Index notation:
![{\displaystyle \sigma _{ij}=2\mu ~\varepsilon _{ij}+\lambda ~\varepsilon _{kk}~\delta _{ij}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a72672b6d6efe9886be225c00423e60bb6623e2f)
If
![{\displaystyle {\begin{aligned}\sigma _{ii}&=2\mu ~\varepsilon _{ii}+\lambda ~\varepsilon _{kk}~\delta _{ii}\\&=2\mu ~\varepsilon _{kk}+3\lambda ~\varepsilon _{kk}\\&=(2\mu +3\lambda )~\varepsilon _{kk}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83b932269735ffc98eec5728456662f6473eeeed)
![{\displaystyle \sigma _{kk}=(2\mu +3\lambda )~\varepsilon _{kk}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a981c668e16cccc3e353745e3863f91fda94b986)
Dummy indices are replaceable.
Index notation:
![{\displaystyle \sigma _{ij}=2\mu ~\varepsilon _{ij}+\lambda ~\varepsilon _{kk}~\delta _{ij}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a72672b6d6efe9886be225c00423e60bb6623e2f)
Multiply by
:
![{\displaystyle {\begin{aligned}\sigma _{ij}~\delta _{ij}&=2\mu ~\varepsilon _{ij}~\delta _{ij}+\lambda ~\varepsilon _{kk}~\delta _{ij}~\delta _{ij}\\\implies \sigma _{jj}&=2\mu ~\varepsilon _{ii}+\lambda ~\varepsilon _{kk}~\delta _{ii}\\\implies \sigma _{kk}&=2\mu ~\varepsilon _{kk}+3\lambda ~\varepsilon _{kk}\\\implies \sigma _{kk}&=(2\mu +3\lambda )~\varepsilon _{kk}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f50e049ccf3d0ea01a6ad364c4037dff9469cfc)
Multiplication by
leads to replacement of one index.
![{\displaystyle A_{ij}~\delta _{kl}=?\qquad A_{ij}~\delta {jl}=?}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db48f757578297bd58a0e38f95d025291935db5d)
Index notation:
![{\displaystyle {\begin{aligned}{\boldsymbol {\sigma }}&=\sigma _{ij}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\{\boldsymbol {\varepsilon }}&=\varepsilon _{ij}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a310fe0a4832cecb29162fc6e2a2938d38f332ca)
From the definition of dyadic product, we can show
![{\displaystyle {\begin{aligned}(\mathbf {a} \otimes \mathbf {b} ):(\mathbf {u} \otimes \mathbf {v} )&=(\mathbf {a} \bullet \mathbf {u} )(\mathbf {b} \bullet \mathbf {v} )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f64c25f5a08d2bf2844e95177b54cf44e25e2184)
Contraction gives:
![{\displaystyle {\begin{aligned}{\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}&=(\sigma _{ij}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}):(\varepsilon _{kl}~\mathbf {e} _{k}\otimes \mathbf {e} _{l})\\&=\sigma _{ij}~\varepsilon _{kl}~(\mathbf {e} _{i}\otimes \mathbf {e} _{j}):(\mathbf {e} _{k}\otimes \mathbf {e} _{l})\\&=\sigma _{ij}~\varepsilon _{kl}~(\mathbf {e} _{i}\bullet \mathbf {e} _{k})(\mathbf {e} _{j}\bullet \mathbf {e} _{l})\\&=\sigma _{ij}~\varepsilon _{kl}~\delta _{ik}~\delta _{jl}\\&=\sigma _{ij}~\varepsilon _{ij}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d0a29d176b221fb03d727ddf1495bf6d07b4922)
Index notation:
![{\displaystyle {\begin{aligned}{\boldsymbol {\varepsilon }}&=\varepsilon _{ij}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\{\boldsymbol {\mathsf {C}}}&=C_{ijkl}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7c2197091916ce960ad60bf0003dd82f71a20d3)
Definition of dyadics products:
![{\displaystyle {\begin{aligned}(\mathbf {a} \bullet \mathbf {b} )\otimes \mathbf {x} &=(\mathbf {b} \bullet \mathbf {x} )\mathbf {a} \\(\mathbf {a} \bullet \mathbf {b} \otimes \mathbf {c} )\otimes \mathbf {x} &=(\mathbf {c} \bullet \mathbf {x} )(\mathbf {a} \otimes \mathbf {b} )\\(\mathbf {a} \bullet \mathbf {b} \otimes \mathbf {c} \otimes \mathbf {d} )\otimes \mathbf {x} &=(\mathbf {d} \bullet \mathbf {x} )(\mathbf {a} \otimes \mathbf {b} \otimes \mathbf {c} )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8fe58cc364383b8811f07e0363d627f8317cab)
We can show that
![{\displaystyle {\begin{aligned}(\mathbf {a} \otimes \mathbf {b} \otimes \mathbf {c} \otimes \mathbf {d} ):(\mathbf {u} \otimes \mathbf {v} )&=((\mathbf {a} \bullet \mathbf {b} \otimes \mathbf {c} \otimes \mathbf {d} )\otimes \mathbf {v} )\bullet \mathbf {u} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f9c88d1b7022d6a464baed6dcee429a9432311)
Contraction gives:
![{\displaystyle {\begin{aligned}{\boldsymbol {\mathsf {C}}}:{\boldsymbol {\varepsilon }}&=(C_{ijkl}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}):(\varepsilon _{mn}~\mathbf {e} _{m}\otimes \mathbf {e} _{n})\\&=C_{ijkl}~\varepsilon _{mn}~(\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}):(\mathbf {e} _{m}\otimes \mathbf {e} _{n})\\&=C_{ijkl}~\varepsilon _{mn}~((\mathbf {e} _{i}\bullet \mathbf {e} _{i}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l})\otimes \mathbf {e} _{n})\bullet \mathbf {e} _{m}\\&=C_{ijkl}~\varepsilon _{mn}~(\mathbf {e} _{l}\bullet \mathbf {e} _{n})(\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k})\bullet \mathbf {e} _{m}\\&=C_{ijkl}~\varepsilon _{mn}~\delta _{ln}(\mathbf {e} _{k}\bullet \mathbf {e} _{m})(\mathbf {e} _{i}\otimes \mathbf {e} _{j})=C_{ijkl}~\varepsilon _{mn}~\delta _{ln}~\delta _{km}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\&=C_{ijkl}~\varepsilon _{kl}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba0862e3577a46de5a3d1284e7c26308c29eac9)
Tensor Product of two tensors:
![{\displaystyle {\begin{aligned}{\boldsymbol {A}}&=A_{ij}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\{\boldsymbol {B}}&=B_{kl}~\mathbf {e} _{k}\otimes \mathbf {e} _{l}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a275d3c733494644de370dc7e8d38b456e059fa)
Tensor product:
![{\displaystyle {\begin{aligned}{\boldsymbol {A}}\otimes {\boldsymbol {B}}&=(A_{ij}~\mathbf {e} _{i}\otimes \mathbf {e} _{j})\otimes (B_{kl}~\mathbf {e} _{k}\otimes \mathbf {e} _{l})\\&=A_{ij}B_{kl}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3c80cfe1e349fe54463857a10c3cf0ab77978cf)
Change of basis: Vector transformation rule
![{\displaystyle v_{i}^{'}=L_{ij}v_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f699b1c45385e3b0ee9535e0172956dd1bf7dba2)
are the direction cosines.
![{\displaystyle {\begin{aligned}L_{11}&=\mathbf {e} _{1}^{'}\bullet \mathbf {e} _{1};&L_{12}&=\mathbf {e} _{1}^{'}\bullet \mathbf {e} _{2};&L_{13}&=\mathbf {e} _{1}^{'}\bullet \mathbf {e} _{3}\\L_{21}&=\mathbf {e} _{2}^{'}\bullet \mathbf {e} _{1};&L_{22}&=\mathbf {e} _{2}^{'}\bullet \mathbf {e} _{2};&L_{23}&=\mathbf {e} _{2}^{'}\bullet \mathbf {e} _{3}\\L_{31}&=\mathbf {e} _{3}^{'}\bullet \mathbf {e} _{1};&L_{32}&=\mathbf {e} _{3}^{'}\bullet \mathbf {e} _{2};&L_{33}&=\mathbf {e} _{3}^{'}\bullet \mathbf {e} _{3}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df915ca832964550491715e921f77f39deb8526c)
In matrix form
![{\displaystyle \mathbf {v} ^{'}=\mathbf {L} ~\mathbf {v} ;~~\mathbf {v} =\mathbf {L} ^{T}~\mathbf {v} ^{'};~~\mathbf {L} \mathbf {L} ^{T}=\mathbf {I} \implies \mathbf {L} ^{T}=\mathbf {L} ^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65ec2aaace53740c3857208a3694d76b59c974d9)
Other common form: Vector transformation rule
![{\displaystyle v_{i}^{'}=Q_{ji}v_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aed53c28f412d02154ce0e70b2f16420ac92198e)
![{\displaystyle {\begin{aligned}Q_{11}&=\mathbf {e} _{1}\bullet \mathbf {e} _{1}^{'};&Q_{12}&=\mathbf {e} _{1}\bullet \mathbf {e} _{2}^{'};&Q_{13}&=\mathbf {e} _{1}\bullet \mathbf {e} _{3}^{'}\\Q_{21}&=\mathbf {e} _{2}\bullet \mathbf {e} _{1}^{'};&Q_{22}&=\mathbf {e} _{2}\bullet \mathbf {e} _{2}^{'};&Q_{23}&=\mathbf {e} _{2}\bullet \mathbf {e} _{3}^{'}\\Q_{31}&=\mathbf {e} _{3}\bullet \mathbf {e} _{1}^{'};&Q_{32}&=\mathbf {e} _{3}\bullet \mathbf {e} _{2}^{'};&Q_{33}&=\mathbf {e} _{3}\bullet \mathbf {e} _{3}^{'}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1d3c6c4298ebc3f9a0d3400505bd69bf1b2ca5)
In matrix form
![{\displaystyle \mathbf {v} ^{'}=\mathbf {Q} ^{T}~\mathbf {v} ;~~\mathbf {v} =\mathbf {Q} ~\mathbf {v} ^{'};~~\mathbf {Q} \mathbf {Q} ^{T}=\mathbf {I} \implies \mathbf {Q} ^{T}=\mathbf {Q} ^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d29c89f5721f76621071e334dfb52b9b437986a)
Change of basis: Tensor transformation rule
![{\displaystyle T_{ij}^{'}=L_{ip}L_{jq}T_{pq}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1191d3b438fea4673f4bbd179c26b40a4165610)
where
are the direction cosines.
In matrix form,
![{\displaystyle \mathbf {T} ^{'}=\mathbf {L} \mathbf {T} \mathbf {L} ^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da3e4c346986f1f3c2cb330290410c1f42a0e93e)
Other common form
![{\displaystyle T_{ij}^{'}=Q_{pi}Q_{qj}T_{pq}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c97e1f3e39ea6b387ff8ead0ac79587f5c0cc4fb)
In matrix form,
![{\displaystyle \mathbf {T} ^{'}=\mathbf {Q} ^{T}\mathbf {T} \mathbf {Q} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc5eefb9fff2cb1fbb720c54583e07b94e3da6b)