CONTINUUM MECHANICS for ENGINEERS

FIGURE E3.5-1
Rotation of axes x 1 and x 2 by 45° about x 3 axis.
Thus, from Eq 3.5-1 expressed in matrix form, a 45° rotation of axes requires
⎡ 1/ 2 1/ 2 0⎤⎡1
3 2⎤⎡1/
2 −1/
2 0⎤
⎢
⎥ ⎢ ⎥ ⎢
⎥
[ σ ij ′ ] = ⎢−1/
2 1/ 2 0⎥⎢
3 1 0
⎥ ⎢1/
2 1/ 2 0⎥
⎢
⎥
⎣
0 0 1
⎦ ⎣
⎢2
0 −2⎦
⎥ ⎢
⎥
⎣
0 0 1
⎦
⎡ 4 0 2 ⎤
⎢
⎥
= ⎢ 0 −2 − 2⎥
⎢
⎣
2 − 2 −2
⎥
⎦
Example 3.5-2
Assume the stress tensor (in ksi) at P with respect to axes Px 1x 2x 3 is
represented by the matrix
[ σ ij]=
⎡ 18 0 −12⎤
⎢
⎥
⎢
0 6 0
⎥
⎣
⎢−12
0 24⎦
⎥
If the axis makes equal angles with the three unprimed axes, and the
axis lies in the plane of , as shown by the sketch, determine the primed
components of assuming is a right-handed system.
′ x1 ′ x2 xx 1′ 3
Px′ x′ x′
1 2 3
Solution
We must first determine the transformation matrix [aij]. Let β be the common
angle which x1′ makes with the unprimed axes, as shown by the sketch. Then
a11 = a12 = a13 = cosβ and from the orthogonality condition Eq 2.5-4 with i = j =
1, cosβ = 1 3 . Next, let φ be the angle between and x3. Then a23 = cosφ =
′ x2