# Isotropic Elastic Material

- Linear elastic materials with Hooke’s law,
- Nonlinear elastic materials with Ogden, Mooney-Rivlin and Arruda-Boyce laws.

## Linear Elastic Material (LAW1)

This material law is used to model purely elastic materials, or materials that remain in the elastic range. The Hooke's law requires only two values to be defined; the Young's or elastic modulus $$E$$, and Poisson's ratio, $\upsilon $. The law represents a linear relation between stress and strain.

## Ogden Materials (LAW42, LAW69 and LAW82)

^{1}. The strain energy $$W$$ is expressed in a general form as a function of $W\left({\lambda}_{1},{\lambda}_{2},{\lambda}_{3}\right)$:

Where, ${\lambda}_{1}$, i_{th} principal stretch

$${\lambda}_{i}=1+{\epsilon}_{i}$$, with $${\epsilon}_{i}$$ being the i_{th} principal engineering
strain

$${\alpha}_{p}$$ and $${\mu}_{p}$$ are the material constants.

$p$ is order of Ogden model and defines the number of coefficients pairs $\left({\alpha}_{p},{\mu}_{p}\right)$.

- If $p$=1, then one pair $\left({\alpha}_{1},{\mu}_{1}\right)$ of material constants is needed andin this case if ${\alpha}_{1}=2$ then it becomes a Neo-hookean material model.
- If $p$=2 then two pairs $\left({\alpha}_{1},{\mu}_{1}\right),\left({\alpha}_{2},{\mu}_{2}\right)$ of material constants are needed and in this case if ${\alpha}_{1}=2$ and ${\alpha}_{1}=-2$ then it becomes a Mooney-Rivlin material model

With:

$$\begin{array}{c}\overline{W}({\overline{\lambda}}_{1},{\overline{\lambda}}_{2},{\overline{\lambda}}_{3})={\displaystyle \sum _{p}\frac{{\mu}_{p}}{{\alpha}_{p}}}\left({\overline{\lambda}}_{1}{}^{{\alpha}_{p}}+{\overline{\lambda}}_{2}{}^{{\alpha}_{p}}+{\overline{\lambda}}_{3}{}^{{\alpha}_{p}}-3\right)\\ U(J)=\frac{K}{2}{(J-1)}^{2}\end{array}$$

Since $${\lambda}_{1}\text{\hspace{0.17em}}\frac{\partial J}{\partial {\lambda}_{i}}=J$$ and $\frac{\partial {\overline{\lambda}}_{j}}{\partial {\lambda}_{i}}=\frac{2}{3}{J}^{-\frac{1}{3}}$ for i=j and $\frac{\partial {\overline{\lambda}}_{j}}{\partial {\lambda}_{i}}=\frac{1}{3}{J}^{-\frac{1}{3}}\frac{{\lambda}_{j}}{{\lambda}_{i}}$ for i≠j

- The rubber is incompressible and isotropic in unstrained state
- The strain energy expression depends on the invariants of Cauchy tensor

- ${\mu}_{2}=-2\cdot {C}_{01}$
- ${\alpha}_{1}=2$
- ${\alpha}_{2}=-2$

The model can be generalized for a compressible material.

### Viscous Effects in LAW42

${\tau}_{\text{i}}$ are relaxation times: ${\tau}_{\text{i}}=\frac{{\eta}_{i}}{{G}_{i}}$

Rate effects are modeled through visco-elasticity using convolution integral using Prony series. This corresponds to extension of small deformation theory to finite deformation.

This viscous stress is added to the elastic one.

- $$M$$
- Order of the Maxwell model
- $$F$$
- Deformation gradient matrix
- $$\overline{F}={J}^{-\frac{1}{3}}F$$
- $$\mathrm{dev}\left(\overline{F}{\overline{F}}^{T}\right)$$
- Denotes the deviatoric part of tensor $$\overline{F}{\overline{F}}^{T}$$

### LAW69, Ogden Material Law (Using Test Data as Input)

This law, like /MAT/LAW42 (OGDEN) defines a hyperelastic and incompressible material specified using the Ogden or Mooney-Rivlin material models. Unlike LAW42 where the material parameters are input this law computes the material parameters from an input engineering stress-strain curve from a uniaxial tension and compression tests. This material can be used with shell and solid elements.

`law_ID`.

