A constitutive model of cold drawing in polycarbonates

Transcription

1 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 of Illinois at Chicago, Chicago, Illinois , USA Received in nal revised form 3 May 1999 Abstract This paper presents a set of constitutive equations to model cold-drawing (necking) in polycarbonates (PC). The model is based on a representation of cold drawing as a double glass transition, i.e., a transition from a glass into a rubbery state, when a certain yield surface in the stress space is reached, and a transition back to the glassy state upon unloading or when a certain molecular orientation (draw ratio) is achieved. The stretching process in the rubbery state is modeled by a hyperelastic extension of the J 2 - ow theory to the nite strain range. An appropriate yield surface and an associative ow rule (de ned via the Kuhn±Tucker optimality conditions) are presented to simulate this process in polycarbonates. The isochoric constraint during double glass transition is treated via an exact multiplicative decomposition of the deformation gradient into volume preserving and spherical parts. Numerical constitutive integration algorithm is based on an operator splitting technique where constraint/consistency during inelastic deformation is enforced via return mapping algorithm. Numerical results are presented to demonstrate the correspondence with the experimental data. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Polycarbonate; Double glass transition; Orientational hardening; Multiplicative nite strain framework 1. Introduction The understanding of physical behavior of engineering thermoplastics and development of appropriate constitutive equations for the analysis and design of load * Corresponding author. Fax: address: amasud@uic.edu (A. Masud) /99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S (99)

2 1140 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139±1157 bearing elements continues to be of considerable interest for engineering applications. Polycarbonate (PC), a transparent amorphous thermoplastic has received considerable attention because of its high impact resistance, and recognition as a load bearing material due to su ciently high glass transition temperature. The characteristics of deformation around a notch in polycarbonates, shear banding at short times and high loads, and micro-cracking at low stresses and long times has been reported by various investigators (see e.g. Boyce and Arruda, 1990; Kim et al., 1993; Stokes and Bushko, 1993; Zhou et al., 1995, and references therein). Over the years, various constitutive models have been proposed to describe this phenomena (see e.g., Hutchinson and Neale, 1983; Neale and Tugcu, 1985; Boyce et al., 1988, 1994; Arruda and Boyce, 1993, Arruda et al., 1993, Tomita and Hayashi, 1993; Wu 1995, and references therein). In a relatively recent development, a novel approach to the modeling of materials that possess multiple natural stress-free con gurations has been presented by Rajagopal and Srinivasa (1995, 1997) and Masud et al. (1997). In these works the crucial role played by the intermediate con guration in the development of constitutive theories is highlighted. These works are based on the notion of existence of an evolution equation for an internal variable that provides the link between the multiple natural stress-free states of the material. This evolution equation in turn de nes the evolution of the inelastic component of the deformation gradient, and therefore, an explicit expression for the ow rule is not required. A thermomechanical analysis of large-scale deformation of polycarbonates has been performed by Chudnovsky and co-workers (see Zhou et al., 1995) where the tensile behavior of polycarbonates is examined as a function of temperature and strain rate. In Zhou et al., (1995) it is proposed that necking phenomena are a special type of transformation involving a three step process, (i) transformation from an isotropic glass to an isotropic rubber when the yield surface in the stress space is reached during loading, (ii) a stretching of the rubbery state, and (iii) a transformation of the stretched rubbery state into an oriented glass upon unloading, or when a certain molecular orientation (draw ratio) is achieved. The above process is named as double glass transition (DGT). Fig. 1 presents a schematic diagram of DGT. The material is initially in a homogeneous, isotropic, glassy state. As it is stretched, it undergoes a phase transition from isotropic glassy state to isotropic rubbery state. This is a nonlinear elastic deformation. Once a material particle has reached isotropic rubbery state, further stretching converts the isotropic rubber to an oriented rubber. During this transformation the initial lamellar structure is deformed and fragmented to give a micro brillar structure. We term it as rubbery state since the infrared observations by Zhou et al. (1995) reveal that a temperature increase accompanies the stretching, similar to that in elastomers. In our constitutive model this transformation is modeled via J 2 ow theory employing operator splitting technique. Since this transition is rather instantaneous and is a metastable state, the proposed stress-transformation surface together with the operator splitting technique seems appropriate and serves as a vehicle to take material particle from one stable branch to the other in the true stress true strain space (see Fig. 4). Once the material has reached the second stable branch termed as oriented rubber, it again

