A 3-D Model of Cold Drawing in Engineering Thermoplastics

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1 Mechanics of Advanced Materials and Structures, 12: , 2005 Copyright c Taylor & Francis Inc. ISSN: print / online DOI: / A 3-D Model of Cold Drawing in Engineering Thermoplastics Arif Masud Department of Civil & Materials Engineering, The University of Illinois at Chicago, Chicago, Illinois, USA This article presents a 3-D multiplicative finite strain model for cold drawing (necking) in engineering thermoplastics. Amorphous as well as semi-crystalline polymers are considered. The notion of intermediate (local) stress-free configuration, that is associated with the multiplicative decomposition of the deformation gradient, is shown to provide an appropriate framework for the modeling and analysis of this class of materials. A hyperelastic stored energy function is written with respect to the intermediate configuration to yield the finite elastic response. It is then combined with the J 2 - flow theory to model the finite inelastic response. The constitutive integration procedure is based on a product formula algorithm with elastic-predictor/inelastic-corrector components. Numerical results are presented to show the behavior of the three dimensional model in the finite strain range. 1. INTRODUCTION This article is an extension of our earlier efforts, Masud and Chudnovsky [1], and Masud [2], where we proposed a finite strain constitutive model for neck propagation and solid-solid state transformation in semi-crystalline polymers and polycarbonates. Emphasis in the present article is on the extension and implementation of the model in a three-dimensional finite element setting. A good review of the various semi-analytical and numerical approaches for the modeling of polymers can be seen in Hutchinson and Neale [3], Neale and Tugcu [4], Tomita and Hayashi [5], Boyce et al. [6, 7], Arruda and Boyce [8, 9], Rao and Rajagopal [10], Reddy and coworkers [11, 12], and references therein. Experimental investigations into large deformation of polycarbonates (Zhou et al. [13], Kim et al. [14]) reveal that necking phenomena in this class of materials is a special type of transformation involving a three step process, usually termed as double glass transition (DGT). The material is initially in homogeneous, isotropic, glassy state. As it is stretched, it undergoes an inter- Received 12 May 2005; accepted 12 July Address correspondence to Arif Masud, Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago, Illinois amasud@uic.edu nal restructuring from isotropic glassy state to isotropic rubbery state. Further stretching converts the isotropic rubber to an oriented rubber. During this transformation the ratio of material stretching within the transformed and untransformed regions remains constant and is considered to be a material property. Accordingly, during neck propagation in this class of materials, the solid-solid interphase boundary translates at essentially a constant stress (called the engineering draw stress and is also considered to be a material property) without further stretching of the material within the transformed (necked) region. Furthermore, the oriented material undergoes additional stretching only after the entire specimen has completely transformed. Experimental studies also reveal that polycarbonates, in general, can be stretched up to % strains, while various semi-crystalline polymers (e.g., polyethylene) can be stretched up to 500% strains. Although micromechanics of cold drawing in semi-crystalline polymers (e.g., polyethylene) is different from that of the amorphous polymers (e.g., polycarbonates), as is discussed in Ward and Hadley [15]. However, from a continuum mechanics perspective the modeling of this general class of materials raises issues that are associated with an appropriate treatment of the large strains. This article aims at addressing these issues in a three-dimensional finite element framework. An outline of the paper is as follows. Section 2 discusses the basic kinematics, strain measures, and a synopsis of the constitutive model proposed in Masud and Chudnovsky [1]. Section 3 discusses the constitutive integration algorithm. Numerical results are presented in Section 4 and conclusions are drawn in Section BASIC KINEMATICS, STRAIN MEASURES AND CONSTITUTIVE MODEL Let 0 R n sd with n sd 2bethe reference placement of a continuum body at time t 0. The deformation map ϕ t : R n sd is a transformation that maps X 0 to x t, where t = ϕ t ( ) denotes the current placement of the body. For a description of various continuum quantities, see, e.g., Marsden and Hughes [16], Belytschko et al. [17], or Simo and Hughes [18]. 457

