## Abstract

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 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)=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.

Original language | English |
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Article number | 77 |

Number of pages | 41 |

Journal | Journal of Nonlinear Science |

Volume | 32 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2022 |

## Keywords

- Ellipticity
- Finite elements
- Hyperelasticity
- Isotropy
- Nonlinear elasticity
- Physics-informed neural networks
- Planar elasticity
- Quasiconvexity
- Rank-one convexity
- Volumetric-isochoric split