Recently, a new carnival called The Giant's Shoulders has been started and the first edition of the same is out at A blog around the clock. A post of mine on the elastic stresses due to inclusions and inhomogeneities made it to the carnival. I am cross-posting the piece here since it might also be of interest to the readers of iMechanica (though I did post a short note earlier here which forms the core of this long post too).

Elastic stress driven morphological instabilities in thin films have been studied extensively: see Jesson et al, J. Elect. Mat., 26, 9, pp., 1039-1047 (1997), for example, for a nice micrograph of rippling in Si-Ge system and a schematic of the explanation for the rippling. In the case of multilayer films, it is also well known that there could be  post-growth morphological changes, and the layer geometry as well as the number of layers play a crucial role apart from the elastic constants in these changes; see the studies on a Gadolinia-Silica system by Sahoo et al for example -- Appl. Surf. Sci., 252, pp. 1520-1537 (2005). In addition, there is also experimental evidence to show that in the case of multilayer films, there could be strong interactions between different layers of films:  see the studies on Si/Ge multilayers by B Rahmati et al, Appl. Phys. A, 62, pp. 575-579 (1996), for example.  Finally, Sridhar et al (Acta Mater., 45, 7, pp. 2715-2733 (1997)), using a linear stability analysis, showed that in embedded single and multilayer films, there could be two dominant modes of break-up, namely, symmetric and anti-symmetric.

The linear stability analyses of the type used by Sridhar et al, however, are not ideal for a detailed quantitative study of the effect of elastic interactions and volume fraction; further, such analyses are also not meant for the study of long term evolution and final break-up of thin films. In a paper of ours (Chirranjeevi et al--submitted to Acta Materialia), we study elastic stress driven morphological instabilities and consequent break-up of thin film assemblies using a phase field model.

Here is a video which might be helpful for those writing grant proposals (Hat tip to Dr. Aparna Bhatnagar, Director, Postdoctoral Affairs, Northwestern University for the email alert ):

Here is an e-book on Engineering Fracture Mechanics ; you can also download a demo version, the preface, and other related stuff from the page. The idea of the book (as described in the page) sounds interesting:

Update: An Open Source Review page has been created. Please feel free to leave links, codes and comments on the page. 

Dear Mechanicians,

I have seen that there is lot of code sharing among the mechanicians at iMechanica; a search for the word "code" for example produces nearly fifty entries, of which, I believe, at least half of the posts are pointers to codes and their sharing.

Here is a post at Directory Aviva listing a dozen U S Laws that every blogger should know.

Recently, the Royal Society Science book prize shortlist was announced; though the shortlisted books cover psychology, evolution, biodiversity, medicine and neurobiology, none in the area of materials or mechanics made it to the list. Or, pick any Best American Science writing volume--there are hardly any articles about materials or mechanics that make it to these anthologies.

Introduction

The blogosphere is abuzz with the latest report of the generalisation of the von Neumann-Mullins grain growth relation to 3 (and N) dimensions by MacPherson and Srolovitz (As an interesting aside, almost all the reports say mathematical structure of beer foam structure resolved, or words to that effect --hence, I also decided to join the bandwagon on that one). I heard Prof. Srolovitz describe the work in a seminar nearly six months ago. Based on my notes of the talk, I would like the explain their work in this post. Curvature in the following refers to mean curvature (and not Gaussian).

JOM is a monthly publication of TMS--The minerals, metals, and materials society. It covers a wide range of materials topics. I expecially like the overview articles, which, in four or five pages pack lots of information. Further, the historical articles about metallurgy and materials in ancient civilizations will interest those of you who like to read about history in general, and science history, in particular.

In an inaugural article in the latest issue of PNAS, George C Schatz writes on Using theory and computation to model nanoscale properties. Here is the abstract of the article:

Modelling and simulation is sometimes said to be the third way of doing science, the first two being theory and experiment; see this essay in Science for example:

In the recent issue of Science, researchers from UCLA (Chung et al) report on an ambient pressure synthesis (using arc melting) of a compound, namely, rhenium diboride, which is superhard. Apparently, the material leaves scratch marks on the surface of diamond. Here is the abstract of the paper:

Here is a video from the Seed magazine called Science in Silico. The video shows results from large scale simulations (and visualization) of fractals, microscopic dynamic processes in ribosomes, structure of viruses, bacterial flagellum, turbulence, explosions, and the modelling of cosmological events.

