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 Free Access

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.

**[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**

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