Rate formulations in nonlinear continuum mechanics

Abstract

In solving problems of geometrically nonlinear structural mechanics, a prominent role is played by formulation of rate equilibrium conditions. In the computational machinery, the evaluation of the stiffness operator provides the trial incremental displacement field as fixed point of an iterative algorithm. The issue is investigated by a new geometric approach to continuum mechanics. Kinematics is described by the motion along a trajectory manifold embedded in the affine four-dimensional space-time. Variational conditions of equilibrium and rate equilibrium are formulated in terms of natural time rates of stress and stretching. The rate elastostatic problem is formulated in the full nonlinear context by adopting a newly contributed rate-elastic constitutive model. The geometric stiffness and forcing operators are expressed in terms of an arbitrary linear spatial connection. It is shown that the adoption of a Levi- Civita connection provides a linear expression of the geometric stiffness involving a curvature term. For bodies in motion in the flat Euclid space with parallel transport by translation, a symmetric expression of the geometric stiffness is obtained, thus extending the standard formula to bodies of any dimensionality.