A class of mixed assumed strain methods and the method of incompatible modes. (English) Zbl 0724.73222

Summary: A three-field mixed formulation in terms of displacements, stresses and an enhanced strain field is presented which encompasses, as a particular case, the classical method of incompatible modes. Within this framework, incompatible elements arise as particular ‘compatible’ mixed approximations of the enhanced strain field. The conditions that the stress interpolation contain piecewise constant functions and be \(L_ 2\)-orthogonal to the enhanced strain interpolation, ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation. The preceding conditions are formulated in a form particularly convenient for element design. As an illustration of the methodology three new elements are developed and shown to exhibit good performance: a plane 3D elastic/plastic QUAD, an axisymmetric element and a thick plate bending QUAD. The formulation described herein is suitable for nonlinear analysis.


74S05 Finite element methods applied to problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
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[1] and , ’A simplified analysis of two plate bending elements–The MITC4 and MITC9 elements’, Proc. Conference NUMETA 87, University College of Swansea, Wales, July 1987.
[2] Bathe, Int. J. Comp. Aided Eng. Software 1 (1984)
[3] , and , ’Triangular elements in plate bending–Conforming and nonconforming solutions’, Proc. First Conference on Matrix Methods in Structural Mechanics, Wright-Patterson ATBFB, Ohio, 1965.
[4] Belytschko, Comp. Methods Appl. Mech. Eng. 23 pp 323– (1986)
[5] The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
[6] Hughes, Int. j. numer. methods eng. 15 pp 1413– (1980)
[7] The Finite Element Method, Prentice-Hall, Englewood Cliffs, New Jersey, 1987.
[8] and , ’Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element’, J. Appl. Mech. ASME, 587-596 (1981). · Zbl 0459.73069
[9] and , ’Implementation of a time dependent plasticity theory into structural computer programs’, in and (eds.), Constitutive Equations in Viscoplasticity: Computational and Engineering Aspects, AMD-20, ASME, New York, 1976.
[10] Nagtegaal, Comp. Methods Appl. Mech. Eng. 4 pp 153– (1974)
[11] Pian, Int. j. numer. methods eng. 20 pp 1685– (1985)
[12] Puch, Comp. Methods Appl. Mech. Eng. 47 pp 331– (1984)
[13] Simo, J. Appl. Mech., ASME 53 pp 51– (1986)
[14] and , Plasticity, Viscoplasticity and Viscoelasticity: Formulation, Algorithms and Numerical Analysis, Springer-Verlag, Berlin, to appear.
[15] Simo, Comp. Methods Appl. Mech. Eng. 48 pp 101– (1985)
[16] Simo, Int. j. numer. methods eng. 22 pp 649– (1986)
[17] Simo, Int. j. numer. methods eng. 26 pp 2161– (1988)
[18] Simo, Comp. Methods Appl. Mech. Eng.
[19] Simo, Comp. Methods Appl. Mech. Eng. 51 pp 177– (1985)
[20] and , An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, New Jersey, 1973.
[21] [1989], Private Communication, Aug. 1989.
[22] Taylor, Int. j. numer. methods eng. 10 pp 1211– (1976)
[23] Taylor, Int. j. numer. methods eng. 22 pp 39– (1986)
[24] ’Calculation of elastic-plastic flow’, in et al. (eds.), Methods of Computational Physics 3, Academic Press, New York, 1964.
[25] , and , ’Incompatible displacement models’, in et al. (eds.), Numerical and Computer Models in Structural Mechanics, Academic Press, New York, 1973.
[26] [1977], The Finite Element Method, 3rd edn, McGraw-Hill, London, 1977.
[27] and , The Finite Element Method, Vol. 1 4th edn, McGraw-Hill, London, 1989.
[28] ’Finite element analysis of cellular structures’, Ph.D. thesis at University of California, Berkeley.
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