An Atomistic-to-Continuum Analysis of Crystal Cleavage in a Two-Dimensional Model Problem

Abstract

A two-dimensional atomic mass spring system is investigated for critical fracture loads and its crack path geometry. We rigorously prove that, in the discrete-to-continuum limit, the minimal energy of a crystal under uniaxial tension leads to a universal cleavage law and energy minimizers are either homogeneous elastic deformations or configurations that are completely cracked and do not store elastic energy. Beyond critical loading, the specimen generically cleaves along a unique optimal crystallographic hyperplane. For specific symmetric crystal orientations, however, cleavage might fail. In this case a complete characterization of possible limiting crack geometries is obtained.

A Derivation of Linearized Griffith Energies from Nonlinear Models

“…Finally, in Section 6.2 we prove a cleavage law and extend the results obtained in [29,30,37] to the case of uniaxial compression, where we essentially follow the proof in [31,37], in particular using a piecewise rigidity result in SBD (see [15]) and a structure theorem on the boundary of sets of finite perimeter (see [25]). It turns out that in the linearized limit the behavior for compression and extension is virtually identical.…”

“…(Also compare a similar discussion before [31,Theorem 2.2].) The present problem in the framework of continuum fracture mechanics with isotropic surface energies is a slightly simplified model of the problem considered in [29,31].…”

“…Our general -limit result can now be applied to also solve boundary value problems of uniaxial compression which is, like the uniaxial tension test, a natural and interesting problem. Hereby we may complete the picture about the derivation of cleavage laws in [29,30].…”

“…As discussed before, the identification of critical loads and the investigation of crack paths is a challenging problem particularly for nonlinear models. The arguments in [29,30,37], where boundary value problems of uniaxial extension for brittle materials were investigated, fundamentally relied on the application of certain slicing techniques and due to the lack of convexity were not adapted to treat the case of compression. Our general -limit result can now be applied to also solve boundary value problems of uniaxial compression which is, like the uniaxial tension test, a natural and interesting problem.…”

Characterization of Optimal Carbon Nanotubes Under Stretching and Validation of the Cauchy–Born Rule

“…We close the section by noting that the scaling of µ frac ,m − µ us in m is typical for atomistic systems with pairwise interactions of Lennard-Jones type and has also been obtained in related models, cf. [5,33,34].…”