Example of an objective vector

Let us look at an example in the context of kinematics. Consider a vector ${\displaystyle d\mathbf {X} }$ in the reference configuration that becomes ${\displaystyle d\mathbf {x} }$ in the deformed configuration. Let us then rotate the vector by an orthogonal tensor ${\displaystyle {\boldsymbol {Q}}}$ so that its rotated form is ${\displaystyle d\mathbf {x} _{r}}$.

Then,

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

The length of the vector ${\displaystyle d\mathbf {x} }$ in terms of ${\displaystyle d\mathbf {X} }$ is given by

${\displaystyle \lVert d\mathbf {x} \rVert _{}=\lVert {\boldsymbol {F}}\cdot d\mathbf {X} \rVert _{}={\sqrt {({\boldsymbol {F}}\cdot d\mathbf {X} )\cdot ({\boldsymbol {F}}\cdot d\mathbf {X} )}}={\sqrt {({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}):(d\mathbf {X} \otimes d\mathbf {X} )}}={\sqrt {{\boldsymbol {C}}:(d\mathbf {X} \otimes d\mathbf {X} )}}}$

Similarly,

${\displaystyle \lVert d\mathbf {x} _{r}\rVert _{}=\lVert {\boldsymbol {Q}}\cdot d\mathbf {x} \rVert _{}={\sqrt {({\boldsymbol {Q}}\cdot d\mathbf {x} )\cdot ({\boldsymbol {Q}}\cdot d\mathbf {x} )}}={\sqrt {{\boldsymbol {\mathit {1}}}:(d\mathbf {x} \otimes d\mathbf {x} )}}={\sqrt {d\mathbf {x} \cdot d\mathbf {x} }}=\lVert d\mathbf {x} \rVert _{}}$

Therefore, the vector ${\displaystyle d\mathbf {x} }$ is objective under rigid body rotation.

Example of a non-objective vector

Let ${\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(\mathbf {X} ,t)}$ and ${\displaystyle \mathbf {x} _{r}={\boldsymbol {Q}}\cdot \mathbf {x} ={\boldsymbol {Q}}\cdot {\boldsymbol {\varphi }}(\mathbf {X} ,t)}$. Then

${\displaystyle \mathbf {v} _{r}={\frac {\partial \mathbf {x} _{r}}{\partial t}}={\boldsymbol {Q}}\cdot {\frac {\partial \mathbf {x} }{\partial t}}+{\dot {\boldsymbol {Q}}}\cdot \mathbf {x} ={\boldsymbol {Q}}\cdot \mathbf {v} +{\dot {\boldsymbol {Q}}}\cdot \mathbf {x} ~.}$

If we compute the length of ${\displaystyle \mathbf {v} _{r}}$ we get

${\displaystyle \lVert \mathbf {v} _{r}\rVert =\lVert \mathbf {v} \rVert +{\sqrt {({\dot {\boldsymbol {Q}}}^{T}\cdot {\dot {\boldsymbol {Q}}}):(\mathbf {x} \otimes \mathbf {x} )}}}$

So the length of the velocity vector ${\displaystyle \mathbf {v} }$ changes if the rate of change of rotation is arbitrary (i.e., not constant or zero). Also the spatial velocity vector does not follow the standard vector transformation rule under rigid body rotation.

Therefore the spatial velocity vector violates objectivity.

Examples of objective 2nd-order tensors

Let now consider the effect of a rotation on the right Cauchy-Green deformation tensor ${\displaystyle {\boldsymbol {C}}}$.

${\displaystyle {\boldsymbol {C}}={\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}\quad \implies \quad {\boldsymbol {C}}_{r}={\boldsymbol {F}}_{r}^{T}\cdot {\boldsymbol {F}}_{r}\quad {\text{where}}\quad {\boldsymbol {F}}_{r}={\boldsymbol {Q}}\cdot {\boldsymbol {F}}}$

Then

${\displaystyle {\boldsymbol {C}}_{r}={\boldsymbol {F}}^{T}\cdot {\boldsymbol {Q}}^{T}\cdot {\boldsymbol {Q}}\cdot {\boldsymbol {F}}={\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}={\boldsymbol {C}}}$

Therefore ${\displaystyle {\boldsymbol {C}}}$ is an non-objective quantity. Similarly we can show that the Lagrangian Green strain tensor ${\displaystyle {\boldsymbol {E}}}$ is objective.

For the spatial deformation tensor ${\displaystyle {\boldsymbol {b}}}$ and the spatial strain tensor ${\displaystyle {\boldsymbol {e}}}$ we can show that

${\displaystyle {\boldsymbol {b}}_{r}={\boldsymbol {Q}}\cdot {\boldsymbol {b}}\cdot {\boldsymbol {Q}}^{T}~;~~{\boldsymbol {e}}_{r}={\boldsymbol {Q}}\cdot {\boldsymbol {e}}\cdot {\boldsymbol {Q}}^{T}}$

These tensors are objective because they follow the standard rules for tensor transformations under rigid body rotations.

