Page 89 - Introduction to Continuum Mechanics

74 Problems

(c) If A,- and n, are the eigenvalues and eigenvectors of U, find the eigenvectors and eigenvec-
tors of V.
2B34. (a) By inspection find an eigenvector of the dyadic product ab
(b) What vector operation does the first scalar invariant of ab correspond to?
(c) Show that the second and the third scalar invariants of ab vanish. Show that this indicates
that zero is a double eigenvalue of ab. What are the corresponding eigenvectors?
2B35. A rotation tensor R is defined by the relations

r
(a) Find the matrix of R and verify that RR = I and det | R| =1.
(b) Find the angle of rotation that could have been used to effect this particular rotation.

2B36. For any rotation transformation a basis e/ may be chosen so that 63 is along the axis of
rotation.

(a) Verify that for a right-hand rotation angle 0, the rotation matrix in respect to the e/ basis
is

(b) Find the symmetric and antisymmetric parts of [R]'.
5
(c) Find the eigenvalues and eigenvectors of R .
(d) Find the first scalar invariant of R.
4
(e) Find the dual vector of R .
(f) Use the result of (d) and (e) to find the angle of rotation and the axis of rotation for the
previous problem.

2B37. (a) If Q is an improper orthogonal transformation (corresponding to a reflection), what
are the eigenvalues and corresponding eigenvectors of Q?
(b) If the matrix Q is

find the normal to the plane of reflection.
2B38. Show that the second scalar invariant of T is