EN224: Linear Elasticity

Division of Engineering

3. 3D Static Boundary Value Problems

Objective: Find elastostatic states in 3D solids with prescribed boundary conditions

This is very difficult to do in general!

A few useful techniques:

(1) Represent the solution in terms of harmonic potentials. We can then generate elastostatic states from any harmonic function, and if we’re very lucky we find the solution to the problem we are interested in. Many useful solutions have been found this way, but the chances of finding a new one are small!

(2) Try superposition and the method of images. This will only solve rather simple problems, but it’s worth a shot.

(3) Use the reciprocal theorem. This is a good way of solving dislocation problems. If you’re very lucky it will give you a general solution for your region, but you need to solve a tricky singular problem first.

(4) Transforms. This is the most powerful approach, as it gives you a formal procedure to follow. Examples include Fourier transforms (good for half-space problems and for the infinite solid), Hankel transforms (good for axisymmetric problems), Mellin transforms (good for quarter-space problems), among others.

(5) Approximate the solution. If the 3D boundary value problem can’t be solved, one can sometimes make progress by reducing the problem to 2D. Examples include plane problems in elasticity, Saint Venant torsion, plates and shell theory.

(6) Use a numerical method. Probably what you will end up doing, when all is said and done!

We will illustrate all these techniques by solving specific boundary value problems. For simplicity, we will generally restrict attention to isotropic, homogeneous solids.

3.1 Papkovich-Neuber Potentials

For 3D problems, the Navier equations are the most convenient representation for the field equations. They are still unwieldy, because we are required to solve three coupled partial differential equations for the three displacement components.

We can simplify the solution by representing u in terms of harmonic potentials. This decouples the equations. There are various ways to do this. The most common approach is to use so called `Papkovich-Neuber potentials’ to represent the solution.

Theorem: Let , be a scalar and a vector field on , satisfying

Then let

and

Then

Proof: Substitute the expression for u and the governing equations for the potentials into the Cauchy-Navier equation and show that it is satisfied (with zero accelerations). This is known in the trade as a `tedious but straightforward calculation’

Completeness of Papkovich-Neuber potentials

We know that Papkovich Neuber potentials, generate elastostatic displacement fields. However, it is no use trying to find potentials for a particular problem unless we are sure that all solutions can be represented in this form. Fortunately, we can show that the potentials are complete, and will generate all elastostatic states.

Theorem: Let u satisfy

Then satisfying (1).

Proof:

Note that we can always find f(x) satisfying

Since there is a general solution to the Poisson equation, namely

We will not prove this just now: for a detailed discussion see Kellog `Foundations of Potential Theory,’ Dover, 1954, or Keener, `Principles of Applied Mathematics,’ Addison Wesley, 1988. We will derive the solution in the next section singular solutions for the infinite solid.

Now, let

Then, eliminating f:

Furthermore,

Giving the correct expression for . In addition,

Observe that

whence

as required.

An observation: Papkovich-Neuber potentials are not unique: several different combinations of potentials may generate the same u. It is easy to see this, because there are four scalar unknowns in but only 3 displacement components!

Stresses derived from Papkovich-Neuber potentials

Substitute the potential representation for u into the stress-strain relations and simplify using the governing equations for the potentials to see that:

Simpler representations derived from Papkovich-Neuber potentials

Papkovich-Neuber potentials are often used to represent 3D solutions. Many other representations are possible, however. Many (but not all) of these are merely special cases of the more general Papkovich-Neuber potentials. For example, Green and Zerna catalog a set of potential representations (known as Boussinesq potentials) that are useful in many applications. None of these representations are complete, but they are considerably simpler than the Papkovich-Neuber potentials. We will follow their notation in listing these representations:

Solution A:

Put , where is another scalar potential, and assume b=0.

Solution B

Put where is another scalar potential, and assume b=0.

Solution C

Put where is another scalar potential, and assume b=0.

Displacements and stresses can be computed from the expressions for Solution B, by replacing all subscripts 3 with 2.

Solution D

Put

Displacements and stresses can be computed from the expressions for Solution B, by replacing all subscripts 3 with 1.

Solution E

Put

Solution F

Put

Displacements and stresses can be computed from the expressions for Solution E, by replacing all subscripts 3 with 2.

Solution G

Put

Displacements and stresses can be computed from the expressions for Solution B, by replacing all subscripts 3 with 1.

Observe that all the representations listed here generate elastostatic states from Harmonic potentials. They are useful because we know a lot about harmonic potentials, so this gives us a procedure for solving a wide range of problems. For example, we will find that we can combine the potentials so as to generate solutions for a surface loaded half-space, using harmonic potentials.