Introduction to 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}

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