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Relation of tensile strength-porosity effects in ceramics to porosity dependence of Young's modulus and fracture energy, porosity character and grain size
Materials Science and Engineering, Al12 (1989) 215-224
215
Relation of Tensile Strength-Porosity Effects in Ceramics to Porosity Dependence of Young's Modulus and Fracture Energy, Porosity Character and Grain Size R. W. RICE*
Naval Research Laboratory, Washington, DC 20375 (U.S.A.) (Received June 28, 1988; in revised form December 7, 1988)
Abstract
The dependence of tensile strength o on the porosity P of ceramics is considered, based on recent theories and observations of failure from pores and emanating microstructural cracks. Extensions of a model for such failure based on a colinear array of pore-crack combinations is used to evaluate such cr dependence and its relationship to the porosity dependence of Young's modulus E and fracture energy 7 as well as the dependence of strength on the grain size G. It is shown that the porosity dependences of o, E and 7 should usually be similar, as is generally observed. It is also shown that the dependences of cr on both P and G are generally separable even if the sizes of the cracks surrounding the pores are related to G. The concept of pore finkage due to connection by cracks following preferred (mainly grain boundary) paths is introduced, and is suggested to be much more common than pore linkage due t0 cracking from overlapping stress fields of closely spaced pores.
1. Introduction
Substantial progress has been made in studying the effects of porosity on mechanical properties, e.g. as reviewed by Rice [1]. Thus, for example, considerable documentation shows that porosity typically has a substantial effect on most mechanical properties. The greatest effects are seen for behavior measured under compression (e.g. compressive strength, hardness and wear), hence the necessity of distinguishing the type of loading being considered i.e. tensile loading in *Present address: W. R. Grace and Co., Columbia, MD 21044, U.S.A. 0921-5093/89/$3.50
our case. Within the tensile loading regime, broad similarity is observed in the porosity dependence of tensile (flexural) strength o, Young's modulus E, fracture energy y, and hence the fracture toughness Kic. The ratio of a property at some volume fraction porosity P to the property at P = 0 can generally be represented as e- bP where e is the Napierian logarithmic base and b is the slope from a semilogarithmic plot of the property vs. P. The value of b is constant for a given type of porosity and property, and often gives very similar, if not identical, values for the above-noted tensile properties of a given set of bodies of varying P. Although substantial progress has been made in modeling such porosity dependence, mainly of E and secondarily of 7 (and hence of Kic ) [1], major changes have occurred in our understanding of tensile failure of porous bodies [1-6], requiring new or modified approaches to modeling tensile strength. Previously, pores were seen simply as stress concentrators accentuating failure from other sources. Now, however, it is generally recognized that pores, if present, are often (if not always) an integral part of the failure causing flaws under tensile loading [1, 2, 7], which was generally not accounted for in previous strength-porosity models. There is thus the need to address pores as an integral part of failure causing flaws in the broader understanding of mechanical property-porosity relationships. Of particular pertinence to this paper are the following basic issues. (1) Is there a sound rationale for the common occurrence of similar porosity trends for o, E and 7; if so, over what range of porosity does it hold? (2) To what extent do pore size, shape and location affect the strength? (3) Is there a sound rationale for porosity and grain size G effects being separable, as commonly © Elsevier Sequoia/Printed in The Netherlands
216 assumed in microstructural studies; if so, over what range of porosity does it hold? All these need to be considered because pores can affect a not only via Kic (through E and ~) but also through flaw geometry size C, and possibly sharpness. E and 7 are interrelated and important because, typically, high temperature exposures increase G and reduce P as well as change pore location (i.e. all the porosity is intergranular at high P and smaller G, and changes to mixed intergranular and intragranular porosity at lower P and larger G). Knowing, for example, whether G and P effects are theoretically separable, and if so over what range, is important, e.g. for extrapolation. Thus, although separation is indicated empirically for microstructures of fine G and fine P that are most readily obtained and most often used for porosity studies, this is less certain for the wide range of porosity achievable. We present a model that addresses the above issues and provides a framework for addressing specific strength-porosity-microstructure relationships. In particular, interrelationships between P and G effects are considered. The model is based on the concept of pores as an integral part of failure causing flaws. It is essentially a modification and expansion of the concepts which Evans and Tappin [2] used for treating pores large compared with the grain size as sources of failure. They adopted Bowie's [3] model of a flaw consisting of a cylindrical hole (radius R with one or two radial cracks of length L (Fig. l(a)) for treating a spherical pore and the immediately adjoining layer of grains as a similar flaw (Fig. l(b)) by simply substituting the flaw shape parameter for a penny crack for that of a slit crack in the Griffith equation. This adaptation is based on the concept that cracks could propagate along grain boundaries emanating from the pore wall until the next layer of grains is encountered. Because the greater difficulty of crack propagation around or through the second layer of grains could arrest the crack at this second layer, the pore plus part of the immediately surrounding grains could constitute the critical flaw when such pores control strength. More recently, models have been developed for spherical pores plus cracks, which are generally a much better approximation of actual pores [4-6, 8, 9] identified as initiating failure. However, it has been proposed that cracks associated with pores may be a function of grain size [7, 8, 10], again raising the issue of the separability of P and G effects. There are some
(B)
R
//~L~L~ Fig. 1. Schematic diagram of pore-crack combinations. (a) Cross-section of cylindrical or spherical void of radius R with one or two (dashed portion) associated axial cracks for a cylindrical void (Bowie'smodel) or a spherical pore with an equatorial crack; in either case the length (or depth) of the crack into the surrounding material is L. (b) Schematic diagram of pore larger than surrounding grains, with possible crack length L (i.e. depth into material) being to the next layer of grains, i.e. 1/2 grain deep. (c) Triple-point pore with L suggested as the length of the adjoining grain boundary facet. (d) Intragranular pore (upper) and grain boundary pore (lower) and suggested L dimensions constrained by meeting of adjacent grains.
differences between the cylindrical and spherical models, as well as between either of these and some of the experimental results [7]. However, these differences are only of moderate numerical values for one or two factors, and are thus noted in the development for various cases.
2.
Theory
2.1. General theory To treat failure from pores, Evans and Tappin [2] adopted the equation for treating a periodic array of colinear planar cracks normal to an applied stress to treating similar arrays of the above cylindrical pore-crack combinations of the same size. They considered the stress cr to cause either failure from a single crack or the linking of two or more cracks in the periodic array. Although failure is the most common case (as discussed later), the lower of the two stresses is given by
a= 2 ~t-anfl} a-e
(1)
217
where fl=(:rtR/2R+ 2), R is the pore radius, 2 is the pore spacing, L is the length of the crack emanating from the pore (which may be related to the grain size (e.g. Fig. 1)), 7 is the fracture energy for the particular case and Op/Oc is the ratio of the stress to propagate a single pore-crack combination to the stress to propagate a crack of dimensions R + L. We first consider the nature of the pores to which the model is applicable. Although Evans and Tappin presented the above concept for large pores and finer grain sizes (R > L, Fig. l(b)), there is no reason why it cannot be applied to other pores and grain sizes. For example, it has been shown that their model can be extended to large non-circular pores by taking R as half the minor axis of an elliptical approximation of the pore in the fracture plane (typically at or near normal to the applied stress) [11, 12]. Although more complicated modification may have to be considered for some extreme pores, these observations allow most pores to be considered and, in particular, the very important cases of typical pores at triple points (Fig. 1(c)) and on grain boundaries (Fig. l(d)), as well as intragranular pores (Fig. l(d)). Similarly, it is not necessary to have intergranular failure around intergranular pores, consistent with the observation that transgranular rather than intergranular failure often occurs around large pores [1, 7, 11, 12]. Treatment of pores as indicated above is sketched in Fig. 1, with approximate values suggested by some models for L is in terms of the grain size G. As discussed elsewhere [6, 7] and later, it may be that, L '~ G with G ,> R, or that L > G for R ,> G. The approximation of a colinear array of through-the-thickness cylinder-crack combinations normal to the stress is valid for several reasons. Firstly, the orientation of pore-crack combinations essentially normal to the stress will be extremely common and clearly represents the expected case of failure. Secondly, many cases involve only a single pore (where the 13 terms cancel) or a few pores whose spacing is determined by other microstructural factors, e.g. grain size. Thus, the periodic assumption either is no longer involved or is reasonably satisfied. For the unusual cases where more pores are involved, an average periodicity or spacing may often be reasonable. Thirdly and more generally, the same periodic crack array approach is the basis of Hasselman's successful thermal shock models [13] where the cracks are not likely to be any
more periodically located than pores are. As for the nature of the pores themselves, much of the effect of porosity on o comes from its effects on E and 7, which do not depend on the above assumptions. Further, in many cases where flaw aspects of pores are important, the effect of the size (and hence most shape) aspects of the pore is negligible. When this is not the case, pore shape-size effects can be handled as for single pores. Use of flaw shape parameters for elliptical, surface or interior flaws [2] instead of slit or through-the-thickness cracks is reasonable and gives good results. However, the Op/Oc term should depend on pore shape. It becomes significantly greater than unity only when L/R decreases below about 0.1 for spherical [6] (as well as cylindrical [3]) pores. Because of uncertainties in other parameters (e.g. R, L and 7), critical tests of the applicability of Op/Oc values based on these models require L/R < 0.05, and hence have generally not been made. Thus, op/oc will generally be treated as negligible, i.e. equal to unity, except as noted. Experimental data [7] generally support oplo~ = 1 for polycrystalline bodies, at least failing from pores on machined surfaces. For glass specimens, Op/Oc is often greater than unity. Incorporating the above modifications, i.e. op/ o c = 1, and elliptical flaws yields
(2)
where Y= 1.25 for interior circular cracks and Y= 1.12 for surface semicircular cracks. Because pore-linking effects due to pore-stress interactions become significant only when the spacing between pore walls is less than their diameter [2], o of eqn. (1) will normally represent the failure stress in the majority of the cases of interest (other cases will be specifically noted). For small fl, tanfl ~ t3 (i.e. when 2/R > 1), so the 13 terms cancel (i.e. single pores control strength). Further, as shown in Table 1, (13/tan 13)1/2 decreases to only about 0.93 at P - 0 . 3 0 so (13/tanfl) 1/2-'~ 1 over the region of greatest interest. Where the 13terms are no longer negligible, they can be estimated using an expression for the spacing of spherical pores, which commonly has the form -
AR(1-P) P
(3)
218 TABLE 1
Values of (titian fl) 1/2 as a function of porosity
Volume fraction porosity P
(fl /tanfl) I/2.
