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The treatment of results for tensile bond strength testing
J. Dent.
1986;
14:
165-l
68
Printed in Great Britain
165
The treatment of results for tensile strength testing J. F. McCabe
and A. W. G. Walls
Dental
University
School,
J. Dent 1986;
14: 165-l
of Newcastle
bond
upon Tyne
68 (Received 23 January 1986;
accepted 14 April 1986)
ABSTRACT Data from tensile bond strength tests of polyalkenoate materials to dentine and of composite resins to acrylic veneers has been analysed using a Weibull distribution
function. This function relates failure probability to stress and generates a value of Weibull Modulus from which the dependability of the bond can be quantified, and enables failure probability at any level of stress to be predicted. In order to generate meaningful data from these tests, in excess of 20 and preferably 30 or more specimens should be tested.
INTRODUCTION Tensile bond strength tests are notorious for producing results having a wide variation (Beech, 1985). Variations may reflect the quality of the ‘bond’, but are also due to experimental variations involved in such tests, including improper alignment during bond rupture such that a degree of torsion or shear is introduced. The situation is complicated by the use of a plethora of different types of testing equipment. The traditional method for presenting data from these tests is by quoting the number of tests, mean bond strength and standard deviation. Hence, when comparing systems considerable emphasis is placed on the mean value. A better method for dealing with the results may be to use a function which relates probability of bond failure to tensile stress. This approach has been shown to offer a suitable means of evaluating fracture processes in other dental materials (McCabe and Carrick, 1986). Weibull described a suitable equation for carrying out this type of analysis (Weibull, 1951). The Weibull equation relates probability of failure, Pf, to stress (Tas follows : pf=’
-eexp_
Weilbull Modulus and has considerable practical importance. A low value of m indicates a wide scatter of results with a long ‘tail’ at low stress levels, while a high value of m indicates a close grouping of fracture stress values. The value of m can be interpreted as giving an indication of the dependability of the material. After carrying out a series of tests on a material to establish values of the coefficients u. and m, it is possible to use the Weibull equation to predict fracture probability at any level of stress or vice versa. The Weibull distribution has proved a useful method for evaluating the fracture behaviour of materials and components used in engineering, where experimentally observed variations in results are often attributed to the presence of faults, defects or porosities (Davies, 1973; Stanley et al., 1973; Creyke et al., 1982). It was decided to test the use of this distribution using two ‘adhesive’ systems used in dentistry where bond strength measurement and prediction are important. First, in assessing the adhesion of polyalkenoate cements to dentine and secondly in evaluating the bond strength of acrylic laminate veneers to composite resins.
(o-au)m ( %o)
Pfis the probability of failure at any level of stress up to a value u and uU, cc and m are constants. au is the lowest stress at which fracture is possible. For most applications it is assumed that a, = o, an assumption that has been found to be applicable previously (Stanley et al., 1973). a,, is a normalizing parameter and has no obvious practical significance. The constant, m, is termed the
MATERIALS
AND
METHODS
For measurements of bond strength of polyalkenoate cement to dentine an encapsulated material (Ketac-Fil; ESPE, Seefeld, FR Germany) was used. One series of tests was carried out in which nine extracted teeth (stored in neutral buffered formalin before use) were used to form dentine/cement bonds for evaluation. The method of tooth preparation, bond formation and tensile bond
