I - Macroscopic Response of Continua with Random Microstructures

Abstract

This chapter discusses the macroscopic response of continua with random microstructures. It discusses the two length scales required to reduce the general theory to an effective modulus theory. The chapter reviews studies relating the effective elastic moduli tensor to statistical descriptions of the microstructure. It presents a result of central importance for predicting the ensemble averaged response field for statistically homogeneous materials. The averaged response field satisfies field equations of the same form as those governing each manifestation of the response field provided one replaces the local, randomly varying property field measure by a nonlocal, deterministic, effective property field measure. The integral converges absolutely in the limit if this rate of decay is faster than r − 3, r being the separation distance, and diverges if the decay rate is slower than this. It is found that for a decay rate equal to r – 3, the integral can be said to be conditionally convergent in the sense that although a limiting value can be achieved, the actual value obtained depends on the shape chosen for the outer boundary before taking the limit.