The balance of angular momentum in an inertial frame can be expressed as:
We assume that there are no surface couples on
or body couples
in
. Recall the general balance equation
![{\displaystyle {\cfrac {d}{dt}}\left[\int _{\Omega }f(\mathbf {x} ,t)~{\text{dV}}\right]=\int _{\partial {\Omega }}f(\mathbf {x} ,t)[u_{n}(\mathbf {x} ,t)-\mathbf {v} (\mathbf {x} ,t)\cdot \mathbf {n} (\mathbf {x} ,t)]~{\text{dA}}+\int _{\partial {\Omega }}g(\mathbf {x} ,t)~{\text{dA}}+\int _{\Omega }h(\mathbf {x} ,t)~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcb120ab19e9001d85370ccb115c509cec72cf33)
In this case, the physical quantity to be conserved the angular momentum
density, i.e.,
.
The angular momentum source at the surface is then
and the angular momentum source inside the body
is
. The angular momentum and moments are
calculated with respect to a fixed origin. Hence we have
![{\displaystyle {\cfrac {d}{dt}}\left[\int _{\Omega }\mathbf {x} \times (\rho ~\mathbf {v} )~{\text{dV}}\right]=\int _{\partial {\Omega }}[\mathbf {x} \times (\rho ~\mathbf {v} )][u_{n}-\mathbf {v} \cdot \mathbf {n} ]~{\text{dA}}+\int _{\partial {\Omega }}\mathbf {x} \times \mathbf {t} ~{\text{dA}}+\int _{\Omega }\mathbf {x} \times (\rho ~\mathbf {b} )~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9534f6c944fcf047c42808b058382a74472902)
Assuming that
is a control volume, we have
![{\displaystyle \int _{\Omega }\mathbf {x} \times \left[{\cfrac {\partial }{\partial t}}(\rho ~\mathbf {v} )\right]~{\text{dV}}=-\int _{\partial {\Omega }}[\mathbf {x} \times (\rho ~\mathbf {v} )][\mathbf {v} \cdot \mathbf {n} ]~{\text{dA}}+\int _{\partial {\Omega }}\mathbf {x} \times \mathbf {t} ~{\text{dA}}+\int _{\Omega }\mathbf {x} \times (\rho ~\mathbf {b} )~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ca847be882330cb6c1a321f005b781df40b594)
Using the definition of a tensor product we can write
![{\displaystyle [\mathbf {x} \times (\rho ~\mathbf {v} )][\mathbf {v} \cdot \mathbf {n} ]=[[\mathbf {x} \times (\rho ~\mathbf {v} )]\otimes \mathbf {v} ]\cdot \mathbf {n} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f22c59227a981b5b650444352ebac9523c13aa6f)
Also,
. Therefore we have
![{\displaystyle \int _{\Omega }\mathbf {x} \times \left[{\cfrac {\partial }{\partial t}}(\rho ~\mathbf {v} )\right]~{\text{dV}}=-\int _{\partial {\Omega }}[[\mathbf {x} \times (\rho ~\mathbf {v} )]\otimes \mathbf {v} ]\cdot \mathbf {n} ~{\text{dA}}+\int _{\partial {\Omega }}\mathbf {x} \times ({\boldsymbol {\sigma }}\cdot \mathbf {n} )~{\text{dA}}+\int _{\Omega }\mathbf {x} \times (\rho ~\mathbf {b} )~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/153c16fc3456ff8953048198236fb4e6e45ff5c0)
Using the divergence theorem, we get
![{\displaystyle \int _{\Omega }\mathbf {x} \times \left[{\cfrac {\partial }{\partial t}}(\rho ~\mathbf {v} )\right]~{\text{dV}}=-\int _{\Omega }{\boldsymbol {\nabla }}\bullet [[\mathbf {x} \times (\rho ~\mathbf {v} )]\otimes \mathbf {v} ]~{\text{dV}}+\int _{\partial {\Omega }}\mathbf {x} \times ({\boldsymbol {\sigma }}\cdot \mathbf {n} )~{\text{dA}}+\int _{\Omega }\mathbf {x} \times (\rho ~\mathbf {b} )~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92dbb7a4a6558b131fdf8edcf9a64628aa123f50)
To convert the surface integral in the above equation into a volume
integral, it is convenient to use index notation. Thus,
![{\displaystyle \left[\int _{\partial {\Omega }}\mathbf {x} \times ({\boldsymbol {\sigma }}\cdot \mathbf {n} )~{\text{dA}}\right]_{i}=\int _{\partial {\Omega }}e_{ijk}~x_{j}~\sigma _{kl}~n_{l}~{\text{dA}}=\int _{\partial {\Omega }}A_{il}~n_{l}~{\text{dA}}=\int _{\partial {\Omega }}{\boldsymbol {A}}\cdot \mathbf {n} ~{\text{dA}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6761546fca01eba628fb6269c078a100936c3f8)
