Elasticity/Solution strategy for Prandtl stress function

The equation ${\displaystyle \nabla ^{2}{\phi }=-2\mu \alpha }$ is a Poisson equation.

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

${\displaystyle \phi =\phi _{p}+\phi _{h}\,}$

where ${\displaystyle \phi _{p}\,}$ is a particular solution and ${\displaystyle \phi _{h}\,}$ is the solution of the homogeneous equation.

Examples of particular solutions are, in rectangular coordinates,

${\displaystyle \phi _{p}=-\mu \alpha x_{1}^{2}~~;~~\phi _{p}=-\mu \alpha x_{2}^{2}\,}$

and, in cylindrical co-ordinates,

${\displaystyle \phi _{p}=-{\frac {\mu \alpha r^{2}}{2}}}$

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

${\displaystyle z=x_{1}+ix_{2}=re^{i\theta }\,}$

Thus,

${\displaystyle \phi _{h}={\text{Re}}(f(z))~~{\text{or}}~~\phi _{h}={\text{Im}}(f(z))}$

Suppose ${\displaystyle f(z)=z^{n}\,}$. Then, examples of ${\displaystyle \phi _{h}\,}$ are

${\displaystyle \phi _{h}=C_{1}r^{n}\cos(n\theta );\phi _{h}=C_{2}r^{n}\sin(n\theta );\phi _{h}=C_{3}r^{-n}\cos(n\theta );\phi _{h}=C_{4}r^{-n}\sin(n\theta )}$

where ${\displaystyle C_{1}\,}$, ${\displaystyle C_{2}\,}$, ${\displaystyle C_{3}\,}$, ${\displaystyle C_{4}\,}$ are constants.

Each of the above can be expressed as polynomial expansions in the ${\displaystyle x_{1}\,}$ and ${\displaystyle x_{2}\,}$ 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.