There is also a check for compression, since the material degradation and viscoelasticity will lead to compressive stress i f the strain is taken back to zero in each cycle. Compression is avoided because of the risk of a slip in the grips. Also, the

26

damage mechanisms and the effect of damage could be different in such a combined compression-tension test.

During the first step of the loading ramp, the functional relationship is the same as for a tensile test described previously. The second step of the ramp is unloading with a negative strain rate, i.e. -e. The time dependence of stress now is a sum of two integrals:

a, (/) = é [S(t - T)dr - £ \E(t - T)dr U<'<U (40)

When the loading direction is turned again at time ;,wheno~0 = 1 MPa, the functional

relationship turns out to be:

CT, (r) = £ }E(t - r)dz - £ \E(t - r)dr +e j£(t - r)dr ,tx<t<t2 (41)

Every new turn will lead to adding one additional integral. I f Eq. (40) is integrated, the result becomes:

cr,(0=(l-A) y

h<t<h

From Eq. (41) follows:

C0(2t0-t) + YiCkTk{2é"-')^ -e-" r ' -1) (42)

C T , ( / ) = (1-D,) 2 |é

U<t<ti

C0 (2t0 - 2t, +1) + 2 C, r, (1 - 2e(,'-'),T' + 2e"'"Vr' - e"/T') (43)

The average strain to failure in tests from [19] was 1.1% using é = 2xl(T4 s"1 and a strain increment of 0.1%, meaning 11 cycles. For 11 cycles the functional relationship outlined by the above turns is:

CT, ( ? ) = (1 - A ) 2 k f c (2'o - 2/, + 2t2 - 2?3 +... + 2t20 -1)] +

+ ( i - A ) y

tio<t< ti

YjCkrk(2e{,'~')'h -2e{'<-'):T' +2e"^"'^ -2e,h~'}'T' + ... + 2e(h'-)/T' -e~"T' -1) k

(44)

The first seven turning times as well as the complete strain ramp are seen in Figure 12. Matlab was used for the simulation of the cyclic test.

27

In order to understand the importance of the phenomena viscoelasticity and damage development we first consider the viscoelastic response, assuming that damage does not develop. The result is seen in Figure 13. It appears that residual strain is developing, but the hysteresis is rather small and the stress at strain-to-failure is greatly overestimated. However, the predicted residual strain is reversible and caused by the creep of the material, due to the positive value of the average applied load. It could be said that experimentally measured curves may also contain reversible effects if the specimen was unloaded before failure. There is no mechanism for irreversible strains in the model.

120

Strain [%]

Figure 13. Result of simulated cyclic test using only viscoelastic material model (no damage development) at strain ratee = 2x10"4sec'1.

I f simulating an almost instant test - here the strain rate used isé = 1 s"1 - corresponding to virtually no viscoelastic effects but allowing for damage, the result is different, see Figure 14. No hysteresis or residual strains are present, only stiffness degradation leading to an inelastic stress-strain curve. The stress level corresponding to experimental strain at failure is reasonably correct compared with test data.

28

70

Strain [%]

Figure 14. Result of simulated cyclic test at very high strain rate (t = 1 sec1) meaning virtually no viscoelasticity.

The result of the simulation using the viscoelastic-damageable material model developed here and strain rates = 2xl0"4is seen in Figurel5, together with a typical test result. Hysteresis and residual strain are predicted similarly as seen in tests. However the model underestimates the residual strain which could be due to non-linear viscoplastic or non-linear viscoelastic effects as well as cracks that do not close due to local plastic deformations. The shape of the predicted curve is more non-linear compared to tests. This is the same phenomenon as seen in the tensile test. Again, the final failure point can not be predicted because the model is not taking into account the mechanisms appearing close to failure, such as fiber breaks, fiber bundle debonding and pull-out of longitudinally oriented bundles.

29

80 t

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Strain [%]

Figurel5. Simulation and test of cyclic loading response of SMC A composite material at strain rates = 2x10"4sec1.

For FE-simulations of structures of this type of material, it is of interest to have an incremental formulation of the model. Incremental formulation exists in the literature for orthotropic viscoelasticity, see for example [29], and our future research includes expanding such a model to incorporate damage.

5 CONCLUSIONS

1. A material model that considers the SMC composite as linear-viscoelastic material with evolving damage was suggested to explain the nonlinear stress-strain behavior and the observed damage accumulation with increasing stress levels. The simplifying assumption in the model is that damage development may be considered as an elastic process and hence depends only on the maximum stress experienced by the material. This allows for damage quantification in terms of stiffness reduction in quasi-static tensile loading and unloading testing. Then the time- and damage-dependent viscoelastic functions of the composite are described as a product of damage- and time-dependent terms, where the time-dependence of the viscoelastic behavior is described by creep compliance functions of undamaged composite. Hence, the damage-dependent term serves as a scaling factor.

2. This model requires a very simple methodology for material characterization: a) creep tests of undamaged material to determine the time-dependent functions; b) stepwise quasi-static tensile tests with increasing maximum stress to

30

establish the damage evolution law. The success of the approach was demonstrated in creep tests using specimens with different degrees of damage made of SMC material with hollow glass spheres and toughening additive for low density or improved damage resistance. The second validation experiment - tensile tests with constant strain rate showed that the model can predict the rate dependence and other general trends. However, the predicted deviations from linearity were too large in the case of fast loading and too small in slow loading tests. This was explained by the effect of time on damage evolution, which is ignored in the model and should be studied more deeply in the future. The time effect appears important at high loading levels.

3. Finally, the model was used for qualitative description of the stepwise loading and unloading tests. Al l the basic features were captured. However, residual strain in experiments was significantly larger than simulated. This indicates non-linear viscoelasticity, or that irreversible strains develop in addition to viscoelasticity and damage. Perhaps these irreversible strains could be described as viscoplasticity.

4. To gain a deeper understanding of the mechanisms of damage process, a micro-mechanism (transverse cracks in bundles, debonding of bundles etc) based damage evolution analysis is planned. Finally, for a general load case the model will be formulated for incremental analysis and used in finite element analysis.

ACKNOWLEDGEMENTS

The authors wish to thank Reichhold A/S in Norway for supplying the test material used in the experimental work. This research is funded by the vehicle research program (PFF) in Sweden and Volvo Car Corporation.

31

REFERENCES

1 H.G.Kia, Sheet Molding Compounds - Science and Technology, Pub. Hanser, Munich (1993).

2 S.P. Fernberg and L.A. Berglund, J. Comp. Sei. Tech., 61, 2445 (2002).

3 D. Hull, An introduction to composite materials, Cambridge solid state science (1981).

4 R.L. McCullough, In: L.A. Carlsson and J.W. Gillespie eds., Delaware Composites Design Encyclopedia, Volume 2, Technomic Publishing Co., Inc., Lancaster, Pa, pp. 93-142 (1990.)

5 M. Oldenbo, D. Mattsson, J. Varna and L.A. Berglund, d. Reinf. Plast. Comp., 23, 37 (2004).

6 G.W. Ehrenstein, Polymerie materials - Science and technology, Pub. Hanser, Munich (2001.)

7 J. Fitoussi, G. Guo and D. Baptiste, Comp. Sei. Tech., 58, 759 (1998).

8 J.D. Eshelby, Proc. Roy. Soc. A, 241, 376 (1957).

9 G. Guo, J. Fitoussi and D. Baptiste, Int. d. Dam. Mech., 6, 278 (1997).

10 J. Kabelka, L. Hoffmann and G. Ehrenstein, d. Appl. Pol. Sei., 62, 181 (1996).

I I P . Ladeveze, In: R. Talreja ed, Damage Mechanics of Composite materials, Elsevier Science B.V., pp. 117-138 (1994).

12 J. Lemaitre and J. Dufailly, Eng. Fract. Meek, 28, 643 (1987).

13 M-L. Dano, F. Maillette, G. Gendron and B. Bissonnette, Proc. Cancom 2001: Third Canadian Int. Comp. Conf, Quebec, Canada, Aug 21-24, pp. 9-17 (2001).

