Английская Википедия:Hyperelastic material

Stress–strain curves for various hyperelastic material models.

Шаблон:Continuum mechanics A hyperelastic or Green elastic material^[1] is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density function. The hyperelastic material is a special case of a Cauchy elastic material.

For many materials, linear elastic models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose stress-strain relationship can be defined as non-linearly elastic, isotropic and incompressible. Hyperelasticity provides a means of modeling the stress–strain behavior of such materials.^[2] The behavior of unfilled, vulcanized elastomers often conforms closely to the hyperelastic ideal. Filled elastomers and biological tissues^[3]^[4] are also often modeled via the hyperelastic idealization.

Ronald Rivlin and Melvin Mooney developed the first hyperelastic models, the Neo-Hookean and Mooney–Rivlin solids. Many other hyperelastic models have since been developed. Other widely used hyperelastic material models include the Ogden model and the Arruda–Boyce model.

Hyperelastic material models

Saint Venant–Kirchhoff model

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 <math display="block">\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}</math> where <math>\mathbin{:}</math> is tensor contraction, <math>\boldsymbol{S}</math> is the second Piola–Kirchhoff stress, <math>\boldsymbol{C} : \R^{3 \times 3} \to \R^{3 \times 3}</math> is a fourth order stiffness tensor and <math>\boldsymbol{E}</math> is the Lagrangian Green strain given by <math display="block">\mathbf E =\frac{1}{2}\left[ (\nabla_{\mathbf X}\mathbf u)^\textsf{T} + \nabla_{\mathbf X}\mathbf u + (\nabla_{\mathbf X}\mathbf u)^\textsf{T} \cdot\nabla_{\mathbf X}\mathbf u\right]\,\!</math> <math>\lambda</math> and <math>\mu</math> are the Lamé constants, and <math>\boldsymbol{\mathit{I}}</math> is the second order unit tensor.

The strain-energy density function for the Saint Venant–Kirchhoff model is <math display="block">W(\boldsymbol{E}) = \frac{\lambda}{2}[\text{tr}(\boldsymbol{E})]^2 + \mu \text{tr}\mathord\left(\boldsymbol{E}^2\right)</math>

and the second Piola–Kirchhoff stress can be derived from the relation <math display="block"> \boldsymbol{S} = \frac{\partial W}{\partial \boldsymbol{E}} ~. </math>

Classification of hyperelastic material models

Hyperelastic material models can be classified as:

phenomenological descriptions of observed behavior
- Fung
- Mooney–Rivlin
- Ogden
- Polynomial
- Saint Venant–Kirchhoff
- Yeoh
- Marlow
mechanistic models deriving from arguments about underlying structure of the material
- Arruda–Boyce model^[5]
- Neo–Hookean model^[1]
- Buche–Silberstein model^[6]
hybrids of phenomenological and mechanistic models
- Gent
- 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 <math>(\lambda_1, \lambda_2, \lambda_3)</math>: <math display="block">

W = f(\lambda_1) + f(\lambda_2) + f(\lambda_3) \,.
</math>

Stress–strain relations

Compressible hyperelastic materials

First Piola–Kirchhoff stress

If <math>W(\boldsymbol{F})</math> is the strain energy density function, the 1st Piola–Kirchhoff stress tensor can be calculated for a hyperelastic material as <math display="block">

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

</math> where <math>\boldsymbol{F}</math> is the deformation gradient. In terms of the Lagrangian Green strain (<math>\boldsymbol{E}</math>) <math display="block">

\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}} ~.
</math>

In terms of the right Cauchy–Green deformation tensor (<math>\boldsymbol{C}</math>) <math display="block"> \boldsymbol{P} = 2~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}} \qquad \text{or} \qquad P_{iK} = 2~F_{iL}~\frac{\partial W}{\partial C_{LK}} ~. </math>

Second Piola–Kirchhoff stress

If <math>\boldsymbol{S}</math> is the second Piola–Kirchhoff stress tensor then <math display="block">

\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}} ~.
</math>

In terms of the Lagrangian Green strain <math display="block">

\boldsymbol{S} = \frac{\partial W}{\partial \boldsymbol{E}} \qquad \text{or} \qquad
S_{IJ} = \frac{\partial W}{\partial E_{IJ}} ~.
</math>

In terms of the right Cauchy–Green deformation tensor <math display="block">

\boldsymbol{S} = 2~\frac{\partial W}{\partial \boldsymbol{C}} \qquad \text{or} \qquad
S_{IJ} = 2~\frac{\partial W}{\partial C_{IJ}} ~.
</math>

