Abstract
Typical implementations of peridynamics use a constant or tapered micromodulus (or influence) function, the choice of which has been shown to have a large impact on the dispersion relation. In this work, a method for computing micromodulus function values at discretized points within a node’s horizon is presented for linearized peridynamics. The technique involves constructing a system of equations representing the desired dispersion relation and solving for the micromodulus function coefficients at discretized node locations. Both 1D and 2D formulations are presented. A straightforward implementation of the method results in negative coefficients, which improve wave propagation accuracy, but results in unstable solutions of fracture problems using a bond-breakage scheme. Two methods for addressing this issue are discussed: A hybrid method that uses a constant micromodulus function after damage has occurred at a node, and a constrained solution that results in only positive coefficients. The dispersion properties of the method are examined in detail, including the numerical anisotropy in 2D. Finally, results for wave propagation in 1D and 2D, static fracture, and dynamic fracture are given.
Similar content being viewed by others
References
Azdoud Y, Han F, Lubineau G (2014) The morphing method as a flexible tool for adaptive local/non-local simulation of static fracture. Comput Mech 54(3):711–722
Bažant ZP, Luo W, Chau VT, Bessa MA (2016) Wave dispersion and basic concepts of peridynamics compared to classical nonlocal damage models. J Appl Mech 83(11):111004
Bobaru F, Duangpanya M (2010) The peridynamic formulation for transient heat conduction. Int J Heat Mass Transf 53(19-20):4047–4059
Butt SN, Timothy JJ, Meschke G (2017) Wave dispersion and propagation in state-based peridynamics. Comput Mech 60(5):725–738
Chen YM (1975) Numerical computation of dynamic stress intensity factors by a lagrangian finite-difference method (the hemp code). Eng Fract Mech 7(4):653–660
Diyaroglu C, Oterkus E, Oterkus S, Madenci E (2015) Peridynamics for bending of beams and plates with transverse shear deformation. Int J Solids Struct 69:152–168
Foster JT, Silling SA, Chen WW (2010) Viscoplasticity using peridynamics. Int J Numer Methods Eng 81(10):1242–1258
Foster JT, Silling SA, Chen W (2011) An energy based failure criterion for use with peridynamic states. Int J Multiscale Comput Eng 9(6):675–688
Gerstle W, Silling S, Read D, Tewary V, Lehoucq R (2008) Peridynamic simulation of electromigration. Computers Materials & Continua 8(2):75–92
Gu X, Zhang Q, Huang D, Yv Y (2016) Wave dispersion analysis and simulation method for concrete shpb test in peridynamics. Eng Fract Mech 160:124–137
Ha Y, Bobaru F (2009) Traction boundary conditions in peridynamics: a convergence study. Tech. rep., Technical report, Department of Engineering Mechanics, University of Nebraska–Lincoln, Lincoln, Nebraska
Ha YD, Bobaru F (2010) Studies of dynamic crack propagation and crack branching with peridynamics. Int J Fract 162(1-2):229–244
Han F, Lubineau G, Azdoud Y (2016) Adaptive coupling between damage mechanics and peridynamics: a route for objective simulation of material degradation up to complete failure. J Mech Phys Solids 94:453–472
Harari I (1997) Reducing spurious dispersion, anisotropy and reflection in finite element analysis of time-harmonic acoustics. Comput Methods Appl Mech Eng 140(1-2):39–58
Harari I, Turkel E (1995) Accurate finite difference methods for time-harmonic wave propagation. J Comput Phys 119(2):252–270
Jabakhanji R, Mohtar RH (2015) A peridynamic model of flow in porous media. Adv Water Resour 78:22–35
Kalthoff J (1973) On the propagation direction of bifurcated cracks. In: Proceedings of an international conference on dynamic crack propagation. Springer, pp 449–458
Liu ZL, Menouillard T, Belytschko T (2011) An XFEM/Spectral element method for dynamic crack propagation. Int J Fract 169(2):183–198
Madenci E, Oterkus E (2014) Peridynamic theory and its applications, vol 17. Springer, Berlin
Oterkus S, Madenci E, Agwai A (2014) Fully coupled peridynamic thermomechanics. J Mech Phys Solids 64:1–23
Ouchi H, Katiyar A, York J, Foster JT, Sharma MM (2015) A fully coupled porous flow and geomechanics model for fluid driven cracks: a peridynamics approach. Comput Mech 55(3):561–576
Seleson P (2014) Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations. Comput Methods Appl Mech Eng 282:184–217
Seleson P, Littlewood DJ (2016) Convergence studies in meshfree peridynamic simulations. Computers & Mathematics with Applications 71(11):2432–2448
Seleson P, Parks M (2011) On the role of the influence function in the peridynamic theory. Int J Multiscale Comput Eng 9(6):689–706
Seleson P, Beneddine S, Prudhomme S (2013) A force-based coupling scheme for peridynamics and classical elasticity. Comput Mater Sci 66:34–49
Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175– 209
Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83(17-18):1526–1535
Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88(2):151–184
Tam CK, Webb JC (1993) Dispersion-relation-preserving finite difference schemes for computational acoustics. J Comput Phys 107(2):262–281
Weckner O, Abeyaratne R (2005) The effect of long-range forces on the dynamics of a bar. J Mech Phys Solids 53(3):705–728
Weckner O, Emmrich E (2005) Numerical simulation of the dynamics of a nonlocal, inhomogeneous, infinite bar. J Comput Appl Mech 6(2):311–319
Weckner O, Silling SA (2011) Determination of nonlocal constitutive equations from phonon dispersion relations. Int J Multiscale Comput Eng 9(6):623–634
Wildman RA, Gazonas GA (2014) A finite difference-augmented peridynamics method for reducing wave dispersion. Int J Fract 190(1-2):39–52
Wildman RA, OGrady JT, Gazonas GA (2017) A hybrid multiscale finite element/peridynamics method. Int J Fract 207(1):41–53
Zimmermann M (2005) A continuum theory with long-range forces for solids. PhD thesis, Massachusetts Institute of Technology
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wildman, R.A. Discrete Micromodulus Functions for Reducing Wave Dispersion in Linearized Peridynamics. J Peridyn Nonlocal Model 1, 56–73 (2019). https://doi.org/10.1007/s42102-018-0001-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42102-018-0001-0