In a recent Progress Article in Nature Materials, David Taylor, Jan G Hazenberg, and T Clive Lee write about the damage that human bones sustain as a result of cyclic stresses under which they operate and the repair mechanisms that operate.

From this Modern Mechanix scan of a 4 page article from the 1924 issue of Popular mechanics, we learn about Lincoln, the inventor!

A typical two phase microstructure consists of a topologically continuous `matrix' phase in which islands of `precipitate' phase are embedded. Usually, the matrix phase is also the majority phase in terms of volume fraction. However, sometimes this relationship between the volume fraction and topology is reversed, and this reversal is known as phase inversion. Such a phase inversion can be driven by an elastic moduli mismatch in two-phase solid systems. In this paper (submitted to Philosophical magazine), we show phase inversion, and the effect of the elastic moduli mismatch and elastic anisotropy on such inversion.

During solid-solid phase transformations elastic stresses arise due to a difference in lattice parameters between the constituent phases. These stresses have a strong influence on the resultant microstructure and its evolution; more specifically, if there be externally applied stresses, the interaction between the applied and the transformation stresses can lead to rafting.

Rafting is the preferential coarsening of (dilatationally) misfitting precipitates in a direction parallel (P-type) or perpendicular (N-type) to an applied stress. In the materials literature, it is sometimes argued that rafting is an elasto-plastic phenomenon, and that plastic pre-strains are essential for rafting. In this paper (which we have submitted to Acta Materialia) we show that purely elastic stress driven rafting is a distinct possibility.

Resonance is a journal of science education published by the Indian Academy of Sciences for the past twenty years or so. As the introduction page states,

It is well known that the algebra associated with edge dislocations can be forbidding. As Prof. Frank (of the Frank-Read source fame) noted once,

• I found all that elasticity mathematics rather difficult, but I found it easier to concentrate on the screw dislocation, with only one displacement variable, instead of two for the edge. So I became particularly fond of the screw dislocation. Mott and Nabarro liked to work with edge dislocations., because they liked two-dimensional diagrams. I was less afraid than they were of the third dimension, and more afraid of algebra.

Even the great Eshelby called the displacement field expressions of an edge dislocation field "rather forbidding expressions" in a pedagogical paper that he wrote in 1966.

This paper, published in the British Journal of Applied Physics (the abstract of which is given below), describes a process to obtain the elastic stress fields of the edge dislocation using what Eshelby calls a wedge dislocation:

My googling today brought me to this treasure trove of write-ups in mechanics:

This site contains informal (usually rough draft) technical notes and tutorials on topics in mechanics. The sophistication is at the first or second year graduate level. These write-ups include:

• TUTORIALS: straightforward primers on particular topics.
• MYTH BUSTERS: Misconceptions in mechanics
• DUSTY CORNERS: little-known or interesting aspects of mechanics issues.
• BACK DOORS: "Better ways" to do common tasks.

The write-ups are limited to topics that are too well-known to be published (in journals) but not known enough to be easily found in the literature.

Suppose if somebody asked you the following question, and more importantly, wanted the answer to an accuracy of 100-digits:

• Problem A: A particle at the center of a 10 x 1 rectangle undergoes Brownian motion (i.e., two-dimensional random walk with infinitesimal step lengths) until it hits the boundary. What is the probability that it hits at one of the ends rather than at one of the sides?

Or, this question (again, demanding the answer to an accuracy of 100-digits):

• Problem B: A square plate [-1,1]x[-1x1] is at a temperature u = 0. At time t=0 the temperature is increased to u=5 along one of the four sides while being held at u=0 along the other three sides, and heat then flows into the plate according to u_t = \nabla u. When does the temperature reach u=1 at the center of the plate?

You might be tempted to answer back "What's the point?", and, probably, you are justified. However, if you did that, you would have missed some wonderful oppurtunity to learn (and/or teach), among other things, the following lessons:

Journal of Materials (JOM) of The Minerals, Metals, and Materials Society (TMS) is running a survey on the greatest moments in materials science and engineering; the voting deadline is January 5, 2007.

Recently, during one of my net searches, I came across this page of RPI, where I learnt about a couple of numerical mechanics software which might be of interest to some of you.

FMDB:

As for the effort toward the scalable engineering simulations on distributed environements, we addressed this challenge by developing a distributed mesh data management infrastructure that satisfies the needs of distributed domain of applications.

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,