6-Influence of Test Method on Failure Stress of Brittle Dental Materials

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1990 69: 1791J DENT RESS. Ban and K.J. Anusavice

Influence of Test Method on Failure Stress of Brittle Dental Materials

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Influence of Test Method on Failure Stress of Brittle Dental Materials

S. BAN and K.J. ANUSAVICE'

Department of Dental Materials Science, School of Dentistry, Aichi-Gakuin University, Chikusa-ku, Nagoya 464, Japan; and 'Department ofDental Biomaterials, College of Dentistry, University of Florida, Gainesville, Florida 32610-0446

A bi-axial flexure test (piston-on-three-balls), a four-point flex-ure test, and a diametral tensile test were used to measure thefailure stress of four brittle dental materials: zinc phosphatecement, body porcelain, opaque porcelain, and visible light-cured resin composite. Furthermore, the fracture probabilityof the bi-axial test specimens was predicted from the resultsof the four-point flexure test, with use of statistical fracturetheory. Bi-axial failure stresses calculated from an equationdeveloped by Marshall (1980) exhibited no significant differ-ence for zinc phosphate cement as a function of piston size,specimen thickness, presence or absence of a stress-distribut-ing film, and loading rate. The four-point flexure strength val-ues of zinc phosphate cement and opaque porcelain weresignificantly lower (p0.05) than the corresponding mean bi-axialstrength values. The diametral tensile strength of all materialswas significantly lower than the bi-axial flexure strength. Themean bi-axial flexure strengths of zinc phosphate cement andopaque porcelain were much higher than the theoretical valuespredicted from surface flaw theory, while the strength valuesfor body porcelain and resin composite were comparable withthose determined from the four-point flexure test. These resultsdemonstrate that the strength of zinc phosphate cement de-pends not only upon the geometric factors, but also upon sam-ple preparation conditions.

J Dent Res 69(12):1791-1799, December, 1990

Introduction.

Mechanical strength is an important factor that controls theclinical success of dental restorations. Usually, complex stressdistributions that are induced by compressive, tensile, and shearstresses are present in most specimens under practical condi-tions. It is extremely difficult to induce a pure stress of a singletype in a body. In general, tensile strength is easily determinedfor ductile materials such as metals. For convenience, com-pressive strength is often measured for brittle materials suchas porcelains, cements, amalgams, and resin composites.However, brittle materials are much weaker in tension than incompression, and tensile stresses in some materials are (incertain cases) larger than compressive stress (Anusavice andHojjatie, 1987). Therefore, tensile strength is generally con-sidered as the more meaningful property for these brittle ma-terials (compared with compressive strength) for assessmentof the failure potential of dental restorations, especially in thepresence of critical surface flaws.

To determine the tensile strength for brittle dental materials,the diametral tensile test has been used frequently (Earnshawand Smith, 1966; Williams and Smith, 1971; Powers et al.,

1976). The traditional tensile test has rarely been used forbrittle materials (Bowen and Rodriguez, 1962; Zidan et al.,1980) because of the difficulty associated with gripping andaligning the specimens. The diametral tensile test provides asimple experimental method for measurement of the tensilestrength of brittle materials. However, the complex stress dis-tribution developed in the specimen can lead to various modesof fracture. If the specimen deforms significantly before fail-ure, the data may not be valid. Zidan et al. (1980) suggestedthat the diametral tensile test cannot be considered reliable fordental resinous materials. Chiang and Tesk (1989) demon-strated that a correction of the stress calculation equation fordiametral tension is needed with double cleft fracture.

The main advantage of the flexure test is that a state of puretension can be established on one side of the specimen (Ber-enbaum and Brodie, 1959). Three-point and four-point flexuretests have also been used for strength evaluation of single-component brittle materials (Bryant and Mahler, 1986; Sod-erholm, 1986) and metal-ceramic structures (Coffey et al.,1988). For these uni-axial flexure tests, the principal stress onthe lower surfaces of the specimens is tensile, and it is usuallyresponsible for crack initiation in brittle materials. However,undesirable edge fracture (which can increase the variance ofthe failure stress value) can occur. Furthermore, these methodswere designed for engineering materials that are usually as-sociated with relatively large specimens. For brittle dental ma-terials, construction of such specimens is not usually convenientbecause suitable quantities of dental restorative materials arenot often available to prepare a sufficient number of specimensfor assessment of statistically significant differences. Further-more, the residual stress states due to polymerization shrinkageor thermal contraction difference and the flaw characteristicsinduced in large specimens may not be representative of thosethat are present in smaller clinical restorations.

