Strength modeling using Weibull distributions

Transcription

1 Joural of Mechaical Sciece Techology Joural of Mechaical Sciece Techology 22 (2008) 247~254 Stregth modelig usig Weibull distributios Saralees Nadarajah,* Samuel Kotz 2 School of Mathematics, Uiversity of Machester, Machester M60 QD, U.K. 2 Departmet of Egieerig Maagemet Systems Egieerig, The George Washigto Uiversity, Washigto DC 20052, U.S.A. (Mauscript Received March 8, 2006, Revised April 24, 2008; Accepted April 24, 2008) Abstract The two-parameter Weibull distributio is the most popular model for material stregth. However, it may ot be a good model for all materials over a wide rage of sizes. I this ote, a comprehesive review of the kow variatios of the two-parameter Weibull distributio is provided to help providig better modelig. Over 20 variatios are reviewed. The appropriateess of the variatios is discussed as models for brittle versus ductile stregth. A compariso study of a selectio of the variatios is also provided. It is hoped that this review will also serve as a importat referece ecourage developmets of further variatios of the two-parameter Weibull distributio. Keywords: Material stregth; Statistical modelig; Weibull distributio Itroductio The most widely used distributio for characterizig material stregth is the two parameter Weibull distributio. If X is a rom variable represetig some material stregth the the probability distributio fuctio (pdf) the cumulative distributio fuctio (cdf) of the Weibull distributio ca be expressed as px ( ) = λ( λx) exp{ ( λx) } () Px ( ) = exp{ ( λx) }, (2) * Correspodig author. Tel.: , Fax.: address: saralees.adarajah@machester.ac.uk DOI 0.007/s respectively, for x > 0, λ > 0 > 0. The parameters λ are referred to as the scale shape parameters, respectively. The Rayleigh Levy distributios are the particular cases of ()-(2) for = 2 = /2, respectively. The th momet of X associated with ()-(2) is E( X ) = λ Γ (( + ) / ). While ()-(2) has bee show to be a good approximatio for the distributio of average stregth of may brittle materials, it may ot be a good represetatio for all materials over a wide rage of sizes. This pheomeo is especially sigificat because it might result i a serious stregth overestimatio or uderestimatio. This problem could be remedied by usig oe of the may geeralizatios of ()-(2) that have bee proposed i the statistical literature. The aim of this ote is to help this by providig a comprehesive review of the kow variats of ()-(2). Over 20 variatios of ()-(2) are reviewed-formulas for the distributio fuctios the momets are give for each. Apart from helpig to provide better modelig, it is hoped that this review will serve as a importat referece ecourage developmets of further geeralizatios of ()-(2). We also discuss the appropriateess of the variatios as models for brittle versus ductile stregth provide a compariso study of a selectio of the variatios usig six data sets o fracture toughess. 2. Reflected Weibull distributio If X has the Weibull distributio give by () the -X is said to have the reflected Weibull distribu-

