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Cracking at grain boundaries in polycrystalline brittle materials
WO~-~~~O/S~/~O~~J~S-O~SIZ.OO,~~ 0 Copyright 1981 Pergmon Press Ltd
Alw Ml%,llurglc,: Vol. 29. pp. 16% 10 1702. 1981 kinwd in Gent Britain. All rights rcrerved
CRACKING AT GRAIN BOUNDARIES POLYCRYSTALLINE BRITTLE MATERIALS
IN
R. W. DAVIDGE Materials Development Division, Building 552, AERE Harwell, Oxon. OX1 1 ORA, U.K. (Received
24 February
1981)
Abstract-The cracking of grain boundary facets in polycrystalline materials showing anisotropic thermal expansion behaviour is controlled by several microstructural factors in addition to the intrinsic thermal and elastic properties. Of specific interest are the relative orientation of the two grains meeting at the facet, and the size of the facet; these factors thus introduce two statistical aspects to the problem. The criteria for facet fracture are critically reviewed. The statistical factors are then introduced to give quantitative data on crack density vs temperature. The theory is compared briefly with limited experimental measurements of Young’s modulus for various rocks as a function of temperature. R4aum&La fissuration des facettes intergranulaires dans les materiaux polycristallins B dilatation thermique anisotropc est contr&e par plusieurs facteurs microstructuraux, en plus des prop&&s thermiques et tlastiques intrins&ques. L’orientation relative des deux grains adjacents sur la facette et la dimension de la facette jouent un r6lc intCressant; ces facteurs donnent une double dimension statistique a cc problbme. Les critbres pour la rupture aux facettes sont pa&s en revue de man&e critique. Les facteurs statistiques sont alors introduits pour fournir des don&es quantitatives sur la densit de fissures en fonction de la temp&rature. Nous comparons rapidement cette thCorie avec quelques mesures exp&rimentales sur la variation du module d’Young de diverses roches en fonction de la temptrature. Zusammenfaasuag-Die RiDbildung an Korngrenzfacetten in polykristallinen Materialien, die anisotropes thermisches Ausdehnungsverhalten aufweisen, ist zusiitzlich zu intrinsischen therm&hen und elastischen Eigenschaften durch einige mikrostrukturelle Factoren bestimmt. Von besonderem Interesse. sind die gegenseitige Orientierung der beiden an der Facette .zusammenstoBcnden Khmer und die GrbBe der Facette. Diese Faktoren bringen zwei statist&he Aspekte in das Problem. Die Bedingungen Rir Facettenbruch werden kritisch zusammengestellt. Die statist&hen Faktoren werden dann benutzt, urn quantitative Angaben iibcr die RiBdichte in Abhingigkeit von der Temperatur zu machen. Die Theorie wird mit den bcgtenzten experimentellen Messungen der Temperaturabhangigkeit des Elastizitiitsmoduls verschiedener Gesteine kurz verglichen.
1. INTRODUCTION The cracking that occurs at grain boundaries due to anisotropic thermal expansion in polycrystals has received much experimental and theoretical attention over the last two decades [l-S]. In hexagonal single phase polycrystals for example, such as alumina or beryllia, where there is differential expansion between the a- and c-axes, stresses and strains are set up around grain boundaries and these increase as the temperature falls from the initial fabrication temperature. The maximum differential grain boundary strain E,,,~~is simply proportional to the maximum difference in expansion coefficients (Aa,,,,,) times the temperature difference (AT) between ambient and that below which plastic effects become negligible (often assumed * 1ooo”c). t.1
= Aa,,,.,AT
(1)
For the complete fracture of a single grain boundary facet two conditions must be satisfied. The elastic strain energy round the facet must at least equal the energy required to fracture the boundary: and a flaw
of sufficient severity must be present to give a high enough stress concentration for fracture to proceed. Either condition may be critical. There is general agreement that for a given AT a critical facet size is
where yti is the grain boundary surface energy and E is Young’s modulus; K is a constant. Facets with 1 2 1, may fracture depending on the relative orientation of the adjacent grains. Because of the practicalities of the situation, experimental observations are usually concerned with the observation of microcracks in various materials in the as-fabricated condition at ambient temperatures. Preparation of samples with a range of controlled grain size can thus lead to the identification of the critical grain size for which fracture is observed for specific materials. The experimental data show that equation (2) is of the correct qualitative form. Quantitative interpretation is however difficult because neither y,* nor the initial flaw size are known pre-
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cisely. Similar considerations should apply to assemblages of multiphase poiycrystalline ceramics, where the different phases will have different expansion properties in addition to individual phases showing anisotropic behaviour. The aim of the current paper is to develop some aspects of the problem in more detail including the energy available for cracking; and grain orientation and size effects on a statistical basis. In this way predictions can be made about the quantitative effects of cracking on mechanical properties. The new theory is compared with experimental data for the variation of Young’s modulus with temperature for rocks; good agreement is obtained, but more detailed experimental data are required to give a critical comparison. 2. FRACIIJRE AT A GRAIN BOUNDARY FACET
x
We will reconsider first the original model due to Clarke [I], on which was based a two-dimensional fracture mechanics calculation. Whilst this is valid in its essential features there are a number of problems associated with it: the model is only two dimensional; the fracture mechanics criterion developed is not the critical one for complete fracture of a facet; and a more accurate approximation to the stress distribution of the stress round the boundary may be used. 2.1 Energy balance calculations for complete propagation ofa facet crack The model is shown in Fig 1 for a facet of average
lb)
Approximation
Fig. 1. (a) Model of grain boundary facet. (b) Stress distribution in xy section for u, as function of distance from xz (facet) plane. Equating equations (4) and (5) gives for the critical facet size 1, 1c=
88y,,(l IT-2
- Y’)~ (6)
diameter 1.The stress in the x direction (03 falls with For the two dimensional case the equivalent exincrease in distance from the boundary, and Clarke followed the data of Nye [t?rlfor the analogous case of pression is identical to [6]. the distribution of stresses round a glide band, Fig. 1. 2.2 Fracture mechanics calculations for initiation of Clarke assumed that the stress fell linearly with infacet crack crease in y to zero at y = l/2, but reexamination of In the fracture mechanics approach for crack inNye’s graphs shows that a much better approximation is a linear fall in stress to zero at y = 0.2751, itiation used by Clarke, the equivalent expression for 1 is N 24y&e2. The modified stress expression used in bearing in mind also that the strain energy is proporthe current work would give I- 44y&‘. The imtional to the square of the stress. This ideal stress distribution must be disturbed near the three-grain portant point however is that the crack initiation crijunctions but as this will vary widely between juncterion is not critical compared with the above crack tions the simple stress distribution only will be conpropagation criterion, i.e. initiation is easier than sidered. The maximum stress at the facet boundary, propagation. The more recent calculations of amX is Evans [S] for fracture initiation involving a more Ekmsx complex two-dimensional model gives 1% 32yJfi2, a,,=-. (3) in reasonable agreement with the modified Clarke cal2/l - A’ v is Poisson’s ratio. culation above. Realistic models for crack initiation in The elastic strain energy (V,) in three dimensions the three-dimensional case are still to be developed associated with the facet is thus contained in an but the energy release rate for cracks that are small approximately cylindrical volume of diameter 1 and compared with the facet area is likely to be rather less than that for the twodimensional models because of height 0.55 1. the greater constraint of the surrounding material Ill2 0.55 1 Ee,&,n13 u,=-_.-= (4) (and lower release rate of energy). v2)2. 2E4 3 176(1 2.3 Conclusion The energy required to fracture the facet (V,) is
u:,..
