Elastic stresses due to inclusions and inhomogeneities

Crystallanity

Almost all the metals and alloys that are used in practical applications are crystalline — that is, the atoms or molecules that make up the metal or alloy are arranged periodically in space. For the sake of simplicity (and, without loss of generality), in this post, I am assuming that this periodic arrangement can be built up of cubes — the corners and face centers of which are populated by atoms. This specific crystal structure is known as face centered cubic (fcc).

The periodic arrangement of atoms/molecules in a crystal results in many important and interesting properties. One of them is the lattice parameter, which, in our case, is the size of the cube, or the distance between the centers of two atoms that are occupying the cube corners.

The elasticity also follows from the crystalline structure rather naturally. In a crystalline solid, the atoms act as if they are connected by springs, and the crystalline structure that a particular metal or alloy chooses is dictated, at some level, also by these “springs” and their strength. Hence, in a crystal, if you try to move any atom from its equilibrium position (determined by the lattice parameter), it tries to go back; if it can’t, the springs that attach it to the other atoms are either stretched or compressed; these stretchings and compressions are what make the crystal elastic. As soon as the forces on these atoms (which moved them away from their equilibrium positions) are removed, the atoms go back to their original positions. The elastic constants of a material tells us, for a given force, how much these atoms can be strained.

The crystal structure, by definition, makes the solid anisotropic (that is, if you sit on an atom and look at different directions, its properties are different); hence, the elastic constants naturally inherit the anisotropy of the underlying crystalline structure. In the case of cubic crystals, the elastic constants are obviously cubic anisotropic. What this means in practical terms is that, if you look in the directions of cube edges and the cube diagonals, the elastic properties are different; more specifically, either the cube edge direction or the diagonal direction is elastically softer as compared to the other; that is, for a given force, the atoms in the softer direction are relatively more pliable.

Dual phases

Many of the metallic materials used in practical applications are not only alloys (that is, they consist of more than one type of elements) and polycrystalline (that is, each material consists of several crystallites), but also consist of more than one phase (that is, consists of solid material that has different physical properties). The different phases and the different combination of crystallites give rise to a wide variety of interesting microstructural features (features initially noticed at the micrometre scale, and hence the name), which, in turn, give rise to several interesting (and some times important) properties to the material. Thus, it is no wonder that a large fraction of materials scientists and engineers are interested in studying the microstructural features, their effects on properties and ways of tuning both.

For this particular post, I am going to consider a specific model alloy which consists of two elements — nickel and aluminium (in practice, several other elements too — but for our purposes, it is sufficient to deal with the alloy as if it consists of only these two elements). It also consists of two phases; one of them, the nickel rich phase, has fcc crystal structure; the other, which consists of the specific fraction of three nickel atoms for each aluminium atom, crystallises in a structure that is very close to fcc called $L1_2$; in fact, it is the fcc structure, except that the aluminium atoms prefer to occupy the cube corners while nickel atoms occupy the face centers. Naturally, these two phases have different lattice parameters and elastic constants; however, since both these phases are cubic crystalline structure based, the elastic anisotropy is the same for both phases. The typical microstructure in this material consists of cuboids of the $L1_2$ phase (precipitates) distributed in the fcc material (matrix), making it look like a miniature version of bricks ($L1_2$) and mortar (fcc) in masonry.

Misfit and elastic inhomogeneity

In the case of dual phase materials like the one described above, the complexity of the microstructure of the materials also leads to several other important materials properties and parameters, of which, two are of specific interest to us. One is known as the misfit, which gives the difference in lattice parameter between the matrix and precipitate phases (normalised by the matrix phase). The second property is called inhomogeneity — that is, at different parts of the material, the properties (specifically, the elastic constants) are different (since the phases are different).

