damage fiber exp-linear

Module: solid

Category: material

Type string: "damage fiber exp-linear"

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	[]
c3	c3	0	[]
c4	c4	0	[]
c5	c5	0	[]
lambda	lambda	0	[]
fiber			[]

Description

This continuous damage formulation uses a slightly modified damage formulation. The total damage is here defined as,

\[ D=D_{1}+D_{2} \]

Here, \(D_{1}\) defines the same damage term as defined above. The \(D_{2}\) is defined as follows,

\[ D_{2}=D_{3}\left[a\left(\mathrm{exp}\left(b\beta\right)-1\right)+c\left(\mathrm{exp}\left(d\beta\right)-1\right)\right] \]

The material parameters a and c control the magnitude of the continuous damage of the two phenomena: 1) slow, constant increased in damage, 2) sharp jump in damage near failure. The other material parameters, b and d, control the rate of continuous damage accumulation for the two phenomena mentioned above, and is a function of the collagen discontinuous damage. Furthermore,

\[ D_{3}\left(\gamma\right)=\frac{D_{3\infty}}{1+exp\left(-\left(\gamma-\gamma_{0}\right)/r_{\gamma}\right)} \]

Here, \(D_{3\infty}\) is the maximum achievable discontinuous damage possible for the model, \(\gamma_{0}\) determines the shape of the curve, and \(r_{\gamma}\) controls the rate of discontinuous damage accumulation.

The effective strain-energy function is given by,

\[ \Psi^{0}=\sqrt{I_{4}} \]

and

\[ m\left(P\right)=\left\{ \begin{array}{cc} c_{3}\left(\exp\left(-c_{4}\right)\left(Ei\left(c_{4}\left(P+1\right)\right)-Ei\left(c_{4}\right)\right)-\log\left(P+1\right)\right), & P\leqq\lambda^{*}+1\\ c_{5}P+c_{6}\log\left(P+1\right)+d & P>\lambda^{*}+1 \end{array}\right. \]

Here, \(d\) is determined by requiring continuity at \(\lambda^{*}\).