13 Toughening Mechanisms 13.1 INTRODUCTION The notion of designing tougher materials is not a new one. However, it is only in recent years that scientists and engineers have started to develop the fundamental understanding that is needed to guide the design of tougher materials. The key concept in this area is the notion of shielding the crack tip(s) from applied stress(es). When this is done, higher levels of remote stresses can be applied to a material before fracture-critical conditions are reached. Various crack-tip shielding concepts have been identified by research- ers over the past 30 years. These include: 1. Transformation toughening 2. Twin toughening 3. Crack bridging 4. Crack-tip blunting 5. Crack deflection 6. Crack trapping 7. Microcrack shielding/antishielding 8. Crazing Copyright © 2003 Marcel Dekker, Inc. The above toughening concepts will be introduced in this chapter. However, it is important to note that toughening may also occur by some mechanisms that are not covered in this chapter (Fig. 13.1). In any case, the combined effects of multiple toughening mechanisms will also be discussed within a framework of linear superposition of possible synergistic interactions between individual toughening mechanisms. 13.1.1 Historical Perspective Toughening concepts have been applied extensively to the design of compo- site materials. Hence, before presenting the basic concepts and associated equations, it is important to note here that even the simplest topological forms of composite materials are complex systems. In most cases, these incorporate interfaces with a wide range of internal residual stresses and FIGURE 13.1 Crack-tip shielding mechanisms. Frontal zone: (a) dislocation cloud; (b) microcrack cloud; (c) phase transformation; (d) ductile second phase. Crack-wake bridging zone: (e) grain bridging; (f) continuous-fiber brid- ging; (g) short-whisker bridging; (h) ductile second phase bridging. From B. Lawn, reprinted with permission from Cambridge University Press.) Copyright © 2003 Marcel Dekker, Inc. thermalexpansionmisfit.Also,mostoftheexpressionspresentedinthis chapterare,atbest,scalinglawsthatcapturetheessentialelementsof complexbehavior.Inmostcases,theexpressionshavebeenverifiedby comparingtheirpredictionswiththebehaviorofmodelmaterialsunder highlyidealizedconditions.However,duetotherandomfeaturesinthe topologiesoftheconstituentparts,theagreementbetweenthemodelsand experimentsmaybelimitedwhentheconditionsaredifferentfromthose capturedbythemodels(Argon,2000). Inanycase,therearetwotypesoftougheningapproaches.Theseare generallyreferredtoasintrinsicandextrinsictoughening.Inthischapter, intrinsictougheningisassociatedwithmechanisticprocessesthatareinher- enttothenormalcracktipandcrackwakeprocessesthatareassociated withcrackgrowth.Incontrast,extrinsictougheningisassociatedwithaddi- tionalcracktiporcrackwakeprocessesthatareinducedbythepresenceof reinforcementssuchasparticulates,fibers,andlayers.Availablescalinglaws willbepresentedforthemodelingofintrinsicandextrinsictoughening mechanisms.SelectedtougheningmechanismsaresummarizedinFig.13.1. 13.2TOUGHENINGANDTENSILESTRENGTH Inmostcases,tougheninggivesrisetoresistance-curvebehavior,asdis- cussedinChap.11.Inmanycases,theassociatedmaterialseparationdis- placements are large. This often makes it difficult to apply traditional linear and nonlinear fracture concepts. Furthermore, notch-insensitive behavior is often observed in laboratory-scale specimens. Hence, it is common to obtain expressions for the local work of rupture, ÁW, and then relate these to a fracture toughness parameter based on a stress intensity factor, K,oraJ- integral parameter. If we now consider the most general case of material with an initiation toughness (energy release rate) of G i and a toughening increment (due to crack tip or crack wake processes) of ÁG, then the overall energy release rate, G c , may be expressed as: G c ¼ G i þ ÁG ð13:1Þ Similar expressions may be obtained in terms of J or K. Also, for linear elastic solids, it is possible to convert between G and K using the following expressions: G ¼ K 2 =E 0 ð13:2aÞ or K ¼ ffiffiffiffiffiffiffiffiffi E 0 G p ð13:2bÞ Copyright © 2003 Marcel Dekker, Inc. where E 0 ¼ E for plane stress conditions, E 0 ¼ E=ð1 À 2 Þ for plane strain conditions, E is Young’s modulus, and is Poisson’s ratio. In scenarios where the material behaves linear elastically in a global manner, while local material separation occurs by nonlinear processes that give rise to long-range disengagement, it is helpful to relat e the tensile strength and the work of fracture in specific traction/separation (T/S) laws. An example of a T/S law is shown in Fig. 13.2(a). These are mapped out in front of the crack, as is shown schematically in Fig. 13.2(b). In the T/S law [Fig. 13.2(a)], the rising portion corresponds to the fracture processes that take the material from an initial state to a peak traction corresponding to the tensile strength, S. The declining portion SD corresponds to the fracture processes beyond the peak stat e, and the total area under the curve corresponds to the work of rupture of the mate- rial. It is also important to note here that the way in which the T/S laws affect the fracture processes ahead of an advancing crack can be very com- FIGURE 13.2 Schematic illustration of (a) traction/separation (T/S) across a plane and (b) T/S law mapped in front of a crack of limited ductility. (From Argon, 2000.) Copyright © 2003 Marcel Dekker, Inc. plex.Inanycase,foraprocesszoneofsize,h,cracklength,a,andwidth,W, theconditionforsmall-scaleyieldingisgivenbyArgonandShack(1975)to be CTOD c (h<a<Wð13:3Þ WhereCTOD c isthecriticalcracktipopeningdisplacementandtheother variableshavetheirusualmeaning.Forfiber-reinforcedcomposites,the CTOD c is$1À2mm(ThoulessandEvans,1988;Budianskyand Amazigo,1997),whileinthecaseoffiber-reinforcedcements,itisusually oftheorderofafewcentimeters.Consequently,verylargespecimensare neededtoobtainnotch-sensitivebehavioronalaboratoryscale.Failureto uselargeenoughspecimensmay,therefore,leadtoerroneousconclusions onnotch-insensitivebehavior. 13.3REVIEWOFCOMPOSITEMATERIALS AnoverviewofcompositematerialshasalreadybeenpresentedinChaps9 and10.Nevertheless,sincemanyofthecrack-tipshieldingmechanismsare knowntooccurincompositematerials,itisimportanttodistinguish betweenthetwomaintypesofcompositesthatwillbeconsideredinthis chapter.Thefirstconsistsofbrittlematriceswithstrong,stiffbrittlerein- forcements,whilethesecondconsistsprimarilyofbrittlematriceswithduc- tilereinforcements.Verylittleattentionwillbefocusedoncompositeswith ductilematricessuchasmetalsandsomepolymers. Inthecaseofbrittlematrixcompositesreinforcedwithalignedcon- tinuousfibers,thetypicalobservedbehaviorisillustratedinFig.13.3for tensile loading. In this case, the composite undergoes progressive parallel cracking, leaving the fibers mostly intact and debonded from the matrix. At the so-called first crack strength, mc , the cracks span the entire cross-sec- tion, and the matrix contribution to the composite stiffness is substantially reduced. Eventually, the composite strength resembles the fiber bundle strength, and there is negligible load transfer between the matrix and the fibers. This leads to global load sharing, in which the load carried by the broken weak fibers is distributed to the unbroken fibers. In the case of unrestrained fracture of all fibers, there would be only limited sliding/rubbing between broken fiber ends and the loosely attached matrix segments. This will result in the unloading behavior illustrated in Fig. 13.3(a). The associated area under the stress–strain curve would correspond to the work of stretching the intact fibers and the work of matrix cracking. However, this does not translate into fracture toughness improvement or crack growth resistance. Copyright © 2003 Marcel Dekker, Inc. For toughening or crack growth resistance to occur, the crack tip or crack wake processes must give rise to crack-tip shielding on an advancing crack that extends within a process zone in which the overall crack tip stresses are reduced. The mechanisms by which such reductions in crack tip stresses (crack tip shielding) can occur are described in the next few sections. 