I.
Objective
The primary objective of this lab was to analyze how several different specimens reacted under
increasing uniaxial load. Samples of Aluminum 6061-T6, Steel 4140, PS-4 Rubber and Polycarbonate were
subjected to increasing axial loads using a mechanical testing machine. In performing these experiments, we as
engineers were able to experience first-hand the derivations of materials formulas that are largely empirical. By
reviewing the respective stress vs. strain graphs, we can easily calculate the Young’s modulus, 0.2% offset
yield, and ultimate tensile strength of the material.
II.
Experimental Methods:
2.1
Aluminum 6061-T6
We performed our first test on a cylindrical rod of Aluminum 6061-T6. The tensile testing machine that
we used was called ADMAT (Advanced Materials Tester), which has a maximum tensile load of about 10,000
Newtons. The aluminum specimen was a cylindrical rod, characterized by its larger diameter for the top and
bottom portions (~12.7 mm) and a thinner centralized diameter (~6 mm). It was constructed in this way so that
deformation would take place in the center and not the outsides, where the instrument clamps would make it
difficult to measure. After we secured the specimen in the machine, we calibrated the extensometer, which
ultimately measures local deformation in the central region of the specimen. We initialized the test, and
ADMAT began applying a steadily-increasing uniaxial load. Numerical data was uploaded to a computer in the
form of a Force (N) vs. Deformation (mm) plot, and also in raw data file that can be exported into Microsoft
Excel for further review. The machine continued to apply uniaxial load until the specimen experienced failure.
This failure occurred in the narrow central section, as expected, within the region of the extensometer’s knives.
2.2
4140 Steel
We tested two different specimens of steel using the mechanical testing apparatus. The first specimen
was a cylindrical rod of steel that had been annealed by the lab technician prior to the test. This process of
annealing consisted of heating the specimen in an oven to alter its mechanical properties. The annealed
specimen had an initial central diameter of 6.15 mm. The second specimen was a cylindrical rod of as-received
steel that had not been altered prior to testing. The as-received specimen had an initial diameter of 6 mm. We
tested the annealed specimen first. We clamped the specimen into the machine, calibrated the extensometer, and
initialized. We followed a similar procedure afterwards for the piece of as-received steel. Both specimens were
subjected to increasing uniaxial loads until failure. Results of both tests were uploaded numerically and in the
form of a Force vs. Elongation graph on the computer.
2.3
PS-4 Rubber
The specimen of rubber was cut from a thin, flat polymer sheet, so instead of a circular cross-section,
it had a rectangular cross-section (3.14 mm x 1mm). The width of the specimen was greater on the top and the
bottom with a narrower central-region, so that deformation would be localized to the center. This particular test
was performed by a smaller, more accurate electronic tensile-tester (Zwick/ Roell Z 2.5) with a maximum
tensile load of about 2,500 Newtons. We measured deformation in this specimen by using a camera that was
calibrated to measure the differences in positions of markings drawn on the specimen itself. Once the test was
initialized, the machine underwent a cyclic pattern of loading/unloading, and data was uploaded to the computer
in a Force (N) vs. Elongation (mm) graph. The specimen was not tested until failure.
2.4
Polycarbonate
In order to analyze the mechanical properties of Polycarbonate, we used a cylindrical specimen with a
similar shape to that of the aluminum and steel specimens. The cross-sectional area at both ends was greater
than the area of the center so that deformation would be constrained to the narrower central region. The initial
diameter of this central region was approximately ~9mm. We clamped the specimen into place on the tensile
testing apparatus and calibrated the extensometer. We placed the extensometer towards the very center where
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the specimen was likely to deform, and the test was initialized. The machine applied a steadily increasing
uniaxial load, but the rate at which load increased was much slower so that the specimen would experience
maximum deformation. Numerical data of the test was uploaded to a computer, as well as a graph of Force vs.
Elongation of the specimen. Uniaxial load continued to increase steadily until the specimen experienced failure.
III.
