Consider a ridig pillar ontop of a elastic substrate. Applying a moment to the pillar will lead to elastic deformation of the substrate. If the pillar is infinitly large in diameter, then this problem is the same as an infinitely sharp crack, considering the symmetry of the crack problem, i.e. there are square root singularities. However, the infinitly large diameter assumtion does not hold if the global rotation of the substrate under the pillar is of interest, because the both sides of the pillar interact.

To identify a coordinate system: If the moment is applied along the y-axis and the z-axis is the axial direction of the pillar, then the maxium stresses in the pillar will be along the x-axis.  

The lecture notes of the two courses I taught at Stanford University during the last two quarters, "ME 340 Elasticity" and "ME 334 Introduction to Statistical Mechanics", are available in PDF format online at:

  http://micro.stanford.edu/~caiwei/me340/

  http://micro.stanford.edu/~caiwei/me334/

Perhaps it could be useful to you.

The dried raisin, the crushed soda can, and the collapsed bicycle inner tube exemplify the nonlinear mechanical response of naturally curved elastic surfaces with different intrinsic curvatures to a variety of different external loads. To understand the formation and evolution of these features in a minimal setting, we consider a simple assay: the response of curved surfaces to point indentation.

This paper studies the invariance of balance of
energy for a system of interacting particles under groups of
transformations. Balance of energy and its invariance is first
examined in Euclidean space. Unlike the case of continuous media,
it is shown that conservation and balance laws do not follow
from the assumption of invariance of balance of energy under
time-dependent isometries of the ambient space. However, the
postulate of invariance of balance of energy under arbitrary
diffeomorphisms of the ambient (Euclidean) space, does yield
the conservation laws. These ideas are then extended to the case
when the ambient space is a Riemannian manifold. Pairwise

Dear All,

the solution of an elastic half space subjected to any generalized load may be seen as the solution of another elastic problem, that is elastic space with an infinite length hole when the hole radius goes to infinite.

Is there a "general" solution for that elastic problem? For general I mean a solution that can be used e.g. like a kernel in a convolution operation.

I've expanded the Navier equation in cylindrical coordinate (radial, theta, z), with a Fourier approach; assigning periodicity to theta variable and the "square summability" along the z-direction, the problem is reduced to the r-direction.

Hello,

 I am looking for a job in applied mechanics in California, to start in or around July 2008.  I am trained (Ph.D.) in applied mathematics, modeling, and computational mechanics (of biological tissues), general continuum mechanics, constitutive modeling, and optimal control.  Greatly interested in dynamic modeling and analysis, stress / thermal analysis, and modeling problems related to design, e.g. medical devices.  Will be glad to be more involved in design.

 My CV, including my contact information, can be found at  http://math.uci.edu/~sadovsky/docs/cv-02-2008.pdf

 If seeking such candidates, please email me.  Thanks!

One file is attached.

Professor Y. C. Fung, Professor Emeritus of Bioengineering at UC San Diego's Jacobs School of Engineering, is the recipient of the Fritz J. and Dolores H. Russ Prize of 2007.

The Russ Prize is presented biannually to an outstanding candidate in the field of bioengineering who has made significant contributions to improving the human condition through research, development, teaching, or management. The recipient receives a $500,000 cash award and an engraved gold medallion.

I have a question making me no sleep. In the elastic theory we have a separable Lagrangian L = T(v) + W(u) since we can write the internal elastic energy as a function of displacements. What happens if we use a rate form for the strain and the stress? Can we write the potential energy in terms of just velocities? If that is the case the stationary path of the Lagrangian reduces to:

 d/dt (dT/dv - dW/dv)  = 0

Dear all,

Infinity asked me for posting more information about one of our papers. It was published in 2006 in Rubber Chemistry and Technology and proposes a comparison and a ranking of 20 different hyperelastic constitutive models for rubber (from the Mooney model (1940) to the micro-sphere model (2004)) in the incompressible case.

Marckmann G. et Verron E., Comparison of hyperelastic models for rubberlike materials, Rubber Chemistry and Technology, 79(5), 835-858, 2006.

