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Fracture stability, R-curves and strength variability
Acta metoll. Vol. 36, No. 3, pp. 555-562, 1988 Printed in Great Britain. All rights reserved
Copyright
FRACTURE STABILITY, R-CURVES STRENGTH VARIABILITY
0
oool-6160188 53.00 + 0.00 1988 Pergamon Journals Ltd
AND
R. F. COOK and D. R. CLARKE IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. (Received
10 June 1987)
Abstract-Variability in fracture strength is examined as a function of the increase in fracture resistance with crack length. The driving force for fracture is modelled generally as the sum of two components, one stabilizing and the other destabilizing crack propagation, leading to a crack extension force displaying a minimum as a function of crack length. The resistance to crack extension, the R-curve, is modelled as an increasing power law. By invoking the equilibrium and instability conditions for fracture, strength is thus shown to have a simple power law dependence on the magnitude of the stabilizing component of the crack driving force. The magnitude of this power law dependence is shown to be inversely related to the rate of increase of the R-curve. This latter result implies that although strengths will increase for a given flaw size in the presence of an R-curve, a more important ramification is that the related increased crack stabilization leads to diminished distributions of strengths for given ranges of flaw sizes. An expression is derived quantifying this conclusion, showing an increasing relation between the Weibull modulus of a strength distribution and the slope of the R-curve. The results substantiate the empirical finding that appropriate R-curve behavior is an effective means of achieving high, narrow strength distributions, independent of randomly introduced flaws. R&un&Nous examinons la variabilite de la resistance globale a la rupture en fonction de l’augmentation de la resistance a la rupture avec la longueur de la fissure. La force motrice de la rupture est en general modelisee comme la somme de deux composantes, l’une stabilisant et l’autre destabilisant la propagation des fissures, ce qui conduit a une force d’ouverture des fissures qui prisente un minimum en fonction de la longueur des fissures. Nous modihsons la resistance a I’ouverture des fissures (la courbe R) par une loi de puissance croissante. En invoquant les conditions d’tquilibre et d’instabilite de la rupture, nous montrons ainsi que la resistance depend de la valeur de la composante stabilisante de la force motrice de la fissure, selon une loi de puissance simple. Cette dtpendance est inversement proportionnelle a la vitesse d’accroissement de la courbe R. Ce dernier risultat implique que--quoique la resistance croisse pour une taille don&e du defaut en presence dune courbe R-la stabilisation accrue des fissures qui lui est associb conduit a une repartition limit&e de resistances pour un domaine don& de tailles de defauts. Nous en deduisons une expression qui quantifie cette conclusion et montre une relation croissante entre le module de Weibull dune repartition de resistances et la pente de la courbe R. Les rtsultats justifient les conclusions empiriques qu’un comportement de courbe R approprie est un moyen efficace pour realiser des repartitions de resistances hautes et ttroites, independantes des dtfauts introduits au hasard. Zusanunenfaasung-Die Veriinderung der Bruchfestigkeit wird in Abhlngigkeit von dem Anstieg im Bruchwiderstand mit der RiBlange untersucht. Die treibende Kraft fur den Bruch wird allgemein mit einer Summe aus zwei Komponenten modellhaft beschriebcn: eine Komponente stabihsiert, die andere destabilisiert den RiBfortschritt. Dieses fiihrt zu einer Kraft fiir den Rillfortschritt, welche ein Minimum in der Abhlngigkeit von der RiBlange aufweist. Die Kurve Widerstand iiber der RiBausdehnung, die sogenannte R-Kurve, wird mit einem ansteigenden Potenzgesetz beschrieben. Unter Beriicksichtigung der Gleichgewichts- und Instabilitatsbedingungen fur den Bruch wird gezeigt, dal3 die Festigkeit mit einem einfachen Potenzgesetz von der Hohe der stabilisierenden Komponente der treibenden Kraft fiir den Bruch abhlngt. Das Smrke dieser Potenzgesetz-Abhlngigkeit hangt umgekehrt mit dem Anstieg der R-Kurve zusammen. Dieses Ergebnis wiederum bedeutet, dal3 letztlich die damit zusammenhlngende verbesserte RiBstabilisierung zu einer engeren Verteilung der Festigkeit fur einen gegebenen Bereich von Spaltgr6Ben fiihrt. Ein Ausdruck zur Quantilizierung dieser Folgerung wird abgeleitet, welcher auf einen zunehmend engeren Zusammenhang mit dem Weibull-Parameter einer Festigkeitsverteilung mit der Steigung der R-Kurve hinweist. Die Ergebnisse untermauem den empirischen Befund, da13ein geeignetes R-Kurvenverhalten ein wirksames Mittel ist, eine enge Verteilung hoher Festigkeiten unabhangig von zufalhg eingefiihrten Spalten zu erhalten.
1. INTRODUCTION
to as “R-curve”
behavior,
as the fracture resistance of crack length c, instead of remaining constant. As the measured fracture resistante usually shows a decreasing tendency for increased toughening with increasing crack propagation, ie the toughness appears to tend to an upper plateau,
R increases as a function
Many materials, including metals [l], ceramics [2], polymers [3], and composites [4], exhibit increasing resistance to fracture, increasing toughness, with crack extension. This behavior is frequently referred
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COOK and CLARKE:
FRACTURE STRENGTH VARIABILITY AND R-CURVES
fracture data can usually be well described by an empirical power law R a c2’ with 2r < 1. This particularly simple form permits fracture resistance data to be easily compared and also allows for relatively straightforward analysis of toughness functions in the prediction of component strengths. The fracture strength of a component depends on both the material toughness properties and the nature and magnitude of the strength controlling flaw. Kendall et al. recently examined strength variation in the presence of an R-curve [S]: by describing the fracture resistance by a power law and modelling the dominant Aaws as simple cracks under uniform applied loading Kendall et al. predicted that an R-curve would narrow the range of component strengths for a given range of flaw sizes. The analysis of Kendall et al. is of restricted utility however, as the simple stress distribution used by them leads to a monotonic increase with crack length in the driving force for fracture which is not frequently encountered for typical strength controlling flaws. Here we examine the strength properties of materials described by power law R-curves using a more general strength controlling flaw. An additional driving force for crack extension will be superposed on the uniform loading used earlier by Kendall et al. The additional driving force takes the form of a localized loading, which leads to a decreasing driving force for fracture with increasing crack length. The net driving force for fracture then has the feature of decreasing at small crack lengths, passing through a minimum, and increasing at large crack lengths. Non-monotonic crack extension forces such as this are seen in many mate~al/flaw systems. For instance, Brown and Ogin [6] show that cracks propagating out of persistent slip bands, formed during the cyclic straining of plastic materials, metals or ionic crystals, have this form of net crack extension force. In this case the short-range, decreasing component arises from localized bands of intense slip which carry much of the plastic strain in the stressed component. Marshall and Lawn [7j show that cracks generated by sharp particle contacts on the surfaces of brittle materials, ceramics or semiconductors, also have this type of non-monotonic driving force. In these cases the localized plastic deformation zone at the contact leads to a residual stress field which decreases as the crack propagates into the surrounding material [8-111. A crack centered on an imbedded inclusion with differing elastic or thermal expansion properties from the surrounding matrix will be influenced by a similarly decreasing stress field as shown by Swain [ 121 and Green [ 131. Hence, the driving force for fracture will exhibit a minimum during extension of the crack from the inclusion as the matrix is stressed. Specimen geometry alone may be enough to lead to a driving force for fracture which exhibits a minims. Cracks in flexed components, which are comparable in length with the component thickness, or which propagate through components of varying dimen-
sions, undergo similar variations in crack driving force on crack extension [14]. (We also note that some standard fracture mechanics test geometries, such as the single edge-notched beam, chevron-notched beam, tapered cantilever beam and the double torsion specimen, are susceptible to this latter variation [15,16].) We will begin by defining the equilibrium criteria for stable and unstable fracture and the crack extension and resistance forces. This allows the determination of strengths under various loading conditions in the presence of the increasing R-curve. The variability of the strengths thus derived is then considered, using the modulus of the Weibull distribution, m as a statistical indicator of the width of a strength distribution. An analytic expression for the Weibull modulus, m, as a function of the slope parameter of the R-curve, r, will be determined. As well as modelling the behavior of flaws more typically strength controlling in brittle materials, our analysis provides insight into general questions of crack stability in the presence of increasing fracture resistance and the usefulness of power-law R-curves for describing the properties of real materials. 2. FRACTURE MECHANICS Here we first define stable and unstable equilibrium fracture criteria and com~nent “strength”. Second, the specific forms of the crack driving force, consisting of both local and uniform components, and the crack resistance are presented. The following sections thence consider the strength of components containing flaws under local, uniform and combined loadings. 2.1, Equilibrium fracture criteria The driving force for crack propagation will be characterized here by the stress intensity factor, K, which is a measure of the amplitude of the stress field at a crack tip. Within the elastic limit (i.e. for truly brittle materials) R is related to the mechanical energy release rate for virtual crack advance, G, by K = (GE)“‘, where E is the Young’s modulus [l-3]. We will find the stress intensity factor more useful than the mechanical energy release rate here, as stress intensity factors arising from separate sources of loading on a crack are simply additive. The quantity analogous to the stress intensity factor which characterizes the resistance to crack extension is the toughness, T (written as 1% in the earlier literature) [l-3]. Toughness is a measure of the critical crack tip stress for crack advance and is related to the work necessary for incremental crack extension, R, by T = (RE)“‘. Equilibrium for the fracture system is obtained when the mechanical energy released for a virtual crack advance equals the work used to create new crack surfaces, G = R. In terms of the stress intensity factor and toughness the equilibrium condition is K = T
(equilibrium).