13.4 TRANSFORMATION TOUGHENING In 1975, Garvie et al. (1975) discovered that the tetragonal (t) phase of zirconia can transform to the monoclinic (m) phase on the application of a critical stress. Subsequent work by a number of researchers (Porter et al., 1979; Evans and Heuer, 1980; Lange, 1982; Chen and Reyes-Morel, 1986; FIGURE 13.3 Schematic (a) progressive matrix cracking in a fiber-reinforced composite subjected to larger strain to fracture than the brittle matrix, leaving composite more compliant, and (b) macrocrack propagating across fibers at < mc with three matrix cracks in the process zone. (From Argon, 2000.) Copyright © 2003 Marcel Dekker, Inc. Rose,1986;Greenetal.,1989;Soboyejoetal.,1994;LiandSoboyejo,2000) showedthatthemeasuredlevelsoftougheningcanbeexplainedlargelyby modelsthatweredevelopedinworkbyMcMeekingandEvans(1982), Budianskyetal.(1983),AmazigoandBudiansky(1988),Stumpand Budiansky(1989a,b),HomandMcMeeking(1990),Karihaloo(1990), andStam(1994). Theincreaseinfracturetoughnessoncrackgrowthwasexplained readilybyconsideringthestressfieldatthecracktip,aswellasthecrack wakestressesbehindthecracktip.Thelatter,inparticular,areformedby priorcrack-tiptransformationevents.Theygiverisetoclosuretractions thatmustbeovercomebytheapplicationofhigherremotestresses,Fig. 13.4(a).Asthecracktipstressesareraised,particlesaheadofthecracktip undergostress-inducedmartensiticphasetransformations,atspeedscloseto thatofsound(Greenetal.,1989).Theunconstrainedtransformationyields adilatationalstrainof$4%andashearstrainof$16%,whicharecon- sistentwiththelatticeparametersofthetetragonalandmonoclinicphases, Fig.13.4(a)andTable13.1. Theearlymodelsoftransformationtougheningweredevelopedby McMeekingandEvans(1982)andBudianskyetal.(1983).Thesemodels didnotaccountfortheeffectsoftransformation-inducedshearstrains, whichwereassumedtobesmallincomparisonwiththoseofdilatational strains.Theeffectsofdeformation-inducedtwinningwereassumedtobe smallduetothesymmetricnatureofthetwinvariantswhichgiveriseto straincomponentsthatwerethoughttocanceleachotherout,Figs13.4(c) and13.4(d).However,subsequentworkbyEvansandCannon(1986), Reyes-MorelandChen(1988),Stam(1994),SimhaandTruskinovsky (1994),andLiandSoboyejo(2000)showedthattheshearcomponents mayalsocontributetotheoverallmeasuredlevelsoftoughening. Forpurelydilatanttransformation,inwhichthetransformations resultinpuredilatationwithnoshear,thedependenceofthemeanstress, P m ,onthedilationalstressisillustratedinFig.13.5.Inthisfigure, P c m is the critical transformation mean stress, B is the bulk modulus and F is the volume fraction of transformed phase. For a purely isotropic solid, G is given by G ¼ E=½2ð1 þ Þ and B ¼ E=½3ð1 À2Þ. Stress-induced phase transformations can occur when P m > P c m . They can also continue until all the particles are fully transformed. Furthermore, during transformation, three possible types of behavior may be represented by the slope B in Fig. 13.5. When B < À4G=3, the transfor- mation occurs spontaneously and immediately to completion. This behavior is termed supercritical. When B > À4G=3, the behavior is subcritical, and the material can remain stable in a state in which only a part of the particle is transformed. This transformation also occurs gradually without any Copyright © 2003 Marcel Dekker, Inc. FIGURE 13.4 (a) Schematic illustration of transformation toughening; (b) the three crystal structures of zirconia; (c) TEM images of coherent tetragonal ZrO 2 particles in a cubic MgO–ZrO 2 matrix; (d) transformed ZrO 2 particles near crack plane—n contrast to untransformed ZrO 2 particles remote from crack plane. [(c) and (d) are from Porter and Heuer, 1977.] Copyright © 2003 Marcel Dekker, Inc. jumps in the stress or strain states. Finally, when B ¼À4G=3, the material is termed critical. This corresponds to a transition from subcritical to super- critical behavior. Budiansky et al. (1983) were the first to recognize the need to use different mathematical equations to characterize the physical responses of subcritical, critical, and supercritical materials. The governing equations for subcritical behavior are elliptic, so that the associated stress and strain fields are smooth. Also, the supercritical transformations are well described by hyperbolic equations that allow for discontinuities in the stress and strain fields. The stress–strain relations are also given by Budiansky et al. (1983) to be TABLE 13.1 Lattice Parameters (in nanometers) Obtained for Different Phases of Zirconia at Room Temperature Using Thermal Expansion Data l 1 l 2 l 3 Cubic 0.507 0.507 0.507 