Impact on HDPE and PVC plates – Experimental tests and numerical simulations

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International Journal of Impact Engineering 37 (2010) 580–598

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International Journal of Impact Engineering

journal homepage: www.elsevier .com/locate/ i j impeng

Impact on HDPE and PVC plates – Experimental tests and numerical simulations

Rafael Traldi Moura a, Arild H. Clausen b,*, Egil Fagerholt b, Marcılio Alves a, Magnus Langseth b

a Group of Solid Mechanics and Structural Impact, University of Sao Paulo, Av. Prof. Mello de Moraes, 2231, Sao Paulo, SP, 05508-030, Brazilb Structural Impact Laboratory (SIMLab), Centre for Research-based Innovation and Department of Structural Engineering, Norwegian University of Science and Technology, RichardBirkelands vei 1A, NO-7491 Trondheim, Norway

a r t i c l e i n f o

Article history:Received 3 February 2009Received in revised form4 December 2009Accepted 20 December 2009Available online 4 January 2010

Keywords:Material propertiesDigital image correlationPoisson’s ratioRate sensitivitySplit-Hopkinson pressure barPneumatic acceleratorLS-DYNA

* Corresponding author.E-mail address: [email protected] (A.H. Claus

0734-743X/$ – see front matter � 2010 Elsevier Ltd.doi:10.1016/j.ijimpeng.2009.12.004

a b s t r a c t

This paper presents first material tests on HDPE and PVC, and subsequently impact tests on plates madeof the same materials. Finally, numerical simulations of the plate impact tests are compared with theexperimental results. A rather comprehensive series of mechanical material tests were performed todisclose the behaviour of PVC and HDPE in tension and compression. Quasi-static tests were carried outat three rates in compression and two in tension. Digital image correlation, DIC, was used to measure thein-plane strains, revealing true stress–strain curves and allowing to analyze strain-rate sensitivity andisotropy of Poisson’s ratio. In addition, dynamic compression tests were carried out in a split-Hopkinsonpressure bar. Quasi-static and dynamic tests were also performed on clamped plates made of the samePVC and HDPE materials, using an optical technique to measure the full-field out-of-plane deformations.These tests, together with the material data, were used for comparative purposes of a finite elementanalysis. A reasonable agreement between experimental and numerical results was achieved.

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1. Introduction

Material scientists often define three classes of materials:metals, ceramics and polymers [1]. Considering polymers, they arecommonly grouped as thermoplastics, thermosets and elastomers(rubbers). The main difference between those is related to theirchemical structure. All polymers consist of long molecule chains,where carbon atoms constitute the back-bone of the chain. Inthermoplastics, there are no covalent bondings between the chains.There are only weak van der Waals bonds between atoms inneighbouring chains. In addition, the chains have a preferred coiledstate, resulting in the so-called entanglements where the moleculesare winded around each other [2]. This molecule structure is thereason why many thermoplastics turn out to be very ductile, butneither stiff nor strong. The two other groups of polymers, i.e.thermosets and elastomers, are different in chemical structurebecause of covalent bondings, or cross-links, between carbon atomsin neighbouring chains. These affect their mechanical properties.Thermosets, having up to one cross-link for every 20 carbon atoms[2], are rather stiff, while elastomers, with only one cross-link forevery 1000 carbon atoms, have great flexibility. Among these three

en).

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groups, thermoplastics are the far most important one. They arelight, easy to process, rather cheap, some of them are extremelyductile, resistant to corrosion and, depending on additives andexposure, they may also be rather sustainable.

Nowadays, polymers find new applications in many branches ofindustry and structural engineering, and they occur in productswhich are subjected to mechanical loading in some way. Inparticular, the use of thermoplastics subjected to impact loading inaccidental situations is increasing. Some examples include auto-motive industry, pipes and tubes, containers, packaging devicesand mobile phone covers. Even as protective armour againstmissiles, thermoplastics, and in particular polycarbonate (PC), havefound application [3]. This calls also for a rather careful design ofthose components or structures involved, in accordance withcurrent engineering practise and the application of a numerical toolsuch as the finite element method. Within this context, reliablematerial models for thermoplastics are a challenge. There has beena substantial research effort during the last decades, with ther-moplastics models still in need of further improvement [4]. Anecessary pre-requisite for the development of new constitutivemodels, however, is a database of material tests which can be usedfor validation, but also provide information on what kind of phys-ical behaviour the model should be able to represent. Moreover,a set of well-defined component tests is required for validation, andwe apply plates for this purpose.

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Table 1Material data as provided by SIMONA.

Material r

(kg/m3)sy

(MPa)3y

(%)3b

(%)E(MPa)

Tmin

(�C)Tmax

(�C)a

(K�1)

PE-HWU 950 22 9 300 800 �50 70 1.8� 10�4

PVC-TF 1430 53 3 20 3000 �10 60 0.8� 10�4

R.T. Moura et al. / International Journal of Impact Engineering 37 (2010) 580–598 581

Considering plates made of non-metallic materials, there area vast number of references presenting tests and/or simulations ofcomposite plates. Limiting the review to transverse loading ofplates made of thermoplastics, the amount of investigations ismore limited. Many of them have been concerned with ballisticsituations. Wright et al. [3] shot missiles against polycarbonate (PC)plates of different thicknesses and also performed quasi-static tests.Applying a high-speed camera, they found that there were differ-ences in the deformed profile of the plate between the quasi-staticand dynamic cases, being caused by inertia effects. Their mainemphasis, however, was on projectile nose shape and fracturemode. Also, using PC plates, now under different degradationconditions, Weber and Miguez Suarez [5] fired 0.38 inchesprojectiles. Rosenberg et al. [6] studied the ricochet of projectileshitting an inclined target plate, and included also some numericalsimulations describing the polymer material with an equation ofstate. Testing ABS plates at speeds ranging from 0.1 to 1000 mm/s,Dean and Wright [7] presented force-time curves, showing that themodulus and strength increased with the load rate. They alsoprovided some numerical predictions employing Abaqus anda Drucker–Prager model, showing good agreement with experi-mental results. Jimenez et al. [8] presented force-time curvesobtained from plates made of different polystyrene (PS) qualities.The plates were hit by a falling weight system, and the results werecompared with those from an analytical model. Bourne et al. [9]investigated the dynamic response of four different polymerssubjected to impact from copper disks.

