Page 28 - Introduction to Continuum Mechanics

Part B Components of a Tensor 13

Example 2B1.5
Let T be a tensor that transforms the specific vectors a and b according to
Ta = a+2b, Tb = a-b
Given a vector c = 2a+b, find Tc.

Solution. Using the linearity property of tensors
Tc = T(2a+b) = 2Ta+Tb = 2(a+2b)+(a-b) = 3a+3b
2B2 Components of a Tensor

The components of a vector depend on the base vectors used to describe the components.
This will also be true for tensors. Let ej_, 63, ©3 be unit vectors in the direction of the xi~, X2~,
jt3-axes respectively, of a rectangular Cartesian coordinate system. Under a transformation T,
these vectors, e l5 62, e 3 become Te ls Te2, and Te3. Each of these Te/ (/= 1,2,3), being a vector,
can be written as:

or

It is clear from Eqs. (2B2.1a) that

or in general

The components TJJ in the above equations are defined as the components of the tensor T.
These components can be put in a matrix as follows:
T T
TII n 13
[T] = 7^! r 2 r 23
2
Til 732 ^33
This matrix is called the matrix of the tensor T with respect to the set of base vectors
r
e
e
e
r e
i i> 2, s} ° l /} f° short. We note that, because of the way we have chosen to denote the
components of transformation of the base vectors, the elements of the first column are
components of the vector Tej, those in the second column are the components of the vector
Te2, and those in the third column are the components of Te 3.