Hyperelastic material#First Piola.E2.80.93Kirchhoff stress

The simplest hyperelastic material model is the Saint Venant–Kirchhoff model which is just an extension of the geometrically linear elastic material model to the geometrically nonlinear regime. This model has the general form and the isotropic form respectively

$$\backslash begin\{align\}$$

\boldsymbol{S} &= \boldsymbol{C} : \boldsymbol{E} \\

\boldsymbol{S} &= \lambda~ \text{tr}(\boldsymbol{E})\boldsymbol{\mathit{I}} + 2\mu\boldsymbol{E} \text{.}

\end{align}

where $\backslash mathbin\{:\}$ is tensor contraction, $\backslash boldsymbol\{S\}$ is the second Piola–Kirchhoff stress, $\backslash boldsymbol\{C\}\; :\; \backslash R^\{3\; \backslash times\; 3\}\; \backslash to\; \backslash R^\{3\; \backslash times\; 3\}$ is a fourth order stiffness tensor and $\backslash boldsymbol\{E\}$ is the Lagrangian Green strain given by

$$\backslash mathbf\; E\; =\backslash frac\{1\}\{2\}\backslash left[\; (\backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u)^\backslash textsf\{T\}\; +\; \backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u\; +\; (\backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u)^\backslash textsf\{T\}\; \backslash cdot\backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u\backslash right]\backslash ,\backslash !$$

$\backslash lambda$ and $\backslash mu$ are the Lamé constants, and $\backslash boldsymbol\{\backslash mathit\{I\}\}$ is the second order unit tensor.

The strain-energy density function for the Saint Venant–Kirchhoff model is

$$W(\backslash boldsymbol\{E\})\; =\; \backslash frac\{\backslash lambda\}\{2\}[\backslash text\{tr\}(\backslash boldsymbol\{E\})]^2\; +\; \backslash mu\; \backslash text\{tr\}\backslash mathord\backslash left(\backslash boldsymbol\{E\}^2\backslash right)$$

and the second Piola–Kirchhoff stress can be derived from the relation

$$\backslash boldsymbol\{S\}\; =\; \backslash frac\{\backslash partial\; W\}\{\backslash partial\; \backslash boldsymbol\{E\}\}\; ~.$$

Hyperelastic material models can be classified as:

1. phenomenological descriptions of observed behavior
2. * Fung
3. * Mooney–Rivlin
4. * Ogden
5. * Polynomial
6. * Saint Venant–Kirchhoff
7. * Yeoh
8. * Marlow
9. mechanistic models deriving from arguments about the underlying structure of the material
10. * Arruda–Boyce model{{cite journal|last1=Arruda|first1=E.M.|last2=Boyce|first2=M.C.|title=A three-dimensional model for the large stretch behavior of rubber elastic materials|journal=J. Mech. Phys. Solids|volume=41|pages=389–412|year=1993|doi=10.1016/0022-5096(93)90013-6|s2cid=136924401 |url=https://hal.archives-ouvertes.fr/hal-01390807/file/MACB.pdf }}
11. * Neo–Hookean model
12. * Buche–Silberstein model{{cite journal|last1=Buche|first1=M.R.|last2=Silberstein|first2=M.N.|title=Statistical mechanical constitutive theory of polymer networks: The inextricable links between distribution, behavior, and ensemble|journal=Phys. Rev. E|volume=102|pages=012501|year=2020|issue=1 |doi=10.1103/PhysRevE.102.012501|pmid=32794915 |arxiv=2004.07874 |bibcode=2020PhRvE.102a2501B |s2cid=215814600 }}
13. hybrids of phenomenological and mechanistic models
14. * Gent
15. * Van der Waals

Generally, a hyperelastic model should satisfy the Drucker stability criterion.

Some hyperelastic models satisfy the Valanis-Landel hypothesis which states that the strain energy function can be separated into the sum of separate functions of the principal stretches $(\backslash lambda\_1,\; \backslash lambda\_2,\; \backslash lambda\_3)$:

W = f(\lambda_1) + f(\lambda_2) + f(\lambda_3) \,.

If $W(\backslash boldsymbol\{F\})$ is the strain energy density function, the 1st Piola–Kirchhoff stress tensor can be calculated for a hyperelastic material as

\boldsymbol{P} = \frac{\partial W}{\partial \boldsymbol{F}} \qquad \text{or} \qquad P_{iK} = \frac{\partial W}{\partial F_{iK}}.

where $\backslash boldsymbol\{F\}$ is the deformation gradient. In terms of the Lagrangian Green strain ($\backslash boldsymbol\{E\}$)

\boldsymbol{P} = \boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}} \qquad \text{or} \qquad P_{iK} = F_{iL}~\frac{\partial W}{\partial E_{LK}} ~.

