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International Journal of Impact Engineering 62 (2013) 35e47

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International Journal of Impact Engineering

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

A constitutive description of the rate-sensitive response ofsemi-crystalline polymers

H. Pouriayevali*, S. Arabnejad, Y.B. Guo, V.P.W. ShimImpact Mechanics Laboratory, Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore

a r t i c l e i n f o

Article history:Received 9 January 2013Received in revised form3 May 2013Accepted 15 May 2013Available online 2 June 2013

Keywords:Visco-hyperelasticityViscoplasticityThermo-mechanical modelHigh rate deformation

* Corresponding author. Tel.: þ65 6516 2228; fax: þE-mail addresses: pouriayevali.habib@nus.edu.s

habibpouriayevali@gmail.com (H. Pouriayevali).

0734-743X/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.ijimpeng.2013.05.002

a b s t r a c t

A constitutive model is proposed to describe the quasi-static and high rate large deformation response ofsemi-crystalline polymers. This model is developed based on an elasticeviscoelasticeviscoplasticframework, to predict the temperature and rate-dependent response of an incompressible semi-crystalline polymer, Nylon 6. Material samples are subjected to high rate compressive and tensileloading using Split Hopkinson Bar devices, and they exhibit a temperature increase, which induces aphase change at the glass-transition temperature. The material parameters in the constitutive model,such as the yield stress, stiffness and viscosity coefficients, are proposed as functions of temperature andstrain rate. This study aims to formulate a thermodynamics-based model with minimum parameters, forimplementation in FEM software (ABAQUS) by the writing of a user-defined material subroutine(VUMAT). The model is validated via comparison with compressive and tensile experimental test resultsfor material response at different temperatures and deformation rates. It shows good potential indescribing the thermo-mechanical response of Nylon 6, and in predicting the dynamic behavior ofpolymeric material.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Polymericmaterials arewidely used because ofmany favourablecharacteristics, such as ease of forming, durability, recyclability, andrelatively lower cost and weight. Their effective application re-quires a good understanding of their thermo-mechanical responseover a wide range of deformation, loading rates and temperature.The dynamicmechanical properties of polymers are of considerableinterest and attention, because many products and components aresubjected to impacts and shocks, and need to accommodate themand the energies involved.

The largest group of commercially used polymers are semi-crystalline, comprising crystalline and amorphous phases. Thecrystalline phases are defined by three-dimensional regions asso-ciated with the ordered folding and/or stacking of adjacent chains.Crystallinity makes a polymer rigid, but reduces ductility.Conversely, amorphous phases are made up of randomly coiled andentangled chains; these regions are softer andmore deformable [1].A semi-crystalline polymer can be viewed as a composite with rigidcrystallites suspended within an amorphous phase. This is

65 6779 1459.g, Pouriayevali@yahoo.com,

All rights reserved.

desirable, because it combines the strength of the crystalline phasewith the flexibility of its amorphous counterpart. When a semi-crystalline polymer is subjected to large deformation, the defor-mation is first accommodated by rearrangement of chains in theamorphous region; this occurs for any degree of strain. Whendeformation becomes sufficiently large, intra-crystal sliding andfragmentation of crystalline blocks are induced, and these result inconsiderable energy dissipation [2e5]. It is well known that thelarge-strain response of semi-crystalline polymers displays rateand temperature dependence linked to irreversible deformationthat follows yield, and is accompanied by strain hardening [6e15].

A comprehensive review of constitutive descriptions of semi-crystalline material has been presented by Holmes and his co-workers [7]; it is noted that material behavior is commonlydescribed via two general approaches e micromechanics andmacromechanics. At the micro level, material is modeled by twostructural phases e crystalline and amorphous. An anisotropicarrangement is assumed for molecular chains and their rear-rangement is activated by thermal energy. Drozdov et al. have doneconsiderable work on the micromechanics of semi-crystallinepolymers, particularly on the kinetics of chain rearrangement,when an activated chain is separated from a junction and mergedwith another one [16e18]. Micro-scale modeling is accompanied byconsiderable difficulty in associating the mechanical response ofpolymers with complex micro-scale structure.

Nomenclature

X, x Position vectors in the reference and deformedconfigurations

F, J ¼ def (F) Deformation gradient tensor and its determinantU, V, R Right and left stretch tensors, rigid rotation tensorC, B Right and left CauchyeGreen deformation tensorsIi Principal invariants of CauchyeGreen deformation

(i ¼ 1, 2, 3)I Second order identity tensorsS; s;J Second PiolaeKirchhoff and Cauchy stresses, driving

force tensorL, D, W Velocity gradient, stretch and spin rate tensors_l;N Magnitude and direction of stretch rate tensorj;W Helmholtz free energy densities per unit reference

volume

4; h Internal energy and entropy density per unit referencevolume

Gij;Di;g;H;Q Material constantsL;G Hardening functionsEa, Tg Almansi strain tensor, Glass-transition temperatureu;m Inelastic work fraction generating the heat, viscosity

termE; s; 3 Elastic modulus, one-dimensional stress and strainU;p Temperature-dependent scalar functionsc Stress in friction slider

Super and subscriptse, v, ve, p, y, D, V Elastic, viscous, viscoelastic, plastic, yield,

deviatoric, volumetricten, com Tension, compressionT Tensor transpositionA, B Hyperelastic elements

H. Pouriayevali et al. / International Journal of Impact Engineering 62 (2013) 35e4736

In macro-scale modeling, it is assumed that material deforma-tion can be described by an isotropic phase, which is a homogeni-zation of the crystalline and amorphous phases [19]. Awide varietyof rheological frameworks, which are combinations of elastic,viscoelastic and viscoplastic components are proposed to describethis. Elastic springs, viscous dashpots, friction sliders, etc, areidealized elements employed to capture a wide spectrum of re-sponses [7e9,20e22].

