Posts Tagged ‘The giant’s shoulders’

The third Giant’s Shoulders: five days to go

September 10, 2008

As noted earlier, the third Giant’s Shoulders carnival will be hosted here on the 15th of this month; so, send your entries to me at this email address (preferably on or before 14th of September 2008):

gs dot carnival dot guru at gmail dot com

I am looking forward to a carnival with huge number of entries which will keep the readers occupied for at least a month. So, send as many entries as you can and on as varied subjects as you can.

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The third Giant’s Shoulders!

August 23, 2008

I am very happy to announce the hosting of the third Giant’s shoulders blog carnival here at Entertaining research; this is the first carnival that I would be hosting; I am thrilled to be in the company of Coturnix and Lay Scientist, who hosted the first and second editions, respectively.

The third edition will be published on the 15th of September, 2008; so, send your entries to me at this email address (preferably on or before 14th of September 2008):

gs dot carnival dot guru at gmail dot com

I am looking forward to many interesting submissions on varied subjects!

Shapes of crystallites and their evolution

August 11, 2008

If you can look into the seeds of time,
And say which grain will grow and which will not,
Speak then to me, …

— Shakespeare (Macbeth)

Grain growth in polycrystalline solids

Most of the metallic and ceramic materials in use usually consist of many crystallites of differing orientations; such materials are classified as polycrystalline and the crystals of differing orientations that make up the polycrystalline aggregate are known as grains.

In a polycrystalline sample that is held at high enough temperatures, the total number of grains will decrease (with a corresponding increase in the mean size of the surviving grains). This process of known as grain growth.

The shapes and sizes of grains as well as their evolution when the sample is held at high enough temperatures is a problem of both industrial importance and academic interest; it is of industrial importance because many of the material properties of interest are determined by the underlying grain structure; it is of academic interest since grain growth is but one example of a large class of phenomena in which a material system reduces the area of internal interfaces in an effort to minimize its free energy.

There exists an important result for the growth of (idealized) grains in two dimensions (2D), which can be stated as follows:

The rate of change of area of a grain is determined purely by its topology; if it has more than six sides, it would grow; if it has less than six sides, it would shrink; and, if the number of sides are six, such a grain would neither grow nor shrink.

Or,

\frac{\partial S}{\partial t} \propto (n-6)

where, S is the area of the grain, t is the time, and n is the number of sides.

This is known as the Neumann-Mullins law following its postulation and derivation by Neumann [1] and Mullins [2].

In this post, I want to discuss these two papers [1,2], some of the history behind the contribution of Neumann, and the recent extension of this result by MacPherson and Srolovitz [3] to three dimensional (3D) systems.

A bit of history: the role of Cyril Stanley Smith

Though the name of C S Smith is not associated with the Neumann-Mullins law, he played a key role in its formulation. Smith, as I understand, delivered the Institute of Metals lecture in 1948 titled Grains, phases and interfaces: An interpretation of microstructure [4]. Unfortunately, I do not have access to this piece at the moment; however, the impact of this work is clear from the Foreword to the proceedings of the seminar called Metal Interfaces held during the thirty-third national metal congress and exposition, Detroit, October 13 to 19, 1951 [5]:

There was a sharp spurt in metallurgical research in this general field [surfaces or interfaces] as a result of a lecture in 1948 by Cyril Stanley Smith which brought new ideas to bear on an old and important topic. The new ideas and data resulting from vigorous and productive research on metal interfaces in the past three years are the basis of this volume.

The contribution of Neumann is but a two page written discussion on the paper of Smith in this volume titled Grain shapes and other metallurgical applications of topology — reminding ourselves of the times when conference proceedings, and the written discussion in them were considered as proper venues for reporting original and far-reaching research findings.

As I see it, the main contribution of C S Smith is in (a) drawing attention to the possible applications of topological ideas to metallurgical problems; (b) setting the grain growth problem as belonging to the same class as cell structures in biological tissues, soap froths, foams, and problems on space filling; and, (c) using models of soap froths as near exact geometric models of grain growth in metals.

