Analytic study of plastic necking instabilities during plane tension tests

In the present article, we develop a linear analysis of necking plastic instabilities during plane-strain tension tests on metallic plates. We search for the growth rate θ of perturbations δ υ → = exp ( θ t ) × F → ( space variables ) of the velocity field in the “ground flow” (i.e. the uniform mean plastic flow), periodic along the tension direction, symmetrical with respect to the median plane of the plate, increasing exponentially. This supposes implicitly a separation of time scale between the instability and the ground flow. The materials considered here are metals which are supposed to be homogeneous, isotropic, incompressible, elastoviscoplastic with associated flow rule, and to satisfy the Von Mises plasticity criterion. Damage and heat conduction are neglected. n viscous materials, the analysis shows that the first instabilities, which are long wavelength ones, appear when the applied force is maximum: we retrieve the classical Considèreʹs criterion. In the absence of viscous effects, the analysis also establishes a condition for the absence of a “cutting” wavelength, defined as the minimal wavelength below which shorter wavelengths are all stable. This condition is satisfied when thermal softening dominates work hardening, and coincides with the one of the instantaneous occurrence of infinitely dense networks of shear bands, inclined at 45° with respect to the tension axis, as established in the framework of the well-known bifurcation analysis of Hill and Hutchinson. We observe these networks in finely meshed numerical simulations carried out with a hydrodynamics code. It is seen that the bands develop more rapidly and densely as the mesh in this code is refined. Refining the mesh introduces short wavelength perturbations, which are very unstable. other hand, when a cutting wavelength does exist, mesh-insensitive simulations can be produced, and the growth-rates determined analytically are retrieved in the numerical simulations.