Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

# Questions

### Quick questions

You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!

1. Let a = (2,3,5) and b = (-1, 2, -4).
Calculate (the scalar product)

2. Let a = (2,3,5) and b = (-1, 2, -4).
Calculate the tensor product).

3. Let,  and . Find the matrix products AB, BA, AC, CAB and the determinants |A|, |AB| and |BA|.

4. What is the transformation matrix for a rotation through angle θ about Ox2?

5. A vector is first rotated through angle θ about Ox2 and then through φ about Ox1. What is the combined transform matrix?

6. Show that a pure shear stress field  can be represented as a pure normal stress field by rotating through 45° about the vertical axis.

7. The conductivity tensor of a crystal is found to be . Show that the crystal does not conduct in one direction and find this direction relative to the lab basis.

8. For the stress state with normal stresses σxx = 50 GPa, σyy = -70 GPa, σzz = 20 GPa and shear stresses σxy = 30 GPa, σxz = 45 GPa and σyz = 60 GPa, find the normal stress on the (263) plane in a cubic system.

9. Graphite has a layered hexagonal structure with cell dimensions a = 0.246 nm and c = 0.679 nm and has electrical conductivities parallel and perpendicular to the planes of σparallel = 1.02 × 105 S m-1 and σperpendicular = 0.24 × 105 S m-1 respectively.
A sample is mounted such that an electric field is set up along the [112] direction. The current density is measured parallel to the electric field.
By first constructing the conductivity tensor and the direction cosine vector, find the expected current density in this direction for a 100 V m-1 electric field.