Evolution of plastic anisotropy in amorphous polymers during finite straining

The large strain deformation response of amorphous polymers results primarily from orientation of the molecular chains within the polymeric material during plastic straining. Molecular network orientation is a highly anisotropic process, thus the observed mechanical response is strongly a function of the anisotropic state of these materials. Through mechanical testing and material characterization, the nature of the evolution of molecular orientation under different conditions of state of strain is developed. The role of developing anisotropy on the mechanical response of these materials is discussed in the context of assessing the capabilities of several models to predict the state of deformation-dependent response. A three-dimensional rubber elasticity spring system that is capable of capturing the state of deformation dependence of strain hardening is used to develop a tensorial internal state variable model of the evolving anisotropic polymer response. This fully three-dimensional constitutive model is shown to be successfully predictive of the true stress vs. true strain data obtained in our isothermal uniaxial compression and plane strain compression experiments on amorphous polycarbonate (PC) and polymethylmethacrylate (PMMA) at moderate strain rates. A basis is established for providing the polymer designer with the ability to predict the flow strengths and deformation patterns of highly anisotropic materials. A companion paper by Arruda, Boyce, and Quintus-Bosz [in press] shows how the model developed herein is used to predict various anisotropic aspects of the large strain mechanical response of preoriented materials. Additional work has been done to extend the model to include the effects of strain rate and temperature in Arruda, Jayachandran, and Boyce [in press].     

A physically-based constitutive model for anisotropic damage in rubber-toughened glassy polymers during finite deformation

The present work focuses on the development of a physically-based model for large deformation stress-strain response and anisotropic damage in rubber-toughened glassy polymers. The main features leading to a microstructural evolution (regarding cavitation, void aspect ratio, matrix plastic anisotropy and rubbery phase deformation) in rubber-toughened glassy polymers are introduced in the proposed constitutive model. The constitutive response of the glassy polymer matrix is modelled using the hyperelastic-viscoplastic model of [Boyce et al., 1988] and [Boyce et al., 2000]. The deformation mechanisms of the matrix material are accounted for by two resistances: an elastic-viscoplastic isotropic intermolecular resistance acting in parallel with a visco-hyperelastic anisotropic network resistance, each resistance being modified to account for damage effects by void growth with a variation of the void aspect ratio. The effective contribution of the hyperelastic particles to the overall composite behaviour is taken into account by treating the overall system in a composite scheme framework. The capabilities of the proposed constitutive model are checked by comparing experimental data with numerical simulations. The deformation behaviour of rubber-toughened poly(methyl methacrylate) was investigated experimentally in tension at a temperature of 80 °C and for different constant true strain rates monitored by a video-controlled technique. The reinforcing phase is of the soft core-hard shell type and its diameter is of the order of one hundred nanometers. The particle volume fraction was adjusted from 15% to 45% by increments of 5%. The stress-strain response and the inelastic volumetric strain are found to depend markedly on particle volume fraction. For a wide range of rubber volume fractions, the model simulations are in good agreement with the experimental results. Finally, a parametric analysis demonstrates the importance of accounting for void shape, matrix plastic anisotropy and rubber content.     

Plastic flow in oriented glassy polymers

A manufactured product often possesses residual texture which was either incidentally or deliberately acquired during its processing history. This is particularly true for the case of polymers, where the ability to easily preferentially preorient the material in specific directions is exploited in order to obtain a higher strength product. Specific examples include synthetic fibers, and biaxially-oriented films and containers. The response of the preoriented/textured product to normal service life loading conditions will differ considerably from that of a product composed of isotropic material. This paper addresses the issue of the effects of texture on the deformation behavior of glassy polymers. Here, the physically-based constitutive model of Boyce, Parks, and Argon describing the rate, temperature, and pressure dependent inelastic deformation of initially isotropic glassy polymers is used to model the effects of preorientation (i.e., initial texture), via the use of appropriate initial conditions on internal state variables. The model is then utilized in an analysis of the effects of texture on the yield of glassy polymers and the shear localization which normally follows yielding in oriented polymers. These results are compared with trends found in experiments as reported in the literature. The effectiveness of the proposed model for the present application is also compared with earlier models of yielding of anisotropic materials such as Hill's criterion.     

