It is with great sadness that I report the passing away of Prof. Don Carlson. The link below describes his life and work.

http://mechse.illinois.edu/content/news/article.php?article_id=410

This paper presents a generalization of traditional continuum approaches to liquid crystals and
liquid crystal elastomers to allow for dynamically evolving line defect distributions. In analogy with
recent mesoscale models of dislocations, we introduce fields that represent defects in orientational
and positional order through the incompatibility of the director and deformation ‘gradient’ fields.

 (in Computational Methods for Microstructure-Property Relationships," Springer. Edited by Somnath Ghosh and Dennis Dimiduk)

Dislocation mediated continuum plasticity: case studies on modeling scale dependence, scale-invariance, and directionality of sharp yield-point

Claude Fressengeas, Amit Acharya, Armand Beaudoin

(in Journal of the Mechanics and Physics of Solids)

Johannes Zimmer, Karsten Matthies, Amit Acharya

We consider an exact reduction of a model of Field Dislocation Mechanics to a scalar problem in one spatial dimension and  investigate the existence of static and slow, rigidly moving single or collections of planar screw dislocation walls in this  setting. Two classes of drag coefficient functions are considered,  namely those with linear growth near the origin and those with  constant or more generally sublinear growth there. A mathematical  characterisation of all possible equilibria of these screw wall  microstructures is given. We also prove the existence of travelling   wave solutions for linear drag coefficient functions at low wave  speeds and rule out the existence of nonconstant bounded travelling   wave solutions for sublinear drag coefficients functions. It turns  out that the appropriate concept of a solution in this scalar case   is that of a viscosity solution. The governing equation is not  proper and it is shown that no comparison principle holds. The   findings indicate a short-range nature of the stress field of the  individual dislocation walls, which indicates that the nonlinearity  present in the model may have a stabilising effect.

(in Journal of the Mechanics and Physics of Solids)

Nonsingular, stressed, dislocation (wall) profiles are shown to be 1-d equilibria of a non-equilibrium theory of Field Dislocation Mechanics (FDM). It is also shown that such equilibrium profiles corresponding to a given level of load cannot generally serve as a traveling wave profile of the governing equation for other values of nearby constant load; however, one case of soft loading with a special form of the dislocation velocity law is demonstrated to have no ‘Peierls barrier’ in this sense. The analysis is facilitated by the formulation of a 1-d, scalar, time-dependent, Hamilton-Jacobi equation as an exact special case of the full 3-d FDM theory accounting for non-convex elastic energy, small, Nye-tensor dependent core energy, and possibly an energy contribution based on incompatible slip. Relevant nonlinear stability questions, including that of nucleation, are formulated in a non-equilibrium setting. Elementary averaging ideas show a singular perturbation structure in the evolution of the (unsymmetric) macroscopic plastic distortion, thus pointing to the possibility of predicting generally rate-insensitive slow response constrained to a tensorial ‘yield’ surface, while allowing fast excursions off it, even though only simple kinetic assumptions are employed in the microscopic FDM theory. The emergent small viscosity on averaging that serves as the small parameter for the perturbation structure is a robust, almost-geometric consequence of large gradients of slip in the dislocation core and the persistent presence of a large number of dislocations in the averaging volume. In the simplest approximation, the macroscopic yield criterion displays anisotropy based on the microscopic dislocation line and Burgers vector distribution, a dependence on the Laplacian of the incompatible slip tensor and a nonlocal term related to a Stokes-Helmholtz-curl projection of an ‘internal stress’ derived from the incompatible slip energy.

This post is in response to the imechanica request

http://www.imechanica.org/node/5034

 (a separate post, as I have to attach notes - it would be really nice to be able to attach documents to imechanica comments)

 Attached are hand-written notes I have used to implement the Network B for the Bergstrom-Boyce model. They were written for my use only, so if it seems stream-of-consciousness at times, don't blame me. The details should all be there, though.

 It is a formulation in principal stretches and takes care of the Jacobian in situations with eigenvalue coalescence (double coalescence in part III, triple coalescence in pdf of ppt).

I post some (hand-written) notes on compatibility conditions for both small and finite strains that I have used for helping me in lecturing. These may be useful for our student friends on imechanica. I also post a paper on compatibility conditions for the Left Cauchy-Green field in three dimensions as well as the paper by Janet Blume on the same subject.

The criteria of Cermelli and Gurtin (2001, J. Mech. Phys. Solids) for choosing a geometric dislocation tensor in finite plasticity are reconsidered. It is shown that physically reasonable alternate criteria could just as well be put forward to select other measures; overall, the emphasis should be on the connections between various physically meaningful measures as is customary in continuum mechanics and geometry, rather than on criteria to select one or another specific measure.

The main idea is the following: a most natural mathematical setup for considering the motion of the void-solid interface of an expanding void is that of the traveling wave. Thus, a theory for macroscopic damage evolution may be suspected as being a homogenized version of basic theory that has such wave phenomena as an essential ingredient. This paper is a first step in probing such questions. 

I came across some content I used to have on my webpage a long time ago.

http://www.ce.cmu.edu/~amita/webpage_misc.html

Hopefully it is inspirational for that bright, young, graduate student waiting in the wings to usher in the revolution that shows us how to solve some of the outstanding theoretical problems of solid mechanics that we seem to put off  e.g. concrete, quantitative methods for calculating time-dependent microstructure in plasticity and its effect on time-dependent macroscopic properties.....

For those who read the link, do not miss Faraday's quote and Rota's "Ten lessons...'

A technique for setting up generalized continuum theories based on a balance law and nonlocal thermodynamics is suggested. The methodology does not require the introduction of gradients of the internal variable in the free energy. Elements of a generalized damage model with porosity as the internal variable are developed as an example.

A field theory of dislocation mechanics and plasticity is illustrated through new results at the nano, meso, and macro scales. Specifically, dislocation nucleation, the occurrence of wave-type response in quasi-static plasticity, and a jump condition at material interfaces and its implications for analysis of deformation localization are discussed.

 The Koiter-Sanders-Budiansky bending strain measure and a nonlinear generalization

 We know from strength of materials that non-uniform stretching of fibers along the cross section of a beam produces bending moments. But does this situation necessarily correspond to a 'bending' deformation? For that matter, what do we exactly mean kinematically when we talk about a bending deformation?

To make the question more concrete, consider a cylinder that expands uniformly along all radial rays. Does this deformation of the cylinder correspond to bending? I think it is fair to say that most would say that this is purely a stretching deformation with no bending. But then, what is precisely a bending deformation?

Two papers are attached related to randomness discussion.

Attached is a paper outlining ideas for averaging autonomous dynamics, based on a dynamical systems point of view.

People interested in computational multiscale modeling, especially of the sequential kind, as well as nonequilibrium statistical mechanics may find these ideas useful.