ASTM E111-04(2010)
(Test Method)Standard Test Method for Young's Modulus, Tangent Modulus, and Chord Modulus
Standard Test Method for Young's Modulus, Tangent Modulus, and Chord Modulus
SIGNIFICANCE AND USE
The value of Young's modulus is a material property useful in design for calculating compliance of structural materials that follow Hooke's law when subjected to uniaxial loading (that is, the strain is proportional to the applied force).
For materials that follow nonlinear elastic stress-strain behavior, the value of tangent or chord modulus is useful in estimating the change in strain for a specified range in stress.
Since for many materials, Young's modulus in tension is different from Young's modulus in compression, it shall be derived from test data obtained in the stress mode of interest.
The accuracy and precision of apparatus, test specimens, and procedural steps should be such as to conform to the material being tested and to a reference standard, if available.
Precise determination of Young's modulus requires due regard for the numerous variables that may affect such determinations. These include (1) characteristics of the specimen such as orientation of grains relative to the direction of the stress, grain size, residual stress, previous strain history, dimensions, and eccentricity; (2) testing conditions, such as alignment of the specimen, speed of testing, temperature, temperature variations, condition of test equipment, ratio of error in applied force to the range in force values, and ratio of error in extension measurement to the range in extension values used in the determination; and (3) interpretation of data (see Section 9).
When the modulus determination is made at strains in excess of 0.25 %, correction should be made for changes in cross-sectional area and gage length, by substituting the instantaneous cross section and instantaneous gage length for the original values.
Compression results may be affected by barreling (see Test Methods E9). Strain measurements should therefore be made in the specimen region where such effects are minimal.
SCOPE
1.1 This test method covers the determination of Young's modulus, tangent modulus, and chord modulus of structural materials. This test method is limited to materials in which and to temperatures and stresses at which creep is negligible compared to the strain produced immediately upon loading and to elastic behavior.
1.2 Because of experimental problems associated with the establishment of the origin of the stress-strain curve described in 8.1, the determination of the initial tangent modulus (that is, the slope of the stress-strain curve at the origin) and the secant modulus are outside the scope of this test method.
1.3 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.
1.4 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 and health practices and determine the applicability of regulatory requirements prior to use.
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Designation: E111 − 04 (Reapproved 2010)
Standard Test Method for
Young’s Modulus, Tangent Modulus, and Chord Modulus
ThisstandardisissuedunderthefixeddesignationE111;thenumberimmediatelyfollowingthedesignationindicatestheyearoforiginal
adoptionor,inthecaseofrevision,theyearoflastrevision.Anumberinparenthesesindicatestheyearoflastreapproval.Asuperscript
epsilon (´) indicates an editorial change since the last revision or reapproval.
This standard has been approved for use by agencies of the U.S. Department of Defense.
1. Scope rials at Room Temperature
2 E21TestMethodsforElevatedTemperatureTensionTestsof
1.1 This test method covers the determination of Young’s
Metallic Materials
modulus, tangent modulus, and chord modulus of structural
E83Practice for Verification and Classification of Exten-
materials.Thistestmethodislimitedtomaterialsinwhichand
someter Systems
to temperatures and stresses at which creep is negligible
E231Method for Static Determination of Young’s Modulus
comparedtothestrainproducedimmediatelyuponloadingand
of Metals at Low and Elevated Temperatures (Withdrawn
to elastic behavior.
1985)
1.2 Because of experimental problems associated with the
E1012Practice for Verification of Testing Frame and Speci-
establishment of the origin of the stress-strain curve described
men Alignment Under Tensile and Compressive Axial
in 8.1, the determination of the initial tangent modulus (that is,
Force Application
the slope of the stress-strain curve at the origin) and the secant
2.2 General Considerations—While certain portions of the
modulus are outside the scope of this test method.
standards and practices listed are applicable and should be
1.3 The values stated in SI units are to be regarded as
referred to, the precision required in this test method is higher
standard. No other units of measurement are included in this
than that required in general testing.
standard.
1.4 This standard does not purport to address all of the
safety concerns, if any, associated with its use. It is the
3. Terminology
responsibility of the user of this standard to establish appro-
3.1 Definitions:
priate safety and health practices and determine the applica-
3.1.1 accuracy—the degree of agreement between an ac-
bility of regulatory requirements prior to use.
cepted standard value of Young’s modulus (the average of
many observations made according to this method, preferably
2. Referenced Documents
by many observers) and the value determined.
2.1 ASTM Standards:
3.1.1.1 Increasedaccuracyisassociatedwithdecreasedbias
E4Practices for Force Verification of Testing Machines
relativetotheacceptedstandardvalue;twomethodswithequal
E6Terminology Relating to Methods of MechanicalTesting
bias relative to the accepted standard value have equal accu-
E8Test Methods for Tension Testing of Metallic Materials
racyevenifonemethodismoreprecisethantheother.Seealso
E9Test Methods of Compression Testing of Metallic Mate-
bias and precision.
3.1.1.2 The accepted standard value is the value ofYoung’s
modulus for the statistical universe being sampled using this
This test method is under the jurisdiction of ASTM Committee E28 on
method. When an accepted standard value is not available,
Mechanical Testing and is the direct responsibility of Subcommittee E28.04 on
Uniaxial Testing. accuracy cannot be established.
Current edition approved Sept. 15, 2010. Published January 2011. Originally
3.1.2 bias, statistical—a constant or systematic error in test
approved in 1955. Last previous edition approved in 2004 as E111–04. DOI:
results.
10.1520/E0111-04R10
3.1.2.1 Bias can exist between the accepted standard value
ThistestmethodisarevisionofE111–61(1978),“Young’sModulusatRoom
Temperature” and includes appropriate requirements of E231–69(1975), “Static
andatestresultobtainedfromthistestmethod,orbetweentwo
Determination of Young’s Modulus of Metals at Low and Elevated Temperatures”
test results obtained from this test method, for example,
to permit the eventual withdrawal of the latter method. Method E231 is under the
between operators or between laboratories.
jurisdiction of ASTM-ASME Joint Committee on Effect of Temperature on the
Property of Metals.
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 last approved version of this historical standard is referenced on
the ASTM website. www.astm.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E111 − 04 (2010)
3.1.3 precision—the degree of mutual agreement among 5. Significance and Use
individual measurements made under prescribed like condi-
5.1 The value of Young’s modulus is a material property
tions.
useful in design for calculating compliance of structural
3.1.4 Young’s modulus—the ratio of tensile or compressive
materials that follow Hooke’s law when subjected to uniaxial
stress to corresponding strain below the proportional limit (see
loading (that is, the strain is proportional to the applied force).
Fig. 1a).
5.2 For materials that follow nonlinear elastic stress-strain
3.1.4.1 tangent modulus—theslopeofthestress-straincurve
behavior, the value of tangent or chord modulus is useful in
at any specified stress or strain (see Fig. 1b).
estimating the change in strain for a specified range in stress.
3.1.4.2 chord modulus—the slope of the chord drawn be-
tween any two specified points on the stress-strain curve (see 5.3 Sinceformanymaterials,Young’smodulusintensionis
Fig. 1c). different from Young’s modulus in compression, it shall be
3.2 For definitions of other terms used in this test method, derived from test data obtained in the stress mode of interest.
refer to Terminology E6.
5.4 Theaccuracyandprecisionofapparatus,testspecimens,
and procedural steps should be such as to conform to the
4. Summary of Test Method
material being tested and to a reference standard, if available.
4.1 Auniaxial force is applied to the test specimen and the
force and strain are measured, either incrementally or continu- 5.5 Precise determination of Young’s modulus requires due
ously. The axial stress is determined by dividing the indicated regard for the numerous variables that may affect such deter-
force by the specimen’s original cross-sectional area. The minations. These include (1) characteristics of the specimen
appropriate slope is then calculated from the stress-strain such as orientation of grains relative to the direction of the
curve, which may be derived under conditions of either stress, grain size, residual stress, previous strain history,
increasing or decreasing forces (increasing from preload to dimensions, and eccentricity; (2) testing conditions, such as
maximum applied force or decreasing from maximum applied alignment of the specimen, speed of testing, temperature,
force to preload). temperature variations, condition of test equipment, ratio of
FIG. 1 Stress-Strain Diagrams Showing Straight Lines Corresponding to (a) Young’s Modulus, (b) Tangent Modulus, and (c) Chord
Modulus
E111 − 04 (2010)
6.3 Loading Fixtures—Gripsandotherdevicesforobtaining
andmaintainingaxialalignmentareshowninTestMethodsE8
and E9. It is essential that the loading fixtures be properly
designed and maintained. Procedures for verifying the align-
ment are described in detail in Practice E1012. The allowable
bending as defined in Practice E1012 shall not exceed 5%.
6.4 Extensometers—Class B-1 or better extensometers as
described in Practice E83 shall be used. Corrections may be
applied for proven systematic errors in strain and are not
considered as a change in class of the extensometer. Either an
averaging extensometer or the average of the strain measured
by at least two extensometers arranged at equal intervals
aroundthecrosssectionbeused.Iftwoextensometersareused
on other than round sections, they shall be mounted at ends of
an axis of symmetry of the section. If a force-strain recorder,
strain-transfer device, or strain follower is used with the
extensometer, they shall be calibrated as a unit in the same
manner in which they are used for determination of Young’s
modulus. The gage length shall be determined with an accu-
racy consistent with the precision expected from the modulus
determination and from the extensometer.
NOTE 1—The accuracy of the modulus determination depends on the
precision of the strain measurement. The latter can be improved by
increasing the gage length. This may, however, present problems in
maintaining specimen tolerances and temperature uniformity.
