Residual stresses and stored elastic energy of composites and polycrystals

Residual Stresses in heterogeneous materials may arise because of differential or anisotropic thermal expansion of constituents. The paper is concerned with thermoelastic solids whose material properties fluctuate on the microscopic scale. Rigorous general relations between stored elastic energy and statistical averages (mean values and fluctuations) of residual stresses are derived. These results are applied to two-phase composites and to materials where the fluctuations of elastic constants can be neglected. One obtains exactly the stored energy, certain conditional mean values and the covariance matrix of the residual stresses. Under the assumptions of statistical homogeneity and isotropy, the results hold for any type of heterogeneous microstructure.