In this lecture, two different topics of considerable interest to composite materials and structural mechanics communities are discussed. The first topic deals with the development of locking-free shell finite elements with thickness stretch for the nonlinear analysis of isotropic and laminated composite shells [1,2]. The shell finite elements are based on seven-parameter and twelveparameter shell theories, and account for change in shell thickness and 3-D constitutive relations. The element is found to be robust (i.e., does not require any ad-hoc approaches like reduced integration and stabilization to eliminate thickness, shear, and membrane locking). The second topic is concerned with a graph-based finite element framework (GraFEA) for the study of fracture in solids [3-6]. The major difference between GraFEA and other continuum-based approaches to study fracture is that instead of placing the focus on the elements and introducing a displacement discontinuity either between or inside the elements, GraFEA focuses on nodes and the distance between the nodes. Fracture is merely introduced by breakage of the edges (or any link between any two distinct nodes). Fracture and breakage of an edge is introduced into GraFEA using the idea of weakest link statistics, where the edge-based failure criterion is imposed directly on the discretized body. The fracture criterion chosen in our study is a strain-based criterion; however, this is not a limitation of the method and other fracture criteria can be applied. Numerical examples are presented to illustrate the robustness of the shell element and the workings of GraFEA.
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