A model of neck formation on a rod under tension

Stability of equilibrium of a circular cylinder under homogeneous axial stretching is investigated in the frame of 3-D nonlinear elasticity. The axisymmetric buckling modes describing developing of a "neck" on the stretched rod are studied. The isotropic incompressible material of the rod is described through the logarithmic strain tensor. The constitutive equations for the rod correspond to the power-law hardening of elastic-plastic materials. Solving the linearized stability equations of the stretched cylinder, we find the spectrum of critical values of longitudinal deformation and buckling eigenmodes of the rod. The bifurcation modes relating with the neck formation arise when the elongation of the rod insignificantly exceeds the maximum point on the diagram of stretching. It is noted that different buckling modes have close eigenvalues. This accumulation of the eigenvalues describes formation of the neck as a result of the superposition of many buckling modes. Similar results were established for a stretched rectangular beam under plane deformation [1]