In this paper, the nonlinear three-dimensional (3D) stress analysis of shell structures in buckling and snapping problems is presented. The exact geometry or geometrically exact (GeX) hybrid-mixed four-node solid-shell element is developed using a sampling surfaces (SaS) method. The SaS formulation is based on the choice of N SaS parallel to the middle surface to introduce the displacements of these surfaces as basic shell unknowns. The SaS are located at the Chebyshev polynomial nodes (roots of the Chebyshev polynomial of degree N), that is, the outer surfaces are not included into a set of SaS. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the through-the-thickness distributions of displacements, strains and stresses leads to an efficient higher-order shell formulation. The incremental equilibrium equations are solved by the Newton–Raphson method combined with the Crisfield arc-length algorithm. The tangent stiffness matrix is evaluated by effective 3D analytical integration. As a result, the proposed GeX hybrid-mixed solid-shell element exhibits superior performance for coarse meshes and allows much larger load increments than is possible with existing displacement-based solid-shell elements. This can be useful for the 3D stress analysis of thin and thick shells in different states such as pre-buckling, bifurcation and post-buckling.

本文提出了壳结构在屈曲和折断问题中的非线性三维（3D）应力分析。使用采样表面（SaS）方法开发出精确的几何形状或几何精确的（GeX）混合混合四节点固体壳单元。SaS公式基于平行于中间表面的N SaS的选择，以引入这些表面的位移作为基本壳层未知数。将SAS位于切比雪夫多项式节点（的程度切比雪夫多项式的根Ñ，这是），外表面不包括成一组的SA。如此选择未知数，随后使用N级的拉格朗日多项式在位移，应变和应力的整个厚度分布中，–1导致有效的高阶壳公式。增量平衡方程是通过牛顿-拉夫森方法和克里斯菲尔德弧长算法求解的。切线刚度矩阵通过有效的3D分析集成进行评估。结果，与现有的基于位移的固体单元相比，建议的GeX混合混合固体单元在粗网格上表现出优异的性能，并允许更大的载荷增量。这对于薄壳和厚壳在不同状态（例如预屈曲，分叉和后屈曲）的3D应力分析很有用。