Morrey’s Conjecture for the Planar Volumetric-Isochoric Split: Least Rank-One Convex Energy Functions

Abstract

AbstractWe consider Morrey’s open question whether rank-one convexity already implies quasiconvexity in the planar case. For some specific families of energies, there are precise conditions known under which rank-one convexity even implies polyconvexity. We will extend some of these findings to the more general family of energies $$W{:}{\text {GL}}^+(n)\rightarrow \mathbb {R}$$ W : GL + ( n ) → R with an additive volumetric-isochoric split, i.e. $$\begin{aligned} W(F)=W_{\mathrm{iso}}(F)+W_{\mathrm{vol}}(\det F)={\widetilde{W}}_{\mathrm{iso}}\bigg (\frac{F}{\sqrt{\det F}}\bigg )+W_{\mathrm{vol}}(\det F)\,, \end{aligned}$$ W ( F ) = W iso ( F ) + W vol ( det F ) = W ~ iso ( F det F ) + W vol ( det F ) , which is the natural finite extension of isotropic linear elasticity. Our approach is based on a condition for rank-one convexity which was recently derived from the classical two-dimensional criterion by Knowles and Sternberg and consists of a family of one-dimensional coupled differential inequalities. We identify a number of “least” rank-one convex energies and, in particular, show that for planar volumetric-isochorically split energies with a concave volumetric part, the question of whether rank-one convexity implies quasiconvexity can be reduced to the open question of whether the rank-one convex energy function $$\begin{aligned} W_{\mathrm{magic}}^{+}(F)=\frac{\lambda _{\mathrm{max}}}{\lambda _{\mathrm{min}}}-\log \frac{\lambda _{\mathrm{max}}}{\lambda _{\mathrm{min}}}+\log \det F=\frac{\lambda _{\mathrm{max}}}{\lambda _{\mathrm{min}}}+2\log \lambda _{\mathrm{min}} \end{aligned}$$ W magic + ( F ) = λ max λ min - log λ max λ min + log det F = λ max λ min + 2 log λ min is quasiconvex. In addition, we demonstrate that under affine boundary conditions, $$W_{\mathrm{magic}}^+(F)$$ W magic + ( F ) allows for non-trivial inhomogeneous deformations with the same energy level as the homogeneous solution, and show a surprising connection to the work of Burkholder and Iwaniec in the field of complex analysis.