We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solution for the theory including $O(h^5)$ terms. The energy allows us to show the coercivity for terms up to order $O(h^5)$ and the convexity of the energy. Secondly, we consider only that part of the energy including $O(h^3)$ terms. In this case the obtained minimization problem is not the same as those previously considered in the literature, since the influence of the curved initial shell configuration appears explicitly in the expression of the coefficients of the energies for the reduced two-dimensional variational problem and additional mixed bending-curvature and curvature terms are present. While in the theory including $O(h^5)$ the conditions on the thickness $h$ are those considered in the modelling process and they are independent of the constitutive parameter, in the $O(h^3)$-case the coercivity is proven on some more restrictive conditions under the thickness $h$.

中文翻译：

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各向同性 Cosserat 壳模型包括高达 $O(h^{5})$ 的项。第二部分：最小化器的存在

我们展示了几何非线性各向同性弹性 Cosserat 6 参数壳模型的全局极小值的存在。主要定理的证明基于变分计算的直接方法，主要使用非线性应变和曲率测量中能量的凸性。我们首先展示了包含 $O(h^5)$ 项的理论解的存在性。能量使我们能够显示高达 $O(h^5)$ 阶项的矫顽力和能量的凸性。其次，我们只考虑包含 $O(h^3)$ 项的那部分能量。在这种情况下，获得的最小化问题与之前在文献中考虑的问题不同，因为弯曲初始壳构型的影响明确地出现在简化的二维变分问题的能量系数表达式中，并且存在额外的混合弯曲曲率和曲率项。虽然在包含 $O(h^5)$ 的理论中，关于厚度 $h$ 的条件是在建模过程中考虑的，并且它们与本构参数无关，但在 $O(h^3)$-case在厚度 $h$ 下的一些更严格的条件下证明了矫顽力。