Journal of Dental Research http://jdr.sagepub.com/
Influence of Test Method on Failure Stress of Brittle Dental Materials S. Ban and K.J. Anusavice J DENT RES 1990 69: 1791 DOI: 10.1177/00220345900690120201 The online version of this article can be found at: http://jdr.sagepub.com/content/69/12/1791
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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 of Dental Biomaterials, College of Dentistry, University of Florida, Gainesville, Florida 32610-0446
A bi-axial flexure test (piston-on-three-balls), a four-point flexure test, and a diametral tensile test were used to measure the failure stress of four brittle dental materials: zinc phosphate cement, body porcelain, opaque porcelain, and visible lightcured resin composite. Furthermore, the fracture probability of the bi-axial test specimens was predicted from the results of the four-point flexure test, with use of statistical fracture theory. Bi-axial failure stresses calculated from an equation developed by Marshall (1980) exhibited no significant difference for zinc phosphate cement as a function of piston size, specimen thickness, presence or absence of a stress-distributing film, and loading rate. The four-point flexure strength values of zinc phosphate cement and opaque porcelain were significantly lower (p0.05) than the corresponding mean bi-axial strength values. The diametral tensile strength of all materials was significantly lower than the bi-axial flexure strength. The mean bi-axial flexure strengths of zinc phosphate cement and opaque porcelain were much higher than the theoretical values predicted from surface flaw theory, while the strength values for body porcelain and resin composite were comparable with those determined from the four-point flexure test. These results demonstrate that the strength of zinc phosphate cement depends not only upon the geometric factors, but also upon sample preparation conditions. J Dent Res 69(12):1791-1799, December, 1990
Introduction. Mechanical strength is an important factor that controls the clinical success of dental restorations. Usually, complex stress distributions that are induced by compressive, tensile, and shear stresses are present in most specimens under practical conditions. It is extremely difficult to induce a pure stress of a single type in a body. In general, tensile strength is easily determined for ductile materials such as metals. For convenience, compressive strength is often measured for brittle materials such as porcelains, cements, amalgams, and resin composites. However, brittle materials are much weaker in tension than in compression, and tensile stresses in some materials are (in certain cases) larger than compressive stress (Anusavice and Hojjatie, 1987). Therefore, tensile strength is generally considered as the more meaningful property for these brittle materials (compared with compressive strength) for assessment of the failure potential of dental restorations, especially in the presence of critical surface flaws. To determine the tensile strength for brittle dental materials, the diametral tensile test has been used frequently (Earnshaw and Smith, 1966; Williams and Smith, 1971; Powers et al., Received for publication February 27, 1990 Accepted for publication August 2, 1990 This study was supported by NIDR Grant DE 06672.
1976). The traditional tensile test has rarely been used for brittle materials (Bowen and Rodriguez, 1962; Zidan et al., 1980) because of the difficulty associated with gripping and aligning the specimens. The diametral tensile test provides a simple experimental method for measurement of the tensile strength of brittle materials. However, the complex stress distribution developed in the specimen can lead to various modes of fracture. If the specimen deforms significantly before failure, the data may not be valid. Zidan et al. (1980) suggested that the diametral tensile test cannot be considered reliable for dental resinous materials. Chiang and Tesk (1989) demonstrated that a correction of the stress calculation equation for diametral tension is needed with double cleft fracture. The main advantage of the flexure test is that a state of pure tension can be established on one side of the specimen (Berenbaum and Brodie, 1959). Three-point and four-point flexure tests have also been used for strength evaluation of singlecomponent brittle materials (Bryant and Mahler, 1986; Soderholm, 1986) and metal-ceramic structures (Coffey et al., 1988). For these uni-axial flexure tests, the principal stress on the lower surfaces of the specimens is tensile, and it is usually responsible for crack initiation in brittle materials. However, undesirable edge fracture (which can increase the variance of the failure stress value) can occur. Furthermore, these methods were designed for engineering materials that are usually associated with relatively large specimens. For brittle dental materials, construction of such specimens is not usually convenient because suitable quantities of dental restorative materials are not often available to prepare a sufficient number of specimens for assessment of statistically significant differences. Furthermore, the residual stress states due to polymerization shrinkage or thermal contraction difference and the flaw characteristics induced in large specimens may not be representative of those that are present in smaller clinical restorations. Recently, the bi-axial flexure test has been used frequently for the determination of fracture characteristics of brittle materials. The measurement of the strength of brittle materials under bi-axial flexure conditions rather than uni-axial flexure is often considered more reliable, because the maximum tensile stresses occur within the central loading area and spurious edge failures are eliminated. This allows slightly warped specimens to be tested and produces results unaffected by the edge condition of the specimen. This feature makes the method suitable for assessment of the effects of surface conditions on strength. A wide variety of loading arrangements has been developed for bi-axial flexure tests: (1) ring-on-ring (Kao et al., 1971), (2) piston-on-ring (Wilshaw, 1968), (3) ball-on-ring (McKinney 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, the fifth option was used, since it is suitable for slightly warped and small specimens such as brittle dental materials, and excellent results by this method have been reported previously for some glasses and ceramics. The bi-axial flexure strength is determined by support of a disc specimen on three metal spheres positioned at equal distances from each other and from the center of the disc. The load is applied to the center of the 1791
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Code
Type
ZP BP
Zinc Phosphate Cement
OP RC
Body Porcelain Opaque Porcelain Resin Composite
J Dent Res December 1990 TABLE 1 MATERIALS TESTED Brand Name Orthodontic Cement with Fluoride White Body #2 White Opaque #4 Herculite XR
Manufacturer
Ormco, Glendora, CA J.F. Jelenko & Co., Armonk, NY J.F. Jelenko & Co. Kerr Manufacturing Co., Romulus, MI
TABLE 2
SPECIMEN DIMENSIONS (mm) Four-point Flexure Test 1 w
ZP BP OP RC D,
Bi-axial Flexure Test D t 0.8 0.1 12.7 0.1 S M 13.6 ± 0.1 1.3 ± 0.1 33.7 + 0.5 6.8 ± 0.1 L 2.6 + 0.2 15.8 ± 0.1 14.4 ± 1.0 1.9 ± 0.3 4.9 ± 0.2 28.9 ± 0.3 14.4 + 0.1 2.0 - 0.1 29.8 + 0.2 5.0 ± 0.4 1.2 + 0.1 3.1 + 0.1 13.7 ± 0.2 36.0 ± 0.2 diameter of specimen; t, thickness of specimen; 1, length of specimen; w, width of specimen; S,
opposite surface by a flat piston. Kirstein and Woolley (1967) demonstrated that stresses in a thin, circular aluminum-alloy plate were independent of angular orientation and the number of supports. Wachtman et al. (1972) reported that bi-axial flexure strength values for eight types of alumina show that a coefficient of variation of about 7% can be achieved by testing five specimens and that different laboratories generally obtain good agreement on strength values. Marshall (1980) modified the stress equation to correct for geometry effects and demonstrated that the stress calculated from this modified equation for two types of glasses was in good agreement with the stress measured from strain gauges. Pletka and Wiederhorn (1982) compared the failure characteristics of specimens fractured in bi-axial flexure tests with those fractured by means of a conventional four-point flexure test. Morena et al. (1986) investigated the dynamic fatigue of dental porcelain using a bi-axial flexure test. Usually, disc specimens (12-50 mm in diameter and 1-3 mm in thickness) are used for these tests. They can be easily made under typical restorative conditions. Furthermore, the flat surface of the test specimen can be easily controlled by conventional metallographic polishing methods and typical dental finishing techniques. The objectives of this study were: (1) to test the hypothesis that the bi-axial flexure test reduces the variance of fracture strength values for brittle dental materials with various levels of homogeneity, compared with the four-point flexure test and the diametral tensile test, and (2) to analyze the influence of specimen geometry on the mean fracture strength values for brittle 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 phosphate cement, a feldspathic body porcelain, a feldspathic opaque porcelain, and a visible-light-cured resin composite (Table 1). Zinc phosphate cement specimens were prepared by conventional techniques according to the solubility test of ADA Specification No. 8 for zinc phosphate cement. The powder/liquid ratio was 3/1 (1.5 g/0.5 mL). Approximately 0.5 mL of cement of standard consistency was placed on a flat glass plate. Three ring sizes (inner diameter/thickness ratios of 13/0.6, 14/1.2, and 16/2.4 mm) were placed in the soft cement, and another glass plate was used to press the cement into a disk. Three
Diametral Tensile Test t D
t 6.7 ± 0.2
6.2 ± 0.1
4.3 ± 0.2 7.7 ± 0.1 4.3 ± 0.4 7.8 ± 0.1 3.1 + 0.1 6.2 + 0.0 small; M, medium; and L, large.
3.2 ± 0.1
3.5 + 0.5 3.3 ± 0.3 3.2 ± 0.1
minutes after the mix was started, the glass plates and cements were placed in a humidor at 37°C for one h. After removal from the humidor, the specimens were separated from the glass and stored in water at 37°C for 24 h. Porcelain specimens were prepared by normal fabrication procedures. A slurry of porcelain powder was vibrated and condensed into a mold 16 mm in diameter and 2 mm in depth. The discs were fired in a dental oven (Mark IV, J.M. Ney Co., Bloomfield, CT) at a heating rate of 55°C/min under vacuum to 982°C followed by a 90-second holding time in air. The specimens were removed from the furnace and rapidly cooled in ambient air by natural convection. The porcelain discs were ground from 120-grit to 600-grit papers and polished with 1->Lm and 0.3-gm A1203 powder on a metallographic polishing wheel. Visible-light-cured resin composite specimens were made as follows: Approximately 0.5 mL of resin paste was placed on a flat glass plate, 1.0 mm in thickness. A flexible ring, approximately 14 mm in inner diameter and 1.2 mm in thickness, was placed in the paste, and another glass plate was used to press the resin paste into a disc. Light emitted from a fiber optic handpiece (Translux, Kulzer & Co., Bad Homburg, Germany) passed through the glass plates for a total of 200 s so that adequate polymerization of each side would be ensured. After light irradiation, specimens were separated from the glass plates and stored in water at 37°C for 24 h before being tested. Specimens for the four-point flexure and the diametral tensile tests were also prepared in a similar manner. The final dimensions of the specimens are listed in Table 2. Determination of fracture strength. -The bi-axial flexure test apparatus is described in the ASTM Standard F394 for biaxial flexure testing of ceramic substrates. In this study, the dimensions of the apparatus were smaller than those described in the ASTM standard so that the small specimen size of brittle dental materials typically used in dental restorations would be accommodated. As shown in Fig. 1, specimens were supported on three steel spheres (3.2 mm in diameter) equally spaced along a diameter of 10 mm. For zinc phosphate cement specimens, loading was applied by a steel piston (with flat areas of 1.2 mm and 1.6 mm in diameter ground along the surface of contact) until fracture occurred. For the porcelain and resin composite specimens, 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
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FAILURE STRESS OF BRITTLE DENTAL MATERIALS
equations were based on the bi-axial flexure test developed by Wachtman et al. (1972), and original equations derived by Kirstein and Woolley (1967). The failure stress, a, can be expressed as: A p / t2 (1) and A = (3/4¶r) [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, a is the radius of the support circle, b is the radius of disc specimen, 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 ro is the radius of the piston at the surface of contact. If a series of nearly identical specimens is tested, the factors A and t (which depend only on the dimensions and Poisson's ratio) are constant, so the equation reduces to a simple proportionality between load and stress. In this study, the strength values were calculated with Poisson's ratio values of 0.35 for cement, 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 temperature using a thin plastic film (about 50 pum in thickness) between the piston and the upper surface of the specimen, to assist in obtaining uniform loading over the surfaces of the discs. Crosshead loading rates of 0.1 and 1 mm/min were applied by a universal testing machine (Instron Universal Testing Machine Model 1125, Instron Corp., Canton, MA). For investigation of the effects of piston diameter, disc diameter, and loading rate 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 specimens were 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 maximum tensile stress was calculated by the equation: uJ = PL/wt2 (3) where P is the applied load at failure, L is the length of outer span, w is the width of the specimen, and t is the thickness of the specimen. For the diametral tensile test, cylindrical specimens were tested at a cross-head loading rate of 0.1 mm/min. The maximum tensile stress for the diametral test is given by the equation: -=2 P / rrDt (4) where P is the applied load at failure, D is the diameter of the specimen, and t is the thickness of the specimen. From these data, the mean value, standard deviation, and coefficient of variation were calculated. Fracture surfaces representative of each group of specimens were characterized by means of scanning electron microscopy (SEM: JSM-35C, JEOL Ltd., Tokyo, Japan). Fracture statistics. -The failure strengths of brittle materials are statistically distributed as a function of the homogeneity of the material (Ritter, 1986). One commonly used statistic for the description of this distribution is the Weibull distribution, which is given by: Pf = 1-exp [-(-/U)m] (5) where Pf is the fracture probability defined by the relation Pf=i/(N+1), i is the rank in strength, N denotes the total number of specimens in the sample, m is the shape parameter,
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which is called the Weibull modulus, and u,,, is the scale parameter or characteristic strength. Higher values of Weibull modulus correspond to a higher level of homogeneity of the material. Most ceramics are reported to have m values in the range of 5 to 15, whereas metals, which fail in a ductile manner, have m values in the range of 30 to 100 (Johnson, 1983). The Weibull modulus, characteristic strength, and strength at a predicted failure level of 5% were obtained with use of a computer program designed to carry out the Weibull analysis from the fracture data. This analytical method is very popular because of its ease of application. However, the Weibull approach is not based on physical principles, but is based on statistical concepts. Therefore, as pointed out by several investigators (Giovan and Sines, 1979; Shetty et al., 1983; Lamon and Evans, 1983), the Weibull analysis has some limitations that challenge its ability to predict failure of components having complex geometries and which are subjected to a multi-axial stress state. This problem may be crucial for components used in dental restorations having complex geometries and subjected to multi-axial stress states. The elemental strength approach (Evans, 1982) represents a more physical analysis of failure, based essentially on the premise
p
Load Piston 2ro
jXTjj~~~ _Specimen Support
Ball Bearing
Specimen Holder
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, the thickness of the specimen; and r, the radius of the piston at the surface of contact.
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TABLE 3 BI-AXIAL FLEXURE TEST RESULTS FOR ZINC PHOSPHATE CEMENT Condition Specimen Diameter LoadLoading Meant of Piston distributing Speed No. of Strength Size* Film Tests Code (mm) (mm/min) (MPa) Yes 0.1 M 1.2 10 18.1 ZP 10 Yes 0.1 19.0 1.2 ZP1 S 16.9 Yes 0.1 10 L 1.2 ZP2 0.1 10 19.2 Yes M 1.6 ZP3 17.3 1.2 No 0.1 10 ZP4 M 17.5 Yes 1.0 10 M 1.2 ZP5 60 18.0 Total *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 can be characterized by their flaw-extension stress. A multi-axial elemental strength model was derived by Lamon and Evans (1983) and applied to specific test geometries, providing an experimental demonstration of practical requirements for a multiaxial analysis of fracture. However, these approaches are mathematically complex and require extensive numerical analysis even for simple stress states. Shetty et al. (1983) used the Barnett-Freudenthal approximation (Batdorf, 1977) for comparison of bi-axial data with uni-axial data from three- and four-point flexure tests. In the present study, the failure probability of brittle specimens subjected to bi-axial flexure was calculated from the results of the four-point flexure test, by use of the following statistical fracture theory. Eq. (5) can be converted to the following equation:
Result Standard
Coefficient
Deviation
of Variation 0.073 0.098
(MPa) 1.3 1.9 2.6 2.1 2.5 2.0 2.2
0.155
0.108 0.142 0.113 0.121
Weibull introduced this transformation of scale parameters to relate the multi-axial stress state to the uni-axial stress condition. Eq. (9) can be integrated in closed form giving: (11) Bs = (or a2) (oru/or)m Ls where o,,L is the maximum stress at the center of the disc specimen and L. is the loading factor,
L,
=
(ot + P)/(m + 1)Oc
3
(12)
and a
=
3P(3+v)a2/8
t2oal
(13) (14)
P = 3 P (1 +3v) a2/8 t2oL From Eqs. (6) and (11), In In [1/(1 - Pf)] m[ln (oa) - Cj] (15) where C= In (o,) - ln(ir a2 L,)/m (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) (18) Lv ((x + 3)/(m + 1)2 a and In In [1/(1 -Pf)] m[ln (a,,) CQ (19) where C= In (or,) - ln (orr a2t/2 L,)/m (20) With these equations, statistics parameters for the bi-axial test can be predicted from the four-point flexure test. The loading factor is one convenient parameter for this purpose, because it incorporates the stress-state and stress-gradient effects into the fracture statistics. The loading factors decrease with increasing Weibull modulus, reflecting the influence of stress gradients. The volume loading factors are smaller than the surface loading factors because of the additional stress gradient in the axial direction. In the present study, the failure probability for the bi-axial test of four brittle dental materials was predicted from the Weibull parameters for the four-point flexure test by means of these equations. =
Pf
=
1
-
exp [-B]
(6)
where B is the risk of rupture. For multi-axial stress states, B is 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 normal tensile stress at an arbitrary angle relative to the principal stresses crl, 02, and O3; and dwo an elemental area on a unit solid sphere. The geometric variables used to describe and dw are defined in an orientation relative to principal stresses. Eq. (7) is evaluated by integration of those portions of the unitsphere where orn is tensile. Weibull assumed that B was zero for orientations for which oa, was compressive. For the uniaxial stress case, the two-parameter form is n (cn) (8) (3nlcJno) 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. According to the Barnett-Freudenthal approximation (Batdorf, 1977), the principal stresses are assumed to act independently. This assumption leads to the following equation for the risk of rupture in 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. For the uniform-pressure-on-disc specimen, the stress state is biaxial tension, with a,= 0r, U2 =ot, and C3= 0. ao, is related to cr,, through the equation ao-" = [(2m + 1)/2 'j]u "(m(10) o,,
=
=
=
-
=
Results. The mean bi-axial flexure strength values of zinc phosphate cement as a function of six different test conditions are listed
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FAILURE STRESS OF BRITTLE DENTAL MATERIALS
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TABLE 4
COMPARATIVE DATA FOR THE THREE TEST METHODS
Specimen Bi-axial flexure test ZP* BP OP RC Four-point flexure test ZP BP OP RC Diametral tensile test ZP BP OP RC *Same data as in Table 3
Ratio of Strength to Bi-axial Flex-
No. of Tests
Meant Strength (MPa)
Standard Deviation (MPa)
Coefficient of Variation
10 10 10 10
18.1 52.4 75.6 103.9
1.3 5.6 8.6 15.6
0.073 0.106 0.114 0.150
1 1 1 1
10 10 10 10
6.8 48.4 52.4 98.4
0.6 5.8 10.2 11.4
0.088 0.125 0.195 0.116
0.375 0.923 0.693 0.947
4.5 22.4 23.8 24.6
0.5 8.7 6.0 5.8
0.088 0.391 0.253 0.234
0.249 0.427 0.315 0.236
10 10 10 10 for group ZP.