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

Cauchy stress

Similarly, the Cauchy stress is given by <math display="block">

\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} ~.
</math>

In terms of the Lagrangian Green strain <math display="block">

\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} ~.
</math>

In terms of the right Cauchy–Green deformation tensor <math display="block">

\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} ~.
</math>

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:^[7] <math display="block">

\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}} ~.
</math>

Incompressible hyperelastic materials

For an incompressible material <math>J := \det\boldsymbol{F} = 1</math>. The incompressibility constraint is therefore <math>J-1= 0</math>. To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form: <math display="block">W = W(\boldsymbol{F}) - p~(J-1)</math> where the hydrostatic pressure <math>p</math> functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola–Kirchhoff stress now becomes <math display="block"> \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}} ~.

</math> 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 <math display="block"> \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} ~.

</math>

Expressions for the Cauchy stress

Compressible isotropic hyperelastic materials

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 <math display="block">W(\boldsymbol{F})=\hat{W}(I_1,I_2,I_3) = \bar{W}(\bar{I}_1,\bar{I}_2, J) = \tilde{W}(\lambda_1,\lambda_2, \lambda_3),</math> then <math display="block">\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} </math> (See the page on the left Cauchy–Green deformation tensor for the definitions of these symbols).

Шаблон:Math proof </math> where <math>\boldsymbol{C} = \boldsymbol{F}^T\cdot\boldsymbol{F}</math> is the right Cauchy–Green deformation tensor and <math>\boldsymbol{F}</math> is the deformation gradient. The Cauchy stress is given by <math display="block">

\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
</math>

where <math>J = \det\boldsymbol{F}</math>. Let <math>I_1, I_2, I_3</math> be the three principal invariants of <math>\boldsymbol{C}</math>. Then <math display="block">

\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}} ~.
</math>

The derivatives of the invariants of the symmetric tensor <math>\boldsymbol{C}</math> are <math display="block">

\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}

</math> Therefore, we can write <math display="block">

\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} ~.
</math>

Plugging into the expression for the Cauchy stress gives <math display="block">

\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]

</math>

Using the left Cauchy–Green deformation tensor <math>\boldsymbol{B}=\boldsymbol{F}\cdot\boldsymbol{F}^T</math> and noting that <math>I_3 = J^2</math>, we can write <math display="block">

\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}}~.
</math>

For an incompressible material <math>I_3 = 1</math> and hence <math>W = W(I_1,I_2)</math>.Then <math display="block">

\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})
</math>

Therefore, the Cauchy stress is given by <math display="block">

\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}}~.
</math>

where <math>p</math> is an undetermined pressure which acts as a Lagrange multiplier to enforce the incompressibility constraint.

If, in addition, <math>I_1 = I_2</math>, we have <math> W = W(I_1) </math> and hence <math display="block">

\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}}
</math>

In that case the Cauchy stress can be expressed as <math display="block"> \boldsymbol{\sigma} = 2\frac{\partial W}{\partial I_1}~\boldsymbol{B} - p~\boldsymbol{\mathit{1}}~. </math> }}

Шаблон:Math proof:=J^{-1/3}\boldsymbol{F}</math>, 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 <math>\bar{\boldsymbol{B}} := \bar{\boldsymbol{F}}\cdot\bar{\boldsymbol{F}}^T=J^{-2/3}\boldsymbol{B}</math>. The invariants of <math>\bar{\boldsymbol{B}}</math> are <math display="block">\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}</math> 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 <math>J</math> into the fray to describe the volumetric behaviour.

To express the Cauchy stress in terms of the invariants <math>\bar{I}_1, \bar{I}_2, J</math> recall that <math display="block">

\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} ~.
</math>

The chain rule of differentiation gives us <math display="block">\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} </math> Recall that the Cauchy stress is given by <math display="block">

\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}}~.
</math>

In terms of the invariants <math>\bar{I}_1, \bar{I}_2, J</math> we have <math display="block">

\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}}~.
</math>

Plugging in the expressions for the derivatives of <math>W</math> in terms of <math>\bar{I}_1, \bar{I}_2, J</math>, we have <math display="block">\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}</math> or, <math display="block"> \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} </math> In terms of the deviatoric part of <math>\boldsymbol{B}</math>, we can write <math display="block">\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}
</math>

