Biaxialhiultiaxial Fatigue and Fracture
Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoii (Eds.)
0 Elsevier Science Ltd. and ESIS. All rights reserved. 243
THE MULTIAXIAL FATIGUE STRENGTH OF SPECIMENS CONTAINING SMALL DEFECTS
Masahiro END0
Department of Mechanical Engineering, Fukuoka Universiv, Jonan-ku, Fukuoka 814-0180, Japan
ABSTRACT
A criterion for multiaxial fatigue strength of a specimen containing a small defect is proposed.
Based upon the criterion and the & parameter model, a unified method for the prediction of the fatigue limit of defect-containing specimens is presented. In making this prediction, no fatigue testing is necessary. To validate the prediction procedure, combined axial and torsional loading fatigue tests were carried out using smooth specimens as well as specimens containing holes of diameters ranging from 40 to 500 pm which acted as artificial defects. These tests were conducted under in-phase loading condition at R = -1. The materials investigated were annealed 0.37 % carbon steel, quenched and tempered Cr-Mo steel, high strength brass and nodular cast irons. When the fatigue strength was influenced by a defect, the fatigue limit was determined by the threshold condition for propagation of a mode I crack emanating from the defect. The proposed method was used to analyze the behavior of the materials, and good agreement was found between predicted and experimental results. The relation between a smooth specimen and a specimen containing a defect is also discussed with respect to a critical size of defect below which the defect is not detrimental.
KEYWORDS
Multiaxial loading, fatigue thresholds, small defects, small cracks, 6 parameter model, steels, brass, cast irons.
INTRODUCTION
Over a number of years a great deal of effort has been expended in the attempt to establish reliable predictive methods for the determination of the fatigue strength under both uniaxial and multiaxial loading conditions. However, prior to the 1970’s, the methods proposed did not provide a useful mean for the analysis of materials which contained either non-metallic inclusions or small flaws that are usually encountered in engineering applications. This was in part because most of the proposed methods were applicable only to two-dimensional cracks or
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notches of simple shapes, whereas actual inclusions are often of a three-dimensional irregular shape. In addition, whereas the large crack problem has attracted attention in fatigue studies since the birth of fracture mechanics, the behavior of small cracks could not be analyzed in a similar way, for their behavior has been found to be anomalous with respect to large cracks, as pointed out by Kitagawa and Takahashi [I]. These authors, in the first quantitative characterization of the fatigue threshold behavior of small cracks, showed that the value of &h
decreased with decreasing crack size. This finding led the development of many subsequent studies on small or short cracks. Since their initial work a number of models and predictive methods for the determination of the fatigue strength of defect-containing components have been proposed, although most of these have dealt only with uniaxial fatigue. These models have been reviewed in detail by Murakami and Endo [2]. Research has shown [3,4] that the fatigue strength of metal specimens containing small defects above a critical size is essentially determined by the fatigue threshold for a small crack emanating from the defect. Based upon this consideration, Murakami and Endo [4] used linear elastic fracture mechanics (LEFM) to propose a geometrical parameter, G, which quantifies the effect of a small defect. Using this parameter they succeeded in deriving a simple equation 151 for predicting the fatigue
strength of metals containing small defects. Subsequently, this model, referred to as the 6
parameter model, has been successfully employed in the analysis of a number of uniaxial fatigue problems which dealt with small defects and inhomogeneities [6,7].
