Crushing of an Elastic-Plastic Ring Between Rigid Plates With and Without Unloading

Abstract

Load-deflection curves are computed for an elastic-plastic ring that is slowly crushed between frictionless, rigid plates (platens). The ring is assumed to be inextensional with plane sections remaining plane and to obey a bi-linear stress-strain law with isotropic hardening. These assumptions lead to a local nonlinear moment-curvature relation identical to that developed by Liu et al. When inserted into the exact equation for moment equilibrium, this constitutive relation yields a second-order, nonlinear ordinary differential equation for the angle α between the deformed centerline of the ring and the horizontal. The numerical solution of this equation, which uses a combined penalty-continuation method, along with an auxiliary equation relating the vertical deflection to α, leads to overall load-deflection curves that depend on two dimensionless parameters, λ and μ. The first is the ratio of the plastic modulus to the elastic modulus; the second measures the ratio of plastic to elastic effects. As μ→0, the overall load-deflection curve of Frish-Fay for the elastica is recovered; as μ→∞, that of DeRuntz and Hodge for a rigid-perfectly plastic ring is recovered. Three scenarios are considered: I0, in which an initially straight, stress-free beam is bent elastically into a ring and then crushed; II0, in which an initially stress-free ring is crushed; and III0, in which an initially straight beam is bent first elastically and then elastically-plastically into a ring and then crushed. Results for scenario II0 are shown to agree well with experiments of Reddy and Reid if λ=0.01 and μ=10 and 20 and with experiments of Avalle and Goglio if λ=0.02 and μ=11. In scenarios I0 and II0, the effects of unloading prove to be small, reinforcing a similar conclusion of Liu et al., who considered the large-deflection of an elastic-plastic cantilever under a tip load. If no unloading is assumed, a more analytical treatment is possible, as shown in the second part of the present paper. The model predicts that the ring always remains in full contact with the platens, in agreement with recent experiments by Avalle and Goglio on annealed aluminum tubes. Pull-away from the platens also observed in experiments is ascribed to end effects which cannot be modeled by a one-dimensional beam theory. However, it is argued that, even if there is pull-away, the effect on the overall force-deflection relation must be small because in both cases the forces exerted by the platens are concentrated at the ends of the contact region. Moving pictures of successive stages of deformation of the ring showing the formation of plastic loading and unloading zones in all three scenarios may be found on the web site www.people.virginia.edu/∼jgs/ring.html.