Accuracy of Quasicontinuum Approximations near Instabilities

The formation and motion of lattice defects such as cracks, dislocations, or grain boundaries, occurs when the lattice configuration loses stability, that is, when an eigenvalue of the Hessian of the lattice energy functional becomes negative. When the atomistic energy is approximated by a hybrid energy that couples atomistic and continuum models, the accuracy of the approximation can only be guaranteed near deformations where both the atomistic energy as well as the hybrid energy are stable. We propose, therefore, that it is essential for the evaluation of the predictive capability of atomistic-to-continuum coupling methods near instabilities that a theoretical analysis be performed, at least for some representative model problems, that determines whether the hybrid energies remain stable up to the onset of instability of the atomistic energy. We formulate a one-dimensional model problem with nearest and next-nearest neighbor interactions and use rigorous analysis, asymptotic methods, and numerical experiments to obtain such sharp stability estimates for the basic conservative quasicontinuum (QC) approximations. Our results show that the consistent quasi-nonlocal QC approximation correctly reproduces the stability of the atomistic system, whereas the inconsistent energybased QC approximation incorrectly predicts instability at a significantly reduced applied load that we describe by an analytic criterion in terms of the derivatives of the atomistic potential.