On modeling the micro-indentation response of an amorphous polymer

Interest in instrumented indentation experiments as a means to estimate mechanical properties has grown rapidly in recent years. Although numerous nano/micro-indentation experimental studies on polymeric materials have been reported in the literature, a corresponding methodology for extracting material property information from the experimental data does not exist. This situation for polymeric materials exists primarily because baseline numerical analyses of sharp indentation using appropriate large deformation constitutive models for the nonlinear viscoelastic–plastic response of these materials appear not to have been previously reported in the literature. An existing, widely used theory for amorphous polymers (e.g. [Boyce, M., Parks, D., Argon, A.S., 1988. Large inelastic deformation of glassy polymers. Part 1: Rate dependent constitutive model. Mechanics of Materials 7, 15–33; Arruda, E.M., Boyce, M.C., 1993. Evolution of plastic anisotropy in amorphous polymers during finite straining. International Journal of Plasticity 9, 697–720]) has been recently found to lack sufficient richness to enable one to quantitatively reproduce the major features of the indentation load-versus-depth curves for some common amorphous polymers [Gearing, B.P., 2002. Constitutive equations and failure criteria for amorphous polymeric solids. Ph.D. thesis, Massachusetts Institute of Technology].This study develops a new continuum model for the viscoelastic–plastic deformation of amorphous polymeric solids. We have applied the constitutive model to capture salient features of the mechanical response of the amorphous polymeric solid poly(methyl methacrylate) (PMMA) at ambient temperature and stress states under which this material does not exhibit crazing. We have conducted compression-tension strain-controlled experiments, as well as stress-controlled compression-creep experiments, and these experiments are used to calibrate the material parameters in the constitutive model for PMMA.We have implemented our constitutive model in a finite-element computer program, and using this finite-element program we have simulated micro-indentation experiments on PMMA. We show that our constitutive model and finite element simulations reproduce the experimentally-measured indentation load-versus-depth response with reasonable accuracy.     

Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility

In this article we investigate several models contained in the literature in the case of near-incompressibility based on invariants in terms of polyconvexity and coerciveness inequality, which are sufficient to guarantee the existence of a solution. These models are due to Rivlin and Saunders, namely the generalized polynomial-type elasticity, and Arruda and Boyce. The extension to near-incompressibility is usually carried out by an additive decomposition of the strain energy into a volume-changing and a volume-preserving part, where the volume-changing part depends on the determinant of the deformation gradient and the volume-preserving part on the invariants of the unimodular right Cauchy–Green tensor. It will be shown that the Arruda–Boyce model satisfies the polyconvexity condition, whereas the polynomial-type elasticity does not. Therefore, we propose a new class of strain-energy functions depending on invariants. Moreover, we focus our attention on the structure of further isotropic strain-energy functions.     

Cyclic stress-softening model for the Mullins effect in compression

This paper models the cyclic stress softening of an elastomer in compression. After the initial compression the material is described as being transversely isotropic. We derive non-linear transversely isotropic constitutive equations for the elastic response, stress relaxation, residual strain, and creep of residual strain in order to model accurately the inelastic features associated with cyclic stress softening. These equations are combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic stress softening for a transversely isotropic, hyperelastic material, in particular a carbon-filled rubber vulcanizate. To establish the validity of the model we compare it with two test samples, one for filled vulcanized styrene–butadiene rubber and the other for filled vulcanized natural rubber. The model is found to fit this experimental data extremely well.     

An endochronic plasticity formulation for filled rubber

The nature of elastomeric material demands the consideration of finite deformations, nonlinear elasticity including damage as well as rate-dependent and rate-independent dissipative properties. While many models accounting for these effects have been refined over time to do better justice to the real behavior of rubber-like materials, the realistic simulation of the elastoplastic characteristics for filled rubber remains challenging.The classical elastic-ideal-plastic formulation exhibits a distinct yield-surface, whereas the elastoplastic material behavior of filled rubber components shows a yield-surface free plasticity. In order to describe this elastoplastic deformation of a material point adequately, a physically based endochronic plasticity model was developed and implemented into a Finite Element code. The formulation of the ground state elastic characteristics is based on Arruda and Boyce (1993) eight-chain model. The evolution of the constitutive equations for the nonlinear endochronic elastoplastic response are derived in analogy to the Bergström–Boyce finite viscoelasticity model discussed by Dal and Kaliske (2009).     

