This paper describes a method for modifying explicit finite difference equations for the dynamic motion of a continuum to produce stress relaxation equations for static stress equilibrium. The equations modified are those of Wilkins, but the method is applicable to other finite difference codes. An artificial "stress diffusion" equation is used, in which successive displacements are toward stress equilibrium. Results of a two-dimensional numerical calculation are compared with an analytic solution.
Most methods for static equilibrium problems are restricted to the linear case. Explicit finite difference methods commonly used for dynamic problems can accept quite general material descriptions, such as plastic yielding and tensile cracking. This paper describes a simple modification that converts an explicit dynamic method, such as Wilkins' [1], to a method for static equilibrium allowing the same generality of material description. The modification is especially useful for finding a static stress distribution as an initial condition for a dynamic problem.
In Wilkins' procedure, a two-dimensional continuous medium is divided up into quadrilateral zones. The mesh is Lagrangian, meaning that the intersections of the grid lines, called grid points in this paper, move with the medium, and the quadrilateral zones distort with the medium. Displacements (or spatial coordinates) and velocity are defined at grid points, while stress and strain components are defined in the interiors of zones.
Four distinct steps are performed for each zone in each computational cycle. 202