The motion of a body
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Initial orthonormal basis:
![{\displaystyle ({\boldsymbol {E}}_{1},{\boldsymbol {E}}_{2},{\boldsymbol {E}}_{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4ecbd750409c01452c0776c935439d0032fe39)
Deformed orthonormal basis:
![{\displaystyle (\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/762706eceeab3d638bac64a40440ffaf7764307e)
We assume that these coincide.
![{\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(\mathbf {X} ,t)=\mathbf {x} (\mathbf {X} ,t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71276954339dc82bcbdd15934a4a1d09381ba99c)
![{\displaystyle {\begin{aligned}{\boldsymbol {F}}&={\frac {\partial {\boldsymbol {\varphi }}}{\partial \mathbf {X} }}={\boldsymbol {\nabla }}_{o}{\boldsymbol {\varphi }}\\&={\frac {\partial \mathbf {x} }{\partial \mathbf {X} }}={\boldsymbol {\nabla }}_{X}{\boldsymbol {\varphi }}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4213a0f18591eaf46585f96a239a4bfacf94dbe)
Effect of
:
![{\displaystyle {\begin{aligned}d\mathbf {x} _{1}&={\boldsymbol {F}}\bullet d\mathbf {X} _{1}~;&d\mathbf {x} _{2}&={\boldsymbol {F}}\bullet d\mathbf {X} _{2}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f40c4c7d9c0b91406214f8532f892d37e75ba617)
Dyadic notation:
![{\displaystyle {\boldsymbol {F}}=F_{iJ}~\mathbf {e} _{i}\otimes {\boldsymbol {E}}_{J}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d3e4807dc3af5696c5ffb572d7189ed1881f8f)
Index notation:
![{\displaystyle F_{iJ}={\frac {\partial x_{i}}{\partial X_{J}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9cdac1fa214af53ea906b87e953406e7829627b)
The determinant of the deformation gradient is usually denoted by
and is a measure of the change in volume, i.e.,
![{\displaystyle J=\det {\boldsymbol {F}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/484773fcbe525f7e1c1c28fc7e3bce7e77cb9bf6)
Forward Map:
![{\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(\mathbf {X} ,t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b222810e295e6c3090992394914191723ea4406d)
Forward deformation gradient:
![{\displaystyle {\boldsymbol {F}}={\frac {\partial \mathbf {x} }{\partial \mathbf {X} }}={\boldsymbol {\nabla }}_{o}{\boldsymbol {\varphi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a4d2e30e28fae4612a116762c7aeff56c0cb07)
Dyadic notation:
![{\displaystyle {\boldsymbol {F}}=\sum _{i,J=1}^{3}{\frac {\partial x_{i}}{\partial X_{J}}}~\mathbf {e} _{i}\otimes {\boldsymbol {E}}_{J}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d209e303e7df33c09681dbb9053e67dffa873f7)
Effect of deformation gradient:
![{\displaystyle d\mathbf {x} ={\boldsymbol {F}}\bullet d\mathbf {X} ={\boldsymbol {\varphi }}_{*}[d\mathbf {X} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/713a1b1c9ed4308419a4ab4f71798cd1ce9d85c6)
Push Forward operation:
![{\displaystyle {\boldsymbol {\varphi }}_{*}[\bullet ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abf24ff735450b28309aac3d548750c794487a62)
= material vector.
= spatial vector.
Inverse map:
![{\displaystyle \mathbf {X} ={\boldsymbol {\varphi }}^{-1}(\mathbf {x} ,t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a805aa8aa00bc834e34e6f73c2961d6a0366df29)
Inverse deformation gradient:
![{\displaystyle {\boldsymbol {F}}^{-1}={\frac {\partial \mathbf {X} }{\partial \mathbf {x} }}={\boldsymbol {\nabla }}{\boldsymbol {\varphi }}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68faf0dae90694aa6239eb25450a2de3f7c4e04c)
Dyadic notation:
![{\displaystyle {\boldsymbol {F}}^{-1}=\sum _{i,J=1}^{3}{\frac {\partial X_{I}}{\partial x_{j}}}~{\boldsymbol {E}}_{I}\otimes \mathbf {e} _{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02adc755227f0577bf2c385110dbad703488b382)
Effect of inverse deformation gradient:
![{\displaystyle d\mathbf {X} ={\boldsymbol {F}}^{-1}\bullet d\mathbf {x} ={\boldsymbol {\varphi }}^{*}[d\mathbf {x} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/338931a885b45a4e2085dca70d44a4640270f165)
Pull Back operation:
![{\displaystyle {\boldsymbol {\varphi }}^{*}[\bullet ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eafa2b8d8f39890a595387e5b697cabc27cf8623)
= material vector.
