Rank‐one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two‐dimensional case

Abstract

We show that in the two-dimensional case, every objective, isotropic and isochoric energy function which is rank-one convex on GL + (2) is already polyconvex on GL + (2). Thus we negatively answer Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasiconvexity. Our methods are based on different representation formulae for objective and isotropic functions in general as well as for isochoric functions in particular. We also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor. , especially the concepts of polyconvexity (i.e. convexity in terms of minors) of an energy function W : GL + (n) → R, n ∈ N, quasiconvexity (which is tantamount to the weak lower semicontinuity of the energy functional ϕ → W (∇ϕ) dx on appropriate Sobolev spaces) and rank-one convexity (or Legendre-Hadamard ellipticity). Polyconvexity, in particular, has been an important notion in continuum mechanics and a cornerstone of the direct methods of the calculus of variations since its introduction to elasticity theory in John Ball's seminal paper [2,3]. It is well known that the implications polyconvexity ⇒ quasiconvexity ⇒ rank-one convexity hold for arbitrary dimension n. However, it is also known that rank-one convexity does not imply polyconvexity in general, and that for n > 2 rank-one convexity does not imply quasiconvexity. The question whether rank-one convexity implies quasiconvexity in the two-dimensional case is considered to be one of the major open problems in the calculus of variations. Charles B. Morrey conjectured in 1952 that the two are not equivalent [4], i.e. that there exists a function W : R 2×2 → R which is rank-one convex but not quasiconvex. Isochoric energy functionsIn [5], we present a condition under which rank-one convexity implies polyconvexity (and thus quasiconvexity), thereby further complicating the search for a counterexample: any function W : GL + (2) → R which is isotropic and objective (i.e. bi-SO(2)-invariant) as well as isochoric is rank-one convex if and only if it is polyconvex. A function W : GL + (2) → R is called isochoric ifSuch energy functions play an important role in nonlinear elasticity theory [6,7], where an additive volumetric-isochoric splitof the elastic energy potential W into an isochoric part W iso and a volumetric part W vol is oftentimes assumed. This constitutive requirement is equivalent [8] to the existence of a function p :where σ(V ) denotes the Cauchy-stress tensor corresponding to the left Biot stretch tensor V = √ F F T . Our approach is based on certain representation formulae for isochoric energy functions: it is well known that any objective, isotropic function can be expressed in terms of the singular values of F , i.e. that there exists a uniquely determined function

The Quasiconvex Envelope of Conformally Invariant Planar Energy Functions in Isotropic Hyperelasticity

“…The following proposition summarizes the main results from Martin et al (2017) and completely characterizes the generalized convexity of conformally invariant functions on GL + (2). Proposition 2.13 (Martin et al 2017, Theorem 3.3) Let W : GL + (2) → R be conformally invariant, and let g : (0, ∞) × (0, ∞) → R, h : (0, ∞) → R and : [1, ∞) → R denote the uniquely determined functions with…”

“…for given : [1, ∞) → R and ϕ 0 : → R 2 . Since K(a R ∇ϕ) = K(∇ϕ a R) = K(∇ϕ) for all a > 0 and all R ∈ SO(2), the distortion function K is conformally invariant, and indeed every conformally invariant energy W on GL + (2) can be expressed in the form W (F) = (K(F)), see Martin et al (2017).…”

“…which is convex on (0, ∞) if W is rank-one convex and thus, in particular, monotone on [1, ∞) due to symmetry considerations (Martin et al 2017).…”

“…A recent contribution (Martin et al 2017) introduced a number of criteria for generalized convexity properties (including quasiconvexity) of so-called conformally invariant functions (or energies) on the group GL + (2) of 2 × 2-matrices with positive determinant, i.e., functions W : GL + (2) → R with…”

Quasiconvex relaxation of isotropic functions in incompressible planar hyperelasticity

“…Compared to functions defined on the full matrix space R n×n , the restricted domain of the energy W poses additional challenges with respect to these convexity properties (a number of which were famously addressed and solved by John Ball in his seminal 1977 paper [4,5]), but also allow for obtaining some significantly simplified criteria. In particular, under the additional assumptions of objectivity and isotropy, a large number of necessary and sufficient criteria for rank-one convexity and polyconvexity of energy functions on GL + (n) and SL(n) have been given in the literature [1,3,11,13,20,21,23,25,28,29].…”