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

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

www.elsevier.com/locate/ijplas

International Journal of Plasticity 21 (2005) 145160to demonstrate the improved modeling capabilities of VBO for solid polymer deformation.

The unied 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 inuenced by the level of

the stress, the tests are performed at dierent 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 modied

viscoplasticity theory based on overstress.

2004 Elsevier Ltd. All rights reserved.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 nal 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 modied viscoplasticity theory based on overstress (VBO). In

addition, VBO used by Krempl and Ho [An overstress model for solid polymer deformationE-mail address: colako@alum.rpi.edu.

0749-6419/$ - see front matter 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijplas.2004.04.004

146 O.U. Colak / International Journal of Plasticity 21 (2005) 1451601. Introduction

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

elasticviscoplastic 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 rst 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

signicant recovery at zero stress.

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

behavior. In addition, signicant 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 eect on the mechanical

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

havior depends on the molecular chain exibility, entanglement and on dierences 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 dicult. 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 ow 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 unloadingcurve is highly nonlinear. The unloading curves show less strain rate dependence

O.U. Colak / International Journal of Plasticity 21 (2005) 145160 147than loading curves when the loading and unloading strain rates have the same

magnitude.

3. Yield behavior is signicantly aected 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 signicant. 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 modied to represent accurate mechanical behavior of

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

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

such as uniaxial and multiaxial monotonic and cyclic loading. In addition, a detailedknowledge of the inuences 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 ow. Two basic resistances to deformation are dened 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 nite 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 viscoelasticplastic constitutive equation takes

account of the eect 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 unied 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

behav

and t

Re

148 O.U. Colak / International Journal of Plasticity 21 (2005) 145160attempt to compare metallic and polymeric inelastic deformation behavior. Con-

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

polymers: loading at dierent strain rates, nonlinear relation betwe