Polyconvexity for incompressible inversion-symmetric energies of Valanis-Landel type
Abstract
{Let λi = νi(F) denote the three singular values of the deformation gradient F ∈ GL+(3).} We consider the family of incompressible isotropic energies $ W_ψ(F)=\sum_i ψ\left(|\!\logλ_i|\right)$ with ψ:[0,∞)↦ℝ. Set g(s)=ψ(arcoshs/2) for all s≥2. If g has a convex and non-decreasing extension g̅ to [0,∞), then Wψ is the restriction to SL(3) of the explicit polyconvex function \ {$ F\mapsto\sum_i \bar g\left(ν_i(F+{\rm Cof} F)\right). $} The proof uses the identity νi(F+Cof F)=λi+λi−1, up to permutation, on SL(3) and Ball's convexity theorem for functions of the singular values. We also give a direct proof along rank-one lines contained in SL(3) and derive a convenient one-dimensional differential sufficient condition. In particular, $ F\mapsto\sum_i e^{\log^2\!λ_i} $ is rank-one convex on SL(3) and possesses the stated polyconvex extension. A simple-shear computation shows that scalar convexity in log λi alone is insufficient; the quadratic Hencky energy ∑i log2λi is not rank-one convex on SL(3).
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BibTeX
@misc{ghiba2026polyconvexity,
title = {{Polyconvexity for incompressible inversion-symmetric energies of Valanis-Landel type}},
author = {Ionel-Dumitrel Ghiba and Maximilian P. Wollner and Patrizio Neff},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18660},
eprint = {2609.18660},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18660},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Polyconvexity for incompressible inversion-symmetric energies of Valanis-Landel type
AU - Ionel-Dumitrel Ghiba
AU - Maximilian P. Wollner
AU - Patrizio Neff
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18660
UR - https://arxiv.org/abs/2609.18660
AB - {Let λ_i = ν_i(F) denote the three singular values of the deformation gradient F ∈ GL^+(3).} We consider the family of incompressible isotropic energies $ W_ψ(F)=\sum_i ψ\left(|\!\logλ_i|\right)$ with ψ:[0,∞)↦ℝ. Set g(s)=ψ(arcoshs/2) for all s≥2. If g has a convex and non-decreasing extension g̅ to [0,∞), then W_ψ is the restriction to SL(3) of the explicit polyconvex function \ {$ F\mapsto\sum_i \bar g\left(ν_i(F+{\rm Cof} F)\right). $} The proof uses the identity ν_i(F+Cof F)=λ_i+λ_i^−1, up to permutation, on SL(3) and Ball's convexity theorem for functions of the singular values. We also give a direct proof along rank-one lines contained in SL(3) and derive a convenient one-dimensional differential sufficient condition. In particular, $ F\mapsto\sum_i e^{\log^2\!λ_i} $ is rank-one convex on SL(3) and possesses the stated polyconvex extension. A simple-shear computation shows that scalar convexity in log λ_i alone is insufficient; the quadratic Hencky energy ∑_i log^2λ_i is not rank-one convex on SL(3).
ER -