3 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139± Fig. 1. Schematic of double glass transition. behaves elastically. However, unlike rubber, on unloading it does not go to zero strain state, rather it releases some elastic strain and then freezes the remaining strain, thus transforming to an oriented glassy state. The objective of this paper is to develop a set of constitutive equations that can model the above described ``double'' glass transition in polycarbonates. In order to keep the model simple, we are postponing the consideration of thermal and viscoelastic e ects to a follow-up paper. An outline of the paper is as follows. Section 2 gives a brief synopsis of the experimental observations of the necking behavior and presents experimentally obtained stress±strain plots. Section 3 presents strain measures, ow rule and a set of constitutive equations. Section 4 discusses the operatorsplitting methodology employed for the numerical integration of constitutive equations. Numerical simulation of necking in uniaxial tension is reported in Section 5 and conclusions are drawn in Section Experimental observations The present work is based on an experimental study conducted on two polycarbonates with di erent molecular weights. Dumbbell tensile test specimens were machined from the compression molded sheets of polycarbonates as well as from the necked portion of the cold drawn specimens. These dumbbell specimens were pulled with initial strain rates of four orders of magnitude range from s 1 to 3.0 s 1 at temperatures in the range of 23±115 C ( 0.5 C). To obtain very low strain rates, the specimens were held at various constant loads and the strain rate was taken as the reciprocal of the time to necking. At a temprature of 23 C and under very low strain rates, the draw-stress leveled o at approximately 48 MPa. This value is termed as the drawing stress or the equilibrium yield stress. The natural draw-ratio l n is

4 1142 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139±1157 de ned as the ratio of the length of the drawn part of the specimen upon unloading to its original length. Remark 1: It was observed that l n increases only slightly from 1.65 to 1.72 over the temperature range from 23 to 115 C. Consequently, in the present model we ignore the weak temperature dependence of the draw ratio l n. Fig. 2 shows the engineering stress±strain curve for PC at 23 C and at the initial strain rate of s 1. The specimen was unloaded and then reloaded three times during the necking process. The engineering yield stress y is taken as the maximum of the engineering stress at low strains. This stress is associated with the point at which a signi cant amount of micro-shear band formation occurs (Ma et al., 1989; Kim et al., 1993; Zhou et al., 1995). On further straining, the shear bands coalesced to form a well-de ned neck. Necking then continues under essentially constant drawing stress dr by transformation of undrawn material into the drawn state across the neck boundary. The strain at which necking began, " dr, was obtained by extrapolation of dr to the engineering stress±strain curve before y. The unloading and reloading test showed that the initial recovery on unloading was fast, followed by a delayed recovery, as expected for a visco-elastic material. The subsequent reloading showed that a small amount of yielding occurred before neck growth resumed. The amount of yielding observed on reloading is proportional to the amount of strain recovery allowed in the specimen. Finally, as the neck propagated through the entire narrow portion of the specimen an upturn in the engineering stress was observed. Only a part of the engineering stress±strain curve between the origin and yield point re ects the tensile behavior of the original (undrawn) PC. The rest of engineering stress±strain curve shown in Fig. 2 does not represent true material behavior since it Fig. 2. The engineering stress±strain response for polycarbonates (PC) at 23 C at an initial strain rate of s 1.

5 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139± is obtained from a heterogeneous specimen consisting of the undrawn and drawn materials. Thus it is more convenient to discuss the deformation of PC from a material particle perspective, in contrast with the specimen geometry-dependent representation of the engineering stress±strain curves. The actual path dependency of the strain in the process of drawing could not be determined experimentally because of the rather spontaneous nature of transformation. To reconstruct the true stress±strain behavior of PC, the drawn material was subjected to the same testing procedure as the original one. The tensile modulus of the drawn PC is higher than that of the undrawn one by about 25% and there is a signi cant change in the stress±strain response after loading beyond the initial true drawing-stress dr. The behavior of the drawn material during unloading and reloading is also shown in Fig. 3, further exemplifying the di erences in stress±strain curves of drawn and undrawn PC when compared with Fig. 2. A composite picture of PC behavior based on tests shown in Figs. 2 and 3 is given in Fig. 4. The stress±strain behavior of the undrawn and drawn materials are indicated by solid lines, while the dotted line a b is assumed to be the path followed during the glass transformation. Thus, in light of Fig. 4 and as discussed in Zhou et al. (1995) as well, the thermomechanical response of initial and drawn PC can be envisioned as follows. On loading, an isotropic glassy material ( state) reaches yield stress (strain), followed by a transition to a rubbery mesostate. The material then is stretched into an oriented rubbery state. Only when the entire specimen is transformed, a material can be further stretched by continuing along the a b curve of Fig. 4. On unloading, the material transforms from the oriented rubbery state to the oriented glassy state ( state). Remark 2. It has been reported in the literature (Zhou et al., 1995) that on necking the strain of the undrawn material on one side of the neck boundary suddenly decreases Fig. 3. The tensile behavior of the drawn PC.

6 1144 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139±1157 Fig. 4. The process of cold-drawing as transitions!!. from 1.06 to 1.03, whereas in the drawn material the strain jumps to 1.7. The natural draw-ratio l n is, therefore, calculated as 1.7/1.03. Remark 3. Although the in uence of the visco-elastic nature of the PC cannot be completely ignored, all the essential features of the true stress±strain response have been accounted for in this discussion. 3. Constitutive model Let 0 R 3 be the reference placement of a continuum body in the 3D space at time t 0. We refer t =' t () as the current placement of the body. The deformation map ' t :!R 3 is a transformation from X2 to x in 2 t. Motivated by the above mechanical description of PC, i.e. an amorphous engineering thermoplastic, we assume a local multiplicative split of the deformation gradient FX into elastic and inelastic (or plastic) components. (see e.g. Lee, 1969; Nemat-Nassar, 1983). F ˆ F e F i where F e is the elastic component of the deformation gradient while F i is the inelastic component. The inelastic component F i de nes an additional or intermediate con guration t as shown in Fig. 5, and is designed such that it enforces the isochoric constraint during inelastic deformations. Furthermore, F e 1 is assumed to unload elastically the stresses on the neighborhood ' O X in the current con- guration. We denote the quantities in the reference con guration by upper case letters, in the intermediate con guration with superposed bars and in the current con guration by lower case letters. 1

7 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139± Fig. 5. Schematic diagram of material, spatial and intermediate con guration. It is important to realize that like many other engineering materials, polycarbonate can exists in two di erent stress-free (natural) con gurations, i.e. the a- state and the g-state in Fig. 4. In either state, it is a stable material. As pointed out in the introduction, a theory for stress-induced phase transformation for materials that possess multiple natural con gurations has been presented by Rajagopal et al. (1995, 1997). We wish to state that the present developments accommodate this notion of multiple stress-free (natural) con gurations. To see this, we refer to Fig. 4 and de ne a deformation map '^ that represents a complete micromechanic transformation of the material from the isotropic glassy state, i.e. the a-state to the oriented glassy state i.e. the g-state. The gradient of this transformation is represented as G. We will utilize this information in Section 3.1 to establish the link between the present development and the theory of multiple natural con gurations by Rajagopal et al. (1995, 1997). We now present the constitutive model in the spatial framework. The strain tensors associated with the spatial con guration are e ˆ 1 g b 1 ; ee ˆ 1 g be

8 1146 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139±1157 where e and e e are the total and elastic spatial strain tensors, g is the spatial metric tensor, while b ˆ FF T and b e =F e F et are the total and elastic left Cauchy-Green tensors, respectively. The inelastic or plastic strain tensor e i associated with the spatial description can now be de ned as e i ˆ e e e ˆ 1 2 be 1 b 1 3 where b e 1 plays the role of inelastic/plastic metric tensor in the spatial con guration. We employ Lagrange multipliers and optimality conditions along with an associative ow rule to model the inelastic response in the polycarbonates. The constraint conditions then follow from the principle of maximum plastic dissipation as the classical Kuhn±Tucker optimality conditions. We assume that the thermodynamic state is characterized by the variables fg; b e 1 ; q; Fg, where q is a set of internal state variables characterizing the inelastic (or plastic) response. In addition, we assume an uncoupled free energy potential in the internal variables q of the form g; b e 1 ; q; F ˆ g; b e 1 ; F q 4 The uncoupled hyperelastic stored energy function with uncoupled pressure relative to the intermediate or the unloaded con guration is de ned as g; b e 1 ; F ˆ UJ 1 2 I b e I b e 3 ˆ 1 2 K log J where I b e ˆ J 2 3 b e : g ˆ b eij gij; K is the bulk modulus and >0 is the shear modulus. The Kircho stress tensor can now be obtained g; b e 1 ; F ˆ 6 K log Jg J 2 3 dev b e Š 7 s ˆ dev Š 8 where dev Š ˆ 1 3 tr g represents the deviatoric part of the indicated argument in the spatial con guration. The Von Mises yield condition can be written as f p g; b e 1 q; F :ˆ kk s i 2=3 e 40 9 where q ˆ e i ;e i p is the equivalent inelastic (or plastic) ow and kkˆ s s : s. Using Eq. (8) in (9) we get the yield condition expressed in the strain space as 5

