ASTM C1683-10(2019)
(Practice)Standard Practice for Size Scaling of Tensile Strengths Using Weibull Statistics for Advanced Ceramics
Standard Practice for Size Scaling of Tensile Strengths Using Weibull Statistics for Advanced Ceramics
SIGNIFICANCE AND USE
5.1 Advanced ceramics usually display a linear stress-strain behavior to failure. Lack of ductility combined with flaws that have various sizes and orientations typically leads to large scatter in failure strength. Strength is not a deterministic property, but instead reflects the intrinsic fracture toughness and a distribution (size and orientation) of flaws present in the material. This standard is applicable to brittle monolithic ceramics which fail as a result of catastrophic propagation of flaws. 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 the Weibull distribution. This standard is restricted to the two-parameter formulation.
5.3 Tensile and flexural test specimens are the most commonly 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 solutions for the effective volume and effective surfaces and the Weibull material scale factor are included for these configurations. This practice also incorporates size-scaling methods for C-ring test specimens for which numerical approaches are necessary. A generic approach for arbitrary shaped test specimens or components that utilizes finite element analyses is presented in Annex A3.
5.4 The fracture origins of failed test specimens can be determined using fractographic analysis. The spatial distribution of these strength-controlling flaws can be over a volume or an area (as in the case of surface flaws). This standard allows for the conversion of strength parameters associated with either type of spatial distribution. Length scaling for strength-controlling flaws located along edges of a test specimen is not covered in this practice.
5.5 The scaling of strength with size in accordance with the Weibull model is based on several key assumptions (5). It is assumed that th...
SCOPE
1.1 This standard practice provides methodology to convert fracture strength parameters (primarily the mean strength and the Weibull characteristic strength) estimated from data obtained with one test geometry to strength parameters representing other test geometries. This practice addresses uniaxial strength data as well as some biaxial strength data. It may also be used for more complex geometries proved that the effective areas and effective volumes can be estimated. It is for the evaluation of Weibull probability distribution parameters for advanced ceramics that fail in a brittle fashion. Fig. 1 shows the typical variation of strength with size. The larger the specimen or component, the weaker it is likely to be.
1.5 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.
1.5.1 The values stated in SI units are in accordance with IEEE/ASTM SI 10.
1.6 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.
1.7 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
General Information
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Standards Content (Sample)
This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: C1683 − 10 (Reapproved 2019)
Standard Practice for
Size Scaling of Tensile Strengths Using Weibull Statistics
for Advanced Ceramics
This standard is issued under the fixed designation C1683; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope also assumes that the flaw population is stable with time and
that no slow crack growth occurs.
1.1 This standard practice provides methodology to convert
fracture strength parameters (primarily the mean strength and
1.4 This practice includes the following topics:
the Weibull characteristic strength) estimated from data ob-
Section
tained with one test geometry to strength parameters represent-
Scope 1
Referenced Documents 2
ing other test geometries. This practice addresses uniaxial
Terminology 3
strength data as well as some biaxial strength data. It may also
Summary of Practice 4
be used for more complex geometries proved that the effective
Significance and Use 5
Probability of Failure Relationships 6
areas and effective volumes can be estimated. It is for the
Test Specimens with Uniaxial Stress States—Effective 7
evaluation of Weibull probability distribution parameters for
Volume and Area Relationships
advancedceramicsthatfailinabrittlefashion.Fig.1showsthe
Uniaxial Tensile Test Specimens 7.1
Rectangular Flexure Test Specimens 7.2
typical variation of strength with size. The larger the specimen
Round Flexure Test Specimens 7.3
or component, the weaker it is likely to be.
C-Ring Test Specimens 7.4
Test Specimens with Multiaxial Stress States—Effective 8
1.2 As noted in Practice C1239, the failure strength of
Volume and Area Relationships
advanced ceramics is treated as a continuous random variable.
Pressure-on-Ring Test Specimens 8.1
Anumberoffunctionsmaybeusedtocharacterizethestrength Ring-on-Ring Test Specimens 8.2
Examples—Converting Characteristic Strengths 9
distribution of brittle ceramics, but the Weibull distribution is
Report 10
the most appropriate, especially since it permits strength
Precision and Bias 11
scaling for the size of specimens or component. Typically, a Keywords 12
Combined Gamma Function for Round Rods Tested Annex A1
number of test specimens with well-defined geometry are
in Flexure
broken under well-defined loading conditions. The force at
Components or Test Specimens with Multiaxial Annex A2
Stress Distributions
whicheachtestspecimenfailsisrecordedandfracturestrength
Components or Test Specimens with Complex Annex A3
calculated. The strength values are used to obtain Weibull
Geometries and Stress Distributions
parameter estimates associated with the underlying population
1.5 The values stated in SI units are to be regarded as
distribution.
standard. No other units of measurement are included in this
1.3 This standard is restricted to the assumption that the
standard.
distribution underlying the failure strengths is the two-
1.5.1 The values stated in SI units are in accordance with
parameter Weibull distribution with size scaling. The practice
IEEE/ASTM SI 10.
