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# Traveling Wave solutions for a quasilinear model of Field Dislocation Mechanics

(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.

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## Comments

## Can this approach be extended to amorphous alloys?

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

## amorphous materials and continuous defects

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

## traveling waves and equilibria in dislocation mechanics

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.