Introduction to Elasticity/Solution strategy for Prandtl stress function

From Wikiversity

Jump to: navigation, search

[edit] 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.
Retrieved from "http://en.wikiversity.org/wiki/Introduction_to_Elasticity/Solution_strategy_for_Prandtl_stress_function"