It is noted that there are few macro-scale models that charac-terize comprehensively the three-dimensional thermo-mechanicalresponse of polymers and incorporate rate and temperaturedependence, as well as a yield criterion and post-yield hardening;existing formulations generally consist of descriptions of quasi-static behavior [23e27]. It would therefore be useful to formulatea constitutive description, based on a different framework, todescribe the behavior of polymeric material at high rates ofdeformation. Compared to existing models, the proposed model isdirected at capturing both the rate and temperature-dependentresponses observed in high rate experiments. Consequently, thisstudy aims to propose a constitutive model to predict the responseof an incompressible semi-crystalline material, Nylon 6, undercompressive and tensile quasi-static and dynamic deformation. Thematerial studied displays an increase in temperature under highrate deformation, which gives rise to a phase transition when theglass-transition temperature is exceeded. The proposed model is

Fig. 1. Schematic diagram of proposed elasticeviscoelasticeviscoplastic model;

formulated on a thermodynamics basis, using a macromechanicsapproach, and defines elastic-viscoelasticeviscoplastic behavior,coupled with post-yield hardening. Material parameters, such asstiffness coefficients, viscosity and hardening, are cast as functionsof temperature, as well as degree and rate of deformation. Simpleforms of these material parameters are sought, to precludeinvolvement in the complexity of polymer molecular structuredetails. The constitutive model proposed is implemented in an FEMsoftware (ABAQUS) via a user-defined material subroutine(VUMAT) and validated by compressive and tensile tests conductedat different temperatures and deformation rates.

2. Constitutive model

A constitutive model is formulated to predict the quasi-staticand dynamic response of semi-crystalline polymers. It is sche-matically depicted in Fig. 1, and captures large-strain deformationassociated with elasticeviscoelasticeviscoplastic behavior. Theproposed model comprises two groups of idealized mechanicalcomponents connected in series; the first captures rate-dependentreversible behavior and is modeled by the hyperelastic element A,which acts in parallel with a visco-hyperelastic component thatconsists of a hyperelastic element B and a viscous element definedby a viscosity coefficient mve. The second cluster describes irre-versible rate-dependent response and is activated when the stress

p, e, v, ve denote the plastic, elastic, viscous and viscoelatic components.

H. Pouriayevali et al. / International Journal of Impact Engineering 62 (2013) 35e47 37

in the material exceeds the yield value sc. It has two parallel ele-ments e a friction slider that defines sc and a viscous elementdefined by the coefficient mp.

2.1. Kinematic considerations

Consider a deformable body comprising material points termedparticles. A typical particle is identified or labelled by its positionvector X in the reference configuration. After deformation, theparticle moves to a new position defined by the vector x in thecurrent configuration. The deformation gradient F ¼ vx=vX relatesthese quantities. The right and left polar decompositions of F andthe right and left CauchyeGreen deformation tensors are given by:

F ¼ RU ¼ VR;C ¼ FTF ¼ U2;B ¼ FFT ¼ V2 (2.1)

Multiplicative decomposition of the deformation gradient facili-tates analysis, and the total deformation is composed of elastic,viscoelastic and plastic components [28e30]. In line with theschematic diagram in Fig. 1, decomposition of the deformationgradient F, the right CauchyeGreen deformation tensor C and thevelocity gradient L ¼ _FF�1 ¼ �F _F

�1are affected as follows:

F ¼ FeFp ¼ FveFp ¼ FeveF

vveFp

Cve ¼ Ce ¼ FTeFe ¼ F�T

p CF�1p

Ceve ¼ FeT

veFeve ¼ Fv�T

ve CveFv�1

ve ¼ Fv�T

ve F�Tp C F�1

p Fv�1ve

L ¼ _FF�1 ¼ Le þ FeLpF�1e ¼ Leve þ Fe

veLvveF

e�1

ve þ FveLpF�1ve

(2.2)

where, Fe, Feve, F

vve and Fp are the respective deformation gradient

tensors associated with deformation of the hyperelastic element A,hyperelastic element B, viscous dashpot connected to element Band friction slider in the viscoplastic cluster. J, Je and Jeve denote thedeterminants of the deformation gradient tensors F, Fe and Feve. It isassumed that inelastic deformation is isochoric (Section 2.5); thusdetðFpÞ ¼ detðFv

veÞ ¼ 1 and J ¼ Je ¼ Jeve.