In terms of the right Cauchy–Green deformation tensor ($\backslash boldsymbol\{C\}$)

$$\backslash boldsymbol\{P\}\; =\; 2~\backslash boldsymbol\{F\}\backslash cdot\backslash frac\{\backslash partial\; W\}\{\backslash partial\; \backslash boldsymbol\{C\}\}\; \backslash qquad\; \backslash text\{or\}\; \backslash qquad\; P\_\{iK\}\; =\; 2~F\_\{iL\}~\backslash frac\{\backslash partial\; W\}\{\backslash partial\; C\_\{LK\}\}\; ~.$$

If $\backslash boldsymbol\{S\}$ is the second Piola–Kirchhoff stress tensor then

\boldsymbol{S} = \boldsymbol{F}^{-1}\cdot\frac{\partial W}{\partial \boldsymbol{F}} \qquad \text{or} \qquad S_{IJ} = F^{-1}_{Ik}\frac{\partial W}{\partial F_{kJ}} ~.

In terms of the Lagrangian Green strain

\boldsymbol{S} = \frac{\partial W}{\partial \boldsymbol{E}} \qquad \text{or} \qquad

S_{IJ} = \frac{\partial W}{\partial E_{IJ}} ~.

In terms of the right Cauchy–Green deformation tensor

\boldsymbol{S} = 2~\frac{\partial W}{\partial \boldsymbol{C}} \qquad \text{or} \qquad

S_{IJ} = 2~\frac{\partial W}{\partial C_{IJ}} ~.

The above relation is also known as the Doyle-Ericksen formula in the material configuration.

Similarly, the Cauchy stress is given by

\boldsymbol{\sigma} = \frac{1}{J}~ \frac{\partial W}{\partial \boldsymbol{F}}\cdot\boldsymbol{F}^\textsf{T} ~;~~ J := \det\boldsymbol{F} \qquad \text{or} \qquad

\sigma_{ij} = \frac{1}{J}~ \frac{\partial W}{\partial F_{iK}}~F_{jK} ~.

In terms of the Lagrangian Green strain

\boldsymbol{\sigma} = \frac{1}{J}~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}}\cdot\boldsymbol{F}^\textsf{T} \qquad \text{or} \qquad

\sigma_{ij} = \frac{1}{J}~F_{iK}~\frac{\partial W}{\partial E_{KL}}~F_{jL} ~.

In terms of the right Cauchy–Green deformation tensor

\boldsymbol{\sigma} = \frac{2}{J}~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^\textsf{T} \qquad \text{or} \qquad

\sigma_{ij} = \frac{2}{J}~F_{iK}~\frac{\partial W}{\partial C_{KL}}~F_{jL} ~.

The above expressions are valid even for anisotropic media (in which case, the potential function is understood to depend implicitly on reference directional quantities such as initial fiber orientations). In the special case of isotropy, the Cauchy stress can be expressed in terms of the left Cauchy-Green deformation tensor as follows:Y. Basar, 2000, Nonlinear continuum mechanics of solids, Springer, p. 157.

\boldsymbol{\sigma} = \frac{2}{J}\frac{\partial W}{\partial \boldsymbol{B}}\cdot~\boldsymbol{B} \qquad \text{or} \qquad

\sigma_{ij} = \frac{2}{J}~B_{ik}~\frac{\partial W}{\partial B_{kj}} ~.

For an incompressible material $J\; :=\; \backslash det\backslash boldsymbol\{F\}\; =\; 1$. The incompressibility constraint is therefore $J-1=\; 0$. To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:

$$W\; =\; W(\backslash boldsymbol\{F\})\; -\; p~(J-1)$$

where the hydrostatic pressure $p$ functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola–Kirchhoff stress now becomes

\boldsymbol{P}=-p~J\boldsymbol{F}^{-\textsf{T}} + \frac{\partial W}{\partial \boldsymbol{F}}

= -p~\boldsymbol{F}^{-\textsf{T}} + \boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}}

= -p~\boldsymbol{F}^{-\textsf{T}} + 2~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}} ~.

This stress tensor can subsequently be converted into any of the other conventional stress tensors, such as the Cauchy stress tensor which is given by

\boldsymbol{\sigma}=\boldsymbol{P}\cdot\boldsymbol{F}^\textsf{T} =

-p~\boldsymbol{\mathit{1}} + \frac{\partial W}{\partial \boldsymbol{F}}\cdot\boldsymbol{F}^\textsf{T}

= -p~\boldsymbol{\mathit{1}} + \boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}}\cdot\boldsymbol{F}^\textsf{T}

= -p~\boldsymbol{\mathit{1}} + 2~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^\textsf{T} ~.

