FATIGUE OF RESTORATIVE MATERIALS

G. Baranl*K. Boberick2J. McCool3

'College of Engineering, Temple University, 1947 N. 12th Street, Philadelphia, PA 19122, USA; 2School of Dentistry, Temple University; and 3Penn State Great Valley; *corresponding author,GRBaran@astro.temple.edu

ABSTRACT: Failure due to fatigue manifests itself in dental prostheses and restorations as wear, fractured margins, delami-nated coatings, and bulk fracture. Mechanisms responsible for fatigue-induced failure depend on material ductility: Brittlematerials are susceptible to catastrophic failure, while ductile materials utilize their plasticity to reduce stress concentrations atthe crack tip. Because of the expense associated with the replacement of failed restorations, there is a strong desire on the partof basic scientists and clinicians to evaluate the resistance of materials to fatigue in laboratory tests. Test variables includefatigue-loading mode and test environment, such as soaking in water. The outcome variable is typically fracture strength, andthese data typically fit the Weibull distribution. Analysis of fatigue data permits predictive inferences to be made concerningthe survival of structures fabricated from restorative materials under specified loading conditions. Although many dental-restorative materials are routinely evaluated, only limited use has been made of fatigue data collected in vitro: Wear of materi-als and the survival of porcelain restorations has been modeled by both fracture mechanics and probabilistic approaches. A needstill exists for a clinical failure database and for the development of valid test methods for the evaluation of composite materi-als.

Key words. Fatigue, composites, glass ionomers, ceramics, compomers.

(I) IntroductionMost materials, when subjected to stress or strain over a

period of time, will fail in a process termed "fatigue".Failure may manifest itself as fracture, loss of compliance, oras wear, and is often influenced by environmental factors. Themode of stress or strain application may be static (remainingconstant with time), dynamic (applied at some constant rate),or cyclic (stress or strain magnitude varying with time). Theelapsed time before failure depends on the magnitude of theapplied stress or strain; however, for some materials, a lowerlimit of stress or strain exists below which the material may besaid to possess infinite life.

One can easily imagine how, in the oral environment, thematerials used to fabricate prostheses are subjected to fatigue(Wiskott et al., 1995) and, as a result, fail or exhibit wear (Reid etal., 1990; Sakaguchi et al., 1992). To differentiate between andamong materials, and to obtain guidance leading to restorationsthat are fatigue-resistant, investigators conduct fatigue tests inthe laboratory (Braem et al., 1994a). Tests can be used to evalu-ate specific designs, to obtain fundamental material properties,or to gather data needed to predict lifetimes. In this review, wewill first discuss the mechanisms responsible for fatigue failurein brittle and quasi-brittle materials (such as polymers, poly-mer-based composites, and ceramics), describe fatigue testmethods and techniques for data analysis, and, finally, surveythe literature related to specific classes of restorative and pros-thetic materials. The fatigue behavior of ductile metallic materi-als does not represent a significant design challenge for bio-medical researchers or engineers, and the interested reader isreferred to other sources (Davidson and Lankford, 1992).

Fatigue is a process involving nucleation, propagation, andcoalescence of cracks. An example of a fracture mechanicsapproach to the study of fatigue crack growth is portrayed in Fig.1, where the rate of crack growth per cycle "N" (da/dN) is plot-ted vs. AK (change in stress intensity factor). Three distinctregions are evident: region I, where rapid crack growth followscrack nucleation; region II, where the crack grows more slowly(this region is identified with subcritical crack growth); andregion III, where rapid crack propagation and coalescence are theimmediate precursor to failure. Although crack nucleation is stillnot well-understood, this event generally occurs in regions ofstress concentration, such as scratches or grain boundaries on thesurface, or voids in the interior. In region II, the crack propagationdirection tends to be perpendicular to the tensile axis, and the rateof propagation correlates well with the stress intensity. Here, therate of crack growth obeys the Paris law (Paris et al., 1961):

da/dN = AAKm (1)

where "A" and "m" are empirically determined constants.Several other crack-growth equations have been developed,including power law and exponential forms (Jakus et al., 1981).

(II) Fatigue Mechanisms(IIA) INORGANIC GLASSES AND CERAMICS

According to the Griffith theory, brittle materials such as glassesand ceramics contain pre-existing flaws formed during process-ing. These flaws, which may be present on the surface or withinthe volume of the specimen, serve as nuclei for cracks. The sizes

350Crit Rev Oral Biol Med 12(4)350-360 (2001)350 Crit Rev Oral Biol Med 12(4):350-360 (2001)

of these cracks and the critical stress intensity factor define thefracture strength for these materials: At stresses exceeding thefracture stress, cracks propagate catastrophically, while at stresslevels below the fracture stress, cracks can heal. This conclusion,as well as experimental difficulties associated with the testing ofbrittle materials, had encouraged the belief that glasses andceramics do not exhibit fatigue. However, it is now recognizedthat these materials, particularly glasses with an appreciable sil-ica content, are subject to static fatigue mediated by water, whichcontributes to stable subcritical crack growth in region I (in Fig.1); the effects of moisture in region II are less pronounced. Cracksmay also propagate in a stable manner, and not confound theGriffith theory by virtue of R-curve behavior, where materialsexhibit a critical stress intensity factor that increases more rapid-ly than the applied stress intensity as the crack elongates.

Cyclic fatigue effects have been observed in various crys-talline ceramics, where failure times during cyclic stress appli-cation were shorter than in static fatigue at similar stress levels,even though the materials exhibit no plasticity. Instead, irre-versible deformation occurred as the result of frictional grainboundary sliding, wedging of crack faces by debris, and phasetransformations (Suresh, 1998). Though ultimate failure is gen-erally assumed to proceed from the largest single flaw orientedin a favorable position with respect to the tensile axis, an alter-native concept of a "crack band" or smeared crack has beenused in finite element modeling of damage zone growth inceramic restorations (Peters et al., 1993).

(IIB) POLYMERSMost polymers may be regarded as quasi-brittle materials withlimited plasticity at the crack tip. Permanent deformation isaccommodated by either shear banding or crazing, with craz-ing representing a more brittle deformation mode. Duringcyclic loading of these materials, cracks advance in a processcalled Discontinuous Crack Growth (DCG), wherein damageaccumulates at a craze in front of the crack tip, and propagatesafter the crack-opening displacement reaches some criticalvalue. Evidence of DCG is found on fracture surfaces, butbecause of this damage-accumulation mechanism specific forpolymers, the spacing is considerably larger than between thestriations found on metallic fatigue surfaces, where the spacingcan be directly linked to the distance of crack advance per cycle.The markings indicative of discontinuous crack propagationare not found in cross-linked polymers (Clark et al., 1993).

Crack-growth data of the da/dN vs. AK variety for poly-mers are similar to those shown in Fig. 1, but the correlation ofgrowth rates with material properties is not as it is for metals,where fatigue crack-growth rates may be normalized byYoung's modulus (Courtney, 2000). On the other hand, crack-growth rates in polymers correlate with toughness and crys-tallinity, and can be affected by changes in polymer microstruc-ture. Attempts have also been made to modify the Paris law forpolymers, particularly because AK is derived from an essential-ly elastic analysis, while polymer fatigue crack propagation isaccompanied by shear yielding or craze formation (Chow andLu, 1990). Therefore, the modified equations typically incorpo-rate the J-integral, a parameter used with non-linear materials.

During cyclic loading, polymers are likely to strain-soften,and the greater the polymer ductility, the greater the degree ofsoftening. At high cycling frequencies, polymers experience hys-teretic heating, which can lead to decreases in fatigue life; in-creasing polymer molecular weight and the degree of cross-link-

iIgda IogIAr

II II I

log AKFigure 1. Classic depiction of crack propagation rate da/dN as afunction of the difference in stress intensity factor AK. The slope of thestraight line portion in region 11 yields the Paris law constant.

ing increase fatigue life. Some polymers can exhibit both ductileand brittle behavior, depending on specimen thickness, loadingrate, and the geometry of the induced flaw (Sehanobish et al.,1993). Changes in the mechanism of fatigue failure can berevealed by changes in the shape of the "stress to number ofcycles 'til failure" (S-N) curve (Matsumoto and Gifford, 1985).

(IIC) COMPOSITESComposites typically consist of high-modulus, brittle reinfor-cing fibers or particles dispersed in a quasi-brittle polymermatrix. Maximum strengthening of these engineered materialsoccurs when load is transferred from the matrix to the reinforc-ing phase. Depending on the type of reinforcement (e.g., partic-ulate or fibrous), the direction of load application, the strengthsof the various phases, and the interfacial strength, several mech-anisms may participate in fatigue-induced damage of compos-ite materials. These include matrix cracking, matrix deforma-tion, void formation, multidirectional cracking, filler debond-ing, and filler failure. Consequently, scatter in fatigue data forcomposites is greater than in monolithic materials, where typi-cally a single damage mechanism is presumed to be active. Thechoice of dominant mechanism is also influenced by the modeof load application: In cyclic fatigue, voids are more likely toform at the fiber-matrix interface than during monotonic load-ing (Horst and Spoormaker, 1997). Debonding at the filler-matrix interface occurs in particulate-reinforced composites atlow static stresses, producing a rough fracture surface, while at

higher applied stresses that cause higher crack velocities, thecrack propagates through filler and matrix, resulting in a

smooth fracture surface (Cantwell and Roulin-Moloney, 1988).In fiber-reinforced materials, cracks do not propagate for

long within the matrix before reaching the fiber interface. There,the crack may bifurcate and travel for considerable distancesalong the interface, since the angular distribution of stresses

(determining crack propagation direction) at the crack tip in

composites is determined by microstructure, not by the direction

35112)4) 350-360 (2001)Grit Rev Oral Biol MedCrit Rev Oral Biol Med12(4):350-360 (2001)

C

log N (Cycles Till Failure)Figure 2. Stress (5) vs. number of cycles to failure (N) curves for rep-resentative polymers. (a) An endurance or fatigue limit; (b) noendurance limit; (c) typical of a polymer which does not craze at thecrack tip.

of the applied load. Similarly, the speed of crack propagation isdetermined not simply by the stress intensity, but also bymicrostructure: The strength gradient at the interface betweenmatrix and filler will determine the crack-growth rate ratherthan the crack propagation rate determined for the matrix alone.Following local matrix and interface failure surrounding a dis-persed fiber, the fiber itself ruptures; then, load is transferred toneighboring fibers, which in turn rupture. After a critical densi-ty of single-fiber failures is attained, the fracture of the bodytakes place. Failures may also be localized within a specificdomain, and this damage is termed "brush-like cracking", fromwhich ultimate failure of the body proceeds (Bolotin, 1999).

When sufficient microcrack damage has accumulated viathe mechanisms described above, a macrocrack is initiated, andits presence changes the compliance of the bulk composite. Thischange in compliance is often useful in defining fatigue life, sincethe load-bearing capacity of the composite structure deteriorateswell before actual failure through the specimen; strength vs.cycles to failure (S/N) data can exhibit more than one change incompliance (Dyer and Isaac, 1998). Final failure can take placeover a wide variation in final crack sizes; at higher applied stress-es, short cracks are responsible for failure, while at low stresses,long cracks are responsible for failure. Interestingly, fiber-rein-forced composites generally show lower fatigue resistance incompression than in tension, because of cooperative buckling ofadjacent fibers and matrix shear (Suresh, 1998).

There is a concern that, because of the wide variety offatigue damage modes occurring within composites, crack-growth rate measurements are not an appropriate designapproach for predicting lifetimes of composites (Reifsnider,1980). Simply stated, there usually are a great many crackswhich form in the matrix and in the reinforcement. The propa-gation rates and directions of these microcracks are continu-ously modified during the fatigue process as a result of changesin the distribution of internal stresses. In addition, theories ofcrack propagation were developed for isotropic materials. In

composites, the reinforcement materials have not "one"strength, but rather a statistical strength distribution furtheraccentuated by the fact that particulate fillers are typically notmonodisperse, and fiber diameters vary. Therefore, the sizeeffect needs to be considered in dealing with the strength of thereinforcing phase (Reifsnider, 1980).

(III) Test MethodsFatigue behavior may be characterized by several laboratorytest methods. Kelly (1995) has provided a useful review of themany test protocols, test variables, and analytical approachesthat influence the strength values collected during fatigue test-ing. When a test is planned, an a priori decision is made con-cerning the desired information: crack-growth rate data or phe-nomenological lifetime data. If growth rate data are of interest,reference can be made to several standard procedures, such asthe American Society for Testing and Materials (ASTM) speci-fications (ASTM, 1986, 1987a,b, 1993). Here, standard speci-mens (with or without a pre-crack) are loaded, and the rate ofcrack advance is monitored; eventually, a plot similar to that inFig. 1 is generated, and the data in region II may be fit toexpressions such as that developed by Paris et al. (1961).

The phenomenological approach is characterized by thesubjection of either standard specimens or surrogate structuresto cyclic stresses or strains, and the outcomes presented as plotsof u, Ao, (Umax/crmin (the R ratio), E or AE vs. log N (Conway andSjodahl, 1991). Test variables include temperature, cycling fre-quency and waveform shape, and environment. The resultingdata plots may take several shapes, as shown in Fig. 2.Materials may exhibit an endurance limit, as in Fig. 2a, or not,as in Fig. 2b; polymers which do not craze, such as nylon,appear as in Fig. 2c. Cyclic bending fatigue studies of resinsused as matrix materials in dental composites revealed nofatigue or endurance limits (Baran et al., 1998).

Specimens may be cyclically tested in tension-tension, flex-ure, torsion, shear, or compression. Contact fatigue, such as thatinduced by cyclic loading of an indenter into the surface of amaterial, has also been used because of its perceived relevancefor the study of wear processes. The indentation causes tensilestresses at the contact edges, which can lead to crack formation(Gioannakopoulos et al., 1999). This technique has been usedwith ceramic materials, where it is found that the damage modedepends on material microstructure; in homogeneous ceramics,a cone-shaped crack (Hertz crack) emanating from the circle ofcontact can ensue, while in heterogeneous ceramics, a diffusedamage zone driven by shear stresses forms underneath theindenter (Kim et al., 1999). Strength degrades more rapidly if aquasi-plastic zone forms than if only a Hertz crack is present.After damage accumulates during cyclic fatigue at sufficientlyhigh stresses, a second, radial, crack system forms which severe-ly degrades specimen strength (Jung et al., 2000). Indentation byblunt indenters is presented as being a more "realistic" loadingmode, since it incorporates compressive stresses as well as ten-sile stresses in propagating the crack. It is also important to notethat the size effect needs to be considered in the choice of inden-ter size in the testing of polymers or polymer-based composites;indenters with diameters smaller than some material-dependentcritical diameter will not elicit cone crack formation, but ratherwill allow the material to behave plastically (Baran et al., 1994).

A useful technique for determining crack velocity parame-ters in brittle materials without directly measuring crack-growthrates is the dynamic fatigue test. Here, specimen strength is

352 Crit Rev Oral Biol

12(4):350-360 (2001)352 Crit Rev Oral Biol Med

determined as a function of stressing rate, and it is assumed thata power-law relationship holds between these two variables.Specimens used may either contain a flaw of known size, such asthat induced by controlled indentation, thereby permitting theinitial flaw size effects on strength degradation to be studied, orpossess intrinsic flaws only (Cook et al., 1985). Although thedynamic stressing technique has been proven valuable for thestudy of both homogeneous and heterogeneous ceramics, in con-junction with this test, it is useful to determine the inert strengthof the material, i.e., the inherent, non-fatigued stress at failure. Itis not always possible to be certain that the inert strength has infact been measured, or to know what stressing rate is requiredfor the test. In fact, some ceramics exhibit a drop in fracturestrengths at high stressing rates (Wang et al., 1993). Dynamicfatigue testing is preferred over fracture-mechanics-based crackpropagation tests, because it yields more conservative lifetimeestimates, and because the flaws causing failure more closelysimulate those encountered in "real life"-i.e., they are smaller intest specimens without artificially induced flaws (Pletka andWiederhom, 1982). Lifetimes predicted by the use of data fromlong-crack (a 3 mm) experiments are longer than those pre-dicted from short-crack (a < 250 ,um) experiments (EswaraPrasad and Bhaduri, 1988).

Failure predictions performed as described in Ritter (1978)and based on dynamic stressing data usually do not correspondto those obtained from cyclic fatigue test data. One cause is thepresence of asperities on the crack-face surfaces; these enhancethe tensile and shear stresses at the crack tip during the compres-sion-tension cycle (Wang et al., 1993). Because different mecha-nisms are involved in crack propagation in these two modes ofstress application, crack-growth rates in ceramics measured incyclic tests are not predictable by the use of static crack-growthrates. The constantA in the Paris law is also different when deter-mined by static fatigue vs. dynamic fatigue; apparently, the sever-ity of the flaw formed during static fatigue-testing is larger thanthat formed during dynamic fatigue (Ritter et al., 1979).

Still another method of fatigue-testing is the so-called stair-case method. Here, the investigator pre-selects a cyclic lifetime,then cycles the material at some stress level. If the specimenfails, a lower stress is chosen, and the experiment is repeated; ifthe specimen survives to the selected number of cycles, the nextexperiment is performed at a higher stress level (Draughn,1979). The concern with this method is that its use virtuallyimplies a fatigue limit: The assumption of a fatigue limit, i.e., astress level below which no specimens will fail, is non-conserv-ative and not useful for statistically based lifetime predictionefforts. Additionally, the pre-determined cycle limit is seldomrationalized, and the choice of a "low" limit will precludeobservation of changes in fatigue mechanism. Cross-overbehavior, in which some materials perform best at high stresslevels while others perform best at low stress levels, such as hasbeen observed in compression testing of polymer-based com-posites (Draughn, 1979), could remain unobserved if the cyclelimit were to be set too low. Furthermore, this method has beenoriginally developed for the analysis of so-called sensitivityexperiments, and assumes that data are normally distributed,while the strength data for brittle materials typically fit theWeibull distribution (Dixon and Mood, 1948). However, thereare materials for which the staircase method provides a usefulmeans of determining the fatigue limit (McCabe et al., 1990).

When cyclic tests in flexure are performed with polymersor polymer-based composites, it is likely that creep will occur,

particularly after many cycles at low stress levels. Clamping thespecimen and causing it to reverse (in displacement control), orstressing at a constant R, can compensate for specimen creepand change in compliance (Braem et al., 1994b). An additionaladvantage to displacement control testing of polymers is thatthe stress decreases continually, and therefore hysteretic heat-ing does not take place (Hertzberg and Manson, 1980).

(IV) Analysis of DataWhen materials fail in a brittle manner, the stress at which rup-ture occurs is not constant but must be regarded as a randomvariable. Similarly, under a cyclically varying load, the numberof cycles 'til failure varies randomly and has a mean value thatincreases as the value of the stress range applied to the materialdiminishes. A distribution-free, or non-parametric, approach,known as the Cox proportional hazards model (Cox, 1972), hasbeen used for the characterization of and testing for differencesamong the lives in cyclic fatigue of various glass ionomers(Nakajima et al., 1996). However, if the data follow a specificknown probability model, there is some loss of efficiency whenthe Cox model is used compared with the use of methodsapphlcable to the known parametric model (Lawless, 1982).

The more usual approach in the evaluation of tests of den-tal materials is to adopt a probability distribution to describethe variation in breaking strength or life. Various probabilitydistributions have been proposed and evaluated as a model forbreaking strength, but the literature shows the two-parameterWeibull model to be the overwhelming choice. The physicaljustification for the Weibull distribution is based on the fact thatit is an asymptotic distribution of smallest extremes (Tobias andTrindade, 1995). This property is often referred to as the "weak-est link" property. It establishes that if the time to failure of a

component may be regarded as the shortest of the times to fail-ure associated with a large number of independent potentialfailure sites (such as pre-existing flaws), then under fairly gen-eral conditions, the component failure time distribution willtend toward the Weibull.

Alternatives to the Weibull distribution must be consid-ered for materials which do not exhibit a size effect or whichhave an inverse size effect wherein the strength increases withvolume (Kittl and Diaz, 1988). Such behavior can be observedduring testing in compression, where the propagation of a sin-gle crack does not always lead to failure (de Jayatilaka andTrustrum, 1977). An explanation of methods used to arrive atthe most appropriate distribution to describe the strengths offiber-reinforced composites compiled in S-N testing has beenrecently provided by Lee et al. (1997).

There are two forms of the Weibull distribution in currentuse, distinguished by the appearance of either two or three con-stants in the mathematical expression for the distribution. If a

random variable X follows the two-parameter Weibull model,the probability that a random observed value will not exceed a

specified value x is expressible as:

Prob[X < x] = F(x) = 1- exp [- (r)o] (2)

F(x) is called the cumulative distribution function (CDF). Thequantities -q and 1 are parameters, with -q having the same

units as those in which X is measured (e.g., MPa, cycles, etc.); ,Bis dimensionless and is commonly called the shape parameteror the dispersion parameter. -q is called the scale parameter or

the characteristic value of the Weibull distribution. The term

12(4. 350 360 (200fl Crit Rev Oral Biol Med353353Crit Rev Oral Biot Med1 2(4):350-360 200 1 )

"characteristic value" derives from the fact that when x = ,F(x) = 1 - 1/e 0.632 regardless of [. For any other value of x,F(x) depends upon 3.

In the three-parameter version of the Weibull, a location orthreshold parameter y is introduced constraining the values ofX to exceed y. The resultant CDF is:

Prob[X < x] = F(x) = - exp [-(x) ];x >y (3)

A three-parameter Weibull analysis best fit the fatigue strengthdata of fiber-reinforced composites (Lamela et al., 1997).

Our focus will be on statistical methods applicable to thetreatment of data arising from two-parameter Weibull popula-tions. A property of the Weibull family that makes it a naturalchoice as a reliability and fracture strength model is the closureproperty, according to which the smallest of several identicalWeibull variables is itself a Weibull variable. This property, aspecial case of the weakest-link property cited above, makes itconvenient to describe a "size effect" in structural materials. Ifthe fracture strength of a structure of volume V0 follows aWeibull distribution, then the fracture strength of a structure ofgreater volume V under similar loading will follow a Weibulldistribution, with the scale parameter reduced by the fraction

vVO

Thus, with a Weibull representation, test results on small speci-mens of size V0 can be readily used to predict failure probabilityon large specimens of size V. Correspondingly, if failure initiatesat the surface rather than within the material volume, a similarscaling applies to the relative areas, i.e., the scale parameter forcomponents having surface area A is reduced by the fraction

from the scale parameter of components having area A0.Taking logarithms twice transforms Eq. (2) to:

In In ( 1 F)= 3lnx - 81n7 (4)

In this form, the left-hand side is linear in In x, with a slopeequal to the shape parameter [3 and intercept equal to -[3 In -.This is the basis of a method of estimation of the parametersbased on a random sample of the values of x. Arranging thevalues in a sample of size n in ascending order, x(l) < x(2) <'A'.x(n),and estimating the probability that x is less than x as F(x(i))yields n pairs of points that may be substituted into Eq. (4) andto which a straight line may be fit by various methods. Thereare several choices of F(x( ) in use, with only slight differencesamong them (Nelson, 1982). Least-squares estimates are fre-quently used to fit the straight line. In so doing, it is logical touse ln x as the response variable and ln ln[ /(1 - F)] as the inde-pendent variable, since the values of F are fixed by the samplesize, and the values of ln x are known only after the data are inhand. In practice, the opposite choice is often made, resultingin a somewhat different fit. Gurvich et al. (1997) used the vol-ume ratio factor discussed above to adjust strength valuestaken with different-sized specimens to a common volume forleast-squares estimation. The process was iterative, in that theadjustment depends on the estimated value of [3.

The conditions for least-squares estimation are not strictly

applicable in fitting probability distributions, in that theordered sample values are not statistically independent of eachother. Generalized least-squares estimates known as "best lin-ear unbiased (BLUE) estimates" overcome this shortcoming,but the numerical values needed for implementation are avail-able only for comparatively small sample sizes (Nelson, 1982).We have not found any instance of BLUE estimation beingapplied to fracture strength or cycles-to-failure data in the den-tal materials literature. Still another estimation method some-times used in practice is known as the method of moments. Themean and variance of the sample values are equated to the cor-responding expression in terms of the population parameters,and the resultant two equations are solved for -q and [3.

The estimation method favored in the statistical commu-nity and used by many dental materials researchers (Drum-mond and Miescke, 1991; Fernandez-Saez et al., 1993) is themethod of maximum likelihood. The maximum likelihoodestimate ,Bof the Weibull shape parameter is the solution of thefollowing non-linear transcendental equation (Lawless, 1982;Kittl and Diaz, 1988):

(5)

where x1...xn are the values observed in a sample of size n. Analternate form exists for the case where the sample is censored.This occurs most often when x represents life and testing is sus-pended before all of the longest-lived items are observed. Thesolution of Eq. (5) is too difficult for hand calculations whenmore than two failures have occurred. However, it is readilyprogrammed for implementation on a computer or program-mable calculator, and software is widely available commercial-ly for maximum likelihood estimation under the Weibullmodel. With Eq. (5) having been solved for ,f, the maximumlikelihood estimate of the scale parameter -q is calculated as:

(6)

Exact confidence interval estimates of the Weibull shape andscale parameters may be computed by the use of percentagepoints of certain "pivotal" functions determined by MonteCarlo sampling for sample sizes of interest. A two-sided 90%confidence interval on [3 is computed as

v0 95(n) v0.05(n)where v0 05(n) and '0U95(n) denote the 5th and 95th percentilesof the distribution of the random variable v(n). Note that theratio of the upper to lower ends of the confidence intervaldepends only on sample size. Given appropriate tables of per-centage points of v(n), one may determine the sample sizeneeded for a prescribed ratio of the upper to lower ends of aconfidence interval. Similarly, a two-sided 90% confidenceinterval for q is computed as:

[exp {-( u-95)) ] < q < [exp -( U005())]A (8)

where u(n) is another pivotal function. Percentage points of uand v for uncensored samples may be found in Thoman et al.

354~~~~~~~~~~~GrtRvOa ltMd1(13030(01

(7)

354 Crit Rev Oral Biol Med 12(4):350-360 (2001)

1-1 A 1 1 n n A

:..x 0A+ I.Xi i Inxi = 0

171 1 8 nI

1

.= Ix 18

n

(1969). Again, sample size may be determined for a prescribedprecision of estimation as expressed in terms of the ratio of theupper to lower ends of a confidence interval for q. A full dis-cussion of sample size determination for Weibull-distributedrandom variables is given in McCool and Baran (1999).

When sets of Weibull data, such as the observed fracturestrength in samples of different materials, are compared, theusual practice in materials testing is to use a logarithmic trans-formation of the sample observations and standard normal dis-tribution methodology such as the analysis of variance andmultiple-comparison methods such as Tukey's test (Mont-gomery, 1997). Multiple comparison techniques, however, havebeen developed for direct comparison of Weibull scale andshape parameter estimates (McCool, 1975; Baran et al., 1998).

The equivalent of one-way ANOVA but with a Weibull-distributed response has been developed and is applicablewhen the Weibull shape parameter is the same for all sets ofdata, a restriction comparable with the constant varianceassumption of ordinary ANOVA (McCool, 1979). More recent-ly, comparable methodology has been developed for the analy-sis of two-way data layouts to assess the effect of each of twofactors and their interaction on a Weibull-distributed response.This methodology was applied to the assessment of the effectsof soaking and silanation on the fracture strength of a dentalcomposite material (McCool and Baran, 1999).

Linear regression or ordinary least-squares analysis is usedextensively in research on biomaterials. It has been used todeduce the constants A and m in the Paris crack-growth ratelaw (Truong et al., 1990). The logarithm of the measured crack-growth rate, da/dN, is a linear function of the logarithm of thestress intensity range AK, with an intercept equal to the loga-rithm ofA and slope equal to m. Regression analysis is similar-ly used to deduce the material constants that control crackgrowth from the results of static dynamic and cyclic fatiguetests (see, for example, Ritter, 1978; Pletka and Wiederhorn,1982; Fairhurst et al., 1993; Twiggs et al., 1995).

(V) Applications of Fatigue Data Analysis

(VA) LIFE PREDICTIONBecause the eventual goal of fatigue data collection is to predictmaterial or structural lifetimes, an understanding of the vari-ous issues involved is important. Recent reviews on this subjectwere offered by Ritter (1995a,b). Lifetime predictions rely oncalculations involving Weibull strength parameters as well ason the crack-growth exponent. The reliability of these predic-tions is strongly dependent on the number of test specimens.At least 50 are recommended, and, if dynamic fatigue tests areused, a stressing rate range of at least 3 decades, with 30 speci-mens at each stressing rate, is usual (Ritter et al., 1981). Stillmore specimens are needed to achieve highly reliable cyclic orstatic fatigue data.

If the lifetimes of structures such as dental restorations are tobe predicted, a finite element model of the structure under loadmust be developed and a statistical methodology incorporatingmaterial behavior applied to the structure. An effort utilizing anequivalent stress approach without taking fracture mechanics orfatigue data into account successfully predicted the failure stressof a composite notched-beam specimen from previouslyacquired material property data (van der Varst et al., 1993). Analternative failure prediction methodology which included frac-

ture mechanics (Batdorf theory) was developed, but it also didnot possess the ability to predict lifetimes by the use of fatiguedata (Nemeth et al., 1990). However, a subsequent version calledCARES/LIFE incorporated crack-growth and fatigue analyses,making lifetime prediction under cyclic loading possible(Nemeth et al., 1996). This latter version has been used for relia-bility analysis of metal-ceramic crowns, with dynamic fatiguedata obtained from feldspathic porcelain utilized to predict fail-ures within the body porcelain component of a porcelain-fused-to-metal restoration (Sokolowski et al., 1996). CARES/LIFE hasalso verified the size effects of dry and soaked composites, indi-cating that this approach is useful for lifetime prediction effortsof dental composites, materials which are somewhat less brittlethan inorganic glasses and ceramics (Baran et al., 1999).

(VB) WEARFatigue crack propagation rates have been correlated with wearrates, particularly when wear occurs by the delamination mech-anism (Fleming and Suh, 1977). Many dental restorations aresubjected to contact forces that can be resolved into subsurfacestresses, causing crack propagation (Braem et al., 1986). Althoughfatigue appears to be responsible for at least some types of wear,only a few studies have attempted to correlate crack propagationbehavior with clinical wear, even though these have been rela-tively successful (Truong and Tyas, 1988; Truong et al., 1990). Thewear of dental restorative materials may also be related to cracksoriginating on the surface. The surface "damage zone" observedby silver-staining techniques prestumably weakens the compos-ite's resistance to fatigue. An indenter sliding on such surfacesinitiates surface cracks, which then propagate into the bulk of thematerial, and eventually link up to form surface pits that are evi-dence of wear (Zhang et al., 1999). Subsurface cracks are drivenby shear stresses, and the angular distribution of KII determinesthe direction of crack propagation.

A novel rolling-ball device that minimized friction at theindenter/material interface showed that fatigue life varies withfiller volume fraction, and appears to be optimized at filler con-centrations in the range of 30 to 50 vol% (McCabe et al., 1997).At low-volume fractions, the explanation could be the relative-ly high cyclic deformations which occur beneath the rollingball, eventually leading to microcracking. At extremely highvolume fractions, there may be an element of increased brittle-ness, leading to more rapid crack propagation. Studies showedthat the resin-modified glass ionomers exhibit poor wear char-acteristics, with conventional glass ionomers being subject tohigh early wear, apparently as a result of subsurface fatiguephenomena (de Gee et al., 1996).

(VI) Restorative Materials

(VIA) DENTAL CERAMICSFeldspathic dental porcelain exhibits fatigue (a reduction instrength) after cyclic loading by both blunt and Vickers (sharp)indenters. During loading by blunt indenters, the stress levelsused caused only Hertz cracks to form, while the sharp inden-ters formed a zone of plastic deformation underneath theindenter (White, 1993; White et al., 1995). These results conflictwith reports from other investigators who found close agree-ment between dynamic and cyclic fatigue test results withporcelain in cases where specimens were directly loaded, sug-gesting that no major damage occurred during cycling

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(Fairhurst et al., 1993; Twiggs et al., 1995). Crack growth inporcelains is influenced by tempering procedures as well as bycorrosion in an aqueous environment, and may be modeled bymeans of a modified power law crack-growth formulation(Anusavice and Lee, 1989; Anusavice et al., 1991). The principaleffect of tempering was to inhibit crack formation, rather thanto slow crack propagation. Discrepancies between growth ratesmeasured by means of indentation cracks caused by a Vickersindenter and crack-growth rates derived from dynamic fatiguedata in glasses were first thought to be due to an eventualdecay of residual stresses surrounding the indent (Han et al.,1989). However, subsequent studies have indicated that post-indentation crack growth is unique and that the plateau region(when the crack originating from the indent ceases to propa-gate) could be characterized as a fatigue limit. Therefore,dynamic testing is preferred for the evaluation of fatigue para-meters in glasses (Choi and Salem, 1992).

Increased resistance against fatigue failure could beachieved by a reduction in processing-related flaws or porosityin the structure as well as by a reduction in surface irregulari-ties (Chen et al., 1999).

(VIB) POLYMER-BASED COMPOSITESParticulate reinforcements initially toughen epoxy resin-basedcomposites by crack-front pinning; further toughening can takeplace through crack blunting, but the effectiveness of pinning isreduced by breakdown of the particle/matrix interface(Spanoudakis and Young, 1984a). Therefore, optimal strengthproperties are obtained in composites where the particle/matrixadhesion is good (Spanoudakis and Young, 1984b). With poorlybonded particles, the maximum tensile stress is at the equatorsof the particles, with fracture surfaces exhibiting clean particlesfree of matrix material. In composites with well-bonded parti-cles, the maximum stress is in the matrix above and below thepoles of the particle, and fracture surfaces show a layer of matrixcovering the particle following failure (Spanoudakis and Young,1984b). In these composites, the crack propagation directiondepends on the elastic constants of the particle and matrix(Spanoudakis and Young, 1984b). Composites with considerablylow or high filler content (< 60% or > 80% by weight) have sig-nificantly lowered tensile fatigue resistance when evaluated bythe staircase method (Htang et al., 1995).

In an evaluation of composite restorative materials classi-fied according to filler particle size, observation of the fracturesurface revealed inter- and intra-particle fracture for the smallparticle and hybrid composites (Drummond, 1989). Micro-filled composites fractured between and through pre-poly-merized particles. Further study of the fracture characteristicsof composite materials would be better served by an investi-gation of slow crack growth and the effects of microdefects.

Aging in water or aqueous fluids decreases the fatigueresistance of polymer-based composites as a result of wateruptake by the matrix (whereby the matrix is plasticized), andhydrolysis of the silane bonding agent. Water also leaches outfiller elements and induces filler failures. Debonded fillers mayact as stress concentrators, multiplying the numbers of poten-tial crack-growth sites (Braem et al., 1994b). The relationshipbetween storage time and decrease in strength is logarithmic(Yamamoto and Takahashi, 1995).

If it is true that fatigue plays a significant role in wear, thena fracture mechanics approach to crack propagation couldyield important predictive information related to the lifetime of

a material. In a study of fatigue crack propagation in pre-cracked dental composites carried out using static and dynam-ic methods, investigators found that the difference in stressintensities between dry and water-soaked specimens was dueto a change in modulus. Aging also increased the constant "m"of the Paris law, and increasing the degree of cure increasedresistance to fracture. After Evans' method was applied (Evans,1974) to predict lifetime, it appears that good wear resistancecould be achieved by a combination of high fracture toughness,small inherent flaw size, high crazing stress, and high fatigueresistance (Truong and Tyas, 1988; Truong et al., 1990).

Water sorption affects mechanisms of crack propagationin dental composites, with dry specimens fracturing in astick-slip mode, while soaked specimens exhibit stable slowcrack growth, with the filler/matrix interface offering lessresistance to crack growth after water sorption (Marcos-Montes and Draughn, 1987). The properties of the matrix, thefiller, and the filler volume fraction determine the failureproperties of the composite at high crack velocities, while atlow velocities the stability of the filler/matrix interface is thecontrolling factor (Marcos-Montes and Draughn, 1987).Strength data for dental composites aged in water for fivedistinct time periods were best analyzed by the two-parame-ter Weibull model for each separate aging time. For cyclic-fatigue data, a three-parameter accelerated failure timemodel with a Weibull baseline distribution was most appro-priate (Drummond and Miescke, 1991).

Glass fiber reinforcements increase the fatigue resistance ofcomposite resin (Vallittu et al., 1994a; Freilich et al., 1998) andacrylic resin removable partial dentures (Vallittu, 1993, 1996;Vallittu et al., 1994b, 1995; Vallittu and Narva, 1997) by acting asthe stress-bearing component and by activating crack-stoppingor crack-deflecting mechanisms (Rudo and Karbhari, 1999).Factors affecting mechanical properties include the types offibers used (DeBoer et al., 1984; Mullarky, 1985; Ekstrand et al.,1987; Berrong et al., 1990; Gutteridge, 1992; Vallittu and Docent,1999), the direction and pattern design of the fiber reinforcement(Vallittu, 1998; Rudo and Karbhari, 1999; Vallittu and Docent,1999), and uniform pre-impregnation (wetting) of the fiber withresin (Vallittu, 1995; Vallittu and Docent, 1999). Silanated glassfibers are often used because of the well-documented goodadherence of treated glass fibers to the polymer matrix, therebyfacilitating stress transfer from matrix to fiber and improvedesthetics (Takagi et al., 1996). Because the mechanical propertiesof fiber composites depend on the direction of fibers in the poly-mer matrix, the reinforcing effects of continuous unidirectionalfibers are anisotropic, in contrast to woven fibers, which rein-force in two directions, providing orthotropic mechanical prop-erties (Vallittu and Docent, 1999). In lay-ups containing a largenumber of fibers in the loading direction, fatigue at high appliedstresses tends to be dominated by fiber properties, while at lowapplied stresses, matrix-related damage mechanisms have timeto initiate and develop. Lay-ups containing few or no fibers inthe load direction have matrix-dependent damage mechanismsoccurring regardless of the applied stress level (Dyer and Isaac,1998). Increasing the toughness of the matrix material improvesthe lifetime and fatigue strength of the composite.

(VIC) GLASS IONOMERS AND RESIN IONOMERSCombining glass ionomers with resin composites has resultedin hybrid materials called polyacid modified resin composites(compomers), and resin-modified glass ionomers. These new

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materials can have significantly differ-ent properties and characteristics whencompared with the parent materials.The combination of low clinical failurerate and ease of manipulation suggeststhat compomer materials are acceptablefor Class II restorations of primaryteeth (Peters et al., 1996; Papagiannouliset al., 1999), for class III and V restora-tions in permanent teeth, but not instress-bearing restorations of perma-nent teeth (Denehy and Vargas, 1996;Prati et al., 1998; Tyas, 1998). Glass iono-mers exhibit brittleness and poor frac-ture toughness (McKinney et al., 1987;Attin et al., 1996; Gladys et al., 1997),limiting their use to non-stress-bearingareas such as cervical/abrasion lesionsand restorations of primary teeth(Abdalla and Alhadainy, 1997; Abdallaet al., 1997).

A comparison among new hybridrestorative materials, conventional glassionomers, and light-cured resins showedthat the flexural fatigue limit (deter-mined by the staircase method) of hybridrestorative materials is comparable with

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that of micro-filled composites (Gladys et al., 1997). Interestingly,the traditional glass ionomers may exhibit a crack-healingresponse (Davidson, 1994). Metal additives to glass-ionomermaterials elicited no significant improvement in static mechanicalproperties, but fatigue resistance improved (Nakajima et al.,1996). Fatigue testing of notched three-point flexure specimenswas carried out to 100,000 cycles, with 5 specimens at each load,and the investigators noted wide scatter in the data which werefit to an exponential regression model. Fractographic examina-tion of the fatigue surfaces showed smooth crack faces whencrack propagation was rapid and rough crack faces in the case ofslow crack propagation. The diversity in structure and composi-tion among composites, glass ionomers, and resin-reinforcedglass ionomers does not allow findings from one product to beextrapolated to other similar products, and neither the elasticproperties nor strength data are accurate predictors of the fatiguebehavior of dental restorations (Braem et al., 1995; Gladys et al.,1998). The discrepancy in behavior among nominally similarproducts is shown in Fig. 3, which depicts the results of dynamicfatigue tests. First, it is evident that the two compomer materialsexhibit the classic straight-line dynamic fatigue relationship com-mon to brittle materials; yet, it is also obvious that one compomermaterial undergoes an enhancement in strength after aging for sixmonths in distilled water, while the other does not.

Encapsulated glass-ionomer materials have improvedphysical properties because of decreases in the inclusion ofvoids, which normally initiate cracks on the surface or withinthe volume of a specimen. A reduction of the internal defectsproduced during the mixing stage produces a higher finalstrength (Cattani-Lorente et al., 1993).

(VII) AfterwordMany instances of fatigue-related failures of restorative andbiomaterials have not been discussed in this review. For exam-ple, delayed bonding failures in porcelain-, hydroxyapatite- or

bioactive glass-fused-to-metal restorations or implants can bedue to slow crack propagation along the metal-ceramic inter-face driven by residual stresses (Ritchie et al., 1993). Prostheticcomponents supported by implants may undergo fatigue andfail prematurely (Patterson and Johns, 1992). The marginal frac-ture of amalgam restorations is commonly attributed to fatigue(Arola et al., 1999). In the orthopedic environment, the materialmost susceptible to fatigue is ultra-high-molecular-weightpolyethylene (Elbert et al., 1994), which, when present on artic-ulating surfaces of load-bearing implants, undergoes wear andgenerates microscopic debris particles believed to be eventual-ly responsible for peri-implant osteolysis and implant loosen-ing (Wang et al., 1993). Finally, there is the problem of fatigue innatural tissues. Fracture in osteoporotic bone represents asevere clinical problem, and studies related to flaw develop-ment and crack propagation in bone are highly relevant in thedevelopment of treatments for the disease (Griffin et al., 1999).

Many opportunities continue to exist for further researchrelated to fatigue in the restorative dental materials reviewedhere. Although there is a wealth of in vitro fatigue data avail-able, a missing component in relating these data to the clinicalexperience is the examination of retrieved failed prostheses orrestorations that would verify the cause of failure, corroboratethe predicted lifetime, and identify the site of the flaw leadingto failure. An obvious problem, too, in comparisons of theresults of various laboratory studies is the disparity in testmethods and data reduction. Additionally, not all studies areplanned to provide the same information: Those intended tofacilitate lifetime prediction involve standardized specimenpreparation with statistically based experimental design, andshould include fractographic examination to identify the loca-tion of the flaw responsible for failure, so that the appropriatescaling parameter (volume or surface area) can be used in mod-eling. Testing of surrogates for restorative structures should bebased on clinical documentation of failure and be coupled with

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stress analyses so that potential solutions can be identified.The interpretation of fatigue studies of particulate and

fiber-reinforced composite materials continues to be problem-atic: These materials occur in great variety, and are subject tostresses that vary widely, depending on the site of placement.The fact that fracture mechanics approaches to the study ofcrack propagation may not be valid for heterogeneous mate-rials, and that the finite element or similar stress-analyticalmodeling of composites is difficult, suggests that develop-ment of valid laboratory test methods for these materials is anongoing priority.

For these reasons, perhaps the safest course in designingfor fatigue is to design materials based on known fundamentalprinciples. For example, the inherent flaw population should bereduced in size and in quantity, so that restoration lifetime maybe extended by increasing the time needed for crack nucleationand propagation. Means of inhibiting crack propagation, suchas controlling reinforcing phase microstructure and modifyingthe matrix-filler interphase, should also be explored.

AcknowledgmentPortions of the work done by the authors were supported by DE-09530.

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360 Crit Rev Oral Biol Med 12(4):350-360(2001)

360 Crit Rev Oral Biol Med l12(4):350-360 (2001l)