14 L. Ye, Comp. Sei. Tech., 36, 339 (1989).

15 M-L. Dano, G. Gendron and H. Mir, J. Therm. Comp. Mater., 15, 169 (2002).

16 C L . Chow and J. Wang, Int. d. Fract., 33, 3 (1987).

17 T.L. Ingham, V.C. Shah, T. Botts and R.D. Anderson. SAE Tech. paper series, no 980982 (1998).

18 B.V. Gregl, L.D. Larson, M. Sommer and J.R. Lemkie, SAE Tech. paper series, no 1999-01-0980(1999).

19 M . Oldenbo, S.P. Fernberg and L.A. Berglund, Comp. A, 34, 875 (2003).

20 Y.J. Weitsman, Mech. Comp. Mater., 38, 381 (2002).

32

21 R.S. Kumar and R. Talreja, Mech. Mater., 35, 463 (2003).

22 D. Hull, An introduction to composite materials, Cambridge Solid State Science Series (1981).

23 B. von Bemstoff and G.W. Ehrenstein, d. Mater. Sei., 25, 4087 (1990).

24 F. Aymerich and P. Priolo, Key Eng. Mater., 144, 179 (1998).

25 S.S. Wang and E. S.-M. Chim, / Comp. Mat., 17, 114(1983).

26 S.S. Wang, H. Suemasu and E.S.-M. Chim, Eng. Fract. Mech., 25, 829 (1986).

27 R.M. Hinterhoelzl, Int. J. Mater. Prod. Tech., 17, 121 (2002).

28 P.-O. Hagstrand and R.W. Rychwalski, Int. d. Proc, 14, 196 (1999).

29 M.A. Zocher, S.E. Groves and D.H. Allen, Int. J. Num. Meth. Eng., 40, 2267 (1997).

33

A 2D constitutive model for FE-simulations of SMC composite accounting for linear

viscoelasticity and damage development

M. Oldenbo1'2'' and J. Varna2

1) Volvo Car Corporation, Exterior Engineering, dept 93610, PV3C2 SE-405 31 Göteborg, Sweden

2) Luleå University of Technology, Division of Polymer Engineering SE-971 87 Luleå, Sweden

Submitted to International doumal for Numerical Methods in Engineering

ABSTRACT A model accounting for linear viscoelasticity and micro damage evolution in SMC is described. An incremental 2D formulation suitable for FE-simulation is derived and implemented in FE-solver ABAQUS. The formulation and subroutine is validated with analytical results and experimental data in a tensile test with constant strain rate. Close to the "yield point", the simulation is ended due to numerical problems. FE-simulation of a 4-point bending test is performed using shell elements. The result is compared with linear elastic solution and test data using a plot of maximum surface strain in compression and traction versus applied force. Development of damage in compression is assumed to be negligible. Simulation and test results are in very good agreement regarding the slope of the load-strain curve and the slope change.

KEY WORDS Sheet moulding compound, Viscoelasticity, Damage mechanics, Finite element analysis, Strength

* Corresponding author. Address: Volvo Car Corp., Dept. 93610, PV3C2 , SE-405 31 GÖTEBORG,

S W E D E N . Tel .: +46-31-32-56-924; Fax.: +46-31-59-10-34; E-mail: moldenbo@volvocars.com

1

INTRODUCTION This paper is based on a constitutive model for non-linear behaviour of SMC composite accounting for linear viscoelasticity and micro damage developed by Oldenbo and Varna [1]. This model consider the material as initially in-plane linear viscoelastic and with orthotropic damage development. The model uses continuum damage mechanics theory by Chow and Wang [2]. It is applied to a low density SMC composite material containing toughening additive. This material has been previously studied, see [3,4]. The approach is based on the assumption that damage and viscoelastic effects may be separated, that is, damage development is considered in elastic formulation and used as a scaling factor of the viscoelastic response. The relaxation functions of the damaged composite are represented as a product of two terms representing damage and viscoelasticity, respectively. The model has been validated in unidirectional creep tests of samples with different degree of damage and applied analytically to tensile tests with different strain rates.

Viscoelastic-damageable material models for short fiber composites scarcely exist in the literature and, to the best of our knowledge, not at all for SMC composites if excluding the work described above. An example of this kind of model applied to swirl-mat composite is the model by Weitsman et al 2002 [5] who used a power law to describe viscoelasticity and a scaling factor, for damage to model unidirectional creep. Kumar et al 2003 [6] have introduced a pseudo strain energy defined in Laplace domain to describe viscoelasticity of laminates with different degrees of damage. Hinterhoelzl [7] has implemented a linear viscoelastic-isotropic damage model for reinforced plastics in FE-program ABAQUS. Fitoussi, Guo and Baptiste [8,9] have presented an elastic-damageable material model based on a statistical failure criterion for the identified failure mechanism of interfacial shear failure. This model was implemented in ABAQUS and applied to simple structures. Zocher et al. [10] have made a general incremental formulation suitable for orthotropic viscoelasticity.

The objective of the present paper is to expand the model developed previously [1] to 2D and to derive an incremental formulation suitable for FE-simulations using shell theory. It is implemented in a commercial FE-solver and the results are validated with analytic solution for constant strain rate tensile test and by comparing test data for four-point bended specimens with a multi element model for the simulations.

2D MATERIAL MODEL FOR FINITE ELEMENT APPLICATIONS

In this paper we are not using the convention that repeating indexes mean summation.

2D formulation of the model

The stress-strain relationship for in plane components of a linear viscoelastic-damageable material is:

de i , j = 1,2,6 (1)

2

Here D denotes the damage parameters D, and D2 and C.;.(?,Z))is the damage reduced

relaxation matrix for the case of plane stress:

C„(t,D) =

Cu(t,D)Cl2(t,D) 0

CJt,D) CJt,D) 0

0 0 CJt,D)

(2)

The relation between damage parameters and the relaxation matrix is taken from the continuum damage mechanics [2]:

Cil(t,D) = [M]-'[ciJ(t)lM]T (3)

where Cr(t) is relaxation stiffness matrix of undamaged material and [M] is damage effect

tensor:

M-,

1

1-A

svm

1-A (4)

After matrix multiplication C.. (?) may be written explicitly: Cv (t,D) =

(l-DJ2Cu(t) (l-Dx)(\-D2)Cn(t) 0

(\-DJ(\-D2)Cn(t) (\-D2)-C22(t) 0

0 0 (l-Dx)(\-D2)CJt)

(5)

Damage evolution is considered as an elastic process. It depends only on the maximum stress o(ek(t)))or strain statee(e k(t))during the service life.

D} =Dj(£) 7 = 1,2 (6)

The above means that C. (t,D) may be presented as

Cr (?,D) = b.. (e)Cjj (?) (no summation over repeating index) (7)

The time dependency is modelled with Prony series as

c ^ q + X q v " 1 - (8)

3

Incremental form of constitutive equations In an incremental formulation of the constitutive model stress, strain and damage state is known at time t and has to be calculated at time t + At. In a strain controlled analysis e(t + At) is set and a(t + At,D)is to be calculated. Eq. (1) for time instant f + Aris

de, a,(t + At,D) = Y. ]Cv(t + At-T,D)-j£dT (9)

We define:

te,(tt) = £,«k+J-£i(tk) [Aai(tk) = a,(tM)-crl(tk)

(10)

Integrating Eq. (9) and solving for Ao(tt) in Eq. (10) give the result below. The integration

procedure is given in Appendix 1. The analysis is an extension of [10] to also include damage.

Ao-,(?i) = Xö, (^)Af : ( ( / Ä ) + AcT,'i(rt) j

Here é, 7 =C, y +f, y

And

( I D

(12)

c; + Yc"'—{\- -Aa r... \

^" A?" 1 ' (13)

C-el(tk) + Yd^lt"^,(tk) (14)

dt\

3e,

A<rf (tk) = - X bt (tk)£ (1 - e» r"' K Uk)

(15)

(16)

CT- (tk) = <r*"r" er"' OV,) + (1 - ^ )C; £ A£t ( / w ) (17)

A local coordinate system coinciding with the directions of principal stress is used. The relaxation matrix components are calculated from£(0 and v(t) considering the undamaged material as in-plane isotropic [11]:

4

C22(t) = Cu(t)

v « £ ( 0 (18)

'-W \-v(tf

20+KO)

£(/)and v(t)are calculated point by point using a quasi-elastic approach and unidirectional creep data from ref [1]. The calculatedE(t)and v(/)curves are seen in Figure 1.

ra

—

o

9400

9200

9000

8800

8600

M 8400 C 3

£ 8200

8000

7800

0 Young's modulus

Poisson's ratio

0 Young's modulus

Poisson's ratio

•

a. D •

X

• •

• •

D D d o

10 100

Time [Seconds] 1000

0.35

0.34

0.33

0.32

0.31 2

0.3 O V5

0.29 -g

0.28

0.26

0.25

10000

Figure 1. Graph showing evolution of relaxation modulus and Poisson's ratio of undamaged material.

The Poisson's ratio is decreasing during the first 100 seconds and then stabilising. A possible reason for this behaviour is a initially not prefectly unidirectional stress field in the test specimen. It might be better to use the asymptotic value in calculations but at this point it was decided to use the not modified data. The resulting relaxation matrix components are seen in Figure 2 together with the transverse shear relaxation moduli (see below). The corresponding Prony series, see Eq (8), coefficients and relaxation times based on these curves are seen in Table 1.

5

12000

0 -I , , , i

1 10 100 1000 10000

Time [Seconds]

Figure 2. Relaxation stiffness matrix ( plane stress assumed) components for undamaged SMC composite with random fiber orientation.

Table 1 Prony series relaxation times and coefficients.