Results
Lab Data Conversions
As stated earlier, Lab data from each test was uploaded in the form of a Force (N) vs. Elongation (mm)
graph. In order to obtain Engineering Stress vs. Engineering Strain relations, the following equations were used:
Load
 engineering  fffffffffffffffff
A
[1]
L @L0
 engineering  ffffffffffffffffffff
L0
[2]
In addition, True Stress vs. True Strain relations can be derived from known properties (as well as
Engineering Stress and Engineering Strain) using the following relationships:
 true 
b
c
Pfffff ffffffff
P fffff
l
 A   eng 1   eng
A A0 l 0
L
 true  R d Z
L0
f
[3]
g
b
c
dL
L
ffffffff
fffffff
 ln
 ln 1   eng
L
L0
[4]
True Stress and True Strain differ from their Engineering counterparts in that True Stress and Strain take into
account the instantaneous cross-section and length, as opposed to the undeformed cross-section and length.
Formulas [3] and [4] assume that volume stays constant. Graphs of Engineering Stress vs. Engineering Strain, as
well as True Stress and True Strain, for each material are attached to the end of this lab report and will be
referenced throughout.
3.1
Aluminum 6160 T-6
As is evident in Figure A-1, the aluminum specimen initially deformed linearly with respect to the
applied force. This occurred because the material itself was in the elastic regime, and molecular bonds remained
intact but continued to flex under increasing uniaxial tension. If the load was suddenly removed while the
material was still in this regime, the specimen would have returned back to its original length. However, around
9000 N, Figure A-1 shows that the trend is no longer linear, and bonds are beginning to separate at the
molecular level. The specimen begins to visibly deform (as made evident by the formation of a localized neck)
and weakens under increasing load. The near vertical drop that Figure A-1 shows around 10,000 N is the
machine unloading and allowing the specimen to equilibrate. A similar trend can be noted in Figure A-2, which
displays the Engineering Stress vs. Engineering Strain of the material. As the machine unloads, the specimen
returns along the same linear slope under which it deformed initially. This is because the Young’s Modulus
(measure of the stiffness of a material) is a property of the material itself and does not change when the
specimen deforms. Young’s Modulus, represented by the gradient of the linear portion of Figure A-2, can be
calculated using the following formula:
F* A 0
 engineering
FL0
E  fffffffffffffffffffffffffffff  fffffffffffffffffff  fffffffffffffffffffff
 engineering L* L0 A0 L0
2
[5]
Once load is re-applied to the specimen, Figure A-2 shows that a maximum occurs. This maximum, which will
be discussed further in the Discussion subsection, is the Ultimate Tensile Strength, or UTS , of the material:
F max
UTS  fffffffffffff
A0
[6]
This is the maximum stress that the material can sustain. After this is reached, the material begins to elongate
more at decreasing load until, ultimately, it fails. This failure occurs at a deformation of about ~2.95 mm in
Figure A-1, and at an engineering strain of about ~.116 in Figure A-2.
3.2
As-Received 4140 Steel
Results of the Force vs. Elongation relation for the As-Received Steel test can be seen in Figure B-1.
The specimen elongates linearly until about 27 kN of Force, when it enters the plastic regime. After this point,
the specimen will no longer return to its original length if the load is taken away. It is also after this point that
the specimen experiences elongation under (relatively) constant loading conditions. Notice that around 1.5 mm
of elongation (around 0.12 on the extensometer data set on Figure B-2), the graph follows a downward line with
the same slope as the initial linear slope experienced by the specimen in its elastic regime. This is a result of the
machine unloading. The specimen unloads and equilibrates with respect to its Young’s Modulus [5]. If the
specimen had been unloaded completely, the line would have intersected the abscissa at around 1.4 mm,
signifying that the specimen had been permanently elongated by 1.4 mm. Instead, load was reapplied until it
reached the constant load that it had been applying before, and the specimen undergoes continual tensile stress
until it ultimately fails.
3.3
Annealed 4140 Steel
Close investigation of Figure C-1, which represents the relationship between Force and Elongation for
the annealed steel specimen, reveals surprising results. Even though it is made from the same material as asreceived steel, it behaves differently once it begins to plastically deform. At around 15 kN, the specimen exits
the plastic regime, and we see that it goes through a period of parabolic deformation until it begins to deform
under constant load (slope = 0). This is largely due to the annealing process, which will be discussed more indepth in the Discussion subsection. Figure C-1 and Figure C-2 both document an unloading/reloading cycle
shortly after the specimen begins to plastically deform, and as expected, it traces back a line with the same slope
as its Young’s Modulus [5] in Figure C-2. As stated earlier, the specimen eventually begins to deform with near
constant loading conditions, reaches UTS [6], deforms under decreasing uniaxial load, and finally fails.
