Dislocation Dynamics as Gradient Descent in a Space of Currents

Research output: Chapter in Book/Report/Conference proceedingChapterResearchpeer-review

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.
Original languageEnglish
Title of host publicationAdvances in Mechanics of Materials and Structural Analysis
Place of PublicationCham
PublisherSpringer
Pages207 - 221
Number of pages14
ISBN (Electronic)978-3-319-70563-7
ISBN (Print)978-3-319-70562-0
DOIs
Publication statusPublished - 5 Jan 2018

Publication series

NameAdvanced Structured Materials
Volume80

Fields of Expertise

  • Advanced Materials Science

Cite this

Hochrainer, T. (2018). Dislocation Dynamics as Gradient Descent in a Space of Currents. In Advances in Mechanics of Materials and Structural Analysis (pp. 207 - 221). (Advanced Structured Materials; Vol. 80). Cham: Springer. https://doi.org/10.1007/978-3-319-70563-7_9

Dislocation Dynamics as Gradient Descent in a Space of Currents. / Hochrainer, Thomas.

Advances in Mechanics of Materials and Structural Analysis . Cham : Springer, 2018. p. 207 - 221 (Advanced Structured Materials; Vol. 80).

Research output: Chapter in Book/Report/Conference proceedingChapterResearchpeer-review

Hochrainer, T 2018, Dislocation Dynamics as Gradient Descent in a Space of Currents. in Advances in Mechanics of Materials and Structural Analysis . Advanced Structured Materials, vol. 80, Springer, Cham, pp. 207 - 221. https://doi.org/10.1007/978-3-319-70563-7_9
Hochrainer T. Dislocation Dynamics as Gradient Descent in a Space of Currents. In Advances in Mechanics of Materials and Structural Analysis . Cham: Springer. 2018. p. 207 - 221. (Advanced Structured Materials). https://doi.org/10.1007/978-3-319-70563-7_9
Hochrainer, Thomas. / Dislocation Dynamics as Gradient Descent in a Space of Currents. Advances in Mechanics of Materials and Structural Analysis . Cham : Springer, 2018. pp. 207 - 221 (Advanced Structured Materials).
@inbook{a9e104ba443d4db295e296109ce711d4,
title = "Dislocation Dynamics as Gradient Descent in a Space of Currents",
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.",
author = "Thomas Hochrainer",
year = "2018",
month = "1",
day = "5",
doi = "10.1007/978-3-319-70563-7_9",
language = "English",
isbn = "978-3-319-70562-0",
series = "Advanced Structured Materials",
publisher = "Springer",
pages = "207 -- 221",
booktitle = "Advances in Mechanics of Materials and Structural Analysis",

}

TY - CHAP

T1 - Dislocation Dynamics as Gradient Descent in a Space of Currents

AU - Hochrainer, Thomas

PY - 2018/1/5

Y1 - 2018/1/5

N2 - 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.

AB - 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.

U2 - 10.1007/978-3-319-70563-7_9

DO - 10.1007/978-3-319-70563-7_9

M3 - Chapter

SN - 978-3-319-70562-0

T3 - Advanced Structured Materials

SP - 207

EP - 221

BT - Advances in Mechanics of Materials and Structural Analysis

PB - Springer

CY - Cham

ER -