NOTICE: This is the author s version of a work that was accepted for publication in the Engineering Fracture Mecahnics. Changes resulting from the

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1 NOTICE: This is the author s version of a work that was aepted for publiation in the Engineering Frature Meahnis. Changes resulting from the publishing proess, suh as peer review, editing, orretions, strutural formatting, and other quality ontrol mehanisms may not be refleted in this doument. Changes may have been made to this work sine it was submitted for publiation. A definitive version was subsequently published in Engineering Frature Meahnis 74 (2007) doi:0.06/j.engframeh

2 Frature statistis of eramis Weibull statistis and deviations from Weibull statistis R. Danzer a,b,,, P. Supani a,b, J. Pasual a, T. Lube a a Institut für Struktur- und Funktionskeramik, Montanuniversität Leoben, Peter Tunner Strasse 5, A-8700 Leoben, Austria b Materials Center Leoben, Franz Josef Strasse 3, A-8700 Leoben, Austria While guest researher at Max Plank Institute of Metal Researh, Heisenbergstrasse 5 D Stuttgart. Germany Abstrat Weibull statistis is up to now the bakbone in the mehanial design proedure of erami omponents but is not generally valid. Different situations that deviate are presented, e.g. materials having a multimodal flaw distribution or having an inreasing R - urve. It is shown by means of Monte Carlo simulations that on the basis of the standardised testing proedure (30 speimens) is not possible to deide whether a material is Weibull distributed or not. Measuring the strength of speimens with different size allows after reasonable experimental effort- the determination of the strength distribution in a reliable way and in a wide range of parameters. Keywords Brittle failure, Weibull statistis, Monte Carlo Corresponding author. Tel: : Fax: address: isfk@unileoben.a.at 2

3 . Introdution Frature of brittle materials (e.g. eramis) usually initiates from flaws [], whih are distributed in the material. The strength of the speimen depends on the size of the major flaw, whih varies from speimen to speimen. Therefore, the strength of brittle materials has to be desribed by a probability funtion (statistis) [2-5]. It follows from the experiments that the probability of failure inreases with load amplitude and with size of the speimens [2, 3, 6]. The first observation is trivial. The seond observation follows from the fat that it is more likely to find a major flaw in a large speimen than in a small one. Therefore the mean strength of a set of large speimens is smaller than the mean strength of a set of small speimens. This size effet on strength is the most prominent and relevant onsequene of the statistial behaviour of strength in brittle materials. For more than sixty years Weibull proposed a statistial theory of brittle frature [7, 8]. His fundamental assumption was the weakest link hypothesis, i.e. the speimen fails, if its weakest volume element fails. Using some empirial arguments neessary to obtain a simple and good fitting to his experimental data, he derived the so-alled Weibull distribution funtion, whih - in its simplest form, for a uniaxial homogenous tensile stress state (stress amplitude σ) and for speimens of the volume, V, - is given by: m V σ ( ) F σ, V = exp. () V0 σ 0 The Weibull modulus m is a measure for the satter of strength data: the wider the distribution is the smaller is m. σ 0 is a harateristi strength value and V 0 the hosen normalising volume. Up to these days the Weibull distribution funtion is the basis of the state of the art in mehanial design proedure of erami omponents. It should be noted that in almost every experimental study on frature statistis of eramis is laimed that the data are Weibull distributed. But this is not neessarily true, beause -for that type of statement- the sample size is too small in any ase 2. To show this is a major goal of this paper. The strength testing of eramis and the determination of the Weibull distribution are standardised [0, ]. Following the standards the Weibull distribution funtion has to be measured on a sample of at least 30 speimens (due to high mahining osts, larger samples Eq. defines the so alled two-parameter form of the Weibull distribution. Weibull also proposed a three parameter form, where σ in Eq. is replaed by the expression ( σ σ u ). For stresses below a tensile threshold stress σ failure annot our. Although this threshold stress reflets the wishes of erami engineers for reliable u omponents, the two parameter form of the Weibull distribution seems to reflet the observed behaviour of erami materials better as the three parameter form (see for example Lu et al [9]). For speimens with signifiant internal stress fields, the threshold stress may reflet the influene of these stresses. For a given set of data the two parameter equation gives an upper limit and makes therefore a safe reliability analysis possible. 2 The same statement is also valid for the three parameter form. Espeially the fit of the strength stress - if it is based on a small sample - is in general not stable. 3

