characterization of 2-dimensional crack propagation behavior by simulation and analysis

International Journal of Fracture 75: 247-259, 1996. 247 © 1996 KluwerAcademic Publishers. Printed in the Netherlands.

Characterization of 2-dimensional crack propagation behavior by simulation and analysis

BYUNG-NAM KIM, SHUICHI WAKAYAMA and MASANORI KAWAHARA Department of Mechanical Engineering, Tokyo Metropolitan University, 1-I Minami-Ohsawa, Hachioji-City, Tokyo 192-03, Japan

Received 20 January 1995; accepted 22 May 1995

Abstract. Crack propagation behavior in 2-dimensional polycrystals is simulated and analyzed as a function of the fracture toughness of the grain boundary. The path of a crack impinging on a grain boundary is determined by the competition theory between intergranular and transgranular propagation. With decreasing boundary toughness, the tendency of intergranular propagation increases and the apparent fracture toughness of the polycrystal decreases. The results of the 2-dimensional analysis are compared with the simulation, and the advantages and limitations are discussed. The grain boundary toughness is evaluated by comparing the simulated crack paths with direct observations, resulting in a reasonable value for alumina ceramics, The fracture behavior is characterized in a macro-scale by the percentage of transgranular fracture and also in a micro-scale by the distribution of crack deflection angles.

1. Introduction

What is the fundamental fracture mechanism of non-transforming ceramic polycrystals? For this subject, several mechanisms, such as microcracking, grain bridging and crack deflection, have been proposed. Microcracking is a mechanism resulting from residual stresses at grain boundaries and is well known as a toughening mechanism in ceramics. According to the analytical and experimental studies for alumina ceramics, a critical grain size of about 10,-~ 100 #m is required to generate microcracks before unstable fracture [1-4]. Grain bridging also requires large grains for the effective formation of a wake zone [5, 6]. However, when the 'fundamental' mechanism is considered, crack propagation in an elementary polycrystal, i.e. composed of a grain and a grain boundary only, has to first be examined. Residual stress and grain size are a secondary factor to be taken into account.

Observing the crack path in ceramic polycrystals of small grain size, we readily find the characteristics of inter-/transgranular propagation with crack deflection, while the other two mechanisms assume a plane crack. One of the authors analyzed the crack deflection behavior in brittle polycrystals by considering the effect of 3-dimensional (3D) crack propagation. According to the model, the crack path at triple and quadruple junctions of grain boundaries is constrained by adjacent crack planes, and it was concluded that crack deflection at the junctions is the fundamental fracture mechanism of polycrystalline materials [7]. In this study, the crack propagation behavior in 2-dimensional (2D) polycrystals is simulated with the inter- /transgranular crack deflection mechanism, and the fracture toughness and the percentage of transgranular fracture are evaluated as a function of the grain boundary toughness. Since the previous 3D analysis is restricted to the initial propagation of a relatively simple-shaped crack due to analytical limitations of crack tip stress, the simulation of continuous propagation of long cracks may be helpful to understand the fundamental nature of fracture phenomenon. In

248 B.-N. Kim et al.

addition, the 2D analysis of the initial crack propagation is conducted and compared with the simulated results.

2. Determination of crack path

The propagation of the deflected crack in polycrystals is a mixed mode fracture. Although several different criteria have been proposed for the mixed mode fracture, the most commonly used criteria are the maximum tangential stress criterion [8], minimum strain energy density criterion [9] and maximum energy release rate criterion [10-12]. In this simulation, the maximum energy release rate criterion is employed because it is consistent with the original energy balance concept of Griffith. Specifically, among the proposed criteria of the maximum energy release rate, the Nuismer criterion [I 1] is used to determine the crack path mainly due to the simple expressions.