Koiter's PhD thesis, dated 1945, gave the birth of post-buckling analysis, and quantified the notion of imperfection sensitivity.  He wrote the thesis in Dutch.  An excellent English translation is free online.  (Direct URL:  http://www.gl.iit.edu/wadc/DigitalCollection/1970/AFFDLTR70-025.pdf)

I am happy to recommend the following book for your general reading.

Ranganath, G.S., ``Mysterious Motions and other Intriguing Phenomena in Physics," Hyderabad, India: Universities Press (2001)

This blog focuses on viscoelasticity (http://en.wikipedia.org/wiki/Viscoelasticity)

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:

an interesting puzzle for fun:

Lame’s classical solution for an elastic 2D plate, with a hole of radius a and uniform tensile stress applied at the far field, gives a stress concentration factor (SCF) of two at the edge of the hole. This SCF=2 is independent of the hole radius.

Consider what happened to this concentration factor if the radius a approaches infinitely small. The SCF is independent of a, so it remains equal to two even when the hole disappears.

This is inconsistent with what one would expect physically, namely that the limit a->0 should be the same as when the plate is whole without a hole and has no stress concentration.

Henry.

A free surface in a multi-layer can experience an undulation due to surface diffusion during fabrication or etching process. In order to analyze the undulation, the elasticity solution for the undulating film is needed. Considering the undulation as a perturbation of a flat surface, a boundary value problem for 2D elasticity is formulated. The solution procedure is straightforward, but very lengthy especially for a multi-layer.

Singular elastic stress fields are generally developed at sharp re-entrant corners and at the end of bonded interfaces between dissimilar elastic materials. This behaviour can present difficulties in both analytical and numerical solution of such problems. For example, excessive mesh refinement might be needed in a finite element solution.

Williams (1952) pioneered a method for determining the strength of the dominant singularity by expressing the local field as an asymptotic expansion. The same method has since been used for a variety of situations leading to singular points, including bonded dissimilar wedges and frictionless or frictional contact between bodies with sharp corners.

Kluwer has published the second edition of my book `Elasticity'. It contains five new chapters and a much wider selection of end-of-chapter problems than the first edition. See below for the Table of Contents and the Preface. A sample chapter can be downloaded here.

The ISBN number is 1-4020-0964-X (Hardback) and 1-4020-0966-6 (Paperback)

Using the Griffith energy method for analysis of cavitation under hydrostatic tension we conclude that the critical tension tends to infinity when the cavity radius approaches zero (IJSS, 2006, doi: 10.1016/j.ijsolstr.2006.12.022). The conclusion is physically meaningless, of course. Moreover, if we assume that the failure process occurs at the edge of the cavity then the critical tension should be length-independent for small but finite cavities while the Griffith analysis always exhibits length-dependence. The main Griffith idea - introduction of the surface energy - is controversial because it sets up the characteristic length, say, surface energy over volume energy. By no means is this approach in peace with the length-independent classical continuum mechanics.

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,

I have not read the above-mentioned paper, as I have never been able to find it. However it is said to be "a brilliantly insightful paper which was to lay the foundations of modern elasticity." However, I believe it is also noteworthy for being one of the major contributions by a female mechanician prior to the modern era. For a great biography of Sophie Germain, including a fantastic quote from a letter from Carl Gauss on discovering that she was female--and not "Monsieur Le Blanc"--visit this site (from which the above quote, on the impact of her paper, came).

There are no female mechanicians listed on http://en.wikipedia.org/wiki/Mechanicians but I believe it could be argued that Germain deserves a mention!

Several people have suggested that iMechanicians compile a set of classics in mechanics. Given the mission of iMechanica (to use the Internet to enhance communications among mechanicians, and to pave a way to evolve all knowledge of mechanics online), it seems fitting for us to facilitate the communication with mechanicians of all times, and to embrace publications of all times.

I'm adding "classics" as a tag featured at the top of iMechanica. You will have to interpret for yourself what you consider to be a classic. For me, a classic should have stood the test of time (say greater than 20 years) and have influenced me deeply and directly. I should have read it and used it in my own work.