Computational aspects of the mechanics of complex materials

Bodies with exotic properties display material substructural complexity from nano to meso-level. Various models have been built up in condensed matter physics to represent the behavior of special classes of complex bodies. In general, they fall within the setting of an abstract model building framework which is not only a unifying structure of existing models but—above all—atool to construct special models of new exotic materials. We describe here basic elements of this framework, the one ofmultifield theories, trying to furnish a clear idea of the subtle theoretical and computational problems arising within it. We present the matter in a form that allows one to construct appropriate algorithms in special cases of physical interest. We discuss also issues related to the construction of compatible and mixed finite elements in linearized setting, the extension of extended finite element methods to analyze the influnce of material substructures on crack growth, the evolution of sharp discontinuity surfaces in complex bodies. Concrete examples of complex bodies are also presented with a number of details.