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.


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


Zbl 1365.82001
Full Text: DOI


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