2 248 S. Nadarajah S. Kotz / Joural of Mechaical Sciece Techology 22 (2008) 247~254 tio. The pdf the cdf are give by p( x) ( x) = λ λ exp{ ( λx) } (3) Px ( ) = exp{ ( λx) } (4) respectively, for < x < 0, λ > 0 > 0. The associated mea the variace are λ Γ ( + / ) 2 2 λ { Γ (+ 2/ ) Γ (+ / )}, respectively. For further details of this distributio see Cohe []. 3. Double Weibull distributio A distributio made up by piecig () (3) together is the double Weibull distributio [2]. The pdf the cdf are give by px ( ) = ( λ / 2) λx exp{ λx } (5) (/ 2)exp{ λx}, if x< 0, Px ( ) = (6) (/ 2)exp{ λx}, if x 0, respectively, for < x <, > 0 λ > 0. The th momet correspodig to (5) is: λ Γ (( + ) /, if is eve, E( X ) = 0, if is odd. 4. Log Weibull distributio If X has theweibull distributio give by () the logx is said to have the logweibull distributio. The pdf the cdf take the forms x a x a px ( ) = exp exp exp b b b x a Px ( ) = exp exp, b (7) (8) respectively, for < x <, < a <, b > 0. The mea the variace of this distributio 2 are a γ b π b 2 /6, respectively, where γ deotes the Euler s costat. For further details see White [3]. 5. Iverse Weibull distributio If X has the Weibull distributio give by () the /X is said to have the iverse Weibull distributio. The pdf the cdf take the forms px ( ) x = λ exp{ ( x/ λ) } (9) Px ( ) = exp{ ( x/ λ) }, (0) respectively, for x > 0, > 0 λ > 0. The th momet correspodig to (9) is: E( X ) = λ Γ(( ) / ). This distributio is also referred to as the complemetary or the reciprocal Weibull distributio. For further details see Drapella [4] Mudholkar Kollia [5]. 6. Trucated Weibull distributio A doubly trucated versio of () is give by the pdf the cdf px ( ) qx ( ) = () Pb ( ) Pa ( ) Px ( ) Pa ( ) Qx ( ) = (2) Pb ( ) Pa ( ) respectively, for 0 a< x< b<, where px ( ) Px ( ) are give by () (2), respectively. The associated th momet is give by exp{( λb) } E( X ) = λ Γ +,( λb) exp{ ( λa) } Γ +, ( λa), where Γ (, ) is the complemetary icomplete gamma fuctio defied by ( a, x) a t exp( t) dt. x Γ = For further details about this distributio see McEwe Parresol [6]. 7. Marshall Olki s Weibull distributio Marshall Olki [7] proposed a exteded Weibull distributio give by the pdf νλ( λx) exp{ ( λx) } px ( ) = (3) ( ν)exp{ ( λx) } for x > 0, ν > 0, λ > 0 > 0. The Weibull distributio i () is the particular case of (3) for

3 S. Nadarajah S. Kotz / Joural of Mechaical Sciece Techology 22 (2008) 247~ ν =. The cdf the th momet associated with (3) are: exp{ ( x) } Px ( ) = ν λ ( )exp{ ( x ) ν λ } k λ ( v) E( X ) = Γ, / k= 0 ( k + ) respectively, with the latter beig valid for ν. 8. distributio The pseudo Weibull distributio due to Voda [8] is give by the pdf + px ( ) = xλ { Γ ( + / )} exp{ ( λx) } (4) for x > 0, > 0 λ > 0. Note that this form is obtaied by multiplyig () by a additioal x term. The cdf the th momet correspodig to (4) are : γ( + /, ( λx) ) Px ( ) = Γ ( + / ) Γ ( + ( + ) / ) E( X) =, / λ Γ ( + / ) respectively, where γ (, ) deotes the icomplete gamma fuctio defied by γ x ( a, x ) t a = exp( t ) dt distributio Stacy [9] proposed a distributio with the pdf c c c px ( ) = c { Γ( )} x exp{ ( x/ ) } (5) for x > 0, c > 0, > 0 > 0. The Weibull distributio i () arises as the particular case of (5) for =. The cdf the th momet correspodig to (5) are: c x Px ( ) = { Γ( )} γ, E( X ) = Γ ( + / c)/ Γ ( ), respectively. A further extesio of (5) due to Ghitay [0] has the pdf give by ( ) ( m )/ + λ m 2 λ x + x x px ( ) = exp Γλ (( m ) /, ) for x > 0, > 0, > 0, λ > 0, m> 0 > 0, where Γ (, ) is defied by λ a y exp( y) Γ λ ( a, x) = dy. x λ ( y+ x) 0. Expoetiated Weibull distributio Mudholkar et al. [] itroduced the expoetiated Weibull distributio give by the pdf the cdf px ( ) a x exp{ ( x) }[ exp{ ( x) }] a Px ( ) = [ exp{ ( λx) }], (7) respectively, for x > 0, a > 0, > 0 λ > 0. The Weibull distributio i () is the particular case of (6) for a =. The th momet associated with (6) is give by λ = λ λ (6) ( a) i E( X) = aλ Γ +, ( + )/ i= 0 i!