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International Journal of Plasticity 21 (2005) 145–160

Modeling deformation behavior of polymers withviscoplasticity theory based on overstress

Ozgen U. Colak

Department of Mechanical Engineering, Yildiz Technical University, Istanbul 34349, Turkey

Received in final revised form 4 March 2004

Available online 14 May 2004

Abstract

The nonlinear strain rate sensitivity, multiple creep and recovery behavior of polyphenylene

oxide (PPO), which were explored through strain rate-controlled experiments at ambient

temperature by Khan [The deformation behavior of solid polymers and modeling with the

viscoplasticity theory based overstress, Ph.D. Thesis, Rensselaer Polytechnic Institute, New

York], are modeled using the modified viscoplasticity theory based on overstress (VBO). In

addition, VBO used by Krempl and Ho [An overstress model for solid polymer deformation

behavior applied to Nylon 66, ASTM STP 1357, 2000, p. 118] and the classical VBO are used

to demonstrate the improved modeling capabilities of VBO for solid polymer deformation.

The unified model (VBO) has two tensor valued state variables, the equilibrium and kinematic

stresses and two scalar valued states variables, drag and isotropic stresses. The simulations

include monotonic loading and unloading at various strain rates, multiple creep and recovery

at zero stress. Since creep behavior has been found to be profoundly influenced by the level of

the stress, the tests are performed at different stresses above and below the yield point. Nu-

merical results are compared to experimental data. It is shown that nonlinear rate sensitivity,

nonlinear unloading, creep and recovery at zero stress can be reproduced using the modified

viscoplasticity theory based on overstress.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Viscoplasticity; PPO; Creep; Rate sensitivity; Recovery

E-mail address: [email protected].

0749-6419/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijplas.2004.04.004

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146 O.U. Colak / International Journal of Plasticity 21 (2005) 145–160

1. Introduction

The determination of deformation mechanisms and the modeling of the visco-

elastic–viscoplastic or elasto-viscoplastic behavior of polymeric materials have re-

cently received considerable interest due to the increased use of polymers in a broad

range of applications, including the electronic systems, aerospace and automotiveindustries and consumer appliances. Particularly in critical load bearing applica-

tions, high performance thermoplastics are replacing the metallic materials. There-

fore, they are expected to exhibit the same reliability and predictability as metallic

materials. To ensure reliability and predictability, the structural components, which

are subjected to severe loading conditions and environment, require a lifetime

analysis prior to production. The first step in this analysis is the inelastic analysis,

which provides information about stresses and strains as a function of position and

time during manufacturing and service time. During design process, to estimate theprecise deformation behavior of these materials, the experimental results and the

constitutive models are needed. The complexity of the mechanical behavior requires

a comprehensive model of the polymer that can reproduce nonlinear strain rate

dependency, nonlinear unloading, pressure sensitive yielding, cyclic softening and

significant recovery at zero stress.

It is well known that polymers exhibit strain rate and temperature-dependent

behavior. In addition, significant creep and relaxation can be observed even at room

temperature. Material behavior of polymers can change from brittle to visco-plasticdepending upon loading conditions and temperature. Their behavior can be ex-

plained in terms of their microstructures. Polymers can have either amorphous or

semi-crystalline structure. The degree of crystallinity and the size and distribution of

the crystallites in a semi-crystalline polymer have a large effect on the mechanical

properties of these materials. If the polymer has amorphous structure, inelastic be-

havior depends on the molecular chain flexibility, entanglement and on differences in

the structure of the molecular chains. Molecular structures can be linear, branched,

cross-linked and network. Linear long molecular chains have backbone bonds,which permit rotation but little extension. In cross-linked polymers, adjacent linear

chains provide additional rigidity. At temperatures well below the glass transition,

long molecular chains are rigid and resulting a brittle character. At high tempera-

tures, backbone bonds rotate and allow molecules partially disentangle and move

relative to one another. As a result, a viscoelastic and viscoplastic behavior can be

observed, Bardenhagen et al. (1997).

The complexity of polymeric behavior and the problems obtaining relevant ex-

perimental data make the constitutive model development difficult. The followingproperties observed in polymers should be considered to develop an appropriate

constitutive model.

1. Material behavior can be highly nonlinear and strain rate and temperature depen-

dent. The flow stress increases nonlinearly with an increase of the loading rate; a 10-

fold increase in the loading rate does not yield a 10-fold increase in the stress level.

2. Unloading curve is nonlinear. In comparison with metals, the shape of unloading

curve is highly nonlinear. The unloading curves show less strain rate dependence

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O.U. Colak / International Journal of Plasticity 21 (2005) 145–160 147

than loading curves when the loading and unloading strain rates have the same

magnitude.

3. Yield behavior is significantly affected by hydrostatic pressure. In metallic mate-

rials, it is assumed that inelastic deformation is incompressible which means that

yield stress is pressure independent. However, the existence of free volume around

the molecules of the polymers makes the polymer deformation behavior hydro-static pressure dependent.

4. Recovery at zero stress is significant. For metals, the recovery at zero stress is

small, but for polymers, it is quite large and dependent on the prior loading rate.

Considering the behaviors listed above, the constitutive models developed for

metallic materials need to be modified to represent accurate mechanical behavior of

polymers. To develop an experimentally based constitutive model, the mechanical

response of polymers needs to be investigated under different loading conditions,

such as uniaxial and multiaxial monotonic and cyclic loading. In addition, a detailedknowledge of the influences of temperature and strain rate is essential.

In recent years, a number of constitutive equations have been developed to de-

scribe the time-dependent mechanical behavior of polymeric materials, Boyce et al.

(1988, 2000); Krempl and Bordonaro (1995); Hasan and Boyce (1995); Bardenhagen

et al. (1997); Takashi et al. (1997); Yang and Chen (2001); Khan and Zhang (2001);

Krempl (1998a,b); Krempl and Ho (2000); Ho and Krempl (2002); Krempl and

Khan (2003); Colak et al. (2003); Van Dommelen et al. (2003); Drozdov and Yuan

(2003); Drozdov and Christiansen (2003); Ahzi et al. (2003).A micromechanically based constitutive model for elasto-viscoplastic deformation

of polymeric materials has been developed by Boyce et al. (1988). Thus the change in

the deformation mechanism with temperature and the microstructural constituents

can easily be accommodated. The model by Boyce et al. (1988, 2000) is based on the

macromolecular structure of amorphous polymers and the micromechanicsm of

inelastic flow. Two basic resistances to deformation are defined as intermolecular

resistance occurring in parallel with a network resistance. The network resistance is

modeled as a network orientation and molecular relaxation process acting togetherto accommodate deformation. The model used by Ahzi et al. (2003) is based on the

constitutive models presented by Boyce et al. (2000) for the finite deformation stress–

strain behavior of PET above the glass transition temperature. Unlike the work by

Boyce et al. (2000), strain induced crystallization is accounted explicitly.

Tang et al. (2001) proposed a model to simulate the nonlinear deformation re-

sponse of high impact polystyrene (HIPS) under uniaxial tensile loading with dif-

ferent constant strain rates. The viscoelastic–plastic constitutive equation takes

account of the effect of craze damage as well. The volume dilatation is used tocharacterize the craze damage of HIPS under tensile loading.

The viscoplasticity theory based on overstress for polymers (VBOP) which has

been derived from a unified state variable theory for metallic materials is applied by

Krempl and Ho (2000) to model nonlinear rate sensitivity and unloading, cyclic

softening and recovery behavior of Nylon 66. The simulations have shown that the

overstress theory is capable of modeling the behavior of Nylon 66. The stress-

controlled loading and unloading behavior are also successfully predicted. In

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148 O.U. Colak / International Journal of Plasticity 21 (2005) 145–160

another paper, Ho and Krempl (2002) modified the viscoplasticity theory based on

overstress (VBO) model by introducing an augmentation function, b into the equi-

librium stress evolution equation. The rate-dependent strain softening and hardening

behavior of polymethylmethacrylate (PMMA) are modeled through the decreasing

and the subsequently increasing augmentation function.