(1)
COOK and CLARKE:
FRACTURE
STRENGTH
VARIABILITY
AND R-CURVES
557
We note that this condition is equivalent to finding a point of zero derivative in the potential energy of the system as a function of crack length [17]. The stabiiity of the system is obtained by considering whether the potential energy of the system is a mi~mum or a maximum at this point. Hence, taking derivatives with respect to crack length of equation (1) we gain the conditions for stable and unstable equilibrium dK/dc < dT/dc corres~nding
(stability)
(2)
to a minimum in potential energy, and dKldc > dT/dc
(instability)
(3)
corresponding to a maximum in potential energy. In stable equilibrium the system returns to its initial configumtion for ~rturbations in the crack length, the rate of return determined by the kinetics of nonequilibrium crack propagation. When the equilibrium is unstable the crack spontaneously propagates away from the initial configuration, in the direction of the perturbation. If the kinetics of non-equilibrium propagation are rapid and the ~rt~bation of the system is in the direction of increasing crack length catastrophic failure of the component may ensue. It is this latter case which interests us here, as the “strength” of a component is observed when an applied stress perturbs a crack beyond an unstable equilib~um position, in conjunction with rapid nonequilibrium crack propagation kinetics. 2.2. Crack driving and resistance forces Here the net stress intensity factor K is the sum of two components, one arising from a uniform applied stress, K* and the other arising from a localized loading, K,,
CRACK LENGTH, c
Fig. I. (a) Plot of stress intensity factor as a function of crack length for a crack under uniform applied stress, lu,, and localized stress, &, (logarithmic coordinates). The solid line is the net stress intensity factor [equation (4)] and the dotted lines the asymptotic limits [equations (5) and (6)]. (b) Plot of toughness as a function of crack length for a
power law R-curve [equation (7)] (logarithmic coordinates).
the net K decreases to a minimum. On further crack extension the applied stress intensity factor begins to K = K&$ K,. (4) dominate and the net K increases with crack length. The applied stress intensity factor takes the form We see that K, is a stabilizing influence on the fracture system and & a destabilizing influence (equations 2, K8 = 8&r,&$9 (5) 3, 5, 6). where o-, is the uniform applied stress, c is the The choice of the residual stess intensity factor crack length and II/ is a numerical crack geometry applicable to a contact flaw for the decreasing parameter (e.g. $ = rcli2 for an imbedded linear crack, component of K provides the contact load P as a and $ = 2/7~“~ for an imbedded circular crack 1141). convenient measure of the magnitude of the stabilizing stress field and hence flaw size. However, any We model the localized component by the residual stress intensity factor arising from the elastic/plastic localized loading on a crack will stabilize the system deformation field of a sharp particle contact in much the same way. For a crack arising at a persistent slip band (PSB) in a plastic material K, = xP/c”= (6) Eshelby (in an appendix to Ref. [6]) shows that the where P is the peak contact Load, x is a numerical stress intensity factor arising from the PSB is of the constant dependent on the contact geometry and the same form as equation (6) with XP replaced by terms elastic/plastic properties of the material, and r is a in the strain and width of the PSB and r = 1 appronumerical constant characterizing the geometry of priate to a linear geometry. For a crack centered the localized loading (e.g. r = 1 for line force center on an imbedded inclusion which has undergone a loading of a linear crack and r = 3 for point force volume change on phase transformation or with center loading of a circular crack) [7, 11, 141. Figure differing elastic or thermal expansion properties from I(a) plots K from equations (4x6) as a function of the surrounding matrix, xP in equation (6) would be c (r = 3). At small crack lengths the residual stress replaced by expressions in the volume change, elastic moduli or thermal expansion coefficients, and the intensity factor dominates and as the crack extends
558
COOK and CLARKE:
FRACTURE
STRENGTH
VARIABILITY
AND R-CURVES
toughness, T = T& R-curve behavior, z > 0 increases the possibility of system stability (equations 2,7). 2.3. Cracks under center loading For zero applied stress, Q, = 0, the net stress intensity factor is simply given by the residual stress intensity factor, K = K, and the equilibrium condition is K, = T. As dK,/dc < dT/dc for all c, this equilibrium is stable (equations 2, 6, 7) as shown in Fig. 2(a). For a given contact load P there is a unique crack length c, at which the system remains stable to perturbations in the crack length. Larger or smaller values of P will result in variations in co, but there is no contact load at which the crack will propagate spontaneously. Increasing the contact load from P to P, [Fig. 2(a)] simply increases the stable crack length. The crack length c, is obtained by equating equations (6) and (7) and inverting to give c0 = (,Pd’/TO)*“*‘+‘?
(8)
Increasing z both decreases c, for a given value of P and decreases the variability in c,, for variations in P. 2.4. Cracks under uniform loading
CRACK LENGTH, c
Fig. 2. (a) Plot demonstrating
stable equilibrium. A crack
driven by a decreasing residual stress intensity factor, 4, and resisted by an increasing toughness function, Z’, has a stable length co for a given contact load P. (b) Plot demonstrating unstable equilibrium. A crack driven by an increasing applied stress intensity factor, KS, becomes unstable at a given uniform applied stress a, for a given crack length c,,. (c) Plot demonstrating a transition from stable to unstable equilibrium. A crack driven by both stress intensity factors, K = K, + K, undergoes stable precursor growth from c, to c* as the applied stress is increased from 0 to Q*, whereupon unstable failure ensues,
inclusion dimension [12, 131. For changes in component geometry which stabilize the crack, the parameter $ in equation (5) is altered to become a function of the crack length and component dimensions, rather than a constant [14]. As stated we assume here a power law dependence for the fracture resistance, and hence toughness, on crack length T = T~(c/d~
(c >, d)
(7)
For uniform loading of a crack of length c, in the absence of a residual stress field, x = 0, the net stress intensity factor is given by the applied stress intensity factor, K = K,, and the ~uilib~um condition is & = T. For 7 CI l/2 we see that the opposite case is obtained from that above, here the system is unstable as dK,/dc > dT/dc for all c. Hence, there is a value of the applied stress, cr,, at which the crack will propagate spontaneously for perturbations in the crack length beyond c,, or in the applied stress above cr,. Impositions of applied stresses smaller than a, [say u,, Fig, 2(b)] will not cause failure, as although K, is destabilizing, the system is not in equilibrium for the given crack length (theoretically the crack should close for o, < a, but metastable bond rupture processes usually prevent this). The magnitude of the strength a, is obtained by equating equations (5) and (7) and inverting to give cr,,= TOcsz - ‘)‘2/dr$.
