P8-1
EXPERIMENT P-8
UNIAXIAL TENSILE TEST
*You need to bring a disk (not USB key) for this lab
Objective
To familiarize the students with the measurements of mechanical properties (yield
strength, ultimate tensile strength, %elongation, ductility) of engineering materials via the
uniaxial tensile test.
Background
The uniaxial tension test is widely used to provide information on the strength and plastic
properties of materials. In this test the sample of the material is elongated by an uniaxial load.
The axial load, F, and the change of the sample’s length, Δl , are recorded. The test is
performed on a tensile test press equipped with load and displacement sensors and recording
devices. Fig. 1 illustrates a typical, F(Δl ), tensile curve for a ductile material.
Fig. 1 Tension load - extension curve
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The actual range of the load and displacement depends on the material and dimensions of
the sample. In order to compare different materials the effect of sample dimensions is eliminated
by translating load into force per unit cross-sectional area - tensile stresses, σ, and displacement
into relative elongation - tensile strain, ε. Fig. 2 illustrates a typical stress-strain curve, σ(ε), for
an elastic plastic material. The basic mechanical properties pertaining to the strength and
ductility of the materials are referred to in terms of stresses and strains. Particularly, so called
effective stress and strain utilized in strength analysis, translate any general three dimensional
stress-strain state to values equivalent to the uniaxial tensile test data, σ, and ε . The stress strain
curve shown in fig. 2 contains several characteristic points usually listed in standard material
data sheets. The slope of the initial linear portion of the curve, σ(ε), is referred to as Young’s
modulus, E, which is the elastic stiffness of the material under normal stress. Upon unloading
from a stress-strain state within this range of deformation, the sample fully recovers its initial
dimensions, i.e the deformation is purely elastic, εel. Some materials may exhibit a non-linear
elasticity, i.e. the value of Young’s modulus, E, may not be constant, yet there is full elastic
shape recovery upon unloading.
Fig. 2 Stress-strain curve, elastic-plastic material model
If, during the loading stage the strain exceeds a specific limit, upon unloading the initial
shape of the sample is not fully recovered, and some portion of the total loading strain, ε,
becomes permanent. This permanent strain is referred to as plastic strain, εpl . The plastic strain,
εpl, cannot be induced without being preceded by the elastic deformation, εel , The stress level at
which the plastic deformation is initiated is referred to as the yield stress, Yo , of the material.
Any structural application of a given material should insure that the maximum value of the
effective stress remains below the yield stress, otherwise under the expected loading the shape of
the structure would become permanently distorted due to plastic deformation. Most materials do
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not exhibit the presence of a distinct point, Yo, on the curve, σ(ε). For materials without the
distinct point, Yo , the yield stress is usually defined as the stress level at which the permanent,
plastic, strain is 0.2%. In the material data sheets this kind of yield stress is referred to as R0.2.
Different national standards also list other definitions of the yield stress such as R0.1, R0.02, etc.,
which refer to 0.1% and 0.02% plastic elongation at the yield point. Beyond the yield point the
deformation is elastic-plastic. This range of deformation is of primary importance for forming
technology applications in which different mechanical components are shaped by plastic strains.
In the forming technology applications the plastic strain may become of two or more orders of
magnitude greater than the elastic. Often, in the analysis of these applications, the elastic
properties are neglected, and the material behavior is simplified by the so called rigid-plastic
material model , for which the curve , σ(εpl), begins at the initial yield stress, Yo(εpl =0), and
ignores the elastic strain, fig. 3.
Fig. 3 Stress-strain curve, rigid plastic material model
In the elastic-plastic range of deformation the level of stresses may increase with the
magnitude of strains. This increase is referred to as strain hardening. The strain hardening
indicates that the material is gaining strength with the amount of the induced plastic deformation.
If a strain hardened material is unloaded from the elastic-plastic range of deformation and
reloaded again, the plastic deformation will not resume at the stress level indicated by the initial
yield point, Yo or R0.2, but at the stress level near or at the level, σ(εpl), reached just before the
unloading.
In the uniaxial tension test, only one from the triple, [σ1, σ2, σ3], of principal stress
components is non-zero - the axial stress, σ = σ1, while the remaining two components associated
with the planes orthogonal to the tension direction are zero, σ1=σ2 =0. The elongation of the
sample is measured by the axial strain, ε =ε1. However, as the volume of the sample is conserved,
while the sample is being elongated its width and thickness or in general its cross-sectional area
decrease, and therefore, none of the principal strains is zero, ε1>0, ε2<0 and ε3<0. In the elasticplastic range a single stress component, σ1, results in six principal strain components, three
elastic strains, [ε1, ε2, ε3]el , and three plastic strains, [ε1, ε2 ε3]pl . The axial stress and strain are
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used as the reference, σ(ε), but in general the deformation in tensile test has all the same stress
and strain characteristics as any other deformation process.
The uniaxial tension stress-strain state exists as long as the sample is being deformed
uniformly. The initial increase of the tension load, F(Δl ), shown in fig. 1 is due to the strain
hardening (monotonic increase of the stress level of the, σ(ε), curve). However, the axial stresses,
σ = σ1, are carried by the decreasing cross-section area, A, of the sample. At some point the effect
of stress increase on the tension force becomes equal to the effect of the cross-section area
decrease. At this point the force reaches a maximum value, Fmax.. Any further elongation of the
sample results in the drop of force and elastic unloading of the previously stressed material with
the exception of one zone, which due to micro-structural or dimensional defects has the lowest
load carrying capacity. At this stage the overall extension of the sample length results in
localization of the plastic deformation only in the weakest zone referred to as “the neck”. The
stress state in the neck changes to a tri-axial tension, which is caused by the non-co-linearity of
the neck profile with the sample axis. Essentially the stress state in the neck is undefined and
only the portion of, σ(ε), within the uniform elongation range is considered valid.
Stress and strain measures
There are two basic stress and strain measures used in material data sheets: the
engineering and the true measure. The engineering stress and strain measures are obsolete;
nevertheless many national standards still utilize these traditional measures. The engineering
stress, σeng , is defined as the ratio of the instantaneous tensile force, Fi , to the initial crosssection area of the sample, Ao,
F
σ eng = i ,
(1)
Ao
and the engineering percent strain, e%, as the % ratio of the length increase, Δl , to the initial
length, lo, of the sample:
e% =
Δl
100% .
lo
(2)
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Fig. 4 Engineering stress-strain curve
However, during the tensile test the cross-section area, A, of the sample decreases due to
elongation while each subsequent increase of sample length, dl , takes place over an already
elongated sample length, l. It is evident that by neglecting the changes in the cross-section area,
A, and sample length, l, the engineering measures are not representative of the actual strain and
stress state of the material. The stress-stain curve expressed in the engineering measures is
shown in fig. 4. Apparently, based on the engineering stress definition (1) the value of
engineering stress decreases past the point marked with the symbol U.T.S . This is misleading; in
reality the true stresses continuously increase.
The so-called true stress and strain measures define the stress and strain state correctly.
The true stress, σ, is defined as the ratio of the instantaneous applied force, Fi, to the
instantaneous cross-section area, Ai:
F
σ= i
(3)
Ai
and true strain, ε, is defined as the product of integration, given by:
li dl
⎛l ⎞
(4)
ε =∫
= ln⎜⎜ i ⎟⎟ ,
lo l
⎝ lo ⎠
where, lo, is the initial and, li, is the instantaneous length of the sample. The true strain is also
often referred to as the logarithmic strain.
Stress-strain curve for plastic deformation
The relationship, σ(εpl), for true stress and strain measures is referred to as the stressstrain curve. For many materials the experimental, σ(εpl), data can be interpolated by an
exponential function in the form:
σ = K(εo + εpl) n
(5)
where K, εo and n are material constants. These constants are determined by fitting a curve
expressed by equation (5) between experimental points obtained from the tensile test. Usually the
curve-fitting algorithm neglects the small elastic deformation. The constant, K , is referred to as
flow stress constant . It represents the stress level for εo + εpl = 1.0 and its value indicates the
overall stress level at which the material is deformed plastically. The constant, εo , is referred to
as initial strain offset. This constant shifts the exponential curve (5) to a position at which for,
εpl=0, (the beginning of plastic deformation) the stress level is equal to the initial yield stress , σ
= Yo.. The constant, n , is referred to as the strain hardening exponent. It indicates the rate of
material strain hardening. On a logarithmic scale graph, fig. 5, the stress-strain curve, σ(εpl),
represented by equation (5) becomes a straight line with the slope n,
log( σ) = n log ( εo + εpl ) + log(K).
(6)
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Fig. 5 Stress-strain curve in logarithmic scale
Coefficient of anisotropy, r.
An isotropic material exhibits identical properties in all the directions in its volume. In
general, an anisotropic material is characterized by different magnitude of its characteristic
values such as Young’s modulus, yield stress, strain hardening etc. depending on the orientation
of the loading direction in the material space. Particularly in the sheet forming technology
applications (for example, forming automotive body panels) a desirable deformation behavior is
such that the material resists deformation in its thickness direction but is easily deformable in its
plane. This behavior promotes shaping of the surface over the localized thinning and ultimately,
splitting, of the sheet during forming. The uniaxial tension test is used to evaluate this property
of the sheet products by means of the coefficient of anisotropy, r, defined as the ratio of the true
strain, ε2 = εw, measured in the direction of sample’s width over the strain, ε3 = εt, measured in
the thickness direction:
r = εw /εt .
(7)
The coefficient of anisotropy, r , is determined for three different directions 0o, 45o and 90o in
the sheet plane with respect to the rolling direction of the sheet, r0, r45, r90 .
Tensile test data sheet
Standard tensile test data sheet provides the following data:
Sample dimensions - National standards list several standard dimensions of the tensile
test samples. They are different for flat, bar and wire products. The 4:1 gage
length/width type sample used in the laboratory comply with the ISO/ASTM
recommendations for flat (sheet) products with recommended gage length of 60mm (2.25
in) and width 12.5mm (0.5 in),
ME 3M02 – Experiment P8: Uniaxial Tensile Test
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Young’s modulus, E,
Yield stress ,Yo or R0.2 , - expressed as engineering stresses,
Ultimate temsile strength U.T.S. - expressed as engineering stresses,
U.T.S = Fmax / Ao,
Total elongation, et%, - obtained by putting a fractured sample back together and
measuring the % of total length change, ef% = Δlmax / lo. $ 100%,
Reduction of area, q, A.R. or %At , - % of the area reduction of the fractured section
Additional data :
Plastic stress-strain curve parameters K, εo , n , ( true measures ),
Uniform elongation, εu or %εu , - maximum elongation of the material outside the neck
expressed in true or engineering strain measures,
Coefficient of anisotropy, r0, r45, r90 (used for flat products only).
Laboratory experiment
The objective of the experiment is to generate a complete material data sheet for a flat
product. The experiment will be performed on a standard tensile testing machine equipped with
load cells, displacement and strain gages. The T.A. will give instructions pertaining to the testing
machine and data acquisition system operations, but the students themselves will conduct the
actual test. Prior to the experiment, the students should plan the steps of the experimental
procedure. As part of the experimental procedure the students should perform full calibration of
all the gages. A set of weights and a device equipped with micrometer screw will be provided to
perform the calibration of the load and displacement sensors. The laboratory report should
include:
After lab, before write up session:
1. Create a graph of true stress/strain for each sample. Use extensometer data to calculate
strain.
2. Create a graph showing both displacement versus time and extension versus time for one
of the samples. Why did we measure both of these parameters? What is their
significance? What does the graph tell you?
3. Complete the property calculations sheets
4. Complete the material data tables
5. Discuss the results of the labs. Were the yield strengths of the materials what were
expected? Why or why not? Justify your results with other sources of information.
Repeat for Young’s Modulus. Make sure to indicate your source. Do these values make
sense according to the rolling directions of the samples?
ME 3M02 – Experiment P8: Uniaxial Tensile Test
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6. Create an appendix of calibration data and graphs.
Write Up Session:
You will be given some questions based on what you did in the lab, the only preparation
you need is to bring your brain, creativity and your completed pre lab. Good luck!
Experiment safety
The participants of the experiment must wear safety glasses.
All participants of the this lab experiment must be familiar and follow the Standard
Operating Procedure, (SOP), entitled “P5 Plastic properties of sheet metal (Instron 1140)”. The
full text of the SOP is available in JHE 314.
ME 3M02 – Experiment P8: Uniaxial Tensile Test
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Property Calculations (Sample Calculations):
Sample Rolled in 0° Direction:
Property
Formula
Young’s Modulus
Ultimate Tensile Strength
Note: Please indicate on graph also
UTS = Fmax/Ao
Total Elongation
et% = (∆lo / lo) X 100%
Reduction of Area
Flow Stress Constant
Strain Hardening Exponent
Initial Strain Offset
Uniform Elongation
Coefficient of Anisotropy
Calculation
ME 3M02 – Experiment P8: Uniaxial Tensile Test
Sample Rolled in 45° Direction:
Property
Formula
Young’s Modulus
Ultimate Tensile Strength
Note: Please indicate on graph also
UTS = Fmax/Ao
Total Elongation
et% = (∆lo / lo) X 100%
Reduction of Area
Flow Stress Constant
Strain Hardening Exponent
Initial Strain Offset
Uniform Elongation
Coefficient of Anisotropy
P8-10
Calculation
ME 3M02 – Experiment P8: Uniaxial Tensile Test
Sample Rolled in 90° Direction:
Property
Formula
Young’s Modulus
Ultimate Tensile Strength
Note: Please indicate on graph also
UTS = Fmax/Ao
Total Elongation
et% = (∆lo / lo) X 100%
Reduction of Area
Flow Stress Constant
Strain Hardening Exponent
Initial Strain Offset
Uniform Elongation
Coefficient of Anisotropy
P8-11
Calculation
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Material Data Tables:
Sample Rolled in 0° Direction:
Parameter
Sample Dimensions Initial
Standard
Material Data
Sheet
Final
Additional
Parameters
Young’s Modulus
Yield Stress
Ultimate Tensile Strength
Total Elongation
Reduction of Area
Flow Stress Constant
Strain Hardening Exponent
Initial Strain Offset
Uniform Elongation
Coefficient of Anisotropy
Value
Overall length
Overall width
Overall
thickness
Gage length
Gage width
Gage thickness
Overall length
Overall width
Overall
thickness
Gage length
Gage width
Gage thickness
E
Yo or R0.2
UTS
e t%
q, AR or %At
K
n
εo
εu
r
60 mm
12.5 mm
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Sample Rolled in 45° Direction:
Standard
Material Data
Sheet
Parameter
Sample Dimensions Initial
Final
Additional
Parameters
Young’s Modulus
Yield Stress
Ultimate Tensile Strength
Total Elongation
Reduction of Area
Flow Stress Constant
Strain Hardening Exponent
Initial Strain Offset
Uniform Elongation
Coefficient of Anisotropy
Value
Overall length
Overall width
Overall
thickness
Gage length
Gage width
Gage thickness
Overall length
Overall width
Overall
thickness
Gage length
Gage width
Gage thickness
E
Yo or R0.2
UTS
e t%
q, AR or %At
K
n
εo
εu
r
60 mm
12.5 mm
ME 3M02 – Experiment P8: Uniaxial Tensile Test
Sample Rolled in 90° Direction:
Parameter
Sample Dimensions Initial
Standard
Material Data
Sheet
Final
Additional
Parameters
Young’s Modulus
Yield Stress
Ultimate Tensile Strength
Total Elongation
Reduction of Area
Flow Stress Constant
Strain Hardening Exponent
Initial Strain Offset
Uniform Elongation
Coefficient of Anisotropy
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Value
Overall length
Overall width
Overall
thickness
Gage length
Gage width
Gage thickness
Overall length
Overall width
Overall
thickness
Gage length
Gage width
Gage thickness
E
Yo or R0.2
UTS
e t%
q, AR or %At
K
n
εo
εu
r
60 mm
12.5 mm