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A nonlocal plasticity formulation for the material point method. (English) Zbl 1253.74021
Summary: A new multi-variate fixed-point iteration scheme is devised for solving the coupled dynamic integral equations governing nonlocal plasticity using the material point method (MPM). Novel use of the MPM grid for particle-particle communications results in a simple and efficient, matrix-free method. Moreover, a straightforward method for deriving a convergence criterion for this method is developed and applied to two classical verification problems that are well known to be mesh dependent with a local model, but are shown to be mesh-independent with the new nonlocal MPM formulation.

MSC:
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
Software:
Uintah
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[1] Sulsky, D., A particle method for history-dependent materials, Comput. methods appl. mech. engrg., 118, 1-2, 179, (1994) · Zbl 0851.73078
[2] Bardenhagen, S.; Kober, E., The generalized interpolation material point method, Comput. model. engrg. sci., 5, 6, 477-495, (2004)
[3] Steffen, M.; Wallstedt, P.; Guilkey, J.; Kirby, R.; Berzins, M., Examination and analysis of implementation choices within the material point method (mpm), CMES - comput. model. engrg. sci., 31, 2, 107-127, (2008)
[4] Guilkey, J.; Weiss, J., Implicit time integration for the material point method: quantitative and algorithmic comparisons with the finite element method, Int. J. numer. methods engrg., 57, 9, 1323-1338, (2003) · Zbl 1062.74653
[5] Sulsky, D.; Kaul, A., Implicit dynamics in the material-point method, Comput. methods appl. mech. engrg., 193, 12-14, 1137-1170, (2004) · Zbl 1060.74674
[6] York, A.; Sulsky, D.; Schreyer, H., The material point method for simulation of thin membranes, Int. J. numer. methods engrg., 44, 1429-1456, (1999) · Zbl 0971.74079
[7] Love, E.; Sulsky, D., An energy-consistent material-point method for dynamic finite deformation plasticity, Int. J. numer. methods engrg., 65, 1608-1638, (2005) · Zbl 1111.74047
[8] Nairn, J., Material point method calculations with explicit cracks, Comput. model. engrg. sci., 4, 649-663, (2003) · Zbl 1064.74176
[9] Guo, Y.; Nairn, J., Three-dimensional dynamic fracture analysis using the material point method, Comput. model. engrg. sci., 16, 141-155, (2006)
[10] Ma, S.; Zhang, X.; Qiu, X., Comparison study of mpm and sph in modeling hypervelocity impact problems, Int. J. impact engrg., 36, 272-282, (2009)
[11] J. Burghardt, B. Leavy, J. Guilkey, Z. Xue, R. Brannon, Application of uintah-mpm to shaped charge jet penetration of aluminum, IOP Conference Series: Materials Science and Engineering, vol. 10, 2010, p. 012223 (9 pp.). http://dx.doi.org/10.1088/1757-899X/10/1/012223.
[12] Bažant, Z.P.; Zubelewicz, A., Strain-softening bar and beam: exact non-local solution, Int. J. solids struct., 7, 7, 659-673, (1988) · Zbl 0639.73040
[13] de Borst, R.; Sluys, L.; Muhlhaus, H.; Pamin, J., Fundamental issues in finite element analyses of localization of deformation, Engrg. comput., 10, 99-121, (1993)
[14] Erigen, A.C., On nonlocal plasticity, Int. J. engrg. sci., 19, 1461-1474, (1981) · Zbl 0474.73028
[15] Bažant, Z.P.; Lin, F.-B., Non-local yield limit degradation, Int. J. numer. methods engrg., 26, 1805-1823, (1988) · Zbl 0661.73041
[16] T. Belytschko, M. Kulkarni, On imperfections and spatial gradient regularization in strain softening viscoplasticity, in: Failure criteria and analysis in dynamic response, presented at the Winter Annual Meeting of the American Society of Mechanical Engineers, vol. 107, Dallas, TX, USA, 1990, pp. 1-5.
