1、Designation: C1683 10 (Reapproved 2015)Standard 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 ca
2、se of revision, the 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 (primar
3、ily 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 data. It may alsobe used for more complex geom
4、etries 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. The larger the specimenor component, the
5、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 distribution isthe most appropriate especially
6、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 strength calculated. Thestrength values are us
7、ed 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 scaling. The practicealso assumes that the flaw p
8、opulation 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 6Test Specimens with Uniaxial Stress StatesEffectiveVolum
9、e 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-Ring Test Specimens 8.1Ring-on-Ring Test Specimens
10、 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 Specimens with ComplexGeometries and Stress Distribu
11、tionsAnnex 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, associat
12、ed 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.1This practice is under the jurisdiction of ASTM Committee C28 on AdvancedCeramics and is the direct res
13、ponsibility of Subcommittee C28.01 on MechanicalProperties and Performance.Current edition approved Jan. 1, 2015. Published April 2015. Originallyapproved in 2008. Last previous edition approved in 2010 as C1683 10. DOI:10.1520/C1683-10R15.Copyright ASTM International, 100 Barr Harbor Drive, PO Box
14、C700, West Conshohocken, PA 19428-2959. United States12. Referenced Documents2.1 ASTM Standards:2C1145 Terminology of Advanced CeramicsC1161 Test Method for Flexural Strength of AdvancedCeramics at Ambient TemperatureC1211 Test Method for Flexural Strength of AdvancedCeramics at Elevated Temperature
15、sC1239 Practice for Reporting Uniaxial Strength Data andEstimating Weibull Distribution Parameters for AdvancedCeramicsC1273 Test Method for Tensile Strength of MonolithicAdvanced Ceramics at Ambient TemperaturesC1322 Practice for Fractography and Characterization ofFracture Origins in Advanced Cera
16、micsC1323 Test Method for Ultimate Strength of AdvancedCeramics with Diametrally 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 C
17、eramics at Ambient TemperatureE6 Terminology Relating to Methods of Mechanical 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
18、statistical terms, terms related tomechanical testing, and terms related 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-poin
19、t 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 specimend = thickness of a flexur
20、e 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 pressure-on-ring or ring-on-ring
21、 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 a three-point flexure testspecime
22、nm = 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 specimenVE= effective volume of a tes
23、t 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 specimenVCR= gage volume of a C-ring test
24、specimen = uniaxial tensile stressmax= maximum tensile stress in a test specimen at fracture1, 2, 3= principal stresses (tensile) at the integrationpoints in any finite element0= Weibull material scale parameter (strength relative tounit size)= Weibull characteristic strengthT= Weibull characteristi
25、c strength of a uniaxial tensile testspecimenB4= Weibull characteristic strength for a four-point flexuretest specimenB3= Weibull characteristic strength for a three-point flex-ure test specimenCR= Weibull characteristic strength for a C-ring test speci-men2For referenced ASTM standards, visit the A
26、STM 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.3The boldface numbers in parentheses refer to the list of references at the end ofthis standard.FIG. 1 S
27、trength Scales with SizeC1683 10 (2015)2POR= Weibull characteristic strength for a pressure-on-ring test specimenROR= Weibull characteristic strength for a ring-on-ringtest specimen* = an arbitrary, assumed estimate of the Weibull materialscale factor = mean strengthT= mean strength for a uniaxial t
28、ensile test specimenB4= mean strength for a four-point flexure test specimenB3= mean strength for a three-point flexure test specimenCR= mean strength for a C-ring test specimenPOR= mean strength for a pressure-on-ring test specimenROR= mean strength for a ring-on-ring test specimen = angle in a C-r
29、ing test specimen = 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,
30、 on the basis of assumptions listed in 5.5. Using theexisting fracture strength data, estimates of the Weibull char-acteristic strength , 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 loadi
31、ngsizes and geometries, and and m to calculate the Weibullmaterial scale parameter 0. The Weibull characteristic strength, the mean strength , or the Weibull material scale factor 0,may be scaled to alternative test specimen geometries. Finally,a report citing the original test specimen geometry and
32、 strengthparameters, as well as the size scaled Weibull strength param-eters 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 largescatt
33、er 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 propagation
34、 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 spe
35、cimens 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 incl
36、uded 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 fractur
37、e 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 eithertype o
38、f 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 type
39、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 isotro
40、pic 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 compos
41、ite 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 consequen
42、ce 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 flaw
43、s 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 guidanc
44、e 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 lattertwo t
45、erms.) 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 from test data are in
46、 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 strength parameters
47、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 inC1683 10 (2015)35.6 may also contribute to variations between computed andobserved strength-size sc
48、aling 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
49、 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 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 t
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