Successive Linear Approximation for Large Deformation—Instability of Salt Migration

Abstract

In continuum mechanics, concerning the motion of a body, besides the Lagrangian and the Eulerian descriptions, the description relative to the present configuration of the body instead of the fixed reference configuration has been known as the relative motion description. Although the relative motion description is mostly ignored in the formulation of boundary value problems, it is interesting to consider such a formulation for problems in general for solid bodies. In doing so, there is an advantage that when the time increment from the present state is small enough, the nonlinear constitutive equation can be linearized relative to the present configuration, so that the resulting boundary value problem becomes linear. We can then propose a linear algorithm for large deformation, by building up successive small incremental deformation problem at every time step in the deformation process. In fact, the proposed method is a process of repeated applications of the well-known “small deformation superposed on finite deformation” in the literature. As an application of the proposed numerical method, we consider instability of a two-layered solid body of a denser material on top of a lighter one. This problem is widely known to geoscientist in sediment-salt migration as salt diapirism. In the literature, this problem has often been treated as Rayleigh–Taylor instability in viscous fluids instead of solid bodies. As an example, we propose a viscoelastic solid material model from constitutive theories of continuum mechanics, and present results of numerical simulations of sediment-salt migration which exhibit some main characteristics of salt diapirism as observed by geophysicists.