Dent Mater 8:100-104, March, 1992
Effect of loading conditions on bi-axial flexure strength of dental cements S. Ban', J. Hasegawa 1, K.J. Anusavice 2 Department ofDental Materials Science,SchoolofDentistry, Aichi-Gakuin University,Nagoya, Japan 2Departmentof Dental Biomaterials, CollegeofDentistry, UniversityofFlorida, Gainesville,FL, USA
Abstract. A ball-on-three-ball bi-axial flexure test was used to evaluate the flexure strengths of five kinds of dental cements (zinc phosphate, polycarboxylate, glass ionomer, silicate, and zinc oxide-eugenol cements). The bi-axial flexure strengths of zinc phosphate cement discs of variable dimensions were also measured to determine the size effects of test specimens. Finite element stress analyses were performed on disc specimens which were subjected to both a concentrated load and a uniformly distributed load. The observed values were compared with the theoretical value calculated by the finite element method. A ballon-three-ball bi-axial flexure test could not be used for thicker specimens, since the fracture origin of these specimens is more likely to occur on the loaded side. The bi-axial flexure test has been used frequently to determine the fracture characteristics of brittle materials. The measurement of the strengths of brittle materials under bi-axial rather than uni-axial flexure conditions 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 (Wachtman et al., 1972). Morena et al. (1986) used a piston-on-three-ball bi-axial flexure measurement for dental porcelains. Ishizaki and Kameda (1988) used a ball-on-ring bi-axial flexure measurement for light-cured resin composites. The results of pistonon-three-ball bi-axial flexure measurements for zinc phosphate cement, porcelain, and resin composite were previously reported (Ban and Anusavice, 1990). The bi-axial flexure strength was determined by a disc specimen supported on three metal spheres positioned at equal distances from each other and from the center of the disc. The load was applied to the center of the opposite surface by a flat piston. Compared with the four-point flexure test, the results demonstrated that the bi-axial flexure test was suitable for evaluation of the flexure strength of brittle dental materials which are used for clinical restorations having a small volume and dimensions. However, smaller and thinner discs are required when specimens of luting cements are tested. Shetty et al. (1980) evaluated three bi-axial flexure test designs: ball-on-ring, piston-on-three-ball, and ring-on-ring. They found that only the ball-on-ring loading was satisfactory because uncertainties exist about fracture stresses in the other two cases. Their support ring was made with 12 balls. Kirstein and Woolley (1967) demonstrated that stresses in a thin circular aluminum alloy plate were independent of angular orientation and the number of supports. Therefore, a
100 Ban et aL/Bi-axial flexure strength of dental cements
simpler ball-on-three-ball system was used in this study. The improved bi-axial flexure test was applied to measure the flexure strength of five types of dental cements (zinc phosphate, polycarboxylate, glass ionomer, silicate, and zinc oxide-eugenol). The strength of zinc phosphate cement as a function of disc size was also measured. Furthermore, finite element stress analyses were performed on these disc specimens which were subjected to both concentrated and uniformly distributed loading conditions. This investigation was undertaken to study effects of loading conditions on the bi-axial flexure strengths of dental cements. MATERIALS AND METHODS Disc specimens were prepared from zinc phosphate, polycarboxylate, glass ionomer, silicate, and zinc oxideeugenol cements (Table 1). Specimens were prepared by use of conventional techniques according to the solubility test of ADA Specification No. 8 for zinc phosphate cement. A volume of cement of standard consistency sufficient to fill a given ring mold was placed on a flat glass plate. One of five sizes of rings [inner diameter (mm)/thickness (mm) ratios of $1 = 8/0.2, S2= 8/0.5, $3 = 14/1.2, S4= 16/2.4, and $5 = 20/4.4] was placed in the soft cement and another glass plate was used to press the cement into a disk. Three minutes after the mix was started, the glass plates and cements were placed in a humidor at 37°C for one hour. After removal from the humidor, the specimens were separated from the glass and stored in water at 37°C for 24 h. The final dimensions of the specimens tested are listed in Table 2. A Ball-on-Three-Ball Bi-axial Flexure Test. Specimens were supported on three steel spheres (1 mm in diameter) equally spaced along either a small support circle diameter (SC) of 6.8 mm or a large support circle diameter (LC) of 11.4 ram, shown schematically as 2a in Fig. 1. Ten specimens of each condition were center-loaded with a steel ball (1 mm in diameter) until fracture occurred. For a zinc phosphate cement, eight combinations of specimen size and the support circle were used. Disc $1 and $2 were tested with a small support circle (SC), while $3, $4, and $5 discs were tested for both SC and LC support conditions. For other cements, $3 specimens were tested for the LC support condition. Tests were carried out in water at room temperature. A cross-head loading rate of 0.5 mm/min was applied by a universal testing machine (Instron Universal Testing Machine 1125, Instron Corp., Canton, MA, USA). Failure Stress Analyses. The failure stress, c, at the center of the lower surface was calculated by use of equations developed by Marshall (1980) and Shetty et al. (1980). The
TABLE 1: MATERIALS*USED
Cement Name
Brand Name
Code
Lot No. (Powder)
Lot No. (Liquid)
P/L ratio
Zinc Phosphate
Elite Cement 100
ZP
140761
180681
1.5 g/0.50 mL
Glass Ionomer
Fuji Ionomer type I
GI
270682
290681
1.8 g/1.00 g
Polycarboxylate
Liv Cenera
PC
010461
070461
2.0 g/l.00 g
Silicate
Lumicolor Cement
SI
050381
030461
1.2 g/0.40 mL
Zinc oxide-eugenol
Eugenol Cement
EG
220861
070761
1.0 g/0.22 mL
*G-C Dental IndustrialCorp., Tokyo,Japan failure stress, ~, can be expressed as:
AP
(1)
t2
(Ritter, 1986). One commonly used statistic for describing this distribution is the Weibull distribution with the equation:
and
[ Iol I
A=(4n)3 2(l+v)ln
1
+(l-v)
/ (l+v)
(2)
where P is the applied load at failure, v is Poisson's ratio, a is the radius of the support circle (SC radius = 3.4 mm or LC radius = 5.7 mm), b is the radius of disc specimen (Table 2), t is the thickness of the disc specimen (Table 2), and r ois the radius of the ball used on the loading surface (0.5 mm). Instead of a flat piston, as was used in a previous study (Ban and Anusavice, 1990), a steel ball bonded to the piston was used in this study. For small r o values such as that used in this study: ro=~(l'6r~+t2) -0"675t (3) where r0* is an equivalent radius of contact between the loading ball and the disc specimen, where the loading can be considered to be uniform (Shetty et al., 1980). In this study, the strength values were calculated based on a Poisson's ratio value of 0.35 for each cement type (Craig, 1989). The failure strengths of brittle materials are statistically distributed as a function of the homogeneity of the material
Load Piston ~ 2 r o
Specimen
Pf = 1 - exp -
where P f is the fracture probability defined by the relation
Pf-- i / (N+ 1), where i is the rank in strength, N denotes the total number of specimens in the sample, m is the shape parameter (called the Weibull modulus) and ~¢ is the scale parameter or characteristic strength. Prom the raw fracture data, the Weibull modulus and characteristic strength were calculated on a personal computer (Macintosh II, Apple Computer Inc., Cupertino, CA). A statistical approach, such as Weibull analysis, would require the testing of a great number of specimens, particularly for brittle materials (McCabe and Carrick, 1986) and for subsequent calculations of fatigue parameters used for lifetime prediction analyses. However, a smaller population of specimens (10) was considered appropriate for the present study since the purpose of the analyses in the present study was to compare the relative strengths of cements and the relative effects of specimen shape and size. Finite Element Analyses. Finite element stress analyses were performed on disc specimens that were subjected to both a concentrated load and a uniformly distributed load. A computer program (FPLANE) was used for two-dimensional elastic analysis as developed by Miyoshi (1985). A finite element mesh was generated to use higher-order elements, including six-node triangles and eight-node isoparametric quadrilaterals. A typical finite element layout used in this analysis is shown in Fig. 2. The finite element mesh is more refined in the region of loading to allow for a more accurate representation of the high-stress gradients at this site. The model was divided into 135 nodes and 40 elements, by use of a two-dimensional axisymmetric representation of the structure. The shape of the specimen in this model has been modifiedto process rotational symmetry. Suehisoparametric elements have achieved widespread popularity because of their versatility and efficiency, particularly where complicated geometries are to be modeled (Fenner, 1987). An
Support Ball Bearing
TABLE 2: DIMENSIONSOF SPECIMEN (mm)
Code Specimen Holder
Fig. 1. Schematic illustration of a ball-on-three-ball loading system.
(4)
Diameter (2b)
Thickness(t)
S1
8.0 - 8.1
0.19 - 0.35
S2
7.9 - 8.0
0.38 - 0.55
S3
13.2 -
13.7
0.99 - 1.29
S4
15.6 -
15.9
2.35 - 2.49
Ss
19.8 -
19.9
4.00 - 4.39
Dental Materials~March 1992
1111
Height
(mnn Specimen size S4 Support circle SC
4.0 Load P 3.0
D
Element LE o
2.0 . .
.
/
.,-. --=
1.0
-1-
J
_=
o==
j
-2-
m
2.0 Element SE
6.0
8.0
//~//
EG Radial Distance (mm) Fig. 2. Finite element model of the specimen size S4with the support circle SC. Element LE is the innermostelement, 0.2-mm radius and 0.l-ram thickness, on loaded (upper) side. Element SE is the innermost one on support (bottom) side.
elastic modulus of 13.7 GPa and Poisson's ratio of 0.35 for a zinc phosphate cement (Craig, 1989) were used for this calculation. The stress values were determined at the centroid of each element, and the principal stresses were determined along the radial plane. The observed mean load value which caused fracture of the specimens was used for each calculation for the eight combinations of specimen size and support circle diameter. So that the effects of loading conditions on the bi-axial flexure strength could be examined, stress analyses were performed on the center loading system with a concentrated point loading and a distributed loading across a 0.4 mm diameter region. RESULTS
A Ball-on-Three-Ball Bi-axial Flexure Test. Summarized in Table 3 are the mean bi-axial flexure strengths, standard deviations, and coefficients of variation for eight sizes of zinc phosphate cement specimens and those for the four other cements. Weibull moduli and characteristic strengths are also given for these materials. The bi-axial flexure strength of the S~ discs for zinc phosphate cement and the support circle SC was significantly lower than that of the other specimens from others (p < 0.05). Furthermore, the Weibull
-3
PC
ZP
GI Sl
0 In ~
(MPa)
Fig. 3. Weibull plots of bi-axial flexure strength data for five kinds of dental cements tested with specimen size S3at the support circle LC.
modulus was higher than that of the other materials. Shown in Fig. 3 are Weibull plots of the bi-axial flexure strength data for the five types of dental cements with specimen size S 3 at the support circle LC. The mean strength of EG cement was lower than that of any other cement. The mean strength of GI was not significantly different from SI (p > 0.05), and these were significantly greater than the values for the other three cements (p < 0.05). The Weibull modulus for PC cement showed the lowest value, which corresponds to the largest scatter of data. Shown in Fig. 4 are SEM micrographs for typical fracture surfaces of the five zinc phosphate cement discs tested on the SC support circle. The arrows indicate the most likely sites of crack initiation. Fracture surfaces for specimens S4and S 5 showed a depression created by the steel ball on the loaded surface. Furthermore, these surfaces showed a radial propagation of cracks from the depression. Fracture surfaces for all specimens exhibited a porous structure, especially for the larger S4 and S 0 specimens. It is difficult for their influence on fractures to be assessed. However, cracks developed through these pores were observed on the fracture surface. Therefore, the fracture origin was judged to have occurred at
TABLE 3: RESULTSOF BI-AXlAL FLEXURESTRENGTHANALYSES
Cement ZP
Size
S~ S2 S3 S4 Ss S3 S4 Ss
Support Circle Diam (mm) 6.8 6.8 6.8 6.8 6.8 11.4 11.4 11.4
Mean Strength (MPa) 17.7 15.1 18.3 16.5 12.4 15.9 15.2 15.0
S.D. (MPa) 4.4 2.4 3.8 2.8 1.5 3.2 3.6 3.0
Coefficient of Variation (%) 24.9 15.9 20.7 17.2 12.1 20.1 23.4 19.9
Weibull Modulus 3.5 4.3 4.2 4.4 7.1 4.3 3.2 4.5
Characteristic Strength (Mea) 19.5 15.9 19.8 17.4 13.1 17.2 15.6 16.5
G!
S3
11.4
23.9
2.6
10.7
7.7
24.9
PC
S3
11.4
10.5
4.0
38.4
2.1
11.5
SI
S3
11.4
24.4
2.6
10.8
9.6
25.7
EG
S3
11.4
1.3
0.2
16.3
4.5
1.4
102 Ban et al./Bi-axial flexure strength of dental cements
I
the depression site (upper surface). On the other hand, fracture surfaces for $1, $2, and S 3revealed no depression, and the fracture origin was judged to be on the free, unloaded surface. Finite Element Analyses. Shown in Figs. 5 and 6 are calculated principal stresses of element LE on the loaded side and SE on the support side for the disc specimens of zinc phosphate cement subjected to concentrated and distributed loading conditions for the support circles SC and LC. Under the concentrated loading condition, the calculated stresses of element LE on the loaded sides of the S, and S 2 discs were compressive in nature, while the $3, $4, and S5 discs were under tension, as shown in Fig. 5. All specimens subjected to a distributed loading condition sustained compressive stresses. These stresses increased with increasing specimen thickness. The stresses for the support circle SC were slightly larger than those for the LC support condition. Element SE on the support sides of the disc specimens under both loading conditions sustained tensile stresses, and the tensile stresses of all specimens were similar in magnitude under both loading conditions, as shown in Fig. 6. Under the concentrated loading condition, the tensile stresses of element LE on the loaded sides of specimens S 4 and S 5were larger than those of element SE on the support sides of these specimens. Therefore, it appears that the fracture origin for discs S4 and S~ is located around the depression on the loaded side because of the larger tensile stresses on the loaded side compared with those on the support side.
I
1 mm
Fig. 4. SEM micrographs for fracture surfaces of five disc sizes for the zinc phosphate cements tested on a support circle SC.
Element SiS2
S3 I
400
LE
Element
$I 4
S, 5
Sll
93 i
20
Tensile
..o
300'
CONCENTRATED
/
LC
S 5 i
~
15"
100 13. v
[1 ...................................................................................
SE
$4 i
CONCENTRATED LOADING (SC)
sc
200
~-
S, 2
DISTRIBUTED (LC}
LOADING (LC)
10-
== -100 DISTRIBUTED LOADING (SC)
DISTRIBUTED'~
-200 300
400
LC
5"
Compressive , 1
, 2
, 3
Specimen thickness
, 4
(mm)
Fig. 5. Calculated principal stresses of element LE, the innermost element on the loaded side, for disc specimens of zinc phosphate cement subjected to concentrated and distributed loading conditions.
i
,
i
=
1
2
3
4
Specimen thickness
(mm)
Fig.& Calculated principal stresses of element SE, the innermost element on the support side, for the disc specimens of zinc phosphate cement subjected to concentrated and distributed loading conditions.
Dental Materials~March 1992
103
DISCUSSION For a concentrated loading condition, as in loading by a 0.5 mm ball, the stress can be determined by use of an equivalent radius r0*which can be calculated by Eq. 3. Marshall (1980) applied this equation for the flat piston-on-three-ball test design and found that, even with a relatively flat test piece such as float glass, imperfect contact between rigid loading rings and the test surface can lead to significant errors in the strength evaluated from bending equations. This error increases for more severely warped test pieces. Various methods have been described for obtaining uniform loading. According to ASTM standard F394-78 (reapproved in 1984) for determination of the bi-axial flexure strength of ceramic substrates, a pad of non-rigid material (polyethylene sheet) should be used between the piston and the upper surface of the specimen to ensure uniform loading over the surfaces of the discs. However, in a previous study (Ban and Anusavice, 1990), no significant differences (p > 0.05) were found between the strengths of zinc phosphate cement with or without a thin plastic film. Furthermore, it is difficult to apply a thin film between a small piston and a small disc specimen when testedinwater. Thus, a ball-on-three-ball loading system was selected as a more convenient test design for the present study. Shetty et al. (1980) reported that for a concentrated loading condition, as in loading by a ball, the stress can be calculated by use of an equivalent radius ro* of the loading ball in Eq. 3 and this formula may be used for values of r o < 0.5t. In the present study, r o was 0.5 mm. Thus, the disc thickness must be greater than 1 mm. This means that only disc sizes $3, $4, and $5 can be used for this case. Basically, the equation used for stress calculation is based on the condition that the maximum tensile stress at failure develops at the center of the support surface. However, S4 and S 5 discs exhibited a fracture origin on the loaded side. The results of simulation by finite element stress analysis indicate that the tensile stresses of element LE on the loaded sides of discs $4 and $5 under the concentrated loading condition were larger than those of element SE on the support sides of these specimens, as shown in Figs. 5 and 6. Therefore, it is concluded that the stresses which developed on the support side were not significantly affected by the loading condition, while the stresses on the loaded side were dependent on the loading condition. The stress distributions within the thick specimens were strongly affected by the loading condition since these specimens are easily fractured when a Hertzian-type load distribution develops within brittle, weak materials such as dental cements. Therefore, we conclude that discs of size $3 are the only size suitable for these stress calculations. Furthermore, this specimen size for dental cements is nearly ideal since a uniform mix of cement for larger specimens is difficult to achieve and smaller-size specimens are difficult to remove from the ring used for making disc specimens. Thus, we used size $3 disc specimens for evaluation of the bi-axial flexure strengths of the other dental cements. The disc thickness is one of the most important factors in the determination of bi-axial flexure strength. The ASTM standard describes the test fLxture for supporting and loading specimens. Furthermore, a minimum thickness value is required to limit the central deflection of the specimen to onehalf the specimen thickness at fracture. Kao et al. (1971) reported a numerical stress analysis method for chemically
104 Ban et aL/Bi-axial flexure strength of dental cements
strengthened glass plate tested with ring-on-ring loading by considering large non-linear deflections and radial surface stresses. Their results indicate that stress magnifications at the loading point are significant when the plate deflection exceeds one-half the specimen thickness. The minimum specimen thickness depends on test-loading rate, Young's modulus, and Poisson's ratio. The present study has shown that a maximum thickness limit must also be established for smaller test specimens, such as those typically used to evaluate dental materials.
ACKNOWLEDGMENT This investigation was supported in part by NIDR Grants DE 06672 and DE 09307. Received January 8, 1991/Accepted August 26, 1991 Address correspondence and reprint requests to: K.J. Anusavice Department of Dental Biomaterials College of Dentistry University of Florida Gainesville, Florida, 32610 USA
REFERENCES Ban S, Anusavice KJ (1990). Influence of test method on failure stress of brittle dental materials. J Dent Res 69:1791-1799. Craig RG (1989). Restorative dental materials. 8th ed. St. Louis: C.V. Mosby Co., 81. Fenner DN (1987). Engineering stress analysis. West Sussex, England: Ellis Horwood Limited, 132-164. Ishizaki H, Kameda Y (1988). The bending strength of visible light cured posterior composite resins using one point concentrated load test. Jpn J Conserv Dent 31:107121 (in Japanese). Kao R, Perrone N, Capps W(1971). Large-deflection solution of the coaxial-ring-circular-glass-plate flexure problem. J A m Ceram Soc 54:566-571. KirsteinAF, Woolley RM (1967). Symmetrical bending ofthin circular elastic plates of equally spaced point supports. J Res Natl Bur Stds 71C:1-10. Marshall DB (1980). An improved biaxial flexure test for ceramics. A m Ceram Soc Bull 59:551-553. McCabe JF, Carrick TE (1986). A statistical approach to the mechanical testing of dental materials. Dent Mater 2:139142. Miyoshi T (1985). FEM/BEM programing by MS-FORTRAN. Tokyo: Science, 43-95 (in Japanese). Morena R, Beaudreau GM, Lockwood PE, Evans AL, Fairhurst CW (1986). Fatigue of dental ceramics in a simulated oral environment. J D e n t R e s 65:993-997. Ritter JE (1986). Fracture:reliability criteria for brittle materials. In: BeverMB, editor. Encyclopedia of materials science and engineering. Oxford: Pergamon Press, 1852-1858. Shetty DK, Rosenfield AR, McGuire P, Bansal GK, Duckworth WH(1980). Biaxial flexure tests for ceramics. Ceramic Bull 59:1193-1197. Wachtman JB Jr, Capps W, Mandel J (1972). Biaxial flexure tests of ceramic substrates. J Mater 7:188-194.
