ASTM E2245-11(2018)
(Test Method)Standard Test Method for Residual Strain Measurements of Thin, Reflecting Films Using an Optical Interferometer (Withdrawn 2023)
Standard Test Method for Residual Strain Measurements of Thin, Reflecting Films Using an Optical Interferometer (Withdrawn 2023)
SIGNIFICANCE AND USE
5.1 Residual strain measurements are an aid in the design and fabrication of MEMS devices. The value for residual strain can be used in Young's modulus calculations.
SCOPE
1.1 This test method covers a procedure for measuring the compressive residual strain in thin films. It applies only to films, such as found in microelectromechanical systems (MEMS) materials, which can be imaged using an optical interferometer, also called an interferometric microscope. Measurements from fixed-fixed beams that are touching the underlying layer are not accepted.
1.2 This test method uses a non-contact optical interferometric microscope with the capability of obtaining topographical 3-D data sets. It is performed in the laboratory.
1.3 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.
1.4 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.
WITHDRAWN RATIONALE
This test method covers a procedure for measuring the compressive residual strain in thin films. It applies only to films, such as found in microelectromechanical systems (MEMS) materials, which can be imaged using an optical interferometer, also called an interferometric microscope. Measurements from fixed-fixed beams that are touching the underlying layer are not accepted.
Formerly under the jurisdiction of Committee E08 on Fatigue and Fracture, this test method was withdrawn in November 2023. This standard is being withdrawn without replacement due to its limited use by industry.
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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:E2245 −11 (Reapproved 2018)
Standard Test Method for
Residual Strain Measurements of Thin, Reflecting Films
Using an Optical Interferometer
This standard is issued under the fixed designation E2245; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope Atomic Force Microscope at Subnanometer Displacement
Levels Using Si(111) Monatomic Steps (Withdrawn
1.1 This test method covers a procedure for measuring the
2015)
compressive residual strain in thin films. It applies only to
films, such as found in microelectromechanical systems 2.2 SEMI Standard:
MS2Test Method for Step Height Measurements of Thin
(MEMS) materials, which can be imaged using an optical
Films
interferometer,alsocalledaninterferometricmicroscope.Mea-
surements from fixed-fixed beams that are touching the under-
3. Terminology
lying layer are not accepted.
3.1 Definitions:
1.2 This test method uses a non-contact optical interfero-
3.1.1 The following terms can be found in Terminology
metricmicroscopewiththecapabilityofobtainingtopographi-
E2444.
cal 3-D data sets. It is performed in the laboratory.
3.1.2 2-D data trace, n—a two-dimensional group of points
1.3 This standard does not purport to address all of the
that is extracted from a topographical 3-D data set and that is
safety concerns, if any, associated with its use. It is the
parallel to the xz-or yz-plane of the interferometric micro-
responsibility of the user of this standard to establish appro-
scope.
priate safety, health, and environmental practices and deter-
3.1.3 3-D data set, n—a three-dimensional group of points
mine the applicability of regulatory limitations prior to use.
with a topographical z-value for each (x, y) pixel location
1.4 This international standard was developed in accor-
within the interferometric microscope’s field of view.
dance with internationally recognized principles on standard-
ization established in the Decision on Principles for the
3.1.4 anchor, n—in a surface-micromachining process, the
Development of International Standards, Guides and Recom-
portion of the test structure where a structural layer is inten-
mendations issued by the World Trade Organization Technical
tionally attached to its underlying layer.
Barriers to Trade (TBT) Committee.
3.1.5 anchor lip, n—in a surface-micromachining process,
the freestanding extension of the structural layer of interest
2. Referenced Documents
around the edges of the anchor to its underlying layer.
2.1 ASTM Standards:
3.1.5.1 Discussion—In some processes, the width of the
E2244Test Method for In-Plane Length Measurements of
anchor lip may be zero.
Thin, Reflecting Films Using an Optical Interferometer
3.1.6 bulk micromachining, adj—a MEMS fabrication pro-
E2246Test Method for Strain Gradient Measurements of
cess where the substrate is removed at specified locations.
Thin, Reflecting Films Using an Optical Interferometer
3.1.7 cantilever, n—a test structure that consists of a free-
E2444Terminology Relating to Measurements Taken on
standing beam that is fixed at one end.
Thin, Reflecting Films
E2530Practice for Calibrating the Z-Magnification of an 3.1.8 fixed-fixed beam, n—a test structure that consists of a
freestanding beam that is fixed at both ends.
3.1.9 in-plane length (or deflection) measurement, n—the
This test method is under the jurisdiction ofASTM Committee E08 on Fatigue
experimental determination of the straight-line distance be-
and Fracture and is the direct responsibility of Subcommittee E08.05 on Cyclic
Deformation and Fatigue Crack Formation. tween two transitional edges in a MEMS device.
Current edition approved May 1, 2018. Published May 2018. Originally
ɛ1
approved in 2002. Last previous edition approved in 2011 as E2245–11 . DOI:
10.1520/E2245–11R18
2 3
For referenced ASTM standards, visit the ASTM website, www.astm.org, or The last approved version of this historical standard is referenced on
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM www.astm.org.
Standards volume information, refer to the standard’s Document Summary page on For referenced Semiconductor Equipment and Materials International (SEMI)
the ASTM website. standards, visit the SEMI website, www.semi.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2245−11 (2018)
3.1.9.1 Discussion—This length (or deflection) measure- 3.2 Symbols:
ment is made parallel to the underlying layer (or the xy-plane 3.2.1 ForCalibration:σ =themaximumoftwouncali-
6same
of the interferometric microscope).
brated values (σ and σ ) where σ is the standard
same1 same2 same1
deviation of the six step height measurements taken on the
3.1.10 interferometer, n—a non-contact optical instrument
physical step height standard at the same location before the
used to obtain topographical 3-D data sets.
data session and σ is the standard deviation of the six
same2
3.1.10.1 Discussion—The height of the sample is measured
measurementstakenatthissamelocationafterthedatasession
along the z-axis of the interferometer. The x-axis is typically
σ =the certified one sigma uncertainty of the physical
aligned parallel or perpendicular to the transitional edges to be cert
step height standard used for calibration
measured.
σ =the standard deviation of the noise measurement,
noise
3.1.11 MEMS, adj—microelectromechanical systems.
calculated to be one-sixth the value of R minus R
tave ave
3.1.12 microelectromechanical systems, adj—in general,
σ =the standard deviation of the surface roughness
Rave
this term is used to describe micron-scale structures, sensors,
measurement, calculated to be one-sixth the value of R
ave
actuators,andtechnologiesusedfortheirmanufacture(suchas,
σ =the standard deviation in a ruler measurement in the
xcal
silicon process technologies), or combinations thereof.
interferometric microscope’s x-direction for the given combi-
3.1.13 residual strain, n—in a MEMS process, the amount
nation of lenses
of deformation (or displacement) per unit length constrained
σ =the standard deviation in a ruler measurement in the
ycal
within the structural layer of interest after fabrication yet
interferometric microscope’s y-direction for the given combi-
before the constraint of the sacrificial layer (or substrate) is
nation of lenses
removed (in whole or in part).
cal =the x-calibration factor of the interferometric micro-
x
scope for the given combination of lenses
3.1.14 sacrificial layer, n—a single thickness of material
cal =the y-calibration factor of the interferometric micro-
that is intentionally deposited (or added) then removed (in
y
scope for the given combination of lenses
whole or in part) during the micromachining process, to allow
freestanding microstructures. cal =the z-calibration factor of the interferometric micro-
z
scope for the given combination of lenses
3.1.15 stiction, n—adhesion between the portion of a struc-
cert=the certified (that is, calibrated) value of the physical
turallayerthatisintendedtobefreestandinganditsunderlying
step height standard
layer.
ruler =the interferometric microscope’s maximum field of
x
3.1.16 (residual) strain gradient, n—a through-thickness
view in the x-direction for the given combination of lenses as
variation (of the residual strain) in the structural layer of
measured with a 10-µm grid (or finer grid) ruler
interest before it is released.
ruler =the interferometric microscope’s maximum field of
y
3.1.16.1 Discussion—If the variation through the thickness
view in the y-direction for the given combination of lenses as
inthestructurallayerisassumedtobelinear,itiscalculatedto
measured with a 10-µm grid (or finer grid) ruler
be the positive difference in the residual strain between the top
scope =the interferometric microscope’s maximum field of
x
andbottomofacantileverdividedbyitsthickness.Directional
view in the x-direction for the given combination of lenses
information is assigned to the value of “s.”
scope =the interferometric microscope’s maximum field of
y
3.1.17 structural layer, n—a single thickness of material
view in the y-direction for the given combination of lenses
present in the final MEMS device.
x =the calibrated resolution of the interferometric micro-
res
3.1.18 substrate, n—thethick,startingmaterial(oftensingle
scope in the x-direction
crystalsiliconorglass)inafabricationprocessthatcanbeused
z¯ =the uncalibrated average of the six calibration
6same
to build MEMS devices.
