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FATIGUE CRACK GROWTH MODEL RANDOM2 USER MANUAL
Prepared by :
Lola Boyce, Ph.D., P.E. Thomas B. Lovelace
APPENDIX 1 of Annual Report of Project Entitled
Development of Advanced Methodologies for Probabilistic Constitutive Relationships
of Material Strength Models
NASA G m t NO. NAG 3-867
Prepared for :
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Lewis Research Center Cleveland, OH 44135
The Division of Engineering The University of Texas at San Antonio
San Antonio, TX 78285 January, 1989
TABLE OF CONTENTS
SECTION PAGE
1.0 Introduction ............................................ 1
2.0 TheoreticalBackground. ................................... 2
3.0 InputDa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.0 Sample Problem ......................................... 6
5.0 References ............................................ 9
6.0
7.0 Appendix B: RANDOM2 Sample Problem: Source, Input and Output Files . . 11
8.0 Appendix C: IMSL Subroutine Calls from RANDOM2 ............... 33
9.0 Appendix D: SAS/GRAPH Program for RANDOW. ............... 34
Appendix A: Physical Quantities, Symbols, and Units ................ 10
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1.0 INTRODUCTION
This User Manual documents the FORTRAN program RANDOM. RANDOM2 is based on hcture mechanics using a probabilistic fatigue crack growth model. It predicts the random lifetime of an engine component to reach a given crack size (see Section 2.0, Theoretical Background).
Included in this Manual are details regarding the theoretical background of RANDOM2, input data instructions and a sample problem illustrating the use of m M 2 . Appendix A gives information on the physical quantities, their symbols, FORTRAN names, and both SI and U.S. Customary units. Appendix B includes photocopies of the actual computer printout carresponding to the sample problem. Appendices C and D detail the IMSL, Ver. 10 1, subroutines and functions called by RANDOM2 and a SAWGRAPH 2 program that can be used to plot both the probability density function @.df.) and the cumulative distribution function (cdf.).
1
2.0 THEORETICAL BACKGROUND
Fatigue crack growth data are usually presented as cycles, N, to reach a particular crack length, a. The initial crack size is ai. It is generally accepted that under constant amplitude alternating stress, fatigue crack growth can be related to stress intensity through a first order differential equation.3
WdN = C(AIQrn
where C is a material parameter, m is a material property (often a constant) and AK is the stress intensity range. Stress intensity range is given by
AK = Y A f i
where Y is a constant dependent upon component and crack geometry and ACJ is the constant amplitude alternating stress. Therefore, equation (1) can be written as
WdN = C ( Y A f i ) r n
Equation (2) can be integrated, from the initial crack length, ai, to the final crack length, af, to yield N, the number of cycles. The result is
1 1 N = CY?
(3)
Thus, equation (3) gives the "cycles to reach a given crack length."
Metallurgical evidence indicates that casting pores play a significant role in the high- cycle fatigue life of cast nickel base-superalloys, especially at high temperatures? The location and size of these fatigue crack-initiating pores vary greatly from one aerospace propulsion system component to another. This accounts for the large variability in fatigue life and leads to consideration of fatigue crack growth as a random phenomenon.
