## INITIAL_DAMAGE_RANDOM

#### Initial conditions

*INITIAL_DAMAGE_RANDOM
entype, enid, $a$, $b$, $D_{max}$, $R$, cid, did

### Parameter definition

Variable
Description
entype
Entity type
options: M, P, PS
enid
Entity ID
$a$
Defect distribution parameter
$b$
Defect distribution parameter
$D_{max}$
Maximum initial damage
$R$
cid
ID of a CURVE or FUNCTION defining yield stress (sigy0) as function of initial damage
default: not used
did
Damage variable ID
default: did = 1

### Description

This command is used to define a randomly distributed initial damage. A distribution function $f(D)$ describes the number of defects per unit volume of matter.

$f(D) = \left\{ \begin{array}{cc} a \cdot e^{-b D} & D \leq D_{max} \\ 0 & D > D_{max} \end{array} \right.$

Note that the maximum initial damage cannot be larger than $D_{max}$. The number of defects $N$ per unit volume of matter in the range $D_0$ to $D_{max}$ can be calculated by integrating $f(D)$ from $D_0$ to $D_{max}$:

$N = \displaystyle{\int_{D_0}^{D_{max}}} f(D) \mathrm{d} D$

Based on the assumed damage distribution $f(D)$ one can show that the probability $p$ of having at least one initial defect larger than or equal to $D_0$ in a volume $v$ is:

$p = 1 - \mathrm{e}^{-N \cdot v}$

This probability expression can be used to assign an initial damage level to each integration point in the model. The damage level is obtained by solving the expression for $D_0$ (given a random number $p$ and an integration point volume $v$).

Some failure models work with multiple damage variabeles. The parameter did can then be used to specify which damage variable to initialize.