Hyperelastic Materials

Hyperelastic material models describe the behavior of materials that can undergo large elastic deformations under loading, returning to their original shape when the load is removed. For hyperelastic materials, such as rubber, the strain-stress relationship is nonlinear.

Strain Energy Potential

The strain-stress relationship for hyperelastic materials can be expressed in terms of a strain energy density function, which is a frame invariant potential defined as:

Figure 1. EQUATION_DISPLAY

Ψ (F) = Ψ (C) = Ψ (E) = Ψ (U)

(4528)

where $F$ is the deformation gradient (Eqn. (4428)), $C$ is the right Cauchy Green strain (Eqn. (4446)), $E$ is the Green-Lagrange strain (Eqn. (4445)), and $U$ is the right stretch tensor (Eqn. (4429)).

The general strain-stress relationship for hyperelastic materials can be written as:

Figure 2. EQUATION_DISPLAY

S = 2 \frac{\partial Ψ}{\partial C}

(4529)

where $S$ is the 2nd Piola Kirchhoff stress (Eqn. (4435)).

The second derivative of the strain energy potential defines the material tangent as a 4th order tensor:

Figure 3. EQUATION_DISPLAY

ℂ = 2 \frac{\partial S}{\partial C} = 4 \frac{\partial^{2} Ψ}{\partial C \partial C}

(4530)

For incompressible and nearly incompressible materials, the strain energy potential can be split into a deviatoric and a volumetric part:

Figure 4. EQUATION_DISPLAY

Ψ = Ψ^{d} (C^{d}) + Ψ^{v} (J)

(4531)

The volumetric part depends only on the volume ratio, $J = d e t (F)$ . The deviatoric part can be expressed in terms of the invariants $I_{1}^{d}, I_{2}^{d}$ of $C^{d}$ (See Tensor Invariants) or in terms of the principal stretches $λ_{k}^{d}; k = 1, 2, 3$ of the modified right stretch $U^{d}$ (Eqn. (4448)).

When modeling thermal expansion, Simcenter STAR-CCM+ replaces the volume ratio $J$ with the elastic volume ratio $J_{e}$ :

Figure 5. EQUATION_DISPLAY

J_{e} = \frac{J}{J_{T}} = \frac{J}{{(1 + ε_{T})}^{3}}

(4532)

where $ε_{T}$ is the isotropic thermal strain (see Eqn. (4452)).

Simcenter STAR-CCM+ provides Neo-Hookean, Mooney-Rivlin, and Ogden models that provide polynomial expressions for the strain energy potential. The coefficients for the polynomial terms are determined by curve fitting the polynomials to material test data. Simcenter STAR-CCM+ also provides a calibration tool that calculates the coefficients for any of the hyperelasticity models based on known modes of deformation: uniaxial, biaxial, shear, and volumetric. For more information, see Hyperelastic Material Calibration.

Neo-Hookean Model

The Neo-Hookean material model [868] is the simplest hyperelastic model and is an extension of Hooke's law for large deformations. This model is suitable for compressible or nearly incompressible materials, and should be used to approximate materials in the absence of more specific material data.

In this case, the strain energy potential is given by:

Figure 6. EQUATION_DISPLAY

Ψ = c_{10} (I_{1}^{d} - 3) + \frac{k_{b}}{2} {(J - 1)}^{2}

(4533)

where $I_{1}^{d}$ is the first invariant of $C^{d}$ (Eqn. (5215)), $k_{b}$ is the initial bulk modulus, and $c_{10}$ is:

Figure 7. EQUATION_DISPLAY

c_{10} = \frac{μ}{2} = \frac{E}{4 (1 + ν)}

(4534)

where $μ$ is the initial shear modulus.

This material model is typically accurate for strains $< 20 %$ .

Mooney-Rivlin Model

The Mooney-Rivlin material model [868] is a specialization of the general polynomial hyperelastic model, with strain energy potential:

Figure 8. EQUATION_DISPLAY

Ψ = \sum_{i, j = 0}^{N} [c_{i j} {(I_{1}^{d} - 3)}^{i} {(I_{2}^{d} - 3)}^{j}] + \frac{k_{b}}{2} {(J - 1)}^{2}

(4535)

where $I_{1}^{d}$ and $I_{2}^{d}$ are the first and second invariants of $C^{d}$ (Eqn. (5215) and Eqn. (5216)) and $k_{b}$ is the initial bulk modulus. The coefficients $c_{i j}$ are obtained by curve fitting the model to experimental stress-strain curves.

Simcenter STAR-CCM+ provides 2-term, 5-term, and 9-term Mooney-Rivlin models. The optimal number of terms can be assessed from experimental measures of the stress-strain curve of the material. In general, the 2-term model is suitable for strain-stress curves that do not have inflection points. The 5-term model is suitable for strain-stress curves that have one inflection point. The 9-term model is suitable for strain-stress curves that have two inflection points.

