Mat Res Innovat (2003) 7:10–18DOI 10.1007/s10019-002-0217-z

Abstract In many applications of polymers, impact per-formance is a primary concern. Impact tests experimen-tally performed on molding prototypes yield useful datafor a particular structural and impact loading case. But, itis generally not practical in terms of time and cost to ex-perimentally characterize the effects of a wide range ofdesign variables. A successful numerical model for im-pact deformation and failure of polymers can provideconvenient and useful guidelines on product design andtherefore decrease the disadvantages that arise frompurely experimental trial and error. Since the specimengeometry and loading mode for multiaxial impact testprovides a close correlation with practical impact condi-tions and can conveniently provide experimental data,the first step of validating a numerical model is to simu-late this type of test. In this paper, we create a finite ele-ment analysis model using ABAQUS/Explicit to simu-late the deformation and failure of a glassy ABS (acrylo-nitrile-butadiene-styrene) polymer in the standardASTM D3763 multiaxial impact test. Since polymers of-ten exhibit different behavior in uniaxial tensile andcompression tests, the uniaxial compression or tensiletests are generally not representative of the three-dimen-sional deformation behavior under impact loading. A hy-drostatic pressure effect (controlled by the parameter γ)is used to generalize a previously developed constitutivemodel (“DSGZ” model) so that it can describe the entirerange of deformation behavior of polymers under anymonotonic loading modes. The generalized DSGZ modeland a failure criterion are incorporated in the FEA modelas a user material subroutine. The phenomenon ofthermomechanical coupling during plastic deformation isconsidered in the analysis. Impact load vs. displacement

and impact energy vs. displacement curves from FEAsimulation are compared with experimental data. The re-sults show good agreement. Finally, equivalent stress,strain, strain rate and temperature distributions in thepolymer disk are presented.

Keywords Polymer · Impact · Failure · Constitutivemodel · Stress–strain curves · Thermomechanical coupling · Finite element analysis

Introduction

Due to good thermal and electrical insulation properties,low density, high resistance to chemicals and ease ofmanufacturing, polymers have increasingly been appliedin applications where impact performance is a primaryconcern [1, 2, 3, 4]. There are several types of standardtests to evaluate the impact strength of polymers, inwhich the most commonly used are Charpy and Izod [5].Although the two types of tests provide some informa-tion on the relative impact resistance of materials, theparticular specimen geometry requirements of both testsmake the test results difficult to relate to practical poly-mer design models. However, the specimen geometryand loading mode for a multiaxial impact test provide aclose correlation with practical impact conditions [6].

Impact tests experimentally performed on moldingprototypes yield useful data for a particular structuraland impact loading case. But, it is generally not practi-cal in terms of time and cost to experimentally charac-terize the effects of a wide range of design variables. Asuccessful numerical model for impact deformation andfailure of polymers can provide convenient and usefulguidelines on product design and therefore decrease thedisadvantages that arise from purely experimental trialand error. Since the multiaxial impact test has a closecorrelation with practical impact conditions and canconveniently provide experimental data, the first step ofvalidating a numerical model is to simulate this type oftest.

Y. Duan · A. Saigal (✉) · R. GreifDepartment of Mechanical Engineering, Tufts University, Medford, MA 02155, USA,e-mail: anil.saigal@tufts.eduTel.: +1-617- 6272549, Fax: +1-617- 6273058

M. A. ZimmermanLucent Technologies, North Andover, MA 01845, USA,

O R I G I N A L A RT I C L E

Yiping Duan · Anil Saigal · Robert GreifMichael A. Zimmerman

Modeling multiaxial impact behavior of a glassy polymer

Received: 20 August 2002 / Revised: 4 November 2002 / Accepted: 5 November 2002 / Published online: 31 January 2003© Springer-Verlag 2003

