- Email: [email protected]
Application of effective stress intensity factor crack closure model to evaluate train load sequence effects on crack growth rates
theoretical and
applied frac re n~chanics ELSEVIER
Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
Application of effective stress intensity factor crack closure model to evaluate train load sequence effects on crack growth rates D.Y. Jeong
*
U.S. Department of Transportation, Research and Special ProgramsAdministration, VolpeNational Transportation Systems Center, Cambridge, MA 02142, USA
Abstract An effective stress intensity factor crack closure model is studied in relation to results of laboratory spectrum crack growth tests on compact tension specimens (CTS) fabricated from rail steel. Comparison of model predictions with test results for crack growth life is effected by means of an analysis of a center-cracked tension panel (CCT) subjected to an equivalent stress spectrum. The trends of the model predictions and test results agree as to the effect of changing cycle order in the spectrum, but the actual effect on crack growth life in the laboratory tests is found to be much stronger than the effect predicted by the crack closure model using the effective stress intensity factor.
1. Introduction R e p e a t e d loading of a rail by the wheels of passing trains causes metal fatigue, which often leads to formation and propagation of a crack in the body of the rail. The amount of crack growth due to a particular wheel load depends on the history of previous loads, as well as the magnitude of the wheel load. Evidence for this load sequence effect in rail fatigue exists in the results of field tests of rail crack growth conducted on the Facility for Accelerated Service Testing (FAST) at the Transportation Test Center (TTC) in Pueblo, CO [1]. Additional evidence has been
* Tel. + 1 617 494 3654, fax + 1 617 494 3066.
found in laboratory tests which simulate the rail crack growth environment imposed by the FAST train and by other train makeups representing revenue service [2]. The research on fatigue crack growth has been long and extensive. However, the mechanics of load sequence effects on crack growth behavior are still not well understood. The cause for acceleration or retardation in fatigue crack growth behavior has been attributed to a p h e n o m e n o n known as crack closure. This p h e n o m e n o n was first studied in fatigue experiments on 2219-T851 aluminum alloy specimens [3,4]. It has been experimentally observed that the surfaces of a crack in a test specimen are fully open at maximum load, and as the load decreases, the crack surfaces come into contact at some load greater than the minimum, even if the entire stress cycle is in
0167-8442/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 8 4 4 2 ( 9 4 ) 0 0 0 4 7 - 6
44
D.Y. Jeong / Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
tension. In the ensuing cycle, the crack remains closed until the load reaches some value above the minimum. No crack growth is assumed to occur when the crack is closed. Another way to view the phenomenon is to consider the effects of local plasticity around the crack. Residual plastic strains develop in the wake of the advancing crack. The plastic flow in the crack wake is assumed to account for retardation, and the plasticity ahead of the crack tip is assumed to account for acceleration. Since the early experiments [3,4], the study of crack closure has remained largely experimental or phenomenological. Methods for calculating the crack opening stress have addressed constant amplitude loading, for the most part. These methods can be divided into two groups: analytical and those based on finite elements. For example, the finite element method was used in [5] to predict the crack opening stress for a center-cracked tension (CCT) specimen. However, the mesh size was found to have a strong influence on the results. An analytical model was developed in [6] based on a modified Dugdale plastic strip model of the crack-tip plastic zone [7]. Although other analytical models have been developed [8-10], none consider the influence of three dimensional constraint on the crack closure behavior. In the analytical model described in [6], the constraint parameter was assumed to vary depending on whether plane stress or plane strain conditions prevailed. While fatigue load sequence effects were studied in [11] by application of the critical available energy density and incremental theory of plasticity, this work explores the numerical stability of the effective stress intensity factor crack closure model [6] for assessing the crack opening stress and subsequent crack growth in a center-cracked panel of finite width. Two different load spectra were considered which describe the stress history produced in a rail head by two different train makeups: one representing field test conditions (referred to as the FAST makeup) and another representing revenue service conditions (referred to as the General Freight makeup). Furthermore, two different load sequences were considered for each spectrum: one where the actual load se-
quence was applied (referred to as the Real Sequence Order or RSO) and the other where the stresses are rearranged in an order of descending magnitude; i.e., maximum stresses are applied first and the minimum stresses applied last (referred to as Decreasing Maximum Stress or DMS). Standard compact tension specimens (CTS) were tested in [12] using these different spectra and load sequences. However, the analytical crack closure model could not be directly applied to the CTS because it was found to be numerically unstable when supplied with finite element approximations for the elastic displacement field at the plastic zone boundary. In order to obtain some quantitative results, the crack closure model was applied to a center-cracked tension (CCT) specimen. The CCT specimen has a much simpler configuration than the CTS and has the advantage of an exact analytical description for it deformation pattern t. The present crack closure model was found to be numerically stable when supplied with the analytical CCT specimen deformation. Although an exact comparison between prediction and experiment cannot be made, a qualitative comparison may be made.
2. Analytical
model
The analytical model used in this study was based on the model developed in [6] for a center-cracked panel. The model assumes that the rate of crack growth depends on opening and closure of the crack surfaces or crack closure. A p a r a m e t e r known as the crack opening stress is defined as measure of this opening and closure of the crack faces. Thus, the analysis is divided into two parts: the crack opening stress calculation and the crack growth computation. The specimen is assumed to have three distinct regions: - a plastic zone ahead of the crack tip; - a plastic zone in wake of the advancing crack;
t The center-cracked tension (CCT) specimen has generally been used in applications of crack closure to crack propagation analyses in the aerospace industry.
