Previous |  Up |  Next


plane elastostatics; stress equilibrium finite element; slab analogy; choice of the degrees of freedom; normal trace; stress tensor; complementary energy functional; existence; convergence
The fundamental problem in the application of the principle of complementary energy is the construction of suitable subsets that approximate the set of all statically admissible fields satisfying both the conditions of equilibrium inside the body and the static boundary conditions. The notion "slab analogy" is motivated and the interface conditions for the Airy stress function are established at the contact of two domains. Some spaces of types of conforming equilibrium stress elements, which can be obtained by slab analogy, are investigated. A weak version of the Castigliano principle is established and the approximate variational problem is defined by using equilibrium stress fields. Some subspaces of equilibrium stress elements are introduced and a priori error estimates in the $L^2$-norm (provided the solutions are smooth enough) and convergence results are obtainded from the well-known results for compatible finite elements.
[1] J. Nečas I. Hlaváček: Mathematical theory of elastic and elasto-plastic bodies: An introduction. Elsevier Sci. Publ. Соmр., Amsterdam, Oxford, New York, 1981. MR 0600655
[2] P. G. Ciarlet: The finite element method for elliptic problems. North-Holland Publ. Соmр., Amsterdam, New York, Oxford, 1978. MR 0520174 | Zbl 0383.65058
[3] B. M. Fraeijs de Veubeke: A course in elasticity. Springer Verlag, New York, Heidelberg, Berlin, 1979. MR 0533738
[4] V. Girault P. A. Raviart: Finite element approximation of the Navier-Stokes equations. Springer-Verlag, Berlin, Heidelberg, New York, 1979. MR 0548867
[5] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. MR 0227584
[6] I. Hlaváček: Variational principles in the linear theory of elasticity for general boundary conditions. Apl. Mat. 12 (1967), 425-448. MR 0231575
[7] I. Hlaváček: Convergence of an equilibrium finite element model for plane elastosiatics. Apl. Mat. 24 (1979), 427-457. MR 0547046
[8] I. Hlaváček: Some equilibrium and mixed model in the finite element method. Banach Center Publ., Vol. 3, 1975, 147-165. MR 0514379
[9] J. Haslinger I. Hlaváček: Convergence of a finite element method based on the dual variational formulations. Apl. Mat. 21 (1976), 43 - 65. MR 0398126
[10] J. Haslinger I. Hlaváček: Contact between elastic bodies - III. Dual finite Element Analysis. Apl. Mat. 26 (1981), 321-344. MR 0631752
[11] M. Křížek: Equilibrium elements for the linear elasticity problem. Variational - difference methods in math. phys., Moscow, 1984, 81 - 92.
[12] G. Sander: Applications of the dual analysis principle. Proceedings of the IUTAM Symp. on High Speed Computing of Elastic Structures, Congrés et Colloques Do l'Université de Liege (1971), 167-207.
[13] B. F. Veubeke G. Sander: An equilibrium model for plate bending. Internat. J. Solids and Structures 4 (1968), 447-468. DOI 10.1016/0020-7683(68)90049-8
[14] V. B. Watwood B. J. Hartz: An equilibrium stress field model for the finite element solutions of two-dimensional elastostatic problems. Internat. J. Solides and Structures 4 (1968), 857-873. DOI 10.1016/0020-7683(68)90083-8
[15] C. Johnson B. Mercier: Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math. 30 (1978), 103-116. DOI 10.1007/BF01403910 | MR 0483904
[16] M. Křížek: An equilibrium finite element method in three-dimensional elasticity. Apl. Mat. 27 (1982), 46-75. MR 0640139
[17] M. Křížek: Conforming equilibrium finite element methods for some elliptic plane problems. R.A.I.R.O. Analyse numérique, vol. 17, No. 1, 1983, 35-65. MR 0695451
[18] D. J. Allman: On compatible and equlibrium models with linear stresses for stretching of elastic plates. Energy methods in finite element analysis. John Wiley& Sons Ltd. Chichester, New York, Brisbane, Toronto, 1979. MR 0537002
[19] B. F. de Veubeke O. C. Zienkiewicz: Strain energy bounds in finite element analysis by slab analogy. J. of Strain Analysis, 1967, vol. 2, No. 4.
[20] I. Babuška K. Rektorys F. Vyčichlo: Mathematische Elastizitätstheorie der ebenen Probleme. Akademieverlag, Berlin 1960. MR 0115343
[21] J. H. Michell: On the direct determination of stress in an elastic solid with application to the theory of plates. Proc. Lond. math. Soc. 1899, 31, 100.
Partner of
EuDML logo