EN224: Linear Elasticity
Division of Engineering
7. Complex Variable Methods for Plane Elastostatics
Airy functions have been used to find many useful solutions to plane elastostatic boundary value problems. The method does have some limitations, however. The biharmonic equation is not the easiest field equation to solve, for one thing. Another limitation is that displacement components are difficult to determine from Airy functions, so that the method is not well suited to displacement boundary value problems.
We found the complex variable methods led to a systematic procedure for solving anti-plane shear problems. It is natural to attempt to develop a similar approach for more general plane problems. It turns out that this can indeed be done, although the results are rather more complex (in one sense of the word) than those for anti-plane shear.
7.1 Complex Variable Representation of Plane Elastostatic Solutions
Let be a plane elastostatic state.
Furthermore, let be analytic functions on the plane region .
Define a complex displacement field
Then a displacement field generated from
satisfies the two dimensional Cauchy-Navier equation
Furthermore, the stress state follows as
Conversely, let u satisfy the Cauchy-Navier equation
then there exist analytic functions such that
Begin by noting some preliminary identities. Consider an analytic function f(z) with
With these identities, we may write
where we have used the Cauchy-Riemann conditions
Now, begin with the proof. Consider the Cauchy-Navier equation
Using the identities listed above, this may be expressed as
Integrating (using the second identity of eq. 1)
where f(z) is an analytic function. It is convenient to set
where is another analytic function. Now
Here, the first two terms are the particular integral for our differential equation; the third term is the homogeneous solution, noting that
The negative sign in front of was introduced for convenience.
This proves our assertion (we can scale the potentials to account for the shear modulus term, if we choose). Thus, the displacement field for a plane elastostatic boundary value problem may be derived from a pair of complex potentials; just as we found that the displacement field for anti-plane shear problems could be derived from a single analytic function. We still need to work out how to actually find these potentials.