Numerical approaches for investigating quasiconvexity in the context of Morrey's conjecture
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Deciding whether a given function is quasiconvex is generally a difficult task. Here, we discuss a number of numerical approaches that can be used in the search for a counterexample to the quasiconvexity of a given function W. We will demonstrate these methods using the planar isotropic rank-one convex function W_ magic^+(F)=λ_ max/λ_ min-logλ_ max/λ_ min+log F=λ_ max/λ_ min+2logλ_ min , where λ_ max≥λ_ min are the singular values of F, as our main example. In a previous contribution, we have shown that quasiconvexity of this function would imply quasiconvexity for all rank-one convex isotropic planar energies W:GL^+(2)→ℝ with an additive volumetric-isochoric split of the form W(F)=W_ iso(F)+W_ vol( F)=W_ iso(F/√( F))+W_ vol( F) with a concave volumetric part. This example is therefore of particular interest with regard to Morrey's open question whether or not rank-one convexity implies quasiconvexity in the planar case.
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