Mechanical Properties of Engineered Materials 2008 Part 14 doc

In any case, for a process zone of size, h, crack length, a, and width, W,the condition for small-scale yielding is given by Argon and Shack 1975 tobeCTODc h < a < W ð13:3ÞWhere CTODcis

Various crack-tip shielding concepts have been identified by ers over the past 30 years These include:

The above toughening concepts will be introduced in this chapter However,

it is important to note that toughening may also occur by some mechanismsthat are not covered in this chapter (Fig 13.1) In any case, the combinedeffects of multiple toughening mechanisms will also be discussed within aframework of linear superposition of possible synergistic interactionsbetween individual toughening mechanisms

13.1.1 Historical Perspective

Toughening concepts have been applied extensively to the design of site materials Hence, before presenting the basic concepts and associatedequations, it is important to note here that even the simplest topologicalforms of composite materials are complex systems In most cases, theseincorporate interfaces with a wide range of internal residual stresses and

compo-FIGURE 13.1 Crack-tip shielding mechanisms Frontal zone: (a) dislocationcloud; (b) microcrack cloud; (c) phase transformation; (d) ductile secondphase 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.)

thermal expansion misfit Also, most of the expressions presented in thischapter are, at best, scaling laws that capture the essential elements ofcomplex behavior In most cases, the expressions have been verified bycomparing their predictions with the behavior of model materials underhighly idealized conditions However, due to the random features in thetopologies of the constituent parts, the agreement between the models andexperiments may be limited when the conditions are different from thosecaptured by the models (Argon, 2000).

In any case, there are two types of toughening approaches These aregenerally referred to as intrinsic and extrinsic toughening In this chapter,intrinsic toughening is associated with mechanistic processes that are inher-ent to the normal crack tip and crack wake processes that are associatedwith crack growth In contrast, extrinsic toughening is associated with addi-tional crack tip or crack wake processes that are induced by the presence ofreinforcements such as particulates, fibers, and layers Available scaling lawswill be presented for the modeling of intrinsic and extrinsic tougheningmechanisms Selected toughening mechanisms are summarized inFig 13.1

In most cases, toughening gives rise to resistance-curve behavior, as cussed inChap 11 In many cases, the associated material separation dis-placements are large This often makes it difficult to apply traditional linearand nonlinear fracture concepts Furthermore, notch-insensitive behavior isoften observed in laboratory-scale specimens Hence, it is common to obtainexpressions for the local work of rupture,W, and then relate these to afracture toughness parameter based on a stress intensity factor, K, or a J-integral parameter

dis-If we now consider the most general case of material with an initiationtoughness (energy release rate) of Gi and a toughening increment (due tocrack tip or crack wake processes) of G, then the overall energy releaserate, Gc, may be expressed as:

Similar expressions may be obtained in terms of J or K Also, forlinear elastic solids, it is possible to convert between G and K using thefollowing expressions:

or

K ¼ ffiffiffiffiffiffiffiffiffi

E0Gp

ð13:2bÞ

where E0¼ E for plane stress conditions, E0¼ E=ð1  2Þ for plane strainconditions, E is Young’s modulus, and is Poisson’s ratio.

In scenarios where the material behaves linear elastically in a globalmanner, while local material separation occurs by nonlinear processes thatgive rise to long-range disengagement, it is helpful to relate the tensilestrength 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 mappedout 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 thefracture processes that take the material from an initial state to a peaktraction corresponding to the tensile strength, S The declining portion

SD corresponds to the fracture processes beyond the peak state, and thetotal 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 lawsaffect the fracture processes ahead of an advancing crack can be very com-

FIGURE 13.2 Schematic illustration of (a) traction/separation (T/S) across aplane and (b) T/S law mapped in front of a crack of limited ductility (FromArgon, 2000.)

plex In any case, for a process zone of size, h, crack length, a, and width, W,the condition for small-scale yielding is given by Argon and Shack (1975) tobe

CTODc h < a < W ð13:3ÞWhere CTODcis the critical crack tip opening displacement and the othervariables have their usual meaning For fiber-reinforced composites, theCTODc is  12 mm (Thouless and Evans, 1988; Budiansky andAmazigo, 1997), while in the case of fiber-reinforced cements, it is usually

of the order of a few centimeters Consequently, very large specimens areneeded to obtain notch-sensitive behavior on a laboratory scale Failure touse large enough specimens may, therefore, lead to erroneous conclusions

on notch-insensitive behavior

An overview of composite materials has already been presented inChaps 9

and10.Nevertheless, since many of the crack-tip shielding mechanisms areknown to occur in composite materials, it is important to distinguishbetween the two main types of composites that will be considered in thischapter The first consists of brittle matrices with strong, stiff brittle rein-forcements, while the second consists primarily of brittle matrices with duc-tile reinforcements Very little attention will be focused on composites withductile matrices such as metals and some polymers

In the case of brittle matrix composites reinforced with aligned tinuous fibers, the typical observed behavior is illustrated in Fig 13.3 fortensile loading In this case, the composite undergoes progressive parallelcracking, leaving the fibers mostly intact and debonded from the matrix Atthe so-called first crack strength, mc, the cracks span the entire cross-sec-tion, and the matrix contribution to the composite stiffness is substantiallyreduced Eventually, the composite strength resembles the fiber bundlestrength, and there is negligible load transfer between the matrix and thefibers This leads to global load sharing, in which the load carried by thebroken weak fibers is distributed to the unbroken fibers

con-In the case of unrestrained fracture of all fibers, there would be onlylimited sliding/rubbing between broken fiber ends and the loosely attachedmatrix 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 orcrack growth resistance

For toughening or crack growth resistance to occur, the crack tip orcrack wake processes must give rise to crack-tip shielding on an advancingcrack that extends within a process zone in which the overall crack tipstresses are reduced The mechanisms by which such reductions in cracktip stresses (crack tip shielding) can occur are described in the next fewsections.

Rose, 1986; Green et al., 1989; Soboyejo et al., 1994; Li and Soboyejo, 2000)showed that the measured levels of toughening can be explained largely bymodels that were developed in work by McMeeking and Evans (1982),Budiansky et al (1983), Amazigo and Budiansky (1988), Stump andBudiansky (1989a, b), Hom and McMeeking (1990), Karihaloo (1990),and Stam (1994).

The increase in fracture toughness on crack growth was explainedreadily by considering the stress field at the crack tip, as well as the crackwake stresses behind the crack tip The latter, in particular, are formed byprior crack-tip transformation events They give rise to closure tractionsthat must be overcome by the application of higher remote stresses, Fig.13.4(a) As the crack tip stresses are raised, particles ahead of the crack tipundergo stress-induced martensitic phase transformations, at speeds close tothat of sound (Green et al., 1989) The unconstrained transformation yields

a dilatational strain of  4% and a shear strain of  16%, which are sistent with the lattice parameters of the tetragonal and monoclinic phases,Fig 13.4(a) andTable 13.1

con-The early models of transformation toughening were developed byMcMeeking and Evans (1982) and Budiansky et al (1983) These modelsdid not account for the effects of transformation-induced shear strains,which were assumed to be small in comparison with those of dilatationalstrains The effects of deformation-induced twinning were assumed to besmall due to the symmetric nature of the twin variants which give rise tostrain components that were thought to cancel each other out, Figs 13.4(c)and 13.4(d) However, subsequent work by Evans and Cannon (1986),Reyes-Morel and Chen (1988), Stam (1994), Simha and Truskinovsky(1994), and Li and Soboyejo (2000) showed that the shear componentsmay also contribute to the overall measured levels of toughening

For purely dilatant transformation, in which the transformationsresult in pure dilatation with no shear, the dependence of the mean stress,P

m, on the dilational stress is illustrated inFig 13.5.In this figure,Pc

misthe critical transformation mean stress, B is the bulk modulus and F is thevolume fraction of transformed phase For a purely isotropic solid, G isgiven by G ¼ E=½2ð1 þ Þ and B ¼ E=½3ð1  2 Þ

Stress-induced phase transformations can occur when P

m>Pc

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 < 4G=3, the mation occurs spontaneously and immediately to completion This behavior

transfor-is termed supercritical When B > 4G=3, the behavior transfor-is subcritical, andthe material can remain stable in a state in which only a part of the particle

is transformed This transformation also occurs gradually without any

FIGURE13.4 (a) Schematic illustration of transformation toughening; (b) thethree crystal structures of zirconia; (c) TEM images of coherent tetragonalZrO2 particles in a cubic MgO–ZrO2 matrix; (d) transformed ZrO2 particlesnear crack plane—n contrast to untransformed ZrO2 particles remote fromcrack plane [(c) and (d) are from Porter and Heuer, 1977.]

jumps in the stress or strain states Finally, when B ¼ 4G=3, the material istermed critical This corresponds to a transition from subcritical to super-critical behavior.

Budiansky et al (1983) were the first to recognize the need to usedifferent mathematical equations to characterize the physical responses ofsubcritical, critical, and supercritical materials The governing equations forsubcritical behavior are elliptic, so that the associated stress and strain fieldsare smooth Also, the supercritical transformations are well described byhyperbolic equations that allow for discontinuities in the stress and strainfields The stress–strain relations are also given by Budiansky et al (1983) tobe

TABLE13.1 Lattice Parameters (in nanometers)

Obtained for Different Phases of Zirconia at

Room Temperature Using Thermal Expansion

Source: Porter et al (1979).

FIGURE 13.5 Schematic illustration of transformation toughening (FromStam, 1994.)

Eij¼ 1

2G _SSijþ 1

3B_X

ij _P

mij,_

P

m¼ _P

pp=3, and _EEij 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 oftransformable inclusions embedded coherently in an elastic matrix (referred

to by index M) If we represent the microscopic quantities in the continuumelement with lower case characters, the macroscopic quantities are obtainedfrom the volume averages over the element The relationship betweenmacroscopic stresses ð _EijÞ and microscopic stresses is, therefore, given byX

Furthermore, considerable effort has been expended in the ment of a theoretical framework for the prediction of the toughening levelsthat can be achieved as a result of crack tip stress-induced transformations(Evans and coworkers, 1980, 1986; Lange, 1982; Budiansky and coworkers,

develop-1983, 1993; Marshall and coworkers, develop-1983, 1990; Chen and coworkers,

1986, 1998) These transformations induce zone-shielding effects that areassociated with the volume increase ( 35% in many systems) that occursdue to stress-induced phase transformations from tetragonal to monoclinicphases in partially stabilized zirconia

For simplicity, most of the micromechanics analyses have assumedspherical transforming particle shapes, and critical transformation condi-tions that are controlled purely by mean stresses, i.e., they have generallyneglected the effects of shear stresses that may be important, especially whenthe transformations involve deformation-induced twinning phenomena(Evans and Cannon, 1986), although the possible effects of shear stressesare 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 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

trans-crack-tip A transformation zone, akin to the plastic zone in ductile als, is thus developed as a crack propagates through a composite reinforcedwith transforming particles This is illustrated schematically inFig 13.4(a).The size of the transformation zone associated with a Mode I crackunder small-scale transformation conditions has been studied (McMeekingand Evans, 1982) Based on the assumption that the transformation occurswhen the mean stress level at the crack tip exceeds a critical stress value ð T

materi-CÞ,McMeeking and Evans (1982) estimated the zone size for an idealized case

in which all the particles within the transformation zone are transformed.Following a similar procedure, Budiansky et al (1983) give the followingequation for the estimation of the height of the transformation zone (Fig.13.4(a):

h ¼

ffiffiffi3

p

ð1 þ Þ2

12

K T C

!2

ð13:7Þ

where h is the half-height of the transformation wake, K is the far-field stressintensity factor, and is Poisson’s ratio For purely dilational transforma-tion, the toughening due to the transformation can also be expressed as(Budiansky et al., 1983):

Kt¼0:22Ecf"T

C

ffiffiffihp

Kt¼0:22Ec"T

C

1 

ðh 0

f ðx Þ

2 ffiffiffiffix

where f ðxÞ is a mathematical function that represents the fraction of formed zirconia as a function of distance, x, from the crack The criticaltransformation stress necessary to achieve the transformation can beexpressed as a function of the total Gibb’s free energy associated with thetransformation from tetragonal to monoclinic phase This may be estimatedfrom (Becher, 1986):

C¼ G

"T C

ð13:10Þ

where T

C is the critical stress, and G is the Gibb’s free energy of thetransformation The above expression does not account for the effect ofthe enthalpy terms in the equivalent Kirchoff circuits for the transformationnor for the potential residual stresses that can be induced as a result of thethermal expansion mismatch between the constituents of composites rein-forced with partially stabilized zirconia particles (Soboyejo et al., 1994).Depending on the thermal expansion coefficients, the zirconia particlesmay be subjected to either mean tension or compression In general, how-ever, if the mean stress is compressive, the far-field applied stress necessaryfor transformation will increase On the other hand, the existence of tensilemean 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 formation of ZrO2particles is modified by the radial residual stress, m Themodified critical condition for transformation is thus given by (Becher,1986):

For simplicity, the above discussion has focused on the toughening due topure dilatational effects associated with stress-induced phase transforma-tions However, in reality, the shear stresses play a role that may sometimescause significant differences between the experimental results and the theo-retical predictions (Evans and Cannon, 1986) Since the shear strains asso-ciated with stress-induced phase transformations may be as high as  14%,

it may be necessary to assess the effects of shear strains in the estimates oftoughening This has been estimated by Lambropoulus (1986) in an approx-imate analysis that gives

Kt¼ 055EcVf"T

c

ffiffiffihp

where the transformations are induced by critical principal strains, and thetransformation strains also develop in that direction In this model, twin-ning is assumed to be induced by the shear stresses in the transformingparticles Also, the model assumes that there is no coupling between thecrack tip fields and the development of the transformation zone The initialwork of Lambropoulus (1986) and Budiansky et al (1983) has been fol-lowed by subsequent work by Stump (1991), Budiansky and Truskinovsky(1993), Simha and Truskinovsky (1994), Stam (1994), and Li et al (2000)

13.5 CRACK BRIDGING

Crack bridging is illustrated schematically in Fig 13.6 The bridgingreinforcements restrict opening of cracks, and thus promote shielding ofthe crack tip The effective stress intensity factor at the crack tip is, there-fore, lower than the remote/applied stress intensity factor In the case ofstiff elastic fibers, interfacial sliding may occur during crack bridging Thetailoring of the interface to optimize frictional energy dissipation is, there-fore, critical This section will concentrate initially on bridging by ductilereinforcements This will be followed by a focus on crack bridging by stiffelastic whiskers/fibers, as well as a section on debonding/fiber pull-out

FIGURE 13.6 Schematic illustration of crack bridging by (a) ductile particlesand (b) stiff elastic whiskers

13.5.1 Bridging By Ductile Phase

An energy approach may be used to explain the toughening due to ductilephase reinforcement Within this framework, ductile phase toughening bycrack bridging may be attributed to the plastic work required for the plasticstretching of the constrained ductile spherical particles For small-scale brid-ging in which the size of the bridging zone is much smaller than the cracklength, the increase in strain energy,GSS, due to the plastic work requiredfor the stretching of the ductile phase is given by (Soboyejo et al., 1996,Ashby et al., 1989; Cao et al., 1989; Shaw and Abbaschian, 1994; Kajuch etal., 1995; Bloyer et al., 1996, 1998; Lou and Soboyejo, 2001):

where Vf is the volume fraction of ductile phase that fails in a ductilemanner (note that the actual reinforcement volume fraction is f ), C is aconstraint parameter which is typically between 1 and 6, y is the uniaxialyield stress, and  is a plastic stretch parameter The small-scale bridginglimit may also be expressed in terms of the stress intensity factor, K:

KSS¼ ðK2þ E0f 0tÞ1

ð13:14Þwhere Kssis the steady-state (or plateau) toughness, K0is the crack-initia-tion toughness (typically equal to brittle matrix toughness), is E0¼ E=ð1 

2Þ for plane strain conditions, and E0¼ E for plane stress conditions (where

E is Young’s modulus and is Poisson’s ratio), f is the volume fraction, 0isthe flow stress of ductile reinforcement, t is equivalent to half of the layerthickness, and is the work of rupture, which is equal to the area under theload–displacement curve

For small-scale bridging, the extent of ductile phase toughening mayalso be expressed in terms of the stress intensity factor This gives theapplied stress intensity factor in the composite, Kc, as the sum of the matrixstress intensity factor, Km, and the toughening component due to crackbridging,Kb The fracture toughness of the ductile-reinforced compositesmay thus be estimated from (Budiansky et al., 1988; Tada et al., 1999):

Kc¼ Kmþ Kb¼ Kmþ

ffiffiffi2



r

V

ðL 0

... strain of  4% and a shear strain of  16%, which are sistent with the lattice parameters of the tetragonal and monoclinic phases,Fig 13.4(a) andTable 13.1

con-The early models of transformation... data-page="8">

FIGURE13.4 (a) Schematic illustration of transformation toughening; (b) thethree crystal structures of zirconia; (c) TEM images of coherent tetragonalZrO2 particles in a cubic MgO–ZrO2... general, the level of crack tip shielding due to stress-induced formations is related to the transformation zone size and the volume frac-tion of particles that transform in the regions of high-stress