Grain boundary toughness effects on crack propagation in brittle polycrystals

ELSEVIER

Materials Science and Engineering A205 (1996) 110-116

A

Grain boundary toughness effects on crack propagation in brittle polycrystals P. Lipetzky 1, W. Kreher Max-Planck-Gesellschafi, Arbeitsgruppe Mechanik heterogener FestkOrper, Hallwachsstrafle 3, D-01069 Dresden, Germany Received 14 December 1994; in revised form 23 June 1995

Abstract

Quasi-static crack advance has been statistically analysed as a function of interfacial toughness using a randomized two-dimensional, polycrystalline model. The average macroscopically observable toughness is calculated to be a strong function of interfacial toughness if the interfaces, grain boundaries in this case, have a fracture resistance less than approximately 1/2 that of the grains. Alternatively, the relative amount of crack deflection, as indicated by the percentage of interfacial failure, is a relatively homogeneous function of the interfacial toughness if interfacial fracture resistance is higher than about 0.3/'in t. Combining these predictions gives a quantitative measure of both the expected material strength and resulting fracture surface roughness. Results exhibit good agreement with a similar analytical model which considers a deterministic approach for a similar geometry. Finally, the method employed here is general enough to be used for any polycrystalline ceramic.

Keywords: Grain boundary toughness; Crack propagation; Brittle polycrystals; Statistical analysis

I. Introduction

Composite materials are receiving much attention in the ceramics community because of their potential for increased toughness, strength and thermal shock resistance, although monolithic, polycrystalline materials cannot be ignored owing to their comparatively low cost, processing simplicity and steadily improving mechanical properties. One example of this is aluminium oxide and alumina-based composites. While it is true that reinforced alumina has many mechanical advantages, single-phase, sintered alumina remains the centre of much attention for both load-bearing and wear-resistant applications [1-4]. With the fairly wide range of compositions and processing conditions, for monolithic ceramics it is clear that the fundamental strength and fracture properties of such polycrystals should also exhibit a wide range of values. The analysis of crack advance and specimen failure

Present address: Rensselaer Polytechnic Institute, Department of Materials Engineering, Troy, NY 12180-3590, USA 0921-5093/96/$15.00 © 1996 - - Elsevier Science S.A. All rights reserved SSD1 0921 - 5093(95)09988-3

in any real material necessarily involves an evaluation of the stress state that exists in the neighbourhood of a crack tip as well as the various levels of fracture resistance which the crack tip encounters. Cracks may propagate through polycrystals in both transgranular and integranular modes. This of course depends on the corresponding toughness values of the grains and the grain boundary interfaces. There are several models investigating the fracture process from a deterministic viewpoint, that is considering some specific local configurations [5]. However, it seems useful in order to approximate actual material fracture more closely to consider also random, rather than deterministic crack tip positions with various levels of interracial toughness. Previous researchers have also considered statistical analysis of crack advance in brittle materials. For example, Chudnovsky and Kunin have addressed the problem of a crack growing in a brittle solid which contains randomly fluctuating strength properties [6]. While this work is informative it is not well suited to polycrystalline ceramics, and it is not aimed at predicting or interpreting quantities which are readily measured experimentally. Another work by Michopoulos

P. Lipetzky, W. Kreher / Materials Science and Engineering A205 (1996) 110-116

and Theocaris analyses both the deterministic and statistical aspects of crack propagation [7]. Again, their comprehensive analysis is very valuable, although it is not aimed at interpreting the early stages of crack advance and fracture. Furthermore, there are several investigations where the fracture process in heterogeneous materials is analysed from the viewpoint of scaling phenomena and fractal properties [8]. However, at present the physical interpretation of parameters used in those theories is still in an unsatisfactory state. In a very recent work specifically concerned with alumina, Kim and Kishi performed an analysis of polycrystalline fracture based on both grain boundary and crystal toughness [9]. Their model contains both deterministic and stochastic elements. That is to say, the crack which they consider is able to depart at random from its initial plane, although it is forced to cross a regular array of identically sized and shaped grains. An investigation of this sort reveals a great deal about crack propagation in polycrystals, but certain aspects of the real fracture process remain to be addressed. The current investigation is therefore undertaken to augment the existing literature by statistically determining the effects of interfacial toughness on early stages of fracture in alumina or other similarly brittle, polycrystalline materials with randomly oriented and sized grains. Consider now some experimental results which point to the need for a clear understanding and quantification of fracture in polycrystals as affected by grain boundary toughness. An excellent example exists for the case of indentation cracks. Indentation techniques have been thoroughly analysed in recent years as a means to determine not only hardness and elastic modulus but crack initiation and propagation characteristics as well [10 13]. As applied to alumina these methods must be critically examined. For example, fracture features and crack paths can be affected by grain size, although evidence exists both in support of and in opposition to this conclusion [14,15]. Other researchers have studied alumina and several similar materials and concluded that grain boundary strength also plays a significant role [16-19]. Of equal importance is perhaps crack velocity-dependent fracture features [20-24]. The apparent inconsistencies are all certainly linked to the fact that compositions and processing methods were not fully consistent. In addition, there is strong evidence to support an R curve behaviour, and thus crack length also becomes a variable [25]. These investigators could all benefit from a clear, quantitative understanding of the effects of grain boundary toughness on overall fracture characteristics.