1.6 This standard does not purport to address all of the
This practice is under the jurisdiction of ASTM Committee C28 on Advanced
Ceramics and is the direct responsibility of Subcommittee C28.01 on Mechanical
safety concerns, if any, associated with its use. It is the
Properties and Performance.
responsibility of the user of this standard to establish appro-
Current edition approved July 1, 2019. Published July 2019. Originally approved
priate safety, health, and environmental practices and deter-
in 2008. Last previous edition approved in 2015 as C1683 –10 (2015). DOI:
10.1520/C1683-10R19. mine the applicability of regulatory limitations prior to use.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
C1683 − 10 (2019)
3.2 For definitions of other statistical terms, terms related to
mechanical testing, and terms related to advanced ceramics
used in this practice, refer to Terminologies E6, E456, and
C1145, or to appropriate textbooks on statistics (1-4).
3.3 Nomenclature:
A = gage area of a uniaxial tensile test specimen
T
A = gage area of a four-point flexure test specimen
B4
A = gage area of a three-point flexure test specimen
B3
A = gage area of a pressure-on-ring test specimen
POR
A = gage area of a ring-on-ring test specimen
ROR
A = gage area of a C-ring test specimen
CR
b = thickness of a C-ring
b = width of a flexure test specimen
d = thickness of a flexure test specimen
D = diameter of a round flexure test specimen
D = overall diameter of a ring-on-ring disk test specimen
D = loading (inner) ring diameter, ring-on-ring disk speci-
L
men
FIG. 1 Strength Scales with Size
D = support ring diameter, ring-on-ring or pressure-on-ring
S
disk specimen
1.7 This international standard was developed in accor-
h = thickness of pressure-on-ring or ring-on-ring disk test
dance with internationally recognized principles on standard-
specimen
ization established in the Decision on Principles for the
k = load factor
Development of International Standards, Guides and Recom-
L = length of the gage section in a uniaxial tensile test
gs
mendations issued by the World Trade Organization Technical
specimen
Barriers to Trade (TBT) Committee.
L = length of the inner span for a four-point flexure test
i4
specimen
2. Referenced Documents
L = length of the outer span for a four-point flexure test
o4
2.1 ASTM Standards:
specimen
C1145 Terminology of Advanced Ceramics
L = length of the outer span for a three-point flexure test
o3
C1161 Test Method for Flexural Strength of Advanced
specimen
Ceramics at Ambient Temperature
m = Weibull modulus
C1211 Test Method for Flexural Strength of Advanced
P = probability of failure
f
Ceramics at Elevated Temperatures
r = inner radius of a C-ring
i
C1239 Practice for Reporting Uniaxial Strength Data and
r = outer radius of a C-ring
o
Estimating Weibull Distribution Parameters forAdvanced
t = thickness of a C-ring
Ceramics
R = radius of the support ring for pressure-on-ring
s
C1273 Test Method for Tensile Strength of Monolithic
R = radius of the pressure-on-ring disk specimen
d
Advanced Ceramics at Ambient Temperatures
S = effective surface area of a test specimen
E
C1322 Practice for Fractography and Characterization of
V = effective volume of a test specimen
E
Fracture Origins in Advanced Ceramics
V = gage volume of a pressure-on-ring test specimen
POR
C1323 Test Method for Ultimate Strength of Advanced
V = gage volume of a ring-on-ring disk test specimen
ROR
Ceramics with Diametrally Compressed C-Ring Speci-
V = gage volume of tensile test specimen
T
mens at Ambient Temperature
V = gage volume of a four-point flexure test specimen
B4
C1366 Test Method for Tensile Strength of Monolithic
V = gage volume of a three-point flexure test specimen
B3
Advanced Ceramics at Elevated Temperatures
V = gage volume of a C-ring test specimen
CR
C1499 Test Method for Monotonic Equibiaxial Flexural
σ = uniaxial tensile stress
Strength of Advanced Ceramics at Ambient Temperature
σ = maximum tensile stress in a test specimen at fracture
max
E6 Terminology Relating to Methods of Mechanical Testing
σ , σ , σ = principal stresses (tensile) at the integration
1 2 3
E456 Terminology Relating to Quality and Statistics
points in any finite element
σ = Weibull material scale parameter (strength relative to
3. Terminology
unit size)
3.1 Unless otherwise noted, the Weibull parameter estima-
σ = Weibull characteristic strength
θ
tiontermsandequationsfoundinPracticeC1239shallbeused.
σ = Weibull characteristic strength of a uniaxial tensile test
θT
specimen
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
Standards volume information, refer to the standard’s Document Summary page on The boldface numbers in parentheses refer to the list of references at the end of
the ASTM website. this standard.
C1683 − 10 (2019)
σ = Weibullcharacteristicstrengthforafour-pointflexure for the effective volume and effective surfaces and the Weibull
θB4
test specimen material scale factor are included for these configurations.This
σ = Weibull characteristic strength for a three-point flex- practice also incorporates size-scaling methods for C-ring test
θB3
ure test specimen specimens for which numerical approaches are necessary. A
σ = WeibullcharacteristicstrengthforaC-ringtestspeci-
generic approach for arbitrary shaped test specimens or com-
θCR
men ponents that utilizes finite element analyses is presented in
σ = Weibull characteristic strength for a pressure-on-
Annex A3.
