Remarks on rank‐one convexity and quasiconvexity for planar functions with an additive volumetric–isochoric split

Abstract

AbstractWe study convexity properties of isotropic energy functions in planar nonlinear elasticity in the context of Morrey's conjecture, which states that rank‐one convexity does not imply quasiconvexity in the two‐dimensional case. Recently, it has been shown that for the special case of isochoric energy functions on GL+(2) = {F ∈ ℝ2×2 | det F > 0}, i.e. for any isotropic function W : GL+(2) → ℝ with W(aF) = W(F) for all a > 0, these two notions of generalized convexity are, in fact, equivalent. Here, we consider the more general case of functions on GL+(2) with an additive volumetric–isochoric split of the form urn:x-wiley:16177061:media:PAMM202200185:pamm202200185-math-0001 with an isochoric function Wiso on GL+(2) and a function Wvol on (0, ∞). In particular, we investigate the importance of the function urn:x-wiley:16177061:media:PAMM202200185:pamm202200185-math-0002 and its convexity properties; here, λmax ≥ λmin > 0 are the ordered singular values of the deformation gradient F ∈ GL+(2). This function arises naturally as an “extremal” case in the class of volumetric–isochorically split energies with respect to rank‐one convexity.