Elasticity/Transformation example 1

Example 1

Derive the transformation rule for second order tensors ($T_{ij}^{\text{'}}=l_{ip}{l}_{jq}{T}_{pq}$). Express this relation in matrix notation.

Solution

A second-order tensor $𝐓$ transforms a vector $𝐮$ into another vector $𝐯$. Thus,

$𝐯=𝐓𝐮=𝐓\bullet 𝐮$

In index and matrix notation,

$\text{(1)}{v}_i=T_{ij}{u}_i\leftrightarrow v_p=T_{pq}{u}_q\text{or,}\left[v\right]=\left[T\right]\left[u\right]$

Let us determine the change in the components of $𝐓$ with change the basis from (${𝐞}_1,{𝐞}_2,{𝐞}_3$) to (${𝐞}_1^{\text{'}},{𝐞}_2^{\text{'}},{𝐞}_3^{\text{'}}$). The vectors $𝐮$ and $𝐯$, and the tensor $𝐓$ remain the same. What changes are the components with respect to a given basis. Therefore, we can write

$\text{(2)}{v}_i^{\text{'}}=T_{ij}^{\text{'}}{u}_i^{\text{'}}\text{or,}{\left[v\right]}^{\text{'}}={\left[T\right]}^{\text{'}}{\left[u\right]}^{\text{'}}$

Now, using the vector transformation rule,

$\begin{array}{cc}\text{(3)}{v}_i^{\text{'}}& =l_{ip}{v}_p;u_i^{\text{'}}=l_{ip}{u}_p\text{or,}{\left[v\right]}^{\text{'}}=\left[L\right]\left[v\right];{\left[u\right]}^{\text{'}}=\left[L\right]\left[u\right]\\ v_q& =l_{iq}{v}_i^{\text{'}};u_q=l_{iq}{u}_i^{\text{'}}\text{or,}\left[v\right]={\left[L\right]}^T{\left[v\right]}^{\text{'}};\left[u\right]={\left[L\right]}^T{\left[u\right]}^{\text{'}}\end{array}$

Plugging the first of equation (3) into equation (2) we get,

$\text{(4)}{l}_{ip}{v}_p=T_{ij}^{\text{'}}{u}_i^{\text{'}}\text{or,}\left[L\right]\left[v\right]={\left[T\right]}^{\text{'}}{\left[u\right]}^{\text{'}}$

Substituting for $v_p$ in equation~(4) using equation~(1),

$\text{(5)}{l}_{ip}{T}_{pq}{u}_q=T_{ij}^{\text{'}}{u}_i^{\text{'}}\text{or,}\left[L\right]\left[T\right]\left[u\right]={\left[T\right]}^{\text{'}}{\left[u\right]}^{\text{'}}$

Substituting for $u_q$ in equation (5) using equation (3),

$\text{(6)}{l}_{ip}{T}_{pq}{l}_{iq}{u}_i^{\text{'}}=T_{ij}^{\text{'}}{u}_i^{\text{'}}\text{or,}\left[L\right]\left[T\right]{\left[L\right]}^T{\left[u\right]}^{\text{'}}={\left[T\right]}^{\text{'}}{\left[u\right]}^{\text{'}}$

Therefore, if $𝐮\equiv \left[u\right]$ is an arbitrary vector,

$l_{ip}{T}_{pq}{l}_{iq}=T_{ij}^{\text{'}}\Rightarrow T_{ij}^{\text{'}}=l_{ip}{l}_{jq}{T}_{pq}\text{or,}{\left[T\right]}^{\text{'}}=\left[L\right]\left[T\right]{\left[L\right]}^T$

which is the transformation rule for second order tensors.

Therefore, in matrix notation, the transformation rule can be written as

${\left[T\right]}^{\text{'}}=\left[L\right]\left[T\right]{\left[L\right]}^T$