” = 2Jr1*y,* I -. 4
(5)
Until three-dimensional crack initiation analyses are produced the crack propagation criterion [6] will
DAVIDGE:
CRACKING IN BRIITLE MATERIALS
Fig. 2. Variation of thermal expansion coefficient with angle 0.
0
15
ANGLE
be used to predict the critical condition for cracking. Implicit in this assumption is the presence of a suitable crack source at the facet. It is interesting to note that for the related case of separate spherical particles in a matrix the total energy balance approach [7] gives fortuitously the same limiting criterion as a finite element fracture mechanics calculation for fracture initiation [8]. This could prove true also for grain boundary facet fracture. 3. GRAIN ORIENTATION
EFF’EC’I-S
For illustration we discuss materials typified by the hexagonal crystal structures where the coefficient of thermal expansion in any direction of the basal plane is a, and in the orthogonal direction a, (The theory can however be readily extended to other crystal structures where for example the expansion along three crystallographic axes may be different.) The expansion coefficient a, at an arbitrary angle w to the c-axis in a section normal to the basal plane is [9] a,
= a, cosb
+ ar, sin’
0 = a, + (a,
-
a,bs2
0,
(7)
as shown in Fig. 2. Consider two grains joined at a grain boundary facet. In general the c-axes of the two grains lie at
1697
30
L5
BETWEEN
60
C-AXES
75
OF GRAINS1
90 0 I
Fig. 4. Relative magnitude of maximum grain boundary strain as a function of orientation of two grains with c-axes in the facet plane.
arbitrary angles /I and y to the facet. For simplicity we examine first the case when the c-axes he in the facet plane, i.e. B = y = 0, Fig. 3. The acute angle between the c-axes is 8. The direction of the maximum strain corresponds to the maximum separation of the ellipses at symmetrical positions defined by o* in Fig. 3. The value of ce+ is found by maximising 1Aal = [a< cos2 o + a,, sin’ o] - [aC cm2(w + a,
+
0)
sin2 (0 + e)]
= (a, - a,)[cos2
w -
cm2
(0
+ O)].
(8)
Differentiating with respect to w gives
4Aal -
do
= (a, - a,)[sin 20~ - sin 2(w + e)];
(9)
equating to zero gives o* = (4S-f7/2), which locates IAal,.. The variation of the relative magnitude of the maximum strain with 0 is shown in Fig. 4. For the general situation when the c-axes of the two grains are not ‘in the grain boundary plane the two ellipses representing the thermal expansion in the boundary plane are of different size. The minor axes are always qua1 to a, (there must be an ‘a’ direction in any arbitrary plane) and the major axes vary between a, and a0 Fig. 5. The values of the maximum expansion coefficients in the grain boundary plane are given by: %,X1 = a,cos2/3 + a,sin2/I = a, + (a, - a,Jcos2 G,.
2 = a, cos2
y + a,
sin2 y
= a, + (a, - a,)cos2
Fig. 3. Orientation dependence of the differential thermal expansion coefficient (Aa) for two grains with c-axes in the grain boundary facet plane.
/3
y.
(10)
The case /I = y = 0” is that considered above, while for B = y = 90” (c-axes normal to the facet) there is zero grain boundary strain. The difference in expansion coefficients in an
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DlFFERENTlAL THERMAL EXPANSION I ANI~~PY FACTOR.A 1
Fig. 5. Orientation dependence of the differential thermat
expansion coefficient for two grains of random orientation. arbitrary direction o from the e,,,, a direction is now &xl=
[~rcos~W+a~sin~W] - [a& 1 coS2(W + e) + a, sin2(o + @J = (%NX1- tr,)cos~ u - (64,* - @&os2(0 + ff).
Substitut~g gives IhI
for a_,
and h,
(11)
2 from equation (10)
= (a, - LYJ[cos’ /? co9 w - cosa y cos2(0 + @J.
Differentiating as before shows that &,, sin 2.oP -2--a, sin2(w*+8)=~,1-a,=cos2
(12) occurs when cos2 y
(13)
For a particular comb~ation of &7,y and 8 it is now possibie to calculate IAal,. The value of o* is first obtained from equation (13) and then substituted in equation (12) to give MaI,,. Although fl, y and 8 can take any value, not all values of /I and y are equally likely. The c-axis of one gram can be regarded as a vector as the radius of a hemisphere with the diametral plane coincident with the grain boundary facet, Fig 6. The end of the vector is equally likely to lie at any position on the hemisphere. Thus for a given value of fl, the relative probability that the vector is at this angle is proportional
Fig. 7. Relative probability for the range of possibIe values for differential thermal expansion.
to 27rRcos jl/2nR = cos /?, (and similarly for the angle y for the second grain). Each value of IA&,., calculated as above thus needs to be associated with a probability frequency factor F = (co@cosyE. To estimate the statistical variation of values of I~al, over all values for /3, y and 8 we can thus proceed as follows. Random values for j?, y and 8 are generated and o*, corresponding to @a/-, calculated from equation (13). Vahtes for lA!xl,from equation (12), are expressed in terms of an anisotropy factor A = [cos*&os”w - cos2vcos’(o + @J which can vary between 0 and 1. Each value of A is then allocated to one of fifty intervals 0 to 0.020.02 to 0.04 etc. The values of the probability f’requency factors F for each of the A values in each interval are then summed to give a histogram showing relative probability P for the grain boundary n&orientation for the range of A values. Figure 7 shows data for 100,000 random ~mbinations of 4, y and B. The si~ifi~nt results are that the highest value for P (P,,,*,J is associated with the maximum value of A N 1; and that fortuitously, P falls linearly from P,,,_ at A = 1 according to P = P,,(O.13 + 0.87A);0.35 > A > 1.00
This conclusion reflects the size of the intervals chosen for A, 0.02. Clearly P must fall to zero at A = 1 and the precipitous drop in P is indicated in Fig. 7. However, a more detailed analysis for much smaller intervals for A indicates that the drop in P occurs very close to A = 1 at A -+ 0.999. This ultimate deviation from linearity is thus real but of only academic interest. Finally, there are minor inffexions in the curve for 0 > A > 0.35.
