ASM Metals HandBook Vol. 8 - Mechanical Testing and Evaluation
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copper bars under DRX conditions. A computational method based on both glide and DRX modeling was
developed. In this method, the texture is first determined by glide modeling until a critical strain is reached, at
which DRX sets in. It was shown that the principal effects of DRX on texture development and the resulting
free-end effect (shortening) can be predicted reasonably accurately.
The changes in length of a twisted bar during straining result in a continuous change in the specimen cross-
sectional area. Thus, if the true shear stress versus shear strain curve is required, then instantaneous values of
specimen dimensions must be used in computing shear stress and shear strain from the measured torque and
angle of twist. Since the shear stress is proportional to r
-3
and the shear strain is proportional to r/ℓ (Eq 12, 21),
the use of initial values of r and ℓ in calculating the shear stress and shear strain curve will generate an error of
15 and 6%, respectively, if a length change of 10% took place (Ref 25). On the other hand, if the length of the
specimen is held constant the developed axial stresses will range from 2 to 20% of the developed shear stress.
In this case, the ratio (
fx
/ ) of the effective stress in the fixed-end condition to that in the free-end condition is
in the range from 1.0 to 1.01 for face-centered cubic metals, and in the range from 1.0 to 1.08 for body-centered
cubic metals (Ref 19, 20).
References cited in this section
7. S. Kalpakjian, Mechanical Processing of Materials, D. Van Nostrand, 1967, p 31
9. J.A. Bailey, S.L. Haas, and M.K. Shah, Int. J. Mech. Sci., Vol 14, 1972, p 735
17. R.R. Hill, The Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1950, p 317
18. J.A. Bailey, S.L. Haas, and K.C. Naweb, J. Basic Eng. (Trans. ASME), March 1972, p 231
19. F. Montheillet, M. Cohen, and J.J. Jonas, Acta Metall., Vol 32, 1984, p 2077–2089
20. F. Montheillet, P. Gilormini, and J. J. Jonas, Acta Metall., Vol. 33, 1985, p 2126–2136
21. J.J. Jonas, Int. J. Mech. Sci., Vol 35, 1993, p 1065–1077
22. L.S. Toth and J.J. Jonas, Scr. Metall., Vol 27, 1992, p 359–363
23. L.S. Toth, J.J. Jonas, D. Daniel, and J.A. Bailey, Textures and Microstructures, Vol 19, 1992, p 245–
262
24. J.J. Jonas and L.S. Toth, Scr. Metall., Vol 27, 1992, p 1575–1580
25. S.L. Semiatin, G.D. Lahoti, and J.J. Jonas, Mechanical Testing, Vol 8, ASM Handbook, American
Society for Metals, 1985, p 154
Fundamental Aspects of Torsional Loading
John A. Bailey, North Carolina State University;Jamal Y. Sheikh-Ahmad, Wichita State University
Testing Equipment
A typical torsion testing machine consists of a drive system, a test section, torque and rotational displacement
transducers, and a rigid frame. A rigid frame that is capable of allowing accurate alignment of the various
torsion machine components is necessary for twisting the torsion specimen accurately around its axis with no
superimposed flexural loading. For this purpose, Culver (Ref 26) and Kobayashi (Ref 27), among others, used a
lathe bed for constructing their torsion testing machine because of its high rigidity and precision-machined
slides. In addition to the components mentioned above, a heating chamber with vacuum or inert gas
environment is required when tests are conducted at high temperatures. A variety of torsion testing machines
have been designed and built, and an excellent review of some of these machines is given in Ref 27 and 28.
More information on torsion testing is also provided in the article “Shear, Torsion, and Multiaxial Testing” in
this Volume.
