×

zbMATH — the first resource for mathematics

A finite element method for analyzing localization in rate dependent solids at finite strains. (English) Zbl 0688.73042
Summary: The finite element method for localization analysis is generalized to account for finite deformations and for material rate dependence. Special shape functions are added to the finite element basis to reproduce band- like localized deformation modes. The amplitudes of these additional modes are eliminated locally by static condensation. The performance of the enhanced element is illustrated in a problem involving shear localization in a plane strain tensile bar. Solutions based on the enhanced element are compared with corresponding results obtained from the underlying compatible isoparametric quadrilateral element and from crossed-triangular and uniformly reduced integration elements. In the finite deformation context, the enhanced element solution is not very sensitive to the precise specification of initial orientation of the additional band-like modes. The enhanced element formulation described here can be used for a broad range of rate independent and rate dependent material behaviors in two dimensional and three dimensional problems.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74B99 Elastic materials
74C99 Plastic materials, materials of stress-rate and internal-variable type
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ortiz, M.; Leroy, Y.; Needleman, A., A finite element method for localized failure analysis, Comp. methods appl. mech. engrg., 61, 189-214, (1987) · Zbl 0597.73105
[2] Chang, Y.W.; Asaro, R.J., An experimental study of shear localization in alunimum-copper single crystals, Acta metall., 29, 241-257, (1981)
[3] Costin, L.S.; Chrisman, E.E.; Hawley, R.H.; Duffy, J., On the localization of plastic flow in mild steel tubes under dynamic torsional loading, (), 90-100, (Bristol and London)
[4] Cox, T.B.; Low, J.R., An investigation of the plastic fracture of AISI 4340 and 18 nickel-200 grade maraging steels, Metall. trans., 5, 1457-1470, (1974)
[5] Vardoulakis, I., Bifurcation analysis of the triaxial test on sand samples, Acta mech., 32, 35-54, (1979) · Zbl 0402.73086
[6] Waversik, W.R.; Brace, W.F., Post-failure behavior of a granite and a diabase, Rock mech., 3, 61-85, (1971)
[7] Van Mier, J.G.M., Strain-softening of concrete under multiaxial loading conditions, () · Zbl 1342.74003
[8] Tvergaard, V.; Needleman, A.; Lo, K.K., Flow localization in the plane strain tensile test, J. mech. phys. solids, 29, 2, 115-142, (1981) · Zbl 0462.73082
[9] Prevost, J.H.; Hughes, T.J.R., Finite-element solution of elastic-plastic boundary value problems, ASME J. appl. mech., 48, 1, 69-74, (1981) · Zbl 0449.73085
[10] Prevost, J.H., Localization of deformations in elastic-plastic solids, Internat J. numer. analyt. methods geomech., 8, 187-196, (1984) · Zbl 0532.73071
[11] Willam, K.J.; Bicanic, N.; Sture, S., Constitutive and computational aspects of strain softening and localization in solids, (), 233-252
[12] Tvergaard, V., Ductile fracture by cavity nucleation between larger voids, J. mech. phys. solids, 30, 265-286, (1982) · Zbl 0491.73118
[13] Needleman, A.; Tvergaard, V., Finite element analysis of localization in plasticity, (), 95-267
[14] Hadamard, J., Leçons sur la propagation des ondes et LES équations de L’hydrodynamique, (1903), Chp. 6 Paris · JFM 34.0793.06
[15] Hill, R., Acceleration waves in solids, J. mech. phys. solids, 10, 1-16, (1962) · Zbl 0111.37701
[16] Mandel, J., Conditions de stabilité et postulat de Drucker, (), 58-68
[17] Thomas, T.Y., Plastic flow and fracture in solids, (1961), Academic Press New York · Zbl 0081.39803
[18] Rice, J.R., The localization of plastic deformation, (), 207-220
[19] Hughes, T.J.R., Generalization of selective integration procedures to anisotropic and nonlinear media internat, J. numer. methods engrg., 15, 1413-1418, (1980) · Zbl 0437.73053
[20] Strang, G.; Fix, G.J., An analysis of the finite element method, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0278.65116
[21] Irons, B.M., Numerical integration applied to finite element methods, () · Zbl 0208.53303
[22] Taylor, R.L.; Beresford, P.J.; Wilson, E.L., A non-conforming element for stress analysis, Internat J. numer. methods engrg., 10, 1211-1219, (1976) · Zbl 0338.73041
[23] Pietruszcak, S.T.; Mroz, Z., Finite element analysis of deformation of strain softening materials, Internat J. numer. methods engrg., 17, 327-334, (1981) · Zbl 0461.73063
[24] Vardoulakis, I., Rigid granular plasticity model and bifurcation in the triaxial test, Acta mech., 49, 57-79, (1983) · Zbl 0522.73085
[25] Green, A.E.; Zerna, W., Theoretical elasticity, (1968), Oxford University Press Oxford · Zbl 0155.51801
[26] Budiansky, B., Remarks on theories of solid and structural mechanics, (), 77-83 · Zbl 0202.24801
[27] 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)
[28] Hill, R.; Hutchinson, J.W., Bifurcation phenomena in the plane tensile test, J. mech. phys. solids, 23, 239-264, (1975) · Zbl 0331.73048
[29] Pan, J.; Saje, M.; Needleman, A., Localization of deformation in rate sensitive porous plastic solids, Int. J. fracture, 21, 261-278, (1983)
[30] Needleman, A., Finite elements for finite strain plasticity problems, (), 387-436
[31] Belytschko, T.; Tsay, C.S., A stabilization procedure for the quadrilateral plate element with one-point quadrature internat, J. numer. methods engrg., 19, 405-419, (1983) · Zbl 0502.73058
[32] Leroy, Y.; Ortiz, M., Finite element analysis of transient localization phenomena in frictional solids internat, J. numer. analyt. methods geomech., 13, 53-74, (1989)
[33] Pierce, D.; Shih, C.F.; Needleman, A., A tangent modulus method for rate dependent solids, Comp. struct., 18, 875-887, (1984) · Zbl 0531.73057
[34] Nagtegaal, J.C.; Parks, D.M.; Rice, J.R., On numerically accurate finite element solutions in the fully plastic range, Comput. methods appl. mech. engrg., 4, 153-177, (1974) · Zbl 0284.73048
[35] S. Nemat-Nasser, D.T. Chung and L.M. Taylor, Phenomenological modelling of rate dependent plasticity for high strain rate problems, CEAM Report, University of California, San Diego.
[36] Hughes, T.J.R., The finite element method, (1987), Prentice-Hall Englewood Cliff, NJ
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.