Correlation between Engineering Stress-Strain and True Stress-Strain Curve

Abstract

The most commonly accepted method in evaluation of the mechanical properties of metals would be the tension test. Its main objective would be to determine the properties relevant to the elastic design of machines and structures. Investigation of the engineering and true Stress-strain relationships of three specimens in conformance with ASTM E 8 – 04 is the aim of this paper. For the purpose of achieving this aim, evaluation of values such as ultimate tensile strength, yield strength, percentage of elongation and area reduction, fracture strain and Young's Modulus was done once the specimens were subjected to uniaxial tensile loading. The results indicate that the properties of steel materials are independent from their thickness and they generally yield and fail at the same stress and strain values. Also, it is concluded that the maximum true stress values are almost 15% higher than that of the maximum engineering stress values while the maximum true strain failure values are 1.5% smaller than the maximum engineering strain failure values.

Full text

American Journal of Civil Engineering and Architecture, 2014, Vol. 2, No. 1, 53-59
Available online at http://pubs.sciepub.com/ajcea/2/1/6
© Science and Education Publishing
DOI:10.12691/ajcea-2-1-6
Correlation between Engineering Stress-Strain and True
Stress-Strain Curve
Iman Faridmehr1, Mohd Hanim Osman1,*, Azlan Bin Adnan2, Ali Farokhi Nejad2, Reza Hodjati1, Mohammad
Amin Azimi1
1Faculty of Civil Engineering, Universiti Teknologi, Malaysia, Skudai, Johor. Malaysia
2Faculty of Mechanical Engineering, Universiti Teknologi, Malaysia, Skudai, Johor. Malaysia
*Corresponding author: mhaim@utm.my
Received January 25, 2014; Revised February 28, 2014; Accepted March 05, 2014
Abstract The most commonly accepted method in evaluation of the mechanical properties of metals would be the
tension test. Its main objective would be to determine the properties relevant to the elastic design of machines and
structures. Investigation of the engineering and true Stress-strain relationships of three specimens in conformance
with ASTM E 8 – 04 is the aim of this paper. For the purpose of achieving this aim, evaluation of values such as
ultimate tensile strength, yield strength, percentage of elongation and area reduction, fracture strain and Young's
Modulus was done once the specimens were subjected to uniaxial tensile loading. The results indicate that the
properties of steel materials are independent from their thickness and they generally yield and fail at the same stress
and strain values. Also, it is concluded that the maximum true stress values are almost 15% higher than that of the
maximum engineering stress values while the maximum true strain failure values are 1.5% smaller than the
maximum engineering strain failure values.
Keywords: tension testing, mechanical properties of metals, stress-strain relationships, uniaxial tensile
Cite This Article: Iman Faridmehr, Mohd Hanim Osman, Azlan Bin Adnan, Ali Farokhi Nejad, Reza Hodjati,
and Mohammad Amin Azimi, “Correlation between Engineering Stress-Strain and True Stress-Strain Curve.”
American Journal of Civil Engineering and Architecture, vol. 2, no. 1 (2014): 53-59. doi: 10.12691/ajcea-2-1-6.
1. Introduction
The significant role of the mechanical properties in
evaluation of the fundamental properties of engineering
materials along with development of new materials and
also quality control of materials for application in design
and construction cannot be neglected. Once a material is
meant to be applied within an engineering structure that
will be subject to loading, it must be guaranteed that the
material possesses enough strength and rigidity to
withstand the loads it will experience in its service life.
Consequently, a number of experimental techniques have
been developed by engineers for the purpose of
mechanical testing of engineering materials subjected to
tension, compression, bending or torsion loading. The
tension test would be the most common type of test
incorporated in measurement of the mechanical properties
of materials. Provision of basic design information on
materials strength is the purpose of conduction of tension
test. Also, tension test is considered as an acceptance test
for the specification of materials. Major parameters such
as the tensile strength (UTS), yield strength or yield point
(σy), elastic modulus (E), percent elongation (ΔL%) and
the reduction in area (RA%) are obtained during the
tension test in order to describe the stress-strain curve.
Other parameters including toughness, resilience and
Poisson’s ratio (ʋ) can also be calculated through the use
of this testing technique. Conduction of the tensile test is
done through application of longitudinal or axial load at a
specific extension rate to a standard tensile specimen with
known dimensions, including the gauge length and cross
sectional area perpendicular to the load direction, until
failure. In order to calculate the stress and strain, the
applied tensile load and extension are recorded during the
test.
A range of universal standards provided by professional
societies such as American Society of Testing and
Materials ASTM E8 [1], British standard [2], JIS standard
[3] and DIN standard [4] provide tests based on
preferential purposes. Each standard may contain a variety
of test standards suitable for different materials,
dimensions and fabrication histories. For instance, ASTM
E8 [1] is a standard test method for tension testing of
metallic materials and ASTM B557 [5] is a standard test
method of tension testing and also for casting aluminum
and magnesium alloy products.
2. Research Methodology
2.1. Test Specimens
Figure 1 illustrates the preparation of a standard
specimen in a round or a square section along the gauge
length in conformance with ASTM E8 [1]. For the
specimens being firmly gripped during testing, sufficient

