EN224: Linear Elasticity
Division of Engineering
5.5 Solving Anti-Plane Shear Problems using Conformal Mapping
You are unlikely to make a living by solving problems involving circular disks or holes subjected to prescribed displacements or tractions. Fortunately, it turns out that these solutions may be used to generate solutions to far more complex problems, using the idea of Conformal Mapping. You probably still wont be able to make a living by selling the solutions to these problems to the general public, but its a step in the right direction.
Definition of a conformal mapping
An analytic function maps a point on the complex plane to an image . The mapping is said to be conformal at any point where. If , the mapping is not invertible at and is a branch point of the inverse map.
You may or may not be able to find a conformal mapping that will enable you to solve a particular problem. The Riemann mapping theorem guarantees that any two simply connected domains may be mapped conformally onto one another, but it doesnt tell us how to actually find the mapping.
Since m is an analytic function, we can expand it as a Taylor series (for a bounded, simply connected region) or a Laurent series (for an annular region or a problem involving an infinite solid with a hole). Thus, we can set
Given enough boundary points, one can in principle solve for enough coefficients in the mapping to approximate a boundary with arbitrary accuracy.
Many useful mappings have been found by guesswork, luck, or using extraordinary intuition. Here are a few examples.
Map a region outside a circle to a region outside an ellipse
Map a region outside a circle to a region outside a hypotrochoid
Map a region inside a circle to a region inside an epitrochoid
Suppose we need to solve a boundary value problem involving a solid which has a complicated boundary. For anti-plane shear problems, this means we need to find an analytic function with either the real part of f (for a displacement BVP) or the imaginary part of f (for a traction boundary value problem) prescribed on the boundary.
We can usually only find f(z) directly if the boundary is particularly simple: a circle or a straight line, for example. So if we need a solution to a problem with a complicated boundary, we try to map the boundary onto a simpler shape.
Then, we try to find
that will satisfy our boundary conditions.
Note that we can call another complex function: say
Then, we try to find satisfying the appropriate conditions on the simpler, mapped boundary. Finally, our solution is given by
Example: Elliptical hole in an infinite plate
Consider an infinite solid, which contains an elliptical hole with semiaxes (a,b) 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.
As before, we superpose a state of uniform shear stress and a correction to account for the traction free boundary conditions on the hole. Thus we seek a function f(z) such that
To simplify the boundary conditions we map the ellipse to the unit circle, using a map from our list of conformal mappings. Let
Then, we must find a function satisfying
We can solve this problem by expanding g as a Laurent series, as before.
To satisfy conditions at infinity, we must set . To find the remaining coefficients, expand the series and set on the hole boundary
Comparing coefficients, we find that
Hence, on the plane, the solution is
We can express this in terms of z, although in practice it is usually more convenient to leave the solution in terms of , since this automatically generates a nice curvilinear coordinate system. Thus, the out of plane displacement field follows as
Computing stress fields using conformal mapping
It is straightforward to compute the stress field as well. We could differentiate the displacement field, but instead, we will use a convenient shortcut provided by the conformal mapping procedure.
Recall that if then we may compute the stresses from
Now, recall that
Apply this to our elliptical hole problem:
We can compute the stress concentration factor directly: the greatest stress occurs at where
Note that setting collapses the ellipse to a crack with length 4R, so we can solve this problem too.