Example 1
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Derive the transformation rule for second order tensors
(T i j ′ = l i p l j q T p q {\displaystyle T_{ij}^{'}=l_{ip}l_{jq}T_{pq}} ). Express this relation in matrix notation.
Solution
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A second-order tensor T {\displaystyle \mathbf {T} } transforms a vector u {\displaystyle \mathbf {u} } into another vector v {\displaystyle \mathbf {v} } .
Thus,
v = T u = T ∙ u {\displaystyle \mathbf {v} =\mathbf {T} \mathbf {u} =\mathbf {T} \bullet \mathbf {u} } In index and matrix notation,
(1) v i = T i j u i ↔ v p = T p q u q or, [ v ] = [ T ] [ u ] {\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} } with change the basis
from (e 1 , e 2 , e 3 {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}} ) to (e 1 ′ , e 2 ′ , e 3 ′ {\displaystyle \mathbf {e} _{1}^{'},\mathbf {e} _{2}^{'},\mathbf {e} _{3}^{'}} ). The vectors u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } , and
the tensor T {\displaystyle \mathbf {T} } remain the same. What changes are the components with respect
to a given basis. Therefore, we can write
(2) v i ′ = T i j ′ u i ′ or, [ v ] ′ = [ T ] ′ [ u ] ′ {\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) v i ′ = l i p v p ; u i ′ = l i p u p or, [ v ] ′ = [ L ] [ v ] ; [ u ] ′ = [ L ] [ u ] v q = l i q v i ′ ; u q = l i q u i ′ or, [ v ] = [ L ] T [ v ] ′ ; [ u ] = [ L ] T [ u ] ′ {\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) l i p v p = T i j ′ u i ′ or, [ L ] [ v ] = [ T ] ′ [ u ] ′ {\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}} in equation~(4) using equation~(1),
(5) l i p T p q u q = T i j ′ u i ′ or, [ L ] [ T ] [ u ] = [ T ] ′ [ u ] ′ {\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}} in equation (5) using equation (3),
(6) l i p T p q l i q u i ′ = T i j ′ u i ′ or, [ L ] [ T ] [ L ] T [ u ] ′ = [ T ] ′ [ u ] ′ {\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]} is an arbitrary vector,
l i p T p q l i q = T i j ′ ⇒ T i j ′ = l i p l j q T p q or, [ T ] ′ = [ L ] [ T ] [ L ] T {\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
[ T ] ′ = [ L ] [ T ] [ L ] T {\displaystyle \left[T\right]^{'}=\left[L\right]\left[T\right]\left[L\right]^{T}}