Neo-Hookean model

The Neo-Hookean hyperelastic constitutive model.¹

Parameters

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

External variables

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

Internal variables

• None.

Helmholtz free energy density

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

Cauchy stress

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

Cauchy tangent stiffness

$${\mathcal{T}}_{ijkL}(\mathbf{F})=\frac{\mu }{J^{5/3}}\left({\delta }_{ik}{F}_{jL}+{\delta }_{jk}{F}_{iL}-\frac{2}{3}{\delta }_{ij}{F}_{kL}-\frac{5}{3}{B}_{ij}^{\prime }{F}_{kL}^{-T}\right)+\frac{\kappa }{2}\left(J+\frac{1}{J}\right){\delta }_{ij}{F}_{kL}^{-T}$$

1. R.S. Rivlin, Philos. Trans. R. Soc. London, Ser. A 240, 459 (1948). ↩