θPOR
ring test specimen
5.4 The fracture origins of failed test specimens can be
σ = Weibull characteristic strength for a ring-on-ring
θROR
determined using fractographic analysis. The spatial distribu-
test specimen
tionofthesestrength-controllingflawscanbeoveravolumeor
σ* = an arbitrary, assumed estimate of the Weibull material
an area (as in the case of surface flaws). This standard allows
scale factor
fortheconversionofstrengthparametersassociatedwitheither
σ¯ = mean strength
type of spatial distribution. Length scaling for strength-
σ¯ = mean strength for a uniaxial tensile test specimen
T
controlling flaws located along edges of a test specimen is not
σ¯ = mean strength for a four-point flexure test specimen
B4
covered in this practice.
σ¯ = mean strength for a three-point flexure test specimen
B3
σ¯ = mean strength for a C-ring test specimen
CR 5.5 The scaling of strength with size in accordance with the
σ¯ = mean strength for a pressure-on-ring test specimen
Weibull model is based on several key assumptions (5).Itis
POR
σ¯ = mean strength for a ring-on-ring test specimen
ROR assumed that the same specific flaw type controls strength in
θ = angle in a C-ring test specimen
the various specimen configurations. It is assumed that the
ν = Poisson’s ratio
material is uniform, homogeneous, and isotropic. If the mate-
rial is a composite, it is assumed that the composite phases are
4. Summary of Practice
sufficiently small that the structure behaves on an engineering
4.1 The observed strength values of advanced ceramics are
scale as a homogeneous and isotropic body. The composite
dependent on test specimen size, geometry, and stress state.
must contain a sufficient quantity of uniformly distributed,
This standard practice enables the user to convert tensile
randomly oriented reinforcing elements such that the material
strength parameters obtained from one test geometry to that of
is effectively homogeneous. Whisker-toughened ceramic com-
another, on the basis of assumptions listed in 5.5. Using the
posites may be representative of this type of material. This
existing fracture strength data, estimates of the Weibull char-
practice is also applicable to composite ceramics that do not
acteristic strength σ , and the Weibull modulus m, are calcu-
θ exhibit any appreciable bilinear or nonlinear deformation
latedinaccordancewithrelatedPracticeC1239fortheoriginal
behavior. This standard and the conventional Weibull strength
test geometry. This practice uses the test specimen and loading
scaling with size may not be suitable for continuous fiber-
sizes and geometries, and σ and m to calculate the Weibull
θ reinforced composite ceramics. The material is assumed to
materialscaleparameterσ .TheWeibullcharacteristicstrength
fracture in a brittle fashion, a consequence of stress causing
σ , the mean strengthσ¯, or theWeibull material scale factorσ ,
θ 0 catastrophic propagation of flaws. The material is assumed to
may be scaled to alternative test specimen geometries. Finally,
be consistent (batch to batch, day to day, etc.). It is assumed
a report citing the original test specimen geometry and strength
that the strength distribution follows a Weibull two-parameter
parameters, as well as the size-scaled Weibull strength param-
distribution. It is assumed that each test piece has a statistically
eters is prepared.
significant number of flaws and that they are randomly
distributed. It is assumed that the flaws are small relative to the
5. Significance and Use
specimen cross section size. If multiple flaw types are present
5.1 Advanced ceramics usually display a linear stress-strain
and control strength, then strengths may scale differently for
behavior to failure. Lack of ductility combined with flaws that
each flaw type. Consult Practice C1239 and the example in 9.1
have various sizes and orientations typically leads to large
forfurtherguidanceonhowtoapplycensoredstatisticsinsuch
scatter in failure strength. Strength is not a deterministic
cases. It is also assumed that the specimen stress state and the
property, but instead reflects the intrinsic fracture toughness
maximum stress are accurately determined. It is assumed that
and a distribution (size and orientation) of flaws present in the
the actual data from a set of fractured specimens are accurate
material. This standard is applicable to brittle monolithic
and precise. (SeeTerminology E456 for definitions of the latter
ceramics which fail as a result of catastrophic propagation of
two terms.) For this reason, this standard frequently references
flaws. Possible rising R-curve effects are also not considered,
other ASTM standard test methods and practices which are
butareinherentlyincorporatedintothestrengthmeasurements.
known to be reliable in this respect.
5.2 Two- and three-parameter formulations exist for the
5.6 Even if test data has been accurately and precisely
Weibull distribution. This standard is restricted to the two-
measured, it should be recognized that the Weibull parameters
parameter formulation.
determined from test data are in fact estimates. The estimates
5.3 Tensile and flexural test specimens are the most com- can vary from the actual (population) material strength param-
monly used test configurations for advanced ceramics. Ring- eters. Consult Practice C1239 for further guidance on the
on-ring and pressure-on-ring test specimens which have multi- confidence bounds of Weibull parameter estimates based on
axial states of stress are also included. Closed-form solutions test data for a finite sample size of test fractures.
C1683 − 10 (2019)
5.7 When correlating strength parameters from test data wher
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