31.80 Angewandte Mathematik
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Meshfree methods (MMs) such as the element free Galerkin (EFG)method have gained popularity because of some advantages over other numerical methods such as the finite element method (FEM). A group of problems that have attracted a great deal of attention from the EFG method community includes the treatment of large deformations and dealing with strong discontinuities such as cracks. One efficient solution to model cracks is adding special enrichment functions to the standard shape functions such as extended FEM, within the FEM context, and the cracking particles method, based on EFG method. It is well known that explicit time integration in dynamic applications is conditionally stable. Furthermore, in enriched methods, the critical time step may tend to very small values leading to computationally expensive simulations. In this work, we study the stability of enriched MMs and propose two mass-lumping strategies. Then we show that the critical time step for enriched MMs based on lumped mass matrices is of the same order as the critical time step of MMs without enrichment. Moreover, we show that, in contrast to extended FEM, even with a consistent mass matrix, the critical time step does not vanish even when the crack directly crosses a node.
We perform both classical molecular dynamics simulations and beam model calculations to investigate the Young's modulus of kinked silicon nanowires (KSiNWs). The Young's modulus is found to be highly sensitive to the arm length of the kink and is essentially inversely proportional to the arm length. The mechanism underlying the size dependence is found to be the interplay between the kink angle potential and the arm length potential, where we obtain an analytic relationship between the Young's modulus and the arm length of the KSiNW. Our results provide insight into the application of this novel building block in nanomechanical devices.