Material time derivatives are needed for many updated Lagrangian formulations
of finite element analysis.
Recall that the motion can be expressed as
![{\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(\mathbf {X} ,t)\quad {\text{or}}\qquad \mathbf {X} ={\boldsymbol {\varphi }}^{-1}(\mathbf {x} ,t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8981fc8d8f7ae27ca1926f6171e00b55e9143d71)
If we keep
fixed, then the velocity is given by
![{\displaystyle \mathbf {V} (\mathbf {X} ,t)={\frac {\partial {\boldsymbol {\varphi }}}{\partial t}}(\mathbf {X} ,t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5579560774aa9199cc0ea1cb973d3e591861010d)
This is the material time derivative expressed in terms of
.
The spatial version of the velocity is
![{\displaystyle \mathbf {v} (\mathbf {x} ,t)=\mathbf {V} ({\boldsymbol {\varphi }}^{-1}(\mathbf {x} ,t),t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f7196da74e323bfba5edb17bd9070c206da83f)
We will use the symbol
for velocity from now on by slightly abusing the
notation.
We usually think of quantities such as velocity and acceleration as spatial
quantities which are functions of
(rather than material quantities which
are functions of
).
Given the spatial velocity
, if we want to find the acceleration
we will have to consider the fact that
, i.e., the
position also changes with time. We do this by using the chain rule. Thus
![{\displaystyle {\cfrac {D\mathbf {v} (\mathbf {x} ,t)}{Dt}}=\mathbf {a} (\mathbf {x} ,t)={\frac {\partial \mathbf {v} (\mathbf {x} ,t)}{\partial t}}+{\frac {\partial \mathbf {v} (\mathbf {x} ,t)}{\partial \mathbf {x} }}\cdot {\frac {\partial {\boldsymbol {\varphi }}(\mathbf {X} ,t)}{\partial t}}={\frac {\partial \mathbf {v} }{\partial t}}+{\boldsymbol {\nabla }}\mathbf {v} \cdot \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/24219be1e204d62de3181fd4ace4ff0a54f486b5)
Such a derivative is called the material time derivative expressed
in terms of
. The second term in the expression is called the
convective derivative..
Let the velocity be expressed in spatial form, i.e.,
.
The spatial velocity gradient tensor is given by
![{\displaystyle {\boldsymbol {l}}:={\frac {\partial \mathbf {v} (\mathbf {x} ,t)}{\partial \mathbf {x} }}={\boldsymbol {\nabla }}\mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d33af359e697c4faa10aa8cc07569e7e17b192)
The velocity gradient
is a second order tensor which can expressed as
![{\displaystyle {\boldsymbol {l}}=l_{ij}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}={\frac {\partial v_{i}}{\partial x_{j}}}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b29083efef855d75fef3acababa3e8696affb8b7)
The velocity gradient is a measure of the relative velocity of two points in the current configuration.
Recall that the deformation gradient is given by
![{\displaystyle {\boldsymbol {F}}={\frac {\partial {\boldsymbol {\varphi }}}{\partial \mathbf {X} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75b44488f78b2ce21665a22336380b56417269d4)
The time derivative of
(keeping
fixed) is
![{\displaystyle {\dot {\boldsymbol {F}}}={\frac {\partial }{\partial t}}~\left({\frac {\partial {\boldsymbol {\varphi }}}{\partial \mathbf {X} }}\right)={\frac {\partial }{\partial \mathbf {X} }}~\left({\frac {\partial {\boldsymbol {\varphi }}}{\partial t}}\right)={\frac {\partial \mathbf {v} }{\partial \mathbf {X} }}={\boldsymbol {\nabla }}_{\circ }\mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc0f9cb50e4d319d49c98bed36a6c37e13650c0c)
Using the chain rule
