Elasticity/Transformation example 1

Example 1
Derive the transformation rule for second order tensors ($$T^{'}_{ij} = l_{ip} l_{jq} T_{pq}$$). Express this relation in matrix notation.

Solution
A second-order tensor $$\mathbf{T}$$ transforms a vector $$\mathbf{u}$$ into another vector $$\mathbf{v}$$. Thus,

\mathbf{v} = \mathbf{T}\mathbf{u} = \mathbf{T}\bullet\mathbf{u} $$ In index and matrix notation,
 * $$\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] $$ Let us determine the change in the components of $$\mathbf{T}$$ with change the basis from ($$\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3$$) to ($$\mathbf{e}_1^{'},\mathbf{e}_2^{'},\mathbf{e}_3^{'}$$). The vectors $$\mathbf{u}$$ and $$\mathbf{v}$$, and the tensor $$\mathbf{T}$$ remain the same. What changes are the components with respect to a given basis. Therefore, we can write
 * $$\text{(2)} \qquad

v^{'}_i = T^{'}_{ij} u^{'}_i ~\text{or,}~ \left[v\right]^{'} = \left[T\right]^{'} \left[u\right]^{'} $$ Now, using the vector transformation rule,
 * $$\begin{align}\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{align}$$ Plugging the first of equation (3) into equation (2) we get,
 * $$\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$$ in equation~(4) using equation~(1),
 * $$\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$$ in equation (5) using equation (3),
 * $$\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]^{'} $$ Therefore, if $$\mathbf{u} \equiv \left[u\right]$$ is an arbitrary vector,

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

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