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ME 330 Engineering MaterialsLectures 2-3

Tensile Properties• Elastic properties• Yield-point behavior• Plastic deformation• True vs. Engineering stress • Stress-strain curves• Fracture surfaces• Hardness Testing

Please read chapters 1 (Lecture 1) & 6 (Lecture 2)Please read chapters 1 (Lecture 1) & 6 (Lecture 2)

Where We Are Going...

• Engineers design products to carry loads, transmit forces, etc.

• Characterize a material’s behavior through properties– Measure properties in lab test … extrapolate behavior to

different scenario– Alternative is proof testing everything!

• Basic mechanical testing– Look for response to applied forces

• Apply load, measure deformation• Indent surface, measure hardness

– Quantify words like “strong”, “ductile”, “hard”, etc

Basic Mechanical TestsTension

Most common mechanical testGage section reduced to ensure deflection hereLoad cell measures applied load Extensometer ensures l measured from gage region

CompressionSimilar to tensile testGood for brittle specimens … hard to gripOften much different properties in compression

TorsionTest of pure shearMember twisted by angle , calculate shear strainMeasure applied torque, calculate shear stress

Bending

In all cases, a displacement is applied and you measure loadCalculate stress from measured loadCalculate strain from change in gage length

Tension Test

Measure load and displacement

Compute stress and strain

Review of Stress and Strain

• Stress: force per unit area

• Traditional units: MPa or ksi• Ao is original area• A is instantaneous area

• Strain: “relative” change in length

• Dimensionless quantity• Lo is original length (“gage length”)• L is instantaneous length

Often interested in measuring force and deformation in a size independent manner

AreaForce

LengthLength

oAF:gEngineerin

o

oLLL:gEngineerin

Ao

Lo

: TFTrueA

oT L

Lln:True A

LFrom dT=dL/L

Relation Between Stress & Strain

Tension (+) Compression (-)

Typical Stress-Strain Curves

0.1 10 100

ceramics

metals

polymers

(M

Pa)

(%)

Stre

ngth

Ductility

Stiffness

Energy Absorption

Elastic Plastic

Yield Today, we’ll talk about the different: Regions in stress-strain spaceProperties important to design

Elastic Region & Properties

~0.1 10 100

ceramics

metals

polymers

(M

Pa)

(%)

Stiffness

ElasticElastic region: proportional stress and strainStiffness = Modulus of Elasticity ductility

Elastic Material BehaviorS

tress

(MP

a)

Strain (%)

Stre

ss (M

Pa)

Strain (%)

Linear Non-linear

1

2

secant modulus @ 1

tangent modulus @ 2E

Elastic region: strain returns to zero when stress removedElastic Modulus (E) - measure of stiffness

Elastic BehaviorS

tress

(MP

a)

Strain (%)

Stre

ss (M

Pa)

Strain (%)

linear non-linear

E Secant Modulus

Tangent Modulus

Atomic Level Effects on Modulus

F

F

Many metals

Most ceramics

F F Most polymers

• Strength of interatomic bonds: stiffness of springs• Atomic packing: springs per unit area

Atomistic Origins of ElasticityForce

Atomic separation, r

Force

r

oo rr2

2

rr drd

drdFE

Strong bonding,stiff

Weak bonding,compliant

ro Energy(r)dr

d)r(F

Final Notes on Stiffness• Interatomic bonding

– Ceramics - Ionic & Covalent– Metals - Metallic & Covalent– Polymers - Covalent &

Secondary• Packing

– Ceramics & Metals • Highly ordered crystals• Dense packing

– Polymers• Randomly oriented chains• Loosely packed

• Temperature effects– Effect depends on types of

bonds– As temperature increases,

modulus decreases

Material E (GPa)

Silicon Carbide 475Ceramics Alumina 375

Glass 70

Steel 210Metals Brass 97

Aluminum 69

PVC 3.3

Polymers Epoxy 2.4LDPE 0.23

(M

Pa)

(%)

Ceramics

Metals

E

Polymers

Elastic Constitutive Relationfor 1-D Tensile Loading (linear materials)

• Hooke’s Law: Stress and strain are directly related by modulus of elasticity,

• Poisson’s ratio: Strain perpendicular to applied load is related to the axial strain,