Drive Systems. Most of the differences between existing torsion testing machines lie in the type of drive system
used. The drive system is required to provide sufficient power to twist the test specimen at a constant rotational
speed. Electric drive systems were used in most of the early torsion testing machines, such as the one shown in
Fig. 5 (Ref 9). The electric drive system consists of an electric motor, gearbox or hydraulic reducers, a
flywheel, and a clutch and brake system. The torsion machine shown here uses a 2.2 kW induction motor and a
drive train consisting of two planetary gear reducers and three gear pairs and is capable of providing 24
different rotational speeds in the range from 0.0115 to 1745 rpm. A flywheel is required in this system in order
to maintain approximately constant rotational speed at clutch engagement and during specimen twisting. A
pneumatic disk clutch that is activated by a three-way solenoid provides quick transmission of torque from the
flywheel to the test section. An inherent problem in this type of drive system is the lack of positive engagement
between the drive system and the test section, which causes loss of rotational speed because of slippage,
especially at high rates of twisting. This problem can be avoided to some extent by using a positive engagement
mechanical “dog or ramp” clutch (Ref 26, 27) or an electromagnetic clutch (Ref 29) that allows shear strain
rates of the order to 300 and 1000 s
-1
, respectively, to be achieved.
Fig. 5 Torsion testing machine. (a) Drive section. C, coupling; F, flywheel; M electric
motor; O, output shaft; P, pillow block; AG, gear pair; GP, interchangeable gear pair;
PR, planetary reducer; TB, timing belt drive. (b) Test section. H1, H2, specimen holders;
I, low inertia coupling; L, linear bearing: P, pillow block; S, specimen; S1, S2, shafts; T,
transducer; V, solenoid value; W, water jacket; CL, clutch; FN, furnace; IC, input shaft;
OC, output shaft; ST, surge tank
The use of a hydraulic drive system abolishes the need for a drive train and clutch and brake system, thus
eliminating the inherent problems associated with these components. A typical hydraulic drive system consists
of a hydraulic motor (Ref 30) or a hydraulic rotary actuator (Ref 31), a source of hydraulic power in the form of
pressurized oil, and servocontrollers for controlling the flow of oil by means of servovalves. The torque in the
hydraulic drive system is provided by the pressurized oil as it pushes against a set of rotary vanes. The
advantage of this system is that it can be accurately controlled in a closed-loop arrangement so that the
prescribed loading history can be obtained. Because angular displacement in a rotary actuator system is limited
to a fraction of a revolution, special torsion specimen designs with a short gage length and a large gage
diameter are required to achieve high values of shear strain and strain rate.
The test section in most torsion testing machines consists of a pair of grips for attaching both ends of the
specimen to the torsion machine and a furnace for heating the specimen. One end of the test specimen is
attached to the output side of the loading train. The other end of the specimen is rigidly mounted to the torque
and axial force transducers. Grips are usually designed in such a way as to eliminate the relative motion
between the test specimen and the testing machine. The grips shown in Fig. 6 utilize a chuck-type design with
three moving jaws. When the test is conducted at elevated temperatures, special care must be taken to reduce
the conduction of heat from the specimen to the drive train and the load cells. This is usually done by
constructing a cooling water jacket around the grips. In addition, the grip design shown in Fig. 6 uses a thick
ceramic disk as a thermal insulator.
Fig. 6 Detailed view of a torsion test specimen holder with a three-jaw chuck
It is known that torsional straining of metals induces axial stresses when the testing is conducted under fixed-
end conditions (i.e., specimen ends are constrained axially). Similarly, when testing is conducted under free-end
conditions (i.e., one end of the specimen is allowed to move axially), the specimen undergoes lengthening or
shortening depending on the workpiece material and testing temperature. These axial effects are inherent in
torsion testing because they are associated with texture development and evolution in the workpiece material, as
discussed in the previous section. Therefore, it is necessary when conducting a torsion test to monitor these
changes and account for their effects when determining the effective stress-strain relationships from
experimental data. Conducting a test under free-end conditions is more difficult than under fixed-end conditions
because of the difficulties in designing grips or fixtures that allow both a rigid reaction to the torsional load
applied and a true free movement axially. In the torsion machine discussed previously, a linear bearing
mechanism was used to ensure free-end movement of the specimen. This mechanism is shown in Fig. 7. It
consists of three case-hardened steel shafts press fitted 120° apart in the free end of the specimen holder. The
three shafts slide freely inside three pairs of linear bearings press fitted 120° apart in the fixed end of the holder.