54 American Journal of Civil Engineering and Architecture
length and a surface condition must be provided for both
ends of the specimens. Table 1 demonstrates the
standardized initial length Lo which varies with the
diameter (Do) or the cross-sectional area (Ao) of the
specimen. This is due to the fact that the % elongation
might be underestimated if the gauge length is too long in
this case [6]. For the purpose of producing the final
specimen to be tested, any kind of heat treatments should
be applied on the test specimen prior to machining. This
was done to avoid surface oxide scales that could behave
as stress concentration which could eventually affect the
final tensile properties as a result of premature failure.
Some exceptions including the surface hardening or
surface coating on the materials might exist. Application
of these processes must be done after specimen machining
to gain the tensile properties results that entail the actual
specimen surface conditions.
Figure 1. Standard Tensile Specimens
Table 1. Dimensional Relationships of Tensile Specimens Used in
Different Countries
Type specimen
United
State
(ASTM)
Great
Britain Germany
0
0
(
sheet )
LA
4.5 5.65 11.3
0
0
(
Rod )
LD
4.0 5.0 10.0
The specimens selected for this study were machined in
conformance with ASTM (?) with a thickness of 6mm,
8mm and 10mm respectively as shown in Figure 2.
Figure 2. Test specimens
2.2. Test Equipment
The equipment applied for the conduction of tensile
testing includes both simple devices and complicated
controlled systems. The most commonly used equipment
for this test would be the so-called universal testing
machines driven by mechanical screw or hydraulic
systems. A relatively simple screw-driven machine
incorporating two screws for the purpose of applying the
load along with a hydraulic testing machine incorporating
the pressure of oil in a piston to supply the load are
illustrated in Figure 3. Application of these types of
machines is not only for tension, but also for compression,
bending and torsion tests. A variety of loads, strain, or
testing machine motions (stroke) are provided by a more
modernized closed-loop servo-hydraulic machine through
the use of a combination of actuator rod and piston. The
majority of the machines used nowadays in tests are
connected to a computer-controlled system that could
graphically display the load and extension data along with
the stress and strain calculations. Extensometer is another
equipment incorporated in this test. An extensometer is
defined as a device applied to measure changes in the
length of an object used in tensile tests and stress-strain
measurements. Finally, the device used to measure the
distance between two opposite sides of an object is called
a caliper.
Figure 3. Schematics Showing of a Screw Driven Machine and a Hydraulic Testing Machine

