AI 中文总结
该研究受各向同性非线性弹性理论启发，证明对称向量函数的向量单调性等价于其诱导的各向同性张量函数的矩阵单调性，推广Chandler Davis定理，还讨论可逆性条件并推导强Baker-Ericksen不等式。
AI 中文摘要
受各向同性非线性弹性理论中经典本构不等式的启发，我们研究形如Σ_f: Sym(n)→Sym(n)的各向同性张量函数的单调性，其中Σ_f满足对任意正交矩阵Q∈O(n)，有Σ_f(Q^T diag(λ₁,…,λₙ)Q)=Q^T diag(f₁(λ₁,…,λₙ),…,fₙ(λ₁,…,λₙ))Q；此处f=(f₁,…,fₙ):ℝⁿ→ℝⁿ为对称向量函数，即对任意置换π:{1,…,n}→{1,…,n}，满足f_i(λ_{π(1)},…,λ_{π(n)})=f_{π(i)}(λ₁,…,λₙ)，其中Sym(n)表示n阶对称矩阵空间，O(n)为正交群，diag(λ₁,…,λₙ)为对角元为λ₁,…,λₙ的对角矩阵。我们证明f在ℝⁿ上的向量单调性等价于其诱导的各向同性张量函数Σ_f在Sym(n)上的矩阵单调性，该结果推广了针对凸标量各向同性函数的Chandler Davis定理，且独立于Rodney Hill对该等价性的原始证明。我们还讨论了各向同性矩阵函数的简单可逆性条件，最后证明：自然状态下，柯西应力映射V↦σ(V)的单射性、连续可微性，以及sym Dσ(1₁)的正定性，可推出强Baker-Ericksen不等式。
英文摘要
Motivated by classical constitutive inequalities in isotropic nonlinear elasticity theory, we investigate the monotonicity of isotropic tensor functions of the form \[ Σ_f\colon\mathrm{Sym}(n)\to\mathrm{Sym}(n)\,,\quad Σ_f(Q^T\mathrm{diag}(λ_1,\dotsc,λ_n)\, Q) = Q^T\mathrm{diag}(f(λ_1,\dotsc,λ_n))\, Q \quad\forall\;Q\in\mathrm{O}(n) \] with a vector function $f=(f_1,\dotsc,f_n)\colon\mathbb{R}^n\to\mathbb{R}^n$ which is symmetric, i.e.\ satisfies \[ f_i(λ_{π(1)},\dotsc,λ_{π(n)}) = f_{π(i)}(λ_1,\dotsc,λ_n) \] for any permutation $π\colon\{1,\dotsc,n\}\to\{1,\dotsc,n\}$, where $\mathrm{Sym}(n)$ denotes the space of symmetric $n\times n$ matrices, $\mathrm{O}n$ is the orthogonal group and $\mathrm{diag}(λ_1,\dotsc,λ_n)$ is the diagonal matrix with diagonal entries $λ_1,\dotsc,λ_n\in\mathbb{R}$. We prove that vector-monotonicity of $f$ on $\mathbb{R}^n$ is equivalent to matrix-monotonicity of the induced isotropic tensor function $Σ_f$ on $\mathrm{Sym}(n)$. Our results generalize the Chandler Davis theorem for convex scalar isotropic functions and are obtained independently of Hill's original proof of this equivalence. We also discuss simple invertibility conditions for isotropic matrix functions. We conclude by showing that injectivity of the Cauchy stress $V\mapstoσ(V)$, continuous differentiability, and positive definiteness of $\mathrm{sym}\,\mathrm Dσ(1\!\!\!\:1)$ in the natural state imply the strong Baker-Ericksen inequalities.
