Evolution equation of Lie-type for finite deformations, time-discrete integration, and incremental methods

Abstract

While the position and shape of a deformed body take place in the usual three-dimensional Euclidean space ${\mathbb{R}^3}$, a corresponding progress of the deformation tensor makes up a trajectory in the space of all symmetric positive-definite matrices ${Sym^+(3,\mathbb{R})}$—a negatively curved Riemannian symmetric manifold. In this context, we prove that a well-known relation ${\partial\mathbf{C}_t=2\mathbf{F}^T\mathbf{d}\mathbf{F}}$ between deformation rate ${\partial\mathbf{C}_t}$ and symmetric velocity gradient ${\mathbf{d}}$, via deformation gradient ${\mathbf{F}}$, can be actually interpreted as an equation of Lie-type describing evolution of the right Cauchy–Green deformation tensor ${\mathbf{C}_t}$ on the configuration space ${Sym^+(3,\mathbb{R})}$. As a consequence, this interpretation leads to geometrically consistent time-discrete integration schemes for finite deformation processes, such as the Runge–Kutta–Munthe-Kaas method. The need to solve such an equation arises from an incremental numerical modelling of deformations of nonlinear materials. In parallel, the exposition is accompanied by an analysis of evolution of the deformation gradient ${\mathbf{F}_t}$ on the general linear group of all non-singular matrices ${GL(3,\mathbb{R})}$.