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

Martn I. Idiart * Articles on similar topics can be found in the following collections Receive free email alerts when new articles cite this article - sign up in the box at the top right-hand corner of the article or click here - Subject collections Email alerting service To subscribe to Proc. R. Soc. A go to: http://rspa.royalsocietypublishing.org/subscriptions BY MARTN I. IDIART1,2,* The elastoplastic response of polycrystalline metals and minerals above their brittle ductile transition temperature is idealized here as rigidperfectly 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 Castaeda (Idiart & Ponte Castaeda 2007 Proc. R. Soc. A 463, 907924 (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. 1. Motivation The elastoplastic response of polycrystalline metals and minerals above their brittleductile transition temperature is to a great extent dictated by the morphology, lattice orientation and elastoplastic response of each individual single-crystal grain composing the aggregate. Relating the macroscopic response with the microscopic properties is necessary to estimate the deformation-induced plastic anisotropy that develops in these materials when subjected to large deformations, a problem relevant to various engineering applications such as metal-forming processessee the monograph by Kocks et al. (1998). Very often the response of these materials is idealized as elastically rigid and plastically Bounding the plastic strength non-hardening. Within this the so-called rigidperfectly plastic model, the above problem reduces to finding the macroscopic yield surface of the polycrystal given the yield surface at the single-crystal level and the statistics of the morphology and orientation distributions of the grains. Owing to their inherent microstructural randomness, cognate polycrystalline solids do not exhibit a single response but ahopefully narrowrange of responses. Therefore, one can either develop estimates that yield a single representative response or derive bounds for the entire range of possible responses. This work is concerned with bounds. Bounds are also useful for two additional reasons: they provide benchmarks to test estimates and they can be used as estimates themselves. The simplest bounds for the yield surface of polycrystalline solids are the outer bound of Taylor (1938) and the inner bound of Reuss (1929). Their extremal character was proved by Bishop & Hill (1951). These elementary bounds are obtained by assuming uniform strain-rate and uniform stress fields in the classical minimum energy principles, and make use of one-point microstructural statistics only. They have proved useful in the context of high-symmetry polycrystalline solidslike face-centred cubic solidswhere the heterogeneity contrast is low, but as crystal anisotropy increases their predictions diverge and become highly inaccurate. This fact has motivated the development of refined bounds that can incorporate higher order microstructural statistics. Most of those bounds have been derived in the context of nonlinear viscoplasticity, which includes rigid-perfect plasticity as a limiting case. The first bounds for polycrystalline solids that account for two-point statistics were derived by Dendievel et al. (1991) via the nonlinear HashinShtrikman procedure initially proposed by Willis (1983) and developed further by Talbot & Willis (1985). A more general method inspired by the linear-comparison procedure of Ponte Castaeda (1991) was later proposed by deBotton & Ponte Castaeda (1995) and developed further by Ponte Castaeda & Suquet (1998). This linear-comparison method allows the use of any available bound for linearly elastic polycrystals to produce bounds for nonlinear viscoplastic polycrystals, thus having the potential of incorporating higher order statistics. The optimal linearization is obtained via suitably designed variational principles. This linear-comparison method was applied to cubic and hexagonal polycrystals by Nebozhyn et al. (2000, 2001) and Liu & Ponte Castaeda (2004), who showed that the linear-comparison predictions for rigidperfectly plastic polycrystals improve significantly over the elementary predictions of Taylor and Reuss, especially when crystal anisotropy is large. In addition, their results served to demonstrate the inconsistency of an incremental theory of polycrystalline plasticity proposed by Hill (1965) and Hutchinson (1976), which was found to violate the bounds. More recently, Lebensohn et al. (2011) have reported linearcomparison bounds for (two-phase) voided polycrystals and have shown that tangent theories of polycrystalline plasticity such as those proposed by Molinari et al. (1987), Lebensohn & Tom (1993) and Bornert et al. (2001) can also violate bounds. Idiart & Ponte Castaeda (2007a) have recently shown that the linearcomparison methods of deBotton & Ponte Castaeda (1995) and Ponte Castaeda & Suquet (1998) make implicit use of a relaxation in their linearization scheme which weakens the resulting bounds. Eliminating this relaxation leads to sharper bounds but increases the computational complexity. Idiart & Ponte Castaeda (2007a,b) derived non-relaxed linear-comparison bounds and applied them to a model material consisting of a porous single crystal with cylindrical symmetry under anti-plane loading. They found that the improvement over the relaxed bounds increased with increasing number of crystal slip systems, being as much as 10 per cent in some extreme cases. Motivated by these developments, the method of Idiart & Ponte Castaeda (2007a) is used in this work to bound the plastic strength of a fairly general class of polycrystalline solids made of cubic and hexagonal single crystals with varying degrees of plastic anisotropy. In addition to providing the sharpest bounds available to date for polycrystalline plastic solids, these results are used to evaluate the performance of the non-relaxed linear-comparison method in the context of realistic material systems. 2. The polycrystalline solid model Polycrystals are idealized here as random aggregates of perfectly bonded single crystals (i.e. grains). Individual grains are (...truncated)