Structure-Sensitive Properties of Materials

Abstract

The division between “structure-sensitive properties”, such as brittle fracture and “structure-insensitive properties”, such as the effective elastic moduli of a weakly disordered materials is a rather murky one. There are many properties which lie on the border between these limits and some properties which are of one sort on one length scale while being of the other sort on longer length scales. In this paper, we make the definition of “structure-sensitive” to mean that the property in question depends on the length scale in a “non-linear” or “non-Euclidian” manner. We take as three examples: the tensile fracture strength of inhomogeneous materials; the elastic moduli and topology of stress-bearing paths near the rigidity threshold; and the topology of minimal surfaces in disordered systems. These properties and others which are “structure-sensitive” are not amenable to treatment by usual “homogenization” methods and so require a new set of tools and ideas in their analysis. As well as large scale computation, the ideas of scaling, self-similarity, fractals, and extreme statistics provide a sound basis for the analysis of these problems. In this paper, we discuss a simple analytic method for finding the scaling behavior of the tensile fracture stress of disordered networks (section II), and then outline some innovative numerical methods for finding the scaling behavior of: the rigid backbone in central force systems (section II), and the topology of minimal surfaces relevant to the plasticity of random materials and the topology of fracture surfaces (section III). The latter sections profit from results in graph theory and combinatorial optimisation.