Analysis of bridging stress effect of polycrystalline alumina using double cantilever beam method

Acta mcuer. Vol. 45, No. 8. pp. 3445-3457. 1997 6 1997 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved

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ANALYSIS OF BRIDGING STRESS EFFECT OF POLYCRYSTALLINE ALUMINA USING DOUBLE CANTILEVER BEAM METHOD KEE-SUN SOHNT, SUNGHAK LEEI_ and SUNGGI BAIK Department

of Materials Science and Engineering, Pohang University of Science and Technology, Pohang, Korea 790-784 (Received 4 April 1996; accepted 10 October 1996)

Abstract-The in situ crack opening displacement (COD) with respect to the distance behind the stationary crack tip was measured in three alumina ceramics with different grain size. An analytical model which can describe the relationship between the bridging stress and the microstructure and a double cantilever beam (DCB) analysis which includes the bridging stress effect are developed in order to analyze the experimental results. The present model shows that the distribution of grain size as well as the average grain size are key microstructural factors that determine the bridging stress function. The results indicate that the shape of the bridging stress distribution is closely related to the grain size distribution. The effect of the bridging stress on the COD is taken into consideration in the DCB analysis. The crack closure due to bridging stress is calculated using the conventional power law relation and the new distributive bridging stress function developed in the present study. Q 1997 Acta Metallurgica Inc.

where PM is the peak stress, 24, is the crack opening displacement at the end of the bridging exponent. The zone, and n is a strain-softening value of n has been reported to be 1 for the

1. INTRODUCTION

Investigation during the last decade has revealed R-curve behavior in monolithic and composite ceramic materials [ 1, 21. The R-curve behavior in monolithic polycrystalline alumina seems to be entirely due to the grain interface bridging in the crack wake [3-61. Recently, the attention of many investigators has been focused on the quantitative interpretation of bridging phenomena [7-221. In order to perform such research, most researchers stress function, which have used the bridging

represents traction acting on both sides of the crack wall owing to bridging with respect to the crack opening displacement or the distance behind the crack tip. When strain-softening behavior is a dominant toughening mechanism, the bridging stress function has a tail-like form decaying gradually from the peak stress at the crack tip [7-91. It is proper to adopt the tail-like bridging stress function, since the major mechanism of the grain interface bridging is a frictional pull-out at grain boundaries [5, lo]. In this case, the following empirical power law form has been predominantly used: n

( >

P(u) = PM 1 - 5

tJointly appointed with Center for Advanced Aerospace Materials, Pohang University of Science and Technology, Pohang, Korea 790-784.

grain bridging in a monolithic alumina [lo] and for the frictional fiber pull-out in ceramic composites [ll]. On the other hand, for steel-fiber-reinforced cement-based composites [12], n is equal to 2. However, it is presumed that n is more microstructurally sensitive than inferred by the above categorization [16, 181. Using the empirical power law form. many researchers have determined the bridging stress distribution in an indirect manner, except Reich1 and Steinbrech [13] who have directly measured the average bridging stress in a polycrystalline alumina to be 29 MPa using a modified short-DCB specimen with a rear notch. For example, Hu et al. [ 141recently estimated the magnitude of the bridging stresses by considering the difference between the optically measured crack length and that predicted through compliance computations. By this method, the peak bridging stress and the strain-softening exponent in a monolithic alumina were determined to be 56 MPa Invoking the Barenblatt and 2.1, respectively. relation

[23] for loaded

cracks,

Riidel

et al.

[15]

characterized the bridging stress distribution by in situ SEM observations of CODS in loaded compact tension specimens. and determined a peak stress of 70 MPa with n = 2.5. Unlike the cases mentioned above, Lawn and Mai [9] and Steinbrech et al. [16] did not determine n by 3445

3446

SOHN et al.:

ANALYSIS OF BRIDGING STRESS

regression procedures. Instead, they computed the maximum bridging stress for various predetermined n. Lawn and Mai [9] determined the maximum stresses to be 25 MPa for n = 0, 40 MPa for n = 1, and 55 MPa for n = 2. Using an iterative process similar to that of Lawn and Mai [9], Steinbrech et al. [16] figured the maximum stresses to be 33 MPa for n = 2, 46 MPa for n = 3, and 60 MPa for n = 4. More recently, Hay and White [17, 181 introduced a direct experimental procedure, i.e. the postfracture tensile test, to obtain the stress distribution as a function of COD in various monolithic ceramics,

P(y) = ;

Phzf;

i=2

P(~) =

2. CRACK CLOSURE DUE TO GRAIN INTERFACE

BRIDGING

Sohn et al. [25] developed the bridging stress function based on Bennison and Lawn’s model [lo]. It includes parameters such as average grain size and shape, grain boundary energy, level of internal stress, and sliding friction coefficient. The key point of Sohn et al.‘s model [25] is that the grain size distribution is involved in the bridging stress function. The resultant equation of Sohn et al.‘s model [25] is as follows:

when

(2)

: i=N- I

and the peak stress in an alumina was estimated to be approximately 3446 MPa. In addition, they first introduced a grain size distribution to the bridging stress obtained from the postfracture test, suggesting that the grain size distribution was related to the bridging stress distribution. Taking the two distributions independently, they could overlay the two graphs and found that the bridging stress distribution from the power law curve fits matched the cumulative area fraction with respect to the grain size. In the present study a new bridging stress function, which takes into account the grain size distribution, is derived, and at the same time a new double cantilever beam (DCB) analysis is developed in order to verify this function and to get the exact bridging stress from COD measurements. Although a similar method using a compact tension (CT) type specimen has been already reported by Rode1 et al. [15] and Tsai et al. [24], the present method is developed on the basis of the double cantilever beam (DCB) theory taking into account the crack closure induced by the grain interface bridging. The new bridging stress function has been tested using DCB specimens of three kinds of alumina having different grain size distributions. The DCB specimens were loaded with the in situ loading device in a SEM chamber, and the crack openings were measured with a video monitor attached to the SEM. The bridging stress distribution was obtained from the result of the COD measurement using numerical fitting procedures. The analytical results correlating the strain-softening exponent in the power law relation with the grain size distribution have been confirmed quite reasonably in coarse grained alumina.

