Finite deformation tensors
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In continuum mechanics, finite deformation tensors are tensors that are used to measure deformation. They are used when the deformation is not small, as is commonly the case in mechanics of rubber, plastics and viscoelastic fluids. For small deformations see strain tensor.
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Deformation gradient tensor
Deformation gradient tensor F is defined as:
- <math> \mathbf { F } = \nabla \mathbf {x} =\frac {\partial \mathbf{x}} {\partial \mathbf {x^\prime}} </math>
or
- <math>F_{i,j} = \frac {\partial x_i} {\partial x_j^\prime}</math>
where <math> \mathbf {x} </math> are the coordinates of a point in deformed state and
- <math> \mathbf {x \prime} </math> are coordinate of a point in undeformed state.
By doing so we assume that <math> \mathbf {x} </math> can be expressed as a differentiable function of <math> \mathbf {x^ \prime} </math> and time t:
- <math> \mathbf {x} = \mathbf {x} (\mathbf {x ^\prime},t) </math>
this will not be the case if a crack develops in the deformed body.
If we have a small vector <math>d \mathbf {x ^\prime} </math> in the undeformed body, then the correspondent vector in the deformed body <math>d \mathbf {x } </math>can by calculated as
- <math>d\mathbf {x } = \mathbf { F } d\mathbf {x ^\prime}</math>
The deformation gradient tensor keeps information about both the true deformation of the body, and solid body rotation. Usually in fluid mechanics we want to treat separately the true deformation and the rotation.
Finger tensor
The deformation gradient tensor F can be expressed as a product of a symmetric tensor V for true deformation and an antisymmetric tensor R for rotation:
- <math>\mathbf{F}=\mathbf{V} \mathbf{R}</math>
As superposition of rotation and the inverse rotation leads to no change (<math>\mathbf{R}\mathbf{R^T}=\mathbf{1}</math>) we can exclude the rotation by multiplying F by its transpose:
- <math>\mathbf{B}=\mathbf{F}\mathbf{F^T}=\mathbf{V}\mathbf{V^T}</math>
This tensor is named the Finger tensor, after J. Finger (1894) .
By definition:
- <math>B_{i,j}=\sum_{k=1..3}\frac {\partial x_i} {\partial x_k^\prime} \frac {\partial x_j} {\partial x_k^\prime}</math>
Physically speaking, this tensor gives us the local changes in area within a sample:
- <math>\mu^2=\mathbf{n} \mathbf{B} \mathbf{n} </math>,
where <math>\mu</math> is the ratio of undeformed surface to the deformed surface and <math>\mathbf{n}</math> is the normal vector to the surface.
Cauchy-Green tensor
If we reversed the order of multiplication in the formula for the Finger tensor (above) we would get the Cauchy-Green tensor:
- <math>\mathbf{C}=\mathbf{F^T}\mathbf{F}</math>
or
- <math>C_{i,j}=\sum_{k=1..3}\frac {\partial x_k} {\partial x_i^\prime} \frac {\partial x_k} {\partial x_j^\prime}</math>
The tensor is named after Augustin Louis Cauchy and George Green.
Physically the Cauchy-Green tensor gives us the local change in distances due to deformation:
- <math>\alpha^2=\mathbf{n^\prime}\mathbf{C}\mathbf{n^\prime}</math>
where <math>\alpha</math> is the ratio of lengths of a vector in deformed and undeformed states and <math>\mathbf{n^\prime}</math> is the direction of the vector in undeformed state.
Examples
Uniaxial extension of an incompressible material
<math>\mathbf{F}=\begin{bmatrix} \alpha & 0 & 0 \\ 0 & \alpha^{-0.5} & 0 \\ 0 & 0 & \alpha^{-0.5} \end{bmatrix}</math>
<math>\mathbf{B}=\mathbf{C}=\begin{bmatrix} \alpha^2 & 0 & 0 \\ 0 & \alpha^{-1} & 0 \\ 0 & 0 & \alpha^{-1} \end{bmatrix}</math>
Simple shear
<math>\mathbf{F}=\begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}</math>
<math>\mathbf{B}=\begin{bmatrix} 1+\gamma^2 & \gamma & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}</math>
<math>\mathbf{C}=\begin{bmatrix} 1 & \gamma & 0 \\ \gamma & 1+\gamma^2 & 0 \\ 0 & 0 & 1 \end{bmatrix}</math>
Solid body rotation
<math>\mathbf{F}=\begin{bmatrix} \cos \theta & \sin \theta & 0 \\ - \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}</math>
<math>\mathbf{B}=\mathbf{C}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \mathbf{1}</math>
Source
- C. W. Macosco Rheology: principles, measurement and applications, VCH Publishers, 1994, ISBN 1-56081-579-5
