On the passage from atomistic systems to nonlinear elasticity theory

Julian Braun; Bernd Schmidt

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arXiv · 1107.4155

On the passage from atomistic systems to nonlinear elasticity theory

Julian Braun·Bernd Schmidt

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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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Julian Braun, Bernd Schmidt. 2011-07-21. On the passage from atomistic systems to nonlinear elasticity theory. https://arxiv.org/abs/1107.4155

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