Analysis of composite structures with peridynamics

Numerical prediction of crack growth is an important problem in computational mechanics. The difficulty in this problem arises from the basic incompatibility of cracks with the partial differential equations that are used in the classical theory of solid mechanics. The spatial derivatives needed for these partial differential equations to make sense do not exist on a crack tip or surface. Any numerical method derived from these equations inherits this difficulty in modeling cracks. In spite of the progress that has been made in developing meshfree methods aimed at modelling fracture such meshfree techniques generally require some special method of evaluating the spatial derivatives on each crack surface. They also require supplemental relations that govern the initiation of cracks, as well as their growth velocity and direction. These relations must be applied along each crack tip, leading to inherent complexity of the method, particularly when multiple cracks occur and interact in three dimensions. It is also possible to construct constitutive models that lead asymptotically to localization in a continuum but these techniques do not entirely avoid the need for special treatment of a crack after it forms. As an attempt at improving this situation, a theory of solid mechanics has been proposed that does not require spatial derivatives to be evaluated within a body. This theory, known as the peridynamic theory, instead uses integral equations. The objective is to reformulate the basic mathematical description of solid mechanics in such a way that the identical equations hold either on or off of a discontinuity such as a crack. In particular in this thesis I wanted to highlight the potential of PDDO (Peridynimac differential operator) that I used to describe the RZT (Refined zig zag theory). In the first chapter I briefly introduced the concept of peridynamics while in the second chapter I went into more detail regarding differential operators. Finally in the third chapter I applied PDDO to the RZT.