Hyperelastic materials

Neo-Hookean hyperelastic law, using a Cauchy stress tensor $\boldsymbol{\sigma}$, stress in the current configuration.

(Quasi-)incompressibility is treated by a volumetric/deviatoric multiplicative split of the deformation gradient, i.e. $\bar{\mathbf{F}} = J^{-1/3}\mathbf{F}$. Hence the deviatoric potential is based on reduced invariants of $\bar{\mathbf{b}} =\bar{\mathbf{F}}\bar{\mathbf{F}}^T$.

$$W\left(I_1,I_2,J\right) = \bar{W}\left(\bar{I_1},\bar{I_2}\right) + K f\left(J\right) = C_1\left(\bar{I_1} - 3\right) + \frac{k_0}{2}\left[ \left(J-1\right)^2 + \ln^2 J\right]$$

Neo-Hookean hyperelastic law, using a Cauchy stress tensor $\boldsymbol{\sigma}$, stress in the current configuration.

Here, the TEMP parameter is not relevant anymore.

Mooney-Rivlin hyperelastic law, using a Cauchy stress tensor $\boldsymbol{\sigma}$, stress in the current configuration.

(Quasi-)incompressibility is treated by a volumetric/deviatoric multiplicative split of the deformation gradient, i.e. $\bar{\mathbf{F}} = J^{-1/3}\mathbf{F}$. Hence the deviatoric potential is based on reduced invariants of $\bar{\mathbf{b}} =\bar{\mathbf{F}}\bar{\mathbf{F}}^T$.

$$W\left(I_1,I_2,J\right) = \bar{W}\left(\bar{I_1},\bar{I_2}\right) + K f\left(J\right) = C_1\left(\bar{I_1} - 3\right) + C_2\left(\bar{I_2} - 3\right)+ \frac{k_0}{2}\left[ \left(J-1\right)^2 + \ln^2 J\right]$$

Mooney-Rivlin hyperelastic law, using a Cauchy stress tensor $\boldsymbol{\sigma}$, stress in the current configuration.

Here, the TEMP parameter is not relevant anymore.

Holzapfel-Gasser-Ogden (invariant-based) anisotropic hyperelastic law, using a Cauchy stress tensor $\boldsymbol{\sigma}$, stress in the current configuration. This model is particularly suited to predict the response of biological tissues.

(Quasi-)incompressibility is treated by a volumetric/deviatoric multiplicative split of the deformation gradient, i.e. $\bar{\mathbf{F}} = J^{-1/3}\mathbf{F}$. Hence the deviatoric potential is based on reduced invariants of $\bar{\mathbf{b}} =\bar{\mathbf{F}}\bar{\mathbf{F}}^T$.

The strain-energy density function $W$ is expressed as the sum of an isotropic (=matrix) and anisotropic (=fibers) contribution: $$W\left(\bar{I}_1,\bar{I}_4,J \right) = W_{iso}\left(\bar{I}_1,J \right) + W_{ani}\left(\bar{I}_1,\bar{I}_4\right)$$

The isotropic contribution takes the form of a generalized Neo-Hookean model: $$W_{iso}\left(\bar{I}_1,J \right) = C_1\left(\bar{I}_1 -3\right) +K f\left(J\right) = C_1\left(\bar{I}_1 -3\right) +\frac{k_0}{2}\text{ln}^2 J$$

The anisotropic contribution to the strain energy density function writes: $$W_{ani}\left(\bar{I}_1,\bar{I}_4\right) = \frac{k_1}{2k_2} \sum_{\alpha=1}^n \left[ e^{k_2\left<E_\alpha \right>^2} - 1 \right] = \frac{k_1}{2k_2} \sum_{\alpha=1}^n \left[ e^{k_2\left<d(\bar{I}_1-3)+(1-3d)(\bar{I}_4^\alpha - 1)\right>^2} - 1 \right],$$ where $k_1$[MPa] and $k_2$[-] are material parameters characterizing all fiber families in the material. $d\in[0,~\frac{1}{3}]$ is a parameter accounting for fiber dispersion, with $d=0$ corresponding to perfectly aligned fibers whilst $d=\frac{1}{3}$ corresponds to randomly oriented fibers (isotropic response). The model adds up to three different families of fibers ($\alpha \leq 3$), with their initial orientation given by $\mathbf{a}^\alpha = \left[a_{\alpha x},~a_{\alpha y},~a_{\alpha z} \right]$. Fiber directions do not have to be orthogonal.

