Page 32 - Introduction to Continuum Mechanics

Part B Sum of Tensors 17

or

We can concisely derive Eq. (2B3.1a) using indicial notation as follows: From a = a/Cj, we
get Ta = Tfl/e,- = a/Te/. Since Te, = 7)/ey, (Eq. (2B2.1b)), therefore,

i.e.,

Eq. (2B3.1d) is nothing but Eq. (2B3.1a) in indicial notation. We see that for the tensorial
equation b = Ta, there corresponds a matrix equation of exactly the same form, i.e., [b] = [T][a].
This is the reason we adopted the convention that Tej = T^i+7*2162+ 73163, etc. If we had
t
e
adopted the convention Te^ = 7ne 1+7 1262+^I3 3' etc -' tnen we would have obtained
7*
[b]=[T] [a] for the tensorial equation b = Ta, which would not be as natural.
Example 2B3.1
Given that a tensor T which transforms the base vectors as follows:

Tej = 2e 1-6e 2+4e 3
T02 = 3ej+462-63
Te 3 = -26J+62+263
How does this tensor transform the vector a = ej+262+363?
Solution. Using Eq. (2B3.1b)
b
i\ [2 3 -2] fll [2"
b 2 = -6 4 1 2 = 5
[ 4 -1 2J [3J [8
b 3
or
b = 2e 1+5e 2+8e 3

2B4 Sum of Tensors
Let T and S be two tensors and a be an arbitrary vector. The sum of T and S, denoted by
T + S, is defined by: