Yeoh model

The Yeoh hyperelastic constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The extra moduli ${\mu }_n$ for $n=2\dots N$.

External variables

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

Internal variables

• None.

Notes

• The Yeoh model reduces to the Neo-Hookean model when ${\mu }_n\to 0$ for $n=2\dots N$.

Helmholtz free energy density

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

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\sum _{n=1}^N\frac{n{\mu }_n}{J}{\left[\mathrm{tr}({\mathbf{B}}^{\ast })-3\right]}^{n-1}{{\mathbf{B}}^{\ast }}^{\prime }+\frac{\kappa }{2}\left(J-\frac{1}{J}\right)1$$

Cauchy tangent stiffness

$$\begin{array}{rl}{\mathcal{T}}_{ijkL}(\mathbf{F})=& \sum _{n=1}^N\frac{n{\mu }_n}{J^{5/3}}{\left[\mathrm{tr}({\mathbf{B}}^{\ast })-3\right]}^{n-1}\left({\delta }_{ik}{F}_{jL}+{\delta }_{jk}{F}_{iL}-\frac{2}{3}{\delta }_{ij}{F}_{kL}-\frac{5}{3}{B}_{ij}^{\prime }{F}_{kL}^{-T}\right)\\ & +\sum _{n=2}^N\frac{2n(n-1){\mu }_n}{J^{7/3}}{\left[\mathrm{tr}({\mathbf{B}}^{\ast })-3\right]}^{n-2}{B}_{ij}^{\prime }{B}_{km}^{\prime }{F}_{mL}^{-T}+\frac{\kappa }{2}\left(J+\frac{1}{J}\right){\delta }_{ij}{F}_{kL}^{-T}\end{array}$$

1. O.H. Yeoh, Rubber Chem. Technol. 66, 754 (1993). ↩