Page 47 - Introduction to Continuum Mechanics

32 Tensors

Solution. The primed components are related to the unprimed components by
Eq. (2B13.1a)
' ij ~ \2mv>2ni*mn
Thus,
=
'ii QmiQni* mn
But, Q miQ ni = d mn (Eq. (2B10.2c)), therefore,
'ii ~ ®mn*mn ~ 'mm
i.e.,

+
=
T
T\\ + 7*22+ TB 7ll+ 22 ^33
We see from Example 2B13.1, that we can calculate all nine components of a tensor T with
respect to e,' from the matrix [T] e ., by using Eq. (2B13.1c). However, there are often times
when we need only a few components. Then it is more convenient to use the Eq. (2B2.2)
(TIJ = e/ -Tej) which defines each of the specific components.
In matrix form this equation is written as:

T
where [e'] denotes a row matrix whose elements are the components of e/ with respect to the
basis {e/}.

Example 2B13.3

Obtain T[i for the tensor T and the bases e/ and e/ given in Example 2B13.1
Solution. Since ej = 62, and 62 = -e l5 thus

TU = ei-Tei = e 2-T(- ei) =-e 2-T ei = -T 2l = -1
Alternatively, using Eq. (2B13.4)
TO i oi [-ii r o"

7i2 = [«il mtel = [0,1,0] 1 2 0 0 = [0,1,0] -1 = -1
0 0 IJ [ OJ I 0
2B14 Defining Tensors by Transformation Laws

Equations (2B12.1) or (2B13.1) state that when the components of a vector or a tensor with
:
respect to {e,-} are known, then its components with respect to any {e,} are uniquely deter-
mined from them. In other words, the components a,- or 7^- with respect to one set of {e/}