Modeling of the Mechanical Response and Evolution of Optical Anisotropy in Deformed Polyaniline.
S. AHZI 
R. V. GREGORY 
The mechanical response under large deformation at room temperature of amorphous polyaniline (PANI) is simulated using an elastic-viscoplastic model for large strain of glassy polymers. The evolution of optical anisotropy under deformation is also simulated by coupling the mechanical anisotropy due to molecular alignment to the optical properties. Several homogeneous deformation tests such as tension, compression and shear are considered. For the evolution of the mechanical anisotropy and the associated optical anisotropy, we compare the results from three network models. Our predicted results are compared to experimental observations and good agreement is found.
The significant potential technological applicability of polyaniline (PANI) stems from its satisfactory environmental stability, high degree of electrical conductivity with reasonable mechanical properties and its ease of processing into films and fibers. Polyaniline base is a unique material that may exist in several oxidation states ranging from fully reduced to fully oxidized. The half-oxidized form is electrically conducting when doped with protonic acids . Commercial interest in polyaniline, its relative ease of processing into films and fibers [1, 2], and its systematic conversion from base to conducting polymer makes polyaniline an ideal model material. Development of a general approach for coupling polyaniline's macroscopic behavior with microscopic molecular parameters would have immediate application to several processing technologies.
In the present work, we propose to study the mechanical response and optical properties during solid state forming of polyaniline, such as the drawing process. For the modeling of the mechanical response, we propose the use of the elastic viscoplastic model of Boyce et al.  which describes the constitutive behavior of glassy polymers under large deformations. The evolution of the mechanical anisotropy due to molecular alignment during deformation is modeled using the approach of Arruda and Boyce  and Wu and Van Der Giessen . For the modeling of optical properties, we propose to couple the mechanical anisotropy to the optical anisotropy as in the work of Kuhn and Grun , Arruda and Przybylo , and Wu and Van Der Giessen . This coupling permits the prediction of the evolution of the refractive indices under large deformations.
While our current application is limited to homogeneous deformation processes such as tension, compression, plane strain compression and simple shear, it is also possible to extend the proposed approach to complex processing methods by implementing the constitutive models in a finite element code.
2. EXPERIMENTAL RESULTS
Uniaxial tension and refractive indices tests were carried out respectively by Tzou  and Gregory and Samuels . Here, we briefly recapitulate the technique and apparatus used to synthesize and characterize the tensile sample.
The polyaniline film was chemically synthesized as previously reported by Chiang and MacDiarmid . Molecular weights were estimated with gel permeation chromatography as Mn = 35000, Mw = 83000. Once the polyaniline was synthesized, the N,N'-dimethyl propylene urea (DMPU) was chosen as the processing solvent for the preparation of the film casting. The cast film was prepared by casting a 7 wt% solution of polyaniline in DMPU onto glass substrates and allowing to dry for 10 days at room temperature.
The mechanical testing was done on an Instron universal testing instrument with a hydraulic clamping system. Testing was conducted in a conditioned laboratory (25[degrees]C) with a gauge length setting of 1cm and a head cross speed of 5mm/min. Tensile samples were cut into 1 cm X 1 cm X 5 cm strips and loaded longitudinally between hydraulic jaws. Figure 1 shows the stress-strain response under uniaxial loading. We note that the general behavior can be decomposed into an elastic part, yielding followed by softening, and strain hardening due to molecular alignment, which accompanies plastic deformation. This is a general characteristic of the mechanical response in glassy polymers. We note that no crystallinity is observed in the PANI samples used in this test .
The three-dimensional refractive indices of the freestanding PANI films were characterized with a modified prism-wave guide coupler (Metricon PC-2010) at 1550 nm. The technique was previously developed and tested on a range of different freestanding and spin coated polymer films . Figure 2 shows the experimental results where refractive indices are reported against the stretch [[lambda].sub.3] in the tensile direction as reported by Gregory and Samuels . Since only two data points are available for each of the refractive indices, the evolution of these indices seems to be linear. For a better description of their evolution, additional data points at intermediate stretches are needed. In this paper, we focus on modeling and simulation and compare our results to these limited data points available.