2 458 A. MASUD Although the micro-mechanical description of deformation is different in amorphous polymers as compared to that in the semicrystalline polymers, we view the resulting deformation to be composed of an elastic/recoverable component and an inelastic component. As such we assume a local multiplicative split of the deformation gradient into an elastic component F e, and an inelastic component F i. F = F e F i (1) The inelastic component of the deformation gradient is associated with orientational hardening and defines an additional or an intermediate configuration t as shown in Figure 1. For notational clarity, we denote the quantities in the reference configuration with upper case letters, in the intermediate configuration with superposed bars and in the current configuration with lower case letters. It is now well documented that the notion of stress-free intermediate configuration has an important significance in the development of constitutive models [1, 18 21]. In addition, it provides a link between the continuum based theories for inelasticity (that presume the existence of a flow rule, see, e.g., Simo and Hughes [18]), and the constitutive theories that are based on the idea of multiple natural (stress-free) configurations of the materials (see e.g., Rajagopal and Srinavasa [20, 21]). These later theories involve the specification of an evolution equation for the internal variable that defines the evolution of the stress-free intermediate configuration. Accordingly, in these theories an explicit definition of a flow rule is not specified. We first present a synopsis of the constitutive model developed in Masud and Chudnovsky [1]. We assume that the thermodynamic state is characterized by the variables {g, b e 1, q, F}, where g is the spatial metric tensor, b e is the elastic left Cauchy Green deformation tensor, and q is a set of internal state variables for the inelastic response. We write the uncoupled free energy potential in terms of the internal variables as φ ( g, b e 1, q, F ) = φ ( g, b e 1, F ) + χ(q) (2) where χ(q) isthe potential for internal hardening. b e = F e F et and b e 1 = F T C i F 1.Itisimportant to note that b e 1 plays the role of inelastic metric tensor. We now consider the following uncoupled hyperelastic stored energy function with uncoupled pressure relative to the intermediate or the unloaded configuration. φ ( g, b e 1, F ) = 1 2 K (log J) µ ( Ī b e 3 ) (3) where Ī b e = b e : g = J 2 3 b e : g = b e ijḡ ij, K is the bulk modulus (K = λ + 2 µ) and µ > 0isthe shear modulus. The 3 Kirchhoff stress tensor is obtained via the hyperelastic stressstrain relation τ = K logj g + µ dev[ b e ] (4) where b e = J 2 3 b e, i.e., elastic left Cauchy Green deformation tensor defined with respect to the intermediate configuration, and dev[ ] = ( ) 1 tr ( )g is an operator that represents the deviatoric part of the indicated argument in the spatial configuration. 3 The general form of von Mises yield condition, which is being used here as the transformation condition, can be expressed as f ( g, b e 1, q, F ) := s 2/3 κ(ē i ) 0 (5) A schematic diagram of material, spatial and intermediate configu- FIG. 1. rations. where ē i is the equivalent inelastic flow. The stress deviator is s = dev[τ], and its norm is defined as s = s : s. κ(ē i ) represents the transformation surface for polymers and is assumed to represent the state of stress at a material particle during transition from the initially unoriented state into an oriented one. Figure 2 shows a schematic diagram of the double glass transition. σ α dr and σγ dr stand for the true drawing stress in the initial and the drawn state, i.e., σ α dr is the initiating stress for neck formation, while σ γ dr is the yield stress at which material completes transformation to the oriented state. The engineering draw stress that sustains a stable neck propagation is constant. The intercept of the two stable branches α and γ with the horizontal axis is termed as the natural draw ratio. The transformation functions or the yield surfaces for the two material types are proposed as follows:

