指数化的Hencky-对数应变能。部分II：强制性、平面多凸性和最小值的存在性
The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity, planar polyconvexity and existence of minimizers
摘要：
本文考虑一族各向同性的体积-等容解耦应变能$$ F\mapsto W_{\rm eH}(F):=\widehat{W}_{\rm eH}(U):=\left\{\begin{array}{lll} \fracμ{k}\,e^{k\,\|{\rm dev}_n\log {U}\|^2}+\fracκ{2\hat{k}}\,e^{\hat{k}\,[{\rm tr}(\log U)]^2}&\text{if}& {\rm det}\, F>0,\\ +\infty &\text{if} &{\rm det} F\leq 0, \end{array}\right.\quad $$，它是基于Hencky对数（真实、自然）应变张量$\log U$，其中$μ>0$是无穷小剪切模量，$κ=\frac{2μ+3λ}{3}>0$是无穷小体积模量，$λ$是第一拉梅常数，$k,\hat{k}$是无量纲参数，$F=\nabla φ$是变形梯度，$U=\sqrt{F^T F}$是右拉伸张量，而${\rm dev}_n\log {U} =\log {U}-\frac{1}{n} {\rm tr}(\log {U})\cdot 1\!\!1$是应变张量$\log U$的偏量部分（投影到无痕张量上）。对于小的弹性应变的能量减少到一阶的经典的二次Hencky能量$$ F\mapsto W{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U):=μ\,\|{\rm dev}_n\log U\|^2+\fracκ{2}\,[{\rm tr}(\log U)]^2, $$，这是已知的是不是秩一凸。本文的主要结果是：在平面弹性静力学中，$W_{_{\rm eH}}$族的能量对于$k\geq \frac{1}{3}$，$\widehat{k}\geq \frac{1}{8}$是多凸的，推广了以前关于其秩一凸性的一个结果.我们的方法使用了一个明智的应用程序Steigmann的多凸性标准的基础上表示的能量拉伸张量$U$的主要不变量。这些能量也满足合适的生长和结晶条件。我们制定的平衡方程，我们证明的存在性极小的变分法的直接方法。
英文摘要：
We consider a family of isotropic volumetric-isochoric decoupled strain energies $$ F\mapsto W_{\rm eH}(F):=\widehat{W}_{\rm eH}(U):=\left\{\begin{array}{lll} \fracμ{k}\,e^{k\,\|{\rm dev}_n\log {U}\|^2}+\fracκ{2\hat{k}}\,e^{\hat{k}\,[{\rm tr}(\log U)]^2}&\text{if}& {\rm det}\, F>0,\\ +\infty &\text{if} &{\rm det} F\leq 0, \end{array}\right.\quad $$ based on the Hencky-logarithmic (true, natural) strain tensor $\log U$, where $μ>0$ is the infinitesimal shear modulus, $κ=\frac{2μ+3λ}{3}>0$ is the infinitesimal bulk modulus with $λ$ the first Lamé constant, $k,\hat{k}$ are dimensionless parameters, $F=\nabla φ$ is the gradient of deformation, $U=\sqrt{F^T F}$ is the right stretch tensor and ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n} {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part (the projection onto the traceless tensors) of the strain tensor $\log U$. For small elastic strains the energies reduce to first order to the classical quadratic Hencky energy $$ F\mapsto W{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U):=μ\,\|{\rm dev}_n\log U\|^2+\fracκ{2}\,[{\rm tr}(\log U)]^2, $$ which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family $W_{_{\rm eH}}$ are polyconvex for $k\geq \frac{1}{3}$, $\widehat{k}\geq \frac{1}{8}$, extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann's polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor $U$. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations and we prove the existence of minimizers by the direct methods of the calculus of variations.
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