When the deflected portion of a crack is infinitesimally small with respect to the total crack length, the stress intensity factors at the deflected crack tip (kl, kit) can be written in terms of those of the main crack (Kb Kn) [ 13, 14]. First-order solutions yield

kl = f l l (0)KI + f21(O)Kn, (la)

ku = fI2(O)Kl + f22(O)Kn, (lb)

where 0 is the deflection angle as described in Figure l(a), and fij (0) are the angular functions analogous to the angular dependence of normal and shear stresses in the near-tip field [15]. Crack advance is assumed to be governed by the strain energy release rate G, represented as [111

1 2 c ( o ) = + (2)

where E / is the elastic modulus of the material. The stress intensity factors at the complicated deflected crack tip ( t ( I , t(II) are obtained

numerically under uniaxial tension by the body force method (BFM) developed by Nisitani [16]. tQ and Ku are obtained as a function of the applied tensile stress (tr). On the other hand, the intrinsically deflected crack in polycrystals is often regarded as a plane crack of length 2a due to analytical convenience, and the apparent stress intensity factor (Ka) is calculated with the imaginary plane crack. In order to evaluate the apparent fracture resistance of the deflected crack, the ratios between the real and the apparent stress intensity factor are taken to be al (= KI/t(~) and a2 (= KII/Ka). Then, from (2), the G(O) can be written as

G(0) -- "~-'Tfl (0, cq, a2), (3)

where fl (0, a l , a2) is a function representing the deflection effect of a crack that is equal to or less than unity.

When a crack impinges on a grain boundary, competition occurs between penetration and deflection, as shown in Figurel(c). Whether the crack will pass through the boundary or be deflected into it depends on the impinging angle on the grain boundary (0b), the stress intensity factors at crack tip and fracture toughness of both grain (Keg) and grain boundary (Kcb) [7].

2-dimensional crack propagation behavior

(a) kl kH KI K.

g l l

249

(b) o,o° "'02

Kl ~ l kn

(c)

Crack

i

deflec~

Kx Ku 0b K ~ J penetration

Grain boundary

Figure I. (a) Single crack deflection, (b) double crack deflection and (c) crack impinging on grain boundary. Competition of crack path occurs between penetration and deflection.

Let G~b (= I(~b/Et), the critical energy release rate of grain boundaries, be constant in the material, and Gcg (= K~g/E I) be the critical energy release rate of an isotropic grain. Then the crack is deflected into the grain boundary when

G(0b) = Gob. (4)

Substituting (3) into (4), the apparent fracture toughness (Ko~), when the crack deflection occurs, is given by

l~cb ](ca -~ ~/fl(Ob, Ctl~ C~2i" (5)

In a similar manner, when the crack advances through the grain in the direction of 0g, K~. is given by

]~ca -~- ~cg I f 1 (0g, cq, O~2)

(6)

For a crack impinged on a grain boundary, the propagation occurs in the direction of the lower K~a of (5) and (6). In the following simulation, by applying this simple competition theory


of crack path, the crack propagation behavior and the fracture toughness are examined as a function of I(cb.

3. Simulation of inter-/transgranular crack propagation

In the numerical analysis of IQ and KII by the BFM, the 2D polycrystal is assumed to be elastically isotropic under plane stress condition but the fracture toughness of the grain and grain boundary is different. The crack path is determined by the two competitive effects, that is, the reduction of stress intensity due to deflection and the reduced toughness of grain boundary. It is also assumed that fracture occurs by the unit of a grain, so that the crack tip propagates from one grain boundary to another grain boundary and is always located at the grain boundaries.

The 2D microstructure for the simulation was obtained from a micrograph of a thermally etched surface of alumina, by reading the coordinates of the vertices of respective grains. The alumina sample was made by hot-pressing at 1500 °C for 1 h, and a relative density of 99.5 percent and an average grain size 4 #m was obtained.

3.1. PROPAGATION OF ARBITRARY SHORT CRACK

As an initial crack for simulation, a short intergranular crack of 2ao was introduced at two different sites in the center of the polycrystal, as illustrated in Figure 2, the plane of which exists on grain boundaries. (Here, the crack length is the length normal to the stress axis). There is no interaction between cracks, although the two cases are represented in the same microstructure. The simulation was conducted within the polycrystal shown in Figure 2. When the crack tip arrived at the end of the polycrystal, the propagation of the other tip was simulated without a break.

