Tensors of Higher-Order

A third-order tensor is a linear transformation from the space of second-order tensors into vectors or vectors into second-order tensors, and can be represented as

$$\mathbf{A} = A_{ijk}\hat\mathbf{e}_i\otimes\hat\mathbf{e}_j\otimes\hat\mathbf{e}_k.$$ (2.8.4.1)

Under a change of basis (2.8.3.9), the transformation rule for its components can be derived as follows.

$$\mathbf{A} =\ A^\prime_{ijk}\hat\mathbf{e}^\prime_i\otimes \hat\mathbf{e}^\prime_j\otimes \mathbf{e}^\prime_k$$

$$=\ A^\prime _{ijk} Q_{il}\hat\mathbf{e}_l\otimes Q_{jm}\hat\mathbf{e}_m\otimes\mathbf{Q}_{kn}\hat \mathbf{e}_{kn}$$

$$=\ A^\prime_{ijk}Q_{il}Q_{jm}Q_{kn} \hat\mathbf{e}_l\otimes \hat\mathbf{e}_m\otimes \hat\mathbf{e}_n$$

$$=\ A_{lmn} \hat\mathbf{e}_l\otimes \hat\mathbf{e}_m\otimes \hat \mathbf{e}_n.$$ (2.8.4.2)

Thus

$$A_{lmn} = Q_{il}Q_{jm}Q_{kn}A^\prime_{ijk}$$ (2.8.4.3)

or

$$A^\prime_{ijk} = Q_{il}Q_{jm}Q_{kn}A_{lmn}.$$ (2.8.4.4)

A fourth-order tensor is a linear transformation from the space of second-order tensors to second-order tensors, and has the representation

$$\mathbf{C} = C_{ijkl}\hat\mathbf{e}_i\otimes \hat\mathbf{e}_j\otimes \hat\mathbf{e}_k\otimes \hat\mathbf{e}_l.$$ (2.8.4.5)

Under the transformation (2.8.3.9) its components will transform as follows.

$$C^\prime_{ijkl} = Q_{ip}Q_{jq} Q_{kr}Q_{ls}C_{pqrs}.$$ (2.8.4.6)