For isotropic hyperelastic materials, the Cauchy stress can be expressed in terms of the invariants of the left Cauchy–Green deformation tensor (or right Cauchy–Green deformation tensor). If the strain energy density function is $$W(\backslash boldsymbol\{F\})=\backslash hat\{W\}(I\_1,I\_2,I\_3)\; =\; \backslash bar\{W\}(\backslash bar\{I\}\_1,\backslash bar\{I\}\_2,\; J)\; =\; \backslash tilde\{W\}(\backslash lambda\_1,\backslash lambda\_2,\; \backslash lambda\_3),$$ then

$$\backslash begin\{align\}$$

\boldsymbol{\sigma} & =

\frac{2}{\sqrt{I_3}}\left[\left(\frac{\partial\hat{W}}{\partial I_1} + I_1~\frac{\partial\hat{W}}{\partial I_2}\right)\boldsymbol{B} - \frac{\partial\hat{W}}{\partial I_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] + 2\sqrt{I_3}~\frac{\partial\hat{W}}{\partial I_3}~\boldsymbol{\mathit{1}} \\[5pt]

& = \frac{2}{J}\left[\frac{1}{J^{2/3}}\left(\frac{\partial\bar{W}}{\partial \bar{I}_1} + \bar{I}_1~\frac{\partial\bar{W}}{\partial \bar{I}_2}\right)\boldsymbol{B} -

\frac{1}{J^{4/3}}~\frac{\partial\bar{W}}{\partial \bar{I}_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right]

+ \left[\frac{\partial\bar{W}}{\partial J} - \frac{2}{3J} \left(\bar{I}_1~\frac{\partial\bar{W}}{\partial \bar{I}_1} + 2~\bar{I}_2~\frac{\partial\bar{W}}{\partial \bar{I}_2}\right)\right] ~\boldsymbol{\mathit{1}} \\[5pt]

& = \frac{2}{J} \left[\left(\frac{\partial\bar{W}}{\partial \bar{I}_1} + \bar{I}_1~\frac{\partial\bar{W}}{\partial \bar{I}_2}\right)\bar{\boldsymbol{B}} -

\frac{\partial\bar{W}}{\partial \bar{I}_2}~\bar{\boldsymbol{B}} \cdot\bar{\boldsymbol{B}} \right] + \left[\frac{\partial\bar{W}}{\partial J} - \frac{2}{3J}\left(\bar{I}_1~\frac{\partial\bar{W}}{\partial \bar{I}_1} + 2~\bar{I}_2~\frac{\partial\bar{W}}{\partial \bar{I}_2}\right)\right] ~\boldsymbol{\mathit{1}} \\[5pt]

& = \frac{\lambda_1}{\lambda_1\lambda_2\lambda_3}~\frac{\partial\tilde{W}}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 + \frac{\lambda_2}{\lambda_1\lambda_2\lambda_3}~\frac{\partial\tilde{W}}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \frac{\lambda_3}{\lambda_1\lambda_2\lambda_3}~\frac{\partial\tilde{W}}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3

\end{align}

(See the page on the left Cauchy–Green deformation tensor for the definitions of these symbols).

{{math proof

| title = Proof 1

| proof = The second Piola–Kirchhoff stress tensor for a hyperelastic material is given by

$$\backslash boldsymbol\{S\}\; =\; 2~\backslash frac\{\backslash partial\; W\}\{\backslash partial\; \backslash boldsymbol\{C\}\}$$

where $\backslash boldsymbol\{C\}\; =\; \backslash boldsymbol\{F\}^T\backslash cdot\backslash boldsymbol\{F\}$ is the right Cauchy–Green deformation tensor and $\backslash boldsymbol\{F\}$ is the deformation gradient. The Cauchy stress is given by

\boldsymbol{\sigma} = \frac{1}{J}~\boldsymbol{F}\cdot\boldsymbol{S}\cdot\boldsymbol{F}^T

= \frac{2}{J}~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^T

where $J\; =\; \backslash det\backslash boldsymbol\{F\}$. Let $I\_1,\; I\_2,\; I\_3$ be the three principal invariants of $\backslash boldsymbol\{C\}$. Then

\frac{\partial W}{\partial \boldsymbol{C}} =

\frac{\partial W}{\partial I_1}~\frac{\partial I_1}{\partial \boldsymbol{C}} +

\frac{\partial W}{\partial I_2}~\frac{\partial I_2}{\partial \boldsymbol{C}} +

\frac{\partial W}{\partial I_3}~\frac{\partial I_3}{\partial \boldsymbol{C}} ~.

The derivatives of the invariants of the symmetric tensor $\backslash boldsymbol\{C\}$ are

\frac{\partial I_1}{\partial \boldsymbol{C}} = \boldsymbol{\mathit{1}} ~;~~

\frac{\partial I_2}{\partial \boldsymbol{C}} = I_1~\boldsymbol{\mathit{1}} - \boldsymbol{C} ~;~~

\frac{\partial I_3}{\partial \boldsymbol{C}} = \det(\boldsymbol{C})~\boldsymbol{C}^{-1}

Therefore, we can write

\frac{\partial W}{\partial \boldsymbol{C}} =

\frac{\partial W}{\partial I_1}~\boldsymbol{\mathit{1}} +

\frac{\partial W}{\partial I_2}~(I_1~\boldsymbol{\mathit{1}} - \boldsymbol{F}^T\cdot\boldsymbol{F}) +

\frac{\partial W}{\partial I_3}~I_3~\boldsymbol{F}^{-1}\cdot\boldsymbol{F}^{-T} ~.