Fatigue life directly relates to casting pore size, and pore size can be used to determine initial crack size, ai. Thus, utilizing principles of both probabilistic analysis and fatigue crack growth, a quantitative probabilistic constitutive relationship between fatigue life and fracture mechanics parameters can be developed. Using the "randomized equation" approach, the fatigue crack growth model, given by equation (3) has the following form:
N = f(C,m,Ao,ai,apY) (4)
2
or, in general,
N = f(Xi), i = 1 ,...., 6, ( 5 )
where the Xi are the six independent variables in equations (3) and (4). Equation (3) is "randomized'' by assuming the first four variables in equation (4) to be random. Assuming a small crack in a relatively large component leads to assuming Y = 1.0, a deterministic value. A deterministic final crack size was chosen since experimental evidence indicated that it was relatively unimportant3
Probabilistic analysis, via simulation, yields the distribution of the dependent random variable, cycles, N. A probability density function @.d.f.) of cycles is generated using the maximum penalized likelihood method. Maximum penalized likelihood generates the pdf . estimate using the method of maximum likelihood together with a penalty function to smooth it.5
3
3.0 INPUTDATA
Data input for RANDOM2 is user friendly and easy to manipulate (see, for example, the file entitled NORMAL.INP, in Section 4.0). The first five lines of input have the same format, namely 2E12.4, and the last two lines differ. The last two lines of input have the formats 13,2XJ3,2X,2El2.4,2X,I3 and 13, respectively. A brief line by line description is given along with an example for each line (Note: the ruler is to aid the user in formatting and is not a part of the input). A table listing the physical quantities, their units and symbols is given in Appendix A.
1. Random Number Generator Seed, ISEED, and Sample Size, NTOT
ExAMPm
123456789012345678901234567890 1 40
2. MaterialProperty,RMM
EXAMPLE:
123456789012345678901234567890 28 .OE-01 1.4E-01
3. Initial Crack Size (pore Diameter), RAI
EXAMPLE:
123456789012345678901234567890 300 .OE-06 45.OE-06
4. Material Property, RCC
EXAMPLE:
123456789012345678901234567890 2.20E-11 0.22E-11
5. Stress Range, DELSIG
EXAMPLE:
123456789012345678901234567890 6.2E+02 6.2E+01
4
6. The DESPL parameters are NODE, INIT, ALPHA, EPS, MAXIT and are entered in that order as follows:
EXAMPLE:
1234567890123456789012345678901234567890 2 1 0 50 .OE-01 10.OE-05 3 0
7. The DESPL parameter, IOPI', is entered as follows:
EXAMPLE:
1234567890 2
5
4.0 SAMPLE PROBLEM FOR RANDOM2
The objective of this program is to predict the random lifetime, to reach a given crack size for an engine component. The theory is based on hcture mechanics, using a probabilistic fatigue crack growth model (see Section 2.0, Theoretical Background). RANDOM;! input parameters are given in Table Al.l . Note that the first four parameters are random. Their means and standard deviations are input by the user. The last two parameters, Af and Y, are deterministic and are fixed internally by the program. They are equal to the values shown in Table Al. l .
Table Al. 1 RANDOM2 sample problem input (SI units)
FORTRAN Name
RMM
AI
RCC
DELSIG
AF
YY
Dishbution Type
Mean
28.OE-01
3OO.OE-06
2.20E- 1 1
6.2E+02
2.OE-03
1 .o
Standard Deviation (value) (% of Mean)
1.4E-0 1 (5%)
45.OE-06 (15%)
0.22E- 1 1 (10%)
6.2E+01 (10%)
NIA
NIA
The input is entered in the following format in a file entitled NORMAL.INP.
1234567890123456789012345678901234567890 1 40
28.OE-01 1.4E-01 300 .OE-06 45.03-06 2.20E-11 0.22E-11 6.2E+02 6.2E+01
21 0 5O.OE-01 10.OE-05 30 2
6
Execution of RANDOM2 (source code entitled NR2.FOR) produces an output file entitled RANDM22 giving intermediate results (see Appendix B). Execution also produces the plotfiles OUT1 and OUT2 (see Appendix B). These files are used to plot the X and Y axes of the probability density function @.d.f.) and the cumulative distribution function (c.d.f.), respectively, generated by RANDOM2. The plots are drawn from the plotfiles by the SAS/GRAPH graphing program (see Appendix C). These plots for the sample problem are shown in Figures A 1.1 and A 1.2.
This same sample problem has been reported in Boyce and Chamis.6 There, however, it utilized U.S. Customary units and an older version of RANDOM2 (IMSL Version 9.2 subroutines).
