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COHESIVE ZONE MODELLING OF FRACTURE IN POLYBUTENE

L. Andena a,*, M. Rink a, J.G. Williams b

a Dipartimento di Chimica, Materiali e Ingegneria chimica “G. Natta”, Politecnico di Milano, Piazza

Leonardo da Vinci 32, 20133 Milano, Italy

b Department of Mechanical Engineering, Imperial College London, Exhibition Road, South Kensington,

London SW7 2AZ, United Kingdom

ABSTRACT

Fracture properties of isotactic polybutene-1 have been investigated. Fracture tests have been conducted

and relevant properties at initiation have been determined according to linear elastic fracture mechanics.

Two distinct fracture mechanisms have been identified, one of them causing partial instability during

crack propagation. Numerical modelling has been performed using a cohesive zone approach. In

particular, the identification of suitable cohesive laws has been tried using parametric identification and

two different experimental methods. Results suggest that two different cohesive laws may be needed in

order to describe crack initiation and crack propagation.

KEYWORDS

Polybutene, fracture, cohesive zone, identification, FEM

1. INTRODUCTION

Isotactic polybutene-1 (PB) is increasingly being used for the manufacturing of piping systems to be used

in heating and plumbing installations. According to the Polybutene Piping Systems Association (PBPSA),

PB is considered to have many advantages over competitive and more traditional polymers, for example

* Corresponding author Tel: +390223993207 Fax: +390270638173

Email address: luca.andena@polimi.it

Manuscript

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with respect to its long-term mechanical performance at high temperatures. There are extensive studies on

the transition which occurs between its two crystalline forms (I and II) after melting [1,2]. However, the

literature concerning the mechanical properties of PB is very scarce [3].

Cohesive zone modelling has proven to be a powerful method to describe fracture of adhesives and tough

polymers. For this reason it has been chosen to study the fracture behaviour of PB. The cohesive zone

approach is used to describe the material behaviour in the zone which is ahead of the crack. To do so a

constitutive law is defined, which correlates the stresses in this process zone (tractions) with the relevant

opening displacement (separation). An open issue is the determination of cohesive laws able to describe

different materials. One way of doing it is by means of parameter identification, assuming a general shape

for the law (i.e. linear, bi-linear, polynomial, etc.) and determining its parameters from experimental tests,

using optimization procedures. Recently a direct measurement technique has been proposed by Williams

[4-6]: the cohesive law can be obtained performing tensile tests on circumferentially notched specimens.

The nearly uniform d istribution of stresses across the section allows for the determination of the tractions

as a function of the locally measured separation.

A new indirect method [7,8] has been more recently developed, which can be applied to any kind of

fracture test. It has been applied to the three point bending configuration in which the stresses across the

section are not uniform. Therefore, the proposed method uses an iterative procedure based on a finite

element (FE) model. Although indirect, the identification procedure doesn’t require an a priori definition

of a shape for the cohesive law as traditional parameter identification methods do.

2. EXPERIMENTAL DETAILS

The material investigated is a pipe grade PB kind ly supplied in the form of pellets by Basell Polyolefins.

The pellets have been compression moulded into 170x120x10 mm plates. After cooling from the melt, PB

crystallizes in form II, which is characterised by tetragonal symmetry. This form is unstable at room

temperature and spontaneously evolves into form I, which has an hexagonal lattice. To allow for

completion of the trans ition, specimens have been cut and machined at least 15 days after moulding [3],

and then tested.

Pure Mode I (opening) conditions have been attained using Compact Tensile (CT), Single Edge Notched

Bending (SENB) and Circumferentially Notched Tensile (CNT) configurations. Notches have been

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introduced by means of razor tapping, razor sliding and using a single point cutting tool on a lathe for CT,

SENB and CNT respectively. For each configuration the most suitable notching technique has been

chosen according to geometry requirements (e.g. axial symmetry) and laboratory expertise. Great care has

been taken while performing the operation in order to ensure proper alignment of notches and to avoid

damage to the specimen.

Fig. 1 shows a sketch of the three geometries while relevant dimensions are listed in Table 1. Tests have

been performed at room temperature on screw-driven electro-mechanic dynamometers. Constant

crosshead speeds of 1 and 10 mm/min have been used for CT and SENB specimens; in the case of CNT

the tests have been run at 0.05 mm/min.

3. RESULTS

For both CT and SENB configurations partially unstable crack propagation has been observed. The load

vs. displacement curves are quite irregular after the peak load, with many small bumps and a few sudden

drops (see Fig. 2). These drops were accompanied by clear “tick” sounds during the tests.

