Effective thermal expansion of heterogeneous materials with application to low temperature environments

This paper presents direct formulations of the effective thermal linear expansion (TLE) and the effective coefficient of thermal linear expansion (CTLE) of heterogeneous materials, or multi-constituent composites, with temperature dependent constituent properties and an arbitrary initial incompatible eigenstrain field. The effective properties are expressed in terms of the stress and strain concentrators. For bi-phase composites results are expressed in terms of the effective elastic properties rather than the concentrators. These developments are based on the linear theory of uncoupled thermoelasticity. An example is presented for niobium (Nb) fibers embedded in a copper (Cu) matrix at cryogenic temperatures. It is shown that this composite achieves negative CTLE despite the CTLE of both Nb and Cu are strictly greater than zero. In addition, it is shown that the presence of an initial field of incompatible eigenstrains is capable of causing anisotropic thermal expansion coefficients in an otherwise macroscopically isotropic material of isotropic constituents. Due to the form equivalence of the governing equations the developments which are presented are also applicable to the area of moisture swelling.