Page 350 - Introduction to Continuum Mechanics

334 The Elastic Solid

m
If the angle of twist K is very small, then Ii and /2 and therefore <PI and <f>2 &y be regarded
as independent of R and the integrals can be integrated to give

and

We see therefore, that if the bar is prevented from extension or contraction (i.e., ^3 = 1),
then twisting of the bar with a K approaching zero, gives rise to a small axial force N which
*y
approaches zero with K . On the other hand if the bar is free from axial force (i.e., N = 0), then
*J
as K approaches zero, there is an axial stretch A 3 such that (A3 -1) approaches zero with K ,
Thus, when a circular bar is twisted with an infinitesimal angle of twist, the axial stretch is
negligible as was assumed earlier in the infinitesimal theory.
From Eqs. (5.38.18) and (5.38.19), we can obtain

Equation (5.38.20) is known as "Rivlin's Universal relation". This equation gives, for small
twisting angle, the torsional stiffness as a function of A 3 , the stretch in the axial direction. We
see, therefore, that the torsional stiffness can be obtained from a simple-extension experiment
which measures N as a function of the axial stretch A 3 .