4. EmCTS
Fig. 6. Possible positions in space of a c-axis vector R.
(14)
OF GRAIN SHAPE AND SIZE VARIATIONS
The average polycrystalline solid contains grains with a wide variation in geometrical form and size. The number of facets on individu~ grams is typically P-18, with the number of edges per facet typically
DAVIDGE:
CRACKING
IN BRITTLE MATERIALS
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The critical failure criterion is given by equation (6) where the strain is defined in equation (1). Generally however the strain is less than the maximum value, according to the factor A (equations (12) and (14) and Fig. 7) , = 88 Ygdl - v2Y E(AAa,,,AT)* ’
(16)
It is convenient to consider the ratio I//,,,,, = I* and the ratio AT/AT,,,, = T* where ATmi,,is the minimum temperature change to fracture the largest facet (I,,,,,) with the maximum misorientation of neighbouring grains (A = 1). Note that Fig. 8. Distribution of facet sizes in polycrystalline magnesia. The numbers on the graph indicate the percentage of pA=T*l
facets larger than the size indicated.
3-8 [lo]. The grain size may vary by one or two orders of magnitude. Much useful information about the distribution of grain sixes and shapes is given in the papers by Aboav and Langdon [ll] who analysed large numbers of grains in magnesia. Their data showed that a normal Gaussian distribution was obtained when the number of grains of a particular diameter was plotted against the square root of the diameter; this held also for the variation in the number of grain sides for a particular side length plotted against the square root of the length of the side. Roth distributions have the same mathematical form given by: N = N.exp{
-ci[(3’”
- 111)
(15)
where N is the number of grains or sides of length x within a given range. x,,, is the value of x when N has the maximum value N, a is a constant. For a particular dense material of grain size w 10 w a was 243 for grains and 2.05 for grain sides; the average number of sides per grain was 5.85 and thus the grain size is close to twice the facet size. (The factor of two difference would be expected for hexagonal grain sections.) These data are replotted on a linear scale in Fig. 8 which shows the equivalent distribution of facet widths expressed in terms of the most frequent grain size (e) of the material. The most frequent facet size is half 6 and the largest observed facet size is twice 6. Care is needed in defining average grain size. For magnesia the median grain size is 5 1.1 G, the mean grain size 5 1.3 6, and 50% by volume is occupied by grains smaller or larger than _ 1.8 6. 5. CRACK DENSITY As A FUNCI’ION OF TEMPERATURE
We are now in a position to combine the effects described in the previous sections and predict how the number of cracks will vary with temperature. A.M. 29110-c
=
88 jb*(l - v2)2 E(ALY,,,A.T,~,)’I,,
= I’
(17)
This ‘first-cracking criterion’ (T* = I* = A = 1) is the one generally considered in the literature, but our interest lies in values of T* > I where both smaller and non-ideally oriented facets may fracture. The variation of facet size is given in Fig. 8 and the distribution of A in Fig. 7. The general situation is thus that cracking occurs at a small proportion of small facets and an increasing proportion of large facets. The number and size of cracked facets can be calculated as follows. The facet size distribution graph is sectioned into a number of equal sectors containing a specific proportion of facets, say 1 or 5%. The facets in each sector are treated as all having a size at the minimum of the sector boundary. Table 1 illustrates the details of the calculation, using ten sectors, corresponding to potential cracking of loo/, of the facets. From Fig. 8, the largest loO/, of facets have P > 0.590; for A = 1,1*T*2 = 1 and this corresponds to T* = 1.301. Elecause all facets in the tenth sector are assumed to have I* = 0.590, then none will crack because the proportion of grains P at A = 1 is zero (Fig. 7). To estimate the number of cracked facets in the other sectors it is convenient to replot the data in Fig, 7 on a cumulative basis starting from large values of A. Consider the first sector where p = 0.825; cracking occurs when T* > 1.101 which is 0.846 times the actual value 1.301. Thus all facets with A > 0.846 will crack which from Fig. 9 gives a fraction of 0.265. The cracked fraction of facets decreases with decreasing facet size and the numerical data are in Table 1.0.112 of these largest facets are cracked but this is only 0.011 of the total. Equivalent calculations can be performed to cover the whole range of facet s&s and for this case 20 sectors of 5% were chosen. Figure 10 and Table 2 give the data for each sector in terms of the threshold condition. For example under conditions to fracture potentially half the total facets (up to sector 10) 0.53 of the facets in the first sector have fractured and summing for each sector shows that 13.7% of all facets have fractured.
DAVIDGE:
CRACKING
IN BRITTLE MATERIALS
IFRACTION
THRESNU.0 CCfdOlTlON - OF FOTEMIALLY CRACKED FXETSI
-
Fig. 10. Cracked facets in each sector for sector 1 (upper curve) to sector 19 (low curve) for each threshold condition.
