Numerical Approaches for Investigating Quasiconvexity in the Context of Morrey’s Conjecture

Abstract Deciding whether a given function is quasiconvex generally difficult task. Here, we discuss number of numerical approaches that can be used in the search for counterexample to quasiconvexity W . We will demonstrate these methods using planar isotropic rank-one convex $$\begin{aligned} W_\textrm{magic}^+(F)=\frac{\lambda _\textrm{max}}{\lambda _\textrm{min}}-\log \frac{\lambda _\textrm{min}}+\log \det F=\frac{\lambda _\textrm{min}}+2\log \lambda _\textrm{min}\,, \end{aligned}$$ W magic + ( F ) = λ max min - log det 2 , where $$\lambda _\textrm{max}\ge _\textrm{min}$$ ≥ are singular values F , as our main example. In previous contribution, have shown this would imply all energies $$W:{\text {GL}}^+(2)\rightarrow {\mathbb {R}}$$ : GL → R with an additive volumetric-isochoric split form W(F)=W_\textrm{iso}(F)+W_\textrm{vol}(\det F)={\widetilde{W}}_\textrm{iso}\bigg (\frac{F}{\sqrt{\det F}}\bigg )+W_\textrm{vol}(\det F) iso vol ~ concave volumetric part. This example therefore particular interest regard Morrey’s open question or not convexity implies case.