= spatial vector.
Push forward and pull back
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Motion:
![{\displaystyle {\begin{aligned}x_{1}&={\cfrac {1}{4}}(18+4X_{1}+6X_{2})\\x_{2}&={\cfrac {1}{4}}(14+6X_{2})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73ac19dd781bd27348b621eff7ba174a20d637d1)
Deformation Gradient:
![{\displaystyle F_{ij}={\frac {\partial x_{i}}{\partial X_{j}}}\implies \mathbf {F} ={\frac {1}{2}}{\begin{bmatrix}2&3\\0&3\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b673601779c3bd52a1ab2d59dcae1b47977c691)
Inverse Deformation Gradient:
![{\displaystyle \mathbf {F} ^{-1}={\cfrac {1}{3}}{\begin{bmatrix}3&-3\\0&2\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ace9327f71e26c280fbf51ea9d9ab9233ba91988)
Push Forward:
![{\displaystyle {\begin{aligned}{\boldsymbol {\varphi }}_{*}[{\boldsymbol {E}}_{1}]&=\mathbf {F} {\begin{bmatrix}1\\0\end{bmatrix}}={\begin{bmatrix}1\\0\end{bmatrix}}\\{\boldsymbol {\varphi }}_{*}[{\boldsymbol {E}}_{2}]&=\mathbf {F} {\begin{bmatrix}0\\1\end{bmatrix}}={\begin{bmatrix}1.5\\1.5\end{bmatrix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/652db2dcb31ef3607763298cdab252d58a0be2a6)
Pull Back:
![{\displaystyle {\begin{aligned}{\boldsymbol {\varphi }}^{*}[\mathbf {e} _{1}]&=\mathbf {F} ^{-1}{\begin{bmatrix}1\\0\end{bmatrix}}={\begin{bmatrix}1\\0\end{bmatrix}}\\{\boldsymbol {\varphi }}^{*}[\mathbf {e} _{2}]&=\mathbf {F} ^{-1}{\begin{bmatrix}0\\1\end{bmatrix}}={\begin{bmatrix}-1\\2/3\end{bmatrix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f07d9e3ad19c9b64b119f090c564767fe1ea7dd1)
Recall:
![{\displaystyle d\mathbf {x} _{1}={\boldsymbol {F}}\bullet d\mathbf {X} _{1}~;~~d\mathbf {x} _{2}={\boldsymbol {F}}\bullet d\mathbf {X} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c050f27b1def0f93d0ecd25145344e4d9bda1e13)
Therefore,
![{\displaystyle d\mathbf {x} _{1}\bullet d\mathbf {x} _{2}=({\boldsymbol {F}}\bullet d\mathbf {X} _{1})\bullet ({\boldsymbol {F}}\bullet d\mathbf {X} _{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa09e4854378b269d0fa19ef0b00deee01918a3)
Using index notation:
![{\displaystyle {\begin{aligned}d\mathbf {x} _{1}\bullet d\mathbf {x} _{2}&=(F_{ij}~dX_{j}^{1})(F_{ik}~dX_{k}^{2})\\&=dX_{j}^{1}~(F_{ij}~F_{ik})~dX_{k}^{2}\\&=d\mathbf {X} _{1}\bullet ({\boldsymbol {F}}^{T}\bullet {\boldsymbol {F}})\bullet d\mathbf {X} _{2}\\&=d\mathbf {X} _{1}\bullet {\boldsymbol {C}}\bullet d\mathbf {X} _{2}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7e3503f2604c302e4e64bafb0d77c2325019a3)
Right Cauchy-Green tensor:
![{\displaystyle {\boldsymbol {C}}={\boldsymbol {F}}^{T}\bullet {\boldsymbol {F}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b44db8a0997468c43b9f6bffdf5b490dfc20f3)
Recall:
![{\displaystyle d\mathbf {X} _{1}={\boldsymbol {F}}^{-1}\bullet d\mathbf {x} _{1}~;~~d\mathbf {X} _{2}={\boldsymbol {F}}^{-1}\bullet d\mathbf {x} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4806eb9922ea38255bacfb41386a243b21e96b46)
