Cauchy-Born rule

Using the Cauchy-Born rule in conjunction with the Johnson embedded-atom potential, compute the energy of a shearing deformation on (111) planes in the [110] direction. Show that the energy has the form given in fig. 2.14. 

Lastly, computational efficiency needs to be discussed. However complete the formulation of the coarse-grained alternative to MD methodology is up to here, additional approximations are required to make it computationally efficient. To begin with, it is assumed that the thermally averaged positions of the constrained atoms can be expressed as a finite-element interpolation of the positions of the representative atoms, i.e., using finite-element shape functions. This is analogous to the procedure followed in the standard QC method to determine the instantaneous positions of the nonrepresentative atoms. Moreover, the computation of Vcg is noticibly expedited when both the local harmonic approximation and the Cauchy-Born rule are taken into account. Under such circumstances, Vcg becomes... 

To bypass the limitations of the Cauchy-Born rule, in 2006, Lu et al. proposed a more involved scheme to couple standard DFT to quasi-continuum calculations. In their method, the part of the system far away from the zone of interest is described using a classical (nonquantum) quasi-continumn approach (see discussion above on QC for details), i.e., considering both local (continuum) and nonlocal (atomistic) terms. Classical potentials (EAM in the applications presented) are used to evaluate the energy within the QC calculations. A third region is considered as well, covering the part of the system that needs a more detailed description. It is in this region that density functional theory is used. 

Studies of Validity of the Cauchy-Born Rule by Direct Comparison of Continuum and Atomistic Modeling. 

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