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CHAPTER 6:
MECHANICAL PROPERTIES
ISSUES TO ADDRESS...
• Stress and strain: What are they and why are
they used instead of load and deformation?
• Elastic behavior: When loads are small, how much
deformation occurs? What materials deform least?
• Plastic behavior: At what point do dislocations
cause permanent deformation? What materials are
most resistant to permanent deformation?
• Toughness and ductility: What are they and how
do we measure them?
1
1
Chapter 6: Mechanical Properties of Metals
6.1 Introduction
Why Study the Mechanical Properties of Metals ?
It is important for engineers to understand
– How the various mechanical properties are measured, and
– What these properties represent
The role of structural engineers is to determine stresses and
stress distributions within members that are subjected to welldefined loads
– By experimental testing
– Theoretical and mathematical stress analysis.
Design structures/components using predetermined materials
such that unacceptable levels of deformation and/or failure will
not occur.
2
6.2 Concepts of
Stress and Strain
Static load changes
relatively slowly with
time
Applied uniformly
over a cross-section or
surface of a member.
Tension
Compression
Shear
Torsion
3
6.2 Concepts of Stress and Strain (Contd.)
TENSION TEST
Most common mechanical stress-strain test
Used to ascertain several mechanical properties that are important in design
A specimen is deformed, usually to fracture, with a gradually increasing
tensile load that is applied uniaxially along the long axis of the specimen.
A standard specimen is shown in Figure 6-2.
4
6.2 Concepts of Stress and Strain (Contd.)
The specimen is mounted by its ends
into the holding grips of the testing
apparatus (Figure 6-3).
Tensile testing machine
– To elongate the specimen at a
constant rate
– To continuously and
simultaneously measure the
instantaneous load and the
resulting extension
– Load using load cell
– Extension using extensometer
Takes few minutes and is destructive.
5
6.2 Concepts of Stress and Strain (Contd.)
Engineering Stress (s) = Instantaneous applied
load (F) / Original Area (Ao)
F
s
A0
Unit: MPa, GPa, psi
Engineering strain (e)
li l0 l
li = instantaneous length
e
l0
l0
lo = original length
COMPRESSION TESTS
Similar to tensile test, compressive load
Sign convention, compressive force is taken negative
stress negative
Since lo > li , negative strain
6
6.2 Concepts of Stress and Strain (Contd.)
SHEAR AND TORSIONAL TESTS
Shear stress : t = F / Ao
• F: Load or force imposed
parallel to the upper and
lower faces
• Ao: shear or parallel area
Shear strain (g) is defined as the
tangent of the strain angle q.
7
6.2 Concepts of Stress and Strain (Contd.)
GEOMETRIC CONSIDERATIONS OF
THE STRESS STATE
Stress is a function of orientations of the
planes
1 cos 2q
s s cos q s (
)
2
sin 2q
t s sin q cos q s (
)
2
2
8
ELASTIC DEFORMATION
1. Initial
2. Small load
3. Unload
bonds
stretch
return to
initial
F
Elastic means reversible!
2
9
ELASTIC DEFORMATION
6.3 Stress-Strain Behavior
Elastic deformation:
– Non-permanent,
completely reversible,
conservative
– Follow same loading and
unloading path
Linear elastic deformation
Hooke’s Law
– Modulus of elasticity or
Young’s Modulus
stiffness or a material’s
resistance to elastic
deformation
s Ee
10
6.3 Stress-Strain Behavior (Contd.)
11
Nonlinear Elastic
Behavior
Gray cast iron,
concrete, many
polymers
Not possible to
determine a
modulus of
elasticity
– Either tangent
or secant
modulus is
normally used.
12
6.3 Stress-Strain Behavior
(Contd.)
On an atomic scale, macroscopic
elastic strain is manifested as
small changes in the interatomic
spacing and the stretching of
interatomic bonds.
E is a measure of the
resistance to separation of
adjacent atoms
Modulus is proportional to the
slope of the interatomic forceseparation curve (Fig 2.8a) at
equilibrium spacing
dF
E
dr ro
13
6.3 Stress-Strain Behavior (Contd.)
With increasing
temperature, the modulus
of elasticity diminishes
Shear stress and strain
are proportional to each
other:
Shear
or
t Gmodulus
g
modulus of rigidity (
Table 6.1)
14
6.4 Anelasticity
Up to this point, it is assumed that
– Elastic deformation is time-independent
– An applied stress produces an instantaneous elastic strain
– Strain remains constant over the period of time the stress is maintained
– Upon release of the load, strain is totally recovered (immediately returns
to zero)
In most engineering materials, there will also exist a time-dependent elastic
strain component , i.e.
– elastic deformation will continue after stress application
– Upon load release some finite time is required for complete recovery
– Loading and unloading path are different
Anelasticity : time-dependent elastic behavior
For metals, the anelastic component is normally small and neglected.
For some polymers, it is significant and known as viscoelastic behavior
(Sec. 16.7)
15
6.5 Elastic Properties of Materials
Poisson’s ratio
ey
ex
ez
ez
E = 2G(1 + n)
Example 6.1
Example 6.2
16
PLASTIC DEFORMATION
For most metals, elastic deformation persists only to
strains of about 0.005
Plastic deformation
– Stress not proportional to strain (Hooke’s law cease
to be valid)
– Permanent
– Nonrecoverable
– Non-conservative
Transition from elastic to plastic deformation
– Gradual for most metals
– Some curvature results at the onset of plastic
deformation
17
PLASTIC DEFORMATION (METALS)
1. Initial
2. Small load
3. Unload
F
Plastic means permanent!
linear
elastic
linear
elastic
plastic
18
3
PLASTIC (PERMANENT) DEFORMATION
(at lower temperatures, T < Tmelt/3)
• Simple tension test:
19
14
Plastic deformation (Contd.)
