Abstract
Recent progress in continuum dislocation dynamics (CDD) has been achieved through the construction of a local density approximation for the dislocation energy and the derivation of constitutive laws for the average dislocation velocity by means of variational methods from irreversible thermodynamics. Individual dislocations are driven by the Peach–Koehler-force which is likewise derived from a variational principle. This poses the question if we may expect that the averaged dislocation state expressed through the CDD density variables is driven by a variational gradient of the average energy, as is assumed in irreversible thermodynamics. In the current contribution we do not answer this questions, but rather present the mathematical framework within which the evolution of discrete dislocations is literally understood as a gradient descent. The suggested framework is that of de Rham currents and differential forms. We briefly sketch why we believe the results to be useful for formulating CDD theory as a gradient flow.
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Notes
- 1.
Note that in [9] there is a mistake in the anti-symmetry condition below, where the exponent of \(-1\) is erroneously given as p and not pq.
- 2.
Regions without holes are called non-periphractic by Gurtin [7], a term which goes back to Maxwell [22]. In terms of modern topology this means that the second Betti number is zero, such that all closed two-forms are exact, saying that they may be obtained from some one-form (a potential) by exterior differentiation.
- 3.
Compare [24], where Felix Otto puts it as follows: ‘The merit of the right gradient flow formulation of a dissipative evolution equation is that it separates energetics and kinetics. The energetics endow the state space M with a functional E, the kinetics endow the state space with a Riemannian geometry via the metric tensor g.’
- 4.
If the mobility is for instance taken to be zero in the climb-direction, \(M^{ij}\) is not positive definite. In this case the tangent space to dislocations is restricted to variations within the glide plane, such that only the in-plane component of the Peach–Koehler force matters. If one then assumes M to be positive definite for vectors in the glide plane, all the following ideas remain valid.
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Acknowledgements
This paper is dedicated to Professor Reinhold Kienzler, who was a mentor for me during my time as ‘Juniorprofessor’ the Universtät Bremen, on the occasion of his official retirement.
I moreover gratefully acknowledge funding by the German Science Foundation DFG within the DFG Research Unit ‘Dislocation based plasticity’ FOR 1650 under project HO 4227/5-1.
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Hochrainer, T. (2018). Dislocation Dynamics as Gradient Descent in a Space of Currents. In: Altenbach, H., Jablonski, F., Müller, W., Naumenko, K., Schneider, P. (eds) Advances in Mechanics of Materials and Structural Analysis. Advanced Structured Materials, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-70563-7_9
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