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Cyclic fatigue crack growth from indentation flaw in silicon nitride: Influence of effective stress ratio
Acta metall, mater. Vol. 42, No. 1I, pp. 3837-3842, 1994
Pergamon
0956-7151(94)E0154-9
Copyright © 1994ElsevierScienceLtd Printed in Great Britain.All rights reserved 0956-7151/94$7.00+ 0.00
CYCLIC F A T I G U E CRACK G R O W T H F R O M I N D E N T A T I O N FLAW IN SILICON NITRIDE: I N F L U E N C E OF EFFECTIVE STRESS RATIO GUEN CHOI, 1 SUSUMU
H O R I B E 2 and Y O S H I K U N I K A W A B E l
tNational Research Institute for Metals, Tsukuba Laboratories, 1-2-1, Sengen, Tsukuba city, Ibaraki 305 and 2Department of Materials Science and Engineering, Waseda University, 3-4-1, Ohkubo, Sinjuku, Tokyo 169, Japan (Received 28 January 1994)
Abstract--The characteristics of cyclic fatigue crack growth in a silicon nitride which hardly shows rising R-curve behavior were investigated by using specimens having an indentation-induced flaw. V-shaped behavior was observed in the relation between the maximum stress intensity factor Kmaxand crack growth rate. When a crack growth rate is plotted as a function of the effective maximum stress intensity factor Ken.~a~ which takes account of the residual crack opening stress component K~in addition to Km~x,crack growth behavior is expressed by a power law relationship for the region above a crack length of about 0.37 mm in this experiment. However, it deviates from this relation below the critical length. Such crack growth behavior is explicable by introducing the term of the effective stress ratio R~n defined as (Kmi n + Kr)/(Kma x + Kr). Crack growth behavior in the terms of Ketr, max and R~r is expressed by a unique growth law, irrespective of crack length, i.e. d a / d N = C (K~fr.~x)3°(AKon)5,where AK~fr= (I --Reff)geff,max. Such behavior is consistent with that in the specimen containing a through-thickness crack. This equation indicates that fatigue crack growth rate in the present material is particularly sensitive to K~g.m~x,and less sensitive to AKen.
1. INTRODUCTION The specimens containing long through-thickness cracks such as CT (compact tension test) specimens and DT (double torsion) specimens have been used to determine crack growth under static and cyclic loads, Since these specimens usually have much longer cracks than intrinsic flaws or a crack, it is uncertain whether these characteristics determined by long cracks can represent them in natural flaws. It has been reported that cyclic fatigue crack growth behavior in small cracks is different from that in long cracks [1, 2]. Contrary to this, indentation cracks closely resemble the naturally occurring damage which ceramic surfaces experience during surface finishing and service. The indentation cracks have been thus often used in place of natural flaws in the study on small crack growth behavior [3-6]. However, the crack growth behavior from the indentation flaw in ceramics has not yet been studied in sufficient detail, particularly cyclic fatigue crack growth behavior, Some recent studies have reported that an indentation crack under static and cyclic loads shows anomalous growth behavior, that is, a crack growth rate decreases with increasing the maximum stress intensity factor Kmax [3-6]. AS recognized by Hoshide et al. [5], the anomalous growth behavior can be explained by the term of the effective maximum stress intensity
factor Ke~,maxdefined as the summation of the maximum stress intensity factor Kmax and the residual stress component due to indentation load Kr. However, a relatively large scatter in these data when crack growth rate is plotted as a function of K0g,max tends to be observed particularly in the early crack growth region [5]. Most likely, such a large scatter seems to indicate that a crack growth rate is associated with another factor besides Ke~,max-Recently, it has been found that the crack growth behavior under cyclic loads in some ceramics strongly depends upon the stress ratio R defined as the ratio of the minimum stress intensity factor Kmin to the maximum stress intensity factor Kmax [3, 4, 7]. However, for the specimen with an indentation crack, since the residual stress component Kr decreases with increasing crack size, it is expected that the effective stress ratio Re~ actually experienced by the crack tip will vary with crack size, although the applied R is constant. Such a variation o f / L ~ may give rise to a large scatter mentioned above. In this study, in order to clarify the influence of Re~ crack growth behavior under cyclic loads in silicon nitride was investigated by using the specimen with the surface crack introduced by indentation load. This material which hardly shows R-curve behavior was chosen to reduce a crack size effect on crack growth.
