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 that it is not generally positive definite, even when all of its eigenvalues are positive. Using a combination of rigorous analysis and numerical experiments, we investigate in detail for which choices of “function spaces” the QCF operator is stable, uniformly in the size of the atomistic system.