Matrix analysis of three-dimensional elastic media - small and large displacements

Abstract

The paper describes a broad generalization of the matrix displacement technique for analyzing three-dimensional elastic media, both in the small and large displacement ranges. The body is approximated by an assembly of tetrahedrons, for which a simplified kinematic pattern is prescribed. Using the novel device of a natural definition for component stresses, nodal loads, and the elemental stiffness, a concise expression is established for the Cartesian stiffness of an arbitrary tetrahedron, from which the stiffness of the complete system is obtained. The solution of the small displacement problem is then straightforward. To extend the theory to large displacements, the concept of an additional or geometrical stiffness is introduced which represents the effects of change of geometry on the equilibrium conditions. The specification of natural nodal loads is thereby most helpful, yielding the surprisingly simple result that the geometrical stiffness of a tetrahedron is identical to that of a six-bar pin-join ted framework, whose members form a geometrically equivalent tetrahedron. The large displacement problem may now be solved by a straightforward procedure based on a step by step linearization. The theory covers the joint presence of external loads and thermal strains.