166
J. Dent.
1986;
14:
No. 4
strength testing are given elsewhere (Walls, 1986). This article is mainly concerned with the method of statistical analysis of the results. Five batches of tensile bond strength specimens were produced from the nine teeth making a total of 45 results. Only 42 results were actually obtained due to factors such as exposure of the pulp on one tooth after grinding to remove a fresh dentine surface. Bond strengths were measured at 24 h, following storage in an oven at 37°C and 100 per cent relative humidity. In a second series of tests 20 specimens (2 batches of 10) were produced using teeth which had been stored in distilled water before use. The bonds were formed in the same way but the specimens were stored at 37°C and 100 per cent relative humidity for 1 week before testing. In both cases bond strengths were measured using an Instron Universal tester (model 1095) at a cross-head speed of 0.5 mm/min. Bond strengths of composite resin (Mastique Luting Composite, LD Caulk Co., Milford, DE, USA) to acrylic laminate veneers (Mastique Acrylic Veneers, LD Caulk Co.) were measured using a similar method (Walls et al. 1985). Thirty specimens (3 batches of 10) were prepared for testing, from which 26 results were obtained. Four results were discarded because rupture occured at low stress levels within the composite resin. The bond strengths were measured at 1 week using an Instron Universal tester at a cross-head speed of 0.5 mm/mm. RESULTS
AND DISCUSSION
The results of the three groups of tests are given in Tables I, H and ZZZand the plots of failure probability against stress are given in Fig. 1. Cumulative failure probability (P,) is calculated from the expression
where N is the total number of specimens under consideration in the sample group and n is the rank number of the specimen. The weakest specimen in a sample group of N specimens is given rank 1 and the strongest rank N. To illustrate how this is carried out: if 20 repeat tests are performed the weakest, with failure stress c,, is given a rank number of 1, while the strongest, with failure stress azO, is given a rank number of 20. In the sample of 20 specimens all (100%) had fractured at or below a stress of azO.When applying the data to a wider population the normal practice of using N -l- 1 instead of N for calculating probabilities is used (Weibull, 195 1). Hence, the probability of fracture at a stress azOin a large population is P = 20420 + l), or O-95 (95%). This approach, therefore, allows for the fact that greater strengths than those measured are possible. As N becomes greater the expressions Pf = nlN and Pf = nl N + 1 become insignificantly different, so the sample more truly represents the whole population. The experimental data was fitted to the Weibull
distribution using a linear least-squares analysis of 1n 1n (1 - Pf) against 1n a from which the coefficients m and a0 can be readily obtained. Calculation of the correlation coefficient for In 1n (1 - Pf) v. 1n c by regression analysis indicates how well the data fits the Weibull equation. Reference to Tables I, II andZZZshows the low values of Weibull Modulus obtained for these tensile bond strength tests (0.89 + 5.60 for cumulative groups). This indicates a wide variation in test results, a factor also reflected in the high standard deviation values obtained. The wide variation is not unexpected and is to a certain extent accepted for this type of test. The values of correlation coefficient shown in the tables indicated how well the experimental data fits the Weibull equation. In tests on other materials Weibull Modulus values of around 7-10 are normal for brittle ceramics, whereas much higher values of around 20-100 are possible for materials which undergo a level of plastic deformation before fracture. A previous study on dental plaster (McCabe and Carrick, 1986) yielded Weibull Modulus values of between 3.68 and 14.20 for nine groups of 12 specimens. Thus it is apparent that the results of tensile bond strength tests are very variable. For this type oftest it is probable that at least 20 and preferably 30 or more test points are necessary in order to make a reasonable assessment of material properties (McCabe and Carrick, 1986). The higher values of Weibull Modulus obtained for the composite resin/acrylic veneer bond strength tests are probably indicative of a more ductile mode of failure in the resin-based system than that seen in the polyalkenoate/ dentine system. This assumes that errors in alignment during testing are similar in the two tests. This assumption is probably valid, since both were tested in the same test rig. If an assumption is made that most of the variation in results for these tests is due to experimental design, and that higher values of bond strength more closely approach the ‘true value’, some indication of the attainable bond strength may be given by considering the stress at which one would expect a 90 per cent chance of bond rupture (a,., ). For the separate groups of polyalkenoate/dentine specimens tested at one day (Table I), it can be seen that this predicted value varies from 2.26 to 5.08 MPa. For the cumulative results on the same specimens, a realistic value of around 3 MPa emerges as a value of the attainable bond strength at day 1 for this polyalkenoate to dentine. The results show that under the test conditions there is a 1 per cent chance of bond failure at relatively low values of stress (~c.c, = O-015 MPa for 34 specimens). Close inspection of the results in Table Z reveals evidence of the need to consider relatively large sample groups for this type of test. One or two abnormal results in a group of 10 specimens may give a misleading impression. The large scatter of results also makes distinction between experimental groups difficult when statistical analysis based on mean values is carried out. When Weibull
McCabe and Walls: Tensile bond strength tests
167
Table /. Weibull analysis for polyalkenoate dentine bond strengths (24-h values, teeth stored in neutral buffered formalin prior to use)
No. in group
Group no.
