A multi-scale modeling of surface effect via the modified boundary Cauchy–Born model

In this paper, a new multi-scale approach is presented based on the modified boundary Cauchy–Born (MBCB) technique to model the surface effects of nano-structures. The salient point of the MBCB model is the definition of radial quadrature used in the surface elements which is an indicator of material behavior. The characteristics of quadrature are derived by interpolating data from atoms laid in a circular support around the quadrature, in a least-square scene. The total-Lagrangian formulation is derived for the equivalent continua by employing the Cauchy–Born hypothesis for calculating the strain energy density function of the continua. The numerical results of the proposed method are compared with direct atomistic and finite element simulation results to indicate that the proposed technique provides promising results for modeling surface effects of nano-structures. ► A multi-scale approach is presented to model the surface effects in nano-structures. ► The total-Lagrangian formulation is derived by employing the Cauchy–Born hypothesis. ► The radial quadrature is used to model the material behavior in surface elements. ► The quadrature characteristics are derived using the data at the atomistic level.