Page 90 - Introduction to Continuum Mechanics

Tensors 75

by expanding this equation.
2B39. Using the matrix transformation law for second-order tensors, show that the third scalar
invariant is indeed independent of the particular basis.
2B40, A tensor T has a matrix

(a) Find the scalar invariants, the principle values and corresponding principal directions of
the tensor T.
(b) If 111,112,113 are the principal directions, write [T] n..

(c) Could the following matrix represent the tensor T in respect to some basis?

2B41. Do the previous Problem for the matrix

2B42. A tensor T has a matrix

u u z,
Find the principal values and three mutually orthogonal principal directions.
2B43. The inertia tensor 1 0 of a rigid body with respect to a point o, is defined by

where r is the position vector, r= \ r| ,p- mass density, I is the identity tensor, and dV is a
differential volume. The moment of inertia, with respect to an axis pass through o, is given by
l nn = n • I 0n, (no sum on n), where n is a unit vector in the direction of the axis of interest.

(a) Show that 1 0 is symmetric.
(b) Letting r = jcej+^+z^, write out all components of the inertia tensor \ 0.
(c) The diagonal terms of the inertia matrix are the moments of inertia and the off-diagonal
terms the products of inertia. For what axes will the products of inertia be zero? For which
axis will the moments of inertia be greatest (or least)?