Abstract
Symmetric quasiconvexity plays a key role for energy minimization in geometrically linear elasticity theory. Due to the complexity of this notion, a common approach is to retreat to necessary and sufficient conditions that are easier to handle. This article focuses on symmetric polyconvexity, which is a sufficient condition. We prove a new characterization of symmetric polyconvex functions in the two- and three-dimensional setting, and use it to investigate relevant subclasses like symmetric polyaffine functions and symmetric polyconvex quadratic forms. In particular, we provide an example of a symmetric rank-one convex quadratic form in 3d that is not symmetric polyconvex. The construction takes the famous work by Serre from 1983 on the classical situation without symmetry as inspiration. Beyond their theoretical interest, these findings may turn out useful for computational relaxation and homogenization.
Highlights
Variational models based on energy minimization principles are known to yield good descriptions of the elastic materials in nonlinear elasticity, and have inspired new mathematical developments in the calculus of variations over the last decades
We investigate functions that are polyconvex when composed with the linear projection of Rd×d onto the subspace of symmetric matrices Sd×d, that is, f : Sd×d → R such that Rd×d F → f (Fs) is polyconvex. We call such functions symmetric polyconvex, see Definition 2.1 for more details. This notion has applications in the geometrically linear theory of elasticity, which results from nonlinear elasticity theory by replacing the requirement of frame-indifference with the assumption that the elastic energy density is invariant under infinitesimal rotations, see e.g., [7,8,12]
Let us explain in more detail our new mathematical results, which were inspired by the following examples of functions defined on Sd×d : (i) ε → det ε is not symmetric polyconvex in d = 2, 3; (ii) ε → − det ε is symmetric polyconvex in d = 2, but not in d = 3; (iii) ε → −(cof ε)ii, i = 1, 2, 3 is symmetric polyconvex in d = 3, while ε → cof ε is not
Summary
Variational models based on energy minimization principles are known to yield good descriptions of the elastic materials in nonlinear elasticity (so-called hyperelastic materials), and have inspired new mathematical developments in the calculus of variations over the last decades. We call such functions symmetric polyconvex, see Definition 2.1 for more details This notion has applications in the geometrically linear theory of elasticity, which results from nonlinear elasticity theory by replacing the requirement of frame-indifference with the assumption that the elastic energy density is invariant under infinitesimal rotations, see e.g., [7,8,12]. As we will discuss further below, these conditions will facilitate finding explicit symmetric polyconvex functions that can serve as interesting energy densities in continuum mechanics or that provide lower bounds in the relaxation of models for materials with microstructures. A deeper understanding of symmetric polyconvexity may be useful in further applications of the translation method (see Section 1.4 below for more details)
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