4 are hardly tested in the daily testing pratie). The range of measured failure probabilities, F, inreases with the sample size [2] and is - for a sample of 30 tests - very limited (between ~ /60 and ~ 59/60). To determine the design stress, the measured data have - in general - to be extrapolated to the tolerated failure probability of the omponents (~ /0 6 ) and to the (effetive) volume respetively, whih often results in a very large extrapolation span [2]. In the last fifty years a signifiant amount of researh has been made to give Weibull's empirial theory a more fundamental basis [3-2]. The paper of Kittl and Diaz [22] gives an exellent overview on the former developments. Freudenthal [5] showed for homogenous and brittle materials that, if the flaws do not interat (i.e. if they are sparsely distributed), the probability of failure only depends on the number of destrutive flaws, speimen of size and shape, S, [ ] ( σ ) exp ( σ ) F S N, S N, S, ourring in a =. (2) In this equation, N, S denotes the mean number of destrutive (ritial) flaws in a large sample (i.e. the value of expetation). Jayatilaka and Trustrum [3] demonstrated in their noteworthy paper, that, for a brittle and homogeneous material, the distribution of the strength data is aused by the distribution of sizes and orientations of the flaws, and that a Weibull distribution of strength will be observed for flaw populations with a monotonially dereasing density of flaw sizes. Danzer et al. [8-20] applied their ideas to flaw populations with any size distribution and to speimens with an inhomogeneous flaw population. Then, the strength distribution strongly depends on the shape of the flaw distributions in the material. Again, it was neessary to assume that a speimen fails if any flaw initiates frature (the weakest link hypothesis), and if there is no interation between flaws. On the basis of these ideas a diret orrelation between the flaw size distribution and the satter (statistis) of strength data an be reognised. In this paper orrelations between flaw size distribution and frature statistis are disussed. It will be shown that, for many reasons, the frature statistis may stritly deviate from the Weibull statistis, Eq.. It also will be shown by means of Monte Carlo simulations that these deviations an hardly be deteted on the basis of small samples but that the testing of speimens of different size may offer a way out of that dilemma. 2. Flaw size distribution and strength distribution For design purposes the expliit dependene of the number of destrutive flaws and the applied load has to be known. If not differently mentioned a homogeneous tensile stress field (stress amplitude σ ) is assumed in the following disussions. For simpliity and without loss of generality it is also assumed that the flaws an be desribed by a single parameter a whih haraterises their size, (e.g. the rak radius) and that the rak like flaws are perpendiularly oriented to the stress diretion (an example how to aount for orientation effets is given in 4

5 the paper of Jayatilaka and Trustrum [3]). The relative frequeny of flaws sizes (flaw size a ) an be desribed by a funtion g(a, r) [defets/m 4 ], whih may in general also depend on the position vetor r. (see Fig. ) Defet density g(a) [a.u.] g(a)=g 0 (a/a 0 ) -r a (σ) Flaw size a [a.u.] Fig. : Relative density of flaw sizes versus flaw size. The dashed line shows a typial distribution, the full line the behaviour neessary for a Weibull distribution. The dashed area gives the density of destrutive flaws. The left bound shifts to the left if the stress inreases. A failure riterion onnets the size of the flaw with a ritial load. E.g. the Griffith/Irwin riterion predits that rak like flaws gets ritial, if their stress intensity fator K = σy π a exeeds the frature toughness K I [3, 4]: K K I, (3) Y is a geometri fator, whih for flaws whih are small ompared to the speimen size is of the order of one. The ritial rak size is then: 2 KI a =. (4) π Y σ Fig. shows an example for a relative flaw size density funtion versus the flaw size (dashed a σ will ause failure. Their density is given line). All flaws larger than the ritial value ( ) by the integral (shaded area): n, r d a. (5) ( σ ) = g( a, r ) a ( σ ) It should be noted that the lower integration limit would shift to the left if the stress inreases. Therefore, giving a higher density of destrutive flaws at higher stresses. The mean number of destrutive flaws an be found by integration over the volume 5

