Modeling Finite Deformation Behavior of Semicrystalline Polymers under Uniaxial Loading Unloading

Modeling Finite Deformation Behavior of Semicrystalline Polymers under Uniaxial Loading Unloading NECMI DUSUNCELI* Department of Mechanical Engineering, Aksaray University, 68100 Aksaray, Turkey ABSTRACT: The aim of this work is to investigate the finite deformation behavior of polymeric materials under monotonic loading unloading. The strain rate sensitivity behaviors of polymeric materials were modeled using viscoplasticity theory based on an overstress (VBO) model. The modeling capability of the VBO model was improved to describe the nonlinear stress strain behavior of the fully inelastic flow region in loading at small strain level. In this model, the tangent modulus (E t ) is taken nonlinearly to simulate this polymeric material behavior. The numerical results were compared to the experimental data in the literature. These results were in good agreement with the experimental data. KEY WORDS: polymeric materials, modeling, strain rate sensitivity. INTRODUCTION POLYMERIC MATERIALS ARE used in large range of applications in aerospace and automotive industries, in electronic systems, and consumer appliances. Due to this increased use, much research has been focused on understanding the behavior of polymeric materials under different loading conditions. The mechanical behaviors (viscoelastic and *E-mail: dusunceli@gmail.com JOURNAL OF ELASTOMERS AND PLASTICS Vol. 42 July 2010 347 0095-2443/10/04 0347 18 $10.00/0 DOI: 10.1177/0095244310368126 ß The Author(s), 2010. Reprints and permissions: http://www.sagepub.co.uk/journalspermissions.nav

348 N. DUSUNCELI viscoplastic) of polymeric materials are very complex and depend on how products made from polymeric materials are used. The challenge when designing products to be made from polymeric materials is the prediction of performance in the long term, that is to accurately determine the amount of deformation in advance, that is, at the design stage. In recent years, a number of constitutive equations have been developed to represent the mechanical behavior of polymeric materials. These equations are founded on the two main approaches of micro- and macromechanical modeling. The micromechanical constitutive equations are based on the physics of the deformation mechanism. The macromechanical constitutive equation utilizes a modified theory of linear viscoelasticity and viscoplasticity to model nonlinear mechanical behavior by introducing a stress- or strain-based time scale via the invariants of the stress or strain. Determining the complex mechanical behaviors of polymeric materials is a difficult task therefore, there is a need to develop the existing constitutive equations. In the modeling of polymeric materials there are various experimental considerations that need to be taken into account. The points listed below are some of the widely observed experimental properties concerning mechanical behavior of polymeric materials. 1. They are highly nonlinear, strain rate, and temperature dependent. The flow stress increases nonlinearly with an increase of the loading rate. 2. The unloading curve is nonlinear and shows less strain rate dependence than the loading curve when the loading and unloading strain rates have the same magnitude. 3. The yield behavior is significantly affected by hydrostatic pressure. 4. Recovery at zero stress is significant. Furthermore, creep and relaxation are also observed at room temperature [1 3]. In recent years a number of constitutive equations have been developed to describe the viscoelastic viscoplastic behavior of polymeric materials. These studies have been conducteed by Popellar et al. [4], Lai and Bakker [5], Bardenhagen et al. [6], Ariyama and Koneko [7], Zhang and Moore [8], Nikolov and Doghri [9], Nikolov et al. [10], Hasan and Boyce [11], Boyce et al. [12], Ahzi et al. [13], and Makardi et al. [14], van Dommelen et al. [15], Bergstrom et al. [16], Ayoub et al. [17], Kitagawa and Takagi [18], Krempl and Ho [19], Ho and Krempl [20], Colak and Dusunceli [21], and Dusunceli and Colak [22]. The Schapery thermodynamics representation was further developed by Popellar et al. [4] using relaxation data and verified using the

