Fatigue-crack Propagation in Metals The results of cyclic-tension tests are examined in the light of linear elastic fracture-mechanics concepts and two simple crack-propagation laws are assessed
byJ. C. Radon and L. E. Culver
ABSTRACT--Cyclic-tension tests between constant limits of stress-intensity factor and at constant speed have been conducted on aluminum alloy RR 58 and mild steel BS 15. The resulting crack propagation was monitored against the number of cycles. The results, together with those obtained by other workers on similar metals, have been examined in the light of linear elastic fracture-mechanics concepts and two simple crack-propagation laws assessed. A typical linear relationship between crack length and number of cycles was observed, the relationship between crock growth and stress intensity was found to exhibit three separate regions and a law based on mean levels of the stress intensity found to adequately describe the results. There was evidence of o threshold value of stress intensity below which fatigue cracks may not grow.
List of Symbols a B C E G
= = = = = Kc = z~K = Km =
K1 = N = P =
crack length net fracture width compliance Young's modulus energy-release rate critical stress-intensity factor r a n g e of s t r e s s - i n t e n s i t y f a c t o r = Kmax - - Kmia mean stress-intensity factor stress-intensity factor for mode 1 n u m b e r of cycles load
R = Kmin/Kmax ~, =
Kmax 2 --
gmin
2 =
2•K
"
Km
= Poisson's r a t i o c, m, n, fl = c o n s t a n t s
Introduction F a t i g u e is e s s e n t i a l l y t h e i n i t i a t i o n and g r o w t h of c r a c k s caused b y the a p p l i c a t i o n of cyclic loading. 1. C. Radon and L. E. Culver are Research Fellow and Senior Lecturer, respectively, Department of Mechanical Engineering, Imperial College, London SW7, England.
A s t u d y of e a c h r e g i m e is t h e r e f o r e n e c e s s a r y if a c o m p l e t e u n d e r s t a n d i n g of t h e p h e n o m e n o n is to b e obtained. F o r m a n y y e a r s f a t i g u e d a t a w e r e p r e s e n t e d in t h e f o r m of t h e w e l l k n o w n S - N c u r v e s a n d g a v e a u s e f u l g u i d e for m a t e r i a l selection. A great d e a l of r e s e a r c h is, h o w e v e r , n o w d i r e c t e d t o w a r d s a d e t a i l e d i n v e s t i g a t i o n of s u b c r i t i c a l c r a c k g r o w t h f r o m t h e i n i t i a t i o n p e r i o d up to t h e c r i t i c a l c r a c k size. T h e d e v e l o p m e n t of l i n e a r e l a s t i c f r a c t u r e m e c h a n i c s ( L E F M ) has led to t h e s u c c e s s f u l a p p l i c a t i o n of the c o n c e p t s in f a t i g u e , 1,2 s t r e s s - c o r r o s i o n c r a c k i n g 3 and c r e e p c r a c k i n g . 4 I n L E F M , t h e c r i t e r i o n of f r a c t u r e is t h e a t t a i n m e n t of a c r i t i c a l s t r e s s - i n t e n s i t y f a c t o r K c , a p r o p e r t y of t h e m a t e r i a l . S i n c e K is t h e o n l y p a r a m e t e r charact e r i z i n g the stress field in t h e r e g i o n of a c r a c k tip, it m i g h t b e e x p e c t e d to a p p l y to a n y c r a c k - p r o p a g a t i o n situation. D u r i n g t h e last decade, a n u m b e r of f a t i g u e - c r a c k - g r o w t h l a w s h a v e b e e n p r o p o s e d , the basic and p r o b a b l y m o s t w i d e l y k n o w n of w h i c h is d u e to Paris, 1 da/dN = C(AK) m (1) where
a is t h e c r a c k l e n g t h , N is t h e n u m b e r of cycles, h K is t h e r a n g e of s t r e s s - i n t e n s i t y f a c t o r
a n d C and m a r e c o n s t a n t s d e p e n d e n t u p o n t h e m a t e rial tested and the environment. M o r e r e c e n t l y it has b e e n s u g g e s l e d 5 as a r e s u l t of w o r k on p o l y m e r s that, w i t h i n t h e l o a d i n g range tested, t h e m e a n l e v e l of K, Km a n d n o t j u s t t h e r a n g e ~XK is a significant p a r a m e t e r a n d this l e d to a c r a c k p r o p a g a t i o n m o d e l of t h e f o r m d a / d N = fl~n
(2)
w h e r e ~ = K m a x 2 - - Kmin 2 = 2 A K 9 a n d fl a n d n a r e constants. T h e p u r p o s e of t h e p r e s e n t p a p e r is to r e p o r t on w o r k u n d e r t a k e n to o b t a i n a n d e x a m i n e e m p i r i c a l e v i d e n c e on t h e c r a c k - p r o p a g a t i o n c h a r a c t e r i s t i c s of
Experimental Mechanics [ 105
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Fig. 1--Typical fatigue data for RR 58
two metals, a l u m i n u m alloy RR 58 and mild steel, in the light of LEFM concepts and the above laws.
