The Stochastic Micromechanical Approach to the Response Behaviour of Engineering Materials

Abstract

In the past several decades, the micro mechanical approach has been recognized as a promising tool for the description of the response behaviour of engineering materials with the inclusion of the so-called “local” or “microstructural” effects. The microstructure of a class of such materials, however, is discrete in the sense of being heterogeneous and/or discontinuous. In view of this fact, the so-called “deterministic micromechanics”, that are based on the concepts of continuum mechanics, could no longer be accepted for the interpretation of the experimental results concerning the behaviour of discrete materials. It has been, therefore, increasingly appreciated that a more appropriate representation of discrete materials would only be achieved by including the random characteristics of the real microstructure. Further, the response behaviour of such microstructure is often both time- and loading history-dependent. Thus, the pertaining deformation process and its space- and timeevolutions are expected to be stochastic in character. In this context, the establishment of the connection between the response behaviour of the individual elements of the microstructure, their interactions, and the observable macroscopic behaviour would be an essential requirement. The fulfilment of the latter seems possible (Axelrad, 1993, and Axelrad and Haddad, 1998) by the introduction of the principles of set theory, together with the concepts of measure theory and topology. Thus, in the stochastic micromechanical formulations, continuum mechanics concepts are generally replaced by considerations of microstructural response variables in the form of discrete statistical functions. The latter are established within well-defined “measuring scales” defining the levels of observation into the material system.