In this paper layered shells subjected to static loading are considered. The displacements of the Reissner–Mindlin theory are enriched by a an additional part. These so-called fluctuation displacements include warping displacements and thickness changes. They lead to additional terms for the material deformation gradient and the Green–Lagrangian strain tensor. Within a nonlinear multi-field variational formulation the weak form of the boundary value problem accounts for the equilibrium of stress resultants and couple resultants, the local equilibrium of stresses, the geometrical field equations and the constitutive equations. For the independent shell strains an ansatz with quadratic shape functions is chosen. This leads to a significant improved convergence behaviour especially for distorted meshes. Elimination of a set of parameters on element level by static condensation yields an element stiffness matrix and residual vector of a quadrilateral shell element with the usual 5 or 6 nodal degrees of freedom. The developed model yields the complicated three-dimensional stress state in layered shells for elasticity and elasto-plasticity considering geometrical nonlinearity. In comparison with fully 3D solutions present 2D shell model requires only a fractional amount of computing time.

中文翻译：

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用于分析几何和材料非线性壳的高级壳模型

在本文中，考虑了承受静载荷的分层壳。Reissner-Mindlin 理论的位移由一个附加部分丰富。这些所谓的波动位移包括翘曲位移和厚度变化。它们导致材料变形梯度和格林-拉格朗日应变张量的附加项。在非线性多场变分公式中，边界值问题的弱形式解释了应力合力和耦合合力的平衡、应力的局部平衡、几何场方程和本构方程。对于独立壳应变，选择具有二次形状函数的 ansatz。这会显着改善收敛行为，尤其是对于扭曲的网格。通过静态压缩消除单元级别的一组参数会产生单元刚度矩阵和具有通常 5 或 6 个节点自由度的四边形壳单元的残差向量。开发的模型在考虑几何非线性的弹性和弹塑性分层壳中产生复杂的三维应力状态。与全 3D 解决方案相比，目前的 2D 壳模型只需要一小部分计算时间。