3 3-D MODEL OF COLD DRAWING IN ENGINEERING THERMOPLASTICS 459 where γ is the consistency parameter and f is the transformation condition. The persistency condition is γ f ( g, b e 1, q, F ) = 0 (10) We can write the rate equation for the evolution of internal variable vector q in the spatial description as the derivative of q relative to flow associated with the spatial velocity field. L v q = γ h ( g, b e 1, q, F ) (11) FIG. 2. A schematic diagram of the state transition phenomenon. Transformation function for polycarbonates (amorphous polymer) κ PC (ē i ) = σ α dr + ( σ γ dr dr) σα (1 exp( δ1 ē i )) ( σ σ γ ) dr (1 exp( δ1 ē i )) σ γ dr exp( σ δ 2 (ē i ln(λ n ))) where σ α dr, σγ dr, σ and λ n are material specific constants for polycarbonate. Transformation function for Polethylene (semi-crystalline polymer) κ PM (ē i ) = σ α dr + ( σ γ dr dr) σα (1 exp( δ1 ē i )) + ( σ γ dr ) σα dr exp (δ (ēi )) ln(λ n ) 2 ln(λ n ) where once again σ α dr, σγ dr and λ n are material specific constants for polyethylene. In Eqs. (6) and (7), λ n is the maximum residual strain under uniaxial state, also termed as the natural draw-ratio. δ 1 and δ 2 are the constitutive parameters. σ in (6) represents the maximum asymptotic stress that a material particle can develop in the oriented state. Remark: The micro-mechanics of cold drawing in semicrystalline polymers is different from that of amorphous polymers [15]. However, the objective of the present work is to address the issues related to large strains that arrise at the continuum level in this class of materials, and to present a corresponding numerical integration scheme to model cold drawing in this general class of materials. The loading-unloading conditions can be expressed in Kuhn- Tucker form as (6) (7) γ 0, f ( g, b e 1, q, F ) 0 (8) γ f ( g, b e 1, q, F ) = 0 (9) where h(g, b e 1, q, F) isthe gradient of the transformationsurface/hardening law. Since in the present study we are using the von Mises criteria, the evolution equation for inelastic flow is expressed as ē i = 2/3 γ (12) The associative flow rule obtained via maximum plastic dissipation (see, e.g., Simo and Hughes [18]) can be expressed in the spatial description as the Lie derivative of τ i L v τ i = µdev[l v b e ] = 2 3 µ tr[ b e ] γ n (13) where n = s/ s is the normal to the transformation surface. As mentioned earlier, the continuum based theories for inelasticity presume the existence of a flow rule. On the other hand the constitutive theories based on the notion of multiple-natural stress-free configurations of the materials involve the specification of an evolution equation for an internal variable that in turn defines the evolution of the inelastic component of the deformation gradient. The numerical scheme being presented here can easily be applied to the latter framework by defining a scalar internal variable δ that represents the fraction of the drawn or transformed state. Due to the interpretation of δ, werequire that 0 δ 1. The two limit cases are { 0 the material is in the undrawn state (α-state) δ = 1 the material is in the fully drawn state (γ-state) (14) A simple definition of this bounded, nondimensional internal parameter δ(ē i ) (0, 1) which is a function of the inelastic flow ē i is presented as follows. δ(ē i ) = 1 (ln(λ n) ē i ) ln(λ n ) (15) where λ n is the natural draw ratio, which is a material property. Following Rajagopal et al. [20, 21], we can define the kinetic