A linear differential transformer is used to continuously monitor the relative motion between the free end and
fixed end of the linear bearing mechanism.
Fig. 7 Detailed view of a linear bearing mechanism used to ensure free-end movement of
the specimen in torsion testing
Load and Displacement Transducers. It was shown previously that knowledge of the applied torque and
rotational displacement are sufficient to calculate the state of shear stress and shear strain based on specimen
geometry. It is of interest, therefore, to measure both the applied torque and rotation continuously during the
torsion test. It is also necessary to monitor the axial force induced by fixed-end testing or the change in
specimen length induced in free-end testing. The measurement of axial and torsional loads are performed using
various types of reaction-load transducers that utilize foil strain gages as the load-sensing element. These load
cells are conveniently mounted at the fixed end of the torsion machine. Rotational displacement is measured
electrically using a variable resistor or a differential capacitor or optically using photoelectric devices in
combination with a perforated disk or an optically encoded shaft. The torsion machine uses a perforated disk
with 120 holes equally spaced around its circumference. A photo transistor detects the holes and sends an
electric pulse to the control panel for conditioning and amplification. The output signal is recorded
simultaneously with the torque signal generated by the load cell using a chart recorder or a storage oscilloscope.
Torsion Specimens. A wide range of specimen sizes and geometries have been used in the past, and a standard
size or geometry has not been agreed upon. A good survey of various specimen designs used in torsion testing
is given in Ref 27. A typical torsion specimen is composed of a uniform cylindrical gage section, two shoulders
for clamping into the machine grips, and two fillets to connect the gage section to the shoulders. Solid and
hollow gage sections have been used. Thin-walled specimens that have a hollow gage section and a wall
thickness that is a small fraction of the radius of the section offer the possibility of homogeneous stress and
strain states in the gage section. However, because of their tendency toward torsional buckling, solid specimens
are preferred over hollow specimens for large deformation studies. The problem of torsional buckling in thin-
walled specimens can be suppressed, to some extent, by shortening the gage length. The range of ratios of gage
length to gage radius (ℓ/r) for specimens reported in the literature varies from 0.67 to 8 (Ref 28). A small
length-to-radius ratio is preferred because it provides an increase in the maximum shear strain and shear strain
rate for a given rotational displacement and rotational speed, respectively.
In torsion specimen design, care must be taken to maintain uniform plastic deformation throughout the gage
section. This can be achieved by maintaining a truly uniform cross section, proper polishing of the outside
surface to eliminate stress raisers such as scratches, and by providing properly sized fillets. In addition, bulky
shoulders as compared with the gage section will help to constrain the plastic deformation in the gage section.
White (Ref 32) performed an analysis of the plastic deformation in a thin-walled specimen using the finite
element method. He reported that plastic deformation extends into the transition region between the gage
section and the grips. The fraction of the total torsional displacement that is experienced by the gage section
was determined. Knowledge of this factor allows for the correct evaluation of shear strain in the gage section.
Khoddam and coworkers (Ref 33) studied the effect of plastic deformation outside the gage section on the
analysis of shear stress and shear strain. They suggested that an effective gage length be used in the calculation
of stress and strain, instead of the actual length. For a power law constitutive equation, the effective length was
found to be a function of specimen geometry and the coefficients in the constitutive equation.
Another factor that may affect the uniformity of deformation during twisting is the temperature rise in the gage
section that is caused by plastic deformation. At high rates of deformation, and especially for materials with
low thermal conductivity, the temperature rise may lead to localized flow and shear banding (Ref 34, 35, and
36). The temperature distribution in torsion specimens can be determined numerically as described in previous
work. Once the temperature distribution in the gage section is known, the effect of temperature on flow stress
can be determined (Ref 36). Zhou and Clode (Ref 37) used the finite element method to study the effect of
specimen design on temperature rise in the gage section of an aluminum specimen. Their work showed the
effect of temperature rise can be minimized by proper specimen design, but it cannot be totally eliminated.