Mean fracture stress ([MPaj o)
Standard deviation s.d.
Normalizing parameter
0.97 I.1 3 0.94 1.34 1.05 0.97 I.06 1.02 1.10 1.09
Separate test groups 1 2 : 3 9 4 9 5 8
Stress for 1% chance of fracture
Stress for 90% chance of fracture
(oJ
Weibull Modulus (ml*
Correlation toe fficien t (r)
@OO I IMPaJI
0.51 0.77 0.97 0.65 0.64
0.92 1.31 1.14 I.55 1.23
1 .16(.15) 1.14(.12) 0.56(.08) 1.9 1(.26) 1.37(.15)
0.92 0.93 0.86 0.87 0.94
0.02 0.02 0.0003 0.14 0.04
2.46 2.71 5.08 2.40 2.26
0.51 0.68 0.80 0.77 0.75
0.92 I.21 I.22 1.34 1.31
1.16(.15)
0.92
1.35(.04)
0.98 0.89 0.88 0.89
0.02 0.04 0.007 0.015 0.02 1
2.46 2.25 3.07 3.03 2.78
lw
IMW
Cumulative test groups
(l-5)
42
0*90(.07) 1.02(*07) 1 .I l(O6)
*Standard error of Weibull Modulus in parentheses.
2
-$ % ; ’ 5 z IL
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
2
4
6
8 Tensile
10 stress
12 14 (MPa)
16
18
20
Fig. 7. Plot of fracture probability against tensile stress. 0, Polyalkenoate/dentine bonds-24-h values, teeth stored in neutral buffered formalin before use. 0, Polyalkenoate/ dentine bonds-7-d values, teeth stored in distilled water prior to use. ?? , Composite resin/acrylic veneer bonds-7-d tests. The lines indicate the best least-squares fit to the Weibull distribution.
analysis is performed differences in behaviour between systems become apparent by visual inspection of the fracture probability against stress plots (Fig. I). Statistical comparisons between test groups may be performed by using means and standard deviations and also by comparing the values of Weibull Modulus and the standard errors of these values. In Table I, for example, the results for sample groups 2 and 3 indicate no significant difference between the mean fracture stress values. There is, however, a very significant difference between the values of Weibull Modulus (1.14 and 0.56, highly significant at the 95 per cent confidence limits) indicating that the results of the two test groups are not showing similar behaviour. In Table I, only test groups 1 and 2 can be considered similar when a statistical analysis of both mean fracture stress and Weibull Modulus values is performed. For cumulative test groups it can be seen that changes in Weibull Modulus become insignificant in the region between 25 test specimens and 34 test specimens (P > 0.05 at this stage). This suggests that one
must use around 30 specimens in order to achieve results from which meaningful conclusions can be drawn. Table 11 shows that the predicted attainable bond strength for 1 week’s storage in water is around 2.5 MPa (oe9). There is less chance of failure at low stress levels for these specimens since the 1 per cent failure level (a,,, ) is now 0.34 MPa, compared with 0.015 MPa for the 1-day specimens. Results for the composite resin/acrylic veneer system (Table UZ’) indicate a higher level of attainable bond strength (a,., = 15 MPa) with less chance of fracture at low stress levels (a,,, N 5 MPa). One potential danger of using the Weibull analysis in the manner described above is that it involves extrapolation to the extremes or even beyond the extremes of the collected experimental data. It is difficult to speculate on the validity of this exercise, although it is commonly used to predict the safety limits of components in engineering. The extrapolation procedure is most valid when the experimental data closely tits the Weibull distribution as shown by values of correlation coefficient which approach unity. The Weibull method of analysis highlights many areas of tensile bond strength testing which could be improved. First, it is apparent that test groups of up to 10 specimens, commonly used by research workers, are inadequate for generating results on which meaningful predictions of performance can be made. Secondly, the Weibull analysis of experimental data enables due account to be taken of experimentally introduced variables such as imperfect alignment by selecting a high value for failure probability (say oe+,) for comparing bond strength data. Thirdly, the value of the Weibull Modulus, m, gives an indication of the reliability or dependability of the bond. Fourthly, it is possible to predict that even for materials giving a reasonable high mean bond strength value there is a finite measurable probability of failure occurring at relatively low stress levels. This explains why one occasionally
168
Table //. Weibull use)
Group no. Separate 1 2
1986; 14: No. 4
J. Dent.
analysis
No. in group
dentine
bond strengths
(-/-day
values,
teeth
stored
in distilled
Stress for
Mean fracture stress ([MPa] a)
Standard deviation s.d.
Normalizing parameter IoJ
1.57 I.73
0.57 0.68
1.79 1.98
1.65 I.57
0.63 0.57
1.86 1.79
test groups IO 10
Cumulative
for polyalkenoate
water
Stress for 90% chance of fracture
Correlation coefficient (r)
_ 1% chance
2.39(.17) 2.40(.28)
0.96 0.90
0.26 0.29
2.54 2.80
2.68(.15) 2.39(.17)
0.96 0.95
0.34 0.26
2.54
Weibull Modulus (ml*
of fracture (co o 1 IMW
prior to
lw
9 (MW
test groups
(112,
20 IO
*Standard error of Weibull Modulus in parentheses.
Table l/l. Weibull
Group no. Separate
for comDosite
resin acrylic veneer
bond strengths
(7-dav
tests)
Mean fracture stress ([MPa] a)
Standard deviation s.d.
Normalizing parameter
II.71 12.78 1 1.82
2.81 2.19 1.01
13.00 13.35 12.37
4.86 3.63 9.90(
(.23) (.32) 1.03)
0.97 0.94
3.66 5.18 7.77
15.85 16.36 13.45
test groups 9 12.78
2.19
13.35
4.86
(.32)
0.97
5.18
15.85
2.54 2.19
12.90 13.10
4.62 5.60
(.I 8) 5)
0.98
4.84 5.67
14.97 15.69
No. in group
foJ
Weibull Modulus (ml*
Correlation coefficient (r)
Stress for 1% chance of fracture ho
1 IMW
Stress for 90% chance of fracture bo-9
IMPall
test groups
: 3 Cumulative
analysis
: 8
::
1 1.99 I.94
*Standard error of Weibull Modulus in parentheses.
observes failures for materials which apparently have good properties. The use of Weibull analysis may introduce a degree of realism into standard specification tests for tensile bond strength. The preceding discussion indicates that such tests require the use of around 30 specimens to ensure meaningful results but this may be a necessary price to pay in order to avoid producing unrepresentative or misleading data.
CONCLUSION
Large variations in results which are normally encountered in tensile bond strength tests require that the data be treated with caution and that unwarranted claims or comparisons are not based on small sample groups. The method of Weibull analysis described in this paper offers a convenient means of predicting failure probability/ stress relationships. Meaningful results can only be achieved if at least 20 and preferably 30 or more test specimens are used.
References Beech D. R. (1985) Bonding of restorative resins to dentine. In: Vanherle G. and Smith D. C. (eds), International Symposium on Posterior Composite Resin Dental Restorative Materials. The Netherlands, Peter Szulc, pp.
231-237. Creyke W. E. C., Sainsbury I. E. J. and Morrell R. (1982) Design with Non-Ductile Materials. London, Applied Science, pp. 83-89. Davies D. C. S. (1973) A statistical approach to engineering designs in ceramics. Proc. Br. Ceram. Sot. 22, 429-452. McCabe J. F. and Carrick T. E. (1986) A statistical approach to the mechanical testing of dental materials. Dent. Mater. (in the press). Stanley P., Fessler H. and Sivill A. D. (1973) An engineer’s approach to the prediction of failure probability of brittle components. Proc. Br. Ceram. Sot. 22, 453-487. Walls A. W. G. (1986) A clinical and laboratory investigation of adhesive restorative materials. PhD Thesis, University of Newcastle upon Tyne. Walls A. W. G., McCabe J. F. and Murray J. J. (1985) The bond strength of composite laminate veneers. J. Dent. Res. 64, 1261-1264.
Weibull W. (195 1) A statistical distribution function of wide applicability. J. Appl. Mech. 18, 293-297.
Correspondence should be addressed to: Dr J. F. McCabe, Department of Prosthodontics, Newcastle upon Tyne, Framlington Place, Newcastle upon Tyne, NE2 4BW.