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Some work has been done with respect to modelingmultiaxial impact tests and developing numerical modelsfor impact deformation of polymers during the last twodecades [7, 8, 9]. Nimmer [7] created a FEA (finite ele-ment analysis) model to simulate a fixed velocity punc-ture test of Bisphenol A-polycarbonate disk, where aconstant bilinear stress–strain curve was used to approxi-mate the material behavior. Good agreement wasachieved between model prediction and experimentalload vs. deflection data for deflections up to four timesthe thickness of the test disk. By using the G’Sell-Jonasconstitutive model, Billon and Haudin [8] numericallyexplored the effects of specimen thickness, friction be-tween striker and specimen disk, and quality of theclamping device during a multiaxial impact test on poly-propylene, a semicrystalline polymer. They indicatedthat the material response during impact is difficult toanalyze since the deformation is not homogeneous andnot isothermal. After validating the application of theG’Sell-Jonas constitutive model to a semicrystallinepolymer polyamide 12 for a large range of strain ratesduring uniaxial tensile test, Schang et al. [9] developed aFEA model to simulate the multiaxial impact test. Modelpredictions were compared with experimental data forthe history of impact load. Even though good agreementwas achieved at the beginning of the load vs. timecurves, the maximum impact load was largely overesti-mated. They pointed out that this result occurred becauseeither the strain rate dependence in the constitutive mod-el was not correctly taken into account, or the tensilecharacterizations were not representative of the impactsituation. All of the aforementioned simulations are forimpact deformation of polymers.

At least two material models: a stress–strain constitu-tive model and a failure model are required in numericalsimulation of destructive impact events. Both the materi-al models play a significant role on the reliability and ac-curacy of simulation results. Using a failure criterion anda new stress–strain constitutive model (generalized“DSGZ” model) which describe the stress–strain consti-tutive relationship of polymers under any monotonicloading modes, a FEA model is formulated in this paperusing ABAQUS/Explicit to simulate the deformation andfailure of a glassy ABS (acrylonitrile-butadiene-styrene)polymer, which has a glass transition temperature ofaround 378 K, in the standard ASTM D3763 multiaxialimpact test.

Relation to previous work

Two major science and engineering databases weresearched for previous work that is relevant to the topic ofcurrent paper. The first database is ISI (Institute for Scientific Information) SCI-EXPANDED (Science Cita-tion Index Expanded–1945-present), and the second dat-abase is Ei Compendex® (1970-present). The followingkeyword sequences were used in search of the two dat-abases: multiaxial impact test or multiaxial impact tests,

impact and polymer and FEA, impact and polymers andFEA, constitutive model, stress–strain curves, deforma-tion mode and polymers, and thermomechanical cou-pling and polymers. Also, books and Ph.D. thesis weresearched that talked about the mechanical behavior ofpolymers, constitutive modeling, and failure of plasticsor polymers. Based on the literature search, we foundthat some work has been done with respect to modelingmultiaxial impact tests and developing numerical modelsfor impact deformation of polymers during the last twodecades, and the research work that was most relevantwere cited in this paper.

However, the topic of developing numerical modelsfor impact deformation of polymers and modeling multi-axial impact test of polymeric materials is far from suffi-ciently explored. It is well known that at least astress–strain constitutive model and a failure criterionare required in numerical modeling of destructive impactevents. Both these material models play a significant roleon the reliability and accuracy of simulation results. Inthe open literature that we searched, some authors used aconstant bilinear stress–strain curve to approximatepolymer behavior, and some authors used the G’Sell-Jonas constitutive model, which can not describe strain-softening behavior of polymers, characterized by tensiletest to approximate polymer behavior. In general, thesesimulation results are accurate at small deflections, butdeviate substantially at relatively large deflections.

Using concepts from the Johnson-Cook model,G’Sell-Jonas model, Brooks model and Matsuoka model,we developed a phenomenological constitutive model(called “DSGZ” model) to uniformly describe the entirerange of deformation behavior of both glassy and semi-crystalline polymers under monotonic compressive load-ing. However, unlike metals, polymers often exhibit dif-ferent behavior in uniaxial tensile and compression tests.The uniaxial compression or tensile tests are generallynot representative of the three-dimensional (multiaxial)deformation behavior under impact loading. The defor-mation mode should be considered. In this paper, we usea hydrostatic pressure effect (controlled by parameter γ)to generalize the “DSGZ” model so that it can describethe entire range of deformation behavior of polymers un-der any monotonic loading modes. We consider thiswork an important “innovation” in polymer modeling.The generalized DSGZ model was applied in modelingthe multiaxial impact behavior of a glassy ABS polymer.Impact load vs. displacement and the impact energy vs.displacement curves from simulation fit well with exper-imental data up to failure.