In the polycrystal of Kcb = 0.7Keg, the crack propagates mainly by intergranular mode, while transgranular fracture occurs at large Ob as in case 2. However, even though Ob is large, it is possible for intergranular fracture to occur depending on the value of KI and 1QI as well as the sign of Km as observed from the right portion of the crack in case 2. Figure 3(a) represents the variations of Kca with respect to crack extension. Kca represents a large variation depending on Ob at each propagating step, but represents a tendency to increase slightly. However, judging from all the simulated results, it is concluded that crack deflection itself does not contribute to the rising fracture toughness with crack extension (R-curve). The same conclusion can be inferred from the analytical results of invariant relative stress intensity factors ( a l , a2) regardless of the number of deflections, when a crack propagates in a zigzag pattern with the same deflection angle [17, 18].

Since the critical impinging angle for crack deflection decreases with increasing K~b, the probability for transgranular fracture increases in proportion to Kc6, while only intergranular fracture occurs when K~b <<, 0.4K~g. When the crack advances transgranularly, K~a has a value close to K~g, as illustrated in Figure 3(a). However, the value of Kca for the extension of an inclined crack should be a little higher than that of K~g according to (6). The slightly lower Kca is thought to result from the numerical error of the BFM. Similar results are shown in the case of K~b = 0.9K~g. In this study, the relative variation of stress intensity factors due to deflection is emphasized, rather than accuracy.

Figure 3(b) represents the fracture stress (a~) for respective crack extensions, normalized with the initial fracture stress (a~o) defined as K~Jx/(Trao). ac decreases continuously from


2"

(a) Kcb=0.7Kcg

(b) Kcb=0.8Kcg

251

(c) Kcb=0.9Kcg

Figure 2. Simulation of a crack path for two short cracks at different grain boundary toughness.

initial extension, indicating that the stable growth of such short cracks does not occur under inert conditions. Once the crack begins to grow, unstable propagation would occur. In contrast to the short crack, however, the stable growth of a relatively long crack is expected to occur to some extent, because the decreasing rate of o-c becomes lower and the local increase of fracture resistance can increase ~rc over the value required for the previous extension, as illustrated in the case of l(c~ = 0.7K~g in Figure 3(b).

3.2. EVALUATION OF GRAIN BOUNDARY TOUGHNESS

From the above simulation, the crack path is found to be strongly dependent on the value of Kcb. A few methods have been proposed to estimate the value quantitatively based on the

6 • ' , ; 4 ~ I I

o i", ll \1\ \

--Or-- Kcb=0.9Kcg 0.6 - - e - - Kcb=0.8Kcg

(a) ~ Kcb=0.7Kcg 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 5 I0 15


~ J

¢ , ) e - ,

t -

O [--

kt.

O

<

1.0

Crack Extension, &dao

• • •' " , " ' i , • , " * ! • ' " ' " ' " • •

~ Kcb=0.9Kcg 8 0.8 ~I~ - - e - - Kcb=0.SKcg ~ f ~ '-'.~ Kcb=0.7Kcg

0.6-

0.4" -

,= " 0.2

(b)

0.0 . . . . . . . . . . . . . . . . . . . . . 0 5 10 15

Crack Extension, Aa/ao

Figure 3, Variation of (a) apparent fracture toughness and (b) fracture stress with respect to crack extension for the case I in Figure 2.

percentage of transgranular fracture (Tf) [ 19, 20]. However, the comparison of the simulated crack path with the experimentally observed one may give an altemative way to estimate the value of Kcb.