where PM is the peak stress,J is the area fraction of grains the size of which is about ipm, N is the maximum grain size, C, is the bridging zone size, and y is the distance behind the crack tip. Now we develop a new DCB analysis to obtain the bridging stress distribution. In the analysis, we first consider the influences of the grain interface bridging on the crack opening of the loaded DCB specimen. There is no doubt that the grain interfaces bridging around the cracked region interrupts the crack opening. Accordingly, a certain amount of crack closure could not be avoided in the coarse alumina which shows a prominent bridging effect. The crack closure due to the grain bridging is derived by accommodating either of the two bridging stress functions within the DCB analysis, i.e. the empirical power law function and our distributive bridging stress function given in equation (2). The DCB specimen has been generally used for investigating the fracture properties of various ceramic materials [2629]. The crack opening displacement (COD) and the strain energy of loaded DCB specimens caused by bending of the beam, were first derived by Benbow and Roesler [30] and by Gilman [3 11.Gillis and Gilman [32] developed a more advanced derivation including the shear of the beam as well as the bending, whereas Gross and Srawley [33] analyzed the DCB technique by a method entirely different from that used by Gillis and Gilman [32]: instead of deriving the strain energy of the configuration, they obtained an elastic solution of the configuration. Wiederhorn et al. [34] refined the Gillis and Gilman derivation by evaluating the unknown constants and found that Gillis and Gilman’s

SOHN

ef al.:

ANALYSIS

approach is consistent with that of Srawley and Gross. Unlike the above analyses, wherein each arm of the DCB specimen is treated as a built-in cantilever beam, the end of which is fixed at a rigid wall, Kanninen [35] developed an augmented DCB system by treating the beam as partly free and partly supported by an elastic foundation. It is noted that the beam-on-elastic foundation model takes account of the deformation in the area beyond the crack tip. On the other hand, Freiman et al. [36] analyzed the DCB specimen in three parts: the cantilever part, the ideal plastic zone, and the beam-on-elastic foundation. All the analyses mentioned above were derived on the well-developed theoretical basis, so their results would be consistent if the conventional specimen were adopted. In fact, as we calculated the compliances of each result with the specimen dimensions used in the present study, the compliances as a function of crack length were almost identical regardless of the routes through which they were derived. Hence, Wiederhorn’s result obtained by refining Gillis and Gilman’s analysis is adopted in the present study. According to the Wiederhorn’s solution [34], we can get the COD with respect to the position (distance from the loading point) as follows:

+ g

(I. - x) + S(L-X)

OF BRIDGING

3447

STRESS

theoretically and analytically, using the power law function and the distributive bridging stress function, respectively. 2.1. Power lnw bridging stress ,function Considering the one arm of the DCB specimen to be a cantilever beam shown in Fig. 1, the power law relation as a function of the distance from the free end of the beam (.Y), is expressed as the following form:

q = bP, (y+

I:,;

where P, is the peak stress at the crack tip (fixed end of the beam in Fig. I), C, is the bridging zone size, L is the total crack length (beam length in Fig. l), and b is the specimen thickness; q is the load-intensity function [37], which could be regarded simply as the by load per unit length. So q can be obtained multiplying the bridging stress by the specimen thickness (b). Strictly speaking, the term “load” seems to be improper. In fact, there only exists the constraining force (bridging force) against the externally applied load at the loading point. As shown in Fig. 1, the beam system has a singularity point at a distance a from the free end. The singularity point is caused by introducing a machined notch. Thus, equation (5) has to be reduced

(3)

where Us is the crack opening displacement, E the elastic modulus, G the shear modulus, b the specimen thickness, h the specimen half height, L the total crack length, P the load at specimen end, and x the distance from the loading point. Recently, a larger grain size has occasionally been considered in order to attain a prominent R-curve through effective grain interface bridging, since the R-curve behavior of polycrystalline alumina is highly desirable for practical applications. In this case, measured COD data are lower than those calculated by equation (3) while the data in alumina with very fine grains are in good agreement with the calculated values. The deviation in coarse alumina is attributed to the crack closure caused by the grain interface bridging in the crack wake. Thus, the closure term (AzL)should be incorporated in equation (3) to predict the exact COD values in coarse grained alumina. Hence, the real COD value is given as the following form: u = tin, + Au.

(4)

Matching the measured and calculated COD data, the exact bridging stress distribution can be obtained via numerical fitting procedures. In the following sections, we developed the two closure terms

Fig. 1. Schematic diagram of DCB system bridging

loaded by a stress. This system is adopted to calculate the bridging stress using the power law relation.

3448

SOHN et al.:

ANALYSIS OF BRIDGING STRESS

to singularity functions; q, and q2 are the singularity functions (Fig. 1) that were chosen among the family of conventional singularity functions [37]:

q=e+q*=

_bPM[l -(q+

[l -($y+3]+%

x

[l-(~+l~J(~~+‘.

l)J

The singularity function could be expressed as brackets, ()“. When n > 0 the notation ( )” has the following significance: if the expression in the brackets is negative (i.e. if x < a), the singularity function is zero; if the expression in the brackets is positive (i.e. if x > a), then brackets in the singularity function become ordinary brackets. The integration law for these functions follows ordinary integration. The shearing force (V) is derived using the relation, q = dV/dx, and the boundary condition is V = 0 at x = 0:

X (@++‘+hP,(~+

x

l)“(x-a).