This study explores material and component responses. Twopolymeric materials were chosen: polyethylene of high density(HDPE) and polyvinyl chloride (PVC). Plates of 5 mm and 10 mmthickness made of these materials were tested under central quasi-static and impact loading, providing an experimental databasewhich is useful for a subsequent benchmarking of material modelsin future research. In addition, material tests at different rates arenecessary, both for revealing the physical behaviour of the materialat hand, and for calibration of the coefficients in the materialmodel. A digital image correlation technique was used in thematerial tests in order to obtain true stress-strain curves. Consid-ering the component tests, emphasis was given to the plasticdeformation process, or, to be more precise, the applied force vs.displacement of the plate’s centre point. Moreover, some of thetests were monitored with a camera-based equipment measuringout-of-plane deformation of the whole plate during the test. Thisinformation is very useful when comparing response from theexperimental test with the numerical predictions. Neither the loadcapacity of the plates nor the ballistic limit is a key issue in thepresent investigation.

This paper is organized as follows. The material test programmeis presented in Section 2, the component tests on the HDPE and PVCplates follow in Section 3, while Section 4 covers some preliminarynumerical simulations of the plates, applying LS-DYNA. Finally, thepaper is rounded off with a discussion and conclusions.

2. Material tests

2.1. Materials and experimental programme

A semi-crystalline polyethylene of high density, HDPE, and anamorphous polyvinyl chloride, PVC, were selected for this study.Both materials were acquired as extruded plates from the Germancompany SIMONA. The producer describes the HDPE material,named PE-HWU, as a black material specially stabilized for exterioruse, with high UV-protection, low water absorption and highchemical and corrosion resistance. Also according to SIMONA, thePVC material, called PVC-TF, has similar properties with respect to

stability, although it may absorb some moisture. The producer doesnot report any particle or fiber reinforcement, but some additivesare introduced for degradation resistance.

The material properties as provided by the manufacturer arepresented in Table 1, where r is density, sy is yield stress, 3y is theelongation at yield, 3b is the elongation at break, and E is Young’smodulus. Further, Tmin and Tmax are the minimum and maximumoperating temperatures, and a is the coefficient of thermalexpansion.

There are several reasons for this choice of materials. Firstly, theexperimental study reported herein is a part of a larger researchproject, involving development of a constitutive model for ther-moplastics and validation of this model by means of well-definedbenchmark tests. HDPE and PVC, being semi-crystalline andamorphous, respectively, cover two main types of thermoplastics.Keeping the intention of model development in mind it was alsochosen to work with generic materials without any reinforcementat this stage of research. Plates are a convenient geometry forseveral purposes, as it is possible to machine material as well ascomponent test samples from a large plate. Finally, HDPE and PVCare both important in engineering applications. Their greatproduction corresponds, respectively, to 15.1% and 14.1% of the totalvolume of polymers produced in the United States in 2001 [10].Among other fields of use, both are also extensively applied inwater and gas pipelines.

The extruded plates delivered from SIMONA had length2000 mm and width 1000 mm. Two different plate thicknesses of5 mm or 10 mm were purchased for both materials. From eachplate, three quadratic pieces of 600 mm� 600 mm were cut, andthese pieces were used in the impact tests. The remaining parts ofthe original plates were retained for material tests or other types ofcomponent tests. All material test specimens presented hereinwere taken from the 10 mm thick plate.

A rather comprehensive material test programme involvingquasi-static and dynamic tests was carried out. The quasi-statictests, employing a servo-hydraulic test machine, were performed inuniaxial tension and compression, see Section 2.2. In addition toestablish true stress-strain curves, possible anisotropy and plasticvolume changes of the material were also explored in these tests. Asplit-Hopkinson pressure bar was used in the dynamic tests, carriedout in compression only, see Section 2.3. All tests were performedat room temperature.

2.2. Quasi-static material tests

2.2.1. IntroductionThe quasi-static material tests were performed under displace-

ment control in a servo-hydraulic Dartec machine with a 20 kN loadcell and an Instron controller. The maximum stroke length of thismachine was 100 mm.

Two different designs of compression specimens denoted C1and C2, see Fig. 1(a), and two kinds of tension specimens, called T1and T2, see Fig. 1(b), were machined. While C1 is a cubic sample(10 mm)3, C2 has length 15 mm in the loading direction. Thedifference between T1 and T2 is the thickness. In order to assess anypossible anisotropy, tension and compression coupons were takenin different angles with respect to the extrusion direction as shown

Fig. 1. Geometry of (a) compression samples and (b) tension samples, with (c) layout of the specimens cut from the original plate. The small geometrical imperfection (fillet) madein the gauge section of the tension samples is not shown.

R.T. Moura et al. / International Journal of Impact Engineering 37 (2010) 580–598582

in Fig. 1(c). The compression samples are denoted C1-x, C1-y orC1-z, where x, y or z define the loading direction in accordance withthe coordinate system defined in Fig. 1(c). The specimens of type C2were made in the x- and y-direction, and are respectively denotedC2-0 and C2-90 in Fig. 1(c). This figure also defines the orientationof the specimens T2-0, T2-45 and T2-90. Clearly, the angle0� corresponds to the x-axis direction.

Moreover, a small geometrical imperfection (fillet) with radius100 mm and depth 0.1 mm was machined in the gauge section ofthe tensile specimens in order to ensure a well-defined onset of theinitial neck. This fillet facilitates optical measurements of defor-mation, yet it is so small that it does not affect the stress and strainfield in the gauge section. A similar approach during tension testson polymers was recently reported by Fang et al. [11].

It is well known that thermoplastics, in contrast to metals, mayexperience volume change during plastic deformation [11–15].Therefore, the standard formulas 3¼ ln(1þ e) and s¼ s(1þ e),relating nominal and true values of strain and stress, are notdirectly applicable for HDPE and PVC. Measurement of transversestrains in addition to longitudinal strains is therefore requiredduring the test. Another feature related to thermoplastics is that

Fig. 2. Complete set-up with

a neck occurs rather early in tension tests, resulting in a non-homogeneous deformation field. A convenient way to handle thesechallenges is to use an optical technique for strain measurements.Images of the sample were obtained during test with a Canon EOS-1 camera, see Fig. 2. A software based on a procedure called digitalimage correlation (DIC) was applied to process the images andsubsequently determine the strains, as described in Section 2.2.

In order to calculate the true stress, the transverse strain on twoperpendicular surfaces of the coupon has to be measured. For thispurpose, the specimens T1-0 and C1, see Fig. 1(c) were used. Withonly one camera and no use of mirrors, the pictures can only givethe strain components at one surface of the specimen, i.e. at onecoordinate plane. By doing two replicate tests, however, where twoperpendicular surfaces of the coupon face the camera, informationon all three normal strain components can be obtained. The idea isillustrated in Fig. 3. This figure shows two replicate compressiontests, using specimen C1, denoted and . Both cubes are loaded inthe z-direction, i.e. in the thickness direction of the original plate.The camera, however, is positioned in different ways in these twotests. The camera faces the xz-surface of coupon in test , and thepictures from this test will give the strain components 3z, 3x and gxz

specimen and camera.

Fig. 3. Replicate tests in compression with two perpendicular camera views.

