Material time derivatives

Material time derivatives are needed for many updated Lagrangian formulations of finite element analysis.

Recall that the motion can be expressed as

${\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(\mathbf {X} ,t)\quad {\text{or}}\qquad \mathbf {X} ={\boldsymbol {\varphi }}^{-1}(\mathbf {x} ,t)}$

If we keep ${\displaystyle \mathbf {X} }$ fixed, then the velocity is given by

${\displaystyle \mathbf {V} (\mathbf {X} ,t)={\frac {\partial {\boldsymbol {\varphi }}}{\partial t}}(\mathbf {X} ,t)}$

This is the material time derivative expressed in terms of ${\displaystyle \mathbf {X} }$.

The spatial version of the velocity is

${\displaystyle \mathbf {v} (\mathbf {x} ,t)=\mathbf {V} ({\boldsymbol {\varphi }}^{-1}(\mathbf {x} ,t),t)}$

We will use the symbol ${\displaystyle \mathbf {v} }$ 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 ${\displaystyle \mathbf {x} }$ (rather than material quantities which are functions of ${\displaystyle \mathbf {X} }$).

Given the spatial velocity ${\displaystyle \mathbf {v} (\mathbf {x} ,t)}$, if we want to find the acceleration we will have to consider the fact that ${\displaystyle \mathbf {x} \equiv \mathbf {x} (\mathbf {X} ,t)}$, i.e., the position also changes with time. We do this by using the chain rule. Thus

${\displaystyle {\cfrac {D\mathbf {v} (\mathbf {x} ,t)}{Dt}}=\mathbf {a} (\mathbf {x} ,t)={\frac {\partial \mathbf {v} (\mathbf {x} ,t)}{\partial t}}+{\frac {\partial \mathbf {v} (\mathbf {x} ,t)}{\partial \mathbf {x} }}\cdot {\frac {\partial {\boldsymbol {\varphi }}(\mathbf {X} ,t)}{\partial t}}={\frac {\partial \mathbf {v} }{\partial t}}+{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {V} ={\frac {\partial \mathbf {v} }{\partial t}}+{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {v} }$

Such a derivative is called the material time derivative expressed in terms of ${\displaystyle \mathbf {x} }$. The second term in the expression is called the convective derivative..

Velocity gradient

Let the velocity be expressed in spatial form, i.e., ${\displaystyle \mathbf {v} (\mathbf {x} ,t)}$. The spatial velocity gradient tensor is given by

${\displaystyle {\boldsymbol {l}}:={\frac {\partial \mathbf {v} (\mathbf {x} ,t)}{\partial \mathbf {x} }}={\boldsymbol {\nabla }}\mathbf {v} }$

The velocity gradient ${\displaystyle {\boldsymbol {l}}}$ is a second order tensor which can expressed as

${\displaystyle {\boldsymbol {l}}=l_{ij}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}={\frac {\partial v_{i}}{\partial x_{j}}}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}}$

The velocity gradient is a measure of the relative velocity of two points in the current configuration.

Time derivative of the deformation gradient

Recall that the deformation gradient is given by

${\displaystyle {\boldsymbol {F}}={\frac {\partial {\boldsymbol {\varphi }}}{\partial \mathbf {X} }}}$

The time derivative of ${\displaystyle {\boldsymbol {F}}}$ (keeping ${\displaystyle \mathbf {X} }$ fixed) is

${\displaystyle {\dot {\boldsymbol {F}}}={\frac {\partial }{\partial t}}~\left({\frac {\partial {\boldsymbol {\varphi }}}{\partial \mathbf {X} }}\right)={\frac {\partial }{\partial \mathbf {X} }}~\left({\frac {\partial {\boldsymbol {\varphi }}}{\partial t}}\right)={\frac {\partial \mathbf {v} }{\partial \mathbf {X} }}={\boldsymbol {\nabla }}_{\circ }\mathbf {v} }$

Using the chain rule

${\displaystyle {\dot {\boldsymbol {F}}}={\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\cdot {\frac {\partial \mathbf {x} }{\partial \mathbf {X} }}={\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\cdot {\frac {\partial {\boldsymbol {\varphi }}}{\partial \mathbf {X} }}={\boldsymbol {l}}\cdot {\boldsymbol {F}}}$

Form this we get the important relation

${\displaystyle {\boldsymbol {l}}={\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}~.}$

Time derivative of strain

Let ${\displaystyle d\mathbf {X} _{1}}$ and ${\displaystyle d\mathbf {X} _{2}}$ be two infinitesimal material line segments in a body. Then

${\displaystyle d\mathbf {x} _{1}={\boldsymbol {F}}\cdot d\mathbf {X} _{1}~;~~d\mathbf {x} _{2}={\boldsymbol {F}}\cdot d\mathbf {X} _{2}}$

Hence,

${\displaystyle d\mathbf {x} _{1}\cdot d\mathbf {x} _{2}=({\boldsymbol {F}}\cdot d\mathbf {X} _{1})\cdot ({\boldsymbol {F}}\cdot d\mathbf {X} _{2})=d\mathbf {X} _{1}\cdot ({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}})\cdot d\mathbf {X} _{2}=d\mathbf {X} _{1}\cdot {\boldsymbol {C}}\cdot d\mathbf {X} _{2}=d\mathbf {X} _{1}\cdot (2~{\boldsymbol {E}}+{\boldsymbol {\mathit {1}}})\cdot d\mathbf {X} _{2}}$

