A discrete crystal model in three dimensions: the line-tension limit for dislocations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911963897921536 |
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| author | Conti, Sergio Garroni, Adriana Ortiz, Michael |
| author_facet | Conti, Sergio Garroni, Adriana Ortiz, Michael |
| contents | We propose a discrete lattice model of the energy of dislocations in three-dimensional crystals which properly accounts for lattice symmetry and geometry, arbitrary harmonic interatomic interactions, elastic deformations and discrete crystallographic slip on the full complement of slip systems of the crystal class. Under the assumption of diluteness, we show that the discrete energy converges, in the sense of $Γ$-convergence, to a line-tension energy defined on Volterra line dislocations, regarded as integral vector-valued currents supported on rectifiable curves. Remarkably, the line-tension limit is of the same form as that derived from semi-discrete models of linear elastic dislocations based on a core cutoff regularization. In particular, the line-tension energy follows from a cell relaxation and differs from the classical ansatz, which is quadratic in the Burgers vector. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_15400 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A discrete crystal model in three dimensions: the line-tension limit for dislocations Conti, Sergio Garroni, Adriana Ortiz, Michael Analysis of PDEs We propose a discrete lattice model of the energy of dislocations in three-dimensional crystals which properly accounts for lattice symmetry and geometry, arbitrary harmonic interatomic interactions, elastic deformations and discrete crystallographic slip on the full complement of slip systems of the crystal class. Under the assumption of diluteness, we show that the discrete energy converges, in the sense of $Γ$-convergence, to a line-tension energy defined on Volterra line dislocations, regarded as integral vector-valued currents supported on rectifiable curves. Remarkably, the line-tension limit is of the same form as that derived from semi-discrete models of linear elastic dislocations based on a core cutoff regularization. In particular, the line-tension energy follows from a cell relaxation and differs from the classical ansatz, which is quadratic in the Burgers vector. |
| title | A discrete crystal model in three dimensions: the line-tension limit for dislocations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.15400 |