For an incompressible material <math>J = 1</math> and hence <math>W = W(\bar{I}_1,\bar{I}_2)</math>.Then the Cauchy stress is given by <math display="block">

\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}}~.
</math>

where <math>p</math> is an undetermined pressure-like Lagrange multiplier term. In addition, if <math>\bar{I}_1 = \bar{I}_2</math>, we have <math> W = W(\bar{I}_1) </math> and hence the Cauchy stress can be expressed as <math display="block"> \boldsymbol{\sigma} = 2\frac{\partial W}{\partial \bar{I}_1}~\bar{\boldsymbol{B}} - p~\boldsymbol{\mathit{1}}~. </math> }}

Шаблон:Math proof = \frac{1}{2\lambda_i}~\boldsymbol{R}^T\cdot(\mathbf{n}_i\otimes\mathbf{n}_i)\cdot\boldsymbol{R}~;~~

i = 1,2,3 ~.
</math>

The chain rule gives <math display="block">\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}
</math>

The Cauchy stress is given by <math display="block">

\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})
</math>

Plugging in the expression for the derivative of <math>W</math> leads to <math display="block">

\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}
</math>

Using the spectral decomposition of <math>\boldsymbol{V}</math> we have <math display="block">

\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.
</math>

Also note that <math display="block"> J = \det(\boldsymbol{F}) = \det(\boldsymbol{V})\det(\boldsymbol{R}) = \det(\boldsymbol{V}) = \lambda_1 \lambda_2 \lambda_3 ~. </math> Therefore, the expression for the Cauchy stress can be written as <math display="block">

\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]
</math>

For an incompressible material <math>\lambda_1\lambda_2\lambda_3 = 1</math> and hence <math>W = W(\lambda_1,\lambda_2)</math>. Following Ogden^[1] p. 485, we may write <math display="block">

\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}}~
</math>

Some care is required at this stage because, when an eigenvalue is repeated, it is in general only Gateaux differentiable, but not Fréchet differentiable.^[8]^[9] A rigorous tensor derivative can only be found by solving another eigenvalue problem.

If we express the stress in terms of differences between components, <math display="block">

\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}
</math>

If in addition to incompressibility we have <math>\lambda_1 = \lambda_2</math> then a possible solution to the problem requires <math>\sigma_{11} = \sigma_{22}</math> and we can write the stress differences as <math display="block"> \sigma_{11} - \sigma_{33} = \sigma_{22} - \sigma_{33} = \lambda_1~\frac{\partial W}{\partial \lambda_1} - \lambda_3~\frac{\partial W}{\partial \lambda_3} </math> }}

Incompressible isotropic hyperelastic materials

For incompressible isotropic hyperelastic materials, the strain energy density function is <math>W(\boldsymbol{F})=\hat{W}(I_1,I_2)</math>. The Cauchy stress is then given by <math display="block">\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} </math> where <math>p</math> is an undetermined pressure. In terms of stress differences <math display="block">

\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}
</math>

If in addition <math>I_1 = I_2</math>, then <math display="block"> \boldsymbol{\sigma} = 2\frac{\partial W}{\partial I_1}~\boldsymbol{B} - p~\boldsymbol{\mathit{1}}~. </math> If <math>\lambda_1 = \lambda_2</math>, then <math display="block"> \sigma_{11} - \sigma_{33} = \sigma_{22} - \sigma_{33} = \lambda_1~\frac{\partial W}{\partial \lambda_1} - \lambda_3~\frac{\partial W}{\partial \lambda_3} </math>

Consistency with linear elasticity

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.

Consistency conditions for isotropic hyperelastic models

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: <math display="block"> \boldsymbol{\sigma} = \lambda~\mathrm{tr}(\boldsymbol{\varepsilon})~\boldsymbol{\mathit{1}} + 2\mu\boldsymbol{\varepsilon} </math> where <math>\lambda, \mu</math> are the Lamé constants. The strain energy density function that corresponds to the above relation is^[1] <math display="block"> W = \tfrac{1}{2}\lambda~[\mathrm{tr}(\boldsymbol{\varepsilon})]^2 + \mu~\mathrm{tr}\mathord\left(\boldsymbol{\varepsilon}^2\right) </math> For an incompressible material <math>\mathrm{tr}(\boldsymbol{\varepsilon}) = 0</math> and we have <math display="block"> W = \mu~\mathrm{tr}\mathord\left(\boldsymbol{\varepsilon}^2\right) </math> For any strain energy density function <math>W(\lambda_1,\lambda_2,\lambda_3)</math> to reduce to the above forms for small strains the following conditions have to be met^[1] <math display="block">\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} </math>

If the material is incompressible, then the above conditions may be expressed in the following form. <math display="block">\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} </math> These conditions can be used to find relations between the parameters of a given hyperelastic model and shear and bulk moduli.

