Statistical analysis of the mechanical properties of composite materials

Transcription

1 Statistical analysis of the echanical properties of coposite aterials E. Barbero, J. Fernández-Sáez, C. Navarro* Departent of Mechanical Engineering, Carlos III University of Madrid, Avda. de la Universidad, 3, 289-Leganés, Madrid, Spain Abstract The Weibull statistic is currently used in designing echanical coponents ade of coposite aterials. This work presents useful forulae to describe the behaviour of the Weibull odulus estiator, which in turn ay be described by eans of a three paraeter Weibull distribution. Expressions for the paraeters of this latter distribution, dependent on the saple size, are also given in the paper, so, the percentage points, published until now in tabular for, ay be directly calculated. Epirical expressions are derived for deterining the A basisandthebbasisaterialpropertiesasafunctionofthesaplesize. Keywords: C. Statistical ethods; B. Mechanical properties; B. Strength; Coposite aterials. Introduction Unlike ost traditional structural aterials, whose echanical behaviour is assued to be hoogeneous and isotropic, echanical properties of coposite aterials exhibit intrinsic statistical dependence. In particular, their strength properties are usually scattered due to their inhoogeneity and anisotropic characteristics and to the brittleness of the atrices and fibres. So, careful statistical analysis is indispensable for the understanding of the echanical characterisation of these aterials. The Weibull statistic [] has been widely used in the recent years to describe the statistical behaviour of the strength properties of any aterials, such as advanced ceraics [2], etallic atrix coposites [3], ceraic atrix coposites [4], and polyeric atrix coposites [5]. The Weibull distribution also describes the fatigue behaviour of aterials and the scatter of the fracture toughness of steels in the ductile brittle transition region, where failure occurs by cleavage[6,7]. The two-paraeter Weibull distribution function is given by: F s exp s s wherefistheprobabilityofruptureoftheaterialunder uniaxial tensile stress s, is the shape paraeter or Weibullodulus,ands isthescaleparaeterofthedistribution.weibullodulus,,isrelatedtothescatterofthe data:thehigherthethelowerthedispersionoffracture stress. It becoes the ost iportant paraeter of the distribution. The scale paraeter is closely related to the ean fracture stress. Since the evaluation of the paraeters of Weibull distribution is ade fro a finite nuber of tests, the estiators of their true values have a statistical character, and thus the uncertainty of the estiation ust be known. In this work a three-paraeter Weibull distribution is also proposed for the estiator of the Weibull odulus. Forulae for the three paraeters defining such a distribution, dependent only on the saple size, are given below. To design structural and echanical coponents, the deterination of the s-values, corresponding to a predefined failure probability, is of great interest to the engineer. These values coincide with their percentiles of the distribution. In particular, the values corresponding to the percentilesof9and99%,estiatedwithaconfidencelevelof 95%, are known [8] as the Abasis and Bbasis aterial property, respectively. These values are obtained fro the estiations of, s, and in ters of two paraeters, P A and P B, which depend only on the saple size and on the estiation ethod used, and are usually obtained frotables[8].inthispaper,tosiplifythecalculusof these paraeters, fitted equations are developed for two estiation ethods: axiu-likelihood and weighted regression.

