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.