A computational model for atomistic-based higher-order continua using the FEM technique

In this paper, an atomistic-based higher-order continuum model is developed in the framework of nonlinear finite element method to present the geometrically nonlinear behavior of nano-structures. In order to model the inhomogeneous deformation within the Cauchy-Born hypothesis, the higher-order CB hypothesis is presented based on a hierarchical multi-scale technique, in which the constitutive model of higher-order continuum is obtained using the derivatives of strain energy density. In order to avoid the use of C1-continuity element, as an alternative procedure, the mixed-type element is utilized employing the nodal deformation gradient as additional degrees of freedom. The relation between the nodal displacement DOFs and the nodal deformation gradient DOFs are enforced using the Lagrange-multipliers method. Finally, the efficiency of the proposed multi-scale technique in capturing the geometrically nonlinear behavior of nano-structures is illustrated through several numerical examples. The influence of higher-order terms in the Cauchy-Born hypothesis is presented by comparing the numerical results with those obtained from the classical CB hypothesis. A study on mesh size dependence of the proposed multi-scale technique is performed for numerical simulations with distinct levels of mesh size.