Page 86 - Introduction to Continuum Mechanics

Tensors 71

(b) Let m=yj(e 1+e2+e3), find the matrix of the linear transformation that corresponds to

this rotation.
(c) Use this linear transformation to find the rotated vector of a = €]_+2*2+3*3-
2B10. (a) Find the matrix of the tensor S that transforms every vector into its mirror image in
a plane whose normal is 62 and then by a 45 right-hand rotation about the e raxis,

(b) Find the matrix of the tensor T that transforms every vector by the combination of first the
rotation and then the reflection of part (a).
(c) Consider the vector e 1+2e2+3e3, find the transformed vector by using the transformations
S, Also, find the transformed vector by using the transformation T.
2B11. a) Let R correspond to a right-hand rotation of angle 6 about the *3-axis.
2
(a)Find the matrix of R .
*?
(b)Show that R corresponds to a rotation of angle 20 about the same axis.
n
(c)Find the matrix of R for any integer n.
2B12. Rigid body rotations that are small can be described by an orthogonal transformation
R = I+eR*, where e-*0 as the rotation angle approaches zero. Considering two successive
2
rotations Rj and R2, show that for small rotations (so that terms containing e can be neglected)
the final result does not depend on the order of the rotations.
2B13. Let T and S be any two tensors. Show that

r
(a) T is a tensor.
T r r
(b)T +$ =(T+S)
r
7 7
(c) (TS) = S ! *.
2B14. Using the form for the reflection in an arbitrary plane of Prob. 2B8, write the reflection
tensor in terms of dyadic products.
2B15. For arbitrary tensors T and S, without relying on the component form, prove that
1 r r 1
(a)(T~ ) =(T )~ .
1
1
1
(b)(TS)~ = S~ T~ .
2B16. Let Q define an orthogonal transformation of coordinates, so that e/ = Q m^ m- Consider
e; • ej and verify that Q miQ mj = d fj.
2B17. The basis e/ is obtained by a 30° counterclockwise rotation of the c/ basis about €3.

:
(a) Find the orthogonal transformation Q that defines this change of basis, i.e., e t = £> mi-e m