Sharp Stability Estimates for the Force-Based Quasicontinuum Approximation of Homogeneous Tensile Deformation

The accuracy of atomistic-to-continuum hybrid methods can be guaranteed only for deformations where the lattice configuration is stable for both the atomistic energy and the hybrid energy. For this reason, a sharp stability analysis of atomistic-to-continuum coupling methods is essential for evaluating their capabilities for predicting the formation of lattice defects. We formulate a simple one-dimensional model problem and give a detailed analysis of the linear stability of the force-based quasicontinuum (QCF) method at homogeneous deformations. The focus of the analysis is the question of whether the QCF method is able to predict a critical load at which fracture occurs. Numerical experiments show that the spectrum of a linearized QCF operator is identical to the spectrum of a linearized energy-based quasi-nonlocal quasicontinuum (QNL) operator, which we know from our previous analyses to be positive below the critical load. However, the QCF operator is nonnormal, and it turns out th...

• Type: Journal
• Year: 2010
• Where: MMAS
• Authors: Matthew Dobson, Mitchell Luskin, Christoph Ortner
• Keywords: Critical Load, Intelligent Agents, MMAS 2010, QCF Operator, Sharp Stability