D.Y. Jeong/ Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
an elastic region in the remainder of the panel. The actual or physical crack has a dimension of 2a (Fig. 1). A fictitious crack of length 2(a + to) is assumed in the model where to is the length of -
the plastic zone ahead of each crack tip. The plastic zone is assumed to be loaded by a uniform flow stress. A modified Dugdale plastic strip model, which assumes small-scale yielding, is used O'max
O'ma x
t
tttt
•~ L-.- -~,,, ~. --<::::::>~a~ ",
t
t
O'f
t
b
b
......
(b) I
%,,
o-~,.
O'mi n
t
t
t
O'f
(a)
t
45
O'mi n
t
t
t
I-- 2d
t
t
t
O'contac t
O'contact
O'contac t
O'contact
t
I--- ~d----4 - - -
(c)
h
(d)
Fig. 1. Schematic of closure model for a center-cracked tension (CCT) specimen. (a) Crack surface displacements at maximum stress. (b) Uniform flow stress distribution along crack tip at maximum stress. (c) Crack surface displacements at minimum stress. (d) Contact stresses on crack surfaces at minimum stress.
D.Y. Jeong / Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
46
to calculate the length of the plastic zone. The plastic zone size is the length at constant flow stress such that the stress intensity factor due to constant flow stress equals the stress intensity factor due to the remotely applied load. In effect, the stress singularity at the fictitious crack tip is removed. For a finite width CCT panel, the plastic zone ahead of the crack tip at maximum stress, (rma×, can be calculated from
tip. Thus, the analysis is performed in a discrete manner. The length of each of the spring elements in the plastic zone ahead of the crack tip is determined by the crack opening displacement calculated at the maximum applied stress: N
L i =O'maxf(Xi) -
~_~ aorfg(xi, Xj), j-I
(2)
where b is the width of the panel, a is the constraint p a r a m e t e r (which has a value of 1 for plane stress or 3 for plane strain), and the flow stress, ~rf, is taken to be the average of the ultimate tensile strength and yield strength of the material. A schematic of the analytical model is shown in Fig. 1.
where f ( x ) and g(x i, %) are influence functions given in the Appendix. Also, N is the number of spring elements assumed in the plastic zones. The lengths of these spring elements are used as crack opening displacements for a crack that is partially loaded along the crack surfaces and remotely loaded by the minimum applied stress. However, the stresses applied to the crack surfaces (or contact stresses) are unknown, and must be found by solving the following system of equations: N,
2.1. Spring elements
E %'g(xi, Xj) =- O'minf(Xi) - t i , i-~l
w--a
--sin
J sin
b-
j
-1
'
(1)
The four-fold symmetry of the CCT specimen allows analysis of half the crack with one crack tip. The plastic zone ahead of and the wake behind the advancing crack tip are divided into several elements that are assumed to behave as rigid-perfectly plastic springs. The spring element at location x~ is assigned a width of 2wi. Referring to Fig. 2, the element widths become smaller the closer they are located to the physical crack
where N t is the total number of spring elements assumed in the model. An iterative solution must be used to find the contact stresses, ~ , which are bound by the flow stress and can only take compressive values. A complete set of bounds on the element stresses can thus be established. For elements in the plastic zone ahead of the crack tip (x i > a): if %. > ao-f, then %. = ao-f; if o) < - of, then % = - of.
O'ma x
O'~i, I
(3)
(4)
For elements along the crack surfaces (xj _< a): if % > O , t h e n % = O ; if ~ < -(re, then % = - o f .
(5)
Note that the constraint parameter, a, is only applied to the tensile stresses; i.e., the stress field is assumed to be uniform through the panel thickness when the crack is closed.
2.2. Effective stress intensity factor range Fig. 2. Crack surface displacements and spring elements along crack line. (a) Maximum applied stress. (b) Minimum applied stress.
Gauss Seidel iteration appears to be an appropriate algorithm for solving the system of equa-
D.Y. Jeong / Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
tions (3). An expression for calculating the crack opening stress, cro is derived by equating the stress intensity factors due to: (1) a remotely applied load that has a value of the minimum stress minus the crack opening stress; and (2) the contact stresses. Thus, for the CCT panel, the crack opening stress is determined from: O'o=O'min -
~
2%'[sin-1X+-sin -1Xy-],
(6)
j= l 'ff
where the summation is performed only for the elements along the physical crack surface, and: sin(~rx~:/b) XJ±=
(7)
sin(ara/b)'
in which (8)
xi ± = xj + wj.
The crack growth calculation is performed using a power law formula of the following functional form: da dN = C¢.AK~, (9) where a is the half crack length for a CCT specimen, Cen is the effective crack growth constant, A Kefr is the effective stress intensity factor range, and m is the crack growth exponent. For a CCT panel, the effective stress intensity factor range is: AK~fr = Aa, fr wa sec
,
(10)
where Acreff is the effective stress range which depends on the crack opening stress. For constant amplitude loading, the effective stress range is determined by: (11)
A O'eff = O'max -- O'o,
where O'max is the maximum variable amplitude loading, range, Ao',ff, varies for each depend on the previous load - If Ormin j < Oroj then
applied stress. For the effective stress cycle, and may also cycle:
Ao'effj = O'maxj -- O'oj.