Recently, Krempl and Khan (2003) presented uniaxial monotonic test data in anattempt to compare metallic and polymeric inelastic deformation behavior. Con-

siderable similarities have been found in the deformation behavior of metals and

polymers: loading at different strain rates, nonlinear relation between loading rate

and stress level. VBO is applied to model creep, relaxation and rate sensitivity of the

amorphous PPO and HDPE. Even though model predicts creep of PPO and HDPE

and the relaxation response of HDPE quite well, unloading behaviors are not re-

produced well. However, improved modeling capabilities of VBO for polymer can be

seen in Ho and Krempl (2002).In this study, VBO is modified to model the behavior of polymeric materials under

different loading conditions. The uniaxial stress–strain diagrams of PPO, which have

been obtained by Khan (2002) at various strain rates, were used to determine the

material constants in the modified version of VBO. The modified VBO, VBO used by

Krempl and Ho (2000), and the classical VBO are employed to demonstrate the im-

proved modeling capability. Numerical experiments included are: (1) uniaxial strain-

controlled loading and unloading with various strain rates, (2) multiple creep at room

temperature and (3) recovery at zero stress. The simulation results are compared tothe experimental data obtained from Krempl and Khan (2003) and Khan (2002).

2. The overstress model

VBO is a rate-dependent unified state variable theory with no yield surface and no

loading/unloading conditions, Krempl (1998a,b). The theory consists of two tensor

valued state variables, which are the equilibrium stress, the kinematic stress and twoscalar valued state variables: isotropic stress and drag stress. Equilibrium stress is the

path-dependent stress that can be sustained at rest after prior inelastic deformation. It

is related to the defect structure of the material. The second state variable, the kine-

matic stress, is the repository for modeling the Bauschinger effect and sets the tangent

modulus at themaximum strain of interest. It is similar to the back stress in the classical

plasticity theories. The isotropic stress, which is a rate-independent contribution to the

stress, is responsible formodeling hardening or softening. The total strain rate is sumof

the elastic and inelastic strain rates, see Krempl (1996) and Colak and Krempl (2003).The flow law for small strain, incompressibility and isotropy is given by

_� ¼ _�el þ _�in ¼ 1þ mCE

_sþ 3

2F

CD

� �s� g

C

� �; ð1Þ

where s and g are the deviatoric part of the Cauchy and the equilibrium stress tensor,

respectively. The quantities E and m are Young’s modulus and Poisson’s ratio, re-

spectively. A superposed dot designates the material time derivative. Square brackets

following a symbol denote ‘‘function of’’. When the stress–strain behavior of

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O.U. Colak / International Journal of Plasticity 21 (2005) 145–160 149

polymeric materials and metallic materials is compared, it can be observed that

elastic stiffness of polymeric materials are changing drastically during unloading. To

construct the change in the elastic stiffness while loading and unloading the function

C is introduced into the equation for the elastic strain rate. This is accomplished by

setting C ¼ 1� kðjG� Kj=AÞa so that C½0� ¼ 1 and is close to zero in the fully es-

tablished inelastic flow region, Krempl (2002). The behavior of C is rate independentand can be adjusted by the material constants k and a. The symbol j j denotes

magnitude of a tensor. Inelastic strain rate is the function of the overstress, o, whichis defined as the difference between the Cauchy and equilibrium stresses, o ¼ s� g. Itis the stress that should be overcome to produce inelastic deformation. C is the

overstress invariant with the dimension of stress defined by

C2 ¼ 3

2ðs� gÞðs� gÞ: ð2Þ

F ½ � is the positive, increasing flow function with the dimension of 1/time and

F ½0� ¼ 0. The flow function is responsible for nonlinear rate dependency. It is given

by F ½ � ¼ BðC=DÞm with B as a universal constant and D is the drag stress, which is

constant in this study.

The equilibrium stress, g, is nonlinear, rate independent and hysteretic. The

growth law for the equilibrium stress has two hardening terms, elastic and inelastic

hardening, followed by the recovery term and finally a term multiplied by rate ofkinematic stress, k. Its evolution equation is given as

_g ¼ W_s

EþWF

CD

� �s� g

C

�� g � k

A

�þ 1

��W

E

�_k: ð3Þ

Another tensor valued state variable is the kinematic stress, k, which is the repository

for the modeling of the Bauschinger effect. A is the isotropic stress, rate-independent

contribution to the stress, which is responsible for modeling hardening or softening.

In the study, it is constant. The shape function, W, has a significant influence on the

transition from the quasi-linear region to fully established inelastic flow. It is given as

W ¼ W1 þc2 �W1

expðc3j _�inj

!; W1 ¼ c1 1

�þ c4

jgjAþ jkj þ C

;

� ��ð4Þ

where c1, c2, c3 and c4 are material constants (see Krempl and Ho (2000)). The

evolution equation for the kinematic stress is

_k ¼ jsjCþ jgj

�Et _�in; ð5Þ

where �Et ¼ Et=ð1� Et=EÞ and Et is the tangent modulus.

