Error estimates for stresses in the finite element analysis of the two-dimensional elasto-plastic problems. (English) Zbl 0899.73532

Summary: In the present paper an error estimate for finite element approximations based on the Haar-Karman variational principle is given. To satisfy exactly equilibrium equations in stresses equilibrium finite elements are used while yield condition is satisfied approximately by means of the penalty function having clear mechanical sense. It is shown that there is some relation between parameters of discretization and penalty which gives a qualified error estimate for stresses.


74S05 Finite element methods applied to problems in solid mechanics
74C99 Plastic materials, materials of stress-rate and internal-variable type
65N15 Error bounds for boundary value problems involving PDEs
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[1] Anzellotti, G.; Giaquinta, M., Manuscripta Math., 32, 101 (1980)
[2] Kohn, R.; Temam, R., Appl. Math. Optim., 10, 1 (1983)
[3] Seregin, G., Zh. Vychisl. Mat. i Mat. Fiz., 25, 237 (1985)
[4] Johnson, C.; Scott, R., A finite element method for problems of perfect plasticity using discontinuous trial functions, (Nonlinear Finite Elem. Anal. Struct. Mech. Proc. Eur. U.S. Workshop. Bochum, 1980. Nonlinear Finite Elem. Anal. Struct. Mech. Proc. Eur. U.S. Workshop. Bochum, 1980, Berlin (1981)), 307-324
[5] Repin, S., Russ. J. Numer. Anal. Model., 9, 33 (1994)
[6] Repin, S., Zh. Vychisl. Mat. i Mat. Fiz., 28, 449 (1988), (in Russian)
[7] Repin, S., USSR Comp. Maths. Math. Phys., 27 (1987), English translation in
[8] Seregin, G., A local Caccioppoli-type estimate for the extremals of variational problems in Hencky plasticity, (Some Applications of Functional Analysis to Problems of Mathematical Physics, Izd. Inst. Math. Akad. Nauk SSSR (1988)), 127-138, (in Russian) · Zbl 0769.73098
[9] Seregin, G., Differentiability properties of the stress tensor in perfect elastic-plastic theory, (Preprint UTM 321 September 1990 (1990), Universita degli Studi di Trento)
[10] Seregin, G., Problems Math. Anal., 11, 51 (1989)
[11] Seregin, G., Diferensial’nye Uravneniyz, 23, 1987 (1981)
[12] Seregin, G., Differensial’nye Uravneniya, 26, 1033 (1990)
[13] Seregin, G., Algebra Analysis, 2, 121 (1990)
[14] Oden, J. T., Finite Elements in Nonlinear Continua (1972), McGraw-Hill: McGraw-Hill New York · Zbl 0235.73038
[15] Ciarlet, P., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam
[16] Hlavacek, I., Aplikace Matematiky, 26, 449 (1981)
[17] Hlavacek, I., Aplikace Matematiky, 31, 486 (1986)
[18] Oden, J. T.; Whiteman, J., Int. J. Engng Sci., 20, 977 (1982)
[19] Mosolov, P.; Miasnikov, V., Mechanics of Rigid Plastic Bodies (1981), Nauka: Nauka Moscow
[20] Johnson, C.; Mercier, B., Numer. Math., 30, 101 (1978)
[21] Duvaut, G.; Lions, J.-L., Les Inequations en Mecanique et en Physique (1972), Dunod: Dunod Paris · Zbl 0298.73001
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