(9Q)
Increasing 7 increases a, for a given value of c, but decreases the variability in c,, for variations in c,. If the crack is the result of a sharp particle contact (with the residual stress annealed or etched away) then c, may be replaced by P by combining equations (8) and (9a) to give o, = (T~/d’lL)(XPd’/T,)‘*‘-“‘“+“.
(9b)
For 7 > l/2 the system is always stable and we shown as a straight line in the logarithmic coordinates of Fig. l(b) (z = 0.25). The amplitude term TO revert to the description of the previous section. Increases in the applied stress in this case simply lead may be regarded as setting the base toughness of the to larger equilibrium crack lengths, with no possibility material in the absence of any toughening mechanisms, and the scaling term d as the spatial extent of of the spontaneous propagation shown in Fig. 2(b). the crack at which toughening begins. The toughen2.5. Cracks under combined loading ing exponent z characterizes the rate at which the For cracks under the influence of both the residual toughness increases (Z = 0 corresponds to constant
COOK and CLARKE:
FRACTURE
STRENGTH
VARIABILITY
AND R-CURVES
c*/c,, = [(r + l)/(l - 2~)]“(~‘+‘).
lo2
559
(14)
Combining equations 9(b) and (13) gives the change in strength by the additional loading of the residual stress field a*/% = [(2r + r)/(r + l)] x [(l + r)/(l - 2~)](~‘-‘)/(~~+‘) . (15)
0.0
I
,
0.1
0.2
I
I
0.5
0.6
I
I
0.3
0.4
Both of these ratios increase from lower bounds set by the geometry of the residual stress field as 7 increases. Setting 7 = 0 and r = 3 leads to the usual indentation/strength equations for materials with no R-curve effects (e.g. glass) containing point contact flaws (e.g. Vickers indentations). Hence from equations (13)-(15) we gain
R-CURVE EXPONENT, 7
Fig. 3. Plot of Weibull modulus, M, as a function of the slope of the R-curve, r (equation (22)], indicating significantly narrower strength distributions for materials possessing rapid increases in toughness with crack length. Note the asymptotic limit at T = l/2, where the slope of the toughness curve T equals the slope of the applied stress intensity factor K,.
and applied stress intensity factors the equilibrium and instability conditions must both be invoked in order to determine the strength. In this case the syste,n is in equilibrium all the time and moves stably from (c = cO, u, = 0) to (c = c*, 0, = LT*)as the applied stress is increased, along the R-curve (or more correctly, T-curve [17]) from T = T(c,) to T = T(c*) as shown in Fig. 2(c). The parameters c*, cr* indicate the critical values of the crack length and the applied stress at which the system passes from stability to instability (7 < l/2 assumed). To determine these parameters we impose the equilibrium condition (equations 1, 46). I(lr~,c”~+ xP/c”* = T&c/d)
(10)
and the instability condition [the derivative of equation (lo)] $(T,c”~ - rXP/c’l’ = 2zT,(c/d)‘. Solving equations gain
(10) and (11) simultaneously
c* = [XP(r + l)d’/T,(l
- 2~)]*“*‘+‘)
CT*= [T,(2z + r)/$d’(r
+ l)]~*(*~-‘)‘~
(11)
we
c */co = 4*13N 2.52 u */as = 3/44’3 N 0.47
(1W (16b) (16~)
noting that equations (16b) and (16c) are the lower limits for equations (14) and (15), respectively, for the case of point geometry. An important result from our analysis to this point is that the strength of a component, Q may be written generally in terms of the slope of the R-curve 7, as u=BP
(2r- 1)/(Zr + r)
(17)
where /l is a material/indenter constant derivable from either equation (9b) or equation (13). We see then that the assumption of a power law R-curve predicts a simple power law dependence of strength on contact load. Increasing the slope of the R-curve, i.e. increasing 7, leads to a weaker dependence of strength on contact load. For the case of point flaws this implies a reduction in the magnitude of the slope of the strengthcontact load response from - l/3 for 7 = 0 [equation (16a)] towards 0 as 7 + l/2. The increased stabilization of the crack by the R-curve is seen explicitly in the increased precursor crack extension prior to failure from 2.52 for 7 = 0 [equation (16b)] towards 03 as 7 + l/2 [equation (14)]. The decreased range of measured strengths for a given range of flaw sizes (contact loads or crack lengths) as the slope of the R-curve is increased will be considered in the next section.
(12)
= [To(2z + r)/lLd’(r + l)] x XP(r + l)d’/T,(l
CJ* = 3 T;‘3/44’311/ (xP)“~
- 2~)](*~~‘)/(~‘+‘) (13)
and we note that both c* and U* have the same dependence on contact load as their counterparts c,, and a,,. Increasing 7 increases Q* for a given value of P but decreases the variability in cr* for variations in P. Combining equations (8) and (12) we have the degree of subcritical, stable equilibrium growth of the crack before failure
3. STRENGTH VARIABILITY A range of dominant flaw sizes in a group of components gives rise to a range of observed strengths described by an extreme value distribution [ 181. The strength distribution we choose here for a group of brittle components is the Weibull distribution [ 18-201
F = 1 - exp[ - k(a/a,)“]
(18)
where F is the cumulative probability of failure of a component at strength 6, gN and m characterize
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COOK and CLARKE:
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the distribution and k is a specimen geometry parameter.? The parameter m is the Weibull modulus and is an inverse measure of the spread of strengths, large values of m imply narrow distributions. The following will determine the variations in m for changes in the R-curve behavior. We begin by noting that the Weibull modulus may be operationally determined by the upper and lower values of the strength distribution
4. DISCUSSION
The major point to be drawn from our considerations on strength variability is that R-curves decrease the width of strength distributions for given ranges of flaw sizes, independent of the absolute magnitude of the toughness, a conclusion also reached by Kendall et al. [S]. Here however, we have extended the generality of the flaw causing failure (locally stressed or not, line or point geometry) and have derived ln( 1 - F,)/ln( 1 - FL) = (a&~,)~ (19) explicit solutions for the change in the Weibull modulus as a function of the local slope of the R-curve where bu and oL are the upper and lower strength [equation (22)]. This latter relationship highlights a values, respectively. The conjugate failure probabilities, F, and FL are related to the number of key point: it is not the existence of thl R-curve per se which results in decreased ranges of strength specimens in the distribution, N, by distributions, but rather the slope of the R-curve at F, = N/(N + 1) (204 the crack lengths of interest. If the R-curve is not steep enough at the appropriate crack lengths, such that FL= l/(N+l) Gob) little stabilization of the cracks occurs, the strength and hence the left side of equation (19) is simply a distribution will not be appreciably narrowed. Where function of the number of specimens in the distribusignificant stabilization is observed we might expect tion and we denote this function by q = q(N). The to observe limited strength distributions. Crack propratio of the strengths-in equation (19) is determined agating out of persistent slip bands in copper single by the range of flaw sizes in the distribution. From crystals show strong stabilization and significant equation (17) we have precursor growth before failure (referred to as “dormant cracks” [6]). The relatively constant number of C”/OL= (p”/p#‘)/Qr+I) (21) loading cycles to failure for these copper specimens where Pu and PL are the contact loads conjugate to suggests that a narrow strength distribution may well the upper and lower strength levels respectively and result for simple monotonic loading of the cracks we note that P, < Pb Combining equations (19)-(21) in the dormant stage. Large-grained polycrystalline gives aluminas demonstrate considerable R-curve behavior [17,21,22]. Strength tests on such materials show m = [In rl W)IWLIPuII UT + r)l(l - 27)l.