Both the misfit and elastic inhomogeneity play a key role in strengthening of alloys; in fact, this is the key process that leads to the superior mechanical properties of superalloys — alloys which are used in aerospace industry and in the making of gas turbines; and, the $L1_2 Ni_3Al$ precipitates in nickel rich fcc is in fact the most important ingredients of nickel-base superalloys.

The Problem

There are several interesting questions that one can ask about the microstructure and its evolution in dual phase (for simplicity, single crystalline) alloy materials of the type described above. In this post, we will ask one such question, namely, what are the elastic stress and strain fields associated with the microstructure in such materials? This question of the elastic stresses and strains is of interest both from (a) the point of view of understanding these materials and their properties, and, (b) from the point of view of using these materials in practical applications; thus, it is no wonder that the two papers that J D Eshelby wrote outlining a process for obtaining these fields (albeit for the case of some special geometries) have become classics in the field.

Both these were published in the Proceedings of the Royal Society of London: Series A. Mathematical and Physical Sciences. The first, published in 1957, is titled The determination of the elastic field of an ellipsoidal inclusion, and related problems [1]. The second, published two years after the first in 1959, is titled The elastic field outside an ellipsoidal inclusion [2]. These two papers are very accessible and are a pleasure to read – and are a must read for anybody who is interested in theoretical materials science.

Of course, Eshelby is not the first scientist to look at the problem; in his first paper, he does list several authors who have discussed many special cases of the problem; however, as Eshelby himself notes, not only is his method rather simple and straight-forward, it is also the most general; this generality allows for easier generalization in applying Eshelby’s methodology for shapes that are not simple and obtaining elastic solutions numerically in such cases. Further, the simplicity of the methodology, and the exact solutions given by him for the special cases also serve another important purpose, namely, that of validating codes written for evaluating stress and strain fields associated with more complex shapes. In fact, Eshelby’s paper also notes several mistakes in the usual expressions for these fields given for certain special cases prior to this work — some by Eshelby himself, and some by masters in the field such as Landau and Lifshitz in their classic textbook. Thus, these two classics not only clarified several important issues, but, in some sense, actually gave the road map for future work in the area of micromechanics of defects in solids for the next several decades; even today, not only the attempts to push Eshelby’s methodology to more and more general (and, more and more complex) situations, but also the efforts to obtain similar, exact analytical expressions for shapes other than those discussed by Eshelby, namely, ellipsoids of revolution, are being pursued.

As with all classics, the appreciation of the work, in the minds of its readers, is always enhanced with a little bit of knowledge about the authors. So, before I proceed with the post, I would like to draw the attention of the readers to the memoir on Eshelby, published by B A Bilby in the Biographical memoirs of Fellows of Royal Society titled John Douglas Eshelby. 21 December 1916 – 10 December 1981 [3]. There is also a short, one page appreciation (more about the man than about his work) of Eshelby, published by Alex D King in the Posterminaries column of the MRS Bulletin in July 1999 [4]; this piece, among other things, tells as to why Eshelby chose to work on theoretical problems instead of doing experiments, and how he came to be awarded FRS; both of these stories are extremely interesting and funny (if a bit apocryphal). Finally, I should also mention the book The coming of materials science by Robert W Cahn [5], which puts this and other classics that Eshelby wrote (Yes; though he wrote only 56 papers or so in his entire career, several of them are considered classics and continue to be studied with great interest – and, at times, with awe and reverence) in the general perspective of growth of materials science as a discipline.

Elastic stresses during phase transformations

The easiest way to imagine as to why there might be elastic stresses and strains in such dual phase materials, and what these stresses or strains might look like, consider a material that is initially in the fcc crystal structure; now, let us assume that one small part of this material transforms into $L1_2$; since this part now has a different lattice parameter and hence a misfit as compared to the original fcc phase, the transformed region either wants to shrink or expand (depending on the sign of the misfit).

If suppose the fcc phase is infinitely rigid, all the strain associated with the expansion or shrinkage will be accommodated by the transformed region; and hence, it will be in a distorted geometry as compared to what it would in free state —that is, if it was not surrounded by this rigid fcc phase.

