A globally defined compressible polyconvex isotropic energy satisfying the true-stress-true-strain monotonicity condition (TSTS-M++)
Abstract
We construct a compressible isotropic energy on GL+(3) combining polyconvexity and global true-stress-true-strain monotonicity (TSTS-M++), answering a question of Wollner, Holzapfel and Neff. For every μ>0 and k>0, the energy Wk(F)=μ/2k[exp(k(lVert FrVert2+3J−1+J−7))−1], J=det F, is polyconvex and strictly rank-one convex. Its Cauchy-stress response satisfies TSTS-M++ globally if and only if k≥1/(8√3). The reference state is stress free with positive infinitesimal shear and bulk moduli. Stress bijectivity has the smaller sharp threshold kB≈0.0083: at equality the stress map is a global homeomorphism with a nondifferentiable inverse, and above it a global smooth diffeomorphism. Thus, for kB≤ k<1/(8√3), the Cauchy-stress map remains globally bijective while TSTS-M++ fails at finite strain. In the TSTS-M++ regime, each prescribed inner radius of a finite plane-strain annulus with a traction-free outer wall has a unique radial equilibrium, and the inner pressure increases smoothly and strictly from zero to infinity. Comparisons with other energies distinguish the roles of polyconvexity, TSTS-M++, reduced convexity and endpoint growth. Under incompressible spherical inflation, global pressure inversion holds for the Demiray family with b≥ b*, including Wk|J=1 for k≥ k* and the stress-free I1-exponential energy, and for the exponentiated Hencky energy when keH>3/8. Incompressible neo-Hooke instead has a limit point. A further compressible I1 energy retains the global constitutive properties and reference moduli of Wk but exhibits two pressure turning points under incompressible spherical inflation, showing that these properties allow a descending pressure branch.
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BibTeX
@misc{bai2026globally,
title = {{A globally defined compressible polyconvex isotropic energy satisfying the true-stress-true-strain monotonicity condition (TSTS-M++)}},
author = {Dongxin Bai and Yunhao Wu and Yong Li and Kai Zhang and Patrizio Neff},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.06026},
eprint = {2609.06026},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06026},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A globally defined compressible polyconvex isotropic energy satisfying the true-stress-true-strain monotonicity condition (TSTS-M++)
AU - Dongxin Bai
AU - Yunhao Wu
AU - Yong Li
AU - Kai Zhang
AU - Patrizio Neff
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.06026
UR - https://arxiv.org/abs/2609.06026
AB - We construct a compressible isotropic energy on GL^+(3) combining polyconvexity and global true-stress-true-strain monotonicity (TSTS-M++), answering a question of Wollner, Holzapfel and Neff. For every μ>0 and k>0, the energy W_k(F)=μ/2k[exp(k(lVert FrVert^2+3J^−1+J−7))−1], J=det F, is polyconvex and strictly rank-one convex. Its Cauchy-stress response satisfies TSTS-M++ globally if and only if k≥1/(8√3). The reference state is stress free with positive infinitesimal shear and bulk moduli. Stress bijectivity has the smaller sharp threshold k_B≈0.0083: at equality the stress map is a global homeomorphism with a nondifferentiable inverse, and above it a global smooth diffeomorphism. Thus, for k_B≤ k<1/(8√3), the Cauchy-stress map remains globally bijective while TSTS-M++ fails at finite strain. In the TSTS-M++ regime, each prescribed inner radius of a finite plane-strain annulus with a traction-free outer wall has a unique radial equilibrium, and the inner pressure increases smoothly and strictly from zero to infinity. Comparisons with other energies distinguish the roles of polyconvexity, TSTS-M++, reduced convexity and endpoint growth. Under incompressible spherical inflation, global pressure inversion holds for the Demiray family with b≥ b_*, including W_k|_J=1 for k≥ k_* and the stress-free I_1-exponential energy, and for the exponentiated Hencky energy when k_eH>3/8. Incompressible neo-Hooke instead has a limit point. A further compressible I_1 energy retains the global constitutive properties and reference moduli of W_k but exhibits two pressure turning points under incompressible spherical inflation, showing that these properties allow a descending pressure branch.
ER -