Inclusions and Inhomogeneities
Eshelby and the inclusion/inhomogeneity problems
Any materials scientist interested in mechanical behaviour would be aware of the contributions of J.D. Eshelby. With less than 20 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,
The transformation problem
A region (the "inclusion") in an infinite homogeneous isotropic elastic medium undergoes a change of shape and size which, but for the constraint imposed by its surroundings (the "matrix"), would be an arbitrary homogeneous strain. What is the elastic state of inclusion and matrix?
The homogeneous strain is known as "eigenstrain" or "transformational strain". In the same paper, Eshelby also introduced the concept of "equivalent inclusion" for solving the transformation problem when the matrix and the region of eigenstrain (the "inhomogeneity") have different elastic constants. These two papers are very accessible and are a pleasure to read - and in case I forgot to mention, these two papers are a must read for anybody who is interested in theoretical materials science.
Thought experiment of Eshelby
The operations that Eshelby used to solve the transformation problem are the following:
Remove the region of interest from the matrix.
Allow it to take the eigenstrain.
Restore the region to its original shape and size by applying suitable surface tractions and put it back into the matrix and rejoin.
Remove the body force on the between the inclusion and matrix by applying an equal and opposite layer of body force.
In step (3), the stress is zero in the matrix and is a known constant in the inclusion. The additional stress introduced in step (4) is found by the integration from the expression for the elastic field of a point force.
The details of these calculations are found in the following (advanced) texts:
(1) Rob Philips, Crystals, defects and microstructures: Modeling across scales , Cambridge University Press (2001), p. 520.
(2) Toshio Mura, Micromechanics of defects in solids, Kluwer academic publishers (1987), p. 74.
(3) A.G. Khachaturyan, Theory of structural transformations in solids, John Wiley & Sons (1983), p. 198.
While Rob philips and Mura invoke the elastic Green function to solve the transformation problem for a single inclusion, Khachaturyan describes the generalisation of the inclusion problem to multiple inclusions/eigenstrains.
Eshelby (with his cuts, strains and weldings) has shown that the transformation problem is equivalent to solving for the equations of elastic equilibrium of a homogeneous body with a known body force distribution.
The Green function approach
For homogeneous bodies with known body force distribution, the equations of elastic equilibrium are solved using the elastic Green function. Rob Philips and Mura describe the Green function approach in great detail.
In case you are interested in looking at some solutions of inclusion problems, Rob Philips describes the radial displacements associated with a spherical inclusion of radius "a" with dilatational eigenstrain obtained using Green functions (See the figure 10.14 on page 524) and indicates that the elastic energy of a spherical inclusion with dilatational misfit scales as the volume of the inclusion. Of course, Mura's classic is a singular testimony to the power of a combination of Green function and the eigenstrain approach of Eshelby.
The Green function approach finally results in the evaluation of elliptic integrals for obtaining the displacements. This is not surprising since we are integrating the body forces over an elliptic geometry (Remember, the inclusions were ellipsoids/ellipses). The gradients of displacement give us the strain - That means we need to differentiate the Green functions. Thus, it is rather cumbersome to calculate the elastic stress, strain, or displacement fields using the Eshelby-Green approach.
The complex variable formalism
However, if we are interested in ellipses and not ellipsoids (that is, 2D problems), it is possible to avoid the cumbersome integrals. In 1960, in a paper published in the Proceedings of Cambridge Philosophical Society, (Volume 57, p. 669), Jawson and Bhargava showed how to avoid the elliptic integrals using a complex variable formulation! The solution of Jawson and Bhargava is based on the following ideas/results:
Eshelby's method involves integrals of point forces on the matrix-inclusion boundary.
The expression for the displacement at any point x due to a point force F acting at the point y being known, the Eshelby problem now reduces to an integration of a continuous distribution of the forces over the inclusion surface.
Green and Zerna (Theoretical elasticity) and Mushkelishvili (Some basic problems of the mathematical theory of elasticity) give the expressions for the required mathematical arsenal for writing down the contour integrals!
Jawson and Bhargava motivated their complex variable formulation by making the following observation:
Although Eshelby has proved some general theorems of great interest, using elegant methods, his solutions involve analytically intractable integrals of a formidable nature.
Eshelby himslef felt the same way, since in his 1959 paper he says:
It has to be admitted that, except in the simplest cases, a calculation of the external field is laborious.
Jawson and Bhargava built their complex variable formalism on the "ingeneous attack" on the transformation problem by Eshelby "utilizing the point-force concept". In view of the "novelty and importance" of the approach, they also give a brief description of Eshelby's arguements - And their description is by far one of the best that I have seen in the literature. Add this paper to your reading list, if you have not done it already!
'Equivalent inclusion' for inhomogeneities
The idea of "equivalent inclusion" is simple - It is one of those nice mathematical tricks where we solve a problem by reducing it to another which has already been solved.
Let the inhomogeneity have an elastic constant that is different from that of the matrix. The idea is to replace the inhomogeneity with an inclusion - The eigenstrain in the hypothetical inclusion is such that it exerts the same actions on the matrix as the original inhomogeneity. Mathematically, finding out the eigenstrain in the equivalent inclusion amounts to solving for a set of three equations in three unknowns (in 2D) or six equations in six unknowns (in 3D).