The contour was devised b e a r i n g in m i n d the well k n o w n relationship
[
Equipment and Test Procedure Test Material~ The materials chosen for this investigation were: (1) the p r e c i p i t a t i o n - h a r d e n a b l e Al-alloy RR 58, which is of practical interest in the aircraft i n d u s t r y and is readily available as large plates. The chemical composition by weight of RR 58 is A1, 93.04 percent; Cu, 2.64; Mg, 1.64; Fe, 1.15; Ni, 1.15; Si, 0.23; Ti, 0.06; Zn, 0.06; and Mn, 0.03. It was received in the form of 7 5 - m m - t h i c k rolled plate which had been strained 2-1/4 percent after solution t r e a t m e n t at 530 ~ C a n d water quenching. Subsequently, each specimen was aged for 30 h in a salt b a t h at 190~ to give peak hardness. The plane of rolling was chosen as the plane of crack propagation, as it has recently been suggested 6 that this was the plane in which cracks grew most readily. (2) Mild steel BS 15, (C, 0.19 percent; Mn, 0.59; Si, 0.02; S, 0.028; P, 0.021; Ni, 0.02; Cr, 0.01 and Mo, 0.02) which is the m a t e r i a l used by Maddox 7 in his study of f a t i g u e - c r a c k - g r o w t h rates i n welded structures. The material was available in the form of 6 - m m - and 12.5-mm-thick plates and from these plates specimens were cut across the rolling direction. This was the same specimen orientation used by Maddox.
Specimen Geometry The contoured d o u b l e - c a n t i l e v e r - b e a m (DCB) specimen used i n this work is described and illustrated in Ref. 6. The overall dimensions of the specim e n were 75 m m high • 230 m m wide, thus p r o v i d ing a useful crack length of approximately 160 mm.
106 ] March 1976
K1-- P where and
E 2B(1 -- ~2)
dC da
]l/2 (3)
P is the applied load, B is the net fracture w i d t h C is the compliance.
The contour gave a constant compliance (dC/da = constant) and for such a specimen, u n d e r a given load, K1 r e m a i n s constant irrespective of crack length. Deep side grooves in the m i d p l a n e of s y m m e t r y were provided to enhance p l a n e - s t r a i n conditions and to help direct crack propagation. After heat t r e a t m e n t each specimen was provided with a machined slot, approximately 40 m m long, with a s w a l l o w - t a i l front which was sharpened w i t h a razor blade to form a crack starter.
Compliance Calibration and Calculation o] Klc The prepared specimen was placed in a n I n s t r o n TT-C testing m a c h i n e of 45-kN load capacity a n d the crack propagated by a f u r t h e r 25-30 m m in air at room temperature, at a constant crosshead speed of 0.05 m m / m i n . Jaw separation and load were recorded automatically and the crack length, a, observed With the aid of a t r a v e l l i n g microscope. A linear variation of compliance with crack l e n g t h was obtained. The p l a n e - s t r a i n fracture toughness K w was calculated using the I r w i n relationships of eq (3) and
K 2 = E G / ( I -- ~2) where G is the energy-release rate.
(4)
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Fig. 2--Fatigue-crackpropagation data for mild steel (BS15)
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Fatigue-crack-propagation Tests on R R 58 A f t e r the initial p r e c r a c k i n g tests on the Instron
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Fig. 3--Crack-growth rate vs./~K. RR58 in air, 0.15 Hz, 21 ~ C
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Fig. 4--Crack-growth rate vs. ~.. RR58 at 0.15 Hz
Experimental Mechanics I 107
tensile fatigue tests p e r f o r m e d u n d e r c o n s t a n t - a m p l i tude load ranges (corresponding to constant K ranges) at a frequency of 10 cycles/rain. At intervals d u r i n g a test on a given specimen after appreciable crack growth had occurred, both the lower and upper limits of K were changed to give different values of the range, hK, and the mean, Km, levels of the elastic s t r e s s - i n t e n s i t y factor. I n this way it was possible t o obtain two or three different values of the p a r a m e ter ~. a n d the corresponding c r a c k - g r o w t h rates, d a / d N from each specimen. The progress of the crack was observed by means of a t r a v e l l i n g microscope. I n order to check w h e t h e r the crack was advancing uniformly, crack-tip observations were made on both sides of the specimen and the crack length t a k e n as the average of the two readings. However, in most cases the crack front was uniform.
