Objectivity is one of the fundamental concepts of continuum mechanics.
Objectivity is another name for frame indifference, i.e., the position of an
observer should not affect any quantities of interest. The concept is an
extension of the idea that rigid body motions should not affect the stress
and strain tensors or the mechanical properties of a material.
A spatial strain tensor is said to transform objectively under rigid body motion if it transforms according to the transformation rules for second order tensors.
Let us look at an example in the context of kinematics. Consider a vector
in the reference configuration that becomes
in the deformed
configuration. Let us then rotate the vector by an orthogonal tensor
so that its rotated form is
.
Then,
![{\displaystyle d\mathbf {x} ={\boldsymbol {F}}\cdot d\mathbf {X} ~;~~d\mathbf {x} _{r}={\boldsymbol {Q}}\cdot d\mathbf {x} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9409284815ffcf8af68e05bd4013a6aa51ff633)
The length of the vector
in terms of
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} )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bae7c567749f5a7c3e12832f54e8900dccda6eff)
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 _{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee93f223423605446f2d9b5eb4ad1b92a574770)
Therefore, the vector
is objective under rigid body rotation.
Let
and
.
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} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f3db016484955e00d45c4ec528aee8123203405)
If we compute the length of
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} )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8010d7494b3191ee34e06c1fb8c4dfdffc6706b1)
So the length of the velocity vector
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.
Let now consider the effect of a rotation on the right Cauchy-Green deformation
tensor
.
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce10c9d299925f08ad544189c8275a0e42913fe0)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df3769d3569e19eedeaaa1f9f74e9e234ad26b99)
Therefore
is an objective quantity. Similarly we can show that the
Lagrangian Green strain tensor
is objective.
For the spatial deformation tensor
and the spatial strain tensor
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70d747454edf86c94cd3d5d9c18073f948ec6269)
These tensors are objective because they follow the standard rules for tensor
transformations under rigid body rotations.
The spatial rate of deformation tensor
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})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5558b64004e3154defa84338bccd6077bd30e4f9)
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}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3494a32eb8a2dc44a9bc2dbe3a86fc4315a474f6)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdcf464a037459aa8adee0341aed62b63477b1e6)
where
Therefore
is an objective quantity.
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77188f62abd8bd876304e2f92a157bf5066e3488)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51ec3a9586f93c5977e808950c35c3b61ece22a8)
Clearly the first term above is not zero for arbitrary rotations. Hence the
spatial velocity gradient tensor is not objective.