2. Model and method Crack propagation in real materials is influenced by

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both random and deterministic quantities; thus, for a model to be appropriate, it must also account for these factors. In the current investigation, we therefore examine the simplified case of a through, planar crack which crosses many microstructural features and intersects a grain with a random position and orientation as shown in Fig. 1. The local deterministic interaction of the crack tip with the individual grains is analysed in terms of the crack driving force and the fracture resistance. According to these assumptions, the driving force is given by the energy release rate G at the tip of a straight crack with a virtual crack increment inclined at an angle 0 with respect to the original crack plane. The fracture resistance in the Griffith sense is given either by the fracture surface energy Fg of the grains or the interfacial fracture energy Fint. Therefore our fracture criterion is

G(O) >~F(O)

(1)

for - n < O < n where F ( O ) = F i n t if 0 = 4 ~ or 0 = ~ - n, or otherwise F = Fg, where the local orientation of the grain boundary with respect to the crack plane is denoted by ~. Possible values of • will be discussed below. The resulting crack propagation direction is thus determined as the angle 0 at which the energy release rate exceeds the fracture resistance (Eq. (1)). The question which can now arise is, what is predicted to occur if there is no unique value of 0 to satisfy Eq. (1)? In the present Monte Carlo simulation, a given crack tip-grain realization, specified by Fg, Fin t and ~, is subjected to incrementally increased external loads from a level at which no crack propagation is possible. Therefore, Eq. (1) is forced to have a unique solution 0 for every realization of 4~. In order to determine the exact crack driving force for a general polycrystal, it is reasonable to neglect the elastic heterogeneity so that the energy release rate can

Crack

Fig. 1. Schematic of the randomized, two-dimensional model. The crack intersects an irregularly shaped and sized grain with virtual crack extension angle of 0 and grain boundary orientation 4~. Average grain diameter is 2R and average edge length is R.

P. Lipetzky, W. Kreher / Materials Science and Engineering A205 (1996) 110-116

112

100

be obtained from the expression for a straight crack in a homogeneous material [26]:

~+l G(O) = ~

3+

cos2

0)\77-o)

(1+3c°s2°)

o

g

_-

where k = 3 - 4v for plane strain, v is Poisson's ratio,/t is the shear modulus, and 0 is the angle measured from the crack plane. In our case,

a

- ~ =30J/m at 2 ~nt 70 J/rn =

K l = K~ = ae(na/2) 1/2

where a e refers to the externally applied, mode I uniaxial stress. Although residual stresses that arise in the material as a result of thermal expansion anisotropy may influence the energy release rate, they will be ignored in this case because in fine-grained materials such as alumina these stresses (typically of the order of 150 MPa) together with the small grain size of less than 10 ~tm give only a negligible contribution to the local stress intensity [27]. However, in two-phase materials the effects of residual stresses have been shown to influence macroscopic fracture resistance significantly [28,29]. The knowledge of internal stress intensity factors and energy release rates is the first step toward the ultimate goal of characterizing the probability and nature of crack growth, but an appropriate determination of the fracture resistance is also necessary before the calculations can begin. The statistical distribution of the angle ~b, which subsequently determines F(O), must therefore be defined. If we assume that the grains resemble regular hexagons with a typical diameter of 2R, then the grain boundary elements in two dimensions have equal length R. If the macrocrack hits a grain with a random orientation and position, the probability of hitting a grain facet oriented at an angle ~(0 < • < n) is simply determined by the projected length of the grain boundary element perpendicular to the crack plane. This is given by R sin ~b so that the normalized probability density is given by p ( ~ ) = ½sin q~