facets per unit volume. Generally, this information is not available but to illustrate the principles involved a calculation is performed below on the assumptions Fig. 9. Data from Fi8. 7 replotted on a cumulative basis. that the grain and facet sixe distributions are those observed by Aboav and Langdon [ll] in magnesia. 6 EFFECTS ON PROPERTIES Since these are controlled primarily by geometrical The above data are essential for predicting the considerations this should represent a reasonable effects of cracking on mechanical and physical general example. properties such as Young’s modulus or thermal conThe magnesk data show for example that for ductivity. Roth these properties vary with cracking in material with G = 10~ (i.e. a mean grain size of a similar way and the equation for Young’s modulus 13 /.@ the total grain density is 6.24 x lOi* m-’ if assuming a random distribution of non-interacting the grains are assumed to be spherical. Thus taking cracks is 1123 the average number of facets per grain as 14 (each Eo shared between two grains)the facet density is (18) E = 1 + 16Nb3/9 4.37 x 10’5m-3.ThedatainFig. lOandTable2can where E. is the modulus of untracked material, N the now be used to calculate Nb3 for each facet size number of microcracks per unit volume and b the sector for various threshold conditions, and details are given in Table 2 for the threshold condition to crack radius The calculation of N is straightforward only when potentially crack half the facets. Here ZNb3 = 0.094 all cracks are of the same size. For real micros&tic- and E = 0.857 Eo. Note that the hugest facets have tures a full analysis of grain size and facet size vari- the more significant effect. Similar results for all ation is required to give the number of grains and thresholds conditions are given in Fig. 11; the calcula0.6 01 02 ANISOTROPY FACTOR IA I
0
Table 1. Fraction (R)ofcracked facets for ten 1% sectors containing the largest facets. 1
2
3
4
5
6
7
8
9
10
P
0.825
0.769
0.728
0.697
0.675
0.653
0.634
0.619
0.603
0.590
F/P10 R
;E 0.265
0876 1.140 6216
0.901 1 172 0.173
0921 1.198 0:148
0’935 1217 0.117
0’951 1237 0:092
0’965 1255 01062
0.977 1271 6037
?g 0:018
:*g O&O
Sector
Tabk 2. Fraction (R) of cracked faats for 10 5% sectors for a threshold condition to crack potentially half the fasts (sectors l-lo), and calculated values of Nb’ (see Fig. 11). Sector
1
2
3
4
5
6
7
8
9
10
P P P/T*10 R
0.675 1.217 0.666 0.530
0.590 1.301 0.712 0.468
0.534 1.368 0.749 0.413
0.488 1.432 0.784 0.364
0.447 1.496 0.819 0.308
0.413 1.557 0.852 0.253
0.378 1.626 0.890 0.197
Nb’
0.036
0.02 I
0.014
0.009
0.006
0.004
0.002
0.350 1.690 0.926 0.136 0.001
0.325 1.754 0.961 0.074 0.001
0.300 I .826 1.000 0.000 O.lXIO
DAVIDGE:
0'
0
CRACKING
1 2 NORMALISED TEMPERATURE DIFFERENCE
IN BRITTLE
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3 I T *I
Fig 11. Variation in Young’s modulus with increasing temperature difference. Numbers on curve indicate threshold facet condition. Dotted curve indicates expected deviation due to cracks in adjacent facets.
tion does not depend on the grain size, provided that the distribution is that assumed above, because Nb3 is
dimensionl~. The data in Fig. 11 are expected to be accurate only at small crack densities when the cracks are relatively isolated. When cracking occurs significantly for adjacent facets the data underestimate the reduction in Young’s modulus. The dotted curve for example shows the sort of deviation expected if all the cracks for the sector 16 situation (35% of all facets cracked) comprised two adjacent facet cracks. 7. COMPARISON EXPERIMENTAL
MATERIALS
WITH DATA
The theory developed above ieads to two parameters that can be compared with experimental data: the minimum temperature change for the generation of the first cracks, and the rate of crack formation (as measured for example by changes in Young’s modulus) with further change in temperature. The two parameters are not independent and the rate of change in modulus with temperature is proportional to the minimum temperature change as indicated by the normal&I universal curve in Fig 11. The first parameter has received much attention in the literature and the main contribution here has heen to refine the calculations [equation (6)]. The second parameter has not been calculated before and there is a dearth of experimental data for comparison purposes. For ceramic materials, that are fabricated at high temperature, the amount of cracking increases as the temperature falls after fabrication and it is difficult to monitor effects except at ambient temperature. Further data could be obtained however from specimens cooled below ambient temperatures. Comparison between experiment and theory can however be made for geological materials where equilibrium is reached not at a high fabrication temperature but under ambient conditions after geolo~~lly long times. Here cracking can be induced by heating
I OO
I
100
1
L
I
ml
3w
03
TEMPERATURE 1°C 1
Fig. 12. Variation of Young’s modulus with increasing temperature for various rocks. (a) Reserve Taconite [13], (b) Dresser Basalt [13), (c) Jasper Quartzite [13], (d) Charcoaf Granite [133, (e) Cornish Granite [143, (f) Welsh Sandstone [IS].
cooling) samples from ambient temperatures. Figure 12 shows the effect of heating on a range of geological samples. (In some cases, where the experimentar data points are sparse, the graphs have been constructed so that the initial regions comprise two straight lines.) All the rocks featured contain anisotropit minerals such as feldspars and quartz and the granites contain both these minerals. The simple theoretical model presented here can be applied to rocks provided that sufficient ~~os~u~ur~ and physical data are known: for instance the frequency and sixes of facets between like and unlike grains, grain boundary fracture energies and the elastic and thermal expansion properties of the constituent minerals. These data are not yet available but the qualitative agreement between the simple theory and the existing data is striking. It is planned to obtain more detailed data on specific rocks to permit further comparisons. (or
Acknowledgements-This work was done as part of the UK programme of research into the burial of radioactive waste for the Commission of the European ~rnm~ity and we thank the Commission for agreement to publish. Thanks are due to J. R. McLaren and I. Titchell for useful discussions and for some of the experimental data for rocks; and to M. H. Rand and A. Rabinovitch for assistance with the calculations in section 3.
RX~REN~ F. J. P. Clarke, Acru metali.12, 139 (1964). R. W. Davidge and G. Tappin, J. Mater. Sci. 3, 297 (1968). J. J. Cleveland and R. C. Bradt, J. Am. Ceram. Sot. 61, 478 (1978). R. W. Rice and R. C. Pohanka, J. Am. Cenun. Sot. 42, 559 (1979). A. G. Evans, Acra meraif.26, 1845 (1978).
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DAVIDGE:
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6. J. F. Nyz Proc. R. Sot. A 200.47 (1950). 7. R. W. Davidge and T. J. Green, J. Mater. Sci. 3, 629 (1968). 8. Y. M. Ito. M. Roscnblatf L. Y. Chcng, F. F. Lange and A. G. Evans, Int. J. Ftcrcr. 17, (1981) in press 9. J. F. Nyc, Physical Propcrries of Crystals Clarcndon Press, Oxford (1957). 10. C. S. Smith, Sciew Am. 190, 58 (1954). 11. D. A. Aboav and T. G. Langdon, Medogrophy 1,333 (1969); 2, 171 (1%9).
1N BRITTLE MATERIALS 12. D. P. H. Ha&man and J. P. Singh, Bull. Am. Cerum. sot. 5& 856 (1979). 13. C. F. Wingquist, U.S. Bureau of Mints Report 7269 (1970). 14. J. R. McLarcn and 1. Titchell. AERE Harwell, to bc published (1981). IS. I. Titchcll and J. R. McLarcn, Unpublished data (1981).