– Maximum (constant volume) : = 0.50– Minimum: = 0– Look at change in volume in a cube of side length, L

– Volume increases during tensile, elastic deformation (if 0.50)

E

z

y

z

x

z

x

0 0 0{ (1 )} { (1 )} { (1 )}xx yy zzLxLxL L x L x L 2

0 0 0 0 0{ (1 )} { (1 )} { (1 )} { (1 )} { (1 )}zz zz zz zz zzL x L x L L x L 3 2 2 30{1 (1 2 ) ( 2) }zz zz zzL 30{1 (1 2 ) }zzL

Elastic Behavior

Elastic Modulus

Elastic Modulus

Poisson’s Ratio

12

12

E

z

y

z

x

allongitudin

transverse

12GE

E

0rrdrdFE

Axial

Shear G

for isotropic material

Elastic +Plastic Properties

0.1 10 100

ceramics

metals

polymers

Stre

ngth

Ductility

(M

Pa)

(%)

Stiffness

Energy Absorption

Elastic Plastic

Yield

Elastic UnloadingS

tress

(MP

a)

Strain (%)plastic elastic

total strain = elastic + plastic

Stress – always elastic, no concept of plastic stress

p

pe

E

E E

Review Stress and StrainEngineering Stress

Engineering Strain

True Strain

True Stress

oAF

1AF

oo

o

LL

LLL

1lnlnln

AA

LL o

oT

Constant Volume 00LAAL

Lo

do

Ao~ L

F

d

A~

Modeling Plastic Deformation:True Stress and Strain

• True stress-strain values for plasticity … takes into account large area changes during plastic deformation

• Can relate true values to engineering values– Valid only for constant plastic deformation– Assuming constant volume, ,

)1ln()L/Lln( oT

L*AL*A oo

AAo

L Lo

* / * oo

o

L LL L

L

)1(*T

o

oT AA

APAP

oL*AL*A

o

o

oL/L*1

L/LL/L*

AAoo

Elastic Constitutive Relationfor Simple Shear

F

F

F

F

Again, stress and strain are directly related, by shear modulus, GG: G

For isotropic materials, shear and elastic modulus are related by: 1G2E

Shear stress:oAF

Ao

Shear strain:

)tan(

Stress & Strain in 3-Dimensions

x

z

yx

zx

z

yxy

zy

yx

yz

xz

x

y

z

xy

xz

yx

yz

zx

zy

x

y

z

xy

xz

yx

yz

zx

zy

Need to relate stress to strain

klijklij C

Originally 9 independent components Cijkl has 81 constants!!Equilibrium indicates ij = ji 6 components 36 constants (most general anisotropic matl)Elastic strain is reversible, so Ci j= Cji 21 constantsBased on crystal symmetry, for cubic crystals 3 constantsFor an isotropic crystal, need only 2 constants to describe 3-D responseRelate 1-D tests to complex loading

1 0 0 0

1 0 0 0

1 0 0 0

2(1 )0 0 0 0 0

2(1 )0 0 0 0 0

2(1 )0 0 0 0 0

x x

y y

z z

xy xy

yz yz

xz xz

E E E

E E E

E E E

E

E

E

3-Dimensional Elastic Stress State

1 0 0 0

1 0 0 0

1 0 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

xy xz

x y z

x xyx yz

y yx y z

z zzyzx

xy xyx y z

yz yz

xyxz xz

yz

xz

E E E

E E E

E E E

GG

G

Isotropic Material

Orthotropic Material

Yield Point

~0.1 10 100

ceramics

metals

polymers

(M

Pa)

(%)

Stiffness

Elastic

Yield Yield point marks the transition from elastic to plastic deformation

(M

Pa)

(%)0.1

y

(M

Pa) (%)0.2

0.2%y

(M

Pa)

(%)

ly

uy

Yield Point Behavior

• Proportional limit marks the end of linearity• Yield point marks the beginning of plastic deformation

– Some materials show an obvious transition, y

– Often need to define 0.2% offset yield, 0.2%y

– Sometime see an upper (uy) and lower (ly) yield stresses occur

• Caused by significant dislocation-solute interaction

• Common in BCC iron based alloys

Plastic Region

~0.1 10 100

ceramics

metals

polymers

(M

Pa)

(%)

Stiffness

Elastic Plastic

Yield Stress is no longer proportional to strainPlastic deformation is permanent, non-recoverable

plastic elastic

E E

p

pe

E

No concept of “plastic stress”

Upon unloading, strain is partitioned between recovered and permanent.