Fatigue-crack-propagation Tests on MiLd Steel After a series of p r e l i m i n a r y tests on 6-ram and 12.5-mm-thick DCB specimens, it was decided that static precracking on the I n s t r o n machine was not possible. The m a i n difficulty was the considerable b e n d i n g of the cantilevers which would clearly i n validate the compliance calibration procedure described earlier. It was, however, established that using 12.5-mm-thick DCB specimens h a v i n g a 75-
m m - l o n g s a w - c u t starter and 3.5-mm-deep side grooves, precracking u n d e r fatigue conditions w a s possible. Hence, all the m i l d - s t e e l specimens were precracked i n this way. The specimens were then fatigue tested in an A v e r y hydraulic testing m a c h i n e of 200-kN load c a p a c i t y in the same m a n n e r as described for RR 58.
Results and Discussion Typical curves showing the relationship b e t w e e n crack l e n g t h a n d the total n u m b e r of cycles are shown in Fig. 1 for the alloy RR 58 and in Fig. 2 for mild steel. I n each case, for a given applied range of stress intensity, ~K, there is a l i n e a r relationship. This linear behavior has previously been established for metals 1 and also for certain polymeric materials s.9 tested at constant frequency. The slopes of curves illustrated in Figs. 1 and 2 were used to obtain corresponding crack-growth rates, da/dN. The f a t i g u e - c r a c k - g r o w t h rates as a function of ~K for RR 58 tested u n d e r various imposed loading conditions are shown i n Fig. 3. I n the first test series hK was varied, b u t Kmin was m a i n t a i n e d at zero t h r o u g h out. The resulting curve m a y be divided into three
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~,( N2 mm 3 ) Fig. 5--Cyclic-crack-growth data for a l u m i n u m alloy (7075-T6); Hudson (1969) z2
108 J March 1976
Fig. 6--Cyclic-crack-growthdata for aluminum alloy (2024-T3); Hudson (1969)12
distinct regions. Region I suggests a fatigue t h r e s h old, the crack-growth rate in Region II can be adequately described by basic equations such as (1) or (2), while Region III shows an even faster growth rate leading to final fracture. The l i n e a r relationship in Region II offers no evidence as to the relative merits of applying eq (1) or eq (2), since in tests i n w h i c h gmln ----- 0, gm and ~K are directly related. Additional tests were made in which K~, -- constant and ~K was allowed to vary. These results are superimposed on Fig, 3 and although each test series gave a typical curve, the curves are displaced, the effect of increasing Km at a constant ~K level being to raise the f a t i g u e - c r a c k - g r o w t h rate in the material. Clearly the simple Paris law quoted as eq (1) would not accommodate such behavior and the results confirm that Kin, as well as ~K, is a significant parameter. The results from the same tests on RR 58 are shown as a f u n c t i o n of the p a r a m e t e r ~ in Fig. 4 and, again, a typical curve showing three regions is produced. A good l i n e a r relationship is seen to exist i n Region II but more significantly, in this region the results from all the tests involving a range of Km values are brought essentially into one curve although they are again displayed in Region I. The law proposed b y Arad, Radon and Culver 11 and quoted as eq (2) would
thus seem to a d e q u a t e l y take account of the effects of m e a n stress as well as the range ~K except in Region I. P r e v i o u s l y published fatigue results~= obtained from h i g h - s t r e n g t h a l u m i n u m alloys 7075T6 and 2024T3 have now been analyzed on the basis of the p a r a m e t e r ~, Figs. 5 and 6, and b e a r i n g i n m i n d the small r a n g e covered b y the tests, they show s i m i l a r features. Results so far obtained from tests on mild steel are shown as a function o~ ~ K in Fig. 7 and of ;~ i n Fig. 8. Also indicated are earlier results by Maddox ~ and G u r ney's scatter b a n d for mild steel ~3 b u t using c e n t e r n o t c h e d - p l a t e specimens. It is e n c o u r a g i n g that the results obtained from DCB specimens are comparable to those obtained from specimens of different geometries. F u r t h e r work needs to be done i n the present series but the curves cover Regions I and II for tests in which Kmj. -" O. Each of the p r e v i o u s l y m e n tioned c r a c k - g r o w t h laws apply e q u a l l y therefore. However, the results from four tests at a n increased value of Kin, indicated in the figures, suggest that m i l d steel will show a similar behavior to RR 58. Figures 9 and 10 show p r e v i o u s l y published results from tests conducted on cold-rolled m i l d steeP4 but
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Fig. 8--Mild steel (BS15), fatigue-crack growth
Experimental Mechanlce I 109
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analyzed in the m a n n e r described above and a c o m parison of these curves shows the same trends and, although f u r t h e r w o r k needs to be done, it is s u g gested that p a r a m e t e r ~. will take account of K m variations m o r e successfully t h a n hK.
