EN224: Linear Elasticity

Division of Engineering

5.3 Solving Anti-Plane Shear Problems using Taylor and Laurent Series

The real advantage of using complex variables in linear elasticity is the machinery associated with the calculus of complex functions.

For example, we can find solutions by expressing our unknown analytic function f(z) as a Taylor or Laurent series, and solving for the coefficients in the series.

Suppose that our solid is the annulus (it may seem silly to try to find solutions for just one geometry, but we will see that a solution for the annulus can be used to find solutions for other solids by means of conformal mapping). Then, all single valued analytic functions on this region may be expressed as

We note in passing that if f were known, the coefficients in the series could be computed from

where C is any contour within the annular region. Note that may be complex.

We can often solve problems by calculating the coefficients in the series expansion.

Example: Hole in an infinite plate.

Consider an infinite solid, which contains a cylindrical hole of radius a centered at the origin. Suppose the solid is subjected to a state of anti-plane shear at infinity, i.e.

Assume the boundary of the hole is free of traction. Find the anti-plane shear solution.

We will try to find an analytic function f(z) such that , chosen so as to satisfy the boundary conditions.

Note that represents a uniform shear strain with magnitude k, and satisfies the conditions at infinity. We therefore assume that

The coefficients in the series must be found using the boundary conditions. To satisfy conditions at infinity, we must set

To satisfy the traction free boundary condition on the boundary of the hole, we must ensure that

Note that

Therefore,

It is helpful to expand this

Comparing coefficients, we conclude that

Hence, the required potential is