A crack of about 2mm long was introduced by bridge-indentation method [21] in the same alumina polycrystal used in the previous simulation, and the crack was observed with scanning electron microscope. The area examined was the center of the crack, about 1 mm distant from the starting point. Since the crack propagated from a free surface, the simulation within the semi-infinite body is preferred. However, the crack path was simulated within an infinite body beca),se the characteristics of crack propagation are expected to be identical for the two conditions. Most of the deflected path for the real crack was flattened except the near-tip portion in order to shorten the calculating time. The error by this approximation is expected to be insignificant.

The simulated crack paths are represented in Figure 4 along with the experimental ones, at which only the propagation of the right tip is simulated• The length of the plane portion of the initial crack is 5 times that of the deflected portion AAq After some intergranular propagations, transgranular fracture occurs at the point B when Kcb 1> 0.5Keg, and Kc~ is

2-dimensional crack propagation behavior 253

C

crack propagalion ~ ~

Figure 4. Crack paths by simulation and experiment in the microstructure of alumina. The point A is the initial crack tip for simulation.

close to Keg, as shown in Figure 5. When Kcb <. 0.4K~g, the crack is deflected into the boundary at the same point, but Kc, is also relatively high due to high angled deflection. At point C, the simulated crack propagates in different directions for K~b = 0.4K~g and 0.6Keg, although the same intergranular fracture occurs. This is because of the different Kn/Kl ratio resulting from the difference of crack shape, even though the near-tip shape is identical. The value of KII / KI at the point C is - 0.120 for Kcb = 0.4Kcg, while -0.163 for K~b = 0.6Kca. This fact indicates that a small change of KII/KI can have remarkable effects on the crack path, as illustrated in Figure 4.

On the other hand, the experimental crack propagates transgranularly at point B, and intergranularly at point C, as described in Figure 4. A simulated path identical to the experimental one was not obtained. In order to have a more realistic simulation, the effect of twist deflection has to be taken into account. In addition, Kcb is expected to vary from grain boundary to grain boundary by the atomic structure and the residual stress due to thermal and elastic mismatch between adjacent grains [1, 22, 23]. However, despite these limitations, the qualitative comparison of the crack path makes it possible to estimate the effective Kcb of alumina, resulting

254 B.-N. Kim et aL

Figure 5. Variation

~' 1 . 2 ~ '

1.0

o " 0.8

0.6

0.4 " " * " " " " ' Kcb = 0.8 Keg

0.2 Kcb = 0.6 Keg . . . . . Kcb = 0.4 Kcg

< 0"00 0.2 0.4 0.6 0.8

Crack Extension, &'da,,

of apparent fracture toughness with respect to crack extension ir~ Figure 4.

in about 0.6Keg. This estimated Kcb corresponds reasonably to the value assumed elsewhere [4] and obtained by the 3D analysis of crack deflection [20].

In addition to the qualitative comparison of the crack path mentioned above, Kcb can also be evaluated by comparing the macroscopic properties. One inspected parameter is the fracture toughness of the polycrystal (KcA). In order to obtain the representative value of KcA for the simulated polycrystal, the average of Kc~ was taken because Kca varies at respective propagating steps. The obtained K~A is represented in Figure 6(a), which increases with K~b and reaches the value of K~g at Kcb = l(cg. It is found that the condition for KcA >1 I(~g, experimental results reported in general, is not accomplished by the 2D simulation, while the analytical results taking into account the 3D effects of crack path made it possible [7]. Hence it can be concluded that the 2D fracture toughness would not play a role in evaluating the fracture toughness of 3D polycrystal as well as Kcb.

The other available parameter is'the percentage of transgranular fracture. The percentage of transgranular fracture from the 2D simulation (TEA) decreases with decreasing Kcb and is in good agreement with the 3D analysis, as illustrated in Figure 6(b), where TIA, not surface ratio, is measured by dividing the number of transgranular fracture steps by the number of total steps for the simulated crack path. The reasonably consistent values indicate that the characteristics of macroscopic crack path are similar between 2D and 3D fracture, and that the value of l(~b for the 3D polycrystal can be estimated by measuring the percentage of transgranular fracture in the 2-dimension.