Plugging into the expression for the Cauchy stress gives

\boldsymbol{\sigma}

= \frac{2}{J}~\left[\frac{\partial W}{\partial I_1}~\boldsymbol{F}\cdot\boldsymbol{F}^T+

\frac{\partial W}{\partial I_2}~(I_1~\boldsymbol{F}\cdot\boldsymbol{F}^T - \boldsymbol{F}\cdot \boldsymbol{F}^T \cdot \boldsymbol{F}\cdot \boldsymbol{F}^T) +

\frac{\partial W}{\partial I_3}~I_3~\boldsymbol{\mathit{1}}\right]

Using the left Cauchy–Green deformation tensor $\backslash boldsymbol\{B\}=\backslash boldsymbol\{F\}\backslash cdot\backslash boldsymbol\{F\}^T$ and noting that $I\_3\; =\; J^2$, we can write

\boldsymbol{\sigma}

= \frac{2}{\sqrt{I_3}}~\left[\left(\frac{\partial W}{\partial I_1} +

I_1~\frac{\partial W}{\partial I_2}\right)~\boldsymbol{B} -

\frac{\partial W}{\partial I_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] +

2~\sqrt{I_3}~\frac{\partial W}{\partial I_3}~\boldsymbol{\mathit{1}}~.

For an incompressible material $I\_3\; =\; 1$ and hence $W\; =\; W(I\_1,I\_2)$.Then

\frac{\partial W}{\partial \boldsymbol{C}} =

\frac{\partial W}{\partial I_1}~\frac{\partial I_1}{\partial \boldsymbol{C}} +

\frac{\partial W}{\partial I_2}~\frac{\partial I_2}{\partial \boldsymbol{C}}

= \frac{\partial W}{\partial I_1}~\boldsymbol{\mathit{1}} +

\frac{\partial W}{\partial I_2}~(I_1~\boldsymbol{\mathit{1}} - \boldsymbol{F}^T\cdot\boldsymbol{F})

Therefore, the Cauchy stress is given by

\boldsymbol{\sigma}

= 2\left[\left(\frac{\partial W}{\partial I_1} +

I_1~\frac{\partial W}{\partial I_2}\right)~\boldsymbol{B} -

\frac{\partial W}{\partial I_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] - p~\boldsymbol{\mathit{1}}~.

where $p$ is an undetermined pressure which acts as a Lagrange multiplier to enforce the incompressibility constraint.

If, in addition, $I\_1\; =\; I\_2$, we have $W\; =\; W(I\_1)$ and hence

\frac{\partial W}{\partial \boldsymbol{C}} =

\frac{\partial W}{\partial I_1}~\frac{\partial I_1}{\partial \boldsymbol{C}} = \frac{\partial W}{\partial I_1}~\boldsymbol{\mathit{1}}

In that case the Cauchy stress can be expressed as

$$\backslash boldsymbol\{\backslash sigma\}\; =\; 2\backslash frac\{\backslash partial\; W\}\{\backslash partial\; I\_1\}~\backslash boldsymbol\{B\}\; -\; p~\backslash boldsymbol\{\backslash mathit\{1\}\}~.$$

}}

{{math proof

| title = Proof 2

| proof = The isochoric deformation gradient is defined as $\backslash bar\{\backslash boldsymbol\{F\}\}:=J^\{-1/3\}\backslash boldsymbol\{F\}$, resulting in the isochoric deformation gradient having a determinant of 1, in other words it is volume stretch free. Using this one can subsequently define the isochoric left Cauchy–Green deformation tensor $\backslash bar\{\backslash boldsymbol\{B\}\}\; :=\; \backslash bar\{\backslash boldsymbol\{F\}\}\backslash cdot\backslash bar\{\backslash boldsymbol\{F\}\}^T=J^\{-2/3\}\backslash boldsymbol\{B\}$.

The invariants of $\backslash bar\{\backslash boldsymbol\{B\}\}$ are

$$\backslash begin\{align\}$$

\bar I_1 &= \text{tr}(\bar{\boldsymbol{B}}) = J^{-2/3}\text{tr}(\boldsymbol{B}) = J^{-2/3} I_1 \\

\bar I_2 & = \frac{1}{2}\left(\text{tr}(\bar{\boldsymbol{B}})^2 - \text{tr}(\bar{\boldsymbol{B}}^2)\right) =

\frac{1}{2}\left( \left(J^{-2/3}\text{tr}(\boldsymbol{B})\right)^2 - \text{tr}(J^{-4/3}\boldsymbol{B}^2) \right) =

J^{-4/3} I_2 \\

\bar I_3 &= \det(\bar{\boldsymbol{B}}) = J^{-6/3} \det(\boldsymbol{B}) = J^{-2} I_3 = J^{-2} J^2 = 1

\end{align}

The set of invariants which are used to define the distortional behavior are the first two invariants of the isochoric left Cauchy–Green deformation tensor tensor, (which are identical to the ones for the right Cauchy Green stretch tensor), and add $J$ into the fray to describe the volumetric behaviour.