ure Strength
tCommon vertical lines indicate no significant statistical difference at a = 0.05.
in Table 3. The mean strength for these conditions was not significantly different at the 95% confidence level when Duncan's multiple range test was used. Although zinc phosphate cement was mixed manually for each specimen, the coefficient of variation fell within a narrow acceptable range (0.073 to 0.155), indicating the adequacy of the experimental data for evaluation of the strength of the cement. The bi-axial flexure strength of zinc phosphate cement was insensitive to specimen size, diameter of the piston, use of a load-distributing film, and loading speed. Fractured specimens could be grouped according to a two-segment or three-segment fracture pattern. However, no relationship between the fracture mode and the strength was observed. Based on these results, the test conditions for specimen group ZP (Table 3) used for zinc phosphate cement were used as the standard test conditions for subsequent bi-axial testing, since the results under these conditions showed the smallest coefficient of variation, and since the mean bi-axial strength represented the mean of the six groups 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 largest mean bi-axial strength. The mean bi-axial flexure strength of opaque porcelain was significantly higher (p0.05) among the diametral tensile strengths of body porcelain, opaque porcelain, and resin composite. The diametral tensile test data revealed that zinc phosphate cement exhibited the lowest coefficient of variation of 0.088, while other materials exhibited relatively larger values of 0.234 to 0.391. Shown in Fig. 2 are Weibull plots of fracture stresses, In In [1/(1 -Ff)] vs. ln u, for the zinc phosphate cement tested according to the three methods. The data points were described by a straight line produced by least-squares fit of the fracture data by use of a computer, and the Weibull modulus, characteristic strength, and strength at a predicted failure level of 5% were also calculated by computer. These results are listed in Table 5. The Weibull moduli for both bi-axial and fourpoint flexure tests were larger than those for the diametral tensile test, except for zinc phosphate cement. The Weibull modulus of zinc phosphate cement exhibited the largest value for each test method. Shown in Table 6 are the failure probability parameters for the bi-axial flexure test of four brittle dental materials derived from the analytical solutions of Eqs. (11) to (20) by use of Weibull analysis results for the four-point flexure test in Table 5. Predicted plots (Bs and B,) for the Weibull analysis of the zinc phosphate cement are presented as dashed lines in Fig. 2. Experimental results for the bi-axial test of zinc phosphate cement and opaque porcelain showed much higher strength than that predicted by surface flaw analysis. The failure probability of both body porcelain and resin composite exhibited good agreement with the values predicted for the surface flaw
condition. Shown in Fig. 3 are SEM images of fracture surfaces produced by bi-axial and four-point flexure stress for zinc phosphate cement, body porcelain, opaque porcelain, and resin composite. The arrows indicate the most likely sites of crack initiation. The fracture surfaces for zinc phosphate cement specimens exhibited a porous structure, especially in the specimens that were prepared for the four-point flexure test, which had much larger pores than those for the bi-axial test. For the specimens tested by four-point flexure, the fracture origin appeared to be located around pores located at the corner of the
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TABLE 5 SUMMARY OF WEIBULL ANALYSIS
2 ZP
Specimen Bi-axial flexure test ZP BP OP RC Four-point flexure test ZP BP
Cc
o.a05
m
(MPa)
(MPa)
10.45 7.54 7.47 6.27
18.6 54.9 80.2 111.6
14.0 37.0 53.9 69.5
Diametral
Four-point
Biaxial
-
I
A CL -
8.95 8.85 4.37 6.88
7.1 50.6 57.1 102.9
5.1 36.2 29.0 66.8
ZP 11.68 BP 2.10 3.25 OP RC 3.39 m, Weibull modulus. or, characteristic strength.
4.7
3.7 6.9 10.5 11.0
OP RC Diametral tensile test
1
1-
-1
-2
25.9 26.2 26.3
cT0.05, strength predicted at the 5% level of failure. fracture surface. The fracture surface of opaque porcelain produced by four-point flexure also showed large pores, which were associated with the crack origin, while that for the biaxial flexure test showed a homogeneous structure having small pores. The fracture surfaces of body porcelain and resin composite for both flexure tests exhibited a similar structure that could be characterized as homogeneous and low in porosity.
Discussion. Effects of test conditions on bi-axialflexure strength. -Maximum tensile stresses produced by bi-axial flexure occur below the central loading area on the bottom surfaces of disc specimens. However, because of the typically high elastic modulus and hardness of brittle materials, any imperfect contact between the rigid loading tool and the test specimen can lead to a substantial deviation from radial symmetry in the stress field and, consequently, to errors in strength measurements. Wachtman et al. (1972) suggested that a layer of polyethylene between piston and test surface would assist in the obtaining of uniform loading at the end of the piston. Based on strain gauge measurements, Marshall (1980) found that the piston applies the load uniformly over its contact area when a film is used. However, in the present study, there were no significant differences (p>0.05) between the strengths of zinc phosphate cement with and without a film. It seems that the specimen surface of zinc phosphate cement, which is covered with a precipitate by reaction between the cement surface and water in the storage chamber, was flattened by the initial contact of the piston and, consequently, developed a uniform loading condition without an intermediate film. Furthermore, if the flat surface of the piston is sufficiently parallel to the test surface, one does not always need to place the film or cushion over the specimen 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 second power of its thickness, as derived in Eq. (1). Furthermore, the stress equation is valid only if the deflection does not exceed about one-half of the plate thickness (Wachtman et al., 1972). Bending is directly proportional to the sustained load and inversely proportional to Young's modulus, whereas it is only slightly dependent on Poisson's ratio. The estimated min-
-
-3 - _ 0.5
B; 1.0
1.5
Bv 2.0
In
r
2.5
3.0
3.5
(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 test methods, and dashed lines represent predictions (Bs, Bv) of bi-axial flexure strength from four-point-flexure-strength data.
imum specimen thickness was calculated according to ASTM Standard F394. Within the stress range (18-180 MPa) encountered in this study and over a Young's modulus range for brittle materials [13.7 GPa for zinc phosphate cement, 16.6 GPa for resin composite, and 69 GPa for porcelain (Craig, 1989)], the estimated minimum thickness was always less than that encountered in the test. Therefore, it is reasonable to assume that the use of this stress equation was valid in the present study. Dimensions of the piston and specimen are included as factors in the stress Eq. (1) for the bi-axial flexure test. The important assumption of this equation is that the specimen structure was homogeneous. However, brittle dental materials used in the present study were not considered as homogeneous materials. For example, zinc phosphate cement contains significant porosity, matrix, and unreacted cement powders, as shown in Fig. 3. However, as shown in Table 3, the bi-axial strengths of the specimens with different dimensions were not significantly different (p>0.05). Although the effects of geometry have been identified in various strength tests (Baratta, 1984; Ikeda et al., 1986; Lamon, 1988), it is concluded that the effect of geometry on bi-axial strength is negligible. Perhaps a more important factor in future refinement of these tests is the design of more uniform load distribution at load application and load-supporting regions to minimize the risk of localized failure at these locations. Some of the variance of measured strength values can be explained on the basis of a few two-segment fractures vs. the more common three-segment fractures. The two-segment fractures may indicate that some of these were initiated at the load-application or loadsupport regions. The strengths of brittle materials generally increase with increasing loading rate. The dependence of strength stressing rate, caused by subcritical crack growth, has been described by Evans (1974) as follows: on
orf
=
where af is fracture stress at
stress rate, C is constant, and
C 6rl/(l + n) a n
(21)
given stressing rate, a is the is a crack-propagation param-
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TABLE 6
PREDICTED FAILURE PROBABILITY PARAMETERS FOR BI-AXIAL FLEXURE STRENGTH BASED ON FOUR-POINT FLEXURE DATA Surface Flaw Volume Flaw In ('rr a2L9)/m L ln [(QT a2tl2)L,]/m ln oC Code L, cv - 0.288 0.002 2.371 0.015 0.020 2.063 ZP 2.083 4.274 0.002 - 0.258 0.045 3.971 BP 4.016 0.017 0.161 0.026 3.984 0.005 -0.221 4.366 OP 4.145 5.062 0.020 4.691 0.003 - 0.304 0.067 RC 4.758
eter. These n values vary appreciably from one material to another (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 of 0.64 and 6.4 MPa/s, respectively. These two loading rates are often used for mechanical testing. It seems that zinc phosphate cement has a small value of n, which implies that it is susceptible 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 test represents a reliable method for determinations of the strength of brittle dental materials, since it is relatively insensitive to test conditions. However, it should be noted that the four-point flexure specimens were fairly large in this study, and some of these 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 alumina was higher than its four-point flexure strength and lower than its three-point flexure strength, and they compared bi-axial strength with predictions based on four-point flexure data. They found that Weibull statistics provided a good description of the size effects on data from the two uni-axial tests, but underestimated the effect of stress bi-axiality. On the other hand, Pletka and Wiederhorn (1978) reported that the four-point flexure strength of magnesium aluminosilicate glass in water was higher than its bi-axial flexure strength in water. Giovan and Sines (1979) showed that the bi-axial flexure strength of dense alumina was 8.5 and 8.1% lower than the uni-axial strength for ground and lapped surfaces, respectively. Pletka and Wiederhorn (1982) showed a consistent relationship between biaxial and four-point flexure strength data for five types of ceramics over a range of stressing rates. In contrast, the present study showed that there was no consistent relationship between uni-axial and bi-axial flexure strength data. The first reason for this inconsistency is the surface condition of the specimens. The surfaces of the specimens contain many 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. The specimens in the present study were polished through 0.3-pLm alumina powder for each porcelain, and zinc phosphate cement and resin composite specimens were flattened by glass plates
during setting. The second reason for the discrepancy is the specimen size effect. The effect of specimen size on four-point flexure strength has been reported by many investigators (Berenbaum and Brodie, 1959; Baratta, 1984; Ikeda et al., 1986; Lamon, 1988). Berenbaum and Brodie (1959) showed that the four-point flexure strength of pure plaster of Paris increased with a decrease in specimen thickness. They suggested that the four-point flexure strength of porous, weak, and brittle materials such as plaster is strongly influenced by specimen size. The surface layer of these chemically setting materials is tougher than their
internal structure, since the surface tends to form a dense structure having few pores during setting under pressure with the mold, as shown in Fig. 3. Furthermore, the properties of mixing materials such as plaster and cement are significantly affected by the total volume of mixing. For preparation of zinc phosphate cement specimens for the four-point flexure test, the mixture of 6 g of powder and 2 mL of liquid was used, while a standard amount of powder and liquid (1.5 g and 0.5 mL) was used for preparation of specimens for the bi-axial flexure test. Larger pores were formed and remained in the specimens for the four-point flexure test, compared with the bi-axial flexure test specimens, because of insufficient mixing and pressure. Thus, the bi-axial testing of zinc phosphate cement yielded much higher strength values than those predicted for both surface-flaw failure and volume-flaw failure, possibly because of differences in the homogeneity of specimens. Opaque porcelain specimens were also relatively inhomogeneous. Compared with body porcelain, opaque porcelain is relatively difficult to condense for large specimen volumes such as that used for the four-point flexure test, since opaque porcelain contains a higher fraction of opacifiers such as zirconium or titanium oxide, which reduce the fluidity of the slurry. The results for body porcelain and resin composite were comparable with those determined from the four-point flexure test, since test specimens for both materials showed a similar structure for both bi-axial and four-point flexure test specimens. These results suggest that this statistical approach demonstrates reasonable agreement between the bi-axial and four-point flexure strengths of brittle specimens with similar flaw characteristics. Comparison with diametral tensile strength.-For the diametral tensile test, it is difficult for ideal loading to be produced along a line when cylindrical specimens are used. A proper load distribution is generally accomplished by placement of a narrow pad of suitable materials between the specimen and the loading platens. For example, in the diametral tensile test procedure for ADA Specification No. 27 for direct filling resins (Council on Dental Materials and Devices, 1977), a thin piece of paper (approximately 0.5 mm thick) wet with water must be inserted between the platens of the testing machine along each side of the specimen. In ADA Specification No. 1 for amalgam, the specimen should be padded with two thicknesses of 0.038-mm aluminum foil on each side. However, it should be noted that the apparent strength changes with the type of padding material and its thickness, because the uniformity of the 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 cement exhibited a small coefficient of variation and a large Weibull modulus, as shown in Tables 4 and 5, whereas other materials exhibited large coefficients of variation in diametral tensile strength and small Weibull moduli, compared with those determined from both flexure tests. Although an advantage of the diametral tensile test is that the maximum tensile stress was not restricted to the surface, the surface effect on the fracture value is large (Rudnick et al., 1963). However, it is difficult to 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. Downloaded from jdr.sagepub.com at UNIVERSITY OF WINDSOR on November 13, 2014 For personal use only. No other uses without permission.
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FAILURE STRESS OF BRITTLE DENTAL MA TERIALS
that the surfaces of zinc phosphate cement specimens were flattened by the initial contact of test platens, thereby yielding a more uniform stress distribution within the specimens. Furthermore, it is well-known that the effect of specimen size on diametral strength is large. Williams and Smith (1971) showed that the diametral tensile strength for a zinc phosphate cement increased with increasing diameter. Thickness-to-diameter ratios used in the present study were 0.51 + 0.01 for zinc phosphate cement, 0.46 + 0.06 for body porcelain, 0.42 ± 0.04 for opaque porcelain, and 0.52 ± 0.01 for resin composite. The strength ratio of body porcelain and opaque porcelain had a relatively large variance due to a large variance in thickness, since these specimens were prepared by being polished. Therefore, it seems that the relatively large variance of diametral strength for both porcelains is attributable to the inhomogeneity of the curved surfaces and a large variance of thickness-todiameter ratio. Although resin composite had a small variance in thickness-to-diameter ratio, it seems that the curved surface was sufficiently inhomogeneous to decrease the variance in strength values. The mean diametral tensile strength of brittle dental materials was significantly lower than the bi-axial and four-point flexure strength values, as shown in Table 4. Flexure stress was enhanced because of the surface compression effect. The fracture strength of brittle materials can best be explained by postulating the presence of flaws distributed randomly throughout the volume. Lamon (1988) showed that tensile and bending tests provided different statistical parameters, suggesting that different populations of flaws controlled the failure in both cases. The flexure test specimens, in which the entire volume is stressed fairly uniformly, should exhibit higher mean strengths than tensile specimens (Rudnick et al., 1963). Thus, specimens with high porosity levels, such as cement, demonstrate low diametral strength values, compared with their bi-axial flexure strength values. For resin composite specimens, a low-degree conversion may have occurred within the deep interior region and along the curved surface, since light irradiation was controlled from the flat ends of the cylindrical specimens. It is well-recognized that the fracture strength of brittle materials depends upon several structural parameters, such as inclusions of voids and cracks (Evans, 1982), flaw location on the surface and within the volume (Lamon and Evans, 1983), the dimensions of specimens (Lamon and Evans, 1983), stress gradients (Ikeda et al., 1986), and the stress state (Lamon and Evans, 1983). The effects of geometry on strength can strongly influence the results of these strength tests, and it is difficult for the "true" tensile strength to be determined. In summary, the bi-axial test is simpler to perform and provides a better simulation of clinically-relevant sample size than that used for other strength tests, since specimen size and the preparation procedures are more similar to clinical conditions for the bi-axial test.
Acknowledgments. The authors gratefully acknowledge Professor Jiro Hasegawa, Aichi-Gakuin University, for his helpful suggestions and Mr. Robert B. Lee, University of Florida, for his technical support in conducting the mechanical tests. REFERENCES ANUSAVICE, K.J. and HOJJATIE, B. (1987): Stress Distribution in Metalceramic Crowns with a Facial Porcelain Margin, JDent Res 66:1493-1498. BARATTA, F.I. (1984): Requirements for Flexure Testing of Brittle Materials. In: Methods for Assessing the Structural Reliability of Brittle Materials, ASTM STP 844, S.W. Freiman and C.M. Hudson, Eds., Philadelphia: American Society for Testing and Materials, pp. 194-222.
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BATDORF, S.B. (1977): Some Approximate Treatments of Fracture Statics for Polyaxial Tension, Int J Fracture 13:5-11. BERENBAUM, R. and BRODIE, I. (1959): Measurement of the Tensile Strength of Brittle Materials, Br JAppl Physics 10:281-287. BOWEN, R.L. and RODRIGUEZ, M.S. (1962): Tensile Strength and Modulus of Elasticity of Tooth Structure and Several Restorative Materials, J Am Dent Assoc 64:378-387. BRYANT, R.W. and MAHLER, D.B. (1986): Modulus of Elasticity in Bending of Composites and Amalgams, J Prosthet Dent 56:243-248. CHIANG, M.Y.M. and TESK, J.A. (1989): Differences: Hertz vs. Finite Element Calculation for Diametral Tensile Strength, J Dent Res 68:341, Abst. No. 1278. COFFEY, J.P.; ANUSAVICE, K.J.; DEHOFF, P.H.; LEE, R.B.; and HOJJATIE, B. (1988): Influence of Contraction Mismatch and Cooling Rate on Flexural Failure of PFM Systems, J Dent Res 67:61-65. Council on Dental Materials and Devices (1977): New American Dental Association Specification No. 27 for Direct Filling Resins, JAm Dent Assoc 94:1191-1194. CRAIG, R.G. (1989): Restorative Dental Materials, 8th ed., St. Louis: C.V. Mosby Co., p. 81. EARNSHAW, R. and SMITH, D.C. (1966): The Tensile and Compressive Strength of Plaster and Stone, Aust Dent J 11:415-422. EVANS, A.G. (1974): Slow Crack Growth in Brittle Materials under Dynamic Loading Conditions, Int J Frac 10:251-259. EVANS, A.G. (1982): Structural Reliability: A Processing-dependent Phenomenon, JAm Ceram Soc 65:127-137. GIOVAN, M.N. and SINES, G. (1979): Biaxial and Uniaxial Data for Statistical Comparison of a Ceramic's Strength, JAm Ceram Soc 62:510-515. IKEDA, K.; IGAKI, H.; and KURODA, T. (1986): Fracture Strength of Alumina Ceramics under Uniaxial and Triaxial Stress, Am Ceram Soc Bull1 65:683-688. JOHNSON, C.A. (1983): Fracture Statics of Multiple Flaw Distribution. In: Fracture Mechanics of Ceramics, Vol. 5, Surface Flaws, Statics, and Microcracking, R.C. Bradt, A.G. Evans, D.P.H. Hasselman, and F.F. Lange, Eds., New York: Plenum Press, pp. 365-386. KAO, R.; PERRONE, N.; and CAPPS, W. (1971): Large-deflection Solution of the Coaxial-Ring-Circular-Glass-Plate Flexure Problem, J Am Ceram Soc 54:566-571. KIRSTEIN, A.F. and WOOLLEY, R.M. (1967): Symmetrical Bending of Thin Circular Elastic Plates of Equally Spaced Point Supports, J Res Natil Bur Stds 71(C):1-10. LAMON, J. (1988): Statistical Approaches to Failure for Ceramic Reliability Assessment, J Am Ceram Soc 71:106-112. LAMON, J. and EVANS, A.G. (1983): Statistical Analysis of Bending Strength for Brittle Solids: A Multiaxial Fracture Problem, JAm Ceram Soc 66:177182. MARSHALL, D.B. (1980): An Improved Biaxial Flexure Test for Ceramics, Am Ceram Soc Bull 59:551-553. McKINNEY, K.R. and HERBERT, C.M. (1970): Effect of Surface Finish on Structural Ceramic Failure, JAm Ceram Soc 53:513-516. MORENA, R.; BEAUDREAU, G.M.; LOCKWOOD, P.E.; EVANS, A.L.; and FAIRHURST, C.W. (1986): Fatigue of Dental Ceramics in a Simulated Oral Environment, J Dent Res 65:993-997. PLETKA, B.J. and WIEDERHORN, S.M. (1978): Fracture Mechanics of Ceramics, Vol. 4, Crack Growth and Microstructure, R.C. Bradt, D.P.H. Hasselman, and F.F. Lange, Eds., New York: Plenum Press, pp. 745759. PLETKA, B.J. and WIEDERHORN, S.M. (1982): A Comparison of Failure Predictions by Strength and Fracture Mechanics Techniques, J Mater Sci 17:1247-1268. POWERS, J.M.; FARAH, J.W.; and CRAIG, R.G. (1976): Modulus of Elasticity and Strength Properties of Dental Cements, JAm DentAssoc 92:588591. RITTER, J.E. (1986): Fracture:Reliability Criteria for Brittle Materials. In: Encyclopedia of Materials Science and Engineering, M.B. Bever, Ed., Oxford: Pergamon Press, pp. 1852-1858. RUDNICK, A.; HUNTER, A.R.; and HOLDEN, F.C. (1963): An Analysis of the Diametral-Compression Test, Mater Res & Stds 3:283-289. SHETTY, D.K.; ROSENFIELD, A.R.; DUCKWORTH, W.H.; and HELD, P.R. (1983): A Biaxial-Flexure Test for Evaluating Ceramic Strength, J Am Ceram Soc 66:36-42. SODERHOLM, K-J.M. (1986): Flexure Strength of Repaired Dental Composites, Scand J Dent Res 94:364-369. WACHTMAN, J.B., Jr.; CAPPS, W.; and MANDEL, J. (1972): Biaxial Flexure Tests of Ceramic Substrates, J Mater 7:188-194. WILLIAMS, P.D. and SMITH, D.C. (1971): Measurement of the Tensile Strength of Dental Restorative Materials by Use of a Diametral Compression Test, J Dent Res 50:436-442. WILSHAW, T.R. (1968): Measurement of Tensile Strength of Ceramics, JAm Ceram Soc 51:111. ZIDAN, O.; ASMUSSEN, E.; and JORGENSEN, K.D. (1980): Tensile Strength of Restorative Resins, Scand J Dent Res 88:285-289.