Effect of loading conditions on bi-axial flexure strength of dental cements S. Ban', J. Hasegawa 1, K.J. Anusavice 2 Department ofDental Materials Science,SchoolofDentistry, Aichi-Gakuin University,Nagoya, Japan 2Departmentof Dental Biomaterials, CollegeofDentistry, UniversityofFlorida, Gainesville,FL, USA
Abstract. A ball-on-three-ball bi-axial flexure test was used to evaluate the flexure strengths of five kinds of dental cements (zinc phosphate, polycarboxylate, glass ionomer, silicate, and zinc oxide-eugenol cements). The bi-axial flexure strengths of zinc phosphate cement discs of variable dimensions were also measured to determine the size effects of test specimens. Finite element stress analyses were performed on disc specimens which were subjected to both a concentrated load and a uniformly distributed load. The observed values were compared with the theoretical value calculated by the finite element method. A ballon-three-ball bi-axial flexure test could not be used for thicker specimens, since the fracture origin of these specimens is more likely to occur on the loaded side. The bi-axial flexure test has been used frequently to determine the fracture characteristics of brittle materials. The measurement of the strengths of brittle materials under bi-axial rather than uni-axial flexure conditions 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 (Wachtman et al., 1972). Morena et al. (1986) used a piston-on-three-ball bi-axial flexure measurement for dental porcelains. Ishizaki and Kameda (1988) used a ball-on-ring bi-axial flexure measurement for light-cured resin composites. The results of pistonon-three-ball bi-axial flexure measurements for zinc phosphate cement, porcelain, and resin composite were previously reported (Ban and Anusavice, 1990). The bi-axial flexure strength was determined by a disc specimen supported on three metal spheres positioned at equal distances from each other and from the center of the disc. The load was applied to the center of the opposite surface by a flat piston. Compared with the four-point flexure test, the results demonstrated that the bi-axial flexure test was suitable for evaluation of the flexure strength of brittle dental materials which are used for clinical restorations having a small volume and dimensions. However, smaller and thinner discs are required when specimens of luting cements are tested. Shetty et al. (1980) evaluated three bi-axial flexure test designs: ball-on-ring, piston-on-three-ball, and ring-on-ring. They found that only the ball-on-ring loading was satisfactory because uncertainties exist about fracture stresses in the other two cases. Their support ring was made with 12 balls. Kirstein and Woolley (1967) demonstrated that stresses in a thin circular aluminum alloy plate were independent of angular orientation and the number of supports. Therefore, a
100 Ban et aL/Bi-axial flexure strength of dental cements
simpler ball-on-three-ball system was used in this study. The improved bi-axial flexure test was applied to measure the flexure strength of five types of dental cements (zinc phosphate, polycarboxylate, glass ionomer, silicate, and zinc oxide-eugenol). The strength of zinc phosphate cement as a function of disc size was also measured. Furthermore, finite element stress analyses were performed on these disc specimens which were subjected to both concentrated and uniformly distributed loading conditions. This investigation was undertaken to study effects of loading conditions on the bi-axial flexure strengths of dental cements. MATERIALS AND METHODS Disc specimens were prepared from zinc phosphate, polycarboxylate, glass ionomer, silicate, and zinc oxideeugenol cements (Table 1). Specimens were prepared by use of conventional techniques according to the solubility test of ADA Specification No. 8 for zinc phosphate cement. A volume of cement of standard consistency sufficient to fill a given ring mold was placed on a flat glass plate. One of five sizes of rings [inner diameter (mm)/thickness (mm) ratios of $1 = 8/0.2, S2= 8/0.5, $3 = 14/1.2, S4= 16/2.4, and $5 = 20/4.4] was placed in the soft cement and another glass plate was used to press the cement into a disk. Three minutes after the mix was started, the glass plates and cements were placed in a humidor at 37°C for one hour. After removal from the humidor, the specimens were separated from the glass and stored in water at 37°C for 24 h. The final dimensions of the specimens tested are listed in Table 2. A Ball-on-Three-Ball Bi-axial Flexure Test. Specimens were supported on three steel spheres (1 mm in diameter) equally spaced along either a small support circle diameter (SC) of 6.8 mm or a large support circle diameter (LC) of 11.4 ram, shown schematically as 2a in Fig. 1. Ten specimens of each condition were center-loaded with a steel ball (1 mm in diameter) until fracture occurred. For a zinc phosphate cement, eight combinations of specimen size and the support circle were used. Disc $1 and $2 were tested with a small support circle (SC), while $3, $4, and $5 discs were tested for both SC and LC support conditions. For other cements, $3 specimens were tested for the LC support condition. Tests were carried out in water at room temperature. A cross-head loading rate of 0.5 mm/min was applied by a universal testing machine (Instron Universal Testing Machine 1125, Instron Corp., Canton, MA, USA). Failure Stress Analyses. The failure stress, c, at the center of the lower surface was calculated by use of equations developed by Marshall (1980) and Shetty et al. (1980). The
TABLE 1: MATERIALS*USED
Cement Name
Brand Name
Code
Lot No. (Powder)
Lot No. (Liquid)
P/L ratio
Zinc Phosphate
