CONTINUUM MECHANICS for ENGINEERS

Determine
(a) the transformation matrix between the two sets of axes, and show
that it is a proper orthogonal transformation.
(b) the equation of the plane x1 + x2 + x3 = 1/ 2 in its primed axes
form, that is, in the form + + = b.
Answer:
bx 1 1 ′ bx 2 2 ′ bx 3 3 ′
⎡1/
2 0 1/ 2⎤
⎢
⎥
(a) [ aij]= ⎢ 1/ 2 1/ 2 −1/
2⎥,
⎢
⎣
−1/
2 1/ 2 1/ 2⎥
⎦
(b) ′ + ′ + = 1 ′
2.20 Making use of Eq 2.4-11 of the text in the form det A = ε ijk A 1iA 2jA 3k
write Eq 2.6-6 as
2x 1
and show by expansion of this equation that
to verify Eq 2.6-8 of the text.
= 0
2.21 For the matrix representation of tensor B shown below,
determine the principal values (eigenvalues) and the principal directions
(eigenvectors) of the tensor.
Answer: λ1 = 17, λ2 = 26, λ3 = –39
(2)
, nˆ = 4eˆ + 7eˆ / 65 ,
2.22 Consider the symmetrical matrix
x 2
T − λδ = ε T −λδ
T λδ T λδ
ij ij ijk 1i1i2j2j3k3k x 3
( ) −
( )
( ) −
( )
3 2 ⎡ 1
⎤
λ − Tiiλ + TiiTjj −TijTji
λ− εijkT1iT2jT3k=
0
⎣⎢ 2
⎦⎥
ˆ ˆ
n e
(1) = 1
⎡17
0 0⎤
⎥
⎢
0 23 28
⎥
⎣
⎢ 0 28 10⎦
⎥
⎢ [ Bij]= −
[ Bij]= ( 2 3 )
⎡ 5
3 0 ⎤
2
2 ⎢ ⎥
⎢0
4 0⎥
⎢ 3
5⎥
⎣
0
2
2⎦
( 2 3)
(3)
nˆ = − 7eˆ + 4eˆ / 65