4 460 A. MASUD constitutive equation for the evolution of the gradient of this transformation G as G δ = (1 δ(ē i ))I + δ(ē i )G (16) G δ now plays the role of the incremental inelastic deformation gradient F i. Remark: It is important to note that the state described by δ(ē i ) = 1 is an important state in the evolution of the micromechanical changes taking place in the material and is designated as δ in Figure CONSTITUTIVE INTEGRATION SCHEME 3.1. Operator Splitting Methodology We have adopted an operator splitting methodology that results in an elastic-predictor inelastic-corrector product formula algorithm [18, 22 24]. In the elastic problem the inelastic flow is assumed frozen. Furthermore the rate of deformation tensor d in the spatial configuration is obtained as the Lie derivative of the total spatial strain tensor e relative to the flow associated with the spatial velocity field v. Consequently, L v e = d (17) L v e i = 0 (18) L v q = 0 (19) In view of Eqs. (17) (19) the elastic algorithm reduces to a geometric update in which the intermediate configuration stays fixed. At this point it is checked if the stress state is admissible. If the stress lies in the admissible regime then we have the solution at the current time. In such a situation we ignore the second part of the algorithm, update the state variables and go to the next load level. In case the state is inadmissible and violates the Kuhn-Tucker complementary conditions, i.e., Eqs. (8) and (9), then a correction to the stress needs to be applied. In the inelastic corrector phase, the elastic deformation is frozen and the inelastic corrector takes place at a fixed current configuration. Consequently, the Lie derivative reduces to ordinary time differentiation. The inelastic problem can be written as L v e = 0 (20) L v e i = γ n ( g, b e 1, q, F ) (21) L v q = γ h ( g, b e 1, q, F ) (22) f ( g, b e 1, q, F ) = 0 (23) where n is the normal to the transformation surface, and h is the gradient of the transformation/hardening law. Equation (21) is the statement of the flow rule for the inelastic component of the rate of deformation tensor, and (22) is the hardening law for the inelastic internal variables q Numerical Integration Algorithm We employ a further multiplicative decomposition of the deformation gradient to achieve a completely uncoupled volumetric-deviatoric response throughout the entire range of deformation. Our treatment of the isochoric deviatoric response follows Simo et al. [25]. Let J = det F F = F e F i = J 1 3 F (24) where F e = J 1 3 I and F i F = J 1 3 F with det F = 1. Remark: In our recent work [26, 27] we have developed stabilized finite element formulations that successfully address the issue of incompressibility constraint without employing any of the special treatments commonly used in the literature. In our follow up work, this multiscale/stabilized formulation will be extended to multiplicative finite strain regime to model cold drawing in thermoplastics. We partition the time interval [0, T ] into an ordered series of time levels. A typical time increment is defined as t = t n+1 t n.weassume that we know the solution at time t n, and we want to compute the solution at t n+1 when subjected to a given incremental displacement u n+1.weintegrate the material description of the flow rule via a backward Euler integration scheme. It is then substituted in the hyperelastic stressstrain relations and pushed forward to the current configuration. The main steps in the integration algorithm are described below Step 1: Configuration Update The incremental deformation map is obtained as ϕ n+1 = ϕ n + u. Consequently, the deformation gradient at t n+1 is obtained as F n+1 = F u F n (25) where F n is the deformation gradient at time t n and F u = [1 + n u] = F n+1 F 1 n is the incremental deformation gradient from configuration n to n Step 2: Elastic Predictor The elastic algorithm is a geometric update in which the intermediate configuration stays fixed. The elastic predictor based on the intermediate configuration is b etr n+1 := F u b e n F T u (26) The trial deviatoric stress in which the inelastic flow is assumed frozen and only geometric change is permitted is