Elite Cement 100
ZP
140761
180681
1.5 g/0.50 mL
Glass Ionomer
Fuji Ionomer type I
GI
270682
290681
1.8 g/1.00 g
Polycarboxylate
Liv Cenera
PC
010461
070461
2.0 g/l.00 g
Silicate
Lumicolor Cement
SI
050381
030461
1.2 g/0.40 mL
Zinc oxide-eugenol
Eugenol Cement
EG
220861
070761
1.0 g/0.22 mL
*G-C Dental IndustrialCorp., Tokyo,Japan failure stress, ~, can be expressed as:
AP
(1)
t2
(Ritter, 1986). One commonly used statistic for describing this distribution is the Weibull distribution with the equation:
and
[ Iol I
A=(4n)3 2(l+v)ln
1
+(l-v)
/ (l+v)
(2)
where P is the applied load at failure, v is Poisson's ratio, a is the radius of the support circle (SC radius = 3.4 mm or LC radius = 5.7 mm), b is the radius of disc specimen (Table 2), t is the thickness of the disc specimen (Table 2), and r ois the radius of the ball used on the loading surface (0.5 mm). Instead of a flat piston, as was used in a previous study (Ban and Anusavice, 1990), a steel ball bonded to the piston was used in this study. For small r o values such as that used in this study: ro=~(l'6r~+t2) -0"675t (3) where r0* is an equivalent radius of contact between the loading ball and the disc specimen, where the loading can be considered to be uniform (Shetty et al., 1980). In this study, the strength values were calculated based on a Poisson's ratio value of 0.35 for each cement type (Craig, 1989). The failure strengths of brittle materials are statistically distributed as a function of the homogeneity of the material
Load Piston ~ 2 r o
Specimen
Pf = 1 - exp -
where P f is the fracture probability defined by the relation
Pf-- i / (N+ 1), where i is the rank in strength, N denotes the total number of specimens in the sample, m is the shape parameter (called the Weibull modulus) and ~¢ is the scale parameter or characteristic strength. Prom the raw fracture data, the Weibull modulus and characteristic strength were calculated on a personal computer (Macintosh II, Apple Computer Inc., Cupertino, CA). A statistical approach, such as Weibull analysis, would require the testing of a great number of specimens, particularly for brittle materials (McCabe and Carrick, 1986) and for subsequent calculations of fatigue parameters used for lifetime prediction analyses. However, a smaller population of specimens (10) was considered appropriate for the present study since the purpose of the analyses in the present study was to compare the relative strengths of cements and the relative effects of specimen shape and size. Finite Element Analyses. Finite element stress analyses were performed on disc specimens that were subjected to both a concentrated load and a uniformly distributed load. A computer program (FPLANE) was used for two-dimensional elastic analysis as developed by Miyoshi (1985). A finite element mesh was generated to use higher-order elements, including six-node triangles and eight-node isoparametric quadrilaterals. A typical finite element layout used in this analysis is shown in Fig. 2. The finite element mesh is more refined in the region of loading to allow for a more accurate representation of the high-stress gradients at this site. The model was divided into 135 nodes and 40 elements, by use of a two-dimensional axisymmetric representation of the structure. The shape of the specimen in this model has been modifiedto process rotational symmetry. Suehisoparametric elements have achieved widespread popularity because of their versatility and efficiency, particularly where complicated geometries are to be modeled (Fenner, 1987). An
Support Ball Bearing
TABLE 2: DIMENSIONSOF SPECIMEN (mm)
Code Specimen Holder
Fig. 1. Schematic illustration of a ball-on-three-ball loading system.
(4)
Diameter (2b)
Thickness(t)
S1
8.0 - 8.1
0.19 - 0.35
S2
7.9 - 8.0
0.38 - 0.55
S3
13.2 -
13.7
0.99 - 1.29
S4
15.6 -
15.9
2.35 - 2.49
Ss
19.8 -
19.9
4.00 - 4.39
Dental Materials~March 1992
1111
Height
(mnn Specimen size S4 Support circle SC
4.0 Load P 3.0
D
Element LE o
2.0 . .
.
/
.,-. --=
1.0
-1-
J
_=
o==
j
-2-
m
2.0 Element SE
6.0
8.0
//~//
EG Radial Distance (mm) Fig. 2. Finite element model of the specimen size S4with the support circle SC. Element LE is the innermostelement, 0.2-mm radius and 0.l-ram thickness, on loaded (upper) side. Element SE is the innermost one on support (bottom) side.
elastic modulus of 13.7 GPa and Poisson's ratio of 0.35 for a zinc phosphate cement (Craig, 1989) were used for this calculation. The stress values were determined at the centroid of each element, and the principal stresses were determined along the radial plane. The observed mean load value which caused fracture of the specimens was used for each calculation for the eight combinations of specimen size and support circle diameter. So that the effects of loading conditions on the bi-axial flexure strength could be examined, stress analyses were performed on the center loading system with a concentrated point loading and a distributed loading across a 0.4 mm diameter region. RESULTS
A Ball-on-Three-Ball Bi-axial Flexure Test. Summarized in Table 3 are the mean bi-axial flexure strengths, standard deviations, and coefficients of variation for eight sizes of zinc phosphate cement specimens and those for the four other cements. Weibull moduli and characteristic strengths are also given for these materials. The bi-axial flexure strength of the S~ discs for zinc phosphate cement and the support circle SC was significantly lower than that of the other specimens from others (p < 0.05). Furthermore, the Weibull
-3
PC
ZP
GI Sl
0 In ~
(MPa)
Fig. 3. Weibull plots of bi-axial flexure strength data for five kinds of dental cements tested with specimen size S3at the support circle LC.
