# MATX42

Bulk Data Entry Defines additional material properties for Ogden, Mooney-Rivlin material for geometric nonlinear analysis. This material is used to model rubber, polymers, and elastomers.

## Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATX42 MID SCUT   TBID FBULK
LAW MU1 ALFA1 MU2 ALFA2 MU3 ALFA3
MU4 ALFA4 MU5 ALFA5
Optional continuation lines for prony value:
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
PRONY G1 T1 G2 T2 G3 T3
G4 T4 G5 T5 etc

## Example

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MAT1 102 10.0   0.495 6.0E-10
MATX42 102
LAW 0.10 2.0 -0.010 -2.0

## Definitions

Field Contents SI Unit Example
MID Material ID of the associated MAT1. 1

No default (Integer > 0)

SCUT Cut-off stress in tension.

Default = 1030 (Real ≥ 0)

TBID Identification number of a TABLES1 to define the bulk function f(J) that scales the bulk modulus versus relative volume. If TBID = 0, f(J) = const. = 1.0.

Default = 0 (Integer ≥ 0)

FBULK Scale factor for bulk function.

Default = 1.0 (Real > 0)

LAW Indicates that material parameters MUi and ALFAi follow.
MUi Parameter ${\mu }_{i}$ . Up to five pairs are permitted.

(Real)

ALFAi Parameter ${\alpha }_{i}$ . Up to five pairs are permitted.

(Real)

PRONY Indicates that prony model parameters Gi and Ti follow.
Gi Parameter Gi for prony model. 8 9

(Real)

Ti Parameter Ti for prony model. 8 9

(Real)

1. The material identification number must be that of an existing MAT1 Bulk Data Entry. Only one MATXi material extension can be associated with a particular MAT1.
2. MATX42 is only applied in geometric nonlinear analysis subcases which are defined by ANALYSIS = EXPDYN. It is ignored for all other subcases.
3. The recommended Poisson's ratio for incompressible material is NU = 0.495. NU is defined on the corresponding MAT1.
4. The strain energy density $W$ is computed using:
(1)
$W=\sum _{p}\frac{{\mu }_{p}}{{\alpha }_{p}}\left({\overline{\lambda }}_{1}^{{\alpha }_{p}}+{\overline{\lambda }}_{2}^{{\alpha }_{p}}+{\overline{\lambda }}_{3}^{{\alpha }_{p}}-3\right)+\frac{K}{2}{\left(J-1\right)}^{2}$

Where, ${\overline{\lambda }}_{i}$ is the ith principal engineering stretch ( ${\overline{\lambda }}_{i}=1+{\epsilon }_{i}$ , ${\epsilon }_{i}$ is the ith principal engineering strain). The Cauchy stress is computed as:

(2)
${\sigma }_{\text{i}}=\frac{{\lambda }_{\text{i}}}{\text{J}}\cdot \frac{\partial \text{W}}{\partial {\lambda }_{\text{i}}}-\text{P}$

with J = $\lambda$ 1 * $\lambda$ 2 * $\lambda$ 3 being the relative volume:

(3)
$\text{J}=\frac{{\rho }_{0}}{\rho }$

The quantity, P is the pressure:

P = K * FBULK * f (J) * (J - 1)

The Bulk Modulus, $\mathrm{K}$ is:(4)
$\text{K}=\mu \cdot \frac{2\left(1+\upsilon \right)}{3\left(1-2\upsilon \right)}$

with the ground shear modulus $\mu$ :

5. An incompressible Mooney-Rivlin material is governed by:

W = C10 (I1 - 3) + C01 (I2 - 3)

Where, Ii is ith invariant of the right-hand Cauchy-Green Tensor. It can be modeled using the following parameters:

$\mu$ 1 = 2 * C10

$\mu$ 2 = -2 * C01

α1 = 2.0

α2 = -2.0

6. Coefficients of the Prony series (Gi, Ti) are used to describe viscous effects using the Maxwell model (which can be described in a simplified manner as a system of n springs with stiffness' Gi and dampers $\eta$ i):
The hyperelastic ground shear modulus, $\mu$ is exactly the long-term shear modulus G∞ in the Maxwell model, and Ti is the relaxation time:(5)
$\mathit{Ti}=\frac{\eta i}{\mathit{Gi}}$

The Gi and Ti values must be positive.

7. Viscous effects are modeled through a convolution integral using Prony series. This is an extension of small strain theory (described in comment 6) to large nonlinear strain case. The Kirchhoff viscous stress is given by:
(6)
${\tau }^{v}={\sum _{i=1}^{M}{G}_{i}\underset{0}{\overset{t}{\int }}e}^{-\frac{t-s}{{\tau }_{i}}}\frac{d}{ds}\left[dev\left(\overline{F}{\overline{F}}^{T}\right)\right]ds$

with, F being the deformation gradient matrix, $\overline{\text{F}}={\text{J}}^{-\frac{1}{3}}\mathrm{F}$ and $\text{dev}\left(\overline{\mathrm{F}}{\overline{\mathrm{F}}}^{\text{T}}\right)$ denotes the deviatoric part of tensor $\overline{\mathrm{F}}{\overline{\mathrm{F}}}^{\mathrm{T}}$ .

The viscous-Cauchy stress is written as:(7)
${\sigma }^{v}\left(t\right)=\frac{1}{J}{\tau }^{v}\left(t\right)$
8. This card is represented as a material in HyperMesh.