An approximate analytical 3-d solution for the stresses and strains in eigenstrained cubic materials

A general solution for the total strains that develop in elastically homogeneous and arbitrarily eigenstrained linear-elastic bodies is derived by means of continuous Fourier transforms (CFT). The solution is specialized to the case of a dilatorically eigenstrained spherical region in an infinite body, both of which are made of the same cubic material. It is shown that, for "slight" cubic anisotropy, all integrations can be performed in closed.form. 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 closed-form solution is investigated numerically by means of discrete Fourier transforms (DFT). It is demonstrated that, even for strongly cubic materials, the closedform solution is still useful in order to perform parameter studies. Finally, the total elastic energy of two eigenstrained spheres in slightly cubic material is calculated in closed-form by means of CFT. The minimum of this energy is determined as a function of relative position of the two inclusions with respect to the crystal axes and it is used to explain the formation of preferred textures in cubic materials. ~) 1998 Elsevier Science. All rights reserved