6 N ( ) n ( σ, ) σ = dv. (6), S r In general the relative frequeny of flaw sizes g(a,r) is unknown. Of ourse the flaw population is stritly related to the prodution proess in most ases. To give some examples, typial volume flaws are large grains, pores resulting from organi inlusions, pressing defets ore remnants of agglomerates [, 23]. Very often flaws also ome into existene by the mahining of the omponent (surfae flaws) and are then stritly related to the diretion of the movement of the mahining tool [24]. In servie inadequate handling may ause ontat damage (e.g. Hertzian raks [25], edge flakes [26]). It is obvious that a smooth and struture less frequeny distribution of flaw size an hardly be expeted. In the following the influene of some typial frequeny distribution funtions on the frature statistis is disussed. 2. Weibull distribution In addition to the assumptions made above it is assumed 3, that a material only ontains homogeneous distributed volume flaws 4 with a relative flaw size density as given in Eq. 7 (ontinuous line in Fig. ): r a g( a) = g0 a. (7) 0 This funtion has only two parameters effetively: the exponent ( r ) and a oeffiient ( r g 0 a ). 0 g 0 and r are material parameters and a 0 is a normalizing length 5. The integral Eq. 5 gives: n ( a ) = g( a ) a r. (8) For a homogeneous material the integral Eq. 6 is trivial. Using Eq. 4 holds: N, S 2( r ) ( σ ) = Vn ( σ ) V σ. By omparing Eq. and Eq. 2 results: m = 2( r ). (9) 3 A similar derivation an be found in many papers. For more details see [3, 27]. 4 Analogue alulations ould be made for surfae raks or edge raks. 5 A material with a relative frequeny of flaw sizes aording to Eq. 7 is alled Weibull material. 6

7 The same behaviour an also be observed for brittle materials having flaws populations with any orientation [3]. In summary, a Weibull distribution of strength an be expeted for erami materials ontaining sparsely distributed flaws, whih have a relative frequeny of sizes whih follows an inverse power law 6. Suh Weibull behaviour an be expeted at least in a limited interval of flaw sizes for many advaned eramis. Weibull statistis is also applied to desribe the behaviour of speimens and omponents, whih are loaded in general without a uniaxial tensile stress but with a loally varying stress field. In this ase the relationships determined for tensile testing remain valid, if the stress in Eq. is replaed by a suitable equivalent stress (to aount for the ation of multi axial stress states,) and the volume is replaed by the effetive volume 7 [7]. Details an be found in standard text books [2-4]. The relative flaw size density an be determined from a known Weibull statistis (Eq. ) in the following way: For a homogeneous material the density of ritial defets is n σ = N σ. Using Eq. and Eq. 2 this expression an be transformed into ( ) ( ) V m σ n ( ) V σ =. Inserting into Eq. 8 and using the Griffith/Irwin failure riterion (Eq. 4) 0 σ 0 and Eq. 9 results: g m 2V K Yσ 0 π I r ( a) = a 0 m. (0) That expression an be used to identify the material properties presented in Eq. 7 (Remember that m = 2 (r-) ). Fig. 2.a gives an example for an experimentally determined Weibull distribution. Shown are standardised 4-point bending tests [0] on a ommerial silion nitride erami. The material and the tests are desribed in [28]. The tests have been performed aording to ENV 843- [0] and evaluated aording to ENV []. The data are plotted in suh a way, that a Weibull distribution is represented by a straight line [7]. It an be reognised that the 30 data points are niely grouped around the Weibull line. Typial frature origins are glassy regions, whih are - in many ases losely related to iron rih inlusions, Fig. 2b. The Weibull modulus of the sample (ontaining 30 tests) is m = 5 and the orresponding material parameter r = 8.5. The lowest measured strength value is σ f = 65 MPa and the 6 The same behaviour an also be observed for materials with arbitrary oriented flaws [3]. 7 The effetive volume of a omponent is the volume of a hypothetial tensile speimen loaded with the maximum equivalent stress amplitude of the omponent and giving the same probability of failure. 7

8 highest is σ f = 905 MPa. With K = [29, 30]) and using Y = I 4.9 MPa m (determined using the SEVNB method 2/π (the geometri fator of a small penny shaped volume rak) the size of the frature origin an be determined using Eq.4: they are 45 µm and 23 µm respetively. Fig 2. shows the related relative flaw size density (Eq. 0) and the data points, whih orrespond to the Weibull representation in Fig. 2.a. (a) (b) Probability of Failure F [%] m= Strength [MPa] () 2 0 Ln Ln [/(-F)] g [m -4 ] r =8.5 V eff =0-4 mm 3 V eff =7.5 mm 3 V eff =0 4 mm a [µm] Fig. 2: 4-point bending strength test results on a silion nitride material: (a) Probability of failure versus strength in a Weibull diagram: the strength data are aligned along straight line and are therefore Weibull distributed, (b) frature origins are glassy regions, whih are often related to iron inlusions, the arrows mark the boundary of the glassy region; and () relationship between the relative frequeny density of flaw sizes and the flaw size. The effetive volume of the speimens is 7.5 mm 3. Also indiated are the expeted regions for the ourrene of frature origins for one sample ontaining 30 speimens having a muh larger volume and another sample ontaining 30 speimens having a muh smaller effetive volume. A method to determine the data points an be found in [27]). It an be reognised that due to the high exponent (r = 8.5) the variation in the density is higher than two orders of magnitude. The expeted interval of the sizes of frature origins for sets of 30 speimens with an effetive 8