6.5 Furnaces or Heating Devices—When determining
Young’s modulus at elevated temperature, the furnace or
heating device used shall be capable of maintaining a uniform
temperature in the reduced section of the test specimen so that
avariationofnotmorethan 61.5°Cfortemperaturesuptoand
including 900°C, and not more than 63.0°C for temperatures
above 900°C, occurs. (Heating by self-resistance is not ac-
cepted.) Minimize temperature variations and control changes
FIG. 2 Load-Deviation Graph withintheallowablelimits,sincedifferencesinthermalexpan-
sion between specimen and extensometer parts may cause
significant errors in apparent strain. An instrumented sample
error in applied force to the range in force values, and ratio of
representative of the real test will demonstrate that the setup
error in extension measurement to the range in extension
meets the above capabilities.
values used in the determination; and (3) interpretation of data
(see Section 9).
6.6 Low-Temperature Baths and Refrigeration Equipment—
When determining Young’s modulus at low temperatures, an
5.6 When the modulus determination is made at strains in
appropriate low-temperature bath or refrigeration system is
excess of 0.25%, correction should be made for changes in
required to maintain the specimen at the specified temperature
cross-sectional area and gage length, by substituting the
during testing. For a low-temperature bath, the lower tension
instantaneous cross section and instantaneous gage length for
rod or adapter may pass through the bottom of an insulated
the original values.
container and be welded or fastened to it to prevent leakage.
5.7 Compression results may be affected by barreling (see
Fortemperaturestoabout−80°C,chippeddryicemaybeused
Test Methods E9). Strain measurements should therefore be
to cool an organic solvent such as ethyl alcohol in the
made in the specimen region where such effects are minimal.
low-temperature bath. Other organic solvents having lower
solidification temperatures, such as methylcyclohexane or
6. Apparatus
isopentane,maybecooledwithliquidnitrogentotemperatures
6.1 Dead Weights—Calibrated dead weights may be used.
lower than−80°C. Liquid nitrogen may be used to achieve a
Any cumulative errors in the dead weights or the dead weight
testing temperature of−196°C. Lower testing temperatures
loading system shall not exceed 0.1%.
may be achieved with liquid hydrogen and liquid helium, but
6.2 Testing Machines—In determining the suitability of a special containers or cryostats are required to provide for
testing machine, the machine shall be calibrated under condi- minimumheatleakagetopermitefficientuseofthesecoolants.
tions approximating those under which the determination is When liquid hydrogen is used, special precautions must be
made. Corrections may be applied to correct for proven taken to avoid explosions of hydrogen gas and air mixtures. If
systematic errors. refrigeration equipment is used to cool the specimens with air
E111 − 04 (2010)
as the cooling medium, it is desirable to have forced air mustbementionedinthereportsection.Iftheintentofthetest
circulation to provide uniform cooling. is to verify the performance of a product, the heat treatment
proceduremaybeomitted.Recordtheconditionofthematerial
NOTE 2—At low temperatures, when using a coolant bath, immersion-
tested, including any heat treatment, in the test report.
type extensometers are recommended.
6.7 Temperature measuring, controlling, and recording in-
8. Procedure
struments shall be calibrated periodically against a secondary
8.1 For most loading systems and test specimens, effects of
standard, such as a precision potentiometer. Lead-wire error
backlash, specimen curvature, initial grip alignment, etc.,
shouldbecheckedwiththeleadwiresinplaceastheynormally
introduce significant errors in the extensometer output when
are used.
applying a small force to the test specimen. Measurements
7. Test Specimens
shall therefore be made from a small force or preload, known
to be high enough to minimize these effects, to some higher
7.1 Selection and Preparation of Specimens—Special care
appliedforce,stillwithineithertheproportionallimitorelastic
shall be taken to obtain representative specimens which are
limit of the material. For linearly elastic materials, the slope of
straight and uniform in cross section. If straightening of the
the straight-line portion of the stress-strain curve shall be
material for the specimen is required, the resultant residual
established between the preload and the proportional limit to
stresses shall be removed by a subsequent stress relief heat
define Young’s modulus. If the actual stress-strain curve is
treatment which shall be reported with the test results.
desired, this line can appropriately be shifted along the strain
7.2 Dimensions—The recommended specimen length (and
axis to coincide with the origin. For nonlinearly elastic
filletradiusinthecaseoftensionspecimens)isgreaterthanthe
materials the tangent or chord modulus may be established
minimum requirements for general-purpose specimens. In
betweentheappropriatestressvaluesonthestressstraincurve.
addition, the ratio of length to cross section of compression
8.2 Measurement of Specimens—Make the measurements
specimens should be such as to avoid buckling (see Test
for the determination of average cross-sectional area at the
Methods E9).
ends of the gage length and at least at one intermediate
NOTE3—Forexamplesoftensionandcompressionspecimens,seeTest
location. Use any means of measuring that is capable of
Methods E8 and E9.
producing area calculations within 1% accuracy.
7.3 For tension specimens, the center lines of the grip
8.3 Alignment—Take special care to ensure as nearly axial
sections and of the threads of threaded-end specimens shall be
loading as possible. The strain increments between the initial-
concentric with the center line of the gage section within close
load and the final-load measurement on opposite sides of the
tolerance
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