The Dental School, University
of
1986;
14:
165-l
68
Printed in Great Britain
165
The treatment of results for tensile strength testing J. F. McCabe
and A. W. G. Walls
Dental
University
School,
J. Dent 1986;
14: 165-l
of Newcastle
bond
upon Tyne
68 (Received 23 January 1986;
accepted 14 April 1986)
ABSTRACT Data from tensile bond strength tests of polyalkenoate materials to dentine and of composite resins to acrylic veneers has been analysed using a Weibull distribution
function. This function relates failure probability to stress and generates a value of Weibull Modulus from which the dependability of the bond can be quantified, and enables failure probability at any level of stress to be predicted. In order to generate meaningful data from these tests, in excess of 20 and preferably 30 or more specimens should be tested.
INTRODUCTION Tensile bond strength tests are notorious for producing results having a wide variation (Beech, 1985). Variations may reflect the quality of the ‘bond’, but are also due to experimental variations involved in such tests, including improper alignment during bond rupture such that a degree of torsion or shear is introduced. The situation is complicated by the use of a plethora of different types of testing equipment. The traditional method for presenting data from these tests is by quoting the number of tests, mean bond strength and standard deviation. Hence, when comparing systems considerable emphasis is placed on the mean value. A better method for dealing with the results may be to use a function which relates probability of bond failure to tensile stress. This approach has been shown to offer a suitable means of evaluating fracture processes in other dental materials (McCabe and Carrick, 1986). Weibull described a suitable equation for carrying out this type of analysis (Weibull, 1951). The Weibull equation relates probability of failure, Pf, to stress (Tas follows : pf=’
-eexp_
Weilbull Modulus and has considerable practical importance. A low value of m indicates a wide scatter of results with a long ‘tail’ at low stress levels, while a high value of m indicates a close grouping of fracture stress values. The value of m can be interpreted as giving an indication of the dependability of the material. After carrying out a series of tests on a material to establish values of the coefficients u. and m, it is possible to use the Weibull equation to predict fracture probability at any level of stress or vice versa. The Weibull distribution has proved a useful method for evaluating the fracture behaviour of materials and components used in engineering, where experimentally observed variations in results are often attributed to the presence of faults, defects or porosities (Davies, 1973; Stanley et al., 1973; Creyke et al., 1982). It was decided to test the use of this distribution using two ‘adhesive’ systems used in dentistry where bond strength measurement and prediction are important. First, in assessing the adhesion of polyalkenoate cements to dentine and secondly in evaluating the bond strength of acrylic laminate veneers to composite resins.
(o-au)m ( %o)
Pfis the probability of failure at any level of stress up to a value u and uU, cc and m are constants. au is the lowest stress at which fracture is possible. For most applications it is assumed that a, = o, an assumption that has been found to be applicable previously (Stanley et al., 1973). a,, is a normalizing parameter and has no obvious practical significance. The constant, m, is termed the
MATERIALS
AND
METHODS
For measurements of bond strength of polyalkenoate cement to dentine an encapsulated material (Ketac-Fil; ESPE, Seefeld, FR Germany) was used. One series of tests was carried out in which nine extracted teeth (stored in neutral buffered formalin before use) were used to form dentine/cement bonds for evaluation. The method of tooth preparation, bond formation and tensile bond