(22) considerably narrower strength distributions in flaw The first term in square brackets on the right side size regions where the R-curve is steeper (small flaws) of equation (22) is only a function of the number of than in regions where the R-curve is shallow (large specimens and the range of flaw sizes and is therefore flaws) [17,23], in agreement with equation (22). a constant for a given sample size and range of Zirconia materials in which the tetragonal phase is specimen treatments. The second term gives the stabilized show R-curve behavior arising from the dependence of the Weibull modulus on the toughness dilatational transformation of tetragonal particles to characteristics of the material. As can be seen m monoclinic in the crack tip stress field 1241. Such depends only on the slope of the R-curve and not on materials show considerably less variation in strength the absolute value of the toughness [Z’, does not with flaw size than materials in which there is little appear in equation (22)]. Figure 3 plots the value of retained tetragonal phase and shallower R-curves m as a function of z from equation (22) using N = 100 [25], again in agreement with equation (22). and PJP,, = 50. Significant increases in the Weibull Increasing the degree of stable crack growth raises modulus are observed as the slope of the R-Curve two interesting further possibilities for those processincreases. We note also that the Weibull modulus at ing brittle materials. First, non-destructive evaluation a given value of z is greater for r = 3 than for r = 1, techniques (NDE) may be more able to recognize especially at low z. This emphasizes that it is the potentially hazardous flaws arising in the processing stabilization of the fracture system (whether by either procedures. Pre-stressing (“proof-testing”) of comthe crack driving force or the crack resistance ponents to grow large stable cracks easily found by influence) which determines strength variability. NDE, from less easily detectable small flaws, may well be possible in materials possessing steeply sloped R-curves [equation (14)]. Second, the emphasis in tThe use.of the two parameter Weibull distribution, as op- processing brittle materials to obtain high strengths posed to the three parameter distribution, is applicable may be shifted away from the refinement of processhere, as no truncation or lower bound on the strength ing methods to remove strength limiting random distribution is imposed. It can also be shown that the flaws (pores, inclusions, regions of discontinuous simple shift in the strength distribution described by the grain growth, scratches, etc.). Instead, advantage three parameter distribution [18] does not influence our conclusions. may be taken of the fact that increasing the slope
COOK and CLARKE:
FRACTURE
STRENGTH
of the R-curve of a material leads to a degree of flaw tolerance in the strength characteristics, i.e. the strength decreases less for a given increase in flaw size [equation (17)]. Large flaws randomly incorporated during processing will have less effect, provided the microstructure of the material has been systematically manipulated to obtain a strong R-curve effect. We also note from Fig. 3 that stabilization of the strength controlling defect itself will aid in refining strength distributions: the stronger stabilization of point flaws over linear flaws [equation (6)] leads to greater values of m for a given R-curve. The implication is that processing to achieve defects with more strongly localized stress distributions [leading to greater values of r in equation (6)] may also decrease strength variability. There is also the added advantage that the resultant strength of the component under such circumstances will not be influenced by non-equilibrium growth of the dominant flaw (in the localized stress field) after processing. If the dominant flaw extends from c0 to a non-equilibrium length ci (>c,,) in the presence of a reactive environment [such that K(ch) < T(ch), Fig. 2(c)] the strength of the component will not alter. Provided ch < c* the necessary precursor stable growth fixes the instability point, and hence strength, of the system (at c*, a*). The assumption of a power-law R-curve for a brittle material leads to simple power-law predictions for the dependence of strength on flaw size. The increased crack stability introduced by the R-curve causes the magnitudes of the slopes of strength vs flaw size plots to be reduced. However, while it is sometimes possible to fit a select portion of a strength vs indentation load plot (in logarithmic coordinates) with a straight line segment of slope less than - l/3 it is generally not possible to describe the entire strength degradation response by a single straight line [23]. Many brittle materials display very distinct curvature in their strength vs indentation load responses, tending to slopes of - l/3 at large indentation loads and load independent strength plateaus at small indentation loads [23]. The observed continuously changing slopes suggest that the underlying toughness changes of these materials cannot be completely described by simple power-law R-curves. The strong tendency to slopes of -l/3 at large indentation loads in many materials is indicative of a tendency to an upper, constant toughness level. This suggests that the major feature of power-law R-curves which renders them unphysical is the lack of accounting for a saturation of the toughening mechanism at large crack lengths. In metallic or polymeric materials where the deformation zone at the crack tip (dislocation plasticity, crazing, voiding, etc.) is not limited in size, a continuously rising R-curve may well be observed until the crack propagates entirely through the component and the inhomogeneous deformation zone reaches the component boundaries [1, 31. In ceramic materials or rocks, the deformation zone causing the toughening (ligamentary bridges,
VARIABILITY
AND R-CURVES
561
phase transformations, microcracking, etc.) is often limited in scale by the material stiffness. The larger elastic moduli of ceramic materials thus sets an upper bound to the lateral extent of the process zone which is frequently much less than the scale of the component [22,24,26]. Hence saturation effects are much more important in ceramic systems and simple powerlaw R-curves may not be effective for description of the entire toughness response. In closing it is noted that direct measurements of toughness are frequently made using large crack lengths. As discussed above power-law fits to toughness data under these circumstances may severely overestimate the toughness which pertains at smaller crack lengths. The use of extrapolations of such power-law fits in the prediction of component strengths, which are controlled by small flaws, may then result in considerable overestimate, notwithstanding the stable growth of the crack prior to failure. Similar overestimates of small flaw strength may occur if power-law fits are made to large indentation load strength data. We see then that the powerlaw R-curve is not consistent with conservative design principles. 5. CONCLUSIONS The strength of a component containing a dominant flaw under the influence of stabilizing and destabilizing stress fields has been presented as a function of the rate of increase in toughness with crack length. The assumption of a power-law R-curve characterizing this increase leads to power-law dependencies for both the stable, equilibrium crack length and the strength on the magnitude of the stabilizing stress field. These dependencies are inversely related to the slope of the R-curve. The implication is that for a given range of flaw sizes, materials with steeper R-curves will show smaller ranges of strengths. The Weibull modulus of a strength distribution is directly related to the slope of the R-curve, independent of the absolute magnitude or spatial scale of the toughness function (and also increases with the degree of localization of the stabilizing stress field). Correlations between R-curve slopes and strength-flaw size plots in both transforming and non-transforming ceramic materials support this conclusion. REFERENCES 1. D. Broek, Elementary Engineering Fracture Mechanics, 3rd edn. Martinus Nijhoff, The Hague (1974). 2. B. R. Lawn and T. R. Wilshaw, Fracture of Brittle Solids. Cambridge Univ. Press (1975). 3. J. G. Williams, Fracture Mechanics of Polymers. Ellis Horwood, Chichester (1974). 4. D. B. Marshall and A. G. Evans, in Fracfure Mechanics of Ceramics (edited by R. C. Bradt, A. G. Evans, D. P. H. Ha&man and F. F. Lange), Vol. 7, pp. I-15. Plenum, New York (1986). __ 5. K. Kendall. N. McN. Alford. S. R. Tan and J. D. Birchall. J. ‘Mater. Res. 1, 120’(1986). 6. L. M. Brown and S. L. Ogin, in Fundamentals of Deformation and Fracture (The Eshelby Memorial Sym-
562
7. 8. 9.
10.
Il. 12. 13.
14.
COOK and CLARKE:
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STRENGTH
posium) (edited by B. A. Biiby, K. J. Miller and J. R. Willis), pp. 501-529. Cambridge Univ. Press (1985). D. B. Marshall and B. R. Lawn, 3. Muter. Sci. 14, 2001 (1979). D. B. Marshali and B. R. Lawn, 3. Am. Ceram. Sue. 64, C-6 (1981). B. R. Lawn, in Fracture Mechanics of Ceramics (edited by R. C. Bradt, A. G. Evans, D. P. H. Ha&man and F. F. Lange), Vol. 5, pp. l-25. Plenum, New York (1983). H. P. Kirchner and E. D. Isaacson, in Fracture Mechanics of Ceramics (edited by R. C. Bradt, A. 0. Evans, D. P. H. Hasselman and F. F. Lange), Vol. 5, pp. 57-70. Plenum, New York (1983). B. L. Symonds, R. F. Cook and B. R. Lawn, 3. Mater. Sci. 18, 1306 (1983). M. V. Swain, 3. Mater. Sci. 16, 151 (1981). D. J. Green, in Fracture Mechanics in Ceramics (edited by R. C. Bradt, A. G. Evans, D. P. H. Has&man and F. F. Lange), Vol. 5, pp. 457-478. Plenum, New York (1983). H. Tada, P. C. Paris and G. R. Irwin, The Stress Analysb of Cracks Handbook. Del Research, St. Louis, MO. (1973).
