2 Threshold condition for fatigue crack propagation under · Fatigue crack propagation under mixed...

Fatigue crack propagation under mixed mode I

and III loading

K. Tanaka, Y. Akiniwa

Department of Mechanical Engineering, Nagoya University, Furo-

cho, Chikusa-ku, Nagoya 464-01, Japan

Abstract

The propagation behavior of a circumferential crack in a cylindrical steelbar was studied under various combinations of cyclic torsional and axialloadings. The threshold condition of the onset of fatigue crackpropagation from a pre-crack was expressed in terms of the stress intensityranges for mode I and III. The J integral was applied to fatigue crackpropagation with excessive plasticity under cyclic loading of mixed modeof I and III.

1 Introduction

Fatigue cracks often show the mixed-mode I and III propagation inpower train shafts subjected to reversed torsional and axial loadingsimultaneously. In comparison with the case of mode I , not manystudies have been conducted on mode III or mixed mode of I and III.For damage torelance design, the direction as well as the rate of crackpropagation should be predicted from the loading condition and materialinhomogeneities.

In the present paper, the propagation behavior of a circumferentialcrack in a cylindrical bar of a carbon steel was studied under variouscombinations of cyclic torque and cyclic axial loading. First, the thresholdcondition of fatigue crack propagation was discussed in terms of the stressintensity factor (SIF). Next, the J integral was applied to fatigue crackpropagation with excessive plasticity under cyclic loading of mixed modeof I and III.

Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533

154 Localized Damage

2 Threshold condition for fatigue crack propagation undermixed mode of I and III

2.1 Models for crack propagation threshold

The propagation path of cracks can deviate from the original crack planeunder mode II or III loading. Figure 1 illustrates the formulation of a tiltcrack facet and a twist crack facet from the original plane crack. Tiltfacets can be formed under mode II loading with or without mode Iloading; twist facets under mode III loading with or without mode Iloading.

Various criteria proposed for brittle fracture under monotonic mixed-mode loading have been applied to the threshold condition of fatiguecrack propagation. They are briefly described as follows with a specialreference to the mixed-mode loading of mode I and III.

Erdogan and Sih [1] proposed the maximum hoop stress criterion forpredicting the tilt angle of the crack path and the brittle fracture loadunder mixed-mode I and II.

Pook [2] proposed the maximum principal stress criterion for fatiguecrack propagation under mixed mode of I and III. The twist facet angle0 is the direction of the maximum principal stress, and is given by

1• - - tan9 l-2v (1)

where Kj and K^ are SIF for mode I and III, and v is Poisson's ratio.The maximum principal stress a is given by

(2)

For the case of mode I , the critical value of a ^ is related to the

Normal of facet

Main crack

(a) Tilt crack formation (b) Twist crack formaiton

Figure 1. Fracture path under mixed mode loading.


Localized Damage 155

threshold value of SIF, K^, as

a =•* 2 (3)

From Eqs. (2) and (3), we have

K = \\ K (l+2v)+J /(l-2v)*+4/ith 2|_ i V i m J (4)

Yates et al. [3] proposed that the tilt angle 0 was determined by themaximum principal stress of the net-section stress

*~2 ( *«) (5)

where r and o are the net-section shear and normal stress. Thethreshold condition is given by the crack-tip opening displacementperpendicular to the facet plane as

(6)

where K is the critical value of SIF for mode I loading.The maximum energy release rate criterion is a logical extension of

Griffith theory. By denoting the SIF value of twist crack facet by k,(d>)and k,,,( 0 ), the energy release rate for crack extension is given by [4]'

,

The k, and k^ values can be approximated by

2v— 1i— — Sin2^ + Armcos2^ (9)

The direction of twist crack is given as the direction of the maximum valueof G( 0 ), and G^ is constant at the onset of crack extension.

Local symmetry criterion, or mode II SIF for a tilt crack facet k,,=0,is proposed for determining the tilt angle for mixed mode of I and II [5].The k^=0 criterion for the case of mixed mode of I and III yields thecrack direction given by Eq. (1), as far as the approximations of Eqs. (8)and (9) are valid.

In the experiment described in the next section, a cylindrical bar with



a circumferential pre-crack is subjected under the combination of axialand torsional loadings. The radius of the bar is 16 mm and the depth ofthe pre-crack is 1.9 mm. For this geometry, the twist angle calculatedfrom Eqs. (1) and (5), and also from G^ model is shown Fig. 2, wherethe mode mixity parameter is defined by

(10)

and v is 0.279. The threshold condition derived from the three models is

40

^ 30

10

0 20 40 60 80

Mode mixity parameter, (3 = tan (K /K ), deg

Figure 2. Facet angle predicted under mixed mode of I and III.