A high-resolution digital camera has been used to perform crack propagation measurements by taking

shots at regular intervals during the tests. The pictures showed the formation of localized regions of

highly stretched material along the crack path. The sudden rupture of these regions has been deemed

responsible for causing small jumps in the propagating crack and the abrupt decreases of the load.

Fractured samples have been analysed using an optical microscope (see Fig. 3): two distinct kinds of

behaviour can be clearly identified. There is a general rough pattern characterised by a shiny appearance,

and a few dark marks, randomly distributed across the section, which are almost flat. These two separate

set of features correspond to different fracture mechanisms. A first one exhibits small scale ductility

giving rise to the rough shiny surface. The other mechanism is clearly associated with the unstable crack

propagation occurring when a localized stretched region fails : in fact the number and size of the dark

marks have been correlated with the number and amount of the drops in the load vs. displacement curves.

This second mechanism is only active during the propagation stage.

The observed phenomenology was very similar for CT and SENB. The only difference was the presence,

in the latter case, of a small kink in the load trace at crack initiation: its origin is not clearly understood

yet but as can be seen in Fig. 4, it is associated to significant blunting at the crack tip, which was less

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evident on CT specimens. Although this kink is a small feature on the overall load vs. displacement

curve, its existence makes the application of the indirect method for the identification of cohesive laws

difficult. This issue will be further discussed in section 6.

Linear elastic fracture mechanics (LEFM) critical parameters have been evaluated according to ISO13586

[9]. Crack onset has been detected optically and critical values of 9 kJ/m2 for the energy release rate GC

and 1.75 MPa√m for the critical stress intensity factor KC have been determined; there is substantial

agreement between the two configurations (CT and SENB). The ISO standard specifies size criteria

which need to be satisfied in order to ensure small scale yield ing and plane strain conditions. Both criteria

are usually simultaneously satisfied if the specimens have standard dimensions and the condition is:

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5.2

⋅>

Y

CKh

σ (1)

where σY is the yield stress for the material, which is about 20 MPa for PB [3]. In the present case the

thickness h should be over 18 mm, which is significantly higher than the actual value of 10 mm.

Therefore, size requirements are not strictly satisfied due to the large extent of the plastic zone for this

material. However, since the same values for the critical fracture parameters are obtained, the different

stress triaxiality ahead of the crack tip for CT and SENB does not play a significant role.

The calculation for GC has been extended beyond crack initiation to get an elastic estimate of the

propagation energy release rate (see Fig. 5). It can be seen that after initiation G increases beyond GC up

to a value which stays approximately constant during the propagation stage.

A thorough investigation of the influence of the sample geometry on fracture properties has also been

conducted. Results are not discussed here but they have been presented in [10]. Rate effects have not been

observed in the range of applied testing speeds (1-10 mm/min).

A preliminary set of tests on CNT samples has been conducted. Unstable crack propagation occurring

after the peak load led to complete fracture before the whole traction-separation law could be observed.

This prevented the application of the direct method for the identification of a cohesive law for polybutene.

The instability is very likely to be caused by the high compliance inherent in the deep notched specimens.

The notch depth cannot be reduced in order to preserve the high level of constraint required by CNT tests

[5].

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4. PARAMETRIC IDENTIFICATION

A first attempt to describe fracture data has been made using a cohesive model proposed by Hadavinia

and other authors [11,12]. A cubic traction-separation law has been assumed:

(2)

where T is the traction and:

(3)

belowabove uu − is the normal displacement discontinuity at the interface. δc and σmax are the two

parameters which fully define the cohesive zone behaviour. The fracture energy Gc can be derived

through a simple integration of the traction-separation law. It can be used in place of one of the other two

parameters, to which it is related by the following equation:

(4)

An elasto-plastic material model with a Mises yield surface has been used to describe the bulk material

outside the cohesive zone. Relevant parameters are shown in Table 2 and they have been taken from

results obtained by Passoni [3]. The overall model has been implemented in a commercial finite element

code.

A parametric study has been conducted to identify cohesive parameters from CT and SENB tests. The

values have been determined so as to obtain the best agreement between the outcome of the numerical

simulations and the experimental load vs. displacement curves. For both configurations it has been

possible to identify a set of parameters giving a very good agreement between numerical and

experimental data, as shown in Fig. 6. However, the two sets differ in the values of both Gc and σmax. The

a priori assumption of a cubic law prevents the identification procedure from obtaining a cohesive law

able to represent the intrinsic (i.e. independent of the testing configuration) material behaviour. Moreover,

c

belowabove uuδ

λ −=

ccG δσ max169

=

)21(4

27 2max λλλσ +−=T

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Gc values identified for both CT and SENB configurations significantly exceed the experimental value of

9 kJ/m².

