Math 275 Homework 3
Due: 11/18/05

Consider an infinite slab of aluminum with periodically spaced rods of infinitely rigid steel (expensive stuff...) running through it. Each rod is rotated clockwise 1 degree with its axis held in place. Solve the equations of elasticity

$$\mu\Delta\mathbf{u}+(\lambda+\mu)\nabla(\nabla\cdot\mathbf{u})=0$$

in the aluminum assuming plane strain deformation and neglecting gravity. The Lamé coefficients for aluminum are $\mu=26\,$GPa, $\lambda=58\,$GPa. Impose displacement boundary conditions at the steel interface, natural boundary conditions at the horizontal surfaces, and periodic boundary conditions on the left and right ends of the computational domain. Use linear or quadratic elements for displacement, preferably the latter. You're not expected to use isoparametric elements on the curved boundaries, but you're welcome to do so if you can. Make plots of the tractions $\sigma_{12}(y)$ and $\sigma_{11}(y)$ along the cross sections shown. Also plot the deformed shape of the upper boundary:

$$\mathbf{r}(t)=(t+u(t,5/2),v(t,5/2)), \qquad (-5\le t\le5).$$

Here is the geometry:

• meshes.zip