Introduction to 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

$\begin{align} \phi_h &= C_1 r^n\cos(n\theta) \\ \phi_h &= C_2 r^n\sin(n\theta) \\ \phi_h &= C_3 r^{-n}\cos(n\theta) \\ \phi_h &= C_4 r^{-n}\sin(n\theta) \end{align}$

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.