(2)

The evaluation of the fracture criterion within the statistical model will be performed by using a Monte Carlo simulation in which many realizations of the local geometry (described by ~) are calculated using a random number generator and taking into account the probability density as given by Eq. (2). Simply worded, this method states that if grains were round in two dimensions every point on the profile could be struck by the crack with an equal probability. The crack extension probability is then calculated as the percentage of total realizations which satisfies the inequality G(O) >~ F(O), and the crack is classified as interfacial or transgranular if it can propagate. Crack extension probability is then correlated to the applied load in order to determine the macroscopic toughness.

0

20

40 60 G e (J/m2)

80

I00

Fig. 2. Failure probability plotted as a function of externally applied energy release rate for three levels of interracial fracture resistance.

Given the assumptions and simplifications as above, a realistic look at a ceramic microstructure is now useful. It is well known that features exist which are quite non-ideal, such as residual glass phase on grain boundaries and triple points as well as voids and microcracks. Furthermore, it is not unusual for individual ceramic grains to have elastic and fracture resistance anisotropy. These apparently neglected features are certainly important and are indirectly accounted for in this model. For example, if a glassy phase in the form of residual sintering additives is distributed in the material, the average grain boundary fracture resistance will be affected. Average /'int is clearly accounted for here. Similarly, if the material contains microcracks or voids, these will also be absorbed in the average /'int factor. With regard to individual grain anisotropy, an actual crack which extends through the thickness of the sample will encounter a global, macroscopic crack resistance which is also considered here, in /'g. Finally, as seen in Fig. 1, it is possible that the crack may intersect a triple grain junction rather than a grain boundary. In such a case, F(~b)~ F ( ~ - n ) as assumed here. The frequency of this event is clearly much lower than the event depicted in Fig. 1 and the nature of such an intersection is not fundamentally different. Therefore such events will not alter the results given here significantly. Again, for purposes of wide applicability, no more specific assumptions are made.

3. R e s u l t s

Crack advance or failure probability as defined above is the probability that a given value of applied ce (and the corresponding K~ or G e) will result in an energy release rate greater than the fracture resistance considering any possible grain boundary orientation. It is understood to be a "failure criterion" because it corresponds to the onset of crack advance. Fig. 2 shows the

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failure probability as a function of applied G e for the cases of Fint = 70 J m - 2 and 30 J m -2 and 10 J m -2. The maximum value of 70 J m -2 is chosen to equal the fracture resistance of the grains. Decreasing the fracture resistance of the interface necessarily influences the failure probability of the polycrystal by opening avenues for crack advance at energy release rates lower than the critical values for the constituent grains. Fig. 2 shows quantitatively that decreasing the crack growth resistance of the interface increases the probability of failure at a given load. These results can be correlated with the macroscopically observable toughness by two methods. The first method is to calculate the probability density function p(~r') based on the relation P(a) =

fo

p(o') da'

where P(~r) is the failure probability as plotted in Fig. 2 for the full range of applied K~ (applied stress). The second method is to calculate the probability density function directly by fixing the grain boundary orientation, realization by realization, and increasing the applied stress until the energy release rate fulfils the failure criterion in Eq. (1). The first-order moment of the probability density function is therefore equal to the average macroscopically observable toughness, or critical applied energy release rate, for a given value of interfacial toughness Ge(/'int), as plotted in Fig. 3. Notice that a G e can be applied that is exactly equal to the critical energy release rate of the components prior to the onset of crack advance when Fin t ~---/'g, as must logically be the case. As shown, the average G e is a strong function of interfacial crack growth resistance until the interfacial toughness exceeds roughly 50% of /'g.

As with any model, the usefulness of these calculations lies in the predictive capabilities. In that sense, it is desirable to calculate experimentally observable

soI 70

60 50 40~ 30 i

20:

f

100

0

:::::::::::::::::::::::::::::::::::::::::

10

20

30

40

50

60

70

80

Fint (J/m2) Fig. 3. Average macroscopic toughness of the modelled polycrystal plotted against interfacial fracture resistance. A weak dependence of overall toughness is seen if /'int ~> l/-g.

,_ 1 0 0 ~ - - . ~ . .