Taking the derivative with respect to ${\displaystyle t}$ gives us

${\displaystyle {\frac {\partial }{\partial t}}(d\mathbf {x} _{1}\cdot d\mathbf {x} _{2})=d\mathbf {X} _{1}\cdot {\frac {\partial {\boldsymbol {C}}}{\partial t}}\cdot d\mathbf {X} _{2}=2~d\mathbf {X} _{1}\cdot {\frac {\partial {\boldsymbol {E}}}{\partial t}}\cdot d\mathbf {X} _{2}}$

The material strain rate tensor is defined as

${\displaystyle {\dot {\boldsymbol {E}}}={\frac {\partial {\boldsymbol {E}}}{\partial t}}={\frac {1}{2}}~{\frac {\partial {\boldsymbol {C}}}{\partial t}}={\frac {1}{2}}~{\dot {\boldsymbol {C}}}}$

Clearly,

${\displaystyle {\dot {\boldsymbol {E}}}={\frac {1}{2}}~{\frac {\partial }{\partial t}}({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}})={\frac {1}{2}}~({\dot {\boldsymbol {F}}}^{T}\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot {\dot {\boldsymbol {F}}})~.}$

Also,

${\displaystyle {\frac {1}{2}}~{\frac {\partial }{\partial t}}(d\mathbf {x} _{1}\cdot d\mathbf {x} _{2})=d\mathbf {X} _{1}\cdot {\dot {\boldsymbol {E}}}\cdot d\mathbf {X} _{2}=({\boldsymbol {F}}^{-1}\cdot d\mathbf {x} _{1})\cdot {\dot {\boldsymbol {E}}}\cdot ({\boldsymbol {F}}^{-1}\cdot d\mathbf {x} _{2})=d\mathbf {x} _{1}\cdot ({\boldsymbol {F}}^{-T}\cdot {\dot {\boldsymbol {E}}}\cdot {\boldsymbol {F}}^{-1})\cdot d\mathbf {x} _{2}}$

The spatial rate of deformation tensor or stretching tensor is defined as

${\displaystyle {\boldsymbol {d}}={\boldsymbol {F}}^{-T}\cdot {\dot {\boldsymbol {E}}}\cdot {\boldsymbol {F}}^{-1}={\frac {1}{2}}~{\boldsymbol {F}}^{-T}\cdot {\dot {\boldsymbol {C}}}\cdot {\boldsymbol {F}}^{-1}}$

In fact, we can show that ${\displaystyle {\boldsymbol {d}}}$ is the symmetric part of the velocity gradient, i.e.,

${\displaystyle {\boldsymbol {d}}={\frac {1}{2}}~({\boldsymbol {l}}+{\boldsymbol {l}}^{T})}$

For rigid body motions we get ${\displaystyle {\boldsymbol {d}}={\boldsymbol {\mathit {0}}}}$.

Spin tensor

The velocity gradient tensor can be additively decomposed into a symmetric part and a skew part:

${\displaystyle {\boldsymbol {l}}={\frac {1}{2}}~({\boldsymbol {l}}+{\boldsymbol {l}}^{T})+{\frac {1}{2}}({\boldsymbol {l}}-{\boldsymbol {l}}^{T})={\boldsymbol {d}}+{\boldsymbol {w}}}$

We have seen that ${\displaystyle {\boldsymbol {d}}}$ is the rate of deformation tensor. The quantity ${\displaystyle {\boldsymbol {w}}}$ is called the spin tensor.

Note that ${\displaystyle {\boldsymbol {d}}}$ is symmetric while ${\displaystyle {\boldsymbol {w}}}$ is skew symmetric, i.e.,

${\displaystyle {\boldsymbol {d}}={\boldsymbol {d}}^{T}~;~~{\boldsymbol {w}}=-{\boldsymbol {w}}^{T}~.}$

So see why ${\displaystyle {\boldsymbol {w}}}$ is called a "spin", recall that

${\displaystyle {\boldsymbol {l}}={\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}}$

Therefore,

${\displaystyle {\boldsymbol {w}}={\frac {1}{2}}({\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}-{\boldsymbol {F}}^{-T}\cdot {\dot {\boldsymbol {F}}}^{T})}$

Also,

${\displaystyle {\boldsymbol {F}}={\boldsymbol {R}}\cdot {\boldsymbol {U}}\quad \implies \quad {\dot {\boldsymbol {F}}}={\dot {\boldsymbol {R}}}\cdot {\boldsymbol {U}}+{\boldsymbol {R}}\cdot {\dot {\boldsymbol {U}}}}$