Finite Deformation Behavior of Semicrystalline Polymers 349 stress strain behavior of polyethylene (PE). This can model the stress strain behavior of PE at different strain rate level except for unloading. An integral representation, also based on Schapery thermodynamics theory, was proposed by Lai and Bakker [5] to explain the creep, recovery, and loading unloading behavior of high density polyethylene (HDPE) at different stress/strain rate levels. They concluded that the model can be also used to describe the process of the preconditioning of semi-crystalline polymers. Bardenhagen [6] proposed a phenomenological constitutive equation for polymeric materials, which offers a general methodology to describe the strain sensitivity, creep, and relaxation behaviors of polymeric materials. This model was derived using a standard linear solid model that was created by replacing the linear spring with an elastic-plastic spring. The constituted model is used to describe loading behavior at different strain rate, relaxation behavior at different stress levels and creep behavior at different strain levels, and good agreement was found between the experimental and simulation results. Ariyama and Koneko [7] studied the simulation of semi-crystalline polymeric materials behavior by evolving a constitutive equation based on a simple overly truss model. The purpose of the model is to describe the cyclic deformation of polypropylene (PP) and a good match can be obtained with experimental results. This model shows the cyclic deformation behavior of PP including the decreasing of the yield point and strain softening behavior. Zhang and Moore [8] developed two uniaxial constitutive equations; the first using the mechanical analogy consisting of one independent spring and six Kelvin elements in series. The second model based on viscoplasticity theory (Bodner s Model) included an inelastic strain rate into state variable. The modeling capability of these models is verified using the nonlinear compressive loading and creep behavior of HDPE at different strain rate/strain levels. The simulation results had a good match with the experimental results. Nikolov and Doghri [9] developed a micro/macro constitutive model to predict the small deformation behavior of HDPE. The micromechanical model is constituted by assuming that the microstructures of semi-crystalline polymer are an aggregate of a randomly oriented composite that represent the crystalline and amorphous phase. In order to model of the overall behavior, different homogenizations schemes were implemented in the models. The simulation results of the model explained the stress strain behavior of semi-crystalline polymers related to the crystalline ratio. Nikolov et al. [10] proposed a multiscaled model to describe the mechanical behavior of semi-crystalline polymers under small deformation. The behavior of crystalline phase

350 N. DUSUNCELI was modeled using a viscoplastic model and the behavior of amorphous phase was cleared using a nonlinear viscoelastic model. The overall deformation of semi-crystalline polymer was described via the Sachs homogenization schemes. The simulation results showed that the model can explain the mechanical behavior of semi-crystalline polymer depending on the crystalline ratio and strain rate. Finally, good agreement was found when the simulation results of the model were compared to loading experimental data of HDPE and MDPE at different stress rate levels. Hasan and Boyce [11] proposed a model based on thermal activation energy dissipation in large strain levels to describe the nonlinear viscoelastic viscoplastic behavior of polymethylmethacrylate (PMMA). This model described the loading unloading behavior of PMMA at different strain rate and temperature and the predictions for creep behavior with material data were determined using stress strain data. Good matches are found with the experimental data. Boyce et al. [12] proposed a model to describe the elastic-viscoplastic loading behavior of polyethyleneterephtalate (PET) at different loading strain rates. This is a micromechanical model that includes the process of molecular orientation and molecular relaxation and contains a deformation mechanism with temperature and a microstructural variable. This model clarified the strain rate and temperature-dependent stress strain behavior of PET at temperatures above glass transition temperatures; however, this model was not good enough to describe the unloading behavior of PET. Ahzi et al. [13] and Makardi et al. [14] developed a micromechanical model to describe large stress strain behavior and strain-induced crystallization behavior of PET at different strain rates and temperatures above the glass transition temperature. This model was based on those created by Boyce et al. [12] and a composite intermolecular resistance was proposed where the amorphous and crystalline phases are considered as different resistances. The simulation results of the model were compared with experimental observation related to the crystalline ratio and good agreement was found. van Dommelen et al. [15] developed an elasto-viscoplastic micromechanical model to simulate the loading unloading, cyclic behavior, and texture evolution of HDPE. This model included a composite unit element represented the crystalline and amorphous phase related to the microstructure of material. A simulation was undertaken with this model for the loading unloading and cyclic behavior at different crystalline ratios. The simulation results were in agreement with the experimental observations in the literature. Finally, a model has been

