We analyze the conditions of necking in a thermoviscoplastic rod subjected to tension in broad ranges of strain rates and temperatures. Numerical calculations take into account complex constitutive relations for the material of the rod, the process of heat transfer in the rod, and the presence of discontinuities. The stability of uniform tension is investigated by the linear analysis of perturbations in a long-wave approximation based on limiting the expansions of perturbations into Fourier series to the first term. The roots of the polynomials obtained in this case are analyzed using the Routh-Hurwitz theory. After this, the results are improved by applying nonlinear analysis. This introduces corrections into the data of linear analysis by taking into account the influence of the amplitude dependence of perturbations on the stability of plastic deformation. The numerical results demonstrate that the length of the wave of perturbations considerably affects the rate of its damping and growth, whereas the evolution of perturbations is practically independent of the initial amplitude. The presence of discontinuities in the deformed rod makes necking much easier but is not a necessary condition of necking
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