Therefore,

${\displaystyle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}=({\dot {\boldsymbol {R}}}\cdot {\boldsymbol {U}}+{\boldsymbol {R}}\cdot {\dot {\boldsymbol {U}}})\cdot ({\boldsymbol {U}}^{-1}\cdot {\boldsymbol {R}}^{T})={\dot {\boldsymbol {R}}}\cdot {\boldsymbol {R}}^{T}+{\boldsymbol {R}}\cdot {\dot {\boldsymbol {U}}}\cdot {\boldsymbol {U}}^{-1}\cdot {\boldsymbol {R}}^{T}}$

and

${\displaystyle {\boldsymbol {F}}^{-T}\cdot {\dot {\boldsymbol {F}}}^{T}=({\boldsymbol {R}}\cdot {\boldsymbol {U}}^{-1})\cdot ({\boldsymbol {U}}\cdot {\dot {\boldsymbol {R}}}^{T}+{\dot {\boldsymbol {U}}}\cdot {\boldsymbol {R}}^{T})={\boldsymbol {R}}\cdot {\dot {\boldsymbol {R}}}^{T}+{\boldsymbol {R}}\cdot {\boldsymbol {U}}^{-1}\cdot {\dot {\boldsymbol {U}}}\cdot {\boldsymbol {R}}^{T}}$

So we have

${\displaystyle {\boldsymbol {w}}={\frac {1}{2}}~({\dot {\boldsymbol {R}}}\cdot {\boldsymbol {R}}^{T}+{\boldsymbol {R}}\cdot {\dot {\boldsymbol {U}}}\cdot {\boldsymbol {U}}^{-1}\cdot {\boldsymbol {R}}^{T}-{\boldsymbol {R}}\cdot {\dot {\boldsymbol {R}}}^{T}-{\boldsymbol {R}}\cdot {\boldsymbol {U}}^{-1}\cdot {\dot {\boldsymbol {U}}}\cdot {\boldsymbol {R}}^{T})}$

Now

${\displaystyle {\boldsymbol {R}}\cdot {\boldsymbol {R}}^{T}={\boldsymbol {\mathit {1}}}\quad \implies \quad {\dot {\boldsymbol {R}}}\cdot {\boldsymbol {R}}^{T}+{\boldsymbol {R}}\cdot {\dot {\boldsymbol {R}}}^{T}={\boldsymbol {\mathit {0}}}}$

Therefore

${\displaystyle {\boldsymbol {w}}={\dot {\boldsymbol {R}}}\cdot {\boldsymbol {R}}^{T}+{\frac {1}{2}}~{\boldsymbol {R}}\cdot ({\dot {\boldsymbol {U}}}\cdot {\boldsymbol {U}}^{-1}-{\boldsymbol {U}}^{-1}\cdot {\dot {\boldsymbol {U}}})\cdot {\boldsymbol {R}}^{T}}$

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 ${\displaystyle {\boldsymbol {w}}}$ is called a spin.

The spin tensor is a skew-symmetric tensor and has an associated axial vector ${\displaystyle {\boldsymbol {\omega }}}$ (also called the angular velocity vector) whose components are given by

${\displaystyle {\boldsymbol {\omega }}={\begin{bmatrix}w_{1}\\w_{2}\\w_{3}\end{bmatrix}}}$

where

${\displaystyle \mathbf {w} ={\begin{bmatrix}0&-w_{3}&w_{2}\\w_{3}&0&-w_{1}\\-w_{2}&w_{1}&0\end{bmatrix}}}$

The spin tensor and its associated axial vector appear in a number of modern numerical algorithms.

Rate of change of volume

Recall that

${\displaystyle dv=J~dV\qquad {\text{where}}~J=\det {\boldsymbol {F}}}$

Therefore, taking the material time derivative of ${\displaystyle dv}$ (keeping ${\displaystyle \mathbf {X} }$ fixed), we have

${\displaystyle {\cfrac {d}{dt}}(dv)={\dot {J}}~dV={\cfrac {\dot {J}}{J}}~dv}$

At this stage we invoke the following result from tensor calculus:

If ${\displaystyle {\boldsymbol {A}}}$ is an invertible tensor which depends on ${\displaystyle t}$ then

${\displaystyle {\cfrac {d}{dt}}(\det {\boldsymbol {A}})=(\det {\boldsymbol {A}})~{\text{tr}}\left({\cfrac {d{\boldsymbol {A}}}{dt}}\cdot {\boldsymbol {A}}^{-1}\right)}$

In the case where ${\displaystyle {\boldsymbol {A}}={\boldsymbol {F}},~J=\det {\boldsymbol {F}}}$ we have

${\displaystyle {\cfrac {d}{dt}}(J)=J~{\text{tr}}\left({\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}\right)}$

or,

${\displaystyle {\dot {J}}=J~{\text{tr}}({\boldsymbol {l}})=J~{\text{tr}}(\mathbf {d} )}$

Therefore,

${\displaystyle {\cfrac {d}{dt}}(dv)={\text{tr}}(\mathbf {d} )~dv}$

Alternatively, we can also write

${\displaystyle {\dot {J}}={\frac {1}{2}}~J~{\boldsymbol {C}}^{-1}:{\dot {\boldsymbol {C}}}}$

These relations are of immense use in numerical algorithms - particularly those which involved incompressible behavior, i.e., when ${\displaystyle {\dot {J}}=0}$.