Recently, the bi-axial flexure test has been used frequentlyfor the determination of fracture characteristics of brittle ma-terials. The measurement of the strength of brittle materialsunder bi-axial flexure conditions rather than uni-axial flexureis often considered more reliable, because the maximum tensilestresses occur within the central loading area and spurious edgefailures are eliminated. This allows slightly warped specimensto be tested and produces results unaffected by the edge con-dition of the specimen. This feature makes the method suitablefor assessment of the effects of surface conditions on strength.A wide variety of loading arrangements has been developedfor bi-axial flexure tests: (1) ring-on-ring (Kao et al., 1971),(2) piston-on-ring (Wilshaw, 1968), (3) ball-on-ring (Mc-Kinney and Herbert, 1970), (4) ring-on-ball (Shetty et al.,1983), (5) piston-on-three-ball (Kirstein and Woolley, 1967),and (6) ring-on-spring (Marshall, 1980). For this study, thefifth option was used, since it is suitable for slightly warpedand small specimens such as brittle dental materials, and ex-cellent results by this method have been reported previouslyfor some glasses and ceramics. The bi-axial flexure strengthis determined by support of a disc specimen on three metalspheres positioned at equal distances from each other and fromthe center of the disc. The load is applied to the center of the

1791

Received for publication February 27, 1990Accepted for publication August 2, 1990This study was supported by NIDR Grant DE 06672.

1792 BAN & ANUSAVICE

TABLE 1MATERIALS TESTED

Code Type Brand Name ManufacturerZP Zinc Phosphate Cement Orthodontic Cement with Fluoride Ormco, Glendora, CABP Body Porcelain White Body #2 J.F. Jelenko & Co., Armonk, NYOP Opaque Porcelain White Opaque #4 J.F. Jelenko & Co.RC Resin Composite Herculite XR Kerr Manufacturing Co., Romulus, MI

TABLE 2SPECIMEN DIMENSIONS (mm)

Bi-axial Flexure Test Four-point Flexure Test Diametral Tensile TestD t 1 w t D t

S 12.7 0.1 0.8 0.1ZP M 13.6 0.1 1.3 0.1 33.7 + 0.5 6.8 0.1 6.7 0.2 6.2 0.1 3.2 0.1

L 15.8 0.1 2.6 + 0.2BP 14.4 1.0 1.9 0.3 28.9 0.3 4.9 0.2 4.3 0.2 7.7 0.1 3.5 + 0.5OP 14.4 + 0.1 2.0 - 0.1 29.8 + 0.2 5.0 0.4 4.3 0.4 7.8 0.1 3.3 0.3RC 13.7 0.2 1.2 + 0.1 36.0 0.2 3.1 + 0.1 3.1 + 0.1 6.2 + 0.0 3.2 0.1D, diameter of specimen; t, thickness of specimen; 1, length of specimen; w, width of specimen; S, small; M, medium; and L, large.

opposite surface by a flat piston. Kirstein and Woolley (1967)demonstrated that stresses in a thin, circular aluminum-alloyplate were independent of angular orientation and the numberof supports.

Wachtman et al. (1972) reported that bi-axial flexure strengthvalues for eight types of alumina show that a coefficient ofvariation of about 7% can be achieved by testing five speci-mens and that different laboratories generally obtain goodagreement on strength values. Marshall (1980) modified thestress equation to correct for geometry effects and demon-strated that the stress calculated from this modified equationfor two types of glasses was in good agreement with the stressmeasured from strain gauges. Pletka and Wiederhorn (1982)compared the failure characteristics of specimens fractured inbi-axial flexure tests with those fractured by means of a con-ventional four-point flexure test. Morena et al. (1986) inves-tigated the dynamic fatigue of dental porcelain using a bi-axialflexure test. Usually, disc specimens (12-50 mm in diameterand 1-3 mm in thickness) are used for these tests. They canbe easily made under typical restorative conditions. Further-more, the flat surface of the test specimen can be easily con-trolled by conventional metallographic polishing methods andtypical dental finishing techniques.

The objectives of this study were: (1) to test the hypothesisthat the bi-axial flexure test reduces the variance of fracturestrength values for brittle dental materials with various levelsof homogeneity, compared with the four-point flexure test andthe diametral tensile test, and (2) to analyze the influence ofspecimen geometry on the mean fracture strength values forbrittle dental materials by use of fracture statistics.

Materials and methods.Specimen preparation. -For bi-axial flexure measurements,

disc specimens were prepared for an orthodontic zinc phos-phate cement, a feldspathic body porcelain, a feldspathic opaqueporcelain, and a visible-light-cured resin composite (Table 1).Zinc phosphate cement specimens were prepared by conven-tional techniques according to the solubility test ofADA Speci-fication No. 8 for zinc phosphate cement. The powder/liquidratio was 3/1 (1.5 g/0.5 mL). Approximately 0.5 mL of cementof standard consistency was placed on a flat glass plate. Threering sizes (inner diameter/thickness ratios of 13/0.6, 14/1.2,and 16/2.4 mm) were placed in the soft cement, and anotherglass plate was used to press the cement into a disk. Three

minutes after the mix was started, the glass plates and cementswere placed in a humidor at 37C for one h. After removalfrom the humidor, the specimens were separated from the glassand stored in water at 37C for 24 h.