2.2. Thermodynamic considerations

The continuum mechanics formulation adopted is based on thefree energy density ej a scalar quantity e which is a function offrame-invariant variables, commonly defined using parameters thatdescribe elastic deformation (Ce, Ceve; Eq. (2.2)), the absolute temper-ature q, as well as internal variables affecting the free energy stored.Generally, the total free energy can be defined as the sum of free en-ergies stored in the different rheological components [7,9,20,25]. Inthis study, the proposed elasticeviscoelasticeviscoplastic constitu-tive model (Fig. 1) is described by the following free energy function:

j ¼ je þ jve ¼ bjeðCe; qÞ þ bjveðCeve; qÞ (2.3)

where je and jve are stored energies in the elastic and viscoelasticelements, respectively. The constitutive equation must satisfyfundamental thermodynamic considerations, and it is assumedthat quasi-static deformation corresponds to a constant tempera-ture process, while high rate deformation is adiabatic. Therefore, inthe absence of heat flux and a heat supply, the balance of energyand Second Law of Thermodynamics (inequality) per unit referencevolume are expressed as [25,31,32].

S :12_C ¼ _4 ¼ _jþ h _qþ _hq; _hq ¼ S :

12_C � _j� h _q � 0 (2.4)

where S is the second PiolaeKirchhoff stress, S : ð1=2Þ _C the work-rate of an externally applied load per unit reference volume,

4 ¼ jþ hqthe internal energy density per unit reference volume, _3,the time derivative of the internal energy; j, the Helmholtz freeenergy density per unit reference volume, and h ¼ �vbj=vq(Eq.(2.8)) is the entropy density per unit reference volume. _j is ob-tained by applying the chain rule to the time derivative of Eq. (2.3).

_j ¼ _je þ _jve ¼ vbjevCe

: _Ce þ vbjve

vCeve

: _Ceve þ

vbjevq

_qþ vbjvevq

_q (2.5)

The chain rule time derivative in Eq. (2.5) is substituted into theinequality in Eq. (2.4) and results in the following simplified form: S � 2F�1

pvbjevCe

F�Tp � 2

�FvveFp

��1vbjve

vCeve

�FvveFp

��T

!

:12_C �

vbjevq

þ vbjvevq

þ h

!_qþ 2F�1

pvbjevCe

F�Tp

: CF�1p

_Fp þ 2�FvveFp

��1vbjve

vCeve

�FvveFp

��T

: C�FvveFp

��1�Fvve_Fp� � 0 (2.6)

A well-accepted argument in the derivation of constitutiveequations [25,27] is applied to Eq. (2.6) to obtain the Second PiolaeKirchhoff stress S and the entropy h expressed respectively in Eqs.(2.7) and (2.8).

S ¼ 2F�1p

vbjevCe

F�Tp þ2

�FvveFp

��1vbjve

vCeve

�FvveFp

��T ¼ SeþSve (2.7)

vbjevq

þ vbjvevq

¼ �h (2.8)

The relationship between the second PiolaeKirchhoff stress Sand the Cauchy stress s in the deformed configuration, is

s ¼ J�1FSFT ¼ se þ sve (2.9)

2.3. Dissipation inequality

The last two terms of the inequality (Eq. (2.6)) are expressed interms of the plastic and viscoelastic work associated with J andJ ve as the driving forces which are power-conjugate with the in-elastic deformation rates �Lp, Lvve (Eq. (2.2)). Substitution of Eqs.(2.7) and (2.8) into Eq. (2.6) results in a simplified form of theinequality as follows:

J : Lp þJ ve: Lvve � 0 (2.10)

where

J ¼ JF�1e sFe ; J ve ¼ JeveF

e�1ve sveFe

ve (2.11)

Taking thematerial to be strongly dissipative, the inequality (Eq.(2.10)) is satisfied by the following:

J : Lp � 0; J ve : Lvve � 0 (2.12)

2.4. Evolution of temperature variation

The entropy h is related to the inelastic work dissipated using acombination of Eqs. (2.4), (2.6)e(2.8) and (2.10) via the following:

�v

vbjevq

þ vbjvevq

!vt

q ¼ _hq ¼ J : Lp þ J ve : Lvve (2.13)

H. Pouriayevali et al. / International Journal of Impact Engineering 62 (2013) 35e4738

This relationship is employed to capture the temperature variationand heat generated by the inelastic work dissipated (Eq. (2.25)).

2.5. Inelastic flow rule

The viscoelastic and plastic deformation are described respec-tively by the velocity gradients Lvve and Lp (Eq. (2.2)), and aregenerally decomposed into inelastic stretch (Dv

ve, Dp) and spin ratetensor ðWv

ve;WpÞ components, which are defined by:

Lvve ¼ 12

�Lvve þ Lv

T

ve� þ 1

2

�Lvve � Lv

T

ve� ¼ Dv

ve þWvve

Lp ¼ 12

�Lp þ LTp

� þ 12

�Lp � LTp

� ¼ Dp þWp

(2.14)

It is commonly assumed that inelastic flow is irrotationalðWv

ve ¼ Wp ¼ 0Þ and incompressible ðtrace Dvve ¼ trace Dp ¼ 0Þ

[25,30,33]. Substitution of Eq. (2.14) into Eq. (2.12) results in theinelastic work dissipated as follows:

