The Heterogeneous Multiscale Method for Dynamics of Solids With Applications to Brittle Cracks
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1229-LL01-02
The Heterogeneous Multiscale Method for Dynamics of Solids With Applications to Brittle Cracks Jerry Zhijian Yang1 and Xiantao Li2 1 School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623 2 Department of Mathematics, Penn State University, University Park, PA 16802 ABSTRACT We present a multiscale method for the modeling of dynamics of crystalline solids. The method employs the continuum elastodynamics model to introduce loading conditions and capture elastic waves, and near isolated defects, molecular dynamics (MD) model is used to resolve the local structure at the atomic scale. The coupling of the two models is achieved based on the framework of the heterogeneous multiscale method (HMM) [6] and a consistent coupling condition with special treatment of the MD boundary condition. Application to the dynamics of a brittle crack under various loading conditions is presented. Elastic waves are observed to pass through the interface from atomistic region to the continuum region and reversely. Thresholds of strength and duration of shock waves to launch the crack opening are quantitatively studied and related to the inertia effect of crack tips. INTRODUCTION In recent years, there has been a growing amount of effort on developing multiscale models that combine atomistic and continuum models. One class of such methods is based on a domain decompositions (DD) strategy, in which one explicitly divides the system into two types of subdomains, modeled separately by atomistic and continuum models. For dynamics problems, the DD approach has to be combined with an appropriate boundary condition for the atomistic model, to take into account the atoms that have been replaced by the continuum description. This idea was first pursued in the work of E and Huang [1] for simplified models, in which efficient boundary conditions were obtained based on minimizing phonon reflection. Later, Wagner and Liu[2] and their coworkers proposed the bridging scale method (BSM), which uses a projection operator to connect molecular dynamics to the continuum elastodynamics model. At the interface, the boundary condition for molecular dynamics was adopted to serve as the coupling condition. Using the same framework, To and Li [3] proposed another type of boundary condition analogous to the perfectly matched layers (PML) used in wave propagation problems. Based on the work of E and Huang, Li and E [4, 5] pursued further systematic study of the variational boundary condition, which compromise between accuracy and feasibility. In the present study, we combine the variational boundary conditions with the general framework proposed in [1]. The novel aspect in our approach is to define a mechanical equilibrium state, around which the linearization is done. On one hand, this greatly simplifies the linearization procedure. On the other hand, it will preserve a mechanical equilibrium state, which in the state case, has be demonstrated as an indication of uniform accuracy.
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