Relaxation Coefficient E(t) v(t) c,.(0 c « ( 0

time [MPa] v(t)

[MPa] [MPa] [MPa] r 0 = °° C 7751 0.2767 8396 2325 3034

r, = 20 C1 462 0.0430 795 669 68 r 2 =100 C 2 314 0.0049 354 132 111

r 3 =103 c 3 267 0.0013 301 99 100

r 4 =10 4 c 4 457 -0.0015 483 116 185

Out-of-plane shear relaxation modulus For FE-simulation of a geometry modelled with shell theory the out-of-plane shear modulus, G 1 3=G J 3 is needed in addition to Eq. (2). However, its influence is generally small and it may be treated separately in a simplified manner. In simulations a constant value ofG,, is used, representing the average values from Figure 2. The shear relaxation

modulus G u(r) is calculated using relaxation functions of the matrix and elastic properties

of the fiber. Elastic properties of the fibers are given in [12], i.e. Young's modulus 72 GPa and Poisson's ratio 0.22. Properties of the matrix are obtained by inverse modelling and the in-plane creep data for SMC composite are used as input. A micromechanical model by Eduljee, McCullough and Gillespie [13,14] for a hypothetical composite with aligned ellipsoidal inclusions has been used in combination with averaging over fiber orientation distribution described by McCullough, Jarzebski and McGee [15]. See for example [12], for details of the application of this model to SMC composite. For any time t, the matrix

6

relaxation functions were calculated using quasi-elastic method as the ones that give the best fit to the composite's in-plane relaxation function. The resulting out-of-plane shear stiffness G13 as a function of time is calculated using the same model [13,14.15,12] and the result is seen in Figure 2.

Determination of damage parameters for the current stress state In a thermodynamics approach the damage evolution is governed by associated thermodynamic forces. The thermodynamic force Y j , associated with a certain damage

variable Z>, may be found using the expression for strain energy 4 / e of the damaged

composite and taking derivatives with respect to this variable:

a y

y,=^r- (19)

V is calculated based on the definition of the strain energy:

and the result is:

CT. 2C, 2 c 7

2 C N vp<* ._

2(1 -D, ) 2 (C 2 2 C U - C 1 2

2 ) 2(1 -D2)2(C22CXX - C 1 2

2 )

Ö " , C T ? C , C T 2

(\-Dx)(l-D2)(C12Clx - C 1 2

2 ) 2 (1 -A) (1 -D 2 )C

The derivatives with respect to the damage parameters are found to be

Y = W ' _ a\C22

(21)

ao, (i-z),)3(c2 2cn-c1 2

2) axa2CX2 a;

{\-Dx)2(\-D2){C22Cu -CX2) 2(1-£> 1)

2(1-Z) 2)C 3

„ W c C T 2 C , ,

(22)

• 11

3D2 (\-D2f(C22Cu-Cu

2)

CTXCT2C}:

(\-Dx)(\-D2)2(C22Cxx-Cx{) 2 ( 1 - D x ) ( \ ~ D 2 f C3

(23)

The evolution law is formulated i f the dependence of damage variables on thermodynamic forces is defined.

Z), = fAt,%) 1 1 Hi (24)

D2=f2(Yx,Y2)

7

Yj denote the most extreme values during the load history. In a general loading case both

damage variables are functions of both associated forces and identification of these functions using test data is not trivial. I f damage parameters are assumed uncoupled the simplified relations are:

D2=f2(Y2)

I f the material is assumed initially planar isotropic the relations in Eq.(24) are fully analogous, i.e. i f the relationZ), = /,(F1,72)is known,/)-, = f2(YuY2) is known as well. Eq. (24) was obtained experimentally for an SMC composite in a previous work [1] based on E-modulus measurements in unidirectional tensile test on specimens with different degrees of damage. The results are for 72=0 and Y{ <0.5 MPa:

D,=a(Y,-Yü)

D, = 0 with a = 0.205 MPa 1 and 70 = 0.0654MPa

D2 * f2(Y]) validates the simplification made in Eq. (25) for the case where only one

thermodynamic force is non zero and value of 7, is moderate. At a maximum strain state,

e(ek (0), the system of Eq. (22) through (24) have to be solved in a iterative manner, for

example using Newton-Raphson iterations, see Appendix 2.

IMPLEMENTATION IN FE-SOLVER Material data are the relaxation stiffness matrix of undamaged composite described with Prony series and the damage development law in Eq. (26). The relaxation stiffness matrix is given directly in the FE-solver ABAQUS input file which includes geometry, load case and description of the FE-analysis to be performed. Transverse shear stiffness is given as a constant calculated from out-of-plane shear relaxation stiffness, G 1 3, for linear elastic orthotropic material [16]:

Kn=K22=^-Gnh (27) 6

Here h is the thickness of the shell.

The material model is implemented in the UMAT subroutine for user-defined material models. The subroutine updates stress and tangent stiffness, i.e. Jacobian, for the next increment. Introduction of the damage tensor/?.. (£) leads to a scaling parameter to the

viscoelastic solution, see Eqs. (13), (15) and (16). To solve Eq. (15) forRiV(tk) where the

derivatives of bAe) for all strains are needed we use finite differences:

8

(28)

with

A ^ M ' ^ - M ' M ) ( 2 9 )

Damage data and strain increments needed to calculate Eq. (29) are stored as solution dependent state variables (SDSV). cr"(tk) in Eq. (17) is solved in a recursive manner and

<y"{ik-\) is also stored as SDSV. The Jacobian used to balance internal and external forces

is:

•'-0 c - r (30)

The damage parameters are updated after updating the stress. This means thatDk_t is used

at tk instead o fD A . Using Dk is possible on expense of calculation time, but gives only a

little improvement i f the increments are sufficiently small.

VERIFICATION OF ABAQUS IMPLEMENTATION

1. Tensile test

In a typical tensile test the loading ramp is unidirectional strain that increases linearly with time:

ex=a ( 3 1 )

In [1] a tensile test was performed on the SMC composite at two constant strain rates. The used constant crosshead speed corresponded to (a) strain rate in a standard test following ASTM D3039 and; (b) very low strain rate (a factor 200 lower than the standard test) . The strain rates for these tests were 2x10~4 s"1 respectively l.OxlO"6 s"1. In addition, tensile tests with these strain rates were simulated using an analytic expression based on the non­linear model explained here, see ref [1]. The model predicts the stress-strain curve with good accuracy until the region close to failure where new mechanisms not accounted for are taking place. In the case of slow strain rate the failure occurs earlier in reality than in model predictions due to time dependent crack growth not accounted for in the model. The focus here is to simulate a tensile test with ABAQUS and the UMAT subroutine and compare with the analytic result in [1] to validate the developed 2D incremental formulation.

The FE-model for the tensile test is reduced to a single ABAQUS S4 element. The boundary condition is that the element is locked in 1 -direction at one end and a constant strain rate (also in 1-direction) is applied to the opposite end. It is seen in Figure 3 that the FE-result and analytical result coincide for both strain rates. With the used time increments the simplification of using the damage state from previous increment is leading only to a negligible error.

9

80 t

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Strain [%]

Figure 3. Tensile test — analytic simulation and FE-analysis at two strain rates.

2. Four-point bending test

A four-point bending test, see Figure 4, has been modelled using an FE-model built from ABAQUS S4 shell elements. The model is seen in Figure 5 together with the deformed shape at the end of simulation (magnified 10 times). Five integration points through the thickness are used. The model is locked in 3-direction at the ends (at the location of the outer supports) and node forces in negative 3-direction are applied at the location of loading noses. The results are compared with test data and the set-up corresponds to ASTM D6272 for a four point bending test. According to the standard for specimen with size 3.025 x 12.7 x 75 mm the support span is 60 mm.

Outer Support Loading noses

Outer Support

20 mm 20 mm 20 mm

T T

J

Figure 4. Scheme of the four-point bending test.

10

Loading rate corresponds to displacement rate 2.2 mm/min at the loading noses. Since the damage parameters are uncoupled, damage development in 2D is described with the following relations fori), andD,:

D, = a(Y.-Y„)

D2=a(Y2-Y0)

where the constants a and Y0 remains the same as in Eq. (26).

3

Figure 5. FE-model for bending test specimen and deformed shape (10 times magnified).

The model ignores any damage development due to compression. The reason for the simplification made is previous research [3,17] which indicates that the compressive strength of SMC is approximately twice the tensile strength. It was not possible to continue the simulation after failure on any surface, limiting the simulation to 1.2% strain (vs. an analytically obtained failure strain of 1.4%). Maximum stress levels at the end of the simulation are found to be 69.9 MPa on the face in traction and 93 MPa on the face in compression. The stress level at the face in traction is in a perfect agreement with the tensile test in Figure 3, as would be expected, and there is a very evident difference in stress level at the face in compression compared to at the face in traction. This is due to the fact that there is a substantial damage development in the face in traction but no damage on the part of the specimen that is in compression. In Figure 6 damage parameter TJ, on the face in traction is seen and the maximum value isZ), =0.189. Due to Poisson's effect we observe saddled surface and the center of the specimen is buckling towards the applied load. As a result the strain in 1 -direction is higher at the edges and is giving slightly higher damage level (parameterDJ there. Damage parameter for direction

11

transverse to the loading, D2, was found to be zero over the surface in compression indicating that the tensile strain in 2-direction is below the onset of damage development.