By the way, Smith’s paper is probably the only one in metallurgical literature which contains a line drawing based on a painting of Picasso.

It is not clear if Neumann was present at the Detroit meeting in 1951. However, what is clear from Smith’s paper is that he did take active interest in involving mathematicians in this kind of project — one of his acknowledgements in this paper, for example, is to Saunders MacLane — so, it is quite possible that Smith circulated a copy of his paper to Neumann too for comments; further, since Smith also worked in the Manhattan project along with Neumann, and since Neumann gave strong political support to the formation of the field of materials science [6], it is not out of place to postulate such a strong working relationship between Neumann and Smith on such a problem of metallurgical interest with a strong mathematical component.

For those of you who are interested in learning more about C S Smith, the man and his work, I can do no better than to point to this June 2006 issue of Resonance, which is a special issue honouring C S Smith.

Before I leave Smith behind, I just want to draw your attention to the reply that Smith gave to the discussions section, in which he expressed a hope and a word of caution:

The discussion by Dr. von Neumann is much appreciated, and his conclusions are as remarkable for their simplicity as they are nonobvious on first consideration of the problem. It is greatly to be hoped that he, or some other mathematician, will be able to deduce similar relations in three dimensions and can combine these with the topological requirements to give the equilibrium distribution of bubbles toward which a froth must tend.

… Metals are not soap bubbles, even though soap bubbles are so useful in illustrating the basic principles, as I hope I have demonstrated in the paper.

While it took another fifty years (and another meeting between a mathematician and a materials scientist) for part of Smith’s hope to be fulfilled, his word of caution continues to give directions to research in this area.

The formulations of Neumann and Mullins

Neumann considered

the changes of bubble-volume due to diffusion, that occur in a two-dimensional bubble-froth.

And, as Neumann pointed out, to first approximation, this diffusion flow is proportional to the pressure difference, and

The pressure difference of the two adjascent bubbles, at a given point P of a wall, is \frac{2 \gamma}{R}, where \gamma is the surface tension of the liquid forming the froth, and R is the radius of curvature of the wall at P.

Further, since it is a bubble-froth, \gamma is a constant throughout the froth. With these given, assuming that the triple junctions in such a forth are at equilibrium (that is, they tend 120 degrees), Neumann obtained the now famous result:

In a two-dimensional bubble-froth the total gas-gain-rate of any bubble is (positively) proportional to n-6, where n is the number of sides of the bubble …

The derivation of Mullins is much closer in spirit to what is given in modern text-books (like that of Gottstein and Shvindlerman [7], for example). Mullins begins his derivation assuming that any boundary segment moves with a velocity v which is given by -M \gamma \kappa, where \kappa is the local curvature, and M is the mobility. And, as he notes, the result of Neumann is but a special case in Mullins’ derivation since

… within each cell of a soap froth, the possibility of a rapid mass flow of air maintains a uniform pressure which in turn causes each film to have a constant mean curvature; within a metal grain there is no possibility of a rapid mass flow and its associated uniformity of pressure so that the motion of any portion of a boundary is governed by local conditions only. Thus the problem of grain boundary motion, according to the curvature rule, is a problem in differential geometry.

Thus, the contribution of Mullins is in showing that the Neumann result holds even if the curvature is not a constant on any given domain wall — local curvature is what matters.

3D generalization of MacPherson and Srolovitz

Apparently, there had been many attempts to generalise Neumann-Mullins to 3D; none of these attempts have been successful in obtaining an exact result. However, recently, Srolovitz and MacPherson have not only managed to generalise the results to 3D, but to any higher dimension. The generalization is achived using the following strategy:

  1. Generalise 2D von Neumann-Mullins to multiply connected domains;
  2. Volume integrate the result; this is equivalent to considering all possible sections of a 3D structure, and doing a von Neumann-Mullins analysis on each, and putting the results together.

The result of the above manipulations is the introduction of a natural measure of length called mean width (which is a Hadwiger measure); and, using such a measure, Neumann-Mullins can be generalised to N-dimensional domain wall networks and the 2D and 3D networks are just the special cases of such a general result. And, as it turns out, in 3D (and in higher dimensions), curvature driven growth is not purely topological.

Here are some more resources on the 3D generalization.

  1. The supplementary information to the Nature article which gives the details of the derivation.
  2. David Kinderlehrer, in a News and Views piece puts the work of MacPherson and Srolovitz in perspective:

    A long-standing mathematical model for the growth of grains in two dimensions has been generalized to three and higher dimensions. This will aid our practical understanding of certain crucial properties of materials.

    He further notes some of the crucial assumptions made in deriving the 3D result and the future direction of research in the area:

    A physical grain network is beset by inhomogeneities and anisotropy, to name just two impediments to ideal growth. Even in the abstract, the effect of such features is unknown: we enter the realm of stochastic analysis through simple combinatorial events such as cell or facet deletion. Such analysis, implemented with automated data acquisition in the laboratory and large-scale simulation at the desk, will be the future direction of the subject.

  3. The Scientific American commentary.

Whither

The Neumann-Mullins result is a classic; not only does it put the problem of grain growth in the same class as that of coarsening in soap froths, namely, curvature driven growth, but also shows the close relationship between problems that are of interest to materials scientists and mathematicians, and how a collaboration between the two can result in startling-ly simple and elegant results. Further, these results also serve as an important benchmark for both experiments and theory. The nearly half-a-century long, hard search for its generalization has brought much progress to the field, and the final result is as satisfactory as the 2D one, and elegant in its own way. However, like all true classics, Neumann and Mullins, even as they maintain their relevance, also lead us in new directions.

In real systems, say, a grain boundary, for example, the boundary energies are anisotropic; the mobilities are not constant; the triple junctions induce drag on the boundary motion; and, all these are experimentally well known. Or, in other words, each of the assumptions made by Neumann, Mullins, MacPherson, and Srolovitz are to be relaxed; and, studies along these lines are pursued, and, I hope I will be able to report on some of them and our own phase field studies in these pages some time soon. In the meanwhile, let Neumann and Mullins engage you.

Related reading

As long time readers of this blog might already have noticed, this post is based on a couple that I wrote on this problem: here and here.

Acknowledgements

It is a pleasure to thank Abi, Peter, Shankara, and Prof. Ranganathan for many useful discussions and their comments on some of my earlier writings on Neumann-Mullins and MacPherson-Srolovitz (though, the mistakes, if any, that remain, are all my own).

References

[1] J von Neumann, in a written discussion to Grain shapes and other metallurgical applications of topology by C S Smith, in Metal Interfaces, American Society for Metals, Cleveland, Ohio, 1952.

[2] W W Mullins, Two-dimensional motion of idealized grain boundaries, Journal of Applied Physics, 27, pp. 900-904, 1956.

[3] R D MacPherson and D J Srolovitz, The von Neumann relation generalized to coarsening of three-dimensional microstructures, Nature 446, pp. 1053-1055, 2007.

[4] C S Smith, Grains, phases and interfaces — an interpretation of microstructure, Transactions, American Institute of Mining and Metallurgical Engineers, 175, pp. 15-51, 1948.

[5] R M Brick, Foreword to Metal Interfaces, American Society for Metals, Cleveland, Ohio, 1952.

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

[7] G Gottstein and L S Shvindlerman, Grain boundary migration in metals: Thermodynamics, kinetics and applications, CRC Press, 1999.

Elastic stresses due to inclusions and inhomogeneities

July 15, 2008
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.

Science fiction contest and commentary on classic papers

June 13, 2008

Abi alerts us about the 2008 Science Fiction contest; here are the rules (only Indian citizens; only one story per person; not more than 6000 words). Best of luck!

If you are not interested in writing fiction, but are still interested in science and writing about it, Greg Laden alerts us about a monthly science blogging event called The Giant’s Shoulders, which idea, is very appealing to me — in fact, I can immediately think of at least three posts that I can write — on Gibbs, on Eshelby and on Cahn!