Fractal geometry in structural analysis problems: A variational formulation for fractured bodies with non-monotone interface conditions

Experimental results indicate that the fracture in many structures is rough and irregular and depends strongly on their microstructures and loading situations. In many cases the problem is to repair structures involving such irregular cracks and interfaces. The aim of this paper is to contribute to the analysis of these problems by using a new geometry, fractal geometry which describes with great accuracy irregular phenomena in nature. According to fractal geometry the interface of a crack can be considered to be the unique ‘fixed point’ of a given iterative function system (IFS) or the graph of a fractal interpolation function. On these interfaces, conditions describing the mechanical behaviour of the bonding material will be assumed to hold. The general case of a mechanical behaviour described by a non-monotone possibly multivalued stress-strain law will be treated. The method that will be developed in the framework of this paper will be an extension of the classical FEM to the case of fractal interfaces.