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Energy and force transition between atoms and continuum in quasicontinuum method

  • Chang, Shu-Wei (Department of Civil Engineering, National Taiwan University) ;
  • Liao, Ying-Pao (Department of Civil Engineering, National Taiwan University) ;
  • Huang, Chang-Wei (Department of Civil Engineering, Chung-Yuan Christian University) ;
  • Chen, Chuin-Shan (Department of Civil Engineering, National Taiwan University)
  • Received : 2013.10.23
  • Accepted : 2013.11.11
  • Published : 2014.03.25

Abstract

We present a full energy and force formulation of the quasicontinuum method with non-local and local transition elements. Non-local transition elements are developed to transmit inhomogeneity from the atomistic to the continuum regions. Local transition elements are developed to resolve the mathematical mismatch between non-local atoms and the local continuum. The rationale behind these transition elements is provided by analyzing the energy and force transitions between atoms and continuum under the Cauchy-Born rule. We show that breakdown of the Cauchy-Born rule occurs for slaved atoms of local elements within the cutoff of non-local atoms. The inadequacy of the Cauchy-Born rule at the transition region naturally leads to the need of atomistic treatment of transition slaved and transition representative atoms. Such an atomistic treatment together with a full or cutoff sampling allows non-local transition elements containing these transition entities to transmit inhomogeneity. Different force formulations for transition representative atoms and pure local representative atoms allow the local transition elements to resolve non-local and local mismatches. The method presented herein is validated by force calculations in an unstressed perfect crystal as well as an unrelaxed grain boundary model. A nanoindentation simulation in 3D is conducted to demonstrate the accuracy and efficiency of the proposed method.

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