On continuum limits of heterogeneous discrete systems modelling cracks in composite materials

Abstract

AbstractIn this paper we consider a periodic heterogeneous chain of atoms with nearest neighbour interactions of Lennard‐Jones type that allow for cracks. We are interested in the continuum limit of the corresponding energy functional as the number of atoms tends to infinity. Therefor we combine variational techniques for passages from discrete to continuous systems with techniques for passages from heterogeneous to homogeneous systems extending earlier work on interaction potentials with polynomial growth to potentials of Lennard‐Jones type.The limiting energy, which is obtained by the method of Γ‐convergence, contains a homogenization formula which captures the underlying heterogeneous structure. Moreover, we consider a rescaled energy functional which provides information about cracks also in the continuum limit. In this case, the limiting energy is a Griffith type energy consisting of a quadratic integral term and a jump contribution, which again show a dependence on the underlying heterogeneity.