Bifurcation analysis of the onset of necking in an elastic/plastic cylinder under uniaxial tension

The Bifurcation problem governing the onset of axisymmetric necking in a circular cylindrical specimen in uniaxial tension is analysed. The specimen is made of an incompressible elastic/plastic material. One end is subject to a prescribed uniform axial displacement relative to the other and both ends are shear free. The true stress at bifurcation is greater than the stress at which the maximum load is attained by an amount which depends on (i) the radius to length ratio of the specimen, (ii) the ratio of the elastic shear modulus to the tangent modulus, and (iii) the derivative of the tangent modulus with respect to the stress. Bifurcation takes place immediately following attainment of the maximum load when the specimen is sufficiently slender.