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

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

Voss, Jendrik (Universität Duisburg-Essen; Technische Uni­ver­si­tät Dort­mund)
Martin, R.P. (TU Delft Structural Integrity & Composites; Universität Duisburg-Essen)
Sander, Oliver (Technische Universität Dresden)
Kumar, Siddhant (TU Delft Team Sid Kumar)
Kochmann, Dennis M. (ETH Zürich)
Neff, Patrizio (Universität Duisburg-Essen)

2022

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.

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

http://resolver.tudelft.nl/uuid:8007394a-c559-4e2c-b8ae-6338e7260ff4

https://doi.org/10.1007/s00332-022-09820-x

0938-8974

Journal of Nonlinear Science, 32 (6)

Institutional Repository

journal article

© 2022 Jendrik Voss, R.P. Martin, Oliver Sander, Siddhant Kumar, Dennis M. Kochmann, Patrizio Neff