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

V Racherla P. Ponte Castaeda () 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 Linear comparison estimates for the effective resistivity of three-dimensional nonlinear polycrystals BY V. RACHERLA1 AND P. PONTE CASTA NEDA2,* 1LMS, Departement de Mecanique, Ecole Polytechnique, 91128 Palaiseau, France 2Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA Estimates for the effective resistivity of nonlinear polycrystals are obtained using the linear comparison homogenization scheme of DeBotton and Ponte Castaneda (DeBotton & Ponte Castaneda 1995 Proc. R. Soc. A 448, 121142). 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, 26132625), which dramatically improves upon the classical Taylor upper bound at large crystal anisotropy. In addition, the estimates also satisfy the HashinShtrikman bounds, which are more restrictive than the GarroniKohn 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. 1. Introduction In this work, we are primarily concerned with the effective resistivity of random nonlinear polycrystals with overall isotropic behaviour. Thus, the statistical distribution of grains is taken to be isotropic in both space and orientation so that the polycrystal behaviour is isotropic even for highly anisotropic and nonlinear single-crystal behaviour. The classical Taylor and Sachs estimates provide easy upper (UB) and lower bounds (LB), for effective resistivity. However, these bounds can be far from optimal as they do not use microstructural information beyond the one-point statistics of the crystal orientations (or crystallographic texture). Methods with demonstrated capabilities to significantly improve upon the classical bounds include: (i) the translation method and (ii) the linear comparison method. The translation method is a powerful method that was developed initially for linear composites (see Milton 2002) and has been used recently to obtain dramatic improvements upon the Taylor UB for some model, two-dimensional, ideally plastic polycrystals (Kohn & Little 1998; Nesi et al. 2000; Goldsztein 2001), as well as for three-dimensional polycrystals in the context of dielectric breakdown (Garroni & Kohn 2003). Although this method is difficult to implement for general types of nonlinearities, and may be of limited value in applications, such as in plasticity, it provides rigorous results against which the predictions of other more general methods may be checked for accuracy. The linear comparison method is a variational procedure that can be used easily and conveniently to extend bounds and estimates for linear composites to nonlinear composites with identical microstructures. It was originally proposed by Ponte Castaneda (1991) for nonlinear composites with isotropic phases, and generalized for viscoplastic polycrystals by DeBotton & Ponte Castaneda (1995). This method can provide improved predictions, over the classical uniform-field bounds, for nonlinear composites by making use of microstructural information beyond the one-point statistics. It was first used to obtain bounds of the Hashin Shtrikman (HS) type for nonlinear polycrystals by DeBotton & Ponte Castaneda (1995). Earlier bounds for polycrystals were given by Willis (1994), using a nonlinear generalization of the HS variational method by Talbot & Willis (1985). However, it is now well understood (Milton 2002) that the HS boundsalthough exact to second order in the heterogeneity contrastare not optimal even for linear polycrystals, especially at large contrast. Owing to this, although improving on the classical bounds, the HS bounds are also not very useful in applications. On the other hand, it has been found that the use of the classical self-consistent (SC) estimateswhich can also incorporate information on crystal orientations and average grain shapestogether with the linear comparison method provides realistic predictions, at least for statistically isotropic nonlinear polycrystals. Thus, it was found that the SC predictions for the effective flow stress of model two-dimensional viscoplastic polycrystals satisfy the KohnLittle bound (Ponte Castaneda & Nebozhyn 1997), and agree reasonably well with the results of full-field numerical simulations (Lebensohn et al. 2004). However, the main advantage of the linear comparison method is that it is quite general and has been foundtogether with a slight generalization, called the second-order methodto be very useful in plasticity ( Nebozhyn et al. 2001; Lebensohn et al. 2007). In this paper, we compute SC estimates for the effective resistivity behaviour of statistically isotropic, three-dimensional, power-law polycrystals, using the linear comparison method. These SC estimates will be shown to satisfy all the bounds, including the recent translation UB of Garroni & Kohn (2003), in the limit of threshold-type behaviour for the electric field. Furthermore, it will be shown that at large crystal anisotropy, the scaling of the SC estimates is actually sharper than that of any of the known bounds. In their paper, Garroni & Kohn (2003) show that the UB for effective resistivity derived using the translation method gives a superior scaling law than a bound derived using the linear comparison method in conjunction with the bound by Avellaneda et al. (1988), which is optimal for linear isotropic polycrystals. It will be shown here that polycrystals with statistically isotropic microstructures (which have isotropic overall behaviour even in the nonlinear case) form only a subclass Effective resistivity of polycrystals of the set of all linear polycrystals with isotropic overall behaviour. This will help explain why the bound derived using the linear comparison method in conjunction with the linear bound of Avellaneda et al. (1988) turns out to be less sharp than the bound derived directly using the translation method. 2. Mathematical formulation of the problem vw N J Z vE x; E; with wx; E Z X crxwrE; 2:1 rZ1 where the c(r) denote the characteristic functions for the phases (i.e. crx (...truncated)