J 0: Dp � 0; J 0

ve : Dvve � 0 (2.15)

where the deviatoric part of the symmetric driving force tensors aredenoted by the following:

J 0 ¼ dev J 0

¼ devðsym J Þ; J 0ve ¼ dev J 0

ve ¼ devðsym J veÞ(2.16)

The stretch rate tensors are related to the corresponding drivingforces as follows:

Dp ¼ _lpNp ¼

���J 0���

mDp

J 0���J 0���; Dv

ve ¼ _lvveN

vve ¼

���J 0ve

���mDve

J 0ve���J 0ve

���(2.17)

where, _lvve and _lp represent the magnitudes of the stretch rate

tensors, Nvve and Np define their directions, and mDve and mDp are the

deviatoric components of the viscosities. In the following section,J 0 is modified in order to incorporate a yield criterion and post-yield hardening behavior.

2.6. Initiation of plastic deformation

In the proposed model, motion of the friction slider element(Fig. 1) defines plastic flow when the driving force J 0 exceeds athreshold. This threshold is made to increase with plastic defor-mation, to capture the hardening observed in experiments, andresults reported in previous work on semi-crystalline materials

bjAC

�Ce; Je

�¼ Ge10

�Ie1 � 3

�þ Ge01�Ie2 � 3

�þ Ge11�Ie1 � 3

��Ie2 � 3

�þ De1ðJe � 1Þ2bjBC

�Ceve ; J

eve

�¼ Ge

ve10

�Ieve1 � 3

�þ Ge

ve01

�Ieve02 � 3

�þ Ge

ve11

�Ieve1 � 3

��Ieve2 � 3

�þ De

ve1

�Jeve � 1

�2 (2.22)

[5,7]. (Appendix A.1 provides details on the formulation of a one-dimensional description of plastic flow based on the proposedmodel). A modified von Mises criterion, accompanied by a hard-ening function L, is proposed as the yield threshold for the three-dimensional constitutive description, and f ðJ 0

; sy þ LÞ which

defines the stress beyond the yield threshold, is expressed using Eq.(A.3) as follows:

f�J 0

; sy þ L�

¼���J 0

���� ffiffiffi23

r �sy þ L

�(2.18)

where sy is the yield stress (a material constant), and L is a frame-invariant scalar that increases with plastic deformation. TheAlmansi plastic strain Ea, which is work-conjugate with the Cauchystress, is employed to define L ¼ bLðBpÞ � 0 via the simplestfunction that exhibits good agreement with experiments (Section2.8.1).

L ¼ H�exp

�kEakQ

�� 1

�Ea ¼ ðI� B�1

p Þ=2; Bp ¼ FpFTp

(2.19)

where H and Q are material constants.The plastic flow Dp defined in Eq. (2.17), is modified to incor-

porate a yield criterion and post-yield hardening behavior, bysubstitution of Eq. (2.18) into Eq. (2.17); it satisfies the dissipationinequality (Eq. (2.15)) as follows:

Dp ¼ _lpNp ¼ < f�J 0

;syþL�>

mDp

J 0��J 0��

J 0: Dp ¼ < f

�J 0

;syþL�>

mDp

���J 0��� � 0

(2.20)

2.7. Helmholtz free energy density for the proposed model

The free energy density is basically a function of frame-invariantparameters. The proposed model (Fig. 1) employs two different free

energy densitiesebjAðCe; Je; qÞ and bjBðCeve; J

eve; qÞewhich correspond

respectively to the hyperelastic elementsA andB. The total free energydefined in Eq. (2.3) is redefined in the following separable form:

j ¼ bjeðCe; qÞ þ bjveðCeve; qÞ

¼ bjA�Ce; Je; q

�þ bjB�

Ceve ; J

eve; q

�¼ bjA

C

�Ce ; Je

�þ bjA

C;qð Je; qÞ

þbjBC

�Ceve ; J

eve

�þ bjB

C;q�Jeve; q

�þ bjqðqÞ (2.21)

where Ce ¼ FTeFe, Fe ¼ Je�1=3Fe, and C

eve ¼ F

eT

ve Feve , F

eve ¼

Je�1=3

ve Feve .

bjAC ðCe; JeÞ and bjB

CðCeve ; J

eveÞ denote the energy stored by

pure elastic deformation and defined using a polynomial formwiththe minimum number of material parameters that providereasonable agreement with experiments (Section 2.8.1).

where, Geij , Geveij , and De1, De

ve1 are material constants and corre-

spond respectively to the shear modulus and bulk modulus. Ie1, Ie2,

and Ieve1; I

eve2are the principal invariants of Ce and C

eve , respectively.bjA

C;qð Je; qÞ, bjBC;qð Jeve; qÞ represents the thermo-elastic portion of

the free energy [24,25,34].

Table 1Model parameters and material coefficients.

Strain rate Compression Tension Unit

Low High Low High

mDvea 2.0 e8 3.5 e4 7.0 e8 5.0 e4 Pa s

mDpa 2.0 e8 2.2 e4 9.0 e7 5.0 e4 Pa s

Ge10b 280.9 �249.8 MPa

Ge01b �215 330 MPa

Ge11b 52.07 67.7 MPa

De1 1.2e4 1.55e4 MPasy

b 35.0 46.0 MPaE 720 958 MPaH 10.0 55.3 MPaU ¼ bUðqÞ 17:198e�2:838q=q0 4:747e�1:557q=q0 q0 ¼ 298 (K)p ¼ bpðqÞ 30;945 e�10:34q=q0 q0 ¼ 298 (K)c 1700 J/kg Kath 7e�5 m/m Kr 1150 kg/m3

Q 0.45g 5u 0.9

a Temperature-dependent parameters modified by multiplication with p.b Temperature-dependent parameters modified by multiplication with U.

H. Pouriayevali et al. / International Journal of Impact Engineering 62 (2013) 35e47 39

bjAC;qð Je; qÞ ¼ �6athDe1ðq� q0ÞðJe � 1ÞB � � � � (2.23)

bjC;q Jeve; q ¼ �6athDe

ve1ðq� q0Þ Jeve � 1

where, ath is the thermal expansion coefficient, and q0the referencetemperature.bjqðqÞ represents the purely thermal portion of the free energy[27,34].bjqðqÞ ¼ cðq� q0Þ � cqln

�q

q0

(2.24)

where c is the specific heat capacity per unit reference volume.In order to capture the temperature variation and heat gener-

ated by inelastic deformation, _q is defined using a combination ofEqs. (2.8), (2.13) and (2.21)e(2.24) as follows:

h ¼ cln�

qq0

þ 6athDe1ðJe � 1Þ þ 6athDe

ve1

�Jeve � 1

�_hq ¼ c _qþ 6athq

�_JeDe1 þ _J

eveD

eve1

�¼ J : Lp þ J ve : Lvve

(2.25)

Thus,

c _q ¼ u�J : Lp þJ ve : Lvve � 6athq

�_JeDe1 þ _J

eveD

eve1

��(2.26)

where 0 � u � 1, and it is assumed that only a portion of theinelastic work generates heat (Table 1).

Fig. 2. Nylon 6 specimens before and after quasi-static deform

The Cauchy stress s in the proposed three-dimensional model(Fig. 1) consists of the hyperelastic stresses of elements A and B, s ¼sA þ sB is thus defined by substituting Eqs. (2.21)e(2.24) into Eqs.(2.7)e(2.9) (see Appendix A.2). In order to keep the number ofparameters to a minimum, g is defined to be a constant that linksthe coefficients of element A with those of element B (Eqs. (2.22)and (2.23)) as follows:

Geveij ¼ gGeij ; De

ve1 ¼ gDe1 (2.27)

where, g is determined by fitting the proposed constitutive equa-tion to experimental data (Section 2.8.1).

2.8. Model calibration and results

Nylon 6 is selected as the semi-crystalline polymer to be stud-ied. The glass-transition temperature Tg ¼ 309 K was determinedusing DMA (Dynamic Mechanical Analysis) testing. The proposedconstitutive model is first cast in one-dimensional form andimplemented in MATLAB, and subsequently in three-dimensionalform for implementation in the finite element analysis softwareABAQUS, via its VUMAT subroutine. The one-dimensional modeland a three-dimensional single-element finite element model arefitted to experimental data for calibration and to predict materialresponse at low and high rates. In Section 2.9, a three-dimensionalmulti-element model is also used to simulate the deformation andforce-displacement response of complex-shaped specimens; theresults are compared with experiments.

2.8.1. Quasi-static tests at room temperatureSamples of Nylon 6 are subjected to quasi-static compression

and tension at room temperature (q ¼ 298 K), using an Instronuniversal testing machine (Fig. 2). An optical technique is used tomeasure the large-strain deformation, whereby visual images ofthe specimen are captured by a video camera and processed usingTEMA software (TrackEye Motion Analysis 3.7). Simultaneously, astrain gauge is also mounted on the specimens tomeasure the axialstrain up to 1%; this yields elastic modulus values ofEcom ¼ 720 (MPa) and Eten ¼ 958 (MPa) for compression and ten-sion, respectively. Fig. 3 shows asymmetry in the experimentalquasi-static large-strain tension and compression response ofNylon 6. It is generally acknowledged that Nylon 6 behaves as anapproximately incompressible polymer [15].

Samples of Nylon 6 are subjected to high strain rate compressionand tension using compressive and tensile Split Hopkinson Bar de-vices (Section 2.8.3). Fig. 4 shows the compressive and tensile re-sponses of thematerial at different strain rates; it is observed that thematerial is relatively rate-insensitive at low strain rates, but

ation: (a) compression, (b) tension, ASTM D638 (type V).

Fig. 3. Experimental quasi-static response of Nylon 6 showing asymmetry in tensionand compression (strain rate of 0.0005/s).