where
represents the
-th component of the vector. Using
the divergence theorem
![{\displaystyle \int _{\partial {\Omega }}{\boldsymbol {A}}\cdot \mathbf {n} ~{\text{dA}}=\int _{\Omega }{\boldsymbol {\nabla }}\bullet {\boldsymbol {A}}~{\text{dV}}=\int _{\Omega }{\frac {\partial A_{il}}{\partial x_{l}}}~{\text{dV}}=\int _{\Omega }{\frac {\partial }{\partial x_{l}}}(e_{ijk}~x_{j}~\sigma _{kl})~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77493e5b88208b312cc5a0613837c3d243860216)
Differentiating,
![{\displaystyle \int _{\partial {\Omega }}{\boldsymbol {A}}\cdot \mathbf {n} ~{\text{dA}}=\int _{\Omega }\left[e_{ijk}~\delta _{jl}~\sigma _{kl}+e_{ijk}~x_{j}~{\frac {\partial \sigma _{kl}}{\partial x_{l}}}\right]~{\text{dV}}=\int _{\Omega }\left[e_{ijk}~\sigma _{kj}+e_{ijk}~x_{j}~{\frac {\partial \sigma _{kl}}{\partial x_{l}}}\right]~{\text{dV}}=\int _{\Omega }\left[e_{ijk}~\sigma _{kj}+e_{ijk}~x_{j}~[{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}]_{k}\right]~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f90ad452cd24431c1d4888681c915f3d60af2b)
Expressed in direct tensor notation,
![{\displaystyle \int _{\partial {\Omega }}{\boldsymbol {A}}\cdot \mathbf {n} ~{\text{dA}}=\int _{\Omega }\left[[{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}]_{i}+[\mathbf {x} \times ({\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }})]_{i}\right]~{\text{dV}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63f419b46eb4e6b16807613a084b6a99b55eef91)
where
is the third-order permutation tensor.
Therefore,
![{\displaystyle \left[\int _{\partial {\Omega }}\mathbf {x} \times ({\boldsymbol {\sigma }}\cdot \mathbf {n} )~{\text{dA}}\right]_{i}==\int _{\Omega }\left[[{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}]_{i}+[\mathbf {x} \times ({\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }})]_{i}\right]~{\text{dV}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84470145fea4e34e440461d1d7f45d5095872b4b)
or,
![{\displaystyle \int _{\partial {\Omega }}\mathbf {x} \times ({\boldsymbol {\sigma }}\cdot \mathbf {n} )~{\text{dA}}==\int _{\Omega }\left[{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}+\mathbf {x} \times ({\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }})\right]~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b55546dbd778a59c52e35745f76e1ce2565215b5)
The balance of angular momentum can then be written as
![{\displaystyle \int _{\Omega }\mathbf {x} \times \left[{\cfrac {\partial }{\partial t}}(\rho ~\mathbf {v} )\right]~{\text{dV}}=-\int _{\Omega }{\boldsymbol {\nabla }}\bullet [[\mathbf {x} \times (\rho ~\mathbf {v} )]\otimes \mathbf {v} ]~{\text{dV}}+\int _{\Omega }\left[{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}+\mathbf {x} \times ({\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }})\right]~{\text{dV}}+\int _{\Omega }\mathbf {x} \times (\rho ~\mathbf {b} )~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/982e73c78686f2976c71e2a0a3b2728261c0918c)
Since
is an arbitrary volume, we have
![{\displaystyle \mathbf {x} \times \left[{\cfrac {\partial }{\partial t}}(\rho ~\mathbf {v} )\right]=-{\boldsymbol {\nabla }}\bullet [[\mathbf {x} \times (\rho ~\mathbf {v} )]\otimes \mathbf {v} ]+{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}+\mathbf {x} \times ({\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }})+\mathbf {x} \times (\rho ~\mathbf {b} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a274d0237a3ba8fc67faabcb7d1bab8b68da71)
or,