Therefore,
![{\displaystyle d\mathbf {X} _{1}\bullet d\mathbf {X} _{2}=({\boldsymbol {F}}^{-1}\bullet d\mathbf {x} _{1})\bullet ({\boldsymbol {F}}^{-1}\bullet d\mathbf {x} _{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c996a26141b233473fdad0ff6866da07aaa42e4)
Using index notation:
![{\displaystyle {\begin{aligned}d\mathbf {X} _{1}\bullet d\mathbf {X} _{2}&=(F_{ij}^{-1}~dx_{j}^{1})(F_{ik}^{-1}~dx_{k}^{2})\\&=dx_{j}^{1}~(F_{ij}^{-1}~F_{ik}^{-1})~dx_{k}^{2}\\&=d\mathbf {x} _{1}\bullet ({\boldsymbol {F}}^{-T}\bullet {\boldsymbol {F}}^{-1})\bullet d\mathbf {x} _{2}\\&=d\mathbf {x} _{1}\bullet ({\boldsymbol {F}}\bullet {\boldsymbol {F}}^{T})^{-1}\bullet d\mathbf {x} _{2}\\&=d\mathbf {x} _{1}\bullet \mathbf {b} ^{-1}\bullet d\mathbf {x} _{2}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e43298aadebdf247bcdb1d369890f86313c0b1f7)
Left Cauchy-Green (Finger) tensor:
![{\displaystyle \mathbf {b} ={\boldsymbol {F}}\bullet {\boldsymbol {F}}^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af8bb24c57dc0090f19acc73786517fae4771252)
![{\displaystyle {\begin{aligned}{\frac {1}{2}}(d\mathbf {x} _{1}\bullet d\mathbf {x} _{2}&-d\mathbf {X} _{1}\bullet d\mathbf {X} _{2})\\&={\frac {1}{2}}d\mathbf {X} _{1}\bullet ({\boldsymbol {C}}-{\boldsymbol {I}})\bullet d\mathbf {X} _{2}\\&=d\mathbf {X} _{1}\bullet {\boldsymbol {E}}\bullet d\mathbf {X} _{2}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba088a4c0fbab79d80f0b5c478fb23b1a1140a86)
Green strain tensor:
![{\displaystyle {\begin{aligned}{\boldsymbol {E}}&={\frac {1}{2}}({\boldsymbol {C}}-{\boldsymbol {I}})\\&={\frac {1}{2}}({\boldsymbol {F}}^{T}\bullet {\boldsymbol {F}}-{\boldsymbol {I}})\\&={\frac {1}{2}}\left[{\boldsymbol {\nabla }}_{o}\mathbf {u} +({\boldsymbol {\nabla }}_{o}\mathbf {u} )^{T}+{\boldsymbol {\nabla }}_{o}\mathbf {u} \bullet ({\boldsymbol {\nabla _{o}\mathbf {u} )^{T}}}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d8cf60e20678a55257c8ad49406f81d80b786d)
Index notation:
![{\displaystyle {\begin{aligned}E_{ij}&={\frac {1}{2}}(F_{ki}~F_{kj}-\delta _{ij})\\&={\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial X_{j}}}+{\frac {\partial u_{j}}{\partial X_{i}}}+{\frac {\partial u_{k}}{\partial X_{i}}}{\frac {\partial u_{k}}{\partial X_{j}}}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85377e4104881d5e77f6d249153f7b223d80cb5c)
![{\displaystyle {\begin{aligned}{\frac {1}{2}}(d\mathbf {x} _{1}\bullet d\mathbf {x} _{2}&-d\mathbf {X} _{1}\bullet d\mathbf {X} _{2})\\&={\frac {1}{2}}d\mathbf {x} _{1}\bullet ({\boldsymbol {I}}-\mathbf {b} ^{-1})\bullet d\mathbf {x} _{2}\\&=d\mathbf {x} _{1}\bullet \mathbf {e} \bullet d\mathbf {x} _{2}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9035a20964231d800c507f176543242c219f2ad7)
