Let's consider a straight rod with arbitrary cross section whose co-ordinate system is as follows: while the axis remains along the axis of the rod, its origin is somewhere inside and the plane can be taken as the undisplaced cross section.. We define the torsion angle as the angle through which the rod is twisted per unit length, so that two neighboring elements distance apart turns through an angle : . is rather small so that when this distance is of the order of the transverse dimension of the rod, it satisfies . Let's estimate the displacements in a small portion of the rod near the origin of the rod. When the radius vector turns through an angle , the displacement of its end is given by

(15.2)

where is the vector whose magnitude is the angle of rotation and direction is that of the axis of rotation i.e. the axis. For points with coordinates relative to the plane the angle of rotation is , hence, the components and of the displacement vector along the and directions respectively are,

(15.3)

It is easy to see that the components of the stress tensor calculated from these displacements satisfy the stress equilibrium relations and the boundary conditions.

In general points undergo displacement along the axis also. Let's assume that for small , this displacement is,

(15.4)

Here is a single valued function, called the torsion function, which signifies that any cross section of the rod does not remain planer bur become curved. Furthermore, the origin of the co-ordinate system which was fixed at a particular point in the plane can now move along the co-ordinate. Then the components of the strain tensor are as follows:

(15.5)

it should be noted that , signifying that torsion does not result in change in volume, it is a pure shear deformation. Components of stress tensor are,

(15.6)

Then the general equation of equilibrium reduces to,

(15.7)

in which substitution of the expressions in (15.5) results in the two dimensional Laplacian: