Bifurcation in compressible elastic/plastic cylinders under uniaxial tension

Abstract

The bifurcation problem of a circular cylinder of elastic/plastic material under uniaxial tension is investigated, with particular reference to the usual engineering criterion that necking is initiated when the load on the specimen reaches a maximum. The material considered is compressible, with a smooth yield surface and associated flow rule. A lower bound analysis shows that for the particular constitutive equation chosen bifurcation cannot occur under a range of loading conditions while the stress is less than a certain value which is itself slightly less than the stress at the maximum load point. Diffuse axisymmetric necking modes under the commonly assumed loading conditions of prescribed axial components of velocity and shear-free traction-rates on the ends are, however, found to be initiated always after maximum load, the delay depending on the same factors shown for an incompressible material in reference [1]. The effect of the elastic compressibility assumption is to reduce the delay for a wide range of geometries, but to increase it for very slender specimens, as compared with the incompressible case. Surface modes are also found, but at stresses of an unrealistically high order of magnitude.