3. Numerical simulations

Khan (2002) conducted the strain-controlled uniaxial loading and unloading

experiments at different strain rates, multiple creep and recovery tests using

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150 O.U. Colak / International Journal of Plasticity 21 (2005) 145–160

polyphenylene oxide (PPO). Experiments were performed at room temperature. In

this study, uniaxial data of PPO generated by Khan (2002) are used to determine the

material constants for the modified version of VBO. The set of material constants is

listed in Table 1 and is used throughout the paper. The numerical simulations in-

clude monotonic loading and unloading at various strain rates, recovery at zero

stress and multiple creep using the modified VBO for polymers, VBO by Krempl andHo (2000) and the classical VBO. Numerical results are compared to experimental

data obtained by Khan (2002) and Krempl and Khan (2003).

3.1. Nonlinear rate dependency and nonlinear unloading

The mechanical characteristics of polymers are highly sensitive to the rate of

deformation, temperature and chemical nature of the environment. In general in-

creasing the temperature produces a decrease in elastic modulus, a reduction intensile strength and an enhancement of ductility. Decreasing the rate of deformation

has the same influence on the stress–strain characteristics as increasing the temper-

ature; that is material becomes softer and more ductile.

Uniaxial loading and unloading experiments performed on PPO at different strain

rates have revealed that PPO exhibits nonlinear rate dependency at room tempera-

ture, see Fig. 1. Upto 2% strain, two tensile curves overlap and then diverge. When

inelastic flow is fully developed, spacing between the stress–strain curves at different

strain rates consequently becomes equidistant. It is observed that the unloadingcurve is highly nonlinear, which is also seen in the data for polytetrafluoroethylene

(Teflon) as reported by Khan and Zhang (2001), polyetheretketone (PEEK) and

Nylon 66, see Bordonaro and Krempl (1993). It shows less strain rate dependence

than the loading curve when the unloading strain rate has the same magnitude as the

loading strain rate.

Stress–strain behavior of PPO under uniaxial loading is closer to that found in

metals than that of polymers. Fig. 1 shows the stress–strain behavior of PPO under

strain-controlled uniaxial loading/unloading at two different strain rates, 1.E) 3 1/sand 1.E) 4 1/s. In this figure, star and plus signs exhibit the experimental data while

Table 1

Material constants

Modulus E ¼ 2800 MPa

Et ¼ 80 Mpa

Shape function c1 ¼ 380 MPa

c2 ¼ 1900 MPa

c3 ¼ 40, c4 ¼ 2

k ¼ 0:48, a ¼ 0:5

Isotropic stress A ¼ 35 MPa

Flow function B ¼ 1 1/s

D ¼ 90 MPa

m ¼ 4:8

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Fig. 1. Predictions of rate dependence of PPO using the classical VBO.

O.U. Colak / International Journal of Plasticity 21 (2005) 145–160 151

solid and dotted lines show the simulation results using the classical VBO (C ¼ 1,

c4 ¼ 0 and the inelastic strain ( _�in) is replaced by the overstress invariant (C) in the

shape function, see Eq. (4)). Since the material constants are determined using the

modified VBO, when these material constants are used with the classical VBO, a

good match with experiments is not obtained. However, considering Fig. 1, it should

not be concluded that this is the modeling capability of the classical VBO. For an-

other set of material constants, stress–strain characteristics of PPO is modeled anddepicted in Fig. 2. It is observed that even though elastic and inelastic region is

captured quite well, there are some deviations in the transition from elastic to in-

elastic region. Also nonlinear unloading behavior cannot be captured using the

classical VBO.

Stress–strain curves of PPO obtained using VBO by Krempl and Ho (2000)

(C ¼ 1) are depicted in Fig. 3. The differences between the modified VBO and

VBO used by Krempl and Ho (2000) are the parameter C, introduced in the

elastic strain rate. In the VBO by Krempl and Ho (2000), the parameter C isequal to one. In the modified version, the function C evolves according to the

state variables; equilibrium, kinematic and isotropic stresses, see Eq. (1). Krempl

and Ho (2000) used a modified equilibrium stress evolution law, which is slightly

different from Eq. (3). However, in this study, the equilibrium stress evolution law

given in Eq. (3) is used throughout the paper. As seen in Fig. 3, non-linear rate

dependency is apparent and modeled well. Especially, loading behavior with the

strain rate of 1.E) 4 1/s is captured well. However, nonlinear unloading is not

reproduced well.

