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The adaptability of a material to a particular use is determined
by its mechanical properties.Properties are affected by
Bonding typeCrystal StructureImperfectionsProcessing
Mechanical Properties
Learning Objectives Define engineering stress and engineering
strain. State Hookes law, and note the conditions under which it is
valid. Given an engineering stressstrain diagram, determine (a) the
modulus
of elasticity, (b) the yield strength (0.002 strain offset), and
(c) the tensile strength, and (d) estimate the percent elongation.
Name the two most common hardness-testing techniques; note two
differences between them. Define the differences between ductile
and brittle materials. State the principles of impact, creep and
fatigue testing. State the principles of the ductile-brittle
transition temperature.
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Types of Mechanical Testing Slow application of stress Allows
dislocations to move to equilibrium positions Tensile testing
Rapid application of stress Ability of a material to absorb
energy as it fails. Does not allow
dislocations to move to equilibrium positions. Impact
testing
Fracture Toughness How does a material respond to cracks and
flaws
Fatigue What happens when loads are cycled?
High Temperature Loads Creep
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Some DefinitionsTensile stress:Where F: force, normal to the
cross-sectional area,A0: original cross-sectional area
0AF=
Shear StressFs: force, parallel to the cross-sectional area A0:
the cross-sectional area
unit of stress: 0AFs=
2mN
areaForce =
1Pa = 1 Nm-2; 1MPa = 106Pa; 1GPa=109Pa
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Engineering StrainNominal tensile strain (Axial strain) 00
0
ll
lll ==
Engineering Shear StrainFor small strain: tan=
Poissons ratio
z
zz l
l0
=
Nominal lateral strain (transverse strain)
x
xx l
l0
=
Poissons ratio:z
x
straintensilestrainlateral
==
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Dilatation (Volume strain)Under pressure: the volume will
change
p
pp
p
V-VVV=
Elastic Behavior of Materials
Hookes Law (Linear Elasticity)
When strains are small, most of materials are linear
elastic.
ETensile: = Shear: = G Hydrostatic: p =
Youngs modulus
Shear modulus
Bulk modulus
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Modulus of Elasticity Metals
Modulus of Elasticity Ceramics
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Modulus of Elasticity - Polymers
Polymers Elastic Modulus (GPa)Polyethylene (PE) 0.2-0.7
Polystyrene (PS) 3-3.4
Nylon 2-4
Polyesters 1-5
Rubbers 0.01-0.1
Physical Basis of Youngs ModulusReview: Inter-atomic forces
(attractive and repulsive forces) dx
dUF =
Define: stiffness
002
2
0 xxxx dxdF
dxUdS == ==
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Assume the strain is small,
)(
)(
000
00
rrNSAF
rrSF
==
0
0
0
0
0
0
0
0
0
0 )( )(
rSE
ErS
rrr
rS
rrr
==
====
QYoungs modulus
Unit area
Where N: number of bonds/unit area, N=1/r02
Stiffness & Youngs Modulus for different bonds
Bonding type S0(Nm-1) E(GPa)
Ionic(i.e: NaCl) 8-24 32-96
Covalent (i.e: C-C)
50-180 200-1000
Metallic 15-75 60-300
Hydrogen 2-3 8-12
Van der Waals 0.5-1 2-4
Material E (GPa)Metals: 60 ~ 400Ceramics: 10 ~ 1000Polymers:
0.001 ~ 10
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Tensile Testing The sample is pulled slowly The sample deforms
and then fails The load and the deformation are measured
Standard tensile specimen
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The load and deformation are easily transform into engineering
stress () and engineering strain ()
A curve stress-strain is obtained
0AF=
00
0
ll
lll ==
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Parameters Obtained From Stress Strain Curvez Strength
Parameters
Modulus of Elasticity Yield Strength Ultimate Tensile Strength
Fracture Strength Fracture Energy
z Ductility Parameters Percent Elongation Percent Reduction
of
Area Strain Hardening
Parameter
Modulus of Elasticity
It is a measure of material stiffness and relates stress to
strain in the linear elastic range.
