The use of friction in the shaping of a flat sheet into a hemisphere

An analysis is given for the use of friction to control the thickness distribution in a hemispherical preform shell. As the friction is changed between a hemispherical punch and an initially flat sheet that is to be formed into a hemispherical shell, the thickness distribution of the resulting shell changes. When the friction is low, the shell tends to be thinner near the pole of the hemisphere than near the equator. As the friction between the sheet and the punch surface increases, the thinning of the shell moves away from the pole toward the equator. The results of the analysis were validated by experiments with a natural rubber sheet that had the deformability necessary to test the predictions of the analysis. The friction distribution necessary to yield a shell of uniform thickness was also derived, and two-zone and three-zone combinations of constant friction that yield approximately uniform thickness in the resulting hemispherical shell were also obtained.