Finite Deformation Behavior of Semicrystalline Polymers 351 proposed to model the loading unloading and cyclic behavior of the HDPE-dependent crystalline ratio. Bergestrom et al. [16] simulated the mechanical behavior of ultra high molecular weight polyethylene at small and finite deformation levels. This is a hybrid model that includes many features from previous theories and compares the numerical results of this model with Arruda-Boyce, Hasan-Boyce, Bergestrom- Boyce, and J2 plasticity theory models. Ayoub et al. [17] conducted series mechanical tests to investigate the mechanical behavior of HDPE at large strain level. The proposed model describes the mechanical behavior of HDPE related to crystalline ratio. This study investigated the influence of the crystalline ratio on the loading and unloading behavior of HDPE. In order to fully model this mechanical behavior, an empirical expression was generated related to the evolution of the overall Young s modulus. The simulation results were compared with the experimental results. Subsequently, this model was carried out to describe relaxation behaviors of HDPE at different strain levels. Moreover, to compare the modeling capability, the simulation results were matched with results from the Boyce and Ahzi models. Drozdov and Christiansen [1] conducted a series of loading unloading and cyclic tests with PP materials and a constitutive equation was developed to model these material behaviors. When the simulation and experimental results were compared a good match was found. In the literature, most of the work on polymeric material behavior uses viscoplasticity theories. In this article, one of the unified state variable theories, the viscoplasticity theory based on overstress (VBO) that does not consist of a yield surface, is used to simulate the mechanical behavior of polymeric materials. Kitagawa and Takagi [18] carried out tests on the mechanical behavior of HDPE at combined tension torsion loading and modeled this using an early version of the VBO model. This study is showed that the modeling capability of the VBO is very good for strain rate sensitivity, and stress jumping behaviors. Krempl and Ho [19] further modified the VBO to model nonlinear rate sensitivity loading and unloading, cyclic softening, and the recovery behavior of Nylon66. The simulation results were a good match with the experimental results, thus showing that the VBO model is capable of modeling the mechanical behavior of polymeric materials. Ho and Krempl [20] then improved the VBO model by including a revised equilibrium stress evolution equation. This modified VBO model is used to describe the positive, negative, and zero rate sensitivity behavior of polymeric materials and the rate-dependent strain softening and hardening behavior of PMMA. Recently, Colak [3], Colak and Dusunceli [21], Dusunceli and Colak [2] made improvements to the VBO

352 N. DUSUNCELI model to describe mechanical behavior of polymeric materials under different loading conditions, including, loading unloading, creep, and relaxation. This model is used to model the mechanical behavior of PPO and HDPE under small and finite deformation levels. These studies indicate that the VBO model has very effective capabilities in describing polymeric material behavior. Dusunceli and Colak [22] improved a viscoplasticity model, which consisted of a modified VBO model to observe the mechanical behavior of polymeric materials. This model describes the mechanical behavior of semi-crystalline polymeric materials in relation to the crystalline ratio. The simulation results matched well with the results of the experimental data in literature. In this study, the small strain, isotropic, viscoplasticity theory based on overstress (VBO) is modified to give an account of the strain hardening and softening behavior of polymeric materials on loading unloading at small strain levels. This modification is performed by considering the tangent modulus to be within nonlinear parameters. The tangent modulus (E t ) is represented as being associated with the plastic strain so it can be a nonlinear and variable versus strain. The loading unloading behavior of the PP and HDPE at different strain levels were successfully simulated including the strain hardening softening behavior in the fully plastic flow region. Good agreement was found when the simulation results were compared to the experimental data by Drozdov and Christiansen [1] and Dusunceli and Colak [2]. THE OVERSTRESS MODEL The mechanical behaviors of materials can be modeled from a macroscopic or microscopic perspective. Macroscopic models range from rate-independent plasticity models to unified state variable theories. In this article, models of polymeric materials are constituted using the macroscopic model. The continuum theories contain viscoplasticity theories, which assume that inelastic deformation is rate dependent even at low homologous temperatures. The viscoplasticity models represented by unified state variable theories do not permit the separation of creep and plasticity. State variables are defined as the macroscopic averages of events associated with microstructure changes and cannot be directly measured or controlled but are useful in defining material behavior. These variables are associated with dislocations and their interaction with other dislocations and grain boundaries. State variables are defined as the macroscopic averages of events associated with microstructure changes and cannot be directly measured or

