On the stability of strain rate dependent structures and solids with applications.

Abstract

This thesis is concerned with the stability of strain-rate dependent solids and structures. This work departs from most of the previous studies on the same topic by recognizing the existence of two physical time scales. The first of these time scales relates to the rate at which the structure is loaded and the second originates from the relaxation time of the strain-rate material which composes the structure of the solid which stability is under investigation. The second chapter of this thesis is devoted to the stability of the Shanley column with discrete and continuous distribution of supports. Stability and instability is determined by the rate of growth or decay of a perturbation added to a trajectory. Linear and non-linear stability results are presented, the later based on numerical simulations. It is shown that the stability results are sensitive to the boundary conditions as well as to the value of the ratio between the two time scales defined in the problem. The third chapter is concerned with the generalization of the Shanley column results to the case of strain-rate dependent, cohesive and frictional solids. The linear and non-linear stability of a rectangular block composed of a material described with a viscoplastic version of the Rudnicki-Rice model are reported. The non-linear stability results are obtained with the finite-element method and the initial evolution of the system is compared with the analytical prediction of the linear stability analysis. The fourth chapter of this thesis deals with the stability of a geological two-layer system with density stratification and under the action of tectonic forces. The Rudnicki-Rice model, with no strain-rate dependency, is adopted for the top layer, referred to as the overburden. The second layer, called the substratum, is modeled as a viscous fluid with Newtonian or non-Newtonian rheology. It is shown that the initially unstable stratified systems evolve towards a new stable equilibrium state in the shape of a fold which amplitude is controlled by the initial in-situ lateral compressive stress. This stabilization, due to a complete elastic unloading in the overburden, is always found if the development of the instability is not accompanied by loss of ellipticity. Loss of ellipticity in the overburden implies the development of chevron-type of structure and no apparent restabilization to a new equilibrium.