166
J. Dent.
1986;
14:
No. 4
strength testing are given elsewhere (Walls, 1986). This article is mainly concerned with the method of statistical analysis of the results. Five batches of tensile bond strength specimens were produced from the nine teeth making a total of 45 results. Only 42 results were actually obtained due to factors such as exposure of the pulp on one tooth after grinding to remove a fresh dentine surface. Bond strengths were measured at 24 h, following storage in an oven at 37°C and 100 per cent relative humidity. In a second series of tests 20 specimens (2 batches of 10) were produced using teeth which had been stored in distilled water before use. The bonds were formed in the same way but the specimens were stored at 37°C and 100 per cent relative humidity for 1 week before testing. In both cases bond strengths were measured using an Instron Universal tester (model 1095) at a cross-head speed of 0.5 mm/min. Bond strengths of composite resin (Mastique Luting Composite, LD Caulk Co., Milford, DE, USA) to acrylic laminate veneers (Mastique Acrylic Veneers, LD Caulk Co.) were measured using a similar method (Walls et al. 1985). Thirty specimens (3 batches of 10) were prepared for testing, from which 26 results were obtained. Four results were discarded because rupture occured at low stress levels within the composite resin. The bond strengths were measured at 1 week using an Instron Universal tester at a cross-head speed of 0.5 mm/mm. RESULTS
AND DISCUSSION
The results of the three groups of tests are given in Tables I, H and ZZZand the plots of failure probability against stress are given in Fig. 1. Cumulative failure probability (P,) is calculated from the expression
where N is the total number of specimens under consideration in the sample group and n is the rank number of the specimen. The weakest specimen in a sample group of N specimens is given rank 1 and the strongest rank N. To illustrate how this is carried out: if 20 repeat tests are performed the weakest, with failure stress c,, is given a rank number of 1, while the strongest, with failure stress azO, is given a rank number of 20. In the sample of 20 specimens all (100%) had fractured at or below a stress of azO.When applying the data to a wider population the normal practice of using N -l- 1 instead of N for calculating probabilities is used (Weibull, 195 1). Hence, the probability of fracture at a stress azOin a large population is P = 20420 + l), or O-95 (95%). This approach, therefore, allows for the fact that greater strengths than those measured are possible. As N becomes greater the expressions Pf = nlN and Pf = nl N + 1 become insignificantly different, so the sample more truly represents the whole population. The experimental data was fitted to the Weibull
distribution using a linear least-squares analysis of 1n 1n (1 - Pf) against 1n a from which the coefficients m and a0 can be readily obtained. Calculation of the correlation coefficient for In 1n (1 - Pf) v. 1n c by regression analysis indicates how well the data fits the Weibull equation. Reference to Tables I, II andZZZshows the low values of Weibull Modulus obtained for these tensile bond strength tests (0.89 + 5.60 for cumulative groups). This indicates a wide variation in test results, a factor also reflected in the high standard deviation values obtained. The wide variation is not unexpected and is to a certain extent accepted for this type of test. The values of correlation coefficient shown in the tables indicated how well the experimental data fits the Weibull equation. In tests on other materials Weibull Modulus values of around 7-10 are normal for brittle ceramics, whereas much higher values of around 20-100 are possible for materials which undergo a level of plastic deformation before fracture. A previous study on dental plaster (McCabe and Carrick, 1986) yielded Weibull Modulus values of between 3.68 and 14.20 for nine groups of 12 specimens. Thus it is apparent that the results of tensile bond strength tests are very variable. For this type oftest it is probable that at least 20 and preferably 30 or more test points are necessary in order to make a reasonable assessment of material properties (McCabe and Carrick, 1986). The higher values of Weibull Modulus obtained for the composite resin/acrylic veneer bond strength tests are probably indicative of a more ductile mode of failure in the resin-based system than that seen in the polyalkenoate/ dentine system. This assumes that errors in alignment during testing are similar in the two tests. This assumption is probably valid, since both were tested in the same test rig. If an assumption is made that most of the variation in results for these tests is due to experimental design, and that higher values of bond strength more closely approach the ‘true value’, some indication of the attainable bond strength may be given by considering the stress at which one would expect a 90 per cent chance of bond rupture (a,., ). For the separate groups of polyalkenoate/dentine specimens tested at one day (Table I), it can be seen that this predicted value varies from 2.26 to 5.08 MPa. For the cumulative results on the same specimens, a realistic value of around 3 MPa emerges as a value of the attainable bond strength at day 1 for this polyalkenoate to dentine. The results show that under the test conditions there is a 1 per cent chance of bond failure at relatively low values of stress (~c.c, = O-015 MPa for 34 specimens). Close inspection of the results in Table Z reveals evidence of the need to consider relatively large sample groups for this type of test. One or two abnormal results in a group of 10 specimens may give a misleading impression. The large scatter of results also makes distinction between experimental groups difficult when statistical analysis based on mean values is carried out. When Weibull
McCabe and Walls: Tensile bond strength tests
167
Table /. Weibull analysis for polyalkenoate dentine bond strengths (24-h values, teeth stored in neutral buffered formalin prior to use)
No. in group
Group no.
