1、Designation: C1683 081Standard Practice forSize Scaling of Tensile Strengths Using Weibull Statisticsfor Advanced Ceramics1This standard is issued under the fixed designation C1683; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, t
2、he year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1NOTEEditorial changes were made throughout in January 2010.1. Scope1.1 This standard practice provides methodology to
3、convertfracture strength parameters (primarily the mean strength andthe Weibull characteristic strength) estimated from data ob-tained with one test geometry to strength parameters represent-ing other test geometries. This practice addresses uniaxialstrength data as well as some biaxial strength dat
4、a. It may alsobe used for more complex geometries proved that the effectiveareas and effective volumes can be estimated. It is for theevaluation of Weibull probability distribution parameters foradvanced ceramics that fail in a brittle fashion. Fig. 1 shows thetypical variation of strength with size
5、. The larger the specimenor component, the weaker it is likely to be.1.2 As noted in Practice C1239, the failure strength ofadvanced ceramics is treated as a continuous random variable.Anumber of functions may be used to characterize the strengthdistribution of brittle ceramics, but the Weibull dist
6、ribution isthe most appropriate especially since it permits strength scalingfor the size of specimens or component. Typically, a number oftest specimens with well-defined geometry are broken underwell-defined loading conditions. The force at which each testspecimen fails is recorded and fracture str
7、ength calculated. Thestrength values are used to obtain Weibull parameter estimatesassociated with the underlying population distribution.1.3 This standard is restricted to the assumption that thedistribution underlying the failure strengths is the two-parameter Weibull distribution with size scalin
8、g. The practicealso assumes that the flaw population is stable with time andthat no slow crack growth occurs.1.4 This practice includes the following topics:SectionScope 1Referenced Documents 2Terminology 3Summary of Practice 4Significance and Use 5Probability of Failure Relationships 6SectionTest S
9、pecimens with Uniaxial Stress StatesEffectiveVolume and Area Relationships7Uniaxial Tensile Test Specimens 7.1Rectangular Flexure Test Specimens 7.2Round Flexure Test Specimens 7.3C-Ring Test Specimens 7.4Test Specimens with Multiaxial Stress StatesEffectiveVolume and Area Relationships8Pressure-on-
10、Ring Test Specimens 8.1Ring-on-Ring Test Specimens 8.2Examples of Converting Characteristic Strengths 9Report 10Precision and Bias 11Keywords 12Combined Gamma Function for Round Rods Testedin FlexureAnnex A1Components or Test Specimens with MultiaxialStress DistributionsAnnex A2Components or Test Sp
11、ecimens with ComplexGeometries and Stress DistributionsAnnex A31.5 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.5.1 The values stated in SI units are in accordance withIEEE/ASTM SI 10.1.6 This standard does not purport to a
12、ddress all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2C1145 Termi
13、nology of Advanced CeramicsC1161 Test Method for Flexural Strength of AdvancedCeramics at Ambient TemperatureC1211 Test Method for Flexural Strength of AdvancedCeramics at Elevated TemperaturesC1239 Practice for Reporting Uniaxial Strength Data andEstimating Weibull Distribution Parameters for Advan
14、cedCeramics1This practice is under the jurisdiction of ASTM Committee C28 on AdvancedCeramics and is the direct responsibility of Subcommittee C28.01 on MechanicalProperties and Performance.Current edition approved Jan. 1, 2008. Published January 2008. DOI: 10.1520/C1683-08.2For referenced ASTM stan
15、dards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken,
16、PA 19428-2959, United States.C1273 Test Method for Tensile Strength of MonolithicAdvanced Ceramics at Ambient TemperaturesC1322 Practice for Fractography and Characterization ofFracture Origins in Advanced CeramicsC1323 Test Method for Ultimate Strength of AdvancedCeramics with Diametrally Compresse
17、d C-Ring Speci-mens at Ambient TemperatureC1366 Test Method for Tensile Strength of MonolithicAdvanced Ceramics at Elevated TemperaturesC1499 Test Method for Monotonic Equibiaxial FlexuralStrength of Advanced Ceramics at Ambient TemperatureE6 Terminology Relating to Methods of Mechanical TestingE456
18、 Terminology Relating to Quality and Statistics3. Terminology3.1 Unless otherwise noted, the Weibull parameter estima-tion terms and equations found in Practice C1239 shall be used.3.2 For definitions of other statistical terms, terms related tomechanical testing, and terms related to advanced ceram