9 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139± f p g; b e 1 ; q; f ˆ J 2 3 devb e k k 2=3 e i The loading/unloading conditions can be expressed in Kuhn±Tucker form as : 50 f g; b e 1 ; q; F : f g; b e 1 ; q; F ˆ 0 12 where g is the consistency parameter and Eq. 12 represents the consistency condition. The evolution equation for the equivalent inelastic ow is e :i ˆ p : 2=3 The associative ow rule obtained from maximum plastic dissipation (see e.g. Lubliner, 1984) can be expressed as J 2 3 dev L b e Š ˆ 2 3 J 2 3 tr b e Š : n where n ˆ s= kkis s the normal to the yield surface and trace of the Lie derivative of b e i.e. tr L b e Š ˆ 0 (see e.g. Simo 1988a,b; Simo and Hughes 1998). Based on our experimental observations of the behavior of PC (as described in Section 2), we present the following state transformation/hardening law that represents the state of stress at a material particle during transition from the initial unoriented state into the oriented one, (i.e. during state in Fig. 4). e i ˆ dr dr dr 1 exp 1 e i dr dr exp 2 e i ln l n where dr, dr, 1, 2 and l n are material speci c constants. l n is related to the socalled maximum residual strain under uniaxial state and is termed as the draw-ratio. It has been shown experimentally by Zhou et al. (1995) that under uniaxial loading conditions, neck formation in the physical specimen occurs at an appropriate initiating stress called the draw-stress dr. Furthermore, dr represents the true draw stress, i.e. dr = l n dr. It is the stress that a material particle develops once it has reached the second stable branch (i.e. the end of state in Fig. 4). A further increase in the stress and strain at the material particle takes place only when the entire specimen undergoes complete transformation, i.e. the neck propagates over the entire specimen. At this point further straining in the transformed material takes place only with a further increase in applied stress. 1 and 2 are the constitutive parameters that help in mapping the transformation/hardening law precisely and accurately onto the experimentally obtained transformation surface as sketched in Fig

10 1148 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139±1157 Remark 4. It is important to note that 1 and 2 are functions of the moduli of elasticity of the original and the transformed material and the slope of the transformation surface, i.e. state in Fig Relation between the unloaded and the natural con gurations In Fig. 4, we introduced a deformation map '^, that represents a complete micromechanic transformation of the material from the isotropic glassy state to the oriented glassy state. Following Rajagopal et al. (1995, 1997), we de ne the kinetic constitutive equation for the evolution of the gradient of this transformation G as G ˆ 1 I G 16 where is a scalar internal variable that represents the fraction of the drawn or transformed state. Due to the interpretation of, we require that The two limit cases are 0 () the material is the undrawn state -state ˆ 17 1 () the material is in the fully drawn state -state It is important to note the constant strain at a point inside the fully-drawn (transformed) region is an important restriction on the material behaviour. The strain at this point stays constant and equal to draw-ratio ``l n '' until the time the neck propagates over the entire domain. Further straining at this point occurs only after the entire specimen has undergone a complete transition. Accordingly, we introduce a bounded, nondimensional internal parameter e i 2 0; 1 which is a function of the inelastic ow e i de ned as e i 1 ln l n e i 18 ln l n For a general physical problem, a given incremental strain causes a change in the geometry in addition to causing a stress induced glass transformation in the material. Under the assumption that there exists a deformation map '^ such that the incremental deformation gradient G can be approximated by the incremental inelastic deformation gradient F i, we can establish a link between the present model and that of Rajagopal et al. (1995, 1997). With G and e i known, G can be evaluated using Eq. 16. Since our constitutive model is based on the existence of uncoupled free energy potential with an uncoupled hyperelastic stored energy function (5) with respect to the unloaded or the intermediate con guration t, its relation with 0 and can be readily established. Consequently, stress in the current con guration can be written as a function of the inelastic deformation gradients from the two natural con gurations of the material. Remark 5. It is important to note that the state described by =1 is an important state in the evolution of the micromechanical changes taking place in the material and is designated as in Fig. 4.

