journal J. Am. Ceram Soc., 73 (21 187-205 (1990) Perspective on the Development of High-Toughness Ceramics

Anthony G. Evans* Materials Department, College of Engineering, University of California, Santa Barbara, California 93106

1. Major Developments (Fig. 2). Secondly, the topic attracted the attention of outstanding scientists who (1) Synopsis have since continued to provide invalua- THE science governing the strength and ble contributions to progress in the field. fracture of structural ceramics has devel- Still another important development was oped from a mostly empirical topic in 1965 the appreciation that the mechanical into a mature discipline that now sets the properties of ceramics could be apprecia- standards in the field of mechanical be- bly enhanced by the incorporation of ce- havior. The intent of this review is to pro- ramic fibers and whiskers. Finally, a recent vide a perspective regarding this discovery that may have a similar substan- evolution, followed by succinct descrip- tive impact on the ceramics field concerns tions of current understanding. The rapid metal-toughenedceramics and compos- developments in the field are considered ites formed by directed metal oxidation. to have commenced upon the first con- certed attempt to apply fracture mechan- (2) Introduction of Fracture ics concepts to ceramics, beginning in the Mechanics middle 1960s (Fig. 1). This allowed a dis- The mathematical framework and the tinction between the separate contribu- experimental techniques of linear elastic tions to strength from the flaws in the fracture mechanics that had developed from the early 1950s and had been exten- material and from the microstructure, as Anthony G. Evans is Alcoa Professor and manifest in the fracture toughness. Anoth- sively applied to metals were first recog- Chair of the Materials Department at the nized to have importance to the ceramic er contribution that accelerated the learn- Unviersity of California at Santa Barbara. ing process was the development of field by Davidge and by Wiederhorn. A native of Wales, he earned his B.S. in These authors examined a variety of ex- indentation techniques, which allowed metallurgy in 1964 and Ph.D. in physical trends in the damage resistance of new perimental approaches and began the metallurgy in 1967, both from Imperial ceramics to be assessed on a routine ba- process whereby the fracture toughness College, London, England. During 1967 sis. However, the most important develop- could be established as a material param- to 1971 he was with the Atomic Energy ment, which originated at about the same eter and eventually be related to micro- Research Establishment, Harwell. In 1971 time, was the discovery of toughened zir- structure. At the same time, initial attempts Evans came to the United States as visit- conia alloys. The ensuing research on were made to separate the contributions ing scholar at the University of California these alloys established two vital prece- to strength, S, from the flaw size, a, and at Los Angeles. He then worked at the dents. Firstly, the introduction of the the toughness, Klc, through the simple National Bureau of Standards from 1972 materials-by-designapproach, which es- application of the Griffith relation to 1974, Rockwell International Science tablished the benefits that accrue from the S= YKlc I fi (11 Center from 1974 to 1978, and the strong collaboration between processing, University of California at Berkeley from characterization, testing, and modeling where Y is a well-documented crack/ 1978 to 1985. He has more than 200 specimen geometry parameter. Although technical papers to his credit. A. H. Heuer-contributing editor it took some time to rigorously accomplish A Fellow of the American Ceramic this objective, the first attempts brought at- Society, Evans is a past chair of the Bas- Manuscript No 197905 Received November 20, tention to the concept of a "microstructural ic Science Division and a past vice presi- 1989, approved December 13, 1989 flaw" and soon provided a focus for dent of the Society. He received the Presented in part at the 91st Annual Meeting of the processing activities concerned with the Society's Coffin Purdy Award in American Ceramic Society. Indianapolis, IN, April 24, Ross 1989 (Orton Memorial Lecture) elimination of the largest flaws. This de- 1980, and was recipient of the Fulrath [Key words toughness, fracture microcracking, zir- velopment also addressed a paradox in Award in 1981. Evans delivered the Ed- conia. R curve ] the field, wherein strength measurements ward Orton, Jr., Memorial Lecture and 'Member, American Ceramic Society were then typically expressed in terms of received the John Jeppson Medal and Award during the 1989 Annual Meeting of the Society. He is a member of the Na- tional lnstitue of Ceramic Engineers.

187 Vol. 13, No. 2 188 Journal of the American Ceramic Society-Evans average microstructural features, such as fracture has been its contribution to our grain size and porosity, whereas differ- understanding of the role of "toughness" ences in strength between test methods in damage tolerance. Specifically, the (e.g., bending and tension) were rational- tougher materials exhibit strengths after in- ized on the basis of weakestlink statistics. dentation that vary less rapidly with inden- Some of the first observations concern- tation load (crack size) than for brittle ing the interactions of cracks with the mi- ceramics (Fig. 3). This behavior reflects the crostructure were made within the next existence of a resistance curve. Indeed, several years. Out of these observations this measure of toughness is probably the came preliminary concepts for fracture re- most useful in terms of the practical ap- sistance mechanisms and the models that plication and durability of ceramics. emerged later. Many of these concepts Many of the indentation methods are can be traced to the work of Lange and only approximate and do not provide the of R. Rice. Also, work at Harwell Labora- quality of fracture resistance data needed tories in the United Kingdom established to rigorously relate toughness to micro- that very high toughness ceramics could structure. The surface flaw methods, in- be made by incorporating carbon fibers. troduced first by Petrovic and Jacobson, This body of work provided a strong moti- seem to be the most precise, provided that Fig. 1. Overall chronological trends in the vation to maintain a high level of activity residual stresses are eliminated by polish- development of high-toughness ceramics. in the field and to seek a better under- ing out the plastic zone. However, none standing of mechanisms and approaches of these methods can be used for the for enhancing the reliability and damage highest toughness materials now tolerance of ceramics. available. Although these prospects were evident, Materials by design progress was slow for two reasons. First- (4) Pivotal Role of Zirconia ly, different methods for measuring frac- The discovery in 1976 that zirconia can ture toughness often gave different results, exhibit high toughness initiated a remark- leading to enquiries about the utility of the able decade of development, culminating fracture toughness as a material parame- in materials having toughness on the or- ter. Secondly, the most acceptable test der of 20 MPa.ml'2. A sequence of ma- methods required moderately large test terial inventions in Australia, Germany, and specimens, whereupon the data emerged the United States involving Garvie, Claus- slowly and were confined to the relatively sen, Lange, and Gupta provided the in- few ceramics amenable to the fabrication centive. Such materials elicited of sufficiently large specimens. One con- outstanding characterization research by sequence of these factors was that the fo- Heuer, Ruhle, and Hannink and some cus of much of the research in the field novel experiments originating with Swain was on the characterization and implica- and Marshall. The results were then ration- tions of slow crack growth, with minimal alized through a clear understanding of attention given to the enhancement of the responsible mechanisms, established touahness. by the micromechanics models of Tie first of the above issues would later McMeeking. Budiansky, Hutchinson, Fig. 2. Scheme involved in the materials-by- be resolved upon the revelation that many Evans, and Marshall. This process oc- design concept. polycrystalline ceramics exhibit resistance- curred in an iterative manner built around curve behavior, as elaborated in Section the understanding that the process-zone I (5). However, it is interestingto note that size was important and that resistance- the resistance-curveeffects were first ob- curve effects are inherent to the served in 1974, but the significance was mechanism. not appreciated. The second issue was addressed upon the introductionof inden- (5) Importance of Resistance Curves tation methods discussed in the following All "tough" ceramics exhibit resistance section. curves. However, this feature of their be- havior did not become apparent until (3) Influence of Indentation Fracture about 1980. The existence of resistance- The introductionof approaches for frac- curve behavior was predicted by ture toughness estimation by using vari- McMeeking and Evans to be an essen- ous indentation methods allowed tests on tial feature of transformation toughening small specimens and, thus, permitted rap- and soon verified experimentally by id probing of the "damage tolerance" of Swain. These predictions placed empha- many different ceramics. These ap- sis on the transformation wake. The more proaches emerged from systematic pi- general importance of the wake was oneering research by Lawn over more demonstrated by some clever experi- than a decade. Of particular importance ments conducted by Steinbrecht, who re- was the discovery that the cracks around vealed that the toughness of Al2O3 is hardness indentationsformed on unload- diminished upon removing the wake by Flaw size, a - ing and, hence, were dominated by sawing. Continued research on this topic residual fields Appreciation of this factor revealed that microcrack toughening ex- Fig. 3. Effects of damage on the strength of low and high toughness ceramics The tear permitted the 'dentificationOf the non- hibits wake effects and that crack bridg- ing modulus IS the slope of the resistance dimensionalparameters that related ing by intact grain is a common feature curve toughness to either the size of the cracks of crack extension in polycrystals and is or the failure stress after indentation also an important contribution to the frac- A more recent benefit of indentation ture resistance. Finally, and most recent- February 1990 Perspective on the Development of High-Toughness Ceramics 189 ly, it has been demonstrated that and Isostatically Hot-PressedAl2O3 Com- toughening by fibers, whiskers, and met- posites," J. Am. Ceram. SOC., 66 [6] al networks all exhibit wake-dominated 396-98 (1983); F. F. Lange and M. Met- Limiting Physical Characteristic resistance-curve characteristics. calf, "Processing-Related Fracture Origins: When rigorously measured and inter- 11, Agglomerate Motion and Cracklike In- preted, resistance-curve behavior ration- ternal Surfaces Caused by Differential Sin- alizes effects of specimen geometry on tering," ibid., 398-406; and F. F. Lange, toughness, crack-size effects, and trends B. I. Davis, and I. A. Aksay, "Processing- in strength. Additionally, in the highest Related Fracture Origins: Ill, Differential toughness materials, linear elastic be- Sintering of Zr02 Agglomerates