In the following section, we will present an elastic-plastic model that was developed for the simulation of mechanical response in glassy polymers [3, 4, 14]. This model captures the different stages of deformation response observed in Fig. 1. Following the work of Kuhn and Grun , James and Guth , Arruda and Przybylo , and Wu and Van Der Giessen , we will also show how this mechanical model is coupled with the evolution of optical properties.
3. CONSTITUTIVE EQUATIONS
3-1. Inelastic Deformation of Glassy Polymers
The initial yielding of the isotropic glassy polymers depends on strain rate, pressure, and temperature . After yielding, these materials often strain soften and subsequently strain harden. These characteristics of yielding in glassy polymers result from two physically distinct sources: the intermolecular resistance to segment rotation and the entropic resistance to molecular alignment [3, 14].
At yield, a glassy polymer must be stressed to exceed its intermolecular resistance to segment rotation. Argon  has developed an expression for the free energy change necessary to produce segment rotation based on a double kink model of a chain segment rotating against the elastic impedance of surrounding chains. The corresponding shear transformation rate, or plastic shear strain rate, is given by:
[[gamma].sup.p] = [[gamma].sub.0] exp[-[As.sub.0]/[theta] (1 - [([[tau].sup.*]/[s.sub.0]).sup.5/6])], (1)
where [[gamma].sub.0] is a reference shear rate, A is a material parameter, [theta] is the absolute temperature; [s.sub.0] = [micro] (0.077/1-[nu]) is the athermal shear strength, [micro] is the elastic shear modulus, [nu] is Poisson's ratio, and [[tau].sup.*] is the effective shear stress. Equation 1 describes the rate and temperature dependence of the intermolecular shear resistance of the glassy polymers. Boyce et al.  modified this relation to incorporate the effects of pressure and strain softening. The athermal shear strength [s.sub.0] is replaced by s + [alpha]' p, where p is the pressure, [alpha]' is the pressure dependence coefficient, and s is the athermal deformation resistance of the material indicating the current state of the structure. In the present work, we neglect pressure dependence ([alpha]' = 0). The evolution of this shear resistance is phenomenologically modeled by Boyce et al.  to decrease with plastic straining until reaching a preferred structure, represented by [s.sub.ss], through the foll owing expression:
s = h(1 - S/[S.sub.ss])[[gamma].sup.p], (2)
where h is the rate of resistance drop with respect to the plastic strain.
The basic macromolecular structure of isotropic glassy polymers consists of randomly oriented molecular chains connected by physical entanglement. Once the material is stressed to the point of overcoming intermolecular resistance, the chains begin to orient themselves in an assumed affine manner. This action decreases the configurational entropy of the system which, in turn, creates an internal network stress state (called back stress). This back stress is a measure of the resistance to plastic flow due to molecular alignment. The back stress has been modeled using statistical mechanics network models of rubber elasticity. Wang and Guth  proposed a non-Gaussian three-chain model network consisting of three orthogonal chains which deform affinely with the imposed bulk deformation. In this model the principal components of the back stress have the form:
[C.sub.R] [square root]N/3 [[[lambda].sub.i] [l.sup.-1] ([[lambda].sub.i]/[square root]N) - 1/3 [[[sigma].sup.3].sub.j=1] [[lambda].sub.j] [l.sup.-1] ([[lambda].sub.j]/[square root]N)], (i = 1,2,3) (3)
where N is the number of rigid chain links between entanglements, [C.sub.R] is the rubbery modulus, [[lambda].sub.j] (j = 1, 2, 3) are the imposed principal stretches and l is the Langevin function employed to account for large stretch behavior. Arruda and Boyce  found that this model did not exhibit the cooperative nature of the chain deformations which must be characteristic of real networks but was instead dominated by the behavior of the chain having the greatest extension. At the same time, they developed an eight-chain model network considering a set of eight chains connecting the central junction point and each of the eight corners of the unit cube. The principal values of the back stress tensor according to this model are:
[[B.sup.8-ch].sub.4] = [C.sub.R] [square root]N/3[[l.sup.-1] ([[lambda].sub.c]/[square root]N)[[[lambda].sup.2].sub.i] - 1/3 I/[[lambda].sub.c]], (4)
[[lambda].sub.c] = 1/[square root]3 [([[[lambda].sup.2].sub.1] + [[[lambda].sup.2].sub.2] + [[[lambda].sup.2].sub.3]).sup.1/2] and I = [[[lambda].sup.2].sub.1] + [[[lambda].sup.2].sub.2] + [[[lambda].sup.2].sub.3].