Porcelain specimens were prepared by normal fabricationprocedures. A slurry of porcelain powder was vibrated andcondensed into a mold 16 mm in diameter and 2 mm in depth.The discs were fired in a dental oven (Mark IV, J.M. NeyCo., Bloomfield, CT) at a heating rate of 55C/min undervacuum to 982C followed by a 90-second holding time in air.The specimens were removed from the furnace and rapidlycooled in ambient air by natural convection. The porcelaindiscs were ground from 120-grit to 600-grit papers and pol-ished with 1->Lm and 0.3-gm A1203 powder on a metallo-graphic polishing wheel.

Visible-light-cured resin composite specimens were made asfollows: Approximately 0.5 mL of resin paste was placed ona flat glass plate, 1.0 mm in thickness. A flexible ring, ap-proximately 14 mm in inner diameter and 1.2 mm in thickness,was placed in the paste, and another glass plate was used topress the resin paste into a disc. Light emitted from a fiberoptic handpiece (Translux, Kulzer & Co., Bad Homburg, Ger-many) passed through the glass plates for a total of 200 s sothat adequate polymerization of each side would be ensured.After light irradiation, specimens were separated from the glassplates and stored in water at 37C for 24 h before being tested.

Specimens for the four-point flexure and the diametral ten-sile tests were also prepared in a similar manner. The finaldimensions of the specimens are listed in Table 2.

Determination of fracture strength. -The bi-axial flexuretest apparatus is described in the ASTM Standard F394 for bi-axial flexure testing of ceramic substrates. In this study, thedimensions of the apparatus were smaller than those describedin the ASTM standard so that the small specimen size of brittledental materials typically used in dental restorations would beaccommodated.As shown in Fig. 1, specimens were supported on three steel

spheres (3.2 mm in diameter) equally spaced along a diameterof 10 mm. For zinc phosphate cement specimens, loading wasapplied by a steel piston (with flat areas of 1.2 mm and 1.6mm in diameter ground along the surface of contact) untilfracture occurred. For the porcelain and resin composite speci-mens, the piston with a diameter of 1.2 mm was used.The failure stress, cr, at the center of the lower surface was

calculated by equations developed by Marshall (1980). These

J Dent Res December 1990

FAILURE STRESS OF BRITTLE DENTAL MATERIALS

equations were based on the bi-axial flexure test developed byWachtman et al. (1972), and original equations derived byKirstein and Woolley (1967). The failure stress, a, can beexpressed as:

A p / t2 (1)and

A = (3/4r) [2 (1 +v) in (a/ro*)+ (1-v) (2 a2-ro*2)/2 b2 + (1 +v)] (2)

where P is the applied load at failure, v is Poisson's ratio, ais the radius of the support circle, b is the radius of disc speci-men, ro* is the equivalent radius given in ro* = (1.6 r02 +2))1/22-0.675t, t is the thickness of the disc specimen, and rois the radius of the piston at the surface of contact.

If a series of nearly identical specimens is tested, the factorsA and t (which depend only on the dimensions and Poisson'sratio) are constant, so the equation reduces to a simple pro-portionality between load and stress. In this study, the strengthvalues were calculated with Poisson's ratio values of 0.35 forcement, 0.28 for porcelain (Anusavice and Hojjatie, 1987),and 0.24 for resin composite (Craig, 1989).

Tests were usually carried out in air at room temperatureusing a thin plastic film (about 50 pum in thickness) betweenthe piston and the upper surface of the specimen, to assist inobtaining uniform loading over the surfaces of the discs. Cross-head loading rates of 0.1 and 1 mm/min were applied by auniversal testing machine (Instron Universal Testing MachineModel 1125, Instron Corp., Canton, MA). For investigationof the effects of piston diameter, disc diameter, and loadingrate on the bi-axial fracture strength of a brittle dental material,an orthodontic zinc phosphate cement was used.

For the four-point flexure test, the rectangular specimenswere supported by two 3-mm-diameter rods set 21 mm apart.The load was applied by two rods that were set 7 mm apart.A cross-head loading rate of 0.2 mm/min was used. The max-imum tensile stress was calculated by the equation:

uJ = PL/wt2

which is called the Weibull modulus, and u,,, is the scaleparameter or characteristic strength. Higher values of Weibullmodulus correspond to a higher level of homogeneity of thematerial. Most ceramics are reported to have m values in therange of 5 to 15, whereas metals, which fail in a ductile man-ner, have m values in the range of 30 to 100 (Johnson, 1983).The Weibull modulus, characteristic strength, and strength ata predicted failure level of 5% were obtained with use of acomputer program designed to carry out the Weibull analysisfrom the fracture data.

This analytical method is very popular because of its easeof application. However, the Weibull approach is not basedon physical principles, but is based on statistical concepts.Therefore, as pointed out by several investigators (Giovan andSines, 1979; Shetty et al., 1983; Lamon and Evans, 1983),the Weibull analysis has some limitations that challenge itsability to predict failure of components having complex geom-etries and which are subjected to a multi-axial stress state. Thisproblem may be crucial for components used in dental resto-rations having complex geometries and subjected to multi-axialstress states.The elemental strength approach (Evans, 1982) represents a

more physical analysis of failure, based essentially on the premise

p

2ro(3)

LoadPiston

where P is the applied load at failure, L is the length of outerspan, w is the width of the specimen, and t is the thickness ofthe specimen.

