EN224: Linear Elasticity

Division of Engineering

3.7 Singular Solutions for the Half-Space

Solutions for infinite solids have found many applications in the field of mechanics of materials. Often, however, we cannot neglect the influence of a solid’s boundaries. Solutions for a half-space are a first step towards understanding the effects of free surfaces. The solutions are also useful to understand the behavior of two contacting solids. Indeed, in contact mechanics, it is common to idealize both contacting solids as half-spaces.

As before, we will begin by finding singular solutions. Specifically, we will compute the fields inside a semi-infinite solid subjected to a point force acting perpendicular to its boundary. (You will solve the problem of a tangentially loaded half-space in Homework#3).

Boussinesq’s problem: point force normal to the surface of a half-space.

Begin by stating the problem carefully.

Find with

These conditions are sufficient to ensure that the solution is unique, and automatically guarantee that moment equilibrium is satisfied, i.e.

Potential representation for a normally loaded half-space

We will solve this problem by finding a particularly convenient potential representation for a normally loaded hallf-space. In fact, we will find a representation that automatically guarantees that the half-space is free of traction acting tangent to its surface.

Examine the list of Boussinesq potentials given in Sect.3.1. Our objective is to somehow find a way of combining the potentials from this list so as to satisfy as many of our boundary conditions as possible.

Recall Solution A, we may generate an elastostatic state from a potential as follows:

In particular

For Solution B:

and for this solution,

Now, we can combine solution A and B so as to guaranteeas follows: introduce a new scalar potential satisfying , and let

Now, add solutions A and B and express the result in terms of the new potential:

Hence

So that the tangential traction vanishes on the surface of the half-space automatically.

Note also that

Thus, to solve problems involving a half-space subjected to a prescribed distribution of normal pressure on its surface, we merely need to find a potential that satisfies

Alternatively, to solve problems involving prescribed normal displacements on the surface of the half-space, we must find a harmonic potential satisfying

Of course, this discussion leaves the question of completeness unresolved: we cannot be certain that all problems involving normally loaded half-spaces can be derived from a potential in this way. We will shortly find a solution of this form for a half-space loaded by a point force: this shows that all problems involving only a distribution pressure on the half-space can be derived from a potential that has this form.

Point Force on a Half-Space.

We are now in a position to compute the solution we need. We will use Fourier transforms, following the procedure developed for the infinite solid.

We will take Fourier transforms of the governing equations with respect to and

(we can’t transform with respect to x3 because we don’t know anything about our potential for , although we could use the closely related Laplace transform if we wanted).

Define

and note that

where we have adopted the convention that Greek indices range from 1 to 2, with summation on repeated indices.

It is straightforward to show that

Where

Now, if the boundary traction is taken to be a point force of magnitude F acting normal to the boundary, the boundary condition is

We also require the stress and displacement to decay appropriately at infinity, as outlined in the beginning of this section.

Note that we have reduced the partial differential equations for to an ordinary differential equation for . Solutions to this equation which decay as have the form

Substituting into the boundary conditions, we see that

The potential may now be computed:

We can find this transform in tables (or find it by integrating the transform of 1/r with respect to x3) to find that

The displacement field follows as:

It is straightforward to evaluate the stress fields by substituting the appropriate derivatives in the potential representation given earlier. The results are too lengthy to be recorded here.

Cerruti’s problem: point force tangent to the surface of a half-space

An exactly analogous procedure may be used to determine the fields in a half-space subjected to a tangential force. For details, see Homework 3.