H. Pouriayevali et al. / International Journal of Impact Engineering 62 (2013) 35e4740

significantly rate-sensitive at high deformation rates. Therefore, inthe proposed model (Fig. 1), it is assumed that the hyperelasticelement B does not contribute to thematerial response at the loweststrain rate of 0.0005/s tested; i.e. the dashpot connected to element Bhas sufficient time to be fully stretched. Fig. 5 shows the intersectionsof tangent lines which are taken to define the yield stress sy thattriggers plastic deformation and motion of the friction slider (Fig. 1)[27]. The response of the material before yield is considered non-linearly elastic and modeled by a hyperelastic description proposedby Pouriayevali et al. [35], and details are provided in Appendix A.3.This model is implemented in MATLAB and fitted to the uniaxialcompressive and tensile responses of the material at strain rates of

Fig. 4. Compressive and tensile responses o

Fig. 5. Identification of yield stress marking onset of plastic deform

0:0005=s to obtain the material constants Geij for the energy densityfunction bjA

C (Eq. (2.22), Table 1). With the value of Geij determined,the three-dimensional model is fitted to the large-strain materialresponse at low strain rates (Fig. 6) to obtain the parameters mDp , m

Dve,

Q, H, De1 and g (see Eqs. (2.17), (2.19), (2.22), (2.27) and Table 1).

2.8.2. Quasi-static tests at high temperatureNylon 6 has temperature-dependent material properties and

the material parameters vary with temperature. Fig. 7 shows thequasi-static response of the material at temperatures below andabove Tg. Two temperature-dependent functions, U and p, areemployed to fit the proposed model to experiments, where isdefined by an exponential function which is commonly used todecrease material stiffness with temperature [27, 36]. The materialconstants Geij and sy are multiplied by U (Table 1). Similarly, p isdefined by an exponential function derived empirically, andfrequently employed to describe Newtonian viscosity [37e39]; mDpand mDve are multiplied by p (Table 1).

2.8.3. Deformation at high strain ratesThe responses at high strain rates (Fig. 8) are predicted using the

proposed constitutive model with parameter values obtained byfits with experimental data at low strain rates (Sections 2.8.1 and2.8.2). Fitting of the constitutive model to experimental test re-sults for low strain rates of 0:0005=s is presented in Fig. 6 andduplicated in Fig. 9. Themodel incorporating low strain rate tensionand compression parameters obtained according to Sections 2.8.1and 2.8.2 (Table 1) is fitted respectively to dynamic test results

f Nylon 6 at low and high strain rates.

ation, for compression and tension at strain rate of 0.0005/s.

Fig. 6. Comparison between test data and proposed model for tension and compression.

Fig. 7. Comparison between experimental tension and compression data with the proposed model for a strain rate of 0.0005/s and different temperatures.

Fig. 8. High speed photographic images of dynamic deformation of Nylon 6 specimen; (a) and (b): before deformation; (c) 52% compressive engineering strain at 163 ms after startof loading with strain rate of e3200/s, (d) 17% tensile engineering strain at 267 ms after commencement of loading with strain rate of 630/s.

H. Pouriayevali et al. / International Journal of Impact Engineering 62 (2013) 35e47 41

Fig. 9. Comparison between tension and compression test data for different strain rates with proposed model.

Fig. 10. Proposed values for mDp to describe compressive loading at (a) constant temperature; q ¼ 298 K; (b) different temperatures at low strain rates; (c) different temperatures athigh strain rates.

H. Pouriayevali et al. / International Journal of Impact Engineering 62 (2013) 35e4742

Fig. 11. Infrared images showing temperature increase for high rate deformation: (a) before compression; (b) 5 K increase after 21% engineering strain at �1030/s; (c) before tensileloading; (d) 1.2 K increase after 15% engineering strain at 320/s.

H. Pouriayevali et al. / International Journal of Impact Engineering 62 (2013) 35e47 43

for tensile strain rates of 350=s and 800=s, and compression at�2000=s (Fig. 9). It was found that only mDp and mDve are rate-dependent, and their values obtained for low strain rate experi-ments (Section 2.8.1) must be modified for high strain rates(Table 1). Fig. 10 shows the values of mDp proposed for differenttemperatures and strain rates. The model with compression andtension parameters for high strain rates (Table 1) is able to predictthe responses for compression at strain rates of �980=s and�3200=s, and tension at strain rates of 150=s and 630=s (Fig. 9).

There is a significant temperature increase during high ratedeformation due to considerable viscoelastic and viscoplastic work;Fig. 9 shows softening in the material response at a strain rate of e3200/s because of this. This behavior is reasonably well describedby the proposed model. Fig. 11 depicts the temperature changeduring high rate deformation, via images captured by an infraredcamera, and Fig. 12 shows comparisons between the temperaturevariations predicted by the model and experimental results.

Fig. 12. Comparison between temperature variation

2.9. Multi-element FEM model of a complex-shaped specimen e

short thick walled tube

The quasi-static and high strain rate responses of a complex-shaped specimens are observed experimentally and simulated byFEM. Experimental results are compared with the FEM model tovalidate the potential of the proposed constitutive model in pre-dicting the response of structures made of Nylon 6. Fig. 13 showsthe geometry of a short thick walled tube with a circular holedrilled across its diameter. A three-dimensional ABAQUS modelemploying the compression parameters (Table 1) and C3D4 ele-ments is used to predict the quasi-static and high strain rateresponse to axial compression. Fig. 13 compares the geometry ofthe specimen with the FEM model for quasi-static compression,while Fig. 14 depicts the temperature profile from the FEM modeland that captured experimentally for high rate deformation.Experimental force-displacement data are also compared with

s predicted by model and experimental data.

