Simultaneous Stretching and torsion of a solid neo-Hookean cylinder:

In this lecture we will consider the behavior of a naturally-straight, highly extensible, elastic, cylindrical filaments and rod which is subjected to large extensional and torsional strains. The rod initially extends but finally leads to the formation of solenoidal structures as shown by the optical micrograph (e).

Let us say that the cylinder of length  and radius  is stretched to extension ratio  by applying a dead weight  and is twisted to torsional density  by applying a torque . Since ,  The extensional energy of the cylinder is obtained by multiplying the strain energy density function by the volume of the rod:

(24.1)

Similarly, the torsional energy of the cylinder is obtained as:

(24.2)

where, the torsional rigidity is defined as .
Hence the total elastic energy of the rod is obtained by summing these two energies,  

+

(24.3)