Advanced elasticity/Objectivity in kinematics

Objectivity of kinematic quantities

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

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

Then,

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

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

\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,

\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 $d\mathbf {x}$ is objective under rigid body rotation.

Example of a non-objective vector

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

\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 $\mathbf {v} _{r}$ we get

\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 $\mathbf {v}$ 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 ${\boldsymbol {C}}$ .

{\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

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

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

{\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 $\mathbf {d}$ transforms at

{\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,

{\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,

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