Table 3Experimental programme in compression. Directions are defined in Fig. 1(c).

Coupon Loaddirection

Nominalstrain rate

Replicates Intention

C1-x x 0.001 2 To check if material is isotropic withrespect to the loading directionand transverse deformation

C1-y y 0.001 2C1-z z 0.001 2

C2-0 x 0.0002 1 To evaluate strain rate sensitivityin x-direction (extrusion direction)C2-0 x 0.001 1

C2-0 x 0.005 1

C2-90 y 0.0002 1 To evaluate strain rate sensitivityin y-direction (transverse in-planedirection)

C2-90 y 0.001 1C2-90 y 0.005 1


at this surface via the DIC technique. Similarly, strain components3z, 3y and gyz are obtained from test . Subsequently, the longitu-dinal strain 3z can be applied as a common reference for both tests,allowing to check whether the transversal strains 3x and 3y havea similar evolution as function of 3z. In other words, the tworeplicate tests were employed to check if the deformation istransverse isotropic. All three loading directions in compressionwere checked in this way, involving a total of 6 samples for eachmaterial. A similar approach was taken in tension, now applyingspecimen T1 and considering only the extrusion direction.

The specimens C2-0 and C2-90 were used to investigate thestrain-rate sensitivity in compression, thereby covering the extru-sion direction and the in-plane transverse direction. The strain-ratesensitivity in tension was evaluated with specimen T2-0. Finally,specimens T2-0, T2-45 and T2-90 were used to check the isotropyin the extrusion plane. Details of the test programme are outlined inTables 2 and 3. The same programme was employed for bothmaterials. The scatter between replicate tests was small.

2.2.2. Digital image correlation (DIC)DIC is an optical method for measuring displacement fields on

an object surface by comparing the pictures of the sample atdifferent deformation stages. The pictures have to be obtained bya digital CCD or CMOS camera with high resolution for accuratedeformation measurements. Before each test, a random patternwith good contrast was applied to the surface of the specimen. Thispattern deforms with the sample, as shown in Fig. 4(a) for a tensilespecimen and in Fig. 4(b) for a compression specimen. The testimages were obtained with a Canon EOS-1 camera. The software 7D[16] processed the digital pictures by applying a mesh to thespecimen and calculating the displacements as it finds the positionof the mesh nodes when minimizing a digital image correlationfactor. From the displacements of each node of the element, it ispossible to determine the in-plane Green–Lagrange strain tensor E.

The first image is divided in a grid of square elements, eachrepresented in a local, natural coordinate system (x,y) ˛ (0,1) andwith side length Lp. The subsequent images are compared to thefirst one using a grey level correlation coefficient. The initial image

Table 2Experimental programme in tension. Directions are defined in Fig. 1(c).

Coupon Loaddirection

Nominalstrainrate (s�1)

Replicates Intention

T1-0 x 0.001 2 To check if material is isotropicwith respect to transversedeformation

T2-0 x 0.001 1 To check if material is isotropicwith respect to in-planeloading direction

T2-45 45� 0.001 1T2-90 y 0.001 1

T2-0 x 0.001 1 To evaluate strain rate sensitivityin x-direction (extrusion direction)T2-0 x 0.01 1

grey level is given by the discrete function f ðx; yÞ. After deforma-tion, the function becomes f *ðx*; y*Þ ¼ f *½xþ uðx; yÞ; yþ vðx; yÞ�,with uðx; yÞ and vðx; yÞ being the displacement fields in the x- and y-direction, respectively, for a given grid. For the sample addressed inFig. 4, the x-axis is assumed to be oriented in the coupon’s longi-tudinal direction. The displacement field can be expressed in termsof the displacements of the corners (points A, B, C and D) and thesize of a square element grid, see Fig. 4, by

uðx; yÞ ¼ ðduB � duAÞx� xA

LPþ ðduD � duAÞ

y� yA

LP

þ ðduC � duB þ duA � duDÞL2

P

ðx� xAÞðy� yAÞ þ duA ð1Þ

vðx; yÞ ¼ ðdvB � dvAÞx� xA

LPþ ðdvD � dvAÞ

y� yA

LP

þ ðdvC � dvB þ dvA � dvDÞL2

P

ðx� xAÞðy� yAÞ þ dvA (2)

where duA;duB; duC ;duD are the incremental displacementcomponents in x-direction of the corners, and dvA; dvB; dvC ;dvD arethe corresponding y-direction incremental displacement compo-nents. From the established displacement field in Eqs. (1) and (2), itis possible to calculate the components of Green–Lagrange’s straintensor E, represented by

Exx ¼vuvxþ 1

2

�vuvx

�2

þ�

vv

vx

�2!; (3)

Eyy ¼vv

vyþ 1

2

�vuvy

�2

þ�

vv

vy

�2!

(4)

and

Exy ¼ Eyx ¼12

�vuvyþ vv

vxþ vu

vxvuvyþ vv

vxvv

vy

�: (5)

In the uniaxial case, the Green–Lagrange strain 3G reads

3G ¼L2 � L2

0

2L20

; (6)

where L0 and L are the initial and deformed length in a givendirection. The true (logarithmic) strain 3 can now be found as

3 ¼ lnLL0¼ lnð23G þ 1Þ1=2¼ 1

2lnð23G þ 1Þ: (7)

Provided that the transverse deformation is isotropic, we have3G;width ¼ 3G;thickness ¼ 3GT , where 3GT is the transverse Green–Lagrange strain. The true stress can now be determined from

Fig. 4. Photos of (a) tension specimen and (b) compression specimen, both with random pattern painted and a representative square element of a DIC grid.


s ¼ FA¼ F

A0

1ð23GT þ 1Þ (8)

where F is the force, and A and A0 are the current and initial cross-section area, respectively.

2.2.3. Tension testsThe main purpose with the tension tests was to obtain true

stress–strain curves at different rates. A necessary pre-requisitewas to explore the ratio between the width and thickness strains.Two replicate tests were therefore performed with load applied inthe x-direction, and, respectively, recording images of the xy and xzplanes. The rather thick specimen T1, see Fig. 1(b), was used. Thecross-head velocity of the test machine was 0.033 mm/s in thesetwo tests, giving a nominal strain rate of 10�3 s�1.

By means of the digital photos and the DIC technique, thelongitudinal strain 3x was calculated in different sections along thespecimen’s gauge section. By plotting 3x from different sections asfunction of time, it was seen that the strain evolves fastest in thesection where the small geometrical imperfection was machined,and a constant cold-drawing strain level was therefore first ach-ieved there. Thus, all stress and strain curves presented fromtension tests are taken from this section with the initial fillet. This isalso in accordance with the idea behind the seven markers’ tech-nique proposed by G’Sell et al. [17].

Fig. 5. Longitudinal (thick lines) and transverse (thin lines) strains as function of tim

Longitudinal and transverse strains obtained from the filletsection of the two replicate tests are shown in Fig. 5(a) and (b) forHDPE and PVC, respectively. In the HDPE tests, there is no differ-ence between the transverse strains, see Fig. 5(a). At first sight,Fig. 5(b) suggests some anisotropy in the PVC material. A closerlook, however, reveals that the difference between strains in the yand z directions in Fig. 5(b) probably is due to some scatter betweenthe two replicate tests. This is demonstrated in Fig. 6(a), showingthat the force - displacement curve is slightly different in the twoparallel tests. It turns out that the ratio between the longitudinaland transverse strains is similar in the two tension tests of PVC, seeFig. 6(b). Therefore, transverse deformation isotropy with respectto tension in the extrusion direction seems to be a soundassumption for both materials. Consequently, it is sufficient tomeasure one of the transverse strain components, and Eq. (8) canbe used to calculate the true stress.

After demonstrating the presence of isotropic transversedeformation, tests were carried out to evaluate the isotropy withrespect to the load direction within the extrusion plane, and toassess the strain rate effect. Three directions (0�, 45� and 90�) andtwo strain rates (10�3 and 10�2 s�1, corresponding to cross-headvelocity of respectively 0.033 and 0.33 mm/s) were explored. Now,specimens of type T2, see Fig. 1(b), were applied, only monitoringthe xy-surface with the camera. True stress-strain curves obtainedfrom tests involving different loading direction and rates are shown

e from tension tests at nominal strain rate of 10�3 s�1 for (a) HDPE and (b) PVC.

Fig. 6. Replicate tests on PVC in tension. (a) Force–displacement curves and (b) longitudinal strain–transverse strain curves.


in Figs. 7 and 8, respectively, for both materials. Eqs. (7) and (8)were used to calculate the true strain and true stress, respectively.The small circles in Figs. 7 and 8 address the yield stress. All yieldstresses from the HDPE tests were found by means of the Considereconstruction [18], while the peak values were taken as the yieldstress for PVC.

It can be seen from Fig. 7(a) that HDPE shows almost no direc-tion dependency. PVC behaves differently, see Fig. 7(b), being softerin the extrusion direction than in the transverse direction, and withthe 45� direction as an intermediate case. The PVC specimensfractured in the tension tests. For HDPE in tension, a limiting testcondition was the maximum stroke length of 100 mm of the testmachine, which was not sufficient to strain these samples tofracture.

Another important observation from Fig. 7 is that PVC experi-ences a true-stress softening effect, which is not present for HDPE.This softening is probably connected to the evolution of volumetricstrains. Fig. 5(a) shows that the longitudinal strain for HDPE is

Fig. 7. True stress–strain curves in different load directions from tension tests at nominal

approximately twice the transverse strain, thus, the volume changefor HDPE in tension is not significant. On the other hand, Figs.5(b)and 6(b) reveal that the transverse strain is smaller than half thelongitudinal strain for PVC, resulting in a volume increase. Thesoftening of PVC is directly influenced by the volume changebecause it affects the denominator of Eq. (8). Also, voids wereobserved in the ruptured sections of the PVC samples after fracture,and the volume change seems therefore to be closely related toa damage effect.

Fig. 8 shows that the stress-strain curves of both materials aredependent on strain rate, with PVC being a bit more sensitive thanHDPE. It should be noticed, however, that the set-up applied in thetests presented herein allowed for rather small rates only, withina range of one decade. Also, the lack of smoothness for the curvesobtained at strain rate of order 0.01 s�1 is related to a camera limitof one frame per 5 s.

Fig. 9 shows the relation between the longitudinal and trans-versal strains at the two applied rates, implying that Poisson’s ratio

strain rate of 10�3 s�1 for (a) HDPE and (b) PVC. The points B are applied in Fig. 19.

Fig. 8. True stress–strain curves from tension tests at different nominal strain rates in the extrusion direction for (a) HDPE and (b) PVC. The points B are applied in Fig. 19.


n is the gradient of these curves. It does not seem that there is anysignificant rate dependency on n, and consequently neither on thevolume change.

2.2.4. Compression testsApplying the cubic specimen C1, six tests were used to evaluate

the load direction dependency and the relation between longitu-dinal and transverse strains. They were divided in three pairs ofreplicate tests, each pair subjected to a different load direction x, yor z in the coordinate system defined in Fig. 1(c). As for tension, DICwas employed in all tests to determine the longitudinal andtransverse strains. These six tests were performed using a constantvelocity of 0.01 mm/s of the machine cross-head in order to obtaina nominal strain rate of about 10�3 s�1. Also here, the local strainrate was obtained from the DIC calculations.

Although a layer of grease was applied to the surface betweenthe sample and the support platens, some friction was present,causing some scatter in the strain field on the specimen’s surface.

Fig. 9. True transverse strain vs. true longitudinal strain from

Keeping in mind that the coupons had a regular geometry withoutany imperfection, it was therefore not possible a priori to identifythe section along the specimen to use in the strain calculations.Thus, the distributions of longitudinal and transverse strains wereestablished over the whole length of the compression sample for allpictures. The most representative section was selected as the onewith the highest longitudinal strain in the beginning of the test. Inorder to avoid disturbances from the corners, the homogeneity ofboth normal strain components across the section was alsoconsidered. All curves to be presented from the compression testswere taken from the chosen representative section.

Figs. 10–12 show longitudinal and transverse strains as a func-tion of time in the representative section for load applied in the x, yand z direction, respectively. Again, both materials seem to have anisotropic transverse deformation behaviour under uniaxial loading,and Eq. (8) can therefore be used to determine the true stress alsoin the compression tests. Similar to the tension tests shown in theprevious section, the longitudinal strains for HDPE are about twice

tension tests at different rates for (a) HDPE and (b) PVC.

Fig. 10. Longitudinal (thick lines) and transverse (thin lines) strains as function of time from compression tests with load in the x-direction and nominal strain rate of 10�3 s�1 for (a)HDPE and (b) PVC.


the transverse ones, implying conservation of volume duringplastic deformation. The transverse strains for PVC are compara-tively smaller, which indicates a negative volumetric strain for thismaterial in compression. The true stress-strain curves for speci-mens compressed in different directions are shown in Fig. 13. Thetriangles, addressing the yield stress in the z-direction, are used inthe calibration of a rate-sensitivity model, see Section 2.4. Similar asfor the tension tests, Considere’s construction gives the yield stressfor HDPE, and the yield stress for PVC is simply the maximumstress.

True stress–strain curves obtained at three different rates, nowusing samples with geometry C2, see Fig. 1(a), are presented in Figs.14 and 15 for load applied, respectively, in the longitudinal and in-plane transverse direction. The strain-rate sensitivity is similar tothe one observed in tension. Also here, the yield stresses are indi-cated with symbols. As both materials are slightly anisotropic,different symbols are applied for the two directions.

