Rate formulations in nonlinear continuum mechanics

In solving problems of geometrically non-linear structural mechanics, a prominent role is played by formulation of rate equilibrium condi- tions. 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 contin- uum 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.