For the diametral tensile test, cylindrical specimens weretested at a cross-head loading rate of 0.1 mm/min. The max-imum tensile stress for the diametral test is given by the equa-tion:

-=2 P / rrDt (4)where P is the applied load at failure, D is the diameter of thespecimen, and t is the thickness of the specimen. From thesedata, the mean value, standard deviation, and coefficient ofvariation were calculated. Fracture surfaces representative ofeach group of specimens were characterized by means of scan-ning electron microscopy (SEM: JSM-35C, JEOL Ltd., To-kyo, Japan).

Fracture statistics. -The failure strengths of brittle mate-rials are statistically distributed as a function of the homo-geneity of the material (Ritter, 1986). One commonly usedstatistic for the description of this distribution is the Weibulldistribution, which is given by:

Pf = 1-exp [-(-/U)m] (5)where Pf is the fracture probability defined by the relationPf=i/(N+1), i is the rank in strength, N denotes the totalnumber of specimens in the sample, m is the shape parameter,

jXTjj~~~ _Specimen

SupportBall Bearing

SpecimenHolder

Fig. 1-Schematic illustration of piston-on-three-ball bi-axial flexure test.a, the radius of the support circle; b, the radius of disc specimen; t, thethickness of the specimen; and r, the radius of the piston at the surfaceof contact.

Vol. 69 No. 12 1 793


TABLE 3BI-AXIAL FLEXURE TEST RESULTS FOR ZINC PHOSPHATE CEMENT

Condition ResultSpecimen Diameter Load- Loading Meant Standard Coefficient

of Piston distributing Speed No. of Strength Deviation ofCode Size* (mm) Film (mm/min) Tests (MPa) (MPa) VariationZP M 1.2 Yes 0.1 10 18.1 1.3 0.073ZP1 S 1.2 Yes 0.1 10 19.0 1.9 0.098ZP2 L 1.2 Yes 0.1 10 16.9 2.6 0.155ZP3 M 1.6 Yes 0.1 10 19.2 2.1 0.108ZP4 M 1.2 No 0.1 10 17.3 2.5 0.142ZP5 M 1.2 Yes 1.0 10 17.5 2.0 0.113Total 60 18.0 2.2 0.121

*Dimensions of S, M, and L specimens are indicated in Table 2.tCommon vertical lines indicate no significant difference at ox = 0.05.

that a specimen body contains a distribution of cracks that canbe characterized by their flaw-extension stress. A multi-axialelemental strength model was derived by Lamon and Evans(1983) and applied to specific test geometries, providing anexperimental demonstration of practical requirements for a multi-axial analysis of fracture. However, these approaches aremathematically complex and require extensive numerical analysiseven for simple stress states. Shetty et al. (1983) used theBarnett-Freudenthal approximation (Batdorf, 1977) for com-parison of bi-axial data with uni-axial data from three- andfour-point flexure tests.

In the present study, the failure probability of brittle speci-mens subjected to bi-axial flexure was calculated from theresults of the four-point flexure test, by use of the followingstatistical fracture theory. Eq. (5) can be converted to the fol-lowing equation:

Pf = 1 - exp [-B] (6)

where B is the risk of rupture. For multi-axial stress states, Bis defined at any point in a stressed body as

dB = f, n(orn)dw (7)

where n(o-,) is a characteristic material function; o(n is the nor-mal tensile stress at an arbitrary angle relative to the principalstresses crl, 02, and O3; and dwo an elemental area on a unitsolid sphere. The geometric variables used to describe o,, anddw are defined in an orientation relative to principal stresses.Eq. (7) is evaluated by integration of those portions of the unit-sphere where orn is tensile. Weibull assumed that B was zerofor orientations for which oa, was compressive. For the uni-axial stress case, the two-parameter form is

Weibull introduced this transformation of scale parameters torelate the multi-axial stress state to the uni-axial stress condi-tion. Eq. (9) can be integrated in closed form giving:

Bs = (or a2) (oru/or)m Ls (11)

where o,,L is the maximum stress at the center of the disc speci-men and L. is the loading factor,

L, = (ot + P)/(m + 1)Oc 3and

a = 3P(3+v)a2/8 t2oalP = 3 P (1 +3v) a2/8 t2oL

From Eqs. (6) and (11),In In [1/(1- Pf)] = m[ln (oa) - Cj]

whereC= In (o,) - ln(ir a2 L,)/m

(12)

(13)

(14)

(15)

(16)

In a similar way, the risk of rupture in bi-axial tension, Bv,for failure caused by a volume flaw can be given as follows:

B, = (ir a2t/2)(cr/cr0)n'Lv (17)Lv ((x + 3)/(m + 1)2 a (18)

and

In In [1/(1-Pf)] = m[ln (a,,) - CQ (19)where

C== In (or,) - ln (orr a2t/2 L,)/m (20)n (cn) = (3nlcJno) (8)

where m is a shape parameter, or Weibull modulus, and or,,is a scale parameter, as described previously.

For the disc specimen subjected to uniform loading pressure,bi-axial tension causes failure due to surface flaws. Accordingto the Barnett-Freudenthal approximation (Batdorf, 1977), theprincipal stresses are assumed to act independently. This as-sumption leads to the following equation for the risk of rupturein bi-axial tension, B5, due to surface flaw effects:

B = 2rr fr(ulrcvo)m r dr + 2r fr (alao0)m r dr (9)where or is the radial stress and or, is the tangential stress. Forthe uniform-pressure-on-disc specimen, the stress state is bi-axial tension, with a,= 0r, U2 =ot, and C3=0. ao, is relatedto cr,, through the equation

ao-" = [(2m + 1)/2 'j]u "(m

With these equations, statistics parameters for the bi-axial testcan be predicted from the four-point flexure test. The loadingfactor is one convenient parameter for this purpose, because itincorporates the stress-state and stress-gradient effects into thefracture statistics. The loading factors decrease with increasingWeibull modulus, reflecting the influence of stress gradients.The volume loading factors are smaller than the surface loadingfactors because of the additional stress gradient in the axialdirection. In the present study, the failure probability for thebi-axial test of four brittle dental materials was predicted fromthe Weibull parameters for the four-point flexure test by meansof these equations.

Results.The mean bi-axial flexure strength values of zinc phosphate

cement as a function of six different test conditions are listed


(10)


TABLE 4COMPARATIVE DATA FOR THE THREE TEST METHODS

Ratio ofStrength to

Meant Standard Coefficient Bi-axial Flex-No. of Strength Deviation of ure

Specimen Tests (MPa) (MPa) Variation StrengthBi-axial flexure testZP* 10 18.1 1.3 0.073 1BP 10 52.4 5.6 0.106 1OP 10 75.6 8.6 0.114 1RC 10 103.9 15.6 0.150 1

Four-point flexure testZP 10 6.8 0.6 0.088 0.375BP 10 48.4 5.8 0.125 0.923OP 10 52.4 10.2 0.195 0.693RC 10 98.4 11.4 0.116 0.947

Diametral tensile testZP 10 4.5 0.5 0.088 0.249BP 10 22.4 8.7 0.391 0.427OP 10 23.8 6.0 0.253 0.315RC 10 24.6 5.8 0.234 0.236*Same data as in Table 3 for group ZP.tCommon vertical lines indicate no significant statistical difference at a = 0.05.

in Table 3. The mean strength for these conditions was notsignificantly different at the 95% confidence level when Dun-can's multiple range test was used. Although zinc phosphatecement was mixed manually for each specimen, the coefficientof variation fell within a narrow acceptable range (0.073 to0.155), indicating the adequacy of the experimental data forevaluation of the strength of the cement. The bi-axial flexurestrength of zinc phosphate cement was insensitive to specimensize, diameter of the piston, use of a load-distributing film,and loading speed. Fractured specimens could be grouped ac-cording to a two-segment or three-segment fracture pattern.However, no relationship between the fracture mode and thestrength was observed. Based on these results, the test con-ditions for specimen group ZP (Table 3) used for zinc phos-phate cement were used as the standard test conditions forsubsequent bi-axial testing, since the results under these con-ditions showed the smallest coefficient of variation, and sincethe mean bi-axial strength represented the mean of the sixgroups tested.The fracture strength of the brittle dental materials measured

by the three different test methods is summarized in Table 4.For the bi-axial flexure test, resin composite exhibited the larg-est mean bi-axial strength. The mean bi-axial flexure strengthof opaque porcelain was significantly higher (p0.05) among the diametral tensile strengths of bodyporcelain, opaque porcelain, and resin composite. The dia-metral tensile test data revealed that zinc phosphate cementexhibited the lowest coefficient of variation of 0.088, whileother materials exhibited relatively larger values of 0.234 to0.391.Shown in Fig. 2 are Weibull plots of fracture stresses, In In

[1/(1 -Ff)] vs. ln u, for the zinc phosphate cement tested ac-cording to the three methods. The data points were describedby a straight line produced by least-squares fit of the fracturedata by use of a computer, and the Weibull modulus, char-acteristic strength, and strength at a predicted failure level of5% were also calculated by computer. These results are listedin Table 5. The Weibull moduli for both bi-axial and four-point flexure tests were larger than those for the diametraltensile test, except for zinc phosphate cement. The Weibullmodulus of zinc phosphate cement exhibited the largest valuefor each test method.Shown in Table 6 are the failure probability parameters for

the bi-axial flexure test of four brittle dental materials derivedfrom the analytical solutions of Eqs. (11) to (20) by use ofWeibull analysis results for the four-point flexure test in Table5. Predicted plots (Bs and B,) for the Weibull analysis of thezinc phosphate cement are presented as dashed lines in Fig. 2.Experimental results for the bi-axial test of zinc phosphatecement and opaque porcelain showed much higher strengththan that predicted by surface flaw analysis. The failure prob-ability of both body porcelain and resin composite exhibitedgood agreement with the values predicted for the surface flawcondition.Shown in Fig. 3 are SEM images of fracture surfaces pro-

duced by bi-axial and four-point flexure stress for zinc phos-phate cement, body porcelain, opaque porcelain, and resincomposite. The arrows indicate the most likely sites of crackinitiation. The fracture surfaces for zinc phosphate cementspecimens exhibited a porous structure, especially in the speci-mens that were prepared for the four-point flexure test, whichhad much larger pores than those for the bi-axial test. For thespecimens tested by four-point flexure, the fracture origin ap-peared to be located around pores located at the corner of the

Vol. 69 No. 12 1 795


TABLE 5SUMMARY OF WEIBULL ANALYSIS

Cc o.a05Specimen m (MPa) (MPa)Bi-axial flexure testZP 10.45 18.6 14.0BP 7.54 54.9 37.0OP 7.47 80.2 53.9RC 6.27 111.6 69.5

Four-point flexure testZP 8.95 7.1 5.1BP 8.85 50.6 36.2OP 4.37 57.1 29.0RC 6.88 102.9 66.8

Diametral tensile testZP 11.68 4.7 3.7BP 2.10 25.9 6.9OP 3.25 26.2 10.5RC 3.39 26.3 11.0m, Weibull modulus.or, characteristic strength.cT0.05, strength predicted at the 5% level of failure.

fracture surface. The fracture surface of opaque porcelain pro-duced by four-point flexure also showed large pores, whichwere associated with the crack origin, while that for the bi-axial flexure test showed a homogeneous structure having smallpores. The fracture surfaces of body porcelain and resin com-posite for both flexure tests exhibited a similar structure thatcould be characterized as homogeneous and low in porosity.

Discussion.Effects of test conditions on bi-axialflexure strength. -Max-

imum tensile stresses produced by bi-axial flexure occur belowthe central loading area on the bottom surfaces of disc speci-mens. However, because of the typically high elastic modulusand hardness of brittle materials, any imperfect contact be-tween the rigid loading tool and the test specimen can lead toa substantial deviation from radial symmetry in the stress fieldand, consequently, to errors in strength measurements. Wacht-man et al. (1972) suggested that a layer of polyethylene be-tween piston and test surface would assist in the obtaining ofuniform loading at the end of the piston. Based on strain gaugemeasurements, Marshall (1980) found that the piston appliesthe load uniformly over its contact area when a film is used.However, in the present study, there were no significant dif-ferences (p>0.05) between the strengths of zinc phosphatecement with and without a film. It seems that the specimensurface of zinc phosphate cement, which is covered with aprecipitate by reaction between the cement surface and waterin the storage chamber, was flattened by the initial contact ofthe piston and, consequently, developed a uniform loadingcondition without an intermediate film. Furthermore, if the flatsurface of the piston is sufficiently parallel to the test surface,one does not always need to place the film or cushion over thespecimen center.The thickness of the specimens is one of the most important

factors in the determination of the bi-axial flexure strength,since the calculated stress is inversely proportional to the sec-ond power of its thickness, as derived in Eq. (1). Furthermore,the stress equation is valid only if the deflection does not ex-ceed about one-half of the plate thickness (Wachtman et al.,1972). Bending is directly proportional to the sustained loadand inversely proportional to Young's modulus, whereas it isonly slightly dependent on Poisson's ratio. The estimated min-

2

1 -

CL-

1- -1

-2 -

-3 -_0.5 1.0 1.5 2.0 2.5 3.0 3.5

In r (MPa)Fig. 2-Weibull plots of bi-axial flexure strength, four-point flexure

strength, and diametral tensile strength for zinc phosphate cement (ZP).Solid lines represent regression analyses of the raw data for the three testmethods, and dashed lines represent predictions (Bs, Bv) of bi-axial flexurestrength from four-point-flexure-strength data.

imum specimen thickness was calculated according to ASTMStandard F394. Within the stress range (18-180 MPa) encoun-tered in this study and over a Young's modulus range for brittlematerials [13.7 GPa for zinc phosphate cement, 16.6 GPa forresin composite, and 69 GPa for porcelain (Craig, 1989)], theestimated minimum thickness was always less than that en-countered in the test. Therefore, it is reasonable to assume thatthe use of this stress equation was valid in the present study.

Dimensions of the piston and specimen are included as fac-tors in the stress Eq. (1) for the bi-axial flexure test. Theimportant assumption of this equation is that the specimenstructure was homogeneous. However, brittle dental materialsused in the present study were not considered as homogeneousmaterials. For example, zinc phosphate cement contains sig-nificant porosity, matrix, and unreacted cement powders, asshown in Fig. 3. However, as shown in Table 3, the bi-axialstrengths of the specimens with different dimensions were notsignificantly different (p>0.05). Although the effects of ge-ometry have been identified in various strength tests (Baratta,1984; Ikeda et al., 1986; Lamon, 1988), it is concluded thatthe effect of geometry on bi-axial strength is negligible. Per-haps a more important factor in future refinement of these testsis the design of more uniform load distribution at load appli-cation and load-supporting regions to minimize the risk of lo-calized failure at these locations. Some of the variance ofmeasured strength values can be explained on the basis of afew two-segment fractures vs. the more common three-seg-ment fractures. The two-segment fractures may indicate thatsome of these were initiated at the load-application or load-support regions.The strengths of brittle materials generally increase with

increasing loading rate. The dependence of strength on stress-ing rate, caused by subcritical crack growth, has been de-scribed by Evans (1974) as follows:

(21)

where af is fracture stress at a given stressing rate, a is thestress rate, C is constant, and n is a crack-propagation param-

ZP Diametral Four-point Biaxial

I

A

B; Bv


orf = C 6rl/(l + n)


TABLE 6PREDICTED FAILURE PROBABILITY PARAMETERS FOR BI-AXIAL FLEXURE STRENGTH BASED ON FOUR-POINT FLEXURE DATA

Surface Flaw Volume FlawCode ln o- L, In ('rr a2L9)/m C L ln [(QT a2tl2)L,]/m cvZP 2.083 0.015 0.020 2.063 0.002 - 0.288 2.371BP 4.016 0.017 0.045 3.971 0.002 - 0.258 4.274OP 4.145 0.026 0.161 3.984 0.005 -0.221 4.366RC 4.758 0.020 0.067 4.691 0.003 - 0.304 5.062

eter. These n values vary appreciably from one material toanother (Pletka and Wiederhorn, 1978; Morena et al., 1986).In the present study, there was no significant difference (p > 0.05)between the results determined at a loading rate of 0.1 mm/min and 1 mm/min, which correspond to stressing rates of0.64 and 6.4 MPa/s, respectively. These two loading rates areoften used for mechanical testing. It seems that zinc phosphatecement has a small value of n, which implies that it is suscep-tible to fast crack propagation.The strength values for these different test conditions showed

good agreement with the calculated stress values when Eq. (1)was used. Thus, it is concluded that the bi-axial flexure testrepresents a reliable method for determinations of the strengthof brittle dental materials, since it is relatively insensitive totest conditions. However, it should be noted that the four-pointflexure specimens were fairly large in this study, and some ofthese results may have reflected variations caused by incon-sistencies in mixing.

Comparison with four-point flexure strength.-Shetty et al.(1983) reported that the bi-axial flexure strength of aluminawas higher than its four-point flexure strength and lower thanits three-point flexure strength, and they compared bi-axialstrength with predictions based on four-point flexure data. Theyfound that Weibull statistics provided a good description ofthe size effects on data from the two uni-axial tests, but under-estimated the effect of stress bi-axiality. On the other hand,Pletka and Wiederhorn (1978) reported that the four-point flex-ure strength of magnesium aluminosilicate glass in water washigher than its bi-axial flexure strength in water. Giovan andSines (1979) showed that the bi-axial flexure strength of densealumina was 8.5 and 8.1% lower than the uni-axial strengthfor ground and lapped surfaces, respectively. Pletka and Wied-erhorn (1982) showed a consistent relationship between bi-axial and four-point flexure strength data for five types ofceramics over a range of stressing rates. In contrast, the presentstudy showed that there was no consistent relationship betweenuni-axial and bi-axial flexure strength data.The first reason for this inconsistency is the surface condi-

tion of the specimens. The surfaces of the specimens containmany artificial flaws, since they were ground through 600-grit(Shetty et al., 1983), 400-mesh (Pletka and Wiederhorn, 1978),and 320-grit (Giovan and Sines, 1979) diamond abrasives. Thespecimens in the present study were polished through 0.3-pLmalumina powder for each porcelain, and zinc phosphate cementand resin composite specimens were flattened by glass platesduring setting.The second reason for the discrepancy is the specimen size

effect. The effect of specimen size on four-point flexure strengthhas been reported by many investigators (Berenbaum and Bro-die, 1959; Baratta, 1984; Ikeda et al., 1986; Lamon, 1988).Berenbaum and Brodie (1959) showed that the four-point flex-ure strength of pure plaster of Paris increased with a decreasein specimen thickness. They suggested that the four-point flex-ure strength of porous, weak, and brittle materials such asplaster is strongly influenced by specimen size. The surfacelayer of these chemically setting materials is tougher than their

internal structure, since the surface tends to form a dense struc-ture having few pores during setting under pressure with themold, as shown in Fig. 3. Furthermore, the properties of mix-ing materials such as plaster and cement are significantly af-fected by the total volume of mixing. For preparation of zincphosphate cement specimens for the four-point flexure test, themixture of 6 g of powder and 2 mL of liquid was used, whilea standard amount of powder and liquid (1.5 g and 0.5 mL)was used for preparation of specimens for the bi-axial flexuretest. Larger pores were formed and remained in the specimensfor the four-point flexure test, compared with the bi-axial flex-ure test specimens, because of insufficient mixing and pres-sure. Thus, the bi-axial testing of zinc phosphate cement yieldedmuch higher strength values than those predicted for both sur-face-flaw failure and volume-flaw failure, possibly because ofdifferences in the homogeneity of specimens. Opaque porce-lain specimens were also relatively inhomogeneous. Comparedwith body porcelain, opaque porcelain is relatively difficult tocondense for large specimen volumes such as that used for thefour-point flexure test, since opaque porcelain contains a higherfraction of opacifiers such as zirconium or titanium oxide,which reduce the fluidity of the slurry. The results for bodyporcelain and resin composite were comparable with those de-termined from the four-point flexure test, since test specimensfor both materials showed a similar structure for both bi-axialand four-point flexure test specimens. These results suggestthat this statistical approach demonstrates reasonable agree-ment between the bi-axial and four-point flexure strengths ofbrittle specimens with similar flaw characteristics.

Comparison with diametral tensile strength.-For the diam-etral tensile test, it is difficult for ideal loading to be producedalong a line when cylindrical specimens are used. A properload distribution is generally accomplished by placement of anarrow pad of suitable materials between the specimen and theloading platens. For example, in the diametral tensile test pro-cedure for ADA Specification No. 27 for direct filling resins(Council on Dental Materials and Devices, 1977), a thin pieceof paper (approximately 0.5 mm thick) wet with water mustbe inserted between the platens of the testing machine alongeach side of the specimen. In ADA Specification No. 1 foramalgam, the specimen should be padded with two thicknessesof 0.038-mm aluminum foil on each side. However, it shouldbe noted that the apparent strength changes with the type ofpadding material and its thickness, because the uniformity ofthe tensile stress distribution also changes (Rudnick et al.,1963). Therefore, no pad was used in the present study.

Relative to the diametral tensile test, zinc phosphate cementexhibited a small coefficient of variation and a large Weibullmodulus, as shown in Tables 4 and 5, whereas other materialsexhibited large coefficients of variation in diametral tensilestrength and small Weibull moduli, compared with those de-termined from both flexure tests. Although an advantage ofthe diametral tensile test is that the maximum tensile stress wasnot restricted to the surface, the surface effect on the fracturevalue is large (Rudnick et al., 1963). However, it is difficultto control the surface roughness on curved surfaces. It seems

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Fig. 3-SEM images of fractured surfaces that were subjected to bi-axial and four-point flexure for zinc phosphate cement (ZP), body porcelain (BP),opaque porcelain (OP), and resin composite (RC) specimens.



that the surfaces of zinc phosphate cement specimens wereflattened by the initial contact of test platens, thereby yieldinga more uniform stress distribution within the specimens. Fur-thermore, it is well-known that the effect of specimen size ondiametral strength is large. Williams and Smith (1971) showedthat the diametral tensile strength for a zinc phosphate cementincreased with increasing diameter. Thickness-to-diameter ra-tios used in the present study were 0.51 + 0.01 for zinc phos-phate cement, 0.46 + 0.06 for body porcelain, 0.42 0.04for opaque porcelain, and 0.52 0.01 for resin composite.The strength ratio of body porcelain and opaque porcelain hada relatively large variance due to a large variance in thickness,since these specimens were prepared by being polished. There-fore, it seems that the relatively large variance of diametralstrength for both porcelains is attributable to the inhomogeneityof the curved surfaces and a large variance of thickness-to-diameter ratio. Although resin composite had a small variancein thickness-to-diameter ratio, it seems that the curved surfacewas sufficiently inhomogeneous to decrease the variance instrength values.The mean diametral tensile strength of brittle dental mate-

rials was significantly lower than the bi-axial and four-pointflexure strength values, as shown in Table 4. Flexure stresswas enhanced because of the surface compression effect.

The fracture strength of brittle materials can best be explainedby postulating the presence of flaws distributed randomlythroughout the volume. Lamon (1988) showed that tensile andbending tests provided different statistical parameters, suggestingthat different populations of flaws controlled the failure in bothcases. The flexure test specimens, in which the entire volume isstressed fairly uniformly, should exhibit higher mean strengthsthan tensile specimens (Rudnick et al., 1963). Thus, specimenswith high porosity levels, such as cement, demonstrate low dia-metral strength values, compared with their bi-axial flexure strengthvalues. For resin composite specimens, a low-degree conversionmay have occurred within the deep interior region and along thecurved surface, since light irradiation was controlled from theflat ends of the cylindrical specimens.

It is well-recognized that the fracture strength of brittle ma-terials depends upon several structural parameters, such as inclu-sions of voids and cracks (Evans, 1982), flaw location on thesurface and within the volume (Lamon and Evans, 1983), thedimensions of specimens (Lamon and Evans, 1983), stress gra-dients (Ikeda et al., 1986), and the stress state (Lamon and Ev-ans, 1983). The effects of geometry on strength can stronglyinfluence the results of these strength tests, and it is difficult forthe "true" tensile strength to be determined. In summary, thebi-axial test is simpler to perform and provides a better simulationof clinically-relevant sample size than that used for other strengthtests, since specimen size and the preparation procedures aremore similar to clinical conditions for the bi-axial test.

Acknowledgments.The authors gratefully acknowledge Professor Jiro Hase-

gawa, Aichi-Gakuin University, for his helpful suggestionsand Mr. Robert B. Lee, University of Florida, for his technicalsupport in conducting the mechanical tests.

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