VARIABILITY
AND R-CURVES
1s. B. J. Pletka, E. R. Fuller Jr and B. G. Koepke, m Fracture Mechanics Applied to Brittle Materials (edited by S. W. Freiman), ASTM STP 678, pp. 14-37. Am. Sot. Test. Mat.. Philadelohia (19791. 16. M. Sakai and K: Yomasaki, 3. hm. beram. Sec. 66,371 (1983). 17. Y.-W. Mai and B. R. Lawn, Ann. Rev. Mater. Sci. 16, 415 (1986).
18. R. H. Doremus, 3. uppl. Phys. 54, 193 (1983). W. Weibull, 3. appl. Mech. 18, 293 (1951). ::: G. J. DeSalvo, Theory and Structural Design Applications of WeibuN Statistics. General Westinghouse, Pittsburgh, Pa (1970). 21. R. Knehans and R. Steinbrech, 3. heater. Sci. Lett. 1, 327 (1982).
22. M. V. Swain, 3. Mater. Sci. Left. 5, 1313 (1986). 23. R. F. Cook, B. R. Lawn and C. J. Fairbanks, 3. Am. Ceram. Sot. 68, 604 (1985).
24. M. V. Swain and L. R. F. Rose, 3. Am. Ceram. Sot. 69, 511 (1986).
2s. D. B. Marshall, 3. Am. Ceram. Sot. 69, 173 (1986). 26. P. L. Swanson, C. J. Fairbanks, B. R. Lawn, Y.-W. Mai and B. J. Hockey, J, Am. Ceram. SOC. 70, 279 (1987).
Copyright
FRACTURE STABILITY, R-CURVES STRENGTH VARIABILITY
0
oool-6160188 53.00 + 0.00 1988 Pergamon Journals Ltd
AND
R. F. COOK and D. R. CLARKE IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. (Received
10 June 1987)
Abstract-Variability in fracture strength is examined as a function of the increase in fracture resistance with crack length. The driving force for fracture is modelled generally as the sum of two components, one stabilizing and the other destabilizing crack propagation, leading to a crack extension force displaying a minimum as a function of crack length. The resistance to crack extension, the R-curve, is modelled as an increasing power law. By invoking the equilibrium and instability conditions for fracture, strength is thus shown to have a simple power law dependence on the magnitude of the stabilizing component of the crack driving force. The magnitude of this power law dependence is shown to be inversely related to the rate of increase of the R-curve. This latter result implies that although strengths will increase for a given flaw size in the presence of an R-curve, a more important ramification is that the related increased crack stabilization leads to diminished distributions of strengths for given ranges of flaw sizes. An expression is derived quantifying this conclusion, showing an increasing relation between the Weibull modulus of a strength distribution and the slope of the R-curve. The results substantiate the empirical finding that appropriate R-curve behavior is an effective means of achieving high, narrow strength distributions, independent of randomly introduced flaws. R&un&Nous examinons la variabilite de la resistance globale a la rupture en fonction de l’augmentation de la resistance a la rupture avec la longueur de la fissure. La force motrice de la rupture est en general modelisee comme la somme de deux composantes, l’une stabilisant et l’autre destabilisant la propagation des fissures, ce qui conduit a une force d’ouverture des fissures qui prisente un minimum en fonction de la longueur des fissures. Nous modihsons la resistance a I’ouverture des fissures (la courbe R) par une loi de puissance croissante. En invoquant les conditions d’tquilibre et d’instabilite de la rupture, nous montrons ainsi que la resistance depend de la valeur de la composante stabilisante de la force motrice de la fissure, selon une loi de puissance simple. Cette dtpendance est inversement proportionnelle a la vitesse d’accroissement de la courbe R. Ce dernier risultat implique que--quoique la resistance croisse pour une taille don&e du defaut en presence dune courbe R-la stabilisation accrue des fissures qui lui est associb conduit a une repartition limit&e de resistances pour un domaine don& de tailles de defauts. Nous en deduisons une expression qui quantifie cette conclusion et montre une relation croissante entre le module de Weibull dune repartition de resistances et la pente de la courbe R. Les rtsultats justifient les conclusions empiriques qu’un comportement de courbe R approprie est un moyen efficace pour realiser des repartitions de resistances hautes et ttroites, independantes des dtfauts introduits au hasard. Zusanunenfaasung-Die Veriinderung der Bruchfestigkeit wird in Abhlngigkeit von dem Anstieg im Bruchwiderstand mit der RiBlange untersucht. Die treibende Kraft fur den Bruch wird allgemein mit einer Summe aus zwei Komponenten modellhaft beschriebcn: eine Komponente stabihsiert, die andere destabilisiert den RiBfortschritt. Dieses fiihrt zu einer Kraft fiir den Rillfortschritt, welche ein Minimum in der Abhlngigkeit von der RiBlange aufweist. Die Kurve Widerstand iiber der RiBausdehnung, die sogenannte R-Kurve, wird mit einem ansteigenden Potenzgesetz beschrieben. Unter Beriicksichtigung der Gleichgewichts- und Instabilitatsbedingungen fur den Bruch wird gezeigt, dal3 die Festigkeit mit einem einfachen Potenzgesetz von der Hohe der stabilisierenden Komponente der treibenden Kraft fiir den Bruch abhlngt. Das Smrke dieser Potenzgesetz-Abhlngigkeit hangt umgekehrt mit dem Anstieg der R-Kurve zusammen. Dieses Ergebnis wiederum bedeutet, dal3 letztlich die damit zusammenhlngende verbesserte RiBstabilisierung zu einer engeren Verteilung der Festigkeit fur einen gegebenen Bereich von Spaltgr6Ben fiihrt. Ein Ausdruck zur Quantilizierung dieser Folgerung wird abgeleitet, welcher auf einen zunehmend engeren Zusammenhang mit dem Weibull-Parameter einer Festigkeitsverteilung mit der Steigung der R-Kurve hinweist. Die Ergebnisse untermauem den empirischen Befund, da13ein geeignetes R-Kurvenverhalten ein wirksames Mittel ist, eine enge Verteilung hoher Festigkeiten unabhangig von zufalhg eingefiihrten Spalten zu erhalten.