References cited in this section
9. J.A. Bailey, S.L. Haas, and M.K. Shah, Int. J. Mech. Sci., Vol 14, 1972, p 735
26. R.S. Culver, Exp. Mech., Vol 12, 1972, p 398–405
27. H. Kobayashi, “Shear Localization and Fracture in Torsion of Metals,” Ph.D. thesis, University of
Reading, Reading, U.K., 1987
28. M.J. Luton, Workability Testing Techniques, G.E. Dieter, Ed., American Society for Metals, 1984, p 95
29. T. Vinh, M. Afzali, and A. Roche, Third Int. Conf. Mechanical Behavior of Materials (ICM 3), Vol 2,
K.J. Miller and R.F. Smith, Ed., Pergamon Press, 1979, p 633–642
30. H. Weiss, D.H. Skinner, and J.R. Everett, J. Phys. E, Sci. Instr., Vol 6, 1973, p 709–714
31. U.S. Lindholm, A. Nagy, G.R. Johnson, and J.M. Hoegfeldt, J. Eng. Mater. Technol., Vol 102, 1980, p
376–381
32. C.S. White, J. Eng. Mater. Technol., Vol 114, 1992, p 384–389
33. I. Khoddam, Y.C. Lam, and P.F. Thomson, J. Test. Eval., Vol 26, 1998, p 157–167
34. G.R. Johnson, J. Eng. Mater. Technol., Vol 103, 1981, p 201–206
35. H. Kobayashi and B. Dodd, Int. J. Impact Eng., Vol 8, 1989, p 1–13
36. J.Y. Sheikh-Ahmad and J.A. Bailey, J. Eng. Mater. Technol., Vol 117, 1995, p 255–259
37. M. Zhou and M.P. Clode, Mater. Des., Vol 17, 1996, p 275–281
Fundamental Aspects of Torsional Loading
John A. Bailey, North Carolina State University;Jamal Y. Sheikh-Ahmad, Wichita State University
References
1. C.L. Harmsworth, Mechanical Testing, Vol 8, ASM Handbook, American Society for Metals, 1985, p 62
2. S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill, 1970, p 291
3. W.J.McG. Tegart, Elements of Mechanical Metallurgy, Macmillan, 1967, p 64
4. A. Nadai, Theory of Flow and Fracture of Solids, Vol 1, McGraw-Hill, 1950, p 349
5. G.R. Canova, et al., Formability of Metallic Materials—2000 A.D., STP 753, J.R. Newby and B.A.
Niemeier, Ed., ASTM, 1982, p 189
6. D.S. Fields and W.A. Backofen, Proc. ASTM, Vol 57, 1957, p 1259
7. S. Kalpakjian, Mechanical Processing of Materials, D. Van Nostrand, 1967, p 31
8. E.M. Mielnik, Metal Working Science and Engineering, McGraw-Hill, 1991, p 220
9. J.A. Bailey, S.L. Haas, and M.K. Shah, Int. J. Mech. Sci., Vol 14, 1972, p 735
10. G. Dieter, Mechanical Metallurgy, 2nd ed., McGraw-Hill, 1976, p 353
11. F. Garafalo, Fundamentals of Creep and Creep Rupture of Metals, Macmillan, 1965
12. C.M. Sellars and W.J.McG. Tegart, Int. Met. Rev., Vol 7, 1972, p 1
13. J.J. Jonas, C.M. Sellars, and W.J.McG. Tegart, Met. Rev., Vol 130, 1969, p 14
14. G.R. Johnson and W.H. Cook, Proc. Seventh Int. Symp. Ballistics, 1983, p 541
15. G.R. Johnson, J.M. Hoegfeldt, U.S. Lindholm, and A. Nagy, J. Eng. Mater. Technol. (Trans. ASME),
Vol 105, 1983, p 42
16. G.R. Johnson, J.M. Hoegfeldt, U.S. Lindholm, and A. Nagy, J. Eng. Mater. Technol. (Trans. ASME),
Vol 105, 1983, p 48
17. R.R. Hill, The Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1950, p 317
18. J.A. Bailey, S.L. Haas, and K.C. Naweb, J. Basic Eng. (Trans. ASME), March 1972, p 231
19. F. Montheillet, M. Cohen, and J.J. Jonas, Acta Metall., Vol 32, 1984, p 2077–2089