Alw Ml%,llurglc,: Vol. 29. pp. 16% 10 1702. 1981 kinwd in Gent Britain. All rights rcrerved
CRACKING AT GRAIN BOUNDARIES POLYCRYSTALLINE BRITTLE MATERIALS
IN
R. W. DAVIDGE Materials Development Division, Building 552, AERE Harwell, Oxon. OX1 1 ORA, U.K. (Received
24 February
1981)
Abstract-The cracking of grain boundary facets in polycrystalline materials showing anisotropic thermal expansion behaviour is controlled by several microstructural factors in addition to the intrinsic thermal and elastic properties. Of specific interest are the relative orientation of the two grains meeting at the facet, and the size of the facet; these factors thus introduce two statistical aspects to the problem. The criteria for facet fracture are critically reviewed. The statistical factors are then introduced to give quantitative data on crack density vs temperature. The theory is compared briefly with limited experimental measurements of Young’s modulus for various rocks as a function of temperature. R4aum&La fissuration des facettes intergranulaires dans les materiaux polycristallins B dilatation thermique anisotropc est contr&e par plusieurs facteurs microstructuraux, en plus des prop&&s thermiques et tlastiques intrins&ques. L’orientation relative des deux grains adjacents sur la facette et la dimension de la facette jouent un r6lc intCressant; ces facteurs donnent une double dimension statistique a cc problbme. Les critbres pour la rupture aux facettes sont pa&s en revue de man&e critique. Les facteurs statistiques sont alors introduits pour fournir des don&es quantitatives sur la densit de fissures en fonction de la temp&rature. Nous comparons rapidement cette thCorie avec quelques mesures exp&rimentales sur la variation du module d’Young de diverses roches en fonction de la temptrature. Zusammenfaasuag-Die RiDbildung an Korngrenzfacetten in polykristallinen Materialien, die anisotropes thermisches Ausdehnungsverhalten aufweisen, ist zusiitzlich zu intrinsischen therm&hen und elastischen Eigenschaften durch einige mikrostrukturelle Factoren bestimmt. Von besonderem Interesse. sind die gegenseitige Orientierung der beiden an der Facette .zusammenstoBcnden Khmer und die GrbBe der Facette. Diese Faktoren bringen zwei statist&he Aspekte in das Problem. Die Bedingungen Rir Facettenbruch werden kritisch zusammengestellt. Die statist&hen Faktoren werden dann benutzt, urn quantitative Angaben iibcr die RiBdichte in Abhingigkeit von der Temperatur zu machen. Die Theorie wird mit den bcgtenzten experimentellen Messungen der Temperaturabhangigkeit des Elastizitiitsmoduls verschiedener Gesteine kurz verglichen.
1. INTRODUCTION The cracking that occurs at grain boundaries due to anisotropic thermal expansion in polycrystals has received much experimental and theoretical attention over the last two decades [l-S]. In hexagonal single phase polycrystals for example, such as alumina or beryllia, where there is differential expansion between the a- and c-axes, stresses and strains are set up around grain boundaries and these increase as the temperature falls from the initial fabrication temperature. The maximum differential grain boundary strain E,,,~~is simply proportional to the maximum difference in expansion coefficients (Aa,,,,,) times the temperature difference (AT) between ambient and that below which plastic effects become negligible (often assumed * 1ooo”c). t.1
= Aa,,,.,AT
(1)
For the complete fracture of a single grain boundary facet two conditions must be satisfied. The elastic strain energy round the facet must at least equal the energy required to fracture the boundary: and a flaw
of sufficient severity must be present to give a high enough stress concentration for fracture to proceed. Either condition may be critical. There is general agreement that for a given AT a critical facet size is
where yti is the grain boundary surface energy and E is Young’s modulus; K is a constant. Facets with 1 2 1, may fracture depending on the relative orientation of the adjacent grains. Because of the practicalities of the situation, experimental observations are usually concerned with the observation of microcracks in various materials in the as-fabricated condition at ambient temperatures. Preparation of samples with a range of controlled grain size can thus lead to the identification of the critical grain size for which fracture is observed for specific materials. The experimental data show that equation (2) is of the correct qualitative form. Quantitative interpretation is however difficult because neither y,* nor the initial flaw size are known pre-
1695
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DAVIDGE:
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cisely. Similar considerations should apply to assemblages of multiphase poiycrystalline ceramics, where the different phases will have different expansion properties in addition to individual phases showing anisotropic behaviour. The aim of the current paper is to develop some aspects of the problem in more detail including the energy available for cracking; and grain orientation and size effects on a statistical basis. In this way predictions can be made about the quantitative effects of cracking on mechanical properties. The new theory is compared with experimental data for the variation of Young’s modulus with temperature for rocks; good agreement is obtained, but more detailed experimental data are required to give a critical comparison. 2. FRACIIJRE AT A GRAIN BOUNDARY FACET
x
We will reconsider first the original model due to Clarke [I], on which was based a two-dimensional fracture mechanics calculation. Whilst this is valid in its essential features there are a number of problems associated with it: the model is only two dimensional; the fracture mechanics criterion developed is not the critical one for complete fracture of a facet; and a more accurate approximation to the stress distribution of the stress round the boundary may be used. 2.1 Energy balance calculations for complete propagation ofa facet crack The model is shown in Fig 1 for a facet of average
lb)
Approximation
Fig. 1. (a) Model of grain boundary facet. (b) Stress distribution in xy section for u, as function of distance from xz (facet) plane. Equating equations (4) and (5) gives for the critical facet size 1, 1c=
88y,,(l IT-2
- Y’)~ (6)
diameter 1.The stress in the x direction (03 falls with For the two dimensional case the equivalent exincrease in distance from the boundary, and Clarke followed the data of Nye [t?rlfor the analogous case of pression is identical to [6]. the distribution of stresses round a glide band, Fig. 1. 2.2 Fracture mechanics calculations for initiation of Clarke assumed that the stress fell linearly with infacet crack crease in y to zero at y = l/2, but reexamination of In the fracture mechanics approach for crack inNye’s graphs shows that a much better approximation is a linear fall in stress to zero at y = 0.2751, itiation used by Clarke, the equivalent expression for 1 is N 24y&e2. The modified stress expression used in bearing in mind also that the strain energy is proporthe current work would give I- 44y&‘. The imtional to the square of the stress. This ideal stress distribution must be disturbed near the three-grain portant point however is that the crack initiation crijunctions but as this will vary widely between juncterion is not critical compared with the above crack tions the simple stress distribution only will be conpropagation criterion, i.e. initiation is easier than sidered. The maximum stress at the facet boundary, propagation. The more recent calculations of amX is Evans [S] for fracture initiation involving a more Ekmsx complex two-dimensional model gives 1% 32yJfi2, a,,=-. (3) in reasonable agreement with the modified Clarke cal2/l - A’ v is Poisson’s ratio. culation above. Realistic models for crack initiation in The elastic strain energy (V,) in three dimensions the three-dimensional case are still to be developed associated with the facet is thus contained in an but the energy release rate for cracks that are small approximately cylindrical volume of diameter 1 and compared with the facet area is likely to be rather less than that for the twodimensional models because of height 0.55 1. the greater constraint of the surrounding material Ill2 0.55 1 Ee,&,n13 u,=-_.-= (4) (and lower release rate of energy). v2)2. 2E4 3 176(1 2.3 Conclusion The energy required to fracture the facet (V,) is
u:,..
