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

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. I. General results 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 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 Castaneda 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 stressstrain 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 Castaneda for these specific material systems. 1. Introduction For a linear-elastic constitutive response, there are well-established methods to estimate the effective or overall behaviour of composite materials. These so-called homogenization methods include the variational principles of Hashin & Shtrikman (1963), which can be used to bound the effective modulus tensor of linear-elastic composites. A comprehensive review of this and other works on linear composites is given, for example, in the recent monograph by Milton (2002). On the other hand, for nonlinear (e.g. plastic, viscoplastic, etc.) composites, rigorous methods have not been available until more recently, even though efforts along these lines have been going on for some time, particularly in the context of ductile polycrystals (e.g. Hill 1965; Hutchinson 1976). Making use of a nonlinear extension of the HashinShtrikman (HS) variational principles, due to Willis (1983), the first bounds of the HS type for nonlinear composites were derived by Talbot & Willis (1985). Further applications of this methodology to composites with nonlinear anisotropic phases (e.g. polycrystals) were given by Willis (1994). A more general variational approach making use of the notion of optimally chosen linear comparison composites was proposed by Ponte Castaneda (1991). This approach is not only capable of delivering bounds of the HS type for nonlinear composites, but also, in addition, can be used to generate bounds and estimates of other types, such as self-consistent estimates and three-point bounds (Ponte Castaneda 1992). The connections between these two different approaches were explored by Willis (1992) and Talbot & Willis (1992). Suquet (1993) and Olson (1994) proposed alternative, but equivalent methods for the special classes of power-law and ideally plastic materials, respectively. Suquet (1995) also gave a novel physical interpretation of the variational estimates of Ponte Castaneda (1991) in terms of secant moduli and the second moments of the local fields in the linear comparison composite. In addition, deBotton & Ponte Castaneda (1995) proposed an extension of the variational principle of Ponte Castaneda (1991) to generate bounds for composite materials with (anisotropic) viscoplastic crystalline phases. This variational approach was, in turn, generalized further by Suquet (see Ponte Castaneda & Suquet 1998) to include more general types of anisotropies in the context of nonlinear composites. Finally, it should be mentioned that there is another approach, called the translation method, which has been applied successfully to polycrystals, but thus far only for scalar potential problems (Kohn & Little 1998; Nesi et al. 2000; Goldsztein 2001; Garroni & Kohn 2003). The present work is concerned with an extension of the variational method of Ponte Castaneda (1991, 1992) for composites with nonlinear (viscoplastic) anisotropic phases. It will be shown that this generalization, which is more closely tied to the original formulation, has the capability to give improved bounds relative to the generalizations provided by deBotton & Ponte Castaneda (1995) in the specific context of viscoplastic polycrystals, and the further generalization provided by Ponte Castaneda & Suquet (1998) for more general anisotropies. Thus, attention will be focused on composite materials made of N homogeneous constituents, or phases, that are randomly distributed in a specimen occupying a volume U at a length-scale which is much smaller than the size of the specimen and the scale of variation of the loading conditions. The constitutive behaviour of the nonlinear (viscoplastic) anisotropic phases is characterized by convex strain (strain-rate) potentials w(r) (rZ1, ., N ), satisfying the conditions w(r)(0)Z0 and w(r)(3)/N as j3j/N. The microstructure is defined by characteristic functions c(r) that are equal to 1 if the position vector x is in phase r (i.e. x2U(r)), and 0 otherwise. In this case, the relation between the stress s and the strain (strain-rate) 3 is defined by a local strain potential w, such that vw N s Z v3 x; 3; wx; 3 Z X crxwr3: 1:1 rZ1 The problem is then that of determining the effective behaviour of the composite, which is defined as the relation between the average stress sZ hsi $ (r) will be used to adnendottheevoalvuemraegeavsetrraagines3oZvehr3it.heHceorem, ptohseitsey(mUb)oalsndh$pihaansde rh (iU(r)), respectively. For the class of materials characterized by relations (1.1), the effective behaviour Bounds for nonlinear anisotropic composites is determined by the effective strain potential, defined by N w~3 Z inf hwx; 3i Z inf X crhwr3ir; 32K3 32K3 rZ1 where c(r)Zhc(r)i denotes the volume fraction of phase r; and K3 is the set of kinematically admissible strain fields, such that there is a continuous displacement field u satisfying 3Z 1=2VuC VuT in U, and u Z 3x in vU. It is important to recall that (strict) convexity of the local strain potential w on the local strain 3 implies (strict) convexity of the effective potential w~ in the applied strain 3. The effective stressstrain relation is then known to be given by where it has been assumed that w~ is differentiable, as has been done for the local potential w in expres (...truncated)