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Influence of Test Method on Failure Stress of Brittle Dental Materials S. Ban and K.J. Anusavice J DENT RES 1990 69: 1791 DOI: 10.1177/00220345900690120201 The online version of this article can be found at: http://jdr.sagepub.com/content/69/12/1791
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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 of Dental Biomaterials, College of Dentistry, University of Florida, Gainesville, Florida 32610-0446
A bi-axial flexure test (piston-on-three-balls), a four-point flexure test, and a diametral tensile test were used to measure the failure stress of four brittle dental materials: zinc phosphate cement, body porcelain, opaque porcelain, and visible lightcured resin composite. Furthermore, the fracture probability of the bi-axial test specimens was predicted from the results of the four-point flexure test, with use of statistical fracture theory. Bi-axial failure stresses calculated from an equation developed by Marshall (1980) exhibited no significant difference for zinc phosphate cement as a function of piston size, specimen thickness, presence or absence of a stress-distributing film, and loading rate. The four-point flexure strength values of zinc phosphate cement and opaque porcelain were significantly lower (p0.05) than the corresponding mean bi-axial strength values. The diametral tensile strength of all materials was significantly lower than the bi-axial flexure strength. The mean bi-axial flexure strengths of zinc phosphate cement and opaque porcelain were much higher than the theoretical values predicted from surface flaw theory, while the strength values for body porcelain and resin composite were comparable with those determined from the four-point flexure test. These results demonstrate that the strength of zinc phosphate cement depends not only upon the geometric factors, but also upon sample preparation conditions. J Dent Res 69(12):1791-1799, December, 1990
Introduction. Mechanical strength is an important factor that controls the clinical success of dental restorations. Usually, complex stress distributions that are induced by compressive, tensile, and shear stresses are present in most specimens under practical conditions. It is extremely difficult to induce a pure stress of a single type in a body. In general, tensile strength is easily determined for ductile materials such as metals. For convenience, compressive strength is often measured for brittle materials such as porcelains, cements, amalgams, and resin composites. However, brittle materials are much weaker in tension than in compression, and tensile stresses in some materials are (in certain cases) larger than compressive stress (Anusavice and Hojjatie, 1987). Therefore, tensile strength is generally considered as the more meaningful property for these brittle materials (compared with compressive strength) for assessment of the failure potential of dental restorations, especially in the presence of critical surface flaws. To determine the tensile strength for brittle dental materials, the diametral tensile test has been used frequently (Earnshaw and Smith, 1966; Williams and Smith, 1971; Powers et al., Received for publication February 27, 1990 Accepted for publication August 2, 1990 This study was supported by NIDR Grant DE 06672.
1976). The traditional tensile test has rarely been used for brittle materials (Bowen and Rodriguez, 1962; Zidan et al., 1980) because of the difficulty associated with gripping and aligning the specimens. The diametral tensile test provides a simple experimental method for measurement of the tensile strength of brittle materials. However, the complex stress distribution developed in the specimen can lead to various modes of fracture. If the specimen deforms significantly before failure, the data may not be valid. Zidan et al. (1980) suggested that the diametral tensile test cannot be considered reliable for dental resinous materials. Chiang and Tesk (1989) demonstrated that a correction of the stress calculation equation for diametral tension is needed with double cleft fracture. The main advantage of the flexure test is that a state of pure tension can be established on one side of the specimen (Berenbaum and Brodie, 1959). Three-point and four-point flexure tests have also been used for strength evaluation of singlecomponent brittle materials (Bryant and Mahler, 1986; Soderholm, 1986) and metal-ceramic structures (Coffey et al., 1988). For these uni-axial flexure tests, the principal stress on the lower surfaces of the specimens is tensile, and it is usually responsible for crack initiation in brittle materials. However, undesirable edge fracture (which can increase the variance of the failure stress value) can occur. Furthermore, these methods were designed for engineering materials that are usually associated with relatively large specimens. For brittle dental materials, construction of such specimens is not usually convenient because suitable quantities of dental restorative materials are not often available to prepare a sufficient number of specimens for assessment of statistically significant differences. Furthermore, the residual stress states due to polymerization shrinkage or thermal contraction difference and the flaw characteristics induced in large specimens may not be representative of those that are present in smaller clinical restorations. Recently, the bi-axial flexure test has been used frequently for the determination of fracture characteristics of brittle materials. The measurement of the strength of brittle materials under bi-axial flexure conditions rather than uni-axial flexure is often considered more reliable, because the maximum tensile stresses occur within the central loading area and spurious edge failures are eliminated. This allows slightly warped specimens to be tested and produces results unaffected by the edge condition of the specimen. This feature makes the method suitable for assessment of the effects of surface conditions on strength. A wide variety of loading arrangements has been developed for bi-axial flexure tests: (1) ring-on-ring (Kao et al., 1971), (2) piston-on-ring (Wilshaw, 1968), (3) ball-on-ring (McKinney 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, the fifth option was used, since it is suitable for slightly warped and small specimens such as brittle dental materials, and excellent results by this method have been reported previously for some glasses and ceramics. The bi-axial flexure strength is determined by support of a disc specimen on three metal spheres positioned at equal distances from each other and from the center of the disc. The load is applied to the center of the 1791
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Code
Type
ZP BP
Zinc Phosphate Cement
OP RC
Body Porcelain Opaque Porcelain Resin Composite
J Dent Res December 1990 TABLE 1 MATERIALS TESTED Brand Name Orthodontic Cement with Fluoride White Body #2 White Opaque #4 Herculite XR
Manufacturer
Ormco, Glendora, CA J.F. Jelenko & Co., Armonk, NY J.F. Jelenko & Co. Kerr Manufacturing Co., Romulus, MI
TABLE 2
SPECIMEN DIMENSIONS (mm) Four-point Flexure Test 1 w
ZP BP OP RC D,
Bi-axial Flexure Test D t 0.8 0.1 12.7 0.1 S M 13.6 ± 0.1 1.3 ± 0.1 33.7 + 0.5 6.8 ± 0.1 L 2.6 + 0.2 15.8 ± 0.1 14.4 ± 1.0 1.9 ± 0.3 4.9 ± 0.2 28.9 ± 0.3 14.4 + 0.1 2.0 - 0.1 29.8 + 0.2 5.0 ± 0.4 1.2 + 0.1 3.1 + 0.1 13.7 ± 0.2 36.0 ± 0.2 diameter of specimen; t, thickness of specimen; 1, length of specimen; w, width of specimen; S,
opposite surface by a flat piston. Kirstein and Woolley (1967) demonstrated that stresses in a thin, circular aluminum-alloy plate were independent of angular orientation and the number of supports. Wachtman et al. (1972) reported that bi-axial flexure strength values for eight types of alumina show that a coefficient of variation of about 7% can be achieved by testing five specimens and that different laboratories generally obtain good agreement on strength values. Marshall (1980) modified the stress equation to correct for geometry effects and demonstrated that the stress calculated from this modified equation for two types of glasses was in good agreement with the stress measured from strain gauges. Pletka and Wiederhorn (1982) compared the failure characteristics of specimens fractured in bi-axial flexure tests with those fractured by means of a conventional four-point flexure test. Morena et al. (1986) investigated the dynamic fatigue of dental porcelain using a bi-axial flexure test. Usually, disc specimens (12-50 mm in diameter and 1-3 mm in thickness) are used for these tests. They can be easily made under typical restorative conditions. Furthermore, the flat surface of the test specimen can be easily controlled by conventional metallographic polishing methods and typical dental finishing techniques. The objectives of this study were: (1) to test the hypothesis that the bi-axial flexure test reduces the variance of fracture strength values for brittle dental materials with various levels of homogeneity, compared with the four-point flexure test and the diametral tensile test, and (2) to analyze the influence of specimen geometry on the mean fracture strength values for brittle 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 phosphate cement, a feldspathic body porcelain, a feldspathic opaque porcelain, and a visible-light-cured resin composite (Table 1). Zinc phosphate cement specimens were prepared by conventional techniques according to the solubility test of ADA Specification No. 8 for zinc phosphate cement. The powder/liquid ratio was 3/1 (1.5 g/0.5 mL). Approximately 0.5 mL of cement of standard consistency was placed on a flat glass plate. Three ring sizes (inner diameter/thickness ratios of 13/0.6, 14/1.2, and 16/2.4 mm) were placed in the soft cement, and another glass plate was used to press the cement into a disk. Three
Diametral Tensile Test t D
t 6.7 ± 0.2
6.2 ± 0.1
4.3 ± 0.2 7.7 ± 0.1 4.3 ± 0.4 7.8 ± 0.1 3.1 + 0.1 6.2 + 0.0 small; M, medium; and L, large.
