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On axially symmetric crack and punch problems for a medium with transverse isotropy†

Published online by Cambridge University Press:  24 October 2008

L. E. Payne  and
A. E. Green
L. E. Payne
Affiliation:
Institute for Fluid Dynamics and Applied MathematicsUniversity of Maryland, U.S.A.
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Extract

1. Introduction. The problem of determining the distribution of stress in a medium with transverse isotropy (one axis of elastic symmetry) has recently been considered by Elliott (l, 2) and Shield (3). They have obtained solutions to certain special problems both with and without rotational symmetry. Elliott (2) has employed the method of dual integral equations to solve a number of axially symmetric crack and punch problems. (This crack problem is not to be confused with the torsional crack problem treated by Weinstein (4)). Shield (3) has reduced the general crack and punch problems for a medium with transverse isotropy to corresponding isotropic problems as formulated by Green (5), and has treated a few special examples. For the axially symmetric isotropic crack and punch problems the author (6) has shown that Green's formulation may be simplified, and has obtained in each case a simple usable solution of the general axially symmetric problem without the need for dual integral equations. As one might expect these simplifications may also be realized in the case of a medium with transverse isotropy. In this paper the author, guided by the work of Green (5) and Shield (3), presents a straightforward method for solving the general axially symmetric crack and punch problems for media with transverse isotropy. This method permits one to write down immediately in an appropriate system of curvilinear coordinates the solution to the general axially symmetric problem, whereas the method given by Shield requires an individual treatment of each problem. In fact it would appear to be extremely difficult if not impossible to obtain the complete solution to the general axially symmetric crack or punch problem by the methods suggested by Shield. Since the formulation given in this paper differs from that given by Shield, his equations for transforming the problem in transverse isotropy to an isotropic problem must be altered slightly. The solutions to the general axially symmetric problems for a medium with transverse isotropy then follow immediately from the solutions to the corresponding isotropic problems in (6). We shall therefore find it necessary to repeat some of the arguments and results of (6).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 3 , July 1954 , pp. 466 - 473
Copyright
Copyright © Cambridge Philosophical Society 1954

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References

REFERENCES

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