zbMATH — the first resource for mathematics

A variational multiscale approach to strain localization – formulation for multidimensional problems. (English) Zbl 1011.74069
This paper examines the strain localization by using a multiscale variational formulation. The idea is to compute on the coarse scale, but still retain (on the coarse scale) the effects of fine scale. This extends former one-dimensional approach to weak discontinuities [the authors, ibid. 159, No. 3-4, 193-222 (1998; Zbl 0961.74009)]. This multiscale approach results in a non-local constitutive law at the coarse scale, thus avoiding ill-posed mesh-dependent solutions. Both the variational formulation and numerical implementation are presented. The paper concludes with some numerical examples.

74S05 Finite element methods applied to problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74C99 Plastic materials, materials of stress-rate and internal-variable type
Full Text: DOI
[1] Hughes, T.J.R., Multiscale phenomena Green’s functions the Dirichlet-to-Neumann formulation subgrid scale models bubbles and the origins of stabilized ethods, Comput. meth. appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044
[2] Hughes, T.J.R., A space – time formulation for multiscale phenomena, J. comput. appl. math., 74, 217-229, (1996) · Zbl 0869.65061
[3] Garikipati, K.; Hughes, T.J.R., A study of strain localization in a multiple scale framework – the one-dimensional problem, Comput. meth. appl. mech. engrg., 159, 193-222, (1998) · Zbl 0961.74009
[4] T.Y. Thomas, Plastic Flow and Fracture in Solids, Academic Press, New York, 1961 · Zbl 0095.38902
[5] Hill, R., Acceleration waves in solids, J. mech. phys. solids, 10, 1-16, (1962) · Zbl 0111.37701
[6] J. Mandel, Conditions de Stabilité et Postulat de Drucker, in: J. Kravtchenko, P.M. Sirieys (Eds.), Rheology and Soil Mechanics, IUTAM Symposium, 1964, pp. 58-68
[7] Hill, R.; Hutchinson, J.W., Bifurcation phenomena in the plane tension test, J. mech. phys. solids, 23, 239-264, (1975) · Zbl 0331.73048
[8] J.R. Rice, The localization of plastic deformation, in: W.T. Koiter (Ed.), Theoretical and Applied Mechanics, North Holland, Amsterdam, 1976, pp. 207-220
[9] Asaro, R.J., Micromechanics of crystals and polycrystals, Adv. appl. mech., 23, 1-115, (1983)
[10] C. Truesdell, W. Noll, The Non-linear Field Theories (Handbuch der Physik band III), Springer, Berlin, 1965
[11] K.F. Graff, Wave Motion in Elastic Solids, Dover, New York, 1975 · Zbl 0314.73022
[12] J.E. Marsden, T.J.R. Hughes, Mathematical Foundations of Elasticity, Dover, New York, 1994 · Zbl 0545.73031
[13] Rudnicki, J.W.; Rice, J.R., Conditions for the localization of deformation in pressure-sensitive dilatant materials, J. mech. phys. solids, 23, 371-394, (1975)
[14] Rice, J.R.; Rudnicki, J.W., A note on some features of the theory of localization of deformation, Int. J. solids and structures, 16, 597-605, (1980) · Zbl 0433.73032
[15] K. Willam, Experimental and computational aspects of concrete fracture, in: R. Owen, E. Hinton, Bićanić (Eds.), Proceedings of the International Conference on Computer Aided Analysis and Design of Concrete Structures, Pineridge, Swansea, 1984, pp. 33-70
[16] Ottosen, N.S.; Runesson, K., Properties of discontinuous bifurcation solutions in elasto-plasticity, Int. J. solids and structures, 27, 401-421, (1991) · Zbl 0738.73021
[17] Bigoni, D.; Zaccaria, D., On strain localization analysis of elastoplastic materials at finite strains, Int. J. plasticity, 9, 21-33, (1993) · Zbl 0768.73024
[18] Anand, L.; Kim, K.H.; Shawki, T.G., Onset of shear localization in viscoplastic solids, J. mech. phys. solids, 35, 407-429, (1987) · Zbl 0612.73046
[19] Shawki, T.G.; Clifton, R.J., Shear band formation in thermal viscoplastic materials, Machanics of materials, 8, 13-43, (1989)
[20] Needleman, A., Dynamic shear band development in plane strain, J. appl. Mach., 56, 1-9, (1989)
[21] Batra, R.C.; Liu, D.S., Adiabatic shear banding in plane strain problems, J. mech. phys. solids, 56, 527-534, (1989) · Zbl 0709.73025
[22] J. Hadamard, Leçons sur la Propagation des Ondes et les Equations de l’Hydro-dynamique, Hermann, Paris, 1903
[23] H. Geiringer, Beitrag zum Vollständingen Ebenen Plastizitätsproblem, in: Proccedings of the Third International Congress Applied Mechanics, Stockholm, 1930, pp. 185-190
[24] H. Matthies, G. Strang, E. Christiansen, The saddle point of a differential program energy methods in finite element analysis, in: R. Glowinski, E.Y. Rodin, O.C. Zienkiewicz (Eds.), Wiley, New York, 1979, pp. 309-318
[25] Simo, J.C.; Oliver, J.; Armero, F., An analysis of strong discontinuities induced by softening solutions in rate-independent solids, J. comput. mech., 12, 277-296, (1993) · Zbl 0783.73024