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Normalized mode I stress intensity factor, K^ / K^

Figure 3. Crack propagation threshold under mixed mode of I and III.



shown in Fig. 3. G^ model gives the same prediction as Book's model asfar as K,,/K, is less than 2.05 or 0 is less than 64.0 deg. The threshold ofmode HI, K,, , is equal to K^ for Book's model, K, =0.79 K^ forYates' model, and K, =0.85 K,^ for G^ model. The twist angle formode III is 45 deg for Book's and Yates' model, and 0 deg for G^ model.

2.2 Experimental procedure

A cylindrical bar of carbon steel (JIS S45C) with the diameter of 16 mmwas circumferentially notched, and annealed at 850 C for 1 hr. The yieldstrength was 311 MBa, and the tensile strength was 588 MBa. TheYoung's modulus was 216 GBa and Boisson's ratio was 0.279.

A pre-crack of a depth of 0.40 mm was introduced below acircumferential notch (with a depth of 1.5 mm) by cyclic compression. Allthe specimens were annealed at 650 C for 1 hr before fatigue tests.

The fatigue tests were conducted in a computer-controlled electro-servo hydraulic tension-torsion fatigue testing. Figure 4 shows the loadwave applied. The load ratio of the torque T is R=-l; that of the axialload B is R=0. The frequency is 10 Hz for the torque and 20 Hz for theaxial load. The mode ratio AK,,/AK, adopted for tests was 0, 0.4, 1.0,2.4 and oo, where AK^ and AK, mean the total range of SIF includingreverse loading.

The direction of crack extension was conducted by the d.c. electricalpotential method [6]. The change of the potential deference of 1 # Vcorresponded to the crack extension of about 15 m in our experimentalsystem. The specimens in which the crack extension had been detectedby the d.c. potential method were fractured at liquid nitrogentemperature. The fracture surface was examined with a scanning electronmicroscope (SEM).

(b)

R= -1f = 10 Hz

R= 0f =20 Hz

Figure 4. Loading waves in mixed mode ( I +III) loading.



2.3 Experimental results and discussion

The experimental results of the threshold tests were shown in Fig. 5,where the open circles indicate no-growth and the solid circles indicatecrack growth detected. The threshold range of SIF for mode I is AK^=3.3MPa\Tm, and that for mode III is AK^ =3.0MPa/"m. Theexperimental data can be approximated by a quarter ellipse expressed by

= 1 (11)

• Crack growthO No growth

1/4 ellipse

B.3

0 1 2 3

Mode I stress intensity factor, AK^ MPaV^m

FigureS. Threshold conditions for fatigue crack propagationunder mixed mode( I +III) loading

A B

yz

(a) Positive torque at point (D (b) Negative torque at point

Figure 6. Stress state at crack tip under mode I and III loading.



For the case of torsional loading with R = -l, the principal stressdirection is different between positive torque and negative torque. Figure6 illustrates the switching of the principal stress direction due to thechange of the direction of the applied torque. To apply Book's and Yates'model [2,3] to the wave form (Fig. 4) of the present experiment, it isunknown how to deal with the switching of the principal stress direction.

By overlooking the above switching, AK, and AK^ are substitutedfor K, and K,,, in Eqs. (1) and (4) for Book's model, and in Eqs. (5) and(6) for Yates' model. The relations are shown in Fig. 7, where AK, andAK^j are normalized by AK^. the experimental data seem to fit theprediction by Yates' model.

Figure 8 shows the SEM micrographs of the fracture surface of thespecimen subjected to the cyclic load just above the threshold, where A, B,and C indicate the pre-crack face, the fatigue crack face and the brittlefracture surface, respectively. The face of fatigue crack propagation isinclined to the pre-crack face in Fig. 8(a) with AK^=3.0 MPa/"m, whilethat is parallel to the pre-crack face in Fig. 8(b) with AK^= AK, =2.3MPav^m. According to the model prediction shown in Fig. 2, the tiltangle is around 30 to 40 deg. for the case of AK^/AK^ = 1. Theprediction does not agree with the experimental results. Furtherdevelopments of modelling are necessary to include the effect of variousloading conditions.

% i.okT

O

_ci

-ai13<D

• Crack growthO No growth

Book's modelYates' model

model

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Normalized mode I stress intensity factor, AK^ / AK^

Figure 7. Experimental results compared with model prediction.