11 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139± Operator splitting methodology (in an updated Lagrangian framework) We adopt an operator splitting methodology and an updated Lagrangian framework to develop the solution procedure for elastic±inelastic deformation in polycarbonates. This technique results in an elastic-predictor inelastic-corrector product formula algorithm for the numerical integration of the constitutive equations (see Simo, 1985b). We assume that the solution is known for con guration x n ˆ ' n X at time t n, and we want to compute the solution at time t n 1, given incremental displacement u : ' n! R n sd : The updated con guration and the updated deformation gradient are given as ' n 1 ˆ ' n u ' n 19 F n 1 ˆ F u F n 20 where F u ˆ 1 r n u is the incremental deformation gradient from time t n to t n 1. The issue is to update the state variables associated with mapping ' n X to obtain F e n 1 ; F i n 1 ; q n 1; n 1 associated with con guration 'n 1 at time t n 1. In the computational setting, the updating process is strain driven, therefore con guration ' n 1 and the associated deformation gradient F n 1 can be regarded as given. The stress tensor can be regarded as a dependent variable which is to be computed once the nal state of strains e e and e i are obtained. Following Simo and Ortiz (1985), Ortiz and Simo (1986), and Simo and Hughes (1998), we de ne elastic and inelastic/plastic problems via two problems of evolution Part I: elastic predictor In the elastic problem the inelastic (or plastic) ow is assumed frozen. Furthermore the rate of deformation tensor d in the spatial con guration is obtained as the Lie derivative of the total spatial strain tensor e relative to the ow associated with the spatial velocity eld. Consequently, L e ˆ d L e i ˆ 0 L q ˆ In view of (21)±(23) the elastic algorithm reduces to a geometric update in which the intermediate con guration stays xed. In this part of the algorithm the elastic trial stress can be obtained by mere function evaluation. (See Simo, 1985b for details). 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. (11) and (12), then a correction to the stress needs to be applied as follows.

12 1150 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139± Part II: inelastic/plastic corrector (return mapping) The inelastic/plastic corrector is often referred to as return mapping because it enforces the yield condition in a manner consistent with the assumed ow rule. In the inelastic (or plastic) corrector phase L v e 0, i.e. the elastic deformation is frozen and the inelastic corrector takes place at a xed current con guration. Consequently, the Lie derivative reduces to ordinary time di erentiation. The inelastic/ plastic problem can be written as L e ˆ 0 L e i ˆ : n g; b e 1 ; q; F L q ˆ : h g; b e 1 ; q; F ' g; b e 1 ; q; F ˆ where n is the gradient of the inelastic/plastic ow direction, and h is the gradient of the transformation/hardening law. Eq. (25) is the statement of the ow rule for the inelastic component of the rate of deformation tensor, and (26) is the hardening law for the inelastic/plastic internal variables q. Here, both (25) and (26) are formulated in the strain space. In the solution of Eqs. (24)±(27) the total deformation gradient F n 1 and as a result e n 1 are known and remain xed. Algorithms for the solution of (24)±(27) are referred to as return mapping since their purpose is to restore consistency. In this work, we have used a backward Euler implicit formula for numerical integration of (24)±(27). Once we obtain the stress state compatible with the proposed model, we can solve the resulting nonlinear system of equations via the Newton Raphson method or with any of its variants. After obtaining the converged solution for the current load level we update the state variables and go to the next load level The unloaded con guration We can determine the unloaded con guration from the current deformation gradient F n 1 and the updated stain tensors e e n 1 ; ei n 1. This can be accomplished by pull-back via F e n 1. However, the problem of nding F e n 1 requires the speci cation of an orientation of the intermediate con guration. For the anisotropic case a variety of choices for the orientation of the intermediate con guration have been proposed in the literature (Naghdi and Trapp, 1975; Nemat-Nasser, 1983). Nevertheless we restrict ourselves to the situation where the elastic response, though, may be completely anisotropic, but remains una ected by inelastic (plastic) ow. Hence, the elastic response is characterized by a stored energy function relative to the intermediate con guration which is independent of the inelastic (plastic) deformation gradient. Accordingly, the intermediate con guration is updated by locally inverting