Constitutive modeling

Over the past four decades, much effort has been devot-ed to modeling the stress–strain constitutive relation-ships for polymers [10, 11, 12, 13]. Using concepts fromthe Johnson-Cook model, G’Sell-Jonas model, Brooksmodel and Matsuoka model, the authors [14] developed

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a phenomenological constitutive model (called “DSGZ”model) to uniformly describe the entire range of defor-mation behavior of both glassy and semicrystalline poly-mers under monotonic compressive loading. The DSGZmodel explicitly gives the compressive true stress σc, de-pendence on true strain ε, true strain rate and tempera-ture T and is given by:

(1)

and is defined to be the dimensionless form of

where,

(2)

(3)

and is defined to be the dimensionless form of. The eight material coefficients in this model are

Kc (Pa · sm), C1, C2, C3 (sm), C4, a (K), m, and α.It has been well known that the deformation behavior

of polymers is not only sensitive to strain, strain rate andtemperature but also relate to the deformation mode [15,16, 17, 18]. Unlike metals, polymers often exhibit differ-ent behavior in uniaxial tensile and compression tests.Suppose [s] is the deviatoric part of a true stress tensor[σ]. For materials that exhibit the same behavior in uni-axial tension and uniaxial compression, the hydrostaticpressure part of the stress tensor is generally assumed tohave no effect on the mechanical behavior of the materi-al. The equivalent stress for the stress tensor [σ] is giv-en as,

(4)

However for materials that exhibit different behavior inuniaxial tension and uniaxial compression, the effect ofhydrostatic pressure has to be included. Using this hy-drostatic pressure effect, a generalized DSGZ model isproposed to describe the stress–strain constitutive rela-tionship of polymers under any monotonic loadingmodes, and is given by

(5)

where,

(6)

(7)

. The equivalent stress is defined by Eq. 4, thehydrostatic stress p is defined as and the equivalentstrain is defined as

(8)

where [e] is the deviatoric part of a true strain tensor [ε].The equivalent strain rate is defined as the derivativeof the equivalent strain with respect to time t,

(9)

and γ, called the hydrostatic sensitivity parameter, is amaterial coefficient accounting for the effect of loadingmode. The other eight material coefficients in the gener-alized DSGZ model are K (Pa · sm), C1, C2, C3 (sm), C4, a(K), m and α.

In the generalized DSGZ constitutive model for a uni-axial compression test, the compressive stress σc can bewritten in the form

and for a uniaxial tensile test, the tensile stress σt can bewritten as

Note that Eq. 1 for a uniaxial compression test can be

obtained from Eq. 10 by substituting the term by Kc.The value of γ can be calculated by combining Eqs. 10and 11. For a given strain ε, strain rate and temperatureT, γ has the form

(12)

From Eq. 12, it can be seen that γ is a function of strain ε,strain rate and temperature T over the entire range of de-formation. Because of the failure of polymers during uni-axial tensile tests at low strains in comparison to the largestrains obtained in compression tests, γ cannot be calculat-ed over a large strain range. Therefore γ is calculated atthe yield stress and assumed to be constant in the form

(13)

where σcy is the yield stress in uniaxial compression testand σty is the yield stress in uniaxial tensile test. The val-

(10)

(11)

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ues of the two yielding stresses can be obtained experi-mentally. The other eight material coefficients in thegeneralized DSGZ model can be deduced from uniaxialcompression stress–strain curves following the proce-dures given in reference [14].

Thermomechanical coupling during plastic deformation

Since the mechanical properties of polymers are sensitiveto temperature, an accurate estimate of the temperaturerise during plastic deformation is important. Arruda et al.[4] did a series of uniaxial compression tests in which thespecimen surface temperatures were monitored using aninfrared detector to investigate the relationship betweenstrain rate and temperature rise for the glassy polymerpolymethyl-methacrylate. It was found that the specimenwas nearly isothermal up to a true strain of 0.8 at a strainrate of 0.001/s, but significant temperature rise (around30 °C) were observed up to the same true strain at strainrates of 0.01/s and 0.1/s. The rise of temperature had adramatic effect on stress–strain curves. Rittel [19] embed-ded a small thermocouple in polycarbonate specimendisks to record the transient temperature during impacttests with strain rates ranging from 5000/s to 8000/s.Within a time order of 10–4 s, a true strain of 0.45 was ob-tained and the recorded temperature increased by nearly25 °C. The temperature rose significantly in the softeningregion of the corresponding stress–strain curve. Using afast response infrared radiometer to monitor the surfacetemperature of epoxy specimens in a Split HopkinsonPressure Bar impact test, Trojanowski et al. [20] observedthat there was a temperature increase of approximately50 °C. These experimental results indicate that for poly-mers, the temperature rise is significant during high strainrate large plastic deformation.

For strain rates of 102~103/s, it is reasonable to as-sume that the deformation process is essentially adiabat-ic. The governing equation for the increase of tempera-ture ∆T at each increment of plastic strain is

(14)

where ρ is material mass density, c is heat capacity, β isthe fraction of dissipated plastic energy which convertsinto thermal energy, is the equivalent stress at thebeginning of an increment, is the equivalent stressat the end of the increment and is the increment ofequivalent plastic strain. At each increment of plastic de-formation, the local temperature of the plastic deforma-tion zone will increase by an amount governed byEq. 14. The increase in temperature decreases the equiv-alent stress through the generalized DSGZ model giv-en by Eqs. 5, 6, and 7. This gives a framework to ac-count for the thermomechanical coupling during highstrain rate plastic deformation. The issue is how to de-cide the value of β in Eq. 14 for each increment of strain.Rittel [19] found that β is dependent on strain and strain

rate during plastic deformation of polymers. Since thereis a lack of available experimental data for β vs. strain εand strain rate for the glassy polymer, a constant valueof β = 0.5 is used for all the simulations except other-wise stated. Macdougall and Harding [21] used a similarapproximation in their numerical modeling of the highstrain rate torsion tests on Ti6Al4V bars.

Failure criterion

Polymer failure in the multiaxial impact test is exhibitedby the sudden significant reduction of its load carrying capability. Some significant work has been done with respect to understanding the impact strength and the dy-namic crack propagation process of polymers [22, 23].Through considering failure resulting from crack propaga-tion in Charpy impact test, Brostow [22] proves that theimpact transition temperature, which is highly related toimpact strength, of a polymer is not only related to thepolymeric material property but also related to the dimen-sions of artificial notches and other imperfections that leadto stress concentration. In the same reference, he givesformulas that relate the impact transition temperature tostress concentration factor. Very recently, Brostow et al.[24] numerically simulated the crack initiation and propa-gation in polymer liquid crystals during tensile deforma-tions through a molecular dynamics simulation procedure.

A material failure criterion has to be combined to-gether with the stress–strain constitutive model in orderto simulate the impact failure. To the authors’ knowl-edge, there are few reports found in the open literaturefor numerical simulation of polymer failure in multiaxialimpact tests. Strictly speaking, molding polymer speci-mens always have certain defects, such as scratches.However, unlike the Charpy impact test specimens thatinclude artificial notches, the multiaxial impact test spec-imens do not have any artificial notches and are treatedto include as few defects as possible. Therefore, the ef-fect of stress concentration is not a significant factor inthe multiaxial impact tests. The current simulations in-volve a macroscopic approach to the failure of polymers.

