The adaptability of a material to a particular use is determined by its mechanical properties. Properties are affected by Bonding type Crystal Structure Imperfections Processing Mechanical Properties Learning Objectives Define engineering stress and engineering strain. State Hooke’s law, and note the conditions under which it is valid. Given an engineering stress–strain 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.
Transcript
The adaptability of a material to a particular use is determined by its mechanical properties.Properties are affected by
Learning ObjectivesDefine engineering stress and engineering strain.State Hooke’s law, and note the conditions under which it is valid.Given an engineering stress–strain 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.
Types of Mechanical TestingSlow application of stress
Allows dislocations to move to equilibrium positionsTensile testing
Rapid application of stressAbility of a material to absorb energy as it fails. Does not allow dislocations to move to equilibrium positions.Impact testing
Fracture ToughnessHow does a material respond to cracks and flaws
FatigueWhat happens when loads are cycled?
High Temperature LoadsCreep
Some DefinitionsTensile stress:Where F: force, normal to the cross-sectional area,A0: original cross-sectional area
0AF
=σ
Shear StressFs: force, parallel to the cross-sectional area A0: the cross-sectional area
Engineering Shear StrainFor small strain: θγ tan= θγ ≅
Poisson’s ratio
z
zz l
l
0
Δ=ε
Nominal lateral strain (transverse strain)
x
xx l
l
0
Δ−=ε
Poisson’s ratio:z
x
straintensilestrainlateral
εεν −=−=
Dilatation (Volume strain)Under pressure: the volume will change
p
pp
p
V-ΔV
VVΔ
=Δ
Elastic Behavior of Materials
Hooke’s Law (Linear Elasticity)
When strains are small, most of materials are linear elastic.
σ
ε
ETensile: σ = Ε ε
Shear: τ = G γ
Hydrostatic: – p = κ Δ
Young’s modulus
Shear modulus
Bulk modulus
Modulus of Elasticity Metals
Modulus of Elasticity Ceramics
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 Young’s ModulusReview: Inter-atomic forces (attractive and repulsive forces) dx
dUF =
Define: stiffness
002
2
0 xxxx dxdF
dxUdS == ==
Assume the strain is small,
)(
)(
000
00
rrNSAF
rrSF
−==
−≈
σ
0
0
0
0
0
0
0
0
0
0 )( )(
rSE
ErS
rrr
rS
rrr
==
==−
=−
=
εσ
εεσεQYoung’s modulus
σ σ
Unit area
Where N: number of bonds/unit area, N=1/r02
Stiffness & Young’s 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
Tensile Testing• The sample is pulled slowly• The sample deforms and then fails• The load and the deformation are measured
Standard tensile specimen
• The load and deformation are easily transform into engineering stress (σ) and engineering strain (ε)
• A curve stress-strain is obtained
0AF
=σ00
0
ll
lll Δ
=−
=ε
Parameters Obtained From Stress Strain CurveStrength Parameters
– Modulus of Elasticity– Yield Strength– Ultimate Tensile Strength– Fracture Strength– Fracture Energy
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
ε−εσ−σ
=δ εδ σ
=E
Yielding and Yield Strength•Proportionality 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 σ−ε 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 Strengthσf<<σUTS Due to the definition of Engineering stress and tensile specimen necking.
o
ff AP
=σ
Fracture Energy (Toughness)Is a measure of the work required to cause the material to fracture. Is a function of strength and ductility. Its magnitude is defined by the area under the stress strain curve
Approximated by:f
UTSysG εσ+σ
= *2
∫= f dUε
εσ0
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.
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
εεσ
0Modulus of resilience Ur
If it is in a linear elastic region,
EEU yy
yyyr 221
21 2σσ
σεσ =⎟⎟⎠
⎞⎜⎜⎝
⎛==
DuctilityDuctility is a measure of the degree of plastic deformation at fracture–expressed as percent elongation
–also expressed as percent area reduction
–lO and AO are the original gauge length and original cross-section area respectively–lf 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
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 ** ===σ
ll AA oo =)1(1 ε+=+
δ=
+δ==
oo
o
o
o
AA
ll
l
l
l
)1()1( ε+σ=ε+=σo
T AP
True StrainThe rate of instantaneous increase in the instantaneous gauge length.
