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

Martn I Idiart Pedro Ponte Castaeda () 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 - Email alerting service To subscribe to Proc. R. Soc. A go to: http://rspa.royalsocietypublishing.org/subscriptions Variational linear comparison bounds for nonlinear composites with anisotropic phases. II. Crystalline materials BY MARTIN I. IDIART1,2 AND PEDRO PONTE CASTAN EDA1,2,* 1Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA 2Laboratoire de Mecanique des Solides, C.N.R.S. UMR 7649, Departement de Mecanique, Ecole Polytechnique, 91128 Palaiseau Cedex, France 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 Castaneda. 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 Castaneda, 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. 1. Crystalline phases and polycrystals In part I of this work (Idiart & Ponte Castaneda 2007), bounds have been derived for the effective stress potentials of nonlinear composites made of a fairly general class of anisotropic constituents, satisfying a certain square convexity hypothesis. These bounds were obtained by making use of an appropriate generalization of the linear comparison variational method, introduced by Ponte Castaneda (1991) in the context of composites with isotropic constituents, and will be referred to here as variational bounds. In this second part of the work, we consider the special case of nonlinear composites with crystalline constituents, including polycrystals, which is perhaps the most common type of composite material with anisotropic constituents. It will be shown that the variational bounds are at least as good as an earlier version of the bounds due to deBotton & Ponte Castaneda (1995), which were developed specifically for nonlinear composites with crystalline constituents. We consider a reference single crystal which is capable of undergoing viscoplastic deformation on a set of K preferred crystallographic slip systems. These systems are characterized by the second-order tensors m(k), kZ1, ., K, defined by where n(k) and m(k) are the unit vectors normal to the slip plane and along the slip direction in the k th system, respectively. When the crystal is subjected to an applied stress s, the resolved shear stress acting on the k th slip system is given by t(k)Zs$m(k) and the strain (rate) 3 in the crystal is the superposition of the strain (rates) g(k) on each slip system k (kZ1, ., K ). They are assumed to depend on the resolved shear stress t(k), through a slip potential j(k), such that gk Z j0ktk. For consistency with the hypothesis of square convexity introduced in part I, the potentials j(k) will be assumed here to be convex in the variable t2k (and are therefore also convex in t(k)). A commonly used form for the slip potentials j(k) is the power-law form where mZ1/n (0%m%1) and (t0)(k) are the strain-rate sensitivity and flow stress of the k th slip system, respectively, and g0 is a reference strain rate. Note that the limiting values of the exponent mZ1 and 0 correspond to linear and rigid-ideally plastic behaviours, respectively. In this connection, it is recalled that, even though the slip potentials j(k) are not differentiable in the rigid-ideally plastic case, it is still possible to relate g(k) and t(k) via the subdifferential of convex analysis. Since the phases in a composite made of such crystalline materials may also exhibit different orientations, it is useful to introduce a set of rotation tensors R(r) (rZ1 ., N ). Then, defining phase r as the region occupied by all crystals of a given type and orientation R(r), its constitutive behaviour is characterized by the stress potential trk Z s$ RrT mrkRr : 1:4 It is recalled that a polycrystal is an aggregate of a large number of identical single crystals with different orientations. This special case is included in expression (1.3), provided that all the phase slip systems and potentials be taken identical to each other (mrk Z mk and jrk Z jk). But the definition (1.3) is general enough to include multi-phase polycrystals, as well as composites with crystalline phases, such as the porous crystalline materials considered in 1a. (a ) Variational bounds Since the phase stress potentials (1.3) have been assumed to be square convex, the variational bound given in result 4.4 of part I holds and can be used for the above-defined class of composites with crystalline phases. Result 1.1. The effective stress potential u of an N-phase nonlinear composite e (or polycrystal) with crystalline phase potentials (1.3) is bounded below by ( ) sup ; where the error functions v(r) are given by vr S0r Z susp 12 1 u~0s Z 2 s$S~0 S0s s; 1:7 is the effective stress potential of a linear comparison composite (LCC) with uniform compliance tensors S0r in each of the phases (rZ1 ., N ), and effective compliance tensor S~0 As discussed in remark 4.5 of part I, the bound (1.5) involves a non-smooth, concave optimization problem for the variables S0r, which can be solved by making use of appropriate numerical methods. However, as will be seen in the context of the model problem considered below, the main difficulty in the determination of this bound lies in the computation of the error functions v(r), which involve a non-concave optimization problem that must be solved making use of more sensitive numerical algorithms. r If the slip potentials jk are all of the power-law type (1.2) with the same exponent n, as in the model problem considered in 2, the bound (1.5) admits the following alternative representation, as can be deduced from result 4.6 of part I. Result 1.2. The effective stress potential u of an N-phase, power-law composite e (or polycrystal) with crystalline phase potentials given by equation (1.3), together with equatio (...truncated)