ASTM E111-97
(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
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 use of either initial tangent modulus (that is, the slope of the stress-strain curve at the origin) or secant modulus is not recommended and their determination is outside the scope of this test method.
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 and health practices and determine the applicability of regulatory limitations prior to use.
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Designation: E 111 – 97
Standard Test Method for
Young’s Modulus, Tangent Modulus, and Chord Modulus
This standard is issued under the fixed designation E 111; 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 (e) indicates an editorial change since the last revision or reapproval.
This specification has been approved for use by agencies of the Department of Defense.
1. Scope E 83 Practice for Verification and Classification of Exten-
2 someters
1.1 This test method covers the determination of Young’s
E 231 Method for Static Determination of Young’s Modu-
modulus, tangent modulus, and chord modulus of structural
lus of Metals at Low and Elevated Temperatures
materials. This test method is limited to materials in which and
to temperatures and stresses at which creep is negligible
NOTE 1—General Considerations:While certain portions of the stan-
compared to the strain produced immediately upon loading and
dards and practices listed are applicable and should be referred to, the
precision required in this test method is usually higher than that required
to elastic behavior.
in general testing.
1.2 Because of experimental problems associated with the
establishment of the origin of the stress-strain curve described
3. Terminology
in 8.1, the use of either initial tangent modulus (that is, the
3.1 Definitions:
slope of the stress-strain curve at the origin) or secant modulus
3.1.1 accuracy—the degree of agreement between an ac-
is not recommended and their determination is outside the
cepted standard value of Young’s modulus (the average of
scope of this test method.
many observations made according to this method, preferably
1.3 This standard does not purport to address all of the
by many observers) and the value determined.
safety concerns, if any, associated with its use. It is the
3.1.1.1 Increased accuracy is associated with decreased bias
responsibility of the user of this standard to establish appro-
relative to the accepted standard value; two methods with equal
priate safety and health practices and determine the applica-
bias relative to the accepted standard value have equal accu-
bility of regulatory limitations prior to use.
racy even if one method is more precise than the other. See also
2. Referenced Documents bias and precision.
3.1.1.2 The accepted standard value is the value of Young’s
2.1 ASTM Standards:
3 modulus for the statistical universe being sampled using this
E 4 Practices for Force Verification of Testing Machines
method. When an accepted standard value is not available,
E 6 Terminology Relating to Methods of Mechanical Test-
3 accuracy cannot be established.
ing
3.1.2 bias, statistical—a constant or systematic error in test
E 8 Test Methods for Tension Testing of Metallic Materials
results.
E 9 Test Methods of Compression Testing of Metallic Ma-
3.1.2.1 Bias can exist between the accepted standard value
terials at Room Temperature
and a test result obtained from this test method, or between two
E 21 Test Methods for Elevated Temperature Tension Tests
test results obtained from this test method, for example,
of Metallic Materials
between operators or between laboratories.
3.1.3 precision—the degree of mutual agreement among
This test method is under the jurisdiction of ASTM Committee E-28 on individual measurements made under prescribed like condi-
Mechanical Testing and is the direct responsibility of Subcommittee E 28.03 on
tions.
Elastic Properties and Definitions on Mechanical Testing.
3.1.4 Young’s modulus—the ratio of tensile or compressive
Current edition approved Apr. 10, 1997. Published November 1997. Originally
stress to corresponding strain below the proportional limit of
published as E 111 – 55 T. Last previous edition E 111 – 82 (1996)e .
This test method is a revision of E111 – 61(1978),“ Young’s Modulus at Room
the material (see Fig. 1a).
Temperature” and includes appropriate requirements of E231 – 69(1975), “Static
3.1.4.1 tangent modulus—the slope of the stress-strain
Determination of Young’s Modulus of Metals at Low and Elevated Temperatures”
curve at a specified value of stress or strain (see Fig. 1b).
to permit the eventual withdrawal of the latter method. Method E 231 is under the
jurisdiction of ASTM-ASME Joint Committee on Effect of Temperature on the
Property of Metals.
