Bifurcations and instability in non-linear elastic solids with interfaces

The study of local and global instability and bifurcation phenomena is crucial for many engineering applications in the field of solid mechanics. In particular, interfaces within solid bodies are of great importance in the bifurcation analysis, as they constitute localized zones in which discontinuities or jumps in displacement, strain or stress may occur. Different instability phenomena, heavily conditioned by the presence of interfaces, were analyzed in the present thesis. The first phenomenon that has been considered is the propagation of a shear band, which is a localized shear deformation developing in a ductile material. This shear band, assumed to be already present inside of a ductile matrix material (obeying von Mises plasticity with linear hardening), is modelled as a discontinuity interface following two different approaches. In the first approach, the conditions describing the behavior of a layer of material in which localized strain develop are introduced and implemented in a finite element computer code. A shear deformation is simulated by imposing appropriate displacement conditions on the boundaries of the matrix material, in which the shear band is present and modelled through an imperfect interface, having null thickness. The second approach is based on a perturbative technique, developed for a J2-deformation theory material, in which the shear band is modeled as the emergence of a discontinuity surface for displacements at a certain stage of a uniform deformation process, restricted to plane strain conditions. Both the approaches concur in showing that shear bands (differently from cracks) propagate rectilinearly under shear loading and that a strong stress concentration is expected to be present at the tip of the shear band, two key features in the understanding of failure mechanisms of ductile materials [results of this study have been reported in (Bordignon et al. 2015)]. The second type of interface analyzed in the present thesis is a perfectly frictionless sliding interface, subject to large deformations and assumed to be present within a uniformly strained nonlinear elastic solid. This type of interface may model lubricated sliding contact between soft solids, a topic of interest in biomechanics and for the design of small-scale engineering devices. The analyzed problem is posed as follows. Two elastic nonlinear solids are considered jointed through a frictionless and bilateral surface, so that continuity of the normal component of the Cauchy traction holds across the surface, but the tangential component is null. Moreover, the displacement can develop only in a way that the bodies in contact do neither detach, nor overlap. Surprisingly, this finite strain problem has not been correctly formulated until now, so that this formulation has been developed in the thesis. The incremental equations are shown to be non-trivial and different from previously (and erroneously) employed conditions. In particular, an exclusion condition for bifurcation is derived to show that previous formulations based on frictionless contact or ‘spring-type’ interfacial conditions are not able to predict bifurcations in tension, while experiments (one of which, ad hoc designed, is reported) show that these bifurcations are a reality and can be predicted when the correct sliding interface model is used. Therefore, the presented approach introduces a methodology for the determination of bifurcations and instabilities occurring during lubricated sliding between soft bodies in contact [results of this study have been reported in (Bigoni et al. 2018)].