From as atomic perspective
– Plastic deformation corresponds to the breaking of bonds
with original atom neighbors
– Reforming bonds with new neighbors
– Large number of atoms and molecules move relative to one
another
– Upon removal of stress, they do not return to their original
position
Mechanism of plastic deformation:
– Crystalline Solids:
» accomplished by a process called slip
» Involves the motion of dislocations (Sec 7.2)
– Non-crystalline solids (as well liquids)
» Occurs by a viscous flow mechanism (Sec 13.9)
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YIELD STRENGTH, sy
• Stress at which noticeable plastic deformation has
occurred.
when ep = 0.002
tensile stress, s
sy
engineering strain, e
ep = 0.002
21
15
6.6 Tensile Properties
YIELDING and YIELD STRESS
Typical stress strain behavior (Figure)
– Proportional Limit (P)
– Yielding
– Yield strength
In most cases, the position of yield
point may not be determined
precisely.
Established convention: a straight
line is constructed parallel to the
elastic portion at some specified
strain offset, usually 0.002 (0.2%)
Fig. 6.10a corresponding
intersection point gives yield
strength.
22
6.6 Tensile Properties (Contd.)
Some steels and other materials exhibit the behavior as
shown in Fig 6.10b
– The yield strength is taken as the average stress
that is associate with the lower yield point.
Magnitude of yield strength is a measure of its
resistance to plastic deformation
– Range from 35 MPa to 1400 MPa
– 35 MPa for low-strength aluminum
– 1400 MPa for high-strength steel
23
6.6 Tensile Properties (Contd.)
TENSILE STRENGTH
Tensile strength TS (MPa or psi)
is the stress at the maximum on
the engineering stress-strain curve
All deformation up to this point is
uniform.
Onset of necking at this stress at
some point all subsequent
deformation at this neck.
Range: 50 - 3000 MPa
50 MPa for aluminum
3000 MPa for high strength steel
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DUCTILITY, %EL
• Plastic tensile strain at failure:
L f Lo
%EL
x100
Lo
Adapted from Fig. 6.13,
Callister 6e.
• Another ductility measure:
Ao A f
%AR
x100
Ao
• Note: %AR and %EL are often comparable.
--Reason: crystal slip does not change material volume.
--%AR > %EL possible if internal voids form in neck.
25
19
26
Effect of Temperature
As with modulus of elasticity (E), the magnitudes of both
yield and tensile strengths decline with increasing
temperature
Ductility usually increases with temperature
Figure shown stress-strain behavior of iron
27
RESILIENCE
Resilience is the capacity of a material
to absorb energy when it is deformed
elastically and then, upon unloading, to
have this energy recovered.
Modulus of resilience (Ur)
– Associated property
– Area under the engineering stressstrain curve
– Strain energy per unit volume
required to stress from an unloaded
state to yielding
e
Mathematically,
s y2
1
U r sde s ye y
2
2E
0
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TOUGHNESS
• Energy to break a unit volume of material
• Approximate by the area under the stress-strain
curve.
Engineering
tensile
stress, s
smaller toughness (ceramics)
larger toughness
(metals, PMCs)
smaller toughnessunreinforced
polymers
Engineering tensile strain, e
29
20
TOUGHNESS
A measure of the ability of a material to absorb energy up to
fracture.
Specimen geometry and the manner of load application are
important in toughness determination:
– Notch toughness: dynamic (high strain rate) loading, specimen
with notch (or point of stress concentration) (Sec 8.6)
– Fracture toughness: property indicative of a materials resistance
to fracture when crack is present (Sec 8.5)
For static (low strain rate) condition, modulus of toughness is equal
to the total area under the stress-strain curve (up to fracture ):
For Ductile Material :
For Brittle Material:
1
U T s u e f s y ( 0.2%) s u e f
2
2
U T s ue f
3
30
6.7 True Stress and Strain
Engineering stress-strain curve
beyond maximum point (M) seems
to indicate that the material is
becoming weaker.
– Not true, rather it becomes
stronger.
Since cross-sectional area is
decreasing at the neck reduces
load bearing capacity of the
F
sT
material
Ai
True stress: Actual or current or
l
instantaneous force divided by the
li
A0
D0
dli
e
ln
ln
2
ln
l
A
instantaneous cross-sectional area. T li
0
i
Di
l
True Strain: Change in length per
Ai li A0l0
31
unit instantaneous length
i
0
6.7 True Stress and Strain (Contd.)
Relation between two
definitions
Above equations are valid
only to the onset of necking;
beyond this point true stress
and strain should be
computed from actual load,
area and gauge length.
Schematic comparison in Figure
6.16
– Corrected takes into
account complex stress
state with in neck region.
e T ln( 1 e )
s T s (1 e )
32
6.7 True Stress and Strain (Contd.)
For some metals and alloys, the true stressstrain curve is approximated as
Parameter n
– strain-hardening exponent
– A value less than unity
– Slope on log-log plot
Parameter K
– Known as strength coefficient
– True stress at unit true strain
s T Ke
n
T
33
34
6.8 Elastic Recovery During Plastic Deformation
Upon release of load,
some fraction of total
strain is recovered as
elastic strain
During unloading,
straight path parallel to
elastic loading
Reloading
– Yielding at new yield
strength
35
Solve Examples in Class
– 6.3
– 6.4
– 6.5
– 6.6
– Design Example 6.1
36