There are a variety of proposed macroscopic materialfailure criteria such as maximum tensile stress, maxi-mum shear strain, and maximum strain energy density.Since the polymer investigated in this paper has signifi-cant ductility, the maximum plastic strain failure criteri-on is used. A failure indicator ψ is created and defined as

(15)

where is a prescribed maximum equivalent plasticstrain, and is the increment of equivalent plasticstrain. When the sum of the equivalent plastic strain in-crement at a material point is equal to or greater than theprescribed value of , i.e. when ψ ≥ 1, the materialpoint fails and is permanently removed from future cal-culations.

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Finite element analysis model

In the ASTM D3763 multiaxial impact test, a cylindri-cal striker with hemispherical end is dropped from agiven height onto the center of a clamped polymer disk.Figure 1 shows the geometrical parameters used. Theclamp assembly consists of two circular parallel plateswith a 76 mm diameter hole in the center. Sufficientpressure is applied to prevent slippage of the polymerdisk from the clamp assembly during impact. The strik-er, consisting of a 12.7 mm diameter steel rod with ahemispherical end of the same dimension, is positionedperpendicular to and centered on the clamped disk.During impact, the striker moves down and the polymerdisk is totally penetrated. History of impact load, im-pact energy and displacement of the striker are record-ed.

Figure 2 shows a FEA model for the ASTM D3763multiaxial impact test. Nine hundred eight-noded linearbrick, reduced integration elements with a total of 1519nodes are used to mesh the polymer disk. An analyticalrigid surface is used to model the geometry of the striker.The rigid surface is associated with a rigid body refer-ence node that defines the mass and the motion of thestriker. The boundary conditions are set as fixed supportaround the outer edge of the disk and the six degrees offreedom of those nodes located on the circular edge areset to be zero. The striker can only move along the verti-cal axis, all the other five degrees of freedom of the rigidbody reference node are set to be zero. The friction coef-ficient between the striker and the polymer disk is set tobe a constant value of 0.3.

An algorithm of elastic prediction – plastic correctionis applied to update the stress tensor of each materialpoint. At the end of each increment of strain, the stresstensor [σ]new is calculated. The corresponding equivalentstress is obtained from Eq. 4. Equation 14 is thenapplied to calculate the increase of the local temperatureof polymer. The failure indicator ψ is updated throughEq. 15 and when ψ ≥ 1 the material point fails and ispermanently removed from future calculations. Based onthis framework of calculation, a user material subroutineis developed and applied in the FEA model to implementthe generalized DSGZ constitutive model, the thermo-mechanical coupling model and the maximum plasticstrain failure criterion.

Results and discussion

Using available stress–strain curves of uniaxial tensionand compression tests, we calibrated the nine materialcoefficients in the generalized DSGZ constitutive modelfor the glassy ABS polymer [18]. Table 1 shows the cal-culated material coefficients. The compressive stress σcin a uniaxial compression test can be predicted by Eq. 10for various strain, strain rate and temperature. Figure 3shows comparison of the DSGZ constitutive model pre-diction with available uniaxial compression test data for

Fig. 1 Geometrical configuration and parameters used in ASTMD3763 multiaxial impact test

Fig. 2 FEA model for ASTM D3763 multiaxial impact test

Fig. 3 Comparison of DSGZ model prediction with uniaxial com-pression test data (T=296 K, )

the polymer. It can be seen that the DSGZ constitutivemodel accurately predicts the stress–strain behavior ofthe polymer over a wide range of strains.

As previously stated, a constant value of 0.5 is usedfor the β in Eq. 14 in the simulations of the multiaxial

Fig. 4 Comparison of the FEA model predictions with experimen-tal impact load vs. displacement data (impact velocity: 3 m/s)

Fig. 5 Comparison of the FEA model predictions with experimen-tal impact energy vs. displacement data (impact velocity: 3 m/s)

Fig. 6 Predicted equivalent stress (Pa) distributions on the bottom surface of the polymer disk

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impact test. In order to estimate the importance ofthermomechanical coupling, simulations were performedin which the effect of the thermomechanical coupling isignored by setting β = 0. To evaluate the effect of hydro-static pressure, simulations were done in which the con-stitutive model was calibrated using uniaxial compres-sion data and the parameter γ set to zero. Figure 4 showscomparison of the FEA model predictions with experi-mental impact load vs. displacement data, while Fig. 5shows comparison of the FEA model predictions withexperimental impact energy vs. displacement data. FromFigs. 4 and 5, we can see that the FEA model prediction,with β = 0.5 and γ appropriately taken from Table 1,agree well with the impact load vs. displacement and im-pact energy vs. displacement experimental data of theglassy polymer up to the maximum impact loading (fail-ure). It is also observed that not accounting for β and/orγ in the simulation model tends to overestimate the im-

pact load and impact energy especially at large displace-ments.

