# MAT4

Material Property ElementMAT4 lets you define a hyper-elastic material model for NLFE elements based on the Mooney-Rivlin material model.

## Format

<MAT4
id       = "integer"
mu01     = "real"
mu10     = "real"
nu       = "real"
rho      = "real"
YS       = "real"
/> 

## Attributes

id
Unique material property identification number.
mu01
Material constant ( ${\mu }_{01}$ ).
mu10
Material constant ( ${\mu }_{10}$ ).
nu
Poisson's ratio for the element. Default is 0.49.
rho
Element density.
YS
An elastic limit for strain. Default is 0.0.
YS >= 0.0

## Example

The example demonstrates the definition of a MAT4 element.

<MAT4 id="1" mu01="8e+3"  mu10="2e+3" nu="0.499" rho="7.810e-6" YS="0.125"/>

1. This material element defines a hyper-elastic material that follows the Mooney-Rivlin material model law for the strain energy density function:

$U={\mu }_{10}\left({\overline{I}}_{1}-3\right)+{\mu }_{01}\left({\overline{I}}_{2}-3\right)+\frac{k}{2}{\left(J-1\right)}^{2}$

where

$\mu =2\left({\mu }_{01}+{\mu }_{10}\right)$ is the shear modulus

$k=\frac{2\mu \left(1+v\right)}{3\left(1-2v\right)}$ is the bulk modulus

$v$ is the Poisson's ratio

${\overline{I}}_{1}={J}^{\frac{-2}{3}}{I}_{1}$ , ${\overline{I}}_{2}={J}^{\frac{-4}{3}}{I}_{2}$

${I}_{1}=tr\left(C\right)={r}_{x}^{T}{r}_{x}+{r}_{y}^{T}{r}_{y}+{r}_{z}^{T}{r}_{z}$

${I}_{2}=\frac{1}{2}\left({\left(tr\left(C\right)\right)}^{2}-\left(tr\left({C}^{2}\right)\right)\right)$

$J=\mathrm{det}\left(J\right)={r}_{x}^{T}\left({r}_{y}×{r}_{z}\right)$

Each element must have a unique material identification number.

2. This material model can only be used with a fully parameterized element (BEAM12, QUAD12, TRIA12 and solid elements).
3. mu10 and mu10 are material constants. mu10 can be determined using a tension or compression tests, however, mu01 should be determined using a biaxial test.
4. YS lets you specify a maximum limit on the elastic strain that the component is allowed. If, during the simulation, the component strain (at any element in the component) exceeds this value, MotionSolve issues a warning message.