Figure 6. Contour plot from simulation showing damage parameter D, on the face in traction.

In Figure 7 a simulation using the model is compared with a simulation using a linear elastic material model with the same initial E-modulus. The strains on the surface in traction respectively on the surface in compression in the centre of the specimen are plotted versus the force applied on the loading noses. Both strains on the face in traction as well as the face in compression are plotted. It is evident that the surface strain level is higher using a non-linear model and the progressive degradation of the material is revealed. Comparison of simulated strain at surface in traction and compression show that damage leads to more strain at surface in traction than on the other side in agreement with the result for stress previously discussed.

Comparison with the test data is seen in Figure 8. Also in the test the strain in traction is higher than in compression validating the assumption that damage development in traction is dominant. It is also seen that the slope and magnitude of the simulated and tested response are in good agreement. However, the maximum strain and load in tests are significantly higher than in the simulation ending at 1.2% strain on the surface in traction. In test, the maximum surface strain is 1.9% in average, with a standard deviation of 0.2% for the five tested samples. Such strain at break in tension loading is not realistic for this material. The reason for this result is likely that in bending the material's load bearing capacity is not critically dependent on the surface layer failure. The surface, even though severely damaged, holds together enough to make the strain gage function. The specimen sustains load until a critical number of load bearing bundles is neutralised. From a

12

physical perspective, a neutralised load bearing bundle in SMC material means longitudinal fiber bundles that do not carry load, that is, it is broken or interfacially debonded from the matrix. The feature of SMC composite not to fail in bending even though the surface fails is an advantage compared to material sensitive to failure initiations such as most injection moulded thermoplastics. It would be of interest to study the mechanisms of failure in a bending test from a through the thickness perspective.

200 -FE Linear FE Non-linear

Surface in compression

-1.5 -1 -0.5 0 0.5 1 1.5 Strain [%]

Figure 7. Graph comparing surface strain in the centre of a specimen versus applied force on the loading noses in a simulated bending test, using the developed model and a linear model.

350 -

300

x FE Non-linear

250 -

200 ^ Loa

d [N

]

- Test data \

loo -

\ o -Surface in compression

0^ Surface in traction

-1 0 Strain [%]

Figure 8 Graph comparing surface strain in centre of specimen versus applied force on the loading noses for simulations and tests (5 specimens).

13

CONCLUSIONS

I . A material model accounting for linear viscoelasticity and damage development was applied to SMC composite and described in incremental formulation. A FORTRAN code is written to implement the model in FE-programme ABAQUS through the option of a user defined material model written in a subroutine.

IL Application to constant strain rate tensile tests comparing analytical and FE results prove the accuracy of the incremental formulation and the subroutine. Simulation is possible until strain levels close to failure when too low tangential stiffness (Jacobian) becomes a numerical issue.

III . Surface strains in bending test at both the face in traction as well as the face in compression are simulated with good accuracy compared to the test data. The simulated surface failure strain is lower than the actual failure strain of the structural part, which indicates that the specimen does not fail at the same time as the surface strain is critical. Apparently, the SMC material is not sensitive to failure initiation.

ACKNOWLEDGEMENTS The authors wish to thank Dr Magnus Svanberg and Dr Anders Holmberg at the Swedish Institute of Composites (SICOMP AB) for fruitful discussions and assistance. A lot of time was saved using the incremental formulation of orthotropic viscoelasticity by Zocher et al. which was already implemented in ABAQUS by SICOMP in an earlier project (see [ 18]). Sponsors to this work have been Volvo Car Corporation and VINNOVA in Sweden through the Vehicle Research Program (PFF), grant no 2002-01578.

14

References

1 Oldenbo M, Varna J. A constitutive model for non-linear behaviour of SMC accounting for linear viscoelasticity and micro damage. Polymer Composites (submitted)

2 Chow CL, Wang J. An anisotropic theory of elasticity for continuum damage mechanics. Internationaljournal of Fracture 1987; 33:3-16

3 Oldenbo M, Fernberg SP, Berglund LA. Mechanical behavior of SMC composites with toughening and low density additives, Composites part A 2003; 34:875-885

4 Fernberg SP, Berglund LA. Bridging law and toughness characterisation of CSM and SMC composites. Composites Science and Technology 2002; 61:2445-2454

5 Weitsman YJ. Time-dependent behavior of randomly reinforced polymeric composites. Mechanics of Composite Materials 2002; 38:381-386

6 Kumar RS, Talreja R. A continuum damage model for linear viscoelastic composite materials. Mechanics of Materials 2003; 35:463-480

7 Hinterhoelzl RM. Coupling the theory of viscoelasticity and continuum damage mechanics to model the time dependent behavior of fiber reinforced plastics and particulate composites. International Journal of Materials & Product Technology 2002; 17:121-133

8 Fitoussi J, Guo G and Baptiste D. A statistical micromechanical model of anisotropic damage for SMC composites. Composites Science and Technology 1998; 58:759-763

9 Guo G, Fitoussi J and Baptiste D. Modelling of damage behaviour of a short-fiber reinforced composite structure by the finite element analysis using a micro-macro law. International Journal of Damage Mechanics 1997; 6:278-299

10 Zocher MA, Groves SE, Allen DH. A three-dimensional finite element formulation for thermoviscoelastic orthotropic media. International Journal for Numerical Methods in Engineering 1997; 40:2267-2288

11 Fung YC, Tong P. Classical and computational solid mechanics. World Scientific Publishing Co: Singapore, 2001.

12 Oldenbo M, Mattsson D, Varna J, Berglund LA. Global stiffness of a SMC panel considering process induced fibre orientation, Journal of Reinforced Plastics and Composites 2004; 23:37-49

13 Eduljee RF, McCullough RL, Gillespie JW. The influence of inclusion geometry on the elastic properties of discontinuous fiber composites. Polymer Engineering and Science 1994; 34:352-360

14 Eduljee RF, McCullough RL. Structures and properties of composites. VCH: New York, 1993; 381-474

15 McCullough RL, Jarzebski GJ, McGee SH. Constitutive relationships for sheet molding materials, proceedings of a joint U.S. - Italy Symposium on Composite Materials, June 15-19, 1981.

16 ABAQUS 6.3-1 Documentation, Standard User's manual, Hibbitt, Karlsson & Sorensen, Inc., 2002

15

17 Taggart DG, Pipes RB, Blake RA, Gillespie Jr. JW, Prabhakaran R, Whitney JM. Properties of SMC Composites, Center for Composite Materials, University of Delaware, Newark, Delaware, report no CCM-79-1, 1979

18 Allen DH, Holmberg JA, Ericson M , Lans L, Svensson N, Holmberg S. Modeling the viscoelastic response of GMT structural components. Composites Science and Technology 2001; 61:503-515

16

APPENDIX 1

Incremental formulation Summation over repeating indexes is not assumed in the notation used in this Appendix.

Cy in the stress-strain relationship

j o " T

May be written according to (5) as

CIJ(t,D) = ai(D)aJ(D)CIJ(t)

where C ( , ) - ( ' • ^ Y " , '

( A U )

(A1.2)

(A1.3)

with a,

\-D{

l-D2

V ( i - A ) 0 - A )

(A1.4)

Since D = D(e), where £ represents the worst strain state experienced by the the

composite during its life, functions ai are also aj (e).

Denote b:i {s) = ai (e )oj (e)

Substituting (A1.5) in ( A L I ) and (A1.2) we obtain ' de

a, (t, D) = £ b„ (e) JC, (t-t)-J-dT

(A1.5)

(A 1.6)

£ can be considered as a complex function of the current strain state e, • Hence bu are

monotonously increasing and are also functions of f , which change values only i f the

current strain state is more severe than the worst situation before.

Now denote e) = ei(tk),a- = ai(tk) and introduce:

&£l(tk) = £i*l-£f

A c T , U t ) = c r f + 1 - C T *

b\ =bg(e>)

According to ( A U ) for t = tM

(A1.7)

de,

2 > ' f ( ' +&-T)-^dT+Y,b,(er) J C , ( / T +At-T)~±dT ( A l .

