ME-321: Advanced Mechanics of Solids

1. Write the following equations in expanded from ( {1,2,3}).

2)

2)

0)

,,,,

,

c

uuuub

FTa

2. Expand and simplify the following

aaeb

a

)

)

3. Given that, 2 , with out expanding show that

2)(2)

)23()

b

a

4. Given that, dxudy )( , , with out expanding show that

dxdxuuuudydy ,,,,

5. For the (3 x 3) matrices a (aij) and b (bij), write down the following products in index notation using summation convention

ab, ba, aTb, abT, aaT and aTa

6. Given that qandp are components of a second order and first order tensor

respectively, prove that

qpra ) are components of a third order tensor and

qpsb ) are components of a first order tensor

7. Prove that ij and eijk are isotropic tensors in RCC. Now verify if components of eijk are invariant under a coordinate transformation defined by

332211 ;; xxxxxx , explain your observation

8. If ui are the components of a vector, then prove that ui,j are the components of a second order cartesian tensor.