inhomogeneities

Universal displacements are those displacements that can be maintained, in the absence of body forces, by applying only boundary tractions  for any material in a given class of materials. Therefore, equilibrium equations must be satisfied for arbitrary elastic moduli for a given anisotropy class. These conditions can be expressed as a set of partial differential equations for the displacement field that we call universality constraints.

Eshelby and the inclusion/inhomogeneity problems

Any materials scientist interested in mechanical behaviour would be aware of the contributions of J.D. Eshelby. With 56 papers, Eshelby revolutionised our understanding of the theory of materials. The problem that I wish to discuss in this page is the elastic stress and strain fields due to an ellipsoidal inclusion/inhomogeneity - a problem that was solved by Eshelby using an elegant thought experiment.

In two papers published in the Proceedings of Royal Society (A) in 1957 and 1959 (Volume 241, p. 376 and Volume 252, p. 561) Eshelby solved the following problem ("with the help of a simple set of imaginary cutting, straining and welding operations"): In his own words,