High temperature crack growth in silicon nitride under static and cyclic loading: Short-crack behavior and brittle-ductile transition

Pergamon 0956-7151(95)00307-x

Acta mater. Vol. 44, No. 5, pp. 2079-2092, 1996 Elsevier Science Ltd Acta Metallurgica Inc. Printed in Great Britain

HIGH TEMPERATURE CRACK GROWTH IN SILICON NITRIDE UNDER STATIC AND CYCLIC LOADING: SHORT-CRACK BEHAVIOR AND BRITTLE-DUCTILE TRANSITION? SHM-YU LIU$ and I-WE1 CHEN Department of Materials Science and Engineering, The University of Michigan, Ann Arbor, MI 48109-2136, U.S.A. (Received 26 May 1995)

Abstract-Crack propagation in S&N, at elevated temperatures was investigated using controlled surface flaws. Crack growth was generally slower under cyclic than static loading conditions. Concomitantly, there was a tendency for crack growth rate to initially decelerate despite an increasing driving force, exhibiting the so-called short-crack behavior. This tendency became more pronounced at higher temperatures, lower stress intensity factors, and larger cyclic stress variations. A corresponding transition in the crack profile, from a sharp to a blunt crack, was observed. These phenomena are attributed to evolutions of crack-wake shielding. Specifically, with rising temperature and stress cycling, grain fraction is lowered, triple points are separated and massive grain pullout is triggered. This mode of pullout mechanism is qualitatively distinct from that operating at lower temperatures, and the transition is believed to occur when the slip length of the grain boundary reaches the average half-length of the grain. This picture is supported by a fracture mechanics estimation of crack growth rate, crack opening displacement and characteristic length of the R-curve based on the pullout mechanism.

1. INTRODUCTION

Controlled surface flaws loaded in bending were first employed by Petrovic et al. to study slow crack growth behavior almost two decades ago [l, 21. In their work on static fatigue of a hot-pressed S&N4 at temperatures ranging from 1100 to 13OO”C, two features were reported but not satisfactorily explained [2]. First, high growth rates were found for some cracks at low applied stress intensity factors. Second, cracks of a similar size subject to the same applied stress were found to grow at different rates at different stages of their extension. Not surprisingly, the above observations were accompanied by a very large data scatter. This led the authors to conclude that crack growth rates and applied stress intensity factors could not be correlated by a simple power law. They further postulated that these features originated from the interaction of the small indentation cracks with local material heterogeneities [2]. This postulate and the peculiar crack growth behavior of silicon nitride at high temperatures have remained unexplored since. In retrospect, we believe that the above observations are reminiscent of the so-called short-crack tThis research was sponsored by the U.S. Air Force under Grant No. AFOSR 95-l-01 19. SNOW at Westinghouse Science and Technology Center, Pittsburgh, U.S.A. 2079

behavior [3]. A characteristic of the short-crack behavior is an inverse relationship between crack growth rates and applied stress intensity factors. For metallic materials, short-crack behavior is often observed when the crack length is shorter than the characteristic length of a microstructural feature or the characteristic range of residual stress [3]. For ceramics, short-crack behavior caused by thermal stress fluctuations between grains was reported in coarse-grained MgO-partially-stabilized zirconia (Mg-PSZ) [4]. For very fine-grained Y,O,-stabilized tetragonal zirconia (Y-TZP), in which thermal stress fluctuations are very short range in nature, a welldefined short-crack behavior can still be found for indentation cracks because of residual stresses around the indent [5]. Residual stresses are not expected to be significant at elevated temperatures. Thus, the apparent short-crack behavior observed for silicon nitride in static fatigue at high temperatures must arise from different origins. One possibility could be the increased shielding of a growing crack, in analogy to the very pronounced short-crack behavior observed in fiber-reinforced metal matrix composites which experience fiber bridging and fiber pullout at the crack-wake [6J. Similar short-crack behavior has also been reported for a SiC-whiskerreinforced alumina ceramic composite [7]. Evidence for the operation of enhanced grain bridging and pullout in monolithic ceramics at high

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temperature has been reported [8-121. For example, Mutoh et al. [12] have reported that, for a hot isostatically pressed S&N,, the fracture toughness decreased initially with temperatures up to 12OO”C, then increased rapidly between 1200 and 1275°C and finally followed by a rapid decrease above 1300°C. The bend strength also exhibited a similar trend but the increase during the transition was less remarkable. Since the associated fractographic observations indicated the softening of the intergranular glassy phase which facilitated the subsequent grain bridging and pullout, the enhanced toughness due to crackwake shielding must be attributed to the increased population of grain pullout and not to the decreased magnitude of bridging stress. Such shielding mechanisms can be especially active in cyclic fatigue when the resistance to grain boundary sliding is further reduced, as suggested by our recent study on a hot-pressed S&N, [13]. As a result, a lower crack growth rate may be expected in cyclic loading than in static loading at elevated temperatures. We believe that the enhanced shielding at elevated temperatures and in cyclic loading is central to the understanding of hot fracture behavior of ceramics. A model based on this premise has been recently developed to predict the effects of temperature, strain rate and load cycling on the steady-state fracture toughness [14]. Specifically it was pointed out that since the toughness enhancement is proportional to the product of V, z and h2, where Y = volume fraction of pullout grains, z = interfacial friction and h = slip length, enhanced shielding can arise from a large increase of Seven if r decreases somewhat. Such enhanced shielding should also lead to the shortcrack behavior seen by Petrovic et al. in static fatigue, and this behavior should be even more pronounced in cyclic fatigue at the same temperature. With such expectations, we have performed static and cyclic fatigue tests on a hot-pressed S&N, at temperatures ranging from 1300 to 1400°C and monitored the crack growth responses. The short-crack behavior was indeed found under certain conditions as reported in this paper. These results will be compared with a more detailed fracture mechanics analysis that computes the evolution of the crack growth rate under static and cyclic fatigue conditions assuming reduced grain boundary sliding resistance in the latter case. Together, with the previous analysis of the steady-state toughness, they provide a comprehensive picture of the temperature and cyclic loading effects on the crack growth behavior of structural ceramics at high temperatures. 2. EXPERIMENTAL 2.1. Material