(7)

Then the bending moment (n/r) is also derived in a similar way using the relation, V = dM/dx, and the boundary condition A4 = 0 at x = 0:

When we determined the boundary condition, we omitted contributions owing to both the compromise between a vertical cross-section and a horizontal neutral surface, and the strains in the region of the specimen past the crack tip. Although these contributions have to be considered in the conventional DCB analysis where only the end load exists [32, 381, it could be appropriate to neglect them if the loading condition were quite different from that of the conventional DCB analysis. In the present DCB analysis, the relatively small load (bridging stress distribution) is placed on a wide area of the beam. Such a loading condition does not necessitate compensating for the compromise between vertical and horizontal cross-sections, and cannot produce a significant deformation in the region beyond the crack tip. Consequently, the boundary condition used here is adequate for the present analysis. A second integration, together with the boundary condition that the closure is zero at the crack tip, gives the final result as below:

Au=%@+ A4 =

(10)

I)‘[

(L-a)3x-;
(l’;lf;n+“‘;, [I-(g+L)‘] - (L - uy_L + ; (L - a)4 1

The closure deflection equation is obtained by double integration of the curvature relation d2Au -dX2

-

-;I+&%

bP,(L - a)’ + EZ(n + l)(n + 2)(n + 3)

(9)

where A is the cross-sectional area, G is the shear modulus, and k is a numerical factor close to unity. Substitution and integration, incorporated with the boundary condition that only the rotation due to the shear at the crack tip (fixed end of the beam, x = L) exists, gives:

x

(l- (@=)‘J]+g$

x

(f$+

1)kx-+(L-4'1

kbP,(L - a)’ [+$+1)‘] + AG(n + I)(n + 2) bP,(L - a)’ [l-(T+lr] + EZ(n + l)(n + 2)

x

[(f+>“+2-I]

(11)

SOHN

2.2. Distrihutire

et al.:

ANALYSIS

OF BRIDGING

3449

STRESS

bridging stress jiinction

According to the distributive bridging stress function given in equation (2), the load-intensity function, i.e. the bridging stress function, of each group can be given by

q, = -f;bP,

“‘f;

7

L) + 1

(12) >

which has been derived based on the coordinate system in Fig. 2, and shows the singularity points at the end of the briding zone of each group. It is, however, noted that the load-intensity functions of the groups with larger grains have the singularity point at the notch tip (at a distance a from the free end). Thus, other singularity functions should be introduced to take account of the groups with larger grains, which will be represented by 9? and 92 in equation (13). Superposing the contribution of each group, we get the actual load-intensity function consisting of three terms:

9 = 91 +

qz + 97 = - 2 f;bP, ! =!,

?;

/

L) + 1

9, stands for the contribution of the groups whose bridging zone size (iC, IN) is smaller than the crack length (AC). On the other hand, the contribution of groups with a larger bridging zone (larger grains) than the crack length, is divided into y? and 9:. referred to the principle of singularity. The integer parameters, i,-and i,,, which represent the number of the bridging stress function (load-intensity function) of each group, can be obtained by the position (s) and the notch length (a), respectively, i.e. /, is the number of a bridging stress function whose end is closest to the left side of x, and i, is the number of a bridging stress function whose end is closest to the right side of a (Fig. 2). For example, considering the system having seven groups (N = 7) shown in Fig. 2, i( and i, are equal to 2 and 3, respectively. Using the load-intensity function of equation (13) and the same relation and boundary condition that were given in the previous section where the power law function was adopted, we derived the closure as below:

Y

X

+ 1 i ,f;bPM N(a - L) iC, EI /=/“?*I 24

X

x [(L - a)’ ~ (x -

X

x ./;bphl N(a - L)

Y

a>y-

;I 2 /-/,

Y

-

6

iC,

& i

,f*(L

/=/,,,-,

Fig. 2. Schematic diagram of DCB system loaded by a bridging stress. This system is adopted to calculate the bridging stress using the distributive bridging stress function developed in this study.

x

1_

Lt

”

+ 1 (L - U)‘(L - s) >

- a)‘(L - s)

N(sL) + , iC,

>l i

3450

SOHN

_-

k t

f@%!![(L- a)’

AG t=i,+, 6X,

x [(L - a)’ - (x 3. EXPERIMENTAL 3.1.

et al.:

-

ANALYSIS

(x

u>q.

-

a)3]

(14)

PROCEDURE

Materials

Three different polycrystalline aluminas, each characterized by a unique unimodal grain size distribution, were prepared for the present investigation. The starting powder was 99.997% alumina powder (AKP 3000 alumina, Sumitimo) containing 500 mol p.p.m. MgO. All the samples were sintered to greater than 99% of full density. Figure 3(a) shows the grain size distribution of a fine grained alumina sintered in air for 1 h at 165O”C, and Fig. 3(b) and (c) shows the grain size distributions of coarse aluminas sintered in Ar atmosphere for 10 h at 1750°C and for 12 h at 18OO”C,respectively. As can be seen in Fig. 3, the grain size distribution moved to the right as compared with the average grain size. The reason is that the size distribution is referred to the total area of grains within a specific size region, whereas the conventional average grain size is referred to the number frequency. 3.2. Specimen preparation For the present investigation, we need DCB specimens with natural cracks propagating along the

-’ Fig. 3. Grain

OF BRIDGING

STRESS

center line of the specimen. In fact, it is very difficult to introduce several millimeter-long cracks into ceramic DCB specimens. Most investigators used center-grooved specimens to avoid the deflected crack propagation. Unfortunately, the theoretical analysis in the present study prevents us from introducing the guiding groove into the DCB specimen, although there are several investigations [36, 39,401 which have taken account of the effect of the groove on the result of DCB analysis. According to the result in the previous section, the bridging traction leading to the crack closure acts on the crack surfaces. Accordingly, the narrower the width of the groove, the smaller the magnitude of closure. If the groove were introduced for the purpose of guiding the crack, the magnitude of closure would become too small to be estimated properly. Consequently, we developed a method to prepare the DCB specimens with no guiding groove. The first step is to introduce a deep groove and a blunt notch into a relatively thick specimen using a diamond blade. The groove width and the notch width are 0.4 mm and 0.16 mm, respectively. The second step is to introduce a straight crack propagating from the machined notch tip. Using the crack stabilizer guarantees stable crack growth and enables us to control the crack length at our disposal. The grooved DCB specimen compressed by the stabilizer was loaded in the wedge loading stage mounted on an optical microscope. Introducing the sharp wedge into the specimen mouth, a crack with the intended length could be obtained. The final step is to make a DCB specimen with no groove by eliminating the lower part of the specimen.

64

Grain Size (pm) size distribution

of three polycrystalline aluminas. Their average to be (a) 1.6 pm, (b) 10.2 pm, and (c) 19.0 pm.

grain sizes were measured

SOHN et al.:

II

3.3. COD measurement

4. RESULTS 4.1. Fine grain size

It is generally known that the bridging effect increases with increasing grain size in polycrystalline alumina [41]. Fine grained alumina (average grain size l-2 pm) was reported to indicate a flat R-curve [41]. Figure 4 depicts the measured CODS with respect to distance from the loading point (x) in the fine grained alumina with the grain size distribution shown in Fig. 3(a). The x range over which CODS were measured is about 7 mm. The arrows in Fig. 4 indicate the location of the machined notch tip and crack tip. The symbols in Fig. 4 indicate CODS measured at fixed positions at three different loads, respectively, and the solid lines in Fig. 4 represent Wiederhorn’s solution [34] at each load. The measured data are in good agreement with the curve calculated via Wiederhorn’s solution [34], which was derived on the basis of the perfect linear elastic behavior of brittle ceramic materials. Wiederhorn’s solution [34] cannot take into account non-linear effects such as grain interface bridging in the crack wake. Therefore, the good agreement between the measured and the predicted values suggests that fine grained alumina with about 1.6 pm average grain size 6

4

-

.

P=ZO.l9N

.

P=14.89N Wederhorn'ssolutm

2

!

2

0

15

16

8

t

-

An in situ SEM loading stage [25] was contrived for COD measurement by modifying the 20 kgf tensile stage for a Cambridge 250 MK3. A load was delivered by automatically rotating a knob connected to the outside of the SEM chamber, using the step motor together with the reduction gear and the speed controller. The applied load was monitored by a 20 kgf load cell attached to the loading stage, and recorded by an X-Y plotter. During externally controlled loading, crack interface events were taped on a video recording unit, and intermittently recorded on photographs.

z-

3451

ANALYSIS OF BRIDGING STRESS

17

18

19

20

Distance from the loading point (mm)

21

-

Fig. 4. Measured and predicted crack opening displacements vs the distance from the loading point (behind the crack tip) under three different loads, i.e. 20.19 N, 14.89 N and 9.8 N, of an alumina with an average grain size of 1.6 pm. Solid lines represent Wiederhorn’s solution.

16

17

stress function

18

19

20

Distance from the loading point (mm)

Fig. 5. Measured and predicted crack opening displacements of an alumina with an average grain size of 10.2 pm vs the distance from the loading point under a load of 33.24 N. The thick solid lines represent Wiederhorn’s solution. The thin solid and dotted lines represent the CODS predicted by COD derivation based on the power law function, and the distributive stress function, respectively.

has no significant bridging effect. It is thus concluded from the result that there is no closure due to the grain interface bridging in the fine grained alumina. 4.2. Intermediate

grain size

For the case of alumina with an average grain size N 10 pm [Fig. 3(b)], Fig. 5 shows the measured CODS when the applied load is 33 N. The measured CODS in this alumina deviate clearly from Wiederhorn’s solution [34]. The deviation reflects the closure due to the bridging exhibiting a prominent R-curve behavior. We derived the closure in Section 2 analytically in order to estimate the bridging stress distribution from the measured COD data by matching the formulations with the measured CODS. Two different stress functions were taken as a bridging stress distribution in order to derive the closure-one is the empirical power law form, and the other is the distributive bridging stress function. Consequently, two independent regression procedures were conducted. To facilitate such an analysis, the least squares method was used. The fitting procedure adopted for the power law function was carried out via equation (4) together with equations (3) and (11). In order to compute the exact values of PM, C, and n using the incremental searching method, the first step is to specify their initial values. The peak stress, PM, was increased from 20 MPa to 100 MPa in increments of 1 MPa, the initial bridging zone size (C,) was increased from 5.5 mm to 15 mm in increments of 0.1 mm, and the exponent (n) was increased from 1 to 4 in increments of 0.1. Table 1 shows the final values that this procedure yielded. The ensuing U(X) function is represented as the thin solid curve in Fig. 5. The distributive bridging stress function has a fewer number of parameters to be fitted than the power law function, i.e. the power law exponent n

1 22

15

I

Power law function

-- Distnbutive

14

I

Wiederhon’s solutm

3452

SOHN et al.: Table

1.