(12)
-- If Orrninj > O'oj and ~m~x j < Crma~j-- 1 then Ao'eff j = O'maxj -- O'minj.
(13)
47
- If ~rmin j > Croj and Crmax j > Crrnaxj_ 1 then m AO'effj = [( O'maxy- O'oj )
-- ( O'mini- O'oj )
m ~ l/m ] '
(14) where Crmi. j, croj, and O'max j are respectively the minimum, opening, and maximum stresses for the jth cycle. The crack opening stress is calculated for each cycle in a given load spectrum (one block). Thus, the crack growth rate formula is applied on a cycle-by-cycle basis to determine the increment of crack growth for one spectrum block. An assumption is now made that the average crack opening stress for that spectrum block will be constant for 0.01 in. of crack growth. That is, the number of blocks required to grow the crack 0.01 in. is calculated based on the cycle-by-cycle analysis for one spectrum block.
3. Discussion of results The results of the analytical model are presented in the form of plots showing crack length as a function of the number of cycles or blocks. The half crack length has an initial length of 0.75 in. The different load spectra are applied until the crack extends 0.25 in. The width of the CCT specimen was assumed to be 10 in. The material properties assumed in the present analysis for rail steel are listed in Table 1. The crack opening stress was found to have a weak sensitivity to changes in these material properties. The effective crack growth constant in Eq. (9) is calculated based on an empirical fit of crack
Table 1 Mechanical properties for rail steel [13] Modulus of elasticity E (ksi)
Poisson's ratio v
Yield strength %ha (ksi)
Ultimate strength ~ult (ksi)
30 × 103
0.3
72.8
135
D. Y. Jeong / Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
48
Table 2 Crack growth parameters for closure model
ioo~
Effective crack growth constant
Crack growth exponent
Ceff
m
0.30312X 10 l0
3.655
:
Fast RSO / /
/i •
ooo o 90
.... ght R S O ~ / /
.....
/ /
I I
0 85
Freight
"~ /
•
growth data from experiments performed on rail steel in [14]:
//
//
da dN
= ( C ] R + C 2 ) A K m,
(15)
where C 1 = 1.537 × 10 -m, C 2 = 4.16 x 10 - ~ , and m = 3.655 are the values of the fitted parameters. The crack growth rates in Eqs. (9) and (15) are then equated to define I-R Cef f = ( C l R
-~- C 2 )
]m
] __'~o-~-max H
,
(16)
where the bars represent average quantities, °'maxtt is the largest maximum stress for a given loading spectrum, and R = Ormin//Ormax . The average stress ratios were computed, and the average opening stress was calculated from the compliance measurements of crack closure [12]. Thus, an effective crack growth constant was found for the four load spectra. The average of these four values was used for C~ff in the crack growth calculations. Thus, the assumed crack growth parameters for the crack closure analysis are listed in Table 2. As shown in Fig. 3 and Table 3, the predicted crack growth rate under RSO load sequence is higher than that predicted under DMS for both the FAST and general freight spectra. However, the difference between RSO and DMS is greater for the FAST spectrum than for general freight.
Number of spectrum blocks x 10 I
Fig. 3. Life prediction based on crack closure model.
The D M S / R S O life ratios for the two spectra are 1.16 and 1.05, respectively. Figs. 4 and 5 illustrate the histories of calculated crack opening stress for the FAST and general freight spectra, respectively. In both cases, the opening stress is smaller under the RSO spectrum. Changing to DMS load sequence increases the opening stress by about 10% for the FAST spectrum, and by about 5% for the general freight spectrum. The trend of these results generally agrees with the life trends. Table 4 compares the predicted D M S / R S O life ratios with the results from laboratory tests [12]. The trends are similar, but the difference between RSO and DMS life for the FAST spec-
65
DMS
60
/
'
/
RSO
Table 3 Number of blocks required for crack growth from 0.75 to 1.00 in. Spectrum
Life (blocks)
FAST RSO FAST DMS General freight RSO General freight DMS
1375 1601 4651 4912
/
Crack length a x 10 ~ (in)
Fig. 4. Crack opening stress histories predicted for FAST load spectrum.
D.Y. Jeong / Theoreticaland Applied Fracture Mechanics 22 (1995) 43-50 Table 5 Plastic zone sizes for two crack configurations
6.5
6.0
5.5
--~
.......
_.
r .....
- ...........
,
/-
81.0
910
81.5
9!5
Constraint condition
Center crack (crack length = 2a)
Edge crack (crack length = a)
Plane stress Plane strain
0.22 0.021
0.22 0.025
DMS RSO
5.0 7.5
49
11.0 ~
Crack length a x 10 1 (in)
Fig. 5. Crack opening stress histories predicted for general freight spectrum.
the center-cracked panel and, perhaps, the single edge cracked panel, which have closed form expressions for the necessary stress intensity factors. For instance, the stress intensity factor for an edge crack of length a in a semi-infinite plane loaded by remote stress, o-=, is given by [15] as: K~ = 1.1215tr°~x/~b-.
Table 4 Comparison with experimental results D M S / R S O life ratio
Closure model Experiment (0.7-1.04 in.) Experiment (0.7-1.39 in.)
FAST spectrum
General freight spectrum
1.16 2.00 1.95
1.05 0.87 1.04
trum is much larger in the test results than in the calculations.
