Taylor-SPH Method for Viscoplastic Damage Material
Applied and Computational Mathematics
Volume 4, Issue 3, June 2015, Pages: 162-173
Received: May 8, 2015; Accepted: May 17, 2015; Published: May 29, 2015
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Hajar Idder, Laboratory of Mechanics and Civil Engineering, Faculty of Science and Technology, Abdelmalek Essaâdi University, Tangier, Morocco
Mokhtar Mabssout, Laboratory of Mechanics and Civil Engineering, Faculty of Science and Technology, Abdelmalek Essaâdi University, Tangier, Morocco
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In this paper, we apply the meshless method Taylor-SPH to solve the propagation of shock wave in viscoplastic material coupled to damage. The equations are written in terms of stress and velocity. Taylor-SPH method is based on the Taylor series expansion of stress and velocity and on the corrected SPH approximation. Numerical stability of the method as a function of the smoothing length and the Courant number is analysed in the elastic case. The Taylor-SPH method is used to simulate localization in a one dimensional viscoplastic damage problem. The numerical results show that the Taylor-SPH method is able to model localization phenomena in viscoplastic damage material without lose of hyperbolicity of partial differential equations.
Taylor-SPH, Meshless, Viscoplastic, Damage, Shock Wave, Stability
To cite this article
Hajar Idder, Mokhtar Mabssout, Taylor-SPH Method for Viscoplastic Damage Material, Applied and Computational Mathematics. Vol. 4, No. 3, 2015, pp. 162-173. doi: 10.11648/j.acm.20150403.19
L.B. Lucy, A numerical approach to the testing of fusion process, Astronomical Journal, 82, pp. 1013-1024, 1977.
R.A. Gingold, J.J, Monaghan, Smoothed particles hydrodynamics: Theory and application to nonspherical stars, Monthly Notices of the Royal Astronomical Society, 181, pp. 375-389, 1977.
J.W. Swegle, D.A. Hicks, S.W. Attaway, Smooth particle hydrodynamics stability analysis, J. Comput. Phys. 116, pp. 123–134, 1995.
T. Belytschko, Y. Guo, W.K. Liu, S.P.Xiao, A unified stability analysis of meshless particle methods, Int. J. Numer. Methods Eng. 48, pp. 1359–1400 , 2000.
W.K. Liu, S. Jun, Y.F. Zhang, Reproducing kernel particle methods, Int. J. Numer. Methods Eng. 20 (8–9), pp. 1081–1106, 1995.
P.W. Randles, L.D. Libersky, Smoothed particle hydrodynamics: some recent improvements and applications, Comput. Methods Appl. Mech. Eng. 139, pp. 375–408, 1996.
G.R. Johnson, S.R. Beissel, Normalized smoothing functions for SPH impact computations, Int. J. Numer. Methods Engrg. 39, pp. 2725–2741, 1996.
C.T. Dyka, R.P. Ingel, An approach for tensile instability in smoothed particle hydrodynamics, Comput. Struct. 57, pp. 573–580, 1995.
M. Mabssout, M. Pastor, A Taylor–Galerkin algorithm for shock wave propagation and strain localization failure of viscoplastic continua, Comput. Methods Appl. Mech. Eng. 192, pp. 955–971, 2003.
M. Mabssout, M. Pastor, M.I. Herreros, M. Quecedo, A Runge–Kutta, Taylor–Galerkin scheme for hyperbolic systems with source terms. Application to shock wave propagation in viscoplastic geomaterials, Int. J. Numer. Anal. Methods Geomech. 30 (13), pp.1337–1355, 2006.
M. I. Herreros, M. Mabssout, A two-steps time discretization scheme using the SPH method for shock wave propagation, Comput. Methods Appl. Mech. Eng. 200, pp. 1833–1845, 2011.
M. Mabssout, M. I. Herreros, Taylor-SPH vs Taylor-Galerkin for shock waves in viscoplastic continua, Eur. J. Comput. Mech. 20 (5-6), pp. 281-308, 2011.
M. Mabssout, M. I. Herreros, Runge-Kutta vs Taylor-SPH. Two time integration schemes for SPH with application to Soil Dynamics, App. Math. Modelling, 37(5), pp. 3541-3563, 2013.
P. Perzyna, Fundamental problems in viscoplasticity, Recent Advances in Applied Mechanics. Academic press, New York, 9, pp. 243-377 ,1966.
L. M. Kachanov, Introduction to Continuum Damage Mechanics, Martinus Nijhoff, Dordrecht, the Netherlands, 1986.
D. Krajcinovic, J. Lemaitre, Continuum Damage Mechanics, Theory and applications. Springer.Vienna. 1987.
T. Rabczuk, T. Belytschko, S. P. Xiao, Stable particle methods based on Lagrangian kernels, Comput. Methods. Appl. Mech. Engrg, 193, pp. 1035–1063, 2004.
J.J. Monaghan, J.C. Lattanzio, A refined particle method for astrophysical problems, Astronomy and Astrophysics, 149 , pp. 135 -143, 1985.
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