P. Ponte Castañeda

The effective mechanical properties of nonlinear isotropic composites

Abstract A new variational structure is proposed that yields a prescription for the effective energy potentials of nonlinear composites in terms of the corresponding energy potentials for linear composites with the same microstructural distributions. The prescription can be used to obtain bounds and estimates for the effective mechanical properties of nonlinear composites from any bounds and estimates that may be available for the effective properties of linear composites. The main advantages of the procedure are the simplicity of its implementation, the generality of its applications and the strength of its results. The general prescription is applied to three special nonlinear composites : a porous material, a two-phase incompressible composite and a rigidly reinforced material. The results are compared with previously available results for the special case of power-law constitutive behavior.

Exact second-order estimates for the effective mechanical properties of nonlinear composite materials

Abstract Motivated by previous small-contrast perturbation estimates, this paper proposes a new method for estimating the effective behavior of nonlinear composite materials with arbitrary phase contrast. The key idea is to write down a second-order Taylor expansion for the phase potentials, about appropriately defined phase average strains. The resulting estimates, which are exact to second order in the contrast, involve the “tangent” modulus tensors of the nonlinear phase potentials, and reduce the problem for the nonlinear composite to a linear problem for an anisotropic thermoelastic composite. Making use of a well-known result by Levin for two-phase thermoelastic composites, together with estimates of the Hashin-Shtrikman type for linear elastic composites, explicit results are generated for two-phase nonlinear composites with statistically isotropic particulate microstructures. Like the earlier small-contrast asymptotic results, the new estimates are found to depend on the determinant of the strain, but unlike the small-contrast results that diverge for shear loading conditions in the nonhardening limit, the new estimates remain bounded and reduce to the classical lower bound in this limiting case. The general method is applied to composites with power-law constitutive behavior and the results are compared with available bounds and numerical estimates, as well as with other nonlinear homogenization procedures. For the cases considered, the new estimates are found to satisfy the restrictions imposed by the bounds, to improve on the predictions of prior homogenization procedures and to be in excellent agreement with the results of the numerical simulations.

Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—theory

Abstract This paper is concerned with the development of an improved second-order homogenization method incorporating field fluctuations for nonlinear composite materials. The idea is to combine the desirable features of two different, earlier methods making use of “linear comparison composites”, the properties of which are chosen optimally from suitably designed variational principles. The first method (Ponte Castaneda, J. Mech. Phys. Solids 39 (1991) 45) makes use of the “secant” moduli of the phases, evaluated at the second moments of the strain field over the phases, and delivers bounds, but these bounds are only exact to first-order in the heterogeneity contrast. The second method (Ponte Castaneda, J. Mech. Phys. Solids 44 (1996) 827) makes use of the “tangent” moduli, evaluated at the phase averages (or first moments) of the strain field, and yields estimates that are exact to second-order in the contrast, but that can violate the bounds in some special cases. These special cases turn out to correspond to situations, such as percolation phenomena, where field fluctuations, which are captured less accurately by the second-order method than by the bounds, become important. The new method delivers estimates that are exact to second-order in the contrast, making use of generalized secant moduli incorporating both first- and second-moment information, in such a way that the bounds are never violated. Some simple applications of the new theory are given in Part II of this work.

New variational principles in plasticity and their application to composite materials

In this paper, a variational method for bounding the effective properties of nonlinear composites with isotropic phases, proposed recently by ponte castaneda (J. Mech. Phys. Solids 39, 45, 1991), is given full variational principle status. Two dual versions of the new variational principle are presented and their equivalence to each other, and to the classical variational principles, is demonstrated. The variational principles are used to determine bounds and estimates for the effective energy functions of nonlinear composites with prescribed volume fractions in the context of the deformation theory of plasticity. The classical bounds of Voigt and Reuss for completely anisotropic composites are recovered from the new variational principles and are given alternative, simpler forms. Also, use of a novel identity allows the determination of simpler forms for nonlinear Hashin-Shtrikman bounds, and estimates, for isotropic, particle-reinforced composites, as well as for transversely isotropic, fiber-reinforced composites. Additionally, third-order bounds of the Beran type are determined for the first time for nonlinear composites. The question of the optimality of these bounds is discussed briefly.

Self-consistent modelling of the mechanical behaviour of viscoplastic polycrystals incorporating intragranular field fluctuations

We present a detailed description of the numerical implementation, within the widely used viscoplastic self-consistent (VPSC) code, of a rigorous second-order (SO) homogenization procedure for non-linear polycrystals. The method is based on a linearization scheme, making explicit use of the covariance of the fluctuations of the local fields in a certain linear comparison material, whose properties are, in turn, determined by means of a suitably designed variational principle. We discuss the differences between this second-order approach and several first-order self-consistent (SC) formulations (secant, tangent and affine approximations) by comparing their predictions with exact full-field solutions. We do so for crystals with different symmetries, as a function of anisotropy, number of independent slip systems and degree of non-linearity. In this comparison, the second-order estimates show the best overall agreement with the full-field solutions. Finally, the different SC approaches are applied to simulate texture evolution in two strongly heterogeneous systems and, in both cases, the SO formulation yields results in better agreement with experimental evidence than the first-order approximations. In the case of cold-rolling of low-SFE fcc polycrystals, the SO formulation predicts the formation of a texture with most of the characteristic features of a brass-type texture. In the case of polycrystalline ice, deforming in uniaxial compression to large strain, the SO predicts a substantial and persistent accommodation of deformation by basal slip, even when the basal poles become strongly aligned with the compression direction.