4. Conclusions The results of the analytical model for crack closure and crack growth appear to be encouraging. The analytical model can be modified for other crack configurations if the solutions for stress intensity factor and the crack surface displacements exist under remotely applied loading and for partially loaded cracks. Unfortunately, closed form solutions do not exist for many crack configurations, and alternative computational models must be adopted instead. At present, the only practical applications of the analytical crack closure model appear to be
(17)
Some preliminary calculations have been made for the edge crack geometry. Under plane strain conditions, the plastic zone size was found to be larger for the edge crack than for the CCT panel at the same nominal stress, which was assumed to be one-third the flow stress (see Table 5). The implications of the plastic zone size on the crack opening stress are not known, at this time. However, if the crack opening stress is found to be lower for the edge crack configuration, the fatigue crack will grow faster. Although stress intensity factor solutions exist for a partially loaded edge crack in a semi-infinite medium, the crack surface displacements must also be known in order to determine the influence coefficients, f ( x i) and g(xi, X j). Although the results presented in this paper are encouraging, the question is still open as to whether crack closure is the sole cause for delay in crack growth behavior [16].
Acknowledgements The work reported in this paper was performed while the author, on leave from the U.S. Department of Transportation, was studying at the Institute of Fracture and Solid Mechanics in Lehigh University. The author sincerely acknowledges the support and guidance from Dr. Oscar Orringer and Prof. George C. Sih.
50
D.Y. Jeong / Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
Appendix. Influence functions for center-cracked tension panel T h e influence functions for analyzing the center crack tension (CCT) specimen are:
sec( d ]
2(1 - p2) [
(A.1) and
g( xi, xj) = G( xi, xj) + G ( - x i , xj),
(A.2)
in which
G( xi, x i) 2(1 - u 2) [ -
cosh -
-"¥
d2 --
dlx~-xi[
sin- i X/+ _ sin- l X 7 sin_-~x/~/--~-
X
sec
Xj X i
d I xT- x~
-(x;-x Dcosh-' ×
-~
1
,
s~n-l(-~/d)
]
1 (A.3)
where E is the modulus of elasticity, d = a + to, ~, = 0 for plane stress or v is Poisson's ratio for plane strain. Expressions for Xj +- and xj +- were defined in Eqs. (7) and (8), respectively.
References [1] O. Orringer, Y.H. Tang, J.E. Gordon, D.Y. Jeong, J.M. Morris and A.B. Perlman, Crack propagation life of detail fractures in rails, Final Report, D O T / F R A / ORD-88/13 (1988). [2] D.A. Jablonski, Y.H. Tang and R.M. Pelloux, Simulation
of railroad crack growth life using laboratory specimens, Theoretical and Applied Fracture Mechanics 14, 27-86 (1990). [3] W. Elber, Fatigue crack closure under cyclic tension, Engineering Fracture Mechanics 2, 37-45 (1970). [4] W. Elber, The significance of fatigue crack closure, Damage Tolerance in Aircraft Structures, ASTM STP 486 (American Society for Testing and Materials, Philadelphia PA, 1971) pp. 230-242. [5] J.C. Newman, A finite element analysis of fatigue crack closure, Mechanics of Crack Growth, ASTM STP 590 (American Society for Testing and Materials, Philadelphia PA, 1976) pp. 280-301. [6] J.C. Newman, A crack closure model for predicting fatigue crack growth under aircraft spectrum loading, Methods and Models for Predicting Fatigue Crack Growth Under Random Loading, ASTM STP 748 (American Society for Testing and Materials, Philadelphia PA, 1981) pp. 53-84. [7] D.S. Dugdale, Yielding of steel plates containing slits, Journal of the Mechanics and Physics of Solids 8, 100-108 ( 1960). [8] H.D. Dill and C.R. Saff, Spectrum crack growth prediction method based on crack surface displacement and contact analyses, Fatigue Crack Growth Under Spectrum Loads, ASTM STP 595 (American Society for Testing and Materials, Philadelphia PA, 1976) pp. 306-319. [9] B. Budiansky and J.W. Hutchinson, Analysis of closure in fatigue crack growth, Journal of Applied Mechanics 45, 267-276 (1978). [10] H. Fuhring and T. Seeger, Dugdale crack closure analysis of fatigue cracks under constant amplitude loading, Engineering Fracture Mechanics 11, 99-122 (1979). [11] G.C. Sih and D.Y. Jeong, Fatigue load sequence effect ranked by critical available energy density, Theoretical and Applied Fracture Mechanics 14, 141-151 (1990). [12] D.A. Jablonski and R.M. Pelloux, Effect of train load spectra on crack growth in rail steel, Residual Stress in Rails, Vol. I (Kluwer, 1992) pp. 81-98. [13] J.J. Scutti, R.M. Pelloux and R. Fuquen-Moleno, Fatigue behavior of a rail steel, Fatigue and Fracture of Engineering Materials and Structures 7, 122-135 (1984). [14] B.G. Journet, Fatigue crack propagation under complex loading, Ph.D. Thesis, Department of Materials Science and Engineering, Massachusetts Institute of Technology (1986). [15] R.J. Hartranft and G.C. Sih, Solving edge and surface crack problems by an alternating method, Mechanics of Fracture I: Methods of Analysis and Solutions of Crack Problems (Noordoff, 1973) pp. 179-238. [16] T.T. Shih and R.P. Wei, A study of crack closure in fatigue, Engineering Fracture Mechanics 6, 19-32 (1974).