The silicon nitride studied was a two phase material containing cr’-SiAlON (30 v/o) and p-S&N4 (7Ov/o). It was hot-pressed in our laboratory at 1780°C for 1 h to reach full density. Details of

material design, processing conditions and preliminary mechanical properties of this series of materials were published elsewhere [ 151. Excellent strength has been measured in four-point bending in air at room temperature (1050 MPa) and elevated temperature (750 MPa at 1400°C). Preliminary high temperature fatigue crack growth data of the same material have also been reported [ 131. 2.2. Specimens Bend bars with dimensions of 17mm x 3 mm x 1.2 mm were obtained from the hotpressed material. The long axis of the specimen lies on the hot-pressing plane which is normal to the tensile surface. The surface of the specimen was finely ground, then polished to obtain a good finish. A series of Vickers indents was placed on the tensile surface at a load of 12 kg. These indents were 0.5 mm apart and carefully aligned so that corner cracks emanating from the indents were all parallel/perpendicular to the long axis of the specimen. To remove the effect of residual stress on the crack growth during fatigue testing, the following procedure was followed. The specimen was placed in a four-point bending fixture with an outer span of 14 mm and an inner span of 8.5 mm which covered all the indents. It was then loaded stepwise, at a rate of 5 N/s with 10 s holding time at each stress level, to 125 N. This allowed a typical crack to extend by about 50% of its original length. After this treatment, a 25 pm thick layer, which is about the depth of the indent impression, was removed from the indented surface using a lOOO-grit grinding wheel. Specimens were then polished using 6 pm diamond paste to remove grinding marks and provide a good finish. 2.3. Fatigue testing A modified four-point bending configuration with an inner span of 1.25 mm and an outer span of 13 mm was used in this study. Usually, three indents were placed within the inner span and four other indents on each end outside. The adoption of this configuration facilitated data acquisition over a relatively broad range of crack extension force from one specimen [13]. When the specimen eventually failed from one of the cracks within the inner span, the two surviving cracks within the inner span could be used for critical crack length measurement and microscopy. Meanwhile, cracks outside the inner span experienced lower stresses and provided data for crack growth rates at lower driving forces. Since some shear stresses were also present outside the inner span, a relatively thin thickness of the specimen (1.2 mm) was chosen to minimize the shear stress in this configuration. Cyclic bending tests were conducted using a computer-controlled servo-hydraulic machine (Model 810 from MTS corporation, Minneapolis, MN) operated under load control, with a sinusoidal wave form. The specimen was first heated to the desired temperature

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LIU and CHEN:

at a rate of 60”C/min. The load was then slowly applied at a ramp rate of 1 N/s until the maximum stress level was reached. The R-ratio (minimum stress/maximum stress) was 0.1 and the frequency was 5 Hz. For comparison, static tests at the same temperature and held constant at the maximum stress were also conducted. During the test, the displacement of the loading point was continuously monitored to record the apparent creep strain. All fatigue tests were periodically interrupted and the specimens cooled at a rate of 75”C/min to room temperature for the crack length measurement. The curvature of the specimen was then measured to calculate the actual creep strain. The applied stress intensity factor at the tip of a semi-elliptical surface crack subject to pure bending was calculated from the finite element analysis of Newman et al. [16]. The aspect ratio, a/b, for the semi-elliptical crack was estimated by using an empirical equation recommended by ASTM E-740 [17] a

Fig. 1. Creep curves under static and cyclic loading conditions at (a) 1300, (b) 1350, and (c) 1400°C.

a

;=;+1

where a is the half-crack length of the major axis and d is the specimen thickness. The above equation was found to be in good agreement with the experimental observation in our previous study [S]. The issue of the validity of stress intensity factor as a measure of the driving force for the crack growth will be addressed in Section 5.4. 3. EXPERIMENTAL RESULTS

The accumulated creep strains at the tensile surface at 1300, 1350 and 1400°C are shown in Fig. 1. Here the creep strain was calculated using the curvature of the deformed and unloaded specimen following the standard beam bending theory. (No attempt was made to account for the asymmetry of creep in tension and compression [18].) The strain contribution due to indentation cracks was also estimated by measuring the crack opening area on the tensile surface and normalizing it by the total area of the tensile surface. This contribution was generally found to be less than 5% of the total creep strain. At a comparable peak stress level, time to failure was longer and total accumulated strain was larger in cyclic than in static loading. We interpret this observation as indicating that the unloading portion in cyclic fatigue was ineffective in causing creep and crack growth, so more creep strain was allowed to accumulate before failure. A modified form of the empirical relation originally proposed by Nadai for Table

1300 1350 1400

primary creep was assumed to describe creep data in both static and cyclic fatigue loading conditions [19] g = Ca$r-q.

(2)

After integration [( 1 + q)Ct]“(’ +q)arf.

E =

Here d is the creep strain rate, t is the creep strain, C, and m =p/(l + q). In static loading a__8is the applied stress. In cyclic fatigue with a sinusoidal loading profile, creep strain rate results from the time-integrated effect of the sinusoidal stress. Then simple integration shows that equation (3) still holds provided a,e is expressed as

p and q are constants,

3.1. Creep deformation and stress redistribution

Temp (“C)

Time (hours)

I. Constants

aeff=

![1+R+1-R. I)m‘lrnamax 2

2

sin(2nwt)

= F(m, R )amax (4) where R = a,,,,,,/a,,,, o is frequency, and F(m, R) is the prefactor for converting amax to aef. Using equation (3) to fit creep data over the entire range of stress, strain and load ratio, we obtain the constants p and q listed in Table 1. The fitted curves are also shown in Fig. 1, where the fit is apparently satisfactory except in Fig. l(a) at 333 MPa where accelerated strain accumulation is witnessed after 5 h due to excessive damage development. The outer fiber stresses directly relate to failure. For non-linear material, it drops as non-linear creep for creep equation

P

9

m

c

3.0 2.6 3.5

0.41 0.44 0.41

2.1 1.8 2.5

2.32 x lo-‘* 6.43 x 10-l’ 5.40 x lo--‘*

F(m,

R)”

0.3934 0.4270 0.3646

c( 2.00 1.95 2.05

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proceeds but eventually approaches a steady-state value. An analysis by Fett [ZO]showed that the ratio of the actual stress (a,) to the initial elastic stress (a,) followed the empirical relations m-l r

o’g-

tanh”(m&,/&,)lia

0, m-l 7

cl-

160 -

G(m0,)

&!

60

% m

40

and a E 1.65 +O.l15p

Time (hours)

(6)

where .Q and E, represent creep strain and elastic strain, respectively, m and p are constants from equations (2) and (3), and G(ms,/e,) asymptotically approaches unity at large strain. For a = 2, which is typical for the present study, G(~E,/E,) is greater than 0.99 if m&J&, is larger than 9 according to Fett [ZO]. Thus, once the creep strain exceeds five times the linear elastic strain, the actual stress approaches its steady-state value for the non-linear material

200

(b) 135O”C, Cyclic

h

J 0

Referring to Table 1, we find cc to be 83,85 and 80% of the nominal elastic stresses for 1300, 1350 and 14OO”C,respectively. For the peak loads used in static fatigue, the elastic strains at 1300, 1350 and 1400°C are 1.1 x 10m3, 9 x 10m4, and 6 x 10e4, and the estimated times required to reach the steady-state are 5.2, 100

(a) 13OO”C, Static &

5

.-

60

40

Y 8

40

50

Tize (hok) Fig. 3. Crack extension curve at 1350°Cunder (a) static and (b) cyclic loading.

1.8 and 0.4 h, respectively. These times are relatively short compared to the test durations used. Furthermore, since the majority (85%) of the stress redistribution takes place at a small fraction (10%) of the above time, it is then clear that all of our experiments were essentially conducted at the steady-state stress level given above. 3.2. Fatigue crack growth

I W 2

10

20

ij 0 0

1

0.5

1.5

Time (hours)

0

2

4

6

0

10

12

Time (hours) Fig. 2. Crack extension curve at 1300°C under (a) static and (b) cyclic loading.

Typical crack growth curves obtained at 1300,135O and 1400°C are shown in Figs 2-4. For each temperature, data from both cyclic and static fatigue are provided for different initial stress intensity factors Ki. (The nominal stress was used to calculate KS.) Since the applied stress intensity factor increases with crack growth at a constant applied load, the same data can also be replotted versus maximum applied stress intensity factors as shown in Fig. 5. At 1300°C positive curvatures in Fig. 2 were always observed. This indicates crack growth acceleration. Such a behavior is expected in a constant load experiment because the stress intensity factor increases as the crack grows. This corresponds to the straight line with a positive slope in Fig. 5(a). At 1350°C (Fig. 3), static fatigue crack growth still exhibits a positive curvature, while cyclic crack growth at higher $ had an inflection point. In the latter case, the crack growth rate decreased before the inflection point, then accelerated after the inflection point all the way to failure. This corresponds to the

LIU and CHEN:

10

5

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CRACK GROWTH IN SILICON NITRIDE

for A and n are listed in Table 2. Also shown in Fig. 5 (dashed line) is the predicted cyclic fatigue crack growth rates from the static fatigue data [equation (8)] after taking into account the unloading effect. [Sinusoidal load cycling introduces a correction factor F(n, R) which is identical in form to F(m, R) in equation (4). The calculated F(n, R) is also listed in Table 2.1 In all cases, the measured cyclic fatigue crack growth rates were considerably slower than the predicted ones. This indicates an additional retardation effect of cyclic loading beyond what can be accounted for by the load-correction factor. Such

20

15

Time (hours) 70 E

60

a

50

(4 1 o-7

3

6 'iij40

g al

ti 3 30

2

s20

'r

1300°C 10'8

10-g g

10

f 5

0 0

20

40

60

60

100

120

lo-lo

s 0 10-l'

Time (hours) Fig. 4. Crack extension curve at 1400°Cunder (a) static and (b) cyclic loading. V-shape curve in Fig. 5(b) and the so-called shortcrack behavior. The position of the inflection point moved to a longer time with decreasing K,; at a sufficiently low 4, crack deceleration prevailed for the entire test duration at this temperature. At 1400°C (Fig. 4), growth deceleration in cyclic fatigue at lower Ki was again evident. Moreover, crack growth in static fatigue at lower 4 also appeared to initially decelerate. At higher 4, growth acceleration was again seen in Fig. 4. Correspondingly, a V-shape curve or simply a curve with a negative slope is seen in Fig. 5(c). The crack growth rate data plotted in Fig. 5 show good agreement among different Ki in the growth acceleration stage. Crack growth rates increase monotonically with &,, and the data can be represented by a power-law relationship da - = AK;, dt

~

2

3

6

Kapp(fvllkm)

5

2

6 ia,,

(tvlF&m) 5

@I

(8)

where &, is the maximum apparent stress intensity factor, and A and n are constants. The best-fit values Table 2. Constants for crack growth equation Temp. (“C)

A

n

W, W

1300 (static) 1300 (cyclic)

2.22 x lo-‘2 3.13 x 10-15

5.8 8.4

0.2411

1350 (static) 1350 (cyclic)

1.54 x lo-l*

5.8

0.2411

1400 (static) 1400 (cyclic)

1.40 x 10-P

2

3

4

5

6

7

Kapp (MPaJm) 0.9

0.6040

Crack growth rate versus applied stress intensity factor curves at (a) 1300, (b) 1350 and (c) 1400°C.

Fig. 5.

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3.3. Fractography An examination of the crack tip of the surviving indentation cracks revealed the following features. At 13OO”C,the frontal zone [Fig. 6(a)] contained transverse intergranular cracks that were partially connected with unbroken ligaments. In the crack-wake [Fig. 6(b)], some unbroken ligaments that survived a relatively large opening displacement ( z 0.5 JIm) were also seen. At 14OO”C, numerous microcracks parallel to the main crack were seen even behind the crack tip (Fig. 7). Also, the crack profile is much sharper in static fatigue [Fig. 7(a)] than in cyclic fatigue [Fig. 7(b)]. The density of microcracks was higher in cyclic fatigue. For both static and cyclic fatigue, the residual crack mouth opening displacement was found to increase with decreasing applied stress intensity factor. In this sense, a ductile appearance was acquired. This trend of brittltiuctile

Fig. 6. SEM micrographs of a surviving crack at 1300°C in

the (a) frontal zone and (b) crack-wake. an effect seems to diminish with increasing K. Lastly, note that at 1300 and 135O”C!,the n value is very similar and much higher than unity. At 14OO”C, however, n approaches unity indicating a significant diffusional contribution [21]. This is consistent with our knowledge of this material which shows excellent strength retention (750 MPa or 70% of room temperature strength) up to 1400°C. In the deceleration stage, the crack growth data in Fig. 5 from different K, do not coincide. Here, an inverse dependence of crack growth rate on K was observed, giving a V-shape appearance to these curves. It has been common in the literature to find that crack growth rates with short-crack behavior do not follow a universal curve as in this case [3]. Some cracks with smaller Ki did not have time to develop a complete V-shape curve before the termination of the test when failure occurred elsewhere at the longest crack. This is especially common for the cyclic fatigue test at 1400°C.

Fig. 7. SEM micrographs of surviving cracks at 1400°C (a) static loading and (b) K = 3.11 MPa,/m in K = 3.02 MPa,/m in cyclic loading.

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2085

Fig. 8. SEM micrographs of fracture surface at 1300°C under (a) and (b) static loading and (c) and (d) cyclic loading.

in crack appearance was found to parallel that of crack acceleration-retardation transition. On the fracture surface, no substantial difference in appearance could be distinguished between cyclic and static fatigue failures; in both cases intergranular fracture prevailed [Figs 8(a) and (c)l. (To enhance the contrast, viewing was carried out at a glancing angle to the fracture surface.) Indeed, even the fast fracture surface resembled the fatigue fracture surface in appearance. Occasionally, fractured grains with a relatively long pullout distance (e 0.5 pm) could be found [Figs 8(b) and (d)]. transition

4. MODELING 4.1. Synopsis

24 Monolithic, non-transformable ceramics derive fracture toughness mainly from frictional sliding between grains [22]. During this process, sliding, bridging grains in the crack-wake can support load

and substantially reduce the effective stress intensity factor at the crack tip. We have recently proposed that, at high temperature, grain sliding is facilitated by softening of the grain boundary glassy phase, and the slip length (/t) above or below the crack plane progressively increases with temperature and eventually reaches the grain half-length 1 before grain cleavage occurs at the crack plane [14]. As schematically shown in Fig. 9, in which the bridging grains are shown, in exaggeration, as elongated fibers, grain separation at triple junctions occurs when h > 1 so that grain pullout can follow. A large contribution to crack tip shielding and fracture toughness thus results. We also proposed that cyclic loading can facilitate grain boundary sliding by reduction of grain boundary sliding resistance. Thus, increased crack tip shielding and retarded crack growth may be expected in cyclic fatigue at high temperatures. It is important to appreciate that the enhanced crack tip shielding in

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f

\

(a) hcl

(b) h>!

point in Fig. 10(a) and the V-shaped growth rate curve in Fig. IO(b). Experimentally, we have established a general trend of transition from type I to type II behavior when the applied stress becomes smaller, the test temperature becomes higher, and the loading condition becomes more cyclic. It seems plausible that the transition is caused by the decrease of grain boundary sliding resistance expected for higher temperature and more cyclic loading conditions. In the following, we explore a simple model in which the crack growth kinetics are assumed to be the same in static and cyclic fatigue and follow a power law that depends on the crack tip stress intensity factor. By allowing the latter to evolve depending on the shielding contribution of sliding grains, we will demonstrate that temperature and load cycling (both lowering the grain boundary sliding resistance) can profoundly affect crack-wake shielding and the crack growth rate. 4.2, Problem formulation

\

.

/

Fig. 9. Schematics of (a) grain cleavage when h < 1 and (b) grain separation when h > 1.

this case is not due to increased bridging stress (just the opposite is happening) but rather due to increased population of bridges. This interplay between total shielding contribution, bridge population and shear traction is evident from the previous model calculation shown in Fig. 4 of Ref. [14]. In contrast, at lower temperature, the population of bridges is constant so that stress cycling always reduces total shielding. Since stress cycling has entirely opposite effects at high temperatures and at low temperatures, a reassessment of fracture mechanics is called for. The experimentally observed crack growth behavior can be categorized into two types, schematically shown in Fig. 10. The curves labeled I resemble the conventional crack growth behavior with little or no crack shielding, showing growth acceleration [Fig. 10(a)] and Paris law [Fig. 10(b)]. The curves labeled II show initial growth retardation followed by growth acceleration, giving a sigmoidal crack growth curve [Fig. 10(a)] and a V-shaped da/dt vs K curve [Fig. 10(b)]. The two different behaviors can be rationalized by different amounts of crack-wake shielding. For type I, shielding increase is slower than Qp increase due to crack growth, hence growth acceleration. The opposite is true initially for type II. In the latter case, however, when the grains at the far wake are fully pulled out, steady-state shielding is obtained so that further increase of Kappdue to crack growth causes crack growth to accelerate. This explains the .retardation-acceleration transition at the inflection

Fracture mechanics modeling of crack growth follows the scheme summarized in Fig. 11. The crack opening displacement (COD) of a bare crack (u,rr) is first calculated using the standard elastic solution [23]. The bridging stress (c,,) corresponding to this COD (u) is then determined by the following equation

o-,=;(h-24) t

(a)

Time

2 =

*

(W

a

2 2 f a s ??

Applied Stress Intensity Factor (log) Fig. 10. Schematics of (a) crack extension versus time and (b) crack growth rate versus factor ..- applied stress ^ intensity . curves showing two dltterent types ot crack growtn.

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in this way, the bridging zone builds up until the COD of the bridging grain exceeds a critical pullout distance h, which renders the bridge ineffective by loss of contact with crack faces or by fracture. The crack growth rate may either increase or decrease depending on the evolution of Ktip. In the above calculation, grain boundary sliding resistance z is the main adjustable parameter, along with n’ and A’, for fitting purposes. This follows from a review of the pullout model outlined below [ 141.The stress-displacement relationship for a bridging grain prior to fracture is [24]

Fig. Il. Flow chart of modeling crack growth at high temperature.

where r is the sliding resistance, w is the grain half-width and h - u is the grain length that experiences the sliding friction. (The above relation follows the shear-lag analysis and is well-known.) By assuming that the bridging stress is uniform within the cross-section of each bridging grain but is zero outside, the crack closure (ub) due to all the bridging stresses can be calculated everywhere (see the Appendix). The total COD (utotal) is then obtained by superposition of uapp and ub, and it can be used to determine ub again in the next iteration. This continues until a self-consistent set of stress and displacement is found everywhere. (The above iterative process can also be reformulated into a matrix equation so that the solution of r~,,and utotalcan be obtained at once without iteration.) The shielding contribution (K,) due to bridging stresses is then integrated [23] and the crack tip stress intensity factor (&) determined, again by superposition of Kapp and KS. The crack growth rate that follows a power law on Ktip, analogous in the form to equation (8) (with parameters n’ and A ‘) is computed to find the time for the crack tip to advance to the next bridging grain. This completes one increment of crack growth. In the next increment, the process repeats itself and Table 3. Constants Static fatigue Temp. (“C) 1300 1350 1400

(Mia) 120 100 96

At high temperature where z is small, h can exceed the distance from the grain’s end to the crack plane. Intense stress concentrates at the triple point due to grain boundary sliding then activates grain-end separation to facilitate grain pullout. Allowing the distance from the grain’s end to the crack plane to vary between 0 and 1 (grain half-length) in a statistically random manner, the volume fraction of grains undergoing pullout (and hence contributing significantly to shielding) increases with h (hence t -‘) in the following way

(12) where D is introduced and can be identified as the average spacing between pullout grains. In our calculation, the following material constants have been chosen based on microstructural observations and material properties, w = 0.2 pm, I = 1 pm, Vf = l/3, erg= 1 GPa and E = 300 GPa. Then, equations (11) and (12) dictate that zh = constant and hD = constant. This leaves t as the only adjustable parameter. Its value and those of h and D are listed in Table 3 for the calculations presented below.

for crack growth

modeling

Cyclic fatigue (&a)

(pk) 0.500 0.600 0.625

where crbis the (maximum) stress in the bridged grain at the crack plane, Vf is the volume fraction of grains participating in bridging, y is the distance between the crack plane and the end of the sliding zone, and E is the elastic constant. Before grain fracture, the length of the sliding zone (y - u) increases with ob. This continues until oi, equals the grain strength erg that determines the maximum sliding distance

2.40 2.00 1.92

80 60 50

(A 0.75 1.oo 1.20

(A 1.6 1.2 1.0

Maximum stress (MPa)

A’

n’

F(n’, R)”

3 1 l/286/26 1 26912561235 174/161/134

5.0 x lo-‘2 1.8 x 10-l’ 1.8 x lO-9

5.5 4.7 0.8

0.2477 0.2671 0.6254

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4.3. Numerical calculations At each temperature, we have chosen three maximum stress levels listed in Table 3 for sample calculation. (Nominal stresses are given here, but the actual calculations were done using the steady-state stress after relaxation.) These values are chosen to simulate the experimental conditions studied here. A starting crack length of 155 pm was used. We also let Y = 0.8 1 in the procedure outlined in Fig. 11. For the initial configuration of the crack, we allowed five pre-existing bridging grains in the wake (extending about 5-12 pm from the crack tip depending on D).

(4

4

3

5

6

Kapp (MPaqm)

Kapp W’adm)

This choice was made to reflect the fact that the precracking procedure used at room temperature invariably left some bridging ligaments behind the crack tip. The calculated da/dt versus Kappis shown in Fig. 12. At 1300°C [Fig. 12(a)] static fatigue crack growth rates at all stress levels followed a unique power-law relation throughout the entire Kapprange. In cyclic fatigue, similar results were seen. (Crack growth retardation was barely noticeable at very low Kapp regime.) At 1350°C [Fig. 12(b)] the power-law behavior is still followed for most of the static fatigue data but V-shaped curves are evident in cyclic fatigue. At 1400°C [Fig. 12(c)] V-shaped curves are seen for static fatigue, whereas only crack growth retardation behavior is observed in cyclic fatigue. The above results, both in trend and in numerical value, are in close correspondence to the experimental observations. Note again that this close correspondence was obtained by varying z, A ’ and n’ only. In the case where short-crack behavior was found, the calculated results in our experience were quite sensitive to the configurations of the crack which could be varied by choosing different initial crack length ai, number of pre-existing bridging grains N,, and crack geometry as reflected by the value of Y. With relatively small variations, it was possible to extend or delay, and even suppress, crack growth retardation. This finding can explain the experimental observation of Petrovic et al. [2] of a very large data scatter in their high temperature crack growth measurements. two branches of Specifically, we compared da/dt-K,,, curves in the growth retardation regimes, e.g. Fig. 12(b) in cyclic loading. At the same Kappor ada, the branch with a lower initial stress intensity factor (i.e. a lower applied stress) must grow to a longer crack length than the one with a higher initial stress intensity factor (i.e. a higher applied stress). This in turn allows more crack-wake shielding to develop so that KLi, is actually lower in the former case. Thus, a lower crack growth rate is always associated with the branch with a lower Ki . This trend was observed in all the plots shown in Fig. 12. The daldt-Z&,, dependence is not unique but is crack-geometry dependent when crack-wake shielding is developing. 5. DISCUSSION 5.1. Short -crack behavior In addition to the short-crack behavior due to residual and other short range internal stresses cited

2

Kapp (MPihn) Fig. 12. Simulated crack growth rate versus applied stress intensity factor curves at (a) 1300, (b) 1350 and (c) 1400°C.

in the Introduction, observations of growth retardation followed by growth acceleration in fatigue due to increased crack-wake shielding have been reported before. A notable example is a metal matrix (TiAl) composite reinforced by continuous, unidirectional fibers (Sic) tested at room temperature [6]. Sigmoidal curves of crack extension versus time, similar to those

LIU and CHEN:

CRACK GROWTH IN SILICON NITRIDE

shown in Fig. 4, were seen at high applied stresses. They compared with the unreinforced TiAl which had a concave upward crack extension-time curve and the composite at low applied stresses which had a concave downward crack extension-time curve. These observations were theoretically modeled by McMeeking et al. [25] by estimating the crack tip value of the stress intensity factor amplitude AZ&. It was found that when the applied load amplitude was held fixed, A& was steadily reduced by a frictional constraint on fiber sliding toward an asymptotic value independent of crack length. Thus, if the fatigue crack growth rate followed a Paris law proportional to AK”, as is the case in most metals, crack growth retardation would occur and the crack growth rate would asymptotically approach a steady-state value with bridging fibers extending all the way in the crack-wake. At higher applied load, fibers in the wake eventually failed when the stress on them reached a unique fiber strength. This fiber breakage reduced the shielding contribution and, by load shedding, caused additional fiber breakage in the wake. This resulted in a significant acceleration in the crack growth and a sigrnoidal crack extension-time curve [6]. The above crack growth behaviors are similar in appearance to those observed here. However, the following differences are noted. First, the steady-state crack growth rate is not achievable in our material, because, unlike long fibers, grain pullout is eventually limited by grain length. Second, the crack growth kinetics at the crack tip at high temperature is not likely to depend on AK,,, in our material. (Otherwise faster, not slower, growth would have been obtained in cyclic loading.) Third, the nature of friction in the fiber composite at low temperature is athermal whereas that in grain boundary sliding at high temperature is thermal. Thus, despite the close resemblance between the two sets of observations, the different microstructures, load bearing mechanisms, and crack growth kinetics for metal matrix, fiberreinforced composites and monolithic ceramics at very different test temperatures necessitated new modeling following a different set of physical assumptions as outlined in the previous section. 5.2. Fatigue retardation In addition to the present study, there have been several reports on the observation of fatigue retardation in ceramics tested at high temperatures. This includes our previous work on the same silicon nitride [13] the crack growth experiments on alumina and Sic whisker-containing alumina by Suresh et al. [26,27], the stress-lifetime experiments of Lin et al. on alumina [28], and similar experiments by two independent groups on silicon nitrides [29, 301. In our view [14], these apparently general observations of fatigue retardation can be rationalized using the concept of enhanced crack-wake shielding that is now shown to be consistent with the observation of short crack behavior.

2089

The increased shielding at higher temperature and especially under cyclic loading conditions is a consequence of decreasing interfacial friction. This is directly responsible for the toughness peak and fatigue retardation at elevated temperature [14]. The increased wake-shielding is associated with the increased population of bridging grains that become separate at their triple points when the sliding zone reaches there. When the frictional sliding resistance is sufficiently low, essentially every grain can participate in the action of frictional pullout. This is in contrast to the wake-shielding mechanism that operates at lower temperatures in most polycrystalline ceramics, in which frictional sliding is limited to those intact grains with loose grain boundaries previously broken and unlocked by the meandering main crack during crack deflection [22]. (The population of these grains is of the order of lo%, with the rest of the grains simply fracturing transgranularly along the main crack plane.) In the latter case the reduction of interfacial friction by cyclic loading actually reduces shielding and accelerates crack growth. Such opposite effects of cyclic loading on shielding at low and high temperatures have been previously demonstrated by using steady-state toughness calculations [ 141. The observations of generally decreased crack growth rate and the more pronounced short crack behavior under cyclic loading conditions in the temperature range studied here provides additional support for the above picture. The other possible explanation of fatigue retardation is to resort to enhanced crack tip shielding due to plasticity (or pseudoplasticity such as cavitation, microcracking, etc.). This mechanism, however, is not known to lead to fatigue retardation in the metallurgy literature. Specifically, we note that compared to ceramics, metals have considerably more plasticity but they also share the same crack tip damage mechanisms of cavitation, grain boundary cracking and grain boundary sliding. Yet cyclic loading at elevated temperatures is generally known to enhance and not retard crack growth in metals [31,32]. This implies that the net effect of cyclic loading on crack tip processes is to reduce the crack growth resistance. It follows that the contrasting observations of fatigue retardation in ceramics versus fatigue enhancements in metals is not likely to be due to some crack tip processes but rather due to certain crack-wake processes which are particularly prominent in ceramics. Fatigue retardation in ceramics due to the crackwake mechanism at high temperatures may nevertheless be masked by other damage mechanisms that are exacerbated by load cycling. This appears to be the case in the crack growth experiments of silicon nitride and Sic whisker-containing silicon nitride reported by Suresh et al. [33]. Unlike ours and other reports of fatigue retardation in this class of materials, they found fatigue acceleration when the silicon nitrides were tested in air between 1300 and 1450°C. Since both S&N., and Sic whiskers are known to oxidize

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LIU and CHEN:

CRACK GROWTH IN SILICON NITRIDE

quite severely at such temperatures, it is likely that some damage on crack-wake shielding (through K,) and some enhancement on crack tip kinetics (through A) may be operating in this case. It is further recognized that while such environmental effects make it difficult to investigate the intrinsic fatigue crack growth mechanisms in ceramics, they are of serious concern for material evaluations under service conditions. Thus, a full assessment of fatigue resistance of ceramics, especially non-oxide ceramics, may need to incorporate consideration of environmental effects when appropriate. 5.3. Ductile-brittle transition Another element in the crack-wake shielding mechanism concerns the length of crack extension over which the crack-wake shielding (or, equivalently, the R-curve) rises to the steady-state value. This, along with the toughness peak and fatigue retardation, can be used to rationalize the observation of ductile-brittle transition. The characteristic crack extension is determined by the condition when the COD in the wake equals the sliding length h. (Only grains that have an end-to-crack-plane distance shorter than h can be pulled out, and such pullout is complete when COD = h.) For a crude estimate, we adopt the COD in the elastic fracture mechanics limit [34], COD cc K,,/& where Aa is the distance from the crack tip to the outermost bridging grain in the wake. Then the characteristic length of crack extension, Aassr is found to vary as Aa, cc (h/K)’ CC(&-*. Thus, a more gradual rise in the R-curve toward the steady-state value over a longer Aass is experienced at higher temperatures, under cyclic loading conditions, and with a smaller initial stress intensity factor. This in turn allows a more pronounced retardation of crack growth rate and a fuller development of a V-shaped da/dt-K curve to appear. In the extreme case that Bass becomes sufficiently long, only crack retardation is seen and the fracture will appear “ductile-like”, especially if large deformation has accumulated before Aa,, is reached. Our observations of a more ductile appearance in crack extension curves and in fractography at higher temperatures and under cyclic loading conditions are in support of this picture. The above crack-wake shielding mechanism again offers an explanation of the ductile-brittle transition of ceramics at high temperature. This is an alternative to the conventional explanation which rests on crack tip strain-producing mechanisms such as creep and microcracking. As we argued before, it is not clear how the crack tip processes are actually promoted by cyclic loading conditions, especially in view of the opposite observations of fatigue-enhanced crack growth in metals despite their considerable ductility, cavitation and/or microcracking at elevated temperatures. This comparison again leads us to favor the crack-wake sliding process proposed here as the more likely mechanism that is responsible for the appar-

Fig. 13. Brittkxluctile

transition map for S&N,.

ently ductile behavior of ceramics under cyclic loading. The dependence of the onset of short-crack behavior on temperature, initial stress level and cyclic component of the stress for a given crack contiguration can be summarized in the map depicted in Fig. 13. The parametric space for this map is constituted of temperature, Ki and R-ratio (R = 1 corresponds to static loading). Within the dome of lower stress, higher temperature and lower R-ratio (i.e. larger ACT),type II behavior with short-crack growth should be found, whereas type I behavior is found outside of the dome. The numerical values of e,,, , T and R-ratio are roughly given in Fig. 13 for the S&N, tested in the present study. The boundary of the dome also signifies a brittle-ductile transition. Physically, the transition occurs when the slip length reaches the grain half-length on average. Such a transition has been previously illustrated in the temperature coordinate (see Ref. [14] and prior experimental observations referenced therein) and is now generalized to the space of loading parameters. 5.4. Fracture mechanics Finally, we return to the issue of stress intensity factor as a measure of driving force for crack propagation. As shown previously, we are quite certain that, despite non-linear creep, an essentially constant stress can be maintained in the outer fiber of a bend bar to exert a well-defined applied stress on the crack. However, time-dependent inelastic deformation at high temperature can also cause stress relaxation at the crack tip to render the K field no longer valid. At very high temperatures, the characteristic time t,, for stress relaxation is typically short, and the stress field will eventually evolve into a new steady-state creep field known as C* field [35]. For example, the characteristic time in the work of Suresh et al. on silicon nitride was estimated to be between 100 and 1000 s [33]. Such time is no doubt too short to allow meaningful determination of steady-state crack growth rate under either static or cyclic loading

LIU and CHEN:

CRACK GROWTH IN SILICON NITRIDE

conditions. In this regard, we disagree with the argument that the validity of the K field in cyclic experiments can be assured if the cycle time, defined as l/frequency, is shorter than the characteristic time [33]. This is because the creep strain continues to accrue at the crack tip even in the unloading portion of the cyclic experiment, hence, the K field must also continue to relax regardless of the length of the cycle time. Specifically, noting that the K field stress is largely relaxed when the creep strain exceeds the elastic strain, an estimate of the characteristic time for stress relaxation in cyclic loading can be obtained by substituting the cycle-average of K and C* into the equation of t, [33,35]. It can be readily verified through these estimates that the characteristic times are not significantly different in static and cyclic loading. It then follows that the crack tip K field is always relaxed at very high temperatures regardless of stress-time profiles. Therefore, we believe that the present experiment, those reported in Ref. [33] and probably most other fracture mechanics experiments conducted at very high temperatures all suffer from such uncertainty to various extents. Despite the above problem, we believe that the main conclusions of our study regarding fatigue retardation remain valid at least qualitatively. This can be assured by the following arguments. (a) Since creep deformation is more sensitive to stress than elastic deformation, the completion of creep relaxation is reached sooner at higher stresses than at lower stresses. Thus, the stress field in static loading, which has a higher average stress, is more relaxed than that in cyclic loading. In the absence of any extrinsic fatigue retardation mechanism, this would lead to faster crack growth rates under cyclic loading if such rates are predicted using the cycle-average data of static crack growth. This is contrary to our observation. (b) The crack tip C* field scales with o(a/r)‘@+ ’where 0 is the applied stress, p is the stress exponent of creep rate and u is the crack length (p = 1 gives the K field as expected). For a relatively small p such as 3 found in our work, the difference in features of K field and C* field is relatively small [36] and both scale with the applied stress. (c) For the empirical crack growth law used in this work, equation (8) we expect the driving force for crack growth to scale with stress to the n th power if the K field is used. Assuming crack growth is controlled by the crack tip stresses, the stress proportionality between the K and C* field then implies a similar C*” dependence despite their differences in spatial distribution of the crack tip field. These similarities strongly suggest that the K field is at least proportional to the actual driving force and thus our analysis in terms of the K field probably captures the essential picture of the crack growth phenomenology. In particular, our conclusion of fatigue retardation remains valid at least qualitatively.

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6. CONCLUSIONS (1) Short-crack behavior with a decreasing crack growth rate at an increasing crack length under constant applied load has been observed in silicon nitride at high temperature. The tendency for such a behavior increases with higher temperature, lower applied load and larger cyclic load variations. The crack growth rate under cyclic load is always lower than that under sustained load. (2) The short-crack behavior and fatigue retardation of crack growth can be rationalized by accumulation of crack-wake shielding due to grain pullout. A higher temperature and a cyclic load decrease the grain sliding friction, thus promoting more triple point separation and allowing more grains to pullout. The crack opening displacement and the characteristic length of the R-curve are correspondingly larger, giving rise to a more ductile appearance of the crack profile. These aspects have been illustrated using fracture mechanics computation. (3) A generalized transition from low temperature behavior to high temperature behavior has been delineated in the parametric space of temperature, R-ratio and initial stress intensity factor. The onset of the transition occurs when the slip length of a grain reaches the average grain half-length. Together with the previously reported toughness-strength transition at high temperatures in silicon nitride, alumina and glass ceramics, the present study illustrates this general phenomenon that is apparently widespread among monolithic ceramics. REFERENCES 1.

2. 3. 4.

5

J. J. Petrovic, L. A. Jacobson, P. K. Talty and A. K. Vasudevan, J. Am. &ram. Sot. 58, 113, (1975). M. G. Mendiratta and J. J. Petrovic, J. Am. Ceram. Sot. 61, 226 (1978). See, for example, R. 0. Ritchie and J. Lankford, Small Fatigue Cracks. Metallurgical Society, Warrendale, PA (1986). M. V. Swain, G. H. Schneider and V. Zelizko, in Fatigue of Advanced Materials (edited by R. 0. Ritchie, R. H. Dauskardt and B. N. Cox), pp. 169-192. Materials Component Engineering Publications Ltd, Birmingham, UK (1991). S.-Y. Liu and 1-W. Chen, J. Am. Gram. Sot. 74, 1206 (1991).

6 D. P. Walls, G. Bao and F. W. Zok, Acta metall. 41, 2061 (1993).

7. R. H. Dauskardt, M. R. James, J. R. Porter and R. 0. Ritchie, J. Am. Ceram. Sot. 75, 759 (1992).

8. A. G. Crough and K. H. Jolliffe, Proc: Br. ‘Ceram. Sot. 15, 37 (1970).

9. R. W. Davidge and G. Tannin. Proc. Br. Ceram. Sot. __ 15, 47 (1970):

10. B. J. Dalgleish, A. Fakhr, P. L. Pratt and R. D. Rawlings, J. Mater. Sci. 14. 2605 (1979). 11. S. H. Knickerbocker, A. Zangvil and S: D. Brown, J. Am. Ceram. Sot. 67, 365 (1984).

12. Y. Mutoh, K. Yamaishi, N. Miyahara and T. Oikawa, in Fracture Mechanics of Ceramics (edited bv R. C. Bradt et al.), Vol. 10, pp. 427440. Plenum Press, New York (1992).

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LIU and CHEN:

CRACK GROWTH IN SILICON NITRIDE

13. S.-Y. Liu, I.-W. Chen and T.-Y. Tien, J. Am. Ceram. sot. 77, 137 (1994). 14. I-W. Chen, S.-Y. Liu and D. S. Jacobs, Acta metall. mater. 43. 1439 (1995). 15 S. Boskovic, K.‘J. Lee and T.-Y. Tien, in Materials Research Society Symposium Proceedings, Vol. 287, Silicon Nitride Ceramics (edited by 1-W: Chen, P. F.

Becher. M. Mitomo. G. Petzow and T.-S. Yen). pp. 373-380. Materials Research Society, Pittsburgh: PA (1993). 16. J. C. Newman, Jr and I. S. Raju, Engng Fract. Mech. 15, 185 (1981).

17. ASTM Standard E. 740-80, in ASTM Annual Book of Stamdards, Vol. 3.01, pp. 740-750. American Society of Testing and Materials, Philadelphia, PA (1983). 18. T-J. Chuna, J. Mater. Sci. 21. 165 (1986). 19. A. Nadai,-in The Infuence of Time’ Upon Creep: The Hyperbolic Sine Creep Law, S. Timoshenko Anniversary Volume, New York (1938). 20. T. Fett, J. Mater. Sci. Lett. 6, 967 (1987). 21. H. Riedel, Fracture at High Temperatures. Springer, Berlin, Germany (1987). 22. G. Vekinsis, M. F. Ashby and P. W. R. Beaumont, Acta

APPENDIX For a crack with a uniformly disturbed stress acting between x = 0 and x = b, the COD at any x is

a(~)=$

Jbx

+(b -x)ln-

1.

Jb+Jx %/m

(Al)

where E’ is the effective elastic constant under plane strain condition E’ = E/(1 - v*), and v is the Poisson’s ratio (taken to be 0.25 here). By superposition, the COD for a loaded single bridge (with stress ci) extending between Li and I, from the crack tip is &+Jx

JL,x+(L,-x)ln-

JM -&(r,,)ln$&].

(A.2)

I This gives the COD at the two sides of the ith grain as

metall. 38, 1152 (1990).

23. H. Tada, P. Paris and G. Irwin, The Stress Analysis of Cracks Handbook. Delph Corp., Hellertown, PA (1985). 24. D. B. Marshall and B. N. Cox, Acta metall. 33, 2013 (1985).

25. R. M. McMeeking and A. G. Evans, Mech. Mater. 9, 217 (1990).

26. L. Ewart and S. Suresh, J. Mater. Sci. 27, 5181 (1992). 27. L. X. Han and S. Suresh. J. Am. Ceram. Sot. 72. 1822 (1989). 28. C-J. K. Lin and D. F. Socie, J. Am. Ceram. Sot. 74, 1511 (1991). 29. T. Fett, G. Himsolt and D. Munz, Ad. Ceram. Mater. 1, 17 (1986). 30. M. K. Ferber and M. G. Jenkins, J. Am. Ceram. Sot.

as well as COD at the ith grain due to the traction of another (jth grain) bridge

75, 2453 (1992).

31. S. Suresh, Fatigue of Material. Cambridge University Press, Cambridge, U.K. (1991). 32. R. Raj, in Flow and Fracture at Elevated Temperatures (edited by R. Raj), pp. 215-269. Amer. Sot. Metals, Metals Park, OH (1985). 33. U. Ramamurty, T. Hansson and S. Suresh, J. Am.

-&Li-(~-L&I@]

(A.5)

I &i

Ceram. Sot. 77, 2985 (1994).

34. D. Broek, in Elementary Engineering Fracture Mechanics, 4th Ed., p. 80. Martinus Nijhoff Pub., Dordrecht, Netherlands (1986). 35. H. Riedel and J. R. Rice, Fracture Mechanics Twerfh Conference, ASTM STP 700 (edited by P. C. Paris), pp. 112-130. Amer. Sot. Testing and Mater., Philadelphia, PA (1980). 36. J. Bassani and F. A. McClintock, Int. J. Solids Struct. 17, 479 (1981).

-&-(~-i)ln-$++$].

(A.6)

I

1

The total COD of the ith bridge, taken as the average of the COD at L; and I,, is up = ;I

[U(Lj,) + u(l,)]. J

(‘4.7)