A study that you can look at is by Shergold and Fleck, (2004), Mechanisms of deep penetration of soft solids, with application to the injection and wounding of skin, DOI: 10.1098/rspa.2004.1315.
From the abstract:
Penetration of the soft solid by a flat–bottomed punch is by the formation of a mode–II ring crack that propagates ahead of the penetrator tip. The sharp–tipped punch penetrates by the formation of a planar mode–I crack at the punch tip, followed by wedging open of the crack by the advancing punch. For both modes of punch advance the steady–state penetration load is calculated by equating the work done in advancing the punch to the sum of the fracture work and the strain energy stored in the solid.
and
For both geometries of punch tip, the predicted penetration pressure increases with diminishing punch radius, and with increasing toughness and strain–hardening capacity of solid. The penetration pressure for a flat–bottomed punch is two to three times greater than that for a sharp–tipped punch (assuming that the mode–I and mode–II toughnesses are equal)
A more recent experimental paper that does not use fracture mechanics is: Roesthuis et al. (2011) Mechanics of needle-tissue interaction, DOI: 10.1109/IROS.2011.6094969.
There is a vast amount of work on the subject in the scientific literature; mostly behind pay walls.