Atomistic-to-continuum convergence for quasi-static crack growth in brittle materials

2024/02/05 by Manuel Friedrich, Friedrich, Manuel, Joscha Seutter +1
Engineering · Materials Science · #49J45 #70G75 #74A25 #74R10 #Analysis of PDEs (math.AP) #FOS: Mathematics #High-Velocity Impact and Material Behavior #Metallurgy and Material Forming #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2402.02966

openalex publication_date 2024/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We study the atomistic-to-continuum limit for a model of a quasi-static crack evolution driven by time-dependent boundary conditions. We consider a two-dimensional atomic mass spring system whose interactions are modeled by classical interaction potentials, supplemented by a suitable irreversibility condition accounting for the breaking of atmoic bonding. In a simultaneous limit of vanishing interatomic distance and discretized time step, we identify a continuum model of quasi-static crack growth in brittle fracture featuring an irreversibility condition, a global stability, and an energy balance. The proof of global stability relies on a careful adaptation of the jump-transfer argument by Francfort and Larsen to the atomistic setting.

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