/MAT/LAW42 (OGDEN)

This law is generally used to model incompressible rubbers, polymers, foams, and elastomers. This material can be used with shell and solid elements.

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1. This material model defines a hyperelastic, viscous, and incompressible material specified using the Ogden, Neo-Hookean, or Mooney-Rivlin material models. This law is generally used to model incompressible rubbers, polymers, foams, and elastomers. This material can be used with shell and solid elements.
   LAW42 uses the following strain energy density representation of the Ogden material model.(1)

   $$W\left({\lambda }_1,{\lambda }_2,{\lambda }_3\right)={\displaystyle \sum _{p=1}^5\frac{{\mu }_p}{{\alpha }_p}\left({\overline{\lambda }}_1{}^{{\alpha }_p}+{\overline{\lambda }}_2{}^{{\alpha }_p}+{\overline{\lambda }}_3{}^{{\alpha }_p}-3\right)}+\frac{K}{2}{\left(J-1\right)}^2$$
   Where,

   $W$

   Strain energy density

   ${\lambda }_i$

   i^th principal engineering stretch

   $J$

   Relative volume defined as: $J={\lambda }_1\cdot {\lambda }_2\cdot {\lambda }_3=\frac{{\rho }_0}{\rho }$

   ${\overline{\lambda }}_i=J^{-\frac{1}{3}}{\lambda }_i$

   Deviatoric stretch

   ${\alpha }_p$ and ${\mu }_p$

   Material constants coefficient pairs.

   Up to 5 material constant pairs can be defined.

   The initial shear modulus $\mu$ and bulk modulus ( $K$ ) are given by:(2)

   $$\mu =\frac{{\displaystyle \sum _{p=1}^5{\mu }_p\cdot {\alpha }_p}}{2}$$
   and(3)

   $$K=\mu \cdot \frac{2\left(1+\nu \right)}{3\left(1-2\nu \right)}$$

   Where, $\nu$ is the Poisson's ratio and is only used for computing the bulk modulus.

2. Parameters ${\alpha }_p$ and ${\mu }_p$ must be chosen so that initial shear modulus is $\mu >0$ .

   For material stability, it is required that each material constant pair ${\mu }_p\cdot {\alpha }_p>0$ .

3. Poisson's ratio $\nu$ is used only for computing the bulk modulus, Equation 3.

   For pure incompressible materials, $\nu =0.5$ . This value leads to a finite bulk modulus ( $K$ ). Therefore, the recommended maximum Poisson's ratio for incompressible materials is $\nu =0.495$ .

   Higher values of the Poisson's ratio may lead to a small time step value or divergence in case of implicit and explicit simulations.

4. A particular case of the Ogden material model is the incompressible Mooney-Rivlin model, which can be represented using the following equation for the strain energy density function:(4)
   $$W=C_{10}\left(I_1-3\right)+C_{01}\left(I_2-3\right)$$

   Where, $I_1$ and $I_2$ are the first and second invariants of the right Cauchy-Green Tensor.

   This representation can be derived from the Ogden strain energy density function when:(5)

   $${\mu }_1=2\cdot C_{10}$$

   (6)

   $${\mu }_2=-2\cdot C_{01}$$

   (7)

   $${\alpha }_1=2$$

   (8)

   $${\alpha }_2=-2$$
5. A simple case of the Ogden material model is the Neo-Hookean model represented using the following equation for the strain energy density function:(9)
   $$W=C_{10}\left(I_1-3\right)$$
   Where,

   $I_1$

   First invariants of the right Cauchy-Green Tensor

   $C_{10}$

   Material constant

   This representation can be derived from the LAW42 Ogden strain energy density function when:(10)

   $${\mu }_1=2\cdot C_{10}$$

   (11)

   $${\alpha }_1=2$$
   and(12)

   $${\mu }_2={\alpha }_2=0$$
6. In cases when the bulk modulus of a material is not large enough to prevent compression, LAW42 allows the input of a function (fct_ID_blk) which scale the bulk modulus as a function of the relative volume so that the bulk modulus can be increased to maintain incompressibility.
7. Viscous (rate) effects are modeled in LAW42 using a Maxwell model, which can be described in a simplified manner as a system of n springs with stiffness' $G_i$ and dampers ${\eta }_i$ :

   
   
   Figure 1. Maxwell Model

   The Maxwell model is represented using Prony series inputs ( $G_i,{\tau }_i$ ). The hyperelastic initial shear modulus $\mu$ is the same as the long-term shear modulus $G_{\infty }$ in the Maxwell model, and ${\tau }_i$ is the relaxation time:(13)

   $${\tau }_i=\frac{{\eta }_i}{G_i}$$

   The $G_i$ and ${\tau }_i$ values must be positive.

8. /VISC/PRONY can be used with this material law to include viscous effects.