damage fiber exponential

Module: solid

Category: material

Type string: "damage fiber exponential"

Parameters

Name	Description	Default	Units
density	density	1	[M/L^3]
t0	t0	1e+09	[]
Dmax	Dmax	1	[]
beta_s	beta_s	0	[]
gamma_max	gamma_max	0	[]
D2_a	D2_a	0	[]
D2_b	D2_b	0	[]
D2_c	D2_c	0	[]
D2_d	D2_d	0	[]
D3_inf	D3_inf	0	[]
D3_g0	D3_g0	0	[]
D3_rg	D3_rg	1	[]
k1	k1	0	[]
k2	k2	0	[]
kappa	kappa	0	[]
fiber			[]

Description

The material type for Damage Fiber Exponential is damage fiber exponential.

The effective strain-energy function is given by,

\[ \Psi^{0}=\kappa I_{1}+(1-3\kappa)I_{4} \]

and

\[ m(P)=\frac{k_{1}}{2k_{2}}\left\{ \exp\left(k_{2}\left\langle P\right\rangle ^{2}\right)-1\right\}, \]

where \(a\) denotes the direction of the fibers. So that,

\[ \Psi=\frac{k_{1}}{2k_{2}}\left\{ \exp\left(k_{2}\left\langle \left(1-D_{a}\right)\left(\kappa I_{1}-\left(1-3\kappa\right)I_{4}\right)-1\right\rangle ^{2}\right)-1\right\} . \]

The Cauchy stress then takes on the following form,

\[ \sigma=\frac{2}{J}(1-D)\frac{dm}{dP}\left[\kappa b+(1-3\kappa)I_{4}m\right]. \]

Example:

<solid type="damage fiber exponential"> 
  <k1>1288.97</k1> 
  <k2>400</k2> 
  <kappa>0.2</kappa> 
  <t0>0.9</t0> 
  <Dmax>0.99</Dmax> 
  <beta_s>0.001</beta_s> 
  <gamma_max>6.67</gamma_max> 
  <fiber type="angles"> 
    <theta>-54.94</theta> 
    <phi>90</phi> 
  </fiber> 
</solid>