measurements from which σ is found
6same
z =the uncalibrated positive difference between the av-
3.1.19 support region, n—in a bulk-micromachining
drift
process,theareathatmarkstheendofthesuspendedstructure. erageofthesixcalibrationmeasurementstakenbeforethedata
session (at the same location on the physical step height
3.1.20 surface micromachining, adj—a MEMS fabrication
standard used for calibration) and the average of the six
process where micron-scale components are formed on a
calibration measurements taken after the data session (at this
substratebythedeposition(oraddition)andremoval(inwhole
same location)
or in part) of structural and sacrificial layers.
z =over the instrument’s total scan range, the maximum
lin
3.1.21 test structure, n—acomponent(suchas,afixed-fixed
relative deviation from linearity, as quoted by the instrument
beamorcantilever)thatisusedtoextractinformation(suchas,
manufacturer (typically less than 3%)
the residual strain or the strain gradient of a layer) about a
z =the calibrated resolution of the interferometric micro-
res
fabrication process.
scope in the z-direction
3.1.22 transitional edge, n—the side of a MEMS structure
z¯ =the average of the calibration measurements taken
ave
that is characterized by a distinctive out-of-plane vertical
alongthephysicalstepheightstandardbeforeandafterthedata
displacement as seen in an interferometric 2-D data trace.
session
3.1.23 underlying layer, n—the single thickness of material 3.2.2 For In-plane Length Measurement: α=the misalign-
directly beneath the material of interest. ment angle
3.1.23.1 Discussion—This layer could be the substrate. L=thein-planelengthmeasurementofthefixed-fixedbeam
E2245−11 (2018)
L =thein-planelengthcorrectiontermforthegiventype ε =in determining the combined standard uncertainty
offset r-low
of in-plane length measurement taken on similar structures value for the residual strain measurement, the lowest value for
when using similar calculations and for the given combination ε given the specified variations
r
of lenses for a given interferometric microscope
σ =the in-plane length repeatability standard de-
Lrepeat(samp)'
v1 =one endpoint of the in-plane length measurement
viation (for the given combination of lenses for the given
end
interferometric microscope) as obtained from test structures
v2 =anotherendpointofthein-planelengthmeasurement
end
fabricated in a process similar to that used to fabricate the
x1 =the calibrated x-value that most appropriately lo-
uppert
sample and when the transitional edges face each other
cates the upper corner associated with Edge 1 in Trace t
σ =the relative residual strain repeatability stan-
x2 =the calibrated x-value that most appropriately lo-
repeat(samp)
uppert
darddeviationasobtainedfromfixed-fixedbeamsfabricatedin
cates the upper corner associated with Edge 2 in Trace t
a process similar to that used to fabricate the sample
y =the calibrated y-value associated with Trace a'
a'
R =the calibrated surface roughness of a flat and leveled
y =the calibrated y-value associated with Trace e'
ave
e'
surface of the sample material calculated to be the average of
3.2.3 For Residual Strain Measurement: δ =the
εrcorrection
three or more measurements, each measurement taken from a
relative residual strain correction term
different 2-D data trace
ε =the residual strain
r
R =the calibrated peak-to-valley roughness of a flat and
tave
A =the amplitude of the cosine function used to model the
F
leveled surface of the sample material calculated to be the
first abbreviated data trace
average of three or more measurements, each measurement
A =the amplitude of the cosine function used to model the
S
taken from a different 2-D data trace
second abbreviated data trace
U =the expanded uncertainty of a residual strain measure-
εr
L =thecalibratedlengthofthefixed-fixedbeamifthereare
ment
no applied axial-compressive forces
u =the combined standard uncertainty of a residual strain
cεr
L =the total calibrated length of the curved fixed-fixed
c
measurement
beam (as modeled with two cosine functions) with v1 and
end
u =the component in the combined standard uncertainty
cert
v2 as the calibrated v values of the endpoints
end
calculation for residual strain that is due to the uncertainty of
L =the calibrated length of the cosine function modeling
cF
the value of the physical step height standard used for
the first curve with v1 and i as the calibrated v values of the
end
calibration
endpoints
u =the component in the combined standard uncer-
correction
L =the calibrated length of the cosine function modeling
cS
tainty calculation for residual strain that is due to the uncer-
the second curve with i and v2 as the calibrated v values of
end
tainty of the correction term
the endpoints
u =the component in the combined standard uncertainty
drift
L =the calibrated effective length of the fixed-fixed beam
e'
calculation for residual strain that is due to the amount of drift
calculated as a straight-line measurement between v and v
eF eS
during the data session
n1 =indicative of the data point uncertainty associated with
t
u =the component in the combined standard uncertainty
L
the chosen value for x1 , with the subscript “t” referring to
uppert
calculation for residual strain that is due to the measurement
the data trace. If it is easy to identify one point that accurately
uncertainty of L
locates the upper corner of Edge 1, the maximum uncertainty
u =thecomponentinthecombinedstandarduncertainty
associated with the identification of this point is n1x cal ,
linear
t res x
calculation for residual strain that is due to the deviation from
where n1=1.
