# American Institute of Mathematical Sciences

March  2021, 26(3): 1675-1710. doi: 10.3934/dcdsb.2020178

## Finite element approximation of nonlocal dynamic fracture models

 1 Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA 2 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

* Corresponding author: P. K. Jha

Received  May 2019 Revised  January 2020 Published  March 2021 Early access  June 2020

Fund Project: This material is based upon work supported by the U. S. Army Research Laboratory and the U. S. Army Research Office under contract/grant number W911NF1610456

In this work we estimate the convergence rate for time stepping schemes applied to nonlocal dynamic fracture modeling. Here we use the nonlocal formulation given by the bond based peridynamic equation of motion. We begin by establishing the existence of $H^2$ peridynamic solutions over any finite time interval. For this model the gradients can become large and steep slopes appear and localize when the non-locality of the model tends to zero. In this treatment spatial approximation by finite elements are used. We consider the central-difference scheme for time discretization and linear finite elements for discretization in the spatial variable. The fully discrete scheme is shown to converge to the actual $H^2$ solution in the mean square norm at the rate $C_t\Delta t +C_s h^2/\epsilon^2$. Here $h$ is the mesh size, $\epsilon$ is the length scale of nonlocal interaction and $\Delta t$ is the time step. The constants $C_t$ and $C_s$ are independent of $\Delta t$, and $h$. In the absence of nonlinearity a CFL like condition for the energy stability of the central difference time discretization scheme is developed. As an example we consider Plexiglass and compute constants in the a-priori error bound.