3837
3838
GUEN CHOI et al.: FATIGUE CRACK GROWTH IN SILICON NITRIDE 2. EXPERIMENTAL
2.1. Materials and fatigue test Silicon nitride normally sintered at 1750°C with Y203-MgA104 additives was used for the present work. Figure 1 shows the microstructure of this material. This material has a rod-like grain structure.The sintered materials were ground and lappingpolished to the fatigue specimens, the size of which was 3 x 4 × 40 mm. Before testing, a precrack was introduced at the center of the specimen by the Vickers indentation of 98 N load. This load is selected to reduce uncertainty about crack morphology so that c/b > 2.5, where c is the crack length and b is the half diagonal length of the indent. In this case, the crack morphology is a half-penny radial crack [9, 10]. Cyclic fatigue tests were conducted in four-point bending (outer span 30 mm, inner span 10 mm). In cyclic fatigue a sign-wave loading was used at a stress ratio R = 0.1 and a frequency of 20 Hz, using an electrohydraulic testing system. The crack length was measured by an optical microscope, the magnification x 400, by interrupting the testing and taking the specimen out of the machine. All experiments were carried out at room temperature in air. Fracture toughness was measured using single edge precracked beam (SEPB)specimens with dimensions 3 × 4 x 40 mm [8]. The precrack was introduced by a bridging indentation method and subsequently extended by cyclic loading. The fracture toughness value obtained in this way was 5.8 MPa ~/m. 2.2. R-curve behavior
where al and tr2 are the strength in materials with flaws introduced by indentation load PI and P2, respectively. The tensile side of the specimens with dimensions of about 3 x 4 x 25 mm was indented using various loads between 49 and 490 N. The strength was measured in four-point bending (outer span 20 mm, inner span 10 mm) using a loading rate of 8 MPa/s. 2.3. The evaluation o f the stress intensity factor
For a half-penny radial crack generated by Vickers indentation, the effective stress intensity tor K ~ for a crack length c, indentation load P, applied stress tr is given as follows [12-14] Ke~ = Ka + Kr = ~btrc1/2+ xP/c 3/2
(1)
the facand (3)
where Ka is the external stress component, K r the residual stress component, ~k a crack geometry constant, and X an elastic-plastic constraint constant. From equation (3), if the ~k and X values are known, the Kerrvalue can be obtained. The equilibrium crack length ce at stress a is given by Kef~= ~Oac~/2 + z P /c 3/2 = K¢
The crack growth resistance tests were carried out using the indentation-strength-in-bending (ISB) method. According to Cook and Clarke [11], toughness which characterizes the resistance to crack extension K R on the crack length c is represented by a power function, that is KR = Ko(c/co) ~
with 0 ~<~ ~<0.5. K0 is regarded as the base toughness of the material in the absence of any toughening mechanisms and the term co as the spatial extent of the crack at which toughening begins. And also they suggested that the decrease in strength as function of the Vickers indentation load can be described by the following expression [11] trl/a2 = (pl/p2)(2~-1)/(2~+ 3) (2)
(4)
where K¢ represents the fracture toughness of the material. Equation (4) can be rearranged to give trc~ = (KJ~k)c 3/z- xP/qJ (5) so that a plot of ac 2 vs c 3/2will be a straight line. KcAO is obtained as the slope of the plot. Therefore, if Ke (or ~O) is known, ~O (or Kc) and X is obtained. In order to obtain the relationship between tr and c, the specimen was loaded to cause certain crack propagation. After the crack length was measured, the loading was resumed to a higher value to cause further crack growth. This procedure was repeated until the specimen failed. The crack length was measured by an optical microscope, the magnification x 400, by interrupting the testing and taking the specimen out of the machine at room temperature in air. Since silicon nitride used in this study is relatively insensitive to static fatigue [15], an environmental crack growth effect during tests was neglected. 3. RESULTS AND DISCUSSION 3.1. The estimation o f ~k and ~ values for the indentation crack
Fig. 1. Scanning electron micrograph of a surface etched by molten NaOH.
Figure 2 shows typical tr vs ce data plotted according to equation (5). The relation between ac ~ and c ~.5
GUEN CHOI et al.: FATIGUE CRACK GROWTH IN SILICON NITRIDE
tl X
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3839
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~
Ce 1.5(ml.5)
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Indentation Load (N) Fig. 4. Variation of fracture strength with indentation load.
Fig. 2. The plot ofcrc ~vs c ~,'2for indentation load P = 98 N. is the linear and the slope of this line, KelP, is 6.80 MPa x/m. If the shape factor ¢, is obtained, the Kc and X values can be calculated. However, the ~, values estimated by some researchers are inconsistent with each other. ~k is determined when the variation of crack shape with the increasing crack is known. However, it is difficult to measure the variation of crack shape with the increasing crack in ceramics such as silicon nitride. Using the ~ values obtained in the finite element calculations of Newman and Raju [16] for half-penny surface cracks and the Z value given by Anstis and coworkers [13], Ke~ can be calculated from equation (3). The K ~ value obtained in this way increases with the increase of crack size as shown in Fig. 3. Such R-curve behavior is inconsistent with the result of the ISB test indicating a flat R-curve in Fig. 4. Moreover, the toughness in Fig. 3 mostly exceeds 5.8 MPa x/m of the value obtained using the SEPB specimens. Similar observations are reported in some recent studies [17, 18].
10
.
.
.
~o
oo
.
8] ~
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4 ,
~ , ~ , L , 0.2 0.3 0.4 0.5 Crack size, C(mm) Fig. 3. R-curve generated from equation (5) using the @ values obtained in the finite element calculations of Newman and Raju [16] for half-penny surface cracks and the Z value given by Anstis and coworkers [13]. 42/I I - - Q
~ 0.1
,
(6)
where ~ depends on the response of the material to the indentation, and E and H are an elastic modulus and the Vickers hardness, respectively. For the present material, using X = 0.082, E = 300 GPa and H = 13 GPa, ¢ = 0.017 from equation (6). This value is almost the same as ¢ = 0.016 obtained by Anstis and coworkers [13]. Chantikul et aL [14] have determined an empirical constant r/ (=0.59) by correlating I - S data to conventional fracture toughness for a range of glasses and ceramics. From indentation fracture mechanics (7)
•
•
AMM
Z = ~ (E/H)'~2
t/ = [(256/27)(~J)3~ ]1/4.
