×

Temporally stabilized peridynamics methods for shocks in solids. (English) Zbl 07492681

Summary: The computational methods for the shocks modeling would face two major challenges: (1) the severe damage with large deformations and (2) the intermittent waves. Peridynamics (PD) takes the integral form of its governing equation and shows exceeding advantages in modeling large deformation and severe damage. On the other hand, the propagation of intermittent wave within the PD based numerical system often experiences oscillatory instability. It can be attributed to the instability in time domain and the zero energy mode. Aiming for addressing such issues, the temporally stabilized PD methods are proposed in the present work. The stabilization force component is introduced and the general framework of stabilized PD methods is established. The formulation of the corresponding force state is proposed based on the features of the spurious oscillations. The case studies indicate that the stabilized PD methods are capable of effectively suppressing the nonphysical oscillations and are well-suited for the bond-based as well as the state-based PD formulations.

MSC:

74-XX Mechanics of deformable solids
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Amani, J.; Oterkus, E.; Areias, P.; Zi, G.; Nguyen-Thoi, T.; Rabczuk, T., A non-ordinary state-based peridynamics formulation for thermoplastic fracture, Int J Impact Eng, 87, 1, 83-94 (2016)
[2] Baek, J.; Chen, JS; Zhou, G.; Arnett, KP; Hillman, MC; Hegemier, G.; Hardesty, S., A semi-Lagrangian reproducing kernel particle method with particle-based shock algorithm for explosive welding simulation, Comput Mech, 67, 6, 1601-1627 (2021) · Zbl 1467.74097
[3] Barenblatt, GI, The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks, J Appl Math Mech, 23, 3, 622-636 (1959) · Zbl 0095.39202
[4] Bažant, ZP; Luo, W.; Chau, VT; Bessa, MA, Wave dispersion and basic concepts of peridynamics compared to classical nonlocal damage models, J Appl Mech, 83, 11, 111004 (2016)
[5] Belytschko, T.; Black, T., Elastic crack growth in finite elements with minimal remeshing, Int J Numer Methods Eng, 45, 5, 601-620 (1999) · Zbl 0943.74061
[6] Belytschko, T.; Liu, W.; Moran, B., Nonlinear finite elements for continua and structures (2000), Hoboken: Wiley, Hoboken · Zbl 0959.74001
[7] Bessa, M.; Foster, J.; Belytschko, T.; Liu, WK, A meshfree unification: reproducing kernel peridynamics, Comput Mech, 53, 6, 1251-1264 (2014) · Zbl 1398.74452
[8] Butt, SN; Timothy, JJ; Meschke, G., Wave dispersion and propagation in state-based peridynamics, Comput Mech, 60, 5, 725-738 (2017) · Zbl 1387.74007
[9] Caramana, EJ; Shashkov, MJ; Whalen, PP, Formulations of artificial viscosity for multi-dimensional shock wave computations, J Comput Phys, 144, 1, 70-97 (1998) · Zbl 1392.76041
[10] Dukowicz JK (1985) A general, non-iterative Riemann solver for Godunov’s method. J Comput Phys 61(1):119-137 · Zbl 0629.76074
[11] Foster, JT; Xu, X., A generalized, ordinary, finite deformation constitutive correspondence model for peridynamics, Int J Solids Struct, 141, 245-253 (2018)
[12] Foster, JT; Silling, SA; Chen, WW, Viscoplasticity using peridynamics, Int J Numer Methods Eng, 81, 10, 1242-1258 (2010) · Zbl 1183.74035
[13] Gerstle, W.; Sau, N.; Aguilera, E., Micropolar peridynamic constitutive model for concrete (2007), Toronto: IASMiRT, Toronto
[14] Gibbs JW (1899) Fourier’s series. Nature 59(1539):606 · JFM 30.0240.04
[15] Godunov, SK, A difference scheme for numerical computation of discontinuous solutions of equations in fluid dynamics, Math Sb, 47, 271-306 (1959) · Zbl 0171.46204
[16] Gu, X.; Zhang, Q.; Huang, D.; Yv, Y., Wave dispersion analysis and simulation method for concrete shpb test in peridynamics, Eng Fract Mech, 160, 124-137 (2016)
[17] Han, F.; Liu, S.; Lubineau, G., A dynamic hybrid local/nonlocal continuum model for wave propagation, Comput Mech, 67, 385-407 (2020) · Zbl 07360509
[18] Harten A, Engquist B, Osher S, Chakravarthy SR (1987) Uniformly high order accurate essentially non-oscillatory schemes, iii. In: Yousuff Hussaini M, van Leer B, Van Rosendale J (eds) Upwind and high-resolution schemes, pp 218-290 · Zbl 0652.65067
[19] Hu, W.; Ha, YD; Bobaru, F., Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites, Comput Methods Appl Mech Eng, 217-220, 4, 247-261 (2012) · Zbl 1253.74008
