一个全局定义的多凸各向同性能量，满足真应力-真应变单调性条件（TSTS-M++）
A globally defined polyconvex isotropic energy satisfying the true-stress-true-strain monotonicity condition (TSTS-M++)
- School of Aerospace Engineering and Applied Mechanics, Tongji University（同济大学航空航天与力学学院）
- Department of Chemistry and Bioscience, Aalborg University（奥尔堡大学化学与生物科学系）
- Chair of Nonlinear Analysis and Modelling, University of Duisburg-Essen（杜伊斯堡-埃森大学非线性分析与建模讲席）
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AI总结：
本文构造了一个全局定义的多凸各向同性存储能量函数，证明其在特定参数范围内满足真应力-真应变单调性，并给出应力双射性的尖锐阈值，应用于环的径向平衡唯一性。
AI中文摘要：
多凸性是有限弹性变分存在理论中的标准要素，而真应力-真应变单调性（TSTS-M++）要求正的增量柯西应力响应。这两种本构限制是独立的，Wollner、Holzapfel 和 Neff 留下了以下开放问题：定义在整个 $\mathrm{GL}^{+}(3)$ 上的可压缩各向同性能量能否同时满足两者。我们给出了明确的肯定回答。对于每个 $\mu>0$ 和 $k>0$，存储能量函数 $W_k(F)=\frac{\mu}{2k}\bigl[\exp\bigl(k(\lVert F\rVert^2+3J^{-1}+J-7)\bigr)-1\bigr]$，其中 $J=\det F$，是多凸且严格秩一凸的。其柯西应力响应全局满足 TSTS-M++ 当且仅当 $k\ge 1/(8\sqrt{3})$。在该范围内，每个对称柯西应力对应唯一的正定拉伸，而参考拉伸是无应力的，且具有正的无穷小剪切模量和体积模量。应力双射性具有严格更小的尖锐阈值 $k_{\mathrm{B}}\approx 0.00827233304$：在等号处，应力映射是全局同胚且逆映射不可微；在阈值之上，该映射是全局 $C^{\infty}$ 微分同胚。因此，对于 $k_{\mathrm{B}}\le k<1/(8\sqrt{3})$，映射 $V\mapsto\sigma(V)$ 保持全局双射，而 TSTS-M++ 在有限应变下失效。在 TSTS-M++ 范围内，具有无牵引外壁的有限平面应变环的每个指定内半径都有唯一的径向平衡，其内压从零到无穷大平滑且严格递增。
英文摘要：
Polyconvexity is a standard ingredient in the variational existence theory of finite elasticity, whereas true-stress-true-strain monotonicity (TSTS-M++) requires a positive incremental Cauchy-stress response. These two constitutive restrictions are independent, and Wollner, Holzapfel and Neff left open whether a compressible isotropic energy defined on the whole of $\mathrm{GL}^{+}(3)$ can satisfy both. We give an explicit affirmative answer. For every $μ>0$ and $k>0$, the stored-energy function $W_k(F)=\fracμ{2k}\bigl[\exp\bigl(k(\lVert F\rVert^2+3J^{-1}+J-7)\bigr)-1\bigr]$, $J=\det F$, is polyconvex and strictly rank-one convex. Its Cauchy-stress response satisfies TSTS-M++ globally if and only if $k\ge 1/(8\sqrt{3})$. In this regime every symmetric Cauchy stress corresponds to a unique positive-definite stretch, while the reference stretch is stress free with positive infinitesimal shear and bulk moduli. Stress bijectivity has the strictly smaller sharp threshold $k_{\mathrm{B}}\approx 0.00827233304$: at equality the stress map is a global homeomorphism with a nondifferentiable inverse, and above it the map is a global $C^{\infty}$ diffeomorphism. Thus, for $k_{\mathrm{B}}\le k<1/(8\sqrt{3})$, the map $V\mapstoσ(V)$ remains globally bijective while TSTS-M++ fails at finite strain. In the TSTS-M++ regime, every prescribed inner radius of a finite plane-strain annulus with a traction-free outer wall has a unique radial equilibrium, and its inner pressure increases smoothly and strictly from zero to infinity.
