Sharp rank-one convexity conditions in planar isotropic elasticity for\n the additive volumetric-isochoric split

2020/08/10 by Jendrik Voss, Voss, Jendrik, Ionel‐Dumitrel Ghiba +5
Biochemistry, Genetics and Molecular Biology · Computer Science · Engineering · #26A51 #26B25 #74A10 #74B20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Connective tissue disorders research #Elasticity and Material Modeling #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2008.04188

openalex publication_date 2020/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the volumetric-isochoric split in planar isotropic\nhyperelasticity and give a precise analysis of rank-one convexity criteria for\nthis case, showing that the Legendre-Hadamard ellipticity condition separates\nand simplifies in a suitable sense. Starting from the classical two-dimensional\ncriterion by Knowles and Sternberg, we can reduce the conditions for rank-one\nconvexity to a family of one-dimensional coupled differential inequalities. In\nparticular, this allows us to derive a simple rank-one convexity classification\nfor generalized Hadamard energies of the type W(F)=\(\μ)/(2)\(\‖\nF\‖²)/(\det F)+f(\det F); such an energy is rank-one convex if and only if\nthe function f is convex.\n

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