5 3-D MODEL OF COLD DRAWING IN ENGINEERING THERMOPLASTICS 461 obtained as s tr ] etr n+1 = µdev[ b n+1 (27) At this point it is checked if the stress state lies within the permissible domain, i.e., f tr n+1 := s tr n+1 2/3κ ( ēn) i 0 (28) If the state satisfies the inequality then we set ( ) n+1 = ( ) tr n+1 and skip the second part of the algorithm. However, if the state violates the yield condition then a correction is applied as follows Step 3: Inelastic Corrector Consistency condition (8) requires that the state must lie on the transformation surface. This condition is enforced via the radial return scheme which enforces the constraint in accordance with the Kuhn Tucker optimality conditions (8) (9). In the incremental form the normal n n+1 can be computed from the predictor (trial) values. str n+1 n n+1 = (29) s tr n+1 We now need to compute Ī b e = b e : g = J 2 3 b e : g = 1 etr tr[ b 3 n+1 ]. We set tr [ b e ] ] n+1 = tr [ b etr n+1 (30) where tr[ ] is the trace operator. Using Eq. (30) we can now determine µ in terms of the trial (predictor) state TABLE 1 Material properties for Polycarbonate (Source Zhou et al. [13]) Youngs modulus E 2.2 GPa Poisson s ratio ν 0.3 Draw stress σ α dr 48 MPa Yield stress σ γ dr 60 MPa Ultimate stress σ 120 MPa Draw ratio λ n 1.7 Density ρ 1200 kg/m 3 Exponential coeff. δ 1 3 Exponential coeff. δ NUMERICAL RESULTS As mentioned earlier, emphasis in the present article has been on the implementation of the constitutive model in a threedimensional finite element setting. This section presents numerical simulation of 3-D neck propagation and orientational hardening in polycarbonate and polyethylene rods Example 4.1: Cold-Drawing in Polycarbonate This test problem presents a numerical simulation of neck propagation in a circular rod under displacement control. Table 1 presents the material properties for the polycarbonate at temperature 23 C and for a strain rate of s 1. µ n+1 := 1 3 ] etr µ tr[ b n+1 (31) In our continuous-in-time model, γ is the consistency parameter. In the incremental form this consistency parameter γ n+1 can be computed via a local Newton iteration of the following nonlinear equation (ēi 2/3κ n + ) 2/3γ n+1 s tr n µγn+1 = 0 (32) Once we obtain γ n+1, the updated deviatoric stress and equivalent inelastic strain can be expressed as s n+1 = s tr n+1 2 µ n+1γ n+1 n n+1 (33) ē i n+1 = ēi n + 2/3γ n+1 (34) Having obtained γ n+1 and µ n+1, the intermediate configuration is updated as b e n+1 = b etr n+1 2 µγ n+1 etr tr[ b 3µ n+1] nn+1 (35) Consequently, it turns out that the variables { b e, ē i } constitute the basic database for the algorithm presented above. FIG. 3A. FIG. 3B. FIG. 3C. FIG. 3D. FIG. 3E. 22 percent axial strain. 44 percent axial strain. 66 percent axial strain. 88 percent axial strain. 110 percent axial strain.

6 462 A. MASUD The ratio of length to radius dimensions in the computational specimen are 4:1. Invoking symmetry, only one eighth of the problem is discretized. The computational mesh is composed of 240 eight-node brick elements. Necking is initiated by providing 0.8% reduction in the radius of the rod at one end. Figures 3A 3E present the deformation history of the specimen. For comparison, we have also plotted the outline of the original undeformed configuration. As can clearly be seen in Figures 3A 3C, once the material reaches its draw ratio, further straining of the drawn (transformed) material ceases and the neck starts moving toward the undrawn material in a steady state. In Figure 3D the specimen has undergone 88% axial stretch and neck has propagated over the entire length of the specimen. Figure 3E indicates the initiation of localization in the transformed material that, infact, represents the failure of the material. Figures 4A 4E present the through thickness and surface distribution of the axial stress during the process of neck propagation. We have also performed numerical studies of the sensitivity of the calculations to temporal and spatial refinements. Figure 5 shows the processes of initiation, evolution and stabilization of the neck. The horizontal axis represents the normalized elongation of the test specimen (where normalization is done with respect to the original undeformed length) while the vertical axis represents the normalized radius of the neck (where normalization is done with respect to the initial radius). In order to study the effects of spatial refinement on the response of the model, the pseudo time-step is kept constant for the various mesh configurations and the total displacement is applied in 200 steps. Neck is fully developed at around 20% elongation of the specimen and it stays stable until the entire specimen undergoes state FIG. 4A. 22 percent axial strain. FIG. 4B. 44 percent axial strain.

