Finite Element Convergence for State-Based Peridynamic Fracture Models

doi:10.1007/s42967-019-00039-4

摘要/Abstract

摘要： We establish the a priori convergence rate for fnite element approximations of a class of nonlocal nonlinear fracture models. We consider state-based peridynamic models where the force at a material point is due to both the strain between two points and the change in volume inside the domain of the nonlocal interaction. The pairwise interactions between points are mediated by a bond potential of multi-well type while multi-point interactions are associated with the volume change mediated by a hydrostatic strain potential. The hydrostatic potential can either be a quadratic function, delivering a linear force-strain relation, or a multi-well type that can be associated with the material degradation and cavitation. We frst show the well-posedness of the peridynamic formulation and that peridynamic evolutions exist in the Sobolev space H². We show that the fnite element approximations converge to the H² solutions uniformly as measured in the mean square norm. For linear continuous fnite elements, the convergence rate is shown to be C_tΔt + C_sh²∕ε², where ε is the size of the horizon, h is the mesh size, and Δ_t is the size of the time step. The constants C_t and C_s are independent of Δt and h and may depend on ε through the norm of the exact solution. We demonstrate the stability of the semi-discrete approximation. The stability of the fully discrete approximation is shown for the linearized peridynamic force. We present numerical simulations with the dynamic crack propagation that support the theoretical convergence rate.

关键词: Nonlocal fracture models, Peridynamic, State-based peridynamic, Numerical analysis, Finite element approximation

Abstract: We establish the a priori convergence rate for fnite element approximations of a class of nonlocal nonlinear fracture models. We consider state-based peridynamic models where the force at a material point is due to both the strain between two points and the change in volume inside the domain of the nonlocal interaction. The pairwise interactions between points are mediated by a bond potential of multi-well type while multi-point interactions are associated with the volume change mediated by a hydrostatic strain potential. The hydrostatic potential can either be a quadratic function, delivering a linear force-strain relation, or a multi-well type that can be associated with the material degradation and cavitation. We frst show the well-posedness of the peridynamic formulation and that peridynamic evolutions exist in the Sobolev space H². We show that the fnite element approximations converge to the H² solutions uniformly as measured in the mean square norm. For linear continuous fnite elements, the convergence rate is shown to be C_tΔt + C_sh²∕ε², where ε is the size of the horizon, h is the mesh size, and Δ_t is the size of the time step. The constants C_t and C_s are independent of Δt and h and may depend on ε through the norm of the exact solution. We demonstrate the stability of the semi-discrete approximation. The stability of the fully discrete approximation is shown for the linearized peridynamic force. We present numerical simulations with the dynamic crack propagation that support the theoretical convergence rate.

Key words: Nonlocal fracture models, Peridynamic, State-based peridynamic, Numerical analysis, Finite element approximation

中图分类号:

Prashant K. Jha, Robert Lipton. Finite Element Convergence for State-Based Peridynamic Fracture Models[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 93-128.

