Section 1.13
Solid Mechanics Part III
Kelly
110
1.13 Coordinate Transformation of Tensor Components
It has been seen in §1.5.2 that the transformation equations for the components of a vector
are
j
ij
i
u
Q
u
′
=
, where
[
]
Q
is the transformation matrix.
Note that these
ij
Q
’s are
not the
components of a tensor
– these
s
Q
ij
'
are mapping the components of a vector onto the
components of the
same vector
in a second coordinate system – a (second-order) tensor,
in general, maps one vector onto a different vector.
The equation
j
ij
i
u
Q
u
′
=
is in matrix
element form, and is not to be confused with the index notation for vectors and tensors.
1.13.1
Relationship between Base Vectors
Consider two coordinate systems with base vectors
i
e
and
i
e
′
.
It has been seen in the
context of vectors that, Eqn. 1.5.4,
)
,
cos(
j
i
ij
j
i
x
x
Q
′
≡
=
′
⋅
e
e
.
(1.13.1)
Recal that the
i
’s and
j
’s here are not referring to the three different components of a
vector, but to
different
vectors (nine different vectors in all).
It is interesting that the relationship 1.13.1 can also be derived as follows:
j
ij
j
i
j
i
j
j
i
i
Q
e
e
e
e
e
e
e
Ie
e
′
=
′
⋅
′
=
′
⊗
′
=
=
)
(
)
(
(1.13.2)
Dotting each side here with
k
e
′
then gives 1.13.1.
Eqn. 1.13.2, together with the
corresponding inverse relations, read
j
ij
i
Q
e
e
′
=
,
j
ji
i
Q
e
e
=
′
(1.13.3)
Note that the components of the transformation matrix
[
]
Q
are the components of the
change of basis tensor 1.10.24-25.
1.13.2
Tensor Transformation Rule
As with vectors, the components of a (second-order) tensor will change under a change of
coordinate system.
In this case, using 1.13.3,
n
m
pq
nq
mp
n
nq
m
mp
pq
q
p
pq
j
i
ij
T
Q
Q
Q
Q
T
T
T
e
e
e
e
e
e
e
e
⊗
′
=
⊗
′
=
′
⊗
′
′
≡
⊗
(1.13.4)

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