908 Tetragonal 0.507 0.507 0.516 908 Monoclinic 0.515 0.521 0.531 $ 818 Source: Porter et al. (1979). FIGURE 13.5 Schematic illustration of transformation toughening. (From Stam, 1994.) Copyright © 2003 Marcel Dekker, Inc. E ij ¼ 1 2G _ SS ij þ 1 3B _ XX m ij þ 1 3 f ij ð13:4Þ or _ XX ij ¼ 2G _ EE ij À 1 3 _ EE pp ij þ B _ EE pp À _ ff ij ð13:5Þ where _ EE ij are the stress rates of the continuum element, _ SS ij ¼ _ PP ij À _ PP m ij , _ PP m ¼ _ PP pp =3, and _ EE ij represents the strain rates. For transformations involving both shear and dilatant strains, Sun et al. (1991) assume a continuum element, consisting of a large number of transformable inclusions embedded coherently in an elastic matrix (referred to by index M). If we represent the microscopic quantities in the continuum element with lower case characters, the macroscopic quantities are obtained from the volume averages over the element. The relationship between macroscopic stresses ð _ EE ij Þ and microscopic stresses is, therefore, given by X ij ¼h ij i v ¼ 1 v ð ij dV ¼ f h ij i v 1 þð1 À f Þh ij i V m ð13:6Þ where <> denotes the volume average of microscopic quantities, f is the volume fraction of transformed material. Note that f is less than f m , the volume fraction of metastable tetragonal phase. Furthermore, considerable effort has been expended in the develop- ment of a theoretical framework for the prediction of the toughening levels that can be achieved as a result of crack tip stress-induced transformations (Evans and coworkers, 1980, 1986; Lange, 1982; Budiansky and coworkers, 1983, 1993; Marshall and coworkers, 1983, 1990; Chen and coworkers, 1986, 1998). These transformations induce zone-shielding effects that are associated with the volume increase ($ 3À5% in many systems) that occurs due to stress-induced phase transformations from tetragonal to monoclinic phases in partially stabilized zirconia. For simplicity, most of the micromechanics analyses have assumed spherical transforming particle shapes, and critical transformation condi- tions that are controlled purely by mean stresses, i.e., they have generally neglected the effects of shear stresses that may be important, especially when the transformations involve deformation-induced twinning phenomena (Evans and Cannon, 1986), although the possible effects of shear stresses are recognized (Chen and Reyes-Morel, 1986, 1987; Evans and Cannon, 1986; Stam, 1993; Simha and Truskinovsky, 1994; Li et al., 2000). In general, the level of crack tip shielding due to stress-induced trans- formations is related to the transformation zone size and the volume frac- tion of particles that transform in the regions of high-stress triaxiality at the Copyright © 2003 Marcel Dekker, Inc. [...]... Second-phase particles located in the near-tip field of a propagating crack will perturb the crack path, as shown in Fig 13.10, causing a reduction in the stress intensity The role of in-plane tilting/crack deflection and out-ofplane twisting can be assessed using the approach of Bilby et al (1977) and Cotterell and Rice (1980) The possible tilting and twisting modes are shown schematically in Figs 13 .14( a)–13 .14( c)... the flow stress of ductile reinforcement, t is equivalent to half of the layer thickness, and is the work of rupture, which is equal to the area under the load–displacement curve For small-scale bridging, the extent of ductile phase toughening may also be expressed in terms of the stress intensity factor This gives the applied stress intensity factor in the composite, Kc , as the sum of the matrix... 1996), and Odette et al (1992), for the modeling of large-scale bridging These researchers provide rigorous modeling of crack bridging by using self-consistent solutions of the crack opening pro- Far-field: FIGURE 13.7 Kfab ðyÞ ab % pffiffiffiffiffiffiffiffi 2pr Schematic illustration of spring model of crack bridging Copyright © 2003 Marcel Dekker, Inc FIGURE 13.8 Schematic of elastic–plastic spring load–displacement... displacement at the end of the zone One can equate the maximum crack opening displacement at the end of the bridging zone, umax , to the tensile displacement in the bridging brittle ligament at the point of failure: umax ¼ "lf ldb ð13:21Þ where "lf represents the strain to