( i+ ) which ca be reduced to the simpler form E( X) ( a) a i = λ Γ + ( )/ + i= 0 i!( i+ ) if a is a iteger. For further details of this distributio see Mudholkar Hutso [2], Nassar Eissa [3] Nadarajah Gupta [4].. Xie et al. s weibull distributio Xie et al. [5] proposed a modificatio of the Weibull distributio give by the pdf the cdf x x x px ( ) = λ exp + λ exp (8) Px ( ) exp exp x = λ (9) ε for x > 0, λ > 0, > 0 > 0. The Weibull distributio i () arises as the limitig case of (8) as. Expressios for the momets associated with (8) caot be obtaied i closed form. However, if k/ = is a iteger the the kth momet ca be

4 250 S. Nadarajah S. Kotz / Joural of Mechaical Sciece Techology 22 (2008) 247~254 expressed as ν k k ( λ) Γ( ν, λ) E( X ) = exp( λ) (20) ν for =, 2,..., where the derivative is evaluated as ν 0. For further details of this distributio see Che [6], Tag et al. [7], Wu et al. [8] Nadarajah [9]. 2. distributio Aother modificatio of the Weibull distributio developed by Lai et al. [20] is give by the pdf the cdf p( x) ( vx) x = λ + exp( νx)exp{ λx exp( νx} (2) Px ( ) = exp{ λx exp( νx)}, (22) respectively, for x > 0, λ > 0, > 0 ν > 0. The Weibull distributio i () is the particular case of (2) for ν = 0. The log-webiull distributio i (7) is the particular case of (2) for = Geeralized Weibull distributios Mudholkar Srivastava [2] Mudholkar et al. [] developed three differet geeralized Weibull distributios. The first of these has the pdf the cdf give by λ ( λx) / ν px ( ) = ν( λx) ν( λx) (23) / ν Px ( ) = ν( λx) (24) respectively, for 0 < x < (if ν 0 ), 0 < x < / λν (if ν > 0 ), λ > 0 > 0. The Weibull distributio i () arises as the limitig case of (23) for ν 0. The th momet associated with (23) is: / E( X ) ν = B (/ ν, / + ). The secod geeralizatio is give by the cdf / ν Px ( ) = ν( + x/ ) (25) for < x < (if < 0 ν < 0 ), < x < / ν (if < 0 ν > 0), < x < (if / > 0 ν < 0 ), < x< v (if > 0 ν > 0 ). The iverse Weibull (see Sectio 5) is a particular case of this distributio. The th momet associated with (25) is: k / E( X ) = ( ) ν B(/ ν, / + ). k= 0 k The third geeralizatio discussed i Mudholkar Srivastava [2] Mudholkar et al. [] is give by the cdf { x } exp ( + / ), if 0, Px ( ) = exp{ exp( x)}, if = 0 (26) for < x < (if < 0 ), < x < (if > 0 ) < x < (if = 0 ). For further details about these geeralizatios see Mudholkar Kollia [5]. 4. Four five-parameter weibull distributios Phai [22] developed a four-parameter Weibull distributio give by the cdf { λ } Px ( ) = exp [( x a)/( b x)] (27) for 0 a< x< b<, λ > 0 > 0. A fiveparameter extesio of this give by Kies [23] has the cdf 2 Px ( ) = exp { λ( x a) / b( x) } (28) for 0 a< x< b<, λ > 0, > 0 2 > Mixture Weibull distributios A additive twofold Weibull mixture distributio ca be expressed by the pdf p( x) = p ( x) + ( ) p ( x) (29) 2 for 0< <, where pi, i =, 2 could take the form of ay of the pdfs discussed above. The cdf correspodig to (29) is: Px ( ) = Px ( ) + ( ) P( x), 2 where P i is the cdf correspodig to pi. Multiplicative twofold mixtures ca be defied by the cdfs Px ( ) = PxP ( ) ( x) 2 Px ( ) = { Px ( )}{ P( x)} 2 for 0< < 0< <. Mixtures of more tha two compoets ca be defied i the obvious maer.

