Page 283 - Introduction to Continuum Mechanics

The Elastic Solid 26?

cylindrical, the unit normal to the lateral surface has the form n = n^i + n^ and the
associated surface traction is given by

We require that the lateral surface be traction-free, i.e., t = 0, so that on the boundary the
function <p must satisfy the condition

Equations(5.14.3) and (5.14.4) define a well-known boundary-value problem which is
known to admit an exact solution for the function (p. Here, we will only consider the torsion of
an elliptic cross-section by demonstrating that

gives the correct solution.
Taking A as a constant, this choice of <p obviously satisfy the equilibrium equation [Eq.
(5.14.3)]. To check the boundary condition we begin by defining the elliptic boundary by the
equation

The unit normal vector is given by

and the boundary condition of Eq. (5.14.4) becomes

Substituting our choice of <p into this equation, we find that

f It is known as a Neumann problem