Mean fracture stress ([MPaj o)
Standard deviation s.d.
Normalizing parameter
0.97 I.1 3 0.94 1.34 1.05 0.97 I.06 1.02 1.10 1.09
Separate test groups 1 2 : 3 9 4 9 5 8
Stress for 1% chance of fracture
Stress for 90% chance of fracture
(oJ
Weibull Modulus (ml*
Correlation toe fficien t (r)
@OO I IMPaJI
0.51 0.77 0.97 0.65 0.64
0.92 1.31 1.14 I.55 1.23
1 .16(.15) 1.14(.12) 0.56(.08) 1.9 1(.26) 1.37(.15)
0.92 0.93 0.86 0.87 0.94
0.02 0.02 0.0003 0.14 0.04
2.46 2.71 5.08 2.40 2.26
0.51 0.68 0.80 0.77 0.75
0.92 I.21 I.22 1.34 1.31
1.16(.15)
0.92
1.35(.04)
0.98 0.89 0.88 0.89
0.02 0.04 0.007 0.015 0.02 1
2.46 2.25 3.07 3.03 2.78
lw
IMW
Cumulative test groups
(l-5)
42
0*90(.07) 1.02(*07) 1 .I l(O6)
*Standard error of Weibull Modulus in parentheses.
2
-$ % ; ’ 5 z IL
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
2
4
6
8 Tensile
10 stress
12 14 (MPa)
16
18
20
Fig. 7. Plot of fracture probability against tensile stress. 0, Polyalkenoate/dentine bonds-24-h values, teeth stored in neutral buffered formalin before use. 0, Polyalkenoate/ dentine bonds-7-d values, teeth stored in distilled water prior to use. ?? , Composite resin/acrylic veneer bonds-7-d tests. The lines indicate the best least-squares fit to the Weibull distribution.
analysis is performed differences in behaviour between systems become apparent by visual inspection of the fracture probability against stress plots (Fig. I). Statistical comparisons between test groups may be performed by using means and standard deviations and also by comparing the values of Weibull Modulus and the standard errors of these values. In Table I, for example, the results for sample groups 2 and 3 indicate no significant difference between the mean fracture stress values. There is, however, a very significant difference between the values of Weibull Modulus (1.14 and 0.56, highly significant at the 95 per cent confidence limits) indicating that the results of the two test groups are not showing similar behaviour. In Table I, only test groups 1 and 2 can be considered similar when a statistical analysis of both mean fracture stress and Weibull Modulus values is performed. For cumulative test groups it can be seen that changes in Weibull Modulus become insignificant in the region between 25 test specimens and 34 test specimens (P > 0.05 at this stage). This suggests that one
must use around 30 specimens in order to achieve results from which meaningful conclusions can be drawn. Table 11 shows that the predicted attainable bond strength for 1 week’s storage in water is around 2.5 MPa (oe9). There is less chance of failure at low stress levels for these specimens since the 1 per cent failure level (a,,, ) is now 0.34 MPa, compared with 0.015 MPa for the 1-day specimens. Results for the composite resin/acrylic veneer system (Table UZ’) indicate a higher level of attainable bond strength (a,., = 15 MPa) with less chance of fracture at low stress levels (a,,, N 5 MPa). One potential danger of using the Weibull analysis in the manner described above is that it involves extrapolation to the extremes or even beyond the extremes of the collected experimental data. It is difficult to speculate on the validity of this exercise, although it is commonly used to predict the safety limits of components in engineering. The extrapolation procedure is most valid when the experimental data closely tits the Weibull distribution as shown by values of correlation coefficient which approach unity. The Weibull method of analysis highlights many areas of tensile bond strength testing which could be improved. First, it is apparent that test groups of up to 10 specimens, commonly used by research workers, are inadequate for generating results on which meaningful predictions of performance can be made. Secondly, the Weibull analysis of experimental data enables due account to be taken of experimentally introduced variables such as imperfect alignment by selecting a high value for failure probability (say oe+,) for comparing bond strength data. Thirdly, the value of the Weibull Modulus, m, gives an indication of the reliability or dependability of the bond. Fourthly, it is possible to predict that even for materials giving a reasonable high mean bond strength value there is a finite measurable probability of failure occurring at relatively low stress levels. This explains why one occasionally
168
Table //. Weibull use)
Group no. Separate 1 2
1986; 14: No. 4
J. Dent.
analysis
No. in group
dentine
bond strengths
(-/-day
values,
teeth
stored
in distilled
Stress for
Mean fracture stress ([MPa] a)
Standard deviation s.d.
