Inserted: 14 jan 2009
Last Updated: 27 jul 2011
Journal: Math. Models Methods Appl. Sci.
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 $\Gamma$-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.
Keywords: Gamma-convergence, Asymptotic expansions, Multi-scale modeling, next-to-nearest-neighbour interactions