Page 27 - Introduction to Continuum Mechanics

12 Tensors

Example 2B1.2
Let T be a transformation which transforms every vector into a vector that is k times the
original vector. Is this transformation a tensor?
Solution. Let a and b be arbitrary vectors and a and ft be arbitrary scalars, then by the
definition of T,
Ta = Jta, Tb = fcb, and T(aa+£b) = fc(aa+/3b)
Clearly,
T(aa+£b) = a(ka)+p(kb) = aTa+£Tb
Thus, by Eq. (2B1.2), T is a linear transformation. In other words, it is a tensor.

In the previous example, if fc=0 then the tensor T transforms all vectors into zero. This
tensor is the zero tensor and is symbolized by 0.

Example 2B1.3
Consider a transformation T that transforms every vector into its mirror image with respect
to a fixed plane. Is T a tensor?
Solution. Consider a parallelogram in space with its sides represented by vectors a and b
and its diagonal represented the resultant a + b. Since the parallelogram remains a paral-
lelogram after the reflection, the diagonal (the resultant vector) of the reflected parallelogram
is clearly both T(a + b), the reflected (a + b), and Ta + Tb, the sum of the reflected a and
the reflected b. That is, T(a + b) = Ta + Tb. Also, for an arbitrary scalar a, the reflection
of aa is obviously the same as a times the reflection of a (i.e., T(aa )= aTa) because both
vectors have the same magnitude given by a times the magnitude of a and the same direction.
Thus, by Eqs. (2B1.1), T is a tensor.

Example 2B 1.4
When a rigid body undergoes a rotation about some axis, vectors drawn in the rigid body in
general change their directions. That is, the rotation transforms vectors drawn in the rigid body
into other vectors. Denote this transformation by R. Is R a tensor?
Solution. Consider a parallelogram embedded in the rigid body with its sides representing
vectors a and b and its diagonal representing the resultant a + b. Since the parallelogram
remains a parallelogram after a rotation about any axis, the diagonal (the resultant vector) of
the rotated parallelogram is clearly both R(a + b) , the rotated (a 4- b), and Ra 4- Rb, the
sum of the rotated a and the rotated b. That is R(a + b) = Ra + Rb.A similar argument as
that used in the previous example leads to R(aa )= aRa . Thus, R is a tensor.