3 4
Annual Book of ASTM Standards, Vol 03.01. Discontinued, see 1981 Annual Book of ASTM Standards, Vol 03.01.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
E111–97
FIG. 1 Stress-Strain Diagrams Showing Straight Lines Corresponding to (a) Young’s Modulus, (b) Tangent Modulus, and (c) Chord
Modulus
3.1.4.2 chord modulus—the slope of the chord drawn be- 5.3 Since for many materials, Young’s modulus in tension is
tween any two specified points on the stress-strain curve, different from Young’s modulus in compression, it should be
below the elastic limit of the material (see Fig. 1c). derived from test data obtained in the mode of stressing of
3.2 For definitions of other terms used in this test method,
interest.
refer to Terminology E 6.
5.4 The accuracy and precision of apparatus, test specimens,
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 The test specimen is loaded uniaxially and load and
5.5 Precise determination of Young’s modulus requires due
strain are measured, either incrementally or continuously. The
regard for the numerous variables that may affect such deter-
axial stress is determined, either incrementally or continuously,
minations. These include (1) characteristics of the specimen
by dividing the load value by the specimen’s original cross-
such as orientation of grains relative to the direction of the
sectional area. The appropriate slope is then calculated from
stress, grain size, residual stress, previous strain history,
the stress-strain curve, which may be derived under conditions
dimensions, and eccentricity; (2) testing conditions, such as
of either increasing or decreasing load (increasing from preload
alignment of the specimen, speed of testing, temperature,
to maximum load or decreasing from maximum load to
temperature variations, condition of test equipment, ratio of
preload).
error in load to the range in load values, and ratio of error in
5. Significance and Use extension measurement to the range in extension values used in
the determination; and (3) interpretation of data (see Section
5.1 The value of Young’s modulus is a material property
9).
useful in design for calculating compliance of structural
materials that follow Hooke’s law when subjected to uniaxial 5.6 When the modulus determination is made at strains in
excess of 0.25 %, correction should be made for changes in
loading (that is, the strain is proportional to the applied force).
5.2 For materials that follow nonlinear elastic stress-strain cross-sectional area and gage length, by substituting the
instantaneous cross section and instantaneous gage length for
behavior, the value of tangent or chord modulus is useful in
estimating the change in strain for a specified range in stress. the original values.
E111–97
5.7 Compression results may be affected by barreling (see considered as changing the class of the extensometer. It is
Test Methods E 9). Strain measurements should therefore be recommended that an averaging extensometer or the average of
made in the specimen region where such effects are minimal. the strain measured by at least two extensometers arranged at
equal intervals around the cross section be used. If two
extensometers are used on other than round sections, they
should be mounted at ends of an axis of symmetry of the
section. If a load-strain recorder, strain-transfer device, or
strain follower is used with the extensometer, they should 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 accuracy consistent with the precision
expected from the modulus determination and from the exten-
someter used.
NOTE 2—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
a variation of not more than 61.5°C for temperatures up to and
including 900°C, and not more than 63.0°C for temperatures
above 900°C, occurs. (Heating by self-resistance is not recom-
mended.) Temperature changes within the allowable limits
should be minimized, since differences in thermal expansion
between specimen and extensometer parts may cause signifi-
cant errors in apparent strain.
6.6 Low-Temperature Baths and Refrigeration Equipment—
When determining Young’s modulus at low temperatures, an
appropriate low-temperature bath or refrigeration system is
required to maintain the specimen at the specified temperature
during testing. For a low-temperature bath, the lower tension
rod or adapter may pass through the bottom of an insulated
container and be welded or fastened to it to prevent leakage.