However, in many applications, engineering components are often subjected to multiaxial cyclic loading involving combinations of bending and torsion. A number of studies have been concerned with this topic [8-131, but with the exception of pure torsional fatigue very few studies have been directed at the study of the behavior of small flaws under multiaxial fatigue loading conditions despite the importance of small flaws in design considerations. Nisitani and Kawano [14] performed rotating bending and reversed torsion fatigue tests on 0.36 YO carbon steel specimens which contained defect-like holes of diameters ranging from 0.3 to 2 mm. They reported that the ratio of torsional fatigue limit to bending fatigue limit, q5 = rw Icrw, was about 0.75 and attributed the result to the ratio of stress concentrations at the hole edge at fatigue limits; that is, 3 0 , under bending and 4r , under torsion. (Here s, and uw are the fatigue strengths of specimens containing small flaws in reversed torsion and tension, respectively.) Mitchell [ 151 also predicted q5 = 0.75 for specimens having a hole in the similar way. Endo and Murakami [ 161 drilled superficial holes which simulated defects ranging from 40 to 500 pm in diameter in 0.46 % carbon steel specimens to investigate the effects of smail defects on the fatigue strength in reversed torsion and rotating bending fatigue tests. Based upon the observation of cracking pattern at the holes, they correlated the fatigue strength under torsion with that under bending by comparing the stress intensity factors (SIFs) of a mode I crack emanating from a two-dimensional hole. They predicted 4 = -0.8 for specimens containing a surface hole. In that study, they also observed that there was a critical diameter of a hole below which the defect was not detrimental to the fatigue strength, and that the critical size under reversed torsion was much larger than under rotating bending.
In recent papers [17-191, the fbrther application of the && parameter to multiaxial fatigue problems has been made. Combined axial-torsional fatigue tests were carried out using annealed 0.37 % carbon steel specimens containing a small hole or a very shallow notch [ 171. It was concluded that the fatigue strength was related to the threshold condition for propagation of a mode I crack emanating from a defect, and an empirical method for the prediction of the fatigue limit of a specimen containing a small defect was proposed [17]. Murakami and Takahashi [18] analyzed the fatigue threshold behavior of a small surface crack in a torsional
The Multiaxial Fatigue Strength of Specimens Containing Small Defects 245
shear stress state and extended the use of the & parameter to mixed-mode threshold problems. In addition, Nadot et al. [19] have discussed the extension of Dang Van’s multiaxial fatigue criterion [20] to the defect problem by using the & parameter. Beretta and Murakami [21,22] used numerical analysis to calculate the stress intensity factor (SIF) for a three-dimensional mode I crack emanating from a drilled hole or a hemispherical pit under a biaxial stress state. By comparing with the previous experimental data [ 171, they concluded that the value of SIF at the tip of a crack emanating from a defect determined the fatigue strength of a specimen which contained a small defect above the critical size subjected to combined stresses. The present author [23] subsequently proposed a new criterion for fatigue failure which was also based upon the SIF. This criterion was expressed in the form of an equation which, by including within the criterion the && parameter model, provided a unified method for predicting the fatigue strength of a metal specimen containing a small defect. The applicability of the method was investigated with an annealed steel E231 and nodular cast irons [23,24]. The essence of this approach will be presented in the present paper.
The principal objective of this study is to determine the generality of the author’s predictive method [23] with additional experimental newly obtained data. In the present study the relation between the fatigue strengths of smooth specimens and specimens containing defects in multi- axial fatigue will also be discussed.
BACKGROUND FOR THE PREDICTION OF THE MULTIAXIAL FATIGUE STRENGTH
The &¶meter model
Murakami and Endo [4] have shown that the maximum value of the SIF, Klmax, at the crack front of a variety of geometrically different types of surface cracks can be determined within an accuracy of 10% as a function of & , where the area is the area of a defect or a crack projected onto the plane normal to the maximum tensile stress, see Fig. 1. The expression for KI,,, as a function of area (Poisson’s ratio of 0.3) is:
\ Maximum tensile stress direction
Fig. 1. Definition of area.
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K,, = 0.650a0d=
where a0is the remote applied stress. Thereafter Murakami and Endo [5] employed the Vickers hardness value as the representative material parameter and showed that the threshold level for small surface cracks or defects could be expressed by the following equation for uniaxial loading at the stress ratio, R, of -1 :
AK~,, = 3.3x10-3(~v+120)(Jarea)1/3 (2)
In addition, they found that the fatigue limit could also be expressed as a function of 6 by:
1.43(HY + 120)
6, = (3)
where AK,,, , the threshold SIF range, is in M P a G , aw,the fatigue limit stress amplitude, is in MPa, HV, the Vickers hardness, is in kgf/mmz, and 6 is in pm. Equations (2) and (3) were derived on the basis of LEFM considerations. More recently a justification for the exponents of 1/3 in Eq.(2) and -1/6 in Eq.(3) was provided by McEvily et al. [25] in a modified LEFM analysis which also considered the role of crack closure in the wake of a newly formed crack.
The prediction error involved in the use of Eqs (2) and (3) is generally less than 10 percent for values of 6 less than 1000 pm, and for a wide range of HV [5,6].
Murakami and co-workers [26-281 further extended Eq.(3) to include the location of the defect, i.e., whether it was at the surface or sub-surface, and also to include the effect of mean stress. The generalized expression they developed for the fatigue strength is:
C ( W + 1 2 0 ) I - R =
(Jarea)"6 [ T I
ow = (4)
where the value of C depends on the location of the defect being 1.43 at the surface, 1.41 at a subsurface layer just below the free surface, and 1.56 for an interior defect. The value of the exponent a was related to the Vickers hardness by a = 0.226 + HV x 1 O4 . Equations (2)-(4) are useful for practical applications in that they require no fatigue testing in making predictions.
The & parameter model has been applied to deal with many uniaxial fatigue problems including the effects of small holes, small cracks, surface scratches, surface finish, non-metallic inclusions, corrosion pits, carbides in tool steels, second-phases in aluminum alloys, graphite nodules and casting defects in cast irons, inhomogeneities in super clean bearing steels, gigacycle fatigue, etc. They are summarized in detail in the literature [6,7].
Criterion for multiaxial fatigue failure of defect-containing specimens
The author has previously shown [17,23,24] that, in in-phase combined axial and torsional fatigue loading tests, the fatigue limit for specimens containing small defects is determined by the threshold condition for propagation of a small crack emanating from a defect. The materials investigated were an annealed 0.37 % carbon steel [17,23] and nodular cast irons [23,24].
The Multiaxial Fatigue Strength of Specimens Containing Small Defects 241
Figure 2 shows typical examples of non-propagating cracks observed in those materials at the fatigue limit. The direction of a non-propagating crack is approximately normal to the principal stress, 01, regardless of combined stress ratio, 7/0. Under a stress slightly higher than the fatigue limit, a crack which propagated in a direction normal to q resulted in the failure of the specimen. Based upon such observations, the fatigue limit problem for specimens containing small surface defects when subjected to combined stress loading was considered to be equivalent to a fatigue threshold problem for a small mode I crack emanating from defects in the biaxial stress field of the maximum principal stress, 01, normal to the crack and the minimum principal stress, 0 2 , parallel to the crack.
Consider an mi-symmetric surface defect containing a mode I crack under the remote biaxial stresses, o, and a,, as shown in Fig. 3 . The mode I SIF, ZC1, at the crack tip is given by the
50 pm
2, 31.7"
Axial direction
(a) 0.37% carbon steel; u
175 MPa, 7, = 87.5 MPa [17].
h = 100 pm, u, = (b) FCD700 nodular cast iron; smooth specimen, c, = 7, = 160 MPa [24].
Fig. 2. Small non-propagating cracks emanating from a defect observed at fatigue limit under combined loading.
t t
Y +
L x
A + B
Fig. 3 . A three-dimensional defect leading to a crack subjected to biaxial stress.
248 M. END0
following superposition.
where FIA and FIB are the correction factors for the cases A and B in Fig. 3, respectively, and c
is the representative crack length. It is hypothesized that the threshold SIF range under biaxial stress, A&h,bi, is equal to that under uniaxial stress, L&,mi, or:
AKt,,bi = AK Ih,um . (6)
This criterion has previously been used by Endo and Murakami [ 161 in the correlation of the pure torsional fatigue limit, 7,;the biaxial fatigue limit, with the rotating bending fatigue limit, aw; the uniaxial fatigue limit, for specimens having a small hole at the surface. Based upon this criterion, Beretta and Murakami [2 1,221 predicted that 4, the ratio of the fatigue limit in torsion to that in tension, ie., r, /a,, for a mode I crack emanating from a three-dimensional surface defect under cyclic biaxial stressing should have a value between 0.83 and 0.87. They found that the predicted value of 4 agreed well with previously reported experimental results for various steel and cast iron specimens which contained small artificial defects. For fully reversed loading; Le., R = -1, b&h,bj and AKtbuni were expressed using Eq.(5) as
where 01 and q are the maximum and minimum principal stress amplitudes resulting from the combined stress at fatigue limit, respectively, and ow is the threshold stress amplitude for a mode I crack under tension-compression cyclic loading; that is, the uniaxial fatigue limit of a specimen containing the same sized defect under R = -1 loading.
When the crack length, c, under uniaxial loading is equal to that under biaxial loading, Eq.(6) is reduced to
where k = Fle/F*, and represents the effect of stress biaxiality. If the torsional fatigue limit is designated by rw, since UI = -m = r,, then 4 = rw/aw = 1/(1 - k). Equation (9) as well as Eq.(6) provides a criterion for fatigue failure of specimens containing small defects when subjected to multi-axial loading.
For round-bar specimens subjected to combined axial and torsional loading, Eq.(9) can be expressed as
where and r, are the normal and shear stress amplitudes, respectively, at the fatigue limit under combined loading. Equation (10) is identical in form to Gough and Pollard's "eIlipse arc"
relationship [29], which has been used to fit the experimental data for brittle cast irons and specimens with a large notch [29,30]. The ellipse arc is empirical, and as such it requires fatigue tests for the determination of a, and .,7 In contrast, in the case of small defects, a, can
The Multiaxial Fatigue Strength of Specimens Containing Small Defects 249
be predicted using Eq.(3) without the need for a fatigue test. In addition, the value of 4 can be estimated by stress analysis, as verified by Beretta and Muakami [21,22]. If the average value 4 calculated by Beretta and Murakami is used, i e . , 0.85, &.(lo) becomes:
1.38(rJaW)’ + 0.176(aJq$ + 0.824(aJow) = 1 (1 1) This expression is considered to be applicable to specimens containing a round defect on or near the surface where plane stress condition is satisfied. The use of this expression will be demonstrated in the following.
MATERIALS AND EXPERIMENTAL PROCEDURE
The materials investigated in this study are: steels, nodular cast irons and a high strength brass.
Although experimental data for an annealed steel and the cast irons have previously reported elsewhere [23,24], they will be included below for purposes of discussion. The chemical compositions of the various metals are listed in Table 1. The 0.37 YO carbon steel (JIS S35C) was annealed at 860°C for 1 hour. The Cr-Mo steel (JIS SCM435) was heat-treated by quenching from 860°C followed by tempering at 550°C. The two nodular cast irons (JIS FCD400 and FCD700) have different matrix structures; ferritic for FCD4OO and almost pearlitic for FCD700. The cast irons and the brass were used in the as-received condition. Mechanical properties are given in Table 2, and microstructures are shown in Fig. 4.
Table 1. Chemical composition (wt.%)
C Si Mn P S Cu Ni Cr Mo Mg
S35C steel 0.37 0.21 0.65 0.019 0.017 0.13 0.06 0.14 - SCM435 steel 0.36 0.30 0.77 0.027 0.015 0.02 0.02 1.06 0.18 -
FCD400castiron 3.72 2.14 0.32 0.008 0.018 0.04 - - 0.038 FCD700 cast iron 3.77 2.99 0.44 0.023 0.11 0.47 - - 0.058
cu Pb Fe Zn Mn A1
High strength brass 59.1 0.0030 0.0032 bal. 0.02 1 0.0047
Table 2. Mechanical properties
Tensile Elongation % Vickers hardness strength (Gage length: HV
MPa 80 mm) ( kgf/mm2)
Annealed S35C steel 586 25 160
Quenchedkempered SCM435 steel 1030 14 380
FCD400 cast iron 418 25 190 (femte)
FCD700 cast iron 734 8.0 330 (pearlite)
High strength brass 467 42 110
. ..
,, * . - ' . . .
. .
. .
1 .
The Multiaxial Fatigue Strength of Specimens Containing Small Defects 25 1
(a) For combined axiaVtorsiona1 load test
(b) For tension-compression test and combined axialltorsional load test or reversed torsion test
under do = 112
( 5 0 50
< -'- 140 -'- h
(c) For reversed torsion test
(d) For rotating bending test
Fig. 5. Shapes and dimensions of smooth specimens.