=

"',,

80.

~xx ' \

~ eo. 4020-

o

0

. ---,. -

10

Krell & Blank

- ~

\

Kim&Kishi

20

30

~

=.

40

.

.

.

50

.

.

60

70

. . . .

80

Fint (J/m 2)

Fig. 4. PIF plotted for both analytical and statistical models. Literature results are in good, general agreement with the results calculated here. quantities. The amount of interfacial failure is a logical quantity because fracture surfaces are often characterized by examining the relative amounts of interfacial vs. transgranular fracture. As seen in the model details above, a crack must necessarily be classified as interfacial or transgranular depending on where the fracture criterion is satisfied with respect to the grain boundary orientation. Utilizing this model, the calculated percentages of interfacial failure (PIF) as a function of Fint are plotted in Fig. 4. Additional curves are plotted from the literature and will be discussed below. In addition to the amount of interfacial failure, it is also of interest to determine the topology of the crack path as an indication of possible fracture surface roughness. As shown above in Fig. 1, the crack is assumed to intersect the edge of an irregular polygon with an average diameter of 2R and average edge length of R. If, according to our fracture criterion, the crack advance is classified as transgranular then it continues to propagate in its plane an average distance of 22R until it hits the next self-similar grain boundary. This corresponds to a mean intercept length as calculated using the stereological approximation that all grains are spheres. If the crack propagates in an intercrystalline manner, we assume for simplicity that the average projected crack length is again ~2R between the point of initial deflection and the point where the crack hits the next subsequent grain boundary. The deflected crack propagates in the interface an average distance of ½R and the next segment of the crack is parallel to the original crack plane. Therefore an average distance of ½R sin q5 exists between the "original" crack plane and the plane of the parallel section in the next crack advance increment. (Use of the normalized probability density function, Eq. (2), with the two approximate grain geometries, hexagons and spheres, shows that the overall average crack deflections are (7c/8)R and ½R respectively. This small differ-

P. Lipetzky, W. Kreher / Materials Science and Engineering A205 (1996) 110-116

114

12.0 10.0 8.0 6.0

•£

4.0

() (D

2.0

Fint = 60 J/ms

0.0

"(3 ~P

.N

~

-

"

~

"

-

"

. -

-

~

-2.o

O

E

-4.o

E

-6.0

k_ 0

-8.0

-I0.0 -12.0

0

50

100

150

200 crack

250

300

350

400

450

500

propagation steps

Fig. 5. Calculated results for fracture surface profile shown for three levels of/'int- Crack height relative to the initial crack plane is normalized by the average grain radius R and the average "crack step" length is ~2R. Surface roughness is indicated by R S = 1.13, 1.02 and 1.00 for /'int = 20 J m -2, 40 J m -2 and 60 J m 2 respectively.

ence is within the limits of the other model approximations.) Examples of crack path segments over many grains for Fi,, = 20 J m -e, 40 J m-2 and 60 J m -2 given these assumptions are plotted in Fig. 5. The corresponding surface roughness parameters Rs are also given as 1.13, 1.02 and 1.00 respectively with Rs defined as [30] 4 R s = -~z (R

L --

1) + 1

where RL is the actual crack length as sketched in Fig. 5 normalized to the projected crack length per grain. In addition, as an indication of fracture surface roughness, the calculated standard deviations for these curves are about 4.4R, 0.93R and 0.34R respectively.

4. Discussion

The significance of the results given here is that the trends shown in Fig. 3 and 4 may reasonably be combined to predict an optimum grain boundary toughness, which is controlled by composition, for a given application. For example, changing the amount of interfacial or intergranular additives can alter Fi,, and PIF (crack deflection) which has clear implications for fracture surface roughness and macroscopic crack resistance. Although only a simplified model is used here, Fig. 5 shows that, for various /'int values, Rs can

