Energetic solutions to rate-independent large-strain elasto-plastic evolutions driven by discrete dislocation flow

Abstract

This work rigorously implements a recent model, introduced in [34], of large-strain elasto-plastic evolution in single crystals where the plastic flow is driven by the movement of discrete dislocation lines. The model is geometrically and elastically nonlinear, that is, the total deformation gradient splits multiplicatively into elastic and plastic parts, and the elastic energy density is polyconvex. There are two internal variables: The system of all dislocations is modeled via 1-dimensional boundaryless integral currents, whereas the history of plastic flow is encoded in a plastic distortion matrix-field. As our main result we construct an energetic solution in the case of a rate-independent flow rule. Besides the classical stability and energy balance conditions, our notion of solution also accounts for the movement of dislocations and the resulting plastic flow. Because of the path-dependence of plastic flow, a central role is played by so-called "slip trajectories", that is, the surfaces traced out by moving dislocations, which we represent as integral 2-currents in spacetime. The proof of our main existence result further crucially rests on careful a-priori estimates via a nonlinear Gronwall-type lemma and a rescaling of time. In particular, we have to account for the fact that the plastic flow may cause the coercivity of the elastic energy functional to decay along the evolution, and hence the solution may blow up in finite time.

Elasto-plastic evolution of single crystals driven by dislocation flow

“…It is therefore the goal of the present work to derive a full model of elasto-plastic evolution driven by (discrete) dislocation motion based on a novel geometric language that can be readily translated into a mathematically rigorous framework. The companion works [106,107] carry out this translation and prove the first existence theorem for solutions to the full evolutionary system in the rate-independent case.…”

Transport of currents and geometric Rademacher-type theorems

“…This equation follows more or less directly from the Reynolds transport theorem for 1dimensional quantities. A theory of plasticity based on dislocation transport are the field dislocation mechanics developed by Acharya, see, for instance, [13] and [2], and the recent variational model in [26,34,33]. Further, also the movement of membranes in a medium can be described by a suitable transport equation, which, when formulated for the normal vector α t = α(t, ) : R d → R d to the surface, reads formally as d dt α t + ∇( α t • b t ) = 0.…”

“…In fact, this is the approach taken in the modelling of dislocation movements contained in [26,34,33]. There, the geometric derivative can be identified with the (normal) dislocation velocity, which is a key quantity in any theory of plasticity driven by dislocation motion.…”

“…Before stating our main results, it is worthwhile to note that the primary goal of our analysis does not lie in proving existence and uniqueness of solutions (even though we prove such a theorem for the sake of completeness), but in analysing the structure and properties of solutions, such as rectifiability and (geometric) differentiability, as well as in understanding the relation between weak solutions and space-time solutions. Indeed, in applications the existence of solutions, usually to much more complicated systems with the transport equation being only one ingredient, can often be established in a variational way by minimising over all possible paths of currents, similar to the classical minimising movement scheme; for instance, this is the approach taken in [33]. On the other hand, the questions investigated in the present work go to the heart of the physical modelling, such as establishing the existence of the geometric derivative (which is the "slip velocity" in the theory of dislocations) and establishing in what sense one may understand weak solutions to (1.1) as "physical".…”

“…The second point is that one can define the "variation" of a path of integral or normal currents in several different ways and we compare these different ways in Section 5. It is then a consequence of the (non-trivial) Rectifiability Theorem 6.1 that the space-time variation and the essential variation with respect to F I are in fact equal, answering in particular a question left open in [34,33].…”