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Lattice Boltzmann simulation of antiplane shear loading of a stationary crack. (English) Zbl 1464.74411
Summary: In this work, the lattice Boltzmann method is applied to study the dynamic behaviour of linear elastic solids under antiplane shear deformation. In this case, the governing set of partial differential equations reduces to a scalar wave equation for the out of plane displacement in a two dimensional domain. The lattice Boltzmann approach developed by G. Yan [J. Comput. Phys. 161, No. 1, 61–69 (2000; Zbl 0969.76076)] is used to solve the problem numerically. Some aspects of the scheme are highlighted, including the treatment of the boundary conditions. Subsequently, the performance of the lattice Boltzmann scheme is tested for a stationary crack problem for which an analytic solution exists. The treatment of cracks is new compared to the examples that are discussed in Guangwu’s work [loc. cit.]. Furthermore, the lattice Boltzmann simulations are compared to finite element computations. Finally, the influence of the lattice Boltzmann relaxation parameter on the stability of the scheme is illustrated.
74S99 Numerical and other methods in solid mechanics
74R10 Brittle fracture
Full Text: DOI
[1] Chen, S.; Doolen, GD, Lattice boltzmann method for fluid flows, Annu Rev Fluid Mech, 30, 329-364, (1998) · Zbl 1398.76180
[2] Abas, A.; Gan, ZL; Ishak, MHH; Abdullah, MZ; Khor, SF, Lattice Boltzmann method of different BGA orientations on I-type dispensing method, PLoS ONE, 11, 1-26, (2016)
[3] Frank, X.; Funfschilling, D.; Midoux, N.; Li, H., Bubbles in a viscous liquid: lattice boltzmann simulation and experimental validation, J Fluid Mech, 546, 113-122, (2006) · Zbl 1097.76058
[4] Chopard, B.; Marconi, S., Lattice Boltzmann solid particles in a lattice boltzmann fluid, J Stat Phys, 107, 23-37, (2002) · Zbl 1126.82324
[5] Chopard, B.; Luthi, PO, Lattice Boltzmann computations and applications to physics, Theor Comput Sci, 217, 115-130, (1999) · Zbl 0914.68139
[6] Yan, G., A lattice Boltzmann equation for waves, J Comput Phys, 161, 61-69, (2000) · Zbl 0969.76076
[7] Frantziskonis, GN, Lattice Boltzmann method for multimode wave propagation in viscoelastic media and in elastic solids, Phys Rev E, 83, 066703, (2011)
[8] Xiao, S., A lattice Boltzmann method for shock wave propagation in solids, Comm Numer Meth Eng, 23, 71-84, (2007) · Zbl 1149.74351
[9] Chopard B, Luthi P, Marconi S (1998) A lattice Boltzmann model for wave and fracture phenomena. arXiv:cond-mat/9812220
[10] Kwon, Y.; Hosoglu, S., Application of lattice Boltzmann method, finite element method, and cellular automata and their coupling to wave propagation problems, Comput Struct, 86, 663-670, (2008)
[11] Bhatnagar, PL; Gross, EP; Krook, M., A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys Rev, 94, 511-525, (1954) · Zbl 0055.23609
[12] Welander, P., On the temperature jump in a rarefied gas, Ark Fys, 7, 507, (1954) · Zbl 0057.23301
[13] Sterling, JD; Chen, S., Stability analysis of lattice Boltzmann methods, J Comput Phys, 123, 196-206, (1996) · Zbl 0840.76078
[14] Irwin, GR, Analysis of stresses and strains near the end of a crack traversing a plate, J Appl Mech, 24, 361-364, (1957)
[15] Burridge, R., The numerical solution of certain integral equations with non-integrable kernels arising in the theory of crack propagation and elastic wave diffraction, Philos Trans R Soc A, 265, 353-381, (1969) · Zbl 0194.28002
[16] Thau, SA; Lu, T-H, Diffraction of transient horizontal shear waves by a finite crack and a finite rigid ribbon, Int J Eng Sci, 8, 857-874, (1970) · Zbl 0216.50902
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