Problem Set 2 Due September 27, 2001

EMch 540—Introduction to Continuum Mechanics Name:
Problem Set 2
Due September 27, 2001
Professor Costanzo Fall 2000
Problem 1
Determine the spectrum, the characteristic spaces and the spectral decomposition for the
following tensor:
T = m ⊗ n + n ⊗ m,
where m and n are two mutually orthogonal unit vectors.
Problem 2
Let A and Q be a symmetric and an orthogonal tensor, respectively. Show that that the
spectrum of A is the same as that of the tensor QAQ T . Furthermore, show that if e is an
eigenvector of A, then Qe is an eigenvector of QAQ T corresponding to the same eigenvalue.
Problem 3
Let {e1, e2, e3} and {g1, g2, g3} be two distinct orthonormal right-handed bases in V such
that gi = 3 j=1 Tijej, where [Tij] is an invertible matrix.
1. Show that the tensor Q = 3 i,j=1 Tji ei ⊗ ej is a rotation such that
gi = Qei;
2. Let {vi} and {ˆvk} be the components of a vector v relative to the {ei} and {gk} bases,
respectively. Show that
ˆvi = Tij vj. (1)
3. Let [Sij] and [ ˆ Smn] be the components of a tensor S relative to the {ei ⊗ ej} and
{gm ⊗ gn} bases, respectively. Show that
ˆSij = TimTjn Smn. (2)
NOTE: In the traditional approach to tensor algebra, Cartesian vectors and (second order)
tensors are DEFINED as objects whose components behave as indicated in Eqs. (1) and
(2), respectively, when the underlying Cartesian coordinate frame undergoes the rotation
implied by the matrix [Tji]. In other words, one can define tensors by defining the behavior
of its components under changes of coordinate frames. According to this view, a N-th
order Cartesian tensor is an object characterized by N indices whose components obey the
following transformation law:
Âi1,i2,...,iN = Ti1j1 Ti2j2 . . . TiN jN
Aj1,j2,...,jN , (3)
[Tmn] being the orthogonal matrix which defines the coordinate transformation in question.

EMch 540—Introduction to Continuum Mechanics Name:
Problem 4
Let F = RU = VR denote the right and left polar decompositions of F ∈ Lin + , where Lin +
is the collection of all tensors with positive determinant.
1. Show that U and V have the same spectrum (ω1, ω2, ω3).
2. Identify the relationship between the eigenvectors ei of U and the eigenvectors fi of V
corresponding to the eigenvalue ωi.
3. Prove that F and R admit the following representations:
3
F = ωifi ⊗ ei,
3
R = fi ⊗ ei,
i=1
i=1
where ei and fi are, respectively, the eigenvectors of U and V corresponding to the
eigenvalue ωi.
Hint: start by recalling that the identity tensor can be represented using the vectors fi
or ei, that is, for example, I = 3 i=1 fi ⊗ fi. Then, combine this result with a relation
ship that expresses fi in terms of ei.