Gent model

The Gent hyperelastic constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The extensibility $J_m$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The Gent model reduces to the Neo-Hookean model when $J_m\to \infty$.

Helmholtz free energy density

$$a(\mathbf{F})=-\frac{\mu J_m}{2}\ln \left[1-\frac{\mathrm{tr}({\mathbf{B}}^{\ast })-3}{J_m}\right]+\frac{\kappa }{2}\left[\frac{1}{2}\left(J^2-1\right)-\ln J\right]$$

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\frac{J^{-1}\mu J_m{{\mathbf{B}}^{\ast }}^{\prime }}{J_m-\mathrm{tr}({\mathbf{B}}^{\ast })+3}+\frac{\kappa }{2}\left(J-\frac{1}{J}\right)1$$

Cauchy tangent stiffness

$$\begin{array}{rl}{\mathcal{T}}_{ijkL}(\mathbf{F})=& \frac{J^{-5/3}\mu J_m}{J_m-\mathrm{tr}({\mathbf{B}}^{\ast })+3}[{\delta }_{ik}{F}_{jL}+{\delta }_{jk}{F}_{iL}-\frac{2}{3}{\delta }_{ij}{F}_{kL}+\frac{2{B_{ij}^{\ast }}^{\prime }{F}_{kL}}{J_m-\mathrm{tr}({\mathbf{B}}^{\ast })+3}\\ & -\left(\frac{5}{3}+\frac{2}{3}\frac{\mathrm{tr}({\mathbf{B}}^{\ast })}{J_m-\mathrm{tr}({\mathbf{B}}^{\ast })+3}\right)J^{2/3}{B_{ij}^{\ast }}^{\prime }{F}_{kL}^{-T}]+\frac{\kappa }{2}\left(J+\frac{1}{J}\right){\delta }_{ij}{F}_{kL}^{-T}\end{array}$$

1. A.N. Gent, Rubber Chem. Technol. 69, 59 (1996). ↩