![{\displaystyle {\mathbf {x} }\times {\left[{\frac {\partial }{\partial t}}(\rho ~\mathbf {v} )-{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} \right]}=-{\boldsymbol {\nabla }}\bullet [[\mathbf {x} \times (\rho ~\mathbf {v} )]\otimes \mathbf {v} ]+{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e586875f0f88d07feea25cf85101e990178520)
Using the identity,
![{\displaystyle {\boldsymbol {\nabla }}\bullet (\mathbf {u} \otimes \mathbf {v} )=({\boldsymbol {\nabla }}\bullet \mathbf {v} )\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )\cdot \mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/404939824430b3b2506784c13b54b0037b3068a0)
we get
![{\displaystyle {\boldsymbol {\nabla }}\bullet [[\mathbf {x} \times (\rho ~\mathbf {v} )]\otimes \mathbf {v} ]=({\boldsymbol {\nabla }}\bullet \mathbf {v} )[\mathbf {x} \times (\rho ~\mathbf {v} )]+({\boldsymbol {\nabla }}[\mathbf {x} \times (\rho ~\mathbf {v} )])\cdot \mathbf {v} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b70c77e5c432d9e244c1f19d6b180f7522ec99)
The second term on the right can be further simplified using index
notation as follows.
![{\displaystyle {\begin{aligned}\left[({\boldsymbol {\nabla }}[\mathbf {x} \times (\rho ~\mathbf {v} )])\cdot \mathbf {v} \right]_{i}=\left[({\boldsymbol {\nabla }}[\rho ~(\mathbf {x} \times \mathbf {v} )])\cdot \mathbf {v} \right]_{i}&={\frac {\partial }{\partial x_{l}}}(\rho ~e_{ijk}~x_{j}~v_{k})~v_{l}\\&=e_{ijk}\left[{\frac {\partial \rho }{\partial x_{l}}}~x_{j}~v_{k}~v_{l}+\rho ~{\frac {\partial x_{j}}{\partial x_{l}}}~v_{k}~v_{l}+\rho ~x_{j}~{\frac {\partial v_{k}}{\partial x_{l}}}~v_{l}\right]\\&=(e_{ijk}~x_{j}~v_{k})~\left({\frac {\partial \rho }{\partial x_{l}}}~v_{l}\right)+\rho ~(e_{ijk}~\delta _{jl}~v_{k}~v_{l})+e_{ijk}~x_{j}~\left(\rho ~{\frac {\partial v_{k}}{\partial x_{l}}}~v_{l}\right)\\&=[(\mathbf {x} \times \mathbf {v} )({\boldsymbol {\nabla }}\rho \cdot \mathbf {v} )+\rho ~\mathbf {v} \times \mathbf {v} +\mathbf {x} \times (\rho ~{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} )]_{i}\\&=[(\mathbf {x} \times \mathbf {v} )({\boldsymbol {\nabla }}\rho \cdot \mathbf {v} )+\mathbf {x} \times (\rho ~{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} )]_{i}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4148cf4fcf6b5d82eb8a468113e63b80d807025)
Therefore we can write
![{\displaystyle {\boldsymbol {\nabla }}\bullet [[\mathbf {x} \times (\rho ~\mathbf {v} )]\otimes \mathbf {v} ]=(\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} )(\mathbf {x} \times ~\mathbf {v} )+({\boldsymbol {\nabla }}\rho \cdot \mathbf {v} )(\mathbf {x} \times \mathbf {v} )+\mathbf {x} \times (\rho ~{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} )]~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd3375983353cb8f31d347e28e43d42f7d41e74)
The balance of angular momentum then takes the form
![{\displaystyle {\mathbf {x} }\times {\left[{\frac {\partial }{\partial t}}(\rho ~\mathbf {v} )-{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} \right]}=-(\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} )(\mathbf {x} \times ~\mathbf {v} )-({\boldsymbol {\nabla }}\rho \cdot \mathbf {v} )(\mathbf {x} \times \mathbf {v} )-\mathbf {x} \times (\rho ~{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} )+{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03b64ea0396cc7d14f257f20a6f5081f54581fef)
or,
![{\displaystyle {\mathbf {x} }\times {\left[{\frac {\partial }{\partial t}}(\rho ~\mathbf {v} )+\rho ~{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} -{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} \right]}=-(\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} )(\mathbf {x} \times ~\mathbf {v} )-({\boldsymbol {\nabla }}\rho \cdot \mathbf {v} )(\mathbf {x} \times \mathbf {v} )+{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dfcb07fb8a30ab939b075da6ac99dafaccf440e)