Almansi strain tensor:
![{\displaystyle {\begin{aligned}\mathbf {e} &={\frac {1}{2}}({\boldsymbol {I}}-\mathbf {b} ^{-1})\\&={\frac {1}{2}}({\boldsymbol {I}}-{\boldsymbol {F}}^{-T}\bullet {\boldsymbol {F}}^{-1})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb2e14a7e7a4577e14c7022af6d99d6a6db0109)
Index notation:
![{\displaystyle {\begin{aligned}e_{ij}&={\frac {1}{2}}(\delta _{ij}-F_{ki}^{-1}~F_{kj}^{-1})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/390732b2e55a4d337918f0f5fde5114a789c9eba)
Recall:
![{\displaystyle d\mathbf {x} _{1}\bullet \mathbf {e} \bullet d\mathbf {x} _{2}=d\mathbf {X} _{1}\bullet {\boldsymbol {E}}\bullet d\mathbf {X} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76d67c65991ce152dddd99107ca88e7d4786494a)
Now,
![{\displaystyle {\begin{aligned}d\mathbf {x} _{1}\bullet \mathbf {e} \bullet d\mathbf {x} _{2}&=({\boldsymbol {F}}\bullet d\mathbf {X} _{1})\bullet \mathbf {e} \bullet ({\boldsymbol {F}}\bullet d\mathbf {X} _{2})\\&=d\mathbf {X} _{1}\bullet ({\boldsymbol {F}}^{T}\bullet \mathbf {e} \bullet {\boldsymbol {F}})\bullet d\mathbf {X} _{2}\\&=d\mathbf {X} _{1}\bullet {\boldsymbol {E}}\bullet d\mathbf {X} _{2}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f1a02ca45df520fc0307f484acc0529b680261)
Therefore,
![{\displaystyle {\begin{aligned}{\boldsymbol {E}}&={\boldsymbol {F}}^{T}\bullet \mathbf {e} \bullet {\boldsymbol {F}}\\\implies \mathbf {e} &={\boldsymbol {F}}^{-T}\bullet {\boldsymbol {E}}\bullet {\boldsymbol {F}}^{-1}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff46bf9dfb5289290a72bf8cf352b17236a3255d)
Push Forward:
![{\displaystyle \mathbf {e} ={\boldsymbol {\varphi }}_{*}[{\boldsymbol {E}}]={\boldsymbol {F}}^{-T}\bullet {\boldsymbol {E}}\bullet {\boldsymbol {F}}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/343ab8184fc12658afc4dd85752b5b107f10e0ab)
Pull Back:
![{\displaystyle {\boldsymbol {E}}={\boldsymbol {\varphi }}^{*}[\mathbf {e} ]={\boldsymbol {F}}^{T}\bullet \mathbf {e} \bullet {\boldsymbol {F}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc21be32672cb75ab1991dfca79c183ee6ecda0e)
We often need to compute the derivative of
with respect to the deformation gradient
. From tensor calculus we have, for any second order tensor
![{\displaystyle {\cfrac {\partial }{\partial {\boldsymbol {A}}}}(\det {\boldsymbol {A}})=\det {\boldsymbol {A}}~{\boldsymbol {A}}^{-T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/585564fa301f476f309e26aab3f0716513908779)
Therefore,
![{\displaystyle {\cfrac {\partial J}{\partial {\boldsymbol {F}}}}=J~{\boldsymbol {F}}^{-T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c34aac19fa2236b7a86745d9302eb33deba802c0)
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The derivative of J with respect to the right Cauchy-Green deformation tensor (
) is also often encountered in continuum mechanics.