E B 0
2 3 4 5 6
LOG OF CYCLES
Fig. A 1.1 p.d.f. of log of mechanical cycles for fatigue crack growth model, using maximum penalized likelihood.
7
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2 3 4 5
LOG OF CYCLES
Fig. A1.2 c.d.f. of log of mechanical cycles for fatigue crack growth model, using maximum penalized likelihood.
6
8
5.0 REFERENCES
IMSL, "STAT/LIBRARY, FORTRAN Subroutines for Statistical Analysis", Houston, Texas, 1987.
2 SAS Institute, Inc. SAWGRAPH User's Gu ide. Vers ion 5 Edition, Cary, NC: SAS Institute, Inc., 1985, p. 596.
3 Kozin, F. and Bogdanoff, J.K., "A Critical Analysis of Some Probabilistic Models of
4 Hoffeler, W., "High-Cycle Fatigue-Life of the Cast Nickel Base-Superalloys in 738 LC and IN 939," Me -al T-CQ 'ons A, Vol. 13A, July, 1982, pp. 1245-1255.
5 Scott, D.W., "Nonparametric Probability Density Estimation by Optimization Theoretic Techniques," NASA CR- 147763, April, 1976.
Fatigue Crack Growth," Vol. 14, 1981, pp. 55-89.
Boyce, L. and Chamis, C.C., "probabilistic Constitutive Relations for Cyclic Material Strength Models," Pmc- 29th Structures. Structural Dvnamics and Materials Conference, Williamsburg, VA, 1988.
9
6.0 APPENDIX A
PHYSICAL QUANTITIES, SYMBOLS, AND UNITS
The physical quantities, their symbols, and units for the fatigue crack growth model are given in the following table.
Table A1.2 Physical quantities, symbols, and units for fatigue crack growth model for RANDOM2
Physical Quantity
Material property
Initial Crack Size
Material Property
Alternating Stress
Final Crack Size
Geometry Dependent Constant
Theory
m
Ai
C
A 0
Af
symbol
Y
FORTRAN Name
RMM
RAI
RCC
DELSIG
AF
YY
1 0
units SI U.S.
m/cyclelM Pa m in/cycle/ksi in
m in
micycle in/cycle
M Pa ksi
m in
(dimensionless)
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7.0 APPENDIX B
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8.0 APPENDIX C
IMSL SUBROUTINE CALLS FROM RANDOM2
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1. RNSET - Initializes a random seed for use in the IMSL random number generators.
2. RNNOR - Generates pseudorandom numbers from a standard normal distribution using an inverse CDF method.
3. RNLNL - Generates pseudorandom numbers from a lognormal distribution.
4. DESPL - Performs nonparametric probability density function estimation by the penalized likelihood method.
5. GCDF - Evaluates a general continuous cumulative distribution function given ordinates of the density.
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9.0 APPENDIX D
SAMPLE SAWGRAPH (VER. 5.16) PROGRAM FOR RANDOM2
data a; INFJLE '0UTl.CPR' FIRSTOBS=zinput x y; GOPTIONS DEVICE=HW470; proc gplot;
axisl label=(h=l f=simplex 'LOG OF CYCLES') value=(h=l f=simplex);
axis2 value=(h=l f=simplex) label=none; plot y*x / haxis=axiSl vaxis=axis2; TITLE H=l A=90 F=SIMPLEX 'PROBABILITY DENSITY FUNCTION; symbol i=spline v=square;
data B; INFJLE '0Un.CPR' FIRSTOBS=2;input x y; proc gplot;
axisl label=(h=l f=simplex 'LOG OF CYCLES') value=(h=l &simplex);
axis2 value=(h=l f=simplex) label=none; plot y*x / haxis=axis 1 vaxis=axis2; "LE H=l A=90 F=SIMPLEX 'CUMULATIVE DISTRIBUTION FUNCITON; symbol i=spline v=square;
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