Fig. 11. Longitudinal (thick lines) and transverse (thin lines) strains as function of time fromHDPE and (b) PVC.

The ratio between transverse and longitudinal strains atdifferent directions and strain rates in compression is shown inFig. 16. For neither HDPE nor PVC, Poisson’s ratio exhibits any strainrate or direction dependence.

2.3. Dynamic compression tests

2.3.1. IntroductionThe dynamic stress-strain curves of the two materials were

obtained with a split-Hopkinson pressure bar (SHPB), see Fig. 17.The rig consists of an input and a transmission bar, which both weremade of polymer materials in order to avoid a large impedancemismatch between the bars and the sandwiched sample. A strikerhits the end of the input bar, generating a pulse of intensity 3i. Thispulse has a length twice the striker and propagates through theinput bar until it reaches the bar-sample interface, where a part ofthe pulse is reflected with intensity 3r and the other part continues

compression tests with load in the y direction and nominal strain rate of 10�3 s�1 for (a)

Fig. 12. Longitudinal (thick lines) and transverse (thin lines) strains as function of time from compression tests with load in the z direction and nominal strain rate of 10�3 s�1 for (a)HDPE and (b) PVC.


through the sample. At the next interface, separating the sampleand the transmission bar, the pulse is once again partly reflectedand partly transmitted. The last part propagates in the transmissionbar with intensity 3t. The strain gauges used to measure thesepulses are placed close to the central position of each bar, therebyavoiding a superposition of the incident and reflected pulse in theinput bar.

The subsequent treatment of the sampled pulses relies on threemajor assumptions: (i) the waves propagating in the bar can bedescribed by one-dimensional wave propagation theory; (ii) thestress and strain fields in the specimen are uniform in its axialdirection; and (iii) the specimen inertia effects and the frictioneffect in the compression test are negligible. A ring-shaped spec-imen was used to diminish the errors related to assumptions (ii)and (iii) [19]. Assumption (i) is a challenge, however, becausea material dispersion and attenuation are introduced to the pulsepropagation by the visco-elasticity and damping of the polymer

Fig. 13. True stress-strain curves in different load directions from compression tests at nomi

bars. The consequence is a difference between the pulse in the bar-specimen interface and the pulse measured by the gauges. Tohandle this, the signals sampled by the strain gauges were con-verted to the frequency domain by means of the Fast-FourierTransform, and thereafter transferred to the bar-specimen inter-face. Bacon [20] studied experimentally how the phase velocity andattenuation coefficient change with frequency variation. Althoughone-dimensional visco-elastic wave propagation theory wasassumed, both geometrical and material dispersion effects wereincluded in the results of his experimental tests. Moreover, Young’smodulus in the frequency domain can be found using the visco-elastic frequency equation proposed by Zhao et al. [21]. Thisequation is based on three-dimensional visco-elastic wave propa-gation theory.

Applying these frequency domain methods to treat the phasevelocity and Young’s modulus, the stress and velocity of the barwere calculated at the bar-sample interface also in the frequency

nal strain rate of 10�3 s�1 for (a) HDPE and (b) PVC. The points 6 are applied in Fig. 19.

Fig. 14. True stress-strain curves from compression tests at different nominal strain rates in the x-direction for (a) HDPE and (b) PVC. The points � are applied in Fig. 19.


domain. These signals were thereafter converted back to the timedomain and used to determine the force F and displacement u atboth ends of the sample. Finally, the true strain 3S and true stress sSin the specimen were obtained by

3SðtÞ ¼ ln�

1þ u2ðtÞ � u1ðtÞLS

�(9)

sSðtÞ ¼F

AðtÞ ¼F

ASe23T ðtÞ¼ F

ASe2n3SðtÞ (10)

where LS, A(t) and AS are, respectively, the initial length and thecurrent and initial cross-section area of the specimen, 3T is thetransverse true strain, n is the sample’s Poisson’s ratio in the plasticregime, and 1 and 2 represents the interface between input bar andspecimen and between specimen and output bar, respectively. Figs.9 and 16 indicate that Poisson’s ratio is not rate-dependent, and itwas therefore assumed that n is the same at dynamic test condi-tions as measured in the quasi-static tests.

Fig. 15. True stress-strain curves from compression tests at different nominal strain rat

2.3.2. Set-up and resultsApplying the rig shown in Fig. 17, the HDPE specimens were

tested with bars made of PMMA, while the PVC material was testedwith PVC bars. All specimens were ring-shaped. The nominaldimensions were 13.10 mm external diameter, 9.4 mm internaldiameter and 2.1 mm thickness. The dynamic tests were carried outin the thickness, or z, direction of the material. All input andtransmission bars were 1.5 m long, with a diameter of 24.42 mm forPMMA and 25.34 mm for PVC. The strikers had the same diameters,while their lengths were 21.44 mm for the PMMA striker and20 mm for the PVC striker. The density of the bar materials was1187 kg/m3 for PMMA and 1497 kg/m3 for PVC.

Two pairs of strain gauges (Excel PA-13-062AB-120L) are glueddiametrically opposed to each other at the middle section of eachbar, and they are connected by a half-bridge to a Fylde FE-H395-TAhigh-speed transducer amplifier. The amplified signal is fed to a NIPCI-6110, 12-Bit, 5 MS/s/ch board. Finally, a MATLAB functioncontrols and analyse the data, applying the procedure outlined inthe previous section.

es in the y-direction for (a) HDPE and (b) PVC. The points þ are applied in Fig. 19.

Fig. 16. True transverse strain vs. true longitudinal strain from compression tests at different rates for (a) HDPE and (b) PVC.


HDPE were tested at two levels of dynamic strain rate, whilefour levels were investigated for PVC. The strain rates rangedfrom 1800 to 4800 s�1. Four tests were carried out at each levelof rate, and the strain rate differed by approx �50 s�1 withinthese replicates. True stress-strain curves obtained from thedynamic tests are shown in Fig. 18. The indicated strain rates arethe mean rates observed during the four replicates. As usual insplit-Hopkinson tests, the elastic part of the curves must beconsidered with care because there is no force equilibriumbetween the specimen interfaces in this test first phase. Theshadows represent the maximum and minimum values withinall four tests at each rate, while the curve itself is the meanvalue. The stress values used for calibration of the rate coeffi-cients in the numerical model, see Section 2.4, are indicated witha triangle.