To calculate the derivative of
with respect to
, we recall that (for any second order tensor
)
![{\displaystyle {\frac {\partial {\boldsymbol {C}}}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}}={\frac {\partial }{\partial {\boldsymbol {F}}}}({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}):{\boldsymbol {T}}=({\boldsymbol {\mathsf {I}}}^{T}:{\boldsymbol {T}})\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot ({\boldsymbol {\mathsf {I}}}:{\boldsymbol {T}})={\boldsymbol {T}}^{T}\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot {\boldsymbol {T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de19bd2b195be2f89226fd9263843be08a679503)
Also,
![{\displaystyle {\frac {\partial J}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}}={\frac {\partial J}{\partial {\boldsymbol {C}}}}:({\frac {\partial {\boldsymbol {C}}}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}})={\frac {\partial J}{\partial {\boldsymbol {C}}}}:({\boldsymbol {T}}^{T}\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot {\boldsymbol {T}})=\left[{\boldsymbol {F}}\cdot {\frac {\partial J}{\partial {\boldsymbol {C}}}}\right]:{\boldsymbol {T}}+\left[{\boldsymbol {F}}\cdot \left({\frac {\partial J}{\partial {\boldsymbol {C}}}}\right)^{T}\right]:{\boldsymbol {T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1729186785d49848ea9d533ddd8ed2ea4a59fc5e)
From the symmetry of
we have
![{\displaystyle {\frac {\partial J}{\partial {\boldsymbol {C}}}}=\left({\frac {\partial J}{\partial {\boldsymbol {C}}}}\right)^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbb78fb23945e5efe1fcadfca2abda79aee24b0)
Therefore, involving the arbitrariness of
, we have
![{\displaystyle {\frac {\partial J}{\partial {\boldsymbol {F}}}}=2~{\boldsymbol {F}}\cdot {\frac {\partial J}{\partial {\boldsymbol {C}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a3c226eb69aebb3d352a27df0031224e795fa12)
Hence,
![{\displaystyle {\frac {\partial J}{\partial {\boldsymbol {C}}}}={\frac {1}{2}}~{\boldsymbol {F}}^{-1}\cdot {\frac {\partial J}{\partial {\boldsymbol {F}}}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77c2e7730c937b7afb13dc35bc754b528ff746af)
Also recall that
![{\displaystyle {\frac {\partial J}{\partial {\boldsymbol {F}}}}=J~{\boldsymbol {F}}^{-T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb996c16de6229ae7917042de3abb43838fbede1)
Therefore,
![{\displaystyle {\frac {\partial J}{\partial {\boldsymbol {C}}}}={\frac {1}{2}}~J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {F}}^{-T}={\cfrac {J}{2}}~{\boldsymbol {C}}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0455b14e7cf80e2a3b31e6b3e2284302aefa8b05)
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In index notation,
![{\displaystyle {\frac {\partial J}{\partial C_{IJ}}}={\cfrac {J}{2}}~C_{IJ}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ae654a0606047202868fa4e233a3ba6e11a54b)
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Derivative of the inverse of the right Cauchy-Green tensor
[edit | edit source]
Another result that is often useful is that for the derivative of the inverse of the right Cauchy-Green tensor (
).
Recall that, for a second order tensor
,
![{\displaystyle {\frac {\partial {\boldsymbol {A}}^{-1}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}=-{\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\cdot {\boldsymbol {A}}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e5e866ad297ebfcde5a7e75ff5c1d91891df7eb)
In index notation
![{\displaystyle {\frac {\partial A_{ij}^{-1}}{\partial A_{kl}}}~T_{kl}=B_{ijkl}~T_{kl}=-A_{ik}^{-1}~T_{kl}~A_{lj}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa47fd7922f73d28b304e84c3478cbfa9f2bcb1)
or,
![{\displaystyle {\frac {\partial A_{ij}^{-1}}{\partial A_{kl}}}=B_{ijkl}=-A_{ik}^{-1}~A_{lj}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff2dec1366f33cd679d5a2672bf8ad63c470b68)
Using this formula and noting that since
is a symmetric second order tensor, the derivative of its inverse is a symmetric fourth order tensor we have
![{\displaystyle {\frac {\partial C_{IJ}^{-1}}{\partial C_{KL}}}=-{\frac {1}{2}}~(C_{IK}^{-1}~C_{JL}^{-1}+C_{JK}^{-1}~C_{IL}^{-1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32fe0753743f3cde7a37f930685382a65a627638)
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