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Fig. 2. Simulation of stress–strain behavior of PPO using the classical VBO for another set of material

data.

Fig. 3. Predictions of rate dependence of PPO using VBO by Krempl and Ho (2000).

152 O.U. Colak / International Journal of Plasticity 21 (2005) 145–160

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O.U. Colak / International Journal of Plasticity 21 (2005) 145–160 153

The best simulation results are obtained by the presently formulated VBO model

as depicted in Fig. 4. This nonlinear rate sensitivity is primarily modeled by the flow

function. Introducing the function, C into the elastic strain rate formulation and

using the shape function given in Eq. (4) which is different from that of the classical

VBO improved the modeling capabilities of VBO for solid polymers. Nonlinear

unloading behavior of PPO is reproduced quite well. As shown in Fig. 4, nonlinearrate dependency is much more predominant in loading that that of unloading and is

captured well.

3.2. Recovery at zero stress

Due to the visco-elastic nature of polymers, partial recovery of strain at zero stress

is observed. A PPO specimen was loaded to 10% strain and immediately unloaded at

the same strain rates, 1.E) 3 1/s and 1.E) 4 1/s. Then, a recovery period of about 1 hwas introduced, Khan (2002). Prediction of recovery at zero stress using the classical

VBO with the strain rates, 1.E) 3 1/s and 1.E) 4 1/s, are depicted in Figs. 5 and 6,

respectively. In these simulations, the material parameters obtained from uniaxial

data are used. Some deviations are observed especially for strain rate 1.E) 3 1/s. At

the strain rate 1.E) 4 1/s, prediction and experimental data are close to each other.

When the amount of strain recovered with the prior unloading rate, 1.E) 3 1/s and

the prediction results of classical VBO is compared, it is observed that experimentally

observed strain recovery is around 1.5% and prediction result is approximately 2%.

Fig. 4. Simulations of rate dependence of PPO using the modified VBO.

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Fig. 5. Prediction of recovery at zero stress using the classical VBO, strain rate 1.E) 3 1/s.

Fig. 6. Prediction of recovery at zero stress using the classical VBO, strain rate 1.E) 4 1/s.

154 O.U. Colak / International Journal of Plasticity 21 (2005) 145–160

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O.U. Colak / International Journal of Plasticity 21 (2005) 145–160 155

This difference is much smaller for strain rate of 1.E) 4 1/s (recovered strain: 1.25%

(experiment) and 1.2% (prediction)). Even though the amount of recovered strain is

close to experimental data, since the strain level at zero stress is different for ex-

periments and predictions, recovered strain versus time curves do not match the

experimental data well.

In the next simulations, the VBO by Krempl and Ho (2000) is used and predictionresults are given in Figs. 7 and 8 for strain rates 1.E) 3 1/s and 1.E) 4 1/s. Similar to

the prediction results obtained by the classical VBO, predictions and experimental

data do not match since the strain differs in the beginning of recovery test. However,

amount of the recovered strain is close to experimental data.

When the modified VBO is used for the simulation of recovery at zero stress with

the prior unloading rates, 1.E) 3 1/s and 1.E) 4 1/s, experimentally observed strain

recovery is reproduced quite well and shown in Fig. 9. Strain rate dependency at the

recovered strain is observed. The amount of strain recovered after unloading to zerostress nonlinearly increases with the increase of the magnitude of the prior unloading

rate.

3.3. Multiple creep

A serious challenge when designing products to be made from polymeric materials

is the prediction of performance over long time periods of time. The amount of

deformation after short- and long-term loading has to be known reasonably

Fig. 7. Prediction of recovery at zero stress using VBO by Krempl and Ho (2000), strain rate¼ 1.E) 3 1/s.

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Fig. 8. Prediction of recovery at zero stress using VBO by Krempl and Ho (2000), strain rate¼ 1.E) 4 1/s.

Fig. 9. Prediction of recovery at zero stress using the modified VBO.