be significantly influenced. Therefore decreasing F i n t and the promotion of crack deflection may be necessary if higher toughness is desirable [31-34]. In fact, careful experimental observations have shown that crack deflection is necessary for the activation of bridging mechanisms [35]. However, decreasing F i n t t o o extensively sharply lowers the average G e that a specimen can sustain before crack advance begins. As shown in Fig. 3, significant global weakening occurs if interfacial fracture resistance decreases below about 30 J m -2 or 1 about ~Fg. In contrast, Fig. 4 reveals a significant increase in PIF as interfacial toughness decreases from 1 Fg to about aFg. These two factors must be considered together owing to their offsetting effects. The curve given in Fig. 3 can thus be seen as a first approximation of a lower bound because other toughening mechanisms such as crack deflection, crack bridging and process or wake zone microcracking will only increase the values calculated here. However, this analysis strictly applies only for quasi-static crack growth which is not influenced by environmental effects or crack growth kinetic effects, such as in the initial stages of crack advance and failure. The results given here will now be compared with other numerical models in order to verify the accuracy of this statistical theory. Fig. 4 shows the variation in the amount of interfacial fracture predicted here compared with that given by Krell and Blank using an analytical model proposed by Cotterell and Rice

P. Lipetzky, W. Kreher / Materials Science and Engineering A205 (1996) 110 116

[18,36]. Although the approaches are very different, the agreement is quite good. This comparison is of particular interest because both models apply to a long, straight quasi-static crack with a small increment of the crack tip extending out of the main crack plane in a homogeneous medium. Further comparison with the data of Kim and Kishi is also shown [9]. Their results TtT for a transgranular crack approaching a grain boundary as in Fig. 1 exhibit a percentage of interfacial failure which exceeds that calculated by both the current model and Krell and Blank. This is perhaps a result of the geometrically regular arrangement of identical grains assumed in their model or an effect of the third dimension. However, the model of Kim and Kishi contains several ambiguous approximations which are not clear in the context of the current work. While it is difficult to determine which results are correct without experimental verification, the three-dimensional model results show that the PIF jumps from 0% to about 40% for only a 20% decrease in Fiot, which seems intuitively rather high. Experimental verification of the current results is not as simple as the preceding numerical comparison. As stated above, experimental results for factors such as PIF are dependent on many variables: environment, crack velocity, grain size, cracking technique, material composition and sintering conditions. Such results are often in apparent contradiction. Finding the data to fit our model should therefore be possible but would not be convincing in a general sense. For example, Kim and Kishi also display a comparison between numerical and experimental results but point out that slow crack growth and specimen geometry influence these results [9]. The current calculations for G e, PIF, R s and standard deviations of the crack level as functions of/'int as in Fig. 3, 4 and 5 respectively are therefore best viewed as definitive, quantitative trends rather than absolute values which will fit every possible material and test condition. For example, Rs and the above-mentioned standard deviations are indicators of fracture surface roughness which subsequently affects crack growth resistance, although no general correlation has been established between these factors in all materials [31 341. The final point of analysis for this model is in regard to the location of the crack tip which has been assumed to end in its weakest configuration, at the grain boundary interface as shown in Fig. 1. Physically, the crack tip may be in a grain or on the interface. Notice that when the crack ends in a grain, the most stable configuration, it will not advance if the driving force is less than 70 J m 2. However in a statistical sense through the thickness of the sample many such "'weakest link" configurations will exist. When the sample and thus the entire crack front are loaded to a given critical energy release rate, the portions ending within

115

grains will remain stationary while the weakest segments will fail. This initiation of crack front propagation will effectively "unpin" the crack so that crack segments which end in grains very near interfaces will propagate into the boundary and further catalyse crack advance. The configuration considered here is therefore the most critical.

5. Summary and conclusion The stochastic analysis of crack growth presented here shows that macroscopic failure analysis of heterogeneous materials cannot be complete without considering microscopic crack advance as affected by both local and global variables. Fluctuations in the fracture resistance of individual components and interfaces have been quantitatively linked to macroscopic toughness. Results show that the effective overall toughness is at least as high as 85% of the component toughness until the critical energy release rate of the interface decreases to less than roughly 50% of the component crack growth resistance. This decreased overall toughness is directly attributed to the effects of the lower interfacial critical energy release rates. The results given here can be viewed as an effective lower bound to material toughness because many active toughening mechanisms, such as crack deflection and branching, have been neglected. However, potentially counteracting factors which will act to destabilize the crack such as kinetic and environmental effects are also being neglected. To verify further the results of this model, materials (not necessarily alumina) with a known interfacial fracture resistance should be tested under conditions of stable crack advance and the resulting fracture surface analysed.

Acknowledgements P.L. gratefully acknowledges financial support as a visiting scientist from the Max-Planck-Gesellschaft and Professor E. Arzt. We also thank Dr. A. Krell for many helpful discussions.

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