7 3-D MODEL OF COLD DRAWING IN ENGINEERING THERMOPLASTICS 463 FIG. 4C. 66 percent axial strain. FIG. 4D. 88 percent axial strain. FIG. 4E. 110 percent axial strain. transformation (as shown in Figures 4A D). Once the entire rod has transformed, a further increase in loading causes localization in the oriented material which infact indicates the failure of the material. Here this failure is depicted via the appearance of the second neck at around 95% normalized stretch. (To actually model fracture in the material, one would need to include an appropriate fracture model as well). Figure 6 presents a numerical study of the sensitivity of the formulation to temporal refinement. In this test we have used the intermediate mesh with 240 brick elements. Once again we see a very stable response for

8 464 A. MASUD FIG. 5. Neck propagation. Numerical study of the sensitivity of calculations with respect to mesh refinement. the initiation, development, propagation and stabilization of the neck for various time steps. Table 2 presents history of the Newton iterations required in the various time steps to achieve convergence for the 240 element mesh with the largest step size, i.e., t = Table 3 presents the reduction in residual norm for some typical time steps in this simulation which clearly show the quadratic rate of convergence of the scheme Example 4.2: Comparative Study of Neck Stabilization In this test case we have performed comparisons with the published results in the literature. Figure 7 presents the process FIG. 6. Neck propagation. Numerical study of the sensitivity of calculations with respect to temporal refinement. of initiation, propagation and stabilization of the neck for the 240 element mesh case. Total load is applied in 200 equal steps. Material properties for this simulation are taken from Neale et al. [4]. The constitutive parameters are selected such that the transformation surface for the proposed model conforms to the transformation surface given in [4]. We have also plotted the axial strain ε xx (0, 0) which also shows a good correlation with the published results. Figure 8 presents a comparison of the normalized neck with Tomita et al. [5] for 270 K temperature. Material properties have been taken from [5]. Once again we have used the intermediate mesh with 240 element, which is in fact a very crude mesh for this type of problem. Still a decent correlation with the published results is attained. TABLE 2 Average number of iterations for various time steps Step 1 Step 2 Step 3 14 Step Step Step No. of Iter TABLE 3 Residual norm for various steps Iter. No. Step 1 Step 2 Step 11 Step 40 Step E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 08

9 3-D MODEL OF COLD DRAWING IN ENGINEERING THERMOPLASTICS 465 FIG. 9. True-stress vs. natural-strain plot. FIG. 7. Normalized neck and axial strain, Neale et al. [4]. of s 1. Figure 9 presents the true-stress versus the natural-strain response that conforms to the material properties given in Table 1 together with the definition of the transformation surface given by Eq. (7) (see [28]). Figure 10 presents the normalized neck for the entire range of deformation. We have used three different meshes to study the effect of spatial refinement which shows virtually same response in the stabilized regime. Figures 11A 11E present the deformation history of the specimen together with the surface stress distribution of axial stress during neck propagation. Once the material reaches its draw ratio, further straining of the drawn (transformed) material ceases and the neck simply starts moving toward the undrawn material at essentially a constant stress. FIG. 8. Normalized neck, Tomita et al. [5] Example 4.4: Stretching of a Polyethylene Plate with a Cylindrical Hole The last numerical example presents a 3-D stress state and simulates the stretching of a polyethylene plate with a 4.3. Example 4.3: Cold-Drawing of Polyethylene This numerical simulation presents neck propagation in a polyethylene rod. Table 4 presents the material properties for the polyethylene at temperature 25 C and for a strain rate TABLE 4 Material properties for Polyethylene (Source Chudnovsky et al. [28]) Youngs modulus E 50 MPa Poisson s ratio ν 0.3 Draw stress σ α dr 10 MPa Yield stress σ γ dr 40 MPa Draw ratio λ n 5.3 Exponential coeff. δ Exponential coeff. δ 2 10 FIG. 10. Normalized neck.