2 2. Theoretical background Several procedures are available for the deterination of the Weibull distribution paraeters. Fro the axiu likelihood ethod, the estiators of the Weibull paraeters, ^and ^s ;shouldsatisfythefollowingequations: X iˆn iˆ and lns i n X iˆn lns i s i ^ n^ C s i ^ A iˆ X iˆn iˆ 2! Xiˆn = ^ ^s n s ^ i 3 iˆ Although Eq.(2) is non-linear, it has a unique positive solution[9], and ay be solved by the Newton Raphson iteration technique or by any other ethod. Eq. () becoes a straight line if a double logarithic transforation is ade, i.e. ln ln F s lns lns TheFvaluesareassignedonthebasisoftheithpositionofa value aong the n ordered s-values foring the saple. Thus the general linear regression ethod ay be used to obtain ^and ^s : Several estiators of F can be considered F i i :5 n F i F i i :3 n :4 i n 4 5a 5b 5c Different authors[,] have pointed out the convenience of using a weight function, W i, in perforing the linear regression. Bergan [] proposed the weight factor given by W i F i ln F i Š 2 and Faucher and Tyson[] suggested the following: W i 3:3F i 27:5 F i :25 Š Using the general linear regression ethod, fro the n pairs of values(s F), the estiators of the Weibull paraeters can be calculated as ^ P W i P W i z i y i P W i z i P W i y i P W i P W i z 2 i P W i z i and ^s exp a^ where z i lns i y i ln ln F i a P Wi y i ^ P W i z i P Wi 9 2 Finally, applying the oents ethod, in which the saple oents are equated to those of the distribution, the estiatorsofands arethesolutionsof s S and ^s G ^ G 2^ G 2 ^ =2 3 ^s G ^ 4 where sandsare,respectively,theeanandthestandard deviation of the experiental data, and G is the Gaa function. Eq.(3) ay be solved using the Newton Raphson procedure. Thoanetal.[2]showedthatiftheaxiu likelihoodethodwereused,thevariables ^=and ^ln ^s =s aredistributedindependentlyofthetruevaluesofands, andtheyhavethesaedistributionas ^ and ^ ln ^s ; respectively,whichcorrespondto and s :The percentage points of these distributions were calculated by Thoan et al.[2] fro Monte Carlo siulations. For a two-paraeter Weibull distribution, the A-basis, s A,andB-basisaterialproperties, s B,ightbeobtained [8]as = ^ s A ^s ln 5 P A = ^ s B ^s ln 6 P B UsuallythevaluesoftheparaetersP A andp B arecalculated by eans of the axiu-likelihood ethod and, so far,theirvalueshavebeenpresentedinatabularfor[8]. HereinthevaluesofP A andp B aybecoputedas P A : exp C P B : exp C 7a 7b wheretheconstantsc. andc. are,respectively,the95 2

3 2, siulations percentagepointsofthepivotalvariables ^ln ^s : =s : and ^ln ^s : =s : ;i.e.shouldverify ^s Pr ^ln : C s : :95 8a : ^s Pr ^ln : s : Evaluation of average, standard deviation and percentiles of the pivotal variable C : Rando nuber generator Unifordistribution,R,with<R< Obtainingofasapleofsizen { σ,..., σ σ i,..., n} Weibull distribution ( = = ) ; σ Estiation progra Estiatedvaluesof, σ o and σ π (, σ, ) o σ p Construction of the pivotal variables p, ln σ σp (for2,values) Fig.. Flow chart of the siulation procedure. :95 8b Fernández-Sáez et al. [3] showed that the variable ^ln ^s p =s p isdistributedindependentlyofthetruepercentilevalueand,barberoetal.[4]haveprovedthat,inorder to estiate the percentiles of the two-paraeter Weibull distribution, the best ethod is the weighted regression one,usingeq.(5c)toestiatef i,andeq.(7)tocalculate W i,ifsaplesizeisabove7,oreq.(6)otherwise. Table Average and standard deviation of variable ^=; according to Eq.