No abstract

We show existence of solutions for the equations of static atomistic nonlinear elasticity theory on a bounded domain with prescribed boundary values. We also show their convergence to the solutions of continuum nonlinear elasticity theory, with energy density given by the Cauchy-Born rule, as the interatomic distances tend to zero. These results hold for small data close to a stable lattice for general finite range interaction potentials. We also discuss the notion of stability in detail.

In linearised continuum elasticity, the elastic strain due to a straight dislocation line decays as O(r −1 ), where r denotes the distance to the defect core. It is shown in [7] that the core correction due to nonlinear and discrete (atomistic) effects decays like O(r −2 ).In the present work, we focus on screw dislocations under pure anti-plane shear kinematics. In this setting we demonstrate that an improved decay O(r −p ), p > 2, of the core correction is obtained when crystalline symmetries are fully exploited and possibly a simple and explicit correction of the continuum far-field prediction is made.This result is interesting in its own right as it demonstrates that, in some cases, continuum elasticity gives a much better prediction of the elastic field surrounding a dislocation than expected, and moreover has practical implications for atomistic simulation of dislocations cores, which we discuss as well.

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.

<p style='text-indent:20px;'>We derive <i>von-Kármán plate theory</i> from three dimensional, purely atomistic models with classical particle interaction. This derivation is established as a <inline-formula><tex-math id="M1">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-limit when considering the limit where the interatomic distance <inline-formula><tex-math id="M2">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> as well as the thickness of the plate <inline-formula><tex-math id="M3">\begin{document}$ h $\end{document}</tex-math></inline-formula> tend to zero. In particular, our analysis includes the <i>ultrathin</i> case where <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon \sim h $\end{document}</tex-math></inline-formula>, leading to a new <i>von-Kármán plate theory for finitely many layers</i>.</p>