Analytical solution for elastic fields caused by eigenstrains in a half-space and numerical implementation based on FFT

Modern engineering design often faces severe challenges in accommodating impurities and imperfections of materials in the presence of considerable thermal expansion and plastic deformation. Based on micromechanics, a versatile and effective approach for such non-linear problems can be conceived by employing an inclusion model. This paper reports on the derivation of explicit integral kernels for the elastic fields due to eigenstrains in an elastic half-space. The domain integrations of these kernels result in analytical solutions to stresses and deformations. After dividing each general kernel into four groups, the integration is resolved into three-dimensional convolutions and correlations, which can be numerically processed with algorithms based on fast Fourier transform (FFT) to enable efficient and accurate numerical computations. The analytical solution corresponding to a cuboidal inclusion (a rectangular parallelepiped domain) is obtained in an explicit closed-form and is utilized to determine influence coefficients. The present solution and numerical implementation can be used as building blocks for analyzing arbitrarily distributed thermal strains, plastic strains and material inhomogeneities, as demonstrated by solving an illustrative example of elasto-plastic contact.