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Finite element analysis of Cauchy-Born approximations to atomistic models. (English) Zbl 1275.82002

The authors introduce and analyze atomistic Cauchy-Born models. The atomistic potential \[ \Phi^a(y)=\varepsilon^d\sum_{l\in\mathcal{L}}\sum_{\eta\in\mathcal{R}}\phi_{\eta}(\overline{D}_{\eta} y_l) \] is considered, where \(\mathcal{L}=\{l=(l_1,\dotsc,l_d)\in \mathbb{Z}^d\cap[-N_1-1,N_1]\times \dotsb \times[-N_d-1,N_d]\}\), \(\Omega=\{x\in[x_{-N_1-1},x_{N_1}]\times \dotsb \times[x_{-N_d-1},x_{N_d}]\}^o\) (the interior of the set), \(W^{s,p}(\Omega,\mathbb{R}^d)\) is the usual Sobolev space of functions \(y:\Omega\rightarrow \mathbb{R}^d\), \(\phi_\eta\) are functions defined on \(\mathbb{R}^d\backslash \{0\}\), \(\overline{D}_{\eta}( y_l)=\frac{y_{l+\eta}-y_l}{\varepsilon}\) with \(l,l+\eta \in \mathcal{L}\) and \(\mathcal{R}\) is a given finite set of interaction vectors. It is shown that the consistency error of the models considered both in energy and in dual \(W^{1,p}\) type norms is \(\mathcal{O}(\varepsilon^2)\), where \(\varepsilon\) denotes the interatomic distance in the lattice.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
81V45 Atomic physics
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