Behavior of solids in the vicinity of singular points

Abstract

The classical theory of linear elasticity predicts spurious phenomena, such as the self-intersection of matter, in the vicinity of interior points of anisotropic solids, corners and crack tips. The self-intersection phenomenon, by its turn, is associated with the violation of the kinematical condition $J>0$, where $J$ is the determinant of the deformation gradient, near to these points. One way to impose $J>0$ combines the classical theory of linear elasticity with the imposition of this constraint by means of a Lagrange multiplier technique. The associated constrained minimization problem is highly nonlinear and, in general, requires a numerical solution. We have used this constrained minimization theory together with a penalty formulation in the theoretical and numerical investigation of problems for which we have assumed {\it a priori} that the solution is radially symmetric with respect to an interior point of the solid. Another way to impose $J>0$ consists of using an appropriate nonlinear elastic model that avoids self-intersection in regions where this phenomenon occurs according to the classical theory and that follows classical models away from these regions. Both ways require differentiability of the displacement field almost everywhere in the solid. Recently, theories based on interactions among material points of a body were proposed to model the behavior of solids in the vicinity of singularities, such as crack tips. These theories do not use classical concepts of deformation and stress, allowing therefore the modeling of discontinuous displacement fields. This project will allow the continuation of our investigations on singular problems by combining classsical theories of solid mechanics with non-classical theories, such as the quasi-continuum and peridynamic theories. In particular, we intend to employ a penalty formulation in the study of bi-dimensional problems without {\it a priori} considerations about symmetry of the displacement field. The results obtained from this investigation will be compared with results obtained from previous investigations and with results available in the literature. The project will also allow my participation in activities of the thematic year on {\it Simulating Our Complex World: Modeling, Computation and Analysis}, which will be sponsored by the Institute for Mathematics and its Applications (IMA) at the University of Minnesota (UMN) and will treat of subjects related to the theme of this project. (AU)