Peridynamic Modeling With Non-Uniform Discretisation and Variable Horizon

Peridynamics [1] is a non-local continuum mechanics formulation where a material point can interact with other material points which are located at a finite distance with respect to each other. The range of interactions between material points is denoted as “horizon” which is a length scale parameter in peridynamics. Although horizon is a very important parameter, research on how to choose this parameter has been rather limited and mainly depends on suggestions made in the influential paper written by Silling and Askari [2]. They suggested to use a horizon size equivalent to three times of the grid spacing between material points based on the experiences of these researchers. Determination of the size of the horizon is an essential step to obtain accurate results from peridynamic simulations. In this study, the horizon size is determined for both bond-based [2], ordinary-state based [3,4] and non-ordinary state based formulations [3,4,5] for a specific condition where there is no existence of damage in the structure and non-local effects are insignificant. For such a condition, peridynamic solution should converge to the classical continuum mechanics solution as horizon size converges to zero [4]. Therefore, in this case, classical continuum mechanics solution can serve as a reference solution for peridynamics. Comparing peridynamic results against analytical and finite element method solutions will be sufficient to make decisions on the suitable value of horizon size. Several important aspects will be explored. Specifically, for uniform discretization, the size of the horizon with respect to the grid spacing and the size of the solution domain will be determined. For non-uniform discretization, the horizon size that we can use at different parts of the solution domain will be evaluated.

REFERENCES

[1] Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids, 48(1), pp.175-209.

[2] S. A. Silling and E. Askari, E. A meshfree method based on the peridynamic model of solid mechanics. Computers & Structures, Vol. 83(17-18), pp.1526-1535, 2005.

[3] Silling, S.A., Epton, M., Weckner, O., Xu, J. and Askari, E., 2007. Peridynamic states and constitutive modeling. Journal of Elasticity, 88(2), pp.151-184.

[4] E. Madenci and E. Oterkus, Peridynamic theory and its applications, New York: Springer, 2014.

[5] Warren, T.L., Silling, S.A., Askari, A., Weckner, O., Epton, M.A. and Xu, J., 2009. A non-ordinary state-based peridynamic method to model solid material deformation and fracture. International Journal of Solids and Structures, 46(5), pp.1186-1195.