in havior is violated for many of the common A1203/Zr02 Composite," ibid., 407-408. test specimens. Consequently, nonlinear J. R. Rice; p. 191 in Fracture, Vol. 11, approaches are needed to characterize Mathematical Fundamentals. Edited by H. material behavior. Liebowitz. Academic Press, New York, Microstructure, Fabrication, 1968. micromechanics machining (6) Discovery of Composites R. W. Rice, "Mechanisms of Toughen- Fig. 4. Aspects for achieving high-reliability High-work-of-fracture-fiber-reinforced ing in Ceramic Matrix Composites," ceramics. ceramics consisting of carbon fiber, rein- Ceram. Eng. Sci. Proc., 2 [7-81 661-701 forced glasses, and glass-ceramicswere (1981). first demonstrated in 1972, accompanied R. W. Steinbrech and A. H. Heuer, "R- by the first complete model of composite Curve Behavior and the Mechanical behavior. However, activities on ceramic Properties of Transformation-Toughened composites were not pursued vigorously ZrO2-ContainingCeramics," Mater. Res. until 1983, when Prewo and Brennan an- Soc. Symp. Proc., 60,469-81 (1986). nounced a Nicalont Sic-fiber-reinforced M. V. Swain, "Inelastic Deformation of glass-ceramic composite. This announce- Mg-PSZ and Its Significance for Strength- ment established the possibility that ce- Toughness Relationships of Zirconia- ramic composites having high resistance Toughened Ceramics," Acta Metall., 33, at elevated temperature might be possi- 2083-91 (1985). ble. A major activity on ceramic-matrix A. V. Virkar and R. L. K. Matsumoto, composites has ensued and is still in pro- "Ferroelastic Domain Switching as a gress. Another related discovery was that Toughening Mechanism in Tetragonal Zir- A1203 could be toughened by SIC conia," J. Am. Ceram. Soc., 69 110) whiskers. Together, these materials have C-224-C-226 (1 986). provided the basis for the concepts of toughening by brittle reinforcements, in- II. Toughening Mechanisms volving the debonding and sliding proper- ties of the reinforcementlmatrixinterfaces. (1) General Features The objective of research concerned (7) Bibliography with structural ceramics is the generation f Crack advance, Aa Actual references to the work alluded of materials having high reliability. To a0 to above can be located in Section 11, achieve this objective, there are two fun- which covers the various mechanisms in damentally different approaches (Fig. 4): Fig. 5. Resistance-curve behavior charac- teristically encountered in tough ceramics some detail. The reader is also referred to flaw control and toughening. The fiaw con- K, the seven-volume series Fracture Mechan- IS the fracture resistance and Aa IS the crack trol approach accepts the brittleness of the advance ics of Ceramics, edited by Bradt, Evans, material and attempts to control the large Hasselman, and Lange (Plenum Press, extreme of processing flaws. The tough- New York, 1974-1983), which contains ening approach attempts to create micro- much of the chronology, plus the follow- structures that impart sufficient fracture ing basic references: resistance (Fig. 5) that the strength be A. G. Evans and A. H. Heuer, comes insensitive to the size of flaws (Fig. "REVIEW-Transformation Toughening in 3). The former has been the subject of Ceramics: Martensitic Transformations in considerable research that identifies the Crack-Tip Stress Fields," J. Am. Ceram. most detrimental processing flaws, as well SOC.,63 [5-61 241-48 (1980). as the processing step responsible for R. C. Garvie, R. H. J. Hannink, and R. those flaws.'-3 The latter has emerged T. Pascoe, "Ceramic Steel?," Nature (Lon- more recently, and has the obvious ad- don), 258, 703-704 (1975). vantage that appreciable processing and A. H. Heuer, "Transformation Toughen- postprocessingdamage can be tolerated ing in Zr02-ContainingCeramics," J. Am. without compromising the structural relia Ceram. SOC.,70 \lo] 689-98 (1987). bility.4.5 A. H. Heuer, F. F. Lange, M. V. Swain, The resistance of brittle solids to the and A. G. Evans (eds.), "Transformation propagation of cracks can be strongly in- Toughening," J. Am. Ceram. Soc., 69 [3] fluenced by microstructure and by the use 169-298 (1986); ibid., 69 [7] 511-84 of various reinforcements. The intent of the (1986). present paper is to provide a succinct re J. W. Hutchinson, Harvard University view of the known effects of microstruc- Report No. 58 (1974). ture and of reinforcements on fracture F. F. Lange, "Processing-Related Frac- resistance. In most cases, toughening ture Origins; I, Observations in Sintered results in resistance-curvecharacteristics (Fig. 5), wherein the fracture resistance tNippon Carbon Co., Tokyo, Japan svsternaticallv increases with crack exten- 190 Journal of the American Ceramic Society-Evans Vol. 13, No. 2 sion. The resulting material strengths then ever the length of the nonlinear zone is ap- depend on the details of the resistance preciably larger than the spacing between curve and the initial crack lengths,s,' such the relevant microstructuralentities. Such that toughness and strength optimization conditions invariably exist when the ma- usually involve different choices of micro- terial exhibits high toughness. The cou- structure. The individual mechanisms in- pling between experiment and theory is clude transformations, microcracking, another prevalent theme, because tough- twinning, ductile reinforcements, fi- ening is sufficiently complex and involves berlwhisker reinforcements, and grain a sufficiently large number of independent bridging. variables that microstructure optimization An underlying principle concerns the only becomes practical when each of the essential role of nonlinearity, as manifest important modes has been described by in mechanisms of dissipation and energy a rigorous model, validated by experiment storage in the material, upon crack propa- (Fig. 2). gation. Consequently, the potent toughen- The known mechanisms can be con- ing mechanisms can be modeled in terms veniently considered to involve either a of stresddisplacement constitutive laws for process zone or a bridging zone (Fig. 7). Fig. 6. Schematic diagram illustrating non- representative volume elements (Fig. 6). The former category exhibits a toughen- linear hysteretic elemental response and as- Furthermore, the toughening can be ex- ing fundamentally governed by a critical sociations with enhanced toughness. plicitly related to stressldisplacement hys- stress for the onset of nonlinearity, $, in teresis (Fig. 6), as will be elaborated for elements near the crack and by the as- each of the important mechanisms. The sociated stress-free strain, ~i.8-11The general philosophy thus adopts the con- resulting stress-strain hysteresis of those cept of homogenizing the properties of the elements within a process zone then yields material around the crack and then for- a steady-state toughness given by11 mulating a constitutive law that character- izes the three-dimensional material response. Models that discretize micro- where AgC is the increase in the critical structure details in two dimensions are strain energy release rate when the crack typically less rigorous, because three- is long,$ f the volume fraction of the dimensional interactions along the crack toughening agent, and h the width of the front are not readily described, and be- process zone in steady state.9,QTransfor- cause multiple calculations are needed to mation, microcrack, and twin toughening eliminate artifacts of the discretization. The are mechanisms of this type. homogenization approach rigorously The bridging category exhibits tough- describes the toughening behavior when- ening governed by hysteresis along the crack surface,'3~14induced by intact ma Fig. 7. Schematic diagram illustrating both $Short cracks and cracks without fully developed terial ligaments process-zoneand bridging-zonemechanisms process zones give smaller changes in toughness as of toughening elaborated in the discussion of resistance curves U*

where 2u is the crack opening, 2u* the opening at the edge of the bridging zone, Table 1. Tough Ceramics t the tractions on the crack surfaces ex- erted by the intact toughening agent (Fig. Highest 7), and f the area fraction of reinforce- toughness Exemplary Mechanism fMPa.rn1'3 materials Limitation ments along the crack plane. Ductile rein- forcements, as well as whiskers, fibers, Transformation -20 ZrO2 (MgO) and large grains, toughen by means of HfO2 Tsolid. The stresses are most where KO is the fracture resistance of the presence of a dilatational transformation zone conveniently expressed in terms of the material in the transformation zone im- for a supercritical transformation. mean stress, 5, the deviatoric stresses, mediately ahead of the crack tip, where- s,, =a,, - d,, 0, and/or the equivalent upon, the observed toughness is stress, ue=v2,and their variation Kc = KO+ AK, during transformation. Such relations are, (1 1) in fact, the equivalent of the more familiar where AKc is the quantity - AK, evaluat- constitutive laws used to describe plastic ed at the fracture criticality. deformation and crack-tip fields in elas- tic/plastic solids. A viable constitutive law (iv) Toughening and Resistance must account for the effect of particle size Curves and shape upon the incident of transfor- There are two equivalent analytical mation by allowing only a certain fraction, methods for determining trends in tough- 6 of the solid to actually transform at a giv- ness: one based on stress intensity factors, en imposed stress, ai. The simplest yield AK, and the other on conservation in- criterion has the form23 tegrals A9. Both approaches predict the same behavior, but have different utility when various aspects of toughening re- quire elucidation. Consequently, both ap- proaches are briefly described. The where R and T are experimentally deter- approach based on stress intensity factors mined, normalizing parameters that de- indicates that the magnitude of AK clear- Fig. 13. Change in normalized toughness pend on f, E~ and particle size ly depends on the shape of the zone and as a function of normalized transformation Experiments per!ormed on partially stabi- on the components of the transformation strain lized zirconia (PSZ) and tetragonal zirco- strain tensor. Initially, it is pertinent to con- nra polycrystals (TZP) are consistent with sider a crack in an untransformed parent, this transformation condition 23 However, such that a frontal zone develops as the February 1990 Perspective on the Development of High- Toughness Ceramics 193 load is imposed. Subsequently, a steady- Few toughened materials exhibit super- statezone of uniform width over the crack critical transformation. Generally, there is surfaces (Fig. 10) is examined. The AK lev- a gradient in the fraction of transformed els are different for these two zone con- material with the transformation zone, with figurations, resulting in R-curve effects.9 the largest fraction present near the crack For the stationary crack, when the long plane. Such conditions are referred to as range strain field of the transformed parti- subcritical. For this case, the toughness cles is purely dilatationa1.f the mean can be obtained by integrating the meas- stress dictates the shape of the transfor- ured transformed fraction over the width mation zone, which then has the shape of the zone.g.10 Also, in some materials, the depicted in Fig. 10. For this condition, transformation is reversible. Such transfor- AK= 0. For the steady-stateconfiguration mations can produce toughening, provid- and when the transformation does not re- ed that hysteresis is involved.8 verse