User login


You are here

Traveling Wave solutions for a quasilinear model of Field Dislocation Mechanics

Amit Acharya's picture

(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.

PDF icon tw-plastic_preprint_accepted.pdf713.46 KB


Min Yi's picture

Dear Amit 

Seeing what you do in your work, I connect it to my present research. I am interest in amorphous alloys, whose atomic structures are long-range disorder and small-range order thus leading to dislocation definition  unfit in some extent and different plastic behavior from crystalline alloys, for example, without strain hardening feature. But shear bands are also  important characteristics to discribe and determine their plastic behavior and some researchers utilize dislocation model to establish the relationship between shear band and plastic behavior and explain some special phenomenon. Now I don't know whether it will be right for me to step into dislocation model to research amorphous alloys. I think you understand dislocation model more deeply than me and I want you to give me some tips. Thanks!

                                                                                                  Yours sincerely

                                                                                                               Min Yi

Amit Acharya's picture

Dear Min,

I would strongly encourage you to understand the continuous distribution of defect literature  in relation to amorphous material research. This is because in essence the theory does not tie itself to the notion of a crystal dislocation for basic geometric definitions. As long as you can define elastic response for a material the theory will work - and when it works, there is very little as powerful.

For a beautiful review of the deep geometric aspects have a look at a 2008 Rev. of Modern Physics article by Kleman and Friedel. You should also look at beautiful older work on geometric aspects and elasticity of amorphous materials (the polytope model of glass) by David Nelson and my CMU colleague Mike Widom.

 Apart from its stunning intrinsic beauty, the continuous theory of defects also has a wide range of applications that are waiting to be milked, in my opinion. Let me give you an example (that I am myself trying to learn with experimentalists and mechanicians) - unlike in plasticity or superconductivity where the elastic response of the underlying matrix containing defects can be, at some level of rational modeling, approximated by linear elasticity, in liquid crystals or polymers this is absolutely impossible. But these materials contain defects (dislocations, disclinations, focal conics) that otherwise move around and interact . The upshot is that you simply cannot pull of a discrete dislocation/vortex model for the dynamics of the defects - then the field model is the only option, and an absolutely spectacular one.A great advantage of the approach is that it does not require you to deal with discontinuous fields whose derivatives are involved in the theory in nonlinear fashion, which usually is an insurmountable headache from the nonlinear PDE point of view.

Of course all of this this requires a very good mechanics background, an at least decent background in mathematics, and a real understanding of deep connections and analogies  as well as the root deficiencies of models that might give you an opportunity to fix. These things are usually very subtle, so you must also keep in mind that even though you will have to learn a lot and use only a fraction of that, you have to resist the temptation to window-dress what is known in more complicated language.

Basically, you want to soar like an eagle, but keep your eyes on the ground - and once you have put in  the hard work, don't let anyone tell you what is possible and what is not - trust me, you will get a lot of advice like that.

So you asked for tips - there you have it, and sorry if it sounds pontificating. Hope it helps.

best wishes,

 - Amit

Amit Acharya's picture

This paper has been accepted to appear in the J. Mech. Phys Solids. The revised version has been uploaded.

In the revision, a special effort thas been made to make the mathematics as accessible as possible. Also, an interesting comparison with some physically relevant equilibria of the corresponding Ginzburg-Landau model is made.

Subscribe to Comments for "Traveling Wave solutions for a quasilinear model of  Field Dislocation Mechanics"

Recent comments

More comments


Subscribe to Syndicate