,~ ¢~
Figure 4 shows the result of the ISB test [19]. The z value in equation (2) obtained from the slope of straight lines of this figure is nearly zero in the material. Such a result implies that this material hardly shows R-curve behavior. Hence, it is expected that Kc of this material is constant regardless of crack size. If K¢ is determined, the ~k and Z values can be calculated from the slope and intercept of the plot in Fig. 2. Using K¢ = 5.8 MPa ~/m determined using the SEPB specimens, the ~k and Z values obtained in this way are 0.85 and 0.082, respectively. Generally the parameter X is given by [13]
Inserting ~ =0.85 and ¢ = 0.017 into equation (7) yields r / = 0.56, which is in reasonable concurrence with the empirical constant ~/= 0.59. Therefore, it is supposed that the ~, and Z values obtained in the present work is reasonable. The values were thus used in the estimation of the stress intensity factor of an indentation crack.
3.2. Crack growth behavior under cyclic loading Figure 5 shows crack growth rates as a function of Kmax. The approximate crack growth length for each
3840
GUEN CHOI et al.: FATIGUE CRACK GROWTH IN SILICON NITRIDE
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O (0.20-0.24mm) (0.25-0.37mm) /x (0.37-0.51mm) • (0.51-1.17mm)
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0.4 0.6 0.8 Crack size (mm)
I
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Fig. 5. Fatigue crack growth rate vs the maximum applied stress intensity factor,
Fig. 7. Variation of the effective stress ratio with crack length using data in Fig. 6.
datum is indicated. The growth rate is reduced with increasing the stress intensity in the early region, In the region above a certain crack length, the normal fatigue crack propagation behavior gradually resumes and positive dependence on the applied stress intensity is again obeyed. Similar V-shaped behavior has been also observed in some ceramic materials [3, 5, 6]. Hoshide and coworkers [5] have shown that such anomalous growth behavior has been attributed to residual crack opening stress due to plastic deformation at the indentation, since the crack growth rate is correlated to the effective maximum stress intensity K+er.max( = Kmax+ Kr). Figure 6 shows crack growth rates as a function of K+~.maxin the present material. In the case of the crack length longer than a = 0.37 mm, the crack growth rate is represented by an unique line, but below a = 0.37 mm crack growth rate data deviate from this line. Such results are likely
to imply that the crack growth rate is not an unique function of K+~,max. It is, so far, uncertain why the crack growth behavior deviates in the early crack growth stage as shown in Fig. 6. Recently, it has been reported that the crack growth rate under cyclic loading in some ceramic materials is closely related to the stress ratio R [3, 4, 7]. In the case of such a specimen with indentation crack in which residual crack opening stress exists, the effective stress ratio Rea actually experienced by crack tip should not be equal to R and vary as the variations of the crack size or the applied stress, since the component of tensile residual stress Kr due to indentation load depends on crack size as known from equation (3) and also is related to Kmi, as well as Kmax. The effective stress ratio R+~ under a tensile residual stress field can be written as follows Res = (Kmin "F K r)/(Km.~ + K r). (8)
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10-6
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Figure 7 shows the variation of Re~ with crack size. The R ~ plot is not a unique curve, because the applied load was sometimes changed in the present experiment. From this figure it is indicated that the Re~ values are larger than the applied stress ratio R = 0.1 and vary between 0.1 and 0.6. Moreover, Ren decreases with the increase of crack size, particularly showing the rapid decrease with the crack extension in the early crack growth region. Such a variation of the R ~ value with crack length might give rise to deviation as shown in Fig. 6. As in metallic materials, cyclic fatigue crack growth rates can be expressed by a conventional Paris law relationship da/dN
10-121
2
3
4
5 6 7 8 910
Keff,max(MPaml/2) Fig. 6. Fatigue crack growth rate vs the effective maximum applied stress intensity factor,
= B
(AK)m
(9)
where B and the exponent m are experimentally measured scaling constants dependent on the materials and environmental conditions. However, it has been recently found that the fatigue crack growth rate
GUEN CHOI et al.: FATIGUE CRACK GROWTH IN SILICON NITRIDE 10 -4
~" -G 10-e
. , ~ (0.20-0.24mm) (0.25-0.37mm)
,
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A (0.37-0.51 mm) • (0.51-1.17mm)
o Oo ~ ~o
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....
1 0 -~
~" ~ 10_10
Using equation (12), when q = 5, for different R ~ ratios crack growth rate data normalized by (l - Re~)5 with Ken,m~xare represented by a single line, as shown in Fig. 8. The m value obtained from the slope is about 35. From this result, the crack growth relationship can finally be written as follows, irrespective of crack size da / d N = C (Keu. . . . )3o(Ages) 5
6"
v--
3841
(13)
where AKe~ = (1 - R~)Ke~.... . Since silicon nitride in this study hardly shows R-curve behavior as shown in Fig. 5, it is expected that crack growth behavior is " independent of crack length. The fact that the crack size dependence on crack growth behavior does not 10-12 2I 3I 4I 5 6 7 8 910 exist in crack size ranging from 0.2 to 1.2mm as shown in Fig. 8, is consistent with the fiat R-curve Keff,rnax(MPam1/2) behavior in this material. Moreover, it has been found that the same crack growth relationship is Fig. 8. Normalized crack growth rate vs the effective maximum stress intensity factor, obtained by using the SEPB (single edge precracked beam) specimen, as shown in Fig. 9 [20]. Such results support the validity of the crack growth relationship in some ceramic materials depends very strongly on in equation (13). Therefore, it is evident that the the maximum stress intensity factor and modestly on deceleration of the crack growth rate in the early the stress intensity factor range [3, 4, 7]. To examine crack growth stage as shown in Fig. 6 is attributed to the specific dependence on Kin,x and AK, the following the variation of the Rea values with increasing crack equation containing both these terms has been used length, particularly in the early stage showing steep [3, 4, 7] variation. d a / d N = C (gmax)P(Ag) q (10) The crack growth law presented by equation (10) has been reported in other ceramics such as silicon where C is a constant equal to B ( 1 - R ) p and carbide whisker-reinforced alumina composites (p + q ) = m . (A1203-SiCw) [4,7], a yttria-stabilized tetragonal AK = (1 - R )Kin,,, thus equation (10) is given by zirconia polycrystal (Y-TZP) [3]. As indicated in d a / d N = C(Kma,)"(1 - - R ) q. (11) these studies, the exponent p is considerably higher than q, although toughening mechanisms in these Equation (11) can be rewritten in the terms of Ko~.m~x materials are different from each other. Namely, the and R~e as follows growth rates in these ceramics are sensitive to the d a / d N = C (Keff, max)rn(l - - Ruff) q. (12) tensile stress state ahead of a crack tip and Kma x o r Ke~. . . . and relatively insensitive to AK or AKin, in contrast to fatigue behavior in metallic materials 10 . 4 , , , , , , , where generally the reverse is true. It is likely that • R=0.1 such dependence is a characteristic feature of cyclic [] R=0.5 fatigue crack growth behavior in ceramic materials.