5. INDIRECT METHOD

The a priori choice of a shape for the cohesive law can give rise to a potential transferability problem.

The cohesive model in this case is a purely phenomenological model which is determined by matching

results of a particular experiment with the numerical analyses. The consequence is a loss of generality; the

extension to an arbitrary geometry is not guaranteed because the cohesive law does not necessarily

describe the real physical fracture processes. With this in mind, results obtained on CT and SENB tests

using parametric identification are not surprising.

A direct identification method could have been used as an alternative approach to determine the cohesive

law but preliminary CNT tests on PB haven’t been successful yet.

A third approach has been considered, based on a hybrid experimental/numerical method which has been

previously applied to amorphous polymers [7,8]. This indirect method has been used for the

determination of a cohesive law from the SENB tests, without the need to assume a predefined shape. The

method uses a finite element model in conjunction with experimental local separation and macroscopic

load data.

The separation at the crack tip has been measured with a video extensometer. This device (a VE5000 by

Trio Sistemi e Misure, Italy) can accurately follow the relative displacements of four markers placed in a

square pattern very close to the crack tip (as shown in Fig. 4). The crack tip opening displacement can

then be obtained through an interpolation of the opening displacement of the two couples of markers

above and below the crack tip.

Unlike with the CNT configuration, in the case of SENB it is not straightforward to derive the local

tractions from the macroscopic load. The exact stress distribution along the process zone depends on the

actual material cohesive law, which is the objective of the identification procedure. Since local tractions

at the crack tip cannot be directly measured, a numerical tool is needed. A finite element model of the

SENB sample has been used for this purpose, with interface elements placed along the crack path to

implement the cohesive zone model. In order not to restrain the shape of the cohesive law, a linear

stepwise function has been chosen. Provided the number of linear segments is large enough, any shape

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can be described with good approximation. Our implementation used 30 segments, with a total of 60

parameters available (slope and length of each segment).

The actual identification procedure consists in an iterative scheme which uses experimental local

separation and macroscopic load data up to crack initiation. F irstly, an initial slope for the cohesive law is

guessed. This slope is positive as it is assumed to represent some kind of initial elastic response of the

material inside the cohesive zone. A constant step size for the opening displacement at the crack tip node

(CTOD) is chosen for the model. The CTOD is the numerical equivalent of the measured local separation.

The finite element model is run using an arclength algorithm [13] to control the CTOD and get the

macroscopic load as an output. After the first iteration this output is compared with the experimental load

corresponding to a measured separation equal to the CTOD imposed in the FE model. If the two values

differ, the segment slope is adjusted and the step repeated until the difference between experimental and

numerical load is cut down to a specified tolerance. At this stage, the first segment of the cohesive law up

to a separation value equal to the imposed CTOD has been identified. A new step is then performed, i.e.

the CTOD is increased. If the increase is small enough, only a very limited number of interface nodes

close to the crack tip will increase their opening displacement beyond the CTOD value at the previous

step: the behaviour of most of the interface nodes will be described by the part of the cohesive law which

has already been identified. Again, the slope of the “new” segment of the cohesive law will be adjusted so

that the load calculated with the FE model equals the corresponding experimental value. Then again the

CTOD is increased and the procedure is repeated until the whole traction-separation law is identified, i.e.

until the traction drops to zero. When this critical condition is reached at the crack tip node, crack

initiation occurs according to the FE model. The scheme of a single iteration step is illustrated in Fig. 7.

The cohesive law identified using this method is rate-independent but the procedure may be applied to

tests conducted at several speeds, thus deriving a set of rate dependent cohesive laws for each material.

The same approach has been used in [6].

Fig. 8 shows the identified cohesive laws obtained for two different samples using the indirect method.

The two laws differ s lightly; it is thought that the identification procedure could be improved by

enhancing the accuracy of the separation measurements. This can be done with better optics and lighting

conditions. Nevertheless the associated fracture energy is almost identical. This value is in much better

agreement with the experimental GC (9 kJ/m²) than the relevant value from the cohesive law obtained

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using parametric identification on a SENB sample, also shown in Fig. 8. Apparently the segment size

used for the purpose of the identification is not very important, provided a sufficient number of segments

is taken to avoid a coarse representation of the cohesive law.

A video camera has been used to detect crack initiation time during the tests. If the plot of separation

(measured with the video extensometer) is entered with this initiation time, an experimental value of the

critical separation can be obtained. This value, indicated by the dotted line in Fig. 8, is very small

compared to the specimen size. This justifies the use of a cohesive zone approach as in this case the

cohesive energy (i.e. the area underlying the traction-separation curve) truly represents the energy release

rate. The critical separation predicted by the indirect method is quite accurate, especially when compared

with δC of the SENB cubic law.