Finite Deformation Behavior of Semicrystalline Polymers 353 controlled but are useful in defining material behavior. These variables are associated with dislocations and their interaction with other dislocations and grain boundaries. VBO was developed by Krempl and his co-workers for use with metallic materials and the standard linear solid (SLS) model was taken as a basis for the development of VBO in early 1980s. The mechanical behaviors of polymeric materials are different from the metallic material behavior for example: hysteresis loop, between loading and unloading curve tangential angel, cyclic hardening behavior, and cyclic softening. Although the VBO model has modeled some important mechanical behaviors of polymeric materials (loading unloading, creep, relaxation, etc.), the constitutive equation has not modeled the capability of different properties of mechanical behavior. This situation required the modification of the VBO model, which is a rate-dependent unified state variable theory with no yield surface and no loading/unloading conditions [23]. The theory consists of two tensor-valued state variables: equilibrium stress (G), the kinematic stress (K) and two scalar-valued state variables: isotropic stress (A) and drag stress (D). All the tensorial quantities are denoted by bold letter. According to Krempl [24] the total strain rate is sum of the elastic and inelastic strain rates and the flow law for small strain, incompressibility and isotropy is given by: _e ¼ _e el þ _e in ¼ 1 þ CE _s þ 3 2 F s g, ð1þ D where s and g are the deviatoric part of the Cauchy (r) and the equilibrium stress (G) tensor, respectively. The quantities E and v are the Young s modulus and the Poisson s ratio, respectively. A superposed dot designates the material time derivative. The square brackets following a symbol denote function of. Inelastic strain rate is function of the overstress (o), which is the difference between Cauchy and equilibrium stresses (o ¼ s g). VBO theory depends on the overstress concept with being the overstress invariant with the dimension of stress defined by: 2 ¼ 3 ð 2 ðs gþ : ðs gþ Þ: ð2þ The nonlinear unloading behavior is described by setting C ¼ 1 ðjg Kj=AÞ so that C[0] ¼ 1 and is close to zero in the fully established inelastic flow region. l and a are the material constants. Nonlinear rate sensitivity is produced by introducing the flow function

354 N. DUSUNCELI F[ ] into the inelastic strain rate equation 1, given above. It is positive and increases with the dimension of 1/time and F[0] ¼ 0. It is given by F½ Š¼Bð =DÞ m with B as a universal constant and D being the drag stress. The equilibrium stress is the path-dependent stress that can be sustained at rest after prior inelastic deformation. It is related to defects in the structure of material. The equilibrium stress G, is nonlinear, rate independent, and hysteretic. Its evolution equation in deviatoric form is given as: _g ¼ _s E þ F s g D g k þ 1 _k A E where k is the deviatoric kinematic stress, which is the repository for the modeling of the Bauschinger effect. A is the isotropic stress, a rateindependent contribution to the stress. Constant isotropic stress is used in this study. One of the material parameters in the model is the shape function, W which has a significant influence on the transition from the quasi-linear region to fully established inelastic flow. In the classical VBO, the shape function is defined as function of the invariant of overstress. In this study, the shape function, modified by Krempl and Ho [19] is used and given as:! C 2 1 ¼ 1 þ exp C 3 " in ð4þ jgj 1 ¼ C 1 1 þ C 4 A þ jkjþ where C 1, C 2, C 3, C 4, n, and are material constants. The second state variable, the kinematic stress (K) is the repository for modeling the Bauschinger effect and sets the tangent modulus at the maximum strain of interest. It is similar to the back stress in classical plasticity theories. The evolution equation for the kinematic stress in deviatoric form is: ð3þ jj _k ¼ E þ jgj t _e in ð5þ where E t ¼ E t =ð1 E t =EÞ and E t is the tangent modulus. The kinematic stress sets the tangent modulus E t at the maximum inelastic strain of interest. The tangent modulus can be positive, zero, and negative.