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.80 0.90 0.95 0.97 0.98 0.99
0.998 0.992 0.982 0.969 0.952 0.933 0.911 0.886 0.858 0.823 0.756 0.667 0.554 0.399 0.284 0.221 0.181 0.128 0.000
1.00
aft = (3riP)/2P + 4 using 2 = (AR( 1 - P )/P) (with A = 4/3).
where P is the volume fraction porosity and A is a constant, e.g. 4/3. (A recent estimate by Wachtman [14] indicates that the difference in 2 values for various forms of eqn. (3), e.g. for different A values, is relatively moderate above P = 0.3. At high values of P (i.e. above 0.5), 2 is likely to average a fraction of the grain size.) Two major cases cover the most common porosity situations of interest: R < L and R > L. Considering these, as well as R = L and mixed cases, gives a broader perspective of strengthporosity relationships and addresses key issues discussed in the previous section. Table 2 represents a summary of the various cases and their results. 22. Case I: R < L with 2/t? > 1 We first consider the case R < L with 2/R > 1, which covers much of the common and important cases of grain boundary and intragranular pores (i.e. Figs. l(c) and l(d)). For P < 0.3, eqn. (1) reduces to a= V
(4)
(For this case, ap/a c of eqn. (1) is theoretically about unity, even for cylindrical pores.) When only a single pore is involved (Fig. 1), ), should typically be about the fracture energy along preferred fracture paths in single crystals or along grain boundaries [15]. In fine-grain bodies where,
except for sporadic, large isolated pores, singlepore causes of failure are probably uncommon, L may be about G/2 or could be several times G. In large-grain bodies where this mechanism should be more common, L may be more than (;/2, as discussed later. A particularly important extension of Case 1 is the linking of nearby pores because of their cracks impinging (mainly along grain boundaries) rather than because of their stress interaction. Examples of pores which are too far apart to link by stress interaction but which can link by crack impingement are sketched in Fig. 2. This impingement increases L, e.g. G <~L <<,(2-4)G, increasing 7, which will approach the polycrystalline value when enough grains are involved [15]. In this case, strength will not depend directly on pore size or shape, but only indirectly through the dependence of E and 7 on these factors. When L is related to grain size G, eqn. (4) gives an inverse square root dependence on grain size. The specifics of this dependence vary with the exact L/G ratio at various failure origins and may be modified by some grain size dependence of 7 [16, 17]. When 7 has no significant grain size dependence, then the grain size dependence of strength (via L ) can be separated from the porosity dependence as follows. Because 7 = 70fT(P) and E= EofE(P ) where 70 and E 0 are the zero porosity fracture energy and Young's modulus respectively, then =
= Yf(P)
(5)
where f(P)=(fT(P)fE(P))l/2. When ~, has significant dependence on G, this is commonly due to crystalline anisotropy effects [16, 17] which arise from microcracking. Such microcracking also depends on porosity which could preclude a true separation of G and P effects. However, because porosity will often suppress microcracking, the G dependence of 7 may often become rather weak in the presence of all but small amounts of porosity. Thus, whether or not L depends on G, the effects of P and G on o are separate. The P trends shown by eqn. (5) should also be noted. Polycrystalline fracture energies typically follow a similar trend with porosity as Young's modulus does [1], so when such fracture energies are applicable, e.g. for the important case of pore linking due to crack impingement, fy(P)'-~f~(P)~fo(P). (However, fy(P) can be
219 TABLE 2
Summary of various porosity cases
Case
Applicable equation a
Effects of P and G separable b
Direct strength effects of pore Size
Relationship of
Applicability
fo(P) to fE ( P )c
Shape
Y(E7/L) j/2
Yes
No
No
Limited mostly to pore linking by crack impingement. May decrease as stress volume increases Limited to cases where R is Large G: fr(P) <~fo(P)/> (fE(P))u2 much greater than other flaw sizes, e.g. very large-grain bodies. May decrease as stressed volume increases
Y(Ev/R) '/2
Yes
Yes
Yes
fy(P)~fy(P)=fE(P)
Frequent occurrence over a wide range of stressed volumes for limited P
(3) R = L
Y(Ev/R + L )1/2
Yes
Yes
Yes
fo(P)~f~(P)-~fE(P)
Frequent occurrence over a wide range of stressed volumes for limited and especially medium P
(4) Mixed Cases 1 and 2
Y(E7/R) j/2
Yes
Yes
Yes
fo(P)'~fr(P)=fe(P)
Frequent occurrence over a wide range of stressed volumes for limited and especially medium P
(5) R < L
Y(ET/C) U2
Yes
Yes
Yes
fo(P)-~fr(P)~fE(P)
Frequent for inhomogeneous low to medium P, especially as stressed volume increases
fl/tanfl(E7/C) 1/2
Yes
Some, but often limited
fo(P)generally becomes progressively less than fy(P) as P increases
Dominant for large P. Tends to increase as stressed volume increases, e.g. for medium P
Small or medium G: fo(P) = fE(P) (1) R < L (2/R) > l
(2) R> L
(A/R)> 1
(A/R)
<
1
(6) (A/R) < 1
Cis the size of flaw from pore linking. bAssuming y does not depend significantly on G. "Although fo(P)~ fE(P) for Cases 1-5, this similarity will often decrease in that order (except for Case 1 with large G). Failure due to other flaws or microplastic mechanisms will generally increase the similarity between fo(P) and fE(P ). a
s o m e w h a t greater than fE(P) in s o m e cases [1]. However, as possible fE(P) and fT(P) differences are not large, the statement fa(P ) = fiE( P )f7(P ))l/2 is generally true.) T h u s in this important instance of Case 1, the similar trends of E - P a n d o - P are explained. However, when failure initiates f r o m only one p o r e smaller than one of or all the surrounding grains, y should a p p r o a c h single-crystal or grain b o u n d a r y values and hence can have a reduced (and ultimately zero) d e p e n d e n c e on P. This would reduce the d e p e n d e n c e of a on P, e.g. in larger-grain bodies, and hence reduce but not destroy the similarity of E - P and a - P trends, i.e. in the extreme f , ( P ) = (fE(P))1/2. In addition to being consistent with the above general trend of the porosity d e p e n d e n c e of
strength, this case shows that substantial variation can occur in the porosity d e p e n d e n c e due to three factors. Firstly, there is the variation in that can occur with the n u m b e r of pores of the fracture origin and the size of associated grains. Secondly, there is the variation in E due to variation in the p o r e density in the b o d y which should also vary 7 [1]. Thirdly, there is the effect of p o r e location, as shown in Figs. l(c) and (d) and eqn. (4). Triple-point pores are normally the most severe, especially for single-pore sources of failure, as they typically have at least one emanating grain b o u n d a r y at a substantial angle to the stress (Bowie's solution for one emanating crack does not differ substantially f r o m the two-crack solution) and L m a y be larger, e.g. greater than or equal to G / 2 . However, m a n y pores along grain
220 A
C
D
Fig. 2. Schematic diagram of pore linkage by impingement of cracks. (a) Triple-point pores linked by cracks along grain boundaries. (b) Triple-point pore and grain boundary pore linked by grain boundary cracks. (c) Triple-point (upper) and grain boundary pore (lower) linked to intragranular pore by impingement of their grain boundary crack and transgranular crack respectively. (d) Linkage of small pore (e.g. triple-point void as shown here) to a large void. Similar effects could occur with intragranular pores (voids).
