On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime

Manuel Friedrich; Bernd Schmidt; Manuel Friedrich; Bernd Schmidt

doi:10.3934/nhm.2015.10.321

Networks and Heterogeneous Media

2015, Volume 10, Issue 2: 321-342. doi: 10.3934/nhm.2015.10.321

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On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime

Manuel Friedrich ^{1
,},
Bernd Schmidt ^{1
,}

1.
Universität Augsburg, Institut für Mathematik, Universitätsstr. 14, 86159 Augsburg

Received: 01 March 2014 Revised: 01 October 2014
Primary: 74R10, 49J45, 70G75.

We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of $\Gamma$-convergence. We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either homogeneous elastic deformations or configurations that are completely cracked generically along a crystallographic line. As applications we discuss cleavage properties of strained crystals and an effective continuum fracture energy for magnets.
- Brittle materials,
- variational fracture,
- atomistic models,
- discrete-to-continuum limits,
- free discontinuity problems
Citation: Manuel Friedrich, Bernd Schmidt. On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime[J]. Networks and Heterogeneous Media, 2015, 10(2): 321-342. doi: 10.3934/nhm.2015.10.321

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Abstract

We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of $\Gamma$-convergence. We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either homogeneous elastic deformations or configurations that are completely cracked generically along a crystallographic line. As applications we discuss cleavage properties of strained crystals and an effective continuum fracture energy for magnets.

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