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Hyperelasticity

This chapter covers the hyperelastic constitutive laws supported by FrontISTR. For details on model selection and input specifications, see the functions section 03_material.

Phenomenon (Reversible Large-Deformation Response)

TBD (to be developed into full text in the next phase).

Constitutive Framework

For an isotropic hyperelastic material, the elastic potential energy is obtained from an isotropic response from the initial unstressed state, and can be expressed as a function of the principal invariants \((I_1, I_2, I_3)\) of the right Cauchy-Green deformation tensor \(C\), or of the principal invariants \((\overline{I}_1, \overline{I}_2, \overline{I}_3)\) of the deformation tensor with the volumetric change removed, that is, as \(W = W(I_1, I_2, I_3)\) or \(W = W(\overline{I}_1, \overline{I}_2, \overline{I}_3)\).

The constitutive equation for a hyperelastic material is defined by the relationship between the second Piola-Kirchhoff stress and Green-Lagrange strain, and the Total Lagrange method is applied to its deformation analysis.

Once the elastic potential energy \(W\) is known, the second Piola-Kirchhoff stress and stress-strain relationship can be calculated as follows.

\[ S = 2\frac{\partial W}{\partial C} \]
\[ \mathbb{C} = 4 \frac{\partial^2 W}{\partial C \partial C} \]

The elastic potential energies \(W\) of the hyperelastic models adopted by FrontISTR are listed below.

Neo-Hookean Hyperelastic Model

The Neo-Hookean hyperelastic model extends an isotropic linear law (Hooke's law) so that it can be applied to large-deformation problems. Its elastic potential is as follows.

\[ W = C_{10} (\overline{I}_1 - 3) + \frac{1}{D} (J - 1)^2 \]

Here, \(C_{10}\) and \(D\) are material constants.

Mooney-Rivlin Hyperelastic Model

\[ W = C_{10} (\overline{I}_1 - 3) + C_{01} (\overline{I}_2 - 3) + \frac{1}{D} (J - 1)^2 \]

Here, \(C_{10}\), \(C_{01}\), and \(D\) are material constants.

Mooney-Rivlin Anisotropic Hyperelastic Model

\[ W = C_{10} (\overline{I}_1 - 3) + C_{01} (\overline{I}_2 - 3) + \frac{1}{D} (J - 1)^2 + C_{42} (\overline{I}_4 - 1)^2 + C_{43} (\overline{I}_4 - 1)^3 \]

Here, \(C_{10}\), \(C_{01}\), and \(D\) are material constants common to the isotropic model, while \(C_{42}\) and \(C_{43}\) are material constants specific to anisotropy.

Arruda-Boyce Hyperelastic Model

\[ \begin{aligned} W &= \mu \left[ \frac{1}{2}(\overline{I}_1 - 3) + \frac{1}{20\lambda_m^2}(\overline{I}_1^2 - 9) + \frac{11}{1050\lambda_m^4}(\overline{I}_1^3 - 27) \right. \\ &\quad \left. + \frac{19}{7000\lambda_m^6}(\overline{I}_1^4 - 81) + \frac{519}{673750\lambda_m^8}(\overline{I}_1^5 - 243) \right] + \frac{1}{D} \left( \frac{J^2 - 1}{2} - \ln J \right) \end{aligned} \]
\[ \mu = \frac{\mu_0}{1 + \dfrac{3}{5\lambda_m^2} + \dfrac{99}{175\lambda_m^4} + \dfrac{513}{875\lambda_m^6} + \dfrac{42039}{67375\lambda_m^8}} \]

Here, \(\mu_0\), \(\lambda_m\), and \(D\) are material constants.