In the final part of the cohesive law identified with the indirect method there is a narrow peak which is

closely related to the kink observed at initiation on the experimental load curves. Since the identification

procedure makes use of load data, it is clear that its final result is influenced by the kink, whose presence

hasn’t clearly been explained so far.

6. DISCUSSION

The experimental load vs. displacement curve and the one obtained with the FE model using the cohesive

law identified with the indirect method have been compared in order to validate the proposed

identification scheme. This comparison is significant since displacement data are not considered by this

scheme which only uses load-separation data instead. As shown in Fig. 9 a very good agreement is

obtained just up to the kink in the load trace, after which there is a steep decrease in the numerical curve

that does not reproduce the behaviour observed in the real experiment. A possible explanation may be

given by looking back at the evaluation of G shown in Fig. 5. Following crack initiation, the energy

release rate increases up to a value which is significantly higher, suggesting that two distinct levels of

fracture energy may characterise initiation and propagation. As the identification procedure only

considers data up to initiation, it cannot capture the propagation behaviour.

This hypothesis is corroborated by the analysis of the simulations of the SENB tests performed using the

cubic law obtained from parametric identification. As shown in F ig. 6 this law gives a good overall

agreement between the experimental and calculated load vs. displacement curves but this is quite obvious

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since this agreement is the very criterion used for its identification. As the shape of load vs. displacement

curve is largely determined by the propagation stage, the parametric identification procedure is quite

insensitive to the initiation behaviour. In fact, the energy associated with this law is very close to the

estimated propagation value of the fracture energy. As a consequence, the cubic cohesive law’s

parameters (fracture energy and critical separation) strongly disagree with the corresponding experimental

measurements at initiation. The law identified using the ind irect method has proven significantly more

accurate in describing crack initiation.

Ideally a single cohesive law should be able to capture the overall fracture behaviour as the apparent

toughness increase observed during crack propagation would be caused by extensive plastic deformation

occurring outside the process zone [14]. However in the case of PB this instance has not been observed

while two fracture mechanisms have been reported (see section 3), one of them only active during

propagation. For this reason separate cohesive laws should be used to reproduce the initiation and

propagation behaviour. The difference between the two fracture energy levels is the net contribution

associated to the formation of localized stretched zones during crack propagation and their subsequent

rupture. The complex interplay of the two mechanisms during the propagation stage could require

incorporation of rate effects both in the bulk and in the cohesive zone description.

7. CONCLUSIONS

The experiments performed on PB highlighted the existence of two fracture mechanisms. The main

mechanism is characterised by a small-scale ductile behaviour and a high surface roughness. During crack

propagation a second mechanism is also active. It is associated to the formation of highly stretched

localized regions. The sudden rupture of these regions causes partial instability during crack propagation

and generates flat areas on the fracture surface. During propagation the fracture energy apparently

increases to a value which is higher than the critical value at initiation.

A cohesive zone approach has been adopted to model the fracture behaviour of PB. Three different

methods have been used for the purpose of identifying a suitable cohesive law for the material.

A direct measurement technique on CNT samples hasn’t given any useful results because unstable crack

propagation occurred before completion of the tests. Nevertheless, these results are preliminary and

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further investigation is ongoing in order to clarify the reasons of this behaviour, possibly with the aid of

numerical simulations.

An alternative approach relies on numerical methods. In this work a finite element method has been used.

Two identification techniques have been adopted: a purely numerical parametric identification and a more

recently developed hybrid technique. Parametric identification performed on the basis of CT and SENB

experiments failed to identify an intrins ic cohesive law, as the result of the procedure depends on the

testing configuration. Provided different values of the cohesive parameters are chosen for the two

configurations, the overall experimental behaviour can be reproduced quite well using a two-parameter

cubic law. However, the identified values of the critical fracture energy and separation performed at crack

initiation do not agree with experimental measurements. A significantly better agreement has been

obtained by using the hybrid technique on SENB samples. The cohesive law thus identified reproduces

very well the experimental behaviour up to initiation, but fails to do so for the propagation stage.

The critical analysis of the observed fracture behaviour of PB and the results of the cohesive zone

modelling approach suggest that separate cohesive laws may be needed in order to give an accurate

description of crack initiation and crack propagation.

REFERENCES

[1] Chatterjee AM. Butene polymers. In: Encyclopaedia of Polymer Science and Engineering, 1985. p.

590.