9 4 volume of V = 0 mm 3 and V = 0 4 mm respetively are also indiated in Fig. 2. For eff eff larger speimens, the interval of the sizes of the frature origins shifts to the right and for smaller speimens to the left. The width of the interval is determined by the sample size. This simple evaluation shows, that the Weibull distribution desribes the situation in materials with a speial (but often ourring; at least in a limited range of parameters) flaw size distribution. If Weibull statistis is used to alulate a tolerable strength for a low failure probability, extrapolations out of the empirially investigated parameter range has to be made (to the densities of larger flaws). 2.2 Unimodal flaw size distributions It is lear from the preeding setions that the frature statistis of brittle materials is strongly related to the flaw population. The Weibull distribution is the most prominent example for a unimodal distribution. But it an only be valid at intermediate flaw sizes. At one hand side for a L ( L being the size of the speimen) it holds g ( a) > 0 and the same holds for the number of destrutive flaws. On the other hand side for a 0 holds g ( a). Both are not realisti assumptions. There exist also other limitations of the Weibull theory. From Eq. it follows that the strength of a sample inreases with dereasing sample size. For the same probability of failure it holds: m m Vσ = V2σ 2 () Therefore in the mean the ritial flaw size is smaller in small speimens than in large ones and the density of ritial flaws in small speimens is higher. In a Weibull material the relative frequeny density may even be so high that the flaws interat. Then the weakest link theory is not longer appliable [27]. It an be assumed that under the ation of tensile stresses the flaws grow together and form larger flaws, whih limit the strength of small speimens [27]. A similar behaviour is expeted to our in brittle porous materials. A different behaviour ours for materials ontaining flaws of a single size (or if their size is narrow peaked, Fig. 3.a) Then the probability of failure is a step funtion of the stress amplitude. It is zero ( F = 0 ) for σ < σ and it is F = exp( V n ) for σ σ Fig. 3.b). The ritial stress σ is the failure stress for a speimen ontaining flaws aording to the failure riterion (Eq. 3) and n is the density of ritial flaws. It is obvious that the probability of failure is smaller than one, even for very high stresses and for speimens with a very high volume. This reflets the fat that in the theory of brittle frature only brittle failure from sparsely distributed flaws is taken into aount and that there remains always some 9

10 probability that a single speimen does not ontain one ritial flaw. Hunt and MCarthy [6] disussed that ase and laimed that at high enough stresses other failure modes (e.g. plastiity indued failure) must ome into existene. This ase is also indiated in Fig. 3.b (vertial dotted line). (a) (b) Defet density g(a) [a.u.] g(a)=g 0 δ(a-a ) Probability of failure F V > V 2 > V 3 V V 2 V 3 plastiity indued failure a 0 Flaw size a [a.u.] σ Stress σ [a.u.] σ y Fig. 3: Narrow peaked (delta funtion) flaw distribution: (a) relative flaw size density (density of defets: g 0 = [m -4 ]; a 0 = 26. [µm]). (b) Corresponding frature statistis. The probability of finding a flaw in a speimen depends on the speimen size and the same does the probability of failure. Note that the probability of failure does not reah one in any ase. But if the stresses get to high additional failure mehanisms beome important (e.g. plastiity indued failure), whih limit the strength of the material. Then the probability of failure is one at stresses higher than the yield strengthσ. y 2.3 Bi- and multi modal flaw size distributions Flaws are inhomogeneities in the mirostruture, whih result from the proessing, the mahining or the handling of the speimens. Examples in erami materials are inorgani (Fig. 2.b) or organi inlusions, hard or hollow agglomerates, badly sintered grain boundaries, large grains or raks arising from the mahining [23]. It is obvious that eah individual flaw population will have its typial size distribution. This has strit onsequenes on the frature statistis. This aspet will be disussed with two examples. The first example ours quite often when testing brittle materials. In a sample of speimens made from a brittle material two types of frature origins are found frequently: surfae flaws (arising from an inadequate mahining) and volume flaws (arising from the material proessing). The number of ritial flaws per speimens is N, S ( σ ) = N, σurfae( σ ) + N, volume( σ ) and eah population an be evaluated separately. An example for the resulting frature statistis is shown in Fig. 4. For the alulation it has been assumed that the volume as well as the surfae flaws are Weibull distributed. For the volume flaws the population of the 4-point bending test data shown in Fig. 2 is used. For the surfae flaws the Weibull modulus is hosen 0