17

and de.

af =Yibij(ef)lciJ(tk-r)-^dT (Al.9)

Using Eqs. (A1.8)-(A1.9) in definition of Aa,(tk), in Eq. (Al.7), give:

't d£ '» fe Aa, (tk) = Xb v (ef+l) J C9 (tk + At - T)-±dr -£ b, (ef) Jc, (tt -T)-£dT +

J o de,

+ Z^( £ / + ' ) j C^+At-t^dr

We can use the approximation

i'de, Ae,(tk)

Denote i?,*, = — 1

bT = bf+JjRfJAe,(tk)

Substitute Eq. (A1.13) inEq. (ALIO) => '* de

Aa, (tk ) = X 6* J[c„ (/, + At -1) - Cs (tk - T)]-£ dt +

i t /

de, + X i I ^ ' 7 ' A ^ ft ) [ tø + A t ~ T ) ^ d T +

de. +H\bu + Z 4 A £ / f t ) [ tøft + A ' - T ) ^ d T

(ALIO)

( A i . i i )

(Al.12)

(Al.13)

( A l . 14)

We analyze the integrals 7,, I2 and J3 in (Al .14)

7, may be written as

C/T

' de Define: <r,7(/) = \c™eH'-T>,T™ —Ldx

o ^

Eq. (A 1.15) becomes: /, = ^ [ e -4 " r » - l j t r ^ / j

Now first term in Eq. (A 1.14) can be rewritten and denoted as A of :

A Crf(?J = - X ^ Z ( 1 - e " A " r " ' K f t )

(A1.15)

(A1.16)

(A1.17)

(A1.18)

is

It can be shown (see [10]) that the following recursive expression may be used to calculate o~™(tk).

a; (tk \=ehl1 CT; ( f M ) + (1 - e^"^ )C™ ± Aekf ( A 1 . 1 9 )

Consider J2 = \C„(tk + Ar - r )— ' - d r in E q . (A1.14) J rir dr

Substitute Eq. (A 1.3) in expression for/,to get:

dr / 2 = c ^ + X e - ^ | c ; ( A 1.20)

Eq. (A20) may be written as

i2=c;£*+Y,e-A"T»o-;(tk) (A1.21)

de, Consider I} = \CJtk + At - r ) — L d z in E q . (A14)

,J dr Substitute Eq. ( A 1.3) in expression for / 3 to get:

de

dr (A 1.22)

Presenting e} in interval tk <t< tM in linearised form:

ei(r) = ej(tk) + Re

J(t-r)H{T-tk)

where

j - {£. (T)}=Re

jH(r-tk) + R] (t-tk)S(T-tk)

Eq. (A1.25) inEq. (A1.22) =>

/ , = C , J / ^ ; A / + Z C ™ \e^+A-z),x-Rc

jdT =

dej

~dt

Ae,

At

The third term in Eq. (A 1.14) now is

[Keeping only linear term with respect to Ae., i.e. neglecting As, • Ae ]

C - N Y " ^ -< l <• v : ) ^ . ( ^ ) =

(A1.23)

(A 1.24)

(A 1.25)

= crJA£j(tk-)+YJc;tm I c-/?;dk# = | c - +Ec^( i -" ; ' "k(g (AI.26)

(A 1.27)

(A1.28)

19

Using Eqs. (A1.17)-(A1.18), (A1.21) and (A1.28) it is possible to write Eq. (A1.14):

Aa, (tk) = ACT," (tk) + X \ Z 4 A e , (tk)

+ 1

C-ek

j+Y.e'M""^(h)

(A1.29)

I f changing the sequence of summation in the second term Eq. (A 1.29) becomes:

Aa,(tk) = Aaf{tk)+Y.\LR Ae,(lk)-

c;+Y.c;^a-e-^)Aej(tk)

Denote:

r - 2 >

and 0 -C

The final result is

AGl(tk) = YJQij(tk)Ae]{tk) + AtT?(tk)

(A1.30)

(A1.31)

(A1.32)

(A1.33)

(A1.34)

A P P E N D I X 2, Newton-Raphson iteration to obtain updated damage state Since the material is initially in-plane isotropic and the damage parameters are uncoupled updating of D2 for a new stress state is analogous to updating of £>, which is shown here. In the general Newton-Raphson iteration equation to obtain the updated damage parameter A :

D r > = n » _ / ( A ) / ' ( A )

(A2.1)

/(£>,) is obtained i f eliminating 7, from 7, =—— {see Eq. (22)} by using D, (7,) {see Eq 3D,

(26)} obtained experimentally. The result is:

/ (D 1 ) = (D,+a7 0 ) ( l -D 1 )3

aal (l-DJ aG\02Cn{\-DJ aa;C22

2(1-Z) 2)C„ (\-D2)(C22Cu-Cx\) C22Cn-C2

2

= 0

/ ' ( A ) is easily found since / ' ( A ) = g/XA) 3A

(A2.2)

(A2.3)

In the calculations above forD, it is also needed to know the damage parameter A • In the

calculation D2 from the previous time step is used.

20

Stiffness reduction in SMC composites with evolving damage

M. Oldenbo1'2, P. Lundmark2 and J. Varna2' * 1) Volvo Car Corporation, Exterior Engineering, dept 93610, PV3C2

SE-405 31 Göteborg, Sweden 2) Luleå University of Technology, Division of Polymer Engineering

SE-971 87Luleå, Sweden

Submitted to Composites Science and Technology

Abstract The development of damage in SMC composites is analyzed using model laminates with bundle structure and varying orientation with respect to the applied load. Bundle cracks along the fibers and matrix cracks transverse to the applied load were identified as the basic damage mechanisms which were described by a damage evolution law for damage simulations of SMC composites. The damage effect on stiffness was evaluated using laminate analogy: SMC composite is replaced by quasi-isotropic laminate with through-the-thickness cracks in layers. Stiffness reduction is simulated using a model for laminates with damage in arbitrary layers which describes the reduction in terms of crack face displacements, developed earlier. The predicted elastic modulus versus applied strain curves are in a very good agreement with experimental data for SMC composites.

1 Introduction It is common to describe the damage evolution in short fiber composites (SFC) in tensile loading using a damage parameter linked to stiffness reduction [1,2,3,4]. The degree of damage is presented as a function of maximum stress (or strain) applied. The damage growth in sheet moulding compound (SMC) composite, measured as modulus reduction, is in form of evolving micro-cracking in a highly heterogeneous material. Hour and Sehitoglu [5] linked the non-linear behaviour of SMC to the formation of multiple cracks and increasing crack density with growing applied stress. Final failure in SMC composite is considered to occur when various small damage entities distributed in the material join and form macro-cracks [8], Transversely oriented fibre bundles have been reported as initiating sites for cracks in SMC [6,7]. In general, crack initiation in composites may occur by fibre/matrix debonding, matrix/filler debonding and at fibre ends [5,8,9]. Cracks in in-plane isotropic SMC and short fiber

' Corresponding author. Address: Division of Polymer Engineering, Lulea University of Technology, SE-971 87 LULEA, SWEDEN. Tel.: -46-920-49-16-49; Fax.: +46-920-49-10-84; E-mail: janis.varna@sirius.luth.se

1

composites loaded in uniaxial tension develop mainly transverse to the loading direction, see [7,10,11,12]. However, for materials with bundle structure, high degree of fiber orientation and with a high fiber content, cracks have been reported to tend to follow the main fiber direction, see the studies by Wang et al. [8,13] on fatigue damage evolution of short fiber composites. Development of large matrix cracks due to difference in stiffness between bundles and matrix coupled with shear stress has been discussed [11].

Jiao et al. [14,15] have concluded that the damage modes for standard SMC (Std-SMC) can be determined by the assessment of the acoustic emission (AE) wave frequency. Cracking is more likely to propagate in regions were the matrix dominate or where fibers are aligned in normal direction to the applied load. From AE analysis two basic initiation mechanisms could be determined: (a) Interfacial debonding of CaCC>3 particles at moderate loading and (b) debonding between the fibers and resin, at higher loading. Interfacial debonding of CaC03 at low loading level has been observed using in-situ microscopy [7] for Std-SMC material, as well.

The stiffness reduction has been modeled using laminate analogy by Kabelka et al.[16]. In their approach the fiber bundles were modeled as unidirectional (UD) plies followed by orientation averaging of the fiber orientation distribution. The model predicted failure in bundles as a function of orientation to the applied unidirectional load using failure criteria for the UD-plies which contain transverse tensile strength and in-plane shear strength of the bundles as parameters.

In the present paper a simple model for stiffness reduction of SMC composite, based on crack density as function on maximum strain history and bundle orientation angle is derived and validated. A [0,90,0] and a [0,±60,0] bundle structure laminate is used to develop the damage evolution model. Two typical crack types (bundle cracks and through-the-layer cracks) were identified and quantified using these laminates. A quadratic failure criterion which accounts for interaction between transverse and shear stresses in the bundle is presented and used for damage state simulation in bundles with an arbitrary orientation. A simple approach is presented that is accounting for both damage modes in the stiffness reduction model. The stiffness modeling is based on laminate analogy, replacing the bundle structure of the SMC composite by quasi-isotropic layered composite with layer properties calculated using the average fiber content in Hashin's model [17,18]. Then, the stiffness reduction model developed in [19], which is valid for general laminates with intralaminar cracks in layers, is used. The simulated elastic modulus reduction as function of strain is compared with test data and predictions based on several modifications of the ply-discount model.