20. F. Montheillet, P. Gilormini, and J. J. Jonas, Acta Metall., Vol. 33, 1985, p 2126–2136
21. J.J. Jonas, Int. J. Mech. Sci., Vol 35, 1993, p 1065–1077
22. L.S. Toth and J.J. Jonas, Scr. Metall., Vol 27, 1992, p 359–363
23. L.S. Toth, J.J. Jonas, D. Daniel, and J.A. Bailey, Textures and Microstructures, Vol 19, 1992, p 245–
262
24. J.J. Jonas and L.S. Toth, Scr. Metall., Vol 27, 1992, p 1575–1580
25. S.L. Semiatin, G.D. Lahoti, and J.J. Jonas, Mechanical Testing, Vol 8, ASM Handbook, American
Society for Metals, 1985, p 154
26. R.S. Culver, Exp. Mech., Vol 12, 1972, p 398–405
27. H. Kobayashi, “Shear Localization and Fracture in Torsion of Metals,” Ph.D. thesis, University of
Reading, Reading, U.K., 1987
28. M.J. Luton, Workability Testing Techniques, G.E. Dieter, Ed., American Society for Metals, 1984, p 95
29. T. Vinh, M. Afzali, and A. Roche, Third Int. Conf. Mechanical Behavior of Materials (ICM 3), Vol 2,
K.J. Miller and R.F. Smith, Ed., Pergamon Press, 1979, p 633–642
30. H. Weiss, D.H. Skinner, and J.R. Everett, J. Phys. E, Sci. Instr., Vol 6, 1973, p 709–714
31. U.S. Lindholm, A. Nagy, G.R. Johnson, and J.M. Hoegfeldt, J. Eng. Mater. Technol., Vol 102, 1980, p
376–381
32. C.S. White, J. Eng. Mater. Technol., Vol 114, 1992, p 384–389
33. I. Khoddam, Y.C. Lam, and P.F. Thomson, J. Test. Eval., Vol 26, 1998, p 157–167
34. G.R. Johnson, J. Eng. Mater. Technol., Vol 103, 1981, p 201–206
35. H. Kobayashi and B. Dodd, Int. J. Impact Eng., Vol 8, 1989, p 1–13
36. J.Y. Sheikh-Ahmad and J.A. Bailey, J. Eng. Mater. Technol., Vol 117, 1995, p 255–259
37. M. Zhou and M.P. Clode, Mater. Des., Vol 17, 1996, p 275–281
Uniaxial Tension Testing
John M. (Tim) Holt, Alpha Consultants and Engineering
Introduction
THE TENSION TEST is one of the most commonly used tests for evaluating materials. In its simplest form, the
tension test is accomplished by gripping opposite ends of a test item within the load frame of a test machine. A
tensile force is applied by the machine, resulting in the gradual elongation and eventual fracture of the test item.
During this process, force-extension data, a quantitative measure of how the test item deforms under the applied
tensile force, usually are monitored and recorded. When properly conducted, the tension test provides force-
extension data that can quantify several important mechanical properties of a material. These mechanical
properties determined from tension tests include, but are not limited to, the following:
• Elastic deformation properties, such as the modulus of elasticity (Young's modulus) and Poisson's ratio
• Yield strength and ultimate tensile strength
• Ductility properties, such as elongation and reduction in area
• Strain-hardening characteristics
These material characteristics from tension tests are used for quality control in production, for ranking
performance of structural materials, for evaluation of newly developed alloys, and for dealing with the static-
strength requirements of design.