Fig. 6. Hole geometries.
reversed (R = -1) loading and a sinusoidal waveform. The combined stress ratios of shear to normal stress amplitude, do, were chosen to be 0, 1/2,1,2 and oc) . For the tension-compression tests, in order to eliminate bending stresses each specimen was equipped with four strain gauges to facilitate proper alignment in the fixtures.
The nominal stresses were defined as
o = ~ P / ( I ~ D ' ) for tension-compression (12)
CT = 32M, /(a3) for rotating bending (13)
T = 16M, / ( z D 3 ) for reversed torsion (14) where CT is the normal stress amplitude, z is the torsional shear stress amplitude, P is the axial load amplitude, Mb is the bending moment amplitude, Mt is the torsional moment amplitude and D is the specimen diameter. The fatigue limits under combined stress are defined as the combination of the maximum nominal stresses, ra and 0,under which a specimen endured 10'
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cycles for a fixed value of do. The minimum increment in stress level in determining the fatigue limit was 5 MPa for greater value of o and r, except for Cr-Mo steels where 10 MPa was used.
RESULTS AND DISCUSSION
Behavior of small cracks at the threshold level
Figures 7 and 8 show the non-propagating cracks emanating from a hole at the fatigue limit of hole-containing Cr-Mo steel and brass specimens. As seen in these figures, the direction of non-
50 pm
(a) Tension-compression; - ow = 340 MPa.
1
Axial direction
50 pm
-
(b) Combined loading; oa = ra = 200 MPa.
Axial direction
50 pm (c) Pure torsion; rw = - 320 MPa
Fig. 7. Small non-propagating cracks emanating from hole at fatigue limit of quenchedtempered SCM435 steel; & = 83 pm (d = 90 pm).
The Multiaxial Fatigue Strength of Specimens Containing Small Defects 253
200 pm
(a) Combined loading; - ua = Za = 70 MPa
Axial direction
(b) Pure torsion; rW = 1 15 MPa
Fig. 8. Small non-propagating cracks emanating from hole at fatigue limit of high strength brass; d = 500 pm.
propagating cracks is approximately normal to the principal stress, q, regardless of the combined stress ratio, do. At a stress level just above the fatigue limit, the failure of the hole- containing specimens resulted from the continuous propagation of a crack emanating from the hole. These observations with Cr-Mo steel and brass further indicate that, as in the case of
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n cd
I
0.37 % carbon steel [17,23] and nodular cast irons [23,24] previously reported, the fatigue limit is determined by the threshold condition for a mode I crack emanating from a defect and propagating in the direction normal to ~1
It has also been noted that holes 40 and 100 pm in diameter did not lower the fatigue strength of annealed 0.37 % carbon steels, under the special condition of a large combined stress ratio, do. Those hole-containing specimens did not break as the result of a crack emanating from the hole, but rather from a crack which initiated from the smooth area at a stress level just above the fatigue limit. The critical size of a non-detrimental defect will be discussed below.
Experimental fatigue limit:
- 0 Smooth
Comparison of experimental and predicted results for hole-containing specimens
Figures 9-11 show the comparison of the experimental data with the predictions for hole- containing specimens for two steels and a high strength brass. The predicted curves were obtained by substituting 6 and Hvinto Eq.(3) and by inserting the value of a, into Eq.( 1 1).
The data points indicated by arrows in Fig. 9 are for the results of specimens containing a non- detrimental hole. Figure 12 presents the comparison for those three materials in a different manner. This figure has been normalized by dividing the values of o, and T~ on the axes by the uniaxial fatigue limit, a,, which was predicted by Eq.(3). In order to compare the predictions only with the experimental results controlled by a defect, in this figure, the results for the non- detrimental holes were not included. It is seen that the predictions for the fatigue limit of hole- containing specimens subjected to combined stress agree quite well with the experimental data.
It is also emphasized that no experimental data were needed in making these predictions.
Prediction: 4
0 100 200 300
Axial stress amplitude, a , MPa
Fig. 9. Comparison of predicted and experimental results for annealed S35C steel specimens.