Incrementation: Part One

Abstract

The focus here is on the analysis and solution of transient, or time-dependent, problems. Time-dependent solutions are normally based on time-stepping algorithms wherewith solutions to increments of the solution variables, i.e., deformation gradient, strain and stress, are obtained. Typically, at each point in time where the incremental values are obtained, they are added to the previously accumulated values of the corresponding variables, thus yielding the summed incremental values for the current solution point. This procedure is continued, marching step-by-step timewise as far as it is desired that the solution progresses. The solution algorithm employed must be unconditionally stable and accurate. The procedure described is additive, where the incremental deformation gradient is primary; there is also a multiplicative approach wherein the relative deformation gradient is the primary variable. The relationship between the incremental deformation gradient and the relative deformation gradient is defined. A numerical example demonstrating the relationship between the incremental deformation gradient and the relative deformation gradient is presented.