To express the Cauchy stress in terms of the invariants $\backslash bar\{I\}\_1,\; \backslash bar\{I\}\_2,\; J$ recall that

\bar{I}_1 = J^{-2/3}~I_1 = I_3^{-1/3}~I_1 ~;~~

\bar{I}_2 = J^{-4/3}~I_2 = I_3^{-2/3}~I_2 ~;~~ J = I_3^{1/2} ~.

The chain rule of differentiation gives us

$$\backslash begin\{align\}$$

\frac{\partial W}{\partial I_1} & =

\frac{\partial W}{\partial \bar{I}_1}~\frac{\partial \bar{I}_1}{\partial I_1} +

\frac{\partial W}{\partial \bar{I}_2}~\frac{\partial \bar{I}_2}{\partial I_1} +

\frac{\partial W}{\partial J}~\frac{\partial J}{\partial I_1} \\

& = I_3^{-1/3}~\frac{\partial W}{\partial \bar{I}_1}

= J^{-2/3}~\frac{\partial W}{\partial \bar{I}_1} \\

\frac{\partial W}{\partial I_2} & =

\frac{\partial W}{\partial \bar{I}_1}~\frac{\partial \bar{I}_1}{\partial I_2} +

\frac{\partial W}{\partial \bar{I}_2}~\frac{\partial \bar{I}_2}{\partial I_2} +

\frac{\partial W}{\partial J}~\frac{\partial J}{\partial I_2} \\

& = I_3^{-2/3}~\frac{\partial W}{\partial \bar{I}_2}

= J^{-4/3}~\frac{\partial W}{\partial \bar{I}_2} \\

\frac{\partial W}{\partial I_3} & =

\frac{\partial W}{\partial \bar{I}_1}~\frac{\partial \bar{I}_1}{\partial I_3} +

\frac{\partial W}{\partial \bar{I}_2}~\frac{\partial \bar{I}_2}{\partial I_3} +

\frac{\partial W}{\partial J}~\frac{\partial J}{\partial I_3} \\

& = - \frac{1}{3}~I_3^{-4/3}~I_1~\frac{\partial W}{\partial \bar{I}_1}

- \frac{2}{3}~I_3^{-5/3}~I_2~\frac{\partial W}{\partial \bar{I}_2}

+ \frac{1}{2}~I_3^{-1/2}~\frac{\partial W}{\partial J} \\

& = - \frac{1}{3}~J^{-8/3}~J^{2/3}~\bar{I}_1~\frac{\partial W}{\partial \bar{I}_1}

- \frac{2}{3}~J^{-10/3}~J^{4/3}~\bar{I}_2~\frac{\partial W}{\partial \bar{I}_2}

+ \frac{1}{2}~J^{-1}~\frac{\partial W}{\partial J} \\

& = -\frac{1}{3}~J^{-2}~\left(\bar{I}_1~\frac{\partial W}{\partial \bar{I}_1}+

2~\bar{I}_2~\frac{\partial W}{\partial \bar{I}_2}\right) +

\frac{1}{2}~J^{-1}~\frac{\partial W}{\partial J}

\end{align}

Recall that the Cauchy stress is given by

\boldsymbol{\sigma}

= \frac{2}{\sqrt{I_3}}~\left[\left(\frac{\partial W}{\partial I_1} +

I_1~\frac{\partial W}{\partial I_2}\right)~\boldsymbol{B} -

\frac{\partial W}{\partial I_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] +

2~\sqrt{I_3}~\frac{\partial W}{\partial I_3}~\boldsymbol{\mathit{1}}~.

In terms of the invariants $\backslash bar\{I\}\_1,\; \backslash bar\{I\}\_2,\; J$ we have

\boldsymbol{\sigma}

= \frac{2}{J}~\left[\left(\frac{\partial W}{\partial I_1}+

J^{2/3}~\bar{I}_1~\frac{\partial W}{\partial I_2}\right)~\boldsymbol{B} -

\frac{\partial W}{\partial I_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] +

2~J~\frac{\partial W}{\partial I_3}~\boldsymbol{\mathit{1}}~.

Plugging in the expressions for the derivatives of $W$ in terms of $\backslash bar\{I\}\_1,\; \backslash bar\{I\}\_2,\; J$, we have

$$\backslash begin\{align\}$$

\boldsymbol{\sigma}

& = \frac{2}{J}~\left[\left(J^{-2/3}~\frac{\partial W}{\partial \bar{I}_1} +

J^{-2/3}~\bar{I}_1~\frac{\partial W}{\partial \bar{I}_2}\right)~\boldsymbol{B} -

J^{-4/3}~\frac{\partial W}{\partial \bar{I}_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right]

+ \\

& \qquad

2~J~\left[-\frac{1}{3}~J^{-2}~\left(\bar{I}_1~\frac{\partial W}{\partial \bar{I}_1}+

2~\bar{I}_2~\frac{\partial W}{\partial \bar{I}_2}\right) +

\frac{1}{2}~J^{-1}~\frac{\partial W}{\partial J}\right]~\boldsymbol{\mathit{1}}

\end{align}

or,

$$\backslash begin\{align\}$$

\boldsymbol{\sigma}

& = \frac{2}{J}~\left[\frac{1}{J^{2/3}}~\left(\frac{\partial W}{\partial \bar{I}_1} +