(2) Paraeter A B C D Average Standarddeviation o 3. Nuerical siulation To obtain the statistical distribution of the pivotal variables ^=and ^ln ^s p =s p ;asiulationprocedure,based onthemontecarloethod,hasbeenused.inthisprocedure (seefig.forbetterunderstanding)asetofnvalueswas generated as = s i s ln 9 R In this work we select and s : Note that, as stated above, the analysis is independent of the true values [2]. R is a rando variable with unifor distribution in the [,] interval. Fro each saple so obtained, {s ;s 2 ;s 3 ; ;s n };estiationsoftheweibullparaeters, ^ and ^s ; and the p-percentile ^s p ; were obtained. Fro these estiations, the variables ^= and ^ln ^s p =s p could also be built. Repeated application of this procedure provides a statistical distribution for the two latter variables. Inthisworkwehavecoputed2,valuesfor ^=and ^ln ^s p =s p for each saple size, which, in turn was progressively increased fro 5 up to 2. ThevaluesofC. andc.,defined byeqs.(8a)and (8b),werealsocoputed,andfrothese,thevaluesofP A and P B were obtained using Eqs. (7a) and (7b), respectively. For the estiation of ^=; the axiu-likelihood ethod (ethod ) was utilised, while the authors have used three estiation ethods for the calculation of P A and P B : axiu-likelihood (ethod ) and weighted regression, with two different weighted functions: Eq.(6) (ethod2),andeq.(7)(ethod3). 4. Paraeter fitting Fro the 2, values of the pivotal variable ^=; its averagevalue, ^= ave wascalculated.inordertofitthis value to the saple size, the following four-paraeter function is proposed: ^ ave C A B 2 ln Dn wherea,b,c,anddarethefitparaetersgivenintable, andnisthesaplesize.asshowninfig.2,thefitseesto beverygood,withaaxiuerrorof.23%.ifparaeter Dweretakenasunity,theincreentoftheaxiuerror would becoe negligible and thus three paraeters would suffice. The average value of the pivotal variable approaches unity(the estiator approach to the true value) when saple sizenincreases,which agreeswiththekhalili etal.[5] results. The standard deviation of the variable ^= was coputed asafunctionofthesaplesize.itsvaluedecreasestozero as the saple size increases, as Khalili et al.[5] observed. 3

4 Average value Nuerical values Epirical fit (Eq. 2) Table 2 Three-paraeter Weibull distribution for the variable ^=; according to Eqs.(2)and(22) Paraeter 2 3 P P P Saple size Fig.2.Variationoftheaverageofthevariable ^=asafunctionofthe saple size, n. Epirical fit according to Eq.(2). To fit the standard deviation we used the sae type of functionasbeforefortheaveragevalue.fig.3showsthe dependence of the standard deviation on the saple size. ThefitparaetersaregiveninTableandtheaxiu error was 2.7%. Percentage points of the pivotal variable were also calculated nuerically. These results, published by Thoan et al. [2] in tabular for, showed that the variable ^= is noral asyptotically p with an average value of and standard deviation :68=n; which eans that the percentiles of the variable could be calculated fro those of a noral distribution. One iproveent to this approach would be to consider that the variable ^= is norally distributed with the average and the standard deviation given by Eq. (2) using the paraeter obtained fro Table. In this paper a three-paraeter Weibull distribution is proposed for the variable ^=; i.e. F exp ^= P P 2 P3 2 wherep,p 2,andP 3 are,respectively,theposition,scaleand for paraeters, that ay be fitted as a function of the Standard Deviation Nuerical values Epirical fit (Eq. 2) Saple Size Fig.3.Variationofthestandarddeviationofthevariable ^=asafunction ofsaplesize,n.epiricalfitaccordingtoeq.(2). saplesize,n,by P i i i2 lnn i3 ; i ;2;3 22 theepiricalparaeters i, i2,and i3 beingthoseshown intable2. This last approach sees to be better than those cited above for the calculation of the percentiles, as shown in Fig.4,whichgivesthevariationoftheerrorwiththesaple size for the.985 percentile of the variable ^= using different ethods: the three-paraeter Weibull distribution; noral p distribution of average and standard deviation :68=n; and finally noral distribution with the average and standard deviation values calculated fro Eq.(2). Also,thevaluesofP A andp B,obtainedusingEqs.(7a) and(7b) fro the nuerical siulation results have been fitted by eans of the following equation: P A orp B M M 2 exp n M 3 23 TheparaetersM,M 2,andM 3 areshownintables3and4. Fig.5showsacoparison ofthenuericalvaluesofp B with the corresponding ones obtained fro Eq.