Till now many researches have accomplished to predict stress intensity
factors during the crack growth in rotary disks and a lot of these researches
base on analytical methods (Roos and Griesinger, 1987). This equation
really is results due to analytical methods that are compared to numerical
analysis results. Also, weighted function method is used to predict stress
intensity factors (Lorenzo and Cartwright, 1994; Schineider and Danzer,
1989). Also, experimental methods are used by researches that it can mention
to standard samples to predict real stress intensity factors (Charls et
al., 1995). In this topic, crack position and number of cracks in
a rotary disk are studied such as radial or peripheral cracks (Wilson
and Meguid, 1995; Isida, 1995; Smith, 1985).
In this study, it will demonstrate how can achieve to algebraic equation
based on statistical resultant with utilizing algebraic methods. This
equation includes some effective parameters for determination of stress
intensity factor of a rotating disk. In this research, we will use a numerical
analysis simulator (FRANC2D software) to predict stress intensity factor
by an emulator (an equation for calculation of SIF) in cracked rotary
disks. Using analytical methods we can show that when we consider geometrical
parameters with unique values, only peculiar physical effective parameter
is Poisson`s ratio.
MATERIALS AND METHODS
Nowadays in optimization researches, reducing computational time and
economic cost and having good accuracy is more important. Emulators technology
is good example for these aims. So, emulators are useful and quick methods
in different optimization problems. In other words, emulators are statistical
functions and display behavior of case study, so, emulators can locate
to predict a simulator`s behavior. Thus, we can use the emulators as a
useful substitution for prediction of behavior of a simulator (Table
In this process, to create an emulator at first step, it must select
valid data, then we can make an emulator base on resultant data from sampling
(design points). Each design point contains effective parameters in problem
and equivalent output data from simulator. In this case, accuracy of the
emulator is important. Indeed accuracy of an emulator depends on effective
parameters in system and number of design points. To make use of emulators
can be obvious optimum designing in complex systems, so decreases costs
and computational time and risk in different problems.
||Comparison between an emulator and a simulator
CRACKED ROTARY DISKS WITH CENTRAL HOLE
Here, an algebraic emulator to predict crack growth behavior in rotary
disks with central hole is considered. We want to find equations for crack
growth behavior in rotary disks with central hole. Before this research,
many equations were designated, but these equations have derived from
analytical methods (Roos and Griesinger, 1987). Initially for study in
crack growth behavior in rotary disks with central hole, a model in FRANC2D
software with three main variables has considered, also effective geometrical
parameters show in Fig. 1 and Table 2.
FRANC2D is skilled software for analyzing crack growth in 2-dimensional
and 3-dimensional geometry. This software was made by mechanical department
in Cornell University and it is free to use. This software is able to
draft geometrical model, mesh generation, loading, creating crack and
show crack growth. Also this software is able to draft crack growth graphs.
Figure 2 shows of FRANC2D menus.
In crack growth study in rotary disks with central hole, 144 tests are
sufficient for extend studies. Totally 144 different samples with specified
effective parameters and equal distances have calculated (Table
2) and these samples have selected to analyze by FRANC2D software.
The sample is a disk that it`s radius and density is unique (i.e., one)
and it`s rotary velocity is 1 rad sec-1. Variation range of
variables has shown in Table 2.
Where, v is the Poisson`s ratio, Ri, inner radius of disk,
Ro, outer radius of disk and L is crack length.
In continue, behavior of each effective parameter is studied. Figure
3 shows extrapolate curve on many test points that stress intensity
factor has shown in terms of radius of the hole variations, also stress
intensity factor in the tip of crack has calculated in FRANC2D software
by finite element method.
Results are derived from finite element analysis in FRANC2D software
show that for Ri/Ro>0.1, Eq. 1
able to predict stress intensity factors in the tip of crack with good
||Schematic view of a rotary disk and effective parameters
in crack growth
||Variation range for each parameter
||A view of FRANC2D software
The main aim is finding equations that have minimum number of
factors and maximum of accuracy. Therefore, CurveExpert 1.3 software was
used because this software plot reputable curves with fitting curves on
experimental data and measures errors and finally selects the best of
all curves that has minimum error and minimum numbers of parameters (Table
2). Results are derived from this modeling show that Harris model
(Eq. 1) is valid for each value of L/Ro. Also, Eq. 2-4 are derived with CurveExpert 1.3 software.
||Stress intensity factor in terms of radius of the hole
||Variations of parameters c and b in terms of parameter
Table 3 shows variant values of coefficients for Eq.
1 due to L/Ro.
Equation 1 shows a relation between stress intensity
factor and (Ri/Ro) and in next stage for minimizing
effective independent variables in Eq. 1 we search a
relation between a, b and c parameters.
||Coefficients relate to Eq. 1 for v
||Variations of parameter a curve in terms of crack
Thus, according as Fig.
4 shows, variables b and c are obtained as functions of a.
Furthermore, according to Eq. 4, parameter a has considered
like Fig. 5 as function of crack length (L/Ro).
Also, Fig. 6 has illustrated stress intensity factor
in terms of radius ratio (Ri/Ro) and crack length
When, Eq. 2, 3 and 4
act on Eq. 1 then Eq. 5 will get, where
L is as (L/Ro) and Ri is as (Ri/Ro).
This recent equation is valid for v = 0.33 and it is obvious that for
different non-unique values, term of ρω2Ro5/2
will multiply on Eq. 5 (Table 4).
The results obtained from emulator and finite element software FRSANC2D
are compared in Table 5.
||Variations of stress intensity factor in terms of variations
of crack length for different radial ratios and v = 0.33
||Coefficients relate to Eq. 5 for v
||Comparison between results of emulator with results
of finite element analyzing (FRANC2D software)
The stress intensity factor variations in due to Poisson`s ratio simulator
compare with emulator accuracy is very small, therefore we ignore of the Poisson`s
ratio variations in calculations. So, Eq. 5 is a suitable
emulator to predict stress intensity factors in the tip of crack for geometrical
variations of cracked rotating disks. In this case study we can calculate stress
intensity factors quickly without using finite element software by replacing
geometrical parameters of crack and length of rotating disk in Eq.
In this research, geometrical parameters variations and Poisson`s ratio
variable are studied and we approached to an algebraic equation to predict
stress intensity factor for crack growth in rotary disks. At first, an
algebraic equation came to hand for v = 0.33, which results showed stress
intensity factor variations in simulator as compared to Emulator accuracy
is very small. So, a good Emulator has been achieved to predict stress
intensity factors in the tip of crack and finally results of simulator
(finite element software FRANC2D) and results of emulator (algebraic equation)
have been compared. The results obtained from two methods are too near
to others, thus the emulator acts very accurate.