Limits of Distributed Dislocations in Geometric and Constitutive Paradigms

Epstein, Marcelo; Kupferman, Raz; Maor, Cy

doi:10.1007/978-3-030-42683-5_8

Cited by 5 publications

(21 citation statements)

References 30 publications

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“…However, the transport equation for kforms requires more regularity on ω. For instance, one can model continuously-distributed dislocation via the so-called Weitzenböck approach (see, e.g., [23] for an overview) and consider the torsion 2-form of a material connection expressing a distribution of dislocations. However, this only makes sense in the smooth setting, and for singular torsions, most notably the torsion concentrated on discrete dislocation lines, one needs to adopt a dual approach, as detailed in the appendix to [26], leading eventually to the transport of singular 1-currents.…”

Section: -Currents: Assume That We Have a Family Of Normal 1-currentsmentioning

confidence: 99%

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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Section: -Currents: Assume That We Have a Family Of Normal 1-currentsmentioning

confidence: 99%

Transport of currents and geometric Rademacher-type theorems

Bonicatto¹,

Nin²,

Rindler³

2022

Preprint

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“…Denoting the total deformation of our specimen as y : Ω ⊂ R 3 → R 3 , for which det ∇y > 0, the commonly used multiplicative Kröner decomposition ∇y = EP splits the deformation gradient into elastic and plastic distortions E, P. Since E, P are not in general deformation gradients themselves, they are henceforth referred to as "distortions". We refer to [20,35,43,45,59,63,68,69,92,93] for justifications and various other aspects of this decomposition. Our description of the crystal in Section 2.3 will in fact give its own justification of the Kröner decomposition based on the crystal scaffold, which is the variable we use to describe the state of the crystal around a point.…”

Section: Introductionmentioning

confidence: 99%

“…In this vein, one could have employed the so-called "geometrical language of continuum mechanics" [22,34,35] to obtain an additional level of consistency (and, perhaps, elegance), but this brings with it further notational complications. We have therefore opted not to pursue this avenue here.…”

Section: Introductionmentioning

confidence: 99%

“…We have therefore opted not to pursue this avenue here. One noteworthy point in this respect is that the dislocation density of continuously distributed (and grain-homogenized) dislocations can be identified with the torsion tensor of an affine material connection [35,47,57,[63][64][65][66][67]76] (which can be seen as the limit of discrete dislocations [35,63,64]).…”

Section: Introductionmentioning

confidence: 99%

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Elasto-plastic evolution of single crystals driven by dislocation flow

Hudson¹,

Rindler²

2021

Preprint

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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, 1-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. However, a comprehensive model linking dislocation dynamics to elaso-plastic evolution has been missing to date. We propose a geometric language, built on the novel 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 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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“…It is modelled as a macroscopic continuum with total deformation y : [0, T ] × Ω → R 3 , for which we require the orientation-preserving condition det ∇y(t) > 0 pointwise in Ω (almost everywhere) for any time t ∈ [0, T ]. We work in the large-strain, geometrically nonlinear regime, where the deformation gradient splits according to the Kröner decomposition [15,26,31,32,37,38,40,41,57,58] ∇y = EP into an elastic distortion E : [0, T ] × Ω → R 3×3 and a plastic distortion P : [0, T ] × Ω → R 3×3 (with det E, det P > 0 pointwise a.e. in Ω).…”

Section: Introductionmentioning

confidence: 99%

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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Limits of Distributed Dislocations in Geometric and Constitutive Paradigms

Cited by 5 publications

References 30 publications

Transport of currents and geometric Rademacher-type theorems

Transport of currents and geometric Rademacher-type theorems

Elasto-plastic evolution of single crystals driven by dislocation flow

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

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