**Division of Engineering**

**Brown University**

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**1. ****Tangentially loaded half-space.**

(1.1) State the appropriate boundary conditions for a half-space subjected to a point

force acting tangent to its surface, as shown in the figure.

(1.2) Examine the list of Boussinesq potentials given at the end of Sect 3.1 of the online

lecture notes. Consider solutions A, B and E, which generate stresses from

harmonic potentials as follows:

Following the procedure outlined in Sect 3.7 for a normally loaded half-space, find a way to combine solutions A, B, and E so as to generate an elastostatic state which automatically satisfies on the surface of the half-space. You should find you can generate the required solution from a single harmonic potential . Set up the boundary conditions that represent a point force at the origin.

(1.3) Take Fourier transforms of the governing equations and boundary conditions

found in (1.2). Hence, deduce that the transform of the required potential satisfies

Use the result for the normally loaded half-space to write down and hence deduce the potential

(1.4) Determine the displacement field for the tangentially loaded half-space.

**2. Spherical inhomogeneity. **Suppose that an
infinite solid with shear modulus and Poisson’s ratio contains a spherical inclusion with shear
modulus . The solid is loaded in uniaxial tension at infinity.
Calculate the stress, strain and displacement in the inclusion (take the
displacement to be zero at the origin)

**3. Energetics of Eigenstrains. **Consider a homogeneous, stress free,
linear elastic solid with elastic constants **. **Suppose
that an eigenstrain distribution is introduced into a bounded subregion of
the solid *B*. Let denote the total strain distribution in the
solid.

Show that the total strain energy of the solid is

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To do this, begin by writing down the strain energy within* B*.
Then, write down an expression for the total strain energy outside *B* in terms of the traction acting on the
boundary of *B*. Then find a way to rearrange the sum of
these two terms into the form given above.

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**4. General
Axisymmetric Contact. **Suppose that two isotropic, linear elastic spheres

with radii and moduli and Poisson’s ratios are pressed into contact.

Assume that if the two spheres did not deform, they would
overlap by a distance *h* as shown in
the figure.

Make the following assumptions:

(1) The radius of the contact area between the two spheres is much smaller than the radius of either sphere.

(2) Both spheres deform as though
they were infinite half-spaces. That is
to say, the radial displacement of a point on the surface of sphere A due to a
point force acting a distance *r *away
on its surface is

(3) Approximate the profile of each sphere by a parabola.

Write down an integral equation for the contact pressure distribution acting between the spheres, in terms of the sphere radii and the elastic constants.

Compare the result with the integral equation that governs
the pressure distribution acting between a rigid sphere and an elastic
half-space. Hence, find expressions for
the radius of the contact area between the spheres, the contact pressure
distribution and the relationship between the load *P* and the approach of remote points on the spheres *h*.