\bar{I}_1~\frac{\partial W}{\partial \bar{I}_2}\right)~\boldsymbol{B} -

\frac{1}{J^{4/3}}~

\frac{\partial W}{\partial \bar{I}_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] \\

& \qquad + \left[\frac{\partial W}{\partial J} -

\frac{2}{3J}\left(\bar{I}_1~\frac{\partial W}{\partial \bar{I}_1}+

2~\bar{I}_2~\frac{\partial W}{\partial \bar{I}_2}\right)\right]\boldsymbol{\mathit{1}}

\end{align}

In terms of the deviatoric part of $\backslash boldsymbol\{B\}$, we can write

$$\backslash begin\{align\}$$

\boldsymbol{\sigma}

& = \frac{2}{J}~\left[\left(\frac{\partial W}{\partial \bar{I}_1} +

\bar{I}_1~\frac{\partial W}{\partial \bar{I}_2}\right)~\bar{\boldsymbol{B}} -

\frac{\partial W}{\partial \bar{I}_2}~\bar{\boldsymbol{B}}\cdot\bar{\boldsymbol{B}}\right] \\

& \qquad + \left[\frac{\partial W}{\partial J} -

\frac{2}{3J}\left(\bar{I}_1~\frac{\partial W}{\partial \bar{I}_1}+

2~\bar{I}_2~\frac{\partial W}{\partial \bar{I}_2}\right)\right]\boldsymbol{\mathit{1}}

\end{align}

For an incompressible material $J\; =\; 1$ and hence $W\; =\; W(\backslash bar\{I\}\_1,\backslash bar\{I\}\_2)$.Then

the Cauchy stress is given by

\boldsymbol{\sigma}

= 2\left[\left(\frac{\partial W}{\partial \bar{I}_1} +

I_1~\frac{\partial W}{\partial \bar{I}_2}\right)~\bar{\boldsymbol{B}} -

\frac{\partial W}{\partial \bar{I}_2}~\bar{\boldsymbol{B}}\cdot\bar{\boldsymbol{B}}\right] - p~\boldsymbol{\mathit{1}}~.

where $p$ is an undetermined pressure-like Lagrange multiplier term. In addition, if $\backslash bar\{I\}\_1\; =\; \backslash bar\{I\}\_2$, we have $W\; =\; W(\backslash bar\{I\}\_1)$ and hence

the Cauchy stress can be expressed as

$$\backslash boldsymbol\{\backslash sigma\}\; =\; 2\backslash frac\{\backslash partial\; W\}\{\backslash partial\; \backslash bar\{I\}\_1\}~\backslash bar\{\backslash boldsymbol\{B\}\}\; -\; p~\backslash boldsymbol\{\backslash mathit\{1\}\}~.$$

}}

{{math proof

| title = Proof 3

| proof = To express the Cauchy stress in terms of the stretches $\backslash lambda\_1,\; \backslash lambda\_2,\; \backslash lambda\_3$ recall that

\frac{\partial \lambda_i}{\partial\boldsymbol{C}} = \frac{1}{2\lambda_i}~\boldsymbol{R}^T\cdot(\mathbf{n}_i\otimes\mathbf{n}_i)\cdot\boldsymbol{R}~;~~

i = 1,2,3 ~.

The chain rule gives

$$\backslash begin\{align\}$$

\frac{\partial W}{\partial\boldsymbol{C}} & =

\frac{\partial W}{\partial \lambda_1}~\frac{\partial \lambda_1}{\partial\boldsymbol{C}} +

\frac{\partial W}{\partial \lambda_2}~\frac{\partial \lambda_2}{\partial\boldsymbol{C}} +

\frac{\partial W}{\partial \lambda_3}~\frac{\partial \lambda_3}{\partial\boldsymbol{C}} \\

& = \boldsymbol{R}^T\cdot\left[\frac{1}{2\lambda_1}~\frac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +

\frac{1}{2\lambda_2}~\frac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 +

\frac{1}{2\lambda_3}~\frac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3\right]\cdot\boldsymbol{R}

\end{align}

The Cauchy stress is given by

\boldsymbol{\sigma} = \frac{2}{J}~\boldsymbol{F}\cdot

\frac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^T =

\frac{2}{J}~(\boldsymbol{V}\cdot\boldsymbol{R})\cdot

\frac{\partial W}{\partial \boldsymbol{C}}\cdot(\boldsymbol{R}^T\cdot\boldsymbol{V})

Plugging in the expression for the derivative of $W$ leads to

\boldsymbol{\sigma} =

\frac{2}{J}~\boldsymbol{V}\cdot

\left[\frac{1}{2\lambda_1}~

\frac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +

\frac{1}{2\lambda_2}~

\frac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 +

\frac{1}{2\lambda_3}~

\frac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3\right]

\cdot\boldsymbol{V}

Using the spectral decomposition of $\backslash boldsymbol\{V\}$ we have

\boldsymbol{V}\cdot(\mathbf{n}_i\otimes\mathbf{n}_i)\cdot\boldsymbol{V} =

\lambda_i^2~\mathbf{n}_i\otimes\mathbf{n}_i ~;~~ i=1,2,3.

