On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth

Julian Braun; Bernd Schmidt; Julian Braun; Bernd Schmidt

doi:10.3934/nhm.2013.8.879

Networks and Heterogeneous Media

2013, Volume 8, Issue 4: 879-912. doi: 10.3934/nhm.2013.8.879

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On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth

Julian Braun ^{1
,},
Bernd Schmidt ^{1
,}

1.
Institut für Mathematik, Universität Augsburg, 86135 Augsburg

Received: 01 November 2012
49J45, 74B20, 74Qxx.

We derive continuum limits of atomistic models in the realm of nonlinear elasticity theory rigorously as the interatomic distances tend to zero. In particular we obtain an integral functional acting on the deformation gradient in the continuum theory which depends on the underlying atomistic interaction potentials and the lattice geometry. The interaction potentials to which our theory applies are general finite range models on multilattices which in particular can also account for multi-pole interactions and bond-angle dependent contributions. Furthermore, we discuss the applicability of the Cauchy-Born rule. Our class of limiting energy densities consists of general quasiconvex functions and the class of linearized limiting energies consistent with the Cauchy-Born rule consists of general quadratic forms not restricted by the Cauchy relations.
- Nonlinear elasticity theory,
- atomistic systems,
- discrete-to-continuum limits,
- Cauchy-Born rule
Citation: Julian Braun, Bernd Schmidt. On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth[J]. Networks and Heterogeneous Media, 2013, 8(4): 879-912. doi: 10.3934/nhm.2013.8.879

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Abstract

We derive continuum limits of atomistic models in the realm of nonlinear elasticity theory rigorously as the interatomic distances tend to zero. In particular we obtain an integral functional acting on the deformation gradient in the continuum theory which depends on the underlying atomistic interaction potentials and the lattice geometry. The interaction potentials to which our theory applies are general finite range models on multilattices which in particular can also account for multi-pole interactions and bond-angle dependent contributions. Furthermore, we discuss the applicability of the Cauchy-Born rule. Our class of limiting energy densities consists of general quasiconvex functions and the class of linearized limiting energies consistent with the Cauchy-Born rule consists of general quadratic forms not restricted by the Cauchy relations.

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