Hyperelasticity¶

This chapter describes the hyperelastic constitutive models supported by FrontISTR. Selection and input details are in the Material Data chapter.

Constitutive Framework¶

For an isotropic hyperelastic material, the strain-energy potential \(W\) is a function of the principal invariants of the right Cauchy-Green deformation tensor \(C\), \((I_1, I_2, I_3)\), or of the modified invariants \((\overline{I}_1, \overline{I}_2, \overline{I}_3)\) that exclude the volume change.

Hyperelastic constitutive equations are written in terms of the second Piola-Kirchhoff stress and the Green-Lagrange strain, and the analysis uses the Total Lagrange formulation.

Given the potential \(W\), the second Piola-Kirchhoff stress and the elasticity tensor are obtained as

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

The supported potentials are listed below.

Neo-Hookean Model¶

The Neo-Hookean model extends the isotropic linear (Hooke's) law to large-deformation problems:

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

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

Mooney-Rivlin Model¶

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

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

Mooney-Rivlin Anisotropic Model¶

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

where \(C_{10}\), \(C_{01}\), and \(D\) are material constants common to the isotropic model, and \(C_{42}\) and \(C_{43}\) are anisotropic material constants.

Arruda-Boyce Model¶

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

\[ \mu = \frac{\mu_0}{1 + \dfrac{3}{5\lambda_m^2} + \dfrac{99}{175\lambda_m^4} + \dfrac{513}{875\lambda_m^6} + \dfrac{42039}{67375\lambda_m^8}}, \]

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