Summary
The variational principles of nonlinear elasticity are applied to a problem of axially symmetric deformation of a uniform circular hyperelastic membrane. The supported edge of the membrane is in a horizontal plane and its radius is equal to that of the undeformed plane reference configuration, so that an initially plane unstretched membrane is subjected to a dead load due to its weight.
It is shown how the stationary complementary energy principle can be used to obtain an accurate approximate solution for the deformation and stress distribution. It is also shown how the potential energy principle can be applied to the problem and how close bounds for an energy functional can be obtained from the two theorems. Numerical results are presented for realistic properties for a rubberlike material and for two strain energy functions, the semi-linear and the neo-Hookean.
Similar content being viewed by others
References
Koiter, W. T.: On the complementary energy theorem in non-linear elasticity. In: Trends in application of pure mathematics (Fichera, G., ed.), pp. 207–232. London: Pitman Publ. 1975.
Lee, S. J., Shield, R. T.: Variational principles in finite elasto-statics. ZAMP31 (4), 437–453 (1980).
Lee, S. J., Shield, R. T.: Applications of variational principles in finite elasticity. ZAMP31 (4), 454–472 (1980).
Haughton, D. M., Ogden, R. W.: On the incremental equations in non-linear elasticity-I. Membrane theory. J. Mech. Phys. Solids26, 93–110 (1978).
Ogden, R. W.: Non-linear elastic deformations. Chichester: Ellis Horwood 1984.
Chen, C., Kong, W. C., Cha, J. Z.: An equality constrained RQP algorithm based on the augmented Lagrangian penalty function. ASME J. Mech. Trans. Autom. Design111, 368–375 (1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Liu, S., Haddow, J.B. & Dost, S. A variational approach to a circular hyperelastic membrane problem. Acta Mechanica 99, 191–200 (1993). https://doi.org/10.1007/BF01177244
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01177244