156 O.U. Colak / International Journal of Plasticity 21 (2005) 145–160

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O.U. Colak / International Journal of Plasticity 21 (2005) 145–160 157

accurately in advance, i.e. at the design stage. During long-term service, creep and

stress relaxation are the main deformation mechanisms that can cause for concern.

Creep response of polymers has been found to differ from that of metals. Creep

strain develops in polymers at relatively low stress levels even at room temperature.

For polymers, the delayed response of polymer chains during deformations is the

cause of creep behavior. Magnitude of stress level relative to the yield point has astrong impact on the creep response in polymers. While primary and secondary creep

may be observed at low stress values, stress levels close to the yield point can induce

primary, brief secondary and tertiary creep.

Since creep behavior has been found to be profoundly influenced by the level of

the stress, creep tests are performed at different stresses, i.e. at low stress, just prior to

yielding and post yield creep, see Fig. 5.15 of Khan (2002). Fig. 10 is reproduced

from Fig. 5.15 of Khan (2002). It exhibits multiple creep tests on a PPO specimen.

The dashed line indicates an uninterrupted stress–strain curve for the same strainrate, i.e. 1.E) 3 1/s. Creep tests are conducted on a specimen at 20.5, 32.5, 47 and 53

MPa stress levels.

The specimen is loaded up to the 20.5 MPa stress level under strain control at

strain rate of 1.E) 3 1/s. After 250 s hold-time, the specimen is loaded at the same

strain rate until next creep stress level, 32.5 MPa. The stress hold-time for the first

three creep tests are 250 s. In the last stress level, hold-time is 148 s. In the inset, the

change in strain versus time is plotted for four creep tests. Primary creep is observed

in each stage of the multiple creep tests as seen in the inset.

Fig. 10. Multiple creep tests on a PPO specimen. The dashed line indicates an uninterrupted stress–strain

curve for the same strain rate, i.e. 1.E) 3 1/s. Reproduced from Fig. 5.15 of Khan (2002).

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158 O.U. Colak / International Journal of Plasticity 21 (2005) 145–160

Multiple creep test is simulated using only the modified version of VBO. Material

constants used in the previous simulations are used here as well. Predictions

of multiple creep test on PPO using the modified VBO for polymers are depicted in

Fig. 11. The inset in Fig. 11 gives the change in the strain–time for each stage of creep

test. It is observed that strain accumulation at 20.5 MPa creep test is much more

than that of the experimental data. Due to this overpredicted creep strain, change inthe strain versus time curves do not match well with experiments. However, when the

amount of strain accumulation in the second, third and last creep tests are compared

to the experimental findings given in Fig. 10, it is seen that amounts of accumulated

strain are close to experimental data.

Overprediction of the strain in the first creep which is performed at 20.5 MPa

stress level can be eliminated by reducing the overstress, which is defined as the

difference between the Cauchy stress and equilibrium stress. By setting the stress to a

constant stress, s0, and stress rate to zero in the flow law, see Eq. (1), creep strain ratecan be obtained as following:

_e ¼ 3

2F

C0

D

� �s0 � gC0

� �: ð6Þ

As seen in Eq. (6), the creep strain rate depends on the overstress, o ¼ s� g. If thedifference between the Cauchy and the equilibrium stresses is reduced, the strain

accumulation at low stress levels can be reduced and a better match with the ex-

perimental data can be obtained.

Fig. 11. Predictions of multiple creep test on PPO using the modified VBO for polymers.

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O.U. Colak / International Journal of Plasticity 21 (2005) 145–160 159

4. Conclusions

The viscoplasticity theory based on overstress (VBO) is modified so that the ex-

perimentally observed nonlinear rate-dependent and nonlinear unloading behavior,

multiple creep and recovery of polyphenylene oxide (PPO) can be reproduced. The

modified VBO, the VBO used by Krempl and Ho (2000), and the classical VBO areemployed to demonstrate the improved modeling capability. The simulation results

are compared with experimental data obtained from Krempl and Khan (2003) and

Khan (2002). It is shown that nonlinear rate sensitivity, unloading behavior, multiple

creep and the strain at zero stress can be modeled quite well when the modified VBO

for polymeric material is used.

Acknowledgements

The author thanks Dr. Fazeel Khan for his permission to use his unpublished

experimental results.

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