Wu and Van Der Geissen  also proposed non-Gausian rubber model in which they account of the spatial orientation distribution of the individual chains in the network by considering a set of infinite chains connecting at the center of a sphere. This model is based on the work of Treloar and Riding  but extends their theory to general 3-D deformations including arbitrary rotations of the principal stretch axes. Wu and Van Der Geissen  demonstrated that combining the 3-chain and 8-chain models can approximate the stress response predicted by their full network model. One possibility is by using the rule of mixture:
[B.sub.i] = (1 - [rho]) [[B.sup.3-ch].sub.i] + [rho][[B.sup.8-ch].sub.i], (5)
where [rho] is related to the maximal principal stretch [[lambda].sub.max] = max([[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3]) by [rho] = [chi] X ([[lambda].sub.max]/[square root]N) in which the factor [chi](0 [less than or equal to] [chi] [less than or equal to] ([square root]N/[[lambda].sub.max])) can be chosen to give the best correlation with full network approach.
3-2. Optical properties
The birefringence [delta][[eta].sub.i-j], or difference in refractive indices in (i-j) plane is related the network polarizability anisotropy ([[beta].sub.i]-[[beta].sub.j]). Following the work of Kuhn and Grun , we have:
[delta][[eta].sub.i-j] = [[eta].sub.i] - [[eta].sub.j] = 2[pi]/3[[eta].sub.av] [([[[eta].sup.2].sub.av] + 2).sup.2] ([[beta].sub.i] - [[beta].sub.j]) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
where [[eta].sub.i] are the components of refractive indices in the principal direction, [B.sub.i] are the principal component of the birefringence (polarization) tensor and [[eta].sub.av] is the mean refractive of the medium.
[[eta].sub.av] = 1/3 ([[eta].sub.1] + [[eta].sub.2] + [[eta].sub.3]). (7)
The three dimensional refractive indices [[eta].sub.i](i = 1, 2, 3) are calculated by combining Eqs 6 and 7. The network polarizibility anisotropy is calculated from the network structure models as listed below.
Following the three chain-model of James and Guth (1943)  for rubber elasticity, the differences of the principal network polarisabilities in plane i-j, according to this model, are given in the form:
[[[beta].sup.3-ch].sub.i] - [[[beta].sup.3-ch].sub.j] =
n([[alpha].sub.1] - [[alpha].sub.2]) [square root]N [-[[lambda].sub.i]/[l.sup.-1]([[lambda].sub.i]/[square root]N) + [[lambda].sub.j]/[l.sup.-1]([[lambda].sub.j]/[square root]N)], (8)
where n is the number of chains per unit volume and [[alpha].sub.1] and [[alpha].sub.2] are respectively the link polarizability along and perpendicular to the chain axis. When the eight-chain model is used, as proposed by Arruda and Przybylo , the principal network polarizabilities in plane i-j are given by:
[[[beta].sup.8-ch].sub.i] - [[[beta].sup.8-ch].sub.j] =
n([[alpha].sub.1] - [[alpha].sub.2])N([[[lambda].sup.2].sub.i] - [[[lambda].sup.2].sub.j]/3[[[lambda].sup.2].sub.c]) [1 - 3[[lambda].sub.c]/[square root]N/[l.sup.-1]([[lambda].sub.c]/[square root]N)]. (9)
The values of the principal components of the network polarizability predicted by the full network model (8) were found between that predicted by the three-chain model and the eight-chain one, for the same values of N and n. Wu and Van Der Geissen (8) have shown that the full integration can be approximated by linear combination of the three-chain and the eight-chain models based on the rule of mixture.