Necking in glassy polymers: Effects of intrinsic anisotropy and structural evolution kinetics in their viscoplastic flow

Two recently proposed developments of the Glass–Rubber constitutive model for glassy polymers treat the viscoplastic deformation as intrinsically anisotropic, and incorporate the kinetics of structural evolution. These features enable the model to capture better the distinctive features of glassy polymers’ constitutive response: post-yield strain-softening and strain-hardening and effects of pre-existing molecular orientation. They have been combined to form a new variant of the model, and the consequences for necking have been explored. Uniaxial extension of prismatic bars was simulated using the finite element method, employing a numerical implementation of the new model, with material parameters of polystyrene. Strain localization predicted with the new model was found to be systematically retarded as compared to predictions with the original (intrinsically isotropic) version of the model, for the same conditions. In particular, the effect of frozen-in molecular orientation was examined. This was found to retard strain localization for stretching parallel to the orientation direction, for both models. But the localization predicted with the new model was always significantly less pronounced than with the original model. Indeed, for sufficiently high pre-orientation (e.g. a uniaxial stretch of 2.2), localization could be effectively prevented with the new model, under conditions when otherwise failure by necking is predicted. Such results can all be explained in terms of a linear stability analysis. They suggest that all previous simulations of necking in glassy polymers made using intrinsically isotropic representations of polymer viscoplasticity may have over-predicted the rate of strain localization.     

Large strain creep analysis of thick-walled cylinders

The finite-strain theory has been used to study the creep behaviour of a thick-walled cylinder under large strains. The analysis is divided into two parts. In part 1 the creep deformation of a thick-walled cylinder of an anisotropic material subjected to internal pressure has been discussed. The effect of the anisotropy has been depicted graphically. It is found that the anisotropy of the material has a significant effect on the axial stress, strain and strain rate. Part 2 of the paper deals with the creep analysis of cylinders of either isotropic or anisotropic materials subjected to combined internal and external pressures. The effect of the anisotropy is found to be similar to that found in part 1. It is seen, however, that the introduction of external pressure results in decreasing the strain rate and thus increasing the life of the cylinder.     

A finite element formulation based on non-associated plasticity for sheet metal forming

In the present paper, a finite element formulation based on non-associated plasticity is developed. In the constitutive formulation, isotropic hardening is assumed and an evolution equation for the hardening parameter consistent with the principle of plastic work equivalence is introduced. The yield function and plastic potential function are considered as two different functions with functional form as the yield function of Hill [Hill, R., 1948. Theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. A 193, 281–297] or Karafillis–Boyce associated model [Karafillis, A.P. Boyce, M., 1993. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J. Mech. Phys. Solids 41, 1859–1886]. Algorithmic formulations of constitutive models that utilize associated or non-associated flow rule coupled with Hill or Karafillis–Boyce stress functions are derived by application of implicit return mapping procedure. Capabilities in predicting planar anisotropy of the Hill and Karafillis–Boyce stress functions are investigated considering material data of Al2008-T4 and Al2090-T3 sheet samples. The accuracy of the derived stress integration procedures is investigated by calculating iso-error maps.     

A generalized coupled viscoplastic-viscodamage-viscohealing theory for glassy polymers

A thermodynamic consistent, small-strain, non-unified model is developed to capture the irregular rate dependency included in the strain controlled inelastic responses of polymers at the glassy state. The model is considered as a generalized Frederick-Armstrong-Philips-Chaboche (FAPC) theory proposed by [Voyiadjis and Basuroychowdhury, 1998] and [Voyiadjis and Abu Al-Rub, 2003] which is based on a von Mises and Chaboche isotropic hardening type viscoplasticity formulation. Using the proposed model, different experimental results are simulated and the range of viscoplastic related material constants are obtained through a parametric study. The thermodynamic framework is used to incorporate the effect of coupling between viscodamage and viscohealing phenomena into the inelastic deformation of glassy polymers. This coupling effect is crucial for polymeric based self healing systems in which different damage mechanisms are active and the efficiency of the healing processes are highly dependent on the damage. The computational aspect for general coupled inelastic-damage-healing processes together with the required solution algorithms are elaborated and the inelastic-damage-healing response of a polymeric based self-healing system is simulated. The proposed viscoplasticity theory constitutes a physically consistent approach to model the irregular mechanical responses of glassy polymers and the viscodamage model provides an exquisite predicting tool to evaluate the ductile damage associated with the large inelastic deformation and low cycle fatigue in polymeric based material systems. In conclusion, a well structured viscohealing theory is formulated for polymeric based self healing systems.     

