Overall equilibrium in the coupling of peridynamics and classical continuum mechanics

Coupling peridynamics based computational tools with those using classical continuum mechanics can be very beneficial, because it can provide a means to generate a computational method that combines the efficiency of classical continuum mechanics with the capability to simulate crack propagation, typical of peridynamics. This paper presents an overlooked issue in this type of coupled computational methods: the lack of overall equilibrium. This can be the case even if the coupling strategy satisfies the usual numerical tests involving rigid body motions as well as uniform and linear strain distributions. We focus our investigation on the lack of overall equilibrium in an approach to couple peridynamics and classical continuum mechanics recently proposed by the authors. In our examples, the magnitude of the out-of-balance forces is a fraction of a percent of the applied forces, but it cannot be assumed to be a numerical round-off error. We show analytically and numerically that the main reason for the existence of out-of-balance forces is a lack of balance between the local and nonlocal tractions at the coupling interface. This usually results from the presence of high-order derivatives of displacements in the coupling zone

Out-of-balance forces in computational methods coupling peridynamics with classical mechanics

Structural engineers should be able to describe all stages of the structural life even those involving crack propagation and branching. However, the description of the fracture phenomena in structural materials is still an open problem. Computational methods based on Classical Continuum Mechanics (CCM) have not been naturally developed to simulate problems involving discontinuities in the displacement field. Therefore, these computational tools have to be equipped with ad hoc extensions to deal with crack propagation problems. Peridynamics (PD) [1-2], was proposed with the aim of including cracks as a natural part of the solution. However, PD is not computationally efficient, due to the non-local nature of the approach and that is a limitation to its practical use. Several researchers are trying to couple computational methods based on CCM with those based on PD to obtain a numerical method having the advantages of both computational techniques and avoids their pitfalls [3, chap.14]. Our proposed coupling approach [4] is realized at the discrete level between the standard displacement version of the Finite Element Method and a meshless version of PD. We observed that even if the coupling method satisfies the usual numerical tests: rigid body motion, uniform and linear strain distribution, under more complex load conditions some out of balance forces could be generated. The paper evaluates the magnitude of the out of balance forces and discusses methods to reduce them