4. Model analysis of 2-dimensional crack propagation

In the above simulation, KcA and TfA were obtained as representative of mechanical property and crack propagation behavior, respectively. The two parameters can also be evaluated analytically on the basis of the 2D crack deflection model. Consider a plane crack impinged on a grain boundary called a transgranular crack [7]. The crack would be deflected into the boundary if [24]

G( O) G~b K2b - - - - 2 ( 7 ) G(O) ) G~ a t~g

2.0

¢o

,x::

o 1.0 H,

0.5

0.0 < 0


3D analysis [7] w-x_ 2D analysis / / " -'-\

0.2 0.4 0.6 0.8

Grain Boundary Toughness, Kcb/Kcg

255

3D analysis [7] .o7/ ,) 80 ~D analysis , ; / /

,,., ~ --o--Simulation 1/1 ~ 60

0 0,2 0.4 0.6 0.8

Grain Boundary Toughness, Kcb/Kcg

Figure 6, Comparison of (a) apparent fracture toughness and (b) percentage of transgranular fracture obtained by simulation and analysis,

Since fl (0, a l , a2) in (3) becomes cos4(0/2) for mode I crack, the critical impinging angle for crack deflection (0c) under mode I loading is given by

O~ = 2 arccos( ~/Kcb/ If~). (8)

The crack impinging on the boundary with the angle lower than 0e would be deflected. When the facet of the grain boundary is assumed to be constant in length and to be distributed uniformly, the probability of the crack impinging on grain boundaries with the angle between 0 and 0 + dO is sin0 dO. Restricting the crack deflection within the range of 0 ~< 0 ~< ~r/2, the macroscopic fracture toughness of the transgranular crack (IQT) can then be represented as

U/2 2 K2T = JO Kc~(O'O)sinOdO' (9)

where Kc~(0, 0) is the apparent fracture toughness when a plane crack impinged at an angle of 0 advances. The value of Kc=(0, 0) takes the lower value of (5) and (6). Solving (9) with (5) and (6) leads to

256 B,-N. Kim et al.

ec rr~/2 K2T = fO I(2~COS-4 ( ~ ) s i n ( O ) d O + Joc I(2usin(g)dg

Since K~b ~< Keg, the maximum KcT is obtained at gc = 0, that is, at K~b =Kcg. Comparing the calculated K~T with the simulated K~A, a relatively good agreement is found despite the apparent difference of stress condition at the crack tip. The transgranular crack was analyzed under mode I loading, while the simulated crack advances always under the mixed mode of I and II. The macroscopic mechanical properties in the 2D cases are not thought to depend strongly on the variation of such loading mode. The same conclusion is also obtained later by the comparison of the analytical results for mixed fracture.

In a similar way, the percentage of transgranular fracture at the propagation of the transgranular crack (TIT) is obtained to be

[ ~r/2 TIT = Tf(0, g) sin(g) dg, (11)

J0

where the value of T](0, 0) is 1 at transgranular and 0 at intergranular propagation. TIT varies linearly in the range of 0.5K~ a <~ t(~b <~ I(~ a, not consistent with the tendency of the simulated results in Figure 6(b).

In order to predict the crack propagation behavior closer to the simulation, the propagation under mixed mode is analyzed in the following. The appropriate crack type for the mixed mode may be the crack advanced by one step from the transgranular, called a mixed crack. Depending on I(c~, 1 - T]:r of the transgranular crack tip would be deflected into the grain boundaries. Then, just the rear plane of the mixed crack tip would be composed of the transgranular crack of TIT and the intergranular of 1 - TIT. Since the transgranular crack straightforward advance does not change the shape, the contribution to the fracture toughness of the mixed crack (KcM) can be calculated from (10). For the contribution by the intergranular portion, the additional propagation of the deflected crack has to be taken into account.