(23). The fits forbothp A andp B areverygood,witherrorsbelow.3%in allthecases.oncep A andp B valuesareknown,thea-basis and B-basis aterial properties ay be obtained fro Eqs. (5) and(6). Error% paraeter Weibull Noral (Average, Variance.68/n) Noral( Average and Standard Deviation by Eq.2 ) Saple Size Fig.4.Errorofestiationofthe.985percentileofthevariable ^=for different ethods. Curve : three-paraeter Weibull distribution. Curve 2: noral of average and variance :68=n: Curve 3: noral of average and variance calculated using Eq.(2). 4

5 Table 3 EstiationofP A asafunctionofsaplesize,accordingtoeq.(23) Method 2 3 R Maxiuerror(%) Table 4 EstiationofP B asafunctionofsaplesize,accordingtoeq.(23) Method 2 3 R Maxiuerror(%) Practical exaples To illustrate the above ethodology two exaples are included in this paper, although it is worth noting that theycanbeappliedintheirintegritytoanykindofcoposite aterial. In the first exaple, the estiation of the confidence intervals of the Weibull odulus for the static flexural strength of a CMC is considered, whereas in the second, the A-basis and B-basis values of the dynaic flexural strength of a CFRP are coputed. In these exaples, the loading rate and the saple size are different. 5.. Exaple : estiation of confidence intervals of Weibull odulus In the first exaple, the confidence intervals of the Weibull odulus (9% and 99% confidence levels), for the static flexural strength of a ceraic atrix coposite, are estiated. The coposite aterial was an aluina(condea APA-.5, 99.97% of purity) atrix reinforced by 3% of zirconia particles(tosoh 3YB), processed by uniaxial pressing under 87MPaandpressurelesssinteringat65 Cfor2h[6]. Three-point bending tests of 5 prisatic speciens of Paraeter P B Nuerical values Epirical fit (eq.23) Saple Size Fig. 5. Variation of the paraeter P B as a function of saple size, n. Epirical fit according to Eq.(23). Table 5 Exaple : flexural strength(mpa) of a 3% zirconia aluina coposite Saple Strength(MPa) Saple Strength(MPa) werecarriedoutusingauniversaltesting achine(instron86)witha5knloadcell.thetests results(flexural strength) are shown in Table 5. Fro these values an estiation of the Weibull odulus and scale paraeters were obtained by the axiu likelihood ethod [4], which gave ^ 5:398; ^s 33:5MPa Fig. 6 gives the experiental results and the fitted values using the Weibull distribution. FrotheparaetersA,B,CandDofTable,thevalues of the average and standard deviation of the pivotal variable arecalculatedbyeq.(2)withn 5were Average of ^ Standard deviation of :5 ^ :252 TofindthepercentagepointsoftheestiatoroftheWeibull odulus, its distribution function, Eq.(2), ay be used. ThevaluesofP i inthisequationwerecoputedfroeq. (22)andtheparaeters, 2 and 3 frotable2.forn 5 the following values were obtained: P :6962 P 2 :457 P 3 :882 Thepercentilesofvariable ^=tobeusedtocalculatethe confidencelevelsofthe99and9%areshownintable6. Fro such values the interval of the variable, correspondingtothe99%confidencelevel,is 2:934 7:487 Table 6 Exaple : percentiles of variable ^=; calculated according Eqs.(2) and (22) Percentile Value Confidence level: 99% Confidence level: 9%

6 Fig. 6. Flexural strength distribution of an aluina 3% zirconia coposite(exaple ) and /9 carbon/epoxy lainate(exaple 2). and for the 9% confidence level 3:523 6: Exaple2:estiationoftheAbasisandBbasis aterial properties In this exaple, the A-basis and B-basis aterial properties (dynaic flexural strength) of a polyeric atrix coposite were calculated. The coposite aterial is a 2.4 thick carbon/epoxy, Hexcel AS4/35-6, [/9] 2S lainate, anufactured by SACESA. Thirty dynaic three-point bending tests were perfored using a CEAST Fractovis drop weight tester. The speciens were 2: beas. The drop weight was Table 7 Exaple 2: flexural strength(mpa) of[/9] carbon/epoxy coposite Saple Strength(MPa) Saple Strength(MPa) kg and its velocity 4.8 /s. The flexural strength results are given in Table 7. For this exaple the estiations of the Weibull odulus and scale paraeters were ^ 2:67; ^s 972:29MPa Fig. 6 gives the experiental results and the fitted values using the Weibull distribution. FroEq.