applied frac re n~chanics ELSEVIER
Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
Application of effective stress intensity factor crack closure model to evaluate train load sequence effects on crack growth rates D.Y. Jeong
*
U.S. Department of Transportation, Research and Special ProgramsAdministration, VolpeNational Transportation Systems Center, Cambridge, MA 02142, USA
Abstract An effective stress intensity factor crack closure model is studied in relation to results of laboratory spectrum crack growth tests on compact tension specimens (CTS) fabricated from rail steel. Comparison of model predictions with test results for crack growth life is effected by means of an analysis of a center-cracked tension panel (CCT) subjected to an equivalent stress spectrum. The trends of the model predictions and test results agree as to the effect of changing cycle order in the spectrum, but the actual effect on crack growth life in the laboratory tests is found to be much stronger than the effect predicted by the crack closure model using the effective stress intensity factor.
1. Introduction R e p e a t e d loading of a rail by the wheels of passing trains causes metal fatigue, which often leads to formation and propagation of a crack in the body of the rail. The amount of crack growth due to a particular wheel load depends on the history of previous loads, as well as the magnitude of the wheel load. Evidence for this load sequence effect in rail fatigue exists in the results of field tests of rail crack growth conducted on the Facility for Accelerated Service Testing (FAST) at the Transportation Test Center (TTC) in Pueblo, CO [1]. Additional evidence has been
* Tel. + 1 617 494 3654, fax + 1 617 494 3066.
found in laboratory tests which simulate the rail crack growth environment imposed by the FAST train and by other train makeups representing revenue service [2]. The research on fatigue crack growth has been long and extensive. However, the mechanics of load sequence effects on crack growth behavior are still not well understood. The cause for acceleration or retardation in fatigue crack growth behavior has been attributed to a p h e n o m e n o n known as crack closure. This p h e n o m e n o n was first studied in fatigue experiments on 2219-T851 aluminum alloy specimens [3,4]. It has been experimentally observed that the surfaces of a crack in a test specimen are fully open at maximum load, and as the load decreases, the crack surfaces come into contact at some load greater than the minimum, even if the entire stress cycle is in
0167-8442/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 8 4 4 2 ( 9 4 ) 0 0 0 4 7 - 6
44
D.Y. Jeong / Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
tension. In the ensuing cycle, the crack remains closed until the load reaches some value above the minimum. No crack growth is assumed to occur when the crack is closed. Another way to view the phenomenon is to consider the effects of local plasticity around the crack. Residual plastic strains develop in the wake of the advancing crack. The plastic flow in the crack wake is assumed to account for retardation, and the plasticity ahead of the crack tip is assumed to account for acceleration. Since the early experiments [3,4], the study of crack closure has remained largely experimental or phenomenological. Methods for calculating the crack opening stress have addressed constant amplitude loading, for the most part. These methods can be divided into two groups: analytical and those based on finite elements. For example, the finite element method was used in [5] to predict the crack opening stress for a center-cracked tension (CCT) specimen. However, the mesh size was found to have a strong influence on the results. An analytical model was developed in [6] based on a modified Dugdale plastic strip model of the crack-tip plastic zone [7]. Although other analytical models have been developed [8-10], none consider the influence of three dimensional constraint on the crack closure behavior. In the analytical model described in [6], the constraint parameter was assumed to vary depending on whether plane stress or plane strain conditions prevailed. While fatigue load sequence effects were studied in [11] by application of the critical available energy density and incremental theory of plasticity, this work explores the numerical stability of the effective stress intensity factor crack closure model [6] for assessing the crack opening stress and subsequent crack growth in a center-cracked panel of finite width. Two different load spectra were considered which describe the stress history produced in a rail head by two different train makeups: one representing field test conditions (referred to as the FAST makeup) and another representing revenue service conditions (referred to as the General Freight makeup). Furthermore, two different load sequences were considered for each spectrum: one where the actual load se-
quence was applied (referred to as the Real Sequence Order or RSO) and the other where the stresses are rearranged in an order of descending magnitude; i.e., maximum stresses are applied first and the minimum stresses applied last (referred to as Decreasing Maximum Stress or DMS). Standard compact tension specimens (CTS) were tested in [12] using these different spectra and load sequences. However, the analytical crack closure model could not be directly applied to the CTS because it was found to be numerically unstable when supplied with finite element approximations for the elastic displacement field at the plastic zone boundary. In order to obtain some quantitative results, the crack closure model was applied to a center-cracked tension (CCT) specimen. The CCT specimen has a much simpler configuration than the CTS and has the advantage of an exact analytical description for it deformation pattern t. The present crack closure model was found to be numerically stable when supplied with the analytical CCT specimen deformation. Although an exact comparison between prediction and experiment cannot be made, a qualitative comparison may be made.
2. Analytical
model
The analytical model used in this study was based on the model developed in [6] for a center-cracked panel. The model assumes that the rate of crack growth depends on opening and closure of the crack surfaces or crack closure. A p a r a m e t e r known as the crack opening stress is defined as measure of this opening and closure of the crack faces. Thus, the analysis is divided into two parts: the crack opening stress calculation and the crack growth computation. The specimen is assumed to have three distinct regions: - a plastic zone ahead of the crack tip; - a plastic zone in wake of the advancing crack;
t The center-cracked tension (CCT) specimen has generally been used in applications of crack closure to crack propagation analyses in the aerospace industry.
