Transcript Document 7421317

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?
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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.
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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.
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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.
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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
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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.
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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
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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
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6.3 Stress-Strain Behavior (Contd.)
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
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.
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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
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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)
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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)

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6.5 Elastic Properties of Materials

Poisson’s ratio
ey
ex
 
ez
ez

E = 2G(1 + n)

Example 6.1
Example 6.2

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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
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PLASTIC DEFORMATION (METALS)
1. Initial
2. Small load
3. Unload
F
Plastic means permanent!
linear
elastic
linear
elastic
plastic

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3
PLASTIC (PERMANENT) DEFORMATION
(at lower temperatures, T < Tmelt/3)
• Simple tension test:
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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
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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.

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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
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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.
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



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
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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
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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
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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
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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 )
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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
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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
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
Solve Examples in Class
– 6.3
– 6.4
– 6.5
– 6.6
– Design Example 6.1
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