American Journal of Civil Engineering and Architecture 55
The universal testing machine incorporated in this test
belongs to to DARTEC Company with capacity 4000 KN
in compression and 5000 KN in tension. The standard
ASTM E4-1998 was used to calibrate the machine with
the ambient temperature is 26.0°C ± 2°C. Figure 4
illustrates the equipment used in this test including the
testing machine, extensometer and caliper respectively
from left to right.
Figure 4. Test Equipment Used in the Test
2.3. Test Procedure
The specimens used in this test are made of low carbon
steel with a thickness of 6mm, 8mm and 10mm
respectively. First, the dimensions of the specimens
(gauge length, thickness and width) are tabulated in Table
2 in order to determine the engineering stress and
engineering strain values. Then, the location of the gauge
length was marked along the parallel length of each
specimen for subsequent necking observations and strain
measurements. Next, the specimen was fit on to the
Universal Testing Machine (UTM) and then, the
extensometer was installed on the specimen. After that the
testing began and load and extension values were recorded.
Finally, the following calculations were done to determine
the material characteristics.
Table 2. Specimen Dimension
Specimen
Width
(mm)
Thickness
(mm)
Cross Section Area
(mm
2
)
Gauge Length
(mm)
1
38
10
380
50
2
38
8
304
50
3
38
6
228
50
2.3.1. Stress and Strain Relationship
Once an external tensile loading is applied on a
specimen, elastic and plastic deformations will be
anticipated. Initially, an elastic deformation will occur to
the metal resulting in a linear relationship of load and
extension. Calculation of the engineering stress and
engineering strain will be done using these two parameters
for the purpose of yielding a relationship as shown in
Figure 5 incorporating equations 1 and 2 as follows:
0
=
EP
σA
(1)
0
00
Δ
−
= =
f
E
LL L
εLL
(2)
Where
σE is the engineering stress.
εΕ is the engineering strain.
P is the external axial tensile load.
Ao is the original cross-sectional area of the specimen.
Lo is the original length of the specimen.
Lf is the final length of the specimen.
The dimension of the engineering stress is Pascal (Pa)
or N/m2 based on the SI Metric Unit while the "psi"
(pound per square inch) dimension could also be
incorporated as well. Once the elastic deformation is in
progress, the engineering stress-strain relationship follows
the Hook's Law and slope of the curve demonstrates the
Young's modulus (E).
=σ
E
ε
(3)
The significance of the Young's modulus calculation
would be in determination of the materials deflection in
engineering applications. For instance, deflection of the
structural beams is known to be a critical factor in
designing the engineering components or structures
including buildings, bridges, ships and etc. Also, spring
constants or Young's modulus values are required for
tennis racket and golf club applications.
Figure 5. Stress-Strain Relationship under Uniaxial Tensile Loading

56 American Journal of Civil Engineering and Architecture
2.3.2. Yield Strength and Ultimate Tensile Strength
Through consideration of the stress-strain curve beyond
the elastic limit, yielding will occur at the beginning of
plastic deformation if the tensile loading continues.
Calculation of the yield stress, σy, is done through
dividing the load at yielding (Py) by the original cross-
sectional area of the specimen (Ao) as demonstrated in
equation 4.
0
=
y
y
P
σA
(4)
The yield point could be located directly from the load-
extension curve of the metals like steel, particularly low
carbon ones, or iron or in molybdenum and polycrystalline
titanium (Figure 6). The yield point elongation
phenomenon indicates the upper yield point followed by a
sudden stress or load reduction until the lower yield point
is reached. Extending of the specimen is continued at the
yield point elongation without a substantial change in the
stress level. Figure 3 indicates the yield strength
determination at 0.2% offset or 0.2% strain which is
carried out through drawing a straight line parallel to the
slope of the stress-strain curve in the linear section,
reaching the intersection on the x-axis at a strain equal to
0.002. An interception among the 0.2% offset line and the
stress strain curve indicates the yield strength at 0.2%
offset or 0.2% strain. The yield strength which is an
indication of the plastic deformation onset is known to be
a significant factor in engineering structural or component
designs where application of safety factors are common
(equation 5). Several considerations are taken into account
for safety factors including the deterioration estimation,
structural components and the consequences of failed
structures such as loss of life, economical losses and etc.
Generally, a safety factor of 2 is required for buildings,
which is quite low since the calculation of loads has been
well understood. A safety factor of 2 for automobiles
along with safety factors of 3 - 4 for pressure vessels are
two examples for this issue.
=
y
w
σ
σSafety Factor
(5)
Beyond yielding, an increase in the stress for permanent
deformation of the specimen occurs through continuous
loading. At this time, the specimen would be strain
hardened or work hardened. Through application of
continuous loading, the maximum point will be reached
for the stress-strain curve, which is the ultimate tensile
strength (σTS) (equation 6). At this stage, the highest stress
before necking could be beard by the specimen. This is
observed through a local reduction in the cross sectional
area of the specimen commonly seen in the center of the
gauge length.
max
0
=
TS P
σA
(6)
Figure 6. Definition of Yield Point
2.3.3. True Stress and True strain
Stress has units of a force measure divided by the
square of a length measure, and the average stress on a
cross-section in the tensile test is clearly the applied force
divided by the cross-sectional area. In a similar manner,
an approximation is made on the strain component along
the long axis of the specimen as the change in length
divided by the original, reference length. This might seem
simple at first glance; however, some choices are still left
to be made. For instance, which are must be chosen for the
cross-sectional area? Which are should be used, the
original one or the current one? Also, should length
changes be compared against the original length of the
specimen? The answer lies in the fact that several types of
stress and stress measurements are done according to our
methodology. Fixed reference quantities are used to
distinguish the engineering stress and strain, mainly the
original cross-sectional area or original length. Such
definitions are accurate in most of the engineering
applications due to fixed values of the cross-sectional area
and length of the specimen while the loads are applied. In
other circumstances such as the tensile test, a substantial
change in the cross-sectional area and the length of the
specimen is expected. In such occasions, the engineering
stress determined using the above definition (as the ratio
of the applied load to the undeformed cross-sectional area)
seems to be an inaccurate measure. Hence, alternative
stress and strain measurement methods are available to
overcome this issue. The following lines are dedicated to
the true stress and true strain discussion.

American Journal of Civil Engineering and Architecture 57
Figure 7. Engineering Stress Measures vs. True Stress Measures
The ratio of the applied load to the instantaneous cross-
sectional area is the definition of the true stress. There
could exist a relationship between the true stress and
engineering stress once no volume change is assumed in
the specimen. Under this assumption;
=
TP
σA
(7)
(
0
=
EP
σA
(1)
0
00
Δ
−
= =
f
E
LL L
εLL
(2))
Where,
A is the reduced cross-sectional area of the specimen
( )
00 00
1= ⇒== = +
T EE
P PL
A.L A .L σ.σ ε
A AL
(8)
True strain is defined as the instantaneous increase rate
in the instantaneous gauge length defined as true strain.
0

= = 

∫L
TdL
εLn
LL
(9)
00
0 00
ΔΔ

+
= ⇒+



LL
TLL
εLn Ln
L LL
(10)
( )
1= +
TE
εLn ε
(11)
In practice, it is noteworthy to mention that the true
stress and strain are basically indistinguishable from the
engineering stress and strain at small deformations (Figure
8). Yet, it should be noted that the true stress could be
much larger than the engineering stress once the strain
increases and the consequently, the cross sectional of the
specimen decreases.
Figure 8. Engineering Stress-Strain Curve vs. a True Stress, True Strain
Curve
2.3.4. Fracture Strength and Strain
Plastic deformation is not uniform after necking and
hence, a decrease in the stress occurs accordingly until
fracture. Calculation of the fracture strength (σ fracture)
could be done from the load at fracture divided by the
original cross-sectional area, Ao, as shown in equation 11.
The fracture strain, εf, is defined as the equivalent strain at
fracture strength. Drawing a straight line starting at the
fracture point of the stress-strain curve parallel to the
slope in the linear relation would give the fracture strain of
the specimen. The fracture strain is indicated through the
interception of the parallel line at the x axis.
0
=
fracture
fracture
P
σA
(12)
2.3.5. Tensile Ductility
The percentage of elongation or percentage reduction in
area represents the tensile ductility of the specimen as
shown in the following equations:
0
100
∆
= ×
L
% Elongation L
(13)
0
00
100 100
−∆
= ×=×
f
AA A
% RA AA
(14)
Where,
Af is the cross-sectional area of specimen at fracture.
3. Results and Discussion
The material characteristics for all the three specimens
are demonstrated in Table 3. Also, a pictorial view of the
failed specimens is demonstrated in Figure 9 and finally,
the engineering and true stress-strain curve are drawn for
the three specimens used in the tests in Figure 10.
According to the stress-strain curves used in this study,
the specimen behaves having a definite spring constant
according to the so-called Hooke’s Law (Eq.3). The
stress-strain curve is linear in this “elastic” region. The
yield point is defined as the point where the linearity ends.
In the stress-strain curve, “E” acts as the slope of the
loading line in the elastic region. As long as loading of the
metal is done within the elastic region, the strains are
totally recoverable and the specimen will return to its
original dimensions as the load is relaxed to zero. Once
the load value exceeds the corresponding yield point,
gross plastic deformation, which is permanent, will occur
to the specimen even the load is returned to zero
afterwards. According to the stress-strain curves obtained
from the tests in this study, it is evident that the low-
carbon steel specimens have a definite yield point (Figure
10). Yet, determination of the offset yield strength was
also done through drawing a parallel line to the elastic
portion of the curve, starting from the 0.2% strain level.
The comparison between the two yield points from direct
observation and the yield offset method revealed similar
results. Furthermore, the ultimate tensile strength values
of the three tested specimens in Table 3 show a 3%
difference among these values while the elongation and
fracture strain values are almost equal.

58 American Journal of Civil Engineering and Architecture
Table 3. Material Characteristics Result at the End of Test
specimen Yield
load
(KN)
Yield
strain
Yield
strength
(N/mm2)
Young
module
(N/mm2)
Maximum
load
(KN)
Ultimate
tensile
strength
(N/mm2)
%Elongation %area of
reduction Fracture
strain
1
119
0.0013
279
2.1E5
110
361
17.5 %
59 %
0.16
2
85
0.0013
282
2.1E5
106
351
18 %
60 %
0.15
3
86
0.0013
275
2.1E5
108
358
18.2 %
62%
0.15
Figure 9. Three failed specimens under tensile loading

American Journal of Civil Engineering and Architecture 59
0
100
200
300
400
500
0 0.05 0.1 0.15 0.2
Str ess ( N/mm2)
Strain (mm/mm)
Samp le 3
Thickness 6 mm
Eng ineering Stre ss-Strain
True S tress-strain
Figure 10. Engineering and True Stress-Strain Curve
4. Conclusions
The Universal Test was conducted on the three test
specimens made of low carbon steel with a thickness of
6mm, 8mm and 10mm respectively and a loading rate of 5
mm/min at the Laboratory of Structures and Materials,
Universiti Teknologi Malaysia (UTM). Based on the
results, it was concluded that the thickness of the
specimens had no effect on the properties of steel
materials mainly yield strength and Young module and
yielding and failure of such specimens occurs at the same
stress and strain values. Also, the maximum engineering
stress values appeared to be 15% lower than that of the
maximum true stress values while the maximum
engineering strain failure values are 1.5% higher than the
maximum true strain failure values.
Acknowledgements
The material required for the specimen fabrication and
financial assistance was provided by Universiti Teknologi
Malaysia (UTM).
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and Materials (ASTM)
[2] Metallic materials - Tensile testing, July 2001, BRITISH
STANDARD.
[3] Japanese Industrial Standards (JIS), J.S.A. (JSA), Editor 2005.
[4] Standard Test Methods for Mechanical Testing of Steel Products-
Metric ASTM A 1058b, 2012, DIN Deutsches Institut für Normung
e. V.
[5] Standard Test Methods for Tension Testing Wrought and Cast
Aluminum- and Magnesium-Alloy Products, A.S.f.T.a. Materials,
Editor 2010, DIN Deutsches Institut für Normung e. V.
[6] Roylance, D., Stress-strain curves. Massachusetts Institute of
Technology study, Cambridge, 2001.