Page 88 - Introduction to Continuum Mechanics

Tensors 73

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(b) Generalize the result of part (a) by considering, the components of a tensor of n order
Rijff.,. Show that /?//*... are components of an (n-2) order tensor.
2B25. The components of an arbitrary vector a and an arbitrary second-order tensor T are
related by a triply subscripted quantity R^ in the manner a/ = RijkTjk for any rectangular
Cartesian basis {61,62,63}. Prove that jRp are the components of a third-order tensor.
2B26. For any vector a and any tensor T, show that
(a) a-1^*8 = 0,
s
(b)a-Ta = a-T a.
2B27. Any tensor may be decomposed into a symmetric and antisymmetric part. Prove that
the decomposition is unique. (Hint: Assume that it is not unique.)
2B28, Given that a tensor T has a matrix

(a) find the symmetric and antisymmetric part of T.
(b) find the dual vector of the antisymmetric part of T.
2B29 From the result of part (a) of Prob. 2B9, for the rotation about an arbitrary axis m by
an angle B,

(a) Show that the rotation tensor is given by R = (l-cay0)(miii)+sin0E , where E is the
antisymmetric tensor whose dual vector is m. [note mm denotes the dyadic product of m with
m].

(b) Find HT , the antisymmetric part of R.
(c) Show that the dual vector for R^ is given by sin0m

2B30. Prove that the only possible real eigenvalues of an orthogonal tensor are A= ± 1.
2B31. Tensors T, R, and S are related by T - RS. Tensors R and S have the same eigenvector
n and corresponding eigenvalues r x and jj_. Find an eigenvalue and the corresponding eigen-
vector of T.
2B32. If n is a real eigenvector of an antisymmetric tensor T, then show that the corresponding
eigenvalue vanishes.

2B33. Let F be an arbitrary tensor. It can be shown (Polar Decomposition Theorem) that any
invertible tensor F can be expressed as F = VQ = QU, where Q is an orthogonal tensor and
U and V are symmetric tensors.
r r
(b) Show that W = FF and UU = F F.