9. CONVENTIONAL STRESS–STRAIN DIAGRAM.
The test results are used to plot the Stress and Strain Diagrams as follows
The nominal or engineering
stress is determined by dividing
the applied load P by the
specimen’s original cross-
sectional area A0. This calculation
assumes that the stress is
constant over the cross section
and throughout the gage length
and is given by
• The nominal or engineering
strain is found directly from the
strain gage reading, or by
dividing the change in the
specimen’s gage length, d, by the
specimen’s original gage length
L0. Thus
10. PROPORTIONAL LIMIT &
ELASTIC LIMIT
The region in light orange,
is referred to as the elastic
region. Here the curve is a
straight line up to the point
where the stress reaches
the proportional limit, σpl.
When the stress slightly
exceeds this value, the
curve bends until the stress
reaches an elastic limit.
For most materials, these
points are very close, and
therefore it becomes rather
difficult to distinguish their
exact values.
11. PROPORTIONAL LIMIT &
ELASTIC LIMIT
What makes the elastic
region unique is that after
reaching σY, if the load is
removed, the specimen will
recover its original shape. In
other words, no damage will
be done to the material
Because the curve is a
straight line up to σpl, any
increase in stress will cause a
proportional increase in
strain.
E represents the slope of the
straight line portion of the curve,
and has the same units as
Stress, [i.e., pascals (Pa),
megapascals (MPa), or
gigapascals (GPa)]
E, the proportionality constant, is
referred to as modulus of
elasticity or Young’s modulus,
12. YIELD STRESS
A slight increase in stress
above the elastic limit will
result in a breakdown of the
material and cause it to deform
permanently.
This behavior is called
yielding, and it is indicated by
the rectangular dark orange
region in Fig. 3–4.
The stress that causes yielding
is called the yield stress or
yield point, σY, and the
deformation that occurs is
called plastic deformation.
13. YIELD STRESS
In some materials such as low-
carbon steels, the yield point is
often distinguished by two
values. The upper yield point
occurs first, followed by a
sudden decrease in load-
carrying capacity to a lower
yield point.
Once the yield point is
reached, the specimen will
continue to elongate (strain)
without any increase in load.
When the material behaves in
this manner, it is often referred
to as being perfectly plastic.
14. STRAIN HARDENING &
ULTIMATE STRESS
When yielding has ended, any
load causing an increase in
stress will be supported by the
specimen, resulting in a curve
that rises continuously until it
reaches a maximum stress
referred to as the ultimate
stress, σu.
The rise in the curve in this
manner is called strain
hardening, identified as the
region in light green.
15. NECKING & FRACTURE
STRESS
This region of the curve due to
necking is indicated in dark
green. Here the stress–strain
diagram tends to curve
downward until the specimen
breaks at the fracture stress,
σf.
Up to the ultimate stress, as the
specimen elongates, its cross-
sectional area will decrease in a
fairly uniform manner over the
specimen’s entire gage length.
16. NECKING & FRACTURE
STRESS
After reaching the ultimate
stress, the cross-sectional
area will then begin to
decrease in a localized region
of the specimen, and so it is
here where the stress begins
to increase.
As a result, a constriction or
“neck” tends to form with further
elongation.
19. TRUE STRESS–STRAIN DIAGRAM.
Instead of always using the
original cross-sectional area
A0 and specimen length L0 to
calculate the (engineering)
stress and strain, we could
have used the actual cross-
sectional area A and
specimen length L at the
instant the load is measured.
The values of stress and
strain found from these
measurements are called
true stress and true strain,
and a plot of their values is
called the true stress–strain
diagram (see upper blue
curve)
The two diagrams appear
condiment when the strain is
small.
21. STRESS–STRAIN DIAGRAM-STEEL
Region Stress Value Strain Value
plastic σpl 241 MPa εpl 0.0012
yield (σY)u 262 MPa (εY)
yield (σY)L 248 MPa (εY)L 0.030 (25 times greater εpl)
ultimate stress σu 434 MPa εu
Fracture σf 324 MPa εf 0.380 (317 times greater than εpl)
22. STRESS-STRAIN BEHAVIOR-DUCTILE & BRITTLE
Ductile Materials. Any material that can be subjected to large
strains before it fractures is called a ductile material . Mild steel is a
typical example and so are most metals.
