© 1948 by Oxford University Press
THE FORMATION AND ENLARGEMENT OF A CIRCULAR HOLE IN A THIN PLASTIC SHEET
( Trinity College Cambridge )
When a circular hole is made in a flat sheet by a conical-headed bullet or by outward radial pressure on its edge, the metal near the hole piles up into a thickened crater. The mechanics of this deformation is discussed. The interest of the problem lies in the fact that the complete strain-history of each element of the sheet has to be calculated. This is because the ratios of the principal stresses at each element of the sheet vary as the deformation proceeds, so that there is no relationship between the stress and total deformation but only between stress and strain increments occurring during a small expansion of the hole.
If b is the radius of the hole at any time, the strain is found to be elastic at points where r, the radius, is > 3.64b. In the annulus 2.21b < r < 3.64b the strain is plastic, but comparable in magnitude with the small elastic strain when r >3.64b. In the annulus b < r < 2.21b there is finite strain. At the edge of the hole the sheet has thickened to 2.61 times the thickness of the sheet.
Experiments made with lead show that the symmetrical deformation contemplated in this analysis does not occur; but an alternative unsymmetrical deformation is produced which calculation shows to require less work, in the ratio 2.6 to 1.0, than the symmetrical mode.