~" ..~ 10-e
~ "
•
R=0.7
,£
>, .o. o
..~ ~" ~" ¢, t
tt)
E" 10 -~ i • z
Z"
~10-1o
h t •
10-1:
:2
:3
4
~ 5~ 6~ 7~ 8910
Kmax (MPamv2) Fig. 9. Normalized crack growth rate vs the maximum stress intensity factor obtained by using single edge precracked beam specimens [20].
4. CONCLUSIONS The crack growth rate under cyclic loading in a silicon nitride which hardly shows rising R-curve behavior was investigated by using specimens having an indentation-induced flaw and the main results obtained are as follows: 1. Fracture toughness, K~c determined using the SEPB specimens, is 5.8 MPa ~/m. On the basis of indentation fracture mechanics, by using this value and the data obtained by measurements of crack length with the applied stress, the indented parameters, the ~ and X values can be obtained. ~ = 0.85 and X = 0.082. 2. The crack growth rate is not controlled by the effective maximum stress intensity factor Kee.... only,
3842
GUEN CHOI et al.: FATIGUE CRACK GROWTH IN SILICON NITRIDE
although the applied stress ratio R is constant. This is attributed to a variation of the effective stress ratio R ~ with the increase o f crack length, which is defined as (Kmit~+ Kr)/(Kmax + Kr) in the terms of the maxim u m stress intensity factor Kmax, the m i n i m u m stress intensity factor Kmin and the residual crack opening stress c o m p o n e n t Kr. 3. A crack growth relationship is given by a power law function, depending strongly upon the effective m a x i m u m stress intensity factor Ke~.max compared to the range o f the effective stress intensity AKe~ , i.e. d a / d N = C (Kef~.... )3°(AKe~) s.
REFERENCES 1. A. A. Steffen, R. H. Dauskardt and R. O. Ritchie, J. Am. Ceram. Soc. 74, 1259 (1991). 2. D. G. Jensen, V. Zelizko and M. V. Swain, J. Mater. Sci. Lett. 8, 1154 (1989). 3. S-Y. Liu and I-W. Chen, J. Am. Ceram. Soc. 74, 1206 (1991). 4. R. H. Dauskardt, M. J. James and R. O. Ritchie, J. Am. Ceram. Soc. 75, 759 (1992). 5. T. Hoshide, T. Ohara and T. Yamada, Int. J. Fract. 37, 47 (1988).
6. M. Yoda, Int. J. Fract. 39, R23 (1989). 7. R. H. Dauskardt, B. J. Dalgleish, D. Yao, R. O. Ritchie and P. T. Becher, J. Mater. Sci. 28, 3258 (1993). 8. T. Nose and T. Fujii, J. Am. Ceram. Soc. 71, 328 (1988). 9. D. K. Shetty, A. R. Rosenfield and W. H. Duckworth, J. Am. Ceram. Soc. 68, c282 (1985). 10. K. Niihara, R. Morena and D. P. H. Hasselman, J. Am. Ceram. Soc. 65, el 16 (1982). l 1. R. F. Cook and D. R. Clarke, Acta metall. 36, 555 (1988). 12. D. B. Marshall and B. R. Lawn, J. Mater. Sci. 14, 2001 (1979). 13. G. R. Anstis, P. Chantikul, B.R. Lawn and D. B. Marshall, J. Am. Ceram. Soc. 64, 533 (1981). 14. P. Chantikul, G. R. Anstis, B. R. Lawn and D. B. Marshall, J. Am. Ceram. Soc. 64, 539 (1981). 15. G. Choi and S. Horibe, J. Mater. Sci. 28, 5931 (1993). 16. J. C. Newman and I. S. Raju, Engng Fract. Mech. 15, 185 (1981). 17. C. W. Li, D. J. Lee and S. C. Lui, J. Am. Ceram. Soc. 75, 1777 (1992). 18. S: M. Smith and R. O. Scattergood, J. Am. Ceram. Soc. 75, 305 (1992). 19. G. Choi, S. Horibe and Y. Kawabe, Acta. metall. Mater. 42, 1407 (1994). 20. G. Choi and S. Hofibe, unpublished.