D.Y. Jeong/ Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
an elastic region in the remainder of the panel. The actual or physical crack has a dimension of 2a (Fig. 1). A fictitious crack of length 2(a + to) is assumed in the model where to is the length of -
the plastic zone ahead of each crack tip. The plastic zone is assumed to be loaded by a uniform flow stress. A modified Dugdale plastic strip model, which assumes small-scale yielding, is used O'max
O'ma x
t
tttt
•~ L-.- -~,,, ~. --<::::::>~a~ ",
t
t
O'f
t
b
b
......
(b) I
%,,
o-~,.
O'mi n
t
t
t
O'f
(a)
t
45
O'mi n
t
t
t
I-- 2d
t
t
t
O'contac t
O'contact
O'contac t
O'contact
t
I--- ~d----4 - - -
(c)
h
(d)
Fig. 1. Schematic of closure model for a center-cracked tension (CCT) specimen. (a) Crack surface displacements at maximum stress. (b) Uniform flow stress distribution along crack tip at maximum stress. (c) Crack surface displacements at minimum stress. (d) Contact stresses on crack surfaces at minimum stress.
D.Y. Jeong / Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
46
to calculate the length of the plastic zone. The plastic zone size is the length at constant flow stress such that the stress intensity factor due to constant flow stress equals the stress intensity factor due to the remotely applied load. In effect, the stress singularity at the fictitious crack tip is removed. For a finite width CCT panel, the plastic zone ahead of the crack tip at maximum stress, (rma×, can be calculated from
tip. Thus, the analysis is performed in a discrete manner. The length of each of the spring elements in the plastic zone ahead of the crack tip is determined by the crack opening displacement calculated at the maximum applied stress: N
L i =O'maxf(Xi) -
~_~ aorfg(xi, Xj), j-I
(2)
where b is the width of the panel, a is the constraint p a r a m e t e r (which has a value of 1 for plane stress or 3 for plane strain), and the flow stress, ~rf, is taken to be the average of the ultimate tensile strength and yield strength of the material. A schematic of the analytical model is shown in Fig. 1.
where f ( x ) and g(x i, %) are influence functions given in the Appendix. Also, N is the number of spring elements assumed in the plastic zones. The lengths of these spring elements are used as crack opening displacements for a crack that is partially loaded along the crack surfaces and remotely loaded by the minimum applied stress. However, the stresses applied to the crack surfaces (or contact stresses) are unknown, and must be found by solving the following system of equations: N,
2.1. Spring elements
E %'g(xi, Xj) =- O'minf(Xi) - t i , i-~l
w--a
--sin
J sin
b-
j
-1
'
(1)
The four-fold symmetry of the CCT specimen allows analysis of half the crack with one crack tip. The plastic zone ahead of and the wake behind the advancing crack tip are divided into several elements that are assumed to behave as rigid-perfectly plastic springs. The spring element at location x~ is assigned a width of 2wi. Referring to Fig. 2, the element widths become smaller the closer they are located to the physical crack
where N t is the total number of spring elements assumed in the model. An iterative solution must be used to find the contact stresses, ~ , which are bound by the flow stress and can only take compressive values. A complete set of bounds on the element stresses can thus be established. For elements in the plastic zone ahead of the crack tip (x i > a): if %. > ao-f, then %. = ao-f; if o) < - of, then % = - of.
O'ma x
O'~i, I
(3)
(4)
For elements along the crack surfaces (xj _< a): if % > O , t h e n % = O ; if ~ < -(re, then % = - o f .
(5)
Note that the constraint parameter, a, is only applied to the tensile stresses; i.e., the stress field is assumed to be uniform through the panel thickness when the crack is closed.
2.2. Effective stress intensity factor range Fig. 2. Crack surface displacements and spring elements along crack line. (a) Maximum applied stress. (b) Minimum applied stress.
Gauss Seidel iteration appears to be an appropriate algorithm for solving the system of equa-
D.Y. Jeong / Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
tions (3). An expression for calculating the crack opening stress, cro is derived by equating the stress intensity factors due to: (1) a remotely applied load that has a value of the minimum stress minus the crack opening stress; and (2) the contact stresses. Thus, for the CCT panel, the crack opening stress is determined from: O'o=O'min -
~
2%'[sin-1X+-sin -1Xy-],
(6)
j= l 'ff
where the summation is performed only for the elements along the physical crack surface, and: sin(~rx~:/b) XJ±=
(7)
sin(ara/b)'
in which (8)
xi ± = xj + wj.
The crack growth calculation is performed using a power law formula of the following functional form: da dN = C¢.AK~, (9) where a is the half crack length for a CCT specimen, Cen is the effective crack growth constant, A Kefr is the effective stress intensity factor range, and m is the crack growth exponent. For a CCT panel, the effective stress intensity factor range is: AK~fr = Aa, fr wa sec
,
(10)
where Acreff is the effective stress range which depends on the crack opening stress. For constant amplitude loading, the effective stress range is determined by: (11)
A O'eff = O'max -- O'o,
where O'max is the maximum variable amplitude loading, range, Ao',ff, varies for each depend on the previous load - If Ormin j < Oroj then
applied stress. For the effective stress cycle, and may also cycle:
Ao'effj = O'maxj -- O'oj.
(12)
-- If Orrninj > O'oj and ~m~x j < Crma~j-- 1 then Ao'eff j = O'maxj -- O'minj.
