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)
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)
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 vs. 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 $μ_{i}$ $μ_{i}$ . Up to five pairs are permitted. (Real)
`ALFAi`	Parameter $α_{i}$ $α_{i}$ . 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)

Comments

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.
MATX42 is only applied in geometric nonlinear analysis subcases which are defined by ANALYSIS = EXPDYN. It is ignored for all other subcases.
The recommended Poisson's ratio for incompressible material is NU = 0.495. NU is defined on the corresponding MAT1.
The strain energy density $W$ $W$ is computed using:
(1)
$W = \sum_{p} \frac{μ_{p}}{α_{p}} ({\bar{λ}}_{1}^{α_{p}} + {\bar{λ}}_{2}^{α_{p}} + {\bar{λ}}_{3}^{α_{p}} - 3) + \frac{K}{2} {(J - 1)}^{2}$ $W = \sum_{p} \frac{μ_{p}}{α_{p}} ({\bar{λ}}_{1}^{α_{p}} + {\bar{λ}}_{2}^{α_{p}} + {\bar{λ}}_{3}^{α_{p}} - 3) + \frac{K}{2} {(J - 1)}^{2}$

Where, ${\bar{λ}}_{i}$ ${\bar{λ}}_{i}$ is the i^th principal engineering stretch ( ${\bar{λ}}_{i} = 1 + ε_{i}$ ${\bar{λ}}_{i} = 1 + ε_{i}$ , $ε_{i}$ $ε_{i}$ is the i^th principal engineering strain). The Cauchy stress is computed as:

(2)
$σ_{i} = \frac{λ_{i}}{J} \cdot \frac{\partial W}{\partial λ_{i}} - P$ $σ_{i} = \frac{λ_{i}}{J} \cdot \frac{\partial W}{\partial λ_{i}} - P$

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

(3)
$J = \frac{ρ_{0}}{ρ}$ $J = \frac{ρ_{0}}{ρ}$

The quantity, P is the pressure:

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

The Bulk Modulus, $K$ $K$ is:(4)
$K = μ \cdot \frac{2 (1 + υ)}{3 (1 - 2 υ)}$ $K = μ \cdot \frac{2 (1 + υ)}{3 (1 - 2 υ)}$

with the ground shear modulus $μ$ $μ$ :
An incompressible Mooney-Rivlin material is governed by:
W = C₁₀ (I₁ - 3) + C₀₁ (I₂ - 3)

Where, I_i is i^th invariant of the right-hand Cauchy-Green Tensor. It can be modeled using the following parameters:

$μ$ $μ$ ₁ = 2 * C₁₀

$μ$ $μ$ ₂ = -2 * C₀₁

α₁ = 2.0

α₂ = -2.0
Coefficients of the Prony series (G_i, T_i) 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' G_i and dampers $η$ $η$ _i):

Figure 1.

The hyperelastic ground shear modulus, $μ$ $μ$ is exactly the long-term shear modulus G∞ in the Maxwell model, and T_i is the relaxation time:(5)
$Ti = \frac{η i}{Gi}$ $Ti = \frac{η i}{Gi}$

The G_i and T_i values must be positive.
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)
$τ^{v} = {\sum_{i = 1}^{M} G_{i} \int_{0}^{t} e}^{- \frac{t - s}{τ_{i}}} \frac{d}{d s} [d e v (\bar{F} {\bar{F}}^{T})] d s$ $τ^{v} = {\sum_{i = 1}^{M} G_{i} \int_{0}^{t} e}^{- \frac{t - s}{τ_{i}}} \frac{d}{d s} [d e v (\bar{F} {\bar{F}}^{T})] d s$

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

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