[[beta].sub.i] = (1 - [rho])[[[beta].sup.3-ch].sub.i] + [rho][[[beta].sup.8-ch].sub.i] (10)
where [rho] is determined in the previous paragraph.
3-3. Three-Dimensional Representation of Deformation
Here we describe the three-dimentional constitutive relation for large elastic-viscoplastic deformation as developed by Boyce et al. (3). The deformation of a material from its isotropic reference configuration at time [t.sub.0] to an actual state at time t is described by the deformation gradient tensor F. This tensor is multiplicatively decomposed into elastic and plastic components, F = [F.sup.e][F.sup.p].[F.sup.e], the elastic part, is taken to be symmetric, and [F.sup.p] represents the plastic part which can be expressed in terms of the plastic stretch tensor, [V.sup.p], according to the polar decomposition: [F.sup.p] = [V.sup.p] R with R the rotation tensor. The principal stretches [[lambda].sub.i] can be deduced from [F.sup.p]. [[[lambda].sup.2].sub.i] are the principal components of [V.sup.[p.sup.2]]. The velocity gradient L is given by L = [FF.sup.-1] = D + W where D is the rate of deformation (symmetric part of L) and W is the spin (antisymmetric part of L]. The plastic velocity gradient is given by:
[L.sup.p] = [F.sup.p] [F.sup.p-1] = [D.sup.p] + [W.sup.p], (11)
where [W.sup.p] is the plastic spin (antisymmetric part of [L.sup.p]) and [D.sup.p] is the plastic deformation rate (symmetric part of [L.sup.p]). The plastic spin [W.sup.p] is set equal zero ([W.sup.p] = 0) and [D.sup.p] is given by the flow rule:
[D.sup.p] = [[gamma].sup.p]N, (12)
where [[gamma].sup.p] is the shear rate given by Eq 1 and N is the normalized driving stress:
N = 1/[square root][[tau].sup.*] [T.sup.*'] (13)
[T.sup.*] = T - 1/J [F.sup.e] [BF.sup.e], .6 (14)
where [T.sup.*] is the driving stress, B is the back stress tensor described in section 2, J is the volume change given by (det[F.sup.e]) and T is the Cauchy stress tensor. The effective shear stress, [[tau].sup.*], is given by:
[[tau].sup.*] = [(1/2 [[T.sup.*'].sub.ij] [[T.sup.*'].sub.ij]).sup.1/2]. (15)
For additional details on this elastic viscoplastic model, the reader may consult the work of Boyce et al. (3).
4. MODEL RESULTS AND COMPARISON WITH EXPERIMENTS
The full network approximation (intermediate model) as well as the three-chain and eight-chain models for rubber elasticity are used in parallel with Boyce et al. (3) model to reproduce the experimental mechanical and optical response under uniaxial tension. We also simulated the response under uniaxial compression, plane strain compression and simple shear.
The boundary conditions for this test are simplified and the velocity gradient is totally prescribed. Since the Poisson ratio ([nu] [approximate] 0.43) is relatively high, we assume isochoric deformation (traceless velocity gradient). This assumption has little effect on our simulated results. The forms of the velocity gradient imposed for the considered tests are:
Uniaxial tension or compression:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
where the positive sign is for the tension and negative sign indicates compression. The considered tests are under constant strain rate [epsilon] = [10.sup.-2][S.sup.-1].
In plane strain compression:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
And for the simple shear:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
Note that the uniaxial tension is used to simulate the drawing process and plane strain compression test simulates the rolling process. The effects of shear that is present in complex processes, such as in rolling, can be characterized by analyzing the simple shear test. The microstructural evolution (molecular orientation) under uniaxial compression is similar to that obtained under biaxial stretching or blow stretching. Therefore, the anisotropy analysis under uniaxial compression may be used to simulate that under biaxial stretching.