1. INTRODUCTION
to as “R-curve”
behavior,
as the fracture resistance of crack length c, instead of remaining constant. As the measured fracture resistante usually shows a decreasing tendency for increased toughening with increasing crack propagation, ie the toughness appears to tend to an upper plateau,
R increases as a function
Many materials, including metals [l], ceramics [2], polymers [3], and composites [4], exhibit increasing resistance to fracture, increasing toughness, with crack extension. This behavior is frequently referred
555
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COOK and CLARKE:
FRACTURE STRENGTH VARIABILITY AND R-CURVES
fracture data can usually be well described by an empirical power law R a c2’ with 2r < 1. This particularly simple form permits fracture resistance data to be easily compared and also allows for relatively straightforward analysis of toughness functions in the prediction of component strengths. The fracture strength of a component depends on both the material toughness properties and the nature and magnitude of the strength controlling flaw. Kendall et al. recently examined strength variation in the presence of an R-curve [S]: by describing the fracture resistance by a power law and modelling the dominant Aaws as simple cracks under uniform applied loading Kendall et al. predicted that an R-curve would narrow the range of component strengths for a given range of flaw sizes. The analysis of Kendall et al. is of restricted utility however, as the simple stress distribution used by them leads to a monotonic increase with crack length in the driving force for fracture which is not frequently encountered for typical strength controlling flaws. Here we examine the strength properties of materials described by power law R-curves using a more general strength controlling flaw. An additional driving force for crack extension will be superposed on the uniform loading used earlier by Kendall et al. The additional driving force takes the form of a localized loading, which leads to a decreasing driving force for fracture with increasing crack length. The net driving force for fracture then has the feature of decreasing at small crack lengths, passing through a minimum, and increasing at large crack lengths. Non-monotonic crack extension forces such as this are seen in many mate~al/flaw systems. For instance, Brown and Ogin [6] show that cracks propagating out of persistent slip bands, formed during the cyclic straining of plastic materials, metals or ionic crystals, have this form of net crack extension force. In this case the short-range, decreasing component arises from localized bands of intense slip which carry much of the plastic strain in the stressed component. Marshall and Lawn [7j show that cracks generated by sharp particle contacts on the surfaces of brittle materials, ceramics or semiconductors, also have this type of non-monotonic driving force. In these cases the localized plastic deformation zone at the contact leads to a residual stress field which decreases as the crack propagates into the surrounding material [8-111. A crack centered on an imbedded inclusion with differing elastic or thermal expansion properties from the surrounding matrix will be influenced by a similarly decreasing stress field as shown by Swain [ 121 and Green [ 131. Hence, the driving force for fracture will exhibit a minimum during extension of the crack from the inclusion as the matrix is stressed. Specimen geometry alone may be enough to lead to a driving force for fracture which exhibits a minims. Cracks in flexed components, which are comparable in length with the component thickness, or which propagate through components of varying dimen-
sions, undergo similar variations in crack driving force on crack extension [14]. (We also note that some standard fracture mechanics test geometries, such as the single edge-notched beam, chevron-notched beam, tapered cantilever beam and the double torsion specimen, are susceptible to this latter variation [15,16].) We will begin by defining the equilibrium criteria for stable and unstable fracture and the crack extension and resistance forces. This allows the determination of strengths under various loading conditions in the presence of the increasing R-curve. The variability of the strengths thus derived is then considered, using the modulus of the Weibull distribution, m as a statistical indicator of the width of a strength distribution. An analytic expression for the Weibull modulus, m, as a function of the slope parameter of the R-curve, r, will be determined. As well as modelling the behavior of flaws more typically strength controlling in brittle materials, our analysis provides insight into general questions of crack stability in the presence of increasing fracture resistance and the usefulness of power-law R-curves for describing the properties of real materials. 2. FRACTURE MECHANICS Here we first define stable and unstable equilibrium fracture criteria and com~nent “strength”. Second, the specific forms of the crack driving force, consisting of both local and uniform components, and the crack resistance are presented. The following sections thence consider the strength of components containing flaws under local, uniform and combined loadings. 2.1, Equilibrium fracture criteria The driving force for crack propagation will be characterized here by the stress intensity factor, K, which is a measure of the amplitude of the stress field at a crack tip. Within the elastic limit (i.e. for truly brittle materials) R is related to the mechanical energy release rate for virtual crack advance, G, by K = (GE)“‘, where E is the Young’s modulus [l-3]. We will find the stress intensity factor more useful than the mechanical energy release rate here, as stress intensity factors arising from separate sources of loading on a crack are simply additive. The quantity analogous to the stress intensity factor which characterizes the resistance to crack extension is the toughness, T (written as 1% in the earlier literature) [l-3]. Toughness is a measure of the critical crack tip stress for crack advance and is related to the work necessary for incremental crack extension, R, by T = (RE)“‘. Equilibrium for the fracture system is obtained when the mechanical energy released for a virtual crack advance equals the work used to create new crack surfaces, G = R. In terms of the stress intensity factor and toughness the equilibrium condition is K = T
(equilibrium).
(1)
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AND R-CURVES
557
We note that this condition is equivalent to finding a point of zero derivative in the potential energy of the system as a function of crack length [17]. The stabiiity of the system is obtained by considering whether the potential energy of the system is a mi~mum or a maximum at this point. Hence, taking derivatives with respect to crack length of equation (1) we gain the conditions for stable and unstable equilibrium dK/dc < dT/dc corres~nding
(stability)
(2)
to a minimum in potential energy, and dKldc > dT/dc
(instability)
(3)
corresponding to a maximum in potential energy. In stable equilibrium the system returns to its initial configumtion for ~rturbations in the crack length, the rate of return determined by the kinetics of nonequilibrium crack propagation. When the equilibrium is unstable the crack spontaneously propagates away from the initial configuration, in the direction of the perturbation. If the kinetics of non-equilibrium propagation are rapid and the ~rt~bation of the system is in the direction of increasing crack length catastrophic failure of the component may ensue. It is this latter case which interests us here, as the “strength” of a component is observed when an applied stress perturbs a crack beyond an unstable equilib~um position, in conjunction with rapid nonequilibrium crack propagation kinetics. 2.2. Crack driving and resistance forces Here the net stress intensity factor K is the sum of two components, one arising from a uniform applied stress, K* and the other arising from a localized loading, K,,
CRACK LENGTH, c
Fig. I. (a) Plot of stress intensity factor as a function of crack length for a crack under uniform applied stress, lu,, and localized stress, &, (logarithmic coordinates). The solid line is the net stress intensity factor [equation (4)] and the dotted lines the asymptotic limits [equations (5) and (6)]. (b) Plot of toughness as a function of crack length for a
power law R-curve [equation (7)] (logarithmic coordinates).
the net K decreases to a minimum. On further crack extension the applied stress intensity factor begins to K = K&$ K,. (4) dominate and the net K increases with crack length. The applied stress intensity factor takes the form We see that K, is a stabilizing influence on the fracture system and & a destabilizing influence (equations 2, K8 = 8&r,&$9 (5) 3, 5, 6). where o-, is the uniform applied stress, c is the The choice of the residual stess intensity factor crack length and II/ is a numerical crack geometry applicable to a contact flaw for the decreasing parameter (e.g. $ = rcli2 for an imbedded linear crack, component of K provides the contact load P as a and $ = 2/7~“~ for an imbedded circular crack 1141). convenient measure of the magnitude of the stabilizing stress field and hence flaw size. However, any We model the localized component by the residual stress intensity factor arising from the elastic/plastic localized loading on a crack will stabilize the system deformation field of a sharp particle contact in much the same way. For a crack arising at a persistent slip band (PSB) in a plastic material K, = xP/c”= (6) Eshelby (in an appendix to Ref. [6]) shows that the where P is the peak contact Load, x is a numerical stress intensity factor arising from the PSB is of the constant dependent on the contact geometry and the same form as equation (6) with XP replaced by terms elastic/plastic properties of the material, and r is a in the strain and width of the PSB and r = 1 appronumerical constant characterizing the geometry of priate to a linear geometry. For a crack centered the localized loading (e.g. r = 1 for line force center on an imbedded inclusion which has undergone a loading of a linear crack and r = 3 for point force volume change on phase transformation or with center loading of a circular crack) [7, 11, 141. Figure differing elastic or thermal expansion properties from I(a) plots K from equations (4x6) as a function of the surrounding matrix, xP in equation (6) would be c (r = 3). At small crack lengths the residual stress replaced by expressions in the volume change, elastic moduli or thermal expansion coefficients, and the intensity factor dominates and as the crack extends
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COOK and CLARKE:
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toughness, T = T& R-curve behavior, z > 0 increases the possibility of system stability (equations 2,7). 2.3. Cracks under center loading For zero applied stress, Q, = 0, the net stress intensity factor is simply given by the residual stress intensity factor, K = K, and the equilibrium condition is K, = T. As dK,/dc < dT/dc for all c, this equilibrium is stable (equations 2, 6, 7) as shown in Fig. 2(a). For a given contact load P there is a unique crack length c, at which the system remains stable to perturbations in the crack length. Larger or smaller values of P will result in variations in co, but there is no contact load at which the crack will propagate spontaneously. Increasing the contact load from P to P, [Fig. 2(a)] simply increases the stable crack length. The crack length c, is obtained by equating equations (6) and (7) and inverting to give c0 = (,Pd’/TO)*“*‘+‘?
