Γ-convergence analysis of the nonlinear self-energy induced by edge dislocations in semi-discrete and discrete models in two dimensions
Abstract
We propose nonlinear semi-discrete and discrete models for the elastic energy induced by a finite system of edge dislocations in two dimensions. Within the dilute regime, we analyze the asymptotic behavior of the nonlinear elastic energy, as the core-radius (in the semi-discrete model) and the lattice spacing (in the purely discrete one) vanish. Our analysis passes through a linearization procedure within the rigorous framework of Γ-convergence.
Funding statement: Roberto Alicandro, Lucia De Luca, and Mariapia Palombaro are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Acknowledgements
The authors thank G. Lazzaroni for interesting and fruitful discussions on discrete nonlinear elasticity models.
References
[1] R. Alicandro and M. Cicalese, Variational analysis of the asymptotics of the XY model, Arch. Ration. Mech. Anal. 192 (2009), no. 3, 501–536. 10.1007/s00205-008-0146-0Search in Google Scholar
[2] R. Alicandro, M. Cicalese and M. Ponsiglione, Variational equivalence between Ginzburg–Landau, XY spin systems and screw dislocations energies, Indiana Univ. Math. J. 60 (2011), no. 1, 171–208. 10.1512/iumj.2011.60.4339Search in Google Scholar
[3] R. Alicandro, L. De Luca, A. Garroni and M. Ponsiglione, Metastability and dynamics of discrete topological singularities in two dimensions: A Γ-convergence approach, Arch. Ration. Mech. Anal. 214 (2014), no. 1, 269–330. 10.1007/s00205-014-0757-6Search in Google Scholar
[4] R. Alicandro, L. De Luca, G. Lazzaroni, M. Palombaro and M. Ponsiglione, Coarse-graining of a discrete model for edge dislocations in the regular triangular lattice, J. Nonlinear Sci. 33 (2023), no. 2, Paper No. 33. 10.1007/s00332-023-09888-zSearch in Google Scholar
[5] R. Alicandro, G. Lazzaroni and M. Palombaro, Derivation of linear elasticity for a general class of atomistic energies, SIAM J. Math. Anal. 53 (2021), no. 5, 5060–5093. 10.1137/21M1397179Search in Google Scholar
[6] M. P. Ariza and M. Ortiz, Discrete crystal elasticity and discrete dislocations in crystals, Arch. Ration. Mech. Anal. 178 (2005), 149–226. 10.1007/s00205-005-0391-4Search in Google Scholar
[7] D. J. Bacon, D. M. Barnett and R. O. Scattergood, Anisotropic continuum theory of lattice defects, Progr. Mater. Sci. 23 (1978), 51–262. 10.1016/0079-6425(80)90007-9Search in Google Scholar
[8] A. Braides, M. Solci and E. Vitali, A derivation of linear elastic energies from pair-interaction atomistic systems, Netw. Heterog. Media 2 (2007), no. 3, 551–567. 10.3934/nhm.2007.2.551Search in Google Scholar
[9] P. Cermelli and G. Leoni, Renormalized energy and forces on dislocations, SIAM J. Math. Anal. 37 (2005), no. 4, 1131–1160. 10.1137/040621636Search in Google Scholar
[10] S. Conti, A. Garroni and R. Marziani, Line-tension limits for line singularities and application to the mixed-growth case, Calc. Var. Partial Differential Equations 62 (2023), no. 8, Paper No. 228. 10.1007/s00526-023-02552-0Search in Google Scholar
[11] G. Dal Maso, M. Negri and D. Percivale, Linearized elasticity as Γ-limit of finite elasticity, Set-Valued Anal. 10 (2002), 165–183. 10.1023/A:1016577431636Search in Google Scholar
[12] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3, Springer, Berlin, 1988. Search in Google Scholar
[13] L. De Luca, Γ-convergence analysis for discrete topological singularities: The anisotropic triangular lattice and the long range interaction energy, Asymptot. Anal. 96 (2016), no. 3–4, 185–221. 10.3233/ASY-151334Search in Google Scholar
[14] L. De Luca, A. Garroni and M. Ponsiglione, Γ-convergence analysis of systems of edge dislocations: The self energy regime, Arch. Ration. Mech. Anal. 206 (2012), no. 3, 885–910. 10.1007/s00205-012-0546-zSearch in Google Scholar
[15] G. Friesecke, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math. 55 (2002), no. 11, 1461–1506. 10.1002/cpa.10048Search in Google Scholar