in the wake, as well as when all par- The equivalent approach for cracking ticles within h are transformed, the plane toughness, based on conservation in- strain zone width for small-sca/edilatational tegrals, dictates that, for the frontal zone, transformation is9,IO the volume elements within the zone do not experience unloading. Consequently, (1 +v)~K, 6 the path-independent J integral applies h= 12n (z) and the relation The corresponding extent of crack shield- J = (1 - Y*)K~/€ (1 7) ing is given by Eq. (5) as pertains for all line contours around the AK= -0.22E f$\/T; /(I -u) (1 3) crack tip. Furthermore,because the elas- tic properties of the transformed and un- The supercritical plane strain toughness transformed materials are essentially the can thus be expressed in the following same, contours around the tip (giving forms: Ktip)and remote from the tip (giving K,) AKc = 0.22E f$fi/(1 - u) yield identical values of K whereupon Ktip=K, and AK=O, as before. Con- AKJKC = (O/I2R)(1 f U)EfE: / F(I - U) versely, when a fully developed zone ex- (1 4) ists (Fig. lo),the material within the zone, behind the crack tip, has experienced un- KJKo=[1 -(fi/1211)(1 +u) loading, and a path-independent J does x EFT/ 3(1 - up (1 5) not apply. In this case, the appropriate If the transformation is not activated by the conservation integral, I, has the same form hydrostatic crack-tip field, but instead oc- as J at the tip curs in shear bands inclined at =n/3 to the crack plane (R>> in Eq. (6)),all of the deleterious transformations in front of the but remote from the tip crack are excluded. Then, if the transfor- mation strain is still hydrostatic, the super- h critical AKc increases to l = [(l - u2) K$E]- 2 I UQdy (1 9) 0 AKc = 0.38E f$fi/(l -u) (16) where U(y) is the residual energy density Zone profi/e considerations are thus of in the wake. Equating the magnitude of prime importance in determining the mag- the conservation integral for the near-tip nitude of the transformation toughening. and remote paths gives It is also noted that, for conditions of plane stress, the zone width is predicted to be h smaller, causing the toughness to dimin- K2, = K&+ [2El(1 - u2)] I Uo/) dy (20) ish. When the transformation zone is large, 0 the zone width may be affected by the previously transformed material and Eq. or (1 2) does not apply. In this case a numeri- h cal solution has revealed the trend in the Aqc=2 I W)dY (21) toughening ratio KJKo with EfETl 3 de- 0 picted in Fig. 13. In particular, it is noted that “lock-up”cccurs when EfE:I8=30. The integral in U(y)can be simply related This phenomenon, first recognized by to the elemental stress-strain curve (Fig. Rose,24 has been elaborated by Amazi- 9). Specifically, material in the process go and Budiansky.15 Specifically, at lock- zone undergoes a complete loading and up, K at the crack tip tends to zero and unloading cycle as the element translates the crack cannot propagate. In practice, from the front to the rear of the crack tip under such conditions, an existing crack during crack advance. Hence, each ele- would arrest and new cracks would form ment in the wake is subject to the residu- and grow, leading to a damage mode of al stress work contained by the hysteresis failure. loop (Fig. 9). Consequently, by appreciat- ing that the wake is subject to a residual (An extensively twinned particle with uniform twins compression, the residual energy densi- and, thereby, no long-range shear strain ty can be readily evaluated aslo 194 Journal of the American Ceramic Society-Evans Vol. 73, No. 2 (B) Microcrack Toughening (I) Basic Features where 6* is the slope of the stress-strain The phenomenon ot microcrack tough- curve of the transforming material. The ening was postulated more than a decade three terms in Eq. (22) derive from the ag0,27-30 and, indeed, a range of materi- areas I, 11, and 111 under the stress-strain als exhibit trends in toughness with parti- curve depicted in Fig. 9. The latter two cle size, temperature, etc., qualitatively terms cancel when consistent with this mechanism. However, the mechanism has only been rigorously 5* = - 2€/3(1+ V) identified and validated for various two- phase ceramics, including AI2O3 tough- - 4Gl3 (23) ened with monoclinic (m)ZrO23' and SIC toughened with TiB2.32 The fundamental and Eq. (21) for the toughening is then premise concerning the mechanism is identical to Eq. (4). This condition for B* depicted in Fig. 17. Microcracks occur applies when the transformation is super- within regions of local residual tension, critical.1 0 caused by thermal expansion mismatch The preceding discussion has referred and/or by transformation.33,34 The exclusively to supercritical transformations, microcracks also occur along the lowest wherein all particles within the zone trans- fracture energy paths and locally relieve form. Subcritical transformations are the residual tension. Consequently, a dila- characterized by a transformation slope tation occurs governed by the volume dis- B*>- 4Gf3. Some typical results deter- placed by the microcrack. Furthermore, mined numerically for the dilatational trans- the microcracks reduce the elastic modu- formation are plotted in Fig. 14.10 lus within the microcrack process zone. Resistance-curve behavior is inevitable The elemental stress-strain curve for a in transformation-toughenedmaterials, be- microcracking solid thus has the form cause initial crack growth is largely un- depicted in Fig. 17. The hysteresis dictat- affected by transformations produced by ed by this curve, when the microcracks the stationary crack. The slope of the are activated by the passage of a macro- resistance curve (the tearing modulus) is crack, constitutes the change in tough- governed by the current geometry of the ness, as elaborated below. However, this zone, which depends on zone evolu- contribution is partially counteracted by a tion. 1215 Consequently, toughening curves degradation of the material ahead of the can only be calculated by applying in- microcrack. The full extent of the degra- cremental crack-growth methods. Such dation is presently unknown. analysis reveals that the zone initially The basic material quantities that allow widens as the crack extends and eventu- microcracking (residual stress and low ally reaches a steady-state crack width fracture energy interfaces) also lead to (Fig. 15). The actual zone morphology grain bridging, as described in Section II and, thus, the shape of the resistance (3).Often, therefore, both microcracking curve, depend on the transformation and bridging operate simultaneously to a criterion and on the flow rule. Calculations relative extent that depends on features of based on a dilatational law indicate a large the microstructure. An important measure- tearing modulus and a peak prior to ment that distinguishes the mechanism is steady state (Fig. 16).12 The peak is most the size of the microcrack process zone. pronounced for the supercritical condition The techniques that have the best capa- when the zone height is large. The exis- bility for characterizing this zone are the tence of the peak is not physically obvi- scanning acoustic microscope (SAM), ous but may be rationalized by which allows measurement of the elastic appreciating that the rising resistance modulus with high spatial resolution,%and curve and the steady-state resistance have transmission electron microscopy (TEM). a separate dependence on the frontal- For example, the SAM technique has zone shape evolution and are thus not uni- recorded a small process zone in a poly- quely connected. The occurrence of a crystalline AI2O3,but a substantial zone in peak fracture resistance coincides with a a glass-ceramic material, whereas TEM peak in zone height. Such zone height has identified microcrack process zones peaks have been observed by experi- in AI203/m-ZrO2and in SiC/TiBp. ment.22 The crack shielding caused by The resistance curves are also depen- microcracks can be conveniently separat- dent on the length of the crack, resulting ed into dilatational and modulus contrjbu- in short crack effects which influence tions. The former depends on the trends in strzngth.12 For crack lengths in process-zonesize and shape, whereas the the range a < 5h, interaction between the latter depends only on the zone shape. zones at the opposite crack tips reduces The dilatational contribution to KtlPfrom the zone shielding. Consequently, the op- the residual field has precisely the same tima in strength and toughness are not form as that associated with transforma- usually coincident. Property optimization tions, but with f~dreplaced by 87. This thus involves consideration of crack-length contribution thus depends on zone size effects, as well as resistance-curve con- and shape, as in the case of transforma- siderations.26 tion toughening. For example, when the February 1990 Perspective on the Development of High-Toughness Ceramics 195 microcrack nucleation condition involves a critical normal stress, the steady-state toughening is34

A& = 0.40EO~fl (24) whereas, for a critical mean stress

AK, = 0.32EO~fi (25) where OT is now the misfit strain caused by microcracks (analogous to f.9. By contrast, the toughening imparted by the modulus reduction depends on zone shape, but not on zone size. Following Hutchinson,% the asymptotic modulus contribution is34 (1 - v)AKJKC= (kf - $)(GIG- 1) + (k2 + $)@GIG- V) (26) where kl and k2 depend on the micro- cracking criterion. Values of AKc/Kc cap be obtained by simply inserting GIG and V into Eq. (26). Typical results for the steady-state toughening are, for a critical normal stress and pemy-shaped micro- cracks

AKJK, =1.42q (27) Fig. 14. Ratio of near-tipto remote stress intensity factors. The dashed line is the asymptotic result for small E: with q being a measure of the microcrack density, defined following Eq. (30). In general, the AKc due to dilatation and modulus effects are not additive: interac- microcracked bodies establish the modu- tion terms are involved. lus effect34 as (ii) Constitutive Law GIG = 1 + (32/45)(1- ~)(5- ~)q/(l - V) The reduction in elastic moduli caused B/B=1 +(16/9)(1 -9) q/(1 -2~) (31) by microcracks, as well as the permanent strain governed by release of the residu- where the bar refers to microcracked al stress, depend upon microstructure. material. Characteristic constitutive laws are illustrat- For particles in residual compression, ed for two important cases consisting of the corresponding constitutive law de- spherical particles subject to either residual pends sensitively on the response of the tension or residual compression. For the interface to the microcrack. For the one former case, the microcracks occur either material studied thus far (A1203/ZrO~), within the particle or at the interface, some debonding occurs at the interface whereas for the latter, the matrix develops and the basic parameters derived for the microcracks. Results for these cases have microcracks are (Fig. 18) Fig. 15. Dependence of relative zone width been derived for materials having on relative crack extension. homogeneous elastic properties prior to 8~ = ~.~(uR/R)TJ/<~ (32) microcracking. For particles subject to where UR is the residual microcrack residual tension, in which penny-shaped opening of the interface, as affected by the microcracks form (Fig. 17), the volume of debond length, d, and each opened microcrack is34 5 = ClR AL' = (16R3/3)( 1 - V~)O!/€ (28) where c is the microcrack length. The where above results may be used in conjunction with Eqs. (25) and (26) to predict 4=2€.$19(1 - U) (29) toughening.

