@phdthesis{Schaeffner2015,
  author    = {Sch{\"a}ffner, Mathias},
  title     = {Multiscale analysis of non-convex discrete systems via \(\Gamma\)-convergence},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-122349},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2015},
  abstract  = {The subject of this thesis is the rigorous passage from discrete systems to continuum models via variational methods. The first part of this work studies a discrete model describing a one-dimensional chain of atoms with finite range interactions of Lennard-Jones type. We derive an expansion of the ground state energy using \(\Gamma\)-convergence. In particular, we show that a variant of the Cauchy-Born rule holds true for the model under consideration. We exploit this observation to derive boundary layer energies due to asymmetries of the lattice at the boundary or at cracks of the specimen. Hereby we extend several results obtained previously for models involving only nearest and next-to-nearest neighbour interactions by Braides and Cicalese and Scardia, Schl{\"o}merkemper and Zanini. The second part of this thesis is devoted to the analysis of a quasi-continuum (QC) method. To this end, we consider the discrete model studied in the first part of this thesis as the fully atomistic model problem and construct an approximation based on a QC method. We show that in an elastic setting the expansion by \(\Gamma\)-convergence of the fully atomistic energy and its QC approximation coincide. In the case of fracture, we show that this is not true in general. In the case of only nearest and next-to-nearest neighbour interactions, we give sufficient conditions on the QC approximation such that, also in case of fracture, the minimal energies of the fully atomistic energy and its approximation coincide in the limit.},
  subject      = {Gamma-Konvergenz},
  language  = {en}
}