[2] Azzurri F. Flores A. Alfonso GC. Baltà Calleja FJ. Polymorphism of isotactic poly(1-butene) as

revealed by microindentation hardness. 1. Kinetics of the transformation. Macromolecules. 2002;35:9069

[3] Passoni P. Frassine R. Pavan A. Small scale accelerated tests to evaluate the creep crack growth

resistance of polybutene pipes under internal pressure. Proceedings of Plastics Pipes XII Milan 2004.

[4] Pandya KC. Williams JG. Measurement of cohesive zone parameters in tough polyethylene. Polym.

Eng. Sci. 2000;40(8):1765-1776.

[5] Pandya KC. Williams JG. Cohesive zone modelling of crack growth in polymers. Part 1 –

Experimental measurement of cohesive law. Plast., Rubber Compos. 2000;29(9):439-446.

[6] Pandya KC. Williams JG. Cohesive zone modelling of crack growth in polymers. Part 2 – Numerical

simulation of crack growth. Plast., Rubber Compos. 2000;29(9):447-452.

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[7] Bianchi S. Corigliano A. Frassine R. Rink M. Modelling of Interlaminar Fracture Processes in

Composites using Interface Elements. Compos. Sci. Technol. (Article in press)

[8] Andena L. Rink M. Fracture of rubber-toughened poly (methyl methacrylate): measurement and study

of cohesive zone parameters. Proceedings of ICF XI Turin 2005.

[9] International Standard Organization. Plastics – Determination of fracture toughness (GIC and KIC) –

Linear elastic fracture mechanics (LEFM) approach. ISO 13586:2000.

[10] Andena L. Frassine R. Rink M. Roncelli M. Thickness effect on fracture behaviour of polybutene.

Proceedings of 4th ESIS TC4 conference Les Diablerets 2005.

[11] Hadavinia H. Kinloch AJ. Williams JG. Finite element analysis of fracture processes in composites

and adhesively-bonded joints using cohesive zone models. In: Advances in Fracture and Damage

Mechanics II, Hoggar, 2002. p.445-450.

[12] Chen J. Crisfield M. Kinloch AJ. Busso EP. Matthews FL. Qiu Y. Predicting progressive

delamination of composite material specimens via interface elements. Mech. Compos. Mater.

Struct.1999;6:301-317.

[13] Corigliano A. Formulation, identification and use of interface models in the numerical analysis of

composite delamination. Int. J. Sol. Struct. 1993;30:2779-2811.

[14] Hutchinson JW. Evans AG. Mechanics of materials: top-down approaches to fracture. Acta Mater.

2000;48:125-135.

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CAPTIONS

Fig.1 – Configurations used for the fracture tests

Fig. 2 – Load vs. displacement curves for two CT samples

Fig. 3 – Fracture surface of a SENB sample

Fig. 4 – Load vs. displacement for a SENB sample. Detail of crack tip blunting at initiation

Fig. 5 – Crack advancement and energy release rate vs. displacement for a CT sample

Fig. 6 – Comparison between experimental data and simulations with identified parameters

Fig. 7 – Iteration step for the determination of the traction-separation law: (a) situation at the beginning of

the step; (b) the separation is increased; (c) a slope is guessed for the new segment of the cohesive law;

(d) the predicted macroscopic load is compared with the experimental value; (e) the slope is adjusted to

reduce the gap between predicted and measured load; (f) the procedure is repeated until the predicted load

equals the experimental value. A new step is performed.

Fig. 8 – Comparison between cohesive laws identified using the indirect method and parametric

identification on SENB samples (experimental GC = 9 kJ/m2); the initiation line indicates the experimental

value of the critical separation as measured by the video extensometer

Fig. 9 – Comparison between experimental data and simulations with cohesive law identified using the

indirect method

Figure 1 Black & WhiteClick here to download high resolution image

Figure 2 Black & WhiteClick here to download high resolution image

Figure 3 Black & WhiteClick here to download high resolution image

Figure 4 Black & WhiteClick here to download high resolution image

Figure 5 Black & WhiteClick here to download high resolution image

Figure 6 colour for the webClick here to download high resolution image

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Figure 8 colour for the webClick here to download high resolution image

Figure 9 Black & WhiteClick here to download high resolution image

Figure 1 colour for the webClick here to download high resolution image

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Table 1 - Nominal dimensions (in mm) of samples

Geometry w W l1 l2 h a R

CT 24 30 28.8 13.2 10 12 3.25

SENB - 20 90 84 10 10 -

CNT - - 3.4 10 - - -

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Table 2 - Material parameters used in the finite element simulations

Young’s modulus 500 MPa

Poisson’s ratio 0.3

Yield stress 20 MPa