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

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

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.

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