Analogy methods to address warping and plasticity in torsion

Abstract

This project is primarily an experimental study on the membrane analogy and the sand-heap analogy associated with plasticity (along with warping) when a prismatic bar (circular or non-circular) is twisted. This is a supplemental lab activity for the junior level Mechanics of Materials course. When a prismatic bar of arbitrary cross-section is twisted, it gets deformed into helical curves as the cross sections rotate during torsion. The analogy between the elastic torsion of a bar and the small deflection of a laterally loaded membrane is what is known as the membrane analogy. The membrane is stretched by a uniform tension and is attached to a die whose edge plane is the same as the cross section of the twisted bar. A uniform lateral pressure is applied to the membrane to produce a deflection. Except for a circular cylindrical bar, the cross-sections become warped and it plays a significant role in structural design. As the torque is increased, the material passes from the elastic to plastic state. There is a limit to the ability of the cross section to withstand increasing torque. This feature is not typically addressed in the junior level Mechanics of Materials course, but we felt the need for it because this deformation can be readily shown in a torsion experiment with a ductile material such as aluminum. When the torque is increased to a critical value, the resultant shear stress attains the yield value 'k' at one or more points, somewhere along the boundary of the cross section. The extension of the membrane analogy to elastic plastic torsion leads to the sand-heap analogy. A sand heap is formed on a base having a cross-section similar to the cross-section under investigation. In the case of membrane analogy, the volume bounded by the deflected membrane is proportional to the applied torque. For the sand heap analogy, the volume of the sand heap is proportional to the limiting torque applied to the specimen. Experimental estimates for the elastic and inelastic torques in twisting circular and non-circular shafts are in reasonable agreement with the values predicted by the membrane analogy and the sand heap analogy.