The stress intensity factors at the deflected crack tip were estimated as in (1), under the assumption of an infinitesimally small deflection, and were used to determine the direction of crack propagation. The same analysis is available in the case of the doubly deflected crack, as described in Figure 1 (b), providing that the effect of the deflected length on stress intensity is assumed to be negligible [7, 25]. This assumption may yield inaccurate values of I(cM, but it is expected that crack propagation behavior is predicted with better consistency compared with that for the transgranular crack. The stress intensity factors at the doubly deflected crack tip (kb k , ) under mode I condition can then be represented as

kl = [fll(O2)fll(O1) + f21(O2)fl2(O1)]l(l, (12a)

/QI = [fl2(g2)fll(Ol) + f22(O2)f12(Ol)]KI, (12b)

where Ol and 02 are the first and the second deflection angle, respectively. Knowing the value of ki and klI, the contribution of the intergranular portion to KcM and

TfM (the percentage of transgranular fracture for a mixed crack) can be estimated approximately in a similar way to (10) and (11). The difference is that the impinging probability


of the intergranularly deflected crack on grain boundaries, is not uniform at an angle unlike the case of the transgranular crack. The crack tip propagated intergranularly is located at a triple junction by the assumption of the grain unit extension of a crack. For the polycrystal composed of regular hexagonal grains, two grain boundaries appear in the direction of ±re/3, and the only possible intergranular path of the crack tip is in the direction of - r e /3 due to the effects of mode II stress intensity. Even when there are variations in the number of grain edges and grain size, the direction of the maximum probability of crack extension would be -7r/3. Thus, it can be considered for simplicity that the competition of the crack path occurs for the intergranular crack tip between the intergranular in the direction of - r e /3 and the transgranular path. The transgranular path is the direction of the maximum energy release rate (dG/ dO = 0).

The apparent fracture toughness K ~ (0,, - 7r / 3 ), when the propagation of the crack deflected at 01 along grain boundaries occurs, is obtained with (5) or (6), after substitution of - r e /3 into 02 of (12). Notice that the K~a (01, -7r /3) is not the fracture toughness when the deflected crack propagates into the direction of - re /3 , the case of the intergranular propagation. Con- sidering the two contributions of both the intergranular and the transgranular portion of the mixed crack, tfcM and Ty;vt can be represented as

"2 , ,'2 l -- T f T 0~0 Oc I az= TJTI T+ I(2(0'--U/3) sin0d0' (13)

1 - T j r fo TyM = TfTTfT + 1 Tf(O,-Tr/3) sinOdO, (14)

respectively. Here, the effects of adjacent crack planes on stress intensity factors are neglected.

The calculated results are shown in Figure 6. The value of Kcm becomes slightly lower than that ofKcT (,YcM ~ 0.9K~T at Kcb = 0.5Kc~), because the intergranular propagation is restricted to the angle of - r r / 3 for the mixed crack, while uniform for the transgranular one. However, the analytical IY~M and TfM are consistent with the simulated -[(cA and TfA, as illustrated in Figure 6(b). Specifically, the fracture toughness represents very good consistency with the simulated results, indicating that the error by the above approximation would not be so significant in the 2D analysis. Also, there is no great difference in the percentage of transgranular fracture between the 2D and the 3D analysis, as discussed previously. The slight difference is expected to result from the effect of twist-deflection in the 3D crack propagation.

The good consistency of macroscopic properties, however, is not sufficient to characterize the microscopic fracture behavior. The macroscopic fracture properties are the results of the respective crack extensions in a microscopic scale. The comparison of the experimentally observed crack path with the simulation may give one way to evaluate the polycrystal quali- tatively in a microscopic scale, as in Figure 4. The crack deflection angle has also been used to account for the microscopic contribution to macroscopic properties such as toughening behavior [20, 26, 27] and to understand the fundamentals of fracture [7]. The microscopic fracture behavior may be described quantitatively by the deflection angle.