Mooney-Rivlin 2-term Model: The 2-term model is a linear extension of the Neo-Hookean model and is suitable for materials that are subject to bi-axial stress states. The strain energy potential is:
Figure 9. EQUATION_DISPLAY

$Ψ = c_{10} (I_{1}^{d} - 3) + c_{01} (I_{2}^{d} - 3) + \frac{k_{b}}{2} {(J - 1)}^{2}$
(4536)

For small strains, $μ = 2 (c_{10} + c_{01})$ .
Mooney-Rivlin 5-term Model: The strain energy potential of the 5-term model is:
Figure 10. EQUATION_DISPLAY

$\begin{matrix} Ψ = & c_{10} (I_{1}^{d} - 3) + c_{01} (I_{2}^{d} - 3) + \\ + c_{11} (I_{1}^{d} - 3) (I_{2}^{d} - 3) + \\ + c_{20} {(I_{1}^{d} - 3)}^{2} + c_{02} {(I_{2}^{d} - 3)}^{2} + \frac{k_{b}}{2} {(J - 1)}^{2} \end{matrix}$
(4537)
Mooney-Rivlin 9-term Model: The strain energy potential of the 9-term model is:
Figure 11. EQUATION_DISPLAY

$\begin{matrix} Ψ = & c_{10} (I_{1}^{d} - 3) + c_{01} (I_{2}^{d} - 3) + \\ + c_{11} (I_{1}^{d} - 3) (I_{2}^{d} - 3) + \\ + c_{20} {(I_{1}^{d} - 3)}^{2} + c_{02} {(I_{2}^{d} - 3)}^{2} + \\ + c_{21} {(I_{1}^{d} - 3)}^{2} (I_{2}^{d} - 3) + \\ + c_{12} (I_{1}^{d} - 3) {(I_{2}^{d} - 3)}^{2} + \\ + c_{30} {(I_{1}^{d} - 3)}^{3} + c_{03} {(I_{2}^{d} - 3)}^{3} + \frac{k_{b}}{2} {(J - 1)}^{2} \end{matrix}$
(4538)

Ogden Model

The Ogden material model [868] is typically used for materials with larger strains. The strain potential is written in terms of the principal stretches $λ_{1}^{d} \leq λ_{2}^{d} \leq λ_{3}^{d}$ , which are the eigenvalues (see Eqn. (5210)) of the modified right stretch tensor $U^{d}$ :

Figure 12. EQUATION_DISPLAY

Ψ = \sum_{i = 1}^{N} Q_{i} (λ_{1}^{d}, λ_{2}^{d}, λ_{3}^{d}) + \sum_{i = 1}^{N} \frac{k_{i}}{2} {(J - 1)}^{2 i}

(4539)

with:

Figure 13. EQUATION_DISPLAY

Q_{i} = \sum_{i = 1}^{3} \frac{μ_{i}}{α_{i}} [{(λ_{k}^{d})}^{α_{i}} - 1]

(4540)

where $λ_{k}^{d}$ is the $k^{t h}$ principal stretch of $U^{d}$ , $μ$ is the initial classical shear modulus, $k_{i}$ is the $i^{t h}$ bulk modulus, $μ_{i}$ is the $i^{t h}$ constant shear modulus, and $α_{i}$ is a dimensionless coefficient.

For small strains:

Figure 14. EQUATION_DISPLAY

\begin{matrix} 2 μ = \sum_{i = 1}^{N} μ_{i} α_{i}; & μ_{i} α_{i} > 0 \end{matrix}

(4541)

Simcenter STAR-CCM+ limits the number of terms to 6. As with the Mooney-Rivlin material model, the coefficients for the Ogden model are found via curve fitting of the model to experimental data.

Hyperelastic Material Calibration

Simcenter STAR-CCM+ provides a calibration tool that calculates the coefficients for any of the hyperelasticity models by fitting the model curve to experimental data.

For the calculation, Simcenter STAR-CCM+ assumes that the material deforms according to one of the following modes of deformation: uniaxial, biaxial, shear, or volumetric. For uniaxial, biaxial, and shear modes, Simcenter STAR-CCM+ assumes that the material is incompressible (that is, the material volume remains constant).

In the calibration tool, you provide experimental data in tabular form. The first column contains the measured deformation, that is, strain or volume ratio, and the second column contains the measured nominal stress. The input type and the deformation tensor for each mode are summarized below.

Uniaxial Deformation

Input type: strain vs uniaxial stress.

For uniaxial deformation, the deformation tensor is:

Figure 15. EQUATION_DISPLAY

F = [\begin{matrix} λ & 0 & 0 \\ 0 & 1 / \sqrt{λ} & 0 \\ 0 & 0 & 1 / \sqrt{λ} \end{matrix}]

(4542)

where $λ = 1 + ϵ$ is the principal stretch, with $ϵ$ denoting the experimental uniaxial strain.

Biaxial Deformation

Input type: strain vs biaxial stress.

For biaxial deformation, the deformation tensor is:

Figure 16. EQUATION_DISPLAY

F = [\begin{matrix} λ & 0 & 0 \\ 0 & λ & 0 \\ 0 & 0 & 1 / λ^{2} \end{matrix}]

(4543)

where $λ = 1 + ϵ$ is the principal stretch, with $ϵ$ denoting the experimental biaxial strain.

Shear Deformation

Input type: shear strain vs shear stress.

For shear deformation, the deformation tensor is:

Figure 17. EQUATION_DISPLAY

F = [\begin{matrix} λ & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 / λ \end{matrix}]

(4544)

where $λ = 1 + ϵ$ is the principal stretch, with $ϵ$ denoting the experimental shear strain.

Volumetric Deformation

Input type: volume ratio vs volumetric stress.

For volumetric deformation, the deformation tensor is:

Figure 18. EQUATION_DISPLAY

F = [\begin{matrix} λ & 0 & 0 \\ 0 & λ & 0 \\ 0 & 0 & λ \end{matrix}]

(4545)

where $λ = J^{1 / 3}$ , with $J$ denoting the experimental volume ratio.