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Dynamics of screw dislocations: A generalised minimising-movements scheme approach†

Part of: Existence theories General theory in ordinary differential equations

Published online by Cambridge University Press:  03 November 2016

GIOVANNI A. BONASCHI ,
PATRICK VAN MEURS  and
MARCO MORANDOTTI
GIOVANNI A. BONASCHI
Affiliation:
ARCA Fondi SGR, Via Disciplini 3, Milano, MI 20123, Italy email: giovanni.bonaschi@gmail.com
PATRICK VAN MEURS
Affiliation:
Faculty of Mathematics and Physics, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan email: pmeurs@staff.kanazawa-u.ac.jp
MARCO MORANDOTTI
Affiliation:
SISSA – International School for Advanced Studies, Via Bonomea, 265, 34136 Trieste, Italy email: morandot@sissa.it
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Abstract

The gradient flow structure of the model introduced in Cermelli & Gurtin (1999, The motion of screw dislocations in crystalline materials undergoing antiplane shear: glide, cross-slip, fine cross-slip. Arch. Rational Mech. Anal.148(1), 3–52) for the dynamics of screw dislocations is investigated by means of a generalised minimising-movements scheme approach. The assumption of a finite number of available glide directions, together with the “maximal dissipation criterion” that governs the equations of motion, results into solving a differential inclusion rather than an ODE. This paper addresses how the model in Cermelli & Gurtin is connected to a time-discrete evolution scheme which explicitly confines dislocations to move at each time step along a single glide direction. It is proved that the time-continuous model in Cermelli & Gurtin is the limit of these time-discrete minimising-movement schemes when the time step converges to 0. The study presented here is a first step towards a generalisation of standard gradient flow theory that allows for dissipations which cannot be described by a metric.

Keywords

Motion of dislocationsgeneralised gradient flowsminimising-movement schemeenergy dissipation inequality

MSC classification

Primary: 49J40: Variational methods including variational inequalities
Secondary: 34A60: Differential inclusions 70F99: None of the above, but in this section
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 4 , August 2017 , pp. 636 - 655
Copyright
Copyright © Cambridge University Press 2016 

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Footnotes

†

G.A.B. kindly acknowledges support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) VICI grant 639.033.008. P.vM. kindly acknowledges the financial support from the NWO Complexity grant 645.000.012. The research of M.M. was partially supported by the European Research Council through the ERC Advanced Grant “QuaDynEvoPro”, grant agreement no. 290888. M.M. is a member of the Progetto di Ricerca GNAMPA-INdAM 2015 “Fenomeni critici nella meccanica dei materiali: un approccio variazionale”.

References

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