Stability and Localization of Deformation in Finitely Strained Solids and Structures : Static and High Strain Rate Dynamic Aspects.

Summary The localization of deformation in a deformed ductile medium is the instability mechanism that causes the final failure. This phenomenon occurs under static as well as dynamic load. It can be found within the materials, in which case it is called material instability, or on the whole structure, in which case it is called geometrical instability. This thesis studies the phenomenon of deformation localization in material or geometrical contexts and with static or dynamic loading conditions. In all cases, a unified tool is used: the evolution of the perturbation with localized support.The first chapter serves as an introduction to the problem of deformation localization in solid mechanics. The second chapter is devoted to the material instability aspect of strain localization in microstructured media under quasi-static compression, as well as its connection to macroscopic ellipticity (the continuum criterion of the presence of a discontinuous deformation field). In this chapter we demonstrate the connection between the homogenized post-bifurcation solution and the presence or absence of a localized deformation field in an infinitely large domain of fiber reinforced composites under compression. The third chapter is devoted to the material instability aspect of the deformation localization under dynamic loading, where the inertia effect becomes non-negligible. In this chapter we study a singular perturbation on an infinitely wide plate under biaxial tension and its influence cones, with a behavior law that loses ellipticity. The fourth chapter is devoted to the geometric instability aspect of deformation localization under dynamic loading, in which we study the dynamic compression of a metal ring under electromagnetic loading. Unlike the quasi-static case, localized deformation domains are observed in the failure mode of the ring.