More information and mathematical derivations, such as the analytical tangent stiffness matrix, can be found in metafor_hgo.pdf.

Neo-Hookean hyperelastic law, using a PK2 tensor.

The potential per unit volume is computed based on the average compressibility over the element, ($\theta$):

$$U^{vol}=\dfrac{k_0}{2} \left[\ln\right(\theta\left)\right]^2$$

The deviatoric potential is computed based on a Cauchy tensor with a unit determinant:

$$U^{dev}=\dfrac{g_0}{2} \left[\text{tr}\right(\hat{\mathbf{C}}\left)-3\right]$$

Logarithmic hyperelastic law, using a PK2 tensor.

The potential per unit volume is computed based on the average compressibility of the element, ($\theta$):

$$U^{vol}=\dfrac{k_0}{2} \left[\ln\right(\theta\left)\right]^2$$

The deviatoric potential is computed based on a Cauchy tensor with a unit determinant:

$$U^{dev}= \dfrac{g_0}{4} \ln \left(\hat{\mathbf{C}}\right):\ln \left(\hat{\mathbf{C}}\right)$$

Logarithmic hyperelastic law, using a PK2 tensor.

The potential per unit volume is computed based on the average compressibility of the element, ($\theta$):

$$U^{vol}=\dfrac{k_0}{2} \left[\ln\right(\theta\left)\right]^2$$

The deviatoric potential is computed based on a Cauchy tensor with a unit determinant:

$$U^{dev}= \dfrac{g_0}{4} \ln \left(\hat{\mathbf{C}}^{el}\right):\ln \left(\hat{\mathbf{C}}^{el}\right)$$

Hyperelastic law, using a PK2 tensor. Its function applied on the strain spectral decomposition is a user law.

The potential per unit volume is computed based on the average compressibility of the element, ($\theta$):

$$U^{vol}=\dfrac{k_0}{2} \left[\ln\right(\theta\left)\right]^2$$

The deviatoric potential is computed based on a hyperelastic user function defined in Viscoelastic laws.

Viscoelastic hyperelastic law, using a PK2 tensor. The law includes a main branch (spring and dashpot in parallel) and one or several Maxwell branches (spring and dashpot in series).

Each branch has its behavior corresponding to a viscoelastic law, supplied by the user.

The potential per unit volume is computed based on the average compressibility of the element, ($\theta$):

$$U^{vol}=\dfrac{k_0}{2} \left[\ln\right(\theta\left)\right]^2$$

The deviatoric potential is computed based on the viscoelastic laws :

$$U^{dev}= U^{dev}_{\text{main,elastic}}\left(\hat{C}\right) + \sum_{Maxwell} U^{dev}_{\text{Maxwell,elastic}}\left(\hat{C}^{\text{el}}\right)$$

The dissipation potential is written as:

$$\Delta t \phi^{dev}= \Delta t \phi^{dev}_{\text{main,viscous}}\left( \exp{\frac{\ln{\Delta\hat{C}}}{\Delta t}} \right) + \sum_{Maxwell} \Delta t \phi^{dev}_{\text{Maxwell,viscous}}\left(\exp{\frac{\ln{\Delta C^{\text{vis}}}}{\Delta t}} \right)$$

where $$\Delta\hat{C} = {\hat{F}^n}^{-T} \hat{C}^{n+1} {\hat{F}^n}^{-1}$$

$$\Delta C^{\text{vis}} = {{F^{\text{vis}}}^n}^{-T} {C^{\text{vis}}}^{n+1} {{F^{\text{vis}}}^n}^{-1}$$

The potentials $U^{dev}_{\text{main,elastic}},~~U^{dev}_{\text{Maxwell,elastic}},~~\phi^{dev}_{\text{main,viscous}},~~\phi^{dev}_{\text{Maxwell,viscous}}$ are hyperelastic functions defined in Viscoelastic laws.