3.2 ± 0.1
3.5 + 0.5 3.3 ± 0.3 3.2 ± 0.1
minutes after the mix was started, the glass plates and cements were placed in a humidor at 37°C for one h. After removal from the humidor, the specimens were separated from the glass and stored in water at 37°C for 24 h. Porcelain specimens were prepared by normal fabrication procedures. A slurry of porcelain powder was vibrated and condensed into a mold 16 mm in diameter and 2 mm in depth. The discs were fired in a dental oven (Mark IV, J.M. Ney Co., Bloomfield, CT) at a heating rate of 55°C/min under vacuum to 982°C followed by a 90-second holding time in air. The specimens were removed from the furnace and rapidly cooled in ambient air by natural convection. The porcelain discs were ground from 120-grit to 600-grit papers and polished with 1->Lm and 0.3-gm A1203 powder on a metallographic polishing wheel. Visible-light-cured resin composite specimens were made as follows: Approximately 0.5 mL of resin paste was placed on a flat glass plate, 1.0 mm in thickness. A flexible ring, approximately 14 mm in inner diameter and 1.2 mm in thickness, was placed in the paste, and another glass plate was used to press the resin paste into a disc. Light emitted from a fiber optic handpiece (Translux, Kulzer & Co., Bad Homburg, Germany) passed through the glass plates for a total of 200 s so that adequate polymerization of each side would be ensured. After light irradiation, specimens were separated from the glass plates and stored in water at 37°C for 24 h before being tested. Specimens for the four-point flexure and the diametral tensile tests were also prepared in a similar manner. The final dimensions of the specimens are listed in Table 2. Determination of fracture strength. -The bi-axial flexure test apparatus is described in the ASTM Standard F394 for biaxial flexure testing of ceramic substrates. In this study, the dimensions of the apparatus were smaller than those described in the ASTM standard so that the small specimen size of brittle dental materials typically used in dental restorations would be accommodated. As shown in Fig. 1, specimens were supported on three steel spheres (3.2 mm in diameter) equally spaced along a diameter of 10 mm. For zinc phosphate cement specimens, loading was applied by a steel piston (with flat areas of 1.2 mm and 1.6 mm in diameter ground along the surface of contact) until fracture occurred. For the porcelain and resin composite specimens, 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
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FAILURE STRESS OF BRITTLE DENTAL MATERIALS
equations were based on the bi-axial flexure test developed by Wachtman et al. (1972), and original equations derived by Kirstein and Woolley (1967). The failure stress, a, can be expressed as: A p / t2 (1) and A = (3/4¶r) [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, a is the radius of the support circle, b is the radius of disc specimen, 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 ro is the radius of the piston at the surface of contact. If a series of nearly identical specimens is tested, the factors A and t (which depend only on the dimensions and Poisson's ratio) are constant, so the equation reduces to a simple proportionality between load and stress. In this study, the strength values were calculated with Poisson's ratio values of 0.35 for cement, 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 temperature using a thin plastic film (about 50 pum in thickness) between the piston and the upper surface of the specimen, to assist in obtaining uniform loading over the surfaces of the discs. Crosshead loading rates of 0.1 and 1 mm/min were applied by a universal testing machine (Instron Universal Testing Machine Model 1125, Instron Corp., Canton, MA). For investigation of the effects of piston diameter, disc diameter, and loading rate 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 specimens were 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 maximum tensile stress was calculated by the equation: uJ = PL/wt2 (3) where P is the applied load at failure, L is the length of outer span, w is the width of the specimen, and t is the thickness of the specimen. For the diametral tensile test, cylindrical specimens were tested at a cross-head loading rate of 0.1 mm/min. The maximum tensile stress for the diametral test is given by the equation: -=2 P / rrDt (4) where P is the applied load at failure, D is the diameter of the specimen, and t is the thickness of the specimen. From these data, the mean value, standard deviation, and coefficient of variation were calculated. Fracture surfaces representative of each group of specimens were characterized by means of scanning electron microscopy (SEM: JSM-35C, JEOL Ltd., Tokyo, Japan). Fracture statistics. -The failure strengths of brittle materials are statistically distributed as a function of the homogeneity of the material (Ritter, 1986). One commonly used statistic for the description of this distribution is the Weibull distribution, which is given by: Pf = 1-exp [-(-/U)m] (5) where Pf is the fracture probability defined by the relation Pf=i/(N+1), i is the rank in strength, N denotes the total number of specimens in the sample, m is the shape parameter,
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which is called the Weibull modulus, and u,,, is the scale parameter or characteristic strength. Higher values of Weibull modulus correspond to a higher level of homogeneity of the material. Most ceramics are reported to have m values in the range of 5 to 15, whereas metals, which fail in a ductile manner, have m values in the range of 30 to 100 (Johnson, 1983). The Weibull modulus, characteristic strength, and strength at a predicted failure level of 5% were obtained with use of a computer program designed to carry out the Weibull analysis from the fracture data. This analytical method is very popular because of its ease of application. However, the Weibull approach is not based on physical principles, but is based on statistical concepts. Therefore, as pointed out by several investigators (Giovan and Sines, 1979; Shetty et al., 1983; Lamon and Evans, 1983), the Weibull analysis has some limitations that challenge its ability to predict failure of components having complex geometries and which are subjected to a multi-axial stress state. This problem may be crucial for components used in dental restorations having complex geometries and subjected to multi-axial stress states. The elemental strength approach (Evans, 1982) represents a more physical analysis of failure, based essentially on the premise
p
Load Piston 2ro
jXTjj~~~ _Specimen Support
Ball Bearing
Specimen Holder
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, the thickness of the specimen; and r, the radius of the piston at the surface of contact.
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TABLE 3 BI-AXIAL FLEXURE TEST RESULTS FOR ZINC PHOSPHATE CEMENT Condition Specimen Diameter LoadLoading Meant of Piston distributing Speed No. of Strength Size* Film Tests Code (mm) (mm/min) (MPa) Yes 0.1 M 1.2 10 18.1 ZP 10 Yes 0.1 19.0 1.2 ZP1 S 16.9 Yes 0.1 10 L 1.2 ZP2 0.1 10 19.2 Yes M 1.6 ZP3 17.3 1.2 No 0.1 10 ZP4 M 17.5 Yes 1.0 10 M 1.2 ZP5 60 18.0 Total *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 can be characterized by their flaw-extension stress. A multi-axial elemental strength model was derived by Lamon and Evans (1983) and applied to specific test geometries, providing an experimental demonstration of practical requirements for a multiaxial analysis of fracture. However, these approaches are mathematically complex and require extensive numerical analysis even for simple stress states. Shetty et al. (1983) used the Barnett-Freudenthal approximation (Batdorf, 1977) for comparison of bi-axial data with uni-axial data from three- and four-point flexure tests. In the present study, the failure probability of brittle specimens subjected to bi-axial flexure was calculated from the results of the four-point flexure test, by use of the following statistical fracture theory. Eq. (5) can be converted to the following equation:
Result Standard
Coefficient
Deviation
of Variation 0.073 0.098
(MPa) 1.3 1.9 2.6 2.1 2.5 2.0 2.2
0.155
0.108 0.142 0.113 0.121
Weibull introduced this transformation of scale parameters to relate the multi-axial stress state to the uni-axial stress condition. Eq. (9) can be integrated in closed form giving: (11) Bs = (or a2) (oru/or)m Ls where o,,L is the maximum stress at the center of the disc specimen and L. is the loading factor,
L,
=
(ot + P)/(m + 1)Oc
3
(12)
and a
=
3P(3+v)a2/8
t2oal
(13) (14)
P = 3 P (1 +3v) a2/8 t2oL From Eqs. (6) and (11), In In [1/(1 - Pf)] m[ln (oa) - Cj] (15) where C= In (o,) - ln(ir a2 L,)/m (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) (18) Lv ((x + 3)/(m + 1)2 a and In In [1/(1 -Pf)] m[ln (a,,) CQ (19) where C= In (or,) - ln (orr a2t/2 L,)/m (20) With these equations, statistics parameters for the bi-axial test can be predicted from the four-point flexure test. The loading factor is one convenient parameter for this purpose, because it incorporates the stress-state and stress-gradient effects into the fracture statistics. The loading factors decrease with increasing Weibull modulus, reflecting the influence of stress gradients. The volume loading factors are smaller than the surface loading factors because of the additional stress gradient in the axial direction. In the present study, the failure probability for the bi-axial test of four brittle dental materials was predicted from the Weibull parameters for the four-point flexure test by means of these equations. =
Pf
=
1
-
exp [-B]
(6)
where B is the risk of rupture. For multi-axial stress states, B is 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 normal tensile stress at an arbitrary angle relative to the principal stresses crl, 02, and O3; and dwo an elemental area on a unit solid sphere. The geometric variables used to describe and dw are defined in an orientation relative to principal stresses. Eq. (7) is evaluated by integration of those portions of the unitsphere where orn is tensile. Weibull assumed that B was zero for orientations for which oa, was compressive. For the uniaxial stress case, the two-parameter form is n (cn) (8) (3nlcJno) 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. According to the Barnett-Freudenthal approximation (Batdorf, 1977), the principal stresses are assumed to act independently. This assumption leads to the following equation for the risk of rupture in 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. For the uniform-pressure-on-disc specimen, the stress state is biaxial tension, with a,= 0r, U2 =ot, and C3= 0. ao, is related to cr,, through the equation ao-" = [(2m + 1)/2 'j]u "(m(10) o,,
=
=
=
-
=
Results. The mean bi-axial flexure strength values of zinc phosphate cement as a function of six different test conditions are listed
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FAILURE STRESS OF BRITTLE DENTAL MATERIALS
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TABLE 4
COMPARATIVE DATA FOR THE THREE TEST METHODS
Specimen Bi-axial flexure test ZP* BP OP RC Four-point flexure test ZP BP OP RC Diametral tensile test ZP BP OP RC *Same data as in Table 3
Ratio of Strength to Bi-axial Flex-
No. of Tests
Meant Strength (MPa)
Standard Deviation (MPa)
Coefficient of Variation
10 10 10 10
18.1 52.4 75.6 103.9
1.3 5.6 8.6 15.6
0.073 0.106 0.114 0.150
1 1 1 1
10 10 10 10
6.8 48.4 52.4 98.4
0.6 5.8 10.2 11.4
0.088 0.125 0.195 0.116
0.375 0.923 0.693 0.947
4.5 22.4 23.8 24.6
0.5 8.7 6.0 5.8
0.088 0.391 0.253 0.234
0.249 0.427 0.315 0.236
10 10 10 10 for group ZP.
ure Strength
tCommon vertical lines indicate no significant statistical difference at a = 0.05.
in Table 3. The mean strength for these conditions was not significantly different at the 95% confidence level when Duncan's multiple range test was used. Although zinc phosphate cement was mixed manually for each specimen, the coefficient of variation fell within a narrow acceptable range (0.073 to 0.155), indicating the adequacy of the experimental data for evaluation of the strength of the cement. The bi-axial flexure strength of zinc phosphate cement was insensitive to specimen size, diameter of the piston, use of a load-distributing film, and loading speed. Fractured specimens could be grouped according to a two-segment or three-segment fracture pattern. However, no relationship between the fracture mode and the strength was observed. Based on these results, the test conditions for specimen group ZP (Table 3) used for zinc phosphate cement were used as the standard test conditions for subsequent bi-axial testing, since the results under these conditions showed the smallest coefficient of variation, and since the mean bi-axial strength represented the mean of the six groups 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 largest mean bi-axial strength. The mean bi-axial flexure strength of opaque porcelain was significantly higher (p0.05) among the diametral tensile strengths of body porcelain, opaque porcelain, and resin composite. The diametral tensile test data revealed that zinc phosphate cement exhibited the lowest coefficient of variation of 0.088, while other materials exhibited relatively larger values of 0.234 to 0.391. Shown in Fig. 2 are Weibull plots of fracture stresses, In In [1/(1 -Ff)] vs. ln u, for the zinc phosphate cement tested according to the three methods. The data points were described by a straight line produced by least-squares fit of the fracture data by use of a computer, and the Weibull modulus, characteristic strength, and strength at a predicted failure level of 5% were also calculated by computer. These results are listed in Table 5. The Weibull moduli for both bi-axial and fourpoint flexure tests were larger than those for the diametral tensile test, except for zinc phosphate cement. The Weibull modulus of zinc phosphate cement exhibited the largest value for each test method. Shown in Table 6 are the failure probability parameters for the bi-axial flexure test of four brittle dental materials derived from the analytical solutions of Eqs. (11) to (20) by use of Weibull analysis results for the four-point flexure test in Table 5. Predicted plots (Bs and B,) for the Weibull analysis of the zinc phosphate cement are presented as dashed lines in Fig. 2. Experimental results for the bi-axial test of zinc phosphate cement and opaque porcelain showed much higher strength than that predicted by surface flaw analysis. The failure probability of both body porcelain and resin composite exhibited good agreement with the values predicted for the surface flaw
condition. Shown in Fig. 3 are SEM images of fracture surfaces produced by bi-axial and four-point flexure stress for zinc phosphate cement, body porcelain, opaque porcelain, and resin composite. The arrows indicate the most likely sites of crack initiation. The fracture surfaces for zinc phosphate cement specimens exhibited a porous structure, especially in the specimens that were prepared for the four-point flexure test, which had much larger pores than those for the bi-axial test. For the specimens tested by four-point flexure, the fracture origin appeared to be located around pores located at the corner of the
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TABLE 5 SUMMARY OF WEIBULL ANALYSIS
2 ZP
Specimen Bi-axial flexure test ZP BP OP RC Four-point flexure test ZP BP
Cc
o.a05
m
(MPa)
(MPa)
10.45 7.54 7.47 6.27
18.6 54.9 80.2 111.6
14.0 37.0 53.9 69.5
Diametral
Four-point
Biaxial
-
I
A CL -
8.95 8.85 4.37 6.88
7.1 50.6 57.1 102.9
5.1 36.2 29.0 66.8
ZP 11.68 BP 2.10 3.25 OP RC 3.39 m, Weibull modulus. or, characteristic strength.
4.7
3.7 6.9 10.5 11.0
OP RC Diametral tensile test
1
1-
-1
-2
25.9 26.2 26.3
cT0.05, strength predicted at the 5% level of failure. fracture surface. The fracture surface of opaque porcelain produced by four-point flexure also showed large pores, which were associated with the crack origin, while that for the biaxial flexure test showed a homogeneous structure having small pores. The fracture surfaces of body porcelain and resin composite for both flexure tests exhibited a similar structure that could be characterized as homogeneous and low in porosity.
Discussion. Effects of test conditions on bi-axialflexure strength. -Maximum tensile stresses produced by bi-axial flexure occur below the central loading area on the bottom surfaces of disc specimens. However, because of the typically high elastic modulus and hardness of brittle materials, any imperfect contact between the rigid loading tool and the test specimen can lead to a substantial deviation from radial symmetry in the stress field and, consequently, to errors in strength measurements. Wachtman et al. (1972) suggested that a layer of polyethylene between piston and test surface would assist in the obtaining of uniform loading at the end of the piston. Based on strain gauge measurements, Marshall (1980) found that the piston applies the load uniformly over its contact area when a film is used. However, in the present study, there were no significant differences (p>0.05) between the strengths of zinc phosphate cement with and without a film. It seems that the specimen surface of zinc phosphate cement, which is covered with a precipitate by reaction between the cement surface and water in the storage chamber, was flattened by the initial contact of the piston and, consequently, developed a uniform loading condition without an intermediate film. Furthermore, if the flat surface of the piston is sufficiently parallel to the test surface, one does not always need to place the film or cushion over the specimen 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 second power of its thickness, as derived in Eq. (1). Furthermore, the stress equation is valid only if the deflection does not exceed about one-half of the plate thickness (Wachtman et al., 1972). Bending is directly proportional to the sustained load and inversely proportional to Young's modulus, whereas it is only slightly dependent on Poisson's ratio. The estimated min-
-
-3 - _ 0.5
B; 1.0
1.5
Bv 2.0
In
r
2.5
3.0
3.5
(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 test methods, and dashed lines represent predictions (Bs, Bv) of bi-axial flexure strength from four-point-flexure-strength data.