t
linearity of the data scan
n2 =indicative of the data point uncertainty associated with
t
the chosen value for x2 , with the subscript “t” referring to u =the component in the combined standard uncertainty
uppert noise
calculation f
...
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation: E2245 − 11 (Reapproved 2018)
Standard Test Method for
Residual Strain Measurements of Thin, Reflecting Films
Using an Optical Interferometer
This standard is issued under the fixed designation E2245; 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 Atomic Force Microscope at Subnanometer Displacement
Levels Using Si(111) Monatomic Steps (Withdrawn
1.1 This test method covers a procedure for measuring the
2015)
compressive residual strain in thin films. It applies only to
2.2 SEMI Standard:
films, such as found in microelectromechanical systems
MS2 Test Method for Step Height Measurements of Thin
(MEMS) materials, which can be imaged using an optical
Films
interferometer, also called an interferometric microscope. Mea-
surements from fixed-fixed beams that are touching the under-
3. Terminology
lying layer are not accepted.
3.1 Definitions:
1.2 This test method uses a non-contact optical interfero-
3.1.1 The following terms can be found in Terminology
metric microscope with the capability of obtaining topographi-
E2444.
cal 3-D data sets. It is performed in the laboratory.
3.1.2 2-D data trace, n—a two-dimensional group of points
1.3 This standard does not purport to address all of the
that is extracted from a topographical 3-D data set and that is
safety concerns, if any, associated with its use. It is the
parallel to the xz- or yz-plane of the interferometric micro-
responsibility of the user of this standard to establish appro-
scope.
priate safety, health, and environmental practices and deter-
3.1.3 3-D data set, n—a three-dimensional group of points
mine the applicability of regulatory limitations prior to use.
with a topographical z-value for each (x, y) pixel location
1.4 This international standard was developed in accor-
within the interferometric microscope’s field of view.
dance with internationally recognized principles on standard-
ization established in the Decision on Principles for the
3.1.4 anchor, n—in a surface-micromachining process, the
Development of International Standards, Guides and Recom-
portion of the test structure where a structural layer is inten-
mendations issued by the World Trade Organization Technical
tionally attached to its underlying layer.
Barriers to Trade (TBT) Committee.
3.1.5 anchor lip, n—in a surface-micromachining process,
the freestanding extension of the structural layer of interest
2. Referenced Documents
around the edges of the anchor to its underlying layer.
2.1 ASTM Standards:
3.1.5.1 Discussion—In some processes, the width of the
E2244 Test Method for In-Plane Length Measurements of
anchor lip may be zero.
Thin, Reflecting Films Using an Optical Interferometer
3.1.6 bulk micromachining, adj—a MEMS fabrication pro-
E2246 Test Method for Strain Gradient Measurements of
cess where the substrate is removed at specified locations.
Thin, Reflecting Films Using an Optical Interferometer
3.1.7 cantilever, n—a test structure that consists of a free-
E2444 Terminology Relating to Measurements Taken on
standing beam that is fixed at one end.
Thin, Reflecting Films
E2530 Practice for Calibrating the Z-Magnification of an 3.1.8 fixed-fixed beam, n—a test structure that consists of a
freestanding beam that is fixed at both ends.
3.1.9 in-plane length (or deflection) measurement, n—the
This test method is under the jurisdiction of ASTM Committee E08 on Fatigue
experimental determination of the straight-line distance be-
and Fracture and is the direct responsibility of Subcommittee E08.05 on Cyclic
tween two transitional edges in a MEMS device.
Deformation and Fatigue Crack Formation.
Current edition approved May 1, 2018. Published May 2018. Originally
ɛ1
approved in 2002. Last previous edition approved in 2011 as E2245 – 11 . DOI:
10.1520/E2245–11R18
2 3
For referenced ASTM standards, visit the ASTM website, www.astm.org, or The last approved version of this historical standard is referenced on
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM www.astm.org.
Standards volume information, refer to the standard’s Document Summary page on For referenced Semiconductor Equipment and Materials International (SEMI)
the ASTM website. standards, visit the SEMI website, www.semi.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2245 − 11 (2018)
3.1.9.1 Discussion—This length (or deflection) measure- 3.2 Symbols:
ment is made parallel to the underlying layer (or the xy-plane
3.2.1 For Calibration: σ = the maximum of two uncali-
6same
of the interferometric microscope). brated values (σ and σ ) where σ is the standard
same1 same2 same1
deviation of the six step height measurements taken on the
3.1.10 interferometer, n—a non-contact optical instrument
physical step height standard at the same location before the
used to obtain topographical 3-D data sets.
data session and σ is the standard deviation of the six
3.1.10.1 Discussion—The height of the sample is measured same2
measurements taken at this same location after the data session
along the z-axis of the interferometer. The x-axis is typically
σ = the certified one sigma uncertainty of the physical
aligned parallel or perpendicular to the transitional edges to be cert
step height standard used for calibration
measured.
σ = the standard deviation of the noise measurement,
noise
3.1.11 MEMS, adj—microelectromechanical systems.
calculated to be one-sixth the value of R minus R
tave ave
3.1.12 microelectromechanical systems, adj—in general,
σ = the standard deviation of the surface roughness
Rave
this term is used to describe micron-scale structures, sensors,
measurement, calculated to be one-sixth the value of R
ave
actuators, and technologies used for their manufacture (such as,
σ = the standard deviation in a ruler measurement in the
xcal
silicon process technologies), or combinations thereof.
interferometric microscope’s x-direction for the given combi-
3.1.13 residual strain, n—in a MEMS process, the amount
nation of lenses
of deformation (or displacement) per unit length constrained
σ = the standard deviation in a ruler measurement in the
ycal
within the structural layer of interest after fabrication yet
interferometric microscope’s y-direction for the given combi-
before the constraint of the sacrificial layer (or substrate) is
nation of lenses
removed (in whole or in part).
cal = the x-calibration factor of the interferometric micro-
x
scope for the given combination of lenses
3.1.14 sacrificial layer, n—a single thickness of material
cal = the y-calibration factor of the interferometric micro-
that is intentionally deposited (or added) then removed (in
y
scope for the given combination of lenses
whole or in part) during the micromachining process, to allow
cal = the z-calibration factor of the interferometric micro-
freestanding microstructures.
z
scope for the given combination of lenses
3.1.15 stiction, n—adhesion between the portion of a struc-
cert = the certified (that is, calibrated) value of the physical
tural layer that is intended to be freestanding and its underlying
step height standard
layer.
ruler = the interferometric microscope’s maximum field of
x
3.1.16 (residual) strain gradient, n—a through-thickness
view in the x-direction for the given combination of lenses as
variation (of the residual strain) in the structural layer of
measured with a 10-µm grid (or finer grid) ruler
interest before it is released.
ruler = the interferometric microscope’s maximum field of
y
3.1.16.1 Discussion—If the variation through the thickness
view in the y-direction for the given combination of lenses as
in the structural layer is assumed to be linear, it is calculated to
measured with a 10-µm grid (or finer grid) ruler
be the positive difference in the residual strain between the top
scope = the interferometric microscope’s maximum field of
x
and bottom of a cantilever divided by its thickness. Directional
view in the x-direction for the given combination of lenses
information is assigned to the value of “s.”
scope = the interferometric microscope’s maximum field of
y
3.1.17 structural layer, n—a single thickness of material
view in the y-direction for the given combination of lenses
present in the final MEMS device.
x = the calibrated resolution of the interferometric micro-
res
3.1.18 substrate, n—the thick, starting material (often single
scope in the x-direction
crystal silicon or glass) in a fabrication process that can be used
z¯ = the uncalibrated average of the six calibration
6same
to build MEMS devices.
measurements from which σ is found
6same
3.1.19 support region, n—in a bulk-micromachining z = the uncalibrated positive difference between the av-
drift
erage of the six calibration measurements taken before the data
process, the area that marks the end of the suspended structure.
session (at the same location on the physical step height
3.1.20 surface micromachining, adj—a MEMS fabrication
standard used for calibration) and the average of the six
process where micron-scale components are formed on a
calibration measurements taken after the data session (at this
substrate by the deposition (or addition) and removal (in whole
same location)
or in part) of structural and sacrificial layers.