11 to be m = 7. For the relationship desribing the speimens with intermediate size (standard bending test speimens) it is assumed, that at σ = σ 0 and V = V0 the ritial surfae flaws our as frequent as the ritial volume flaws. The probability range observable after testing 30 speimens is shaded. In the upper part of this range the volume flaws ause a bend in the strength distribution. The urve onerning the largest speimen (smallest speimen) desribes the behaviour of bending speimens, whih are linearly blown up (saled down). The effetive volume is determined in the usual way [3, 4] and the effetive surfae orresponds to the surfae area underneath the inner loading rolls. Note that for a sample of 30 speimens and for the largest speimen only surfae flaws are responsible for frature ( m = 7), but for the smallest speimen only volume flaws ause frature ( m = 5). This behaviour is often observed in the daily testing pratie [24]. The position of the strutures in the frature statistis (kink) depends on the stress as well as on the size of the speimen. Ln Ln [/(-F)] V (7.5 mm 3 ) V 2 (0-4 mm 3 ) V 3 (0 4 mm 3 ) m= , , -8 m=7 0, Ln (σ /σ norm ) Probability of failure F [%] Fig. 4: Frature statistis of samples ontaining volume defets (steep straight lines) as well as surfae flaws (less steep straight lines). The probability range reognised by a sample of 30 speimens is shaded. It should be mentioned that, in this ase, for small speimens only volume flaws and for large speimens only surfae flaws are relevant for failure. For an intermediate speimen size surfae as well as volume flaws may ause failure. The seond example reflets the behaviour of a material ontaining two different volume flaw populations, the first one (I) orresponds to the population desribed in Fig. 2b and the seond one (II) is the narrowed peaked population of Fig. 3.a. The number of ritial flaws per speimens is: N, S ( σ ) = N, I ( σ ) + N, II ( σ ). The first population auses a Weibull distribution (Fig. 2.a) and the seond a step distribution, Fig. 3.b. The sum of both populations is shown in Fig. 5.a. Also shown is the number of ritial defets versus the defet size. The narrow peaked population auses a knee like struture in the density of ritial defets, whih is also mapped in the frature statistis. For the largest speimens only volume flaws are frature origins in a sample of 30 speimens, for the speimens of intermediate size, both populations may ause failure and for the smallest speimens only the population II is relevant for failure (the orresponding Weibull modulus would be infinite). Note that the Weibull modulus (the slope on the urves in the Weibull diagram) depends on the applied stress.

12 g [m -4 ] (a) g n a [µm] n [m -3 ] Ln Ln [/(-F)] (b) V (7.5 mm 3 ) V 2 (0. mm 3 ) V 3 (000 mm 3 ) m Ln (σ /σ norm ) 99 63,2 0 0, 0,0 Probability of failure F [%] Fig. 5: Example for a bimodal frature statistis: (a) relative frequeny of flaw sizes and density of ritial flaws. (b) Probability of failure versus strength. Different urves refer to different speimen sizes. The shaded area orresponds to the probability range observable after testing 30 speimens. Analogue results an be expeted for other types of bi- modal and multi- modal flaw populations. Strutures in the strength distribution appear due to strutures in the flaw size distribution and vie versa. They our in a typial stress interval and do not depend on the speimen size. It should be noted that in both disussed ases in this setion the strength distribution is not a Weibull distribution. 2.4 Ceramis with inreasing rak resistane urve If the frature toughness inreases after ertain rak extension (R-urve behaviour) some stable rak growth a a 0 + a before frature is possible. Then the Griffith/Irwin frature riterion has to be supplemented by the ondition K / a R / a as demonstrated by frature mehanis. ( R being the rak resistane whih depends on the rak extension a : K I R = K I, 0 + K I ). In this ase the strength depends not only on the initial toughness and flaw size but also on the inrease of toughness and rak length: σ ( ) ( ( )) 2 f = K I, 0 + K I π a0 + a. The inrease in frature toughness, K I as well as the rak extension, a, depend on the shape of the R-urve but also on the size of the initial flaw, a 0. For details see standard textbooks, e.g. [3, 4]. 2

13 5 K IC,0 = 3 MPa m /2 K IC,plateau = 8 MPa m /2 LnLn [/(-F)] V =0 5 mm V =7.5 mm 3 K=K IC,0 V=0-5 mm 3 3 Ln σ/σ norm m = 5 5 K=K IC,plateau Fig. 6: Probability of failure versus strength in a Weibull plot for a material with a pronouned R-urve behaviour: The three solid lines refer to speimens with different volume. The hathed area is the range observable with 30 strength tests. At low (not showed in the plot) and high failure stresses the urves are straight, whih indiates a Weibull behaviour, but in the intermediate stress range, the urves are bent. This auses a stress dependent Weibull modulus and the frature statistis has a non Weibull behaviour. The steep slope of the urves at intermediate stresses indiates a restrited satter of strength dated in that range. The onsequenes of suh behaviour for frature statistis are shown in Fig. 6, where the probability of failure is plotted versus strength in a Weibull diagram 8. The R-urve is seleted in suh a way, that it orresponds to R-urves of published for silion nitride materials [3]. The defet population is that of a Weibull material ( m = 5). The bold urve approahes the dotted Weibull line (for R = K I, 0 ) at the high stress side and the Weibull line (for = ) at the low stress side of the distribution. At intermediate stresses, this R K, plateau distribution moves from one line to the other. This results in a redued satter of strength data (inreased modulus) in this parameter range and auses again a stress dependent modulus. In the used example it reahes its maximum ( m = 3) in the probability range, whih is observable with 30 speimens (shaded area). For speimens with a muh larger volume only the plateau region of the frature toughness is relevant in the plotted stress range. This is benefiial for the strength (see Eq. 3) but has almost no influene on the satter of the strength data (on the modulus). Again strutures in the statistis an be found in the same stress region and do not depend on the speimen size. Again, the strength is not Weibull distributed 2.5 Other effets influening the Weibull modulus and the strength distribution 8 Data used for the onstrution of the diagram: initial frature toughness K I,0 = 3 MPa m /2 ; the R-urve inreases to the plateau value K I, plateau = 8 MPa m /2 aording to ( ), (,,0 ) a λ R = K I pλat K I pλat K I e.the harateristi length λ is 00 µm. The initial defet population is that shown in Fig. 2. 3

14 There exist many other effets, whih may affet the shape of the strength distribution, whih ause a strength and volume dependent Weibull modulus or, in other words, ause a non- Weibull strength distribution. Some examples are mentioned in the following. A simple example is the ase of speimens ontaining internal stresses. Then the internal stresses have to be added up with the applied stresses. For internal ompressive stresses (e.g. surfae stresses, whih are relevant in bending testing) this an ause a threshold stress for failure. This may lead to the behaviour desribed by the tree parameter Weibull distribution. In general internal stresses ause a stress dependent Weibull modulus and lead therefore to a non-weibull strength distribution. Similar effets are also laimed for materials with a nonhomogeneous mirostruture by Danzer et al. [9]. Fett et al. [32] analyzed the ase of steep stress gradients, as they our due to ontat loading or thermal shok loading. In this ase the stress amplitude is not longer (approximately) onstant over the rak length. This has strong influene on the stress intensity fator K, whih in the ase analyzed by Fett et al. [32], gets proportional to the rak length. In onsequene and in a loal sale, the Weibull modulus m gets approximately proportional to the parameter r, whih desribes the distribution density of the initial flaw sizes. Zimmerman [33] analysed the ase of a rak in front of a pore (suh defets are often observed in eramis produed by pressing and sintering spray dried powders; in this ase the pore is overed by many small raks, their size is of the order of the size of the grains). For that ase (pore diameter is large ompared to the grain diameter) the ation of the pore is a rise of about three times the applied stress but the size of the pore has no longer an influene on the strength. Under these onditions the strength depends on the size of the rak in front of the pore, i.e. in general the size of the surrounding grains and no longer on the size of the pore. In onsequene the size effet on strength disappears. Similar behaviour is observed in materials ontaining flaws of suh high densities that interation between flaws gets possible [34]. 2.6 Size effet on strength As mentioned in the introdution the size effet on strength, whih is desribed by Eq., is one of the must important features of the frature statistis of brittle materials Weibull material For the strength data shown in Fig. 2 (standard 4-point bending tests on standard Si 3 N 4 speimens), the onsequenes are shown in Fig. 7.a. Plotted is the harateristi strength of the sample σ 0 versus the effetive volume of the speimens in a double logarithmi sale. The expeted hange of the strength with the volume predited by Eq. is shown by the bold line. The dashed lines give the 90 % onfidene intervals for the Weibull strength predition. 4

15 Also shown are further bending test results of samples ontaining speimens with smaller and larger effetive volume 9. It an be reognised, that they are within the 90 % onfidene limits of the predited behaviour. This fat onfines their onsisteny with the Weibull theory. In this ase the Weibull modulus does not depend on the stress, i.e. in the investigated range of parameters; the material is a Weibull material. But it an be also reognised that the measured strength values are on the borders of the predition limits. The data would fit better to a straight line with slope m = /20. In fat the mean modulus of the other three samples is around 9. It an be onluded that a data fit on the basis of the size effet give more reliable results than a fit to an individual sample of a Weibull distribution. Suh a proedure an be arried out with very small sub-samples for the individual speimen sizes, sine the mean (harateristi) strength of the sample an be determined with a few tests (approximately 0). The typial ritial flaw size is related to the speimen size. By testing different volumes a spread of the flaw size interval is reahed (see Fig. 2.) and the determination of the flaw size density an be done on a wider base Bimodal strength distribution In Fig. 7.b the size effet on strength is demonstrated for the bimodal distribution of Fig. 5. The narrow peaked flaw population II has only influene on the strength in a limited interval of speimen sizes and stresses (in this interval the volumes vary four orders of magnitude). In this range the strength may even be independent of the speimen size. Outside of this range the population II is not relevant and frature is triggered by flaws from population I, whih is that of a Weibull material. The dashed line shows the behaviour of a material, whih only ontains flaws of type I. The effetive volumes of the speimens disussed in Fig. 5 are also indiated in Fig. 7.b. It an be reognised that for speimens with the effetive volume V or V 2 the harateristi strength depends on the flaws of population (II) and for speimens with the effetive volume V 3 it depends on the flaws (I). This example shows that deviations from the Weibull behaviour an be read off a strength volume diagram for the same reason as disussed in the last paragraph R urve behaviour Fig 7. shows the size effet on strength of a material with R-urve behaviour (Fig. 6). Again a deviation of the Weibull behaviour ours, whih in this ase is a onsequene of the ontrolled rak extension and the related inrease in toughness. This effet depends on the initial flaw size. As disussed before that fat relates the strength and the volume of the speimen. Again the material behaves like a Weibull material at very small volumes (where no stable rak extension ours and the toughness keeps its original value). For very large volumes the materials toughness reahes asymptotially the plateau value and again the frature statistis looks like a Weibull statistis. It should be noted that the variation of volumes in the diagram is extremely wide. Suh a wide variation an not our in tehnial appliations. For a realisti range of volumes the disussed example auses an inreased 9 The span lengths orresponding to the smallest effetive volume are 3 and 4.33 mm, for the largest effetive volume are 80 and 40 mm 5

16 apparent modulus (around 30). This orresponds to a redued satter of strength data and is relevant in the design of more reliable materials. In summary, testing speimens of different volumes helps to understand the statistial behaviour of brittle materials. The main advantage to the usual pratie of testing samples ontaining speimens of a unique size is the possibility to ollet information about ritial defets in different sizes range with a limited experimental effort (see Fig 2.b). σ 0 [MPa] (a) /m / V eff [mm 3 ] σ /σ norm (b) V 2 V V [mm 3 ] V 3 4 () σ 0 /σ norm K=K IC,plateau V [mm 3 ] K=K IC,0 Fig. 7: Strength versus speimen s volume in a double logarithmi plot for (a) the Weibull material (shown in Fig. 2), (b) a material with a bimodal flaw population (shown in Fig. 5) and () a material with an R-urve behaviour (shown in Fig. 6). In part (a) experimental bending test results, whih are onsistent with the Weibull theory, are also plotted. 3. Monte Carlo simulations sample and population In the above paragraphs many arguments were given that the frature statistis of a brittle material should in general not follow a Weibull statistis. But it is laimed in almost any experimental work on the strength of eramis, that the strength of a sample is Weibull distributed. Arguments to larify this disrepany will be given in the following. In the following a Monte Carlo simulation tehnique is used to simulate virtual experiments to judge how preise a sample (of 30 speimens) an desribe the whole population: Following the onept of Monte Carlo, random numbers between 0 and are generated. The number of random numbers is equal to the size of the sample. Then - for a given strength distribution - the orresponding strength values are determined (via virtual strength testing, [20]). In this 6