12
12
=
=E
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Yielding and Yield StrengthProportionality Limit (P): Departure
from linearity of the stress-strain curve
Yielding Point Elastic Limit: the turning point which separate
the elastic and plastic regions onset of plastic deformation
Yield strength: the stress at the yielding point.Offset yielding
(proof stress): if it is difficult to determine the yielding point,
then draw a parallel line starting from the 0.2% strain, the cross
point between the parallel line and the curve Tensile Strength
(TS)The stress increases after yielding until a maximum is reached.
It is also known as the Ultimate Tensile Strength (UTS), or Maximum
Uniform Strength.
Prior to TS, the stress in the specimen is uniformly
distributed. After TS, necking occurs with localization of the
deformation to the necking area, which will rapidly go to
failure.
- Fracture Strengthf
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Elastic RecoveryAfter a load is released from a stress-strain
test, some of the total deformation is recovered as elastic
deformation. During unloading, the curve traces a nearly identical
straight line path from the unloading point parallel to the initial
elastic portion of the curve The recovered strain is calculated as
the strain at unloading minus the strain after the load is totally
released.
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ResilienceResilience is the capacity of a material to absorb
energy when it is deformed elastically and then, upon unloading, to
have this energy recovered.
= y dUr 0Modulus of resilience UrIf it is in a linear elastic
region,
EEU yyyyyr 22
121 2 =
==
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DuctilityDuctility is a measure of the degree of plastic
deformation at fractureexpressed as percent elongation
also expressed as percent area reduction
lO and AO are the original gauge length and original
cross-section area respectivelylf and Af are length and area at
fracture
100*)(%0
0
lll f =EL
100*)(%0
0
AAA f=AR
Percentage elongation and percentage area reduction are
UNITLESS
A smaller gauge length will produce a larger overall percentage
elongation due to the contribution from necking. Therefore, the
percentage elongation should be reported with original gauge
length. Percentage reduction is not affected by sample size, thus
it is a better measure of ductility
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Typical mechanical properties for some metals and alloys
True StressTrue stress is the stress determined by the
instantaneous load acting on the instantaneous cross-sectional
areaTrue stress is related to engineering stress:Assuming material
volume remains constant
AA
AP
AA
AP
AP o
oo
oT ** ===
ll AA oo =)1(1 +=+=+==oo
o
o
o
AA
lll
ll
)1()1( +=+=o
T AP
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True StrainThe rate of instantaneous increase in the
instantaneous gauge length.
)1ln(
lnln
ln
+=
+
+=
==
T
oo
o
o
oT
Td
ll
ll
lllll
ll
True Stress-Strain Curve
= F/Ao = (li-lo/lo)
T = F/Ai T = ln(li/lo)
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Strain Hardening Parameter (n)
Strain hardening parameter 0
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Instability in TensionNecking or localized deformation begins at
maximum load, where the increase in stress due to decrease in the
cross-sectional area of the specimen becomes greater than the
increase in the load-carrying ability of the metal due to strain
hardening. This conditions of instability leading to localized
deformation is defined by the condition P = 0.
AP T=0=+= TT AAP TdA
ALL ==
ALLAV oo ==From the constancy-of-volume relationship,
T
T
AA
=
so that at the point of tensile instability
TT
T =
T
TnT
T
TnTT nKnK
=== 1 But
Instability occurs when = n
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The necking criterion can be expressed more explicitly if
engineering strain is used. ( ) T
O
o
TT
T
LL
LLLL
=+=
=== 1
//
+= 1T
T
11+
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Ductile material Significant plastic deformation and energy
absorption (toughness) before fracture.Characteristic feature of
ductile material -neckingBrittle material Little plastic
deformation or energy absorption before fracture. Characteristic
feature of brittle materials fracture surface perpendicular to the
stress.
Fracture Behavior
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SteelBefore and after fracture
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Ductile Fracture (Dislocation Mediated): Extensive plastic
deformation. Necking, formation of small cavities, enlargement of
cavities, formation of cup-and-cone. Typical fibrous structure with
dimples.
