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.
Example of an objective vector
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Let us look at an example in the context of kinematics. Consider a vector
d X {\displaystyle d\mathbf {X} } d x {\displaystyle d\mathbf {x} } Q {\displaystyle {\boldsymbol {Q}}} d x r {\displaystyle d\mathbf {x} _{r}}
Then,
d x = F ⋅ d X ; d x r = Q ⋅ d x {\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 d x {\displaystyle d\mathbf {x} } d X {\displaystyle d\mathbf {X} }
‖ d x ‖ = ‖ F ⋅ d X ‖ = ( F ⋅ d X ) ⋅ ( F ⋅ d X ) = ( F T ⋅ F ) : ( d X ⊗ d X ) = C : ( d X ⊗ d X ) {\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,
‖ d x r ‖ = ‖ Q ⋅ d x ‖ = ( Q ⋅ d x ) ⋅ ( Q ⋅ d x ) = 1 : ( d x ⊗ d x ) = d x ⋅ d x = ‖ d x ‖ {\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 d x {\displaystyle d\mathbf {x} }
Example of a non-objective vector
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Let x = φ ( X , t ) {\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(\mathbf {X} ,t)} x r = Q ⋅ x = Q ⋅ φ ( X , t ) {\displaystyle \mathbf {x} _{r}={\boldsymbol {Q}}\cdot \mathbf {x} ={\boldsymbol {Q}}\cdot {\boldsymbol {\varphi }}(\mathbf {X} ,t)}
v r = ∂ x r ∂ t = Q ⋅ ∂ x ∂ t + Q ˙ ⋅ x = Q ⋅ v + Q ˙ ⋅ x . {\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 v r {\displaystyle \mathbf {v} _{r}}
‖ v r ‖ = ‖ v ‖ + ( Q ˙ T ⋅ Q ˙ ) : ( x ⊗ x ) {\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 v {\displaystyle \mathbf {v} }
Therefore the spatial velocity vector violates objectivity.
Examples of objective 2nd-order tensors
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Let now consider the effect of a rotation on the right Cauchy-Green deformation
tensor C {\displaystyle {\boldsymbol {C}}}
C = F T ⋅ F ⟹ C r = F r T ⋅ F r where F r = Q ⋅ F {\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
C r = F T ⋅ Q T ⋅ Q ⋅ F = F T ⋅ F = C {\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 C {\displaystyle {\boldsymbol {C}}} E {\displaystyle {\boldsymbol {E}}}
For the spatial deformation tensor b {\displaystyle {\boldsymbol {b}}} e {\displaystyle {\boldsymbol {e}}}
b r = Q ⋅ b ⋅ Q T ; e r = Q ⋅ e ⋅ Q T {\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 d {\displaystyle \mathbf {d} }
d r = 1 2 ( l r + l r T ) = 1 2 ( F ˙ r ⋅ F r − 1 + F r − T ⋅ F ˙ r T ) = 1 2 [ ( Q ˙ ⋅ F + Q ⋅ F ˙ ) ⋅ ( F − 1 ⋅ Q T ) + ( Q ⋅ F − T ) ⋅ ( F T ⋅ Q ˙ T + F ˙ T ⋅ Q T ) ] {\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,
d r = 1 2 [ Q ˙ ⋅ Q T + Q ⋅ F ˙ ⋅ F − 1 ⋅ Q T + Q ⋅ Q ˙ T + Q ⋅ F − T ⋅ F ˙ T ⋅ Q T ] = 1 2 [ Q ˙ ⋅ Q T + Q ⋅ l ⋅ Q T + Q ⋅ Q ˙ T + Q ⋅ l T ⋅ Q T ] {\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,
d r = Q ⋅ d ⋅ Q T + 1 2 [ Q ˙ ⋅ Q T + Q ⋅ Q ˙ T ] = Q ⋅ d ⋅ Q T {\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 [ Q ˙ ⋅ Q T + Q ⋅ Q ˙ T ] = ( Q ⋅ Q T ) ⋅ = 0 {\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 d {\displaystyle {\boldsymbol {d}}}
Example of a non-objective 2nd-order tensor
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Let us now look at the velocity gradient tensor by itself. In that case
l r = F ˙ r ⋅ F r − 1 = ( Q ˙ ⋅ F + Q ⋅ F ˙ ) ⋅ ( F − 1 ⋅ Q T ) {\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,
l r = Q ˙ ⋅ Q T + Q ⋅ F ˙ ⋅ F − 1 ⋅ Q T = Q ˙ ⋅ Q T + Q ⋅ l ⋅ Q T {\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.
![{\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)
![{\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)
![{\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)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eccacdec6c91f0f1fbd20140318562a1ea62d3c4)