imum specimen thickness was calculated according to ASTM Standard F394. Within the stress range (18-180 MPa) encountered in this study and over a Young's modulus range for brittle materials [13.7 GPa for zinc phosphate cement, 16.6 GPa for resin composite, and 69 GPa for porcelain (Craig, 1989)], the estimated minimum thickness was always less than that encountered in the test. Therefore, it is reasonable to assume that the use of this stress equation was valid in the present study. Dimensions of the piston and specimen are included as factors in the stress Eq. (1) for the bi-axial flexure test. The important assumption of this equation is that the specimen structure was homogeneous. However, brittle dental materials used in the present study were not considered as homogeneous materials. For example, zinc phosphate cement contains significant porosity, matrix, and unreacted cement powders, as shown in Fig. 3. However, as shown in Table 3, the bi-axial strengths of the specimens with different dimensions were not significantly different (p>0.05). Although the effects of geometry have been identified in various strength tests (Baratta, 1984; Ikeda et al., 1986; Lamon, 1988), it is concluded that the effect of geometry on bi-axial strength is negligible. Perhaps a more important factor in future refinement of these tests is the design of more uniform load distribution at load application and load-supporting regions to minimize the risk of localized failure at these locations. Some of the variance of measured strength values can be explained on the basis of a few two-segment fractures vs. the more common three-segment fractures. The two-segment fractures may indicate that some of these were initiated at the load-application or loadsupport regions. The strengths of brittle materials generally increase with increasing loading rate. The dependence of strength stressing rate, caused by subcritical crack growth, has been described by Evans (1974) as follows: on
orf
=
where af is fracture stress at
stress rate, C is constant, and
C 6rl/(l + n) a n
(21)
given stressing rate, a is the is a crack-propagation param-
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FAILURE STRESS OF BRITTLE DENTAL MATERIALS
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TABLE 6
PREDICTED FAILURE PROBABILITY PARAMETERS FOR BI-AXIAL FLEXURE STRENGTH BASED ON FOUR-POINT FLEXURE DATA Surface Flaw Volume Flaw In ('rr a2L9)/m L ln [(QT a2tl2)L,]/m ln oC Code L, cv - 0.288 0.002 2.371 0.015 0.020 2.063 ZP 2.083 4.274 0.002 - 0.258 0.045 3.971 BP 4.016 0.017 0.161 0.026 3.984 0.005 -0.221 4.366 OP 4.145 5.062 0.020 4.691 0.003 - 0.304 0.067 RC 4.758
eter. These n values vary appreciably from one material to another (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 of 0.64 and 6.4 MPa/s, respectively. These two loading rates are often used for mechanical testing. It seems that zinc phosphate cement has a small value of n, which implies that it is susceptible 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 test represents a reliable method for determinations of the strength of brittle dental materials, since it is relatively insensitive to test conditions. However, it should be noted that the four-point flexure specimens were fairly large in this study, and some of these 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 alumina was higher than its four-point flexure strength and lower than its three-point flexure strength, and they compared bi-axial strength with predictions based on four-point flexure data. They found that Weibull statistics provided a good description of the size effects on data from the two uni-axial tests, but underestimated the effect of stress bi-axiality. On the other hand, Pletka and Wiederhorn (1978) reported that the four-point flexure strength of magnesium aluminosilicate glass in water was higher than its bi-axial flexure strength in water. Giovan and Sines (1979) showed that the bi-axial flexure strength of dense alumina was 8.5 and 8.1% lower than the uni-axial strength for ground and lapped surfaces, respectively. Pletka and Wiederhorn (1982) showed a consistent relationship between biaxial and four-point flexure strength data for five types of ceramics over a range of stressing rates. In contrast, the present study showed that there was no consistent relationship between uni-axial and bi-axial flexure strength data. The first reason for this inconsistency is the surface condition of the specimens. The surfaces of the specimens contain many 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. The specimens in the present study were polished through 0.3-pLm alumina powder for each porcelain, and zinc phosphate cement and resin composite specimens were flattened by glass plates
during setting. The second reason for the discrepancy is the specimen size effect. The effect of specimen size on four-point flexure strength has been reported by many investigators (Berenbaum and Brodie, 1959; Baratta, 1984; Ikeda et al., 1986; Lamon, 1988). Berenbaum and Brodie (1959) showed that the four-point flexure strength of pure plaster of Paris increased with a decrease in specimen thickness. They suggested that the four-point flexure strength of porous, weak, and brittle materials such as plaster is strongly influenced by specimen size. The surface layer of these chemically setting materials is tougher than their
internal structure, since the surface tends to form a dense structure having few pores during setting under pressure with the mold, as shown in Fig. 3. Furthermore, the properties of mixing materials such as plaster and cement are significantly affected by the total volume of mixing. For preparation of zinc phosphate cement specimens for the four-point flexure test, the mixture of 6 g of powder and 2 mL of liquid was used, while a standard amount of powder and liquid (1.5 g and 0.5 mL) was used for preparation of specimens for the bi-axial flexure test. Larger pores were formed and remained in the specimens for the four-point flexure test, compared with the bi-axial flexure test specimens, because of insufficient mixing and pressure. Thus, the bi-axial testing of zinc phosphate cement yielded much higher strength values than those predicted for both surface-flaw failure and volume-flaw failure, possibly because of differences in the homogeneity of specimens. Opaque porcelain specimens were also relatively inhomogeneous. Compared with body porcelain, opaque porcelain is relatively difficult to condense for large specimen volumes such as that used for the four-point flexure test, since opaque porcelain contains a higher fraction of opacifiers such as zirconium or titanium oxide, which reduce the fluidity of the slurry. The results for body porcelain and resin composite were comparable with those determined from the four-point flexure test, since test specimens for both materials showed a similar structure for both bi-axial and four-point flexure test specimens. These results suggest that this statistical approach demonstrates reasonable agreement between the bi-axial and four-point flexure strengths of brittle specimens with similar flaw characteristics. Comparison with diametral tensile strength.-For the diametral tensile test, it is difficult for ideal loading to be produced along a line when cylindrical specimens are used. A proper load distribution is generally accomplished by placement of a narrow pad of suitable materials between the specimen and the loading platens. For example, in the diametral tensile test procedure for ADA Specification No. 27 for direct filling resins (Council on Dental Materials and Devices, 1977), a thin piece of paper (approximately 0.5 mm thick) wet with water must be inserted between the platens of the testing machine along each side of the specimen. In ADA Specification No. 1 for amalgam, the specimen should be padded with two thicknesses of 0.038-mm aluminum foil on each side. However, it should be noted that the apparent strength changes with the type of padding material and its thickness, because the uniformity of the 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 cement exhibited a small coefficient of variation and a large Weibull modulus, as shown in Tables 4 and 5, whereas other materials exhibited large coefficients of variation in diametral tensile strength and small Weibull moduli, compared with those determined from both flexure tests. Although an advantage of the diametral tensile test is that the maximum tensile stress was not restricted to the surface, the surface effect on the fracture value is large (Rudnick et al., 1963). However, it is difficult to 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. Downloaded from jdr.sagepub.com at UNIVERSITY OF WINDSOR on November 13, 2014 For personal use only. No other uses without permission.
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FAILURE STRESS OF BRITTLE DENTAL MA TERIALS
that the surfaces of zinc phosphate cement specimens were flattened by the initial contact of test platens, thereby yielding a more uniform stress distribution within the specimens. Furthermore, it is well-known that the effect of specimen size on diametral strength is large. Williams and Smith (1971) showed that the diametral tensile strength for a zinc phosphate cement increased with increasing diameter. Thickness-to-diameter ratios used in the present study were 0.51 + 0.01 for zinc phosphate cement, 0.46 + 0.06 for body porcelain, 0.42 ± 0.04 for opaque porcelain, and 0.52 ± 0.01 for resin composite. The strength ratio of body porcelain and opaque porcelain had a relatively large variance due to a large variance in thickness, since these specimens were prepared by being polished. Therefore, it seems that the relatively large variance of diametral strength for both porcelains is attributable to the inhomogeneity of the curved surfaces and a large variance of thickness-todiameter ratio. Although resin composite had a small variance in thickness-to-diameter ratio, it seems that the curved surface was sufficiently inhomogeneous to decrease the variance in strength values. The mean diametral tensile strength of brittle dental materials was significantly lower than the bi-axial and four-point flexure strength values, as shown in Table 4. Flexure stress was enhanced because of the surface compression effect. The fracture strength of brittle materials can best be explained by postulating the presence of flaws distributed randomly throughout the volume. Lamon (1988) showed that tensile and bending tests provided different statistical parameters, suggesting that different populations of flaws controlled the failure in both cases. The flexure test specimens, in which the entire volume is stressed fairly uniformly, should exhibit higher mean strengths than tensile specimens (Rudnick et al., 1963). Thus, specimens with high porosity levels, such as cement, demonstrate low diametral strength values, compared with their bi-axial flexure strength values. For resin composite specimens, a low-degree conversion may have occurred within the deep interior region and along the curved surface, since light irradiation was controlled from the flat ends of the cylindrical specimens. It is well-recognized that the fracture strength of brittle materials depends upon several structural parameters, such as inclusions of voids and cracks (Evans, 1982), flaw location on the surface and within the volume (Lamon and Evans, 1983), the dimensions of specimens (Lamon and Evans, 1983), stress gradients (Ikeda et al., 1986), and the stress state (Lamon and Evans, 1983). The effects of geometry on strength can strongly influence the results of these strength tests, and it is difficult for the "true" tensile strength to be determined. In summary, the bi-axial test is simpler to perform and provides a better simulation of clinically-relevant sample size than that used for other strength tests, since specimen size and the preparation procedures are more similar to clinical conditions for the bi-axial test.
Acknowledgments. The authors gratefully acknowledge Professor Jiro Hasegawa, Aichi-Gakuin University, for his helpful suggestions and Mr. Robert B. Lee, University of Florida, for his technical support in conducting the mechanical tests. REFERENCES ANUSAVICE, K.J. and HOJJATIE, B. (1987): Stress Distribution in Metalceramic Crowns with a Facial Porcelain Margin, JDent Res 66:1493-1498. BARATTA, F.I. (1984): Requirements for Flexure Testing of Brittle Materials. In: Methods for Assessing the Structural Reliability of Brittle Materials, ASTM STP 844, S.W. Freiman and C.M. Hudson, Eds., Philadelphia: American Society for Testing and Materials, pp. 194-222.
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