NeckingCavity Formation Cavity coalescence to form a crack,
Crack propagation Fracture
Crack grows 90o to applied stress
45O - maximum shear stress
Scanning Electron Microscopy: Fractographic studies at high
resolution. Spherical dimples correspond to micro-cavities that
initiate crack formation.
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Brittle Fracture (Limited Dislocation Mobility): very little
deformation, rapid crack propagation. Direction of crack
propagation perpendicular to applied load. Crack often propagates
by cleavage- breaking of atomic bonds along specific
crystallographic planes (cleavage planes).
Brittle fracture in
a mild steel
Intergranular fracture: Crack propagation is along grain
boundaries (grain boundaries are weakened or embrittled by
impurities segregation etc.)
Transgranular fracture: Cracks pass through grains. Fracture
surface has faceted texture because of different orientation of
cleavage planes in grains.
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Stress-Strain Behavior of Ceramics
Flexural Strength: the stress at fracture under the bending
tests. Its also called Modulus of rupture, fracture strength, or
the bending strength
3-point Bending tests
3
223
RLFbdLF
ffs
ffs
=
=
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Torsion Test Ductile material twist Brittle material
fractures
GITLP
=LGrG
==
max
max
PolarITr=max
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Impact Test (testing fracture characteristics under high strain
rates)
Notched-bar impact tests are used to measure the impact energy
(energy required to fracture a test piece under impact load), also
called notch toughness. It determines the tendency of the material
to behave in a brittle manner.Due to the non-equilibrium impact
conditions this test will detect differences between materials
which are not observable in tensile test.We can compare the
absorption energy capacity before fracture of different
materials.Two classes of specimens have been standardized for
notched-impact testing, Charpy (mainly in the US) and Izod (mainly
in the UK)
Impact Test Examples
Material Charpy Impact Strength, (Joules)
Steel 20Titanium 20
Aluminum 14Magnesium 6
Low-Grade Plastic 4
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Charpy v-notch TestA 10mm square section material is tested,
having a 45o notched, 2mm deep.
CharpyIzod
hh
Energy ~ h - h
The impact toughness is determined from finding the difference
in potential energy before and after the hammer has fractured the
material. Units are J (Joules) when testing Metals, J/cm2 when
testing polymers (Polymers will stretch, metals will snap).
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As temperature decreases a ductile material can become brittle -
ductile-to-brittle transition.FCC metals show high impact energy
values that do not change appreciably with changes in
temperature.
Ductile-to-brittle transition
BCC metals, polymers and ceramic materials show a transition
temperature, below which the material behaves in a brittle manner.
The transition temperature varies over a wide range of
temperatures. For metals and polymers is between -130 to 93oC. For
ceramics is over 530oC.
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In low alloy and plain carbon steels, the transition temperature
is set to an impact energy of 20J or to the temperature
corresponding to 50% brittle fracture.
Low temperatures can severely embrittlesteels. The Liberty
ships, produced in great numbers during the WWII were the first
all-welded ships. A significant number of ships failed by
catastrophic fracture. Fatigue cracks nucleated at the corners of
square hatches and propagated rapidly by brittle fracture.
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1912: Titanic on its maiden voyage from Southampton April 10,
1912. credit: THE BETTMANN ARCHIVE
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Charpy Samples Steel Fracture Surfaces
It shows the variation in surface fracture morphology from
brittle to ductility (shear fracture) with increasing testing
temperature (C).
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HardnessHardness: a measure of a materials resistance to
localized plastic deformation (eg. Small dent or scratch).
Hardness: Different Techniques1. Scratch hardness 2. Indentation
hardness3. Rebound hardness
Scratch Hardness Early hardness test were based nature minerals
with a scale constructed solely on the ability of one material to
scratch another (Mohs scale German Friedrich Mohs).Mohs scale
ranges from 1 on the soft end for talc to 10 for diamond.
More accurate quantitative hardness techniques have been
developed over the years in which a small indenter is forced into
the surface of the material to be tested under controlled
conditions of load and rate of application.
