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# Kinetics of phase transformations in the peridynamic formulation of continuum mechanics

**Abstract **

We study the kinetics of phase transformations in solids using the peridynamic formulation of continuum mechanics. The peridynamic theory is a nonlocal formulation that does not involve spatial derivatives, and is a powerful tool to study defects such as cracks and interfaces.

We apply the peridynamic formulation to the motion of phase boundaries in one dimension. We show that unlike the classical continuum theory, the peridynamic formulation does not require any extraneous constitutive laws such as the kinetic relation (the relation between the velocity of the interface and the thermodynamic driving force acting across it) or the nucleation criterion (the criterion that determines whether a new phase arises from a single phase). Instead this information is obtained from inside the theory simply by specifying the inter-particle interaction. We derive a nucleation criterion by examining nucleation as a dynamic instability. We find the induced kinetic relation by analyzing the solutions of impact and release problems, and also directly by viewing phase boundaries as traveling waves.

We also study the interaction of a phase boundary with an elastic non-transforming inclusion in two dimensions. We find that phase boundaries remain essentially planar with little bowing. Further, we find a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase

boundaries at the edges of the inclusion while the original phase boundary slows down or stops. Transformation proceeds as the freshly nucleated phase boundaries propagate leaving behind some untransformed martensite around the inclusion.

This paper has appeared in the Journal of the Mechanics and Physics of

Solids, 54 (2006) 1811–1842, and is available at doi:10.1016/j.jmps.2006.04.001

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## Comments

## Peridynamics

Hi Kaushik,

I have some questions for you? First, I am just wondering if you can describe peridynamic theory from your perspective? What exactly it is and what it is not? And my second question is that are you still using peridynamic theory for different type of problems other than phase transformations?

Erkan Oterkus.

## Re: peridynamics

Dear Erkan,

Your first question:

Peridynamics is, very loosely speaking, a continuum analog of pair-potential molecular dynamics. In usual continuum mechanics, one has adjoining volume elements interacting through tractions on the surfaces of these elements. In peridynamics, there are no tractions. Instead there are long range forces between volume elements.

To understand this, consider pair-potential MD. The force on an atom is the sum of the forces between the atom considered and every other atom in the system, ie, force on atom i is

where u_k is the displacement of atom k and f is the inter-atomic force.

In peridynamics, one has a similar equation for the force on an infinitesimal volume element at x. However, as it models a continuum, the sum is replaced by an integral, i.e., the force on an element at point x is

where u(x) is now the displacement field and f is the long-range force between x and x'. (Equations (1)-(4) in Ref. [1] if the text equation isn't clear).

Peridynamics seems an interesting theory for a number of reasons:

1. There's no requirement on the continuity/smoothness of the displacement field (no derivatives of u(x) are used). So it might be a good tool to model processes like fracture or phase transformation.

2. It may be a good way to model long-range effects that are important at small scales. We found that the long-range effects provide some interesting mechanics during nucleation that one doesn't see in phase-field type models (Section 5 and Appendix B of the original reference above).

3. The dispersion relation is similar to atomistic (discrete) models of solids (Section 12 in [1]). In peridynamics, at short wavelengths, the velocity of the waves goes to zero. In atomistic phonon calculations as well, the velocity goes to zero when the wavelength approaches the lattice spacing. This is in sharp contrast to classical continuum (sharp-interface) and phase-field models. In MD models, one doesn't put dissipation into the atomic potentials, but one has short waves with small velocity generated as a phase boundary (or other defect) moves. These are the source of dissipation when one examines the motion at macroscopic levels and are lumped as heat, roughly. We find a very similar mechanism in peridynamics, where we have dissipation at macroscopic scales without putting any dissipation in at the microscopic scale. This is again unlike phase-field modeling where one needs to explicitly put in viscosity at the microscopic level to obtain dissipation at the macroscopic scale. All of this seems to make peridynamics a interesting tool to examine dynamics processes.

Also, in Stewart Silling's original paper formulating peridynamics [1], his introduction and discussion sections give a perspective of the theory.

---------------------------------

Your second question:

In [1], there's a brief example applying peridynamics to fracture. Because of the integral formulation that doesn't require derivatives of the displacement field, there are no continuity or smoothness restrictions on the displacement. This makes it a great tool to look at fracture problems, besides phase transformations.