Fig. 13. (a) Nylon 6 specimen with a complex geometry; (b) ABAQUS model; (c) quasi-static compression of specimen; (d, e) comparison between specimen geometry and model at43% compressive engineering strain.

Fig. 14. Temperature profile corresponding to 0.65 mm compressive deformation at a stain rate of e400/s (Fig. 15) (a) ABAQUS model; (b) infrared image.

H. Pouriayevali et al. / International Journal of Impact Engineering 62 (2013) 35e4744

values generated by the FEM model in Fig. 15. The experimentaldata shows a more compliant response at large strains compared tothe FEM model. This could be attributed to either the complexspecimen geometry or the type of FEM element employed. Thecomplex geometry induces non-uniform shear deformation whichis negligible in the compressive deformation of the simple solidcylindrical specimen used for calibrating the proposed model

Fig. 15. Comparison between experimental force-displacement

(Section 2.8.1). Moreover, the C3D4 tetrahedral elements used arethe only elements that can accommodate the large deformationsinduced in complex geometry specimen, since C3D8R hexahedralelements become excessively distorted. Nevertheless, the com-parison shows that the proposed constitutive model implementedin a three-dimensional FEM model is able to simulate the overallquasi-static and dynamic response of complex-shaped specimens.

data and FEM model for compression of thick walled tube.

H. Pouriayevali et al. / International Journal of Impact Engineering 62 (2013) 35e47 45

2.10. Conclusions

A thermo-mechanical constitutive model for the large defor-mation of polymers is proposed and applied to a semi-crystallinepolymer, Nylon 6, to describe quasi-static and high rate compres-sive and tensile deformation. The model is developed from anelastic-viscoelasticeviscoplastic perspective and a desire to mini-mize the number of model parameters. It is implemented in FEMsoftware (ABAQUS) by the writing of a user-defined material sub-routine using the VUMAT function in ABAQUS. Predictions based onthe model show good agreement with test results for materialresponse at low and high rate deformation. It is noted that themodel requires two sets of parameters to capture the materialbehavior e one for compression and another for tension. The yieldstress, stiffness coefficients and viscosity parameters aretemperature-dependent, and the viscosity parameters are alsofound to be rate-dependent. The present work has focused onquasi-static and dynamic loading and shows that the proposedconstitutive model has good potential in predicting the thermo-mechanical response of complex-shaped components made ofsemi-crystalline polymer. It is envisaged that the model could befurther developed to describe the relaxation, creep, unloading andcyclic loading of polymers.

Acknowledgement

The authors gratefully acknowledge the support provided by anA*Star SERC Project Grant, No. 092 137 0017, as well as the assis-tance of T. L. Goh and C.W. Low in undertaking experimental tests inthe Impact Mechanics Laboratory.

Appendix A

A.1. One-dimensional form of the elastic-viscoelastic-viscoplasticconstitutive model

Post-yield hardening is described using a one-dimensionalelastic-viscoelastic-viscoplastic model (Fig. A.1) [7]. Motion of thefriction slider is activated when the stress sT exceeds a thresholdvalue scsignðsTÞ, sc � 0. The stress components in the various el-ements of the model are governed by the relationships in Fig A.1.

sc ¼ sy þ bG� 3p� � 0 (A.1)

where sy � 0 is the yield stress and amaterial constant, bGð 3pÞ � 0 isa function of the plastic strain and describes hardening. In additionto this threshold stress for plastic deformation, there is an addi-tional component sex that is associated with strain rate, which iscaptured by the viscoplastic dashpot.

sex ¼< f ðsT ; scÞ > sign ðsT Þ (A.2)

where, < f ðsT ; scÞ > is defined by the following.

< f ðsT ; scÞ > ¼ jsT j � sc; jsT j � sc > 00; jsT j � sc � 0 (A.3)

Assumption of a Newtonian dashpot for the viscoplastic compo-nent yields the following expression for the plastic strain rate.

_3p ¼ sexmp

¼ < f ðsT ;scÞ >

mpsign ðsTÞ (A.4)

where mp is a viscosity coefficient. Fig. A.2 shows schematically theresponse of the various elements in the one-dimensional model, fordifferent strain rates; _31 < _32 < _33.

A.2. Derivation of the second PiolaeKirchhoff and Cauchy stressesfor a hyperelastic material

A hyperelastic material is nonlinear and the second PiolaeKirchhoff stress tensor S for such a material is given by

S ¼ 2vWvC

¼ 2�vWvI1

vI1vC

þ vWvI2

vI2vC

þ vWvI3

vI3vC

�(A.5)

where W ¼ cW ðC; qÞ ¼ cW ðI1; I2; I3; qÞis the Helmholtz free en-ergy density per unit reference volume and I1, I2 and I3 are theprincipal invariants of the right and left CauchyeGreen deforma-tion tensors C, B.

