Theory of thermal expansion of composites

Abstract

The problem discussed is to find the overall thermal expansion of a composite consisting of inclusions in a matrix of material with different expansion coefficients and elastic moduli. The volume mismatch causes strain fields. The total strain energy must be a minimum. This problem was solved previously for inclusions which are either spheres or randomly oriented long cylinders. In these simple cases the matrix strain field consists of a short-range shear component and a uniform expansion; the inclusion suffers uniform strain. The matrix is replaced by an effective medium having the average properties of the composite. The overall expansion coefficient could be obtained in closed form. This separation of the strain field into short-range shear and long-range uniform dilation is valid, at least to a good approximation, for all inclusion shapes. Simple expressions can thus be obtained in terms of coefficients which, although not calculated exactly, can be deduced approximately or can be determined empirically. Plasticity can be accounted for by allowing the shear modulus to depend on the temperature and on the maximum shear strain. The size of the inclusions does not enter the theory except through the yield strain, which depends on the extent of the strain field.