On the other hand, it is also possible to imagine that the fcc phase is infinitely compliant as compared to the transformed region; in this case, the transformed region would look exactly like what it would look without the surrounding material; however, all the strain associated with the transformation is now accommodated by the straining of the surrounding fcc phase.

In reality, neither of the phases are infinitely rigid; in fact, a parameter called inhomogeneity ratio $\delta$, can be defined which tells how relatively rigid the transformed region is as compared to the original phase; and, corresponding to $\delta$, the strain due to misfit is distributed in both the phases; thus, the problem is to find the exact value of the stresses and strains at various points given the misfit and the shape of the transforming region.

In fact, even in the case where we assume that the different phases have the same elastic constants and differ only in the lattice parameter, the strain fields are not easily evaluated; and, depending on the geometry of the transformed region, the resulting strain fields might have many subtle and aesthetically pleasing properties (as was pointed out by Eshelby for the first time): for example, for isotropic, circular inclusions, if the eigenstrain is dilatational (no shear components), the shear stresses outside the inclusions are zero; the principal stresses are equal and opposite in sign (and, one of them is discontinuous while the other continuous at the inclusion-matrix boundary).

As the titles of the papers indicate, Eshelby only dealt with a particular class of shapes, namely, ellipsoids of revolution. However, his methodology forms the basis for a numerical evaluation of the strain fields for arbitrary shapes (which is one the reasons why the paper is so essential and influential). Having said that, note that ellipsoids of revolution, contain within them, as special cases, several important shapes that one is interested in general for evaluating many material properties – like needles, spheres, plates, etc. So, the paper and some of the results presented in them are of great use by themselves.

Elastic solutions (using the elegant Eshelbian cuts, strains, and weldings)

The exact problem that Eshelby solved (“with the help of a simple set of imaginary cutting, straining and welding operations”), in his own words, is the following:

• 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.

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 interface 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.

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; he also gave exact expressions for the stress and strain fields provided the region of interest is an ellipsoid of revolution; his methodology has been generalised to arbitrary geometries  and multiple inclusions/inhomogeneities (with the resultant equations being solved numerically, of course) by Khachaturyan and his co-workers [6].

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 [7,8].

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. Solutions for elliptic inclusions and inhomogeneinities are also available at my googlepage; and, sometime soon, I will upload the code used for obtaining these fields, and give a link to the page here.

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.

A digression

By the way, Green is another fascinating figure in the annals of science/mathematics, and I understand that the paper in which Green introduced the idea of the functions that bear his name is a classic by itself.

‘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).

Going beyong Eshelby (at least in 2D): the complex variable formalism

If we are interested in ellipses and not ellipsoids (that is, 2D problems), it is possible to avoid the cumbersome integrals mentioned earlier. In 1960, in a paper published in the Proceedings of Cambridge Philosophical Society, Jaswon and Bhargava showed how to avoid the elliptic integrals using a complex variable formulation [9].

Jaswon 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 himself felt the same way, since in his 1959 paper [2] he says:

• It has to be admitted that, except in the simplest cases, a calculation of the external field is laborious.

Jaswon 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.

The solution of Jaswon 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 [10] and Mushkelishvili [11]  give the expressions for writing down the contour integrals!
A classic, for all times!

Bilby, writing in 1990 [3], notes that the first of Eshelby’s paper discussed here was in the second group of 100 most highly cited papers in all fields of science covered by the Science Citation Index, 1955–1986. However, citations tell only one part of the story. The ideas and results due to Eshelby have become text book material and are being used continually; thus, sometimes references are not made to his paper but to some text book that describes the same methodology.