Conclusions 1. The DCB specimens used are p a r t i c u l a r l y useful in c r a c k - p r o p a g a t i o n studies and linear elastic f r a c t u r e - m e c h a n i c s concepts m a y be successfully used to characterize the f a il u r e processes in h i g h - and m e d i u m - s t r e n g t h metals. 2. F a t i g u e - c r a c k - g r o w t h rates in a l u m i n u m alloy RR 58 and in mild steel are d e p e n d e n t upon both Km and AK. 3. The m a t e r i a l s c o n f o r m b e t t e r to the l a w based on the p a r a m e t e r L = (Kmax 2 -- Kmin2) than on t h e s i m p l er •K law. 4. An increase in K m results in an increase in t h e threshold v a l u e of L below w h ic h fatigue cracks m a y not grow. Acknowledgment
The authors are indebted to C. M. Branco and K. D. Seydali wh o obtained the n e w e x p e r i m e n t a l results quoted d u r i n g t h e i r projects, conducted u n d e r the au thor's supervision, w h i c h w e r e p a r t of th e i r MS r e quirements.
110 I March 1976
Fig. lO--Cold-rolled mild steel, fatigue-crack growt.h z4
RefeTence8 1. Paris, P. C., "'The Fracture Mechanics Approach to Fatigue," Proe. lOth Army Mat. Res. Conf., Syracuse University (1964). 2. Culver, L. E., Burns, D. I. and Borduas, H. F., "Fracture Mechanics Analysis of Fatigue Crack Propagation in Polymethylmethacrylate," Proc. Soc. of Plastics Engineers, Detroit, 233 (1967). 3. Brown, B. F., "'The Application of Fracture Mechanics to Stress Corrosion Cracking," Met. Rev., (13), 129 (1968). 4. Kenyon, 1. L., Webster, G. A., Radon, J. C. and Turner, C. E., "'An Investigation of the Application of Fracture Mechanics to Creep Cracking," Inst. Mech. Eng., Con[. Publ. 13,1-8 (1973). 5. Arad, S., Radon, 1. C. and Culver, L. E., "'Design against Fatigue Failure in Thermoplastics," 1. Eng. Ft. Mech., 4 (3) (1972). 6. Radon, 1. C., ]ohnson, F. A. and Turner, C. E., "Use of the Double Cantilever Beam Test in Fracture Studies," Conf. on Pract. Appl. of Fracture Mechanics to Pressure Vessel Technology, London, paper C7/71, 48-55 (1971). 7. Maddox, S. J., "'A Fracture Mechanics Analysis of the Fatigue Strength of Welded Structures," PhD Thesis, London University (1972). 8. Mukheriee, B., Culver, L. E. and Burns, D. I., "'Growth of Part-through and Through-thickness Fatigue Cracks in Sheet Glassy Plastics," EXPEP.Z~ENTAL~ECHArrXCS, 9 (2), 90-96 (Feb. 1969). 9. Arad, S., Culver, L. E. and Radon, J. C., "'Fatigue Crack Propagation in Polymethylmethacrylate (PMMA); The Effect of Loading Frequency," J. of Mech. Eng. Sci., 14 (5) (1972). 10. Brock, D. and Schiive, L, "'The Influence of Mean Stress on the Propagation of Fatigue Cracks in Al-alloy Sheet," NLR-TRM.2111, Amsterdam (1963). 11. Arad, S., Radon, I. C. and Culver, L. E., "'Fat{gue Crack Propagation in PMMA: The Effect of the Mean Value of Stress Intensity Factor" f. Mech. Eng. Sei., 13 (2) (1971). 12, Hudson, C. M., NASA REP. TND-5390, Langley Res. Lab., U.S.A. (1969). 13. Gurney, T. R., "'The Effect of Mean Stress in Steels," Metal Const., I (2), 91 (1969). 14. Frost, N. E., Pook, L. P. and Denton, K., Eng: Fract. Mech., 3, 109 (1971).