2.4. Calibration of Cowper–Symonds coefficients

As a preliminary numerical approach, material model 89 of thenon-linear finite element code LS-DYNA [22] will be employed inSection 4 of this paper in simulations of the plate impact tests to bepresented in Section 3. Data points from a stress-strain curve isgiven directly as input in this model, thus, a fit of parameters to theobserved stress-strain data is not required. The model handlesstrain-rate sensitivity with the Cowper–Symonds (CS) constitutiveequation [23,24]

Fig. 17. Split-Hopkinson pressure ba

s ¼ s

"1þ

�_3�1=p

#; (11)
D S C
where is sD is the dynamic stress at an elevated strain rate, sS isa quasi-static reference stress, _3 is the strain rate, and C and p arematerial parameters to be determined from experimentalobservations.

The points shown in Fig. 19 represent yield stress vs. strain ratefound from material tests carried out at different strain rates. Forthe quasi-static tests, the abscissa values are the local strain rates asmeasured with the DIC technique. All points in Fig. 19 wereaddressed in Figs. 7, 8, 13–15 and 18. As both materials are slightlyanisotropic in compression, see Fig. 13(a), the three directions x, yand z are identified separately in Fig. 19. Only the extrusion direc-tion is considered in tension.

For both materials, the coefficients C and p were identified fromthe z-direction compression tests by a regression procedure. Thecontinuous curves in Fig. 19 show the resulting Cowper–Symondsmodel for each material as defined by Equation (11). These curvesalso represent the two other directions in compression fairly well.Table 4 shows the obtained coefficients C and p for both materials.The dashed curves, having the same values of C and p, givea satisfactory representation of the strain-rate behaviour for thetension tests. Looking at the quasi-static tests for HDPE, however,these tests seem to over-estimate the rate sensitivity of this

r applied in the dynamic tests.

Fig. 18. True stress–strain curves obtained in z-direction in the dynamic compression tests for (a) HDPE and (b) PVC. The points 6 are applied in Fig. 19.


material. It is therefore important to have dynamic tests in order tocapture the correct strain-rate hardening.

3. Component tests

3.1. Test programme and experimental set-up

Square plates with in-plane dimension 600 mm� 600 mm andthickness 5 mm or 10 mm were used in the component tests, seeFig. 20(a). They were made of the thermoplastics HDPE and PVCintroduced in Section 2.1. The plates were loaded by an impactor atthe centre under quasi-static and dynamic conditions. The clamp-ing system is similar in both types of tests and comprises two stiffrings with free span diameter between the inner rims of 500 mm,which held the plates with 24 pre-stressed M16 bolts along a circle,see Fig. 20(b). The clamping rings and the central part of a severelydeformed HDPE plate are shown in Fig. 21(a), also elucidating theoptical pattern applied on the rear side of the plate by a projector.This optical system, used to measure the transverse deformation of

Fig. 19. Yield stress as function of strain rate as observed in the experimental tests an

the plate, is described in Section 3.2. A cylinder of diameter 60 mmwith a 5 mm rounded end radius, see Fig. 21(b), served as impactornose in all quasi-static and dynamic tests.

For each combination of material (HDPE and PVC) and thickness(5 mm and 10 mm), two quasi-static tests were performed ina servo-hydraulic test rig. The speeds were 1 mm/s and 0.1 mm/s. Aprojector and a camera recording at 4 Hz were also used in order tocapture field measurements of out-of-plane displacements, seeSection 3.2. The out-of-plane displacement field was determined inall four quasi-static tests.

The dynamic impact tests were performed in a pneumaticaccelerator [25,26], consisting of a pressure chamber connected toa vertical accelerator tube, see Fig. 22. The impactor acceleratesdown the tube when the pressure in the chamber is released. Thisimpactor consists of the nose shown in Fig. 21(b) and a central rod.Two strain gauges are glued at diametrically opposite positionsclose to the lower end of this rod. The total mass of the projectileincluding the nose was 18.41 kg. The force in the impactor wasdetermined from the gauge recordings. Further, Newton’s second

d fit of the Cowper–Symonds (CS) constitutive model for (a) HDPE and (b) PVC.

Table 4Cowper–Symonds coefficients.

Material C p

HDPE 119 5.86PVC 252 5.96


law gives the corresponding acceleration, and a subsequent calcu-lation, involving numerical integration, yields the velocity anddisplacement as function of time [25]. The impact velocity imme-diately before the impact was measured by two photo cells [26].Depending on material and plate thickness, this initial velocity wasbetween 4 and 16 m/s in the dynamic tests. Unlike the quasi-staticcase, optical out-of-plane deformation measurements were per-formed only in some of the dynamic tests. Now, digital frames werecollected at 10 kHz. In addition to the projector and the high-speedcamera, a mirror was also used in the dynamic tests, see Fig. 22.Thereby, the camera was kept away from the impact area and wasnot vulnerable to any damage.

3.2. Field measurements of out-of-plane displacements

An optical technique was used to obtain full-field out-of-planedisplacements measurements in some of the tests. This techniquehas earlier been applied in quasi-static and dynamic tests onaluminium plates [27,28], but is now employed also in this study onpolymer plates, experiencing considerably larger deformationsthan the aluminium plates do. A patterned light was projected ontothe rear plate surface, and images were recorded with a camera.Prior to the tests, a calibration procedure allowed to establisha correlation between camera, image and plate coordinate systems.A further description of calibration and image processing isprovided by Grytten et al. [27] and Fagerholt et al. [28].

Three of the photos obtained during a dynamic test of a HDPEplate with thickness 5 mm are presented in Fig. 23(a). Applying theoptical technique, Fig. 23(b) shows the out-of-plane displacementdetermined from the third photo of Fig. 23(a). The in-plane origin ofthe plot in Fig. 23(b) is located at the plate centre. Fig. 23(c) illus-trates the evolution in time of the deformation profile of the plate.The predictions from the optical measurements at the centre of theplate are also compared with displacements determined from thestrain gauge signals. These displacements are marked with an x.The correlation between optical and strain gauge displacementmeasurements is in general good, except that there is a deviation atmaximum deformation, see Fig. 23(c). Thus, it is evident that theoptical technique was able to provide reasonable measurements ofthe displacements of the plate.

3.3. Force–displacement curves

Fig. 24(a)–(d) shows force–displacement curves for the fourcombinations of material and plate thickness. As the main purpose

Fig. 20. (a) Drawing of the plate geometr

of the present study was to establish an experimental database forfuture numerical simulations of the deformation behaviour, frac-ture and exact determination of ballistic limit were not of impor-tance. The plates which actually fractured are identified with an x atthe end of each respective force–displacement curve in Fig. 24, andthese curves are also drawn with a thinner line in the figure. Threeof the quasi-static tests on HDPE avoided fracture because thehydraulic cylinder reached its maximum stroke length of 100 mm.Whether fracture occurred or not in the dynamic tests wasdependent on the impact velocity.

The quasi-static force–displacement curves show that the load,as expected, is nearly doubled when increasing the thickness from5 to 10 mm. Also, the maximum force is about twice as large for thePVC plates than for the HDPE plates. This difference between HDPEand PVC is directly related to the stress-strain curves shown in Figs.7 and 13.

