Cauchy stress in terms of deformation tensors
Compressible neo-Hookean material
For a compressible Rivlin neo-Hookean material the Cauchy stress is given by

where
is the left Cauchy-Green deformation tensor, and

For infinitesimal strains (
)

and the Cauchy stress can be expressed as

Comparison with Hooke's law shows that
and
.
Proof:
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The Cauchy stress in a compressible hyperelastic material is given by
![{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}\left[{\cfrac {1}{J^{2/3}}}\left({\cfrac {\partial {W}}{\partial {\bar {I}}_{1}}}+{\bar {I}}_{1}~{\cfrac {\partial {W}}{\partial {\bar {I}}_{2}}}\right){\boldsymbol {B}}-{\cfrac {1}{J^{4/3}}}~{\cfrac {\partial {W}}{\partial {\bar {I}}_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]+\left[{\cfrac {\partial {W}}{\partial J}}-{\cfrac {2}{3J}}\left({\bar {I}}_{1}~{\cfrac {\partial {W}}{\partial {\bar {I}}_{1}}}+2~{\bar {I}}_{2}~{\cfrac {\partial {W}}{\partial {\bar {I}}_{2}}}\right)\right]~{\boldsymbol {I}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a02006f31ec08f5ff11d32479fcc80ae15ffb0ea)
For a compressible Rivlin neo-Hookean material,

while, for a compressible Ogden neo-Hookean material,

Therefore, the Cauchy stress in a compressible Rivlin neo-Hookean material is given by
![{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}\left[{\cfrac {1}{J^{2/3}}}~C_{1}~{\boldsymbol {B}}\right]+\left[2D_{1}(J-1)-{\cfrac {2}{3J}}~C_{1}{\bar {I}}_{1}\right]{\boldsymbol {I}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/120f858b142ddd9ee4ad00a67d5fb3bb9b99c1ae)
while that for the corresponding Ogden material is
![{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}\left[{\cfrac {1}{J^{2/3}}}~C_{1}~{\boldsymbol {B}}\right]+\left[2D_{1}(J-1)-{\cfrac {2C_{1}}{J}}-{\cfrac {2}{3J}}~C_{1}{\bar {I}}_{1}\right]{\boldsymbol {I}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83d1614008549ca9e50b92d17563b2f0c2bcf4ac)
If the isochoric part of the left Cauchy-Green deformation tensor is defined as , then we can write the Rivlin neo-Heooken stress as
![{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2C_{1}}{J}}\left[{\bar {\boldsymbol {B}}}-{\tfrac {1}{3}}{\bar {I}}_{1}{\boldsymbol {I}}\right]+2D_{1}(J-1){\boldsymbol {I}}={\cfrac {2C_{1}}{J}}\mathrm {dev} ({\bar {\boldsymbol {B}}})+2D_{1}(J-1){\boldsymbol {I}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf8ea695c3fea473897717d472eafdb88642bce)
and the Ogden neo-Hookean stress as
![{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2C_{1}}{J}}\left[{\bar {\boldsymbol {B}}}-{\tfrac {1}{3}}{\bar {I}}_{1}{\boldsymbol {I}}-{\boldsymbol {I}}\right]+2D_{1}(J-1){\boldsymbol {I}}={\cfrac {2C_{1}}{J}}\left[\mathrm {dev} ({\bar {\boldsymbol {B}}})-{\boldsymbol {I}}\right]+2D_{1}(J-1){\boldsymbol {I}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd3215344e359c7134fae8bfe9c0988769c1b56c)
The quantities

have the form of pressures and are usually treated as such. The Rivlin neo-Hookean stress can then be expressed in the form

while the Ogden neo-Hookean stress has the form

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Incompressible neo-Hookean material
For an incompressible neo-Hookean material with

where
is an undetermined pressure.
Cauchy stress in terms of principal stretches
Compressible neo-Hookean material
For a compressible neo-Hookean hyperelastic material, the principal components of the Cauchy stress are given by
![\sigma _{{i}}=2C_{1}J^{{-5/3}}\left[\lambda _{i}^{2}-{\cfrac {I_{1}}{3}}\right]+2D_{1}(J-1)~;~~i=1,2,3](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e8671c7cc3313e7f11cd4b42d7d466d7b4bf69)
Therefore, the differences between the principal stresses are

Proof:
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For a compressible hyperelastic material, the principal components of the Cauchy stress are given by

The strain energy density function for a compressible neo Hookean material is
![W=C_{1}({\bar {I}}_{1}-3)+D_{1}(J-1)^{2}=C_{1}\left[J^{{-2/3}}(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2})-3\right]+D_{1}(J-1)^{2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3080a7a8adbd13388160411dcb3347e301d86d0)
Therefore,
![\lambda _{i}{\frac {\partial W}{\partial \lambda _{i}}}=C_{1}\left[-{\frac {2}{3}}J^{{-5/3}}\lambda _{i}{\frac {\partial J}{\partial \lambda _{i}}}(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2})+2J^{{-2/3}}\lambda _{i}^{2}\right]+2D_{1}(J-1)\lambda _{i}{\frac {\partial J}{\partial \lambda _{i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87611cb53239ecb742121e0e9535183c69fc69b5)
Since we have

Hence,
![{\begin{aligned}\lambda _{i}{\frac {\partial W}{\partial \lambda _{i}}}&=C_{1}\left[-{\frac {2}{3}}J^{{-2/3}}(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2})+2J^{{-2/3}}\lambda _{i}^{2}\right]+2D_{1}J(J-1)\\&=2C_{1}J^{{-2/3}}\left[-{\frac {1}{3}}(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2})+\lambda _{i}^{2}\right]+2D_{1}J(J-1)\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c326c274fc12ae0344245cd5c390a34b06a6dd74)
The principal Cauchy stresses are therefore given by
![\sigma _{i}=2C_{1}J^{{-5/3}}\left[\lambda _{i}^{2}-{\cfrac {I_{1}}{3}}\right]+2D_{1}(J-1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ceaecb06369b0b2869a0bb4180f929d238ef28c9)
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Incompressible neo-Hookean material
In terms of the principal stretches, the Cauchy stress differences for an incompressible hyperelastic material are given by

For an incompressible neo-Hookean material,

Therefore,

which gives

Uniaxial extension
Compressible neo-Hookean material
The true stress as a function of uniaxial stretch predicted by a compressible neo-Hookean material for various values of

. The material properties are representative of
natural rubber.
For a compressible material undergoing uniaxial extension, the principal stretches are

Hence, the true (Cauchy) stresses for a compressible neo-Hookean material are given by

The stress differences are given by

If the material is unconstrained we have
. Then

Equating the two expressions for
gives a relation for
as a function of
, i.e.,

or

The above equation can be solved numerically using a Newton-Raphson iterative root finding procedure.
Incompressible neo-Hookean material
Comparison of experimental results (dots) and predictions for
Hooke's law(1), neo-Hookean solid(2) and
Mooney-Rivlin solid models(3)
Under uniaxial extension,
and
. Therefore,

Assuming no traction on the sides,
, so we can write

where
is the engineering strain. This equation is often written in alternative notation as

The equation above is for the true stress (ratio of the elongation force to deformed cross-section). For the engineering stress the equation is:

For small deformations
we will have:

Thus, the equivalent Young's modulus of a neo-Hookean solid in uniaxial extension is
, which is in concordance with linear elasticity (
with
for incompressibility).
Equibiaxial extension
Compressible neo-Hookean material
The true stress as a function of biaxial stretch predicted by a compressible neo-Hookean material for various values of

. The material properties are representative of
natural rubber.
In the case of equibiaxial extension

Therefore,
![{\begin{aligned}\sigma _{{11}}&=2C_{1}\left[{\cfrac {\lambda ^{2}}{J^{{5/3}}}}-{\cfrac {1}{3J}}\left(2\lambda ^{2}+{\cfrac {J^{2}}{\lambda ^{4}}}\right)\right]+2D_{1}(J-1)\\&=\sigma _{{22}}\\\sigma _{{33}}&=2C_{1}\left[{\cfrac {J^{{1/3}}}{\lambda ^{4}}}-{\cfrac {1}{3J}}\left(2\lambda ^{2}+{\cfrac {J^{2}}{\lambda ^{4}}}\right)\right]+2D_{1}(J-1)\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad7fd4f530a9b5f2bc1a10096975392ba12ae0be)
The stress differences are

If the material is in a state of plane stress then
and we have

We also have a relation between
and
:
![2C_{1}\left[{\cfrac {\lambda ^{2}}{J^{{5/3}}}}-{\cfrac {1}{3J}}\left(2\lambda ^{2}+{\cfrac {J^{2}}{\lambda ^{4}}}\right)\right]+2D_{1}(J-1)={\cfrac {2C_{1}}{J^{{5/3}}}}\left(\lambda ^{2}-{\cfrac {J^{2}}{\lambda ^{4}}}\right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/650a293ebe9af900c3775c1f566a4214c33edc38)
or,

This equation can be solved for
using Newton's method.
Incompressible neo-Hookean material
For an incompressible material
and the differences between the principal Cauchy stresses take the form

Under plane stress conditions we have

Pure dilation
For the case of pure dilation

Therefore, the principal Cauchy stresses for a compressible neo-Hookean material are given by

If the material is incompressible then
and the principal stresses can be arbitrary.
The figures below show that extremely high stresses are needed to achieve large triaxial extensions or compressions. Equivalently, relatively small triaxial stretch states can cause very high stresses to develop in a rubber-like material. Note also that the magnitude of the stress is quite sensitive to the bulk modulus but not to the shear modulus.
The true stress as a function of equi-triaxial stretch predicted by a compressible neo-Hookean material for various values of  . The material properties are representative of natural rubber.
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The true stress as a function of J predicted by a compressible neo-Hookean material for various values of  . The material properties are representative of natural rubber.
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Simple shear
For the case of simple shear the deformation gradient in terms of components with respect to a reference basis is of the form

where
is the shear deformation. Therefore the left Cauchy-Green deformation tensor is

Compressible neo-Hookean material
In this case
. Hence,
. Now,

Hence the Cauchy stress is given by

Incompressible neo-Hookean material
Using the relation for the Cauchy stress for an incompressible neo-Hookean material we get

Thus neo-Hookean solid shows linear dependence of shear stresses upon shear deformation and quadratic dependence of the normal stress difference on the shear deformation. Note that the expressions for the Cauchy stress for a compressible and an incompressible neo-Hookean material in simple shear represent the same quantity and provide a means of determining the unknown pressure
.