Hyper elasticity

Hyper elasticity is used to model rubbers. It should be combined with a total Lagrange formulation for the memory of the material (so use -total or -total_piola for group_materi_memory).

The stresses follow from a strain energy function (with $C_{ij}$ components of the matrix $C$, and where $F$ is the deformation tensor and $U$ is the stretch tensor following from the polar decomposition of the deformation tensor)

$$2 \frac{\partial W}{\partial {C_{ij}}}$$

$$C = F^T F = U^T U$$

Deviatoric contributions

To obtain a purely deviatoric function, the following strain measures are used (with $I_1$, $I_2$ and $I_3$ the first, second and third invariant of the elastic strain matrix $C$ respectively)

$$J_1 = I_1 {I_3}^{\frac{-1}{3}} \; \; \; \; J_2 = I_2 {I_3}^{\frac{-2}{3}}$$

The group_materi_hyper_besseling function reads ( with $K_1$, $K_2$ and $\alpha$ user defined constants)

$$W = K_1 ( J_1 - 3 ) ^ \alpha + K_2 ( J_2 - 3 )$$

The group_materi_hyper_mooney_blatz_ko function reads (with $G$ and $\beta$ user defined constants)

$$W = G * 0.5 * ( I_1 - 3.0 + (2.0/\beta) ( J^{-\beta} - 1. ) );$$

This Blatz-Ko hyperelastic material hardens in compression, and softens slightly in tension; it models a foamlike rubber.

The group_materi_hyper_mooney_rivlin function reads (with $K_1$ and $K_2$ user defined constants)

$$W = K_1 ( J_1 - 3 ) + K_2 ( J_2 - 3 )$$

The group_materi_hyper_neohookean function reads (with $K_1$ a user defined constant)

$$W = K_1 ( J_1 - 3 )$$

The group_materi_hyper_reduced_polynomial function reads (with $K_i$ user defined constants)

$$W = K_i ( J_1 - 3 )^i$$

where a summation over $i = 1,2,\ldots$ is applied.

Volumetric contributions

First we define $J = \sqrt{ I_3 }$. Then a volumetric part can be added to the strain energy.

The group_materi_hyper_volumetric_linear contribution reads:

$$W = \frac{K}{2} (J-1)^2$$

The group_materi_hyper_volumetric_murnaghan contribution reads:

$$W = \frac{K}{\beta} ( \frac{1}{\beta-1} J^{- \beta} + 1 ) J$$

The group_materi_hyper_volumetric_polynomial contribution reads:

$$W = \frac{K_i}{2} (J-1)^{2i}$$

for $i=0,1,\ldots$.

The group_materi_hyper_volumetric_simo_taylor contribution reads:

$$W = \frac{K}{2} ( (J-1)^2 + (ln J)^2 )$$

The group_materi_hyper_volumetric_ogden contribution reads:

$$W = \frac{K}{\beta} ( \frac{1}{\beta} (J^{- \beta} -1) + ln J )$$

As an example, you can combine the group_materi_hyper_mooney_rivlin energy function with the group_materi_hyper_volumetric_linear so that the total strain energy function becomes:

$$W = K_1 ( J_1 - 3 ) + K_2 ( J_2 - 3 ) + \frac{K}{2} (J-1)^2$$

Here the initial shear modulus and bulk modulus are included as:

$${\rm initial ~ shear ~ modulus} = 2 ( K_1 + K_2 )$$

and

$${\rm initial ~ bulk ~ modulus} = K$$

respectively.