ABSTRACT
This chapter reviews the finite element discretization of one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) models of solids, which use only displacements as nodal degrees of freedom. The element degrees of freedom are taken as the displacements at element nodes, referred to a common global Cartesian coordinate system. The shape functions for the 2D and 3D Lagrangian elements are simply products of the shape functions of the 1D element. The most widely used 2D and 3D elements are probably the so-called "serendipity" elements. They have only corner and edge nodes. One of the most important early milestones in the development of finite element technology was the introduction of isoparametric elements. The chapter presents the development of element matrices for a 3D solid continuum element. It also considers a 1D finite element model of a slender hyperelastic bar. When appropriate, 2D idealizations of a full 3D continuum, that is, plane strain and plane stress, are often utilized in practice.
