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# Coupled phase transformations and plasticity as a field theory of deformation incompatibility

(to appear in International Journal of Fracture; Proceedings of the 5th Intl. Symposium on Defect andMaterial Mechanics)

Amit Acharya and Claude Fressengeas

The duality between terminating discontinuities of fields and the incompatibilities of their gradients is used to define a coupled dynamics of the discontinuities of the elastic displacement field and its gradient. The theory goes beyond standard translational and rotational Volterra defects (dislocations and disclinations) by introducing and physically grounding the concept of *generalized disclinations* in solids without a fundamental rotational kinematic degree of freedom (e.g. directors). All considered incompatibilities have the geometric meaning of a density of lines carrying appropriate topological charge, and a conservation argument provides for natural physical laws for their dynamics. Thermodynamic guidance provides the driving forces conjugate to the kinematic objects characterizing the defect motions, as well as admissible constitutive relations for stress and couple stress. We show that even though 'higher-order' kinematic objects are involved in the specific free energy, couple stresses may not be required in the mechanical description in particular cases. The resulting models are capable of addressing the evolution of defect microstructures under stress with the intent of understanding dislocation plasticity in the presence of phase transformation and grain boundary dynamics.

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## Comments

## Finite Deformation Analog

I have attached a presentation that we recently made at the ISDMM 2011 conference in Sevilla. This shows the form of the corresponding finite deformation theory. As we all know, writing these things up is a big pain, but has to be done in due course.

## Re: Coupled phase transformations and plasticity as a field theo

Dear Amit:

Thanks for sharing your interesting work.

Recently, I've been trying to understand the mechanics of distributed disclinations (and have made some progress using Cartan's moving frames). A few questions for you. Are disclinations of any real significance in solids (perhaps a naive question)? I see very few works on disclinations in solids (it's nice to see you're doing some good work) and it seems dislocations have been of much more interest. Why? Have you seen the following book? Zubov, L.M. [1997], Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies, Springer, Berlin. The same author has a couple a recent review articles on disclinations.

Regards,

Arash

## Volterra disclinations inadequate for phase transformations

Arash,

I looked at Zubov. He works on the standard disclination problem/question which is based on the fundamental question asked by Weingarten, and answered by him (W) and Volterra. At finite deformation, I don't know who else are the players on this question, but certainly Zubov has a proof and Casey has a very elegant proof.

At any rate, anything that starts from Weingarten's question will end up with a deformation whose right Cauchy Green field (symmetric strain field) is continuous.

However, from the get-go we are interested in phase transformations which requires the strain/stretch fields to be discontinuous. Thus classical disclinations cannot model phase transformations.

Our 'small deformation' as well as large deformation theories model the physical discontinuities and singularities we are interested in and for that reason the fundamental kinematics and the structure of the theory perforce has to be different from that of classical disclination theory. For phase transformations, the metric tensor itself is not a continuous field, if you will. And we are able to formulate a time-dependent, dissipative model of unrestricted geometric and material nonlinearity. We make sure we can specify the disclination density unambiguously in terms of jumps of elastic deformation 'gradient'.

regards,

- Amit

## Re: Volterra disclinations inadequate for phase transformations

Dear Amit:

I agree with your comments.

I haven't seen any other finite-deformation works on disclinations. Zubov solves a single wedge disclination problem and in his 2011 ZAMM paper an axisymmetric distributed wedge disclination problem (in 2D). Other than these two problems I'm not aware of any exact solutions.

Regards,

Arash

## disclinations in solids

Arash,

You ask the natural (not naive) questions about disclinations. For the

longest time, while working on dislocations, I could not see the

relevance of disclinations in solids. However, once I

understood the duality between terminating discontinuities of fields and

the corresponding incompatibility of their gradients, then its physical

import immediately became clear. So, I now think disclinations are of relevance in solids. Let's talk about it through examples.

First take the classical disclination. A

triple junction in a polycrystal is a

where threeclassical disclinationorientation discontinuities terminate. A triple junction is obviously

real. Claude Fressengeas and his co-workers at Metz (and your colleague, Capulungo, at Georgia Tech) have done some very nice work on classical disclinations, even experimentally.

I hope we have made the physical relevance of the

clear in the paper and presentation I posted. There are many pictures of 5-fold phase boundaries terminating at a junction, a needle may be visualized as two near near parallel phase boundaries terminating by coming together at a g-disclination, a faceted martensitic inclusion may be considered as having edges that are g-disclinations etc. So, these kinematic objects are of significance in solids in the modeling of phase transformations and grain boundaries.g-disclinationFollowing the terminology of misorientation in grain boundaries, let's call the jump of elastic distortion across a phase boundary a misdistortion (clearly, I am running out of terminology here). In general, there is no reason to expect that the misdistortion across a phase boundary should remain fixed at all locations along the phase boundary. Assuming this to be so (theoretically at least it does not seem it can hurt to have this generality), just as we think of a dislocation line as being the boundary of differently slipped regions along a slip plane, similarly the g-disclination line is the boundary of differently misdistorted regions along a phase boundary.

Why has disclination modeling in solids not been popular? I can only hazard a guess here. I think the physical connections with what can be measured and observed and can be unambiguously modeled by disclinations had not been completely understood. I think Fressengeas et al. have made a very good start (especially their experimental work, Beausir, Taupin, Fressengeas, are the authors I believe) and I am sure they will get lots of resistance but the ideas are good and will prevail. On the g-disclination front, we will have to solve problems, but we know of good methods from our dislocations work and the applications are crystal clear, so it is a question of getting things done - not easy, but the way forward is clear and the tools are there (as can be seen from some of Fressengeas et al. recent modeling work, IJSS referred to in our paper).

I should note another point - once one understands what is physically required and sees what appears in the differential geometric formulations, I think going with the curvature as a measure of disclinations amounts to carrying along a bunch of difficult nonlinearities (for the physical problem at hand) which will make life very difficult when it comes to solving anything (as it is, the theory is quite complex) - we point this out in the presentation. The curvature is important if we were asked to solve a problem of geometry, but here we are not asked to solve a problem of geometry, so again this reinforces in my mind the fact that it is the connection between the available mathematical constructs and what is required physically that was not well-understood. Once one understands that, one can create the adapted tools from continuum mechanics.

I have not seen the book by Zubov.

Boy this has become a long response!

regards,

- Amit