Plastic Phenomena

Uniformdeformation

Necking begins:

Localizeddeformation

0dd

(MPa

)

(%)

y2

y1

Upon reloading, stress-strain curvefollows the same path to failure.

Plastic Phenomena

(MPa

)

(%)

True vs. Engineering - Curve

(M

Pa)

(%)

• Decreasing area in plastic regime higher “true” stresses• Once a neck forms,

– Equations are invalid– True curve overpredicts actual stress due to triaxial stress

state

Engineering

True

True vs. Engineering - CurveCompression

Plastic Constitutive Response

• Can approximate relation between true stress-strain curve in constant plastic deformation region by:

– K is the strength parameter– n is the strain-hardening exponent

• 0 n 1• if n = 0, elastic-perfectly plastic response• if n = 1, ideally elastic material• as n increases, achieve more strain hardening

– Typically valid only for some metals and alloys– Termed “power law hardening”

nTT K

(M

Pa)

(%)

Measures of Energy Absorption: Toughness vs. Resilience

(M

Pa)

(%)

Resilience: Ability to absorb energy without

permanent deformation - (elastic only)

Toughness: Total energy absorption capability

of a material - (elastic + plastic)

•Units: Energy per unit volume•Define: Energy stored during deformation •Graphically: Area under - curve

Stress-Strain Properties (cont.)

yyr

y

dU

21

0 Modulus of Resilience

nTT K Stress vs. Strain Eq. uTy for

EEU yy

yr 221 2

Measures of Strength

(MPa

)

(%)

f

Fracture stress, f

0.2%

0.2%y

0.2% offset yield strength, 0.2%y

UTS

Ultimate Tensile Stress, UTS

f

Fracture strain, f (~Ductility)

Measures of Ductility

Percent Elongation: Sensitive to gage length Does not account for necking

100*LLLEL%o

o

Lo L

Area Reduction: Insensitive to gage length Does account for necking Sensitive to cross-section

100*A

AAAR%o

o

AAo

Stress-Strain Properties

% Elongation

% Reduction in Area

100xL

LLEL%o

o

100xA

AARA%o

o

Yield Strength y 0.2% offset or lower yield point

UTS u Highest stress on curve

Proportional limit = highest linear stress

Material Deformation & Fracture

From Callister, p.126

Brittle•Cleavage failure•Flat,rough fracture surface•No necking•Failure in tension

•Ductile•Completely ductile failure necks to a point•Cup-cone fracture surface

•Necking prior fracture•Cavities initiate in neck•Voids coalesce to form crack•Final failure in shear

•Discuss more completely in fracture

Fracture Surfaces


Brittle Ductile

Shear in Tension Test?’

’

’

’

’

’

’

’

’

’

2-D Mohr’s Circle

All stress states on a diameter of this circle are equivalent, just rotation of axes

’

(’/2, ’/2)

Mohr’s CircleGeneralized 2-D Loading

• Stress state (tensor) depends on coordinate frame chosen

• Mathematical construct to ease coordinate transform

• Rotation of in material space is equivalent to 2* in Mohr space– Example: pure shear

• rotate 45º on material unit• rotate 90º on Mohr’s

circle

2xy

22x

yx

2R

2C

2

R

C

x

yxy

-/2/2

Mohr’s Circle Examples

y

x

yx

xy

y

y

y

y

x x= -y

~ 20º

~ 70º

max

~ 10º

~ 35º

min max

= 0 (x ,xy)

(y ,yx)

(y ,0)

(y ,0)(x ,0)

max

45º

max

45º

Failure mode - simple models

f

f

Ductile failure -Tresca criteria

f

f

Brittle failure- Maximum normal stress criteria

More complex failure theory - Von Mises (energy based)

21223

213

212e 2

2

Hardness Testing• Scratch Test - very qualitative

– Mohs• Penetration Tests

– Brinell– Rockwell– Knoop– Vickers

• Hardness testing measures ability to resist plastic deformation– Need to eliminate effect of elastic deformation

• Brinell - load applied for 30 sec• Rockwell - initial preload and differential depth measurement

