Simulation of instabilities in thin nanostructures by a perturbation approach

A new numerical method is proposed to simulate instabilities in thin atomistic structures in quasi-static regime. In contrast with previous approaches based on energy minimization or Newton-Raphson methods, the present technique uses a series expansion of atomistic displacements with respect to a loading path parameter, truncated at high orders. The nonlinear set of equations defined by minimizing the potential energy of the discrete system with respect to nuclei positions is then transformed into a sequence of linear sets of equations, which can be solved efficiently. The solution can be described along very large loading steps without correction, resulting in a significant reduction of matrices to be inverted. Finally, the treatment of limit points and snap-back/snap-through arising when instabilities occur is simplified due to a continuous description with respect to the loading path parameter. The method is applied to the analysis of single carbon atom layers nanostructures like graphene sheets or nanotubes in traction or compression regimes. Accuracy and efficiency of the technique is demonstrated by comparisons with iterative Newton procedures.