failure of the whisker and ldb is the length of the f debonded matrix–whisker interface (Fig 13.10) The strain to failure of the whisker... ð13:22Þ where E l is the Young’s modulus of the reinforcing phase The interfacial debond length depends on the fracture criteria for the reinforcing phase versus that of the interface and can be defined in terms of fracture stress or fracture energy: ldb ¼ ðr l =6 i Þ ð13:23Þ where = represents the ratio of the fracture energy of the bridging ligament to that of the reinforcement–matrix interface From... where denotes the volume average of microscopic quantities, f is the volume fraction of transformed material Note that f is less than f m , the volume fraction of metastable tetragonal phase Furthermore, considerable effort has been expended in the development of a theoretical framework for the prediction of the toughening levels that can be achieved as a result of crack tip stress-induced transformations... where K1 is the fracture toughness of the composite, KIc is the fracture toughness of the matrix, R is the radius of the reinforcement, L is the average distance between the centers of adjacent pinning points (typically the distance between the particles where the crack is trapped, but not neces- Copyright © 2003 Marcel Dekker, Inc FIGURE 13.16 Schematic illustration of (a) crack trapping (From Argon... par where R is the particle radius, L is the particle spacing, Kc is the particle mat toughness, and Kc is the material toughness Equation (13.62) applies to all ratios of R=L, while Eq (13.63) only applies to R=L < 0:25 Experimental evidence of crack trapping has been reported by Argon et al (1994) for toughening in transparent epoxy reinforced with polycarbonate rods Evidence of crack trapping has... to the effects of microcrack distributions on a dominant crack, Figs 13.1(b) and 13.17 The shielding or antishielding due to distributions of microcracks has been modeled by a number of investigators (Kachanov (1986); Rose, 1986; Hutchinson, 1987) In cases where a limited number of microcracks with relatively wide separations are observed ahead of a dominant crack, the shielding effects of the microcracks... compressive, the far-field applied stress necessary for transformation will increase On the other hand, the existence of tensile mean stress will trigger the transformation at a lower level of applied stress As a result of this, the mean stress, m, that is needed to induce the transformation of ZrO2 particles is modified by the radial residual stress, m The modified critical condition for transformation is . Inc. plex.Inanycase,foraprocesszoneofsize,h,cracklength,a,andwidth,W, theconditionforsmall-scaleyieldingisgivenbyArgonandShack(1975)to be CTOD c (h<a<Wð13:3Þ WhereCTOD c isthecriticalcracktipopeningdisplacementandtheother variableshavetheirusualmeaning.Forfiber-reinforcedcomposites,the CTOD c is$1À2mm(ThoulessandEvans,1988;Budianskyand Amazigo,1997),whileinthecaseoffiber-reinforcedcements,itisusually oftheorderofafewcentimeters.Consequently,verylargespecimensare neededtoobtainnotch-sensitivebehavioronalaboratoryscale.Failureto uselargeenoughspecimensmay,therefore,leadtoerroneousconclusions onnotch-insensitivebehavior. 13.3REVIEWOFCOMPOSITEMATERIALS AnoverviewofcompositematerialshasalreadybeenpresentedinChaps9 and10.Nevertheless,sincemanyofthecrack-tipshieldingmechanismsare knowntooccurincompositematerials,itisimportanttodistinguish betweenthetwomaintypesofcompositesthatwillbeconsideredinthis chapter.Thefirstconsistsofbrittlematriceswithstrong,stiffbrittlerein- forcements,whilethesecondconsistsprimarilyofbrittlematriceswithduc- tilereinforcements.Verylittleattentionwillbefocusedoncompositeswith ductilematricessuchasmetalsandsomepolymers. Inthecaseofbrittlematrixcompositesreinforcedwithalignedcon- tinuousfibers,thetypicalobservedbehaviorisillustratedinFig.13.3for tensile. structures of zirconia; (c) TEM images of coherent tetragonal ZrO 2 particles in a cubic MgO–ZrO 2 matrix; (d) transformed ZrO 2 particles near crack plane—n contrast to untransformed ZrO 2 particles. of tensile mean stress will trigger the transformation at a lower level of applied stress. As a result of this, the mean stress, m , that is needed to induce the trans- formation of ZrO 2 particles