Renormalized energy and forces on dislocations

SIAM Journal on Mathematical Analysis

Volumn 37, Issue 4, 2005, Pages 1131-1160

  Cermelli, Paolo ^a   Leoni, Giovanni ^b  

^a UNIVERSITY OF TURIN   (Italy)

^b CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Dislocations; Forces on defects; Variational methods; Variational techniques for singularities

Indexed keywords

DISLOCATIONS; FORCES ON DEFECTS; VARIATIONAL METHODS; VARIATIONAL TECHNIQUES FOR SINGULARITIES;

CRYSTALS; FORCE MEASUREMENT; MATHEMATICAL MODELS; PLASTICITY; PROBLEM SOLVING; VORTEX FLOW;

VARIATIONAL TECHNIQUES;

EID: 33746110773     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/040621636     Document Type: Article

References (30)

1. E. C. AIFANTIS, The physics of plastic deformation, Internat. J. Plasticity, 3 (1987), pp. 211-247.
2. G. ALBERTI, S. BALDO, AND G. ORLANDI, Variational convergence for functional of Ginzburg-Landau type, Indiana Univ. Math. J., 54 (2005), pp. 1411-1472.
3. R. J. ASARO, Micromechanics of crystals and polycrystals, Adv. Appl. Mech., 23 (1983), pp. 1-115.
4. F. BETHUEL, H. BREZIS, B. D. COLEMAN, AND F. HÉLEIN, Bifurcation analysis of minimizing harmonic maps describing the equilibrium of nematic phases between cylinders, Arch. Ration. Mech. Anal., 118 (1992), pp. 149-168.
5. F. BETHUEL, H. BREZIS, AND F. HÉLEIN, Ginzburg-Landau Vortices, Birkhäuser, Basel, 1994.
6. F. BETHUEL, H. BREZIS, AND G. ORLANDI, Asymptotics for the Ginzburg-Landau equation in arbitrary dimensions, J. Funct. Anal., 186 (2001), pp. 432-520.
7. P. CERMELLI AND M. E. GURTIN, The motion of screw dislocations in materials undergoing anti-plane shear: Glide, cross-slip, fine cross-slip, Arch. Ration. Mech. Anal. 148 (1999), pp. 3-52.
8. P. CERMELLI, M. E. GURTIN, AND G. LEONI, Energies for incoherent films: An analytical approach, Interfaces Free Boundaries, 1 (1999), pp. 81-105.
9. C. M. DAFERMOS, Some remarks on Korn's inequality, Z. Angew. Mat. Phys., 19 (1968), pp. 913-920.
10. R. DAUTRAY AND J. L. LIONS, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3, Springer, Berlin, 1988.
11. J. L. ERICKSEN, Remarks concerning forces on line defect, Z. Angew. Mat. Phy., 46 (1995), pp. S247-S271.
12. J. D. ESHELBY, The force on elastic singularity, Phil. Trans. R. Soc. London A, 244 (1951), pp. 87-112.
13. J. D. ESHELBY, The continuum theory of lattice defects, in Solid State Physics 3, F. Seitz and D. Turnbull, eds., Academic Press, New York, 3 1956, pp. 79-144.
14. L. C. EVANS, Weak convergence methods for nonlinear partial differential equations, Regional Conference Ser. Math., 74, American Mathematical Society, Providence, RI, 1990.
15. N. A. FLECK AND J. W. HUTCHINSON, Strain gradient plasticity, Adv. Appl. Mech., 33 (1997), pp. 295-361.
16. D. FRANÇOIS, A. PINEAU, AND A. O ZAOUI, Mechanical Behaviour of Materials I, Kluwer, Dordrecht, The Netherlands, 1998.
17. K. O. FRIEDRICHS, On the boundary-value problems of the theory of elasticity and Korn's inequality, Ann. Math., 48 (1947), pp. 441-471.
18. M. E. GURTIN, Configurational Forces as Basic Concepts of Continuum Physics, Springer-Verlag, New York, 2000.
19. M. E. GURTIN, On the plasticity of single crystals: Free energy, microforces, plastic-strain gradients, J. Mech. Phys. Solids, 48 (2000), pp. 989-1036.
20. M. E. GURTIN, A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations, J. Mech. Phys. Solids, 50 (2002), pp. 5-32.
21. R. L. JERRARD AND H. M. SONER, Dynamics of Ginzburg-Landau vortices, Arch. Ration. Mech. Anal., 142 (1998), pp. 99-125.
22. R. L. JERRARD AND H. M. SONER, The Jacobian and the Ginzburg-Landau energy, Calc. Var. Partial Differential Equations, 14 (2002), pp. 151-191.
23. J. P. HIRTH AND J. LOTHE, Theory of Dislocations, 2nd ed., John Wiley, New York, 1982.
24. C. O. HORGAN, Korn's inequalities and their applications in continuum mechanics, SIAM Rev., 37 (1995), pp. 491-511.
25. D. KUHLMANN-WILSDORF, Theory of plastic deformation: Properties of low energy dislocation structures, Materials Sci. Engrg., A113 (1989), pp. 1-41.
26. F.-H. LIN AND C. LIU, Static and dynamic theories of liquid crystals, J. Partial Differential Equations, 14 (2001), pp. 289-330.
27. F.-H. LIN AND J. X. XIN, On the dynamical law of the Ginzburg-Landau vortices on the plane, Comm. Pure Appl. Math., 52 (1999), pp. 1189-1212.
28. N. I. MUSKHELISHVILI, Some Basic Problems in the Mathematical Theory of Elasticity, Noord-hoff, Groningen, The Netherlands, 1963.
29. F. R. N. NABARRO, Theory of Crystal Dislocations, Clarendon Press, Oxford, UK, 1967.
30. M. PEACH AND J. S. KÖHLER, The forces exerted on dislocations and the stress field produced by them, Phys. Rev., 80 (1950), pp. 436-439.