![{\displaystyle {\dot {\boldsymbol {F}}}={\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\cdot {\frac {\partial \mathbf {x} }{\partial \mathbf {X} }}={\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\cdot {\frac {\partial {\boldsymbol {\varphi }}}{\partial \mathbf {X} }}={\boldsymbol {l}}\cdot {\boldsymbol {F}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ebb03403f5e69d000eff1b9c826082f04219faa)
Form this we get the important relation
![{\displaystyle {\boldsymbol {l}}={\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/320b597311314bbd01fb215b825c7cfe7c673b8f)
Let
and
be two infinitesimal material line segments in a
body. Then
![{\displaystyle d\mathbf {x} _{1}={\boldsymbol {F}}\cdot d\mathbf {X} _{1}~;~~d\mathbf {x} _{2}={\boldsymbol {F}}\cdot d\mathbf {X} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77c402091e08c7e8671800a254d0f5944aa336f3)
Hence,
![{\displaystyle d\mathbf {x} _{1}\cdot d\mathbf {x} _{2}=({\boldsymbol {F}}\cdot d\mathbf {X} _{1})\cdot ({\boldsymbol {F}}\cdot d\mathbf {X} _{2})=d\mathbf {X} _{1}\cdot ({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}})\cdot d\mathbf {X} _{2}=d\mathbf {X} _{1}\cdot {\boldsymbol {C}}\cdot d\mathbf {X} _{2}=d\mathbf {X} _{1}\cdot (2~{\boldsymbol {E}}+{\boldsymbol {\mathit {1}}})\cdot d\mathbf {X} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7907136b005cdf99ffb31d161c8669b56e901ccd)
Taking the derivative with respect to
gives us
![{\displaystyle {\frac {\partial }{\partial t}}(d\mathbf {x} _{1}\cdot d\mathbf {x} _{2})=d\mathbf {X} _{1}\cdot {\frac {\partial {\boldsymbol {C}}}{\partial t}}\cdot d\mathbf {X} _{2}=2~d\mathbf {X} _{1}\cdot {\frac {\partial {\boldsymbol {E}}}{\partial t}}\cdot d\mathbf {X} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c86fa6179406cb2607e041b522081781a57b232)
The material strain rate tensor is defined as
![{\displaystyle {\dot {\boldsymbol {E}}}={\frac {\partial {\boldsymbol {E}}}{\partial t}}={\frac {1}{2}}~{\frac {\partial {\boldsymbol {C}}}{\partial t}}={\frac {1}{2}}~{\dot {\boldsymbol {C}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f56f273e97ba18814ba966e9c2bcfcfd9082339)
Clearly,
![{\displaystyle {\dot {\boldsymbol {E}}}={\frac {1}{2}}~{\frac {\partial }{\partial t}}({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}})={\frac {1}{2}}~({\dot {\boldsymbol {F}}}^{T}\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot {\dot {\boldsymbol {F}}})~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6384db97c5edb0dc162e901feca3111d084bca96)
Also,
![{\displaystyle {\frac {1}{2}}~{\frac {\partial }{\partial t}}(d\mathbf {x} _{1}\cdot d\mathbf {x} _{2})=d\mathbf {X} _{1}\cdot {\dot {\boldsymbol {E}}}\cdot d\mathbf {X} _{2}=({\boldsymbol {F}}^{-1}\cdot d\mathbf {x} _{1})\cdot {\dot {\boldsymbol {E}}}\cdot ({\boldsymbol {F}}^{-1}\cdot d\mathbf {x} _{2})=d\mathbf {x} _{1}\cdot ({\boldsymbol {F}}^{-T}\cdot {\dot {\boldsymbol {E}}}\cdot {\boldsymbol {F}}^{-1})\cdot d\mathbf {x} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4379bf6ab4e4ef02fc9f29cd240147ed3e8074)
The spatial rate of deformation tensor or stretching tensor is
defined as
![{\displaystyle {\boldsymbol {d}}={\boldsymbol {F}}^{-T}\cdot {\dot {\boldsymbol {E}}}\cdot {\boldsymbol {F}}^{-1}={\frac {1}{2}}~{\boldsymbol {F}}^{-T}\cdot {\dot {\boldsymbol {C}}}\cdot {\boldsymbol {F}}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b48904663bd9b17ac7bfbf1c61e7ed531e7e45a)
In fact, we can show that
is the symmetric part of the velocity
gradient, i.e.,
![{\displaystyle {\boldsymbol {d}}={\frac {1}{2}}~({\boldsymbol {l}}+{\boldsymbol {l}}^{T})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e431d6123c0f205a6f62070498befc77605719f1)
For rigid body motions we get
.
Most of the operations above can be interpreted as push-forward
and pull-back operations. Also, time derivatives of these tensors can
be interpreted as Lie derivatives.