10 466 A. MASUD FIG. 11A. 90 percent axial strain. FIG. 11B. 180 percent axial strain. FIG. 11C. 270 percent axial strain. FIG. 11D. 360 percent axial strain. FIG. 11E. 450 percent axial strain. cylindrical hole. Material properties for this test case are presented in Table 4. Due to symmetry, only one-fourth of the plate is discretized and appropriate boundary conditions are imposed along the cut surfaces. The mesh is composed of two hundered eight-node brick elements, and the initial mesh is shown in Figure 12. Figures 13A D present the deformation history along with the superposed axial stress. Once again a stable response is attained in the entire range of deformation. FIG. 12. Initial undeformed mesh. (200 eight-node bricks).

11 3-D MODEL OF COLD DRAWING IN ENGINEERING THERMOPLASTICS 467 FIG. 13A. 10 percent axial stretch. FIG. 13B. 50 percent axial stretch. FIG. 13C. 100 percent axial stretch.

12 468 A. MASUD FIG. 13D. 200 percent axial stretch. 5. CONCLUSIONS We have presented a 3-D multiplicative finite strain framework for the modeling of cold drawing in engineering thermoplastics. The consitutive model is based on a hyperelastic extension of the J 2 flow theory. The notion of the intermediate (local) stress-free configuration, that emanates from a local split of the deformation gradient, is shown to play an important role in the development of the constitutive model. The constitutive integration scheme is based on the notion of operator splitting methodology, leading to a product formula algorithm. The constitutive equations are implemented in a finite element framework using an eight-node hexahedral element. The isochoric constraint is treated via a further decomposition of the deformation gradient into volume-preserving and spherical parts. The proposed numerical solution procedure can equally be applied to constitutive models that, instead of postulating a flow rule, employ the notion of multiple natural stress-free configurations of the material. Numerical tests show the accuracy and the stability of the formulation for modeling the extremely large strains in this class of materials. ACKNOWLEDGEMENTS The author wishes to thank Professor A. Chudnovsky for many helpful discussions. Partial support for this work was provided by DOW Chemical Company, and a grant from NSF-CMS # REFERENCES 1. A. Masud and A. Chudnovsky, A Constitutive Model of Cold Drawing in Polycarbonates, Internat. J. of Plasticity, vol. 15, no. 11, pp , A. Masud, A Multiplicative Finite Strain Finite Element Framework for the Modeling of Semicrystalline Polymers and Polycarbonates, Internat. J. Numer. Methods Engrg., vol. 47, no. 11, pp , J. W. Hutchinson and K. W. Neale, Neck Propagation, J. Mech. Phys. of Solids, vol. 31, no. 5, pp , K. W. Neale and P. Tugcu, Analysis of Necking and Neck Propagation in Polymeric Materials, J. Mech. Phys. of Solids, vol. 33, no. 4, pp , Y. Tomita and K. Hayashi, Thermo-Elasto-Viscoplastic Deformation of Polymeric Bars Under Tension, Internat. J. Solids Structures, vol. 30, no. 2, pp , M. C. Boyce, D. M. Parks, and A. S. Argon, Large Inelastic Deformation of Glassy Polymers, PartI: Rate-Dependent Constitutive Model, Mech. of Materials, vol. 7, pp , Part II: Numerical Simulation of Hydrostatic Extrusion, Mech. of Materials, vol. 7, pp , M. C. Boyce, E. M. Arruda, and R. Jayachandran, The Large Strain Compression, Tension, and Simple Shear of Polycarbonate, Polymer Engineering and Science, vol. 34, pp , E. M. Arruda and M. C. Boyce, Evolution of Plastic Anisotropy in Amorphous Polymers During Finite Straining, Internat. J. of Plasticity, vol. 9, pp , E. M. Arruda, M. C. Boyce, and H. Quintus-Bosz, Effects of Initial Anisotropy on the Finite Strain Deformation Behavior of Glassy Polymers, Internat. J. of Plasticity, vol. 9, pp , I. J. Rao and K. R. Rajagopal, Simulation of the Film Blowing Process for Semicrystalline Polymers, Mechanics of Advanced Materials and Structures, vol. 12, no. 2, pp , A. Rao and J. N. Reddy, Computational Study of Shear-Induced Crystallization in Polymers, Numerical Heat Transfer, Part A: Applications, vol. 34(4), pp , G. T. Lim, H.-J. Sue, M. Wong, A. Moyse, and J. N. Reddy, Mechanical Modeling and Experimental Observation of Surface Damage Phenomena of Polymers, Proceedings of the International Conference on Polyolefins, pp , Houston, Texas, USA, Z. Zhou, A. Chudnovsky, C. P. Bosnyak, and K. Sehanobish, Cold Drawing (Necking) Behavior of Polycarbonates as a Double Glass Transition, Polymer Engrg. & Science, vol. 35, no. 4, pp , A. Kim, L. V. Garrett, C. P. Bosnyak, and A. Chudnovsky, Kinetics and Characterization of the Process-Zone Evolution in Polycarbonates, J. App. Polymer Science, vol. 49, pp , I. M. Ward and D. W. Hadley, An Introduction to Mechanical Properties of Solid Polymers, John Wiley & Sons, Chichester, 1993.

