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 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

unit of stress: 0AFs=τ

2mN

areaForce

=1Pa = 1 Nm-2; 1MPa = 106Pa; 1GPa=109Pa

Engineering StrainNominal tensile strain (Axial strain) 00

0

ll

lll Δ

=−

=ε

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

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 yyyyyr 22

121 2σσσεσ =⎟⎟

⎠

⎞⎜⎜⎝

⎛==

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

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 Td

AA

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 ε

σε

δεδσ

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.

Mohs Hardness Mineral Absolute Hardness1 Talc (Mg3Si4O10(OH)2) 12 Gypsum (CaSO4·2H2O) 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 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.

http://www.wikipedia.org/wiki/Talchttp://www.wikipedia.org/wiki/Gypsumhttp://www.wikipedia.org/wiki/Calcitehttp://www.wikipedia.org/wiki/Fluoritehttp://www.wikipedia.org/w/wiki.phtml?title=Apatite&action=edithttp://www.wikipedia.org/wiki/Orthoclasehttp://www.wikipedia.org/wiki/Quartzhttp://www.wikipedia.org/wiki/Topazhttp://www.wikipedia.org/wiki/Corundumhttp://www.wikipedia.org/wiki/Diamond

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

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

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.

• Stress-strain behavior (Room T):

TS

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

• Use...max

cc aY

Kπ

=σ

( ) ( )B max Amax aa cc σ=σ9 mm112 MPa 4 mm

-- Result:

πaYσKI =

Fracture MechanicsFracture Toughnessc09tf01Stress-Intensity factor (K)Critical Stress Intensity Factor or Fracture ToughnessCompact tension (CT) specimen Design Criteria Against Crack GrowthDesign Example: Aircraft Wing