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 ( $μ_{01}$ $μ_{01}$ ).
mu10: Material constant ( $μ_{10}$ $μ_{10}$ ).
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"/>

Comments

This material element defines a hyper-elastic material that follows the Mooney-Rivlin material model law for the strain energy density function:
$U = μ_{10} ({\bar{I}}_{1} - 3) + μ_{01} ({\bar{I}}_{2} - 3) + \frac{k}{2} {(J - 1)}^{2}$ $U = μ_{10} ({\bar{I}}_{1} - 3) + μ_{01} ({\bar{I}}_{2} - 3) + \frac{k}{2} {(J - 1)}^{2}$

where

$μ = 2 (μ_{01} + μ_{10})$ $μ = 2 (μ_{01} + μ_{10})$ is the shear modulus

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

$v$ $v$ is the Poisson's ratio

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

$I_{1} = t r (C) = r_{x}^{T} r_{x} + r_{y}^{T} r_{y} + r_{z}^{T} r_{z}$ $I_{1} = t r (C) = r_{x}^{T} r_{x} + r_{y}^{T} r_{y} + r_{z}^{T} r_{z}$

$I_{2} = \frac{1}{2} ({(t r (C))}^{2} - (t r (C^{2})))$ $I_{2} = \frac{1}{2} ({(t r (C))}^{2} - (t r (C^{2})))$

$J = \det (J) = r_{x}^{T} (r_{y} \times r_{z})$ $J = \det (J) = r_{x}^{T} (r_{y} \times r_{z})$

Each element must have a unique material identification number.
This material model can only be used with a fully parameterized element (BEAM12, QUAD12, TRIA12 and solid elements).
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.
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.