R is the particle radius and EJ the misfit (iio Toughness Correlation strain between particle and matrix. This Microcracking in various zirconia- volume increase dictates the permanent toughened alumina (ZTA) materials and in strain, such that the microcrack misfit strain SiC/TiBz has been systematically studied becomes by preparing thin foils at various distances, y, from a macrocrack.31.32 For the form- O,=(16/3)(1 -uZ)~~/E (30) er, radial matrix microcracks were ob- where q = N@> is the number density of served. All such radial microcracks Fig. 16. Theoretically predicted ratio of (rela- microcracked particles, with N being the occurred along grain boundaries in the tive) toughness to crack extension. number of microcracked particles per unit A1203. Usually, the interface between the volume. Solutions for the elastic moduli of AI2O3and Zr02 was debonded at the ori- Vol. 73, No. 2 196 Journal of the American Ceramic Society-Evans gin of the microcrack. The detectibility of are similar in magnitude.This is important radial microcracks depended on their in- because, although the former contribution clination with respect to the incoming elec- cannot be readily changed (being tron beam. Trends in the visibility of governed almost entirely by the saturation microcracks upon tilting around one axis microcrack density, qs), the dilatational indicated that tilting in all directions would contribution may be enhanced by increas- be needed to detect each microcrack ing both the process-zone size and the present in the foil. However, tilting in the microcrack density within the zone. This TEM is limited to f 45" in all directions, so may be achieved by control of the m-Zr02 that only 0.3 of the solid angle is covered. particle-size distribution, through its in- Therefore, the fraction of detectable fluence on the nucleation of the microcracks is limited to ~0.3. microcracks. Subject to this detectability limitation, Although progress has been apprecia- large regions of TEM foils of known thick- ble, a number of substantive problems ex- ness have been investigated. One exam- ist in the analysis of microcrack ple is shown in Fig. 19, wherein all toughening, both experimental and theo- microcracks observable under different tilt- retical. Among the problems are a poor ing conditions are marked. It is noted, on fundamental understanding of the degra- the average, that at least two microcracks dation caused by the microcracks direct- emanate from each rn-ZrO2 particle. In ly ahead of the crack front and limited most cases, the microcracks terminate at knowledge of the interactions between the A1203 grain triple junctions. The as- modulus and dilatational contributions to sociated projected length, /, of each crack shielding, as well as experimental microcrack has been measured and relat- microcrack detectability limitations. These ed to the radius, b, of the originating m- topics require further study before Zr02 particle. authoritative conclusions can be reached The utility of microcrack length meas- regarding toughening and before predic- urements is facilitated by defining a tions of toughening trends can be rigorously contemplated. Fig. 17. Basic concepts of microcrack representative geometry and determining toughening. the related microcrack density parameter, q, introduced by Budiansky and O'Con- (3) Bridging Mechanisms ne11.36 Evaluation of the microcrack pro- Contributionsto the toughness caused files observed by tilting suggests that each by reinforcing elements that bridge be- ZrOp particle is circumvented by a radial tween the crack surfaces are convenient- microcrack, consistent with the symmetry ly separated into two types: ductile and of the residual strain field around each par- brittle. The former refers to metal- ticle. The model depicted in Fig. 18 has toughened ceramics (cermets) and relies thus been used for further analysis. on high toughness and ductility to permit The microcrack density determined metal ligaments to exist and to contribute from the frequency distribution of project- to toughness through plastic dissipation. ed microcrack lengths (Fig. 20) diminish- However, when the bridging material is es with distance from the crack plane. A brittle, such that the toughness of the maximum density, qs, adjacent to the bridge is similar to that of the nonbridg- (a) Suggested microcrack configuration crack surface suggests a saturation value, ing (matrix) material, the occurrence of governed by the total Zr02 particle con- bridging is more subtle and requires either microasAnnular Particle tent, as needed to use the shielding for- microstructural residual stress or weak in- $:== ==:I- mulation. The decrease with distance y is terfaces or both. The former may be ap- ---____-- approximately linear, such that preciated by noting that large local Residual residual stresses caused by thermal ex- opening, 6 q-vs(1 -YW (33) pansion mismatchlanisotropy are capable (b) Microcrack model of suppressing local crack propagation where h is the process-zone width. A (Fig. 21) and, thereby, may allow intact similar trend was found for SiC/TiB2.32 ligaments to exist behind the crack front.37 Based on the above results, the residual When these ligaments eventually fail in the strain contribution to the shielding has crack wake, energy is dissipated as been determined as AKd= - 2.5 f 1 acoustic waves and causes toughening MPa.mlQ. The modulus-reduced shield- (Fig. 22). The latter may be understood by (c) Micro mechanics model ing is obtained as AK,= -5+2 recognizing that low fracture energy inter- MPaamlQ. Simple addition of the dilata- faces (andlor grain boundaries) can cause Fig. 18. Microcrack model used to meas- tional and modulus contributions would in- the crack to deflect along those interfaces, ure crack densities and to analyze the changes dicate toughening, AK,, of -7.5 again permitting intact ligaments. As the in crack volume and elastic modulus: (A) MPa.ml'2, sufficient to account fully for the crack extends, further debonding can oc- schematic drawing of a typical configuration, measured toughness, AK,, of -6 cur. Eventually, the bridging material fails, (B) microcrack model used for analysis, and either by debonding around the end or by (C) mechanics model. MPa-ml@. It should be appreciated, however, that the individual contributions fracture. Following failure, frictional sliding to the shielding are relatively large and ad- may occur along the debonded surface. ditivity is not strictly valid. Interaction effects The energy dissipation upon crack propa- should be taken into account before more gation thus includes terms from the ener- rigorous comparisons between theory and gy of the debonded interfaces, the experiment are attempted. It is of interest acoustic energy dissipated upon reinforce to note that the modulus reduction and ment failure, and frictional dissipation dur- dilatational contributions to the shielding ing pullout (Fig. 22). These contributions February 1990 Perspective on the Development of High- Toughness Ceramics 197 lead to a unified model of the fracture resistance of materials that exhibit bridg- ing by brittle ligaments. (A} Ductile Reinforcement Toughening (I} Basic Features Ductile reinforcements may profoundly increase the toughness. One contribution to the toughness derives from crack trap- ping,38 another involves crack bridg- ing,13,39-41 and yet another involves crack shielding and plastic dissipation associat- ed within a plastic zone.42 The material systems that exhibit plasticity-induced toughening have three distinct microstruc- tures: isolated ductile reinforcements, in- terpenetrating networks, and a continuous ductile phase. Isolated ductile reinforce- ment is exemplified by ductile fibers,43-45 by Nb-alloy plates in TiA1,46.50 and, prob- ably, by ferrite in the quasi-cleavage of steels.48 An example of an interpenetrat- ing network is Al-alloy-reinforced A1203 Fig. 19. TEM photograph with all observable microcracks marked. produced by the Lanxide method.49.50 The continuous ductile network case in- cludes most metal-matrixcomposites and, probably, cemented carbides.51 An impor- tant difference between the first two microstructures and the third concerns the the elastic constraint of the matrix, but A potential for plastic strain in the composite should decrease as the crack opens and, furthermore, should depend sensitively on outside the bridging ligaments. Plastic L strain in the first two is limited by elastic the work hardening rate. These simple ca, strains in the elastic network. Consequent- results already provide the important in- E ly, experience and analysis confirm that sight that a peak stress exists at small F m crack bridging is usually the most potent crack openings, such that the plastic dis- Q mechanism when the ceramic phase is sipation is dominated by the large strain continuous, because the only ductile (necking) regime (Fig. 23). The importance regions which experience extensive plas- of interface debonding thereby becomes tic strain are those segments that stretch apparent, because debonds reduce the between the crack surfaces in the bridg- constraint, but increase the plastic stretch ing zone. The plastic dissipation in this prior to failure. Numerical solutions ob- bridging zone can be large and can pro- tained with prescribed initial debonds es- tablish the salient trends (Fig. 23). Notably, vide a major increase in toughness. Con- Distance from crack plane, y (pm) versely, a composite with a continuous the plastic dissipation increases systemat- metal network is subject to nonlinear pow- ically as d/R increases. Furthermore, if the Fig. 20. Trends in the microcrack density er law deformation and can experience debond evolves during crack opening, the parameter, '1 =N, with distance y from the substantial plastic strain within a plastic dissipation is further enhanced. The above crack surface. zone. Consequently, the potential for ap- predictions are generally similar to ex- preciable plastic dissipation within a plas- perimental measurements, which confirm tic process zone is much greater for this large stresses at small u and increased class of microstructure. dissipation when debonding occurs. Bridging Grains Understanding of the toughness gener- The toughness attributed to bridging, Interfaces ated by ductile ligaments is contingent based on Eq. (3),can be reexpressed in upon the law that characterizes the nondimensional form by noting that the Residual strains stress/stretch relation, flu).Insights regard- flow stress scales with the uniaxial yield ing the parameters that affect this relation strength, Y, and that the plastic stretch is can be gained from simplified analytical proportional to the radius of the cross sec- models. Complementary numerical solu- tion of the reinforcing ligaments, R; con- tions then allow determination of specific sequently, the asymptotic toughness is4041 trends. Budiansky and CO-workers13.52 AgCI f YR and Rose53 analyzed srnal/-%a/ebridging, =x (34) in which bridge length is small in relation where x is a "work-of-rupture''parameter to crack length, specimen dimensions, that depends on the critical plastic stretch, and distances from the crack to the speci- u, (or ductility), of the reinforcement and men boundaries. The stresslstretch rela- on the extent of the interface debonding, tion depends strongly on the mode of d. Values of x have been obtained both failure of the ductile ligaments. A small- by calculation and by experiment40 based scale yielding analysis indicates that on determinations the area under the Fig. 21. Ligament formation allowed by t of residual stress. should increase rapidly with initial crack stresdstretch curve for the reinforcing liga- opening. A large-strain necking analysis ments. For a wekbonded interface (d=O) then reveals that, without debonding, the and for ductile ligaments that fail by neck- stress attains high initial levels because of ing to a point, the resultant trend in x with 198 Journal of the American Ceramie Society-Evans Vol. 73, No. 2 wherein the precipitates locally block the passage of dislocation, causing the flow strength to be inversely proportional to the precipitate spacing. Inserting the meas- ured precipitate spacing (kl00 nm), the yield strength is Y=220 MPa. This value is comparable to typical values quoted for Al alloys precipitation hardened with Cu. Stereo measurements have been used to evaluate the plastic stretch to failure, u,. The normalized plastic stretch, u, lR, varied appreciably between ligaments. However, there was no systematic depen- dence on either the ligament dimension, R, ortheaspect ratio: the mean stretch was u,/R= 1.6. The plastic stretch to failure indicates that x is in the range 2 to 3.5. Substituting the above values of Y and x into Eq. (35) and noting that f=0.2 and R=2.0 pm, Ag, is calculated as 170 to 300 J * m-2, Consequently, there is ac- ceptable agreement with the measured value of 150 to 200 J. m-2, Such behavior is consistent with the interpenetratingnet- work microstructure. For cemented carbide materials, the Fig. 22. Schematic indicating the various contributions to the steady-state toughness. contribution from bridging deduced from the measured zone sizes is found to be relatively sma1154,55 (AgC=40J - m-2) for reasonable choices of the flow stress of the Co alloy, whereas the overall meas- ured toughness is g?,=400 J . m-2. work-hardening rate n indicates that x is Consistent with this interpretation, meas- in the range 0.3 to 1. Less-ductile liga- urements of plastic zones indicate zone ments that rupture prematurely by profuse sizes in the range h=40 pm.42 It is sup- hole nucleation have correspondingly posed that dissipation is appreciable in this smaller values of x. Systems subject to de- zone and that this may be the bonding exhibit larger x and approach 8 predominant contribution to the toughness for large dlR. Experimental results have in- of these materials. dicated that trends in Y and d are reflect- ed in a plot of x with rei‘ative plastic stretch (B) FiberMhisker Reinforcement u,/R (Fig. 24). (i) Basic Features (ii) Toughness Correlations Practical ceramic-matrix composites reinforced with continuous fibers exhibit The basic nondimensional solution (Eq. the failureldamage behaviors sketched in 34) can be used both to rationalize tough- Fig. 26. The composite properties are ness measurements and to develop a known to be dominated by the interface, predictive capability. For purposes of the I IcI.7LI” *LIFL”II, _I a and bounds must be placed on the inter- former, Y and n can be inferred either from Fig. 23. Nondimensional stress versus a TEM characterization of the microstruc- face debonding and sliding resistance to have a composite with attractive mechan- stretch behavior and associated “work-of- ture50 or from microhardness measure- rupture” x for various debond lengths. ical properties. The strong dependence of ments (Elliott ef a/.a),while R, u,, and ceramic-matrix composite properties on d can be determined by quantitative SEM the mechanical properties of the interface of the fracture surface.50 Comparison be- generally demands consideration of fiber tween theory and experiment is most coatings andlor reaction product layers. readily achieved by reexpressing Eq. (34) Residual stresses caused by thermal ex- in the form pansion differences are also very im- Aq, = f YRx (u, IR) (35) portant. The specific microstructural parameters To use this result, Y is first obtainem- that govern Mode I failure are the relative lowed by evaluation of x, using u, IR. fibedmatrix interface debond toughness, Thereafter, Aqc is examined for con- rjlrf,the misfit (thermal expansion) strain sistency by comparison with the ex- between fiber and matrix, E; the friction perimentally determined value. For coefficient at the debonded interface, p, Lanxide composites of A1203 reinforced the statistical parameters that characterize with Al, the experimental information is as the fiber strength, So and rn, the matrix follows: The yield strength has been evalu- toughness, rm,and the fiber volume frac- Fig. 24. Work of rupture as a function of ated by noting that the alloy contains Al- tion, f. The prerequisite for toughness is plastic stretch measured for various ductile Cu precipitates which behave as im- that ri /I-,; 114 to allow crack-front de- reinforcements. penetrable obstacles (Fig. 25). Conse- bonding (Fig. 27). Subject to this require- quently, their influence on yielding should ment, the misfit strain must be small be represented by Orowan hardening, (E: 2 3 x 10-3) and preferably negative, February 1990 Perspective on the Development of High- Toughness Ceramics 199 such that the interface is in tension. sites reinforced with SIC fibers reveal that Furthermore, for a continuous fiber, the materials with a C interlayer satisfy de- friction coefficient along the debonded in- bonding requirements (Fig. 29) and also terface should be small. The ideal fiber have small T and thus demonstrate exten- properties include a high median strength sive pullout.60-62 Conversely, composites (large So)and large variability (small m), having a continuous SiOn layer between as needed to encourage large pullout the matrix and fiber exhibit matrix crack lengths. When rjIrf and p are both small, extension through the fiber without experience has indicated that the tensile debonding62.63 (Fig. 30). Experiments on stress-strain behavior illustrated in Fig. these composites and on whisker- 26(A) is obtained. Three features of this reinforced composites also confirm the curve are important: matrix cracking at a strong influence of T. Notably, systems stress uo. fiber-bundle failure at uu, and which debond but do not slide readily (be- pullout. Conversely, larger r,/rfand p cause of