Ductility is specified by % elongation to failure or % reduction in
cross-sectional area (see illustration below from previous lecture
of this course by another instructor).
Brittle Materials. Materials that exhibit little or no yielding before
failure are referred to as brittle materials . Gray cast iron is an
example.
%AR
Ao Af
Ao
x100
%EL
Lf Lo
Lo
x100
23. STRESS-STRAIN BEHAVIOR-DUCTILE & BRITTLE
Some metals, such as
aluminium, do not exhibit
constant yielding beyond
the elastic range, thus such
metals do not have a well-
defined yield point.
To establish their yield
points a graphical
procedure called the offset
method illustrated in the
figure below
(352 MPa)
24. STRENGTH PARAMETERS
The modulus of elasticity is a mechanical property that indicates
the stiffness of a material. Materials that are very stiff, such as
steel, have large values of E (Est = 200 GPa), whereas spongy
materials such as vulcanized rubber have low values (Er=
0.69 MPa).
Values of E for commonly used engineering materials are often
tabulated in engineering codes and reference books.
Representative values are also listed in the back of the book.
E
25. STRAIN HARDENING
If a specimen of ductile material, such
as steel, is loaded into the plastic
region and then unloaded, illustrated
on the stress–strain diagram of the
figure on the right, elastic strain is
recovered as the material returns to its
equilibrium state.
The plastic strain remains, hence the
material gets a permanent set.
Here the specimen is loaded beyond
its yield point A to point A′. Since
interatomic forces have to be
overcome to elongate the specimen
elastically, then these same forces pull
the atoms back together when the
load is removed.
26. STRAIN HARDENING
Consequently, the modulus of
elasticity, E, is the same, and
therefore the slope of line O′A′ is the
same as line OA.
With the load removed, the permanent
set is OO′.
If the load is reapplied, the atoms in
the material will again be displaced
until yielding occurs at or near the
stress A′, and the stress–strain
diagram continues along the same
path as before.
But this new stress–strain diagram,
defined by O′A′B, now has a higher
yield point (A′).
The higher yield point is a
result of strain hardening,
which also cause less
ductility, or a smaller
plastic region, than when it
was in its original state.
27. STRAIN ENERGY
As a material is deformed by an external load, the load will do
external work, which in turn will be stored in the material as internal
energy, which is related to the strains in the material, and so it is
referred to as strain energy .
The energy is often specified as strain energy per unit volume of
material, also referred to as strain-energy density, and expressed
as
For elastic materials, and appling Hooke’s law applies, σ = εP, the
elastic strain-energy density becomes
28. STRAIN ENERGY: MODULUS OF RESILIENCE
If the stress σ reaches the proportional limit, the
strain-energy density, is referred to as the modulus
of resilience.
From the elastic region of the
stress–strain diagram, see figure, ur
is equivalent to the shaded
triangular area under the diagram.
Practically, the ur represents the
largest amount of internal strain
energy per unit volume the material
can absorb without causing any
permanent damage to the material.
This is important when designing
bumpers or shock absorbers.
29. STRAIN ENERGY: MODULUS OF TOUGHNESS
Another important property of a material is the modulus of
toughness, ut. This quantity represents the entire area under the
stress–strain diagram, see figure, and therefore it indicates the
maximum amount of strain-energy the material can absorb just
before it fractures. This is important when designing members that
may be accidentally overloaded.
30. STRAIN ENERGY: MODULUS OF TOUGHNESS
Alloying metals can also change their resilience and toughness.
For example, by changing the percentage of carbon in steel, the
resulting stress–strain diagrams in figure below show how the
degrees of resilience and toughness can be changed
32. END OF TODAY’S LECTURE
Go through all the examples in pages115 to 118
Attempt all the Fundamental Problems on page 119
For your tutorials next week (27/02-03/03/2023), work on
problems 3.1 to 3.12.
Prepare for today’s Test 1 on Pearson platform.
Also note that Quiz 3, is on Wednesday (01 March 2023, at
2000hrs). This is because your class has found Friday to be
inconvenient for any assessment work.
Next lecture will deal with Poisson’s ratio, the shear stress-strain
diagram, and Failure of Materials due to Creep and Fatigue to
conclude our discussion on Mechanical Properties of Materials