`law_ID`=1, Ogden law (Same as LAW42):- $$W\left({\lambda}_{1},{\lambda}_{2},{\lambda}_{3}\right)={\displaystyle \sum _{p=1}^{5}\frac{{\mu}_{p}}{{\alpha}_{p}}\left({\overline{\lambda}}_{1}{}^{{\alpha}_{p}}+{\overline{\lambda}}_{2}{}^{{\alpha}_{p}}+{\overline{\lambda}}_{3}{}^{{\alpha}_{p}}-3\right)}+\frac{K}{2}{\left(J-1\right)}^{2}$$
`law_ID`=2, Mooney-Rivlin law- $$W={C}_{10}\left({I}_{1}-3\right)+{C}_{01}\left({I}_{2}-3\right)$$

After reading the stress-strain curve
(`fct_ID`_{1}), Radioss
calculates the corresponding material parameter pairs using a nonlinear
least-square fitting algorithm. For classic Ogden law,
(`law_ID` =1), the calculated material parameter pairs are ${\mu}_{p}$ and ${\alpha}_{p}$ where the value of $p$ is defined via the `N_pair`
input. The maximum value of `N_pair` = 5 with a default value
of 2.

For the Mooney-Rivlin law (`law_ID` =2), the material parameter ${C}_{10}$ and ${C}_{01}$ are calculated remembering that ${\mu}_{p}$ and ${\alpha}_{p}$ for the LAW42 Ogden law can be calculated using this
conversion.

${\mu}_{1}=2\cdot {C}_{10}$, ${\mu}_{2}=-2\cdot {C}_{01}$, ${\alpha}_{1}=2$ and ${\alpha}_{2}=-2$.

The minimum test data input should be a uniaxial tension engineering stress strain curve. If uniaxial compression data is available, the engineering strain should increate monotonically from a negative value in compression to a positive value in tension. In compression, the engineering strain should not be less than -1.0 since -100% strain is physically not possible.

This material law is stable when (with $p$=1,…5) is satisfied for parameter pairs for all
loading conditions. By default, Radioss tries to fit the
curve by accounting for these conditions
(`I`_{check}=2). If a proper fit
cannot be found, then Radioss uses a weaker condition
(`I`_{check}=1:) which ensures that
the initial shear hyperelastic modulus ($\mu $) is positive.

Once the material parameters are calculated by the Radioss Starter in LAW69, all the calculations done by LAW69 in the simulation are the same as LAW42.

### LAW82

- If $\nu =0$, ${D}_{1}$ should be entered.
- If $\nu \ne 0$, ${D}_{1}$ input is ignored and will be recalculated
and output in the Starter output using:
(23) $${D}_{1}=\frac{3(1-2v)}{\mu (1+v)}$$ - If $\nu =0$ and ${D}_{1}$=0, a default value of $\nu =0.475$ is used and ${D}_{1}$ is calculated using Equation 23

^{5}

Unloading can be represented using an unloading
function, `Fscale`_{unL}, or by
providing hysteresis, `Hys` and shape factor, `Shape`, inputs to a damage model based on energy.

If the unloading function,
`Fscale`_{unL}, is entered,
unloading is defined based on the unloading flag, `Tension` and
the damage model is not used.

## Arruda-Boyce Material (LAW92)

LAW92 describes the Arruda-Boyce material model, which can be used to model hyperelastic behavior. The Arruda-Boyce model is based on the statistical mechanics of a material with a cubic representative volume element containing eight chains along the diagonal directions. It assumes that the chain molecules are located on the average along the diagonals of the cubic in principal stretch space.

Where, Material constant ${c}_{i}$ are:

- $\mu $
- Shear modulus
- ${\mu}_{0}$
- Initial shear modulus

${\lambda}_{m}$ is the limit of stretch which describes the beginning of hardening phase in tension (locking strain in tension) and so it is also called the locking stretch.

Arruda-Boyce is always stable if positive values of the shear modulus, $\mu $, and the locking stretch, ${\lambda}_{m}$ are used.

with ${\overline{\lambda}}_{i}={J}^{-\frac{1}{3}}{\lambda}_{i}$

- Uniaxial test
(32) $\sigma =\lambda \frac{\partial W}{\partial \lambda}=2\mu \left({\lambda}^{2}-\frac{1}{\lambda}\right){\displaystyle \sum _{i=1}^{5}\frac{i\cdot {c}_{i}}{{\lambda}_{m}{}^{2i-2}}}{\left({\overline{\lambda}}_{1}{}^{2}+{\overline{\lambda}}_{2}{}^{2}+{\overline{\lambda}}_{3}{}^{2}\right)}^{i-1}$ with ${\lambda}_{1}=\lambda \text{\hspace{0.17em}}$ and ${\lambda}_{2}={\lambda}_{3}={\lambda}^{-\frac{1}{2}}$, then $\text{\hspace{0.17em}}{\overline{I}}_{1}={\lambda}^{2}+\frac{2}{\lambda}$

and nominal stress is:(33) ${N}^{th}=\frac{\partial W}{\partial \lambda}=2\mu (\lambda -{\lambda}^{-2}){\displaystyle \sum _{i=1}^{5}\frac{i\cdot {c}_{i}}{{({\lambda}_{m})}^{2i-2}}}{\left({\lambda}^{2}+\frac{2}{\lambda}\right)}^{i-1}$ - Equibiaxial test
(34) $\sigma =\lambda \frac{\partial W}{\partial \lambda}=2\mu \left({\lambda}^{2}-\frac{1}{{\lambda}^{4}}\right){\displaystyle \sum _{i=1}^{5}\frac{i\cdot {c}_{i}}{{\lambda}_{m}{}^{2i-2}}}{\left({\overline{\lambda}}_{1}{}^{2}+{\overline{\lambda}}_{2}{}^{2}+{\overline{\lambda}}_{3}{}^{2}\right)}^{i-1}$ with ${\lambda}_{1}={\lambda}_{2}=\lambda \text{\hspace{0.17em}}$ and ${\lambda}_{3}={\lambda}^{-2}$, then ${\overline{I}}_{1}=2{\lambda}^{2}+\frac{1}{{\lambda}^{4}}$

and the nominal stress is:(35) ${N}^{th}=\frac{\partial W}{\partial \lambda}=2\mu (\lambda -{\lambda}^{-5}){\displaystyle \sum _{i=1}^{5}\frac{i\cdot {c}_{i}}{{({\lambda}_{m})}^{2i-2}}}{\left(2{\lambda}^{2}+\frac{1}{{\lambda}^{4}}\right)}^{i-1}$ - Planar test
(36) $\sigma =\lambda \frac{\partial W}{\partial \lambda}=2\mu \left({\lambda}^{2}-\frac{1}{{\lambda}^{2}}\right){\displaystyle \sum _{i=1}^{5}\frac{i\cdot {c}_{i}}{{\lambda}_{m}{}^{2i-2}}}{\left({\overline{\lambda}}_{1}{}^{2}+{\overline{\lambda}}_{2}{}^{2}+{\overline{\lambda}}_{3}{}^{2}\right)}^{i-1}$ with ${\lambda}_{1}=\lambda \text{\hspace{0.17em}},\text{\hspace{0.17em}}{\lambda}_{3}=1\text{\hspace{1em}}$ and ${\lambda}_{2}={\lambda}^{-1}\text{}$, then $\text{\hspace{0.17em}}{\overline{I}}_{1}={\lambda}^{2}+1+{\lambda}^{-2}$

and nominal stress is:(37) ${N}^{th}=\frac{\partial W}{\partial \lambda}=2\mu (\lambda -{\lambda}^{-3}){\displaystyle \sum _{i=1}^{5}\frac{i{c}_{i}}{{({\lambda}_{m})}^{2i-2}}}{\left({\lambda}^{2}+1+{\lambda}^{-2}\right)}^{i-1}$

## Yeoh Material (LAW94)

^{4}is a hyperelastic material model that can be used to describe incompressible materials. The strain energy density function of LAW94 only depends on the first strain invariant and is computed as:

- ${\overline{I}}_{1}={\overline{\lambda}}_{1}^{2}+{\overline{\lambda}}_{2}^{2}+{\overline{\lambda}}_{3}^{2}$
- First strain invariant
- ${\overline{\lambda}}_{i}={J}^{-\frac{1}{3}}{\lambda}_{i}$
- Deviatoric stretch

For incompressible materials with $$i$$=1 only and ${D}_{1}$ are input and the Yeoh model is reduced to a Neo-Hookean model.

The material constant specify the deviatoric part (shape change) of the material and parameters ${D}_{1}$, ${D}_{2}$, ${D}_{3}$ specify the volumetric change of the material. These six material constants need to be calculated by curve fitting material test data. RD-E: 5600 Hyperelastic Material with Curve Input includes a Yeoh fitting Compose script for uniaxial test data. The Yeoh material model has been shown to model all deformation models even if the curve fit was obtained using only uniaxial test data.

The initial shear modulus and the bulk modulus are computed as:

$\mu =2\cdot {C}_{10}$ and $K=\frac{2}{{D}_{1}}$

LAW94 is available only as an incompressible material model.

^{1}Ogden R.W., “Nonlinear Elastic Deformations”, Ellis Horwood, 1984.

^{2}Arruda, E.M. and Boyce, M.C., “A three-dimensional model for the large stretch behavior of rubber elastic materials”, J. Mech. Phys. Solids, 41(2), pp. 389–412, 1993.

^{3}Jörgen Bergström, “Mechanics of solid polymers: theory and computational modeling”, pp. 250-254, 2015.

^{4}Yeoh, O. H., “Some forms of the strain energy function for rubber”, Rubber Chemistry and Technology, Vol. 66, Issue 5, pp. 754-771, November 1993.

^{5}Kolling S., Du Bois P.A., Benson D.J., and Feng W.W., "A tabulated formulation of hyperelasticity with rate effects and damage." Computational Mechanics 40, no. 5 (2007).