The basic principle of the tension test is quite simple, but numerous variables affect results. General sources of
variation in mechanical-test results include several factors involving materials, namely, methodology, human
factors, equipment, and ambient conditions, as shown in the “fish-bone” diagram in Fig. 1. This article
discusses the methodology of the tension test and the effect of some of the variables on the tensile properties
determined. The following methodology and variables are discussed:
• Shape of the item being tested
• Method of gripping the item
• Method of applying the force
• Determination of strength properties other than the maximum force required to fracture the test item
• Ductility properties to be determined
• Speed of force application or speed of elongation (e.g., control of stress rate or strain rate)
• Test temperature
The main focus of this article is on the methodology of tension tests as it applies to metallic materials. Factors
associated with test machines and their method of force application are described in more detail in the article
“Testing Machines and Strain Sensors” in this Volume.
Fig. 1 “Fish-bone” diagram of sources of variability in mechanical-test results
This article does not address the tension testing of nonmetallic materials, such as plastics, elastomers, or
ceramics. Although uniaxial tension testing is used in the mechanical evaluation of these materials, other test
methods often are used for mechanical-property evaluation. The general concept of tensile properties is very
similar for these nonmetallic materials, but there are also some very important differences in their behavior and
the required test procedures for these materials:
• Tension-test results for plastics depend more strongly on the strain rate because plastics are viscoelastic
materials that exhibit time-dependent deformation (i.e., creep) during force application. Plastics are also
more sensitive to temperature than metals. Thus, control of strain rates and temperature are more critical
with plastics, and sometimes tension tests are run at more than one strain and/or temperature. The
ASTM standard for tension testing of plastics is D 638.
• Tension testing of ceramics requires more attention to alignment and gripping of the test piece
*
in the
test machine because ceramics are brittle materials that are extremely sensitive to bending strains and
because the hard surface of ceramics reduces the effectiveness of frictional gripping devices. The need
for a large gripping areas thus requires the use of larger test pieces (Ref 1). The ASTM standard for
tension testing of monolithic ceramics at room temperature is C 1275. The standard for continuous
fiber-reinforced advanced ceramics at ambient temperatures is C 1273.
• Tension testing of elastomers is described in ASTM D 412 with specific instructions about test-piece
preparation, equipment, and test conditions. Tensile properties of elastomers vary widely, depending on
the particular formulation, and scatter both within and between laboratories is appreciable compared
with the scatter of tensile-test results of metals (Ref 2). The use of tensile-test results of elastomers is
limited principally to comparison of compound formulations.
Footnote
*
The term “test piece” is used in this article for what is often called a “specimen” (see “The Test Piece”
in this
article).
References cited in this section
1. D. Lewis, Tensile Testing of Ceramics and Ceramic-Matrix Composites, Tensile Testing, P. Han, Ed.,
ASM International, 1992, p 147–182
2. R.J. Del Vecchio, Tensile Testing of Elastomers, Tensile Testing, P. Han, Ed., ASM International, 1992,
p 135–146
Uniaxial Tension Testing
John M. (Tim) Holt, Alpha Consultants and Engineering
Definitions and Terminology
The basic results of a tension test and other mechanical tests are quantities of stress and strain that are
measured. These basic terms and their units are briefly defined here, along with discussions of basic stress-
strain behavior and the differences between related terms, such as stress and force and strain and elongation.
Load (or force) typically refers to the force acting on a body. However, there is currently an effort within the
technical community to replace the word load with the more precise term force, which has a distinct meaning
for any type of force applied to a body. Load applies, in a strict sense, only to the gravitational force that acts on
a mass. Nonetheless, the two terms are often used interchangeably.
Force is usually expressed in units of pounds-force, lbf, in the English system. In the metric system, force is
expressed in units of newtons (N), where one newton is the force required to give a 1 kg mass an acceleration of
1 m/s
2
(1 N = 1 kgm/s
2
). Although newtons are the preferred metric unit, force is also expressed as kilogram
force, kgf, which is the gravitational force on a 1 kg mass on the surface of the earth. The numerical
conversions between the various units of force are as follows:
• 1 lbf = 4.448222 N or 1 N = 0.2248089 lbf
• 1 kgf = 9.80665 N
In some engineering disciplines, such as civil engineering, the quantity of 1000 lbf is also expressed in units of
kip, such that 1 kip = 1000 lbf.
Stress is simply the amount of force that acts over a given cross-sectional area. Thus, stress is expressed in units
of force per area units and is obtained by dividing the applied force by the cross-sectional area over which it
acts. Stress is an important quantity because it allows strength comparison between tests conducted using test
pieces of different sizes and/or shapes. When discussing strength values in terms of force, the load (force)
carrying capacity of a test piece is a function of the size of the test piece. However, when material strength is
defined in terms of stress, the size or shape of the test piece has little or no influence on stress measurements of
strength (provided the cross section contains at least 10 to 15 metallurgical grains).
Stress is typically denoted by either the Greek symbol sigma, σ, or by s, unless a distinction is being made
between true stress and nominal (engineering) stress as discussed in this article. The units of stress are typically
lbf/in.
2
(psi) or thousands of psi (ksi) in the English system and a pascal (Pa) in the metric system. Engineering
stresses in metric units are also expressed in terms of newtons per area (i.e., N/m
2
or N/mm
2
) or as kilopascals
(kPA) and megapascals (MPa). Conversions between these various units of stress are as follows:
• 1 Pa = 1.45 × 10
-4
psi
• 1 Pa = 1 N/m
2
• 1 kPa = 10
3
Pa or 1 kPa = 0.145 psi
• 1 MPa = 10
6
Pa or 1 MPa = 0.145 ksi
• 1 N/mm
2
= 1 MPa
Strain and elongation are similar terms that define the amount of deformation from a given amount of applied
stress. In general terms, strain is defined (by ASTM E 28) as “the change per unit length due to force in an
original linear dimension.” The phrase change per unit length means that a change in length, ΔL, is expressed
as a ratio of the original length, L
0
. This change in length can be expressed in general terms as a strain or as
elongation of gage length, as described subsequently in the context of a tension test.
Strain is a general term that can be expressed mathematically, either as engineering strain or as true strain.
Nominal (or engineering) strain is often represented by the letter e, and logarithmic (or true) strain is often
represented by the Greek letter ε. The equation for engineering strain, e, is based on the nominal change in
length, (ΔL) where:
e = ΔL/L
0
= (L - L
0
)/L
0
The equation for true strain, ε, is based on the instantaneous change in length (dl) where:
These two basic expressions for strain are interrelated, such that:
ε = ln (1 + e)
In a tension test, the typical measure of strain is engineering strain, e, and the units are inches per inch (or
millimeter per millimeter and so on). Often, however, no units are shown because strain is the ratio of length in
a given measuring system.
This article refers to only engineering strain unless otherwise specified. In a tension test, true strain is based on
the change in the cross-sectional area of the test piece as it is loaded. It is not further discussed herein, but a
detailed discussion is found in the article “Mechanical Behavior under Tensile and Compressive Loads” in this
Volume.
Elongation is a term that describes the amount that the test-piece stretches during a tension test. This stretching
or elongation can be defined either as the total amount of stretch, ΔL, that a part undergoes or the increase in
gage length per the initial gage length, L
0
.
The latter definition is synonymous with the meaning of engineering strain, ΔL/L
0
, while the first definition is
the total amount of extension. Because two definitions are possible, it is imperative that the exact meaning of
elongation be understood each time it is used.
This article uses the term elongation, e, to mean nominal or engineering strain (i.e., e = ΔL/L
0
). The amount of
stretch is expressed as extension, or the symbol ΔL. In many cases, elongation, e, is also reported as a
percentage change in gage length as a measure of ductility (i.e., percent elongation), (ΔL/L
0
) × 100.
Engineering Stress and True Stress. Along with the previous descriptions of engineering strain and true strain, it
is also possible to define stress in two different ways as engineering stress and true stress. As is intuitive, when
a tensile force stretches a test piece, the cross-sectional area must decrease (because the overall volume of the
test piece remains essentially constant). Hence, because the cross section of the test piece becomes smaller
during a test, the value of stress depends on whether it is calculated based on the area of the unloaded test piece
(the initial area) or on the area resulting from that applied force (the instantaneous area). Thus, in this context,
there are two ways to define stress: