Enlargement of a Circular Hole in a Thin Plastic Sheet: Taylor-Bethe Controversy in Retrospect

Abstract

The title problem is solved for an internally pressurised hole within the framework of plane-stress finite-strain plasticity. Material behaviour is modelled by the Tresca-associated flow theory with arbitrary hardening (softening), accounting for thickness changes and elastic compressibility. Three distinct plastic zones emerge successively as deformation progresses, including, as in Bethe's classical solution, a definite corner zone where the normality rule does not apply. It is suggested that the corner phase terminates, by a path selection criterion, when internal pressure encounters less stiffness along an alternative loading path. Mathematical formulation is reduced to ordinary differential equations, two for each plastic phase, with effective stress as independent variable and closed form solutions are found for all three phases. We examine in detail the specific expansion energy required to create a unit of new nominal hole volume and apply the asymptotic value (specific cavitation energy) to ballistic (and residual) velocity predictions. In particular, a closed form solution is derived for the specific cavitation energy accounting for arbitrary stress-strain curves (and compressibility). Illustrative examples are presented over a wide range of material parameters. Tresca model results are compared with plane-stress solutions for associated J 2 deformation and flow theories and with known related plane-strain cavitation fields. Simple scaling between plane-stress-specific cavitation energy and ultimate tensile stress is suggested.