There are some differences between the response in the quasi-static and dynamic tests, see Fig. 24. During the initial phase ofa dynamic test, stress waves are propagating back and forth in theplate until the boundaries are fully activated, and this situation withstress oscillations can also be recognised from the force–displace-ment curves. Moreover, the response seems to be stiffer in thedynamic tests. The maximum force is also, in general, slightlyhigher in the dynamic cases. On the other hand, the maximumdisplacement at fracture is smaller in the dynamic than in thequasi-static tests, indicating a more brittle response at increasedloading rates.

4. Numerical simulations

In order to further explore the data obtained so far, somepreliminary numerical simulations of the component tests wereperformed using the software LS-DYNA, version 971 [18]. The platedescribed in the previous section was modelled as a circularclamped plate of diameter 500 mm. It was discretised with 568075Belytschko–Tsay shell elements. These elements allowed the use ofthe material model described next. The impactor nose, seeFig. 21(b), was modelled with 566594 rigid body solid elements.The finite elements had a side length of 0.5 mm in the contactregion between the impactor and the plate. An automatic generalcontact algorithm with no friction was used to model the contactbetween impactor and plate. The initial velocity was applied to thewhole nose part and gravity effects were included. Fig. 25 depictsa detail of the mesh used in the model, showing the central part ofthe plate and the nose of the impactor.

The material model chosen for the simulations was MAT_PLASTICITY_POLYMER, which is material model number 89 in LS-DYNA [18]. This model requires a true quasi-static stress-straincurve as input. For both materials, the curve obtained in tension inthe extrusion direction at a strain rate of 10�3 s�1 was selected. Themodel can handle strain rate effects via the Cowper–Symonds Eq.(11). On the other hand, it does not allow for any pressure

y. (b) Photo of the clamping system.

Fig. 21. (a) Section view of the clamped ring system and a HDPE plate with severe transverse deformation caused by the impactor’s nose. The optical fringe pattern projected on therear side of the plate is also shown. (b) Nose of the impactor used for quasi-static and dynamic tests.

Fig. 22. Pneumatic accelerator used for dynamic impact tests, including equipmentused for optical field measurements of out-of-plane deformations.


dependency, and the response is therefore similar in tension andcompression. For the clamped plates at hand, the membrane effectensures that the tension mode is dominating, at least for largedeformations.

Force–displacement curves from one dynamic test withoutfracture in each case of plate material and thickness are shown inFigs. 26(a)–(d). There are three curves in each of these figures; oneobtained in the experimental test and two predictions from LS-DYNA. The effect of strain-rate on the numerical results wasexplored by running one simulation with the Cowper–Symondsstrain rate effects activated and one deactivated.

The main trends in the experimental test are captured by thesimulations. The shape of the force–displacement curve is wellrepresented, and the agreement is also good in the phase imme-diately after impact, when stress waves propagate back and forth inthe plate. Further, the simulations where the Cowper–Symondsrelation is included give a rather accurate prediction of themaximum force in all cases. In general, deactivation of the CS rateeffects results in a slightly lower maximum force, but somewhatlarger deformation.

Since none of the experimental tests shown in Fig. 26 resulted infracture, it is also possible to look at the unloading phase. Now, thesimulations without the CS relation provide slightly better resultsthan those with. The unloading stiffness is almost correct, at leastduring the first stage of unloading. The stiffness is reduced,however, during the final stage of unloading in the experimentaltests, and the numerical model is not able to represent this changeof stiffness. Probably, this is a general challenge for most modelsdealing with polymers.

Considering the 5 mm thick PVC plate addressed in Fig. 26(c),a comparison between the deformation fields as found from thefull-field optical measurements and the LS-DYNA simulationswithout and with the Cowper–Symonds relation is shown in Figs.27(a) and (b), respectively. The shape of the deformed plate is ingeneral well captured by both simulations. Again, it is observed thata good agreement is obtained when CS is included, while thesimulations with deactivated CS predict too large deformations.

5. Discussion

A series of mechanical tests were performed on two thermo-plastic materials, HDPE and PVC. Despite of the importance of thesematerials, there is an evident paucity of detailed mechanical datawhich explores issues like anisotropy, softening, volume conser-vation, etc. They have been approached in this paper via a series of

tensile and compressive tests, carried out at quasi-static rates forboth loading cases and also dynamically in compression.

Fig. 23. Optical measurements of transverse deformation. (a) Pictures of a dynamic test at different stages of deformation. (b) Displacement profile at maximum deformation. (c)Comparison of displacements determined with the optical technique (lines) and integration of strain gauge signals (x).


The measurement system in the quasi-static tests includedimage recording of the samples during deformation. Opticalin-plane measurements of strains are essential for determiningthe true stress in a material test sample made of thermoplas-tics because the deformation is non-homogeneous, but alsobecause the volume is susceptible to change during plasticdeformation.

True stress-strain curves for HDPE, addressing different loadingmodes, directions and rates, are shown in Figs. 7(a), 8(a), 13(a),14(a), 15(a) and 18(a). The material has a slight pressure sensitivityas the yield stress is somewhat higher in compression than intension. As the strains increase, this HDPE exhibits a monotonicstrain hardening in both loading modes. For this polymer, the testsat small but finite cross-head speed indicate that the material isstrain rate sensitive, with the same dynamic behaviour in tensionand compression for small strain rates. The rather strong ratedependency is a well-known feature for thermoplastics. Further,HDPE is found to be rather isotropic with respect to transversedeformation as well as strength. Although no microstructuralinvestigation has been carried out in order to make a thoroughcheck, it seems that the extrusion process does not introduce anymajor anisotropy in this material.

PVC has a quite different response. True stress–strain curvesaddressing PVC in tension are shown in Figs.7(b) and 8(b). Clearly,softening is present at relative small strains, immediately after theonset of yielding. As the strain increases, the material exhibitsa continuous strain hardening up to failure. A similar softeningphenomenon is revealed in compression, see Figs. 13(b), 14(b),15(b) and 18(b), but now the stress level is considerably higherthan it was in tension. The softening is probably connected withdilatation, i.e. the evolution of volume strains, and aging mightalso play a role. No mictrostructural measurements were made inorder to have a closer check, yet, macroscopic voids and crackswere observed at the surface of the tension samples at the laststage of deformation before fracture. Thus, it seems that a damagemechanism is present for PVC, which again can explain not only

the macroscopic difference between tension and compression, butalso that the change of volume in general is larger for PVC than itis for HDPE. The tests at small but finite cross-head speed indicatethat PVC is strain-rate sensitive, more in tension than incompression. In the context of constitutive laws, this, as well asthe dilatation effect, poses further challenges which are out of thescope of this article.