Pergamon
0956-7151(94)E0154-9
Copyright © 1994ElsevierScienceLtd Printed in Great Britain.All rights reserved 0956-7151/94$7.00+ 0.00
CYCLIC F A T I G U E CRACK G R O W T H F R O M I N D E N T A T I O N FLAW IN SILICON NITRIDE: I N F L U E N C E OF EFFECTIVE STRESS RATIO GUEN CHOI, 1 SUSUMU
H O R I B E 2 and Y O S H I K U N I K A W A B E l
tNational Research Institute for Metals, Tsukuba Laboratories, 1-2-1, Sengen, Tsukuba city, Ibaraki 305 and 2Department of Materials Science and Engineering, Waseda University, 3-4-1, Ohkubo, Sinjuku, Tokyo 169, Japan (Received 28 January 1994)
Abstract--The characteristics of cyclic fatigue crack growth in a silicon nitride which hardly shows rising R-curve behavior were investigated by using specimens having an indentation-induced flaw. V-shaped behavior was observed in the relation between the maximum stress intensity factor Kmaxand crack growth rate. When a crack growth rate is plotted as a function of the effective maximum stress intensity factor Ken.~a~ which takes account of the residual crack opening stress component K~in addition to Km~x,crack growth behavior is expressed by a power law relationship for the region above a crack length of about 0.37 mm in this experiment. However, it deviates from this relation below the critical length. Such crack growth behavior is explicable by introducing the term of the effective stress ratio R~n defined as (Kmi n + Kr)/(Kma x + Kr). Crack growth behavior in the terms of Ketr, max and R~r is expressed by a unique growth law, irrespective of crack length, i.e. d a / d N = C (K~fr.~x)3°(AKon)5,where AK~fr= (I --Reff)geff,max. Such behavior is consistent with that in the specimen containing a through-thickness crack. This equation indicates that fatigue crack growth rate in the present material is particularly sensitive to K~g.m~x,and less sensitive to AKen.
1. INTRODUCTION The specimens containing long through-thickness cracks such as CT (compact tension test) specimens and DT (double torsion) specimens have been used to determine crack growth under static and cyclic loads, Since these specimens usually have much longer cracks than intrinsic flaws or a crack, it is uncertain whether these characteristics determined by long cracks can represent them in natural flaws. It has been reported that cyclic fatigue crack growth behavior in small cracks is different from that in long cracks [1, 2]. Contrary to this, indentation cracks closely resemble the naturally occurring damage which ceramic surfaces experience during surface finishing and service. The indentation cracks have been thus often used in place of natural flaws in the study on small crack growth behavior [3-6]. However, the crack growth behavior from the indentation flaw in ceramics has not yet been studied in sufficient detail, particularly cyclic fatigue crack growth behavior, Some recent studies have reported that an indentation crack under static and cyclic loads shows anomalous growth behavior, that is, a crack growth rate decreases with increasing the maximum stress intensity factor Kmax [3-6]. AS recognized by Hoshide et al. [5], the anomalous growth behavior can be explained by the term of the effective maximum stress intensity
factor Ke~,maxdefined as the summation of the maximum stress intensity factor Kmax and the residual stress component due to indentation load Kr. However, a relatively large scatter in these data when crack growth rate is plotted as a function of K0g,max tends to be observed particularly in the early crack growth region [5]. Most likely, such a large scatter seems to indicate that a crack growth rate is associated with another factor besides Ke~,max-Recently, it has been found that the crack growth behavior under cyclic loads in some ceramics strongly depends upon the stress ratio R defined as the ratio of the minimum stress intensity factor Kmin to the maximum stress intensity factor Kmax [3, 4, 7]. However, for the specimen with an indentation crack, since the residual stress component Kr decreases with increasing crack size, it is expected that the effective stress ratio Re~ actually experienced by the crack tip will vary with crack size, although the applied R is constant. Such a variation o f / L ~ may give rise to a large scatter mentioned above. In this study, in order to clarify the influence of Re~ crack growth behavior under cyclic loads in silicon nitride was investigated by using the specimen with the surface crack introduced by indentation load. This material which hardly shows R-curve behavior was chosen to reduce a crack size effect on crack growth.
3837
3838
GUEN CHOI et al.: FATIGUE CRACK GROWTH IN SILICON NITRIDE 2. EXPERIMENTAL
2.1. Materials and fatigue test Silicon nitride normally sintered at 1750°C with Y203-MgA104 additives was used for the present work. Figure 1 shows the microstructure of this material. This material has a rod-like grain structure.The sintered materials were ground and lappingpolished to the fatigue specimens, the size of which was 3 x 4 × 40 mm. Before testing, a precrack was introduced at the center of the specimen by the Vickers indentation of 98 N load. This load is selected to reduce uncertainty about crack morphology so that c/b > 2.5, where c is the crack length and b is the half diagonal length of the indent. In this case, the crack morphology is a half-penny radial crack [9, 10]. Cyclic fatigue tests were conducted in four-point bending (outer span 30 mm, inner span 10 mm). In cyclic fatigue a sign-wave loading was used at a stress ratio R = 0.1 and a frequency of 20 Hz, using an electrohydraulic testing system. The crack length was measured by an optical microscope, the magnification x 400, by interrupting the testing and taking the specimen out of the machine. All experiments were carried out at room temperature in air. Fracture toughness was measured using single edge precracked beam (SEPB)specimens with dimensions 3 × 4 x 40 mm [8]. The precrack was introduced by a bridging indentation method and subsequently extended by cyclic loading. The fracture toughness value obtained in this way was 5.8 MPa ~/m. 2.2. R-curve behavior
where al and tr2 are the strength in materials with flaws introduced by indentation load PI and P2, respectively. The tensile side of the specimens with dimensions of about 3 x 4 x 25 mm was indented using various loads between 49 and 490 N. The strength was measured in four-point bending (outer span 20 mm, inner span 10 mm) using a loading rate of 8 MPa/s. 2.3. The evaluation o f the stress intensity factor
For a half-penny radial crack generated by Vickers indentation, the effective stress intensity tor K ~ for a crack length c, indentation load P, applied stress tr is given as follows [12-14] Ke~ = Ka + Kr = ~btrc1/2+ xP/c 3/2
(1)
the facand (3)
where Ka is the external stress component, K r the residual stress component, ~k a crack geometry constant, and X an elastic-plastic constraint constant. From equation (3), if the ~k and X values are known, the Kerrvalue can be obtained. The equilibrium crack length ce at stress a is given by Kef~= ~Oac~/2 + z P /c 3/2 = K¢
The crack growth resistance tests were carried out using the indentation-strength-in-bending (ISB) method. According to Cook and Clarke [11], toughness which characterizes the resistance to crack extension K R on the crack length c is represented by a power function, that is KR = Ko(c/co) ~
with 0 ~<~ ~<0.5. K0 is regarded as the base toughness of the material in the absence of any toughening mechanisms and the term co as the spatial extent of the crack at which toughening begins. And also they suggested that the decrease in strength as function of the Vickers indentation load can be described by the following expression [11] trl/a2 = (pl/p2)(2~-1)/(2~+ 3) (2)
(4)
where K¢ represents the fracture toughness of the material. Equation (4) can be rearranged to give trc~ = (KJ~k)c 3/z- xP/qJ (5) so that a plot of ac 2 vs c 3/2will be a straight line. KcAO is obtained as the slope of the plot. Therefore, if Ke (or ~O) is known, ~O (or Kc) and X is obtained. In order to obtain the relationship between tr and c, the specimen was loaded to cause certain crack propagation. After the crack length was measured, the loading was resumed to a higher value to cause further crack growth. This procedure was repeated until the specimen failed. The crack length was measured by an optical microscope, the magnification x 400, by interrupting the testing and taking the specimen out of the machine at room temperature in air. Since silicon nitride used in this study is relatively insensitive to static fatigue [15], an environmental crack growth effect during tests was neglected. 3. RESULTS AND DISCUSSION 3.1. The estimation o f ~k and ~ values for the indentation crack
Fig. 1. Scanning electron micrograph of a surface etched by molten NaOH.
Figure 2 shows typical tr vs ce data plotted according to equation (5). The relation between ac ~ and c ~.5
GUEN CHOI et al.: FATIGUE CRACK GROWTH IN SILICON NITRIDE
tl X
50
.
.
.
.
.
.
.
.
188, '
.
. . . . . . . .
3839
i
. . . . . . . .
I
.......
80q 70~ 60~ ~; 500
40-
v
~ 400 ca.
o
30
-
20
~ 300 9 -~ ~ 200 iJ_
10 ,
0
1
2
3
4
5
6
,
,
7
8 9 10 [1 x l 0 -6]
~
Ce 1.5(ml.5)
10
'
1
........
10 2
03
Indentation Load (N) Fig. 4. Variation of fracture strength with indentation load.
Fig. 2. The plot ofcrc ~vs c ~,'2for indentation load P = 98 N. is the linear and the slope of this line, KelP, is 6.80 MPa x/m. If the shape factor ¢, is obtained, the Kc and X values can be calculated. However, the ~, values estimated by some researchers are inconsistent with each other. ~k is determined when the variation of crack shape with the increasing crack is known. However, it is difficult to measure the variation of crack shape with the increasing crack in ceramics such as silicon nitride. Using the ~ values obtained in the finite element calculations of Newman and Raju [16] for half-penny surface cracks and the Z value given by Anstis and coworkers [13], Ke~ can be calculated from equation (3). The K ~ value obtained in this way increases with the increase of crack size as shown in Fig. 3. Such R-curve behavior is inconsistent with the result of the ISB test indicating a flat R-curve in Fig. 4. Moreover, the toughness in Fig. 3 mostly exceeds 5.8 MPa x/m of the value obtained using the SEPB specimens. Similar observations are reported in some recent studies [17, 18].
10
.
.
.
~o
oo
.
8] ~
6
4 ,
~ , ~ , L , 0.2 0.3 0.4 0.5 Crack size, C(mm) Fig. 3. R-curve generated from equation (5) using the @ values obtained in the finite element calculations of Newman and Raju [16] for half-penny surface cracks and the Z value given by Anstis and coworkers [13]. 42/I I - - Q
~ 0.1
,
(6)
where ~ depends on the response of the material to the indentation, and E and H are an elastic modulus and the Vickers hardness, respectively. For the present material, using X = 0.082, E = 300 GPa and H = 13 GPa, ¢ = 0.017 from equation (6). This value is almost the same as ¢ = 0.016 obtained by Anstis and coworkers [13]. Chantikul et aL [14] have determined an empirical constant r/ (=0.59) by correlating I - S data to conventional fracture toughness for a range of glasses and ceramics. From indentation fracture mechanics (7)
•
•
AMM
Z = ~ (E/H)'~2
t/ = [(256/27)(~J)3~ ]1/4.
,~ ¢~
Figure 4 shows the result of the ISB test [19]. The z value in equation (2) obtained from the slope of straight lines of this figure is nearly zero in the material. Such a result implies that this material hardly shows R-curve behavior. Hence, it is expected that Kc of this material is constant regardless of crack size. If K¢ is determined, the ~k and Z values can be calculated from the slope and intercept of the plot in Fig. 2. Using K¢ = 5.8 MPa ~/m determined using the SEPB specimens, the ~k and Z values obtained in this way are 0.85 and 0.082, respectively. Generally the parameter X is given by [13]
Inserting ~ =0.85 and ¢ = 0.017 into equation (7) yields r / = 0.56, which is in reasonable concurrence with the empirical constant ~/= 0.59. Therefore, it is supposed that the ~, and Z values obtained in the present work is reasonable. The values were thus used in the estimation of the stress intensity factor of an indentation crack.
3.2. Crack growth behavior under cyclic loading Figure 5 shows crack growth rates as a function of Kmax. The approximate crack growth length for each
3840
GUEN CHOI et al.: FATIGUE CRACK GROWTH IN SILICON NITRIDE
10 -4
'
10-6
O (0.20-0.24mm) (0.25-0.37mm) /x (0.37-0.51mm) • (0.51-1.17mm)
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1
.
.
.
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=
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-8
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=
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o=220MPa
,,'t "
0.2
ix
10 -1
O=190MPa
0.4
1
I
2
I
3
o =160MPa .............................................
I
4
I
I
I
I
I
5 6 7 8910
Kmax(M Pam 1/2)
0
I
I
0.2
I
I
I
I
I
I
0.4 0.6 0.8 Crack size (mm)
I
1
1.2
Fig. 5. Fatigue crack growth rate vs the maximum applied stress intensity factor,
Fig. 7. Variation of the effective stress ratio with crack length using data in Fig. 6.
datum is indicated. The growth rate is reduced with increasing the stress intensity in the early region, In the region above a certain crack length, the normal fatigue crack propagation behavior gradually resumes and positive dependence on the applied stress intensity is again obeyed. Similar V-shaped behavior has been also observed in some ceramic materials [3, 5, 6]. Hoshide and coworkers [5] have shown that such anomalous growth behavior has been attributed to residual crack opening stress due to plastic deformation at the indentation, since the crack growth rate is correlated to the effective maximum stress intensity K+er.max( = Kmax+ Kr). Figure 6 shows crack growth rates as a function of K+~.maxin the present material. In the case of the crack length longer than a = 0.37 mm, the crack growth rate is represented by an unique line, but below a = 0.37 mm crack growth rate data deviate from this line. Such results are likely
to imply that the crack growth rate is not an unique function of K+~,max. It is, so far, uncertain why the crack growth behavior deviates in the early crack growth stage as shown in Fig. 6. Recently, it has been reported that the crack growth rate under cyclic loading in some ceramic materials is closely related to the stress ratio R [3, 4, 7]. In the case of such a specimen with indentation crack in which residual crack opening stress exists, the effective stress ratio Rea actually experienced by crack tip should not be equal to R and vary as the variations of the crack size or the applied stress, since the component of tensile residual stress Kr due to indentation load depends on crack size as known from equation (3) and also is related to Kmi, as well as Kmax. The effective stress ratio R+~ under a tensile residual stress field can be written as follows Res = (Kmin "F K r)/(Km.~ + K r). (8)
10 -4
' ' © (0.20-0.24mm)
'
'
'
' ' '
(0.25-0.37mm) (0.37-0.51 ram)
10-6
•
(0.37-1.17mm) • ~° o ,~
O
o>" 0_ 6 "~ 1
®o
10-I0
Figure 7 shows the variation of Re~ with crack size. The R ~ plot is not a unique curve, because the applied load was sometimes changed in the present experiment. From this figure it is indicated that the Re~ values are larger than the applied stress ratio R = 0.1 and vary between 0.1 and 0.6. Moreover, Ren decreases with the increase of crack size, particularly showing the rapid decrease with the crack extension in the early crack growth region. Such a variation of the R ~ value with crack length might give rise to deviation as shown in Fig. 6. As in metallic materials, cyclic fatigue crack growth rates can be expressed by a conventional Paris law relationship da/dN
10-121
2
3
4
5 6 7 8 910
Keff,max(MPaml/2) Fig. 6. Fatigue crack growth rate vs the effective maximum applied stress intensity factor,
= B
(AK)m
(9)
where B and the exponent m are experimentally measured scaling constants dependent on the materials and environmental conditions. However, it has been recently found that the fatigue crack growth rate
GUEN CHOI et al.: FATIGUE CRACK GROWTH IN SILICON NITRIDE 10 -4
~" -G 10-e
. , ~ (0.20-0.24mm) (0.25-0.37mm)
,
,
A (0.37-0.51 mm) • (0.51-1.17mm)
o Oo ~ ~o
E ~" t rI
....