(13)
47
- If ~rmin j > Croj and Crmax j > Crrnaxj_ 1 then m AO'effj = [( O'maxy- O'oj )
-- ( O'mini- O'oj )
m ~ l/m ] '
(14) where Crmi. j, croj, and O'max j are respectively the minimum, opening, and maximum stresses for the jth cycle. The crack opening stress is calculated for each cycle in a given load spectrum (one block). Thus, the crack growth rate formula is applied on a cycle-by-cycle basis to determine the increment of crack growth for one spectrum block. An assumption is now made that the average crack opening stress for that spectrum block will be constant for 0.01 in. of crack growth. That is, the number of blocks required to grow the crack 0.01 in. is calculated based on the cycle-by-cycle analysis for one spectrum block.
3. Discussion of results The results of the analytical model are presented in the form of plots showing crack length as a function of the number of cycles or blocks. The half crack length has an initial length of 0.75 in. The different load spectra are applied until the crack extends 0.25 in. The width of the CCT specimen was assumed to be 10 in. The material properties assumed in the present analysis for rail steel are listed in Table 1. The crack opening stress was found to have a weak sensitivity to changes in these material properties. The effective crack growth constant in Eq. (9) is calculated based on an empirical fit of crack
Table 1 Mechanical properties for rail steel [13] Modulus of elasticity E (ksi)
Poisson's ratio v
Yield strength %ha (ksi)
Ultimate strength ~ult (ksi)
30 × 103
0.3
72.8
135
D. Y. Jeong / Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
48
Table 2 Crack growth parameters for closure model
ioo~
Effective crack growth constant
Crack growth exponent
Ceff
m
0.30312X 10 l0
3.655
:
Fast RSO / /
/i •
ooo o 90
.... ght R S O ~ / /
.....
/ /
I I
0 85
Freight
"~ /
•
growth data from experiments performed on rail steel in [14]:
//
//
da dN
= ( C ] R + C 2 ) A K m,
(15)
where C 1 = 1.537 × 10 -m, C 2 = 4.16 x 10 - ~ , and m = 3.655 are the values of the fitted parameters. The crack growth rates in Eqs. (9) and (15) are then equated to define I-R Cef f = ( C l R
-~- C 2 )
]m
] __'~o-~-max H
,
(16)
where the bars represent average quantities, °'maxtt is the largest maximum stress for a given loading spectrum, and R = Ormin//Ormax . The average stress ratios were computed, and the average opening stress was calculated from the compliance measurements of crack closure [12]. Thus, an effective crack growth constant was found for the four load spectra. The average of these four values was used for C~ff in the crack growth calculations. Thus, the assumed crack growth parameters for the crack closure analysis are listed in Table 2. As shown in Fig. 3 and Table 3, the predicted crack growth rate under RSO load sequence is higher than that predicted under DMS for both the FAST and general freight spectra. However, the difference between RSO and DMS is greater for the FAST spectrum than for general freight.
Number of spectrum blocks x 10 I
Fig. 3. Life prediction based on crack closure model.
The D M S / R S O life ratios for the two spectra are 1.16 and 1.05, respectively. Figs. 4 and 5 illustrate the histories of calculated crack opening stress for the FAST and general freight spectra, respectively. In both cases, the opening stress is smaller under the RSO spectrum. Changing to DMS load sequence increases the opening stress by about 10% for the FAST spectrum, and by about 5% for the general freight spectrum. The trend of these results generally agrees with the life trends. Table 4 compares the predicted D M S / R S O life ratios with the results from laboratory tests [12]. The trends are similar, but the difference between RSO and DMS life for the FAST spec-
65
DMS
60
/
'
/
RSO
Table 3 Number of blocks required for crack growth from 0.75 to 1.00 in. Spectrum
Life (blocks)
FAST RSO FAST DMS General freight RSO General freight DMS
1375 1601 4651 4912
/
Crack length a x 10 ~ (in)
Fig. 4. Crack opening stress histories predicted for FAST load spectrum.
D.Y. Jeong / Theoreticaland Applied Fracture Mechanics 22 (1995) 43-50 Table 5 Plastic zone sizes for two crack configurations
6.5
6.0
5.5
--~
.......
_.
r .....
- ...........
,
/-
81.0
910
81.5
9!5
Constraint condition
Center crack (crack length = 2a)
Edge crack (crack length = a)
Plane stress Plane strain
0.22 0.021
0.22 0.025
DMS RSO
5.0 7.5
49
11.0 ~
Crack length a x 10 1 (in)
Fig. 5. Crack opening stress histories predicted for general freight spectrum.
the center-cracked panel and, perhaps, the single edge cracked panel, which have closed form expressions for the necessary stress intensity factors. For instance, the stress intensity factor for an edge crack of length a in a semi-infinite plane loaded by remote stress, o-=, is given by [15] as: K~ = 1.1215tr°~x/~b-.
Table 4 Comparison with experimental results D M S / R S O life ratio
Closure model Experiment (0.7-1.04 in.) Experiment (0.7-1.39 in.)
FAST spectrum
General freight spectrum
1.16 2.00 1.95
1.05 0.87 1.04
trum is much larger in the test results than in the calculations.