I1 ¼ traceðBÞ; I2 ¼ 12

�ðtraceðBÞÞ2 � trace

�B2��;

I3 ¼ J2 ¼ detðBÞ vI1vC

¼ I;vI2vC

¼ I1I � C;vI3vC

¼ detðCÞC�1

(A.6)

where I is the identity tensor.The Helmholtz free energy density is assumed to consist of the

deviatoric and volumetric terms of the stored energy. The isochoricterm is defined by F ¼ J�1=3F; I1 ¼ J�2=3I1and I2 ¼ J�4=3I2are theprincipal invariants ofC ¼ F

TF , anddet ðFÞ ¼ 1.Thevolumetric term

is defined using J ¼ detðFÞ, which represents the volume change.Therefore, the strain energy is redefined by:

W ¼ cW ðI1; I2; I3; qÞ ¼ cW�C; J; q

�¼ cW�

I1; I2; J; q�

(A.7)

A combination of Eqs. (A.5)e(A.7) yields the following expres-sions for the second PiolaeKirchhoff stress S and the Cauchy stresstensor s [24,40].

S ¼ 2�vWvI1

I þ vWvI2

ðI1I � CÞ þ vWvI3

I3C�1�

s ¼ J�1FSFT ¼ 2J�1F vWvC F

T

s ¼ 2ffiffiffiI3

p $

��vWvI1

þ I1vWvI2

B� vW

vI2B$B

�þ 2

ffiffiffiffiI3

p �vWvI3

¼ 2

J

��vWvI1

þ I1vWvI2

B� vW

vI2B$B

�þ�vWvJ

� 23J�

I1vWvI1

þ 2I2vWvI2

�I (A.8)

A.3. Incompressible hyperelastic model for uniaxial loading atconstant temperature

Extending Section A.2, an incompressible material is describedF ¼ F , I1 ¼ I1and I2 ¼ I2, because J ¼ 1. A purely elastic strainenergy function which shows good agreement with experimentsconducted at the constant room temperature (Section 2.8.1) can beexpressed using Eq. (2.22) as follows:

W ¼ G10ðI1 � 3Þ þ G01ðI2 � 3Þ þ G11ðI1 � 3ÞðI2 � 3Þþ D1ðJ � 1Þ2

¼ G10ðI1 � 3Þ þ G01ðI2 � 3Þ þ G11ðI1 � 3ÞðI2 � 3Þ � PhðJ � 1Þ(A.9)

where, Ph ¼ �D1ðJ � 1Þis an undetermined hydrostatic pressure,because D1 defines the bulk modulus associated with an infinitevalue for fully incompressible material and (J e 1) ¼ 0 for volume-conserving deformation. �PhðJ � 1Þfunctions as a Lagrange multi-plier to enforce the incompressibility condition.

H. Pouriayevali et al. / International Journal of Impact Engineering 62 (2013) 35e4746

Substitution of Eq. (A.9) into Eq. (A.8) yield the Cauchy stress sfor a fully incompressible hyperelastic material at the constantroom temperature [35,41].

s ¼ �PhI þ a1Bþ a2B$B (A.10)

where

a1 ¼ 2ðvW=vI1 þ I1vW=vI2Þ ¼ 2ðG10 þ G11ðI2 � 3ÞþI1ðG01 þ G11ðI1 � 3ÞÞÞ a2 ¼ �2ðvW=vI2Þ

¼ �2ðG01 þ G11ðI1 � 3ÞÞ (A.11)

For uniaxial loading along the 1-axis, the stretch is defined byl ¼ 1þ 311, where 311 is the engineering strain in the loading di-rection; the principal stretches are thusl1 ¼ l; l2 ¼ l3 ¼ l�1=2. Fand B are defined by:

F ¼24 l 0 00 l�1=2 00 0 l�1=2

35 B ¼ F$FT ¼24 l2 0 0

0 l�1 00 0 l�1

35(A.12)

The stress corresponding to uniaxial loading is:

Fig. A.1. Schematic diagram of a one-dimensio

Fig. A.2. Response of various elements in one-d

s11 ¼ �Ph þ a1 B11 þ a2 B11$B11 (A.13)

The hydrostatic pressure Ph is determined from the conditionthat for uniaxial loading, s22 ¼ s33 ¼ 0, together with fact thatB22 ¼ B11

�1=2.

s22 ¼ s33 ¼ �Pe þ a1 B22 þ a2 B22$B22 ¼ 0 (A.14)

Consequently, the Cauchy stress in the direction of loading isgiven by:

s11 ¼ a1

�B11 � B�1=2

11

�þ a2

�B211 � B�1

11

�(A.15)

A combination of Eqs. (A.11), (A.12) and (A.15) results in thefollowing expression for the uniaxial stretch to the Cauchy stress indirection of loading.

s11 ¼�2l�1� l�3

��$�G10lþ G01 þ G11

��l2 þ 2l�1

�� 3

þ l�l�2 þ 2l� 3

���(A.16)

nal elastic-viscoelastic-viscoplastic model.

imensional model for different strain rates.

H. Pouriayevali et al. / International Journal of Impact Engineering 62 (2013) 35e47 47

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