参考文献

1. Agwai, A., Guven, I., Madenci, E.:Predicting crack propagation with peridynamics:a comparative study. Int. J. Fract. 171(1), 65-78 (2011)
2. Aksoylu, B., Unlu, Z.:Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces. SIAM J. Numer. Anal. 52, 653-677 (2014)
3. Bobaru, F., Hu, W.:The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials. Int. J. Fract. 176(2), 215-222 (2012)
4. Bobaru, F., Foster, J.T., Geubelle, P.H., Geubelle, P.H., Silling, S.A.:Handbook of Peridynamic Modeling. CRC, Boca Raton (2016)
5. Brenner, S., Scott, R.:The Mathematical Theory of Finite Element Methods, vol. 15, 3rd edn. Springer, Berlin (2007)
6. Brezis, H.:Analyse Fonctionnelle, Théorie et Application. Dunod, Paris (1983)
7. Chen, X., Gunzburger, M.:Continuous and discontinuous fnite element methods for a peridynamics model of mechanics. Comput. Methods Appl. Mech. Eng. 200(9), 1237-1250 (2011)
8. Demengel, F.:Functional Spaces for the Theory of Elliptic Partial Diferential Equations. Universitext, 1st edn. Springer, London (2012)
9. Du, Q.:An invitation to nonlocal modeling, analysis and computation. In:Proc. Int. Cong. of Math 2018, Rio de Janeiro, vol. 3, pp. 3523-3552 (2018a)
10. Du, Q.:Nonlocal Modeling, Analysis and Computation. NSF-CBMS Monograph. SIAM, Philadelphia (2018b)
11. Du, Q., Gunzburger, M., Lehoucq, R., Zhou, K.:Analysis of the volume-constrained peridynamic navier equation of linear elasticity. J. Elast. 113(2), 193-217 (2013a)
12. Du, Q., Ju, L., Tian, L., Zhou, K.:A posteriori error analysis of fnite element method for linear nonlocal difusion and peridynamic models. Math. Comput. 82(284), 1889-1922 (2013b)
13. Du, Q., Tian, L., Zhao, X.:A convergent adaptive fnite element algorithm for nonlocal difusion and peridynamic models. SIAM J. Numer. Anal. 51(2), 1211-1234 (2013c)
14. Emmrich, E., Lehoucq, R.B., Puhst, D.:Peridynamics:a nonlocal continuum theory. Meshfree Methods for Partial Diferential Equations VI, pp. 45-65. Springer, Berlin (2013)
15. Foster, J.T., Silling, S.A., Chen, W.:An energy based failure criterion for use with peridynamic states. Int. J. Multiscale Comput. Eng. 9(6), 675-688 (2011)
16. Gerstle, W., Sau, N., Silling, S.:Peridynamic modeling of concrete structures. Nuclear Eng. Des. 237(12), 1250-1258 (2007)
17. Ghajari, M., Iannucci, L., Curtis, P.:A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media. Comput. Methods Appl. Mech. Eng. 276, 431-452 (2014)
18. Guan, Q., Gunzburger, M.:Stability and accuracy of time-stepping schemes and dispersion relations for a nonlocal wave equation. Numer. Methods Part. Difer. Equ. 31(2), 500-516 (2015)
19. Ha, Y.D., Bobaru, F.:Studies of dynamic crack propagation and crack branching with peridynamics. Int. J. Fract. 162(1/2), 229-244 (2010)
20. Jha, P. K., Lipton, R.:Finite element approximation of nonlinear nonlocal models (2017). arXiv preprint arXiv:1710.07661
21. Jha, P.K., Lipton, R.:Numerical analysis of nonlocal fracture models in Hölder space. SIAM J. Numer. Anal. 56, 906-941 (2018a)
22. Jha, P.K., Lipton, R.:Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics. Int. J. Numer. Methods Eng. 114(13), 1389-1410 (2018b)
23. Jha, P. K., Lipton, R.:Small horizon limit of state based peridynamic models. In preparation (2019)
24. Karaa, S.:Stability and convergence of fully discrete fnite element schemes for the acoustic wave equation. J. Appl. Math. Comput. 40(1/2), 659-682 (2012)
25. Lipton, R.:Dynamic brittle fracture as a small horizon limit of peridynamics. J. Elast. 117, 21-50 (2014)
26. Lipton, R.:Cohesive dynamics and brittle fracture. J. Elast. 124(2), 143-191 (2016)
27. Lipton, R., Silling, S., Lehoucq, R.:Complex fracture nucleation and evolution with nonlocal elastodynamics (2016). arXiv preprint arXiv:1602.00247
28. Lipton, R., Said, E., Jha, P.:Free damage propagation with memory. J. Elast. 133(2), 129-153 (2018a)
29. Lipton, R., Said, E., Jha, P.K.:Dynamic brittle fracture from nonlocal double-well potentials:a statebased model. In:Handbook of Nonlocal Continuum Mechanics for Materials and Structures, pp. 1-27 (2018b)
30. Littlewood, D. J.:Simulation of dynamic fracture using peridynamics, fnite element modeling, and contact. In:Proceedings of the ASME 2010 International Mechanical Engineering Congress and Exposition (IMECE) (2010)
31. Macek, R.W., Silling, S.A.:Peridynamics via fnite element analysis. Finite Elem. Anal. Des. 43(15), 1169-1178 (2007)
32. Mengesha, T., Du, Q.:On the variational limit of a class of nonlocal functionals related to peridynamics. Nonlinearity 28(11), 3999 (2015)
33. Ren, B., Wu, C., Askari, E.:A 3d discontinuous galerkin fnite element method with the bond-based peridynamics model for dynamic brittle failure analysis. Int. J. Impact Eng. 99, 14-25 (2017)
34. Silling, S.A.:Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48(1), 175-209 (2000)
35. Silling, S.A., Bobaru, F.:Peridynamic modeling of membranes and fbers. Int. J. Non-Linear Mech. 40(2), 395-409 (2005)
36. Silling, S.A., Lehoucq, R.B.:Convergence of peridynamics to classical elasticity theory. J. Elast. 93(1), 13-37 (2008)
37. Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E.:Peridynamic states and constitutive modeling. J. Elast. 88(2), 151-184 (2007)
38. Silling, S., Weckner, O., Askari, E., Bobaru, F.:Crack nucleation in a peridynamic solid. Int. J. Fract. 162(1/2), 219-227 (2010)
39. Tian, X., Du, Q.:Asymptotically compatible schemes and applications to robust discretization of nonlocal models. SIAM J. Numer. Anal. 52, 1641-1665 (2014)
40. Weckner, O., Abeyaratne, R.:The efect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53(3), 705-728 (2005)