Effective potentials in nonlinear polycrystals and quadrature formulae

This study presents a family of estimates for effective potentials in nonlinear polycrystals. Noting that these potentials are given as averages, several quadrature formulae are investigated to express these integrals of nonlinear functions of local fields in terms of the moments of these fields. Two of these quadrature formulae reduce to known schemes, including a recent proposition (Ponte Castañeda 2015 Proc. R. Soc. A471, 20150665 (doi:10.1098/rspa.2015.0665)) obtained by completely different means. Other formulae are also reviewed that make use of statistical information on the fields beyond their first and second moments. These quadrature formulae are applied to the estimation of effective potentials in polycrystals governed by two potentials, by means of a reduced-order model proposed by the authors (non-uniform transformation field analysis). It is shown how the quadrature formulae improve on the tangent second-order approximation in porous crystals at high stress triaxiality. It is found that, in order to retrieve a satisfactory accuracy for highly nonlinear porous crystals under high stress triaxiality, a quadrature formula of higher order is required.

Fully optimized second-order variational estimates for the macroscopic response and field statistics in viscoplastic crystalline composites

A variational method is developed to estimate the macroscopic constitutive response of composite materials consisting of aggregates of viscoplastic single-crystal grains and other inhomogeneities. The method derives from a stationary variational principle for the macroscopic stress potential of the viscoplastic composite in terms of the corresponding potential of a linear comparison composite (LCC), whose viscosities and eigenstrain rates are the trial fields in the variational principle. The resulting estimates for the macroscopic response are guaranteed to be exact to second order in the heterogeneity contrast, and to satisfy known bounds. In addition, unlike earlier ‘second-order’ methods, the new method allows optimization with respect to both the viscosities and eigenstrain rates, leading to estimates that are fully stationary and exhibit no duality gaps. Consequently, the macroscopic response and field statistics of the nonlinear composite can be estimated directly from the suitably optimized LCC, without the need for difficult-to-compute correction terms. The method is applied to a simple example of a porous single crystal, and the results are found to be more accurate than earlier estimates.

Bounding the plastic strength of polycrystalline voided solids by linear-comparison homogenization techniques

The elastoplastic response of polycrystalline voided solids is idealized here as rigid-perfectly plastic. Bounds on the macroscopic plastic strength for prescribed microstructural statistics and single-crystal strength are computed be means of a linear-comparison homogenization technique developed by Idiart & Ponte Castañeda (2007 Proc. R. Soc. A 463 , 907–924. ( doi:10.1098/rspa.2006.1797 )). Hashin–Shtrikman (HS) and Self-Consistent (SC) results in the form of yield surfaces are reported for cubic and hexagonal polycrystals with isotropic texture and varying degrees of crystal anisotropy. In all cases, the surfaces are smooth, closed and convex. Improvements over earlier linear-comparison bounds of up to 40% are found at high-stress triaxialities. New HS results can even be sharper than earlier SC results for some material systems. In the case of deficient crystals, the SC results assert that voided aggregates of crystals with four independent systems can accommodate arbitrary deformations, those with three independent systems can dilate but not distort, and those with fewer than three independent systems cannot deform at all. We report the sharpest bounds available to date for all classes of material systems considered.

Bounding the plastic strength of polycrystalline solids by linear-comparison homogenization methods

The elastoplastic response of polycrystalline metals and minerals above their brittle–ductile transition temperature is idealized here as rigid–perfectly plastic. Bounds on the overall plastic strength of polycrystalline solids with prescribed microstructural statistics and single-crystal plastic strength are computed by means of a linear-comparison homogenization method recently developed by Idiart & Ponte Castañeda (Idiart & Ponte Castañeda 2007 Proc. R. Soc. A 463 , 907–924 ( doi:10.1098/rspa.2006.1797 )). Hashin–Shtrikman and self-consistent results are reported for cubic and hexagonal polycrystals with varying degrees of crystal anisotropy. Improvements over earlier linear-comparison bounds are found to be modest for high-symmetry materials but become appreciable for low-symmetry materials. The largest improvement is observed in self-consistent results for low-symmetry hexagonal polycrystals, exceeding 15 per cent in some cases. In addition to providing the sharpest bounds available to date, these results serve to evaluate the performance of the aforementioned linear-comparison method in the context of realistic material systems.

Linear comparison estimates for the effective resistivity of three-dimensional nonlinear polycrystals

Estimates for the effective resistivity of nonlinear polycrystals are obtained using the ‘linear comparison’ homogenization scheme of DeBotton and Ponte Castañeda (DeBotton & Ponte Castañeda 1995 Proc. R. Soc. A 448 , 121–142). Computing the effective properties of linear composites, with the same microstructure as the nonlinear composite, is an essential part of this scheme. The classical self-consistent method is employed for this purpose. An important characteristic of these estimates, for polycrystals with field thresholds, is that they satisfy the recent bound of Garroni and Kohn (Garroni & Kohn 2003 Proc. R. Soc. A 459 , 2613–2625), which dramatically improves upon the classical Taylor upper bound at large crystal anisotropy. In addition, the estimates also satisfy the Hashin–Shtrikman bounds, which are more restrictive than the Garroni–Kohn bound at small crystal anisotropy. Interestingly, the scaling exponents for the linear comparison estimates are found to be independent of the constitutive nonlinearity. This last observation provides an explanation for the relative weakness of an earlier linear comparison bound obtained by Garroni and Kohn.

Variational linear comparison bounds for nonlinear composites with anisotropic phases. II. Crystalline materials

In part I of this work, bounds were derived for the effective potentials of nonlinear composites with anisotropic constituents, making use of an appropriate generalization of the linear comparison variational method. In this second part, the special case of nonlinear composites with crystalline constituents is considered. First, it is shown that, for this special but very important class of materials, the ‘variational’ bounds of part I are at least as good as an earlier version of the bounds due to deBotton & Ponte Castañeda. Then, the relative merits of these two types of bounds are studied in the context of a model, two-dimensional, porous composite with a power-law crystalline matrix phase, under anti-plane loading conditions. The results show that, indeed, the variational bounds of part I improve, in general, on the earlier bounds, with the former becoming progressively sharper than the latter as the number of slip systems of the crystalline matrix phase increases. In particular, it is shown that, unlike the bounds of deBotton & Ponte Castañeda, the variational bounds of part I are able to recover the variational bound for composites with an isotropic matrix phase, as the number of slip systems, all having the same flow stress, tends to infinity.

Variational linear comparison bounds for nonlinear composites with anisotropic phases. I. General results

This work is concerned with the development of bounds for nonlinear composites with anisotropic phases by means of an appropriate generalization of the ‘linear comparison’ variational method, introduced by Ponte Castañeda for composites with isotropic phases. The bounds can be expressed in terms of a convex (concave) optimization problem, requiring the computation of certain ‘error’ functions that, in turn, depend on the solution of a non-concave/non-convex optimization problem. A simple formula is derived for the overall stress–strain relation of the composite associated with the bound, and special, simpler forms are provided for power-law materials, as well as for ideally plastic materials, where the computation of the error functions simplifies dramatically. As will be seen in part II of this work in the specific context of composites with crystalline phases (e.g. polycrystals), the new bounds have the capability of improving on earlier bounds, such as the ones proposed by deBotton and Ponte Castañeda for these specific material systems.

A model problem concerning recoverable strains of shape-memory polycrystals

This paper addresses a model problem of nonlinear homogenization motivated by the study of the shape-memory effect in polycrystalline media. Specifically, it numerically computes the set of recoverable strains in a polycrystal given the set of recoverable strains of a single crystal in the two-dimensional scalar (or antiplane shear) setting. This problem shares a direct analogy with crystal plasticity. The paper considers typical or random polycrystals where the grains are generated by a Voronoi tesselation of a set of random points and are randomly oriented. The numerical results show that for such microstructures, the Taylor bound appears to be the most accurate (though pessimistic) bound when the anisotropy is moderate, and that recent Kohn–Little–Goldsztein outer bounds overestimate the recoverable strains when the anisotropy is large. The results also show that the stress tends to localize on tortuous paths that traverse (poorly oriented) grains as the polycrystal reaches its limit of recoverable strain.