Boundary layer energies for nonconvex discrete systems

revised version: August 2010
Lucia Scardia, Anja Schlömerkemper, and Chiara Zanini
(Please use for correspondence this email).

Submission date: 08. Jan. 2009
Pages: 38
published in: Mathematical models and methods in applied sciences, 21 (2011) 4, p. 777-817 
DOI number (of the published article): 10.1142/S0218202511005210
Download preprint: PDF (370 kB)

Abstract:
In this work we consider a one-dimensional chain of atoms which interact through nearest and next-to-nearest neighbour interactions of Lennard-Jones type. We impose Dirichlet boundary conditions and in addition prescribe the deformation of the second and last but one atoms of the chain. This corresponds to prescribing the slope at the boundary of the discrete setting. We compute the -limits of zero and first order, where the latter leads to the occurrence of boundary layer contributions to the energy. These contributions depend on whether the chain behaves elastically close to the boundary or whether there is a crack. This in turn depends on the given boundary data. We also analyse the location of fracture in dependence on the prescribed discrete slopes.