Elasticity/Solution strategy for Prandtl stress function

Solution strategy using the Prandtl stress function
The equation $$\nabla^2{\phi} = -2\mu\alpha$$ is a Poisson equation.

Since the equation is inhomogeneous, the solution can be written as

\phi = \phi_p + \phi_h \, $$ where $$\phi_p\,$$ is a particular solution and $$\phi_h\,$$ is the solution of the homogeneous equation.

Examples of particular solutions are, in rectangular coordinates,

\phi_p = -\mu\alpha x_1^2 ; \phi_p = -\mu\alpha x_2^2 \, $$ and, in cylindrical co-ordinates,

\phi_p = -\frac{\mu\alpha r^2}{2} $$

The homogeneous equation is the Laplace equation $$\nabla^2{\phi}=0$$, which is satisfied by both the real and the imaginary parts of any analytic function $$f(z)\,$$ of the complex variable

z = x_1 + i x_2 = r e^{i\theta} \, $$ Thus,

\phi_h = \text{Re}(f(z)) \text{or} \phi_h = \text{Im}(f(z)) $$

Suppose $$f(z) = z^n\,$$. Then, examples of $$\phi_h\,$$ are

\phi_h = C_1 r^n\cos(n\theta) ; \phi_h = C_2 r^n\sin(n\theta) ;   \phi_h = C_3 r^{-n}\cos(n\theta) ;   \phi_h = C_4 r^{-n}\sin(n\theta) $$ where $$C_1\,$$, $$C_2\,$$, $$C_3\,$$, $$C_4\,$$ are constants.

Each of the above can be expressed as polynomial expansions in the $$x_1\,$$ and $$x_2\,$$ coordinates.

Approximate solutions of the torsion problem for a particular cross-section can be obtained by combining the particular and homogeneous solutions and adjusting the constants so as to match the required shape.

Only a few shapes allow closed-form solutions. Examples are
 * Circular cross-section.
 * Elliptical cross-section.
 * Circle with semicircular groove.
 * Equilateral triangle.