Space-time integral currents of bounded variation

Rindler, Filip

doi:10.48550/arxiv.2109.12447

Cited by 3 publications

(20 citation statements)

References 13 publications

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“…This naturally leads to a theory of integral currents with bounded variation (in time), which was developed in [60]. The required aspects of this theory are recalled in Section 2.4 as the basis upon which our rigorous modeling of dislocations and slip trajectories in Section 3 is built.…”

Section: Dissipationmentioning

confidence: 99%

“…BV-theory of integral currents and deformations. In this section we briefly review some aspects of the theory on space-time currents of bounded variation, which was developed in [60]. In the space-time vector space R 1+d ∼ = R × R d we denote the canonical unit vectors as e 0 , e 1 , .…”

Section: 3mentioning

confidence: 99%

“…Thus, the S H so defined can be understood as deforming T via H into H(1, ) * T . We refer to Lemma 4.3 in [60] for estimates relating to the variation of such homotopical deformations.…”

Section: 3mentioning

confidence: 99%

“…The following compactness theorem for this convergence in the spirit of Helly's selection principle is established as Theorem 3.7 in [60].…”

Section: 3mentioning

confidence: 99%

“…The key result for us in this context is the following "equivalence theorem"; see Theorem 5.1 in [60] for the proof. Proposition 2.7.…”

Section: 3mentioning

confidence: 99%

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

Rindler¹

2021

Preprint

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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.

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Section: Dissipationmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

“…Thus, the S H so defined can be understood as deforming T via H into H(1, ) * T . We refer to Lemma 4.3 in [60] for estimates relating to the variation of such homotopical deformations.…”

Section: 3mentioning

confidence: 99%

“…The following compactness theorem for this convergence in the spirit of Helly's selection principle is established as Theorem 3.7 in [60].…”

Section: 3mentioning

confidence: 99%

“…The key result for us in this context is the following "equivalence theorem"; see Theorem 5.1 in [60] for the proof. Proposition 2.7.…”

Section: 3mentioning

confidence: 99%

See 3 more Smart Citations

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

Rindler¹

2021

Preprint

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Elasto-plastic evolution of single crystals driven by dislocation flow

Hudson

Rindler

2022

Math. Models Methods Appl. Sci.

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This work introduces a model for large-strain, geometrically nonlinear elasto-plastic dynamics in single crystals. The key feature of our model is that the plastic dynamics are entirely driven by the movement of dislocations, that is, [Formula: see text]-dimensional topological defects in the crystal lattice. It is well known that glide motion of dislocations is the dominant microscopic mechanism for plastic deformation in many crystalline materials, most notably in metals. We propose a novel geometric language, built on the concepts of space-time “slip trajectories” and the “crystal scaffold” to describe the movement of (discrete) dislocations and to couple this movement to plastic flow. The energetics and dissipation relationships in our model are derived from first principles drawing on the theories of crystal modeling, elasticity, and thermodynamics. The resulting force balances involve a new configurational stress tensor describing the forces acting against slip. In order to place our model into context, we further show that it recovers several laws that were known in special cases before, most notably the equation for the Peach–Koehler force (linearized configurational force) and the fact that the combination of all dislocations yields the curl of the plastic distortion field. Finally, we also include a brief discussion on how a number of other effects, such as hardening, softening, dislocation climb, and coarse-graining, could be incorporated into our model.

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Transport of currents and geometric Rademacher-type theorems

Bonicatto¹,

Nin²,

Rindler³

2022

Preprint

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The transport of many kinds of singular structures in a medium, such as vortex points/lines/sheets in fluids, dislocation loops in crystalline plastic solids, or topological singularities in magnetism, can be expressed in terms of the geometric (Lie) transport equation d dt Tt + L b t Tt = 0 for a time-indexed family of integral or normal and usually boundaryless k-currents t → Tt in an ambient space R d (or a subset thereof). Here, bt = b(t, •) is the driving vector field and L b t Tt is the Lie derivative of Tt with respect to bt (introduced in this work via duality). Written in coordinates for different values of k, this PDE encompasses the classical transport equation (k = d), the continuity equation (k = 0), as well as the equations for the transport of dislocation lines in crystals (k = 1) and membranes in liquids (k = d − 1). The top-dimensional and bottom-dimensional cases have received a great deal of attention in connection with the DiPerna-Lions and Ambrosio theories of Regular Lagrangian Flows. On the other hand, very little is rigorously known at present in the intermediate-dimensional cases. This work thus develops the theory of the geometric transport equation for arbitrary k and in the case of boundaryless currentsTt, covering in particular existence and uniqueness of solutions, structure theorems, rectifiability, and a number of Rademacher-type differentiability results. The latter yield, given an absolutely continuous (in time) path t → Tt, the existence almost everywhere of a "geometric derivative", namely a driving vector field bt. This subtle question turns out to be intimately related to the critical set of the evolution, a new notion introduced in this work, which is closely related to Sard's theorem and concerns singularities that are "smeared out in time". Our differentiability results are sharp, which we demonstrate through an explicit example of a heavily singular evolution, which is nevertheless Lipschitz-regular in time.

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Space-time integral currents of bounded variation

Cited by 3 publications

References 13 publications

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

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

Elasto-plastic evolution of single crystals driven by dislocation flow

Transport of currents and geometric Rademacher-type theorems

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