Elasticity/Solution strategy for Prandtl stress function

Solution strategy using the Prandtl stress function

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

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

$\varphi ={\varphi }_p+{\varphi }_h$

where ${\varphi }_p$ is a particular solution and ${\varphi }_h$ is the solution of the homogeneous equation.

Examples of particular solutions are, in rectangular coordinates,

${\varphi }_p=-\mu \alpha x_1^2;{\varphi }_p=-\mu \alpha x_2^2$

and, in cylindrical co-ordinates,

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

The homogeneous equation is the Laplace equation ${\mathrm{\nabla }}^2\varphi =0$, 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,

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

Suppose $f(z)=z^n$. Then, examples of ${\varphi }_h$ are

${\varphi }_h=C_1{r}^n\cos (n\theta );{\varphi }_h=C_2{r}^n\sin (n\theta );{\varphi }_h=C_3{r}^{-n}\cos (n\theta );{\varphi }_h=C_4{r}^{-n}\sin (n\theta )$

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.