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 is
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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 Wmagic+(F)=λmaxλmin-logλmaxλmin+logdetF=λmaxλmin+2logλmin,where λmax≥ λmin 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)=Wiso(F)+Wvol(detF)=W~iso(FdetF)+Wvol(detF)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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