)1ln(
lnln
ln
εε
ε
ε
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ+⇒⎟⎟
⎠
⎞⎜⎜⎝
⎛ Δ+=
⎟⎠⎞
⎜⎝⎛ Δ
== ∫
T
oo
o
o
oT
Td
l
l
l
l
l
ll
l
l
l
l
True Stress-Strain Curve
σ = F/Ao ε = (li-lo/lo)
σT = F/Ai εT = ln(li/lo)
Strain Hardening Parameter (n)
Strain hardening parameter 0<n<1
T
T
T
T
ddn
εσ
εσ
=
nTT Kεσ =
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 δP = 0.
AP Tσ=
0=+= TT AAP δσδσδ TdAA
LL εδδ
=−=
ALLAV oo ==From the constancy-of-volume relationship,
T
T
AA
σδσδ
=−
so that at the point of tensile instability
TT
T σδεδσ
=T
TnT
T
TnTT nKnK
εσ
εδεδσ
εσ === −1 But
Instability occurs when εΤ = n
The necking criterion can be expressed more explicitly if engineering strain is used.
( ) TO
o
TT
T
LL
LLLL
σεδεδσ
δεδσ
δδ
δεδσ
δεδε
δεδσ
δεδσ
=+=⎟⎟⎠
⎞⎜⎜⎝
⎛=== 1
//
εδεδ
+=
1Tσσ
σT
ε11+ε
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
SteelBefore and after fracture
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”.
Necking→Cavity 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.
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.
Stress-Strain Behavior of Ceramics
Flexural Strength: the stress at fracture under the bending tests. It’s also called Modulus of rupture, fracture strength, or the bending strength
3-point Bending tests
3
223
RLFbdLF
ffs
ffs
πσ
σ
=
=
Torsion Test• Ductile material twist• Brittle material fractures
GITL
P
=φ
LGrG
φτ
γτ
=
=
max
max
PolarITr
=maxτ
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
Charpy v-notch TestA 10mm square section material is tested, having a 45o notched, 2mm deep.
CharpyIzod
h’h
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).
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.
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.
1912: Titanic on its maiden voyage from Southampton April 10, 1912. credit: THE BETTMANN ARCHIVE
Charpy Samples – Steel Fracture Surfaces
It shows the variation in surface fracture morphology from brittle to ductility (shear fracture) with increasing testing temperature (˚C).
HardnessHardness: a measure of a material’s 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.
Indentation Hardness•Resistance to permanent indentation under static or dynamic loads•ExamplesBrinell 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.
Rebound Hardness•Energy absorbed under impact loads•Examples
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
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 = 10mm–Di = 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
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<BHN<300–1000 kg can be used for 32<BHN<200–750 kg can be used for 24<BHN<150–500 kg can be used for 16<BHN<100
Rockwell Hardness (HR)
Widely used in the USA diamond cone shape indenter is used for hard metals or hard spherical steel ball for softer materials.Different combinations of loads and indenter (Rockwell scale).
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.
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
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.
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.
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.
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 σ 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 πσ=
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:
49
K ≥ KcStress Intensity Factor:--Depends on load & geometry.
Fracture Toughness or Critical SIF:--Depends on the material,
temperature, environment &rate of loading.
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
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.
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:
Compact tension (CT) specimen
single edge notch bend (SENB or three-point bend)
• Crack growth condition:
Yσ πa• Largest, most stressed cracks grow first.
--Result 1: Max flaw sizedictates design stress.
--Result 2: Design stressdictates max. flaw size.
σdesign <
KcY πamax
amax <1π
KcYσdesign
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
K ≥ Kc
amax
σ
no fracture
fracture
amax
σno fracture
fracture
Design Criteria Against Crack Growth
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 =σc• Reducing flaw size pays off.
• Material has Kc = 26 MPa-m0.5Design Example: Aircraft Wing
