1、Designation: C 1683 08Standard Practice forSize Scaling of Tensile Strengths Using Weibull Statisticsfor Advanced Ceramics1This standard is issued under the fixed designation C 1683; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision,
2、the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (e) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This standard practice provides methodology to convertfracture strength parameters (primarily the mean st
3、rength 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 data. It may alsobe used for more complex geometries proved t
4、hat 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. The larger the specimenor component, the weaker it is li
5、kely to be.1.2 As noted in Practice C 1239, 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 distribution isthe most appropriate especially since it permi
6、ts 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 strength calculated. Thestrength values are used to obtain W
7、eibull 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 scaling. The practicealso assumes that the flaw population is s
8、table 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 6Test Specimens with Uniaxial Stress StatesEffectiveVolume and Area Rel
9、ationships7Uniaxial Tensile Test Specimens 7.1SectionRectangular Flexure Test Specimens 7.2Round Flexure Test Specimens 7.3C-Ring Test Specimens 7.4Test Specimens with Multiaxial Stress StatesEffectiveVolume and Area Relationships8Pressure-on-Ring Test Specimens 8.1Ring-on-Ring Test Specimens 8.2Exa
10、mples 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 Specimens with ComplexGeometries and Stress DistributionsAn
11、nex A31.5 The values stated in SI units are to be regarded as thestandard per IEEE/ASTM SI 10.1.6 This standard does not purport to address 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 pra
12、ctices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2C 1145 Terminology of Advanced CeramicsC 1161 Test Method for Flexural Strength of AdvancedCeramics at Ambient TemperatureC 1211 Test Method for Flexural Strength of AdvancedCera
13、mics at Elevated TemperaturesC 1239 Practice for Reporting Uniaxial Strength Data andEstimating Weibull Distribution Parameters for AdvancedCeramicsC 1273 Test Method for Tensile Strength of MonolithicAdvanced Ceramics at Ambient TemperaturesC 1322 Practice for Fractography and Characterization ofFr
14、acture Origins in Advanced CeramicsC 1323 Test Method for Ultimate Strength of AdvancedCeramics with Diametrally Compressed C-Ring Speci-mens at Ambient Temperature1This practice is under the jurisdiction of ASTM Committee C28 on AdvancedCeramics and is the direct responsibility of Subcommittee C28.
15、01 on MechanicalProperties and Performance.Current edition approved Jan. 1, 2008. Published January 2008.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standar
16、ds Document Summary page onthe ASTM website.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.C 1366 Test Method for Tensile Strength of MonolithicAdvanced Ceramics at Elevated TemperaturesC 1499 Test Method for Monotonic Equibiaxial F
17、lexuralStrength of Advanced Ceramics at Ambient TemperatureE6 Terminology Relating to Methods of Mechanical Test-ingE 456 Terminology Relating to Quality and Statistics3. Terminology3.1 Unless otherwise noted, the Weibull parameter estima-tion terms and equations found in Practice C 1239 shall beuse
18、d.3.2 For definitions of other statistical terms, terms related tomechanical testing, and terms related to advanced ceramicsused in this guide, refer to Terminologies E6, E 456, andC 1145, or to appropriate textbooks on statistics (1-4).33.3 Nomenclature:AT= gage area of a uniaxial tensile test spec
19、imenAB4= gage area of a four-point flexure test specimenAB3= gage area of a three-point flexure test specimenAPOR= gage area 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
20、specimend = thickness of a flexure test specimenD = diameter of a round flexure test specimenD = overall diameter of a ring-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
21、 pressure-on-ring or ring-on-ring disk testspecimenk = load factorLgs= length of the gage section in a uniaxial tensile testspecimenLi4= 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
22、a three-point flexure testspecimenm = Weibull modulusPf= probability of failureri= inner radius of a C-ringro= outer radius 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 spec
23、imenVE= effective volume of a test specimenVPOR= gage volume of a pressure-on-ring test specimenVROR= gage volume of a ring-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 specimenV
24、CR= gage volume of a C-ring test specimens = uniaxial tensile stresssmax= maximum tensile stress in a test specimen at fractures1, s2, s3= principal stresses (tensile) at the integrationpoints in any finite elements0= Weibull material scale parameter (strength relative tounit size)su= Weibull charac
25、teristic strengthsuT= Weibull characteristic strength of a uniaxial tensile testspecimen3The boldface numbers in parentheses refer to the list of references at the end ofthis standard.FIG. 1 Strength Scales with SizeC1683082suB4= Weibull characteristic strength for a four-point flex-ure test specime
26、nsuB3= Weibull characteristic strength for a three-point flex-ure test specimensuCR= Weibull characteristic strength for a C-ring testspecimensuPOR= Weibull characteristic strength for a pressure-on-ring test specimensuROR= Weibull characteristic strength for a ring-on-ringtest specimens* = an arbit
27、rary, assumed estimate of the Weibull materialscale factors= mean strengthsT= mean strength for a uniaxial tensile test specimensB4= 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 s
28、trength for a pressure-on-ring test specimensROR= mean strength for a ring-on-ring test specimenu = angle in a C-ring test specimen4. Summary of Practice4.1 The observed strength values of advanced ceramics aredependent on test specimen size, geometry and stress state.This standard practice enables
29、the user to convert tensilestrength parameters obtained from one test geometry to that ofanother, on the 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 r
30、elated Practice C 1239 for theoriginal test geometry. This practice uses the test specimen andloading sizes and geometries, and suand m to calculate theWeibull material scale parameter s0. The Weibull characteristicstrength su, the mean strength s, or the Weibull material scalefactor s0, may be scal
31、ed to alternative test specimen geom-etries. Finally, a report citing the original test specimengeometry 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.