boundaries will be unfavourably oriented, and they typically have shorter potential cracks, e.g. L ~-G/4. For pores within grains, on average, L <~ G/2, which is less than the extremes for triple-point pores. Further, 7 for cracking across a grain is more than twice as high as 7 for cracking along a grain boundary section, and fracture on preferred cleavage planes substantially limits the opportunity of pore linking by crack impingement. The possible smaller value of L, the greater value of ~ and reduced pore rinking are thus other reasons (in addition to grain boundary stress concentrations) previously cited [18] why pores within grains should be less detrimental to strength than pores at grain boundaries. The differing effects of pores of different locations thus should lead to some variations in the effect of porosity on strength. Hence, a body of predominantly triple-point pores (e.g. as-hotpressed to less than theoretical density) should be weaker than one with the same porosity but with many intragranular pores (e.g. a body fired or
annealed at higher temperatures), provided all other variables (e.g. G) are equal, which is very difficult to achieve in practice. Pores, singly or in combination, will be the source of failure only if they are larger than other separate competing flaws. Such flaws are predominantly machine flaws, which are typically a few tens of micrometres in size [1]. Thus, for example, machining flaws will typically be substantially larger than single pores with R < G in most fine-grain bodies. Therefore, only pore linking, mainly due to crack impingement, is realistic for medium- and fine-grain bodies of Case 1, and even this may be limited. However, although the direct applicability of some aspects of Case 1 is generally limited, most of the trends are not. Thus, although some of the variability due to different pore shapes or location on the flaw effects may be removed, some of these effects also enter through E and y. Further, whenever pores do not act as the flaws fo(P) is equal, or similar, to fE(P), SO such similarity trends are reinforced. Also, grain size effects (due to effects on flaw size or 7) are often still separable, as in eqn. (5). Further, the frequently greater ease of cracking along grain boundaries and the shape and related stress concentration should make regions of grain boundary (and especially triple-point) pores preferred areas for machining flaws. Such pores can thus increase the size of such flaws. Similarly, higher densities of such pores may be preferred areas for such pore-enhanced machining flaw introduction.
2.3. Case 2: R > L with 2/R > 1 At the other extreme, R > L (where op/o c typically is greater than unity for pores in glasses [7]), some different results and some similar but more complicated results to those from Case 1 are obtained. With 2 /R > 1, i.e. P< 0.3, only single pores are normally important, so eqn. (1) reduces to
[E \~/2 a 2 YI~ )
(6)
A major difference from Case 1 is that, since R + L .~ R, there is no possible direct dependence on G if 7 has no significant dependence on G. (Again, if y depends on G, P and G effects may not in principle be separable, but may be at least approximately separable in practice.) However, pore size and shape (which also affect R)
221
directly affect Case 2. It is important to note that the y used in this case is that of the material surrounding the pore, which may be substantially higher than that measured for the body. Thus, for example, if a body's only porosity is a substantial number of large isolated pores, the energy for fracture from a large pore would approach or equal that of a dense body rather than the value measured by normal fracture mechanics tests which is lowered by the pores. Young's modulus is typically that of the body, but might be altered at very low L/R ratios because the strain energy, of which E is a critical factor, around the crack would then depend significantly on pore stress concentrations. If pore size remains constant so only the number of pores changes with P, the functional dependence and the rate of strength change with porosity will be due exclusively to Y and E as in Case 1, although the dependence of E and especially of y on P may vary from Case 1. If only the pore size increases with porosity (i.e. the number of pores per unit volume N remains constant), the rate and functional dependence of strength on porosity will depend not only on E and y, but also on R. However, R changes slowly with P, e.g. if uniform spherical pores are assumed
R= ( 3P1~/3 k47tN ] so strength has only a minus one-sixth ( - 1 / 6 ) power dependence on porosity through R in addition to a dependence on R through y and E. As R increases, there may be greater variations in its values (with the maximum value controlling strength). This will also give greater variations in E and possibly y giving greater variation from Case 1, typically a more rapid decrease in strength with increasing P. Normally, both N and R change with porosity so the rate of strength change with porosity for R > L will often be similar to that for R < L, i.e. Case 1. Thus, as in Case 1, there will be similar E-P, y-P and a - P trends. It is important to note that the variable strengths within Case 2 (R > L) can result from the same level of porosity because of variable pore shape and especially pore size. The relative strengths between Cases 1 and 2 for a given porosity depends on two factors. The first is (L/ R) ~/2 which would appear always to be less than unity. The second factor in comparing strengths for Cases 1 and 2 is the combined porosity
dependence of E and y. The porosity dependence of y in Case 2 can be substantially less than that in Case 1, whereas the porosity dependence of E may be somewhat more in Case 2 than in Case 1 [1]. Thus the combined porosity dependence of E and y in Case 2 will often be less than in Case 1, which can partly compensate for the lower value of (L/R)I/2.
2.4. Other porosity cases Three other cases complete the range of possible porosity. The case when R = L will be similar to Case 1, but both R and L must be considered. For example, for 2/R > 1, eqn. (1) reduces to
a -~ y( Ey I ~/2 \R+L]
(7)
(Again, ap/Oc = 1 for cylindrical and spherical pores.) Thus, if L or R or both are related to G, porosity and grain size effects on strength can be expressed as separate functions if y does not depend on G. (If y depends significantly on G, P and G effects may not be separable, as noted earlier.) Effects varying a for a given P will tend to be similar to those for both Cases 1 and 2, but again overall fo(P) will be similar to fE(P ). Combinations of Cases 1 and 2, i.e. pores both large and small compared with the grain size constitute another case. The large pores will dominate the strength; the small pores will affect the strength by extending the cracks around large pores if they are sufficiently close (e.g. Fig. 2), thus increasing L and increasing the porosity dependence of E, and probably that of y. Therefore the overall dependence will be that of Case 2 (but probably with reduced G dependence). Thirdly, 2/R ~< 1 for any of the preceding cases means that eqn. (2), rather than giving the stress for failure, gives the stress for crack linking pores due to pore-pore interactions, not impingement of cracks. This case of close pore spacing can occur at limited porosity levels for fine-grain boundary pores (i. e. Case 1 ) and for inhomogeneous porosity of any of the above cases. This will typically increase the value of L, i.e. since one crack from a pore is nearly as serious as two, the larger crack linking two pores will dominate. Often the combined pore-crack configuration can be treated as a sharp flaw with the dimensions of the configuration, i.e. substituting this size for
222 R, L or R + L. This adds some variability in fo(P) for a given P and to its similarity to fz(P). In such cases, resultant flaw sizes are not likely to be closely related to G, giving little or no G dependence unless Y depends on G. More commonly, 2/R < 1 when P is large (e.g. 0.4 or more). In most such cases there must be a mixture of pore sizes based on simple percolation concepts; large pores will dominate and pore-crack linking will occur. The porosity dependence of strength will still enter through E and 7, but will also enter through fl and the size of the group of pores that makes up the final flaw size (substituting for R + L in eqn. (2)). The variability in the size of the pore cluster thus becomes the dominant factor in varying strength effects. This size, and increasing effects due to fl as P increases, generally leads to fo(P)> rE(P). However, at large P, 7 measured by larger cracks (e.g. as in double cantilever beam tests) may exhibit crack branching leading to large 7 values [8, 9]. But such branching typically requires greater crack propagation than occurs with strengthcontrolling cracks, so f~(P) < f~(P)[19, 20].
2.5. Model limits and interaction with other failure mechanisms It is useful to consider the limits of the model as P--" 0 and as P--" 1.0. Equations (1) and (2) show that if uniform porosity is assumed, a--" 0 as P--" 1.0. Changing pore shape would change the shape of the o - P curve, but not the limit. Inhomogeneous distribution of porosity would make a ~ 0 at P < 1, as it should. At the other extreme, as P ~ 0, eqns. (1) and (2) reduce exactly to eqn. (4). However, other than showing that a as a function of P is well behaved as P--'0, this is not necessarily meaningful, because when P = 0, the mechanism of failure of this theory disappears. Other failure mechanisms become competitive or cooperative with the pore-crack mechanism over a range of porosity as P ~ 0, and become exclusively operative at P = 0. There are two major mechanisms of failure that will come into play as P ~ 0: microplastic and flaw mechanisms. The latter will be other sources of flaws, most commonly surface flaws from machining; internal cracks, e.g. resulting from thermal expansion anisotropy, might also play a role in some cases. Rice [7, 21] has shown that there is considerable competition between
machining flaws and pores in acting as sources of failure. Evans and Tappin [2] and Carniglia [22] have also discussed interaction between the two (i.e. pores near the surface) being linked to surface machining flaws. Failure of some bodies from varying combinations of pores and machining flaws clearly adds to the variability of porosity effects. Competition between flaws (e.g. from machining) and pore-crack combinations affect both the applicability of the different cases and extrapolation of strengths to zero porosity. Extrapolation of strengths to zero porosity when other flaws (e.g. from machining) take over will be meaningful only if the sizes of the pore-crack flaws and the other (e.g. machining) flaws are similar. This can be the case in largergrain bodies where machining flaws are commonly smaller than the grain size, and strength depends on grain size [1, 23]. Thus, in Case 1, with small pores in larger-grain bodies, the strength is also limited by grain size, so extrapolation to zero porosity can often be reasonable. However, when the combination of machining conditions and grain size is such that flaws are bigger than the grains, extrapolation of Case 1 to zero porosity may not be meaningful. In Case 2, for pores which are large compared with the grain size, extrapolation to zero porosity will be meaningful only if R is approximately the same as the flaw size--a situation that may not be too common. When microplastic (e.g. slip or twinning) mechanisms become operative as P-+ 0, eqn. (4) is clearly inappropriate, as the Petch equation is needed instead. However, if pores act only as stress concentrators to enhance slip or twinning mechanisms of failure, then such extrapolation is justified. Because eqn. (4) shows that strength depends only indirectly on porosity through E and 7, grain size and porosity effects are separable, the Petch equation can be multiplied by a porosity correction as assumed by Rice [18] and Carniglia [22]. Such cases are, however, likely to be limited in the range of porosity over which they are applicable, as larger volumes of pores should provide flaws from which brittle fracture will occur in preference to microplastic failure. The range of porosity over which the above correction would hold should be approximately inversely proportional to the yield stress, i.e. greater in NaCI than in MgO. It is clear from the foregoing that strength is not normally controlled by average porosity, but
223 by extremes of p o r e size, .spacing etc. often associated with extremes of grain size. T h u s , to m a k e this t h e o r y predictive instead of only analytical, i.e. to calculate strengths b e f o r e instead of after failure, it is i m p o r t a n t to characterize p o r o s i t y (especially its extremes and associated grain sizes) m u c h m o r e extensively. T h e effect of the v o l u m e u n d e r stress, i.e. of the c o m p o n e n t (or s p e c i m e n size), or the m e a n s of stressing (e.g. true tension vs. flexure) s h o u l d be considered, first o n the p e r t i n e n c e of the different cases, then o n the c o m p e t i t i o n b e t w e e n cases. A s the stressed v o l u m e increases, the probability of finding a larger void or a cluster of voids increases. T h u s , for example, the o c c u r r e n c e of C a s e 1 is decreased, w h e r e a s that of Case 2 or o t h e r cases is increased. A s u m m a r y of the resultant cases, their equations, p o r o s i t y d e p e n d e n c e s and applicability, e.g. as a f u n c t i o n of stressed volume, is s u m m a r i z e d in Table 2.
3. Summary and conclusions A m o d e l b a s e d o n a colinear array o f p o r e c r a c k c o m b i n a t i o n s has b e e n used to evaluate p o r o s i t y effects o n the tensile strengths o f c e r a m ics. G r a i n size effects are either separable or not present (and h e n c e also effectively separable), d u e to possible c r a c k - g r a i n size relationships in m o s t c o m m o n l y e n c o u n t e r e d p o r o sity cases, p r o v i d e d y d o e s n o t d e p e n d o n G. W h e r e y shows m e a s u r a b l e G d e p e n d e n c e , i.e. in cases of substantial crystalline anisotropy, G and P effects o n strength m a y not be separable, at least at limited levels of P. Pore size and shape effects, which c o m m o n l y also correlate with p o r e location, are s h o w n not to directly affect strength w h e n p o r e s are smaller than the grain size, but can otherwise directly affect strength. H o w e v e r , p o r e location can affect strengths. Specifically, grain b o u n d a r y (especially triple-point) p o r e s should be m o r e serious b e c a u s e of their p r o p e n sity to lead to grain b o u n d a r y cracking, and h e n c e the o p p o r t u n i t y for linking of cracks o n intersecting grain b o u n d a r i e s , rather than possible linking of cracks f r o m p o r e s w h o s e stress fields overlap. It was also s h o w n that the p o r o s i t y d e p e n d e n c e of strength, fracture e n e r g y and Y o u n g ' s m o d u l u s will typically be quite similar, but s o m e variations due to y effects can occur. Shifts in trends with mixed (or i n h o m o g e n e o u s ) p o r o s i t y a n d the effects o f s p e c i m e n size are noted.
References 1 R. W. Rice, Microstructural dependence of mechanical properties in R. K. McCrane (ed.), Treatise on Materials Science and Technology, Vol. II, Academic Press, New York, 1977, p. 199. 2 A. G. Evans and G. Tappin, Effects of microstructure on the stress to propagate inherent flaws, Proc. Br. Ceram. Soc., 20 (1972) 275-297. 3 O. L. Bowie, Analysis of an infinite plate containing radial cracks originating at the boundary of an infinite circular hole, Z Math. Phys., 35 ( 1) (1956) 60-71. 4 F. I. Baratta, Stress intensity factor estimates for a peripherally cracked spherical void and a hemispherical surface pit, J. A m . Ceram. Soc., 61 ( 11-12) (1978) 490-93. 5 F. I. Baratta, Correction-stress intensity factor estimates for a peripherally cracked spherical void and a hemispherical surface pit, J. A m . Ceram. Sot., 62 (9-10) (1979) 527. 6 F. I. Baratta, Refinement of stress intensity factor estimates for a peripherally cracked spherical void and a hemispherical surface pit, J. A m . Ceram. So,:., 64 (1) (1981) C3-C4. 7 R.W. Rice, Pores as fracture origins in ceramics, J. Mater Sci., 19(1984) 895-914. 8 A. G. Evans, D. R. Biswas and R. M. Fulrath, Some effects of cavities on the fracture of ceramics: II, spherical cavities, J. Am. Ceram. Soc., 62(1-2)(1979) 101-106. 9 D. J. Green, Stress intensity factor estimates for annular cracks at spherical voids, J. A m . Ceram. Soc., 63 (5-6) (1980) 342-343. 10 J. Heinrich and D. Munz, Strength of reaction-bonded silicon nitride with artificial pores, Am. Ceram. Soe. Bull., 59(12)(1980) 1221-1222. 11 B. Molnar and R. W. Rice, Strength anisotropy in lead zirconate titanate transducer rings, Bull. Am. Ceram. Soc., 52 (6) (1973) 505-509. 12 R. W. Rice, Fractographic identification of strengthcontrolling flaws and microstructure, in R. C. Bradt, D. R H. Hasselman and E E Lange (eds.), Vol. I. Plenum, 1974, pp. 323-345. 13 D. R H. Hasselman, Analysis of the strain at fracture of brittle solids with high densities of microcracks, J. A m . Ceram. Soc., 52 (8) (1967) 458-459. 14 J. B. Wachtman, private communication, 1988. 15 R. W. Rice, S. W. Freiman and J. J. Mecholsky, Jr., The dependence of strength-controlling fracture energy on the flaw-size to grain-size ratio, J. Am. Ceram. Soc., 63 (3-4)(1980) 129-136. 16 R. W. Rice and S. W. Freiman, Grain-size dependence of fracture-energy in ceramics: II, a model for non-cubic materials, J. Am. Ceram. Soc., 64 (6) ( 1981 ) 350-354. 17 R. W. Rice, Elastic anisotropy and the grain size dependence of ceramic fracture energies, J. Mater. Sci., 19 (1984) 1267-1271. 18 R. W. Rice, Strength/grain-size effects in ceramics, Proc. Br. Ceram. Soc., 20 (1972) 205-257. 19 R. W. Rice, K. R. McKinney, C. Cm. Wu, S. W. Freiman and W. J. McDonough, Fracture energy of Si3N4, J. Mater. Sci., 20(1985) 1392-1406. 20 R. W. Rice, Test-microstructural dependence of fracture energy measurements in ceramics, in Fracture Mechanics Methods for Ceramics, Rocks and Concrete, A S T M Spec. Tech. Publ., 745(1981) 96-117.
224 21 R. W. Rice, Ceramic fracture features, observations, mechanisms, and uses, in Fractography of Ceramic and Metal Failures, ASTM Spec. Tech. Publ., 827 (1984) 5-103. 22 S.C. Carniglia, Working model for porosity effects on the uniaxial strength of ceramics, J. Am. Ceram. Soc., 55
(12), (1972)610-618. 23 R. W. Rice, Machining flaws and the strength grain size behavior of ceramics, in The Science of Ceramic Machining and Surface Finishing 11, Nat. Bur. Stand. (U.S.), Spec. Publ., 562 (1979) 429-452.