In Fig. 3 we present the simulated true stress-true strain curves together with the experimental one. In addition to the testing parameters which are the temperature, pressure and strain rate, the models contain other material parameters which are [[lambda].sub.0], A, [S.sub.0], [S.sub.ss], h, N and [C.sub.R], generally determined by fitting each given model to the experimental data. Prior to the yield, the stress-strain curves are described by the elastic properties of the material: the Young's modulus (E = 2200 MPa) and the Poisson's ratio ([nu] = 0.43). The yield and post-yield softening are described by the parameters [S.sub.0] = 104 MPa, [S.sub.ss] = 95 MPa and h = 850 MPa. The dependence on temperature and strain rate is governed by the parameters [[gamma].sub.0] = 65.41 X [10.sup.11] [s.sup.-1] and A = 164.24 K/MPa. The pressure dependency is neglected. The parameters N and [C.sub.R] describe the orientational hardening. The values of N = 9 and [C.sub.R] = 10.8 MPa at room temperature are chosen. Note that [C.sub.R] = nk[theta], were k is the Boltzman constant. [theta] is the temperature and n is the number of chains per unit volume. The value chosen for [C.sub.R] corresponds to n = 2.62 X [10.sup.27] [m.sup.-3].
The material parameters are selected to fit the intermediate network model and the same parameters are used in the eight-chain and three-chain models. All models give identical prediction of the stress-strain curve (Fig. 3) up to [[epsilon].sub.eq] = 0.2 ([[epsilon].sub.eq] is the Mises equivalent strain). Beyond this value, the models predict different values for the stresses. The eight-chain model curve is lower relative the intermediate network model curve, while the three-chain model curve is higher. This is expected since the three-chain model is stiffer than the eight-chain model.
The calculated refractive indices curves by using the three network models sited above are presented in Fig. 4a and compared to the experimental data. The material parameters required to model the evolution of the optical anisotropy are n, N, [alpha] and [[eta].sub.av]. The first two parameters are already determined in the modeling of the mechanical response and are defined above. The mean refractive index of the medium, [[eta].sub.av], is taken to be linearly varying with the applied tensile stretch and its evolution is fitted to the experimental measurement (see Fig. 2). The only parameter which we can vary to match the simulated curves with the experimental one is the link anisotropy [alpha]. This parameter represents the difference between the link polarizability along and that perpendicular to the chain axis and is found to be equal to 15.5 [(10).sup.-29][m.sup.3] according to the intermediate network model. Because of the limited number of experimental data points for the refractive indices, our fitting to experimental results may not be accurate for intermediate stretch values.
Relative to the intermediate network model, the refractive index curve, [[eta].sub.3], in the tensile direction predicted by the eight-chain model is higher, but the one predicted by the three-chain model is lower. In contrast, in the direction perpendicular to the tensile direction, the sample contracts, which leads to the inversion of the eight- and three-chain model curves relative to that predicted by the intermediate network model. The evolution of each refractive index with the tensile stretch from the initial isotropic configuration forms a divergent band. Note that, if we chose the mean refractive index of the medium constant with respect to the tensile stretch [[[eta].sub.av]([lambda]) = [[lambda].sub.av](0)], as it was previously taken in the modeling of rubber optical anisotropy [7, 8], the experimental data deviates slightly from the simulated intermediate model curve, but it will be still between the upper and lower limits of the refractive indices band (see Fig. 4b).
As pointed out by Wu and Van Der Giessen [5, 8], the three-chain model tends to overestimate the optical anisotropy and the mechanical response at large stretch, relative to the Intermediate network model, while the eight-chain model tends to underestimate it. This disagreement at large stretches is associated with the different limit stretches of the complete network [8, 18]. The stretching according to the three-chain model is limited directly by the tensile stretch [[lambda].sub.L] = [square root]N of the chains parallel to the tensile direction, while the overall network limiting stretch for the eight-chain model exceeds [[lambda].sub.L]. This observation is due to the fact that the material parameters are selected to fit the intermediate network model. However, the model parameters [[lambda].sub.L] and [C.sub.R] can be selected to fit the eight-chain or the three-chain model to the experiments.