or,
![{\displaystyle {\mathbf {x} }\times {\left[\rho {\frac {\partial \mathbf {v} }{\partial t}}+{\frac {\partial \rho }{\partial t}}~\mathbf {v} +\rho ~{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} -{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} \right]}=-(\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} )(\mathbf {x} \times ~\mathbf {v} )-({\boldsymbol {\nabla }}\rho \cdot \mathbf {v} )(\mathbf {x} \times \mathbf {v} )+{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc5680a26c0cc0f0edf15f734613c42aef23676f)
The material time derivative of
is defined as
![{\displaystyle {\dot {\mathbf {v} }}={\frac {\partial \mathbf {v} }{\partial t}}+{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a99ae843473e9fa146f060b47d113def2439c05)
Therefore,
![{\displaystyle {\mathbf {x} }\times {\left[\rho ~{\dot {\mathbf {v} }}-{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} \right]}=-\mathbf {x} \times {\cfrac {\partial \rho }{\partial t}}~\mathbf {v} +-(\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} )(\mathbf {x} \times ~\mathbf {v} )-({\boldsymbol {\nabla }}\rho \cdot \mathbf {v} )(\mathbf {x} \times \mathbf {v} )+{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03352108c5b87389318388ebc7f071f3492da17a)
Also, from the conservation of linear momentum
![{\displaystyle \rho ~{\dot {\mathbf {v} }}-{\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}-\rho ~\mathbf {b} =0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e1c44f09b3bcca1fe3a11bf04f453dd1464ff0b)
Hence,
![{\displaystyle {\begin{aligned}0&=\mathbf {x} \times {\cfrac {\partial \rho }{\partial t}}~\mathbf {v} +(\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} )(\mathbf {x} \times ~\mathbf {v} )+({\boldsymbol {\nabla }}\rho \cdot \mathbf {v} )(\mathbf {x} \times \mathbf {v} )-{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}\\&=\left({\frac {\partial \rho }{\partial t}}+\rho {\boldsymbol {\nabla }}\bullet \mathbf {v} +{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} \right)(\mathbf {x} \times \mathbf {v} )-{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4581859fb7f6ee377fb63f93c1b755b7eacedf47)
The material time derivative of
is defined as
![{\displaystyle {\dot {\rho }}={\frac {\partial \rho }{\partial t}}+{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c755826463c2c7fc56f0fd6229cbfc7c16ec69a)
Hence,
![{\displaystyle ({\dot {\rho }}+\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} )(\mathbf {x} \times \mathbf {v} )-{\mathcal {E}}:{\boldsymbol {\sigma }}^{T}=0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb097f3b2944ec89db875190c845a9406b3583da)
From the balance of mass
![{\displaystyle {\dot {\rho }}+\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} =0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29f84fc3709ebd9ec30d1227f386a051192aedbf)
Therefore,
![{\displaystyle {\mathcal {E}}:{\boldsymbol {\sigma }}^{T}=0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d49e1fda822bd60150e3b6be994b1de204372cf8)
In index notation,
![{\displaystyle e_{ijk}~\sigma _{kj}=0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2681be4edb4606fa824444b8da7f9b8173cedaf1)
Expanding out, we get
![{\displaystyle \sigma _{12}-\sigma _{21}=0~;~~\sigma _{23}-\sigma _{32}=0~;~~\sigma _{31}-\sigma _{13}=0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2850bb1e7514a5a799653332024154ecbfcf5013)
Hence,
![{\displaystyle {{\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25d4c853072265d2787a4b531c618a8808a4f0c9)