An Investigation of Elastic-Plastic Torsion in Strain Hardening Materials

Abstract

The elastic-plastic problem of torsion of a solid circular bar of a strain hardening material has been studied in this work. The genesis of this study evolves from an experiment on torsion of circular bars where the torsional load (torque) was gradually increased causing the material to yield. The maximum torque caused the bar to go well into the plastic region. The torque-twist curves were generated for bars of circular cross-section made of aluminum. These curves which represent the material stress-strain curves, display strain-hardening characteristics. Upon load reversal, the materials yield in the reversed loading modes with reduced yield strength values displaying the Bauschinger Effect. The purpose of this investigation is to understand the effects of strain hardening on the torque-twist behavior of the materials under monotonic loading. The usual solutions to the elastic-plastic torsion assume elastic-perfectly plastic material behavior. These solutions are limited to cases where complete cross-section is plastic. This is because the elastic-plastic boundaries are generally difficult to find. Experimental solutions can be obtained readily for torsion of circular bars made of perfectly plastic materials using Nadai’s ingenious sand hill analogy. The torque for the case of sections going fully plastic can be directly obtained by determining the volume of the sand heap formed on a circular base (geometrically like the circular cross section of the bar). The analytical elastic-plastic torsion problem reduces to solving a Poisson’s equation with appropriate boundary conditions. The non-homogeneous part involves plastic strains for the strain hardened material, For the case of fully plastic behavior, the nonhomogeneous part is a constant and is amenable for analytical solutions. For the strain-hardened material, the nonhomogeneous part involves plastic strains. The solution therefore is not straightforward and requires the method of successive elastic solutions or successive approximations. It is to be noted that both the total and incremental theories of plasticity furnish the same solution to the torsion problem provided the material is perfectly plastic. It is reasonable to assume, therefore, that this will be approximately true for the case of strain hardened elastic-plastic material. Torsion experiments were conducted using circular bars of ductile materials such as steel and aluminum that exhibit strain hardening of various degrees. A good correlation results between the experimentally obtained torque twist characteristics with those obtained analytically. Relevant comparisons were also made with the fully plastic behavior obtained from Nadai’s sand heap analogy as the limiting case.