(8)
Increasing z both decreases c, for a given value of P and decreases the variability in c,, for variations in P. 2.4. Cracks under uniform loading
CRACK LENGTH, c
Fig. 2. (a) Plot demonstrating
stable equilibrium. A crack
driven by a decreasing residual stress intensity factor, 4, and resisted by an increasing toughness function, Z’, has a stable length co for a given contact load P. (b) Plot demonstrating unstable equilibrium. A crack driven by an increasing applied stress intensity factor, KS, becomes unstable at a given uniform applied stress a, for a given crack length c,,. (c) Plot demonstrating a transition from stable to unstable equilibrium. A crack driven by both stress intensity factors, K = K, + K, undergoes stable precursor growth from c, to c* as the applied stress is increased from 0 to Q*, whereupon unstable failure ensues,
inclusion dimension [12, 131. For changes in component geometry which stabilize the crack, the parameter $ in equation (5) is altered to become a function of the crack length and component dimensions, rather than a constant [14]. As stated we assume here a power law dependence for the fracture resistance, and hence toughness, on crack length T = T~(c/d~
(c >, d)
(7)
For uniform loading of a crack of length c, in the absence of a residual stress field, x = 0, the net stress intensity factor is given by the applied stress intensity factor, K = K,, and the ~uilib~um condition is & = T. For 7 CI l/2 we see that the opposite case is obtained from that above, here the system is unstable as dK,/dc > dT/dc for all c. Hence, there is a value of the applied stress, cr,, at which the crack will propagate spontaneously for perturbations in the crack length beyond c,, or in the applied stress above cr,. Impositions of applied stresses smaller than a, [say u,, Fig, 2(b)] will not cause failure, as although K, is destabilizing, the system is not in equilibrium for the given crack length (theoretically the crack should close for o, < a, but metastable bond rupture processes usually prevent this). The magnitude of the strength a, is obtained by equating equations (5) and (7) and inverting to give cr,,= TOcsz - ‘)‘2/dr$.
(9Q)
Increasing 7 increases a, for a given value of c, but decreases the variability in c,, for variations in c,. If the crack is the result of a sharp particle contact (with the residual stress annealed or etched away) then c, may be replaced by P by combining equations (8) and (9a) to give o, = (T~/d’lL)(XPd’/T,)‘*‘-“‘“+“.
(9b)
For 7 > l/2 the system is always stable and we shown as a straight line in the logarithmic coordinates of Fig. l(b) (z = 0.25). The amplitude term TO revert to the description of the previous section. Increases in the applied stress in this case simply lead may be regarded as setting the base toughness of the to larger equilibrium crack lengths, with no possibility material in the absence of any toughening mechanisms, and the scaling term d as the spatial extent of of the spontaneous propagation shown in Fig. 2(b). the crack at which toughening begins. The toughen2.5. Cracks under combined loading ing exponent z characterizes the rate at which the For cracks under the influence of both the residual toughness increases (Z = 0 corresponds to constant
COOK and CLARKE:
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VARIABILITY
AND R-CURVES
c*/c,, = [(r + l)/(l - 2~)]“(~‘+‘).
lo2
559
(14)
Combining equations 9(b) and (13) gives the change in strength by the additional loading of the residual stress field a*/% = [(2r + r)/(r + l)] x [(l + r)/(l - 2~)](~‘-‘)/(~~+‘) . (15)
0.0
I
,
0.1
0.2
I
I
0.5
0.6
I
I
0.3
0.4
Both of these ratios increase from lower bounds set by the geometry of the residual stress field as 7 increases. Setting 7 = 0 and r = 3 leads to the usual indentation/strength equations for materials with no R-curve effects (e.g. glass) containing point contact flaws (e.g. Vickers indentations). Hence from equations (13)-(15) we gain
R-CURVE EXPONENT, 7
Fig. 3. Plot of Weibull modulus, M, as a function of the slope of the R-curve, r (equation (22)], indicating significantly narrower strength distributions for materials possessing rapid increases in toughness with crack length. Note the asymptotic limit at T = l/2, where the slope of the toughness curve T equals the slope of the applied stress intensity factor K,.
and applied stress intensity factors the equilibrium and instability conditions must both be invoked in order to determine the strength. In this case the syste,n is in equilibrium all the time and moves stably from (c = cO, u, = 0) to (c = c*, 0, = LT*)as the applied stress is increased, along the R-curve (or more correctly, T-curve [17]) from T = T(c,) to T = T(c*) as shown in Fig. 2(c). The parameters c*, cr* indicate the critical values of the crack length and the applied stress at which the system passes from stability to instability (7 < l/2 assumed). To determine these parameters we impose the equilibrium condition (equations 1, 46). I(lr~,c”~+ xP/c”* = T&c/d)
(10)
and the instability condition [the derivative of equation (lo)] $(T,c”~ - rXP/c’l’ = 2zT,(c/d)‘. Solving equations gain
(10) and (11) simultaneously
c* = [XP(r + l)d’/T,(l
- 2~)]*“*‘+‘)
CT*= [T,(2z + r)/$d’(r
+ l)]~*(*~-‘)‘~
(11)
we
c */co = 4*13N 2.52 u */as = 3/44’3 N 0.47
(1W (16b) (16~)
noting that equations (16b) and (16c) are the lower limits for equations (14) and (15), respectively, for the case of point geometry. An important result from our analysis to this point is that the strength of a component, Q may be written generally in terms of the slope of the R-curve 7, as u=BP
(2r- 1)/(Zr + r)
(17)
where /l is a material/indenter constant derivable from either equation (9b) or equation (13). We see then that the assumption of a power law R-curve predicts a simple power law dependence of strength on contact load. Increasing the slope of the R-curve, i.e. increasing 7, leads to a weaker dependence of strength on contact load. For the case of point flaws this implies a reduction in the magnitude of the slope of the strengthcontact load response from - l/3 for 7 = 0 [equation (16a)] towards 0 as 7 + l/2. The increased stabilization of the crack by the R-curve is seen explicitly in the increased precursor crack extension prior to failure from 2.52 for 7 = 0 [equation (16b)] towards 03 as 7 + l/2 [equation (14)]. The decreased range of measured strengths for a given range of flaw sizes (contact loads or crack lengths) as the slope of the R-curve is increased will be considered in the next section.