” = 2Jr1*y,* I -. 4
(5)
Until three-dimensional crack initiation analyses are produced the crack propagation criterion [6] will
DAVIDGE:
CRACKING IN BRIITLE MATERIALS
Fig. 2. Variation of thermal expansion coefficient with angle 0.
0
15
ANGLE
be used to predict the critical condition for cracking. Implicit in this assumption is the presence of a suitable crack source at the facet. It is interesting to note that for the related case of separate spherical particles in a matrix the total energy balance approach [7] gives fortuitously the same limiting criterion as a finite element fracture mechanics calculation for fracture initiation [8]. This could prove true also for grain boundary facet fracture. 3. GRAIN ORIENTATION
EFF’EC’I-S
For illustration we discuss materials typified by the hexagonal crystal structures where the coefficient of thermal expansion in any direction of the basal plane is a, and in the orthogonal direction a, (The theory can however be readily extended to other crystal structures where for example the expansion along three crystallographic axes may be different.) The expansion coefficient a, at an arbitrary angle w to the c-axis in a section normal to the basal plane is [9] a,
= a, cosb
+ ar, sin’
0 = a, + (a,
-
a,bs2
0,
(7)
as shown in Fig. 2. Consider two grains joined at a grain boundary facet. In general the c-axes of the two grains lie at
1697
30
L5
BETWEEN
60
C-AXES
75
OF GRAINS1
90 0 I
Fig. 4. Relative magnitude of maximum grain boundary strain as a function of orientation of two grains with c-axes in the facet plane.
arbitrary angles /I and y to the facet. For simplicity we examine first the case when the c-axes he in the facet plane, i.e. B = y = 0, Fig. 3. The acute angle between the c-axes is 8. The direction of the maximum strain corresponds to the maximum separation of the ellipses at symmetrical positions defined by o* in Fig. 3. The value of ce+ is found by maximising 1Aal = [a< cos2 o + a,, sin’ o] - [aC cm2(w + a,
+
0)
sin2 (0 + e)]
= (a, - a,)[cos2
w -
cm2
(0
+ O)].
(8)
Differentiating with respect to w gives
4Aal -
do
= (a, - a,)[sin 20~ - sin 2(w + e)];
(9)
equating to zero gives o* = (4S-f7/2), which locates IAal,.. The variation of the relative magnitude of the maximum strain with 0 is shown in Fig. 4. For the general situation when the c-axes of the two grains are not ‘in the grain boundary plane the two ellipses representing the thermal expansion in the boundary plane are of different size. The minor axes are always qua1 to a, (there must be an ‘a’ direction in any arbitrary plane) and the major axes vary between a, and a0 Fig. 5. The values of the maximum expansion coefficients in the grain boundary plane are given by: %,X1 = a,cos2/3 + a,sin2/I = a, + (a, - a,Jcos2 G,.
2 = a, cos2
y + a,
sin2 y
= a, + (a, - a,)cos2
Fig. 3. Orientation dependence of the differential thermal expansion coefficient (Aa) for two grains with c-axes in the grain boundary facet plane.
/3
y.
(10)
The case /I = y = 0” is that considered above, while for B = y = 90” (c-axes normal to the facet) there is zero grain boundary strain. The difference in expansion coefficients in an
DAVIDGE:
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CRACKING
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DlFFERENTlAL THERMAL EXPANSION I ANI~~PY FACTOR.A 1
Fig. 5. Orientation dependence of the differential thermat
expansion coefficient for two grains of random orientation. arbitrary direction o from the e,,,, a direction is now &xl=
[~rcos~W+a~sin~W] - [a& 1 coS2(W + e) + a, sin2(o + @J = (%NX1- tr,)cos~ u - (64,* - @&os2(0 + ff).
Substitut~g gives IhI
for a_,
and h,
(11)
2 from equation (10)
= (a, - LYJ[cos’ /? co9 w - cosa y cos2(0 + @J.
Differentiating as before shows that &,, sin 2.oP -2--a, sin2(w*+8)=~,1-a,=cos2
(12) occurs when cos2 y
(13)
For a particular comb~ation of &7,y and 8 it is now possibie to calculate IAal,. The value of o* is first obtained from equation (13) and then substituted in equation (12) to give MaI,,. Although fl, y and 8 can take any value, not all values of /I and y are equally likely. The c-axis of one gram can be regarded as a vector as the radius of a hemisphere with the diametral plane coincident with the grain boundary facet, Fig 6. The end of the vector is equally likely to lie at any position on the hemisphere. Thus for a given value of fl, the relative probability that the vector is at this angle is proportional
Fig. 7. Relative probability for the range of possibIe values for differential thermal expansion.
to 27rRcos jl/2nR = cos /?, (and similarly for the angle y for the second grain). Each value of IA&,., calculated as above thus needs to be associated with a probability frequency factor F = (co@cosyE. To estimate the statistical variation of values of I~al, over all values for /3, y and 8 we can thus proceed as follows. Random values for j?, y and 8 are generated and o*, corresponding to @a/-, calculated from equation (13). Vahtes for lA!xl,from equation (12), are expressed in terms of an anisotropy factor A = [cos*&os”w - cos2vcos’(o + @J which can vary between 0 and 1. Each value of A is then allocated to one of fifty intervals 0 to 0.020.02 to 0.04 etc. The values of the probability f’requency factors F for each of the A values in each interval are then summed to give a histogram showing relative probability P for the grain boundary n&orientation for the range of A values. Figure 7 shows data for 100,000 random ~mbinations of 4, y and B. The si~ifi~nt results are that the highest value for P (P,,,*,J is associated with the maximum value of A N 1; and that fortuitously, P falls linearly from P,,,_ at A = 1 according to P = P,,(O.13 + 0.87A);0.35 > A > 1.00
This conclusion reflects the size of the intervals chosen for A, 0.02. Clearly P must fall to zero at A = 1 and the precipitous drop in P is indicated in Fig. 7. However, a more detailed analysis for much smaller intervals for A indicates that the drop in P occurs very close to A = 1 at A -+ 0.999. This ultimate deviation from linearity is thus real but of only academic interest. Finally, there are minor inffexions in the curve for 0 > A > 0.35.