modulus was higher than that of the other materials. Shown in Fig. 3 are Weibull plots of the bi-axial flexure strength data for the five types of dental cements with specimen size S 3 at the support circle LC. The mean strength of EG cement was lower than that of any other cement. The mean strength of GI was not significantly different from SI (p > 0.05), and these were significantly greater than the values for the other three cements (p < 0.05). The Weibull modulus for PC cement showed the lowest value, which corresponds to the largest scatter of data. Shown in Fig. 4 are SEM micrographs for typical fracture surfaces of the five zinc phosphate cement discs tested on the SC support circle. The arrows indicate the most likely sites of crack initiation. Fracture surfaces for specimens S4and S 5 showed a depression created by the steel ball on the loaded surface. Furthermore, these surfaces showed a radial propagation of cracks from the depression. Fracture surfaces for all specimens exhibited a porous structure, especially for the larger S4 and S 0 specimens. It is difficult for their influence on fractures to be assessed. However, cracks developed through these pores were observed on the fracture surface. Therefore, the fracture origin was judged to have occurred at
TABLE 3: RESULTSOF BI-AXlAL FLEXURESTRENGTHANALYSES
Cement ZP
Size
S~ S2 S3 S4 Ss S3 S4 Ss
Support Circle Diam (mm) 6.8 6.8 6.8 6.8 6.8 11.4 11.4 11.4
Mean Strength (MPa) 17.7 15.1 18.3 16.5 12.4 15.9 15.2 15.0
S.D. (MPa) 4.4 2.4 3.8 2.8 1.5 3.2 3.6 3.0
Coefficient of Variation (%) 24.9 15.9 20.7 17.2 12.1 20.1 23.4 19.9
Weibull Modulus 3.5 4.3 4.2 4.4 7.1 4.3 3.2 4.5
Characteristic Strength (Mea) 19.5 15.9 19.8 17.4 13.1 17.2 15.6 16.5
G!
S3
11.4
23.9
2.6
10.7
7.7
24.9
PC
S3
11.4
10.5
4.0
38.4
2.1
11.5
SI
S3
11.4
24.4
2.6
10.8
9.6
25.7
EG
S3
11.4
1.3
0.2
16.3
4.5
1.4
102 Ban et al./Bi-axial flexure strength of dental cements
I
the depression site (upper surface). On the other hand, fracture surfaces for $1, $2, and S 3revealed no depression, and the fracture origin was judged to be on the free, unloaded surface. Finite Element Analyses. Shown in Figs. 5 and 6 are calculated principal stresses of element LE on the loaded side and SE on the support side for the disc specimens of zinc phosphate cement subjected to concentrated and distributed loading conditions for the support circles SC and LC. Under the concentrated loading condition, the calculated stresses of element LE on the loaded sides of the S, and S 2 discs were compressive in nature, while the $3, $4, and S5 discs were under tension, as shown in Fig. 5. All specimens subjected to a distributed loading condition sustained compressive stresses. These stresses increased with increasing specimen thickness. The stresses for the support circle SC were slightly larger than those for the LC support condition. Element SE on the support sides of the disc specimens under both loading conditions sustained tensile stresses, and the tensile stresses of all specimens were similar in magnitude under both loading conditions, as shown in Fig. 6. Under the concentrated loading condition, the tensile stresses of element LE on the loaded sides of specimens S 4 and S 5were larger than those of element SE on the support sides of these specimens. Therefore, it appears that the fracture origin for discs S4 and S~ is located around the depression on the loaded side because of the larger tensile stresses on the loaded side compared with those on the support side.
I
1 mm
Fig. 4. SEM micrographs for fracture surfaces of five disc sizes for the zinc phosphate cements tested on a support circle SC.
Element SiS2
S3 I
400
LE
Element
$I 4
S, 5
Sll
93 i
20
Tensile
..o
300'
CONCENTRATED
/
LC
S 5 i
~
15"
100 13. v
[1 ...................................................................................
SE
$4 i
CONCENTRATED LOADING (SC)
sc
200
~-
S, 2
DISTRIBUTED (LC}
LOADING (LC)
10-
== -100 DISTRIBUTED LOADING (SC)
DISTRIBUTED'~
-200 300
400
LC
5"
Compressive , 1
, 2
, 3
Specimen thickness
, 4
(mm)
Fig. 5. Calculated principal stresses of element LE, the innermost element on the loaded side, for disc specimens of zinc phosphate cement subjected to concentrated and distributed loading conditions.
i
,
i
=
1
2
3
4
Specimen thickness
(mm)
Fig.& Calculated principal stresses of element SE, the innermost element on the support side, for the disc specimens of zinc phosphate cement subjected to concentrated and distributed loading conditions.
