COMREL Sample "Fatigue": Program Handling
Input: Formulation of the Failure Function
The user defined functions and
the failure function in the state function window 
See how the name (character identifier) ‘A0’ of
the basic random variable aO (the initial crack length) is imported
from the stochastic model into the failure function by a simple mouse-click. For better readability the failure functions for
the Symbolic Processor are repeated here
DEFFUNC (1) () {check if all random quantities are positive}
= (A0 > 0)*(C > 0)
DEFFUNC (2) () {returns explosion time }
= -0.99 * A0^FUNC(5) / ( FUNC(5) * C*1E-13 * PI^(0.5*m)
* FUNC (99) * Rate )
DEFFUNC (3) () {check on a(t) explosion: returns 1 if no explosion }
= IF (m = 2,1,t < FUNC (2))
DEFFUNC (4) () {check right hand: must be positive}
= FUNC(98) > 0
DEFFUNC (5) () {returns k} = (2 - m) / 2
DEFFUNC (99) () {auxiliary quantity #1}
= (2.808*DeltaS)^m * GAMMA(1+m/2)
DEFFUNC (98) () {auxiliary quantity #2}
= C*1E-13 * FUNC(99) * t * Rate * PI^(0.5*m)
FLIM(1) {failure crit. on the form a_crit - a(t) < 0 }
= IF ( FUNC(1) * FUNC(3) * FUNC(4)
, IF ( ABS(m-2)>SQRT(COGEN)
, (Acrit^FUNC(5) - A0^FUNC(5)) / FUNC(5) - FUNC(98)
, LN(Acrit/A0) - FUNC(98)
)
, SQRT(-1)
)
Notes:
DEFFUNC(1): Note that this auxiliary function presents
an ‘and-gate’.
DEFFUNC(2): For numerical reasons the explosion time is multiplied by
0.99
DEFFUNC(3): Only if m>2, there is a danger of singularity in the failure
function.
DEFFUNC(4): All first four auxiliary user defined function are later
used for checks in If-functions
DEFFUNC(99): This corresponds to eq. (3)
DEFFUNC(98): Observe the scaling of variable C for numerical reasons.
FLIM(1): Just one failure function which employs all user DEFined FUNCtions
Input: The Stochastic Model
The stochastic model window 
This window provides an overview of the stochastic
model for all basic random variables (distribution type, input in moment-
or parameter-form etc.) and also lists all constant (deterministic)
values. In the lower part of the window detailed information about the
selected variable (‘A0’) is given. Here you also define new variables
or constants.
Notes to the Stochastic Model: The slope parameter
C
Starting from the failure events (eq.4), C must
be seen as a load variable. The common practice of assigning a normal
distribution to ln(C) means that C is lognormally distributed. If the
standard deviation s is defined for ln(C) one has a coefficient of variation
(CoV) 
s for C. For larger CoV (> 30%, say) this is problematic. The upper
tail of the density gets very fat for larger CoV allowing for extremely
large values of C. Note that the lognormal density f(x) 1/x exp[-(ln(x))2]. In essence, the lognormal model is not appropriate
for a load-type variable if Cov > 20%, say. Appropriate models are Normal or Weibull for C
(not ln(C)) with CoV =20% to 30%. Note the scaling of C in the failure
function in order to avoid round off errors in the stochastic model
definition for this variable.
Stochastic model: The initial crack size aO
For the very same reasons (load type ...) the lognormal
distribution for aO is not appropriate. Also the exponential
model is not supported by any theoretical arguing. An appropriate model
is the Rayleigh distribution with a shift parameter tau < 0 which
assures aO > 0; tau = 0 leads to a fixed CoV 52%. It then has only one parameter alpha and Mean alpha = (
/2)1/2.
Stochastic model: The stress process S
The stress process S(t) is modelled by a stationary
and ergodic normal narrow band process. For the time invariant case
at hand the stress ranges then are Rayleigh distributed. Integration
over time as required in eq.(1) yields an explicit expression (3), see
DEFFUNC(99). The mean value of the upcrossing rate 
is a constant with name ‘Rate’ and is used for a parameter study. For
the time invariant COMREL-TI the ‘Time’ t is also a constant formally
set to 1. The total number of upcrossings N(T) = t.
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