Abstract
We present a new formulation of a peridynamic model for brittle fracture that incorporates a properly defined bond-breaking rule which leads to a dynamic system of time-dependent differential integral equations having both spatial nonlocal/nonlinear interactions and temporal memory/history dependence. The dynamic system is shown to be well-posed through rigorous mathematical analysis. Its effectiveness in simulating crack propagation in a two dimensional brittle material is also demonstrated.
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Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48(1), 175–209 (2000)
Askari, E., Bobaru, F., Lehoucq, R.B., Parks, M.L., Silling, S.A., Weckner, O.: Peridynamics for multiscale materials modeling. J. Phys. Conf. Ser. 125, 012078 (2008)
Aguiar, A.R., Fosdick, R.: A constitutive model for a linearly elastic peridynamic body. Math. Mech. Solids 19(5), 502–523 (2014)
Silling, S., Weckner, O., Askari, E., Bobaru, F.: Crack nucleation in a peridynamic solid. Int. J. Fract. 162(1), 219–227 (2010)
Oterkus, E., Madenci, E.: Peridynamic analysis of fiber-reinforced composite materials. J. Mech. Mater. Struct. 7(1), 45–84 (2012)
Bobaru, F., Duangpanya, M.: The peridynamic formulation for transient heat conduction. Int. J. Heat Mass Transf. 53(19), 4047–4059 (2010)
Ha, Y.D., Bobaru, F.: Studies of dynamic crack propagation and crack branching with peridynamics. Int. J. Fract. 162(1–2), 229–244 (2010)
Hu, W., Ha, Y.D., Bobaru, F.: Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput. Methods Appl. Mech. Eng. 217, 247–261 (2012)
Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elast. 88(2), 151–184 (2007)
Aksoylu, B., Parks, M.L.: Variational theory and domain decomposition for nonlocal problems. Appl. Math. Comput. 217(14), 6498–6515 (2011)
Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J.: Nonlocal Diffusion Problems. Mathematical Surveys and Monographs, vol. 165. Am. Math. Soc., Providence (2010)
Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23, 493–540 (2013)
Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 56, 676–696 (2012)
Mengesha, T., Du, Q.: The bond-based peridynamic system with Dirichlet type volume constraint. Proc. R. Soc. Edinb. A 144, 161–186 (2014)
Mengesha, T., Du, Q.: Nonlocal constrained value problems for a linear peridynamic Navier equation. J. Elast. 116(1), 27–51 (2014)
Emmrich, E., Weckner, O.: On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity. Commun. Math. Sci. 5(4), 851–864 (2007)
Emmrich, E., Puhst, D.: Well-posedness of the peridynamic model with Lipschitz continuous pairwise force function. Commun. Math. Sci. 11(4), 1039–1049 (2013)
Lipton, R.: Dynamic brittle fracture as a small horizon limit of peridynamics. J. Elast. 117(1), 21–50 (2014)
Mengesha, T., Du, Q.: Characterization of function spaces of vector fields and an application in nonlinear peridynamics. Nonlinear Anal. 140, 82–111 (2016)
Tian, X., Du, Q.: A class of high order nonlocal operators. Arch. Ration. Mech. Anal. 222(3), 1521–1553 (2016)
Emmrich, E., Puhst, D.: A short note on modeling damage in peridynamics. J. Elast. 123(2), 245–252 (2016)
Bobaru, F., Zhang, G.: Why do cracks branch? A peridynamic investigation of dynamic brittle fracture. Int. J. Fract. 196(1–2), 59–98 (2015)
Parks, M., Plimpton, S., Lehoucq, R., Silling, S.A.: Peridynamics with LAMMPS: a user guide. Sandia National Laboratory Report, SAND2008-0135, Albuquerque, New Mexico (2008)
Mengesha, T., Du, Q.: Analysis of a scalar nonlocal peridynamic model with a sign changing kernel. Discrete Contin. Dyn. Syst., Ser. B 18(5), 1415–1437 (2013)
Silling, S.A., Askari, E.: A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83(17), 1526–1535 (2005)
Fosdick, R., Royer-Carfagni, G.: The constraint of local injectivity in linear elasticity theory. Proc. R. Soc. Lond. A, Math. Phys. Eng. Sci. 457, 2167–2187 (2001)
Hale, J.K.: Functional differential equations. In: Analytic Theory of Differential Equations, pp. 9–22. Springer, Berlin (1971)
Wu, J.: Theory and Applications of Partial Functional Differential Equations, vol. 119. Springer, Berlin (2012)
Gajewski, H., Gröger, K., Zacharias, K.: Nonlinear Operator Equations and Operator Differential Equations, p. 336. Mir, Moscow (1978)
Du, Q.: Local limits and asymptotically compatible discretizations. In: Handbook of Peridynamic Modeling, pp. 87–108. CRC Press, Boca Raton (2016).
Tian, X., Du, Q.: Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J. Numer. Anal. 51, 3458–3482 (2013)
Tian, X., Du, Q.: Asymptotically compatible schemes and applications to robust discretization of nonlocal models. SIAM J. Numer. Anal. 52, 1641–1665 (2014)
Parks, M., Littlewood, D., Mitchell, J., Silling, S.A.: Peridigm users’ guide v1.0.0. Sandia Rep. 2012-7800, Sandia National Laboratories, Albuquerque, New Mexico (2012)
Tao, Y., Tian, X., Du, Q.: Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations. Appl. Math. Comput. 305, 282–298 (2017)
Aubin, J.P., Cellina, A.: Differential Inclusions: Set-Valued Maps and Viability Theory. Springer, New York (1984)
Deimling, K.: Multivalued Differential Equations, vol. 1. de Gruyter, Berlin (1992)
Acknowledgements
We thank Dr. Stewart Silling, Prof. Florin Bobaru and Dr. Pablo Seleson for their helpful discussions on the subject. We also thank Prof. Robert Lipton for bringing [21] to our attention.
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Du, Q., Tao, Y. & Tian, X. A Peridynamic Model of Fracture Mechanics with Bond-Breaking. J Elast 132, 197–218 (2018). https://doi.org/10.1007/s10659-017-9661-2
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DOI: https://doi.org/10.1007/s10659-017-9661-2
Keywords
- Peridynamics
- Fracture
- Bond-breaking
- Well-posedness
- Energy decay
- Nonlocal model
- Memory effect
- Crack propagation
- Convergence of numerical simulations