Consistency conditions for incompressible Шаблон:Math based rubber materials

Many elastomers are modeled adequately by a strain energy density function that depends only on <math>I_1</math>. For such materials we have <math> W = W(I_1) </math>. The consistency conditions for incompressible materials for <math>I_1 = 3, \lambda_i = \lambda_j = 1</math> may then be expressed as <math display="block"> \left.W(I_1)\right|_{I_1=3} = 0 \quad \text{and} \quad \left.\frac{\partial W}{\partial I_1}\right|_{I_1=3} = \frac{\mu}{2} \,. </math> The second consistency condition above can be derived by noting that <math display="block">

\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}\,.
</math>

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

References

Шаблон:Reflist

Партнерские ресурсы
Криптовалюты	Обмен криптовалют - www.bestchange.ru Криптовалютная биржа CoinEx Криптовалютная биржа Binance HIVE OS - операционная система для майнинга e4pool - Мультивалютный пул для майнинга.
Магазины	AliExpress — глобальная виртуальная (в Интернете) торговая площадка, предоставляющая возможность покупать товары производителей из КНР; computeruniverse.net - Интернет-магазин компьютеров(Промо код 5 Евро на первую покупку:FWWC3ZKQ);
Хостинг	DigitalOcean - американский провайдер облачных инфраструктур, с главным офисом в Нью-Йорке и с центрами обработки данных по всему миру;
Разное	Викиум - Онлайн-тренажер для мозга Like Центр - Центр поддержки и развития предпринимательства. Gamersbay - лучший магазин по бустингу для World of Warcraft. Ноотропы OmniMind N°1 - Усиливает мозговую активность. Повышает мотивацию. Улучшает память. Санкт-Петербургская школа телевидения - это федеральная сеть образовательных центров, которая имеет филиалы в 37 городах России. Lingualeo.com — интерактивный онлайн-сервис для изучения и практики английского языка в увлекательной игровой форме. Junyschool (Джунискул) – международная школа программирования и дизайна для детей и подростков от 5 до 17 лет, где ученики осваивают компьютерную грамотность, развивают алгоритмическое и креативное мышление, изучают основы программирования и компьютерной графики, создают собственные проекты: игры, сайты, программы, приложения, анимации, 3D-модели, монтируют видео. Умназия - Интерактивные онлайн-курсы и тренажеры для развития мышления детей 6-13 лет SkillBox - это один из лидеров российского рынка онлайн-образования. Среди партнеров Skillbox ведущий разработчик сервисного дизайна AIC, медиа-компания Yoola, первое и самое крупное русскоязычное аналитическое агентство Tagline, онлайн-школа дизайна и иллюстрации Bang! Bang! Education, оператор PR-рынка PACO, студия рисования Draw&Go, агентство performance-маркетинга Ingate, scrum-студия Sibirix, имидж-лаборатория Персона. «Нетология» — это университет по подготовке и дополнительному обучению специалистов в области интернет-маркетинга, управления проектами и продуктами, дизайна, Data Science и разработки. В рамках Нетологии студенты получают ценные теоретические знания от лучших экспертов Рунета, выполняют практические задания на отработку полученных навыков, общаются с экспертами и единомышленниками. Познакомиться со всеми продуктами подробнее можно на сайте https://netology.ru, линейка курсов и профессий постоянно обновляется. StudyBay Brazil – это онлайн биржа для португалоговорящих студентов и авторов! Студент получает уникальную работу любого уровня сложности и больше свободного времени, в то время как у автора появляется дополнительный заработок и бесценный опыт. Автор24 — самая большая в России площадка по написанию учебных работ: контрольные и курсовые работы, дипломы, рефераты, решение задач, отчеты по практике, а так же любой другой вид работы. Сервис сотрудничает с более 70 000 авторов. Более 1 000 000 работ уже выполнено. StudyBay – это онлайн биржа для англоязычных студентов и авторов! Студент получает уникальную работу любого уровня сложности и больше свободного времени, в то время как у автора появляется дополнительный заработок и бесценный опыт.