Compliance contribution tensor of an arbitrarily oriented ellipsoidal inhomogeneity embedded in an orthotropic elastic material

In 1957, Eshelby showed that a homogeneous isotropic ellipsoidal inhomogeneity embedded in an unbounded (in all directions) homogeneous isotropic host would feel uniform strains and stresses when uniform strains or tractions are applied in the far-field. Since then, numerous studies have conducted with the goal of gaining better understanding and improving on the mechanics involved. Compliance contribution tensor H describes the contribution of a single inhomogeneity to the strain produced by uniform remotely applied stress field. Alternatively, one can use the dual stiffness contribution tensor N that gives the extra stress due to inhomogeneity subjected to uniform remote strain field. These tensors serve as the basic building blocks in the calculation of overall elastic properties of heterogeneous materials and are particularly relevant for Maxwell’s scheme. In the past, there has been attempts to evaluate the effect of inhomogeneities on the overall elastic properties of materials with anisotropic matrix, most of which are related to anisotropic materials containing microcracks. For instance, attempts to estimate property contribution tensors using best fit isotropic approximation of the matrix material showed that it may be used for materials with weak anisotropy only.

Overall, there is need to further existing knowledge in this field. In view of the above, Professor Igor Sevostianov from the New Mexico State University together with Dr. Volodymyr Kushch at the Institute for Superhard Materials of the National Academy of Sciences in Ukraine developed a novel approach for approximating the compliance contribution tensor components for an inhomogeneity embedded in an orthotropic matrix. Their work is currently published in the International Journal of Engineering Science.

In their approach, they approximated the original orthotropic material using an elliptically orthotropic one according to the principle of approximate elastic symmetry. Next, they tested the proposed methodology for calculation of the overall thermal properties of anisotropic materials: i.e. they used the affine transformation to transform the second rank tensor obtained at step (A) to the second rank unit tensor. Lastly, they calculated components of the Hills’s tensors in transformed coordinates and then used inverse transformation to get the explicit expression for these tensors in the original orthotropic material.

Generally, the approach was based on approximation of an orthotropic material by one with the special type of symmetry, when fourth rank tensor of elastic stiffness can be expressed in terms of a second rank tensor. The authors showed that the accuracy of the approximation could be evaluated a’priori and, very often, the error produced by such an approximation was smaller than the one typical for experiments on elastic constants measurements.

In summary, the study Professor Igor Sevostianov and Dr. Volodymyr Kushch focused on the analysis of possibilities aimed at finding an approximate closed form solution for an isolated ellipsoidal inhomogeneity embedded in an orthotropic matrix and to calculate overall elastic properties of heterogeneous materials with orthotropic matrix. Overall, the presented approach was based on approximation of the orthotropic material by elliptically orthotropic one for which, as they show, the problem could be solved in closed explicit form.