Having future computational work in mind, we performeda series of transverse impact tests on circular plates made of thesame HDPE and PVC materials that were adressed in the tensionand compression tests. The component test programme involved 8quasi-static and 12 dynamic tests. Fracture was observed in sometests, but this was not any key issue in the present investigationbecause present and near-future comparisons with numericalpredictions will cover the constitutive model only, and not anyfracture criterion. Determination of ballistic limit was therefore notrelevant. On the other hand, about one half of these tests wereinstrumented with a camera system, providing full-field out-of-plane deformation measurements during all stages of the test.When comparing with numerical predictions, this deformationfield is very useful for validation purposes.

As a first and preliminary approach, the existing material model89 in LS-DYNA was employed for some numerical simulations ofone dynamic test of each combination of material (HDPE and PVC)and plate thickness (5 mm and 10 mm). Two simulations werecarried out for each plate, one with and one without rate sensitivitydescribed with the Cowper–Symonds model. In general, the formercase provided predictions of the force–displacement response witha high degree of accuracy. It is as expected that the simulationswithout CS and the strain-rate hardening effect underestimates themaximum force. The lack of this rate-hardening contribution alsoserves to predict too large deformations, see Figs. 26 and 27. On theother hand, the unloading phase was slightly better represented bythe simulations without CS.

Although the agreement between the numerical and experi-mental results in general is good in Fig. 26, there are four main

Fig. 24. Quasi-static and dynamic force–displacement curves at the centre of the plates for (a) HDPE plates with 5 mm thickness, (b) HDPE plates with 10 mm thickness, (c) PVCplates with 5 mm thickness, and (d) PVC plates with 10 mm thickness.


aspects contributing to some uncertainty in the simulations. First ofall, no attempt was made to model the real stiffness of clampingsystem. As indicated in Fig. 20(b), the plates were sandwichedbetween two stiff rings tightened together by a set of 24 M16 bolts.This clamping condition is somewhat different from just restrain-ing the nodes of the finite elements at the plate edge, as it was donewithin LS-DYNA. In the physical tests, the membrane force in theplate is likely to strain the 24 holes around the circumference,thereby reducing the stiffness compared with the numerical model.The importance of this error increases with increasing deformation.Yet, inspection of the plates after the experimental tests did notreveal any severe ovalization of the holes.

The second reason for lack of agreement is the initial deforma-tion of the plate before the test. Originally, the plate had noimperfections, but when mounted between the two clamping ringssome of the plates got an initial curvature. This initial deformationwas captured well by the optical technique measuring out-of-planedisplacements, and the largest one was found to be 4 mm. Theseinitial imperfections were not modelled in the numerical analyses.

Thirdly, the true stress-strain curve obtained from tension testsin the extrusion direction at a rate of 10�3 s�1 was used as input in

the material model, assuming that this curve represents the quasi-static response for tension as well as compression. It is alreadyemphasized that in particular PVC behaves differently in these twoloading modes, and the numerical model, which relies on tensiondata, may therefore be suspected to give too low stresses incompression. It is no doubt that the tension mode is most impor-tant one since the membrane effect suppresses the compressionstresses at increasing deformation. On the other hand, the simu-lations are likely to predict too low forces when the bending modeis dominating. Indeed, this seems to be the situation in the firstphase of the test, see Fig. 26, when the deformation is less thanapprox 20 mm. The deformation level at the onset of the largeincrease in stiffness occurring thereafter is not captured well in allsimulations. The reason for this is unclear, but may be related tovisco-elastic effects.

The last aspect is the treatment of dynamic behaviour in thematerial model. As emphasized, the membrane effect results ina tension stress state in most of the plate, but the parameters of theCowper–Symonds constitutive relation were determined fromcompression tests. Dynamic material tests in tension might resultin slightly different CS coefficients.

Fig. 25. Numerical simulation of the impact of a 5 mm thick plate of PVC immediately after the nose is reflected back by the plate.

Fig. 26. Comparison of force–displacement curves at the centre of the plates obtained from dynamic tests and from LS-DYNA predictions with or without strain-rate effect. (a) HDPEplate with thickness 5 mm and impact velocity of 5.6 m/s, (b) HDPE plate with thickness 10 mm and impact velocity of 8.9 m/s, (c) PVC plate with thickness 5 mm and impactvelocity of 6.9 m/s, and (d) PVC plate with thickness 10 mm and impact velocity of 11.5 m/s.


Fig. 27. Comparison of transverse deformation field found from optical full-field measurements and numerical predictions for a 5 mm thick PVC plate subjected to a 6.9 m/s impact.(a) LS-DYNA without strain-rate effect, and (b) LS-DYNA with strain-rate effect included.


6. Conclusions

One of the main goals of this article is to offer a detailed set ofexperimental test data of HDPE and PVC, including material as wellas component tests. Both materials at hand were purchased froma wholesaler. The obtained data can inspire the development ofnew material models. Also, they can be used to identify the materialcontants required in a numerical model. Finally, the componenttests will serve as benchmark cases during validation.

A standard tensile machine was used to apply the tension orcompression load to the samples in the material tests. The approachused to measure the material response is rather new and is basedon an optical technique where digital image correlation, DIC, isa key feature. The DIC technique is based on image processing, andthe software 7D was used to target the in-plane displacement fieldand calculate the Green and logarithmic strains. Thereby, DIC madepossible to obtain detailed information of the material behaviour inthe necking zone. A set of replicate tests showed that there seemsto be a transverse deformation isotropy of both polymers. Someanisotropy with respect to loading direction was detected, inparticular for PVC in tension. In addition to the quasi-static tests,some dynamic compression tests were carried out on both mate-rials applying a split-Hopkinson pressure bar.

A total of 8 quasi-static and 12 dynamic impact tests were per-formed on quadratic plates made of the HDPE and PVC materialsexplored in the material test programme. In some of these tests, anoptical system was employed in order to obtain full-fieldmeasurements of the transverse deformations. These measure-ments agreed reasonably well with the centre displacement asdetermined directly with the test rig instrumentation, and providevaluable information for validation purposes.

The numerical simulations were of a preliminary character, andapplied the existing material model 89 in LS-DYNA. Two sets ofsimulations were performed, one without and one including rateeffects. The Cowper–Symonds model was used to represent the ratedependency. The overall impression is that the numerical simula-tions with the CS model included provide a rather accurateprediction of the observed behaviour in the experimental tests. Theanalyses without the CS model gave in general a softer response ofthe plate, although the unloading behaviour was slightly bettercaptured in this case.

Acknowledgements

The authors would like to thank the Brazilian research fundingagency FAPESP for the economical support. The authors are alsograteful to Stephane Dumoulin, SINTEF, and Trond Auestad, NTNU,for their laboratory assistance.

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Impact on HDPE and PVC plates – Experimental tests and numerical simulations

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