Thermoelastic fields for a heat exchanger of arbitrary shape in a bi-material infinite plane

Abstract

When a bi-material with two jointed dissimilar half-planes containing an arbitrarily shaped polygonal inclusion is subjected to heat flow, the thermoelastic fields, including temperature and displacement, can be derived by the Green’s function technique with the integral of the source over the inclusion. Using Hadamard’s regularization, the two-dimensional (2D) thermal, elastic, and thermoelastic Green’s functions of two-jointed dissimilar half-planes are firstly derived from the 3D Green’s functions as the corresponding fundamental solutions. The fundamental solutions for semi-infinite and infinite domains can be recovered by adjusting the material constants. Eshelby’s tensors are derived in terms of the biharmonic, harmonic, and two Boussinesq’s displacement potential functions. When a heat exchanger of arbitrary shape is embedded in a matrix with different thermal and mechanical properties, combining a continuously distributed eigen-temperature gradient and eigenstrain field, the dual equivalent inclusion method (DEIM) is applied to handle the material mismatch of thermal conductivity, stiffness, and thermal expansion coefficient, respectively. Therefore, the full thermoelastic fields can be obtained by the integral over the heat exchanger only. The eigen-fields are expanded in the Taylor series referred to the center of the particle, which exhibits tailorable accuracy with uniform, linear or quadratic terms in comparison with the analytical solution for a circular inhomogeneity in the infinite domain. An exact thermoelastic solution of a circular inhomogeneity embedded within the infinite domain is present. The case study of an electric heat cable in the concrete block demonstrates the capability and exactness of the model. The method can be used for a thin film containing a heat exchanger of arbitrary shape as either a heat sink or source.