### Achievement

The lack of overall equilibrium issue in the coupling of peridynamics and classical continuum mechanics has been presented and analyzed. The analysis shows that absence of overall equilibrium results from the lack of balance between the local and nonlocal tractions at the coupling interface. Detailed derivations of the balance between local and nonlocal tractions at coupling interfaces have been presented, along with a practical quantitative way to assess the resulting out-of-balance through computation of the reaction forces. The order of convergence of the net out-of-balance forces in the coupled system, in the limit of vanishing nonlocality, has been obtained via the convergence of the nonlocal traction to the local traction in that limit. This analysis also demonstrates that out-of-balance results from the presence of high-order derivatives of displacements in the coupling zone. Numerical simulations confirm the analysis and demonstrate that the shape of the coupling interface can have a significant impact on the overall out-of-balance level. Computations show that adaptivity on the location of the coupling interface can be used to control the out-of-balance error.

### Significance and Impact

The unavoidable presence of small or large cracks in many aeronautical and aerospace structures still represents a major challenge for engineers who want to simulate a full structural life cycle. Unfortunately, using classical continuum mechanics for damage prediction is challenging due to the presence of spatial derivatives of displacements in the governing equations, which are undefined when the displacement fields are discontinuous. Peridynamics is a nonlocal reformulation of classical continuum mechanics that overcomes this challenge and has been implemented to solve complex problems involving damage initiation and crack propagation. Despite the effectiveness of peridynamics in solving these problems, peridynamics is more computationally expensive than classical continuum mechanics. Consequently, algorithms to couple peridynamics and classical continuum mechanics have been proposed to take advantage of their benefits while avoiding their drawbacks. While several coupling artifacts have been identified and analyzed, the lack of overall equilibrium has been overlooked in the literature.

For the first time, we address the problem of the overall equilibrium in the coupling of peridynamics and classical continuum mechanics. Our studies identify the origin of out-of-balance forces, analyze them, and discuss possible ways to reduce them. The impacts of this work on coupled simulations are twofold. First, the tolerance used in an implicit solution of a coupled computational problem should be carefully chosen: if the tolerance is smaller than the out-of-balance forces, then the computation will not converge. Second, the proper location and shape of the coupling interface in a computational problem can be adaptively defined to control the out-of-balance error. The use of adaptivity can reduce the computational effort considerably with respect to that required by a fully peridynamic simulation and will pave the way to future applications of the coupling of peridynamics and classical continuum mechanics to the solution of many practical problems.

### Research Details

This work concerned the coupling of peridynamics and classical continuum mechanics, focusing on an error given by the lack of overall equilibrium in static problems. This coupling error has been overlooked in the literature. We provided a theoretical analysis describing the reason for the appearance of this spurious effect, and we supported the analysis with numerical simulations. While this paper considered a particular strategy to couple peridynamics and classical continuum mechanics, proposed by the authors, this issue most probably affects other coupling approaches. We observed that a lack of overall equilibrium may occur even if the coupling method satisfies the usual numerical tests for static problems, given by rigid body motions as well as uniform and linear strain distributions. The theoretical analysis and the supporting numerical simulations allow us to conclude:

- The out-of-balance forces are related to the order of the derivatives of displacements in the coupling zone.
- It is easy to evaluate the magnitude of the out-of-balance error by computing the reaction forces.
- In the numerical examples investigated in this paper, the relative out-of-balance error is a fraction of a per cent and reduces as
*δ→0*. - It is usually possible to reduce the out-of-balance error by moving the coupling interface away from regions of high gradients of displacements.
- The numerical examples suggest that the shape of the coupling interface may have a significant impact on the overall out-of-balance error.

### Overview

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 per cent 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.

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Last Updated: April 2, 2021 - 12:47 pm