(23)andTables3and4,theparaetersP A and P B are estiated for a saple size of 3. The values obtained for the three ethods cited in this article are shown in Table 8, which also gives the A-basis and B- basis aterial properties. 6. Suary This work presents useful forulae to analyse the variability of the echanical properties of coposite aterials. The study is focused on the two-paraeter Weibull distribution, currently used to describe statistically the strength properties of any kinds of aterials. In order to obtain the percentage points of the estiator of the Weibull odulus, published until now in tabular for, a three-paraeter Weibull distribution is proposed. Epirical expressions for these three paraeters, dependent only on the saple size, are also given. The A-basis and B-basis aterial Table 8 Exaple 2: estiation of A-basis and B-basis Method P A P B s A (MPa) s B (MPa)

7 properties based on Weibull statistics can be derived fro theestiationsoftheweibullparaetersandthevaluesp A andp B,whichdependonlyonthesaplesizeandthetype ofestiationethod.expressionsforp A andp B,estiated by the axiu likelihood, and two selected weightedregression ethods, are given. Acknowledgeents The authors are indebted to the Fundación Raón Areces (Área de Materiales, IX Concurso Nacional) for the financial support of this research. References [] Weibull W. A statistical function of wide applicability. Journal of Applied Mechanics 95;8: [2] Glaeser AM. The use of transient FGM interlayers for joining advanced ceraics. Coposites Part B: Engineering 997;28B:7 84. [3] FukuiY,YaanakaN,EnokidaY.BendingstrengthofanAl Al 3 Ni functionally graded aterial. Coposites Part B: Engineering 997;28B: [4] Bloyce DM, Ha-Su R, Plucknett KP, Wilkinson DS. Tape casting of SiC platelet reinforced aluina. Advances in Ceraic Matrix Coposites: Ceraic Transactions 993;38: [5] Laela MJ, Fernández A, Reiter H, Viña J. Análisis experiental y estadístico del coportaiento a flexión dináica de lainados de fibradecarbonoyresinaepoxi.actasdelicongresonacionalde Materiales Copuestos,(in Spanish) 995. p [6] Evans AG. Statistical aspects of cleavage fracture in steel. Metallurgical Transactions A 983;4: [7] Berein FM. A local criterion for cleavage fracture of a nuclear pressure vessel steel. Metallurgical Transactions A 983;4: [8] Neal D, Vangel M. Statistical analysis of echanical properties. In: ASM Handbook Coittee(Ed.), Coposites, engineered aterials handbook. ASM International, vol., 987. p [9] McCool JT. Inference on Weibull percentiles and shape paraeter fro axiu likelihood estiated. IEEE Transactions on Reliability 97;R9:2 9. [] Bergan B. Estiation of Weibull paraeters using a weight function. Journal of Materials Science Letters 986;5:6 4. [] Faucher B, Tyson R. On the deterination of Weibull paraeters. Journal of Materials Science Letters 988;7: [2] ThoanDR,BainLJ,AntleCE.Inferencesontheparaetersofthe Weibull distribution. Technoetrics 969;(3): [3] Fernández-Sáez J, Chao J, Durán J, Ao JM. Estiating lower-bound fracture paraeters for brittle aterials. Journal of Materials Science Letters 993;2: [4] Barbero E, Fernández-Sáez J, Navarro C. On the estiation of percentiles of the Weibull distribution. Journal of Materials Science Letters 999;8(7):44 3. [5] Khalili A, Krop K. Statistical properties of Weibull estiators. Journal of Materials Science 99;26: [6] Barbero E. Desarrollo y caracterización ecánica de nuevos ateriales copuestos de atriz ceráica y refuerzo ceráico. PhD thesis, Carlos III University of Madrid, Madrid, 999(in Spanish). 7