[17] Aifantis, E., Strain gradient interpretation of size effects, Int. J. fract., 95, 299-314, (1999)
[18] Bažant, Z.P.; Pang, S.-D., Activation energy based extreme value statistics and size effect in brittle and quasibrittle fracture, J. mech. phys. solids, 55, 1, 91-131, (2007) · Zbl 1173.74004
[19] Engelen, R.A.; Geers, M.G.; Baaijens, F.P., Nonlocal implicit gradient-enhanced elastoplasticity for the modeling of softening behavior, Int. J. plast., 19, 403-433, (2003) · Zbl 1090.74519
[20] Geers, M., Finite strain logarithmic hyperelasto-plasticity with softening: a strongly non-local implicit gradient framework, Comput. methods appl. mech. engrg., 193, 30-32, 3377, (2004) · Zbl 1060.74503
[21] De Borst, R.; Muehlhaus, H.-B., Gradient-dependent plasticity: formulation and algorithmic aspects, Int. J. numer. methods engrg., 35, 3, 521-539, (1992) · Zbl 0768.73019
[22] Peerlings, R.; Geers, M.; de Borst, R.; Brekelmans, W., A critical comparison of nonlocal and gradient-enhanced softening continua, Int. J. solids struct., 38, 7723-7746, (2001) · Zbl 1032.74008
[23] Bažant, Z.P.; Jirasek, M., Nonlocal integral formulations of plasticity and damage: survey of progress, J. engrg. mech., 128, 1119-1149, (2002)
[24] Di Luzio, G.; Bažant, Z., Spectral analysis of localization in nonlocal and over-nonlocal materials with softening plasticity or damage, Int. J. solids struct., 42, 23, 6071-6100, (2005) · Zbl 1119.74437
[25] Stromberg, L.; Ristinmaa, M., Fe-formulation of a nonlocal plasticity theory, Comput. methods appl. mech. engrg., 136, 124-144, (1996) · Zbl 0918.73118
[26] Chen, J.-S.; Wu, C.-T.; Belytschko, T., Regularization of material instabilities by meshfree approximations with intrinsic length scales, Int. J. numer. methods engrg., 47, 7, 1303-1322, (2000) · Zbl 0987.74079
[27] Pan, X.; Yuan, H., Nonlocal damage modelling using the element-free Galerkin method in the frame of finite strains, Comput. mater. sci., 46, 3, 660-666, (2009)
[28] O. Weckner, S. Silling, A. Askari, Dispersive wave propagation in the nonlocal peridynamic theory, in: Proceedings of the ASME International Mechanical Engineering Congress and Exposition, vol. 12, Boston, MA, United states, 2009, pp. 503-504.
[29] Sadeghirad, A.; Brannon, R.M.; Burghardt, J., A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations, Int. J. numer. methods engrg., 86, 12, 1435-1456, (2011) · Zbl 1235.74371
[30] Wallstedt, P.; Guilkey, J., An evaluation of explicit time integration schemes for use with the generalized interpolation material point method, J. comput. phys., 227, 22, 9628-9642, (2008) · Zbl 1148.74047
[31] Wallstedt, P.; Guilkey, J., Improved velocity projection for the material point method, CMES - comput model. engrg. sci., 19, 3, 223-232, (2007) · Zbl 1184.74078
[32] Jirasek, M.; Rolshoven, S., Comparison of integral-type nonlocal plasticity models for strain-softening materials, Int. J. engrg. sci., 41, 13-14, 1553-1602, (2003) · Zbl 1211.74039
[33] Ortega, J.M.; Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, (2000), Society for Industrial and Applied Mathematics Philadelphia, PA, USA · Zbl 0949.65053
[34] Simó, J.; Hughes, T., Computational inelasticity, Interdisciplinary applied mathematics: mechanics and materials, (1998), Springer
[35] de St. Germain, J.; McCorquodale, J.; Parker, S.; Johnson, C., Uintah: a massively parallel problem solving environment, (), 33
[36] Bažant, Z.P., Instability, ductility, and size effect in strain-softening concrete, ASCE J. engrg. mech. div., 102, 2, 331-344, (1976)
[37] Ortiz, M.; Leroy, Y.; Needleman, A., A finite element method for localized failure analysis, Comput. methods appl. mech. engrg., 61, 198-214, (1987) · Zbl 0597.73105
[38] J. Pamin, R. de Borst, Numerical simulation of localization phenomena using gradient plasticity, Heron 1 (1995) 71-92, TNO Built Environment and Geosciences, Delft, and the Netherlands School for Advanced Studies in Construction.
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