Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On convergence of the finite-element method for a class of elastic-plastic solids

Authors: K. S. Havner and H. P. Patel
Journal: Quart. Appl. Math. 34 (1976), 59-68
MSC: Primary 73.65
DOI: https://doi.org/10.1090/qam/449142
MathSciNet review: 449142
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A proof of convergence of the finite-element method in rate-type, quasistatic boundary value problems is presented. The bodies considered may be discretely heterogeneous and elastically anisotropic, their plastic behavior governed by history-dependent, piecewise-linear yield functions and fully coupled hardening rules. Elastic moduli are required to be positive-definite and plastic moduli nonnegative-definite. Precise and complete arguments are given in the case of bodies whose surfaces are piecewise plane.

References [Enhancements On Off] (What's this?)

  • [1] Kerry S. Havner, On convergence of a discrete aggregate model in polycrystalline plasticity, Internat. J. Solids and Structures 7 (1971), 1269–1275 (English, with Russian summary). MR 0342008
  • [2] P. Tong and T. H. H. Pian, The convergence of finite element method in solving linear elastic problems, Int. J. Solids Structures 3, 865-879 (1967)
  • [3] Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR 0443377
  • [4] G. I. Taylor, The distortion of crystals of aluminum under compression, Part II, Proc. R. Soc. Lond. A116, 16-38 (1927)
  • [5] G. I. Taylor, Plastic strain in metals, J. Inst. Metals 62, 307-324 (1938)
  • [6] J. F. W. Bishop and R. Hill, A theoretical derivation of the plastic properties of a polycrystalline face-centred metal, Philos. Mag. (7) 42 (1951), 1298–1307. MR 0045575
  • [7] T. H. Lin, Physical theory of plasticity, in Advances in applied mechanics, Vol. 11 (edited by C.-S. Yih), Academic Press, New York, 255-311 (1971)
  • [8] J. W. Hutchinson, Plastic stress-strain relations of F.C.C. polycrystalline metals hardening according to Taylor's rule, plastic deformation of B.C.C. polycrystals, J. Mech. Phys. Solids 12, 11-33 (1964)
  • [9] J. W. Hutchinson, Elastic-plastic behaviour of polycrystalline metals and composites, Proc. R. Soc. Lond. A319, 247-272 (1970)
  • [10] K. S. Havner and R. Varadarajan, A quantitative study of a crystalline aggregate model, Int. J. Solids Structures 9, 379-394 (1973)
  • [11] K. S. Havner, C. Singh and R. Varadarajan, Plastic deformation and latent strain energy in a polycrystalline aluminum model, Int. J. Solids Structures 10, 853-862 (1974)
  • [12] B. Budiansky and T. T. Wu, Theoretical prediction of plastic strains of polycrystals, Proc. 4th U.S. Nat. Cong. Appl. Mech., ASME, 1175-1185 (1962)
  • [13] W. T. Koiter, Stress-strain relations, uniqueness and variational theorems for elastic-plastic materials with a singular yield surface, Quart. Appl. Math. 11 (1953), 350–354. MR 0059769, https://doi.org/10.1090/S0033-569X-1953-59769-7
  • [14] J. Mandel, Generalization de la théorie de plasticité de W. T. Koiter, Int. J. Solids Structures 1, 273-295 (1965)
  • [15] R. Hill, Generalized constitutive relations for incremental deformation of metal crystals by multislip, J Mech. Phys. Solids 14, 95-102 (1966)
  • [16] G. Maier, A quadratic programming approach for certain nonlinear structural problems, Meccanica 3, 121-130 (1968)
  • [17] G. Maier, "Linear'' flow-laws of elastoplasticity: A unified general approach, Lincei-Rend. Sci. Fis. Mat, e Nat. (8) 47, 267-276 (1969)
  • [18] O. DeDonato and A. Franchi, A modified gradient method for finite element elastoplastic analysis by quadratic programming, Comp. Meth. Appl. Mech. Engng. 2, 107-131 (1973)
  • [19] William Prager, Recent developments in the mathematical theory of plasticity, J. Appl. Phys. 20 (1949), 235–241. MR 0028760
  • [20] D. C. Drucker, Some implications of work hardening and ideal plasticity, Quart. Appl. Math. 7 (1950), 411–418. MR 0034210, https://doi.org/10.1090/S0033-569X-1950-34210-6
  • [21] D. C. Drucker, Variational principles in the mathematical theory of plasticity, Calculus of variations and its applications. Proceedings of Symposia in Applied Mathematics, Vol. VIII, McGraw-Hill Book Co., Inc., New York-Toronto-London, for the American Mathematial Society, Providence, R. I., 1958, pp. 7–22. MR 0093122
  • [22] Kerry S. Havner, A discrete model for the prediction of subsequent yield surfaces in polycrystalline plasticity, Internat. J. Solids and Structures 7 (1971), 719–730 (Russian). MR 0334793

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73.65

Retrieve articles in all journals with MSC: 73.65

Additional Information

DOI: https://doi.org/10.1090/qam/449142
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society