13 3-D MODEL OF COLD DRAWING IN ENGINEERING THERMOPLASTICS J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, New Jersey, 1983, and Dover, New York, T. Belytschko, W. K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons Ltd., Chichester, J. C. Simo and T. J. R. Hughes, Computational Inelasticity, Springer-Verlag, New York, NY, J. C. Simo, On the Computational Significance of the Intermediate Configuration and Hyperelastic Stress Relations in Finite Deformation Elastoplasticity. Mech. of Materials, vol. 4, pp , K. R. Rajagopal and A. R. Srinivasa, On the Inelastic Behavior of Solids Part I: Twinning, Internat. J. of Plasticity, vol. 11, no. 6, pp , K. R. Rajagopal and A. R. Srinivasa, On the Inelastic Behavior of Solids Part II: Energetics Associated with Discontinuous Deformation Twinning, Internat. J. of Plasticity, vol. 13, no. 1/2, pp. 1 35, M. Ortiz and J. C. Simo, An Analysis of a New Class of Integration Algorithms for Elastoplastic Constitutive Relations, Internat. J. Numer. Methods Engrg., vol. 23, pp , J. C. Simo, A Framework for Finite Strain Elastoplasticity Based on Maximum Plastic Dissipation and Multiplicative Decomposition: Part I. Continuum formulation, Comput. Methods in Appl. Mech. Engrg., vol. 66, pp , J. C. Simo, A Framework for Finite Strain Elastoplasticity Based on Maximum Plastic Dissipation and Multiplicative Decomposition: Part II. Computational Aspects, Comput. Methods in Appl. Mech. Engrg., vol. 68, pp. 1 31, J. C. Simo, R. L. Taylor, and K. S. Pister, Variational and Projection Methods for the Volume Constraint in Finite Deformation Elastoplasticity, Comput. Methods in Appl. Mech. Engrg., vol. 51, pp , A. Masud and K. Xia. A New Mixed Finite Element Method for Nearly Incompressible Elasticity, Journal of Applied Mechanics,vol. 72, September A. Masud and K. Xia. A Variational Multiscale Method for Computational Inelasticity: Application to Superelasticity in Shape Memory Alloys, Comput. Methods in Appl. Mech. Engrg. In press. 28. A. Chudnovsky, A. Kim, T. Chen, K. Sehanobish, C. P. Bosnyak,and C. Kao, Thermodynamics of Cold Drawing Phenomena in semicrystalline polymers, Mechanics of Phase Transformations and Shape Memory Alloys, AMD-vol. 189/PVP-vol. 292, 1994.

A constitutive model of cold drawing in polycarbonates

A constitutive model of cold drawing in polycarbonates International Journal of Plasticity 15 (1999) 1139±1157 A constitutive model of cold drawing in polycarbonates Arif Masud*, Alexander Chudnovsky Department of Civil and Materials Engineering, The University

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