Approximate analytical 2D-solution for the stresses and strains in eigenstrained cubic materials

Continuous and discrete Fourier transforms (CFT and DFT, respectively) are used to derive a formal solution for the Fourier transforms of stresses and strains that develop in elastically homogeneous but arbitrarily eigenstrained linear-elastic bodies. The solution is then specialized to the case of a dilatorically eigenstrained cylindrical region in an infinite matrix, both of which are made of the same cubic material with the same orientation of principal axes. In the continuous case all integrations necessary for the inverse Fourier transformation can be carried out explicitly provided the material is 'slightly' cubic. This results in an approximate but analytical expression for the stresses and strains in physical space. Moreover, the stress-strain fields inside of the inclusion prove to be of the Eshelby type, i.e., they are homogeneous and isotropic. The range of validity of the analytical solution is assessed numerically by means of discrete Fourier transforms (DFT). It is demonstrated that even for strongly cubic materials the stresses and strains are quite well represented by the aforementioned approximate solution. Moreover, the total elastic energy of two eigenstrained cylindrical inclusions in slightly cubic material with the same orientation of their principal axes is calculated analytically by means of CFT. The minimum of the energy is determined as a function of the relative position of the two inclusions with respect to the crystal axes and it is used to explain the formation of textures in cubic materials.