(a) AK|=3.0MPa/~m, R = -l

(b)AK,/AK^ = 1.0, AK,=2.3MPa/"m, AK Z. MPa/'m.

Figure 8. Scanning electron micrographs of fracture surface of specimensjust above the threshold.



3 J integral approach to fatigue crack propagation withexcessive plasticity under mixed mode of I and III

3.1 J integral approach

A circumferential crack in cylindrical bars shows coplanar mode IIIpropagation under high cyclic torsional loading. Because of excessiveplasticity, linear elastic fracture mechanics is no longer applicable to thiscase. J integral approach was first applied to mode III propagation byTanaka et al [6]. They estimated the J integral range from the loadingpart of the hysteresis loop of torque versus angle of twist.

Figure 9 illustrates the hysteresis loops of torque versus angle of twistand of load versus displacement obtained under mixed loading of mode I

Crack opening point

\

(a)Tvs.

Crack opening point

(b)Pvs. u.

Figure 9. Evaluation of J integral range.



and III. The point of crack-tip opening was determined from theunloading compliance method. The mode III J integral range, AJ^, isestimated from the loading part above the opening point by

where b is the radius of the ligament of the specimen, AK^^ is the modeIII effective stress intensity range and U^ is the energy corresponding tothe shaded area shown in Fig. 9(a). The mode I J integral is estimatedby

i(13)

where AK^is the mode I effective stress intensity range, Ucorresponds to the shaded area in Fig. 9(b), and AP^, Au, A u^ are alsoshown in Fig. 9(b).

For the case of mixed mode loading, the total range A J can be givenas the sum of AJ, and

3.2 Experimental procedure

The material and the specimen were the same as those described inChapter 2. The notch depth of the specimens used for mode IIIexperiment was 1.5 and 3.5 mm; that for mixed mode and mode Iexperiments was 3.0 mm. All the specimens were pre-cracked by a depthof 0.40 mm under cyclic compression, and then annealed at 650 °C for 1hr before fatigue tests.

The fatigue crack propagation test was conducted with R=-l both fortorque and axial load under in-phase condition. The range of J integral,AJ, was kept constant for mode III crack propagation tests. For theother tests, the torsional angle and axial displacement were controlled.The crack length was measured by the d.c. potential method.

3.3 Experimental results and discussion

The change of crack propagation rate with the crack length for the modeIII case is shown in Fig. 10, where the notch depth t is 1.5 mm and c is thecrack length from the pre-crack. The crack propagation rate decreaseswith crack extension even under a constant A J^ condition. This decreaseis caused by the sliding contact of crack faces as reported by Tschegg [7].The crack propagation rate without contact shielding is obtained byextrapolating the dc/dN vs. c relation to the zero crack length as shown in



<D

1

I

-8

4— >2

cdtoo

oa

ocdV-iU

10,-5

10,-7

10,-9

I I If -1.5 mm

=2.9X10

J I I I0.0 0.2 0.4 0.6 0.8 1.0 1.2

Crack length, c mm

Figure 10. Change of crack propagation rate with crack length.

cd*-H

^O4—1cdbficda2

10

2u 1-9

dc/dN = 1.04X10

Q'dc/dN=7.03x 10'

•

3

%

Mode It =3.0 mmMode IIIt =1.5 mmt =3.5 mm

Mixed mode t =3AJ^IAJi =0.5AJv/AJi =1AJg/AJi =2

0 mm

468 2 468 2 468

J integral range, AJ N/m

Figure 11. Relation between crack propagation rate and AJ .



Fig. 10. The crack propagation rate thus determined is plotted againstAJ^j in Fig. 11. The relation between dc/dN (m/cycle) and AJ (N/m) isapproximated by the following power equation:

and m =1.43.

(15)

This relation is rather independentwhere C =7.03 X 10of the initial notch depth.

For the case of mode I loading, the crack-opening point wasdetermined from the load vs. displacement curve, and AJ, was estimatedby using Eq. (13). The relation between dc/dN (m/cycle) and AJ^ (N/m)was shown with the solid circles in Fig. 11. The relation is expressed by

&-W7V = C,Ay/"' (16)

where C , = 1 .39 x 10 " and m = 1 .71 .In displacement-controlled tests of mixed mode loading, the J integral

ranges of mode I, A J^, and of mode III, AJ^, were nearly constantduring the crack extension of 1 mm from the pre-crack. The ratio of A J^/AJj tested was 0.5, 1, and 2. During this crack extension, the crackpropagation rate was also nearly constant or decreased by a small amount.In Fig. 11, the mean propagation rate is plotted against the total range ofAJ. All the data lie between the relations for mode I and mode III.

