Following the previous research, rather than simple bond-based models combined with high-performance computing in this period we explored alternative formulations for peridynamics that more general but also mathematically more complex. The use of ordinary and non-ordinary state based peridynamics and integration methods (implicit and explicit) was considered.
The non-ordinary state based formulation revealed more complicated, however, allowed to eliminate critical issues related with the surface effects, and establish a more direct relation with the models of the classic continuum solid mechanics.
Unfortunately, the literature regarding the non-ordinary state based peridynamics documents several issues arising from the rank deficiency of the equations namely: zero-energy modes and instability of the dynamic solution. We had the opportunity, of verifying this effects with numerical experiments in 1D and 2D.
An example of the instability effect can be seen in the attached video, showing the issues caused by rank-deficiency when using non-ordinary state based peridynamics. The attached video shows the initial propagation of shock waves (velocity norm) in one half of a notched symmetric beam. Instabilities start to occur when the waves are reflected by the boundaries, and rapidly contaminate the solution thought all the domain.
Those key challenges will be addressed in future while continuing with the development of new constitutive models for concrete under the peridynamics framework.