19、icsused in this guide, refer to Terminologies E6, E456, and C1145,or to appropriate textbooks on statistics (1-4).33.3 Nomenclature:AT= gage area of a uniaxial tensile test specimenAB4= gage area of a four-point flexure test specimenAB3= gage area of a three-point flexure test specimenAPOR= gage are
20、a of a pressure-on-ring test specimenAROR= gage area of a ring-on-ring test specimenACR= gage area of a C-ring test specimenb = thickness of a C-ringb = width of a flexure test specimend = thickness of a flexure test specimenD = diameter of a round flexure test specimenD = overall diameter of a ring
21、-on-ring disk test specimenDL= loading (inner) ring diameter, ring-on-ring disk speci-menDS= support ring diameter, ring-on-ring or pressure-on-ringdisk specimenh = thickness of pressure-on-ring or ring-on-ring disk testspecimenk = load factorLgs= length of the gage section in a uniaxial tensile tes
22、tspecimenLi4= length of the inner span for a four-point flexure testspecimenLo4= length of the outer span for a four-point flexure testspecimenLo3= length of the outer span for a three-point flexure testspecimenm = Weibull modulusPf= probability of failureri= inner radius of a C-ringro= outer radius
23、 of a C-ringt = thickness of a C-ringRs= radius of the support ring for pressure-on-ringRd= radius of the pressure-on-ring disk specimenSE= effective surface area of a test specimenVE= effective volume of a test specimenVPOR= gage volume of a pressure-on-ring test specimenVROR= gage volume of a ring
24、-on-ring disk test specimenVT= gage volume of tensile test specimenVB4= gage volume of a four-point flexure test specimenVB3= gage volume of a three-point flexure test specimenVCR= gage volume of a C-ring test specimens = uniaxial tensile stresssmax= maximum tensile stress in a test specimen at frac
25、tures1, s2, s3= principal stresses (tensile) at the integrationpoints in any finite elements0= Weibull material scale parameter (strength relative tounit size)3The boldface numbers in parentheses refer to the list of references at the end ofthis standard.FIG. 1 Strength Scales with SizeC1683 0812su=
26、 Weibull characteristic strengthsuT= Weibull characteristic strength of a uniaxial tensile testspecimensuB4= Weibull characteristic strength for a four-point flex-ure test specimensuB3= Weibull characteristic strength for a three-point flex-ure test specimensuCR= Weibull characteristic strength for
27、a C-ring testspecimensuPOR= Weibull characteristic strength for a pressure-on-ring test specimensuROR= Weibull characteristic strength for a ring-on-ringtest specimens* = an arbitrary, assumed estimate of the Weibull materialscale factors= mean strengthsT= mean strength for a uniaxial tensile test s
28、pecimensB4= mean strength for a four-point flexure test specimensB3= mean strength for a three-point flexure test specimensCR= mean strength for a C-ring test specimensPOR= mean strength for a pressure-on-ring test specimensROR= mean strength for a ring-on-ring test specimenu = angle in a C-ring tes
29、t specimenn = Poissons ratio4. Summary of Practice4.1 The observed strength values of advanced ceramics aredependent on test specimen size, geometry and stress state.This standard practice enables the user to convert tensilestrength parameters obtained from one test geometry to that ofanother, on th
30、e basis of assumptions listed in 5.5. Using theexisting fracture strength data, estimates of the Weibull char-acteristic strength su, and the Weibull modulus m, are calcu-lated in accordance with related Practice C1239 for the originaltest geometry. This practice uses the test specimen and loadingsi
31、zes and geometries, and suand m to calculate the Weibullmaterial scale parameter s0. The Weibull characteristicstrength su, the mean strength s, or the Weibull material scalefactor s0, may be scaled to alternative test specimen geom-etries. Finally, a report citing the original test specimengeometry
32、 and strength parameters, as well as the size scaledWeibull strength parameters is prepared.5. Significance and Use5.1 Advanced ceramics usually display a linear stress-strainbehavior to failure. Lack of ductility combined with flaws thathave various sizes and orientations typically leads to largesc
33、atter in failure strength. Strength is not a deterministicproperty but instead reflects the intrinsic fracture toughness anda distribution (size and orientation) of flaws present in thematerial. This standard is applicable to brittle monolithicceramics which fail as a result of catastrophic propagat
34、ion offlaws. Possible rising R-curve effects are also not considered,but are inherently incorporated into the strength measurements.5.2 Two- and three-parameter formulations exist for theWeibull distribution. This standard is restricted to the two-parameter formulation.5.3 Tensile and flexural test
