Description of the Atomistic-Continuum Model

The system in the atomistic-continuum model is composed of a matrix, described as a continuum, and an inclusion represented in atomistic detail, as shown in Fig. 26. The matrix is modeled by the finite element method developed by Gusev [97]. The scaling matrix H = [ABC] describes the system under periodic boundary conditions, where A, B, and C are cell vectors [98-100]. A set of nod- 

The key idea to mix two different length scales is to couple the displacement of nodal points on the inclusion boundary with the change of the atomistic scaling matrix via 

In other words, the inclusion boundary follows the homogeneous deformation of the atomistic box. Both the system box H and atomistic box h should be independent degrees of freedom in the model. Instead of considering h as a degree of freedom, the degree of freedom DH is introduced to relate two scaling matrices through 

Consequently, the system can be described by four types of degrees of freedom H,h, S , and s% The total energy of the system is 

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