  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 {\em 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 energy-based 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.