35、specimens are the most com-monly used test configurations for advanced ceramics. Ring-on-ring and pressure-on-ring test specimens which have multi-axial states of stress are also included. Closed-form solutionsfor the effective volume and effective surfaces and the Weibullmaterial scale factor are i
36、ncluded for these configurations. Thispractice also incorporates size scaling methods for C-ring testspecimens for which numerical approaches are necessary. Ageneric approach for arbitrary shaped test specimens or com-ponents that utilizes finite element analyses is presented inAnnex A3.5.4 The frac
37、ture origins of failed test specimens can bedetermined using fractographic analysis. The spatial distribu-tion of these strength controlling flaws can be over a volume oran area (as in the case of surface flaws). This standard allowsfor the conversion of strength parameters associated with eithertyp
38、e of spatial distribution. Length scaling for strength con-trolling flaws located along edges of a test specimen is notcovered in this practice.5.5 The scaling of strength with size in accordance with theWeibull model is based on several key assumptions (5).Itisassumed that the same specific flaw ty
39、pe controls strength inthe various specimen configurations. It is assumed that thematerial is uniform, homogeneous, and isotropic. If the mate-rial is a composite, it is assumed that the composite phases aresufficiently small that the structure behaves on an engineeringscale as a homogeneous and iso
40、tropic body. The compositemust contain a sufficient quantity of uniformly-distributed,randomly-oriented, reinforcing elements such that the materialis effectively homogeneous. Whisker-toughened ceramic com-posites may be representative of this type of material. Thispractice is also applicable to com
41、posite ceramics that do notexhibit any appreciable bilinear or nonlinear deformationbehavior. This standard and the conventional Weibull strengthscaling with size may not be suitable for continuous fiber-reinforced composite ceramics. The material is assumed tofracture in a brittle fashion, a conseq
42、uence of stress causingcatastrophic propagation of flaws. The material is assumed tobe consistent (batch to batch, day to day, etc.). It is assumedthat the strength distribution follows a Weibull two parameterdistribution. It is assumed that each test piece has a statisticallysignificant number of f
43、laws and that they are randomlydistributed. It is assumed that the flaws are small relative to thespecimen cross section size. If multiple flaw types are presentand control strength, then strengths may scale differently foreach flaw type. Consult Practice C1239 and the example in 9.1for further guid
44、ance on how to apply censored statistics in suchcases. It is also assumed that the specimen stress state and themaximum stress are accurately determined. It is assumed thatthe actual data from a set of fractured specimens are accurateand precise. (See Terminology E456 for definitions of the lattertw
45、o terms.) For this reason, this standard frequently referencesother ASTM standard test methods and practices which areknown to be reliable in this respect.C1683 08135.6 Even if test data has been accurately and preciselymeasured, it should be recognized that the Weibull parametersdetermined from tes
46、t data are in fact estimates. The estimatescan vary from the actual (population) material strength param-eters. Consult Practice C1239 for further guidance on theconfidence bounds of Weibull parameter estimates based ontest data for a finite sample size of test fractures.5.7 When correlating strengt
47、h parameters from test datafrom one specimen geometry to a second, the accuracy of thecorrelation depends upon whether the assumptions listed in 5.5are met. In addition, statistical sampling effects as discussed in5.6 may also contribute to variations between computed andobserved strength-size scali
48、ng trends.5.8 There are practical limits to Weibull strength scaling thatshould be considered. For example, it is implicitly assumed inthe Weibull model that flaws are small relative to the specimensize. Pores that are 50 m (0.050 mm) in diameter arevolume-distributed flaws in tension or flexural st
49、rength speci-mens with 5 mm or greater cross section sizes. The same maynot be true if the cross section size is only 100 m.6. Probability of Failure Relationships6.1 General:6.1.1 The random variable representing uniaxial tensilestrength of an advanced ceramic will assume only positivevalues, and the distribution is usually asymmetric about themean. These characteristics limit the use of the normal distri-bution (as well as others) and point to the use of the Weibulland similar skewed distributions. Fig. 2 shows the shape of theWeibull distribution as compared to