Also note that

$$J\; =\; \backslash det(\backslash boldsymbol\{F\})\; =\; \backslash det(\backslash boldsymbol\{V\})\backslash det(\backslash boldsymbol\{R\})\; =\; \backslash det(\backslash boldsymbol\{V\})\; =\; \backslash lambda\_1\; \backslash lambda\_2\; \backslash lambda\_3\; ~.$$

Therefore, the expression for the Cauchy stress can be written as

\boldsymbol{\sigma} =

\frac{1}{\lambda_1\lambda_2\lambda_3}~

\left[\lambda_1~\frac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +

\lambda_2~\frac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 +

\lambda_3~\frac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3

\right]

For an incompressible material $\backslash lambda\_1\backslash lambda\_2\backslash lambda\_3\; =\; 1$ and hence $W\; =\; W(\backslash lambda\_1,\backslash lambda\_2)$. Following Ogden p. 485, we may write

\boldsymbol{\sigma} =

\lambda_1~\frac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +

\lambda_2~\frac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 +

\lambda_3~\frac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3

- p~\boldsymbol{\mathit{1}}~

Some care is required at this stage because, when an eigenvalue is repeated, it is in general only Gateaux differentiable, but not Fréchet differentiable.Fox & Kapoor, Rates of change of eigenvalues and eigenvectors, AIAA Journal, 6 (12) 2426–2429 (1968)Friswell MI. The derivatives of repeated eigenvalues and their associated eigenvectors. Journal of Vibration and Acoustics (ASME) 1996; 118:390–397. A rigorous tensor derivative can only be found by solving another eigenvalue problem.

If we express the stress in terms of differences between components,

\sigma_{11} - \sigma_{33} = \lambda_1~\frac{\partial W}{\partial \lambda_1} - \lambda_3~\frac{\partial W}{\partial \lambda_3} ~;~~

\sigma_{22} - \sigma_{33} = \lambda_2~\frac{\partial W}{\partial \lambda_2} - \lambda_3~\frac{\partial W}{\partial \lambda_3}

If in addition to incompressibility we have $\backslash lambda\_1\; =\; \backslash lambda\_2$ then a possible solution to the problem

requires $\backslash sigma\_\{11\}\; =\; \backslash sigma\_\{22\}$ and we can write the stress differences as

$$\backslash sigma\_\{11\}\; -\; \backslash sigma\_\{33\}\; =\; \backslash sigma\_\{22\}\; -\; \backslash sigma\_\{33\}\; =\; \backslash lambda\_1~\backslash frac\{\backslash partial\; W\}\{\backslash partial\; \backslash lambda\_1\}\; -\; \backslash lambda\_3~\backslash frac\{\backslash partial\; W\}\{\backslash partial\; \backslash lambda\_3\}$$

}}

For incompressible isotropic hyperelastic materials, the strain energy density function is $W(\backslash boldsymbol\{F\})=\backslash hat\{W\}(I\_1,I\_2)$. The Cauchy stress is then given by

$$\backslash begin\{align\}$$

\boldsymbol{\sigma} & = -p~\boldsymbol{\mathit{1}} +

2\left[\left(\frac{\partial\hat{W}}{\partial I_1} + I_1~\frac{\partial\hat{W}}{\partial I_2}\right)\boldsymbol{B} - \frac{\partial\hat{W}}{\partial I_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] \\

& = - p~\boldsymbol{\mathit{1}} + 2\left[\left(\frac{\partial W}{\partial \bar{I}_1} +

I_1~\frac{\partial W}{\partial \bar{I}_2}\right)~\bar{\boldsymbol{B}} -

\frac{\partial W}{\partial \bar{I}_2}~\bar{\boldsymbol{B}}\cdot\bar{\boldsymbol{B}}\right] \\

& = - p~\boldsymbol{\mathit{1}} + \lambda_1~\frac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +

\lambda_2~\frac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \lambda_3~\frac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3

\end{align}

where $p$ is an undetermined pressure. In terms of stress differences

\sigma_{11} - \sigma_{33} = \lambda_1~\frac{\partial W}{\partial \lambda_1} - \lambda_3~\frac{\partial W}{\partial \lambda_3}~;~~

\sigma_{22} - \sigma_{33} = \lambda_2~\frac{\partial W}{\partial \lambda_2} - \lambda_3~\frac{\partial W}{\partial \lambda_3}

If in addition $I\_1\; =\; I\_2$, then

$$\backslash boldsymbol\{\backslash sigma\}\; =\; 2\backslash frac\{\backslash partial\; W\}\{\backslash partial\; I\_1\}~\backslash boldsymbol\{B\}\; -\; p~\backslash boldsymbol\{\backslash mathit\{1\}\}~.$$

If $\backslash lambda\_1\; =\; \backslash lambda\_2$, then

$$\backslash sigma\_\{11\}\; -\; \backslash sigma\_\{33\}\; =\; \backslash sigma\_\{22\}\; -\; \backslash sigma\_\{33\}\; =\; \backslash lambda\_1~\backslash frac\{\backslash partial\; W\}\{\backslash partial\; \backslash lambda\_1\}\; -\; \backslash lambda\_3~\backslash frac\{\backslash partial\; W\}\{\backslash partial\; \backslash lambda\_3\}$$

Consistency with linear elasticity is often used to determine some of the parameters of hyperelastic material models. These consistency conditions can be found by comparing Hooke's law with linearized hyperelasticity at small strains.

