zbMATH — the first resource for mathematics

An implicit gradient model by a reproducing kernel strain regularization in strain localization problems. (English) Zbl 1067.74564
Summary: A reproducing kernel strain regularization (RKSR) as a mathematical generalization of gradient theory and non-local theory for strain localization problems is presented. RKSR introduces a correction of the weight function in the non-local strain by imposition of gradient reproducing conditions. Both continuum and discrete forms of RKSR are presented, and they lead to an implicit representation of gradient models. As such, RKSR provides a gradient type regularization to the localization problem without increasing the order of differentiation in the governing equations. Hence no additional boundary conditions are required, and the need for higher order continuity for the approximation of unknowns in the governing equations is no longer an issue. A von Neumann spectral analysis is employed to study the spectral properties of RKSR of various orders in one dimension. It is shown that RKSR almost duplicates the spectral properties of second and fourth order gradient theories. In summary, RKSR reproduces the regularization properties of gradient methods without dealing with additional boundary conditions or higher order continuity issues.

74R20 Anelastic fracture and damage
74S30 Other numerical methods in solid mechanics (MSC2010)
Full Text: DOI
[1] Askes, H.; Sluys, L.J., A classification of higher-order strain gradient models in damage mechanics, Arch. appl. mech., 72, 171-188, (2002) · Zbl 1065.74004
[2] Askes, H.; Pamin, J.; de Borst, R., Dispersion analysis and element-free Galerkin solutions of second and fourth-order gradient-enhanced damage models, Int. J. numer. methods engrg., 49, 6, 811-832, (2000) · Zbl 1009.74082
[3] Bazant, Z.P.; Cabot, P.G., Nonlocal continuum damage, localization instability and convergence, ASME, J. appl. mech., 55, 287-293, (1988) · Zbl 0663.73075
[4] Bazant, Z.P.; Belytscko, T.; Chang, T.-P., Continuum model for strain softening, J. engrg. mech., 110, 1666-1692, (1984)
[5] Bazant, Z.P.; Jirasek, M., Nonlocal integral formulations of plasticity and damage: survey of progress, J. engrg. mech. ASCE, 128, 1119-1149, (2002)
[6] Belytschko, T.; Bazant, Z.P.; Hyun, Y.-W.; Chang, T.-P., Strain-softening materials and finite-element solutions, Comput. struct., 23, 163-180, (1986)
[7] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent development, Comput. methods appl. mech. engrg., 139, 3-49, (1996) · Zbl 0891.73075
[8] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077
[9] 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, 1301-1322, (2000) · Zbl 0987.74079
[10] Chen, J.S.; Wang, H.P., New boundary condition treatments for meshless computation of contact problems, Comput. methods appl. mech. engrg., 187, 441-468, (1998)
[11] Chen, J.S.; Pan, C.; Wu, C.T.; Liu, W.K., Reproducing kernel particle methods for large deformation analysis of nonlinear structures, Comput. methods appl. mech. engrg., 139, 195-227, (1996) · Zbl 0918.73330
[12] de Borst, R.; Muhlhaus, H.B., Gradient-dependent plasticity: formulation and algorithmic aspects, Int. J. numer. methods engrg., 35, 521-539, (1992) · Zbl 0768.73019
[13] de Borst, R.; Pamin, J., Gradient plasticity in numerical simulation of concrete cracking, Eur. J. mech., A/solids, 15, 295-320, (1996) · Zbl 0858.73069
[14] de Borst, R.; Pamin, R.H.; Peerlings, R.H.J.; Sluys, L.J., On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials, Comput. mech., 17, 130-141, (1996) · Zbl 0840.73047
[15] Feng, J.; Wen, K.; Tian, X.; Chen, Z., Nonlocal softening model and its application to engineering excavation, Engrg. mech. (Chinese journal), 16, 6, 36-43, (1999)
[16] Fusao, O.; Toshihisa, A.; Atsushi, Y., A strain localization analysis using a viscoplastic softening model for Clay, Int. J. plast., 11, 523-545, (1995) · Zbl 0853.73058
[17] Krongauz, Y.; Belytschko, T., Consistent pseudo-derivatives in the meshless methods, Comput. methods appl. mech. engrg., 146, 371-386, (1997) · Zbl 0894.73156
[18] Lade, P.V.; Wang, Q., Analysis of shear banding in true triaxial tests on sand, J. engrg. mech., 127, 8, 762-768, (2001)
[19] Lasry, D.; Belytschko, T., Localization limiters in transient problems, Int. J. solids struct., 24, 581-597, (1988) · Zbl 0636.73021
[20] Lemaitre, J., A continuous damage mechanics model for ductile fracture, J. engrg. mater. tech., 107, 83-89, (1985)
[21] Liu, W.K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. numer. methods engrg., 38, 1655-1679, (1995) · Zbl 0840.73078
[22] Liu, W.K.; Jun, S.; Zhang, Y., Reproducing kernel particle methods, Int. J. numer. methods fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[23] Mühlhaus, H.B.; Vardoulakis, I., The thickness of shear band in granular materials, Geotechnique, 37, 271-283, (1987)
[24] Needleman, A., Material rate dependence and mesh sensitivity in localization problems, Comput. methods appl. engrg., 67, 69-85, (1988) · Zbl 0618.73054
[25] Peerlings, R.H.J.; de Borst, R.; Brekelmans, W.A.M.; de Vree, J.H.P., Gradient enhanced damage for quasi-brittle materials, Int. J. numer. methods engrg., 39, 3391-3403, (1996) · Zbl 0882.73057
[26] Simo, J.C.; Hughes, T.J.R., On the variational foundations of assumed strain methods, ASME J. appl. mech., 53, 51-54, (1986) · Zbl 0592.73019
[27] Simo, J.C.; Ju, J.W., Strain- and stress-based continuum damage models–I. formulation, Int. J. solids struct., 23, 821-840, (1987) · Zbl 0634.73106
[28] L.J. Sluys, Wave propagation, localization and dispersion in softening solids, Ph.D. Thesis, Civil Engineering Department of Delft University of Technology, 1992
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.