Illustrating rotating principal stresses in a materials science course

This work constitutes a laboratory component of a junior level materials science course and illustrates the importance of rotating principal stresses in the design of components such as the automotive crankshaft. The activity is centered on Mohr's circle for biaxial stress situations involving time varying normal and shear stresses. A number of dynamic situations have been considered, namely. (a) sinusoidally varying normal and shear stresses that are in phase, (b) sinusoidally varying normal and shear stresses that are 90° out of phase, (c) constant normal stress and sinusoidally varying shear stress, and (d) sinusoidally varying normal stress and constant shear stress. Employing a graphical approach, the diameter of Mohr's circle (the absolute difference between the two principal stresses) as well as the principal stress directions is monitored. The students see that for certain stress situations the principal stress directions remain unchanged while for others the principal stress directions change with time (rotating principal stresses). In general, the size of Mohr's circle changes with time. The plotting option of the Matlab code has been employed to construct three dimensional plots for the indicated stress situations with the normal and stresses respectively in the x and y directions and time in the z-direction. The plots show how the principal directions change with time, along with the size of Mohr's circle. The students are made aware of the fact that rotating principal stresses play a very important role in designing components that are subjected to biaxial or multiaxial fatigue, such as the crankshaft. Also the diameter of Mohr's circle can be directly related to Tresca or von Mises theories of failure.