Strain Measures in three dimensions[edit | edit source]
Initial orthonormal basis:

Deformed orthonormal basis:

We assume that these coincide.


Effect of
:

Dyadic notation:

Index notation:

The determinant of the deformation gradient is usually denoted by
and is a measure of the change in volume, i.e.,

Push Forward and Pull Back[edit | edit source]
Forward Map:

Forward deformation gradient:

Dyadic notation:

Effect of deformation gradient:
![d\mathbf{x} = \boldsymbol{F}\bullet d\mathbf{X} = \boldsymbol{\varphi}_{*}[d\mathbf{X}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/713a1b1c9ed4308419a4ab4f71798cd1ce9d85c6)
Push Forward operation:
![\boldsymbol{\varphi}_{*}[\bullet]](https://wikimedia.org/api/rest_v1/media/math/render/svg/abf24ff735450b28309aac3d548750c794487a62)
= material vector.
= spatial vector.
Inverse map:

Inverse deformation gradient:

Dyadic notation:

Effect of inverse deformation gradient:
![d\mathbf{X} = \boldsymbol{F}^{-1}\bullet d\mathbf{x} = \boldsymbol{\varphi}^{*}[d\mathbf{x}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/338931a885b45a4e2085dca70d44a4640270f165)
Pull Back operation:
![\boldsymbol{\varphi}^{*}[\bullet]](https://wikimedia.org/api/rest_v1/media/math/render/svg/eafa2b8d8f39890a595387e5b697cabc27cf8623)
= material vector.
= spatial vector.
Push forward and pull back
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Motion:

Deformation Gradient:

Inverse Deformation Gradient:

Push Forward:
![\begin{align}
\boldsymbol{\varphi}_{*}[\boldsymbol{E}_1] & = \mathbf{F}\begin{bmatrix} 1 \\ 0 \end{bmatrix}
= \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\
\boldsymbol{\varphi}_{*}[\boldsymbol{E}_2] & = \mathbf{F}\begin{bmatrix} 0 \\ 1 \end{bmatrix}
= \begin{bmatrix} 1.5 \\ 1.5 \end{bmatrix}
\end{align}](https://wikimedia.org/api/rest_v1/media/math/render/svg/652db2dcb31ef3607763298cdab252d58a0be2a6)
Pull Back:
![\begin{align}
\boldsymbol{\varphi}^{*}[\mathbf{e}_1] & = \mathbf{F}^{-1}\begin{bmatrix} 1 \\ 0 \end{bmatrix}
= \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\
\boldsymbol{\varphi}^{*}[\mathbf{e}_2] & = \mathbf{F}^{-1}\begin{bmatrix} 0 \\ 1 \end{bmatrix}
= \begin{bmatrix} -1 \\ 2/3 \end{bmatrix}
\end{align}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f07d9e3ad19c9b64b119f090c564767fe1ea7dd1)
Cauchy-Green Deformation Tensors[edit | edit source]
Right Cauchy-Green Deformation Tensor[edit | edit source]
Recall:

Therefore,

Using index notation:

Right Cauchy-Green tensor:

Left Cauchy-Green Deformation Tensor[edit | edit source]
Recall:

Therefore,

Using index notation:

Left Cauchy-Green (Finger) tensor:


Green strain tensor:
![\begin{align}
\boldsymbol{E} & = \frac{1}{2}(\boldsymbol{C} - \boldsymbol{I}) \\
& = \frac{1}{2}(\boldsymbol{F}^T\bullet\boldsymbol{F} - \boldsymbol{I}) \\
& = \frac{1}{2}\left[\boldsymbol{\nabla}_o \mathbf{u} + (\boldsymbol{\nabla}_o \mathbf{u})^T
+ \boldsymbol{\nabla}_o \mathbf{u}\bullet(\boldsymbol{\nabla_o \mathbf{u})^T}\right]
\end{align}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d8cf60e20678a55257c8ad49406f81d80b786d)
Index notation:


Almansi strain tensor:

Index notation:

Push Forward and Pull Back[edit | edit source]
Recall:

Now,

Therefore,

Push Forward:
![\mathbf{e} = \boldsymbol{\varphi}_{*}[\boldsymbol{E}] =\boldsymbol{F}^{-T}\bullet\boldsymbol{E}\bullet\boldsymbol{F}^{-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/343ab8184fc12658afc4dd85752b5b107f10e0ab)
Pull Back:
![\boldsymbol{E} = \boldsymbol{\varphi}^{*}[\mathbf{e}] =\boldsymbol{F}^T\bullet\mathbf{e}\bullet\boldsymbol{F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc21be32672cb75ab1991dfca79c183ee6ecda0e)
Derivative of J with respect to the deformation gradient[edit | edit source]
We often need to compute the derivative of
with respect to the deformation gradient
. From tensor calculus we have, for any second order tensor

Therefore,

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Derivative of J with respect to the right Cauchy-Green deformation tensor[edit | edit source]
The derivative of J with respect to the right Cauchy-Green deformation tensor (
) is also often encountered in continuum mechanics.
To calculate the derivative of
with respect to
, we recall that (for any second order tensor
)

Also,
![\frac{\partial J}{\partial \boldsymbol{F}}:\boldsymbol{T} = \frac{\partial J}{\partial \boldsymbol{C}}:(\frac{\partial \boldsymbol{C}}{\partial \boldsymbol{F}}:\boldsymbol{T})
= \frac{\partial J}{\partial \boldsymbol{C}}:(\boldsymbol{T}^T\cdot\boldsymbol{F} + \boldsymbol{F}^T\cdot\boldsymbol{T})
= \left[\boldsymbol{F}\cdot\frac{\partial J}{\partial \boldsymbol{C}}\right]:\boldsymbol{T} +
\left[\boldsymbol{F}\cdot\left(\frac{\partial J}{\partial \boldsymbol{C}}\right)^T\right]:\boldsymbol{T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1729186785d49848ea9d533ddd8ed2ea4a59fc5e)
From the symmetry of
we have

Therefore, involving the arbitrariness of
, we have

Hence,

Also recall that

Therefore,

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In index notation,

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Derivative of the inverse of the right Cauchy-Green tensor[edit | edit source]
Another result that is often useful is that for the derivative of the inverse of the right Cauchy-Green tensor (
).
Recall that, for a second order tensor
,

In index notation

or,

Using this formula and noting that since
is a symmetric second order tensor, the derivative of its inverse is a symmetric fourth order tensor we have

|