215
Relation of Tensile Strength-Porosity Effects in Ceramics to Porosity Dependence of Young's Modulus and Fracture Energy, Porosity Character and Grain Size R. W. RICE*
Naval Research Laboratory, Washington, DC 20375 (U.S.A.) (Received June 28, 1988; in revised form December 7, 1988)
Abstract
The dependence of tensile strength o on the porosity P of ceramics is considered, based on recent theories and observations of failure from pores and emanating microstructural cracks. Extensions of a model for such failure based on a colinear array of pore-crack combinations is used to evaluate such cr dependence and its relationship to the porosity dependence of Young's modulus E and fracture energy 7 as well as the dependence of strength on the grain size G. It is shown that the porosity dependences of o, E and 7 should usually be similar, as is generally observed. It is also shown that the dependences of cr on both P and G are generally separable even if the sizes of the cracks surrounding the pores are related to G. The concept of pore finkage due to connection by cracks following preferred (mainly grain boundary) paths is introduced, and is suggested to be much more common than pore linkage due t0 cracking from overlapping stress fields of closely spaced pores.
1. Introduction
Substantial progress has been made in studying the effects of porosity on mechanical properties, e.g. as reviewed by Rice [1]. Thus, for example, considerable documentation shows that porosity typically has a substantial effect on most mechanical properties. The greatest effects are seen for behavior measured under compression (e.g. compressive strength, hardness and wear), hence the necessity of distinguishing the type of loading being considered i.e. tensile loading in *Present address: W. R. Grace and Co., Columbia, MD 21044, U.S.A. 0921-5093/89/$3.50
our case. Within the tensile loading regime, broad similarity is observed in the porosity dependence of tensile (flexural) strength o, Young's modulus E, fracture energy y, and hence the fracture toughness Kic. The ratio of a property at some volume fraction porosity P to the property at P = 0 can generally be represented as e- bP where e is the Napierian logarithmic base and b is the slope from a semilogarithmic plot of the property vs. P. The value of b is constant for a given type of porosity and property, and often gives very similar, if not identical, values for the above-noted tensile properties of a given set of bodies of varying P. Although substantial progress has been made in modeling such porosity dependence, mainly of E and secondarily of 7 (and hence of Kic ) [1], major changes have occurred in our understanding of tensile failure of porous bodies [1-6], requiring new or modified approaches to modeling tensile strength. Previously, pores were seen simply as stress concentrators accentuating failure from other sources. Now, however, it is generally recognized that pores, if present, are often (if not always) an integral part of the failure causing flaws under tensile loading [1, 2, 7], which was generally not accounted for in previous strength-porosity models. There is thus the need to address pores as an integral part of failure causing flaws in the broader understanding of mechanical property-porosity relationships. Of particular pertinence to this paper are the following basic issues. (1) Is there a sound rationale for the common occurrence of similar porosity trends for o, E and 7; if so, over what range of porosity does it hold? (2) To what extent do pore size, shape and location affect the strength? (3) Is there a sound rationale for porosity and grain size G effects being separable, as commonly © Elsevier Sequoia/Printed in The Netherlands
216 assumed in microstructural studies; if so, over what range of porosity does it hold? All these need to be considered because pores can affect a not only via Kic (through E and ~) but also through flaw geometry size C, and possibly sharpness. E and 7 are interrelated and important because, typically, high temperature exposures increase G and reduce P as well as change pore location (i.e. all the porosity is intergranular at high P and smaller G, and changes to mixed intergranular and intragranular porosity at lower P and larger G). Knowing, for example, whether G and P effects are theoretically separable, and if so over what range, is important, e.g. for extrapolation. Thus, although separation is indicated empirically for microstructures of fine G and fine P that are most readily obtained and most often used for porosity studies, this is less certain for the wide range of porosity achievable. We present a model that addresses the above issues and provides a framework for addressing specific strength-porosity-microstructure relationships. In particular, interrelationships between P and G effects are considered. The model is based on the concept of pores as an integral part of failure causing flaws. It is essentially a modification and expansion of the concepts which Evans and Tappin [2] used for treating pores large compared with the grain size as sources of failure. They adopted Bowie's [3] model of a flaw consisting of a cylindrical hole (radius R with one or two radial cracks of length L (Fig. l(a)) for treating a spherical pore and the immediately adjoining layer of grains as a similar flaw (Fig. l(b)) by simply substituting the flaw shape parameter for a penny crack for that of a slit crack in the Griffith equation. This adaptation is based on the concept that cracks could propagate along grain boundaries emanating from the pore wall until the next layer of grains is encountered. Because the greater difficulty of crack propagation around or through the second layer of grains could arrest the crack at this second layer, the pore plus part of the immediately surrounding grains could constitute the critical flaw when such pores control strength. More recently, models have been developed for spherical pores plus cracks, which are generally a much better approximation of actual pores [4-6, 8, 9] identified as initiating failure. However, it has been proposed that cracks associated with pores may be a function of grain size [7, 8, 10], again raising the issue of the separability of P and G effects. There are some
(B)
R
//~L~L~ Fig. 1. Schematic diagram of pore-crack combinations. (a) Cross-section of cylindrical or spherical void of radius R with one or two (dashed portion) associated axial cracks for a cylindrical void (Bowie'smodel) or a spherical pore with an equatorial crack; in either case the length (or depth) of the crack into the surrounding material is L. (b) Schematic diagram of pore larger than surrounding grains, with possible crack length L (i.e. depth into material) being to the next layer of grains, i.e. 1/2 grain deep. (c) Triple-point pore with L suggested as the length of the adjoining grain boundary facet. (d) Intragranular pore (upper) and grain boundary pore (lower) and suggested L dimensions constrained by meeting of adjacent grains.
differences between the cylindrical and spherical models, as well as between either of these and some of the experimental results [7]. However, these differences are only of moderate numerical values for one or two factors, and are thus noted in the development for various cases.
2.
Theory
2.1. General theory To treat failure from pores, Evans and Tappin [2] adopted the equation for treating a periodic array of colinear planar cracks normal to an applied stress to treating similar arrays of the above cylindrical pore-crack combinations of the same size. They considered the stress cr to cause either failure from a single crack or the linking of two or more cracks in the periodic array. Although failure is the most common case (as discussed later), the lower of the two stresses is given by
a= 2 ~t-anfl} a-e
(1)
217
where fl=(:rtR/2R+ 2), R is the pore radius, 2 is the pore spacing, L is the length of the crack emanating from the pore (which may be related to the grain size (e.g. Fig. 1)), 7 is the fracture energy for the particular case and Op/Oc is the ratio of the stress to propagate a single pore-crack combination to the stress to propagate a crack of dimensions R + L. We first consider the nature of the pores to which the model is applicable. Although Evans and Tappin presented the above concept for large pores and finer grain sizes (R > L, Fig. l(b)), there is no reason why it cannot be applied to other pores and grain sizes. For example, it has been shown that their model can be extended to large non-circular pores by taking R as half the minor axis of an elliptical approximation of the pore in the fracture plane (typically at or near normal to the applied stress) [11, 12]. Although more complicated modification may have to be considered for some extreme pores, these observations allow most pores to be considered and, in particular, the very important cases of typical pores at triple points (Fig. 1(c)) and on grain boundaries (Fig. l(d)), as well as intragranular pores (Fig. l(d)). Similarly, it is not necessary to have intergranular failure around intergranular pores, consistent with the observation that transgranular rather than intergranular failure often occurs around large pores [1, 7, 11, 12]. Treatment of pores as indicated above is sketched in Fig. 1, with approximate values suggested by some models for L is in terms of the grain size G. As discussed elsewhere [6, 7] and later, it may be that, L '~ G with G ,> R, or that L > G for R ,> G. The approximation of a colinear array of through-the-thickness cylinder-crack combinations normal to the stress is valid for several reasons. Firstly, the orientation of pore-crack combinations essentially normal to the stress will be extremely common and clearly represents the expected case of failure. Secondly, many cases involve only a single pore (where the 13 terms cancel) or a few pores whose spacing is determined by other microstructural factors, e.g. grain size. Thus, the periodic assumption either is no longer involved or is reasonably satisfied. For the unusual cases where more pores are involved, an average periodicity or spacing may often be reasonable. Thirdly and more generally, the same periodic crack array approach is the basis of Hasselman's successful thermal shock models [13] where the cracks are not likely to be any
more periodically located than pores are. As for the nature of the pores themselves, much of the effect of porosity on o comes from its effects on E and 7, which do not depend on the above assumptions. Further, in many cases where flaw aspects of pores are important, the effect of the size (and hence most shape) aspects of the pore is negligible. When this is not the case, pore shape-size effects can be handled as for single pores. Use of flaw shape parameters for elliptical, surface or interior flaws [2] instead of slit or through-the-thickness cracks is reasonable and gives good results. However, the Op/Oc term should depend on pore shape. It becomes significantly greater than unity only when L/R decreases below about 0.1 for spherical [6] (as well as cylindrical [3]) pores. Because of uncertainties in other parameters (e.g. R, L and 7), critical tests of the applicability of Op/Oc values based on these models require L/R < 0.05, and hence have generally not been made. Thus, op/oc will generally be treated as negligible, i.e. equal to unity, except as noted. Experimental data [7] generally support oplo~ = 1 for polycrystalline bodies, at least failing from pores on machined surfaces. For glass specimens, Op/Oc is often greater than unity. Incorporating the above modifications, i.e. op/ o c = 1, and elliptical flaws yields
(2)
where Y= 1.25 for interior circular cracks and Y= 1.12 for surface semicircular cracks. Because pore-linking effects due to pore-stress interactions become significant only when the spacing between pore walls is less than their diameter [2], o of eqn. (1) will normally represent the failure stress in the majority of the cases of interest (other cases will be specifically noted). For small fl, tanfl ~ t3 (i.e. when 2/R > 1), so the 13 terms cancel (i.e. single pores control strength). Further, as shown in Table 1, (13/tan 13)1/2 decreases to only about 0.93 at P - 0 . 3 0 so (13/tanfl) 1/2-'~ 1 over the region of greatest interest. Where the 13terms are no longer negligible, they can be estimated using an expression for the spacing of spherical pores, which commonly has the form -
AR(1-P) P
(3)
218 TABLE 1
Values of (titian fl) 1/2 as a function of porosity
Volume fraction porosity P
(fl /tanfl) I/2.
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.80 0.90 0.95 0.97 0.98 0.99
0.998 0.992 0.982 0.969 0.952 0.933 0.911 0.886 0.858 0.823 0.756 0.667 0.554 0.399 0.284 0.221 0.181 0.128 0.000
1.00
aft = (3riP)/2P + 4 using 2 = (AR( 1 - P )/P) (with A = 4/3).
where P is the volume fraction porosity and A is a constant, e.g. 4/3. (A recent estimate by Wachtman [14] indicates that the difference in 2 values for various forms of eqn. (3), e.g. for different A values, is relatively moderate above P = 0.3. At high values of P (i.e. above 0.5), 2 is likely to average a fraction of the grain size.) Two major cases cover the most common porosity situations of interest: R < L and R > L. Considering these, as well as R = L and mixed cases, gives a broader perspective of strengthporosity relationships and addresses key issues discussed in the previous section. Table 2 represents a summary of the various cases and their results. 22. Case I: R < L with 2/t? > 1 We first consider the case R < L with 2/R > 1, which covers much of the common and important cases of grain boundary and intragranular pores (i.e. Figs. l(c) and l(d)). For P < 0.3, eqn. (1) reduces to a= V
(4)
(For this case, ap/a c of eqn. (1) is theoretically about unity, even for cylindrical pores.) When only a single pore is involved (Fig. 1), ), should typically be about the fracture energy along preferred fracture paths in single crystals or along grain boundaries [15]. In fine-grain bodies where,
except for sporadic, large isolated pores, singlepore causes of failure are probably uncommon, L may be about G/2 or could be several times G. In large-grain bodies where this mechanism should be more common, L may be more than (;/2, as discussed later. A particularly important extension of Case 1 is the linking of nearby pores because of their cracks impinging (mainly along grain boundaries) rather than because of their stress interaction. Examples of pores which are too far apart to link by stress interaction but which can link by crack impingement are sketched in Fig. 2. This impingement increases L, e.g. G <~L <<,(2-4)G, increasing 7, which will approach the polycrystalline value when enough grains are involved [15]. In this case, strength will not depend directly on pore size or shape, but only indirectly through the dependence of E and 7 on these factors. When L is related to grain size G, eqn. (4) gives an inverse square root dependence on grain size. The specifics of this dependence vary with the exact L/G ratio at various failure origins and may be modified by some grain size dependence of 7 [16, 17]. When 7 has no significant grain size dependence, then the grain size dependence of strength (via L ) can be separated from the porosity dependence as follows. Because 7 = 70fT(P) and E= EofE(P ) where 70 and E 0 are the zero porosity fracture energy and Young's modulus respectively, then =
= Yf(P)
(5)
where f(P)=(fT(P)fE(P))l/2. When ~, has significant dependence on G, this is commonly due to crystalline anisotropy effects [16, 17] which arise from microcracking. Such microcracking also depends on porosity which could preclude a true separation of G and P effects. However, because porosity will often suppress microcracking, the G dependence of 7 may often become rather weak in the presence of all but small amounts of porosity. Thus, whether or not L depends on G, the effects of P and G on o are separate. The P trends shown by eqn. (5) should also be noted. Polycrystalline fracture energies typically follow a similar trend with porosity as Young's modulus does [1], so when such fracture energies are applicable, e.g. for the important case of pore linking due to crack impingement, fy(P)'-~f~(P)~fo(P). (However, fy(P) can be
219 TABLE 2
Summary of various porosity cases
Case
Applicable equation a
Effects of P and G separable b
Direct strength effects of pore Size
Relationship of
Applicability
fo(P) to fE ( P )c
Shape
Y(E7/L) j/2
Yes
No
No
Limited mostly to pore linking by crack impingement. May decrease as stress volume increases Limited to cases where R is Large G: fr(P) <~fo(P)/> (fE(P))u2 much greater than other flaw sizes, e.g. very large-grain bodies. May decrease as stressed volume increases
Y(Ev/R) '/2
Yes
Yes
Yes
fy(P)~fy(P)=fE(P)
Frequent occurrence over a wide range of stressed volumes for limited P
(3) R = L
Y(Ev/R + L )1/2
Yes
Yes
Yes
fo(P)~f~(P)-~fE(P)
Frequent occurrence over a wide range of stressed volumes for limited and especially medium P
(4) Mixed Cases 1 and 2
Y(E7/R) j/2
Yes
Yes
Yes
fo(P)'~fr(P)=fe(P)
Frequent occurrence over a wide range of stressed volumes for limited and especially medium P
(5) R < L
Y(ET/C) U2
Yes
Yes
Yes
fo(P)-~fr(P)~fE(P)
Frequent for inhomogeneous low to medium P, especially as stressed volume increases
fl/tanfl(E7/C) 1/2
Yes
Some, but often limited
fo(P)generally becomes progressively less than fy(P) as P increases
Dominant for large P. Tends to increase as stressed volume increases, e.g. for medium P
Small or medium G: fo(P) = fE(P) (1) R < L (2/R) > l
(2) R> L
(A/R)> 1
(A/R)
<
1
(6) (A/R) < 1
Cis the size of flaw from pore linking. bAssuming y does not depend significantly on G. "Although fo(P)~ fE(P) for Cases 1-5, this similarity will often decrease in that order (except for Case 1 with large G). Failure due to other flaws or microplastic mechanisms will generally increase the similarity between fo(P) and fE(P ). a
s o m e w h a t greater than fE(P) in s o m e cases [1]. However, as possible fE(P) and fT(P) differences are not large, the statement fa(P ) = fiE( P )f7(P ))l/2 is generally true.) T h u s in this important instance of Case 1, the similar trends of E - P a n d o - P are explained. However, when failure initiates f r o m only one p o r e smaller than one of or all the surrounding grains, y should a p p r o a c h single-crystal or grain b o u n d a r y values and hence can have a reduced (and ultimately zero) d e p e n d e n c e on P. This would reduce the d e p e n d e n c e of a on P, e.g. in larger-grain bodies, and hence reduce but not destroy the similarity of E - P and a - P trends, i.e. in the extreme f , ( P ) = (fE(P))1/2. In addition to being consistent with the above general trend of the porosity d e p e n d e n c e of
strength, this case shows that substantial variation can occur in the porosity d e p e n d e n c e due to three factors. Firstly, there is the variation in that can occur with the n u m b e r of pores of the fracture origin and the size of associated grains. Secondly, there is the variation in E due to variation in the p o r e density in the b o d y which should also vary 7 [1]. Thirdly, there is the effect of p o r e location, as shown in Figs. l(c) and (d) and eqn. (4). Triple-point pores are normally the most severe, especially for single-pore sources of failure, as they typically have at least one emanating grain b o u n d a r y at a substantial angle to the stress (Bowie's solution for one emanating crack does not differ substantially f r o m the two-crack solution) and L m a y be larger, e.g. greater than or equal to G / 2 . However, m a n y pores along grain
220 A
C
D
Fig. 2. Schematic diagram of pore linkage by impingement of cracks. (a) Triple-point pores linked by cracks along grain boundaries. (b) Triple-point pore and grain boundary pore linked by grain boundary cracks. (c) Triple-point (upper) and grain boundary pore (lower) linked to intragranular pore by impingement of their grain boundary crack and transgranular crack respectively. (d) Linkage of small pore (e.g. triple-point void as shown here) to a large void. Similar effects could occur with intragranular pores (voids).
boundaries will be unfavourably oriented, and they typically have shorter potential cracks, e.g. L ~-G/4. For pores within grains, on average, L <~ G/2, which is less than the extremes for triple-point pores. Further, 7 for cracking across a grain is more than twice as high as 7 for cracking along a grain boundary section, and fracture on preferred cleavage planes substantially limits the opportunity of pore linking by crack impingement. The possible smaller value of L, the greater value of ~ and reduced pore rinking are thus other reasons (in addition to grain boundary stress concentrations) previously cited [18] why pores within grains should be less detrimental to strength than pores at grain boundaries. The differing effects of pores of different locations thus should lead to some variations in the effect of porosity on strength. Hence, a body of predominantly triple-point pores (e.g. as-hotpressed to less than theoretical density) should be weaker than one with the same porosity but with many intragranular pores (e.g. a body fired or
annealed at higher temperatures), provided all other variables (e.g. G) are equal, which is very difficult to achieve in practice. Pores, singly or in combination, will be the source of failure only if they are larger than other separate competing flaws. Such flaws are predominantly machine flaws, which are typically a few tens of micrometres in size [1]. Thus, for example, machining flaws will typically be substantially larger than single pores with R < G in most fine-grain bodies. Therefore, only pore linking, mainly due to crack impingement, is realistic for medium- and fine-grain bodies of Case 1, and even this may be limited. However, although the direct applicability of some aspects of Case 1 is generally limited, most of the trends are not. Thus, although some of the variability due to different pore shapes or location on the flaw effects may be removed, some of these effects also enter through E and y. Further, whenever pores do not act as the flaws fo(P) is equal, or similar, to fE(P), SO such similarity trends are reinforced. Also, grain size effects (due to effects on flaw size or 7) are often still separable, as in eqn. (5). Further, the frequently greater ease of cracking along grain boundaries and the shape and related stress concentration should make regions of grain boundary (and especially triple-point) pores preferred areas for machining flaws. Such pores can thus increase the size of such flaws. Similarly, higher densities of such pores may be preferred areas for such pore-enhanced machining flaw introduction.
2.3. Case 2: R > L with 2/R > 1 At the other extreme, R > L (where op/o c typically is greater than unity for pores in glasses [7]), some different results and some similar but more complicated results to those from Case 1 are obtained. With 2 /R > 1, i.e. P< 0.3, only single pores are normally important, so eqn. (1) reduces to
[E \~/2 a 2 YI~ )
(6)
A major difference from Case 1 is that, since R + L .~ R, there is no possible direct dependence on G if 7 has no significant dependence on G. (Again, if y depends on G, P and G effects may not in principle be separable, but may be at least approximately separable in practice.) However, pore size and shape (which also affect R)
221
directly affect Case 2. It is important to note that the y used in this case is that of the material surrounding the pore, which may be substantially higher than that measured for the body. Thus, for example, if a body's only porosity is a substantial number of large isolated pores, the energy for fracture from a large pore would approach or equal that of a dense body rather than the value measured by normal fracture mechanics tests which is lowered by the pores. Young's modulus is typically that of the body, but might be altered at very low L/R ratios because the strain energy, of which E is a critical factor, around the crack would then depend significantly on pore stress concentrations. If pore size remains constant so only the number of pores changes with P, the functional dependence and the rate of strength change with porosity will be due exclusively to Y and E as in Case 1, although the dependence of E and especially of y on P may vary from Case 1. If only the pore size increases with porosity (i.e. the number of pores per unit volume N remains constant), the rate and functional dependence of strength on porosity will depend not only on E and y, but also on R. However, R changes slowly with P, e.g. if uniform spherical pores are assumed
R= ( 3P1~/3 k47tN ] so strength has only a minus one-sixth ( - 1 / 6 ) power dependence on porosity through R in addition to a dependence on R through y and E. As R increases, there may be greater variations in its values (with the maximum value controlling strength). This will also give greater variations in E and possibly y giving greater variation from Case 1, typically a more rapid decrease in strength with increasing P. Normally, both N and R change with porosity so the rate of strength change with porosity for R > L will often be similar to that for R < L, i.e. Case 1. Thus, as in Case 1, there will be similar E-P, y-P and a - P trends. It is important to note that the variable strengths within Case 2 (R > L) can result from the same level of porosity because of variable pore shape and especially pore size. The relative strengths between Cases 1 and 2 for a given porosity depends on two factors. The first is (L/ R) ~/2 which would appear always to be less than unity. The second factor in comparing strengths for Cases 1 and 2 is the combined porosity
dependence of E and y. The porosity dependence of y in Case 2 can be substantially less than that in Case 1, whereas the porosity dependence of E may be somewhat more in Case 2 than in Case 1 [1]. Thus the combined porosity dependence of E and y in Case 2 will often be less than in Case 1, which can partly compensate for the lower value of (L/R)I/2.
2.4. Other porosity cases Three other cases complete the range of possible porosity. The case when R = L will be similar to Case 1, but both R and L must be considered. For example, for 2/R > 1, eqn. (1) reduces to
a -~ y( Ey I ~/2 \R+L]
(7)
(Again, ap/Oc = 1 for cylindrical and spherical pores.) Thus, if L or R or both are related to G, porosity and grain size effects on strength can be expressed as separate functions if y does not depend on G. (If y depends significantly on G, P and G effects may not be separable, as noted earlier.) Effects varying a for a given P will tend to be similar to those for both Cases 1 and 2, but again overall fo(P) will be similar to fE(P ). Combinations of Cases 1 and 2, i.e. pores both large and small compared with the grain size constitute another case. The large pores will dominate the strength; the small pores will affect the strength by extending the cracks around large pores if they are sufficiently close (e.g. Fig. 2), thus increasing L and increasing the porosity dependence of E, and probably that of y. Therefore the overall dependence will be that of Case 2 (but probably with reduced G dependence). Thirdly, 2/R ~< 1 for any of the preceding cases means that eqn. (2), rather than giving the stress for failure, gives the stress for crack linking pores due to pore-pore interactions, not impingement of cracks. This case of close pore spacing can occur at limited porosity levels for fine-grain boundary pores (i. e. Case 1 ) and for inhomogeneous porosity of any of the above cases. This will typically increase the value of L, i.e. since one crack from a pore is nearly as serious as two, the larger crack linking two pores will dominate. Often the combined pore-crack configuration can be treated as a sharp flaw with the dimensions of the configuration, i.e. substituting this size for
222 R, L or R + L. This adds some variability in fo(P) for a given P and to its similarity to fz(P). In such cases, resultant flaw sizes are not likely to be closely related to G, giving little or no G dependence unless Y depends on G. More commonly, 2/R < 1 when P is large (e.g. 0.4 or more). In most such cases there must be a mixture of pore sizes based on simple percolation concepts; large pores will dominate and pore-crack linking will occur. The porosity dependence of strength will still enter through E and 7, but will also enter through fl and the size of the group of pores that makes up the final flaw size (substituting for R + L in eqn. (2)). The variability in the size of the pore cluster thus becomes the dominant factor in varying strength effects. This size, and increasing effects due to fl as P increases, generally leads to fo(P)> rE(P). However, at large P, 7 measured by larger cracks (e.g. as in double cantilever beam tests) may exhibit crack branching leading to large 7 values [8, 9]. But such branching typically requires greater crack propagation than occurs with strengthcontrolling cracks, so f~(P) < f~(P)[19, 20].