For isotropic hyperelastic materials to be consistent with isotropic linear elasticity, the stress–strain relation should have the following form in the infinitesimal strain limit:

$$\backslash boldsymbol\{\backslash sigma\}\; =\; \backslash lambda~\backslash mathrm\{tr\}(\backslash boldsymbol\{\backslash varepsilon\})~\backslash boldsymbol\{\backslash mathit\{1\}\}\; +\; 2\backslash mu\backslash boldsymbol\{\backslash varepsilon\}$$

where $\backslash lambda,\; \backslash mu$ are the Lamé constants. The strain energy density function that corresponds to the above relation is

$$W\; =\; \backslash tfrac\{1\}\{2\}\backslash lambda~[\backslash mathrm\{tr\}(\backslash boldsymbol\{\backslash varepsilon\})]^2\; +\; \backslash mu~\backslash mathrm\{tr\}\backslash mathord\backslash left(\backslash boldsymbol\{\backslash varepsilon\}^2\backslash right)$$

For an incompressible material $\backslash mathrm\{tr\}(\backslash boldsymbol\{\backslash varepsilon\})\; =\; 0$ and we have

$$W\; =\; \backslash mu~\backslash mathrm\{tr\}\backslash mathord\backslash left(\backslash boldsymbol\{\backslash varepsilon\}^2\backslash right)$$

For any strain energy density function $W(\backslash lambda\_1,\backslash lambda\_2,\backslash lambda\_3)$ to reduce to the above forms for small strains the following conditions have to be met

$$\backslash begin\{align\}$$

& W(1,1,1) = 0 ~;~~

\frac{\partial W}{\partial \lambda_i}(1,1,1) = 0 \\

& \frac{\partial^2 W}{\partial \lambda_i \partial \lambda_j}(1,1,1) = \lambda + 2\mu\delta_{ij}

\end{align}

If the material is incompressible, then the above conditions may be expressed in the following form.

$$\backslash begin\{align\}$$

& W(1,1,1) = 0 \\

& \frac{\partial W}{\partial \lambda_i}(1,1,1) = \frac{\partial W}{\partial \lambda_j}(1,1,1) ~;~~

\frac{\partial^2 W}{\partial \lambda_i^2}(1,1,1) = \frac{\partial^2 W}{\partial \lambda_j^2}(1,1,1) \\

& \frac{\partial^2 W}{\partial \lambda_i \partial \lambda_j}(1,1,1) = \mathrm{independent of}~i,j\ne i \\

& \frac{\partial^2 W}{\partial \lambda_i^2}(1,1,1) - \frac{\partial^2 W}{\partial \lambda_i \partial \lambda_j}(1,1,1) + \frac{\partial W}{\partial \lambda_i}(1,1,1) = 2\mu ~~(i \ne j)

\end{align}

These conditions can be used to find relations between the parameters of a given hyperelastic model and shear and bulk moduli.

Many elastomers are modeled adequately by a strain energy density function that depends only on $I\_1$. For such materials we have $W\; =\; W(I\_1)$.

The consistency conditions for incompressible materials for $I\_1\; =\; 3,\; \backslash lambda\_i\; =\; \backslash lambda\_j\; =\; 1$ may then be expressed as

$$\backslash left.W(I\_1)\backslash right|\_\{I\_1=3\}\; =\; 0\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash left.\backslash frac\{\backslash partial\; W\}\{\backslash partial\; I\_1\}\backslash right|\_\{I\_1=3\}\; =\; \backslash frac\{\backslash mu\}\{2\}\; \backslash ,.$$

The second consistency condition above can be derived by noting that

\frac{\partial W}{\partial \lambda_i} = \frac{\partial W}{\partial I_1}\frac{\partial I_1}{\partial \lambda_i} = 2\lambda_i\frac{\partial W}{\partial I_1} \quad\text{and}\quad

\frac{\partial^2 W}{\partial \lambda_i \partial \lambda_j} = 2\delta_{ij}\frac{\partial W}{\partial I_1} + 4\lambda_i\lambda_j \frac{\partial^2 W}{\partial I_1^2}\,.

These relations can then be substituted into the consistency condition for isotropic incompressible hyperelastic materials.

{{Reflist}}

• Cauchy elastic material
• Continuum mechanics
• Deformation (mechanics)
• Finite strain theory
• Ogden–Roxburgh model
• Rubber elasticity
• Stress measures
• Stress (mechanics)

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