The Graves Condition for Incompressible Finite Elasticity

Abstract

7R 3URI 0LOODUG ) %HDWW\ RQ WKH RFFDVLRQ RI KLV UHWLUHPHQW$EVWUDFW The Graves and Weierstrass necessary conditions for minimizers are of central importance in the study of elastic materials that can undergo a change of phase. Proofs of these results applicable to the case of unconstrained materials abound. For the case of an incompressible material, however, only partial results have been established. Here, a complete proof is given. ,1752'8&7,21In 1975, Ericksenòs profoundly influential paper [1] on the equilibrium of bars appeared, and inaugurated a period of intensive and fruitful research into phase transformations in solids. In studies of this sort, the material response is generally modeled by finite elasticity. Phase boundaries are treated as surfaces of discontinuity across which the deformation gradient field experiences large jumps. Such a deformation is regarded as stable if it minimizes the associated energy functional.This research lent fresh relevance to the classical necessary conditions of the calculus of variations for weak and strong relative minimizers. In particular, the Weierstrass necessary condition proved to be of central importance. Its physical import in this particular context is that strong relative minimizers of energy must satisfy the Maxwell relation at any phase boundary they contain. Various proofs of this condition applicable to the case of unconstrained elasticity are to be found in the classical literature of the calculus of variations. A representative one is that of Graves [2]. The case of incompressible elasticity, i.e. of elastic materials that are constrained to undergo only deformations which are isochoric, is more difficult, and is the subject of the present study.The prior literature on the question is rather sparse. Ericksen [3], in a discussion of the Weierstrass condition, stated the form which he thought the analogous condition for the incompressible case ought to take. Abeyaratne [4] considered the equilibrium problem for coexistent phases in incompressible elastic materials without discussing stability. He later [5] concluded a very nice discussion of the Maxwell and related conditions for the unconstrained 0DWKHPDWLFV DQG 0HFKDQLFV RI 6ROLGV 289ï298, 2003