We are trying to formulate a coupled phase transformation/dislocation dynamics problem in the framework of peridynamics, but that is still very preliminary. Stewart Silling, with various coworkers, has applied the theory to fracture and fragmentation problems and he has some very impressive results [2,3,4]. Markus Zimmermann, working with Rohan Abeyaratne, has also done some work on fracture in this theory [5]. Stewart Silling and Florin Bobaru have applied peridynamics to examine polymer networks [6]. There has also been some work on theoretical and algorithmic aspects by Olaf Weckner and coauthors. I can ask Stewart Silling if he's aware of other work using peridynamics.

Regards,

Kaushik

[1] Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys.Solids 48, 175–209.

[2] Silling, S.A., Kahan, S., 2004. Peridynamic modeling of structural damage and failure. In: Conference on High Speed Computing, Gleneden Beach, Oregon, USA.

[3] S. A. Silling. Dynamic fracture modeling with a meshfree peridynamic code. In: K. J. Bathe (ed.), Computational Fluid and Solid Mechanics 2003, Elsevier, Amsterdam, 2003, pp. 641–644.

[4] S. A. Silling and E. Askari. A meshfree method based on the peridynamic model of solid mechanics. Computers & Structures 83 (2005) 17–18, pp. 1526–1535

[5] Zimmermann, M., 2001. A continuum theory with long-range forces for solids. Ph.D. Thesis, Massachusetts Institute of Technology. (http://hdl.handle.net/1721.1/30342)

[6] S.A. Silling and F. Bobaru, "Peridynamic modeling of membranes and fibers", International Journal of Non-Linear Mechanics, 40(2-3): 395-409 (2005), and recent conference proceedings.

## peridynamics vs nonlocal integral theories

Kaushik,

This is quite interesting.....Is there a link between "peridynamics" and the so-called integral nonlocal theories initiated in the early sixties by Kroner, then Edelen? Later, I believe, Eringen extended those extensively. Your description, at least superficially, sounds similar to what I have read in the work of the authors I mention.

## Re: peridynamics vs nonlocal integral theories

Dear Pradeep,

I think there's actually a very similar linear theory that was developed by your colleague Isaak Kunin, while the nonlinear version was independently formulated by Silling. With regard to the work by Eringen, Edelen, Kroner and coworkers, I think you're right that there are many similarities. I'm not very familiar with those theories, but I think the main difference is that they use the strain and define the stress as being a nonlocal integral. In peridynamics, one works directly with the displacement field and the definition of stress is secondary (the constitutive information is contained in the function f in my previous comment above). You might also find Sections 1.2, 3.2 and Appendix A of Markus Zimmermann's thesis interesting.

Regards,

Kaushik

## Kunin's work

Kaushik,

I will try to get hold of Markus's thesis (--I could not access it online). Isaak Kunin does have a theory which he describes as the "quasi-continuum description". Is that what you are referring to?

## email

Dear Pradeep,

I was able to access Markus' thesis from the MIT website. I'll email you a copy.

Kaushik

## Meso/macro scale modeling with Peridynamics

Hi Kaushik,

I've been following the interesting discussion.

I can see how peridynamics can be a nice field-type model for single crystals at the sub micron scale, getting better as one approaches the atomic scale. Am I correct in understanding that there is a big conceptual burden on the design of the vector-valued function 'f' for mesoscopic and macroscopic scale modeling? E.g. suppose you have to model a mesoscale polycrystal as a multicrystal aggregate, much as is often done in polycrystal plasticity modeling with PDE models quite successfully - say texture prediction. I think it is reasonable to say that one cannot use interatomic potentials for this situation. Would a peridynamic model for this situation be easy to set up?

best,

Amit

## modeling polycrystals

Dear Amit,

You're completely correct that a big challenge of building the model is getting the function "f". For our work, we used ad-hoc multiwell structure, because we were more interested in qualitative behavior rather than quantitative. I think Stewart has done a lot more work trying to fit it accurately.

Roughly speaking, I think the peridynamic "f" contains a lot more information than classical continuum material law (which contains only the long-wavelength information). So, given a classical continuum material model, it's often straightforward to find a peridynamic "f" that reproduces that behavior. However, there are usually many such "f" and they will have different behavior at smaller lengthscales. So then one has to think about what/how to put further material information into "f" that is not in the classical continuum model.

