Mechanisms of deep penetration of soft solids, with application to the injection and wounding of skin

Micromechanical models are developed for the deep penetration of a soft solid by a flat–bottomed and by a sharp–tipped cylindrical punch. The soft solid is taken to represent mammalian skin and silicone rubbers, and is treated as an incompressible, hyperelastic, isotropic solid described by a one–term Ogden strain energy function. 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. For the case of a sharp penetrator, this calculation is performed by considering the opening of a plane–strain crack by a wedge using a finite–element approach. Analytical methods suffice for the flat–bottomed punch. In both models the crack dimensions are such that the load on the punch is minimized. 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), in agreement with experimental observations reported in a companion paper.