Session: 09-05-01: Applied Mechanics, Dynamic Systems, Experimental and Computational Methods, Modeling and Virtual Simulations of Dynamic Structures, Advanced Materials and Testing

Paper Number: 147219

147219 - Torsional Response of a Ductile Material Exhibiting Strain Hardening 

The torsion of shafts of circular cross-section is primarily taught for linear elastic conditions in most undergraduate engineering curricula. It is important to understand the resonse of shafts when the torsion causes yielding, and subsequent response when the loading is reversed.  This study is therefore directed to understand such behavior.  We have studied the response in elastic and elastic-plastic domains for a prismatic bar of circular cross sections subjected to torsion. This is an extension of the torsion test where the torques are such that yielding does not take place, and the torque twist characteristic is linear. By exploring the response of the ductile specimen in torsion, the students can see as the material is strained beyond its yield point, more stress is required to provide additional plastic deformation, thus exhibiting work hardening.  The torsion testing usually parallels that of uniaxial tension tests. However, in contrast to uniaxial tension tests, the stresses produced in torsion are not distributed uniformly over the cross section.  During a torsion test, the angle of twist φ and the applied torque T are measured as the test proceeds. For a circular cross-section, in the absence of the other loads, pure shear stress state exists at each point. Torsional elastic shear stresses vary linearly from zero at the axis of twist to a maximum at the extreme fibers. Thus, in a solid circular
bar, when the surface fibers reach the yield shear stress they are, in a sense, supported by elastic interior fibers. Consequently, the elastic resistance of the remainder of the section masks the effect of yielding of the surface fibers during their early stage of yielding. To test the material in torsion the proper test procedure is needed. It involves mounting a shaft into the testing machine, applying torque incrementally and measuring both the applied torque and the corresponding angle of twist. Using the appropriate formulae, relationships, and the measured dimensions, it is possible to determine the shear stress and shear strain on the shaft. Then, the torque vs. angle of twist, and shear stress vs.shear strain can be plotted.  The assumptions made in this experiment include but are not limited to the following:
 The torque is applied along the center of axis of the shaft.
 The material is tested at steady state (absence of strain rate effects).
 Plane sections remain plane after twisting (the circular section conforms to this condition).
An MTS axial and torsion-testing machine was employed to obtain the torque-twist characteristics of the material. This machine has available control modes for torque, strain, and twist. The calibrated ranges for the torques exist in this machine for 565 N-m, 1130 N-m, and 5650 N-m. The calibrated ranges for the twists available in this machine are 5, 10, 25, and 50 degrees. Torsion test was conducted on 6061-T6 aluminum specimen of circular cross section. The diameter of the specimen was 25.4 mm, and the gage length was 203.2 mm. The 1130 N-m range was used for the torque and 25-degree range was used for the twist in the machine.
As the torque was increased gradually, the angle of twist kept increasing steadily. The torque was increased continuously till the specimen yielded and went into the inelastic region leading to a nonlinear torque twist characteristic. The unloading (reversal of twist) was initiated when the torque reached about 900 N-m. The twisting in the reversed direction was carried out until the reversed yielding began and then the experiment was stopped.

When a bar is twisted beyond the elastic range (yielding), there is a non-linear torque-twist behavior. Most ductile materials exhibit strain hardening where in the plastic region, the shear stress monotonically increases with shear strain. In this work, a simplistic assumption is applied first, wherein the equations of linear elasticity are used to construct the shear stress shear strain curve from the experimentally obtained non-linear torque twist curve. Obviously, this is an approximate approach, but it turned out that this method also yields results that are comparable with the “exact” approach. To obtain a better representation of the material behavior, the mechanism of deformation was observed more closely. It was recognized that for an annealed
material (6016-T6 aluminum), there is no well-defined yield point, and the elastic/plastic boundary is therefore absent. Thus, it presents a significant challenge to obtain the shear stress-strain curve from the torque twist curve. A combined analytical and graphical method has been used for this purpose.

Thie experimental study demonstrated the Bauschinger effect where the yield strength in the reversed yielding was observed to be lower than the initial yield.  From the experiment it was possible to extraploate the results for the fully plastic situation and the famous Nadai sand heap analogy could be demonstrated,  The experiment provided detailed understanding of inelastic behavior associated with torsion in circular shafts.

Presenting Author: Somnath Chattopadhyay Cleveland State Unicersity

Presenting Author Biography: Dr. Somnath Chattopadhyay teaches courses on mechanics. design, materials and mamufacuring at Cleveland State University. His research areas are in the areas of mechanics, and materials He has authored a text on pressure vessel design, has been on the ASME Boiler and Pressure Vessel Code Committees, and served as an associate editor of ASME Journal of Pressure Vessel Techno;ogy.

Authors:

Somnath Chattopadhyay Cleveland State Unicersity