2020/03/17 by Irene Fonseca, Fonseca, Irene, Janusz Ginster +3
Computer Science · Engineering · Materials Science · #35K93 #35Q74 #74N05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlocal and gradient elasticity in micro/nano structures #Thermoelastic and Magnetoelastic Phenomena
paper · pdf · doi:10.48550/arxiv.2003.07876
openalex publication_date 2020/03/17 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
It is shown that in core-radius cutoff regularized simplified elasticity\n(where the elastic energy depends quadratically on the full displacement\ngradient rather than its symmetrized version), the force on a dislocation curve\nby the negative gradient of the elastic energy asymptotically approaches the\nmean curvature of the curve as the cutoff radius converges to zero. Rigorous\nerror bounds in H "older spaces are provided.\n As an application, convergence of dislocations moving by the gradient flow of\nthe elastic energy to dislocations moving by the gradient flow of the arclength\nfunctional, when the motion law is given by an H1-type dissipation, and\nconvergence to curve shortening flow in co-dimension 2 for the usual\nL2-dissipation is established. In the second scenario, existence and\nregularity are assumed while the H1-gradient flow is treated in full\ngenerality (for short time).\n The methods developed here are a blueprint for the more physical setting of\nlinearized isotropic elasticity.\n