Normalizing parameter IoJ
1.57 I.73
0.57 0.68
1.79 1.98
1.65 I.57
0.63 0.57
1.86 1.79
test groups IO 10
Cumulative
for polyalkenoate
water
Stress for 90% chance of fracture
Correlation coefficient (r)
_ 1% chance
2.39(.17) 2.40(.28)
0.96 0.90
0.26 0.29
2.54 2.80
2.68(.15) 2.39(.17)
0.96 0.95
0.34 0.26
2.54
Weibull Modulus (ml*
of fracture (co o 1 IMW
prior to
lw
9 (MW
test groups
(112,
20 IO
*Standard error of Weibull Modulus in parentheses.
Table l/l. Weibull
Group no. Separate
for comDosite
resin acrylic veneer
bond strengths
(7-dav
tests)
Mean fracture stress ([MPa] a)
Standard deviation s.d.
Normalizing parameter
II.71 12.78 1 1.82
2.81 2.19 1.01
13.00 13.35 12.37
4.86 3.63 9.90(
(.23) (.32) 1.03)
0.97 0.94
3.66 5.18 7.77
15.85 16.36 13.45
test groups 9 12.78
2.19
13.35
4.86
(.32)
0.97
5.18
15.85
2.54 2.19
12.90 13.10
4.62 5.60
(.I 8) 5)
0.98
4.84 5.67
14.97 15.69
No. in group
foJ
Weibull Modulus (ml*
Correlation coefficient (r)
Stress for 1% chance of fracture ho
1 IMW
Stress for 90% chance of fracture bo-9
IMPall
test groups
: 3 Cumulative
analysis
: 8
::
1 1.99 I.94
*Standard error of Weibull Modulus in parentheses.
observes failures for materials which apparently have good properties. The use of Weibull analysis may introduce a degree of realism into standard specification tests for tensile bond strength. The preceding discussion indicates that such tests require the use of around 30 specimens to ensure meaningful results but this may be a necessary price to pay in order to avoid producing unrepresentative or misleading data.
CONCLUSION
Large variations in results which are normally encountered in tensile bond strength tests require that the data be treated with caution and that unwarranted claims or comparisons are not based on small sample groups. The method of Weibull analysis described in this paper offers a convenient means of predicting failure probability/ stress relationships. Meaningful results can only be achieved if at least 20 and preferably 30 or more test specimens are used.
References Beech D. R. (1985) Bonding of restorative resins to dentine. In: Vanherle G. and Smith D. C. (eds), International Symposium on Posterior Composite Resin Dental Restorative Materials. The Netherlands, Peter Szulc, pp.
231-237. Creyke W. E. C., Sainsbury I. E. J. and Morrell R. (1982) Design with Non-Ductile Materials. London, Applied Science, pp. 83-89. Davies D. C. S. (1973) A statistical approach to engineering designs in ceramics. Proc. Br. Ceram. Sot. 22, 429-452. McCabe J. F. and Carrick T. E. (1986) A statistical approach to the mechanical testing of dental materials. Dent. Mater. (in the press). Stanley P., Fessler H. and Sivill A. D. (1973) An engineer’s approach to the prediction of failure probability of brittle components. Proc. Br. Ceram. Sot. 22, 453-487. Walls A. W. G. (1986) A clinical and laboratory investigation of adhesive restorative materials. PhD Thesis, University of Newcastle upon Tyne. Walls A. W. G., McCabe J. F. and Murray J. J. (1985) The bond strength of composite laminate veneers. J. Dent. Res. 64, 1261-1264.
Weibull W. (195 1) A statistical distribution function of wide applicability. J. Appl. Mech. 18, 293-297.
Correspondence should be addressed to: Dr J. F. McCabe, Department of Prosthodontics, Newcastle upon Tyne, Framlington Place, Newcastle upon Tyne, NE2 4BW.
The Dental School, University
of