4. Conclusions The results of the analytical model for crack closure and crack growth appear to be encouraging. The analytical model can be modified for other crack configurations if the solutions for stress intensity factor and the crack surface displacements exist under remotely applied loading and for partially loaded cracks. Unfortunately, closed form solutions do not exist for many crack configurations, and alternative computational models must be adopted instead. At present, the only practical applications of the analytical crack closure model appear to be
(17)
Some preliminary calculations have been made for the edge crack geometry. Under plane strain conditions, the plastic zone size was found to be larger for the edge crack than for the CCT panel at the same nominal stress, which was assumed to be one-third the flow stress (see Table 5). The implications of the plastic zone size on the crack opening stress are not known, at this time. However, if the crack opening stress is found to be lower for the edge crack configuration, the fatigue crack will grow faster. Although stress intensity factor solutions exist for a partially loaded edge crack in a semi-infinite medium, the crack surface displacements must also be known in order to determine the influence coefficients, f ( x i) and g(xi, X j). Although the results presented in this paper are encouraging, the question is still open as to whether crack closure is the sole cause for delay in crack growth behavior [16].
Acknowledgements The work reported in this paper was performed while the author, on leave from the U.S. Department of Transportation, was studying at the Institute of Fracture and Solid Mechanics in Lehigh University. The author sincerely acknowledges the support and guidance from Dr. Oscar Orringer and Prof. George C. Sih.
50
D.Y. Jeong / Theoretical and Applied Fracture Mechanics 22 (1995) 43-50
Appendix. Influence functions for center-cracked tension panel T h e influence functions for analyzing the center crack tension (CCT) specimen are:
sec( d ]
2(1 - p2) [
(A.1) and
g( xi, xj) = G( xi, xj) + G ( - x i , xj),
(A.2)
in which
G( xi, x i) 2(1 - u 2) [ -
cosh -
-"¥
d2 --
dlx~-xi[
sin- i X/+ _ sin- l X 7 sin_-~x/~/--~-
X
sec
Xj X i
d I xT- x~
-(x;-x Dcosh-' ×
-~
1
,
s~n-l(-~/d)
]
1 (A.3)
where E is the modulus of elasticity, d = a + to, ~, = 0 for plane stress or v is Poisson's ratio for plane strain. Expressions for Xj +- and xj +- were defined in Eqs. (7) and (8), respectively.
References [1] O. Orringer, Y.H. Tang, J.E. Gordon, D.Y. Jeong, J.M. Morris and A.B. Perlman, Crack propagation life of detail fractures in rails, Final Report, D O T / F R A / ORD-88/13 (1988). [2] D.A. Jablonski, Y.H. Tang and R.M. Pelloux, Simulation
of railroad crack growth life using laboratory specimens, Theoretical and Applied Fracture Mechanics 14, 27-86 (1990). [3] W. Elber, Fatigue crack closure under cyclic tension, Engineering Fracture Mechanics 2, 37-45 (1970). [4] W. Elber, The significance of fatigue crack closure, Damage Tolerance in Aircraft Structures, ASTM STP 486 (American Society for Testing and Materials, Philadelphia PA, 1971) pp. 230-242. [5] J.C. Newman, A finite element analysis of fatigue crack closure, Mechanics of Crack Growth, ASTM STP 590 (American Society for Testing and Materials, Philadelphia PA, 1976) pp. 280-301. [6] J.C. Newman, A crack closure model for predicting fatigue crack growth under aircraft spectrum loading, Methods and Models for Predicting Fatigue Crack Growth Under Random Loading, ASTM STP 748 (American Society for Testing and Materials, Philadelphia PA, 1981) pp. 53-84. [7] D.S. Dugdale, Yielding of steel plates containing slits, Journal of the Mechanics and Physics of Solids 8, 100-108 ( 1960). [8] H.D. Dill and C.R. Saff, Spectrum crack growth prediction method based on crack surface displacement and contact analyses, Fatigue Crack Growth Under Spectrum Loads, ASTM STP 595 (American Society for Testing and Materials, Philadelphia PA, 1976) pp. 306-319. [9] B. Budiansky and J.W. Hutchinson, Analysis of closure in fatigue crack growth, Journal of Applied Mechanics 45, 267-276 (1978). [10] H. Fuhring and T. Seeger, Dugdale crack closure analysis of fatigue cracks under constant amplitude loading, Engineering Fracture Mechanics 11, 99-122 (1979). [11] G.C. Sih and D.Y. Jeong, Fatigue load sequence effect ranked by critical available energy density, Theoretical and Applied Fracture Mechanics 14, 141-151 (1990). [12] D.A. Jablonski and R.M. Pelloux, Effect of train load spectra on crack growth in rail steel, Residual Stress in Rails, Vol. I (Kluwer, 1992) pp. 81-98. [13] J.J. Scutti, R.M. Pelloux and R. Fuquen-Moleno, Fatigue behavior of a rail steel, Fatigue and Fracture of Engineering Materials and Structures 7, 122-135 (1984). [14] B.G. Journet, Fatigue crack propagation under complex loading, Ph.D. Thesis, Department of Materials Science and Engineering, Massachusetts Institute of Technology (1986). [15] R.J. Hartranft and G.C. Sih, Solving edge and surface crack problems by an alternating method, Mechanics of Fracture I: Methods of Analysis and Solutions of Crack Problems (Noordoff, 1973) pp. 179-238. [16] T.T. Shih and R.P. Wei, A study of crack closure in fatigue, Engineering Fracture Mechanics 6, 19-32 (1974).