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
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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.
- Kerry S. Havner, On convergence of a discrete aggregate model in polycrystalline plasticity, Internat. J. Solids Structures 7 (1971), 1269–1275 (English, with Russian summary). MR 342008, DOI https://doi.org/10.1016/0020-7683%2871%2990067-9
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)
- 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
G. I. Taylor, The distortion of crystals of aluminum under compression, Part II, Proc. R. Soc. Lond. A116, 16–38 (1927)
G. I. Taylor, Plastic strain in metals, J. Inst. Metals 62, 307–324 (1938)
- 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 45575
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)
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)
J. W. Hutchinson, Elastic-plastic behaviour of polycrystalline metals and composites, Proc. R. Soc. Lond. A319, 247–272 (1970)
K. S. Havner and R. Varadarajan, A quantitative study of a crystalline aggregate model, Int. J. Solids Structures 9, 379–394 (1973)
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)
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)
- 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 59769, DOI https://doi.org/10.1090/S0033-569X-1953-59769-7
J. Mandel, Generalization de la théorie de plasticité de W. T. Koiter, Int. J. Solids Structures 1, 273–295 (1965)
R. Hill, Generalized constitutive relations for incremental deformation of metal crystals by multislip, J Mech. Phys. Solids 14, 95–102 (1966)
G. Maier, A quadratic programming approach for certain nonlinear structural problems, Meccanica 3, 121–130 (1968)
G. Maier, "Linear” flow-laws of elastoplasticity: A unified general approach, Lincei-Rend. Sci. Fis. Mat, e Nat. (8) 47, 267–276 (1969)
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)
- William Prager, Recent developments in the mathematical theory of plasticity, J. Appl. Phys. 20 (1949), 235–241. MR 28760
- D. C. Drucker, Some implications of work hardening and ideal plasticity, Quart. Appl. Math. 7 (1950), 411–418. MR 34210, DOI https://doi.org/10.1090/S0033-569X-1950-34210-6
- 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
- Kerry S. Havner, A discrete model for the prediction of subsequent yield surfaces in polycrystalline plasticity, Internat. J. Solids Structures 7 (1971), 719–730 (Russian). MR 334793, DOI https://doi.org/10.1016/0020-7683%2871%2990089-8
K. S. Havner, On convergence of a discrete aggregate model in polycrystalline plasticity, Int. J. Solids Structures 7, 1269–1275 (1971)
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)
G. Strang and G. J. Fix, An analysis of the finite element method, Prentice-Hall, New Jersey (1973)
G. I. Taylor, The distortion of crystals of aluminum under compression, Part II, Proc. R. Soc. Lond. A116, 16–38 (1927)
G. I. Taylor, Plastic strain in metals, J. Inst. Metals 62, 307–324 (1938)
J. F. W. Bishop and R. Hill, A theoretical derivation of the plastic properties of a polycrystalline face-centered metal, Phil. Mag. (7) 42, 1298–1307 (1951)
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)
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)
J. W. Hutchinson, Elastic-plastic behaviour of polycrystalline metals and composites, Proc. R. Soc. Lond. A319, 247–272 (1970)
K. S. Havner and R. Varadarajan, A quantitative study of a crystalline aggregate model, Int. J. Solids Structures 9, 379–394 (1973)
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)
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)
W. T. Koiter, Stress-strain relations, uniqueness and variational theorems for elastic-plastic materials with a singular yield surface, Quart. Appl. Math. 11, 350–354 (1953)
J. Mandel, Generalization de la théorie de plasticité de W. T. Koiter, Int. J. Solids Structures 1, 273–295 (1965)
R. Hill, Generalized constitutive relations for incremental deformation of metal crystals by multislip, J Mech. Phys. Solids 14, 95–102 (1966)
G. Maier, A quadratic programming approach for certain nonlinear structural problems, Meccanica 3, 121–130 (1968)
G. Maier, "Linear” flow-laws of elastoplasticity: A unified general approach, Lincei-Rend. Sci. Fis. Mat, e Nat. (8) 47, 267–276 (1969)
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)
W. Prager, Recent developments in the mathematical theory of plasticity, J. Appl. Phys. 20, 235–241 (1949)
D. C. Drucker, Some implications of work hardening and ideal plasticity, Quart. Appl. Math. 7, 411–418 (1950)
D. C. Drucker, Variational principles in the mathematical theory of plasticity, Proc. Symposia Appl. Math. 8, McGraw-Hill, New York, 7–22 (1958)
K. S. Havner, A discrete model for the prediction of subsequent yield surfaces in polycrystalline plasticity, Int. J. Solids Structures 7, 719–730 (1971)
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© Copyright 1976
American Mathematical Society