1 0 -~
~" ~ 10_10
Using equation (12), when q = 5, for different R ~ ratios crack growth rate data normalized by (l - Re~)5 with Ken,m~xare represented by a single line, as shown in Fig. 8. The m value obtained from the slope is about 35. From this result, the crack growth relationship can finally be written as follows, irrespective of crack size da / d N = C (Keu. . . . )3o(Ages) 5
6"
v--
3841
(13)
where AKe~ = (1 - R~)Ke~.... . Since silicon nitride in this study hardly shows R-curve behavior as shown in Fig. 5, it is expected that crack growth behavior is " independent of crack length. The fact that the crack size dependence on crack growth behavior does not 10-12 2I 3I 4I 5 6 7 8 910 exist in crack size ranging from 0.2 to 1.2mm as shown in Fig. 8, is consistent with the fiat R-curve Keff,rnax(MPam1/2) behavior in this material. Moreover, it has been found that the same crack growth relationship is Fig. 8. Normalized crack growth rate vs the effective maximum stress intensity factor, obtained by using the SEPB (single edge precracked beam) specimen, as shown in Fig. 9 [20]. Such results support the validity of the crack growth relationship in some ceramic materials depends very strongly on in equation (13). Therefore, it is evident that the the maximum stress intensity factor and modestly on deceleration of the crack growth rate in the early the stress intensity factor range [3, 4, 7]. To examine crack growth stage as shown in Fig. 6 is attributed to the specific dependence on Kin,x and AK, the following the variation of the Rea values with increasing crack equation containing both these terms has been used length, particularly in the early stage showing steep [3, 4, 7] variation. d a / d N = C (gmax)P(Ag) q (10) The crack growth law presented by equation (10) has been reported in other ceramics such as silicon where C is a constant equal to B ( 1 - R ) p and carbide whisker-reinforced alumina composites (p + q ) = m . (A1203-SiCw) [4,7], a yttria-stabilized tetragonal AK = (1 - R )Kin,,, thus equation (10) is given by zirconia polycrystal (Y-TZP) [3]. As indicated in d a / d N = C(Kma,)"(1 - - R ) q. (11) these studies, the exponent p is considerably higher than q, although toughening mechanisms in these Equation (11) can be rewritten in the terms of Ko~.m~x materials are different from each other. Namely, the and R~e as follows growth rates in these ceramics are sensitive to the d a / d N = C (Keff, max)rn(l - - Ruff) q. (12) tensile stress state ahead of a crack tip and Kma x o r Ke~. . . . and relatively insensitive to AK or AKin, in contrast to fatigue behavior in metallic materials 10 . 4 , , , , , , , where generally the reverse is true. It is likely that • R=0.1 such dependence is a characteristic feature of cyclic [] R=0.5 fatigue crack growth behavior in ceramic materials.
~" ..~ 10-e
~ "
•
R=0.7
,£
>, .o. o
..~ ~" ~" ¢, t
tt)
E" 10 -~ i • z
Z"
~10-1o
h t •
10-1:
:2
:3
4
~ 5~ 6~ 7~ 8910
Kmax (MPamv2) Fig. 9. Normalized crack growth rate vs the maximum stress intensity factor obtained by using single edge precracked beam specimens [20].
4. CONCLUSIONS The crack growth rate under cyclic loading in a silicon nitride which hardly shows rising R-curve behavior was investigated by using specimens having an indentation-induced flaw and the main results obtained are as follows: 1. Fracture toughness, K~c determined using the SEPB specimens, is 5.8 MPa ~/m. On the basis of indentation fracture mechanics, by using this value and the data obtained by measurements of crack length with the applied stress, the indented parameters, the ~ and X values can be obtained. ~ = 0.85 and X = 0.082. 2. The crack growth rate is not controlled by the effective maximum stress intensity factor Kee.... only,
3842
GUEN CHOI et al.: FATIGUE CRACK GROWTH IN SILICON NITRIDE
although the applied stress ratio R is constant. This is attributed to a variation of the effective stress ratio R ~ with the increase o f crack length, which is defined as (Kmit~+ Kr)/(Kmax + Kr) in the terms of the maxim u m stress intensity factor Kmax, the m i n i m u m stress intensity factor Kmin and the residual crack opening stress c o m p o n e n t Kr. 3. A crack growth relationship is given by a power law function, depending strongly upon the effective m a x i m u m stress intensity factor Ke~.max compared to the range o f the effective stress intensity AKe~ , i.e. d a / d N = C (Kef~.... )3°(AKe~) s.
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6. M. Yoda, Int. J. Fract. 39, R23 (1989). 7. R. H. Dauskardt, B. J. Dalgleish, D. Yao, R. O. Ritchie and P. T. Becher, J. Mater. Sci. 28, 3258 (1993). 8. T. Nose and T. Fujii, J. Am. Ceram. Soc. 71, 328 (1988). 9. D. K. Shetty, A. R. Rosenfield and W. H. Duckworth, J. Am. Ceram. Soc. 68, c282 (1985). 10. K. Niihara, R. Morena and D. P. H. Hasselman, J. Am. Ceram. Soc. 65, el 16 (1982). l 1. R. F. Cook and D. R. Clarke, Acta metall. 36, 555 (1988). 12. D. B. Marshall and B. R. Lawn, J. Mater. Sci. 14, 2001 (1979). 13. G. R. Anstis, P. Chantikul, B.R. Lawn and D. B. Marshall, J. Am. Ceram. Soc. 64, 533 (1981). 14. P. Chantikul, G. R. Anstis, B. R. Lawn and D. B. Marshall, J. Am. Ceram. Soc. 64, 539 (1981). 15. G. Choi and S. Horibe, J. Mater. Sci. 28, 5931 (1993). 16. J. C. Newman and I. S. Raju, Engng Fract. Mech. 15, 185 (1981). 17. C. W. Li, D. J. Lee and S. C. Lui, J. Am. Ceram. Soc. 75, 1777 (1992). 18. S: M. Smith and R. O. Scattergood, J. Am. Ceram. Soc. 75, 305 (1992). 19. G. Choi, S. Horibe and Y. Kawabe, Acta. metall. Mater. 42, 1407 (1994). 20. G. Choi and S. Hofibe, unpublished.