From Figs. 4 and 5, it is found that the effect ofthermomechanical coupling (β) is small. Even though amaximum temperature rise of 18 °C was obtained in thesimulation, the effect on the predicted impact load andimpact energy is less than 5%. The simulation resultsshow that for materials that exhibit different behaviorin uniaxial tension and uniaxial compression, the effectof hydrostatic stress (γ) has to be included. Neglectingthe hydrostatic pressure effect will lead to a muchgreater error for the simulations of the multiaxial im-pact test. Uniaxial compression or tensile tests taken in-dividually are not representative of the three-dimen-sional deformation behavior of polymers under impactloading. In the application of the generalized DSGZconstitutive model, it is important to experimentallycalibrate the parameter γ.

Fig. 7 Equivalent plastic strain contour maps on the bottom surface of the polymer disk

Table 1 Material coefficients for the glassy ABS polymer

C1 C2 m a (K) K (MPa · sm) C3 (sm) C4 α γ

–0.8 –0.83 0.07 900 0.95 0.0028 7 100 0.25

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Figures 6, 7, 8 and 9 show the simulated equivalentstress distributions, the simulated equivalent plasticstrain contour maps, the simulated temperature distribu-tions, and the contour maps of the strain rates, respec-tively, on the bottom surface of the polymer disk. Thesimulation results show that the highest strain rates areof the order of 102~103/s during most of the deformation.Even though the material coefficients in Table 1 are cal-culated from low strain rates (10–4/s) test data, the cali-brated constitutive model can be used to extrapolate topredict the deformation behavior of the glassy polymerat high strain rates. This has been shown to be true forother polymers such as polymethyl-methacrylate, poly-carbonate and polyamide 12 [14]. From the simulationresults, it is found that most regions of the disk undergoelastic deformation. The plastic deformation and temper-ature rise is localized in the impact region with the maxi-mum values in the center of the disk. However, the max-imum equivalent stress is located in a circular zonearound the striker.

Concluding remarks

A FEA model is created using ABAQUS/Explicit to sim-ulate the standard ASTM D3763 multiaxial impact test.The DSGZ constitutive model is generalized to describethe stress–strain constitutive relationship of polymersunder any monotonic loading modes. The thermome-chanical coupling during high strain rate plastic defor-mation and the failure criteria for polymers are dis-cussed. An ABAQUS/Explicit user material subroutineis developed and applied to implement the generalizedDSGZ constitutive model, the thermomechanical cou-pling model and the maximum plastic strain failure crite-rion in the FEA simulation.

Multiaxial impact test on a glassy ABS polymer wassimulated. The predicted impact load vs. displacementcurve and impact energy vs. displacement curve werecompared with experimental data. They agree well. Theresults indicate that the generalized DSGZ constitutivemodel accurately predicts the stress–strain behavior ofthe polymer over a wide range of strains and it correctlyextrapolates over a large range of strain rates.

For polymers, the uniaxial compression or tensiletests are generally not representative of the three-dimen-sional deformation behavior under impact loading. The

Fig. 8 Predicted temperature (K) distributions on the bottom surface of the polymer disk

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deformation mode should be considered. In the applica-tion of the generalized DSGZ constitutive model, it isimportant to experimentally calibrate the parameter γ sothat hydrostatic stress effects can be included.

Acknowledgements The support of Lucent Technologies, locatedin North Andover, Massachusetts, USA, during this research isgratefully acknowledged.

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Fig. 9 Contour maps of the strain rates (/s) on the bottom surface of the polymer disk