5 S. Nadarajah S. Kotz / Joural of Mechaical Sciece Techology 22 (2008) 247~ Brittle vs Ductile stregth Suppose X, X2,, X are idepedet idetically distributed rom variables, represetig the stregth of items. If the distributio of max ( X, X2,, X ) belogs to the same distributio family of X, X2,, X the the distributio will be a appropriate model for ductile stregth. O the other h, if the distributio of mi ( X, X2,, X ) belogs to the same distributio family of X, X2,, X the the distributio will be a appropriate model for brittle stregth. Of the various Weibull distributios discussed i Sectios to 5, The followig will be appropriate models for brittle stregth: the stardweibull distributio (Eq. (2)); the log Weibull distributio (Eq. (8)); Xie et al. s Weibull distributio (Eq. (9)); Lai et al. s Weibull distributio (Eq. (22)); the geeralized Weibull distributios (Eqs. (24), (25) (26)); the four five parameterweibull distributios (Eqs. (27) (28)). The followig will be appropriate models for ductile stregth: the reflected Weibull distributio (Eq. (4)); the iverse Weibull distributio (Eq. (0)); the expoetiated Weibull distributio (Eq. (7)). The doubleweibull distributio give by (6) ca be used to model both brittle ductile stregths. 7. Compariso study I this sectio, we provide a compariso of the various Weibull distributios discussed i Sectios to 5. We use fracture toughess data from the six differet materials: Bi 2 Sr 2 CaCu 2 O 8+x, Alumia (Al 2 O 3 ), Silico Nitride (Si 3 N 4 ), Sialo (Si 6-x Al x O x N 8-x ), Pyroceram 9606, Titaium Diboride (TiB2). These data are take from the web site: For the iterest of the readers, we have reproduced the data i Table. Some summary statistics of the data are give i Table 2. Note that Bi 2 Sr 2 CaCu 2 O 8+x Sialo (Si 6-x Al x O x N 8-x ) are two of the toughest materials while Titaium Diboride (TiB 2 ) is the lightest material. We fitted eight of theweibull distributios to each of the six data sets: the stardweibull distributio (Eq. ()), iverse Weibull distributio (Eq. (9)), Marshall Olki s Weibull distributio (Eq. (3)), pseudo Weibull distributio (Eq. (4)), distributio (Eq. (5)), expoetiated Weibull Table. Fracture toughess data for six differet materials. Material Material Titaium Diboride (TiB 2 ) Fracture toughess data (i the uits of MPa m /2 ) 3.7,5.75,4.25,6.4,4.87,2.3,5.4,4.6,6,5.2, 5.36 Material 2 Pyroceram ,3.7,2.69,2.4,2.07,2.8,2.5,2.25 Material 3 Sialo (Si 6 x Al x O x N 8 x ) Material 4 Silico Nitride (Si 3 N 4 ) Material 5 Alumia (Al 2 O 3) 3.05,2.9,2.75,2.7,2.65,3.5,3.75,3.8,3.72, 3.52,3.44,3.26,2.99,2.79,3,3.8,3.66,3.2, 3.3,3.5,3.,4.65,3.42,3.38, ,7.2,3.2,4.96,7.8,6.59,4.9,4.,4.5,4.7, 3.2,2.7, ,5,4.9,6.4,5.,5.2,5.2,5,4.7,4,4.5,4.2, 4.,4.56,5.0,4.7,3.3,3.2,2.68,2.77,2.7, 2.36,4.38,5.73,4.35,6.8,.9,2.66,2.6,.68,2.04,2.08,2.3,3.8,3.73,3.7,3.28, 3.9,4,3.8,4.,3.9,4.05,4,3.95,4,4.5,4.5, 4.2,4.55,4.65,4.,4.25,4.3,4.5,4.7,5.5, 4.3,4.5,4.9,5,5.35,5.5,5.25,5.8,5.85,5.9, 5.75,6.25,6.05,5.9,3.6,4.,4.5,5.3,4.85, 5.3,5.45,5.,5.3,5.2,5.3,5.25,4.75,4.5,4.2, 4,4.5,4.25,4.3,3.75,3.95,3.5,4.3,5.4,5, 2.,4.6,3.2,2.5,4.,3.5,3.2,3.3,4.6,4.3,4.3, 4.5,5.5,4.6,4.9,4.3,3,3.4,3.7,4.4,4.9,4.9,5 Material 6 3.2,3.9,2.7,3.2,.9,.2,.8,.4,.8,2.9,2.8, Bi 2 Sr 2 CaCu 2 O 8+x 2.4 Table 2. Summary statistics of fracture toughess data. Mea Media Miimum Maximum Stard devi atio Material Material Material Material Material Material distributio (Eq. (6)), distributio (Eq. (8)) the distributio (Eq. (2)). The fittig of the distributios was performed by the method of maximum likelihood. For example, for fittig the stard Weibull distributio give by () we maximized L(, λ) = λ xi exp λ xi i= i= or equivaletly log L(, λ) = log + log λ + ( ) log xi λ x i i= i= (where x i, i =, 2,..., are the observed fracture toughess) with respect to the two parameters

6 252 S. Nadarajah S. Kotz / Joural of Mechaical Sciece Techology 22 (2008) 247~254 Table 3. Fitted models estimates. Model Parameter estimates NLLH Material : Titaium Diboride (TiB 2 ) Stard Weibull Iverse Weibull Marshall & Olki s Weibull Expoetiated Weibull Material 2: Pyroceram 9606 Stard Weibull Iverse Weibull Marshall & Olki s Weibull Expoetiated Weibull Material 3: Sialo (Si 6 x Al x O x N 8 x ) Stard Weibull Iverse Weibull Marshall & Olki s Weibull Expoetiated Weibull Material 4: Silico Nitride (Si 3 N 4 ) Stard Weibull Iverse Weibull Marshall & Olki s Weibull Expoetiated Weibull Material 5: Alumia (Al 2 O 3 ) Stard Weibull Iverse Weibull Marshall & Olki s Weibull Expoetiated Weibull Material 6: Bi 2 Sr 2 CaCu 2 O 8+x Stard Weibull Iverse Weibull Marshall & Olki s Weibull Expoetiated Weibull ˆ =5.626 ˆλ =0.89 ˆ =2.920 ˆλ =4.047 ˆ =4.596, ˆλ =0.208 ˆ ν =2.800 ˆ =5.00 ˆλ =0.98 ˆ =0.23, ˆ =6.234 ĉ =6.394 ˆ =5.674, ˆ =0.64 â =0.24 ˆ =0.643, ˆ =0.909 ˆλ =0.002 ˆ =0.693, ˆλ =0.808 ˆ ν =3.688x0-6 ˆ =7.584 ˆλ =0.375 ˆ =8.673 ˆλ =2.334 ˆ =5.803, ˆλ =0.408 ˆ ν =2.849 ˆ =7.023 ˆλ =0.384 ˆ =97.34, ˆ =0.005 ĉ =0.742 ˆ =0.989, ˆ =3.675 â =4.638x0 3 ˆ =0.076, ˆ =0.677 ˆλ =0.000 ˆ =.582, ˆλ =0.00 ˆ ν =2.23 ˆ =6.000 ˆλ =0.287 ˆ =9.284 ˆλ =3.073 ˆ =5.049, ˆλ =0.329 ˆ ν =3.744 ˆ =6.57 ˆλ =0.295 ˆ =08.272, ˆ =0.007 ĉ =0.76 ˆ =.206, ˆ =.720 â = ˆ =0.052, ˆ =0.582 ˆλ =0.000 ˆ =7.433, ˆλ =9.22x0-5 ˆ ν =.927x0-6 ˆ =3.303 ˆλ =0.69 ˆ =2.990 ˆλ =4.72 ˆ =2.626, ˆλ =0.98 ˆ ν =2.68 ˆ =2.725 ˆλ =0.94 ˆ =4.008, ˆ =2.44 ĉ =.49 ˆ =2.099, ˆ =0.225 â =2.34 ˆ =0.00, ˆ =0.282 ˆλ =0.007 ˆ =0.73, ˆλ =0.859 ˆ ν =.332x0-6 ˆ =4.965 ˆλ =0.22 ˆ =3.024 ˆλ =3.62 ˆ =3.998, ˆλ =0.238 ˆ ν =2.808 ˆ =4.390 ˆλ =0.225 ˆ =0.80, ˆ =4.959 ĉ =5.690 ˆ =5.504, ˆ =0.206 â =0.837 ˆ =0.007, ˆ =0.393 ˆλ =0.000 ˆ =4.058, ˆλ =0.00 ˆ ν =0.200 ˆ =3.462 ˆλ =0.369 ˆ =2.899 ˆλ =.922 ˆ =2.795, ˆλ =0.428 ˆ ν =2.630 ˆ =2.883 ˆλ =0.48 ˆ =.230, ˆ =2.497 ĉ =3.047 ˆ =3.347, ˆ =0.374 â =.059 ˆ =0.00, ˆ =0.305 ˆλ =0.00 ˆ =3.487, ˆλ =0.03 ˆ ν =9.727x

7 S. Nadarajah S. Kotz / Joural of Mechaical Sciece Techology 22 (2008) 247~ Table 4. Fitted models goodess of fit. Model Stard Weibull Iverse Weibull Marshall & Olki s Weibull Expoetiated Weibull Goodess of fit measure Material Material 2 Material 3 Material 4 Material 5 Material λ. The quasi-newto algorithm lm i the R software package (Deis Schabel, [24]; Schabel et al., [25]; Ihaka Getlema, [26]) was used to maximize the likelihood. A sample program i R for fittig () is illustrated below. Similar programs for fittig the other Weibull distributios ca be obtaied by cotactig the first author. #fittig of the stard Weibull model give by () for data cotai i x f<-fuctio (p) {alpha<-p [] lambda<-p [2] tt< if (alpha>0&lambda>0) tt<--*log (alpha)-*alpha* log (lambda)+ (-alpha)*sum (log (x)) if (alpha>0&lambda>0) tt<-tt+ (lambda**alpha)* sum (x**alpha) retur (tt)} est<-lm (f,p=c (,),iterlim=000) #the maximum likelihood estimates of alpha & lambda retured alpha<-est$estimate [] lambda<-est$estimate [2] The maximum likelihood estimates of the parameters as well as the egative logarithm of the maximized likelihood (NLLH) are give i Table 3 for the eight distributios fitted. The goodess of the fitted models was checked by meas of probability plots. A probability plot cosists of plots of the observed probabilities agaist the probabilities predicted by the fitted model. For example, for the stard Weibull ˆ model, ˆ exp{ ( λx() i ) } was plotted versus (i 0.375)/( +0.25), i =, 2,..., (as recommeded by Blom [27] Chambers et al. [28]), where x (i) are the sorted values of the observed fracture toughess. ˆ For the iverse Weibull model, exp{ ( x ˆ () i / λ) } was plotted versus (i 0.375)/(+ 0.25), i =, 2,...,. A objective measure of the goodess of fit was calculated by takig the sum of the absolute differeces betwee the observed probabilities the probabilities predicted by the fitted model. The values of this measure are show i Table 4. Note that five of the fitted distributios have three parameters while the other three distributios have two parameters. Although ot all of the distributios are ested, those with the same umber of parameters ca be compared by meas of the stard likelihood ratio test. It follows that the best fits are exhibited by the bottom four models: distributio, expoetiated Weibull distributio, Xie et al. s Weibull distributio distributio. This is uderstable because these distributios have more parameters, more recetly developed more flexible tha the other Weibull distributios. There is o evidece to suggest that certai models perform better for the toughest materials or the lightest materials (see Table 2). It is just that the more flexible models perform better irrespective of the stregth of the materials. These observatios are cofirmed by the values of the goodess of fit measure i Table Coclusios We have reviewed over 20 kow variatios of the Weibull distributio which could be used as possible models for material stregth. We have discussed the appropriateess of these distributios as models for brittle versus ductile stregth. We have also provided a compariso study based o six data sets o fracture toughess. We believe that this review will serve as a importat referece, help to provide better modelig ecourage developmets of further variatios of the Weibull distributio. Ackowledgmets The authors would like to thak the editor the three referees for carefully readig the paper for their commets which greatly improved the paper.

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