Field fluctuations and the effective behaviour of micro-inhomogeneous solids

The problem of predicting the effective mechanical properties and response of micro-inhomogeneous solids is revisited. The aim is to highlight the influence of field fluctuations which, as a rule, is neglected by the numerous existing theories that interconnect the micro- and macro-behaviour of such solids. The key observation is that in a heterogeneous solid of random constitution when, say, macroscopic quantities like the mean stress tensor are prescribed, fluctuations always create regions in which considerably higher stresses appear. In these regions either plastic flow or a certain kind of deterioration takes place, which affects the macroscopic behaviour of the solid. The result is that the latter should start exhibiting deviations from linear response from the very beginning of straining, despite the assumed linearity of its constituents. A quantitative approach that takes into account the field fluctuations is proposed and outlined in the lecture. For the simplicity sake, the scalar conductivity problem for a dilute dispersion of spheres, possessing properties different from those of the matrix, is employed in order to illustrate better the basic ideas. The progressive deviation from linearity, when the macroscopic "straining" increases, indeed shows up clearly in the performed analysis.