The mechanical and optical anisotropy developed in polymeric materials at large strains are governed by the evolution of molecular orientation during deformation and have been found to be highly dependent on the state of deformation. To simulate different states of deformation, the material parameters fitted to uniaxial tension are used to predict the response under uniaxial compression, plane strain compression, and simple shear. The mean refractive of the medium, for these tests, is taken constant.
Figures 5a-5b and Figs. 6a-6b show the simulated evolution of mechanical response and optical anisotropy for the uniaxial compression and the plane strain compression respectively. At large deformation, the stress-strain curves predicted by the three models for the uniaxial compression, Fig. 5a, show a closer dispersion and larger locking stretch of the molecular chains in comparison to those predicted by the plane strain compression, Fig. 6a. This behavior is due to differences in the boundary conditions of these tests. The chains have fewer kinematic degrees of freedom under plane strain compression than uniaxial compression. The anisotropy difference between these two tests is also observed in the optical properties. In uniaxial compression, Fig. 5b, the variation of the refractive indices versus the compression ratio is nearly linear. The indices [[eta].sub.1] and [[eta].sub.2], in the plane of the orientation of the molecular chains, are equal due to the axial symmetry and increase with contraction of t he sample to counter balance the decrease of the [[eta].sub.3] in the compression direction. In plane strain compression, Fig. 6b, up to the compression ratio of 0.8 the variation of the refractive indices is nearly linear. At large compression ratios the evolution of the optical anisotropy becomes more important and shows nonlinear behavior. The evolutions of the mechanical and optical anisotropy in simple shear are presented respectively in Figs. 7a and 7b. The Mises equivalent stress ([[sigma].sub.eq]) together with the shear stress ([tau]) times [square root]3 are plotted versus the Mises equivalent plastic strain for different network models (Fig. 7a). The locking of the molecular chains, which is directly related to the failure of the material, is larger for shear stress curves than that shown by the equivalent stress ones. The difference between [[sigma].sub.eq] and [square root]3 [tau] is due to the simplified boundary conditions used to simulate shear where the conditions are imposed on the strain rat e only.
In Fig. 7b we present the evolution of the refractive indices versus the equivalent plastic strain during simple shear test. The curves corresponding to [[eta].sub.1] show an increase while those corresponding to [[eta].sub.2] and [[eta].sub.3] are close and show a decrease with deformation.
In Fig. 8 we present the comparison between the evolution of the mechanical response under different deformation states for the intermediate network model. The strain-hardening trend is relative to the kinematic degrees of freedom of chains under each test. The degrees of freedom for chain alignment is higher for uniaxial compression, where the chains align in the radial direction of the compressed sample, and is lower for uniaxial tension, where the chains align with the tensile direction. The hardening under plane strain compression is closer to that in tension because of the boundary conditions where one of the macroscopic directions is constrained to a zero strain.
We have successfully implemented an elastic-viscoplastic model for large deformation of glassy polymers and used it to compute the mechanical response and the evolution of anisotropy in deformed polyaniline. Three network models are considered and the results from these are compared and discussed. These models produce two bounds and an intermediate prediction for the entropic resistance to yielding due to molecular alignment. Different homogenous deformation tests were simulated. The results of our simulations for the refractive indices under tension are compared to the experimental observations and fair agreement is found. The considered tests are based on the assumption of homogeneous deformation but the proposed modeling can be implemented in a finite element code for the simulation of complex processing methods of PANI in the solid state.
This Work was supported primarily by the ERC Program of the National Science Foundation under Award Number ERC-9731680 with Clemson University.
Center for Advanced Engineering Fibers and Films
(1.) Department of Mechanical Engineering
(2.) School of Textiles, Fiber and Polymer Science Clemson University, Clemson, SC, 29634
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|Author:||MAKRADI, A.; AHZI, S.; GREGORY, R. V.|
|Publication:||Polymer Engineering and Science|
|Date:||Jul 1, 2000|
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