1、Designation: C1683 10Standard 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, th
2、e 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.1. Scope1.1 This standard practice provides methodology to convertfracture strength parameters (primarily the mean stren
3、gth 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 that
4、 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 likel
5、y 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 distribution isthe most appropriate especially since it permits s
6、trength 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 Weibu
7、ll 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 stabl
8、e 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 Specimens with Uniaxial Stress StatesEffectiveVolume and Area
9、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-Ring Test Specimens 8.1Ring-on-Ring Test Specimens 8.2Example
10、s 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 DistributionsAnnex
11、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 address all of thesafety concerns, if any, associated with its
12、 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 Terminology of Advanced CeramicsC1161 Test Method for Flexural Str
13、ength 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 AdvancedCeramics1This practice is under the jurisdiction of ASTM C
14、ommittee C28 on AdvancedCeramics and is the direct responsibility of Subcommittee C28.01 on MechanicalProperties and Performance.Current edition approved Dec. 1, 2010. Published January 2011. Originallyapproved in 2008. Last previous edition approved in 2008 as C1683 081. DOI:10.1520/C1683-10.2For r
15、eferenced 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 standards Document Summary page onthe ASTM website.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700,
16、West Conshohocken, 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 Di
17、ametrally Compressed 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 Mec
18、hanical TestingE456 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 relate
19、d to advanced ceramicsused 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 spe
20、cimenAPOR= 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 specimend = thickness of a flexure test specimenD = diameter of a round flexure test specimenD = overall
21、 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 pressure-on-ring or ring-on-ring disk testspecimenk = load factorLgs= length of the gage section in a u
22、niaxial 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 a three-point flexure testspecimenm = Weibull modulusPf= probability of failureri= inner radius of a C-r
23、ingro= 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 specimenVE= effective volume of a test specimenVPOR= gage volume of a pressure-on-ring test specimenVROR= ga
24、ge 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 specimenVCR= gage volume of a C-ring test specimens = uniaxial tensile stresssmax= maximum tensile stress in a te
25、st 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 characteristic strengthsuT= Weibull characteristic strength of a uniaxial tensile testspecimensuB4= Weibull ch
26、aracteristic strength for a four-point flex-ure test specimensuB3= 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 characte
27、ristic 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 specimensB4= mean strength for a four-point flexure test specimensB3= mean strength for a three-point flexure test speci
28、mensCR= mean strength for a C-ring test specimensPOR= mean strength for a pressure-on-ring test specimen3The boldface numbers in parentheses refer to the list of references at the end ofthis standard.FIG. 1 Strength Scales with SizeC1683 102sROR= mean strength for a ring-on-ring test specimenu = ang
29、le in a C-ring test 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 tha
30、t 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 related Practice C1239 for the originaltest geometry. This practice uses the test spec
31、imen and loadingsizes 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 tes
32、t 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. Lack of ductility combined with flaws thathave various sizes and orientations typicall
33、y 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. This standard is applicable to brittle monolithicceramics which fail as a result of cat
34、astrophic 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 distribution. This standard is restricted to the two-parameter formulation.5.3 Tensile
35、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-form solutionsfor the effective volume and effective surfaces and the Weibullmaterial
36、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 test specimens or com-ponents that utilizes finite element analyses is presented inAnn
37、ex 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 surface flaws). This standard allowsfor the conversion of strength parameters associa
38、ted 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 with theWeibull model is based on several key assumptions (5).Itisassumed that the sam
39、e specific flaw type 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 h
40、omogeneous and isotropic 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
41、 applicable to composite 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
42、 fashion, a consequence 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 statisticallysigni
43、ficant number of flaws 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
44、.1for further guidance 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 definitio
45、ns of the lattertwo terms.) For this reason, this standard frequently referencesother ASTM standard test methods and practices which areknown to be reliable in this respect.5.6 Even if test data has been accurately and preciselymeasured, it should be recognized that the Weibull parametersdetermined
46、from test 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
47、 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 as discussed in5.6 may also contribute to variations between computed andobserved strength-si
48、ze 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 specimensize. Pores that are 50 m (0.050 mm) in diameter arevolume-distributed flaws in tension or fle
49、xural 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 variable representing uniaxial tensilestrength of an advanced ceramic will assume only positivevalues, and the distribution is usually asymmetric about theC1683 103mean. 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
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