Recall that the push-forward of the strain tensor from the material
configuration to the spatial configuration is given by
![{\displaystyle {\boldsymbol {e}}=\phi _{*}[{\boldsymbol {E}}]={\boldsymbol {F}}^{-T}\cdot {\boldsymbol {E}}\cdot {\boldsymbol {F}}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b1aacca393f6689ef43b2514a55ed45ebd9eb5d)
The pull-back of the spatial strain tensor to the material configuration
is given by
![{\displaystyle {\boldsymbol {E}}=\phi ^{*}[{\boldsymbol {e}}]={\boldsymbol {F}}^{T}\cdot {\boldsymbol {e}}\cdot {\boldsymbol {F}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6815825022cb9832dd6ab6fac99a05c729fe3335)
Therefore, the rate of deformation tensor is a push-forward of the
material strain rate tensor, i.e.,
![{\displaystyle {\boldsymbol {d}}={\boldsymbol {F}}^{-T}\cdot {\dot {\boldsymbol {E}}}\cdot {\boldsymbol {F}}^{-1}=\phi _{*}[{\dot {\boldsymbol {E}}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3701a0e514634f13fcd781533391f683f2e8da7)
Similarly, the material strain rate tensor is a pull-back of the rate
of deformation tensor to the material configuration, i.e.,
![{\displaystyle {\dot {\boldsymbol {E}}}={\boldsymbol {F}}^{T}\cdot {\boldsymbol {d}}\cdot {\boldsymbol {F}}=\phi ^{*}[{\boldsymbol {d}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ad93da1ca37c58a0e17548d2c8bbd2b8db8e4a3)
Now,
![{\displaystyle {\boldsymbol {E}}=\phi ^{*}[{\boldsymbol {e}}]\quad \implies \quad {\dot {\boldsymbol {E}}}={\frac {\partial }{\partial t}}\left(\phi ^{*}[{\boldsymbol {e}}]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34a6504a5ccb276877a13474c5731d9a4d2d4c4f)
Also,
![{\displaystyle {\boldsymbol {d}}=\phi _{*}[{\dot {\boldsymbol {E}}}]=\phi _{*}\left[{\frac {\partial }{\partial t}}\left(\phi ^{*}[{\boldsymbol {e}}]\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3a7180e66778e2e97538cb397f8f2015413e39)
Therefore the rate of deformation tensor can be obtained by first pulling
back
to the reference configuration, taking a material time
derivative in that configuration, and then pushing forward the result to
the current configuration.
Such an operation is called a Lie derivative. In general, the Lie
derivative of a spatial tensor
is defined as
![{\displaystyle {\mathcal {L}}_{\phi }[{\boldsymbol {g}}]:=\phi _{*}\left[{\frac {\partial }{\partial t}}\left(\phi ^{*}[{\boldsymbol {g}}]\right)\right]~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad276dbd45b4af11e42bff836044c4c367e2cc5a)
The velocity gradient tensor can be additively decomposed into a symmetric
part and a skew part:
![{\displaystyle {\boldsymbol {l}}={\frac {1}{2}}~({\boldsymbol {l}}+{\boldsymbol {l}}^{T})+{\frac {1}{2}}({\boldsymbol {l}}-{\boldsymbol {l}}^{T})={\boldsymbol {d}}+{\boldsymbol {w}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39951cfa98109aff61016e8f2e945bde1741a8c0)
We have seen that
is the rate of deformation tensor. The quantity
is called the spin tensor.
Note that
is symmetric while
is skew symmetric, i.e.,
![{\displaystyle {\boldsymbol {d}}={\boldsymbol {d}}^{T}~;~~{\boldsymbol {w}}=-{\boldsymbol {w}}^{T}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d426116044222678a52da2aa3f7155ec13e300)
So see why
is called a "spin", recall that
![{\displaystyle {\boldsymbol {l}}={\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4faca789c80c2c85c51b16d9d017a45daf2f6f)
Therefore,
![{\displaystyle {\boldsymbol {w}}={\frac {1}{2}}({\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}-{\boldsymbol {F}}^{-T}\cdot {\dot {\boldsymbol {F}}}^{T})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/396c1076a4b5faa44aa88b68e487cbe193907bd3)
Also,
![{\displaystyle {\boldsymbol {F}}={\boldsymbol {R}}\cdot {\boldsymbol {U}}\quad \implies \quad {\dot {\boldsymbol {F}}}={\dot {\boldsymbol {R}}}\cdot {\boldsymbol {U}}+{\boldsymbol {R}}\cdot {\dot {\boldsymbol {U}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d5b639ded088cae56963ff29ab85933cf623b4)
Therefore,
![{\displaystyle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}=({\dot {\boldsymbol {R}}}\cdot {\boldsymbol {U}}+{\boldsymbol {R}}\cdot {\dot {\boldsymbol {U}}})\cdot ({\boldsymbol {U}}^{-1}\cdot {\boldsymbol {R}}^{T})={\dot {\boldsymbol {R}}}\cdot {\boldsymbol {R}}^{T}+{\boldsymbol {R}}\cdot {\dot {\boldsymbol {U}}}\cdot {\boldsymbol {U}}^{-1}\cdot {\boldsymbol {R}}^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82193f21dae6605ae6bc0fc03bc2e8b0ee701619)