[20] Kilic, B.; Agwai, A.; Madenci, E., Peridynamic theory for progressive damage prediction in center-cracked composite laminates, Compos Struct, 90, 2, 141-151 (2009)
[21] Landshoff R (1955) A numerical method for treating fluid flow in the presence of shocks. Technical report, Los Alamos National Lab NM
[22] Lapidus, L.; Pinder, GF, Numerical solution of partial differential equations in science and engineering (2011), Hoboken: Wiley, Hoboken · Zbl 0584.65056
[23] Madenci, E.; Oterkus, E., Peridynamic theory (2014), Berlin: Springer, Berlin · Zbl 1295.74001
[24] Madenci, E.; Oterkus, S., Ordinary state-based peridynamics for plastic deformation according to von Mises yield criteria with isotropic hardening, J Mech Phys Solids, 86, 192-219 (2016)
[25] Mitchell JA (2011) A nonlocal ordinary state-based plasticity model for peridynamics. Technical report, Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
[26] Roth, MJ; Chen, JS; Danielson, KT; Slawson, TR, Hydrodynamic meshfree method for high-rate solid dynamics using a Rankine-Hugoniot enhancement in a Riemann-scni framework, Int J Numer Methods Eng, 108, 12, 1525-1549 (2016)
[27] Roth, MJ; Chen, JS; Slawson, TR; Danielson, KT, Stable and flux-conserved meshfree formulation to model shocks, Comput Mech, 57, 5, 773-792 (2016) · Zbl 1382.74070
[28] Shu, CW; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J Comput Phys, 77, 2, 439-471 (1988) · Zbl 0653.65072
[29] Silling, SA, Reformulation of elasticity theory for discontinuities and long-range forces, J Mech Phys Solids, 48, 1, 175-209 (2000) · Zbl 0970.74030
[30] Silling SA, Askari A (2014) Peridynamic model for fatigue cracking. Technical report, SAND2014-18590. Sandia National Laboratories, Albuquerque
[31] Silling, SA; Askari, E., A meshfree method based on the peridynamic model of solid mechanics, Comput Struct, 83, 17-18, 1526-1535 (2005)
[32] Silling, SA; Epton, M.; Weckner, O.; Xu, J.; Askari, E., Peridynamic states and constitutive modeling, J Elast, 88, 2, 151-184 (2007) · Zbl 1120.74003
[33] Tupek, M.; Radovitzky, R., An extended constitutive correspondence formulation of peridynamics based on nonlinear bond-strain measures, J Mech Phys Solids, 65, 82-92 (2014) · Zbl 1323.74003
[34] Tupek, MR; Rimoli, JJ; Radovitzky, R., An approach for incorporating classical continuum damage models in state-based peridynamics, Comput Methods Appl Mech Eng, 263, 8, 20-26 (2013) · Zbl 1286.74022
[35] Weckner, O.; Abeyaratne, R., The effect of long-range forces on the dynamics of a bar, J Mech Phys Solids, 53, 3, 705-728 (2005) · Zbl 1122.74431
[36] Wilbraham, H., On a certain periodic function, Camb Dublin Math J, 3, 198-201 (1848)
[37] Wildman, RA, Discrete micromodulus functions for reducing wave dispersion in linearized peridynamics, J Peridynamics Nonlocal Model, 1, 1, 56-73 (2019)
[38] Wildman, RA; Gazonas, GA, A finite difference-augmented peridynamics method for reducing wave dispersion, Int J Fract, 190, 39-52 (2014)
[39] Wilkins, ML, Use of artificial viscosity in multidimensional fluid dynamic calculations, J Comput Phys, 36, 3, 281-303 (1980) · Zbl 0436.76040
[40] Xu, XP; Needleman, A., Numerical simulations of fast crack growth in brittle solids, J Mech Phys Solids, 42, 9, 1397-1434 (1994) · Zbl 0825.73579
[41] Yaghoobi, A.; Chorzepa, MG, Fracture analysis of fiber reinforced concrete structures in the micropolar peridynamic analysis framework, Eng Fract Mech, 169, 238-250 (2017)
[42] Zhang, Q., Finite difference methods for partial differential equations (2017), Beijing: China Science Publishing, Beijing
[43] Zhang, X.; Xu, Z.; Yang, Q., Wave dispersion and propagation in linear peridynamic media, Shock Vib, 9, 9528978 (2019)
[44] Zhou, G.; Hillman, M., A non-ordinary state-based Godunov-peridynamics formulation for strong shocks in solids, Comput Part Mech, 7, 2, 365-375 (2020)
[45] Zhou, X.; Wang, Y.; Shou, Y.; Kou, M., A novel conjugated bond linear elastic model in bond-based peridynamics for fracture problems under dynamic loads, Eng Fract Mech, 188, 151-183 (2018)
[46] Zhu, QZ; Ni, T., Peridynamic formulations enriched with bond rotation effects, Int J Eng Sci, 121, 118-129 (2017) · Zbl 1423.74072
[47] Zimmermann M (2005) A continuum theory with long-range forces for solids. PhD thesis, Massachusetts Institute of Technology
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.