• To measure individual grain hardness, use Knoop or Vickers (lab #8)

Microhardness

Brinell Hardness

D

d

F

22 dDD2D

FBHN

• Large, hard spherical indentor

• Relatively large loads (500-3000 kg)

• Hold load for 30 sec.• Leaves large indent in

specimen• Manually measure

indentation with calibrated microscope

• Single scale for all materials• Takes average hardness

over many grains

Rockwell Hardness

d1

F1

d2

F2

d1

F1

d2

F2

Rockwell B

Rockwell C

• Most common hardness test method• Many scales: 2 important for us:

– Rockwell B- soft materials• Spherical indentor• Low loads (~100 kg)• small indention

– Rockwell C- hard materials• Conical indentor• Slightly higher loads (~150 kg)• Very small indention

• Measures differential penetration depth (initial preload, 10 kg)

• Machines are fully automated• Scale limits 20-100 (HRB, HRC, etc)

– if exceeded, switch test

Conversions & Correlations

• Can convert from one scale to the other - approximately

• Brinell Hardness number (HB) is approximately related to tensile strength by:

• in steels only (empirical relation)

)ksi(HB*5.0)MPa(HB*45.3

UTS

UTS


• Scales are designed for flat specimens– Need “curvature correction” for round

specimens– Avoid specimen edges and other indents

• Specimen thickness must be at least 10x indention depth

Notes on Hardness Testing

Disadvantages“Relatively” nondestructive“Relatively” quantitative

AdvantagesCheapSimple test“Relatively” nondestructive“Relatively” quantitativeCorrelates with tensile strength

Statistical Testing• When conducting experimental

testing, data will vary.• Be aware of your sources of

variability:– Specimen manufacture– Machine

variations/malfunctions– Environmental changes– Improper procedure– Random variables

• In lab, report your statistical differences, don’t hide them.

• For more in-depth analysis, look into IE230.

• Measure of average value:Mean Value

• Measure of scatter: Standard Deviation

• Relative measure of scatter:“Coefficient of variation”

n

xx

n

1ii

1n

xxs

n

1i

2i

xsCv

Thermal Properties• Often design to utilize a material’s thermal properties

– Energy storage– Insulative or Conductive – Use thermally activated switches (beam expands and

closes switch)

• Properties we care most about– Heat Capacity (C)– Conduction (q) – Thermal Expansion (T)

Heat Capacity & Conduction• Heat (Q) and Temperature (T) are related by

• Property can be measured at:– Constant volume, Cv

– Constant pressure, Cp

– Condensed phases (solid in our case) are more often at constant pressure

• Heat always flows from high energy to low

– qx is heat flux, k is thermal conductivity– Metals are excellent conductors due to free electrons– Ceramics and polymers are usually considered insulators

dTdQCCdTdQ

dxdTkqx

Thermal Expansion• Temperature change will induce a change in dimensions

• If a bar is heated while physically constrained, induce a thermal stress

• Thermal expansion coefficient is strongly dependent on material (shape of force vs. atomic separation curve)– Polymers: ~100-200 x 10-6 C-1

– Metals: ~10-20 x 10-6 C-1

– Ceramics: ~1-10 x 10-6 C-1

oflT TTll

oflT

ofleT

TTEE

TT0

l = lo

New Concepts & Terms• Elastic Properties

– Elastic (Young’s) Modulus• Secant Modulus• Tangent Modulus

– Poisson’s ratio– Linear vs. Nonlinear– Isotropic vs. orthotropic

• Yield-point behavior– Proportional limit– 0.2% offset yield strength– Upper & lower yield

• Plastic Deformation– Neck– Uniform vs. localized deformation– Mohr’s circle

• True vs. Engineering stress– Engineering: original area– True: instantaneous area

• Stress-strain curves– Yield strength– Ultimate Tensile Strength– Fracture Strength– Fracture Strain – Toughness, Resilience – Ductility (%AR, %EL)

• Fracture Surfaces– Cleavage– Cup-cone

• Hardness Testing– Rockwell– Brinell

• Statistics (mean, standard deviation)• Thermal Properties

– Heat Capacity– Thermal Expansion– Conduction

Next Lecture ...

• Please read chapters 2 & 3Please read chapters 2 & 3