Dent Mater 8:283-288, July, 1992

The Weibull distribution applied to post and core failure M.C.D.N.J.M. Huysmans, P.G.T. Van Der Varst, M.C.R.B. Peters and A.J.M. Plasschaert Department of Cariology & Endodontology, TRIKON: Institute for Dental Clinical Research, University of Nijmegen, Nijmegen, The Netherlands

Abstract. In this study, data on initial failure loads of direct post and core-restored premolar teeth were analyzed using the Weibull distribution. Restorations consisted of a prefabricated titanium alloy post, and an amalgam, composite or glass cermet core buildup in human upper premolar teeth. The specimens were subjected to compressive forces u ntil failure at angles of 10, 45 and 90 ° to their long axis. The two- and three-parameter Weibull distributions were compared for applicability to the failure load data. For estimation of the parameters of the two-parameter distribution: % (reference stress) and m (Weibull modulus), linear regression was used. In this distribution, it is assumed that the third parameter, % (cut-off stress), equals 0. The Maximum Likelihood (MLH) method was used to estimate all three parameters. It was found that the choice of distribution has a strong influence on the estimated values and that the three-parameter distribution is best fitted for the failure loads in this study. Comparisons were made between the failure probability curves as found by MLH estimation for the different core materials and loading angles. The results indicated that the influence of loading angle on the failure mechanism was stronger than that of core material. INTR00UCTION The classical way to view the strength of materials or structures is a deterministic one. That is, a true strength, a single value that is characteristic of the material or structure, is supposed to exist. In experiments to determine this "true" strength, considerable scatter in the results is usually observed. As this is not considered to be a feature of the material or object itself, it is usually attributed to uncontrollable experimental variables. As a consequence, the second central moment of the experimental data, the standard deviation, is interpreted as indicating the success of standardizing the experimental set-up and procedures. Therefore, standard deviation can be considered to be an indicator of the quality of an experiment or testing method, The deterministic view has become much less popular in the technical sciences. Several observations have made it controversial. If the deterministic view is valid, identical experiments performed on material specimens of different sizes should yield the same results for failure stress. However, it has been shown that for some materials, larger specimens have a lower failure stress compared to smaller ones (Kittl et al., 1990). A difference is also found between 3-and 4-point bending tests on dental composite specimens of equal dimensions (Sander et al., 1986). These systematic differences cannot be explained by random variations in experimental procedures, but by imperfections included in the bulk and at

the surface of every material and structure. Flaws (voids, inclusions, etc.) can cause a material to fail]ong before its ideal strength is reached. Although this ideal strength might be interpreted as "true" strength, for practical purposes it is of little importance. In dentistry, as in engineering, it is not relevant to know what loads a structure might resist ideally, but what loads it almost certainly can resist practically. It is more logical, therefore, to accept imperfections as an integral partofamateria]orstructureandtoaccountfortheirpresence when describingits strength. So, particularly for brittle materials, a new probabilistic approach to mechanical testing has recently been introduced in dental science (McCabe and Carrick, 1986). The distribution of imperfections within a structure is of a probabilistic nature. As a consequence, the strength of the structure itself is ofa probabilistic nature. Several statistical distributions have been used to describe strength. The normal distribution is widely used for this purpose. This is certainly admissible, although McCabe and Carrick (1986) disagree. Correctly interpreted, it represents not merely the distribution of the variability in experimental procedures, but also the distribution of strength within the material population. For materials with a more or less brittle behavior, however, the Weibull distribution is generally preferred for theoretical and practical reasons (Kittl and Diaz, 1988). The basic form of the Weibull equation for cumulative probability density is: Pw~ibun= 1-exp {-[(~-~u)/(~o] m}

(1)

where a is the failure stress and % % and m are the three parameters of the distribution. Of these parameters, (~u is a cut-off stress below which the probability of failure is zero, (~0is a reference or scaling value related to the mean and m is the Weibull modulus or shape parameter. It is often assumed that (~u= 0. This corresponds to an assumption that there is no minimal stress below which failure cannot occur. Parameter estimation can then be performed relatively easily, by a linear regression method. If % v 0, three parameters have to be estimated, and either the Maximum Likelihood(MLH) method or a non-linear least squares method has to be used. The data used in this study are the result of an experiment in which the initial failure loads of direct post and core restorations in human upper premolars were determined. Conventional statistical methods have been used to analyze the data (Huysmans et al., 1992). Additional analysis using the Weibull distribution was desirable because of its superior Dental Materials~July 1992 283

applicability to strength data, and because the occurrence of differences in the shape of the distributions may be thought to indicate different failure processes. The aim of this study was to determine for each of the experimental groups the specific Weibul] distribution shape and to evaluate possible relations to failure processes observed in the laboratory. Material and loading influences were also evaluated. To determine which Weibull distribution should be used, the results of parameter estimates of two parameters, using linear regression, and three parameters, using MLH estimation, were compared first. MATERIALS AND METHODS

Direct post and core restorations in extracted human upper premolar teeth were subjected to strength testing. Since the method of specimen preparation, testing, and the acquired data are discussed elsewhere (Huysmans et al., 1992), only a brief description of the experimental procedure will be given here. Decoronated upper premolar teeth were restored with a prefabricated titanium alloy post (Para Post Plus #4, Whaledent Int., New York, NY, USA) and an amalgam (Dispersalloy, Johnson & Johnson Dental Products Co., East Windsor, NJ, USA), composite (Clearfil Core, Cavex Holland BV, Keur en Sneltjes MFG. Co., Haarlem, The Netherlands) or glass cermet core (Ketac Silver, ESPE GmbH, Seefeld/ Oberbay, FRG). After 1 week, the specimens were subjected to quasistatic loading in displacement control, at 10, 45 or 90 ° to their long axis. For each specimen, load at failure and failure characteristics were recorded. Due to the complexity of the structure, failure stresses could not be calculated, and failure loads were used instead of stresses throughout the study (Kittl and Diaz, 1988). The codes for the parameters of the Weibull distribution were not changed, so that Gu here represents cut-off load and Gois the reference load. Within each of the nine experimental groups, the experimental cumulative failure probability was calculated. This was done by sorting the data in order of increasing failure load and subsequently ranking the results. The experimental cumulative failure probability was calculated using the formula Pfailure---- i / (n + 1) (2) where i is the rank number of the data point and n is the sample size. The experimental PfaJlurewas used to estimate the parameters of the Weibull distribution. Equation (1) can be rearranged and written as: In In (1 / (1 -

Pfailure)}

=

m.ln (o - o u) - m.lno 0

(3)

If it is assumed that u, = 0, the Weibull equation is reduced to a two-parameter equation: In In (1 / (1 -

Prailure)}

=

m-lno - m.lno o

(4)

As O0 is a constant, it is evident that now a linear relation between In ln{1/(1-pfai~u,o)} and lno exists, of slope m and intercept m-lnoo. Using equation (4), m was estimated by linear regression and once it had been established, GOwas calculated. A parameter estimation using the method of Maximum Likelihood (MLH) was also performed. The MLH method can be described as follows: the "probability" offinding a given sample of strength results is defined by the likelihood 284 Huysmans et aL/Weibull statistics and core buildup failure

TABLE 1: RESULTSOFTHEWEIBULLPARAMETERESTIMATION USINGA LINEARREGRESSIONMETHODFORDATAON QUASISTATIC STRENGTHOF DIRECTPOSTANDCORE RESTORATIONSIN PREMOLARS Mean Group,Group failure Standard code size strength(N) deviation(N) o0 (N) % (N) m (-) R2 A10 11 5973 602 0 6237 9.8 0.97 A45 12 940 105 0 989 9.0 0.93 A90 10 355 67 0 384 5.5 0.75

C10 C45 C90

9 9 10

4848 917 445

303 110 73

0 0 0

4997 971 481

15.3 0.95 7.7 0.97 5.5 0.95

G10 9 2756 208 0 2857 12.6 0.98 G45 9 626 57 0 654 10.3 0.85 G90 10 241 35 0 257 6.5 0.94 * Codingof the groups:A: amalgam,C: composite,G: glasscermet,followed by loadingdirectionin degrees. function. The parameters of the distribution are chosen in such a way that this function is maximized, and the sample is "most likely" (Kreyszig, 1970). Using MLH, it is possible to estimate all parameters of the Weibull three-parameter equation (1). The results of the MLH parameter estimation and the three-parameter Weibull distribution were used to construct cumulative probability density curves. To facilitate the cornparison of the results, a rescaling of the load variable was applied when plotting these curves. For each experimental group, the observed failure data o, and the estimates for the cut-off load ou and the reference load u o were expressed in percentages of the mean failure load of that group. The mean failure load of an arbitrary group can be denoted as o .... and let the rescaled variables be f=100 (G/G. . . . ). The rescaled parameters are fu=100 (Gu/a.... ) and f0=100 (Go/o.... ). Since: exp{-[(f-fu)/f0] m} = exp{-[(O-Gu)/G0]m}

(5)

we see that a plot of Prailuroas function of f carries the same information as a plot of Prailure a s function of G. In this way, the absolute values of the failure loads are ignored, and the Weibull probability curves can be compared directly.

RESULTS The results of the parameter estimation by linear regression are given in Table 1. Mean and standard deviation, being more readily interpreted, are added. The results of the MLH estimation are given in Table 2. For all groups, the estimate for the shape parameter m is much lower when ~u is also estimated. The effect this has on the estimates for safety limits can be seen in Table 3 where the load levels for failure probabilities of 5 % and 1% have been calculated using the results of both methods. TheWeibullcumulativefailureprobabilitycurves, asyielded by the MLH method, and the experimental failure probability points for each group are shown in Figs. 1,2and3. Theresults of the normalization procedures can be seen in Figs. 4 and 5. Fig. 4 represents grouping by core material. Differences between the distributions indicate the influence of loading angle on the failure process. Fig. 5 represents grouping by loading angle, thus visualizing the influence of the core material.

TABLE 2: RESULTSOF THE WEIBULL PARAMETERESTIMATIONUSING A MLH ESTIMATIONMETHODFOR THE SAMEDATAAS IN TABLE 1

Group code*

o~ (N)

Oo (N)

m(-)

A10 A45 A90

4582 713 274

1546 252 90

2.8 2.6 1.7

C10 C45 C90

3894 637 271

1061 311 195

4.1 3.1 2.8

2119 512 164

707 127 85

4.1 3.0 3.1

G10 G45 G90 * As for Table 1

TABLE 3: COMPARISONOF THE SAFETYLIMITSAT 5% AND 1% FAILURE PROBABILITY,CALCULATEDWITH THE RESULTSOF LINEAR REGRESSIONAND MLH PARAMETERESTIMATION Load levelfor which Load levelfor which Pfailure= 50/0 (N) Pfailure= 1% (N)

linear Group code* regression A10 4606 A45 711 A90 224

MLH 5117 793 290

linear regression 3901 593 166

MLH 4881 756 280

1

/

~

~

L~

.~ ~ 0.5P

y I

o 2000

4000 60(~0 Failure load (N) Fig. 1. Weibullcumulativefailureprobabilitycurves(lines)andexperimentalfailure probability points for 100 loading direction (A / o: amalgam, C/~7: composite, G/' : glass cermet). l

~ ~

"

oV G

0.5-

C

o A

~"

C10 C45 C90

4115 660 280

4408 756 339

3699 534 208

4240 708 309

G10 G45 G90

2257 490 163

2462 559 197

1983 418 127

2349 539 183

~. o Failure load (N) Fig. 2. Weibullcumulative probability curves (lines) and experimental failure probability points for 45° loading direction. Legend as for Fig. 1.

1 ~

"

°

vv

* As for Table 1

¢~

G



DISCUSSION

The Weibull distribution has the practical advantage of being very flexible. The normal distribution has a shape which is determined by the parameters ~ (mean) and o (standard deviation). The Weibull distribution, however, has three parameters and can therefore be adapted to the experimental data more closely. Moreover, the shape of the distribution can change more markedly as the Weibull modulus m changes.

A ~

v

C

~. o.5-

'~ ~" o loo

, 3o0

I

500

Failure load (N)

Another advantage lies in the very usefulinterpretation of two

Fig.3. Weibullcumuiativefailureprobabilitycurves(lines)andexperimentalfailure probability points for 900 loading direction. Legend as for Fig. 1.

ofthe parameters, Ou and m. Considering the distribution of flaws within a material, m is related to the width of this distribution andis anindicator of material dependability. The inverse of ou, i.e. 1 / ou, is a measure for the largest (and thus failure determining) flaw or crack present in the structure (Freudenthal, 1968). The Weibull distribution is as yet relativelyunknownin dental science, and therefore its parameters are not as familiar as the usual mean and standard deviation. Increasing use will overcome this problem, Parameter estimation can be slightly more complicated for the Weibull distribution. This is especially true if three, instead of two, parameters are needed. However, given the state of the art in numerical mathematics, this is no longer a disadvantage, The group sizes used in this study were relatively small, mainly because of the difficulty of obtaining suitable teeth for

testing. This had a detrimental effect on the reliability of the estimates, and it caused a bias. Evaluation of the variance of the estimated values, especially for the three-parameter distribution, was difficult. Reported methods universally assume one parameter to be known before calculating the bias or variance of the others. However, to give an indication of the error in the estimate of m, the bias of both methods was evaluated. For the linear regression method, the estimate of the Weibull modulus was slightly conservative as a result of the choice of probability estimator, equation (2), in combination with the small sample size (Bergman, 1984). The probability estimator was chosen because it is widely used and adjusts probability values in such a way that failure probabilities for i = 1 and i = n are symmetrically placed: Pfailure(i = 1) = 1 - Pfa~luro(i = n) (Gumbel, 1958). As sample size increases, the Dental Materials~July 1992 285

:"'..'"

/,..

A

C

.':."" #,./."

lO

L ....

45

.....

90

I

I

"'J-

50

100

I

/

G

"'

,.0 °~

O"

:..'.."""

"

.-'.i

.." s* I

150 50

I

I

100

150 50

i

I

100

150

Resealed failure load

Fig. 4. Weibull cumulative failure probability curves for the rescaled failure loads grouped by core material. Letters indicating material groups.

]"

.:'I

"'"

=

4

O e~

P "~

~

/

:,; ~;:

.,= O'

[

50

I

100

A ....

C

.....

o

I

I

150 50

i

.....iI

90

~"

." I

i00

150 5

~)

""

"

I

I

100

150

Rescaled failure load Fig. 5. Weibull cumulative failure probability curves for the rescaled failure loads grouped by loading direction. Numbers indicating degrees of loading direction.

estimateconvergestothetruevalue. Thesamplesizeatwhich the error becomes negligible is reported to be as large as 60 (Sullivan and Lauzon, 1986), and in another study as small as approximatelyl5(Balabaetal., 1990). Usingtheresultsfrom the latter study, we estimated the bias to be about 10 % for a sample size of about 10. The MLH methods results in a slight overestimation of m for small sample sizes (Bain and Englehardt, 1991). Using their method, assuming that o u is known in advance, we calculated a bias of 10 - 15 % for a sample size of approximately 10. Since in our study Ouwas not known in advance, this number should be considered with caution, However, it indicates the order of magnitude of the bias ofm in our study. The estimate of% is generally only very lightly biased (Bain and Englehardt, 1991). Linear regression, as it is used here to determine m, should be considered as a way to draw the best fittingline through the experimental data points, and not a method of statistical analysis. Most of the properties ofthe latter use ofthe method depend on assumptions that are made about the random errors in the X and Y direction: they should be uncorrelated, have zero average value, and the same variance. These assumptions are known not to hold in the application of the regression method to the Weibull distribution (Tobias and Trindade, 1986). When computerized linear regression programs are used, standard errors for m are usually automatically generated. They are sometimes used to calculate confidence bounds for m or to test for significant differences between m values (McCabe and Walls, 1986). For the reason mentioned above, such confidence bounds are not valid for this application, The results of the linear regression method show a good fit of the Weibull equation to the data in general. Correlation 286 Huysmans et aL/Weibull statistics and core buildup failure

coefficient values are high, except for groups G45 and A90. The last group has such low correlation that the theory of an underlying two-parameter Weibull distribution is difficult to maintain. However, before the three-parameter distribution is rejected, the assumption that au = 0 should be verified. Comparison of the results of the two estimation methods showsthatthisassumptionhasanimportantinfluenceonthe value of re. IfGu is estimated as well, a reduction of 50 - 75 % in m value occurs throughout the groups. From the values as estimated for Guby MLH, it is obvious that it is very unlikely that au is indeed as small as 0. As important as the absolute value of Gu, however, is the relative value ou / Go. The larger this ratio is, the less justified the use of(L = 0. Furthermore, if (~u = 0, the size of the largest crack is of the same order of magnitude as the global dimensions of the structure. In material specimens, especially small beams for bending tests, it might be almost possible to justify the assumption, but not in larger structures. The interdependence ofo uand m should invoke caution in the interpretation of m values as they are given in the literature. No comparison of dependability of materials or structures should be made solely on the basis of m value. It should be ascertained what method was used for estimation of the parameters and particularly whether an estimation of ~ was included. Calculations as in Table 3 for 5 % or 1% probability of failure will frequently be used as safety limits. For the 5 % value, the differences between the two methods are not yet marked. Moving toward the end of the distribution, differences increase. This leads to extremely conservative estimates of the safety limit, when assuming o u = 0. With a view to liability, this might be desirable, but unreasonably low estimates are not.

] "~

clo ..--

~ ~.

"

C90core

0.5 .-~ == ~: "~

0

5o

L 1o0 Rescaled failure load

15o

Fig. 6. Weibull cumulative failure probability curve for the rescaled loads for core failure in group C90 compared to the curve for C10. Group codes as for Table 1.

The probability curves as a function of absolute failure load yield more or less the same information as a comparison of group means and standard deviations. A shift in the position of the composite group relative to the amalgam group can be seen in going from 10 to 45 to 90 ° loading. This is a feature thatisratherdifficulttoexplain. Itwouldbelogicaltosuppose that as the loading angle increases, stresses shift from largely compressive to tensile. Composite has a compressive strength that approaches the value for amalgam, but has a much lower tensile strength. Itwouldthusbeexpectedtobemorestrongly affected by the change in loading direction. A possible explanation might be that due to the lower value of Young's modulus ofthe composite material, most ofthe load is diverted to the post, especially if loading is more oblique. This leads to lower stresses in the core material itself, The probability curves for rescaled failure load allow for comparison independent of absolute load values. If we first consider the curves grouped for the three materials (Fig. 4), a close resemblance is seen between amalgam and glass cermet. For both materials, the 10° and 45 ° curves practically coincide, whereas the 90 ° curve has a lesser slope and therefore a larger range. The graph for composite shows a more gradual decrease in slope and increase in range. Fig. 5, grouping the curves by loading direction, shows an interesting feature. For all three loading groups, the curves are remarkably alike. The shape of the Weibull curve can be interpreted as an indicator of a certain failure type or mechanism. So, the results suggest that loading direction is of greater influence on failure mode than the material used. It can be seen in Fig. 4 that there is only a little difference between 10° and 45 °. Beyond that there is a definite change in the nature of the curves. Observations made in the laboratory during and after testing confirm that this corresponds with a change in failure mode. The prevailing failure mode in 10° and 45 ° testing was fracture of the core material. Due to the very high loads at 10°, an overall crushing of the core occurred. Loading under 45 ° generally resulted in a breaking away of a large piece of the core material, and an extensive crushing of the remaining part. Superficially the failure mode in 90 ° loading often resembled that of45 °. A large part of the core was generally broken off. However, in some cases, no core fracture occurred at all and failure consisted solely of a dislodgement of the post, together with the core. Thus, two entirely different failure mechanisms were active. For the composite group, a definitely phased failure could be observed. In 8 specimens of group C90, a dislodgement of the post could be observed during testing, both by the occurrence

of a gap between tooth and core and by a temporary dip in the load as registered by the testing machine. After this dislodgement, the load could rise again until the core fractured. The first maximum was recorded as the post dislodgement load. The mean load for post dislodgement in this group was comparable to the failure loads of those amalgam specimens that failed by dislodgement alone. Although no phased failure could be seen in the amalgam group, a combination of the two failure mechanisms was certainly active: three specimens showed pure dislodgement failure and two core - failed specimens showed radiographic evidence of post dislodgement. From the shape ofthe probability curve for G90, one would conclude that the failure mode was mixed. In this group, however, no dislodgement was observed, and final failure occurred at load levels below the mean failure load at post dislodgement in the other groups. But as the fixating cement in the glass cermet groups was different from that of the other groups, it is stillpossible that mixed failure did occur. For group C90, the failure loads that correspond to core fracture (the same failure mode as for 10° and 45 ° loading) can be isolated and normalized. With these values, Weibull parameter estimation can be performed. The resulting cumulative probability curve is given in Fig. 6, along with the curve for C 10. The similarity is obvious. This supports our assumption about the relationship between the shape of the Weibull curve and failure mechanism. CONCLUSIONS The number of parameters estimated has a distinct influence on the values found for the Weibull parameters. The assumption that ~, = 0 can lead to a considerable overestimation ofthe Weibull modulus m and an underestimation of safety limits. Three parameters should be estimated when c u= 0 is unlikely. The shape of the Weibull cumulative probability curve indicates mechanisms of failure. In this study, changes in the shapes of the curves could be related to failure mechanisms observed in the laboratory. Loading direction influenced failure mode more than the type of core material.

ACKNOWLEDGMENTS The authors gratefully acknowledge Cavex / Kuraray and ESPE GmbH for supplying the materials used in this study.

ReceivedSeptember 3, 1991/AcceptedJune 17, 1992 Addresscorrespondence and reprintrequeststo: M.C.D.N.J.M.Huysmans Department ofCariology & Endodontology TRIKON:Institutefor DentalClinicalResearch UniversityofNijmegen PO Box9101 NL-6500HB Nijmegen, The Netherlands

REFERENCES Bain LE, Englehardt M (1991). Statistical analysis ofreliability and life-testing models. New York: Marcel Dekker Inc., 221-228. Balaba WM, Stevenson LT, Wefers K, Tackle MN (1990).

Dental Materials~July 1992 287

Probability estimators for Weibull statistics of the failure strengths of brittle powder compacts. JMat Sci Letters 9: 648-649. Bergman B (1984). On the estimation ofthe Weibull modulus, J Mat Sci Letters 3: 689-692. Freudenthal AM (1968). Statistical approach to brittle fracture. In: Liebowitz H, editor. Fracture. An advanced treatise. New York: Academic Press, 591-619. Gumbel EJ (1958). Statistics of extremes. New York: Columbia University Press, 29-34. Huysmans MCDNJM, Peters MCRB, Plasschaert AJM, Van der Varst PGT (1992). Failure characteristics of endodontically treated premolars restored with a post and direct restorative material. Int Endodont J 25: 121-129. McCabe JF, Carrick TE (1986). A statistical approach to the mechanical testing of dental materials. Dent Mater 2: 139142.

288 Huysmans et aL/Weibull statistics and core buildup failure

McCabe JF, Walls AWG (1986). The treatment of results for tensile bond strength testing. JDent 14: 165-168. Kittl P, Diaz G (1988). Weibull's fracture statistics or probabilistic strength of materials. Res Mechanica 24: 99-207. Kittl P, Diaz G, Morales M (1990). Determination of Weibull's parameters of flexure strength of round beams of aluminum-copper alloy and cast iron. Theoretical and Applied Fracture Mech 13: 251-255. Kreyszig E (1970). Introductory mathematical statistics. Singapore: John Wiley and Sons, 162-164. Sander J, Solt~sz U, Klaiber B (1986). Zur Festigkeitsprfifung von Ffillungsmaterialien. Laboratory report W8/86, Freiburg: Fraunhofer-Institut ffir Werkstoffmechanik. Sullivan JD, Lauzon PH (1986). Experimental probability estimatorsforWeibullplots.JMatSciLetters5: 1245-1247. Tobias PA, Trindade D (1986). Applied Reliability. New York: Van Nostrand Reinhold Company Inc., 78.

The Weibull distribution applied to post and core failure.

In this study, data on initial failure loads of direct post and core-restored premolar teeth were analyzed using the Weibull distribution. Restoration...
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