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

DoITPoMS Teaching & Learning Packages Tensors in Materials Science Questions
Tensors in Materials Science AimsBefore you start - The basicsIntroductionScalars, Vectors and MatricesWhat is a Tensor?Tensor UsageTensor NotationTransformation of axesPrincipal axesThe representation surfaceMagnitude of a property in a given directionThe effects of crystal symmetrySummaryQuestionsGoing further
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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 Equation(the scalar product)

  2. Let a = (2,3,5) and b = (-1, 2, -4).
    CalculateEquation 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 Equation 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 Equation. 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.

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