Derive the transformation rule for second order tensors
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{\displaystyle T_{ij}^{'}=l_{ip}l_{jq}T_{pq}}
A second-order tensor
T
{\displaystyle \mathbf {T} }
u
{\displaystyle \mathbf {u} }
v
{\displaystyle \mathbf {v} }
v
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T
u
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∙
u
{\displaystyle \mathbf {v} =\mathbf {T} \mathbf {u} =\mathbf {T} \bullet \mathbf {u} }
In index and matrix notation,
(1)
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↔
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{\displaystyle {\text{(1)}}\qquad v_{i}=T_{ij}u_{i}\leftrightarrow v_{p}=T_{pq}u_{q}~{\text{or,}}~\left[v\right]=\left[T\right]\left[u\right]}
Let us determine the change in the components of
T
{\displaystyle \mathbf {T} }
e
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e
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e
3
{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}}
e
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{\displaystyle \mathbf {e} _{1}^{'},\mathbf {e} _{2}^{'},\mathbf {e} _{3}^{'}}
u
{\displaystyle \mathbf {u} }
v
{\displaystyle \mathbf {v} }
T
{\displaystyle \mathbf {T} }
(2)
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or,
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{\displaystyle {\text{(2)}}\qquad v_{i}^{'}=T_{ij}^{'}u_{i}^{'}~{\text{or,}}~\left[v\right]^{'}=\left[T\right]^{'}\left[u\right]^{'}}
Now, using the vector transformation rule,
(3)
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or,
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;
[
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[
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or,
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;
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{\displaystyle {\begin{aligned}{\text{(3)}}\qquad v_{i}^{'}&=l_{ip}v_{p}~;~u_{i}^{'}=l_{ip}u_{p}~{\text{or,}}~\left[v\right]^{'}=\left[L\right]\left[v\right]~;\left[u\right]^{'}=\left[L\right]\left[u\right]\\v_{q}&=l_{iq}v_{i}^{'}~;~u_{q}=l_{iq}u_{i}^{'}~{\text{or,}}~\left[v\right]=\left[L\right]^{T}\left[v\right]^{'}~;\left[u\right]=\left[L\right]^{T}\left[u\right]^{'}\end{aligned}}}
Plugging the first of equation (3) into equation (2) we get,
(4)
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or,
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{\displaystyle {\text{(4)}}\qquad l_{ip}v_{p}=T_{ij}^{'}u_{i}^{'}~{\text{or,}}~\left[L\right]\left[v\right]=\left[T\right]^{'}\left[u\right]^{'}}
Substituting for
v
p
{\displaystyle v_{p}}
(5)
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or,
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{\displaystyle {\text{(5)}}\qquad l_{ip}T_{pq}u_{q}=T_{ij}^{'}u_{i}^{'}~{\text{or,}}~\left[L\right]\left[T\right]\left[u\right]=\left[T\right]^{'}\left[u\right]^{'}}
Substituting for
u
q
{\displaystyle u_{q}}
(6)
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or,
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{\displaystyle {\text{(6)}}\qquad l_{ip}T_{pq}l_{iq}u_{i}^{'}=T_{ij}^{'}u_{i}^{'}~{\text{or,}}~\left[L\right]\left[T\right]\left[L\right]^{T}\left[u\right]^{'}=\left[T\right]^{'}\left[u\right]^{'}}
Therefore, if
u
≡
[
u
]
{\displaystyle \mathbf {u} \equiv \left[u\right]}
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⇒
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or,
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{\displaystyle l_{ip}T_{pq}l_{iq}=T_{ij}^{'}\Rightarrow T_{ij}^{'}=l_{ip}l_{jq}T_{pq}~{\text{or,}}~\left[T\right]^{'}=\left[L\right]\left[T\right]\left[L\right]^{T}}
which is the transformation rule for second order tensors.
Therefore, in matrix notation, the transformation rule can be written as
[
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T
{\displaystyle \left[T\right]^{'}=\left[L\right]\left[T\right]\left[L\right]^{T}}
![{\displaystyle {\text{(1)}}\qquad v_{i}=T_{ij}u_{i}\leftrightarrow v_{p}=T_{pq}u_{q}~{\text{or,}}~\left[v\right]=\left[T\right]\left[u\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b59758773e290bee595dff26fce261df53315b2a)
![{\displaystyle {\text{(2)}}\qquad v_{i}^{'}=T_{ij}^{'}u_{i}^{'}~{\text{or,}}~\left[v\right]^{'}=\left[T\right]^{'}\left[u\right]^{'}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a460e37a41041c61204fbdef591de5ff287be43e)
![{\displaystyle {\begin{aligned}{\text{(3)}}\qquad v_{i}^{'}&=l_{ip}v_{p}~;~u_{i}^{'}=l_{ip}u_{p}~{\text{or,}}~\left[v\right]^{'}=\left[L\right]\left[v\right]~;\left[u\right]^{'}=\left[L\right]\left[u\right]\\v_{q}&=l_{iq}v_{i}^{'}~;~u_{q}=l_{iq}u_{i}^{'}~{\text{or,}}~\left[v\right]=\left[L\right]^{T}\left[v\right]^{'}~;\left[u\right]=\left[L\right]^{T}\left[u\right]^{'}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d41bc89caf0295da98e576e49a52af167b2636c)
![{\displaystyle {\text{(4)}}\qquad l_{ip}v_{p}=T_{ij}^{'}u_{i}^{'}~{\text{or,}}~\left[L\right]\left[v\right]=\left[T\right]^{'}\left[u\right]^{'}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eabd4394eee3a5f9a9b0c5d321b9a23663b2a423)
![{\displaystyle {\text{(5)}}\qquad l_{ip}T_{pq}u_{q}=T_{ij}^{'}u_{i}^{'}~{\text{or,}}~\left[L\right]\left[T\right]\left[u\right]=\left[T\right]^{'}\left[u\right]^{'}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0be33f40efed80708fd5ad3e1de1355d38e5e14a)
![{\displaystyle {\text{(6)}}\qquad l_{ip}T_{pq}l_{iq}u_{i}^{'}=T_{ij}^{'}u_{i}^{'}~{\text{or,}}~\left[L\right]\left[T\right]\left[L\right]^{T}\left[u\right]^{'}=\left[T\right]^{'}\left[u\right]^{'}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fcd28ca6b93530e8ec74bc4ac999ded86a164d3)
![{\displaystyle \mathbf {u} \equiv \left[u\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/450f03c60dbf32f0d4848a07811dff28797c2417)
![{\displaystyle l_{ip}T_{pq}l_{iq}=T_{ij}^{'}\Rightarrow T_{ij}^{'}=l_{ip}l_{jq}T_{pq}~{\text{or,}}~\left[T\right]^{'}=\left[L\right]\left[T\right]\left[L\right]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29753cdc8f5b778080d598f9ddb2ac95f77f004c)
![{\displaystyle \left[T\right]^{'}=\left[L\right]\left[T\right]\left[L\right]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8bce00a2bb3d332d1314d298ddad6e3d8277af8)