z = over the instrument’s total scan range, the maximum
lin
3.1.21 test structure, n—a component (such as, a fixed-fixed
relative deviation from linearity, as quoted by the instrument
beam or cantilever) that is used to extract information (such as,
manufacturer (typically less than 3 %)
the residual strain or the strain gradient of a layer) about a
z = the calibrated resolution of the interferometric micro-
res
fabrication process.
scope in the z-direction
3.1.22 transitional edge, n—the side of a MEMS structure
z¯ = the average of the calibration measurements taken
ave
that is characterized by a distinctive out-of-plane vertical
along the physical step height standard before and after the data
displacement as seen in an interferometric 2-D data trace.
session
3.1.23 underlying layer, n—the single thickness of material 3.2.2 For In-plane Length Measurement: α = the misalign-
directly beneath the material of interest. ment angle
3.1.23.1 Discussion—This layer could be the substrate. L = the in-plane length measurement of the fixed-fixed beam
E2245 − 11 (2018)
L = the in-plane length correction term for the given type ε = in determining the combined standard uncertainty
offset r-low
of in-plane length measurement taken on similar structures value for the residual strain measurement, the lowest value for
when using similar calculations and for the given combination ε given the specified variations
r
of lenses for a given interferometric microscope
σ = the in-plane length repeatability standard de-
Lrepeat(samp)'
v1 = one endpoint of the in-plane length measurement
viation (for the given combination of lenses for the given
end
v2 = another endpoint of the in-plane length measurement interferometric microscope) as obtained from test structures
end
fabricated in a process similar to that used to fabricate the
x1 = the calibrated x-value that most appropriately lo-
uppert
sample and when the transitional edges face each other
cates the upper corner associated with Edge 1 in Trace t
σ = the relative residual strain repeatability stan-
x2 = the calibrated x-value that most appropriately lo-
repeat(samp)
uppert
dard deviation as obtained from fixed-fixed beams fabricated in
cates the upper corner associated with Edge 2 in Trace t
a process similar to that used to fabricate the sample
y = the calibrated y-value associated with Trace a'
a'
R = the calibrated surface roughness of a flat and leveled
y = the calibrated y-value associated with Trace e' ave
e'
surface of the sample material calculated to be the average of
3.2.3 For Residual Strain Measurement: δ = the
εrcorrection
three or more measurements, each measurement taken from a
relative residual strain correction term
different 2-D data trace
ε = the residual strain
r
R = the calibrated peak-to-valley roughness of a flat and
tave
A = the amplitude of the cosine function used to model the
F
leveled surface of the sample material calculated to be the
first abbreviated data trace
average of three or more measurements, each measurement
A = the amplitude of the cosine function used to model the
S
taken from a different 2-D data trace
second abbreviated data trace
U = the expanded uncertainty of a residual strain measure-
εr
L = the calibrated length of the fixed-fixed beam if there are
ment
no applied axial-compressive forces
u = the combined standard uncertainty of a residual strain
cεr
L = the total calibrated length of the curved fixed-fixed
c
measurement
beam (as modeled with two cosine functions) with v1 and
end
u = the component in the combined standard uncertainty
cert
v2 as the calibrated v values of the endpoints
end
calculation for residual strain that is due to the uncertainty of
L = the calibrated length of the cosine function modeling
cF
the value of the physical step height standard used for
the first curve with v1 and i as the calibrated v values of the
end
calibration
endpoints
u = the component in the combined standard uncer-
correction
L = the calibrated length of the cosine function modeling
cS
tainty calculation for residual strain that is due to the uncer-
the second curve with i and v2 as the calibrated v values of
end
tainty of the correction term
the endpoints
u = the component in the combined standard uncertainty
drift
L = the calibrated effective length of the fixed-fixed beam
e'
calculation for residual strain that is due to the amount of drift
calculated as a straight-line measurement between v and v
eF eS
during the data session
n1 = indicative of the data point uncertainty associated with
t
u = the component in the combined standard uncertainty
L
the chosen value for x1 , with the subscript “t” referring to
uppert
calculation for residual strain that is due to the measurement
the data trace. If it is easy to identify one point that accurately
uncertainty of L
locates the upper corner of Edge 1, the maximum uncertainty
u = the component in the combined standard uncertainty
associated with the identification of this point is n1 x cal ,
linear
t res x
calculation for residual strain that is due to the deviation from
where n1 =1.
t
linearity of the data scan
n2 = indicative of the data point uncertainty associated
...
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