Finite Deformation Behavior of Semicrystalline Polymers 355 The comparison of the stress strain behaviors of polymeric and metallic materials reveals that the elastic stiffness of polymeric materials changes drastically during unloading. In order to model this behavior, the VBO model is adapted to construct the change in elastic stiffness while loading and unloading [3]. On the other hand, one of the differences between the stress strain behavior of polymeric and metallic materials are a nonlinear curve at the fully plastic flow region. This region indicates strain hardening and softening behavior in the stress strain behaviors of polymeric materials at small strain levels. To describe the nonlinearity in the full plastic flow region while loading, a new term has been introduced and enables the modeling of nonlinear mechanical behavior at full plastic flow region. The parameter tangent modulus (E t ) is related to plastic strain. This is achieved by setting E t ¼ E t,0 1 1 exp " in. Et,0 is the initial tangent modulus and is the material constant. NUMERICAL SIMULATION AND DISCUSSION The strain rate sensitivity behavior of polymeric materials under monotonic loading unloading at different strain rate levels is modeled using the VBO model that has been modified with nonlinear tangent modulus parameters. The stress strain curve of deformation behavior of polymeric materials is given Figure 1. This study investigates the deformation behavior of PP and HDPE that are semi-crystalline polymeric materials used in broad range of industrial applications. Details of the experimental data for PP obtained by Drozdov and Christiansen [1] and the experimental data for HDPE obtained by Dusunceli and Colak [2] listed below. 1. PP is monotonic loading and unloading under uniaxial tension at different strain rates, that is, 1.7 10 3 and 1.7 10 4 1/s. 2. HDPE is monotonic loading and unloading under uniaxial tension at different strain rates, that is, 1 10 3 and 1 10 4 1/s. Simulations were performed to model the VBO and the modified VBO for PP and HDPE. The VBO model has 13 material constants for the simulation of materials and the modified VBO has 14 material constants for simulation of materials. To determine the material parameters for PP and HDPE uniaxial stress strain data was used at strain rates of 1.7 10 4 and 1 10 4 1/s, respectively. The determined parameters of VBO for HDPE are given in Table 1 and for PP in Table 2. Tables 3 and 4 give the material parameters of the modified VBO model

Stress (MPa) 356 N. DUSUNCELI 35 30 25 Strain hardening Fully plastic flow region Strain softening 20 15 10 5 0 0.00 0.05 0.10 0.15 0.20 0.25 Strain FIGURE 1. Typical stress strain curve of deformation behavior of polymeric materials at small strain level. Table 1. Material parameters of VBO for HDPE. Modulus E ¼ 2000 MPa E t ¼ 2 MPa Shape function C 1 ¼ 80 MPa C 2 ¼ 1400 MPa C 3 ¼ 65 C 4 ¼ 5 n ¼ 1 MPa 1 Flow function B ¼ 3 1/s D ¼ 95 MPa m ¼ 2.7 Isotropic stress A ¼ 20.5 MPa Parameter C l ¼ 0.69 a ¼ 0.25 Table 2. Material parameters of VBO for PP. Modulus E ¼ 2600 MPa E t ¼ 2 MPa Shape function C 1 ¼ 25 MPa C 2 ¼ 1300 MPa C 3 ¼ 35 C 4 ¼ 9 n ¼ 1 MPa 1 Flow function B ¼ 2 1/s D ¼ 60 MPa m ¼ 2.85 Isotropic stress A ¼ 20.5 MPa Parameter C l ¼ 0.78 a ¼ 0.15 for HDFE and PP, respectively. The simulation results from the VBO model (with constant tangent modulus) for PP and HDPE at two strain rates/levels are depicted in Figures 2 4. The simulation results were compared to experimental data obtained by Drozdov and Christiansen [1] and Dusunceli and Colak [2]. The nonlinear rate sensitivity during

Finite Deformation Behavior of Semicrystalline Polymers 357 Table 3. Material parameters of modified VBO for HDPE. Modulus E ¼ 2000 MPa E t,0 ¼ 10 MPa Shape function C 1 ¼ 80 MPa C 2 ¼ 1400 MPa C 3 ¼ 65 C 4 ¼ 5 n ¼ 1 MPa 1 Flow function B ¼ 3 1/s D ¼ 95 MPa m ¼ 2.7 Isotropic stress A ¼ 20.5 MPa Parameter C l ¼ 0.69 a ¼ 0.25 Variable tangent modulus ¼ 30 Table 4. Material parameters of modified VBO for PP. Modulus E ¼ 2600 MPa E t,0 ¼ 15 MPa Shape function C 1 ¼ 25 MPa C 2 ¼ 1300 MPa n ¼1 MPa 1 C 3 ¼ 35 C 4 ¼ 9 Flow function B ¼ 2 1/s D ¼ 60 MPa m ¼ 2.85 Isotropic stress A ¼ 20.5 Mpa Parameter C l ¼ 0.78 a ¼ 0.15 Variable tangent modulus ¼ 13 loading unloading is good except for the fully inelastic flow region especially the strain softening behavior. Since the tangent modulus is constant, a good match with the experimental data on fully inelastic flow region was not obtained. It was observed that although loading and unloading behaviors are described quite well, there are some deviations in the fully inelastic flow region. The loading unloading curves of PP obtained using the modified VBO (tangent modulus is nonlinear) are depicted Figures 5 and 6. The results of the simulation and prediction are well matched with experimental results especially the fully plastic flow region that include strain hardening softening behavior. When the parameter E t is nonlinear, a hyperbolic curve shape in the fully inelastic flow region is observed. Subsequently, simulation and prediction were performed for the mechanical behavior of HDPE using the modified VBO, the tangent modulus being nonlinear. The simulation results were compared to the experimental results of the HDPE and are depicted in Figure 7. It is observed that introducing the nonlinear tangent modulus enables the tracing of the nonlinear stress strain curve at the fully plastic flow region curve. The influence of the nonlinear tangent modulus in the VBO model was examined to model the behavior of the fully inelastic flow region.

358 N. DUSUNCELI 30 25 20 15 17.e 10 4 Exp. 17.e 10 4 Predict. 17.e 10 5 Exp. 17.e 10 5 Sim. Stress (MPa) 10 5 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Strain FIGURE 2. Experimental stress strain curves during loading unloading of PP at different strain rate levels and simulation results of VBO at different strain rate levels when tangent modulus is constant. (Experimental data reproduced from Drozdov and Christiansen [1].) 30 25 20 Stress (MPa) 15 10 0.2 Exp. 0.2 Predict. 0.15 Exp. 0.15 Predict. 5 0 0.00 0.05 0.10 0.15 0.20 0.25 Strain FIGURE 3. Experimental stress strain curves during loading unloading of PP at different strain levels and simulation results of VBO at different strain rate levels when tangent modulus is constant. (Experimental data reproduced from Drozdov and Christiansen [1].)

Stress (MPa) Finite Deformation Behavior of Semicrystalline Polymers 359 30 25 Stress (MPa) 20 15 10 1.e 10 3 1/s Exp. 1.e 10 3 1/s Predict. 1.e 10 4 1/s Exp. 1.e 10 4 1/s Sim. 5 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Strain FIGURE 4. Experimental stress strain curves during loading unloading of HDPE at different strain rate levels and simulation results of VBO at different strain rate levels when tangent modulus is constant. (Experimental data reproduced from Dusunceli and Colak [2].) 30 25 20 15 17.e 10 4 Exp. 17.e 10 4 Predict. 17.e 10 5 Exp. 17.e 10 5 Sim. 10 5 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Strain FIGURE 5. Experimental stress strain curves during loading unloading of PP at different strain rate levels and simulation results of modified VBO at different strain rate levels when tangent modulus is variable. (Experimental data reproduced from Drozdov and Christiansen [1].)

Stress (MPa) 360 N. DUSUNCELI 30 25 20 Stress (MPa) 15 10 0.2 Exp. 0.2 Predict. 0.15 Exp. 0.15 Predict. 5 0 0.00 0.05 0.10 Strain 0.15 0.20 0.25 FIGURE 6. Experimental stress strain curves during loading unloading of PP at different strain levels and simulation results of VBO at different strain rate levels when tangent modulus is variable. (Experimental data reproduced from Drozdov and Christiansen [1].) 30 25 20 15 1.e 10 3 1/s Exp. 1.e 10 3 1/s Predict. 1.e 10 4 1/s Exp. 1.e 10 4 1/s Sim. 10 5 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Strain FIGURE 7. Experimental stress strain curves during loading unloading of HDPE at different strain rate levels and simulation results of modified VBO at different strain rate levels when tangent modulus is variable. (Experimental data reproduced from Dusunceli and Colak [2].)