4. EmCTS
Fig. 6. Possible positions in space of a c-axis vector R.
(14)
OF GRAIN SHAPE AND SIZE VARIATIONS
The average polycrystalline solid contains grains with a wide variation in geometrical form and size. The number of facets on individu~ grams is typically P-18, with the number of edges per facet typically
DAVIDGE:
CRACKING
IN BRITTLE MATERIALS
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The critical failure criterion is given by equation (6) where the strain is defined in equation (1). Generally however the strain is less than the maximum value, according to the factor A (equations (12) and (14) and Fig. 7) , = 88 Ygdl - v2Y E(AAa,,,AT)* ’
(16)
It is convenient to consider the ratio I//,,,,, = I* and the ratio AT/AT,,,, = T* where ATmi,,is the minimum temperature change to fracture the largest facet (I,,,,,) with the maximum misorientation of neighbouring grains (A = 1). Note that Fig. 8. Distribution of facet sizes in polycrystalline magnesia. The numbers on the graph indicate the percentage of pA=T*l
facets larger than the size indicated.
3-8 [lo]. The grain size may vary by one or two orders of magnitude. Much useful information about the distribution of grain sixes and shapes is given in the papers by Aboav and Langdon [ll] who analysed large numbers of grains in magnesia. Their data showed that a normal Gaussian distribution was obtained when the number of grains of a particular diameter was plotted against the square root of the diameter; this held also for the variation in the number of grain sides for a particular side length plotted against the square root of the length of the side. Roth distributions have the same mathematical form given by: N = N.exp{
-ci[(3’”
- 111)
(15)
where N is the number of grains or sides of length x within a given range. x,,, is the value of x when N has the maximum value N, a is a constant. For a particular dense material of grain size w 10 w a was 243 for grains and 2.05 for grain sides; the average number of sides per grain was 5.85 and thus the grain size is close to twice the facet size. (The factor of two difference would be expected for hexagonal grain sections.) These data are replotted on a linear scale in Fig. 8 which shows the equivalent distribution of facet widths expressed in terms of the most frequent grain size (e) of the material. The most frequent facet size is half 6 and the largest observed facet size is twice 6. Care is needed in defining average grain size. For magnesia the median grain size is 5 1.1 G, the mean grain size 5 1.3 6, and 50% by volume is occupied by grains smaller or larger than _ 1.8 6. 5. CRACK DENSITY As A FUNCI’ION OF TEMPERATURE
We are now in a position to combine the effects described in the previous sections and predict how the number of cracks will vary with temperature. A.M. 29110-c
=
88 jb*(l - v2)2 E(ALY,,,A.T,~,)’I,,
= I’
(17)
This ‘first-cracking criterion’ (T* = I* = A = 1) is the one generally considered in the literature, but our interest lies in values of T* > I where both smaller and non-ideally oriented facets may fracture. The variation of facet size is given in Fig. 8 and the distribution of A in Fig. 7. The general situation is thus that cracking occurs at a small proportion of small facets and an increasing proportion of large facets. The number and size of cracked facets can be calculated as follows. The facet size distribution graph is sectioned into a number of equal sectors containing a specific proportion of facets, say 1 or 5%. The facets in each sector are treated as all having a size at the minimum of the sector boundary. Table 1 illustrates the details of the calculation, using ten sectors, corresponding to potential cracking of loo/, of the facets. From Fig. 8, the largest loO/, of facets have P > 0.590; for A = 1,1*T*2 = 1 and this corresponds to T* = 1.301. Elecause all facets in the tenth sector are assumed to have I* = 0.590, then none will crack because the proportion of grains P at A = 1 is zero (Fig. 7). To estimate the number of cracked facets in the other sectors it is convenient to replot the data in Fig, 7 on a cumulative basis starting from large values of A. Consider the first sector where p = 0.825; cracking occurs when T* > 1.101 which is 0.846 times the actual value 1.301. Thus all facets with A > 0.846 will crack which from Fig. 9 gives a fraction of 0.265. The cracked fraction of facets decreases with decreasing facet size and the numerical data are in Table 1.0.112 of these largest facets are cracked but this is only 0.011 of the total. Equivalent calculations can be performed to cover the whole range of facet s&s and for this case 20 sectors of 5% were chosen. Figure 10 and Table 2 give the data for each sector in terms of the threshold condition. For example under conditions to fracture potentially half the total facets (up to sector 10) 0.53 of the facets in the first sector have fractured and summing for each sector shows that 13.7% of all facets have fractured.
DAVIDGE:
CRACKING
IN BRITTLE MATERIALS
IFRACTION
THRESNU.0 CCfdOlTlON - OF FOTEMIALLY CRACKED FXETSI
-
Fig. 10. Cracked facets in each sector for sector 1 (upper curve) to sector 19 (low curve) for each threshold condition.
