** **** EN224: Linear
Elasticity **

Division of Engineering

7. Complex Variable Methods for Plane ElastostaticsAiry 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

Proposition

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

Proof:Substitute.

Conversely,letusatisfy the Cauchy-Navier equationthen there exist analytic functions such that

Proof:

Begin by noting some preliminary identities. Consider an analytic function

f(z)withNow define

so that

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 setwhere is another analytic function. Now

Solve for

Integrate

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.