An 8-chain Model for Rubber-like Materials Accounting for Non-affine Chain Deformations and Topological Constraints

Several industrial applications involve rubber and rubber-like materials, and it is important to be able to predict the constitutive response of these materials. In the present paper, a new constitutive model for rubber-like solids is proposed. The model is based on the 8-chain concept introduced by Arruda and Boyce (J. Mech. Phys. Solids 41, 389–412, 1993) to which two new components are added. Real polymer networks do not deform affinely, and in the proposed model this is accounted for by the inclusion of an elastic spring, acting in series with the representative polymer chain. Furthermore, real polymer chains are not completely free to move, which is modelled by imposing a topological constraint on the transverse motions of the representative polymer chain. The model contains five model parameters and these need to be determined on the basis of experimental data. Three experimental studies from the literature were used to assess the proposed model. The model was able to reproduce experimental data performed under conditions of uniaxial tension, generalised plane deformation, and biaxial tension with an excellent accuracy. The strong predictive abilities together with the numerically efficient structure of the model make it suitable for implementation in a finite element context.     

CAVITATION BIFURCATION FOR COMPRESSIBLE ANISOTROPIC HYPERELASTIC MATERIALS

I. INTRODUCTION The cavitation bifurcation problem, sudden formation and growth of voids in solid materials, haslong attracted much attention because of the fundamental role it plays in the local failure and fractureof materials. For hyperelastic materi…     

Qualitative analysis of strain localization. Part I: Transversely isotropic elasticity and isotropic plasticity

While the conditions for the onset of strain localization in inelastic materials with isotropic elasticity have been extensively studied in recent years, few attempts have been devoted to investigate the effect of anisotropy and in particular the influence of the anisotropic elastic behavior on the localization phenomenon. In Part I of this paper, the localization analysis will be performed for an elastoplastic von Mises material with transversely isotropic elasticity subjected to uniaxial tension. Numerical and analytical results showing the influence of the deviation from isotropic elasticity on the onset of strain localization are presented. The anisotropic elastic behavior may substantially trigger the shear band initiation and modify the corresponding failure pattern. These observations become critical for highly damaged materials. Further developments concerning both transversely isotropic elasticity and plasticity are given in Part II of the paper.     

Incorporating the effects of fabric in the dilatant double shearing model for planar deformation of granular materials

The objective of this paper is to incorporate the effects of fabric and its evolution into the Dilatant Double Shearing Model [Mehrabadi, M.M., Cowin, S.C., 1978. Initial planar deformation of dilatant granular materials. J. Mech. Phys. Solids 26, 269–284] for granular materials in order to capture the anisotropic behavior and the complex response of granular materials in cyclic shear loading. An important consequence of considering the fabric is that one can have unequal shearing rates along the two slip directions. This property leads to the non-coaxiality of the principal axes of stress and strain rate, which is more appropriate for a material that exhibits initial and induced anisotropy. In addition, we employ a fabric-dependent elasticity tensor with orthotropic symmetry. The model developed in this paper also predicts one of the experimentally observed characteristics of granular materials: the gradual concentration of the contact normals towards the maximum principal stress direction.We implement the constitutive equations into ABAQUS/Explicit by writing a user material subroutine in order to predict the strength anisotropy of granular materials in a plane strain biaxial compression test and investigate the mechanical behavior of granular materials under the cyclic shear loading conditions. The predictions from this model show good quantitative agreement with the experiments of [Park, C.S., 1990. Anisotropy in deformation and strength properties of sands in plane strain compression, Masters Thesis, University of Tokyo; Park, C.S., Tatsuoka, F., 1994. Anisotropic strength and deformation of sands in plane strain compression. In: XIII ICSMFE, New Delhi, India; Okada, N., 1992. Energy dissipation in inelastic flow of cohesionless granular media. Ph.D. Thesis, University of California, San Diego].     