Bilby, in the same memoir, also notes that

This work on inclusions and inhomogneities has been applied by others, not only to calculate the stress fields and interactions of inclusions, inhomogeneities, precipitates, twins, martensite plates, cavities and cracks, but also to find the bulk elastic properties of bodies containing distributions of inhomogeneities and cavities and to discuss the properties of polycrystals and composite materials. Eshelby sketched many of these applications and noted also that the method could be applied to find the perturbation caused in a slow viscous flow by the presence of a rigid or deforming ellipsoid. (…) He was particularly interested in the viscous problem because the fact that an ellipsoid remains an ellipsoid under homogeneous deformationmeans that its finite change of shape can be studied without the need to find the details of the complicated flow outside it. This application, which has seen considerable further development, is relevant to the theory of the homogenization of glass, the determination of strain in rocks, the deformation of voids and the flow of suspensions containing rigid or defromable particles.

After nearly 18 years, today, if, had Bilby been writing an appreciation of these papers, he would have included several more fields where his results cotninue to play a crucial role — one of them being the study of microstructural evolution in elastically inhomogeneous solids with defects (to which, Khachaturyan and his co-workers have contributed immensely [6]).

However, as Eshelby himself seems to have noted, the most important reason why these two papers are classics are not for their results, but for the methodology that was developed in them, which continue to be of use fifty years after he published these papers [3]:

However, he liked to regard himself as a humble ‘supplier of tools for the trade’ and often left their detailed use to others.

In the process, this supplier of tools, also made the tools themselves more respectable (as is noted in the context of a prior paper of Eshelby which also uses the imaginary cutting, straining and welding operations) [3]:

Notable in this paper of 1951 is his derivation of results by the use of imaginary cutting, straining and welding operations, a technique that he used frequently with great effect. The method was not regarded as quite respectable by some with a more formal mathematical training.

Finally, as with all classics, these two papers of Eshelby should be read not only because of their relevance and use, but also for their beauty and elegance, which gives the reader so much of pleasure!

References

[1]  J D Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proceedings of the Royal Society of London: Series A. Mathematical and Physical Sciences, 241, p. 376, 1957.

[2] J D Eshelby, The elastic field outside an ellipsoidal inclusion, Proceedings of the Royal Society of London: Series A. Mathematical and Physical Sciences, 252, p. 561, 1959.

[3] B A Bilby, John Douglas Eshelby. 21 December 1916 – 10 December 1981, Biographical memoirs of Fellows of Royal Society, 36, p. 126, 1990.

[4] A H King, Posternimnaries: Lessons from J D Eshelby, M R S Bulletin, p. 80, July 1999.

[5] R W Cahn, The coming of materials science, Pergamon materials series, Elsevier Science Publishers, 2003.

[6] A G Khachaturyan, Theory of structural transformations in solids, John Wiley & Sons (p. 198), 1983; A G Khachaturyan, S Semenovskaya and T Tsakalakos, Elastic strain energy of inhomogeneous solids, Physical Review B, 52, p. 15909, 1992.

[7] R Philips, Crystals, defects and microstructures: Modeling across scales , Cambridge University Press (p. 520), 2001.

[8] T Mura, Micromechanics of defects in solids, Kluwer academic publishers (p. 74), 1987.

[9] M A Jaswon and R D Bhargava, Two-dimensional elastic inclusion problems, Proceedings of Cambridge Philosophical Society, 57, p. 669, 1960.

[10] A E Green and W Zerna, Theoretical elasticity, Oxford University Press, 1968.

[11] N I Muskhelishvili, Some basic problems of the mathematical theory of elasticity, Springer, 1975.

2 Responses to “Elastic stresses due to inclusions and inhomogeneities”

1. biswa Says:

Guru,

This post will get a more appreciative audience if posted on iMechanica. Please consider providing a copy there.

— Biswajit

2. Guru Says:

Dear Biswajit,

I have cross-posted a copy at iMechanica; the core of this piece already exists in iMechanica (in the classics section); I added some preamble to the technical sections and posted it here so that I may submit it for the carnival The Giant’s Shoulders.

Guru