EN224: Linear Elasticity
Division of Engineering
7.6 Solving Half-Plane Problems Using Analytic Continuation
Some of the most interesting boundary value problems in linear elasticity have been solved using the idea of analytic continuation, which reduces many boundary value problems to a so-called Hilbert problem, with a known solution. Examples include the displacement and traction boundary value problem for the half-plane and the disk; contact problems (both for half-spaces and disks); crack problems (including cracks on the interfaces between dissimilar solids); and problems involving dislocations interacting with boundaries.
The method is quite involved and we will not have time to explore it in full. Instead, we will outline the main ideas behind the procedure, and use it to solve a few representative problems.
The Continuation Theorem
Suppose that and are analytic functions in regions and .Suppose that the two regions intersect in a domain R and there exists an infinite sequence of discrete points in R with at least one limit point on which
Then the function
is analytic in the union of and . The function is said to be the analytic continuation of into ; similarly, is the analytic continuation of in.
For our purposes, we will be considering regions that intersect along a line L, andalong L. In this case is analytic in
The idea of analytic continuation provides a powerful tool for solving half-plane problems, and can also be used to solve problems involving regions with circular boundaries. We will illustrate the technique by using it to solve half-plane problems here.
The method of stress continuation for a half-plane
For our first problem, we will attempt to determine the fields inside a half-space subjected to a prescribed distribution of traction on its surface, as shown in the picture.
Suppose that the region of interest is the upper half-plane, which we will refer to as R+.
The problem will be solved using analytic continuation.
Here is the basic idea. In the usual formulation, we need to find two complex potentials,
and . However, when we solve a half-plane problem, the potentials in R- (the lower half-plane) are arbitrary we can choose the potentials in R- in any way we like, without changing the stress and displacement fields in R+. This observation allows us to find an analytic continuation of in R- (i.e. we find a function that is analytic in both R+ and R-), and then using the definition of in R- to replace in R+. Then, instead of having to find two analytic functions in R+, we need to find one potential that is analytic in both R+ and R-, and satisfies certain boundary conditions on the real line.
For example, we will show that the following representation will generate displacement and stress fields in R+ with the surface of the half-space free of tractions. Let be analytic in both R+ and R-, and set
You can check that the surface is free from traction by letting the imaginary part of z approach zero (from above) in the expressions listed above.
The precise way we define in R- is arbitrary. We usually look for a definition that will make the resulting boundary conditions as simple as possible. (The procedure we are using here is a bit like the method we used to solve 3D half-space problems: we tried to superpose a list of potentials that would simplify the boundary conditions. We had to use our insight to find the right way to combine the potentials).
There is a systematic approach you can follow to devise an appropriate continuation, however, which we will illustrate for the case of a traction boundary value problem.
We start with our earlier complex variable formulation. Let be analytic in R+, and set
Now, suppose that some region of the real axis is unstressed, i.e.
Where L is a region on the surface.
In terms of the complex potentials
We will introduce the notation
This boundary condition may be re-written in terms of the functions
which are analytic in R- . Note that some books (Muskhelishvili, for example) use the notation
but in my view this is confusing and we wont use it here.
Now, observe that
Whence, substituting back into the preceding equation and taking complex conjugates:
This condition is equivalent to the statement that the function '(which is analytic in R+) has the same value as the function (which is analytic in R-) on a line segment L. Therefore, by the continuation theorem, these two functions continue one another analytically across the line L. We can therefore think of these two functions as a single complex potential, which is analytic everywhere, and set
We can integrate these equations to see that
We can use this result to find an expression for in R+:
Finally, eliminate from our expression for displacements and stresses in R+ to obtain
Exercise: To check that youve understood the procedure outlined above, try and see whether you can devise a procedure for generating fields in the upper half-plane with zero displacement on the real line. If you follow the procedure we outlined very closely, you should find the following expression for the displacement field:
Challenge: If you found the preceding exercise trivial, try to devise a method of analytic continuation for a circular region (either an infinite solid with a hole, or a disk) with a stress free boundary. In this case, it is helpful to note that if is analytic in |z|<1, then is analytic in |z|>1
We now have a method for solving problems involving a half-plane with a traction free boundary. In fact, this approach can be extended to solve problems involving arbitrary tractions acting on the surface of the half-plane, as we will see in the next section