Secondary Effects in Torsion

In a first approach it is assumed that during torsion the distance between two sections perpendicular to the torsion axis of the rod remains constant. However, in the case of torsion of rubbers or samples whose width is considerably greater than their thickness, this hypothesis is unrealistic. In fact, considering a rectangular section b x d, the longitudinal fibers other than the axis experience an extension (13) when the rod is twisted. For a fiber 

By writing P as a function of the torsion angle by unit length, one has P = bc9a/9z, and from here e (l/8)(6 + J )(9a/9z), 9a/9z being the relative rotation of two separate sections. For a fiber at a distance r from the axis, the strain will be lower  

If the stress is not fully balanced by the clamping constraints, the samples will tend to be shortened by an amount Ae. The value of Ae can be obtained by assuming that the longitudinal extensional force applied to the entire surface of the section is negligible, so 

The elongated fibers tend to incline because of the effect of the rotation, and taking into account that the stress acts along these fibers, it can be separated into components in two directions, parallel and perpendicular to the axis of rotation. In this way, the normal component induces a secondary torque, a, that must be added to the primary one. The value of this torque is given by 

This secondary torque depends on both the relative rotation of the two separate parallel sections da/dz and the distance to the torsion axis. Consequently, for small values of 3 the following approximation holds  

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