Formal solution of quasi-static problems

When the quasi-static problem is defined by a set of differential equations complemented by initial and boundary conditions, the resulting quasi-static solutions may exhibit a limited reach over the time domain. On the other hand, the infinity of equilibrium paths that can be obtained in a general non-linear problem also indicates that a proper definition of the quasi-static solution must be provided. In inelasticity problems, this infinite number of equilibrium paths occur even when no dissipative bifurcations are present. In the present paper, a general solution for quasi-static problems in Solid Mechanics is defined and explored. Special attention is addressed to material non-linearities though geometric non-linearities are also covered by the definition. Earlier concepts of path and state stability are recovered in order to reduce the number of solutions to those that are physically acceptable. The important link with the original dynamic problem is accounted for by enforcing a preferential load direction. The resulting definition relies on a time-objective criterion with straightforward applicability to the most common numerical models. In the final part of the paper, simple 1D problems are used to illustrate some of the concepts introduced in the present developments.