Irodov Problem 1.312

Since one end is fixed all points on that end will be at the same place after the distortion. Consider a line OA on the fixed side and a corresponding parallel line O'B on the free end. After the distortion, B moves to a location B' - this induces a shear strain of magnitude BB'/l. Another point X on AB moves to location X' and so this area experiences a shear XX'/l. Thus, the cylinder experiences different shear strains at different distances from the axis.

Suppose that the point X is at a distance x from the axis of the cylinder. The length of XX' is approximately and so the shear strain experienced at point X is given by,



The energy per unit volume due to this shear strain is given by . Consider an infinitesimally thin cylindrical section of thickness dx at point X. The volume of this infinitesimally thin cylindrical section is . The total energy can be computed by integrating the energy contained in all such infinitesimally thin sections and hence this given by,