Towards a top-down approach in materials science

Materials science or, more specifically, materials characterization represents an extremely vast interdisciplinary research arena involving scientists with very different backgrounds working at many different applications spanning more than ten orders of magnitude in the size of the specimens. From their collective work, some general patterns have emerged, such as scaling, nonclassical nonlinearity, and other 'universalities'. A quest for universal laws is not only interesting 'per se', but can also yield practical applications. If several fields share a common mathematical or conceptual background, a cross-fertilization among them may lead to a quick progress, even if ultimately the specific details of any individual application must be considered independently. The idea behind the proposed top-down approach is of course, not to replace, but to complement current investigations by searching for solutions that often 'mutatis mutandis' already exist, but are confined to a different network of researchers. In the present contribution we start from the conjecture, based on a large amount of experimental observations, of the existence of a nonlinear mesoscopic elasticity universality class. We search for the basic mathematical roots of nonclassical nonlinearity, in order to explain its universality, classify it and correlate it with the underlying meso- or microscopic interaction mechanisms. In our discussions we explicitely consider two quite different kinds of specimens: a two-bonded-elements structure and a thin multigrained bar. It is remarkable that, although the former includes only one interface and the latter very many interstices, the same 'inteaction box' formalism can be applied to both. The generality of the proposed formalism suggests that a similar approach may be adopted in completely different contexts, e.g. in biological, biomedical and social sciences