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, soft tissue and viscoelastic fluids. For small deformations see strain tensor.

Deformation gradient tensor

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Deformation gradient tensor F is defined as:

$\backslash mathbf\; \{\; F\; \}\; =\; \backslash nabla\; \backslash mathbf\; \{x\}\; =\backslash frac\; \{\backslash partial\; \backslash mathbf\{x\}\}\; \{\backslash partial\; \backslash mathbf\; \{x^\backslash prime\}\}$

or

$F\_\{i,j\}\; =\; \backslash frac\; \{\backslash partial\; x\_i\}\; \{\backslash partial\; x\_j^\backslash prime\}$

where $\backslash mathbf\; \{x\}$ are the coordinates of a point in deformed state and $\backslash mathbf\; \{x^\backslash prime\}$ are coordinate of a point in undeformed state.

By doing so we assume that $\backslash mathbf\; \{x\}$ can be expressed as a differentiable function of $\backslash mathbf\; \{x^\; \backslash prime\}$ and time t:

$\backslash mathbf\; \{x\}\; =\; \backslash mathbf\; \{x\}\; (\backslash mathbf\; \{x\; ^\backslash prime\},t)$

this will not be the case if a crack develops in the deformed body.

If we have a small vector $d\; \backslash mathbf\; \{x\; ^\backslash prime\}$ in the undeformed body, then the correspondent vector in the deformed body $d\; \backslash mathbf\; \{x\; \}$can by calculated as

$d\backslash mathbf\; \{x\; \}\; =\; \backslash mathbf\; \{\; F\; \}\; d\backslash mathbf\; \{x\; ^\backslash prime\}$

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 Left Cauchy-Green deformation tensor)

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The deformation gradient tensor F can be expressed as a product of a symmetric tensor V for true deformation and an rotation tensor R for rotation:

$\backslash mathbf\{F\}=\backslash mathbf\{V\}\; \backslash mathbf\{R\}$

As superposition of rotation and the inverse rotation leads to no change ($\backslash mathbf\{R\}\backslash mathbf\{R^T\}=\backslash mathbf\{1\}$) we can exclude the rotation by multiplying F by its transpose:

$\backslash mathbf\{B\}=\backslash mathbf\{F\}\backslash mathbf\{F^T\}=\backslash mathbf\{V\}\backslash mathbf\{V^T\}$

This tensor is named the Finger tensor, after Josef Finger (1894).

By definition:

$B\_\{ij\}=\backslash sum\_\{k=1..3\}\backslash frac\; \{\backslash partial\; x\_i\}\; \{\backslash partial\; x\_k^\backslash prime\}\; \backslash frac\; \{\backslash partial\; x\_j\}\; \{\backslash partial\; x\_k^\backslash prime\}$

Physically speaking, this tensor gives us the local changes in area within a sample:

$\backslash mu^2=\backslash mathbf\{n\}\; \backslash mathbf\{B\}\; \backslash mathbf\{n\}$,

where $\backslash mu$ is the ratio of undeformed surface to the deformed surface and $\backslash mathbf\{n\}$ is the normal vector to the surface.

Cauchy-Green tensor (The right Cauchy-Green deformation tensor)

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If we reversed the order of multiplication in the formula for the Finger tensor (above) we would get the Cauchy-Green tensor:

$\backslash mathbf\{C\}=\backslash mathbf\{F^T\}\backslash mathbf\{F\}$

or

$C\_\{ij\}=\backslash sum\_\{k=1..3\}\backslash frac\; \{\backslash partial\; x\_k\}\; \{\backslash partial\; x\_i^\backslash prime\}\; \backslash frac\; \{\backslash partial\; x\_k\}\; \{\backslash partial\; x\_j^\backslash prime\}$

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:

$\backslash alpha^2=\backslash mathbf\{n^\backslash prime\}\backslash mathbf\{C\}\backslash mathbf\{n^\backslash prime\}$

where $\backslash alpha$ is the ratio of lengths of a vector in deformed and undeformed states and $\backslash mathbf\{n^\backslash prime\}$ is the direction of the vector in undeformed state.

Examples

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Uniaxial extension of an incompressible material

This the case where a specimen is elongated over the x coordinate with the elongation ratio (the ratio of the deformed and undeformed length) of $\backslash alpha=\backslash alpha\_1$ and in the other two dimensions the specimen shrinks, so to keep volume constant ($\backslash alpha\_1\; \backslash alpha\_2\; \backslash alpha\_3\; =1$ or $\backslash alpha\_2=\backslash alpha\_3=\backslash alpha^\{-0.5\}$)

Simple shear

Solid body rotation

See also

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• Piola-Kirchhoff stress tensor, the stress tensor for finite deformations.

Source

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• C. W. Macosko Rheology: principles, measurement and applications, VCH Publishers, 1994, ISBN 1-56081-579-5

Tensors | Continuum mechanics | Non-Newtonian fluids