[16] A. Garroni, G. Leoni and M. Ponsiglione, Gradient theory for plasticity via homogenization of discrete dislocations, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1231–1266. 10.4171/jems/228Search in Google Scholar
[17] A. Giuliani and F. Theil, Long range order in atomistic models for solids, J. Eur. Math. Soc. (JEMS) 24 (2022), no. 10, 3505–3555. 10.4171/jems/1169Search in Google Scholar
[18] J. P. Hirth and J. Lothe, Theory of Dislocations, Krieger Publishing, Malabar, 1982. Search in Google Scholar
[19] S. Luckhaus and L. Mugnai, On a mesoscopic many-body Hamiltonian describing elastic shears and dislocations, Contin. Mech. Thermodyn. 22 (2010), no. 4, 251–290. 10.1007/s00161-010-0142-0Search in Google Scholar
[20] S. Müller, L. Scardia and C. I. Zeppieri, Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations, Analysis and Computation of Microstructure in Finite Plasticity, Lect. Notes Appl. Comput. Mech. 78, Springer, Cham (2015), 175–204. 10.1007/978-3-319-18242-1_7Search in Google Scholar
[21] M. Ponsiglione, Elastic energy stored in a crystal induced by screw dislocations: From discrete to continuous, SIAM J. Math. Anal. 39 (2007), no. 2, 449–469. 10.1137/060657054Search in Google Scholar
[22]
C. Reina and S. Conti,
Kinematic description of crystal plasticity in the finite kinematic framework: a micromechanical understanding of
[23]
C. Reina, A. Schlömerkemper and S. Conti,
Derivation of
[24] E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal. 152 (1998), no. 2, 379–403. 10.1006/jfan.1997.3170Search in Google Scholar
[25] E. Sandier and S. Serfaty, Vortices in the magnetic Ginzburg–Landau model, Progr. Nonlinear Differential Equations Appl. 70, Birkhäuser, Boston, 2007. 10.1007/978-0-8176-4550-2Search in Google Scholar
[26] L. Scardia and C. I. Zeppieri, Line-tension model for plasticity as the Γ-limit of a nonlinear dislocation energy, SIAM J. Math. Anal. 44 (2012), no. 4, 2372–2400. 10.1137/110824851Search in Google Scholar
[27] B. Schmidt, On the derivation of linear elasticity from atomistic models, Netw. Heterog. Media 4 (2009), no. 4, 789–812. 10.3934/nhm.2009.4.789Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Γ-convergence analysis of the nonlinear self-energy induced by edge dislocations in semi-discrete and discrete models in two dimensions
- Characterization of the subdifferential and minimizers for the anisotropic p-capacity
- The best approximation of a given function in L 2-norm by Lipschitz functions with gradient constraint
- Quantitative C 1-stability of spheres in rank one symmetric spaces of non-compact type
- The Yang–Mills–Higgs functional on complex line bundles: Asymptotics for critical points
- A sub-Riemannian maximum modulus theorem
- The least gradient problem with Dirichlet and Neumann boundary conditions
- The 2+1-convex hull of a~finite set
- Non-local BV functions and a denoising model with L 1 fidelity
- Avoidance of the Lavrentiev gap for one-dimensional non-autonomous functionals with constraints
Articles in the same Issue
- Frontmatter
- Γ-convergence analysis of the nonlinear self-energy induced by edge dislocations in semi-discrete and discrete models in two dimensions
- Characterization of the subdifferential and minimizers for the anisotropic p-capacity
- The best approximation of a given function in L 2-norm by Lipschitz functions with gradient constraint
- Quantitative C 1-stability of spheres in rank one symmetric spaces of non-compact type
- The Yang–Mills–Higgs functional on complex line bundles: Asymptotics for critical points
- A sub-Riemannian maximum modulus theorem
- The least gradient problem with Dirichlet and Neumann boundary conditions
- The 2+1-convex hull of a~finite set
- Non-local BV functions and a denoising model with L 1 fidelity
- Avoidance of the Lavrentiev gap for one-dimensional non-autonomous functionals with constraints