Summary
In this paper an investigation is given into the behaviour of the stress singularity which occurs in the linear theory of elasticity at the deformation of a sector plate, if finite deformations are considered. It is assumed that for very small deformations Hooke's law is valid, and only in the neighbourhood of the singularity Hooke's law has to be extended. This extension is not unique. It is shown that for two different strain-energy functions, which have the same asymptotic expansion for infinitesimal deformations, the behaviour of the solutions is quite different. One of the strain-energy functions leads to a bounded solution, while the solution, obtained from the other one becomes singular for the case of contraction. As it cannot be expected that it will be possible to decide on an experimental basis about the right extension, an assertion about the difference in smoothness of solutions to problems in linear and non-linear theory cannot be given. An open question is raised: whether or not the requirement of regularity for this kind of problems in non-linear theory can be posed as a restriction on the admissible energy functions.
Similar content being viewed by others
References
Sansenig, C. B., Non-linear theory for the deformation of pre-stressed circular plates and rings, New York Univ., Courant Inst. of Math. Sc., Rep. I.M.M.-N.Y.U.-326, 1964.
John, F., Comm. Pure Appl. Math. XIII (1960) 239.
Truesdell, C. and R. Toupin, The classical field theories, Handb. der Physik, III, 1, Springer, Berlin, 1960.
Eringen, A. C., Non-linear theory of continuous media, McGraw Hill, New York, 1962.
Biot, M. A., Mechanics of Incremental Deformations, Wiley & Sons, New York, 1965.
Truesdell, C. and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, III, 3, Springer, Berlin, 1965.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Alblas, J.B. On the finite deformation of a sector plate. Appl. Sci. Res. 17, 1–13 (1967). https://doi.org/10.1007/BF00387563
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00387563