Stress-based Atomistic/continuum Coupling: a New Variant of the Quasicontinuum Approximation

The force-based quasicontinuum (QCF) approximation is the principle that lies behind the most commonly used atomistic/continuum hybrid models for crystalline solids. Recent analyses have shown some potential pitfalls of the QCF method, particularly the lack of positive definiteness of the linearized QCF operator and the lack of uniform stability as the number of atoms tends to infinity. We derive a weak variational representation of the QCF operator and identify the origin of these difficulties as the lack of an interface condition on the stresses. This leads us to propose an improved variant of the QCF method that can be understood as a coupling mechanism based on stresses rather than forces.