Skip to main content
×
Home

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

  • GIOVANNI A. BONASCHI (a1), PATRICK VAN MEURS (a2) and MARCO MORANDOTTI (a3)
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.

Copyright
Footnotes
Hide All

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”.

Footnotes
References
Hide All
[1] Alicandro R., De Luca L., Garroni A. & Ponsiglione M. (2014) Metastability and dynamics of discrete topological singularities in two dimensions: A Γ-convergence approach. Arch. Rational Mech. Anal. 214 (1), 269330.
[2] Alicandro R., De Luca L., Garroni A. & Ponsiglione M. (2016) Dynamics of discrete screw dislocations on glide directions. J. Mech. Phys. Solids 92, 87104.
[3] Ambrosio L., Gigli N. & Savaré G. (2008) Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel.
[4] Blass T., Fonseca I., Leoni G. & Morandotti M. (2015) Dynamics for systems of screw dislocations. SIAM J. Appl. Math. 75 (2), 393419.
[5] Blass T. & Morandotti M. (2017) Renormalized energy and Peach-Köhler forces for screw dislocations with antiplane shear. J. Convex Anal., 24 (2), Verify on http://www.heldermann.de/JCA/jcacon.htm
[6] Cermelli P. & Gurtin M. E. (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.
[7] di Bernardo M., Budd C. J., Champneys A. R. & Kowalczyk P. (2008) Piecewise-Smooth Dynamical Systems: Theory and Applications, Vol. 163, Springer Science & Business Media, London.
[8] Filippov A. F. & Arscott F. M. (1988) Differential Equations with Discontinuous Righthand Sides: Control Systems, Vol. 18, Springer Science & Business Media, Dordrecht.
[9] Hull D. & Bacon D. J. (2001) Introduction to Dislocations, Butterworth-Heinemann, Oxford.
[10] Hirth J. P. & Lothe J. (1982) Theory of Dislocations, Wiley, New York.
[11] Mielke A. (2016) On evolutionary Γ-convergence for gradient systems. In: Muntean Adrian, Rademacher Jens D.M., Zagaris Antonios (editors), Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Springer, pp. 187249.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 20 *
Loading metrics...

Abstract views

Total abstract views: 185 *
Loading metrics...

* Views captured on Cambridge Core between 3rd November 2016 - 11th December 2017. This data will be updated every 24 hours.