The spatial rate of deformation tensor ${\displaystyle \mathbf {d} }$ transforms at

${\displaystyle {\boldsymbol {d}}_{r}={\frac {1}{2}}~({\boldsymbol {l}}_{r}+{\boldsymbol {l}}_{r}^{T})={\frac {1}{2}}~({\dot {\boldsymbol {F}}}_{r}\cdot {\boldsymbol {F}}_{r}^{-1}+{\boldsymbol {F}}_{r}^{-T}\cdot {\dot {\boldsymbol {F}}}_{r}^{T})={\frac {1}{2}}~[({\dot {\boldsymbol {Q}}}\cdot {\boldsymbol {F}}+{\boldsymbol {Q}}\cdot {\dot {\boldsymbol {F}}})\cdot ({\boldsymbol {F}}^{-1}\cdot {\boldsymbol {Q}}^{T})+({\boldsymbol {Q}}\cdot {\boldsymbol {F}}^{-T})\cdot ({\boldsymbol {F}}^{T}\cdot {\dot {\boldsymbol {Q}}}^{T}+{\dot {\boldsymbol {F}}}^{T}\cdot {\boldsymbol {Q}}^{T})]}$

or,

${\displaystyle {\boldsymbol {d}}_{r}={\frac {1}{2}}~[{\dot {\boldsymbol {Q}}}\cdot {\boldsymbol {Q}}^{T}+{\boldsymbol {Q}}\cdot {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}\cdot {\boldsymbol {Q}}^{T}+{\boldsymbol {Q}}\cdot {\dot {\boldsymbol {Q}}}^{T}+{\boldsymbol {Q}}\cdot {\boldsymbol {F}}^{-T}\cdot {\dot {\boldsymbol {F}}}^{T}\cdot {\boldsymbol {Q}}^{T}]={\frac {1}{2}}~[{\dot {\boldsymbol {Q}}}\cdot {\boldsymbol {Q}}^{T}+{\boldsymbol {Q}}\cdot {\boldsymbol {l}}\cdot {\boldsymbol {Q}}^{T}+{\boldsymbol {Q}}\cdot {\dot {\boldsymbol {Q}}}^{T}+{\boldsymbol {Q}}\cdot {\boldsymbol {l}}^{T}\cdot {\boldsymbol {Q}}^{T}]}$

or,

${\displaystyle {\boldsymbol {d}}_{r}={\boldsymbol {Q}}\cdot {\boldsymbol {d}}\cdot {\boldsymbol {Q}}^{T}+{\frac {1}{2}}~[{\dot {\boldsymbol {Q}}}\cdot {\boldsymbol {Q}}^{T}+{\boldsymbol {Q}}\cdot {\dot {\boldsymbol {Q}}}^{T}]={\boldsymbol {Q}}\cdot {\boldsymbol {d}}\cdot {\boldsymbol {Q}}^{T}}$

where ${\displaystyle \quad [{\dot {\boldsymbol {Q}}}\cdot {\boldsymbol {Q}}^{T}+{\boldsymbol {Q}}\cdot {\dot {\boldsymbol {Q}}}^{T}]=({\boldsymbol {Q}}\cdot {\boldsymbol {Q}}^{T})^{\cdot }={\boldsymbol {0}}}$

Therefore ${\displaystyle {\boldsymbol {d}}}$ is an objective quantity.

Example of a non-objective 2nd-order tensor

Let us now look at the velocity gradient tensor by itself. In that case

${\displaystyle {\boldsymbol {l}}_{r}={\dot {\boldsymbol {F}}}_{r}\cdot {\boldsymbol {F}}_{r}^{-1}=({\dot {\boldsymbol {Q}}}\cdot {\boldsymbol {F}}+{\boldsymbol {Q}}\cdot {\dot {\boldsymbol {F}}})\cdot ({\boldsymbol {F}}^{-1}\cdot {\boldsymbol {Q}}^{T})}$

or,

${\displaystyle {\boldsymbol {l}}_{r}={\dot {\boldsymbol {Q}}}\cdot {\boldsymbol {Q}}^{T}+{\boldsymbol {Q}}\cdot {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}\cdot {\boldsymbol {Q}}^{T}={\dot {\boldsymbol {Q}}}\cdot {\boldsymbol {Q}}^{T}+{\boldsymbol {Q}}\cdot {\boldsymbol {l}}\cdot {\boldsymbol {Q}}^{T}}$

Clearly the first term above is not zero for arbitrary rotations. Hence the spatial velocity gradient tensor is not objective.