2 Material To produce laminates with [0, ± 9,0] layup, first unidirectional (UD) SMC prepereg was produced from premixed SMC paste and glass fiber roving corresponding to Std-SMC formulation (delivered from from Reichhold AS, Norway). In doing this, a layer of SMC-paste was added to a bed of oriented fibers. The recipe of the paste is given in

2

Table 1 and the thickener is added just before use. Roving R07 glass fibers from Owens Corning were used. The angle of orientation of bundles in the plies was 0 or ±0 degrees, 0=90, 75, 60 or 45 degrees. Plastic sheets in the bottom and sealing at the edges (Tacky tape distributed by Schnee-Morehead) were used to hold the prepreg together during manufacturing. After production, the prepreg was left for a maturing process for 5-10 days similarly as in SMC production. The volume fraction of fibers in the laminates was to some extent controlled by weighting the amount of fibers and paste used in the prepreg. This method is not accurate because of the flow during moulding. More exact values were obtained by burning off the matrix.

A tool to make plates was produced from two large aluminium plates and 2.5 mm steel shims to build a quadric cavity with a side of 25 mm. Hot moulding at 140-160 °C was followed by curing in the same operation for 3 minutes. A hydraulic press (distributed by Pasadena hydraulics, inc) with a maximum capacity of 18tons and 300°C was used. A high-temperature resistant plastic film was used to hinder the moulded plate from sticking to the tool. Laminates of type [0,±0,0] were produced by charging the tool with a suitable stack of UD prepreg.

Std-SMC: plates (SMC 606 from Reichhold AS, Norway) were delivered in size of 2.5 x 300 x 300 mm. This SMC composite contains the same matrix as the produced laminates and 30 weight percent glass fibers from the same Roving type as was used for the laminates. Since tensile testing of samples from the plates showed that x and y directions in the plates had similar E-modulus, the plates were in following considered as in-plane isotropic. The E-modulus of Std-SMC paste was needed to calculate the effective UD properties using the micromechanics model [17,18]. 2 mm plates made of paste of a similar formulation (Std-SMC) from Mecelec et Recyclage, France were tested.

3 Experimental

3.1 Volume fractions

Specimens of size 2.5 x 10 x 100 mm were cut from plates of a [0,90,0] laminate and from random SMC (SMC-R). These specimens were edge-polished using sandpapers of size, in descending order: 120, 320, 600, 800, 1200 and 4000. Final polishing was done with diamond spray of 9, 6 and 3 pm particle size. Volume fractions were measured using image analysis software "Analysis" (distributed by SIS - Soft Imaging System). Both average fiber diameter, D,, and average fiber bundle cross section

area, Ah, was obtained measuring on at least 20 bundles. The volume fraction of fibers

in the bundle, V,h, was calculated from:

(1)

3

where TV, is the number of fibers in the bundle which using the supplier's data sheet

[20] for roving R07 could be calculated tol 80.

Average volume fraction of fibers in the laminates and the SMC-R, V}, was calculated

based on fiber weight fraction, f u s i n g the density of the fiber and other constituents:

Vr=&-W, (2) Pi

Here pc is density of the composite and pf the density of the fibers. Density of the

constituents is taken from [7]. Wf for the laminates was obtained by burning off the

matrix and for SMC-R by using the supplier's data. After burning off the matrix, the inorganic filler in the matrix was removed using diluted acid.

Volume fraction of bundles in the laminates and SMC-R, Vh, was calculated from:

Y ß

The thickness of the 9 layers in the [0,±B,0] laminates was calculated assuming constant V, in the laminates and using the weight fractions of the fibers from 0 and 6

layers, Wf0 respectively WfB. The weight of the layer was obtained by careful

separation of the fiber layers after burning off the matrix and removing the filler. The thickness the 8 layer, he, is calculated as:

W

w n h (4)

la m m are ^ '

3.2 Crack density Specimens of size 2.5 x 20 x 200 mm for damage analysis were produced by cutting them out from the produced laminate plates. Draft polishing (320 paper) of the specimens edges was performed before preloading them. To introduce damage, specimens were preloaded in a screw driven tensile testing machine (Instron 4411) to three strain levels: e = 0.5 %, e = 0.75% and e = 1.0 %. Each specimen was used only at one preloading level. For optical analysis the ends of the original sample in the clamping region were discarded and the middle part of the specimen was cut parallel to the loading direction in to two specimens of size 2.5 x 10 x 100 mm. Polishing was similar as for samples for volume fractions analysis (see above). An optical Microscope (Zeiss Axioscope FS) with a digital video camera attached to it was used for counting the cracks in the samples. The image was reproduced using a PC and the software Mvpilot32 v3.01 (distributed by Matrix Vision). To increase the accuracy of crack counting, the specimens were during microscopy loaded to 10 MPa in tension

4

using the Minimat tester (distributed by Polymer Laboratories, Thermal Sciences Division). This load which contributed to crack opening and the blue ink distributed to the specimens with methanol made the cracks more visible.

3.3 Stiffness reduction and matrix stiffness measurements

Specimens of size 2.5 x 20 x 200 mm were produced by cutting them out from the plates. Three material configurations were tested: SMC-R, [0,90,0] laminate and [0,±60,0] laminate. Six specimens were used for each material configuration. After measuring the undamaged material elastic modulus specimen was loaded to a certain damage state and the modulus was measured again. Two strain levels (damage states) were considered using three specimens for each of them. A hydraulic tensile testing machine (Instron 1272) was used in these tests. E-modulus of the matrix which is needed to calculate UD properties of the homogenized composite with Hashin's composite cylinder assemblage model [17,18] was measured in bending using specimens of size 2 x 20 x 100 consisting only of matrix. A screw driven tensile testing machine (MTS Alliance RF/100) was used.

4 Experimental Results

4.1 Volume fractions

The fiber diameter was found to be 14.1 pm with a standard deviation of 1.3 pm. Based on the fiber diameter and the measured bundle cross-section area, the volume fraction of fibers in the bundles was calculated. The result for laminates (index L) and transverse bundles in SMC-R (index SMC), with standard deviation given in parentheses, is:

The volume fraction of fibers in the bundle around 0.5 has previously been reported in [16] which is in good agreement with our result. Due to differences in moulding the SMC has lower volume fraction of fibers in the bundles. When producing laminates, the flow in the tool was minimized to preserve the fiber orientation, but for SMC normally the charge is not covering more than 50% in order to remove air in the prepreg during moulding. It is likely that bundles are dispersed during the material flow. Another reason for the difference is found in the fact that the laminates contain continuous fibers compared to the SMCs bundles of 25 mm length. A continuous fiber bundle is likely less sensitive to local shear etc, causing the bundle to disperse. However, the standard deviation is rather large and the measured difference in volume fraction fibers between bundles in SMC and laminates is marginal.

The measured fiber weight- and volume fractions in the layers of the analyzed laminates are seen in Table 2. The calculated volume fractions of bundles, Vb and the calculated thickness, h, of laminate and 0 layer is seen in Table 2 as well.

(5)

5

4.2 Crack density

A photograph of a cracked fiber bundle is seen to the left in Figure 1. The crack is localized in the bundle covering its cross section. This type of cracks is called bundle crack in following. In Figure 1 (right), the bundle crack has propagated outside the bundle in the matrix. Since these cracks in laminates often covered the whole thickness of the 6 - layer they are called whole cracks in this paper. Actually they represent a combination of bundle crack and matrix crack. The results of crack density measurements in [0, ± 6,0] laminates is seen in Figure 2. Bundle cracks are running parallel to fibers in the bundle and the crack density is defined by the perpendicular distance between the cracks. Therefore the visual crack density measured on the edge of the specimen has to be divided by sin(0). It is seen that the number of cracks at 0.5% strain is low but increasing up to 0.75% strain. The crack density increases faster between 0.75% strain to 1.0% strain than in the region 0.5% to 0.75% strain. The crack density evolution was approximated by a second order polynomial which is also given for each laminate in Figure 2.

The bundle volume fraction, Vb, in the cracked layers are similar for all laminates,

except for the [0,±75,0] laminate, where the bundle volume fraction is significantly lower, see Table 2. However, the [0,±75,0] laminate has about twice the thickness of the [0,±60,0] laminate or the [0,90,0] laminate which means that the number of bundles per mm in [0,±75,0] laminate is rather similar (slightly larger) to other lay-ups. Therefore the expected crack density (per mm) in [0,±75,0] laminate is rather similar to [0,90,0] laminate which is confirmed in test.

The off-axis layer thickness in the [0,±45,0] laminate is almost 50% higher than in the [0,±60,0] or [0,90,0] laminate used for damage analysis. Since the bundle volume fraction in these laminates is very similar, we conclude that there are more bundles per mm in the 45-layer. However the crack density is much lower. This leads us to conclusion that the tensile stress (or strain) component transverse to the fiber bundles is the main driving force for damage fonnation. This component at fixed applied strain level is much larger in bundles with 90-orientation. However, the number of bundle cracks and whole layer cracks in the [0,±60,0] laminate and the [0,90,0] laminate is very similar which is difficult to explain by the above arguments. Thickness of the transverse layers and bundle volume fraction, Vb, are also similar which makes it likely that the influence of shear strains on the bundle crack formation can not be neglected and that it is the reason for the high crack density in [0, ±60,0] laminate.