[26] J.C. Simo, J. Oliver, A new approach to the analysis and simulation of strain softening in solids, in: Z.P. Bazant, Z. Bittnar, M. Jirásek, J. Mazars (Eds.), Fracture and Damage in Quasibrittle Structures, 1994
[27] F. Armero, K. Garikipati, Recent advances in the analysis and numerical simulation of strain localization in inelastic solids, in: D.R.J. Owen, E. Oñate, E. Hinton (Eds.), Proceedings of the Computational Plasticity IV, CIMNE, Barcelona, 1995, pp. 547-561
[28] Armero, F.; Garikipati, K., An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids, Int. J. solids and structures, 33, 2863-2885, (1996) · Zbl 0924.73084
[29] Oliver, J., Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. part 1: fundamentals, Int. J. numer. methods engrg., 39, 3575-3600, (1996) · Zbl 0888.73018
[30] Pietruszczak, St.; Mróz, Z., Finite element analysis of deformation of strain-softening materials, Int. J. numer. methods engrg., 17, 327-334, (1981) · Zbl 0461.73063
[31] Read, H.E.; Hegemier, G.A., Strain softening of rock soil and concrete – a review article, Mechanics of materials, 3, 271-294, (1984)
[32] Bazant, Z.P., Mechanics of distributed cracking, Appl. mech. rev., 39, 675-705, (1986)
[33] Hillerborg, A.; Modéer, M.; Petersson, P.E., Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement and concrete res., 6, 773-782, (1976)
[34] A. Hillerborg, Numerical methods to simulate softening and fracture of concrete, in: G.C. Sih, A. di Tomasso (Eds.), Fracture Mechanics of Concrete: Structural Application and Numerical Calculation, Martinus Nijhoff, Dordrecht, 1985, pp. 141-170
[35] Bazant, Z.P.; Oh, B.H., Crack band theory for fracture of concrete, Matèriaux et constructions, 39, 155-177, (1983)
[36] Bazant, Z.P.; Belytschko, T.B.; Chang, T.P., Continuum theory for strain softening, J. engrg. mech., 110, 1666-1691, (1984)
[37] Stromberg, L.; Ristinmaa, M., Finite element formulation of a non-local plasticity theory, Comput. methods in appl. mech. engrg., 136, 127-144, (1996) · Zbl 0918.73118
[38] Coleman, B.D.; Hodgdon, M.L., On shear bands in ductile materials, Archives for rational mech. anal., 90, 219-247, (1985) · Zbl 0625.73041
[39] Peerlings, R.H.J.; de Borst, R.; Brekelmans, 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
[40] de Borst, R.; Pamin, J., Some novel developments in finite element procedures for strain gradient-dependent plasticity, Int. J. numer. methods engrg., 39, 2477-2505, (1996) · Zbl 0885.73074
[41] de Borst, R.; Sluys, L.J., Localisation in a Cosserat continuum under static and dynamic loading conditions, Comput. methods appl. mech. engrg., 90, 805-827, (1991)
[42] J. Yu, D. Peric, D.R.J. Owen, An ssessment of the cosserat continuum through the finite element simulation of a strain localisation problem, in: E. Oñate, J. Periaux, A. Samuelsson (Eds.), Springer/CIMNE, Barcelona, 1991, pp. 321-332
[43] Belytschko, T.B.; Fish, J.; Engleman, B.E., A finite element with embedded localization zones, Comput. methods appl. mech. engrg., 70, 59-89, (1988) · Zbl 0653.73032
[44] Ortiz, M.; Leroy, Y.; Needleman, A., A finite element method for localized failure analysis, Comput. methods appl. mech. engrg., 61, 189-214, (1987) · Zbl 0597.73105
[45] R. Larsson, K. Runesson, M. Akesson, Embedded localization band based on regularized strong discontinuity, in: D.R.J. Owen, E. Oñate, E. Hinton (Eds.), Proceedings of Computational Plasticity IV, CIMNE, Barcelona, 1995, pp. 599-609
[46] K. Garikipati, T.J.R. Hughes, A study of strong discontinuities in a multiple scale framework, Int. J. Comput. Civil and Structural Engineering, 1999, to appear in the inaugural issue
[47] Simo, J.C.; Rifai, M.S., A class of mixed assumed strain methods and the method of incompatible modes, Int. J. numer. methods engrg., 29, 1595-1638, (1990) · Zbl 0724.73222
[48] Arunakirinathar, K.; Reddy, B.D., Further results for enhanced strain methods with isoparametric elements, Comput. methods appl. mech. engrg., 127, 127-143, (1995) · Zbl 0862.73056
[49] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods Springer, New York, 1991 · Zbl 0788.73002
[50] J.C. Simo, T.J.R. Hughes, Computational Inelasticity, Springer, 1998 · Zbl 0934.74003
[51] K. Garikipati, On strong discontinuities in inelastic solids and their numerical simulation, Ph.D. Thesis, Stanford University, 1996 · Zbl 0924.73084
[52] Tvergaard, V.; Needleman, A.; Lo, K.K., Flow localization in the plane strain tension test, J. mech. phys. solids, 29, 115-142, (1981) · Zbl 0462.73082
[53] Nix, W.D.; Gao, H., Indentation size effects in crystalline materials: a law for strain gradient plasticity, J. mech. phys. solids, 46, 411-425, (1998) · Zbl 0977.74557
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.