Photomechanics of light-activated polymers

Light-activated polymers are an exciting class of modern materials that respond mechanically when irradiated by light at particular wavelengths. While details of the mechanisms that connect the optical excitation to mechanical behavior are complex and differ from material to material, there is sufficient commonality among them to permit the development of a generalized modeling framework to describe the photomechanics. The features shared by light-activated polymers involve light interacting with the material, which triggers photochemical reactions that alter the structure of the crosslinked polymer network. Many such structural alterations result in an evolution of the polymer network, and subsequent macroscopic deformation. When this process is appropriately executed it can enable a photomechanical shape-memory effect. In this paper, we develop a three-dimensional finite-deformation modeling framework to describe the photomechanical response of light-activated polymer systems. This framework integrates four coupled phenomena that contribute to macroscopic photomechanical behavior: photophysics, photochemistry, chemomechanical coupling, and mechanical deformation. The chemomechanical coupling consists of chemically induced structural alterations of the crosslinked network that result in subsequent deformation. We describe this behavior through a decomposition of the crosslinked network into two components consisting of an original network and a photochemically altered network; both evolve during photomechanical deformation. The modeling framework presented in this paper is sufficiently general that it is applicable to light-activated polymer systems that operate with various mechanisms in each of the four areas. Using this modeling approach, we develop constitutive models for two recently developed light-activated polymer systems [Lendlein, A., Hongyan, J., Junger, O., Langer, R., 2005. Light-induced shape-memory polymers. Nature 434 (7035) 879; Scott, T.F., Schneider, A.D., Cook, W.D., Bowman, C.N., 2005. Photoinduced plasticity in crosslinked polymers. Science 308 (5728) 1615]. For the material developed by Scott and his co-workers we validate our model by measuring and numerically simulating photo-induced stress relaxation and bending deformation and obtain good agreement between measurements and predictions. Finally, we use the model to study the effects of photomechanical parameters (applied strain magnitude, irradiation time and intensity, and photoabsorber concentration) and the behavior of the network evolution rule on the material response.     

HIA: A Hybrid Integral Approach to model incompressible isotropic hyperelastic materials—Part 1: Theory

This paper provides a new constitutive model for rubber-like materials. The model adds to the 8-chain density introduced by Arruda and Boyce, two phenomenological components: an original part made of an integral density and an interleaving constraint part represented by a logarithmic function as proposed by Gent and Thomas. The model contains six rheological parameters connected to the polymer chemistry and to the macroscopic behavior. Four sets of experimental data from the literature are used to identify the rheological parameters and to assess the proposed model. The model is able to reproduce with a good accuracy experimental data performed under different loading conditions such as uniaxial and equibiaxial tension, uniaxial compression, pure and simple shear as well as the Mooney plot.     

Thermal stresses in two- and three-component anisotropic materials

This paper deals with an analytical model of thermal stresses which originate during a cooling process of an anisotropic solid continuum with uniaxial or triaxial anisotropy. The anisotropic solid continuum consists of anisotropic spherical particles periodically distributed in an anisotropic infinite matrix. The particles are or are not embedded in an anisotropic spherical envelope, and the infinite matrix is imaginarily divided into identical cubic cells with central particles. The thermal stresses are thus investigated within the cubic cell. This mulfi-particle-（envelope）-matrix system based on the cell model is applicable to two- and three-component materials of precipitate-matrix and precipitate-envelope-matrix types, respectively. Finally, an analysis of the determination of the thermal stresses in the multi-par- ticle-（envelope）-matrix system which consists of isotropic as well as uniaxial- and/or triaxial-anisotropic components is presented. Additionally, the thermal-stress induced elastic energy density for the anisotropic components is also derived. These analytical models which are valid for isotropic, anisotropic and isotropic-anisotropic multi-particle- （envelope）-matrix systems represent the determination of important material characteristics. This analytical determination includes： （1） the determination of a critical particle radius which defines a limit state regarding the crack initiation in an elastic, elastic-plastic and plastic components; （2） the determination of dimensions and a shape of a crack propagated in a ceramic components; （3） the determination of an energy barrier and micro-/macro-strengthening in a component; and （4） analytical-（experimental）-computational methods of the lifetime prediction. The determination of the thermal stresses in the anisotropic components presented in this paper can be used to determine these material characteristics of real two- and three-component materials with anisotropic components or with anisotropic and isotropic components.