ANALYSIS OF BRIDGING STRESS

The best fitted results using both the power law function and the distributive stress function in two aluminas Power law function

Average grain size (rm)

(IGa)

10.18 18.95

58 59

Distributive stress function

c, (mm)

n

Root mean square error (x 10-3)

5.9 9.5

2.8 3

4.51 4.49

disappears in the distributive bridging stress function by introducing the grain size distribution. The fitting process in this case was carried out via equation (4), together with equations (3) and (14). Notwithstanding the fewer number of parameters to be fitted, the fitting procedure in this case would be more complicated than the case of the power law function because additional considerations should be needed both in selecting i,,, and if and in calculating area fractions (the grain size distribution) at each step of the procedure. The final values of PM and C, in this case were determined in the same manner as presented in the case of the power law function. The final results in this case are shown in Table 1 and the resulting u(x) function is represented as the thin dotted curve in Fig. 5. Figure 6 shows the crack closures which were separated out from the measured CODS. The data points in Fig. 6 were obtained by subtracting the amount calculated via Wiederhorn’s solution [34] from the measured CODS in Fig. 5. The smooth curves represent the values calculated via equation (11) adopting the power law function and via equation (14) adopting the distributive bridging stress function. Figure 6 shows that the curve shapes are slightly different from each other even though both curves show reasonable agreement with the measured values. The curve based on the power law function shows that the closure increases continuously from the crack tip to the machined notch tip. On the other hand, although the curve obtained by adopting the distributive stress function shows a similar tendency as a whole, the closure decreases slightly in the region

&:a)

c, (mm)

Root mean square error (x 10-X)

6.4 8.7

4.49 4.51

51 63

near the machined notch tip. Such an unusual observation in the case of the distributive bridging stress function can be attributed to the unique shape of the grain size distribution shown in Fig. 3(b) which was used in the derivation of equation (14). If the closure predicted by equation (14) decreased even beyond the notch tip, considerable confusion would arise. However, the singularity at the notch tip prevents the closure decreasing beyond the notch tip. 4.3. Coarse grain size As shown in Fig. 3(c) the coarse alumina has a wide grain size distribution, with the maximum grain size approaching about 60pm. Thus, a prominent bridging effect is expected. Figure 7 shows the measured CODS when the applied load is 45 N. The measured CODS in this alumina also deviate from the curve calculated by Wiederhorn’s solution [34], and the deviation is more prominent than the alumina with finer grain sizes. This result confirms again the well-known fact that the bridging effect is dominant in coarse grained alumina. The same fitting procedures yielded the final results shown in Table 1 and Fig. 7. Figure 8 shows the measured closure values and the curves predicted by the power law function and the distributive stress function, respectively. As in the preceding section, the curve obtained using the distributive bridging stress function shows higher curvature than that obtained by the power law

- - - - Distributive

- - - - - Distributive

stress

funcbon

z2 -0

I’ .

#’ .

z

0

3

stress function

2 -0.3 -

-0.2 -0.4

._ -. i-

:/:

,’ .’ ---.?

___ y

F’’ O-

.-

--i- ‘

14

A

I 14

15

16

Distance

from

Fig. 6. Crack closure displacement data in the predicted values distributive

17 the

loading

18 point

19

20

(mm)

data converted from the crack opening Fig. 5. Solid and dotted lines represent using the power law function and the stress function, respectively.

15

16

Distance

from

17 the

loading

I 16 point

I 19

21

(mm)

Fig. 7. Measured and predicted crack opening displacements of an alumina with an average grain size of 19.0 pm vs the distance from the loading point under a load of 45.08 N. The thick solid lines represent Wiederhorn’s solution. The thin solid and dotted lines represent the CODS predicted by COD derivation based on the power law function, and the distributive stress function, respectively.

SOHN

et al.:

ANALYSIS

.

-0.8 -

14

I

I

I

15

16

17

Distance

from the loading point (mm)

I

4

I

18

19

20

Fig. 8. Crack closure data converted from the crack opening displacement data in Fig. 7. Solid and dotted lines represent the predicted values using the power law function and the

distributive stress function, respectively.

function. However, no evidence of the decrease in closure is observed in the region near the machined notch tip. 5. DISCUSSION 5.1. Bridging stress distribution

Most investigators [S-11] have agreed that the bridging stress of monolithic and composite ceramics decreases gradually from the crack tip when the intergranular friction is a dominant bridging mechanism. The power law function has been adopted in order to quantify the bridging stress although some other investigators characterized the bridging stress using either a parabolic empirical function [21] or a uniform bridging stress [13,24]. However, careful in situ SEM observations of crack propagation have provided clear evidence that confirms the existence of a tail-like bridging stress distribution. Figure 9(a) shows a bridging site in the crack wake about 0.5 mm behind the crack tip at an applied stress intensity factor of 4.2 MPa 6, whereas Fig. 9(b) indicates the same bridging site at the same applied stress intensity factor after an additional crack propagation of about 1 mm. There still remains the trace of the severe friction indicated by arrows. The cracks marked “a” and “b” in Fig. 9(a) and (b) right below the bridging site represent the microcracks emanated from the main crack. The shape of the bridging stress distribution can be inferred from the behavior of the microcracks based on the fact that the magnitudes of the microcrack openings might reflect the levels of the bridging traction at the bridging site adjacent to the microcrack. As can be seen in Fig. 9(a), it is obvious that the microcracks “a” and “b” are suffering from the severe opening force generated by the friction and the interlocking on the main crack interfaces. However, some additional crack propagation makes it closed so that the contour of the microcrack becomes obscure [see “b” in Fig. 9(b)]. This clearly shows that the bridging traction acting

OF BRIDGING

STRESS

3453

on the crack wall would be released when the bridging site is distant from the crack tip. Figure 9(a) and (b) should provide good evidence supporting the tail-like bridging stress distribution such as the power law function or the distributive bridging stress function. Using two different bridging stress functions, we obtained two different results for each alumina. In all cases, the results fit the experimental data well, as can be seen in Figs 5 and 7. Figure 10(a) shows that the P(y) determined by using the distributive bridging stress function is almost identical to that determined by the power law function. A similar result was presented for the other coarser alumina [see Fig. 10(b)]. The extracted PM values, e.g. 58, 57, 59 and 63 MPa, are all alike regardless of the grain size distribution and the adopted bridging stress function, and are also similar to those reported previously [8%20]. The major factors which can affect the peak stress are the level of internal residual stress and the intergranular friction coefficient [ 10, 251. In particular, the level of internal residual stress has a decisive influence on the peak stress, i.e. the higher the internal stress level, the higher the peak stress. According to Evans and Clarke’s model [42], the controlling factors that can affect the level of internal residual stress are the grain size and the thermomechanical processing history. All the samples used in the present investigation were made of the same powder

Fig. 9. A series of SEM micrographs at a crack interface of a polycrystalline alumina, showing grain bridging under stress intensity factor levels of 4.2 MPa 6; (a) -0.5 mm and (b) - 1.6mm behind the crack tip. Arrows denote frictional contact points.

3454

SOHN et al.:

ANALYSIS OF BRIDGING STRESS

60 -

Powfu Iaw function(P, = 58 MPa. C_ = 5.9 mm, n = 2.6)

.

. . . . Distributivestressfunction(PM= 57 MPa, C_ = 6.4 mm)

-

can be formulated as below:

in terms of Green’s function [44]

AG = G, - Go =

10 -

0

1

2

3

4

,..__

I

5

6

7

Distance behind the crack tip (mm) ,

70

,

.,

,

,

,

,

,

,

,

IBWfuncllon(PM = 59 MPa. C = 9 5 mm. ” = 3, 00 - - - - Distributivestressfun&on (PM = 63 MPa. C_ = 8.7 mm)

-Power

UrnP(u) du s0

(15)

where G, is the saturated toughness (long crack toughness), Go the initial toughness (intrinsic toughness), urn the crack opening displacement at the end of the bridging zone, and P(u) the bridging stress function. Substituting the power law empirical function and the present model, respectively, for P(u) in equation (15), Green’s function can be changed into the following forms. In the case of the power law empirical function we obtain

(16) while the present model yields

10 -

0

1

2

3

4

5

6

7

8

Distance behind the crack tip (mm)

Fig. 10. Bridging stress functions predicted by the current model (solid line) and the power law function (dotted line) vs the distance from the crack tip of alumina with an average grain size of (a) 10.2 pm and (b) 19.0 pm.

(17) Equating equation (16) with equation (17), we get the following relationship, which correlates the power law exponent (n) with the microstructural parameters utilized in the present model: n==-1.

and annealed out at 1500°C for 2 h after the sintering process so that all specimens could have the same thermo-mechanical processing history. On the other hand, the difference in internal residual stress field induced by different grain sizes could not be avoided. The experimental results in the present study, however, do not show any radical change in peak stress with respect to the grain size. We calculated that, for the two coarse aluminas used in the present study, a difference between the peak stresses of less than 5 MPa could be expected using Evans and Clarke’s model [42]. This number is relatively low enough to be neglected and certainly is within the scattered PM values obtained in the present study (57-63 MPa). Experimental work by Ma and Clarke [43] using the piezospectroscopy technique also shows that there is no significant change in the residual stress over the range of grain sizes higher than N 10 pm. We conjecture that the fact that the grain sizes of the specimens used for the present study are within this range might verify our experimental results. The strain softening exponents which determine the shape of the bridging stress function are our major concern. Once the bridging stress function is established, the saturated toughness increment (AG)

(18)

,$,il; Such a relation makes the present model simpler and at the same time enables us to interpret n quantitatively. Using equation (18), we can obtain the simple power law function including microstructural parameters inferred in the present model as follows: P(u)=PM

1 -E (

(ZN,rn) -I , m= : if;. ,=I >

(19)

According to equation (18), n is related to the ratio of the maximum grain size (N) and the average grain size referred to area (ZZ;“,=zfi). It should be noted here that the average grain size referred to area is quite different from the generally known conventional mean grain size. The average grain size referred to area is usually much greater than the conventional mean grain size. Equation (19) indicates that the strain-softening exponent, which has been considered as a parameter related to bridging mechanism [7-91, also represents the microstructure of the material, i.e. the grain size distribution in monolithic ceramic materials. It is generally known that the grain size distribution of polycrystalline alumina obeys Hillert’s law [40] that maximum grain size can not exceed

SOHN

et al.:

ANALYSIS

twice the mean grain size. If the material has a normal grain size distribution and obeys Hillert’s law [45]. the strain-softening exponent has a value between 2 and 3 according to the present model [eqn (19)l. Table 2 shows the exponents derived from the COD measurement as well as the predicted values via the relation presented in equation (18). The predicted values are almost identical to the values obtained from the COD measurement in both aluminas. Notwithstanding the difference in average grain sizes between the two aluminas, the exponents derived from the grain size distribution are almost the same. Such a result originates from the fact that both aluminas have a similar grain size distribution, i.e. almost the same ratio of maximum size to average size, although the ranges that the distributions cover are quite different. In fact, it should be noted that equation (I 8) can be utilized only when the grain size distribution is unimodal. If a coarse alumina had a wide bimodal distribution of grain size, equations (18) and (19) could not be utilized, i.e the bimodal distribution would partly alter the curvature of a distributive bridging stress function because of the discontinuity between two peaks constituting the whole bimodal distribution. Accordingly, the criterion regarding the correlation between the exponent and the microstructural parameter is confined to unimodal grain size distributions. Fortunately, all the materials prepared in the present investigation represent the unimodal grain size distribution. Of coarse, the original form of the distributive bridging stress function [see equation (2)] could be utilized even in the case of the bimodal distribution. This implies indirectly that the power law function cannot explain the bridging stress of the materials with bimodal grain size distributions. In fact, C, is affected by the applied load as well as the grain size, i.e. C, is not a microstructural parameter any more. C, in the present study, however, stands for the bridging zone size of the equilibrium crack so that it can be regarded as a microstructural parameter of materials. Accordingly, the CODS should have been measured at the equilibrium state, but it would be very difficult to maintain the equilibrium state experimentally during the COD measurement because the crack is on the verge of propagation at the equilibrium state. The only thing we can do to make the equilibrium state is to raise the applied load just enough so that we can approach as close to the equilibrium state as possible.

Table 2. The strain softening exponents obtained from the COD measurement compared with the predicted values in two aluminas Strain hardening exponent Average gram size (pm) IO.18 18.95

Derived from COD measurement

Predicted from equation (18)

2.8 3.0

2.67 2.9

OF BRIDGING

3455

STRESS

3.5 ,

,

0.511 0

"""'I 1

2

3

1 5

4

1.1 6

7

Crack extension (mm) 4.5,.,.,,,,,, 4035-

&. > 5

3.0 -

$

25-

c" 'E

2.0 -

_F r-" 9

1.5 -

-PPowerlawfunctlon(P,=59MPa.C~=95mm,"~3, ---- DIs,nb"twe stressf""tilo" (PM= 63 MPa, cm = 8 7 mm,

1.0 0.5"'

0

j

2

1 4

1 6

11 6

10

Crack extension (mm) Fig. 11. Crack resistance curves predicted by the current model (solid line) and the power law function (dotted line) vs the distance from the crack tip for aluminas with an average grain size of (a) 10.2 pm and (b) 19.0 pm.

The loads adopted for the present study, i.e. 33 and 45 MPa, are the maximum values that we could increase to without any crack propagation at each specimen. Thus, it is likely that the bridging stress distribution obtained in the present study might be overestimated. 5.2. R-curve Once P(y) is established, the toughness increment (AK) can be formulated in terms of Green’s function [21] for our specimen geometry as following. This calculation is possible only when it is assumed that the function P(y) is independent of the specimen and notch geometry, i.e. PM, C, and n depend only on the material and the microstructure. AK(AC) = 2

=$

P(Y) d.v (20) (Co + AC)’ - (Co + AC - y)’ where Co is the initial notch length, AC is the crack extension and y is the distance behind the crack tip. The ensuing curves in Fig. 11(a) and (b) show R-curves calculated via both the distributive bridging stress function and the power law function in the

SOHN et al.: ANALYSIS OF BRIDGING STRESS

3456

coarse alumina used for the present investigation. The fact that the R-curve range and the saturated toughening value in Fig. 1 l(b) is larger than in (a), is analogous to the result in the preceding section. Strictly speaking, the R-curves presented in the present investigation are not real R-curves from which the fracture resistance (or toughness) could be obtained at a certain crack length. They represent just the increment in crack growth resistance. The intrinsic fracture toughness (K,,) of alumina is needed in order to obtain the real R-curves, which can be obtained using the following equation: Z&(AC) = K, + K(AC).

(21)

the Center for Advanced Aerospace Materials is also gratefully acknowledged.

of

REFERENCES 1. Evans, A. G., J. Am. Gram. Sot., 1990, 73, 187. 2. Steinbrech, R. W., in Fracture Mechanics of Ceramics, Vol. 9, ed. R. C. Bradt, D. P. H. Hasselman, D. Munz, M. Sakai and V. Ya. Shavchenko. Plenum Press, New York, 1991, p. 187. 3. Htibner, H. and Jillik, W., J. Muter. Sci. Engng, 1977, 12, 117. 4. Knehans, R. and Steinbrech, R., J. Muter. Sci. Left., 1982, 1, 327. 5. Swanson, P. L., Fairbanks, C. J., Lawn, B. R., Mai, Y. W. and Hokey, B. J., J. Am. Ceram. Sot., 1987,70,

279.

We applied the above relation to the coarser alumina in Fig. 11(b) so that the intrinsic fracture toughness of alumina could be estimated. As can be seen in Fig. 1 l(b), a plateau at 4.2 MPa & is observed for both the power law function and the distributive bridging stress function. This number is plausible compared with the difference between the presumed intrinsic toughness (1.5-2.5 MPa ,/&) of alumina from the literature [9, 10, 16,461 and the lon crack (saturated) toughness (6.38 + 0.25 MPa Jg m) cited from Swain [47] for the alumina similar to those used in the present investigation. This comparison between the R-curve calculated from our results and the cited actual R-curve incorporated with the intrinsic toughness shows that there was no overestimation and the loading condition adopted for the COD measurement was very close to the equilibrium state.

6. Vekinis,

G., Ashby,

7. Foote,

1. The new model in the present study explained the interaction effect between the bridging stress distribution and the local-fracture-controlling microstructure well, providing important information for the systematic interpretation of the microstructure mechanism including the R-curve behavior of a monolithic alumina. of crack 2. A device for in situ observation interfaces and COD measurement in the SEM was developed, and the bridging tractions were estimated from the measured COD data in various alumina DCB specimens with different grain sizes and distributions. 3. The bridging stress obtained using the power law function is almost identical to that made by the distributive stress function in coarse aluminas. The results show that the peak stress is a material constant independent of microstructural effects, but the bridging zone size and the strain softening exponent are dependent upon the grain size and grain size distribution. Acknowledgements-The

authors would like to thank Mr Keemin Sohn of Seoul National University for the numerical calculation. Use of the in situ SEM loading stage

P. W. R.,

R. M. L., Mai, Y. W. and Cotterell,

B., J. Mech.

Phys. Solids, 1986, 34, 593. 8. Cook, R. F., Fairbanks, C. J., Lawn, B. R. and Mai, Y. W., J. Mater. Res., 1987, 2, 345. 9. Lawn, B. R. and Mai, Y. W., J. Am. Ceram. Sot., 1987,

70, 289. S. J. and Lawn, B. R., Acta metall. 1989, 37,

10. Bennison,

2659. 11. Evans,

A. G.,

Heuer,

A. H. and Porter, D. L., in of Waterloo Press, Ontario,

Fracture, Vol. 1. University Canada, 12. Ballarini.

1977, p. 529. _ R.. Sah. S. P. and Keer.

L. M.. , Enp. - Fract.

Mech., lb84; 20, b33. 13. Reichl, A. and Steinbrech,

R. W., J. Am. Ceram. Sot.,

1988, 71, C299. 14. Hu, X.-Z., Lutz, E. H. and Swain, M., J. Am. Cerum. Sot., 1991, 74, 1828. 15. Rodel, J., Kelly, J. G. and Lawn, B. R., J. Am. Ceram.

sot., 1990, 73, 3313. 16. Steinbrech,

6. CONCLUSIONS

M. F. and Beaumont,

Acta metall. muter., 1990, 38, 1151.

R. W., Reichl, A. and Schaarw

Bchter, W.,

J. Am. Ceram. Sot., 1990, 73, 2009. 17. Hay, J. C. and White, K. W., Acta metall. mater., 1992, 40, 3017. 18. Hay, J. C. and White, K. W., J. Am. Ceram. Sot., 1993, 76, 1849. 19. Hsueh, C.-H. and Becher, P., J. Am. Cerum. Sot., 1988, 71, C234. 20. Majumdar, B. S., Rosenfield, A. R. and Duckworth, W. H.. Ew. Fract. Mech.. 1987. 31. 683. 21. Choi, S. R, Salem, J. A. and Sanders, W. A., J. Am. Cerum. Sot., 1992, 75, 1508. 22. Jakus, K. J., Ritter, E. and Schwillinski, R. H., J. Am.

Ceram. Sot., 1993, 76, 33. G. I., Adv. appl. Mech., 1962, 7, 55. Belnap, J. D. and Shetty, D. K., J. Am. Ceram. Sot., 1994, 77, 105. Sohn, K.-S., Lee, S. and Baik, S., J. Am. Ceram. Sot., 1995, 78, 1401. White, K. W. and Kelkar, G. P., J. Am. Ceram. Sot., 1991, 74, 1732. Nicholson, P. S., J. Am. Cerum. Sot., 1990, 73, 1800. With, G. D., J. Am. Ceram. Sot., 1989, 72, 710. Beauchamp, E. K. and Monroe, S. L., J. Am. Ceram. Sot., 1989, 72, 1179. Benbow, J. J. and Roesler, F. C., Proc. Phys. Sot.

23. Barenblatt, 24. Tsai, J.-F., 25. 26. 27. 28. 29. 30.

(Lond), 1957, B70, 201. 31. Gilman, J. J., J. appl. Phys., 1960, 31, 2208. 32. Gillis, P. P. and Gilman, J. J., J. appl. Phys., 1964, 35,

647. 33. Gross, B. and Srawley, J. E., Stress intensity factors by boundary collocation from single-edge-notch specimens subjected to splitting forces. NASA TND-2395, Feb. 1966.

SOHN

ef al.:

ANALYSIS

34. Wiederhorn, S. M., Shorb, A. M. and Moses, R. L.. J. appl. Phys., 1968, 39, 1569. 35. Kanninen, M. F., fnt. J. Fract., 1973, 9, 83. S. W., Mulville, D. R. and Mast, P. W., 36. Freiman, J. Mater. SC;., 1973, 8, 1527. 37. Crandah, S. H., Dahl, N. C. and Lardner, T. J., An Introduction to the Mechanics qf Solids, Chap. 3. McGraw-Hill, Inc.. New York, 1978. 38. Berry, J. P., .I. appl. Ph~,s., 1963, 34, 62. 39. Chow. C. L. and Woo. C. W., Int. J. Fract.. 1980, 16, 121. 40. Wu. C. C., Mckinney, K. R. and Lewis, D., J. An?. Ceram. Sac.. 1984, 67, Cl66. 41. Chantikul. P., Bennison, S. J. and Lawn, B. R., J. Am. <‘ernnt. Sot., 1990. 73, 2419.

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42. Evans, A. G. and Clarke, D. R., in Thermal Stresses in Materials and Structures, ed. D. P. H. Hasselman and R. A. Heller. Plenum Press, New York, 1980, pp. 62948. 43. Ma, Q. and Clarke, D. R., J. Am. Ceram. Sot., 1994, 77, 298. 44. Hilert. M., Acta metall., 1965, 13, 227. 45. Lawn, B. R. and Wilshaw, T. R., Fracture qf Brittle Solids. Cambridge University Press, London. U.K., 1976. 46. Anstis, G. R., Chantikul, P.. Lawn, B. R. and Marshall, D. B.. J. Am. Ceram. Sot., 1981, 64, 533. 47. Swain. M. V.. J. Mater. Sci. Letr.. 1986. 5, 1313.