either a high friction coefficient cause the stress-strain curve to become or morphological irregularity) exhibit small linear (Fig. 26(!3)). The ultimate strength pullout lengths and moderate toughness. then coincides with the propagation of a Such behavior is exemplified by single dominant crack. LASttlSiC composites with oxide inter- Present understanding of debonding is phases and by various whisker-reinforced consistent with the following sequence of materials, respectively. events during matrix crack propagation. Tough composites can be obtained by Initial debonding along the interface at the creating the appropriate interphases crack front requires that ri/rfbe small between the fiber and matrix, either by enough to lie within the debond zone coating or, in situ, by segregationhter- depicted in Fig. 27.56 Furthermore, the ex- diffusion. The most common approach is Fig. 25. Bright-fieldTEM view of the precipi- tent of debonding is typically small when the use of a dual coating: the inner coat- tates in the Al alloy that cause hardening. residual compression exists at the inter- ing satisfies the debonding and sliding re- face, but can be extensive when the in- quirements, while the outer coating terface is in residual tension and the fibers provides protection against the matrix dur- are smooth. Further debonding is usually ing processing. However, the principal induced in the crack wake57 (Fig. 28). The challenge is to identify an inner coating extent of this debonding is governed that has the requisite mechanical proper- largely by the residual field. Residual radial ties while also being thermodynamically tension results in unstable conditions and stable in air at elevated temperatures. encourages the extensive debonding of smooth fibers. Residual compression (ii, Constitutive Laws andlor an irregular fiber morphology The mechanical properties of uniaxial- Fiber bundle failure, o,, cause stable debonding,Se with the extent ly reinforced composites are largely determined by the friction coefficient and governed by the relationships between the the roughness of the debonded interface. opening of a matrix crack, u, and the Subsequent fiber fracture involves the stresses, t, exerted on the crack by the in- statistics of fiber failure,59 subject to an ax- tact bridging fibers and the failed fibers as ial stress governed by the sliding they pull out. The traction t(u)is well-known resistance of the debonded interface. The for composites in which debonding occurs above sequence suggests that, although easily (very small rj/rf)and which also debonding is a prerequisite for high tough- slide easily along the debonded interface ness, the properties of the composite are (small T). For other cases, reliable t(u)func- dominated by the sliding resistance of the tions have yet to be elucidated. Easy E debonded interface, which dictates the debonding and sliding provide a crack- (a) 'Tough' composite major contribution to toughness caused by opening function that depends on the sign pullout. The locations of fiber failure that of the misfit strain, as well as the rough- govern the pullout distributions can be de- ness of the debond interface, through their termined from the stresses on the fibers, effect on the sliding resistance. For in- using concepts of weakest-link statistics. stance, sliding can be described by a Analysis of this phenomenon has been Coulomb friction law performed for composites having debond- ed interfaces subject to a constant sliding T=Pq (36) stress, 7.59 The magnitude of T governs the where p is the friction coefficient and q the load transfer from the fiber to the matrix. nominal residual compression normal to Large values of T cause the fiber stress to the interface. At the simplest level, q is set vary rapidly with the distance from the by the residual misfit strain and the matrix crack and induce fiber failure close fiberkoating morphology, such that T is to the crack, leading to small pullout essentially invariant. In this case, prior to E lengths, h,. Conversely, small T results in the incidence of fiber failure, t and u are (b) 'Brittle' composite large h,. related by17.18 Various observations of crack interac- Fig. 26. Schematic illustrating the range of tions with fibers are supportive of the stress-strain characteristics exhibited by above rules. In particular, experiments on (37) ceramic-matrix composites. glass and glass-ceramic-matrix compo- Equation (37) has been used to describe the mechanical behaviors that are ob- Kithiurn aluminum silicate tained while the fibers are largely intact. 200 Journal of the American Ceramic Society-Evans Vol. 73, No. 2 At another extreme, when all of the fibers fiber and matrix surface separate, leading have failed, the traction on any fiber is59.63 to unrestricted crack opening. Additional complexity is involved when debonding ti = 2f~(hj- u)/R (38) and sliding occur simultaneously. Some preliminary results for such behavior ex- 1.o where hi is the distance from the matrix crack plane at which that fiber failed. ist, but are not addressed here. it is sim- e ply noted that, in such cases, the fracture G More difficult problems to address con- energy of the coating becomes another 0.5 failure cern the incidence and location of fiber failure. This has been regarded as a parameter of interest. problem in weakest-link statistics,a where- The location of fiber failure vis-a-visthe 0 upon the function t(u) depends upon the matrix crack plane is of critical importance I I I I I statistical parameters So and m in addition because this location governs the pullout -1.0 -0.5 0 0.5 1.0 length h,. Both theory59 and experi- Elastic mismatch, a to the variables contained in Eq. (37).The expressions are unwieldy and are not menF suggest that, for aligned reinforce- reproduced here, but can be located in ments, h, is governed by weakest-link the article by Thouless and Evans.59 Some statistics. For the simplest case, wherein trends, expressed in nondimensional form, the sliding stress T remains constant, an are summarized in Fig. 31. The small dis- expression for the mean pullout length has placement behavior is dominated by the been derived as59 intact bridging fibers, whereas the long tail (2h,/R)m" = (1/2rr(m l)m](AdRz) is governed by the pullout of failed fibers. + Another level of complexity is involved x (so/T)mr[(m+ 2)/(m + l)] when the interface stress q~ varies with (40) -1.0 -0.5 0 0.5 1.0 the loads t on the fiber. A simplified result Elastic mismatch, a where r in this expression is the gamma obtained using a modified shear lag ap- function and A. is a reference area for the Fig. 27. Fracture energy requirements for proach suggests the following features58 fibers (usually set equal to 1 m2). Conse- crack-frontdebonding. (Ef= Em = E and vf= urn= u): quently, for aligned fibers, it is evident that U/R=(fJ/p)[f(l+F)/(l - uf)] h R is essentially governed by S~T:high x(l -utuf)/(l-9 (39) fig' er "strength" and low sliding resistance encourage large pullout lengths. Cor- where responding trends for inclined fibers are unknown. f = t/f$E (iiq Matrix Cracking Stress with being the misfit strain which causes the interface to be in residual com- The stress a. at which matrix cracking pression. Equation (39),although approx- occurs has been the most extensively imate, has several salient features. In studied behavior in ceramic-matrix com- particular, as expected, crack opening is posites. For composites in which the inhibited by large values of the friction residual stress normal to the interface, qN, coefficient. Furthermore, as f+l/v, the is tensile and the interface properties can be effectively represented by a unique sliding stress, T, Eq. (37) may be used to derive the lower-bound, steady-state matrix cracking stress."

00 a* P E-E Em where

with p being the axial residual stress in the matrix. This result is independent of the matrix crack length because the crack is "fully" bridged by fibers and applies when the initial crack is larger than the fiber spacing.18 Measurements performed on many different glass- and ceramic- matrix composites (Fig. 32) reinforced with Nicalon fibers have verified that Eq. (28), in fact, provides an adequate repre- sentation of matrix cracking provided that 1220 MPa.62 Values of T in this range have been demonstrated for fibers coat- ed with either C or BN, as elaborated be- Fig. 28. Schematic illustrating the initial debonding of fibers at the crack front, as well as fiber low. For these systems, it is recognized debonding and sliding in the crack wake. that T depends on the misfit strain, and, consequently, for the simple case where- in T and q~ are related through a friction coefficient (Eq. (36)),the matrix cracking February 1990 Perspective on the Development of High- Toughness Ceramics 20 1

Fig. 29. Crack-front debonding in a Nicalon-fiber-reinforcedaluminosilicate-glass-matrix composite.

