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


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.

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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”.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] R. Alicandro , L. De Luca , A. Garroni & M. Ponsiglione (2014) Metastability and dynamics of discrete topological singularities in two dimensions: A Γ-convergence approach. Arch. Rational Mech. Anal. 214 (1), 269330.

[2] R. Alicandro , L. De Luca , A. Garroni & M. Ponsiglione (2016) Dynamics of discrete screw dislocations on glide directions. J. Mech. Phys. Solids 92, 87104.

[4] T. Blass , I. Fonseca , G. Leoni & M. Morandotti (2015) Dynamics for systems of screw dislocations. SIAM J. Appl. Math. 75 (2), 393419.

[6] P. Cermelli & M. E. 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), 352.

[8] A. F. Filippov & F. M. Arscott (1988) Differential Equations with Discontinuous Righthand Sides: Control Systems, Vol. 18, Springer Science & Business Media, Dordrecht.

[9] D. Hull & D. J. Bacon (2001) Introduction to Dislocations, Butterworth-Heinemann, Oxford.

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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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