I'm not familiar with the exact model that is used in polycrystal plasticity. I think modeling polycrystals is straightforward and one rotates the anisotropy of "f". However, applying the idea of the previous paragraph, one can probably build more information into peridynamics, ex. grain boundary structure, that may not be in the PDE model. But it may not be easy to extract and fit this material information in practice. I haven't thought about plasticity at all in this theory so I'm not sure if there will be other difficulties there.

Regards,

Kaushik

## Re: modeling polycrystals

Hi Kaushik,

Thanks.

I can see the advantages in modeling grain boundaries better. The point where I was stuck at is describing the f for say two spatial points, one in one crystal and the other in another when they are not near the grain boundary and yet they do interact by long-range forces and lots of slip has happened in both crystals. Perhaps it requires just another way to look at it to make it simple and I am too stuck in my classical ways of thinking!

At any rate, lots to learn and discuss. Look forward to it.

best,

Amit

## re: modeling slip

Dear Amit,

I think you're completely correct that when there is slip involved, things may get complicated with peridynamics. I expect that may have been the things that Biswajit mentioned below too. I haven't thought about slip at all, but it would certainly be interesting to discuss this with you further next year.

Regards,

Kaushik

## Peridynamics versus phase field for phase transformation

Hi Kaushik,

Very interesting work. In your comment, you just mentioned some difference between the peridynamics and the phase filed method. Since phase filed method is very popular today for simulation of phase transformation problems. Would you please focus on the comparison of peridynamics and phase field method for phase transformation problems ? Could you explain in more details their differences and the merits of peridynamics over phase field?

Jinxiong

## Re: Peridynamics versus phase field for phase transformation

Dear Jinxiong,

The main differences between peridynamics and conventional phase field that come to mind immediately are:

1. Peridynamics is nonlocal. In practical terms, we found in our nucleation analysis, in a peridynamic defect the nucleation behavior depends on the strain in the ambient region that the defect is embedded in. This is not so in phase-field. (Details are in our paper in the section on nucleation and the Appendix).

2. Peridynamics does not impose continuity / smoothness restrictions on the displacement field, that phase-field does. The integral form leads to interesting behavior of discontinuities in peridynamics. Silling, Zimmermann and Abeyaratne (J. Elas 2003) and Weckner and Abeyaratne (J. Mech. Phys. Solids 2005) study this question. Standard phase-field doesn't allow such discontinuities at all.

3. The phonon spectrum / dispersion relation in peridynamics is very similar to atomistics in that short waves are stationary. This is in contrast to conventional phase-field. This may have implications in modeling dynamic processes. In the specific example we studied in peridynamics, we found that despite no dissipation at the microscopic level, the generation of these stationary short waves led to dissipation at the macroscale. In phase-field / regularized models, one usually needs to explicitly put in damping (viscosity).

I don't think that enough is yet known about peridynamics, phase-field, and the processes at the atomistic/MD level, to say conclusively if the points above are merits of peridynamics over phase-field or not.

Regards,

Kaushik

## More on peridynamics

I tried for a while to do dynamic fracture for ductile materials. The main problem that I ran into is that it's hard to map classical plasticity models into peridynamic theory. Does anyone know how work on that sort of mapping is going?

## mapping classical continuum plasticity models to peridynamics

This paper provides a method that you may be able to use:

The DOI (http://dx.doi.org/10.1016/j.ijsolstr.2008.10.02) is not yet linked, look for the paper co-authored by Stewart Silling in the "Articles in Press" section of IJSS.

## Peridynamics - 2

Dear Kaushik,

Thank you very much for the valuable information. Actually, I've been working on peridynamics for about 1 year. As you know, peridynamic community is a pretty small community and peridynamics is a pretty new approach. I think it is important to see different interpretations of the peridynamic theory, so we can understand more about it. Biswajit's question is also very interesting and I'll try to think about this too. I am also thinking about posting a blog in the future about explaining peridynamic theory for beginners. I think this is important because there is no available book on this topic right now.

Regards,

Erkan.

## Re: Peridynamics - 2

Dear Erkan,

That's good to hear. I'm looking forward to reading it, and also any research you post using peridynamics.

Regards,

Kaushik