Constitutive models for porous materials with evolving microstructure

Abstract A constitutive model is developed for the effective behavior of nonlinear porous materials which is capable of accounting, approximately, for the evolution of the materials microstructure under large quasi-static deformations. The model is formulated in terms of an effective potential function for the porous material, which depends on appropriate variables characterizing the state of the microstructure, together with evolution equations for these state variables. For the special case of triaxial loading of an initially isotropic porous material, the appropriate state variables are the porosity and the aspect ratios of the typical void; they serve respectively to characterize the evolution of the size and shape of the pores. The implications of the model are studied in the context of two specific examples: axisymmetric and plane strain loading conditions. It is found that the porosity acts as a hardening mechanism when the material is subjected to boundary conditions resulting in overall hydrostatic compression, and as a softening mechanism for overall hydrostatic tension. On the other hand, the change in shape of the voids is found to have a more subtle influence on the overall behavior of the porous material. Thus the change in shape of the voids has a direct effect which may range from strong softening during void collapse to slight hardening during void elongation, but it also has an indirect effect through its concomitant effect on the evolution of the porosity, which may actually be quite significant. Because of the complex interplay between these hardening-softening mechanisms, the new model is found to yield significantly different predictions, in particular for the onset of localization, than the well-known model of Gurson, which neglects the change in shape of the voids, especially, for low-triaxiality loading conditions. The model, in its present form, is not meant to be used for high-triaxialities for which the Gurson and other associated models are considered to be quite accurate.

A general constitutive theory for linear and nonlinear particulate media with microstructure evolution

Abstract This work is concerned with the development of a constitutive theory for composite materials with particulate microstructures, which is capable of predicting, approximately, the evolution of the microstructure and its influence on the effective response of composites under general three-dimensional finitestrain loading conditions, such as those present in metal-forming operations. In its present form, the theory is general enough to be used for linearly viscous, nonlinearly viscous and perfectly plastic composites with randomly oriented and distributed ellipsoidal inclusions (or pores), which, in the most general case, can change size, shape and orientation. In addition, the “shape” and “orientation” of their center-to-center statistical distribution functions can also evolve with the deformation. To illustrate the key features of the new theory in the context of a simple example, an application is carried out for plane-strain loading of two-phase systems consisting of random distributions of aligned rigid particles in a power-law matrix phase. The results show that the evolution of the relevant microstructural variables, as well as the effective response, depend in a complex fashion on the initial state of the microstructure, as well as on the specific boundary conditions. In particular, it is found that the changes in orientation of the particles provide a mechanism analogous to “geometric softening” in ductile single crystals, which can lead to significant changes in the instantaneous hardening rate of the composite. This is shown to have important consequences for the possible onset of shear localization in the composite.

Variational estimates for the creep behaviour of polycrystals

A variational procedure is developed for estimating the effective constitutive behaviour of polycrystalline materials undergoing high-temperature creep. The procedure is based on a new variational principle allowing the determination of the effective potential function of a given nonlinear polycrystal in terms of the corresponding potential for a linear comparison polycrystal with an identical geometric arrangements of its constituent single-crystal grains. As such, it constitutes an extension, to locally anisotropic behaviour, of the variational procedure developed by Ponte Castañeda (1991) for nonlinear heterogeneous media with locally isotropic behaviour. By way of an example, the procedure is applied to the determination of bounds of the Hashin-Shtrikman type for the effective potentials of statistically isotropic nonlinear polycrystals. The bounds are computed for the special class of untextured FCC polycrystals with isotropic pure power-law viscous behaviour, first considered by Hutchinson (1976), in the context of a calculation of the self-consistent type. The new bounds are found to be more restrictive than the corresponding classical Taylor-Bishop-Hill bounds, and also more restrictive, if only slightly so, than related bounds of the Hashin-Shtrikman type by Dendievel et al. (1991). The new procedure has the advantage over the self-consistent procedure of Hutchinson (1976) that it may be applied, without any essential complications, to aggregates of crystals with slip systems exhibiting different creep rules - with, for example, different power exponents - and to general loading conditions. However, the distinctive feature of the new variational procedure is that it may be used in conjunction with other types of known bounds and estimates for linear polycrystals to generate corresponding bounds and estimates for nonlinear polycrystals.

Asymptotic fields in steady crack growth with linear strain-hardening

Abstract The asymptotic stress and velocity fields of a crack propagating steadily and quasi-statically into an elastic-plastic material are presented. The material is characterized by J 2 -flow theory with linear strain- hardening. The possibility of reloading on the crack flanks is taken into account. The cases of anti-plane strain (mode III), plane strain (modes I and II), and plane stress (modes I and II) are considered. Numerical results are given for the strength of the singularity and for the distribution of the stress and velocity fields in the plastic loading, elastic unloading and plastic reloading regions, as functions of the strain-hardening parameter. An attempt is made to make a connection with the perfectly-plastic solutions in the limit of vanishing strain-hardening.

A second-order homogenization method in finite elasticity and applications to black-filled elastomers

Abstract This work is concerned with the development of an analytical method for estimating the macroscopic behavior of heterogeneous elastic systems subjected to finite deformations. The objective is to generate variational estimates for the effective or homogenized stored-energy function of hyperelastic composites, which will be accomplished by means of a suitable generalization of the “second-order procedure” of Ponte Castaneda (Ponte Castaneda, P., 1996. J. Mech. Phys. Solids 44, 827–862). The key idea in this method is the introduction of an optimally chosen “linear thermoelastic comparison composite,” which can then be used to convert available homogenization estimates for linear systems directly into new estimates for nonlinear composites. To illustrate the use of the method, an application is given for carbon-black filled elastomers and estimates analogous to the well-known Hashin–Shtrikman and self-consistent estimates for linear-elastic composites are generated.