For temperatures to about − 80°C, chipped dry ice may be used
to cool an organic solvent such as ethyl alcohol in the
low-temperature bath. Other organic solvents having lower
FIG. 2 Load-Deviation Graph
solidification temperatures, such as methylcyclohexane or
isopentane, may be cooled with liquid nitrogen to temperatures
lower than − 80°C. Liquid nitrogen may be used to achieve a
6. Apparatus
testing temperature of − 196°C. Lower testing temperatures
6.1 Dead Weights—Calibrated dead weights may be used.
may be achieved with liquid hydrogen and liquid helium, but
Any cumulative errors in the dead weights or the dead weight
special containers or cryostats are required to provide for
loading system shall not exceed 0.1 %.
minimum heat leakage to permit efficient use of these coolants.
6.2 Testing Machines—In determining the suitability of a
When liquid hydrogen is used, special precautions must be
testing machine, it is advisable to calibrate the machine under
taken to avoid explosions of hydrogen gas and air mixtures. If
conditions approximating those under which the determination
refrigeration equipment is used to cool the specimens with air
is made. Corrections may be applied to correct for proven
as the cooling medium, it is desirable to have forced air
systematic errors in load.
circulation to provide uniform cooling.
6.3 Loading Fixtures—Grips and other devices for obtain-
ing and maintaining axial alignment are shown in Test Methods
NOTE 3—At low temperatures, when using a coolant bath, immersion-
E 8 and E 9. It is essential that the loading fixtures be properly type extensometers are recommended.
designed for use at the required temperature, and that they be
7. Test Specimens
properly maintained.
6.4 Extensometers—Class B-1 extensometers as described 7.1 Selection and Preparation of Specimens—Special care
in Practice E 83 shall be used depending on the degree of shall be taken to obtain representative specimens which are
precision required. Corrections may be applied for proven straight and uniform in cross section. If straightening of the
systematic errors in strain. Such corrections shall not be material for the specimen is required, the resultant residual
E111–97
stresses shall be removed by a subsequent stress relief heat line portion of the stress-strain curve should be established
treatment which shall be reported with the test results. between the preload and the proportional limit to define
7.2 Dimensions—In general, it is recommended that the Young’s modulus. If the actual stress-strain curve is desired,
length of specimens (and radius of fillets in the case of tension this line can appropriately be shifted along the strain axis to
specimens) be greater than the minimum requirements for
coincide with the origin. For nonlinearly elastic materials the
general-purpose specimens. In addition, the ratio of length to tangent or chord modulus may be established for stress values
cross section of compression specimens should be such as to
ranging from the appropriate preload to the elastic limit.
avoid buckling (see Test Methods E 9).
8.2 Measurement of Specimens—Make the measurements
for the determination of average cross-sectional area at the
NOTE 4—For examples of tension and compression specimens, see Test
ends of the gage length and at least at one intermediate
Methods E 8 and E 9.
location. Use any means of measuring that is capable of
7.3 For tension specimens, the center lines of the grip
producing area calculations within 1 % accuracy.
sections and of the threads of threaded-end specimens shall be
8.3 Alignment—Take special care to ensure as nearly axial
concentric with the center line of the gage section within close
loading as possible. The strain increments between the initial-
tolerances in order to obtain the degree of alignment required.
load and the final-load measurement on opposite sides of the
If pin-loaded sheet-type specimens are used, the centers of the
specimens should not differ from the average by more than
gripping holes shall be not more than 0.005 times the width of
3 %. For pin-loaded sheet-type tension specimens this degree
the gage section from its center line. For sheet-type specimens,
of alignment can be attained if the gripping holes are located
it may be necessary to provide small tabs or notches for
within the tolerance stated in 7.3.
attaching the extensometer.
8.4 Soaking Time of Specimens at Testing Temperature—
NOTE 5—The effect of eccentric loading may be illustrated by calcu-
After the specimen to be tested has reached the testing
lating the bending moment and stress thus added. For a standard 12.5-mm
temperature, it is necessary to maintain the specimen at the
diameter specimen, the stress increase is 1.5 % for each 0.025 mm of
eccentricity. This error increases to about 2.5 % per 0.025 mm for a 9-mm
testing temperature for a sufficient length of time to attain
diameter specimen and to about 3.2 % per 0.025 mm for a 6-mm diameter
equilibrium conditions in regard to the temperature of the
specimen.
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