A finite element procedure, which utilizes Fourier series, is developed to compute the large. The traction integral is evaluated only on S, since 6u is zero on S.
![Finite element analysis pdf Finite element analysis pdf](/uploads/1/2/5/6/125677403/998336345.png)
Problems in Linear Elasticity and Fields
O.C. Zienkiewicz, ... J.Z. Zhu, in The Finite Element Method: its Basis and Fundamentals (Seventh Edition), 2013
2.2.4.2 Axisymmetric problems
For axisymmetric problems the stress is replaced by (2.15) and body force by . Using the stresses on the element of volume shown in Fig. 2.2 and summing forces in the and directions the two equilibrium equations are given by
(2.23)
By moment equilibrium (angular momentum) we again obtain . Note that in the axisymmetric problem the differential operator on equilibrium is given by
and is not equal to . This difference occurs due to the curvilinear coordinate system. Working with the differential equation form this is a disadvantage, however, for methods used in finite element analysis we will show that the difference disappears and formulations in curvilinear coordinates greatly simplify.
- Adkins, J. E.: Studies in the theory of large elastic deformations. Ph. D. thesis, University of London 1951.Google Scholar
- Jordan, P. F.: Stresses and deformations of the thin-walled pressurized torus. J. Aeron. Sci. 29, 213–225 (1962).Google Scholar
- Sanders, J. L. Jr., Liepins, A. A.: Toroidal membrane under internal pressure. AIAA J. 1, 2105–2110 (1963).Google Scholar
- Kydoniefs, A. D., Spencer, A. J. M.: The finite inflation of an elastic toroidal membrane of circular cross section. Int. J. Engng. Sci. 5, 367–391 (1967).Google Scholar
- Kydoniefs, A. D.: The finite inflation of an elastic toroidal membrane. Int. J. Engng. Sci. 5, 477–494 (1967).Google Scholar
- Li, X., Steigmann, D. J.: Finite deformation of a pressurized toroidal membrane. Int. J. Non-Linear Mech. 30, 583–595 (1995).Google Scholar
- Papargyri–Pegiou, S.: Stability of the axisymmetric analytical and numerical solutions in a thin-walled pressurized torus of compressible nonlinear elastic material. Int. J. Engng. Sci. 33, 1005–1025 (1995).Google Scholar
- Papargyri–Pegiou, S., Stavrakakis, E.: Axisymmetric numerical solutions of a thin-walled pressurized torus of incompressible nonlinear elastic materials. Comput. Struct. 77, 747–757 (2000).Google Scholar
- Green, A. E., Zerna, W.: Theoretical elasticity, 2nd ed. Oxford: The Clarendon Press 1968.Google Scholar
- Green, A. E., Adkins, J. E.: Large elastic deformations, 2nd ed. Oxford: The Clarendon Press 1970.Google Scholar
- Beatty, M. F.: Topics in finite elasticity: hyperelasticity of rubber, elastomers and biological tissues–with examples. Appl. Mech. Rev. ASME 40, 1699-1734 (1987).Google Scholar
- Libai, A., Simmonds, J. G.: The nonlinear theory of elastic shells, 2nd ed. Cambridge: Cambridge University Press 1998.Google Scholar
- Klingbeil, W. W., Shield, R. T.: Some numerical investigations on empirical strain energy functions in the large axisymmetric extensions of rubber membranes. Z. Angew. Math. Phys. (Zamp) 15, 608–629 (1964).Google Scholar
- Foster, H. O.: Very large deformations of axially symmetrical membranes made of neo-Hookean materials. Int. J. Engng. Sci. 5, 95–117 (1967).Google Scholar
- Wu, C. H.: Large finite strain membrane problems. Q. Appl. Math. 36, 347–359 (1979).Google Scholar
- Sagiv, A.: Inflation of an axisymmetric membrane: stress analysis. J. Appl. Mech. ASME 57, 682–687 (1990).Google Scholar
- Roxburgh, D. G.: Inflation of nonlinearly deformed annular elastic membranes. Int. J. Solids Struct. 32, 2041–2052 (1995).Google Scholar
- Oden, J. T., Sato, T.: Finite strains and displacements of elastic membranes by the finite element method. Int. J. Solids Struct. 1, 471–488 (1967).Google Scholar
- Murakawa, H., Atluri, S. N.: Finite elasticity solutions using hybrid finite elements based on a complementary energy principle. Parts 1 and 2. J. Appl. Mech. ASME 45, 539–547 (1978) and 46, 71–77 (1979).Google Scholar
- Wriggers, P., Taylor, R. L.: A fully nonlinear axisymmetrical membrane element for rubber-like materials. Engng. Comput. 7, 303–310 (1990).Google Scholar
- Chang, T. Y. P., Saleeb, A. F., Li, G.: Large strain analysis of rubber-like materials based on a perturbed Lagrangian variational principle. Comput. Mech. 8, 221–233 (1991).Google Scholar
- Gruttmann, F., Taylor, R. L.: Theory and finite element formulation of rubber-like membrane shells using principal stretches. Int. J. Num. Meth. Eng. 35, 1111–1126 (1992).Google Scholar
- Bufler, H., Schneider, H.: Large strain analysis of rubber-like membranes under dead weight, gas pressure and hydrostatic loading. Comput. Mech. 14, 165–188 (1994).Google Scholar
- Khayat, R. E., Perdouri, A.: Inflation of hyperelastic cylindrical membranes as applied to blow moulding: I. axisymmetric case, II. non-axisymmetric case. Int. J. Num. Meth. Engng. 37, 3773–3791 and 3793–3808 (1994).Google Scholar
- De Souza Neto, E. A., Peric, D., Owen, D. R. J.: Finite elasticity in spatial description: linearization aspects with 3-D membrane applications. Int. J. Num. Meth. Eng. 38, 3365–3381 (1995).Google Scholar
- Oden, J. T.: Finite elements of nonlinear continua. London: McGraw-Hill 1972.Google Scholar
- Bathe, K. J.: Finite element procedures. Englewood Cliffs, New Jersey: Prentice Hall 1996.Google Scholar
- Zienkiewicz, O. C., Taylor, R. L.: The finite element method, 5th ed., vol. 2: Solid mechanics. Oxford: Butterworth/Heinemann 2000.Google Scholar
- Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Q. Appl. Math. 2, 164–168 (1944).Google Scholar
- Marquardt, D. W.: An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Indust. Appl. Math. 11, 431–441 (1963).Google Scholar
- Osborne, M. R.: Nonlinear least squares – the Levenberg algorithm revisited. J. Austral. Math. Soc. 19B, 343–357 (1976).Google Scholar
- Moré, J. J.: The Levenberg–Marquardt algorithm: implementation and theory. In: Numerical analysis. Watson, G. A. ed., Lecture Notes in Mathematics vol. 630, pp. 105–116. Berlin: Springer 1978.Google Scholar
- Simo, J. C., Fox, D. D.: On a stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization. Comp. Meth. Appl. Mech. Engng. 72, 267–304 (1989).Google Scholar
- Gallagher, R. H.: Finite element analysis: fundamentals. Englewood Cliffs, NJ: Prentice-Hall 1975.Google Scholar
- Blatz, P. J., Ko, W. L.: Application of finite elasticity to the deformation of rubbery materials. Trans. Soc. Rheol. 6, 223–251 (1962).Google Scholar