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A problem in dynamic plasticity: the enlargement of a circular hole in a flat sheet

Published online by Cambridge University Press:  24 October 2008

W. Freiberger
W. Freiberger
Affiliation:
Clare CollegeCambridge
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Abstract

The problem discussed in this paper is the enlargement of a circular hole in a flat sheet of which the material is in plastic flow under constant shear stress; inertia terms are taken into account in the equilibrium equations so as to permit the occurrence of high velocities.

The treatment is in some ways analogous to that of problems in one-dimensional compressible gas flow. The characteristic equations are seen to represent straight lines under certain initial conditions; this facilitates consideration of any mode of enlargement; in particular, uniform acceleration is illustrated by numerical calculations. There appears a circular shock-front, i.e. a discontinuity in thickness and velocity moving ahead of the hole. The initial conditions for the existence of this ‘simple wave' solution involve a singularity at the centre of the plate.

In general, the characteristic curves will not be straight lines and the equations are best solved step by step. Their characteristic form is well adapted to such a process; this is outlined for the interesting case of initial enlargement with uniform velocity followed by expansion under constant acceleration.

Curves illustrate some of the general results.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 1 , January 1952 , pp. 135 - 148
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

REFERENCES

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