Keywords: extended finite element method, molecular dynamics, bridging domain method, multiscale methods.

To understand the material failure phenomena, there are different methods to model cracking and failure in materials both at the continuum and atomistic scales. Methods which can be applied at the resolution of the lower scales are too expensive and can hardly be used in real engineering problems. This was the motivation of a class of numerical methods called multiscale methods. In these methods, different lengths and time scales are correlated to ensure both accuracy and efficiency.

In this paper, the bridging domain method of Xia and Belytschko [1], is used, which is based on an overlapping domain-decomposition scheme where the strain energies of the atomistic and continuum domain are weighted. The displacement compatibility between atomistic and continuum are enforced by means of Lagrange multipliers in the weak form. In this paper, the partition of unity enriched methods by means of the extended finite element method (XFEM) is used for crack modelling in the continuum domain. It was shown that the spurious wave reflection is a minimum in this kind of coupling method [2].

The implementation of the method is executed by coupling the LAMMPS molecular dynamics simulator [3] with PERMIX (an in-house XFEM code). The Lennard-Jones potential is used to model the interactions in the atomistic domain and the Cauchy-Born method [4] is used to compute the material behaviour in the continuum domain. To show the productivity of the proposed method, two different three dimensional crack examples were modelled. In the first example, a long finite plate is modelled with an edge crack. The plate is loaded in the transverse direction to the crack. In the second example a thin plate is modelled with two edge cracks and both shear and normal loads are applied by means of velocity. In the examples, an atomistic domain is placed on top of the finite element domain, around the crack front.

The centro-symmetry parameter and virial stress tensor in the atomistic region are also computed. It was observed that our three dimensional coupled method is capable of representing the crack and dislocation propagation for much fewer degrees of freedom than a direct full molecular dynamics simulation.

1

S.P. Xiao, T. Belytschko, "A bridging domain method for coupling continua with molecular dynamics", Computer Methods in Applied Mechanics and Engineering, 193, 1645-1669, 2004. doi:10.1016/j.cma.2003.12.053

2

T. Belytschko, M. Xu, "Conservation properties of the bridging domain method for coupled molecular/continuum dynamics", International Journal for Numerical Methods in Engineering, 76, 278-294, 2008.

3

S. Plimpton, "Fast parallel algorithms for short-range molecular dynamics", Journal of Computational Physics, 117(1), 1-19, 1995. doi:10.1006/jcph.1995.1039

4

R.E. Miller, E.B. Tadmor, "The quasicontinuum method: Overview, applications and current directions", Journal of Computer-Aided Materials Design, 9(3), 203-239, 2002.

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