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A hyperelastic or Green elastic material is an 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, incompressible and generally independent of strain rate. Hyperelasticity provides a means of modeling the stress-strain behavior of such materials. The behavior of unfilled, vulcanized elastomers often conforms closely to the hyperelastic ideal. Filled elastomers and biological tissues are also often modeled via the hyperelastic idealization.

Contents

Stress-strain relations

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First Piola-Kirchhoff stress

If W(\boldsymbol{F}) is the strain energy density function, the 1st Piola-Kirchoff 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 \boldsymbol{F} is the deformation gradient. In terms of the Lagrangian Green strain (\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 (\boldsymbol{C})

 \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}} ~.

Second Piola-Kirchhoff stress

If \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}} ~.

Cauchy stress

Similarly, the Cauchy stress is given by

 \boldsymbol{\sigma} = \cfrac{1}{J}~ \cfrac{\partial W}{\partial \boldsymbol{F}}\cdot\boldsymbol{F}^T ~;~~ J := \det\boldsymbol{F} \qquad \text{or} \qquad \sigma_{ij} = \cfrac{1}{J}~ \cfrac{\partial W}{\partial F_{iK}}~F_{jK} ~.

In terms of the Lagrangian Green strain

 \boldsymbol{\sigma} = \cfrac{1}{J}~\boldsymbol{F}\cdot\cfrac{\partial W}{\partial \boldsymbol{E}}\cdot\boldsymbol{F}^T \qquad \text{or} \qquad \sigma_{ij} = \cfrac{1}{J}~F_{iK}~\cfrac{\partial W}{\partial E_{KL}}~F_{jL} ~.

In terms of the right Cauchy-Green deformation tensor

 \boldsymbol{\sigma} = \cfrac{2}{J}~\boldsymbol{F}\cdot\cfrac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^T \qquad \text{or} \qquad \sigma_{ij} = \cfrac{2}{J}~F_{iK}~\cfrac{\partial W}{\partial C_{KL}}~F_{jL} ~.

Cauchy stress in terms of invariants

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(\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), then

 \begin{align} \boldsymbol{\sigma} & = \cfrac{2}{\sqrt{I_3}}\left[\left(\cfrac{\partial\hat{W}}{\partial I_1} + I_1~\cfrac{\partial\hat{W}}{\partial I_2}\right)\boldsymbol{B} - \cfrac{\partial\hat{W}}{\partial I_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] + 2\sqrt{I_3}~\cfrac{\partial\hat{W}}{\partial I_3}~\boldsymbol{\mathit{1}} \ & = \cfrac{2}{J}\left[\cfrac{1}{J^{2/3}}\left(\cfrac{\partial\bar{W}}{\partial \bar{I}_1} + \bar{I}_1~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}\right)\boldsymbol{B} - \cfrac{1}{3}\left(\bar{I}_1~\cfrac{\partial\bar{W}}{\partial \bar{I}_1} + 2~\bar{I}_2~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}\right)\boldsymbol{\mathit{1}} - \right.\ & \qquad \qquad \qquad \left. \cfrac{1}{J^{4/3}}~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] + \cfrac{\partial\bar{W}}{\partial J}~\boldsymbol{\mathit{1}} \ & = \cfrac{\lambda_1}{\lambda_1\lambda_2\lambda_3}~\cfrac{\partial\tilde{W}}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 + \cfrac{\lambda_2}{\lambda_1\lambda_2\lambda_3}~\cfrac{\partial\tilde{W}}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \cfrac{\lambda_3}{\lambda_1\lambda_2\lambda_3}~\cfrac{\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).

Incompressible hyperelastic materials

For an incompressible material J := \det\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(\boldsymbol{F}) - p~(J-1)

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

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

For incompressible isotropic hyperelastic materials, the strain energy density function is W(\boldsymbol{F})=\hat{W}(I_1,I_2). The Cauchy stress is then given by

 \boldsymbol{\sigma} = -p~\boldsymbol{\mathit{1}} + 2\left[\left(\cfrac{\partial\hat{W}}{\partial I_1} + I_1~\cfrac{\partial\hat{W}}{\partial I_2}\right)\boldsymbol{B} - \cfrac{\partial\hat{W}}{\partial I_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right]

Hyperelastic Models

The simplest hyperelastic material model is the Saint Venant-Kirchhoff model which is just an extension of the linear elastic material model to the nonlinear regime. This model has the form

 \boldsymbol{S} = \lambda~ \text{tr}(\boldsymbol{E})\boldsymbol{\mathit{1}} + 2\mu\boldsymbol{E}

where \boldsymbol{S} is the second Piola-Kirchhoff stress and \boldsymbol{E} is the Lagrangian Green strain, and λ and μ are the Lamé constants.

The strain-energy density function for the St. Venant-Kirchhoff model is

 W(\boldsymbol{E}) = \frac{\lambda}{2}[\text{tr}(\boldsymbol{E})]^2 + \mu \text{tr}(\boldsymbol{E}^2)

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

 \boldsymbol{S} = \cfrac{\partial W}{\partial \boldsymbol{E}} ~.

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. Models can be classified as:

1) phenomenological descriptions of observed behavior

2) mechanistic models deriving from arguments about underlying structure of the material

3) hybrids of phenomenological and mechanistic models

  • Gent

References

  • R.W. Ogden: Non-Linear Elastic Deformations, ISBN 0-486-69648-0

See also


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