Dental Materials~March 1992
103
DISCUSSION For a concentrated loading condition, as in loading by a 0.5 mm ball, the stress can be determined by use of an equivalent radius r0*which can be calculated by Eq. 3. Marshall (1980) applied this equation for the flat piston-on-three-ball test design and found that, even with a relatively flat test piece such as float glass, imperfect contact between rigid loading rings and the test surface can lead to significant errors in the strength evaluated from bending equations. This error increases for more severely warped test pieces. Various methods have been described for obtaining uniform loading. According to ASTM standard F394-78 (reapproved in 1984) for determination of the bi-axial flexure strength of ceramic substrates, a pad of non-rigid material (polyethylene sheet) should be used between the piston and the upper surface of the specimen to ensure uniform loading over the surfaces of the discs. However, in a previous study (Ban and Anusavice, 1990), no significant differences (p > 0.05) were found between the strengths of zinc phosphate cement with or without a thin plastic film. Furthermore, it is difficult to apply a thin film between a small piston and a small disc specimen when testedinwater. Thus, a ball-on-three-ball loading system was selected as a more convenient test design for the present study. Shetty et al. (1980) reported that for a concentrated loading condition, as in loading by a ball, the stress can be calculated by use of an equivalent radius ro* of the loading ball in Eq. 3 and this formula may be used for values of r o < 0.5t. In the present study, r o was 0.5 mm. Thus, the disc thickness must be greater than 1 mm. This means that only disc sizes $3, $4, and $5 can be used for this case. Basically, the equation used for stress calculation is based on the condition that the maximum tensile stress at failure develops at the center of the support surface. However, S4 and S 5 discs exhibited a fracture origin on the loaded side. The results of simulation by finite element stress analysis indicate that the tensile stresses of element LE on the loaded sides of discs $4 and $5 under the concentrated loading condition were larger than those of element SE on the support sides of these specimens, as shown in Figs. 5 and 6. Therefore, it is concluded that the stresses which developed on the support side were not significantly affected by the loading condition, while the stresses on the loaded side were dependent on the loading condition. The stress distributions within the thick specimens were strongly affected by the loading condition since these specimens are easily fractured when a Hertzian-type load distribution develops within brittle, weak materials such as dental cements. Therefore, we conclude that discs of size $3 are the only size suitable for these stress calculations. Furthermore, this specimen size for dental cements is nearly ideal since a uniform mix of cement for larger specimens is difficult to achieve and smaller-size specimens are difficult to remove from the ring used for making disc specimens. Thus, we used size $3 disc specimens for evaluation of the bi-axial flexure strengths of the other dental cements. The disc thickness is one of the most important factors in the determination of bi-axial flexure strength. The ASTM standard describes the test fLxture for supporting and loading specimens. Furthermore, a minimum thickness value is required to limit the central deflection of the specimen to onehalf the specimen thickness at fracture. Kao et al. (1971) reported a numerical stress analysis method for chemically
104 Ban et aL/Bi-axial flexure strength of dental cements
strengthened glass plate tested with ring-on-ring loading by considering large non-linear deflections and radial surface stresses. Their results indicate that stress magnifications at the loading point are significant when the plate deflection exceeds one-half the specimen thickness. The minimum specimen thickness depends on test-loading rate, Young's modulus, and Poisson's ratio. The present study has shown that a maximum thickness limit must also be established for smaller test specimens, such as those typically used to evaluate dental materials.
ACKNOWLEDGMENT This investigation was supported in part by NIDR Grants DE 06672 and DE 09307. Received January 8, 1991/Accepted August 26, 1991 Address correspondence and reprint requests to: K.J. Anusavice Department of Dental Biomaterials College of Dentistry University of Florida Gainesville, Florida, 32610 USA
REFERENCES Ban S, Anusavice KJ (1990). Influence of test method on failure stress of brittle dental materials. J Dent Res 69:1791-1799. Craig RG (1989). Restorative dental materials. 8th ed. St. Louis: C.V. Mosby Co., 81. Fenner DN (1987). Engineering stress analysis. West Sussex, England: Ellis Horwood Limited, 132-164. Ishizaki H, Kameda Y (1988). The bending strength of visible light cured posterior composite resins using one point concentrated load test. Jpn J Conserv Dent 31:107121 (in Japanese). Kao R, Perrone N, Capps W(1971). Large-deflection solution of the coaxial-ring-circular-glass-plate flexure problem. J A m Ceram Soc 54:566-571. KirsteinAF, Woolley RM (1967). Symmetrical bending ofthin circular elastic plates of equally spaced point supports. J Res Natl Bur Stds 71C:1-10. Marshall DB (1980). An improved biaxial flexure test for ceramics. A m Ceram Soc Bull 59:551-553. McCabe JF, Carrick TE (1986). A statistical approach to the mechanical testing of dental materials. Dent Mater 2:139142. Miyoshi T (1985). FEM/BEM programing by MS-FORTRAN. Tokyo: Science, 43-95 (in Japanese). Morena R, Beaudreau GM, Lockwood PE, Evans AL, Fairhurst CW (1986). Fatigue of dental ceramics in a simulated oral environment. J D e n t R e s 65:993-997. Ritter JE (1986). Fracture:reliability criteria for brittle materials. In: BeverMB, editor. Encyclopedia of materials science and engineering. Oxford: Pergamon Press, 1852-1858. Shetty DK, Rosenfield AR, McGuire P, Bansal GK, Duckworth WH(1980). Biaxial flexure tests for ceramics. Ceramic Bull 59:1193-1197. Wachtman JB Jr, Capps W, Mandel J (1972). Biaxial flexure tests of ceramic substrates. J Mater 7:188-194.
