Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T18:19:19.284Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-linear optimization of the material constants in Ogden's stress-deformation function for incompressinle isotropic elastic materials

Published online by Cambridge University Press:  17 February 2009

E. H. Twizell  and
R. W. Ogden
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In previous papers, three terms have been included in Ogden's stress-deformation function for incompressible isotropic elastic materials. The material constants have been calculated by elementary methods and the resulting fits to sets of experimental data have been moderately good.

The purpose of the present paper is to improve upon established correlation between theory and experiment by means of a systematic optimization procedure for calculating material constants. For purposes of illustration the Levenberg-Marquardt non-linear least squares optimization algorithm is adapted to determine the material constants in Ogden's stress-deformation function.

The use of this algorithm for three-term stress-deformation functions improves somewhat on previous results. Calculations are also carried out in respect of a four-term stress-deformation function and further improvement in the fit is achieved over a large range of deformation.

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 4 , April 1983 , pp. 424 - 434
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Chadwick, P., Creasy, C. F. M. and Hart, V. G., “The deformation of rubber cylinders and tubes by rotation”, J. Austral. Math. Soc. Ser. B 20 (1977), 62–96.Google Scholar
[2]James, A. G., Green, A. and Simpson, G. M., “Strain-energy functions of rubber I: Characterization of gum vulcanizates”, J. Appl. Polymer Sci. 19 (1975), 2033–2058.Google Scholar
[3]Jones, D. F. and Treloar, L. R. G., “The properties of rubber in pure homogeneous strain”, J. Phys. D: Appl. Phys. 8 (1975), 1285–1304.CrossRefGoogle Scholar
[4]Levenberg, K., “A method for the solution of certain non-linear problems in least squares”, Quart. Appl. Math. 2 (1944), 164–168.Google Scholar
[5]Marquardt, D. W., “An algorithm for least squares estimation of non-linear parameters”, SIAM J. Appl. Math. 11 (1963), 431–441.Google Scholar
[6]Ogden, R. W., “Large deformation isotropic elasticity–on the correlation of theory and experiment for incompressible rubberlike solids”, Proc. Roy. Soc. London Ser. A 326 (1972), 565–584.Google Scholar
[7]Ogden, R. W., “Elastic deformations of rubberlike solids”, in Mechanics of Solids, The Rodney Hill 60th Anniversary Volume (eds. Hopkins, H. G. and Sewell, M. J.), (Pergamon Press, Oxford, 1982).Google Scholar
[8]Shrager, R. I. and Hill, E. Jr, “Non-linear curve fitting in the L1 and L∞ norms”, Math. Comp. 34 (1980), 529–541.Google Scholar
[9]Treloar, L. R. G., “Stress-strain data for vulcanised rubber under various types of deformation”, Trans. Faraday Soc. 40 (1944), 59–70.Google Scholar
[10]Treloar, L. R. G. and Riding, G., “A non-Gaussian thory for rubber in biaxial strain I: Mechanical properties”, Proc. Roy. Soc. London Ser. A 369 (1980), 261–280.Google Scholar