The crack density in the [0,±45,0] is the lowest of all lay-ups for both bundle cracks and whole layer cracks which implies that shear stress (strain) contributes less in damage formation. However, the cracks in 45-bundles are much less open which makes difficult to count them and it is possible that the real number of cracks is larger than the reported.

6

5 Micromechanics modeling of damage evolution and stiffness reduction

5.1 Model for damage evolution

The crack density in a layer with a certain bundle orientation depends on the strain in transverse direction and on the level of in-plane shear strain. Thinking in terms of a strain based failure criteria which includes interaction between transverse and shear strain the condition for bundle cracking can be written as

r . v £2

+ 7X2 = 1 (6)

\£2u J \7\2u)

Since the strength distribution has a statistical nature, cracks do not appear all at once as in the ply discount model. The first crack comes in the weakest bundle and the number of cracks increases with the applied strain. Hence for the first crack the values of £2„ and yl2l< are lower than for the following ones. The cracking process may be

slowed down by crack interaction or by other failure mechanisms taking over at higher loads. Therefore the simple condition Eq. (6) can not be considered as the damage evolution law in SMC composite or in the considered laminates with the bundle structure. Realizing the statistical nature of strength parameters in Eq. (6) we assume here that the ratio between shear and transverse strength remains constant

f v £2u

J\2u, If we introduce the effective strain as

(7)

eeJf = ^s2

2+ky2

2 (8)

the criterion Eq. (6) can be rewritten in form

e*=eu (9)

It means that next crack event takes place when the effective strain in the layer (bundle) reaches a value equal to the transverse strength of the weakest of the still survived bundles. The effective strain defined by Eq. (8) is a simple way to describe the importance and interaction of transverse and shear loading for cracking in a bundle with an arbitrary orientation. The statistical nature of the strength distribution is reflected in the crack density versus effective strain curve, p = p{e"i}) • We suggest that this relationship is obtained from experimental damage evolution data in 90-bundles of the [0,90,0] laminate. For this orientation the shear strain component is zero and the effective strain is equal to the transverse strain. The second degree polynominals that are an approximation to the mean value of the bundle crack density and whole crack density are used to predict damage evolution in other laminates. In order to vise it for laminates with cracks in off-axis layer, the constant k has to be determined. In the presented case

7

the damage evolution data for [0,60,-60,0] laminate were used for this purpose. The simulated cracking evolution in the [0,60,-60,0] laminate with the k value that gave the best fit to the experimental data can be seen in Figure 3. It is assumed that both ±60 layers have the same crack density.

The available tools for stiffness reduction predictions are all based on laminate analogy, replacing the bundle structure with homogenized layers containing cracks going through the whole thickness of the layer. In order to use these tools for stiffness predictions the two different types of crack densities described in section 4.2 have to be replaced by a total crack density. The idea which may be rather correct for thin layers is that each bundle crack is through-the-thickness crack (TT-crack) running parallel to fibers and that each whole crack also is a bundle crack. Then the total density of TT- cracks parallel to fibers in a layer becomes equal to the sum of densities of the bundle cracks and the density of whole cracks.

p = p"""'',+pw,e (10)

Using p = p{eeff) and Eq. (8) with k=0.35 one can predict the total through-the-thickness crack density in any layer of the laminate. Due to lack of information we can not discuss and include in simulations the effect of the volume fraction bundle, Vh and other geometrical parameters on the cracking curve .

The SMC composite is described as the [0+15, +30, ±45, ±60, ±75,90]s laminate. It is further assumed, that in the 0,±15 and ±30 layers damage does not develop. The predicted total crack density in each layer as a function of the applied strain in x-direction can be seen in Figure 4. The damage description is rather accurate except for the 45-degree layer where the damge is overestimated.

Additionally to TT-cracks we have in this simplified schematic picture matrix cracks initiated from bundle cracks. It means that their density is equal to whole crack density. They usually cover (observation result) both ± 6 layers with the crack plane transverse to the load direction.

5.2 Stiffness reduction model The model used for determination of the reduction in stiffness due to damage is a micromechanics based model derived in [19]. In that paper an attempt, similar as performed by Gudmundson [21,22], is presented in the framework of the laminate theory. Stiffness, compliance matrices and thermal expansion coefficients of an arbitrary symmetric laminate with cracks in layers are presented in an explicit form. Derivation of constitutive relationships is following the same routes as in classical laminate theory. The damaged laminate stiffness and thermal expansion coefficient matrices are calculated from the undamaged laminate matrices multiplying it by a matrix which differs from the identity matrix by terms dependent on crack density in layers, stiffness matrix and orientation of these layers and includes a crack face displacement related matrix. The normalized crack face opening and sliding displacements are considered as dependent on the position of the cracked layer

8

(outside or inside cracks) and on the constraint of the surrounding layers in terms of their stiffness and thickness. These dependences are analyzed in [19] and [23] using FEM-calculated crack face sliding displacement profiles in generalized plane strain formulation and presenting the results in form of power laws.

In the derivation, a symmetric laminate subjected to general in-plane loading is considered. To exclude bending effects the laminate is assumed to be symmetric also in the damaged state (crack density is the same in layers with symmetric location with respect to middle-plane). Only in-plane loading is considered. The intralaminar cracks are assumed to run parallel to fibers with a crack plane transverse to the laminate middle-plane and to across the whole widths of the layer. Laminate contains N layers and the k-th layer is characterized by stiffness \o\k , thickness hk and fiber orientation

angle which determines the stress transformation matrix between global and local coordinates \r\ • The line above the matrix and vectors denotes entities in the global

coordinate system. The crack density in a layer is p and the normalized crack density

p t e is defined as

Pk« = h k p k (11)

The expression for the stiffness matrix of the damaged laminate is

k ? r = I V ] + ~ r i . p h , m [TI [u\ [T\ ]Q\ [st r K )1 [& r < 12) (_ hE2 t=\ j

where h is the total laminate thickness, E2 is the modulus transverse to the fiber direction, [S0]

LAM and [Q0]LAM are the compliance and the stiffness matrices of the

undamaged laminate. [U]k is a matrix which contains the normalized crack face opening and sliding displacements

\u\4 (0 0 0 1

0 uL 0 (13)

Here the normalized crack opening displacement is calculated using expression

.Es

A = 0.52 £ = 0.3075 + 0.1652

(14)

2h* n = 0.030667 — - 0 . 0 6 2 6 1 - ^ + 0 . 7 0 3 7 2h. 2/7, {2h,

(15)

9

In Eq. (13), ES

1 S the elastic modulus of the material surrounding the cracked layer in

the direction transverse to the crack plane, h, in Eq. (15) is the thickness of this

material.

The normalized crack face sliding displacement can be with a sufficient accuracy calculated using the following expression

This model will be applied to calculate the stiffness reduction due to following damage mechanisms: a) through-the-thickness cracks (TT-cracks) parallel to bundle direction with density which follows the damage law described in section 6.1; b)matrix cracks going through both + an - off axis layers with the crack density equal to whole crack density. In the latter analysis both off-axis layers will be homogenised first and then the above described methodology applied as for 90-cracks in layer with effective properties..

As an alternative analysis the ply discount model in two modifications will be applied. In the first version the whole stiffness matrix of the damaged layer is assumed to be zero. In the modified version only transverse modulus and shear modulus are assumed zero after failure.

5.3 Stiffness predictions for [0,±8,0] laminate Stiffness reduction predictions based on equations given in section 5.2 were performed for [0,90,0] and [0,60,-60,0] laminates. The result can be seen in Figure 5 and Figure 6 together with experimental data. For cross-ply laminate all the considered cracks are TT-cracks (because the TT-crack is assumed to go through the thickness and coincide with the matrix crack). The model fairly good predicts the reduction of the elastic modulus. In case of the [0,60,-60,0] laminate, see Figure 6, the solid line describes the expected stiffness reduction due to bundle cracks. The interrupted curve shows the summary contribution of TT-cracks and matrix cracks. Predictions are in an excellent agreement with test data. Obviously the effect of matrix cracks is as big as the effect of TT-cracks in bundles. This result gives a certain confidence to the validity of the approach for SMC composite. Predictions based on the ply-discount model assuming zero stiffness of the damaged layer which are also presented in Figure 6 are very conservative.

5.4 Stiffness predictions for SMC

The same procedure is also performed for SMC. First the effect on stiffness of TT-cracks in all layers was analyzed (assuming damage only in 45-, 60-, 75- and 90-layers of the effective laminate) and the result is shown in Figure 7 as a solid line. The experimental stiffness reduction is much larger, which indicates that indeed other damage mechanisms are acting in SMC. Considering matrix cracks (perpendicular to

(17)

10

loading direction) as crossing the ±0 sublaminate and assuming their density equal to whole crack density as measured on the edge we obtain additional stiffness reduction seen in Figure 8. matrix cracks in 75 and 60 layers have very similar effect on stiffness whereas the reduction due to matrix cracks in 45-layers have less effect. The total effect of TT-cracks and matrix cracks on elastic modulus is shown in Figure 7 as interrupted curve. The predictions are in an excellent agreement with experimental stiffness reduction data.

