Introduction to Elasticity/Sample midterm 1

From Wikiversity
Jump to navigation Jump to search

Sample Midterm Problem 1[edit | edit source]

Given:

The vectors , , and are given, with respect to an orthonormal basis , by

Find:

  • (a) Evaluate .
  • (b) Evaluate . Is a tensor? If not, why not? If yes, what is the order of the tensor?
  • (c) Name and define and .
  • (d) Evaluate .
  • (e) Show that .
  • (f) Rotate the basis by 30 degrees in the counterclockwise direction around to obtain a new basis . Find the components of the vector in the new basis .
  • (g) Find the component of in the new basis .

Solution[edit | edit source]

Part (a)[edit | edit source]

Part (b)[edit | edit source]

Part (c)[edit | edit source]

Part (d)[edit | edit source]

Part (e)[edit | edit source]

Because cannot be an even or odd permutation of .

Part (f)[edit | edit source]

The basis transformation rule for vectors is

where

Therefore,

Hence,

Thus,

Part (g)[edit | edit source]

The basis transformation rule for second-order tensors is

Therefore,