Citation: P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178
##### References:
 [1] A. Agwai, I. Guven and E. Madenci, Predicting crack propagation with peridynamics: A comparative study, International Journal of Fracture, 171 (2011), 65-78.  doi: 10.1007/s10704-011-9628-4. [2] B. Aksoylu and Z. Unlu, Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces, SIAM Journal on Numerical Analysis, 52 (2014), 653-677.  doi: 10.1137/13092407X. [3] B. Aksoylu and M. L. Parks, Variational theory and domain decomposition for nonlocal problems, Applied Mathematics and Computation, 217 (2011), 6498-6515.  doi: 10.1016/j.amc.2011.01.027. [4] G. A. Baker, Error estimates for finite element methods for second order hyperbolic equations, SIAM Journal on Numerical Analysis, 13 (1976), 564-576.  doi: 10.1137/0713048. [5] F. Bobaru and W. Hu, The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials, International Journal of Fracture, 176 (2012), 215-222.  doi: 10.1007/s10704-012-9725-z. [6] S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, 3$^rd$ edition, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0. [7] H. Brezis, Analyse fonctionnelle, Théorie et applications, in Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. [8] Z. Chen, D. Bakenhus and F. Bobaru, A constructive peridynamic kernel for elasticity, Computer Methods in Applied Mechanics and Engineering, 311 (2016), 356-373.  doi: 10.1016/j.cma.2016.08.012. [9] K. Dayal, Leading-order nonlocal kinetic energy in peridynamics for consistent energetics and wave dispersion, Journal of the Mechanics and Physics of Solids, 105 (2017), 235-253.  doi: 10.1016/j.jmps.2017.05.002. [10] K. Dayal and K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics, Journal of the Mechanics and Physics of Solids, 54 (2006), 1811-1842.  doi: 10.1016/j.jmps.2006.04.001. [11] P. Diehl, R. Lipton and M. Schweitzer, Numerical verification of a bond-based softening peridynamic model for small displacements: Deducing material parameters from classical linear theory., preprint, Institut für Numerische Simulation, (2016). [12] Q. Du, L. Tian and X. Zhao, A convergent adaptive finite element algorithm for nonlocal diffusion and peridynamic models, SIAM Journal on Numerical Analysis, 51 (2013), 1211-1234.  doi: 10.1137/120871638. [13] Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory, ESAIM: Mathematical Modelling and Numerical Analysis, 45 (2011), 217-234.  doi: 10.1051/m2an/2010040. [14] E. Emmrich, R. B. Lehoucq and D. Puhst, Peridynamics: A nonlocal continuum theory, in Meshfree Methods for Partial Differential Equations VI, Springer, Heidelberg, 2013, 45–65. doi: 10.1007/978-3-642-32979-1_3. [15] J. T. Foster, S. A. Silling and W. Chen, An energy based failure criterion for use with peridynamic states, International Journal for Multiscale Computational Engineering, 9 (2011), 675-688.  doi: 10.1615/IntJMultCompEng.2011002407. [16] F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, 2012. doi: 10.1007/978-1-4471-2807-6. [17] W. Gerstle, N. Sau and S. Silling, Peridynamic modeling of concrete structures, Nuclear Engineering and Design, 237 (2007), 1250-1258.  doi: 10.1016/j.nucengdes.2006.10.002. [18] M. Ghajari, L. Iannucci and P. Curtis, A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media, Computer Methods in Applied Mechanics and Engineering, 276 (2014), 431-452.  doi: 10.1016/j.cma.2014.04.002. [19] M. J. Grote and D. Schötzau, Optimal error estimates for the fully discrete interior penalty DG method for the wave equation, Journal of Scientific Computing, 40 (2009), 257-272.  doi: 10.1007/s10915-008-9247-z. [20] Q. Guan and M. Gunzburger, Stability and accuracy of time-stepping schemes and dispersion relations for a nonlocal wave equation, Numerical Methods for Partial Differential Equations, 31 (2015), 500-516.  doi: 10.1002/num.21931. [21] Y. D. Ha and F. Bobaru, Studies of dynamic crack propagation and crack branching with peridynamics, International Journal of Fracture, 162 (2010), 229-244.  doi: 10.1007/s10704-010-9442-4. [22] D. Huang, G. Lu and P. Qiao, An improved peridynamic approach for quasi-static elastic deformation and brittle fracture analysis, International Journal of Mechanical Sciences, 94 (2015), 111-122.  doi: 10.1016/j.ijmecsci.2015.02.018. [23] P. K. Jha and R. Lipton, Numerical analysis of nonlocal fracture models in Hölder space, SIAM Journal on Numerical Analysis, 56 (2018), 906-941.  doi: 10.1137/17M1112236. [24] P. K. Jha and R. Lipton, Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics, International Journal for Numerical Methods in Engineering, 114 (2018), 1389-1410.  doi: 10.1002/nme.5791. [25] S. Karaa, Stability and convergence of fully discrete finite element schemes for the acoustic wave equation, Journal of Applied Mathematics and Computing, 40 (2012), 659-682.  doi: 10.1007/s12190-012-0558-8. [26] Q. V. Le, W. K. Chan and J. Schwartz, A two-dimensional ordinary, state-based peridynamic model for linearly elastic solids, International Journal for Numerical Methods in Engineering, 98 (2014), 547-567.  doi: 10.1002/nme.4642. [27] R. Lipton, Dynamic brittle fracture as a small horizon limit of peridynamics, Journal of Elasticity, 117 (2014), 21-50.  doi: 10.1007/s10659-013-9463-0. [28] R. Lipton, Cohesive dynamics and brittle fracture, Journal of Elasticity, 124 (2016), 143-191.  doi: 10.1007/s10659-015-9564-z. [29] R. Lipton, E. Said and P. K. Jha, Dynamic brittle fracture from nonlocal double-well potentials: A state-based model, in Handbook of Nonlocal Continuum Mechanics for Materials and Structures, Springer, Cham, 2018, 1–27. doi: 10.1007/978-3-319-22977-5_33-1. [30] R. Lipton, S. Silling and R. Lehoucq, Complex fracture nucleation and evolution with nonlocal elastodynamics, Journal of Peridynamics and Nonlocal Modeling, 1 (2019), 122-130.  doi: 10.1007/s42102-019-00010-0. [31] D. J. Littlewood, Simulation of dynamic fracture using peridynamics, finite element modeling, and contact, in Proceedings of the ASME 2010 International Mechanical Engineering Congress and Exposition (IMECE), Vol. 9, Vancouver, British Columbia, Canada, 2010,209–217. doi: 10.1115/IMECE2010-40621. [32] R. W. Macek and S. A. Silling, Peridynamics via finite element analysis, Finite Elements in Analysis and Design, 43 (2007), 1169-1178.  doi: 10.1016/j.finel.2007.08.012. [33] T. Mengesha and Q. Du, Analysis of a scalar peridynamic model with a sign changing kernel, Discrete Contin. Dynam. Systems B, 18 (2013), 1415-1437.  doi: 10.3934/dcdsb.2013.18.1415. [34] T. Mengesha and Q. Du, On the variational limit of a class of nonlocal functionals related to peridynamics, Nonlinearity, 28 (2015), 3999-4035.  doi: 10.1088/0951-7715/28/11/3999. [35] J. O'Grady and J. Foster, Peridynamic plates and flat shells: A non-ordinary, state-based model, International Journal of Solids and Structures, 51 (2014), 4572-4579.  doi: 10.1016/j.ijsolstr.2014.09.003. [36] S. Silling, O. Weckner, E. Askari and F. Bobaru, Crack nucleation in a peridynamic solid, International Journal of Fracture, 162 (2010), 219-227.  doi: 10.1007/s10704-010-9447-z. [37] S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, Journal of the Mechanics and Physics of Solids, 48 (2000), 175-209.  doi: 10.1016/S0022-5096(99)00029-0. [38] S. A. Silling and R. B. Lehoucq, Convergence of peridynamics to classical elasticity theory, Journal of Elasticity, 93 (2008), 13-37.  doi: 10.1007/s10659-008-9163-3. [39] M. Taylor and D. J. Steigmann, A two-dimensional peridynamic model for thin plates, Mathematics and Mechanics of Solids, 20 (2015), 998-1010.  doi: 10.1177/1081286513512925. [40] X. Tian and Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models, SIAM Journal on Numerical Analysis, 52 (2014), 1641-1665.  doi: 10.1137/130942644.

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##### References:
 [1] A. Agwai, I. Guven and E. Madenci, Predicting crack propagation with peridynamics: A comparative study, International Journal of Fracture, 171 (2011), 65-78.  doi: 10.1007/s10704-011-9628-4. [2] B. Aksoylu and Z. Unlu, Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces, SIAM Journal on Numerical Analysis, 52 (2014), 653-677.  doi: 10.1137/13092407X. [3] B. Aksoylu and M. L. Parks, Variational theory and domain decomposition for nonlocal problems, Applied Mathematics and Computation, 217 (2011), 6498-6515.  doi: 10.1016/j.amc.2011.01.027. [4] G. A. Baker, Error estimates for finite element methods for second order hyperbolic equations, SIAM Journal on Numerical Analysis, 13 (1976), 564-576.  doi: 10.1137/0713048. [5] F. Bobaru and W. Hu, The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials, International Journal of Fracture, 176 (2012), 215-222.  doi: 10.1007/s10704-012-9725-z. [6] S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, 3$^rd$ edition, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0. [7] H. Brezis, Analyse fonctionnelle, Théorie et applications, in Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. [8] Z. Chen, D. Bakenhus and F. Bobaru, A constructive peridynamic kernel for elasticity, Computer Methods in Applied Mechanics and Engineering, 311 (2016), 356-373.  doi: 10.1016/j.cma.2016.08.012. [9] K. Dayal, Leading-order nonlocal kinetic energy in peridynamics for consistent energetics and wave dispersion, Journal of the Mechanics and Physics of Solids, 105 (2017), 235-253.  doi: 10.1016/j.jmps.2017.05.002. [10] K. Dayal and K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics, Journal of the Mechanics and Physics of Solids, 54 (2006), 1811-1842.  doi: 10.1016/j.jmps.2006.04.001. [11] P. Diehl, R. Lipton and M. Schweitzer, Numerical verification of a bond-based softening peridynamic model for small displacements: Deducing material parameters from classical linear theory., preprint, Institut für Numerische Simulation, (2016). [12] Q. Du, L. Tian and X. Zhao, A convergent adaptive finite element algorithm for nonlocal diffusion and peridynamic models, SIAM Journal on Numerical Analysis, 51 (2013), 1211-1234.  doi: 10.1137/120871638. [13] Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory, ESAIM: Mathematical Modelling and Numerical Analysis, 45 (2011), 217-234.  doi: 10.1051/m2an/2010040. [14] E. Emmrich, R. B. Lehoucq and D. Puhst, Peridynamics: A nonlocal continuum theory, in Meshfree Methods for Partial Differential Equations VI, Springer, Heidelberg, 2013, 45–65. doi: 10.1007/978-3-642-32979-1_3. [15] J. T. Foster, S. A. Silling and W. Chen, An energy based failure criterion for use with peridynamic states, International Journal for Multiscale Computational Engineering, 9 (2011), 675-688.  doi: 10.1615/IntJMultCompEng.2011002407. [16] F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, 2012. doi: 10.1007/978-1-4471-2807-6. [17] W. Gerstle, N. Sau and S. Silling, Peridynamic modeling of concrete structures, Nuclear Engineering and Design, 237 (2007), 1250-1258.  doi: 10.1016/j.nucengdes.2006.10.002. [18] M. Ghajari, L. Iannucci and P. Curtis, A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media, Computer Methods in Applied Mechanics and Engineering, 276 (2014), 431-452.  doi: 10.1016/j.cma.2014.04.002. [19] M. J. Grote and D. Schötzau, Optimal error estimates for the fully discrete interior penalty DG method for the wave equation, Journal of Scientific Computing, 40 (2009), 257-272.  doi: 10.1007/s10915-008-9247-z. [20] Q. Guan and M. Gunzburger, Stability and accuracy of time-stepping schemes and dispersion relations for a nonlocal wave equation, Numerical Methods for Partial Differential Equations, 31 (2015), 500-516.  doi: 10.1002/num.21931. [21] Y. D. Ha and F. Bobaru, Studies of dynamic crack propagation and crack branching with peridynamics, International Journal of Fracture, 162 (2010), 229-244.  doi: 10.1007/s10704-010-9442-4. [22] D. Huang, G. Lu and P. Qiao, An improved peridynamic approach for quasi-static elastic deformation and brittle fracture analysis, International Journal of Mechanical Sciences, 94 (2015), 111-122.  doi: 10.1016/j.ijmecsci.2015.02.018. [23] P. K. Jha and R. Lipton, Numerical analysis of nonlocal fracture models in Hölder space, SIAM Journal on Numerical Analysis, 56 (2018), 906-941.  doi: 10.1137/17M1112236. [24] P. K. Jha and R. Lipton, Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics, International Journal for Numerical Methods in Engineering, 114 (2018), 1389-1410.  doi: 10.1002/nme.5791. [25] S. Karaa, Stability and convergence of fully discrete finite element schemes for the acoustic wave equation, Journal of Applied Mathematics and Computing, 40 (2012), 659-682.  doi: 10.1007/s12190-012-0558-8. [26] Q. V. Le, W. K. Chan and J. Schwartz, A two-dimensional ordinary, state-based peridynamic model for linearly elastic solids, International Journal for Numerical Methods in Engineering, 98 (2014), 547-567.  doi: 10.1002/nme.4642. [27] R. Lipton, Dynamic brittle fracture as a small horizon limit of peridynamics, Journal of Elasticity, 117 (2014), 21-50.  doi: 10.1007/s10659-013-9463-0. [28] R. Lipton, Cohesive dynamics and brittle fracture, Journal of Elasticity, 124 (2016), 143-191.  doi: 10.1007/s10659-015-9564-z. [29] R. Lipton, E. Said and P. K. Jha, Dynamic brittle fracture from nonlocal double-well potentials: A state-based model, in Handbook of Nonlocal Continuum Mechanics for Materials and Structures, Springer, Cham, 2018, 1–27. doi: 10.1007/978-3-319-22977-5_33-1. [30] R. Lipton, S. Silling and R. Lehoucq, Complex fracture nucleation and evolution with nonlocal elastodynamics, Journal of Peridynamics and Nonlocal Modeling, 1 (2019), 122-130.  doi: 10.1007/s42102-019-00010-0. [31] D. J. Littlewood, Simulation of dynamic fracture using peridynamics, finite element modeling, and contact, in Proceedings of the ASME 2010 International Mechanical Engineering Congress and Exposition (IMECE), Vol. 9, Vancouver, British Columbia, Canada, 2010,209–217. doi: 10.1115/IMECE2010-40621. [32] R. W. Macek and S. A. Silling, Peridynamics via finite element analysis, Finite Elements in Analysis and Design, 43 (2007), 1169-1178.  doi: 10.1016/j.finel.2007.08.012. [33] T. Mengesha and Q. Du, Analysis of a scalar peridynamic model with a sign changing kernel, Discrete Contin. Dynam. Systems B, 18 (2013), 1415-1437.  doi: 10.3934/dcdsb.2013.18.1415. [34] T. Mengesha and Q. Du, On the variational limit of a class of nonlocal functionals related to peridynamics, Nonlinearity, 28 (2015), 3999-4035.  doi: 10.1088/0951-7715/28/11/3999. [35] J. O'Grady and J. Foster, Peridynamic plates and flat shells: A non-ordinary, state-based model, International Journal of Solids and Structures, 51 (2014), 4572-4579.  doi: 10.1016/j.ijsolstr.2014.09.003. [36] S. Silling, O. Weckner, E. Askari and F. Bobaru, Crack nucleation in a peridynamic solid, International Journal of Fracture, 162 (2010), 219-227.  doi: 10.1007/s10704-010-9447-z. [37] S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, Journal of the Mechanics and Physics of Solids, 48 (2000), 175-209.  doi: 10.1016/S0022-5096(99)00029-0. [38] S. A. Silling and R. B. Lehoucq, Convergence of peridynamics to classical elasticity theory, Journal of Elasticity, 93 (2008), 13-37.  doi: 10.1007/s10659-008-9163-3. [39] M. Taylor and D. J. Steigmann, A two-dimensional peridynamic model for thin plates, Mathematics and Mechanics of Solids, 20 (2015), 998-1010.  doi: 10.1177/1081286513512925. [40] X. Tian and Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models, SIAM Journal on Numerical Analysis, 52 (2014), 1641-1665.  doi: 10.1137/130942644.
Two-point potential $W^\epsilon(S,y - x)$ as a function of strain $S$ for fixed $y - x$
Nonlocal force $\partial_S W^\epsilon(S,y - x)$ as a function of strain $S$ for fixed $y - x$. Second derivative of $W^\epsilon(S,y-x)$ is zero at $\pm \bar{r}/\sqrt{|y -x|}$
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