Finally, the ply discount model predictions are seen in Figure 9. The "classical version" of the model assuming zero stiffness of the damaged layer predicts very large stiffness reduction, see Figure 9a) where also the effect of the damage in each layer is indicated. The model modification which assumes only zero transverse- and shear modulus in the damaged layer is closer to reality but the predictions are still very conservative.

6 Conclusions Damage evolution in SMC composite was analyzed using model composites with bundle structure and varying the bundle orientation. Through-the thickness bundle cracks (TT-cracks) and matrix cracks transverse to the loading were identified using optical microscopy as the basic damage mechanisms.

The damage evolution with increasing applied strain in off-axis bundles was quantified and described in terms of effective strain. The effective strain was defined using the analogy with strain based quadratic failure criterion which includes interaction between transverse and shear strain. This damage evolution law was used to predict damage evolution in SMC laminates which is needed to analyze stiffness reduction with applied strain.

The stiffness of the damaged SMC composite was analyzed using laminate analogy, representing the random bundle distribution in the composite by a [0+15, ±30, ±45, ±60, ±75,90]s laminate. The elastic properties of layers in this model were obtained by homogenizing over the bundle structure. The stiffness reduction in SMC due to TT-cracks and matrix cracks was predicted using previously developed model for damaged laminates with an arbitrary number of cracks in off-axis layers.

The predictions are in a very good agreement with available experimental data which is rather surprising, considering the rather large number of rather rough assumptions used in modeling. Ply discount type of models significantly overestimate the stiffness reduction.

Acknowledgements The authors wish to thank Reichhold AS in Norway for supplying the test material used in the experimental work. This research is funded by Volvo Car Corporation and VINNOVA in Sweden through the vehicle research program (PFF).

11

References

1 Ladeveze P. In: Talreja R editor. Damage Mechanics of Composite materials, Elsevier Science B.V., 1994, pp.117-138.

2 Lemaitre J, Dufailly J. Damage Measurements. Engineering Fracture Mechanics 1987; 28: 643-661.

3 Dano ML, Maillette F, Gendron G, Bissonnette B. Damage modelling of random short glass fiber reinforced composites. In: Proc. Cancom 2001, Quebec, Canada, Aug 21-24, 2001, pp 9-17.

4 Ye L. On fatigue damage accumulation and material degradation in composite materials. Composites Science and Technology 1989; 36:339-350.

5 Hour KY, Sehitoglu H. Damage development in a short fibre composite. Journal of Composite Materials 1993;27:782-805

6 Hull D. An introduction to composite materials. Cambridge solid state science 1981.

7 Oldenbo M, Fernberg SP, Berglund LA, Mechanical behaviour of SMC composites with toughening and low density additives. Composites, Part A 2003; 34:875-885.

8 Wang SS, Chim ESM. Fatigue damage and degradation in random short-fiber SMC composite. Journal of Composite Materials 1983; 17:114-134.

9 Fitoussi J, Guo G, Baptiste D. A statistical micromechanical model of anisotropic damage for SMC composites. Composites Science and Technology 1998; 58:759-763.

10 Dano ML, Gendron G, Mir H. Mechanics of damage and degradation in random short glass fiber reinforced composites. Journal of Thermoplastic Composite Materials 2002; 15:169-177.

11 Von Bernstoff B, Ehrenstein GW. Failure mechanisms in SMC subjected to alternating stresses. Journal of Material Science 1990; 25:4087-4097.

12 Aymerich F, Priolo P. Characterization of a SMC material for bumper applications. Key Engineering Materials 1998; 144:179-190.

13 Wang SS, Suemasu H, Chim ESM. Analysis of fatigue damage evolution and associated anisotropic elastic property degradation in random short-fiber composite. Engineering Fracture Mechanics 1986; 25:829-844.

14 Jiao GQ, Zheng ST, Suzuki M, Iwamoto M. Damage evaluation of sheet moulding compound composite by acoustic emission. Theoretical and Applied Fracture Mechanics 1990;14: 135-140.

15 Suzuki M, Jiao GQ. A study on damage growth of short fiber SMC composites in the case of quasi-static and tensile loading. VI International Congress on Experimental Mechanics. Vol. I , Portland, Oregon; USA; 6-10 June 1988. pp 344-349.

12

16 Kabelka J, Hoffmann L, Ehrenstein GW, Damage process modeling on SMC. Journal of Applied Polymer Science 1996; 62:181-198.

17 Hashin Z, Rosen BW. The elastic moduli of fiber-reinforced materials. Journal of Applied Mechanics 1964;31:223-232.

18 Hashin Z. Analysis of Composite Materials - a Survey. Journal of Applied Mechanics 1983; 50:481-505.

19 Lundmark P, Varna J. Constitutive relationships for damaged laminates in in-plane loading. Submitted to International Journal of Damage Mechanics.

20 Data Sheet: SMC Roving R07EX1, Owens Coming, 2002.

21 Gudmundsson P, Östlund S. First order analysis of stiffness reduction due to matrix cracking. Journal of Composites Material 1992; 26:1009-1030.

22 Gudmundson P, Zang W. A Universal model for thermo-elastic properties of macro cracked composite laminates. International Journal of Solids and Structures 1993; 30:3211-3231.

23 Lundmark P, Varna J. Sliding effect in shear-loaded laminates with transverse cracks in 90-layer. Submitted to Advanced Composites Letters 2004.

13

Table 1. Recipe of the S M C paste (based on supplier's data)

Components Weight percent Unsaturated polyester 23%

resin Low Profile additives 13% Calcium Carbonate 60%

fdler Inhibitor 1.3% Hardener 0.6%

Release agent 1.3% Thickener 0.8%

Table 2. Fiber weight and volume fraction and bundle fractions in the analysed laminates

Laminates analyzed Wf vf vh h [mm] he [mm]

for stiffness reduction he [mm]

[0,±60,0] 0.210 0.148 0.256 2.40 0.44 (x2) [0,90,0] 0.221 0.155 0.269 2.43 0.87 SMC-R 0.30 0.216 0.530 2.43 N/A

Laminates analyzed for crack density

[0,±45,0] 0.216 0.153 0.265 2.67 0.41 (x2) [0,±60,0] 0.235 0.166 0.287 2.51 0.28 (x2) [0,±75,0] 0.140 0.098 0.170 2.55 0.56 (x2) [0,90,0] 0.221 0.155 0.269 2.55 0.32

14

EBB IBS 11 M B

100 um

t i t tia ilt

Figure 1. Picture showing bundle crack (left) and whole layer crack (right) in [0,±G,0] laminate.

Figure 2. Results from the crack density measurements.

15

16

[0,90,0]

1.0 T

+ experimental data

TT-cracks

+

0,7 -I 1 1 i

0,5 0,6 0.7 0.8 0.9 1,0

ex (%)

Figure 5. Model prediction compared with experimental data for the stiffness reduction in [0,90,0] laminate.

0,75 -t 1 i

0,5 0,6 0,7 0,8 0,9 1,0

ex (%)

Figure 6. Model prediction compared with experimental data for the stiffness reduction in [0,60,-60,0] laminate using k = 0.35.

SMC

0.6

0,5

0,5 0.6 0,7 0,8

ex (%) 0.9 1.0

Figure 7. Model prediction compared with experimental data for the stiffness reduction in SMC using k = 0.35.

Matrix cracks in SMC

1

0,99

0,98

0,97

0,96

0,95

0,94

0,93

0,92

0,91

0,9

0,5 0.6

- - 75

--•75+60

- - 75+60+45

0,7 0,8

ex (%)

0,9 1.0

Figure 8. Simulated stiffness reduction in S M C due to matrix cracks related to certain bundle orientation as function of applied strain. Cracks go through both +0 and -8 layers.

i s

Figure 9a: Ply-discount model

ui

ui

1.0

0.9

0.8

0,7

0.6

0.5

0.5 0.6 0.7 0.8

e x(%)

0.9

90-layer broken

•-«--90+75 layers broken

90+75+60 layers broken

90+75+60+45 layers broken

1.0

Figure 9b: Modified Ply-discount model

X

Ui

1.0

0,9

0.7

0.6

0.5

0.5 0.6 0.7 0.8

ex (%)

0.9

— 90-layer broken

-90+75 layers broken

--90+75+60 layers broken

- 90+75+60+45 layers broken

1.0

Figure 9. Ply discount models prediction of stiffness reduction a) assuming zero stiffness of the damaged layer; b) assuming zero transverse and shear modulus.

19

LULE^ UNIVERSITY, OF TECHNOLOGY

Universitetetryckeriet, Luleå