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Derivation of a line-tension model for dislocations from a nonlinear three-dimensional energy: the case of quadratic growth. (English) Zbl 1473.49015

For a general introduction in problems related to dislocations, one recommends the book of P. M. Anderson et al. [Theory of dislocations. 3rd edition. Cambridge: Cambridge University Press (2017; Zbl 1365.82001)] and an extensive article from [D. J. Bacon et al., “Anisotropic continuum theory of elastic defects”, Progr. Mater. Sci. 23, 51–262 (1979)]. In the second section of the present paper one introduces elements related to the model of dislocation, the set of admissible dislocation densities, (h, \(\alpha\))-dilute measures and one presents a compactness result related to the dislocation measures and associated fields with equibounded energies. This result is completed with a \(\Gamma\)-convergence result for the energy functional. Proofs of both results can be found in the fifth section of the article. The third section is dedicated to an extensive analyses of a three-dimensional cell problem in a nonlinear framework. One combines some known results for linear case with known techniques developed for the two-dimensional case. An analysis of the asymptotic behavior of the quadratic energy associated to a sequence of dilute dislocations is performed in the fourth section. Notations used throughout the paper are shown in the Appendix A.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
35Q74 PDEs in connection with mechanics of deformable solids
74B10 Linear elasticity with initial stresses
74N05 Crystals in solids
35B65 Smoothness and regularity of solutions to PDEs

Citations:

Zbl 1365.82001
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