(12)
= [To(2z + r)/lLd’(r + l)] x XP(r + l)d’/T,(l
CJ* = 3 T;‘3/44’311/ (xP)“~
- 2~)](*~~‘)/(~‘+‘) (13)
and we note that both c* and U* have the same dependence on contact load as their counterparts c,, and a,,. Increasing 7 increases Q* for a given value of P but decreases the variability in cr* for variations in P. Combining equations (8) and (12) we have the degree of subcritical, stable equilibrium growth of the crack before failure
3. STRENGTH VARIABILITY A range of dominant flaw sizes in a group of components gives rise to a range of observed strengths described by an extreme value distribution [ 181. The strength distribution we choose here for a group of brittle components is the Weibull distribution [ 18-201
F = 1 - exp[ - k(a/a,)“]
(18)
where F is the cumulative probability of failure of a component at strength 6, gN and m characterize
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COOK and CLARKE:
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the distribution and k is a specimen geometry parameter.? The parameter m is the Weibull modulus and is an inverse measure of the spread of strengths, large values of m imply narrow distributions. The following will determine the variations in m for changes in the R-curve behavior. We begin by noting that the Weibull modulus may be operationally determined by the upper and lower values of the strength distribution
4. DISCUSSION
The major point to be drawn from our considerations on strength variability is that R-curves decrease the width of strength distributions for given ranges of flaw sizes, independent of the absolute magnitude of the toughness, a conclusion also reached by Kendall et al. [S]. Here however, we have extended the generality of the flaw causing failure (locally stressed or not, line or point geometry) and have derived ln( 1 - F,)/ln( 1 - FL) = (a&~,)~ (19) explicit solutions for the change in the Weibull modulus as a function of the local slope of the R-curve where bu and oL are the upper and lower strength [equation (22)]. This latter relationship highlights a values, respectively. The conjugate failure probabilities, F, and FL are related to the number of key point: it is not the existence of thl R-curve per se which results in decreased ranges of strength specimens in the distribution, N, by distributions, but rather the slope of the R-curve at F, = N/(N + 1) (204 the crack lengths of interest. If the R-curve is not steep enough at the appropriate crack lengths, such that FL= l/(N+l) Gob) little stabilization of the cracks occurs, the strength and hence the left side of equation (19) is simply a distribution will not be appreciably narrowed. Where function of the number of specimens in the distribusignificant stabilization is observed we might expect tion and we denote this function by q = q(N). The to observe limited strength distributions. Crack propratio of the strengths-in equation (19) is determined agating out of persistent slip bands in copper single by the range of flaw sizes in the distribution. From crystals show strong stabilization and significant equation (17) we have precursor growth before failure (referred to as “dormant cracks” [6]). The relatively constant number of C”/OL= (p”/p#‘)/Qr+I) (21) loading cycles to failure for these copper specimens where Pu and PL are the contact loads conjugate to suggests that a narrow strength distribution may well the upper and lower strength levels respectively and result for simple monotonic loading of the cracks we note that P, < Pb Combining equations (19)-(21) in the dormant stage. Large-grained polycrystalline gives aluminas demonstrate considerable R-curve behavior [17,21,22]. Strength tests on such materials show m = [In rl W)IWLIPuII UT + r)l(l - 27)l.(22) considerably narrower strength distributions in flaw The first term in square brackets on the right side size regions where the R-curve is steeper (small flaws) of equation (22) is only a function of the number of than in regions where the R-curve is shallow (large specimens and the range of flaw sizes and is therefore flaws) [17,23], in agreement with equation (22). a constant for a given sample size and range of Zirconia materials in which the tetragonal phase is specimen treatments. The second term gives the stabilized show R-curve behavior arising from the dependence of the Weibull modulus on the toughness dilatational transformation of tetragonal particles to characteristics of the material. As can be seen m monoclinic in the crack tip stress field 1241. Such depends only on the slope of the R-curve and not on materials show considerably less variation in strength the absolute value of the toughness [Z’, does not with flaw size than materials in which there is little appear in equation (22)]. Figure 3 plots the value of retained tetragonal phase and shallower R-curves m as a function of z from equation (22) using N = 100 [25], again in agreement with equation (22). and PJP,, = 50. Significant increases in the Weibull Increasing the degree of stable crack growth raises modulus are observed as the slope of the R-Curve two interesting further possibilities for those processincreases. We note also that the Weibull modulus at ing brittle materials. First, non-destructive evaluation a given value of z is greater for r = 3 than for r = 1, techniques (NDE) may be more able to recognize especially at low z. This emphasizes that it is the potentially hazardous flaws arising in the processing stabilization of the fracture system (whether by either procedures. Pre-stressing (“proof-testing”) of comthe crack driving force or the crack resistance ponents to grow large stable cracks easily found by influence) which determines strength variability. NDE, from less easily detectable small flaws, may well be possible in materials possessing steeply sloped R-curves [equation (14)]. Second, the emphasis in tThe use.of the two parameter Weibull distribution, as op- processing brittle materials to obtain high strengths posed to the three parameter distribution, is applicable may be shifted away from the refinement of processhere, as no truncation or lower bound on the strength ing methods to remove strength limiting random distribution is imposed. It can also be shown that the flaws (pores, inclusions, regions of discontinuous simple shift in the strength distribution described by the grain growth, scratches, etc.). Instead, advantage three parameter distribution [18] does not influence our conclusions. may be taken of the fact that increasing the slope
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of the R-curve of a material leads to a degree of flaw tolerance in the strength characteristics, i.e. the strength decreases less for a given increase in flaw size [equation (17)]. Large flaws randomly incorporated during processing will have less effect, provided the microstructure of the material has been systematically manipulated to obtain a strong R-curve effect. We also note from Fig. 3 that stabilization of the strength controlling defect itself will aid in refining strength distributions: the stronger stabilization of point flaws over linear flaws [equation (6)] leads to greater values of m for a given R-curve. The implication is that processing to achieve defects with more strongly localized stress distributions [leading to greater values of r in equation (6)] may also decrease strength variability. There is also the added advantage that the resultant strength of the component under such circumstances will not be influenced by non-equilibrium growth of the dominant flaw (in the localized stress field) after processing. If the dominant flaw extends from c0 to a non-equilibrium length ci (>c,,) in the presence of a reactive environment [such that K(ch) < T(ch), Fig. 2(c)] the strength of the component will not alter. Provided ch < c* the necessary precursor stable growth fixes the instability point, and hence strength, of the system (at c*, a*). The assumption of a power-law R-curve for a brittle material leads to simple power-law predictions for the dependence of strength on flaw size. The increased crack stability introduced by the R-curve causes the magnitudes of the slopes of strength vs flaw size plots to be reduced. However, while it is sometimes possible to fit a select portion of a strength vs indentation load plot (in logarithmic coordinates) with a straight line segment of slope less than - l/3 it is generally not possible to describe the entire strength degradation response by a single straight line [23]. Many brittle materials display very distinct curvature in their strength vs indentation load responses, tending to slopes of - l/3 at large indentation loads and load independent strength plateaus at small indentation loads [23]. The observed continuously changing slopes suggest that the underlying toughness changes of these materials cannot be completely described by simple power-law R-curves. The strong tendency to slopes of -l/3 at large indentation loads in many materials is indicative of a tendency to an upper, constant toughness level. This suggests that the major feature of power-law R-curves which renders them unphysical is the lack of accounting for a saturation of the toughening mechanism at large crack lengths. In metallic or polymeric materials where the deformation zone at the crack tip (dislocation plasticity, crazing, voiding, etc.) is not limited in size, a continuously rising R-curve may well be observed until the crack propagates entirely through the component and the inhomogeneous deformation zone reaches the component boundaries [1, 31. In ceramic materials or rocks, the deformation zone causing the toughening (ligamentary bridges,
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phase transformations, microcracking, etc.) is often limited in scale by the material stiffness. The larger elastic moduli of ceramic materials thus sets an upper bound to the lateral extent of the process zone which is frequently much less than the scale of the component [22,24,26]. Hence saturation effects are much more important in ceramic systems and simple powerlaw R-curves may not be effective for description of the entire toughness response. In closing it is noted that direct measurements of toughness are frequently made using large crack lengths. As discussed above power-law fits to toughness data under these circumstances may severely overestimate the toughness which pertains at smaller crack lengths. The use of extrapolations of such power-law fits in the prediction of component strengths, which are controlled by small flaws, may then result in considerable overestimate, notwithstanding the stable growth of the crack prior to failure. Similar overestimates of small flaw strength may occur if power-law fits are made to large indentation load strength data. We see then that the powerlaw R-curve is not consistent with conservative design principles. 5. CONCLUSIONS The strength of a component containing a dominant flaw under the influence of stabilizing and destabilizing stress fields has been presented as a function of the rate of increase in toughness with crack length. The assumption of a power-law R-curve characterizing this increase leads to power-law dependencies for both the stable, equilibrium crack length and the strength on the magnitude of the stabilizing stress field. These dependencies are inversely related to the slope of the R-curve. The implication is that for a given range of flaw sizes, materials with steeper R-curves will show smaller ranges of strengths. The Weibull modulus of a strength distribution is directly related to the slope of the R-curve, independent of the absolute magnitude or spatial scale of the toughness function (and also increases with the degree of localization of the stabilizing stress field). Correlations between R-curve slopes and strength-flaw size plots in both transforming and non-transforming ceramic materials support this conclusion. REFERENCES 1. D. Broek, Elementary Engineering Fracture Mechanics, 3rd edn. Martinus Nijhoff, The Hague (1974). 2. B. R. Lawn and T. R. Wilshaw, Fracture of Brittle Solids. Cambridge Univ. Press (1975). 3. J. G. Williams, Fracture Mechanics of Polymers. Ellis Horwood, Chichester (1974). 4. D. B. Marshall and A. G. Evans, in Fracfure Mechanics of Ceramics (edited by R. C. Bradt, A. G. Evans, D. P. H. Ha&man and F. F. Lange), Vol. 7, pp. I-15. Plenum, New York (1986). __ 5. K. Kendall. N. McN. Alford. S. R. Tan and J. D. Birchall. J. ‘Mater. Res. 1, 120’(1986). 6. L. M. Brown and S. L. Ogin, in Fundamentals of Deformation and Fracture (The Eshelby Memorial Sym-
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