32、Lack of ductility combined with flaws thathave various sizes and orientations typically leads to largescatter 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. T
33、his standard is applicable to brittle monolithicceramics which fail as a result of catastrophic propagation 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 d
34、istribution. This standard is restricted to the two-parameter formulation.5.3 Tensile and flexural test 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-
35、form solutionsfor the effective volume and effective surfaces and the Weibullmaterial scale factor are included for these configurations. Thispractice also incorporates size scaling methods for C-ring testspecimens for which numerical approaches are necessary. Ageneric approach for arbitrary shaped
36、test specimens or com-ponents that utilizes finite element analyses is presented inAnnex A3.5.4 The fracture 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
37、 surface flaws). This standard allowsfor the conversion of strength parameters associated with eithertype 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
38、with theWeibull model is based on several key assumptions (5).Itisassumed that the material is uniform, homogeneous, andisotropic. If the material is a composite, it is assumed that thecomposite phases are sufficiently small that the structurebehaves on an engineering scale as a homogeneous andisotr
39、opic body. The composite must contain a sufficient quan-tity of uniformly-distributed, randomly-oriented, reinforcingelements such that the material is effectively homogeneous.Whisker-toughened ceramic composites may be representativeof this type of material. This practice is also applicable tocompo
40、site ceramics that do not exhibit any appreciable bilinearor nonlinear deformation behavior. This standard and theconventional Weibull strength scaling with size may not besuitable for continuous fiber-reinforced composite ceramics.The material is assumed to fracture in a brittle fashion, aconsequen
41、ce of stress causing catastrophic propagation offlaws. The material is assumed to be consistent (batch to batch,day to day, etc.). It is assumed that the strength distributionfollows a Weibull two parameter distribution. It is assumed thatthe same specific flaw type controls strength in the variouss
42、pecimen configurations. It is assumed that each test piece hasa statistically significant number of flaws and that they arerandomly distributed. It is assumed that the flaws are smallrelative to the specimen cross section size. If multiple flawtypes are present and control strength, then strengths m
43、ay scaledifferently for each flaw type. Consult Practice C 1239 and theexample in 9.1 for further guidance on how to apply censoredstatistics in such cases. It is also assumed that the specimenstress state and the maximum stress are accurately determined.It is assumed that the actual data from a set
44、 of fracturedspecimens are accurate and precise. (See Terminology E 456for definitions of the latter two terms.) For this reason, thisstandard frequently references otherASTM standard test meth-ods and practices which are known to be reliable in thisrespect.5.6 Even if test data has been accurately
45、and preciselymeasured, it should be recognized that the Weibull parametersdetermined from test data are in fact estimates. The estimatescan vary from the actual (population) material strength param-eters. Consult Practice C 1239 for further guidance on theC1683083confidence bounds of Weibull paramet
46、er estimates based ontest data for a finite sample size of test fractures.5.7 When correlating strength 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
47、 as discussed in5.6 may also contribute to variations between computed andobserved strength-size scaling 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 specimen
48、size. Pores that are 50 m (0.050 mm) in diameter arevolume-distributed flaws in tension or flexural strength 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 variab
49、le 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 a normal distribution. Ifthe random variable representing uniaxial tensile strength of anadvanced ceramic is characterized by a two-parameter Weibulldistribution (see Practice C 1239 for a detailed discussio