13 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139± the stress±strain relations with the elastic part of the deformation gradient alone. Physically, this operation e ectively corresponds to a local unloading of the current con guration. 5. Numerical results The constitutive model presented above is implemented in a 2D nite element framework. Following Simo (1985a) we have used mixed variational approach to deal with the issues associated with incompressibility in the fully integrated elements in the nite deformation regime. In order to test the constitutive equations and the integration algorithm in the nite strain range, we performed a series of uniaxial simulations Test 1: loading/unloading of the material This test case investigates the true-stress true-strain response of the polycarbonate at a typical integration point. Material properties for the polycarbonate are shown Table 1 Material properties for polycarbonate Youngs modulus E 2.2 GPa Poissons ratio 0.3 Draw stress dr 48 MPa Yeild stress dr 60 MPa Draw ratio l n 1.7 Density 1200 kg/m 3 Exponential coe cient 1 3 Exponential coe cient 2 40 Fig. 6. True-stress vs true-strain plot.

14 1152 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139±1157 Fig. 7. Loading-unloading a! b! g states. Fig. 8. Engineering stress vs engineering strain plot. Fig. 9. Problem description.

15 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139± in Table 1, and plane strain conditions are assumed in force. Fig. 6 presents the truestress versus the true-strain response under a displacement control test. This stress± strain response conforms to the material properties given in Table 1 together with the de nition of the transformation/hardening surface as given by Eq. (15). As the strain reaches 70%, i.e. the natural draw ratio of PC, the material is seen to undergo a complete transformation. Now the material is on the second stable branch and further straining results in a sharper increase in the stress value. Fig. 10. (a). Twenty per cent axial strain. (b). Thirty per cent axial strain. (c). Forty per cent axial strain. (d). Sixty per cent axial strain. (e). Seventy per cent axial strain. (f). Seventy ve per cent axial strain.

16 1154 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139±1157 In order to check the unloading behavior of the model, we rst loaded the specimen using the load control technique. The specimen was then unloaded to zero stress state as shown in Fig. 7. This resulted in the transformation of the material into an oriented glass, thereby releasing only the elastic strains. We have also plotted the engineering-stress versus engineering-strain as shown in Fig Test 2: cold-drawing (necking) of polycarbonate We next performed a numerical simulation of neck propagation under displacement control. Invoking symmetry, only one quarter of the problem is discretized (see Fig. 11. Stress-11 for 20% strains. Fig. 12. Stress-11 for 40% strains.

17 A. Masud, A. Chudnovsky / International Journal of Plasticity 15 (1999) 1139± Fig. 13. Stress-11 for 60% strains. Fig. 9). The ratio of length to width dimensions in the computational specimen are 4:1. The computational mesh is composed of 400 four-node elements. Necking is initiated by providing a 2% reduction in the cross-sectional area. Fig. 10(a)±(f) presents the deformation history of the specimen as the neck propagates along the length of the specimen. As can clearly be seen in Fig. 10(a)±(c), 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. For comparison, we have also plotted the outline of the undeformed initial con guration. In Fig. 10(e) the specimen has undergone 70% strains and only a small portion of the untransformed material is left. In Fig. 10(f) the specimen has undergone 75% strains and neck has propagated over the entire length of the specimen. Figs. 11±13 present the axial stress during the process of neck propagation. 6. Conclusions In this paper we have presented a nite strain constitutive model for phase transition in polycarbonates. We have de ned a stress transformation surface that represents the stress state at a material particle during phase transition. A hyperelastic extension of the J 2 - ow theory to the nite strain range is presented in which volume preserving and volumetric responses are uncoupled throughout the entire range of deformation. The isochoric constraint during phase transformation is treated through an exact multiplicative decomposition of the deformation gradient into volume-preserving and spherical parts. We have implemented the nonlinear constitutive equations in a nite element framework using 4-node isoparametric elements. A benchmark problem of cold-drawing in polycarbonates is presented to show the accuracy of the constitutive equations in predicting the physical behavior under stress loading-unloading conditions.

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