3.4
PS-4 Rubber
Rubber is a polymer, so it reacted very differently to increasing uniaxial load than the metals that were
tested. This is because polymers do not exhibit crystallization or structure at the molecular level. Instead, most
polymers are composed of Carbon backbones that are very flexible and, due to this flexibility, are able to fold in
on each-other and exhibit greater bond forces. This flexibility becomes evident in Figure D-1, which is a graph
of the Force (N) vs. Elongation (mm) of the PS-4 rubber specimen. The trend is linear for the most part. What is
particularly unique about this specimen is that it never exits the elastic regime, despite the fact that it
experiments a large deformation. Instead, it exhibits linear behavior in both Force vs. Deformation and
Engineering Stress vs. Engineering Strain (Figure D-2) as the mechanical device cycles through loading and
unloading. This linear behavior is a result of the structural properties of polymers. In this particular test, the
material was not allowed to fail, and accordingly, no UTS [6] was recorded.
3.5
Polycarbonate
Polycarbonate is also an example of a polymer, so its results to uniaxial tension testing are unique as
well. According to Figure E-1, we see that the specimen begins in the elastic regime, represented by the
linearity of the Force/Elongation graph. It obeys Hooke’s law in this regime, as the gradient of the slope remains
constant in Figure E-2. The specimen began to yield as the graph in Figure E-2 reaches non-linearity. The
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specimen began to form a localized neck during the time period represented by the initial downward slope after
yield. As the slope begins to increase, the localized neck propagates outward from the center and the material
itself begins to strain harden, represented by the gradually increasing slop in Figure E-2. This property sets
polycarbonate apart from the metal specimens, as aluminum and steel both failed shortly after the appearance of
a localized neck and do not experience strain hardening after UTS [6] was reached. This phenomenon occurs
because the flexible carbon chains that make up polycarbonate re-orient themselves as strain increases and are
able to establish stable bonds as a result of this orientation. Eventually, the strain becomes to great and carbon
chains can no longer re-orient themselves, resulting in fracture at the molecular level and overall failure of the
material.
IV.
Discussion
4.1
6061-T6 Aluminum
According to Figure A-2, which represents the Engineering Stress [1] vs. Strain [2] curve for the
aluminum specimen, Young’s Modulus [5] comes in at around 68 GPa. This is relatively similar to recorded
values, which are usually between 68 and 71 GPa. The Young’s Modulus for this specimen was deduced by
fitting a line to the linear portion of Figure A-2 and recording the slope**. At around 300 MPa on Figure A-2,
the specimen enters the plastic regime. Shortly thereafter, the line representing 0.2% Offset Yield intersects with
p
the plot. This intersection is the 0.2% Offset Yield Strength (  Y ) which is where a tensile plastic strain (  eng
) of
0.2% is achieved. For the aluminum specimen, the value for  y is around 300 MPa. The maximum depicted in
Figure A-2 (where the slope of the data set is zero) is representative of the UTS [6] of the specimen, which in
this case, is around 349 MPa. Figure A-3 depicts the True Stress vs. True Strain of the aluminum specimen test,
and allows us to visualize the instantaneous stress/strain relationship of the specimen.
The reduction in area of a material often provides information on how ductile that material is. Reduction
of area can be found using the following relationship:
h
i
A0 @A f
q percent j fffffffffffffffffffffffffk C100
Af
`
a
[7]
In the case of aluminum, the percent area reduction is ~52% ( A0 = 29.706 mm2 and A f = 13.9 mm2). Because
this percent area reduction is pretty high, we can classify the material as a relatively ductile material.
4.2
As-Received Steel
Figure B-2 depicts the Engineering Stress vs. Engineering Strain curve for two data sets: extensometer
data and crosshead data. The Young’s Modulus [5] for this annealed steel specimen was roughly 203 GPa,
which is relatively similar to recorded values of 4140 steel (200-220 GPa). The Young’s Modulus was deduced
in a similar manner to that of aluminum, by fitting a line to the linear portion of the Figure B-2 and recording
the slope. For both data sets, we can see the specimen leaving the elastic regime at around 950 MPa of stress.
Shortly thereafter, in relation to the extensometer data set, the line representing 0.2% Offset Yield intersects with
the plot, giving a  y of around 971 MPa. Both data sets reach a maximum value (zero slope) at around 987 MPa,
and this is the value for UTS . We deduce that the percent area reduction [7] is about 58% ( A0 = 28.27 mm2 and
A f = 11.8 mm2). Because the percent reduction is greater than 50%, the material is more ductile than it is
brittle. Figure B-3 depicts the True Stress [3] vs. True Strain [6] of the as-received steel specimen test.
4.3
Annealed Steel
Extensometer and Crosshead data sets for Engineering Stress vs. Engineering Strain for Annealed Steel
are depicted in Figure C-2. The slope of the linearity for Figure C-2 is roughly 225 GPa, which coincides with
**
Fitted lines for the Young’s Modulus of each material can be found in the appendices (Figure Y-#).
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literature values for the Young’s Modulus of steel. The material begins to plastically deform at around 515 MPa
of uniaxial stress, which is about 54% of the as-received specimen. In this case, the 0.2% Offset Yield was  Y =
470 MPa, which is about 48% of the value for  Y for as-received steel. Referring to Figure C-2, the ultimate
tensile strength is UTS = 766 MPa. Although this number is still less than UTS for as-received steel, it is much
greater than 50%. In fact, UTS for annealed steel is about 78% of UTS for as-received steel. This goes to show
that, although annealed steel was weaker over all, it was much more flexible than as-received steel. Once
annealed steel left the elastic regime, it had a greater coefficient of ductility than its as-received counterpart. The
percent area reduction of annealed steal is ~60% ( A0 = 28.27 mm2 and A f = 11.8 mm2). With this in mind, we
can conclude that the ductility of annealed steel is slightly higher than as-received steel. Figure C-3 represents
the True Stress/True Strain relationship of annealed steel.
4.4
PS-4 Rubber
A graph representing the Engineering Stress vs. Engineering Strain of the PS-4 Rubber specimen can be
found on Figure D-2. The Young’s Modulus of this material is about 4.18 MPa, which coincides with recorded
values of rubber (usually around 4 MPa). The specimen never left the elastic regime, and did not fail, so values
for  Y , UTS , and percent area reduction cannot be deduced from this test. Because rubber is a polymer and
undergoes gratuitous strain before failure, we can calculate the stress at 100% engineering strain of the material,
simply by fitting a line on Figure D-2 and recording the value for  eng at 100% strain. In this case, that value is
about 3.32 MPa. A graph for True Stress vs. True Strain for rubber can be found on Figure D-3.
4.5
Polycarbonate
Figure E-2 depicts the  eng vs.  eng for polycarbonate. Inspection of the gradient of the first linear
slope on the graph returns a Young’s Modulus of 2.25 GPa (recorded values: between 2-2.4 GPa). The 0.2%
Offset Yield for this specimen is 28 MPa, and the maximum UTS of the material is 59 MPa (represented by the
hump in the graph created by the transition from a positive to a negative slope). The percent area reduction for
the polycarbonate specimen is 64%, where A0 = 63.6 mm2 and A f = 23.3 mm2. As expected, polycarbonate is
very ductile because it is a polymer. Because the specimen was strained beyond 100%, we can conclude that
 eng at 100% strain is 46.4 MPa. A graph of True Stress vs. True Strain for polycarbonate is available on
Figure E-3.
4.6
Table of Properties
Material
6160-T6 Aluminum
As-Received 4140 Steel
Annealed 4140 Steel
PS4 Rubber
Polycarbonate
V.
Young’s
Modulus (MPa)
68,019
203,452
224,366
4.198
2,253
0.2% Offset Yield
Strength (MPa)
300
917
470
N/A
28
Ultimate Tensile
Strength (MPa)
349
987
766
N/A
59
Percent Area
Reduction at Fracture
52%
58%
60%
N/A
64%
Conclusion
The primary objective to this lab was to explore the derivations of empirical formulas that are
widely used in materials engineering. By studying the reactions of various material specimens under
uniaxial tension, we can derive valuable information on their structural and mechanical makeup.
There’s little use in plugging and chugging certain variables in mechanical engineering problems
without knowing how they were derived. This lab helped us as engineers discover how certain
mechanical properties came to be.
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