The distribution of the deflection angle from the simulation is closer to the 3D results than the 2D, as illustrated in Figure 7, indicating that the simplified 2D analysis has limitations in describing the microscopic fracture behavior. The deviation between the 2D and the simulation becomes large as the intergranular propagation occurs (lower K~b), because the 2D model


2

7

1.0

0.8

0.6

0.4

0.2

0.0

..... A A t * * * * ~ * * * ~ * A A A *

(a)

t ~ " ~ Simulation 1 .... 2D analysis ~ 3D analysis [71 ~

! • - I . . . . . . . . , , ........... i , I

O" 30" 60" 90*

Deflection Angle

l , 0 ............... • , | , , ! , A I I i A A A A

0.8 ( b ) ~

,o 0.6 m .

o / / y - .~ K b=0.6Kcg 0.4

E /~ - ~ Simulation " 0.2 fA~ ~ 2D analysis

3D analysis [7]

0.0 . . . . . . . . . . . . . ' . . . . . 0 ° 30* 60" 90"

Deflection Angle Figure 7. Cumulative probability of deflection angle dependent on grain boundary toughness, The deflection angle was measured with respect to the main propagating direction of the crack.

restricts the direction of the second deflection to -7r /3 , while the 3D analysis accounts for all possibilities of crack deflection angle. Although the 3D analysis is more consistent with the simulation, the~consistency decreases at low 1feb. When the crack propagates intergranularly, the deflection angle is prevailed upon by the spacial structure of grain boundaries. The structural difference between 2D and 3D polycrystals is considered to result in the apparent deviation in Figure 7(b).

5. Conclusions

The simulation of crack propagation was performed on a 2D polycrystal of alumina. The crack extends only intergranularly when Kcb ~< 0.4tfcg, and the percentage oftransgranular fracture increases with increasing t(cb. By comparing the microscopic crack path of the simulation with the experimental ,observation, the value of K~b can be estimated to be approximately 0.6Kog. The fracture toughness and the percentage of transgranular fracture of the 2D polycrystal were also calculated by the model analysis. The analytical results for the mixed crack type show a reasonable consistency with the simulation despite the considerable simplification of fracture


characteristics. In particular, the consistency of the fracture toughness is excellent. The 2D

analysis is considered to be effective in evaluating such macroscopic mechanical properties.

However , since the apparent deviations of crack deflection angle are found between the analysis and the simulation, it is expected that the fracture toughness of the polycrystal is less

sensit ive to the microscopic characteristics, such as the fracture toughness of grain boundaries than the microscopic fracture behavior is.

Acknowledgements

The authors wish to thank Prof. Nisitani o f Kyushu Universi ty for provision of the BFM program.

References

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(1990) 2123-2129. 17. M. Isida, Transactions of the Japan Society of Mechanical Engineers A45 (1979) 306-317. 18. A.A. Rubinstein, Journal of Applied Mechanics 57 (1990) 97-103. 19. I.J. McColm and N.J. Clark, Forming, Shaping and Working of High-Performance Ceramics, Chapman and

Hall, New York (1988) 17-24. 20. B.N. Kim and T. Kishi, Journalofthe Japan Institute of Metals 58 (1994) 1256-1262. 21. T. Nose and T. Fujii, Journal of the American Ceramic Society 71 (1988) 328-333. 22. J.E. Blendell and R.L. Coble, Journal of the American Ceramic Society 65 (1982) 174-178. 23. M. Ortiz and A. Molonari, Journal of the Mechanics and Physics of Solids 36 (1988) 385--400. 24. M.Y. He and J.W. Hutchinson, International Journal of Solids and Structures 25 (1989) 1053-1062. 25. S. Suresh, Metallurgical Transactions 14A (1983) 2375-2385. 26. K.T. Faber and A.G. Evans,Acta Metallurgica 31 (1983) 577-584. 27. G. Pezzotti, I. Tanaka and T. Okamoto, Journal of the American Ceramic Society 73 (1990) 3039-3045.

characterization of 2-dimensional crack propagation behavior by simulation and analysis

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