Hybrid atomistic/continuum mechanics

A hybrid atomistic/continuum mechanics method is established in the Feng, et al. study [70] the deformation and fracture behaviors of carbon nanotubes (CNTs) in composites. The unit cell containing a CNT embedded in a matrix is divided in three regions, which are simulated by the atomic-potential method, the continuum method based on the modified Cauchy-Bom rule, and the classical continuum mechanics, respectively. The effect of CNT interaction is taken into account via the Mori-Tanaka effective field method of micromechanics. This method not only can predict the formation of Stone-Wales (5-7-7-5) defects, but also simulate the subsequent deformation and fracture process of CNTs. It is found that the critical strain of defect nucleation in a CNT is sensitive to its chiral angle but not to its diameter. The critical strain of Stone-Wales defect formation of zigzag CNTs is nearly twice that of armchair CNTs. Due to the constraint effect of matrix, the CNTs embedded in a composite are easier to fracture in comparison with those not embedded. With the increase in the Young s modulus of the matrix, the critical breaking strain of CNTs decreases. 

Two different approaches to modelling fibre fracture are presented the atomistic approach, for those who use the tools of quantum mechanics, and the continuum approach that relies on continuum mechanics. In the coming years, with the advent of nanotechnology, a hybrid approach will probably be needed to deal with nanofibres and biological fibres. 

Due to the sheer number, it is not possible to include in this review all of the proposed hybrid methodologies that deal with the coupling of continuum and atomistic regions in dynamical simulations. However, before leaving this section, we want to mention a few more types of methodologies. First, the multigrid methods,of which the recent work of Waisman and Fish is a good example. Also, the MPM/MD method, where the material point method (MPM) is used instead of the finite-element method (FEM) to couple continuum mechanics to conventional molecular dynamics. 

Miller, R.E. and Tadmor, E.B. (2007) Hybrid continuum mechanics and atomistic methods for simulating materials deformation and failure. MRS Bull., 32, 920-926. 

The effective matrix elements Hfj describe only the intramolecular terms associated with the chemical bonding but do not take into account long range and intermolecular interactions. For instance, the dipolar interaction between a solute and the molecules of a polar solvent are not accounted by the plain EHT matrix elements. Since semiempirical methods are much faster, the limitations imposed by the use of a continuum dielectric model for the solvent, which do not provide a good approximation for the immediate solvation shells in the vicinity of the solute or near the solid surface, can be overcome by atomistic quantum mechanical models for the solvent. Dynamic solvation effects can also be included through the semiempirical models. The hybrid QM/MM methods are also a valuable alternative to describe the dynamic effects of solvents on the quantum dynamics of the solute. The dipoles can be either intrinsic or induced. In the case of polar solvents, the electronic part of the dipole moment produced by the kth solvent molecule is f k f),... 

The hybrid methods which combine quantum-mechanical (QM) and classical descriptions are surely one of the mostly well-suited strategies in this context. Two main families of hybrid methods can be defined according to the model used to describe the classical part of the system. Either continuum or atomistic formulations can be introduced where, in the first case, the classical subsystem is described as a dielectric medium while, in the second case, a Molecular Mechanics (MM) formulation is generally adopted. While QM/continuum methods have been largely and successfully applied to molecular solutes in liquid solutions [2-5], QM/MM formulations have been more often used in the field of structured (biological) environments [6-10] even if the study of chemical reaction dynamics in solution represents another important field of applications of the method [11, 12]. 

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