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On controllable deformations in a mixture of two non-linear elastic solids

Published online by Cambridge University Press:  26 February 2010

T. R. Steel
T. R. Steel
Affiliation:
Center for the Application of Mathematics, Lehigh University, Bethlehem, Pennsylvania, U.S.A.
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Abstract

Recently there has been much interest in theories describing the behaviour of a mixture of two or more continua, which are allowed to diffuse through each other and which interact mechanically and thermally. In the present note we consider such a mixture consisting of two isotropic incompressible non-linear elastic solids; physically this may describe a composite of two rubber-like materials. Such a mixture possesses internal friction between the constituents when deformed (called diffusive resistance or diffusive force), which acts as a damping mechanism. This damping mechanism has a static part and a dynamic part, and we here show that the exact static solutions which exist for an isotropic incompressible non-linear elastic solid are such that the damping mechanism is expressible as a gradient, and hence are also controllable for such a mixture. We also discuss the corresponding dynamic solutions.

From a practical viewpoint, some experiments carried out on vulcanized synthetic rubber-like materials may involve a mixture rather than a pure solid.

Type
Research Article
Information
Mathematika , Volume 16 , Issue 1 , June 1969 , pp. 50 - 56
Copyright
Copyright © University College London 1969

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References

1.Singh, M. and Pipkin, A. C., Z.A.M.P., 16 (1965), 706.Google Scholar
2.Rivlin, R. S., Phil. Trans. Roy. Soc. London A, 241 (1948), 379.Google Scholar
3.Rivlin, R. S., Phil. Trans. Roy. Soc. London A, 242 (1949), 173.Google Scholar
4.Green, A. E. and Shield, R. T., Proc. Roy. Soc. London A, 202 (1950), 407.Google Scholar
5.Adkins, J. E., Green, A. E. and Shield, R. T., Phil. Trans. Roy. Soc. London A, 246 (1953), 181.Google Scholar
6.Ericksen, J. L. and Rivlin, R. S., J. Rat'l Mech. Anal., 3 (1954), 281.Google Scholar
7.Ericksen, J. L. and Rivlin, R. S., Z.A.M.P., 5 (1954), 466.Google Scholar
8.Klingbeil, W. W. and Shield, R. T., Z.A.M.P., 17 (1966), 489.Google Scholar
9.Carroll, M. M., Int. J. Engng. Sci., 5 (1967), 515.CrossRefGoogle Scholar
10.Carroll, M. M., Quart. J. Mech. Appl. Math. (In press.)Google Scholar
11.Coleman, B. D. and Truesdell, C., Z.A.M.M., 45 (1965), 547.Google Scholar
12.Christensen, R. M., J. Appl. Mech. (In press.)Google Scholar
13.Green, A. E. and Naghdi, P. M., Int. J. Engng. Sci., 3 (1965), 231.CrossRefGoogle Scholar
14.Mills, N. and Steel, T. R.. (To be published.)Google Scholar
15.Green, A. E. and Naghdi, P. M., A Note on Mixtures. (To appear.)Google Scholar
16.Green, A. E. and Zerna, W., Theoretical elasticity (Oxford University Press, New York, 1954).Google Scholar
17.Truesdell, C., Arch. Rat'l Mech. Anal., 11 (1962), 106.CrossRefGoogle Scholar
18.Green, A. E. and Steel, T. R., Int. J. Engng. Sci., 4 (1966), 483.CrossRefGoogle Scholar