# zbMATH — the first resource for mathematics

Finite element analysis of plane couple-stress problems using first order stress functions. (English) Zbl 0629.73054
A complementary energy based variational principle, using first order stress functions, is developed for plane linear elastic couple-stress problems. The principle is analogous to that used in a total potential energy based Mindlin/Reissner thick plate bending analysis and as such is a generalization of the classical analogy between plate stretching and plate bending. Traction boundary conditions are enforced using a Lagrange multiplier technique. The resulting $$C^ 0$$ finite element ‘equilibrium stress model’ is validated by investigating the reduction of the stress concentration factor associated with a small hole in a field of uniform tension.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 65K10 Numerical optimization and variational techniques 74S30 Other numerical methods in solid mechanics (MSC2010)
Full Text:
##### References:
 [1] Voigt, Abhandlungen der Koniglichen Gesellschaft der Wissenschaften zu Gottingen 34 (1887) [2] and , Theorie des Corps Deformables, Herman et Fils, Paris, 1909. [3] Koiter, Proc. Royal Netherlands Acad. Sci., Series B LXVII pp 17– (1964) [4] Theory of Micropolar Elasticity, Vol. 2, Fracture, Academic Press, New York, 1968. Chapt. 7. · Zbl 0266.73004 [5] Mindlin, Exper. Mech. 3 pp 1– (1963) [6] Sternber, Int. J. Solids. Struct. 3 pp 69– (1967) [7] Atkinson, Int. J. Solids Struct. 13 pp 1103– (1977) [8] Hermann, Exper. Mech. 12 pp 235– (1972) [9] Lakes, J. Biomech. Eng. 104 pp 6– (1982) [10] Lakes, Int. J. Solids Struct. 22 pp 55– (1986) [11] Banks, Int. J. Solids Struct. 4 pp 15– (1968) [12] ’Mixed finite elements for couple-stress analysis’, Hybrid and Mixed Finite Element Methods, in et al. (eds.), Wiley, New York, 1983, pp. 1-17. [13] Private communication containing material omitted from Ref. 12 but contained in original typescript, Oct. 1984. [14] ’Application of micropolar elasticity to the finite element continuum analysis of articulated structures’, Ph.D. Thesis, Civil Engineering Department, University of California, Davis, U.S.A., 1984. [15] and , Solid Mechanics–A Variational Approach, McGraw-Hill, New York, 1973, Chapt. 7. [16] Pica, Comp. Struct. 11 pp 203– (1980) [17] de Veubeke, J. Strain Anal. 2 pp 265– (1967) [18] Elias, J. Eng. Mech. Div. ASCE 94 pp 931– (1968) [19] Gallagher, Proc. 1st. Int. Conf. on Structural Mechanics in Reactor Technology (SMIRT) 6 pp 443– (1972) [20] ’Stress function approach’, Proc. World Cong. on Finite Element Methods in Structural Mechanics, Bournemouth, U.K., J.1-J51, 1975. [21] de Veubeke, J. Franklin Inst. 302 pp 389– (1976) [22] Dost, Trans. CSME 5 pp 1– (1978) [23] Baluch, J. Struct. Div. ASCE 98 pp 1957– (1972) [24] Malcolm, Int. J. Eng. Sci. 20 pp 1111– (1982) [25] Vallabhan, Int. j. numer. methods eng. 18 pp 291– (1982) [26] ’Finite element analysis of couple-stress problems using a penalty function method for the boundary constraints’, Report C/M/208/85, Department of Civil Engineering, University College of Swansea, Wales, U.K. [27] and , ’Complementary energy revisited’, in et al. (eds), Hybrid and Mixed Finite Element Methods, Wiley, New York, 1983, pp. 433-465. [28] and , Theory of Elasticity, 2nd. edn., McGraw-Hill, New York, 1951. [29] The Finite Element Method, 3rd edn., McGraw-Hill, London, 1977. [30] ’Finite element analysis of couple-stress problems’, Report C/M/193/84, Department of Civil Engineering, University of Swansea, Wales, U.K. [31] Hughes, Comp. Struct. 9 pp 445– (1978) [32] and , ’A study of locking phenomena in isoparametric elements’, in Mathematics of Finite Elements and Applications II, MAFELAP, 1978, Brunel University, 1979, pp. 437-447. [33] Hinton, Comp. Struct. 23 pp 409– (1986) [34] and , Theory of Plates and Shells, 2nd edn., McGraw-Hill, New York, 1969. [35] Bleustein, Int. J. Solids Struct. 3 pp 1053– (1967) [36] and , ’Finite element analysis of couple-stress problems’, in and , (eds.), Int. Conf. on Numerical Methods in Engineering (NUMETA 85) Vol. 2 Swansea, A. A. Balkema, Rotterdam, 1985, pp. 907-12. [37] Farshad, Mech. Res. Comm. 3 pp 399– (1976)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.