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Mohs Hardness Mineral Absolute Hardness1 Talc (Mg3Si4O10(OH)2)
12 Gypsum (CaSO42H2O) 33 Calcite (CaCO3) 94 Fluorite (CaF2) 215
Apatite? (Ca5(PO4)3(OH-,Cl-,F-)) 486 Orthoclase (KAlSi3O8) 727
Quartz (SiO2) 1008 Topaz (Al2SiO4(OH-,F-)2) 2009 Corundum (Al2O3)
40010 Diamond (C) 1500
Indentation HardnessResistance to permanent indentation under
static or dynamic loadsExamplesBrinell Hardness Test (ASTM E 10) -
Commonly used.Rockwell Hardness Test (ASTM E 18) - Commonly used.
Indentor and loads are smaller than with the Brinell test.Vickers
Hardness Test (ASTM E 92) - Similar to Rockwell. Uses a
square-based diamond pyramid for the indentor.Knoop (Tukon)
Hardness Test - used for very thin and/or very small specimens.
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Rebound HardnessEnergy absorbed under impact loadsExamples
Shore Scleroscope (ASTM E 448) - Measures the rebound of a small
pointed device dropped from a 254mm height.Schmidt Hammer -
Measures rebound of a spring loaded hammer. The test has been
correlated with concrete compressive strength.
The fundamental physics of hardness is not yet clearly
understood.All hardness measures are functions of interatomic
forces.There is no single measure of hardness has been devised that
is universally applicable to all materials. Hardness is arbitrarily
defined.
Hardness Some Basic Knowledge
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Brinell Hardness (BHN)A Load applied to a 10mm diameter
ball.Measure diameter of the indentation to the nearest 0.02 mm
under a microscope.Compute the Brinell Hardness Number (BHN)D =
ball diameter (mm) D = 10mmDi = indentation diameter (mm)F = load
(units = kg)
Important BHN Variables
Minimum Brinell hardness for safe testThickness of specimen (mm)
500 kg load 1,500 kg load 300 kg load
2 79 238 4764 40 119 2386 26 79 1598 20 60 11910 16 48 95
Thickness of Specimen:
- Proximity to edge or other test locations: The distance of the
center of the indentation to the edge or from the center of
adjacent indentations 2.5 times the diameter of the
indentation.Applied load:1500 kg can be used for 48
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A. Depth reached by indenter after preliminary test force (minor
load).B. Position of indenter under total test force.C. Final
position reached by indenter after elastic recovery of the
material. D. Position at which measurement is taken.
A minor load (10 kg) is applied firstA major load (60, 100, 150
kg) is applied laterHardness is determined from the difference in
penetration depthSeveral scales are used (A, B, C, etc.)The depth
of the indentation is measured by the machine.No measurement is
made by the operator other than dial reading of hardness.
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Vickers Hardness (HV)Widely used in EuropeA square base diamond
pyramid indenter is used for hard materials. The diagonals of the
square indentation are measured.
Vickers TestOpposing indenter faces are set at a 136 degree
angle to each other2
854.1DFHV =
Knoop TestLong side faces are set at a 172 degree, 30 minute
angle to each
other. Short side faces are set at a 130 degree angle to each
other
Knoop Hardness (HK)
22.14DFHK =
Pyramidal diamond shape indenter
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Correlation between Hardness and Tensile Strength
TS (MPa) = 3.45xBHN
TS (psi) = 500xBHN
Note:No method of measuring hardness uniquely indicates any
other single mechanical property.Some hardness tests seem to be
more closely associated with tensile strength, others with
ductility, etc.
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Fracture MechanicsIt studies the relationships between: material
properties stress levelcrack producing flawscrack propagation
mechanisms
Basic Concepts The measured or experimental fracture strengths
for most brittle
materials are significantly lower than those predicted by
theoretical calculations based on atomic bond energies.
This discrepancy is explained by the presence of very small,
microscopic flaws or cracks that are inherent to the material.
The flaws act as stress concentrators or stress raisers,
amplifying the stress at a given point.