and
![{\displaystyle {\boldsymbol {F}}^{-T}\cdot {\dot {\boldsymbol {F}}}^{T}=({\boldsymbol {R}}\cdot {\boldsymbol {U}}^{-1})\cdot ({\boldsymbol {U}}\cdot {\dot {\boldsymbol {R}}}^{T}+{\dot {\boldsymbol {U}}}\cdot {\boldsymbol {R}}^{T})={\boldsymbol {R}}\cdot {\dot {\boldsymbol {R}}}^{T}+{\boldsymbol {R}}\cdot {\boldsymbol {U}}^{-1}\cdot {\dot {\boldsymbol {U}}}\cdot {\boldsymbol {R}}^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b014294047e69706fffbf362da6655f4621c90)
So we have
![{\displaystyle {\boldsymbol {w}}={\frac {1}{2}}~({\dot {\boldsymbol {R}}}\cdot {\boldsymbol {R}}^{T}+{\boldsymbol {R}}\cdot {\dot {\boldsymbol {U}}}\cdot {\boldsymbol {U}}^{-1}\cdot {\boldsymbol {R}}^{T}-{\boldsymbol {R}}\cdot {\dot {\boldsymbol {R}}}^{T}-{\boldsymbol {R}}\cdot {\boldsymbol {U}}^{-1}\cdot {\dot {\boldsymbol {U}}}\cdot {\boldsymbol {R}}^{T})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74ca79148e5f9f26af05f71c18201366e24e9422)
Now
![{\displaystyle {\boldsymbol {R}}\cdot {\boldsymbol {R}}^{T}={\boldsymbol {\mathit {1}}}\quad \implies \quad {\dot {\boldsymbol {R}}}\cdot {\boldsymbol {R}}^{T}+{\boldsymbol {R}}\cdot {\dot {\boldsymbol {R}}}^{T}={\boldsymbol {\mathit {0}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9edaea60c2ca0fcea98d277d4bbb8d6ae7dc02b)
Therefore
![{\displaystyle {\boldsymbol {w}}={\dot {\boldsymbol {R}}}\cdot {\boldsymbol {R}}^{T}+{\frac {1}{2}}~{\boldsymbol {R}}\cdot ({\dot {\boldsymbol {U}}}\cdot {\boldsymbol {U}}^{-1}-{\boldsymbol {U}}^{-1}\cdot {\dot {\boldsymbol {U}}})\cdot {\boldsymbol {R}}^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7d9f96cbb1e7166ca32ef4077807c695f8da009)
The second term above is invariant for rigid body motions and zero for an
uniaxial stretch. Hence, we are left with just a rotation term. This is
why the quantity
is called a spin.
The spin tensor is a skew-symmetric tensor and has an associated axial vector
(also called the angular velocity vector) whose components are
given by
![{\displaystyle {\boldsymbol {\omega }}={\begin{bmatrix}w_{1}\\w_{2}\\w_{3}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e30876c3c3babf8a5bab537c6a16eb79b07a78ee)
where
![{\displaystyle \mathbf {w} ={\begin{bmatrix}0&-w_{3}&w_{2}\\w_{3}&0&-w_{1}\\-w_{2}&w_{1}&0\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a576d5ebf4985a1adb9f241fb0d599eb06e087cf)
The spin tensor and its associated axial vector appear in a number of modern
numerical algorithms.
Recall that
![{\displaystyle dv=J~dV\qquad {\text{where}}~J=\det {\boldsymbol {F}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f433608406a522ac4c39373c0aa741b8a14ecd1d)
Therefore, taking the material time derivative of
(keeping
fixed),
we have
![{\displaystyle {\cfrac {d}{dt}}(dv)={\dot {J}}~dV={\cfrac {\dot {J}}{J}}~dv}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61360d391023b5064d50c6aad20a3e081d2e8cf5)
At this stage we invoke the following result from tensor calculus:
If
is an invertible tensor which depends on
then
![{\displaystyle {\cfrac {d}{dt}}(\det {\boldsymbol {A}})=(\det {\boldsymbol {A}})~{\text{tr}}\left({\cfrac {d{\boldsymbol {A}}}{dt}}\cdot {\boldsymbol {A}}^{-1}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99f0b30608dce1e31ac5715207191c246489c8e8)
In the case where
we have
![{\displaystyle {\cfrac {d}{dt}}(J)=J~{\text{tr}}\left({\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56f6f3a31a7184bb807770bc63429eae4a9947b2)
or,
![{\displaystyle {\dot {J}}=J~{\text{tr}}({\boldsymbol {l}})=J~{\text{tr}}(\mathbf {d} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a99eaf4617ef322705b7ba39717860e414867b39)
Therefore,
![{\displaystyle {\cfrac {d}{dt}}(dv)={\text{tr}}(\mathbf {d} )~dv}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75db6aacc3da6ce615b32975093b4aa0362d4d4f)
Alternatively, we can also write
![{\displaystyle {\dot {J}}={\frac {1}{2}}~J~{\boldsymbol {C}}^{-1}:{\dot {\boldsymbol {C}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4be2cce4d5a8a82b03c8a0505af8eedf0292c15e)
These relations are of immense use in numerical algorithms - particularly
those which involved incompressible behavior, i.e., when
.