17 way samples of the population an be easily determined. The great advantage of this proedure is, that on the basis of a preisely known population the behaviour of samples an be studied. Fig. 8 shows some virtual strength distributions, whih result from the Monte Carlo proedure desribed above. The used population is the frature statistis shown in Fig. 2. Following the instrutions of the standards, e.g. [], the sample size is N = 30. Of ourse, some of the samples show the Weibull behaviour orresponding to the population (Fig. 8.a), but many of them look like bimodal or other non-weibull distributions. A lassifiation in different types is shown in Fig. 8. The examples are seleted out of 50 sample simulations. Although the differentiation between different types is a little diffuse, Table presents a rough impression about the relationship between sample and the whole population. Sine the distribution of type and e look quite similar and was relatively rare, they have been put into a single lass. Ln Ln [/(-F)] Ln Ln [/(-F)] a) d) b) e) Ln σ /σ norm Ln σ /σ norm Ln σ /σ norm ) f) Fig. 8: Strength distributions generated by Monte Carlo simulations. The dashed line indiates the origin population and the solid lines indiate different types of samples: (a) the sample represents the population quite well, (b) the sample fits a Weibull distribution, but has a quite different modulus and/or mean strength, () the sample seems to follow a bimodal distribution of Fig. 4, (d) the sample ould fit a bimodal distribution as shown in Fig. 5, (e) the sample seems the distribution of an R-urve material as shown in Fig. 6, (f) the sample fits a three parameter distribution ourring due to internal ompressive stresses. The authors have also done Monte Carlo simulations of samples produed from the non Weibull distributions disussed above (bimodal, et.). It was not surprising, that in any ase many samples ould be found whih looked like a Weibull distribution. This problem is also disussed in some of the papers of Lu et al [9, 35]. In the relevant parameter range a lear distintion between different statistial distribution funtions would make the testing of 7

18 several thousand speimens neessary [9]. Under these onditions it is lear that a Weibull distribution an be adjusted to any small sample of a frature statistis of a brittle material. But it should be reognised, that this does not neessarily mean, that the population is a Weibull distribution. Table. Classifiation of samples aording to the types shown in Fig. 8. Sample lass a b d f + e Weibull Weibull bimodal ompressive R-urve or bimodal m* m a m* m narrow peak internal stress wide populations estimated fration [%] a where m* is the modulus of the sample and m the modulus of the population 4. Summary and onlusions The frature statistis of brittle materials reflets the size distribution of flaws in brittle materials. A Weibull statistis only ours under speial onditions, whih are not valid in general. Espeially a Weibull distribution is not expeted to our for brittle materials ontaining bi- or multimodal flaw distributions, surfae and volume flaws, having an R-urve behaviour, showing internal residual stress fields, or having a high defet density. It is also not expeted to our if very small speimens are tested or if the applied stress field presents high gradients. In all these ases the Weibull modulus is no longer a onstant and depends on the applied stress amplitude. The strength distribution is a mapping of the flaw size distribution. The size (volume) of the speimen defines the mean strength and therefore also the mean size of the mapped ritial flaw. The size of the sample defines the width of the mapped flaw interval. This width is relatively small for standardized sample sizes (e.g. N = 30). Therefore the fitting of a strength distribution to data in that (small) interval is relatively unreliable (e.g. for a set of 30 strength tests the typial width of the 90 % onfidene interval of m is ± 30 %). The reliability of the determination of the strength distribution an be improved if the width of the mapped flaw size interval is inreased. This an be done by testing speimens of different size (i.e. effetive volume). Then the determination of the strength statistis (Weibull parameters) is muh more reliable. Published strength data based on one single sample are generally laimed to be Weibull distributed but this needs not be true. In general, they are determined on small samples sine the mahining of speimens is very expensive. On the basis of a small sample it is not possible to deide whether a distribution is a Weibullian or not. 8

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