This localized stress diminishes with distance away from the
crack tip.
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Stress-strain behavior (Room T):
TS
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Fracture Toughness Fracture toughness measures the resistance of
a material to brittle
fracture when a crack or flaw is present. It is a measure of the
amount of stress required to propagate a
preexisting flaw. Flaws may appear as cracks, voids,
metallurgical inclusions, weld
defects, design discontinuities, or some combination thereof.
The occurrence of flaws is not completely avoidable in the
processing, fabrication, or service of a material/component.
It is common practice to assume that flaws are present and use
the linear elastic fracture mechanics (LEFM) approach to design
critical components.
This approach uses the flaw size and features, component
geometry, loading conditions and the fracture toughness to evaluate
the ability of a component containing a flaw to resist
fracture.
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Stress-Intensity factor (K) A parameter called the
stress-intensity factor (K) is used to
determine the fracture toughness of most materials. A Roman
numeral subscript indicates the mode of fracture Mode I fracture is
the condition where the crack plane is normal
to the direction of largest tensile loading. This is the most
commonly encountered mode.
The stress intensity factor is a function of loading, crack
size, and structural geometry. The stress intensity factor may be
represented by the following equation:
KI is the fracture toughness in is the applied stress in MPa or
psia is the crack length in meters or inches Y is the component
geometry factor that is different for each specimen,
dimensionless.
aYKI =
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Critical Stress Intensity Factor or Fracture Toughness All
brittle materials contain a population of small cracks and
flaws
that have a variety of sizes, geometries and orientations. When
the magnitude of a tensile stress at the tip of one of these
flaws exceeds the value of this critical stress, the crack will
propagate. As the size of the crack increases, its SIF becomes
larger leading to failure.
Condition for crack propagation:
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K KcStress Intensity Factor:--Depends on load &
geometry.
Fracture Toughness or Critical SIF:--Depends on the
material,
temperature, environment &rate of loading.
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The value of KIc (Critical SIF) represents the fracture
toughness of the material independent of crack length, geometry or
loading system.
KIc is a material propertySpecimens of a given ductile material,
having standard proportions but different absolute size (
characterized by thickness ) give rise to different measured
fracture toughness. Fracture toughness is constant for thicknesses
exceeding some critical dimension, bo, and is referred to as the
plane strain fracture toughness, KIc.
Role of Specimen Thickness
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KIc : It is a true material property, independent of size. As
with materials' other mechanical properties, fracture toughness is
tabulated in the literature, though not so extensively as is yield
strength for example.
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Plane-Strain Fracture Toughness TestingWhen performing a
fracture toughness test, the most common test specimen
configurations are the single edge notch bend (SENB or three-point
bend), and the compact tension (CT) specimens. It is clear that an
accurate determination of the plane-strain fracture toughness
requires a specimen whose thickness exceeds some critical thickness
(B). Testing has shown that plane-strain conditions generally
prevail when:
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Compact tension (CT) specimen
single edge notch bend (SENB or three-point bend)
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Crack growth condition:
Y a Largest, most stressed cracks grow first.
--Result 1: Max flaw sizedictates design stress.
--Result 2: Design stressdictates max. flaw size.
design < KcY amax
amax < 1Kc
Ydesign
2
K Kc
amax
no fracture
fracture
amax
no fracturefracture
Design Criteria Against Crack Growth
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5555
Two designs to consider...Design A-- largest flaw is 9 mm--
failure stress = 112 MPa
Design B-- use same material-- largest flaw is 4 mm-- failure
stress = ?
Answer: MPa 168)( B =c Reducing flaw size pays off.
Material has Kc = 26 MPa-m0.5Design Example: Aircraft Wing
Use...max
cc aY
K=
( ) ( )B max Amax aa cc =9 mm112 MPa 4 mm
-- Result:
aYKI =
Fracture MechanicsFracture Toughnessc09tf01Stress-Intensity
factor (K)Critical Stress Intensity Factor or Fracture
ToughnessCompact tension (CT) specimen Design Criteria Against
Crack GrowthDesign Example: Aircraft Wing