2.5. Model limits and interaction with other failure mechanisms It is useful to consider the limits of the model as P--" 0 and as P--" 1.0. Equations (1) and (2) show that if uniform porosity is assumed, a--" 0 as P--" 1.0. Changing pore shape would change the shape of the o - P curve, but not the limit. Inhomogeneous distribution of porosity would make a ~ 0 at P < 1, as it should. At the other extreme, as P ~ 0, eqns. (1) and (2) reduce exactly to eqn. (4). However, other than showing that a as a function of P is well behaved as P--'0, this is not necessarily meaningful, because when P = 0, the mechanism of failure of this theory disappears. Other failure mechanisms become competitive or cooperative with the pore-crack mechanism over a range of porosity as P ~ 0, and become exclusively operative at P = 0. There are two major mechanisms of failure that will come into play as P ~ 0: microplastic and flaw mechanisms. The latter will be other sources of flaws, most commonly surface flaws from machining; internal cracks, e.g. resulting from thermal expansion anisotropy, might also play a role in some cases. Rice [7, 21] has shown that there is considerable competition between
machining flaws and pores in acting as sources of failure. Evans and Tappin [2] and Carniglia [22] have also discussed interaction between the two (i.e. pores near the surface) being linked to surface machining flaws. Failure of some bodies from varying combinations of pores and machining flaws clearly adds to the variability of porosity effects. Competition between flaws (e.g. from machining) and pore-crack combinations affect both the applicability of the different cases and extrapolation of strengths to zero porosity. Extrapolation of strengths to zero porosity when other flaws (e.g. from machining) take over will be meaningful only if the sizes of the pore-crack flaws and the other (e.g. machining) flaws are similar. This can be the case in largergrain bodies where machining flaws are commonly smaller than the grain size, and strength depends on grain size [1, 23]. Thus, in Case 1, with small pores in larger-grain bodies, the strength is also limited by grain size, so extrapolation to zero porosity can often be reasonable. However, when the combination of machining conditions and grain size is such that flaws are bigger than the grains, extrapolation of Case 1 to zero porosity may not be meaningful. In Case 2, for pores which are large compared with the grain size, extrapolation to zero porosity will be meaningful only if R is approximately the same as the flaw size--a situation that may not be too common. When microplastic (e.g. slip or twinning) mechanisms become operative as P-+ 0, eqn. (4) is clearly inappropriate, as the Petch equation is needed instead. However, if pores act only as stress concentrators to enhance slip or twinning mechanisms of failure, then such extrapolation is justified. Because eqn. (4) shows that strength depends only indirectly on porosity through E and 7, grain size and porosity effects are separable, the Petch equation can be multiplied by a porosity correction as assumed by Rice [18] and Carniglia [22]. Such cases are, however, likely to be limited in the range of porosity over which they are applicable, as larger volumes of pores should provide flaws from which brittle fracture will occur in preference to microplastic failure. The range of porosity over which the above correction would hold should be approximately inversely proportional to the yield stress, i.e. greater in NaCI than in MgO. It is clear from the foregoing that strength is not normally controlled by average porosity, but
223 by extremes of p o r e size, .spacing etc. often associated with extremes of grain size. T h u s , to m a k e this t h e o r y predictive instead of only analytical, i.e. to calculate strengths b e f o r e instead of after failure, it is i m p o r t a n t to characterize p o r o s i t y (especially its extremes and associated grain sizes) m u c h m o r e extensively. T h e effect of the v o l u m e u n d e r stress, i.e. of the c o m p o n e n t (or s p e c i m e n size), or the m e a n s of stressing (e.g. true tension vs. flexure) s h o u l d be considered, first o n the p e r t i n e n c e of the different cases, then o n the c o m p e t i t i o n b e t w e e n cases. A s the stressed v o l u m e increases, the probability of finding a larger void or a cluster of voids increases. T h u s , for example, the o c c u r r e n c e of C a s e 1 is decreased, w h e r e a s that of Case 2 or o t h e r cases is increased. A s u m m a r y of the resultant cases, their equations, p o r o s i t y d e p e n d e n c e s and applicability, e.g. as a f u n c t i o n of stressed volume, is s u m m a r i z e d in Table 2.
3. Summary and conclusions A m o d e l b a s e d o n a colinear array o f p o r e c r a c k c o m b i n a t i o n s has b e e n used to evaluate p o r o s i t y effects o n the tensile strengths o f c e r a m ics. G r a i n size effects are either separable or not present (and h e n c e also effectively separable), d u e to possible c r a c k - g r a i n size relationships in m o s t c o m m o n l y e n c o u n t e r e d p o r o sity cases, p r o v i d e d y d o e s n o t d e p e n d o n G. W h e r e y shows m e a s u r a b l e G d e p e n d e n c e , i.e. in cases of substantial crystalline anisotropy, G and P effects o n strength m a y not be separable, at least at limited levels of P. Pore size and shape effects, which c o m m o n l y also correlate with p o r e location, are s h o w n not to directly affect strength w h e n p o r e s are smaller than the grain size, but can otherwise directly affect strength. H o w e v e r , p o r e location can affect strengths. Specifically, grain b o u n d a r y (especially triple-point) p o r e s should be m o r e serious b e c a u s e of their p r o p e n sity to lead to grain b o u n d a r y cracking, and h e n c e the o p p o r t u n i t y for linking of cracks o n intersecting grain b o u n d a r i e s , rather than possible linking of cracks f r o m p o r e s w h o s e stress fields overlap. It was also s h o w n that the p o r o s i t y d e p e n d e n c e of strength, fracture e n e r g y and Y o u n g ' s m o d u l u s will typically be quite similar, but s o m e variations due to y effects can occur. Shifts in trends with mixed (or i n h o m o g e n e o u s ) p o r o s i t y a n d the effects o f s p e c i m e n size are noted.
References 1 R. W. Rice, Microstructural dependence of mechanical properties in R. K. McCrane (ed.), Treatise on Materials Science and Technology, Vol. II, Academic Press, New York, 1977, p. 199. 2 A. G. Evans and G. Tappin, Effects of microstructure on the stress to propagate inherent flaws, Proc. Br. Ceram. Soc., 20 (1972) 275-297. 3 O. L. Bowie, Analysis of an infinite plate containing radial cracks originating at the boundary of an infinite circular hole, Z Math. Phys., 35 ( 1) (1956) 60-71. 4 F. I. Baratta, Stress intensity factor estimates for a peripherally cracked spherical void and a hemispherical surface pit, J. A m . Ceram. Soc., 61 ( 11-12) (1978) 490-93. 5 F. I. Baratta, Correction-stress intensity factor estimates for a peripherally cracked spherical void and a hemispherical surface pit, J. A m . Ceram. Sot., 62 (9-10) (1979) 527. 6 F. I. Baratta, Refinement of stress intensity factor estimates for a peripherally cracked spherical void and a hemispherical surface pit, J. A m . Ceram. So,:., 64 (1) (1981) C3-C4. 7 R.W. Rice, Pores as fracture origins in ceramics, J. Mater Sci., 19(1984) 895-914. 8 A. G. Evans, D. R. Biswas and R. M. Fulrath, Some effects of cavities on the fracture of ceramics: II, spherical cavities, J. Am. Ceram. Soc., 62(1-2)(1979) 101-106. 9 D. J. Green, Stress intensity factor estimates for annular cracks at spherical voids, J. A m . Ceram. Soc., 63 (5-6) (1980) 342-343. 10 J. Heinrich and D. Munz, Strength of reaction-bonded silicon nitride with artificial pores, Am. Ceram. Soe. Bull., 59(12)(1980) 1221-1222. 11 B. Molnar and R. W. Rice, Strength anisotropy in lead zirconate titanate transducer rings, Bull. Am. Ceram. Soc., 52 (6) (1973) 505-509. 12 R. W. Rice, Fractographic identification of strengthcontrolling flaws and microstructure, in R. C. Bradt, D. R H. Hasselman and E E Lange (eds.), Vol. I. Plenum, 1974, pp. 323-345. 13 D. R H. Hasselman, Analysis of the strain at fracture of brittle solids with high densities of microcracks, J. A m . Ceram. Soc., 52 (8) (1967) 458-459. 14 J. B. Wachtman, private communication, 1988. 15 R. W. Rice, S. W. Freiman and J. J. Mecholsky, Jr., The dependence of strength-controlling fracture energy on the flaw-size to grain-size ratio, J. Am. Ceram. Soc., 63 (3-4)(1980) 129-136. 16 R. W. Rice and S. W. Freiman, Grain-size dependence of fracture-energy in ceramics: II, a model for non-cubic materials, J. Am. Ceram. Soc., 64 (6) ( 1981 ) 350-354. 17 R. W. Rice, Elastic anisotropy and the grain size dependence of ceramic fracture energies, J. Mater. Sci., 19 (1984) 1267-1271. 18 R. W. Rice, Strength/grain-size effects in ceramics, Proc. Br. Ceram. Soc., 20 (1972) 205-257. 19 R. W. Rice, K. R. McKinney, C. Cm. Wu, S. W. Freiman and W. J. McDonough, Fracture energy of Si3N4, J. Mater. Sci., 20(1985) 1392-1406. 20 R. W. Rice, Test-microstructural dependence of fracture energy measurements in ceramics, in Fracture Mechanics Methods for Ceramics, Rocks and Concrete, A S T M Spec. Tech. Publ., 745(1981) 96-117.
224 21 R. W. Rice, Ceramic fracture features, observations, mechanisms, and uses, in Fractography of Ceramic and Metal Failures, ASTM Spec. Tech. Publ., 827 (1984) 5-103. 22 S.C. Carniglia, Working model for porosity effects on the uniaxial strength of ceramics, J. Am. Ceram. Soc., 55
(12), (1972)610-618. 23 R. W. Rice, Machining flaws and the strength grain size behavior of ceramics, in The Science of Ceramic Machining and Surface Finishing 11, Nat. Bur. Stand. (U.S.), Spec. Publ., 562 (1979) 429-452.