Finite Deformation Behavior of Semicrystalline Polymers 361 This parameter does affect the results and is enabled for modeling the fully plastic flow region in the stress strain curve. The introduction of the nonlinear tangent modulus (E t ) to the VBO improved the modeling capabilities of the VBO in terms of the mechanical behavior of polymeric materials. Furthermore, the nonlinear stress strain behavior of polymeric materials at fully plastic flow region is fairly well reproduced. The VBO model consists of stiff nonlinear ordinary differential equations. The characteristic of these equations are a small change in the input may result in large changes in the output. These equations are numerically integrated using the commercial ordinary differential equation solver program DGEAR of IMSL. The material parameters of the VBO model are determined using genetic algorithm method, for more information about these parameter determination procedures see Dusunceli et al. [25]. CONCLUSIONS The uniaxial strain controlled tests carried on PP and HDPE were simulated using the viscoplasticity theory based on overstress (VBO). Since the stress strain behavior of polymeric materials in the fully plastic flow region (nonlinear) are different from other materials, a nonlinear tangent modulus parameter in the VBO model was used in the simulations and predictions of PP and HDPE. The VBO models the loading unloading behavior very well and the nonlinear tangent modulus parameter improved the modeling capabilities. The modification of the tangent modulus parameter enables a nonlinear curve in the plastic flow region in the stress strain curve. Consequently, the modified VBO model is able to precisely model all the loading unloading behavior of polymeric materials. NOMENCLATURE _e = strain rate (1/s) _s = deviatoric Cauchy stress rate (MPa/s) s = deviatoric Cauchy stress (MPa) G = equilibrium stress (MPa) g = deviatoric equilibrium stress (MPa) o = overstress (MPa) k = deviatoric kinematic stress (MPa)

362 N. DUSUNCELI K = kinematic stress (MPa) E = elasticity modulus (MPa) D = drag stress (MPa) F = flow function (1/s) B = universal constant (1/s) A = isotropic stress (MPa) E t = tangent modulus (MPa) E t,0 = initial tangent modulus (MPa) Greek letters r = Cauchy stress tensor m = Poisson s ratio = overstress invariant (MPa) W = shape function Superscripts el = elastic in = inelastic REFERENCES 1. Drozdov, A.D. and Christiansen, J.C. (2007). Cyclic Viscoplasticity of Solid Polymers: The Effects of Strain Rate and Amplitude of Deformation, Polymer, 48: 3003 3012. 2. Dusunceli, N. and Colak, O.U. (2006). High Density Polyethylene (HDPE): Experiments and Modeling, Mech. Time Depend. Mat., 10: 331 345. 3. Colak, O.U. (2005). Modeling Deformation Behavior of Polymers with Viscoplasticity Theory Based on Overstress, Int. J. Plasticity, 21: 145 160. 4. Popelar, C.F., Popelar, C.H. and Kenner, V.H. (1990). Viscoelastic Material Characterization and Modeling for Polyethylene, Polym. Eng. Sci., 30: 577 586. 5. Lai, J. and Bakker, A. (1995). An Integral Constitutive Equation for Nonlinear Plasto-Viscoelastic Behavior of High-density Polyethylene, Polym. Eng. Sci., 35: 1339 1347. 6. Bardenhagen, S.G., Stout, M.G. and Gray, G.T. (1997). Three dimensional, Finite Deformation, Viscoplastic Constitutive Models for Polymeric Materials, Mech. Mater., 25: 235 253. 7. Ariyama, T. and Kaneko, K. (1995). A Constitutive Theory for Polypropylene in Cyclic Deformation, Polym. Eng. Sci., 35: 1461 1467. 8. Zhang, C. and Moore, I.D. (1997). Nonlinear Mechanical Response of High Density Polyethylene, Part II: Uniaxial Constitutive Modeling. Polym. Eng. Sci., 37: 414 420.

Finite Deformation Behavior of Semicrystalline Polymers 363 9. Nikolov, S. and Doghri, I. (2000). A Micro/Macro Constitutive Model for the Small-deformation Behavior of Polyethylene, Polymer, 41: 1883 1891. 10. Nikolov, S., Doghri, I., Pierard O., Zealouk, L. and Goldberg, A. (2002). Multi-scale Constitutive Modeling of the Small Deformations of Semicrystalline Polymers, J. Mech. Phys. Solids, 50: 2275 2302. 11. Hasan, O.A. and Boyce, M.C. (1995). A Constitutive Model for the Nonlinear Viscoelastic Viscoplastic Behavior of Glassy Polymers, Polym. Eng. Sci., 35: 331 334. 12. Boyce, M.C., Socrate, S. and Llana, P.G. (2000). Constitutive Model for the Finite Deformation Stress Strain Behavior of Poly(ethylene terephthalate) above the Glass Transition, Polymer, 41: 2183 2201. 13. Ahzi, S., Makradi, A., Gregory, R.V. and Edie, D.D. (2003). Modeling of Deformation Behavior and Strain-induced Crystallization in Poly(ethylene terephthalate) above the Glass Transition Temperature, Mech. Mater., 35: 1139 1148. 14. Makradi, A., Ahzi, S., Gregory, R.V. and Edie, D.D. (2005). A Two-phase Self-consistent Model for the Deformation and Phase Transformation Behavior of Polymers above the Glass Transition Temperature: Application to PET, Int. J. Plasticity, 21: 741 758. 15. van Dommelen, J.A.W., Parks, D.M., Boyce, M.C., Brekelmans, W.A.M. and Baaijens, F.P.T. (2003). Micromechanical Modeling of the Elasto- Viscoplastic Behavior of Semi-crystalline Polymers, J. Mech. Phys. Solids, 51: 519 541. 16. Bergstrom, J.S., Kurtz, S.M., Rimnac, C.M. and Edidin, A.A. (2002). Constitutive Modeling of Ultra-high Molecular Weight Polyethylene under Large-deformation and Cyclic Loading Conditions, Biomaterials, 23: 2329 2343. 17. Ayoub, G., Zaïri, F., Naït-Abdelaziz, M. and Gloaguen, J.M. (2009). Modelling Large Deformation Behaviour under Loading unloading of Semicrystalline Polymers: Application to a High Density Polyethylene, Int. J. Plasticity, 26: 329 347. 18. Kitagawa, M. and Takagi, H. (1990). Nonlinear Constitutive Equation for Polyethylene under Combined Tension and Torsion, J. Polym. Sci. Pol. Phys, 28: 1943 1953. 19. Krempl, E. and Ho, K. (2000). An Overstress Model for Solid Polymer Deformation Behavior Applied to Nylon66, ASTM STP, 1357: 118 137. 20. Ho, K. and Krempl, E. (2002). Extension of the Viscoplasticity Theory based on Overstress (VBO) to Capture Non-standard Rate Dependence in Solids, Int. J. Plasticity, 18: 851 871. 21. Colak, O.U. and Dusunceli, N. (2006). Modeling Viscoelastic and Viscoplastic Behavior of High Density Polyethylene (HDPE), J. Eng. Mater-T ASME, 128: 572 578. 22. Dusunceli, N. and Colak, O.U. (2008). Modelling Effects of Degree of Crystallinity on Mechanical Behavior of Semicrystalline Polymers, Int. J. Plasticity, 24: 1224 1242. 23. Krempl, E. (1998). Creep-plasticity Interaction, Lecture Notes, CISM No. 187, Udine, Italy.

364 N. DUSUNCELI 24. Krempl, E. (1996). A Small Strain Viscoplasticty Theory based on Overstress, In: Krausz, A.S. and Krausz, K. (eds), Unified Constitutive Laws of Plastic Deformation, San Diego, Academic Press, pp. 281 318. 25. Dusunceli, N., Colak, O.U. and Filiz, C. (2009). Determination of Material Parameters of a Viscoplastic Model by Genetic Algorithm, Mater. Design, 31: 1250 1255.