Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T16:26:47.525Z Has data issue: false hasContentIssue false
  • English
  • Français

Elliptic inclusions in a stressed matrix

Published online by Cambridge University Press:  24 October 2008

R. D. Bhargava  and
H. C. Radhakrishna
R. D. Bhargava
Affiliation:
Indian Institute of Technology, Kanpur, India
H. C. Radhakrishna
Affiliation:
Indian Institute of Technology, Kanpur, India
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper treats an extension of the problem considered by the authors in a recent paper (1). The minimum energy principle of the classical theory of elasticity was used in the above paper for evaluating the elastic field when an elliptic region (the inclusion, which could be of a material different from the rest) undergoes spontaneous dimensional change in an otherwise unstrained infinite medium (the matrix). By modification of this method, it has been possible to deal with the case when the inclusion is spherical or circular and the matrix is under uniform tension at infinity (2). The present paper deals with the much more general case when the matrix is under tension, at infinity, inclined at any angle to the major axis of the elliptic inclusion. The solution has been possible by the combination of the complex variable method coupled with minimum energy principle and superposition methods of linear elasticity theory. As a consequence we immediately derive almost without further calculation many particular cases, viz. (i) the inclusion problem in a matrix under axial tension parallel to either of the axes, (ii) under all round uniform tension (or pressure) etc. It is obvious that the results for the respective cases of a circular inclusion can be deduced from these results.

It also solves the problem of composite sections under external forces at infinity because of the complete freedom in choosing the elastic constant of the inclusion which can be different from that of the matrix. As a corollary, it solves the problem of a cavity under stress at infinity.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 4 , October 1963 , pp. 821 - 832
Copyright
Copyright © Cambridge Philosophical Society 1963

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Bhargava, R. D. and Radhakrishna, H. C. To appear in Proc. Cambridge Philos. Soc.Google Scholar
(2)Bhargava, R. D. and Radhakrishna, H. C. To appear in Appl. Sci. Res.Google Scholar
(3)Mott, N. F. and Nabarro, F. R. N.Proc Phys. Soc. 52 (1940), 90.CrossRefGoogle Scholar
(4)Eshelby, J. D.Proc. Roy. Soc. London, Ser. A, 241 (1957), 376; 252 (1959), 561.Google Scholar
(5)Jaswon, M. A. and Bhargava, R. D.Proc. Cambridge Philos. Soc. 57 (1961), 669.CrossRefGoogle Scholar
(6)Bhargava, R. D.Appl. Sci. Res. A, 10 (1961).Google Scholar
(7)Love, A. E. H.The mathematical theory of elasticity, 4th ed. (Cambridge, 1927).Google Scholar
(8)Green, A. E. and Zerna, W.Theoretical elasticity (Oxford, 1954).Google Scholar
(9)Sokolnikoff, I. S.Mathematical theory of elasticity (McGraw-Hill; New York, 1956).Google Scholar
(10)Muskhelishvili, N. I.Some basic problems of the mathematical theory of elasticity (Noord-hoff: Groningen, 1953).Google Scholar
(11)Stevenson, A. C.Phil. Mag. (7), 34 (1943), 766.CrossRefGoogle Scholar