stress exhibits a maximum 60 given by17 toughness. The above effects are indica- tive of resistance-cuwe behavior, because 6,x = (2/3)[2fpr,,,/(i+ E/E,)EA~’~ (42) each contribution is only fully realized This result has evident implications for when the fibers fail and pull out. The full material design. details of the increase in the fracture More detailed diagnosis of matrix crack- resistance can be calculated from the ing confirmsthat Eq. (41) is a lower bound crack surface tractions t(u) by applying Eq. for the onset of cracking. Further crack- (3). A useful simplification for further dis- ing occurs as the stress is raised above cussion is the peak (or asymptotic) tough- go, resulting in a periodic crack array (Fig. ness that is obtained when each 33).’6 The crack spacing reaches a satu- mechanism exerts its maximal contribu- ration value, s, when the stress every- tion. At the simplest, physically relevant where in the “matrix blocks” between level, this toughness is given by66 cracks becomes smaller than that applied stress. The magnitude of the saturation Agczfd[S2/E- E(E;)~ crack spacing is governed by the sliding + 4rj/R(1 - f)]+ 2~fhE/R (45) stress, such that and s are related bp T, T The first term is a bridging contribution that derives from the stored strain energy dis- sipated as acoustic waves, with S being Consequently, the matrix cracking stress- the reinforcement “strength.” The second es can be expressed in terms of the crack term is the loss of residual strain energy spacing as caused by matrix crack extension and de bonding. The third term reflects the new “surface area” caused by debonding, Fig. 30. Fracture surface indicating fracture and the fourth term is the pullout contri- through the fiber in a silicalSiC-fibercomposite. The crack spacing clearly provides an ap- bution, dissipated by frictional sliding of the proach for estimating T and for establish- interfaces. ing a self-consistent description of matrix Experience indicates that the residual cracking. strain term is small in systems of practical utility and can often be neglected. The lar- (iv) Toughness gest potential for toughness resides in the In reinforced ceramics which fracture by pullout term, provided that hdR is large. the growth of a single dominant flaw in An extreme range of pullout behaviors is - 0.6 Mode I, there are four effects which in- apparent among the available range of fluence toughness.66 Debonding gener- fiber- and whisker-reinforced ceramics, - 0.4 ates new surface and contributes resulting in wide variations of toughness. positively to toughness. Frictional dissipa- Understanding pullout thus dominates our tion upon pullout results in local heating capability for producing ceramic compo- and again contributes positively. Residu- sites having exceptional toughness. The a/ stresses present in the material are par- dominance of toughness by pullout pro- 012345 tially relieved by matrix cracking and vides a focus for specifying those proper- (UW debonding and thus detract from the ties of the fibers and fiber coatings that Fig. 3,. Nondimensional crack toughness. Finally, when the fibers fail, optimize the frictional dissipation. For stress as a function of crack opening, some of the elastic energy stored in the aligned reinforcements, wherein weakest- fiber is dissipated through acoustic waves link statistics govern the pullout length, an and appears as a positive contribution to explicit dependence of toughening on 202 Journal of the American Ceramic Society-Evans Vol. 73, No. 2 fiber strength and sliding resistance is fects cause the toughness to gradually predicted.59 Physically stated, high fiber build up to the asymptotic values. Conse- strength and low sliding resistance com- quently, the full toughness cannot usually bine to maximize the frictional dissipation be utilized. The toughening rate (tearing by inducing sliding over the largest pos- modulus) has not been broadly studied sible fiber surface area. The inverse de- and trends are relatively unknown. pendence of pullout toughening on T However, some numerical results for emphasizes the need to control and un- pullout-dominated toughness indicate that derstand sliding. Some of the qualitative asymptotic behavior is achieved only af- features are depicted in Fig. 34, illustrat- ter considerable crack extension.67 ing the importance of the fiber morpholo- gy, the misfit strain, and the friction (v) Ultimate Strength When matrix cracking precedes ultimate 600 coefficient. -a Nonaligned fibers and whiskers clearly failure, the uitimate strength coincides with a 500 suppress pullout by virtue of bending fiber-bundle failure.65 A simple estimate of z this strength, based on weakest-link statis- u) 400 strains in the reinforcements that en- u) courage fracture near the matrix crack tics that neglects interaction effects be- 300 plane. Indeed, fibers having low flexural tween failed fibers and ignores the stress -u) 2 200 resilience, such as Sic, tend to fail at the supported by fractured fibers by means .- matrix crack plane when inclined to the of stress transfer from the matrix through E, 100 interface fraction, gives z crack,67.68 whereas C fibers may still pull- 0 out over large distances69 Fiber alignment [1 - (1 - TS/R$m*l] 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 issues associated with pullout thus depend o,=fS exp - Engineering strain (%) on fiber properties. (m + 1)[I - (I - TS/R~)~] When fiber pullout does not contribute -1 i Fig. 32. Tensile stress-strain curves for a to toughness, as in many whisker- (48) range of glass- and glass-ceramic-matrix com- reinforced ceramics, the elastic energy with posites reinforced with Sic (Nicalon) fibers. and debonding energy terms tend to govern the fracture resistance, such that (R&.S)m+l = (&/~xRL~)(~?S~/TS)~ x [l -(1 -Ts/RS)m-q AQczfdS*/f+ 4r; f (d/R)/(I- f ) (46) where L, is the specimen gauge length. To interpret this expression, it is essential The effect of the sliding stress on u,, ap- to appreciate that the debond length d de- pears directly, as well as through its effect pends on the interface fracture energy Ti, on the crack spacing s, while the effect of misfit strain E: and friction coefficient. The residual stress is present through its effect associated relationships are unknown, but on T. The ultimate strength is also expect- dimensional analysis suggests that57.58 ed to be influenced by the residual stress. Specifically, in systems for which the fiber d/R = H(f,R(&,T)*/r,,S/f&,T, 1/p) (47) is subject to residual compression,the ax- where H is a function. The important point ial compression should suppress fiber is that there remains much scope for con- failure and elevate the ultimate strength to trolling toughness by manipulating the in- a level in excess of that predicted by Eq. terface debonding and sliding properties (48). This effect may be estimated by and by maximizing the fiberlwhisker regarding the matrix as clamping onto the strength. fiber and, thus, simply superposing the As already noted, resistance-curve ef- residual stress onto S.

Fig. 33. Periodic matrix cracks formed upon loading above the steady-state matrix cracking stress. February 1990 Perspective on the Development of High- Toughness Ceramics 203 (vi) Propefly Transition are shown in Fig. 35.62 Limited experience Nonlinear macroscopic mechanical be- has indicated that the ratio of median havior on tension is most desirable for strengths scales with the ratio of compo- structural purposes. Analysis of the tran- site ultimate strengths. sition between this regime and the linear Based on the above experimental fea- regime is important, and involves compar- tures that C and BN coatings are effec- ison of the basic trends in the steady-state tive in terms of satisfying crack-front matrix cracking stress, oo, and in the debonding and also providing a low slid- asymptotic fracture resistance, Age. Most ing resistance, whereas amorphous ox- significantly, uo increases but AgC ides are incapable of debonding, it may decreases as T increases. These oppos- be concluded that C and BN are the ing trends with T suggest the existence of preferred coatings. However, both of an optimum T that permits good matrix these materials are susceptible to rapid cracking resistance while still allowing high oxidationa.75 and mostly incapable of per- toughness. More specifically, a property forming their required debonding and slid- transition is expected when the matrix ing functions at elevated temperatures in cracking stress attains the stress needed oxidizing environments. A variety of other for fiber-bundle failure. One estimate on coatings are thus under investigation. the property transition can be obtained by When such coatings have been deve- simply allowing oo to exceed the ultimate loped, some of the more complex crack- strength, ou, whereupon a nondimension- opening and pullout constitutive laws al parameter p which governs the transi- presented may be needed to describe the tion when T is small is mechanical properties of the composite. t (C) Bridging Grains Fig. 34. Schematic of the pullout process for (49) As already noted, bridging grains can debonded fibers indicating the roles of fiber Specifically, when p exceeds a critical morphology, misfit strain, .and friction be induced in two ways:5.76 grain- coefficient. value, brittle behavior initiates. This esti- boundary debonding and residual stress. mate has not been tested, and, further- Low fracture energy grain boundaries can more, alternate parameters may be allow debonding, as in whisker-reinforced conceived. materials, such that toughening involves the same terms described by Eq. (45). In- (vii) Experimental Correlations deed, a trend toward an increase in Uniaxially reinforced ceramic- and steady-state toughness is found in various glass-matrix composites with either C or polycrystalline A1203materials as the grain- BN interlayer generally satisfy crack-front boundary fracture energy decreases.77 debonding requirements, and the materi- Also, elongated grains, which allow larg- als exhibit axial and transverse mechani- er debond lengths, could lead to higher cal properties that accord well with the toughness, as noted over a decade ago above expressions for the matrix cracking for both Si3N478 and AI2O3.79 It is also stress and the ultimate strength.62 To known that, when the grain-boundary frac- achieve these comparisons, each of the ture energy is not low, as in sintered Sic, parameters in Eqs. (41) and (48) has been elongated grains do not lead to higher 8- independently ascertained. As already toughness. However, detailed application mentioned, T can be measured using var- of Eq. (45) has not been attempted. ious techniques: matrix crack spacing,l6,6* Residual stress-induced bridging occurs indentation push-through stress,70,71 and because circumferential compression matrix crack opening hysteresis.61 The causes the crack to circumvent local, high- misfit strain $can also be determined us- ly stressed regions. Furthermore, it has ing a variety of methods: X-ray or neutron been possible to rigorously demonstrate line shifts, offset stresses in the crack open- that such a bridging process increases the ing, and residual displacements obtained steady-state toughness in accordance from nanoindenter tests. Accurate values with37 of the matrix fracture energy rm and of AKc = 2.5fEAaATcR 012345 the constituent elastic properties are also Fiber fracture stress, S (GPa) needed. Furthermore, it is noted that the where / is the volume fraction of highly comparison between theory and experi- stressed grains that result in bridging liga- Fig. 35. Fiber strength distributions ascer- ment for the matrix cracking stress is con- ments and 2R the grain diameter. This in- tained on two composites (refer to Fig. 32 for tingent upon having accurate values of the crease in toughness arises because the corresponding stress-strain curves) using above parameters. ligament failure occurs unstably and ener- fracture mirror measurements Correlations of trends in the ultimate gy is dissipated in this process as acous- strength are primarily contingent upon tic waves, partly negated by some knowledge of the in situ mechanical reduction in residual strain energy (c.f.. the properties of the fibers. Appreciation for first two terms in Eq. (47)). For this process, these properties can be gained by ex- elongated grains are not obviously amining and measuring fracture mirrors beneficial. on the fiber fracture surfaces in the com- posite. Specifically, the distribution of mir- (0) Multiple Mechanisms ror radii can be used to evaluate the axial The preceding sections have described stress S on the fiber of the fracture loca- microstructural issues concerned with tion.72-74 Some typical results for two toughness optimization when a single materials with different ultimate strength mechanism operates. In practice, more Vol. 73, No. 2 204 Journal of the American Ceramic Society-Evans than one mechanism may exist. Conse- 2R. W. Rice, Treatise On Materials Science and quently, interactions between mechanisms Technology, Vol. 11; pp. 191-381. Academic Press, New York, 1978. must be considered, In some instances, 3A. G. Evans, "Structural Reliability: A Processing- the interactions may be highly beneficial Dependent Phenomenon," J. Am. Ceram. Soc., 65 and produce synergism between [3] 127-37 (1982). mechanisms. Such synergism has been 4A. G. Evans, "New Opportunities in the Process- illustrated to exist when both bridging and ing of High-Reliability Structural Ceramics"; pp. process-zone mechanisms operate simul- 989-1 010 in Ceramic Transactions, Vol. 1,Ceramic taneously.15 Conversely, some interaction Powder Science. Edited by G. L. Messing, E. R. effects can be deleterious and reduce the Fuller, Jr., and H. Hausner. American Ceramic so- efficacy of the individual mechanisms. ciety, Westerville, OH, 1989. Synergism is most likely when bridging 5R. F. Cook, 6. R. Lawn, and C. J. Fairbanks, "Microstructure-Strength Properties in Ceramics: I, and process-zone mechanisms interact. Effect of Crack Size on Toughness," J. Am. Ceram Multiplicative interactions are evident in this SOC,68 [ll] 604-15 (1985). case because the crack surface tractions 6D. B. Marshall, "Strength Characteristics of caused by bridging can expand the Transformation-ToughenedZirconia." J. Am. Cefam. process-zonewidth h in the crack wake, SOC.,69 [3] 173-80 (1986). causing an additional increase in shield- 7D. Shetty and J. Wang, "Crack Stability and ing, proportional to this increase in h. Strength Distribution of Ceramics That Exhibit Ris- Straightforward logic indicates that mul- ing Crack-Growth-Resistance [R-Curve] Behavior," tiplicative toughening between bridging J. Am. Ceram. Soc.. 72 [7] 1158-62 (1989). eA. G. Evans and A. M. Cannon, "Toughening of and process-zone effects should occur Brittle Solids by Martensitic Transformations," Acfa when the ratio of the bridging-zone size, Mefall., 34 [5] 761 -800 (1 986). L, to the process-zone width, h, is small, OR. M. McMeeking and A. G. Evans, "Mechanics because bridging generates a new effec- of Transformation-Toughening in Brittle Materiais," tive crack-tip toughness, that causes the J. Am. Ceram. SOC.,65 [5] 242-47 (1982). process-zone size to further increase. 106. Budiansky, J. W. Hutchinson, and J. C. Lam- Calculations of coupled bridging and bropolous, "Continuum Theory of Dilatant Transfor- process-zone effects have been per- mation Toughening in Ceramics," lnf. J. Sohds formed in the case that the bridging trac- StrUCt.. 19 [4] 337-55 (1983). 1lD. B. Marshall, M. D. Drory, and A. G. Evans, tions are constant (Dugdale zone). The "Transformation Toughening in Ceramics"; pp. results reveal two bounds given by15 289-307 in Fracture Mechanics of Ceramics, Vol. 6. Edited by R. C. Bradt, A. G. Evans, F. F. Lange, and KcIKm= b& (514 D. P. H. Hasselman. Plenum Press, NewYork, 1983. the synergistic limit and 120, M. Stump and B. Budiansky, "Crack Growth Resistance in Transformation-ToughenedCeramics," Kc/KM=~b2i-b~-~P (514 lnf. J Solids Sfrucf., 25, 635-46 (1989). 138. Budiansky, J. C. Amazigo, and A. G. Evans, the lower bound, where bp and bb are the "Small-Scale Crack Bridging and the Fracture toughening ratios for the process-zone Toughness of Particulate-Reinforced Ceramics," J. and bridging mechanisms, respectively. Mech. Phys. Solids, 36 [2] 167-87 (1988). Furthermore, and surprisingly, the syner- 14A. G. Evans and R. M. McMeeking, "On Tough- gistic limit was found to be obtained when ening of Ceramics by Strong Reinforcements," Acfa L lh ? 10. Rigorous experimental su bstan- Mefall., 34, 2435-41 (1986). tiation of synergism has yet to be obtained. 15J. C. Amazigo and B. Budiansky, "Interaction of Particulate and Transformation Toughening," J. The interactions between either two Mech. Phys. Solids, 36, 581-95 (1988). bridging-zone mechanisms or two Harvard University Report, MECH-112 (1989). process-zone mechanisms are relatively 16J Aveston, G. A. Cooper, and A. Kelly, "Single unexplored. At the simplest level, two and Multiple Fracture"; pp. 15-26 in Conference bridging mechanisms are additive with Proceedings of the National Physical Laboratory: each AgCgiven by Eq. (3). However, the Properties of Fiber Composites. IPC Science and critical stretch u for one mechanism may Technology Press, Surrey, England, 1971 be affected by the other, whereupon ad- 178. Budiansky, J. W. Hutchinson, and A. G. ditivity may not be realized. Interactions Evans, "Matrix Fracture in Fiber-ReinforcedCeram- ics." J. Mech. Phys. Solids, 34, 167-89 (1986). between process-zone mechanisms are 1aD. B. Marshall, B. N. Cox, and A. G. Evans, "The more difficult to express, because the zone Mechanics of Matrix Cracking in Brittle-Matrix Fiber sizes and the transformation strains are Composites," Acfa Metall., 33, 2013-21 (1985). coupled. Preliminary attempts have been 19L. N. McCarlney. "Mechanics of Matrix Crack- made to examine concurrent transforma- ing in Brittle-Matrix Fiber-ReinforcedComposites," tion and microcrack toughening which in- Proc. R. Soc. London, Series A-Mathematical and dicate that the coupling can result in an Physical Sciences. 409 [1837] 329-50 (1987). augmentation of toughness. zoM. V. Swain and L. R. F. Rose, "Strength Limi- tations of Transformation-Toughened Zirconia Al- loys," J. Am. Ceram. Soc.,69 [7] 511-18 (1986). Z'A. G. Evans, N H. Burlingame. W. M. Kriven, and M. D. Drory, "Martensitic Transformations in Zir- conia Particle Size Effects and Toughening," Acta Mefall., 29, 447-56 (1981). 22R. Dauskhardt, D. K. Veirs, and R. 0. Ritchie, "Spatially Resolved Raman Spectroscopy Study of References Transformed Zones in Magnesia-Partially-Stabilized IF. F. Lange, "Powder Processing Science and Zirconia," J. Am. Ceram. Soc., 72 [7] 1124-30 Technology for Increased Reliability," J. Am. Ceram. (1989). SOC.,72 [l] 3-15 (1989). 231. W Chen and P. E. Reyes-Morel, "Implications February 1990 Perspective on the Development of High-Toughness Ceramics 205 of Transformation Plasticity in ZrO&ontaining Cer- 48W. W. Gerberich and E. Kurman, "New Contri- amics: I, Shear and Dilitation Effects," J. Am. Cer- butions to the Effective Surface-Energy Cleavage," am. Soc.. 69 (31 181-89 (1986). Scr. Metall., 19 [3\ 295-98 (1985). 24L. R. F. Rose, "Kinematical Model for Stress- 49M. S. Newkirk, A. W. Urqhart, and H. R. Zwicker, Induced Transformation around Cracks," J. Am. Cer- J. Mater. Res., 1, 81 (1986). am. Soc.,69 (31 208-12 (1986). SOB. Flinn, M. Ruhle, and A. G. Evans. "Toughen- 25M. Ruhle, N. Claussen, and A. H. Heuer, "Trans- ing in Composites of A1203 Reinforced with At," Acta formation and Microcrack Toughening as Com- Metall., 37 [I 11 3001 -3002 (1 989). plementary Processes in Zr02-Toughened AI2O3," 51V. K. Sarin (ed.), "Science of Hard Materials-3," J. Am. Ceram. Soc., 69 [31 195-97 (1 986). Mater. Sci. 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Out in Brittle Matrix Composites," Mecb. Mafer., 8, 33A. G. Evans and K. T. Faber, "Crack-Growth 1-12 (1989). Resistance of MicrocrackingBrittle Materials," J. Am. 59M. D. Thouless and A. G. Evans, "Effect of Pull- Ceram. Soc., 67 (41 255-60 (1984). Out on the Mechanical Properties of Brittle Matrix MJ. W. Hutchinson, "Crack Tip Shielding by Composites," Acta Metall., 36 (3) 517-22 (1988). Microcracking in Brittle Solids," Acfa Metall., 35 (71 WK. M. Prewo and J. J. Brennan, "Silicon Carbide- 1605-19 (1987). Fiber-ReinforcedGlass-Ceramic-Matrix Composites 35A. Quinten and V. Arnold, "Observation of Sta- Exhibiting High Strength and Toughness," J. Mater. ble Crack Growth Using a Scanning Acoustic Micro- Sci., 17, 1201-206 (1982). scope"; to be published in Mater. Sci. Eng. SlD. B. Marshall and A. G. Evans, "Failure "6. Budiansky and R. J. O'Connell, "Elastic Mechanisms in Ceramic-FiberCeramic-Matrix Com- Moduli of a Cracked Solid," In?.J. Solids Struct , 12 posites," J. Am. Ceram. 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