Figure 12 shows the change of the crack propagation rate at AJ=2X10* N/m with the mode mixity parameter defined by

&E

I•ir

C.o

03ao

U 0 20 40 60

Mode mixity parameter, (3*=tan~\z deg

Figure 12. Change of crack propagation rate with mode mixity.



(a) AJ = 2.6X!0<N/m, AJ,/AJ, =0, dc/dN=1.6x 10* nVcycle.

(b) AJ=2.lxlO'N/m, AJ/AJ^=1, dc/dN=1.8xWnVcycle.

Figure 13. Scanning electron micrographs of fatigue fracture surfaces.



(17)

The bar attached to the data point indicates the range of variation of thecrack propagation rate. The crack propagation rate decreases withincreasing ratio of AJ^/AJ^. The line in the figure is drawn under theassumption that the crack propagation rate is given as the sum of the ratesby mode I and III loadings as

A//'m (18)

Equation (18) seems to express the crack propagation rate under mixedmode loading.

The fatigue fracture surface of all the specimens was macroscopicallyflat and did not show factory-roof type topography. SEM micrographs offatigue fracture surface are shown in Fig. 13, where (a) is for mode III( AJ,,,=2.6 X 10' N/m, dc/dN= 1.6 X 10* m/cycle), and (b) is for a mixedmode(AJ=2.1xiO<N/m, AJ,,/AJ, = 1, dc/dN= 1.8X10* m/cycle). Thecrack propagation direction is from top to bottom. On the fracturesurface made under mode HI, there are no striations. Only rub marksare seen extending perpendicular to the growth direction. On the otherhand, striations can be observed on the fracture surface made undermixed mode loading. Those striations suggest that the mechanism ofmode I crack propagation is operating in mixed-mode propagation.

4 Conclusions

By using a circumferentially cracked steel bar subjected to variouscombinations of cyclic torque and axial loading, the effect of the modemixity of I and III on the threshold condition of the onset of crackpropagation and on the propagation rate of fatigue cracks with excessiveplasticity was studied. The fallowings are the summary of the resultsobtained:(1) The threshold condition of fatigue crack extension was expressed by

where A 1C and AK ^ are the threshold values of the stress intensityI in 111 tnranges for mode I and III.(2) The models proposed by Pook and Yates for the stress ratio R^Ogive conservative estimates for the present experiments for the thresholdunder R=-l.(3) The angle of the fracture facet made at the tip of pre-cracks justabove the threshold did not agree with the maximum principal stressdirection. Further refinement of modelling of the threshold is necessary.



(4) Under high cyclic torque, mode III cracks propagatedmacroscopically coplanar to the pre-crack plane. The crack propagationrate dc/dN (m/cycle) was expressed as a function of the J integral rangeAJ as

where C =7.03 X 10" and m,,,= 1.43.(5) For the case of mode I loading, the relation obtained was

where C = 1 .39 X 1Q-" and m ^ = 1 .71 . The propagation rate is lower undermode III loading than under mode I loading, when compared at thesame A J value.(6) The data of the crack propagation rate under mixed-mode loading asa function of A J lie between the relations for mode I and III.(7) The crack propagation rate under mixed mode loading can beexpressed as

dc/dN = C, AJ/"' + Cm A V*

References

1. Erdogan, F. and Sih, G.C., On the crack extension in plates under planeloading and transverse shear, Trans. ASME, J. Basic Eng., 1963, 85, 519-527.2. Pook, L.P., The fatigue crack direction and threshold behavior of mildsteel under mixed mode I and III loading, Int. J. Fatigue, 1985, 7, 21-30.3. Yates, J.R. and Miller, K.J., Mixed mode ( I +III) fatigue thresholds ina forging steel, Fatigue Fract. Eng. Mater. Struct, 1989,12, 259-270.4. Lawn, B., Fracture of Brittle Solids - Second Edition, Cambridge Univ.Press, 1993.5. Kitagawa, H. and Yuuki, R., Stress intensity factor for branched crackin two-dimensional stress state, Trans. JSME, 1975, 41, 1641-1649.6. Tanaka, K., Akiniwa, Y. and Nakamura, H., J-integral approach tomode III fatigue crack propagation in steel under tortinal loading, FatigueFract. Eng. Mater. Struct., 1996, To appear.7. Tschegg, E.K., Sliding mode crack closure and mode III fatigue crackgrowth in mild steel, Acta me tall., 1983, 9, 1323-1330.


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