facets per unit volume. Generally, this information is not available but to illustrate the principles involved a calculation is performed below on the assumptions Fig. 9. Data from Fi8. 7 replotted on a cumulative basis. that the grain and facet sixe distributions are those observed by Aboav and Langdon [ll] in magnesia. 6 EFFECTS ON PROPERTIES Since these are controlled primarily by geometrical The above data are essential for predicting the considerations this should represent a reasonable effects of cracking on mechanical and physical general example. properties such as Young’s modulus or thermal conThe magnesk data show for example that for ductivity. Roth these properties vary with cracking in material with G = 10~ (i.e. a mean grain size of a similar way and the equation for Young’s modulus 13 /.@ the total grain density is 6.24 x lOi* m-’ if assuming a random distribution of non-interacting the grains are assumed to be spherical. Thus taking cracks is 1123 the average number of facets per grain as 14 (each Eo shared between two grains)the facet density is (18) E = 1 + 16Nb3/9 4.37 x 10’5m-3.ThedatainFig. lOandTable2can where E. is the modulus of untracked material, N the now be used to calculate Nb3 for each facet size number of microcracks per unit volume and b the sector for various threshold conditions, and details are given in Table 2 for the threshold condition to crack radius The calculation of N is straightforward only when potentially crack half the facets. Here ZNb3 = 0.094 all cracks are of the same size. For real micros&tic- and E = 0.857 Eo. Note that the hugest facets have tures a full analysis of grain size and facet size vari- the more significant effect. Similar results for all ation is required to give the number of grains and thresholds conditions are given in Fig. 11; the calcula0.6 01 02 ANISOTROPY FACTOR IA I
0
Table 1. Fraction (R)ofcracked facets for ten 1% sectors containing the largest facets. 1
2
3
4
5
6
7
8
9
10
P
0.825
0.769
0.728
0.697
0.675
0.653
0.634
0.619
0.603
0.590
F/P10 R
;E 0.265
0876 1.140 6216
0.901 1 172 0.173
0921 1.198 0:148
0’935 1217 0.117
0’951 1237 0:092
0’965 1255 01062
0.977 1271 6037
?g 0:018
:*g O&O
Sector
Tabk 2. Fraction (R) of cracked faats for 10 5% sectors for a threshold condition to crack potentially half the fasts (sectors l-lo), and calculated values of Nb’ (see Fig. 11). Sector
1
2
3
4
5
6
7
8
9
10
P P P/T*10 R
0.675 1.217 0.666 0.530
0.590 1.301 0.712 0.468
0.534 1.368 0.749 0.413
0.488 1.432 0.784 0.364
0.447 1.496 0.819 0.308
0.413 1.557 0.852 0.253
0.378 1.626 0.890 0.197
Nb’
0.036
0.02 I
0.014
0.009
0.006
0.004
0.002
0.350 1.690 0.926 0.136 0.001
0.325 1.754 0.961 0.074 0.001
0.300 I .826 1.000 0.000 O.lXIO
DAVIDGE:
0'
0
CRACKING
1 2 NORMALISED TEMPERATURE DIFFERENCE
IN BRITTLE
1701
3 I T *I
Fig 11. Variation in Young’s modulus with increasing temperature difference. Numbers on curve indicate threshold facet condition. Dotted curve indicates expected deviation due to cracks in adjacent facets.
tion does not depend on the grain size, provided that the distribution is that assumed above, because Nb3 is
dimensionl~. The data in Fig. 11 are expected to be accurate only at small crack densities when the cracks are relatively isolated. When cracking occurs significantly for adjacent facets the data underestimate the reduction in Young’s modulus. The dotted curve for example shows the sort of deviation expected if all the cracks for the sector 16 situation (35% of all facets cracked) comprised two adjacent facet cracks. 7. COMPARISON EXPERIMENTAL
MATERIALS
WITH DATA
The theory developed above ieads to two parameters that can be compared with experimental data: the minimum temperature change for the generation of the first cracks, and the rate of crack formation (as measured for example by changes in Young’s modulus) with further change in temperature. The two parameters are not independent and the rate of change in modulus with temperature is proportional to the minimum temperature change as indicated by the normal&I universal curve in Fig 11. The first parameter has received much attention in the literature and the main contribution here has heen to refine the calculations [equation (6)]. The second parameter has not been calculated before and there is a dearth of experimental data for comparison purposes. For ceramic materials, that are fabricated at high temperature, the amount of cracking increases as the temperature falls after fabrication and it is difficult to monitor effects except at ambient temperature. Further data could be obtained however from specimens cooled below ambient temperatures. Comparison between experiment and theory can however be made for geological materials where equilibrium is reached not at a high fabrication temperature but under ambient conditions after geolo~~lly long times. Here cracking can be induced by heating
I OO
I
100
1
L
I
ml
3w
03
TEMPERATURE 1°C 1
Fig. 12. Variation of Young’s modulus with increasing temperature for various rocks. (a) Reserve Taconite [13], (b) Dresser Basalt [13), (c) Jasper Quartzite [13], (d) Charcoaf Granite [133, (e) Cornish Granite [143, (f) Welsh Sandstone [IS].
cooling) samples from ambient temperatures. Figure 12 shows the effect of heating on a range of geological samples. (In some cases, where the experimentar data points are sparse, the graphs have been constructed so that the initial regions comprise two straight lines.) All the rocks featured contain anisotropit minerals such as feldspars and quartz and the granites contain both these minerals. The simple theoretical model presented here can be applied to rocks provided that sufficient ~~os~u~ur~ and physical data are known: for instance the frequency and sixes of facets between like and unlike grains, grain boundary fracture energies and the elastic and thermal expansion properties of the constituent minerals. These data are not yet available but the qualitative agreement between the simple theory and the existing data is striking. It is planned to obtain more detailed data on specific rocks to permit further comparisons. (or
Acknowledgements-This work was done as part of the UK programme of research into the burial of radioactive waste for the Commission of the European ~rnm~ity and we thank the Commission for agreement to publish. Thanks are due to J. R. McLaren and I. Titchell for useful discussions and for some of the experimental data for rocks; and to M. H. Rand and A. Rabinovitch for assistance with the calculations in section 3.
RX~REN~ F. J. P. Clarke, Acru metali.12, 139 (1964). R. W. Davidge and G. Tappin, J. Mater. Sci. 3, 297 (1968). J. J. Cleveland and R. C. Bradt, J. Am. Ceram. Sot. 61, 478 (1978). R. W. Rice and R. C. Pohanka, J. Am. Cenun. Sot. 42, 559 (1979). A. G. Evans, Acra meraif.26, 1845 (1978).
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DAVIDGE:
CRACKING
6. J. F. Nyz Proc. R. Sot. A 200.47 (1950). 7. R. W. Davidge and T. J. Green, J. Mater. Sci. 3, 629 (1968). 8. Y. M. Ito. M. Roscnblatf L. Y. Chcng, F. F. Lange and A. G. Evans, Int. J. Ftcrcr. 17, (1981) in press 9. J. F. Nyc, Physical Propcrries of Crystals Clarcndon Press, Oxford (1957). 10. C. S. Smith, Sciew Am. 190, 58 (1954). 11. D. A. Aboav and T. G. Langdon, Medogrophy 1,333 (1969); 2, 171 (1%9).
1N BRITTLE MATERIALS 12. D. P. H. Ha&man and J. P. Singh, Bull. Am. Cerum. sot. 5& 856 (1979). 13. C. F. Wingquist, U.S. Bureau of Mints Report 7269 (1970). 14. J. R. McLarcn and 1. Titchell. AERE Harwell, to bc published (1981). IS. I. Titchcll and J. R. McLarcn, Unpublished data (1981).