15457907 Stress Strain

Chapter 7: Mechanical Properties

Chapter 7: Mechanical Properties

Why mechanical properties?

ISSUES TO ADDRESS... • Stress and strain: Normalized force and displacements.

• Need to design materials that will withstand applied load and in-service uses for… Bridges for autos and people

Engineering : σ e = Fi / A0

ε e = Δl / l 0

True : σ T = Fi / Ai

MEMS devices

εT = ln(l f / l 0 )

• Elastic behavior: When loads are small. Young ' s Modulus : E

[GPa]

• Plastic behavior: dislocations and permanent deformation

Canyon Bridge, Los Alamos, NM

skyscrapers

Yield Strength : σ YS Ulitmate Tensile Strength : σ TS

[MPa] (permanent deformation) [MPa] (fracture)

• Toughness, ductility, resilience, toughness, and hardness: Define and how do we measure? Space elevator?

Space exploration

• Mechanical behavior of the various classes of materials.

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Pure Tension

Stress and Strain stress

Stress: Force per unit area arising from applied load. Tension, compression, shear, torsion or any combination.

strain

σe =

εe =

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Pure Compression

Fnormal Ao l − lo lo

Elastic σ e = Eε response

Stress = σ = force/area

stress τ e = Fshear

Pure Shear

Ao

Strain: physical deformation response of a material to stress, e.g., elongation.

strain

γ = tan θ

Elastic τ e = Gγ response Pure Torsional Shear 3 MatSE 280: Introduction to Engineering Materials

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Engineering Stress • Tensile stress, s:

Common States of Stress

• Shear stress, t:

• Simple tension: cable

σ=

Ski lift

• Simple shear: drive shaft

F σ= t Ao original area before loading

F Ao

Fs τ = Ao

Stress has units: N/m 2 (or lb/in 2 )

Note: τ = M/AcR here.

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(photo courtesy P.M. Anderson)

MatSE 280: Introduction to Engineering Materials

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Common States of Stress

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Common States of Stress

• Simple compression:

• Bi-axial tension:

• Hydrostatic compression:

Ao

Canyon Bridge, Los Alamos, NM (photo courtesy P.M. Anderson)

Balanced Rock, Arches National Park

Note: compressive structure member (σ < 0).

(photo courtesy P.M. Anderson)

Pressurized tank (photo courtesy P.M. Anderson)

Fish under water

(photo courtesy P.M. Anderson)

σθ > 0 σz > 0

σ h< 0

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Engineering Strain • Tensile strain:

Elastic Deformation

• Lateral (width) strain: δ/2

• Shear strain:

π/2

3. Unload

bonds stretch

δ/2 δL/2

return to initial

δ

θ/2

π/2 - θ

2. Small load

Lo

wo δL/2

1. Initial

F Strain is always dimensionless.

γ = tan θ

F

Linearelastic

Elastic means reversible!

δ

θ/2

Non-Linearelastic

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Plastic Deformation of Metals 1. Initial

2. Small load bonds stretch & planes shear

Strain Testing

3. Unload

• Tensile specimen

planes still sheared

Often 12.8 mm x 60 mm Adapted from Fig. 7.2, Callister & Rethwisch 3e.

extensometer

specimen

gauge length

F F linear elastic

linear elastic

δplastic MatSE 280: Introduction to Engineering Materials

• Tensile test machine

δplastic

δelastic + plastic

Plastic means permanent!

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δelastic

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δ

• Other types: -compression: brittle materials (e.g., concrete) -torsion: cylindrical tubes, shafts.

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Linear Elasticity • Modulus of Elasticity, E:

Example: Hooke’s Law • Hooke's Law:

Units: E [GPa] or [psi]

(also known as Young's modulus)

σ=Eε

(linear elastic behavior)

Copper sample (305 mm long) is pulled in tension with stress of 276 MPa. If deformation is elastic, what is elongation?

• Hooke's Law: σ = E ε

σ

Axial strain

ε

Axial strain

 Δl  σl σ = Eε = E   ⇒ Δl = 0 E  l0  (276MPa)(305mm) Δl = = 0.77mm 110x103 MPa

E Linearelastic

For Cu, E = 110 GPa.

Width strain

Width strain

Hooke’s law involves axial (parallel to applied tensile load) elastic deformation. 13 MatSE 280: Introduction to Engineering Materials

Elastic Deformation 1. Initial

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Mechanical Properties

2. Small load

3. Unload

• Recall: Bonding Energy vs distance plots

bonds stretch return to initial

δ F

F

Linearelastic

Elastic means reversible!

δ

tension Non-Linearelastic

compression Adapted from Fig. 2.8 Callister & Rethwisch 3e.

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Elasticity of Ceramics

Mechanical Properties • Recall: Slope of stress strain plot (proportional to the E) depends on bond strength of metal

And Effects of Porosity

• Elastic Behavior

E= E0(1 - 1.9P + 0.9 P 2)

E larger E smaller

Al2O3

Adapted from Fig. 7.7, Callister & Rethwisch 3e.

Neither Glass or Alumina experience plastic deformation before fracture! 17 MatSE 280: Introduction to Engineering Materials

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Polymers: Tangent and Secant Modulus

Comparison of Elastic Moduli

• Tangent Modulus is experienced in service. • Secant Modulus is effective modulus at 2% strain. - grey cast iron is also an example

• Modulus of polymer changes with time and strain-rate. - must report strain-rate dε/dt for polymers. - must report fracture strain εf before fracture. Silicon (single xtal) 120-190 (depends on crystallographic direction) Glass (pyrex) 70 SiC (fused or sintered) 207-483 Graphite (molded) ~12 High modulus C-fiber 400 Carbon Nanotubes ~1000 Normalize by density, 20x steel wire.

initial E

Stress (MPa)

secant E tangent E

strength normalized by density is 56x wire.

%strain 1

2

3

4

5 …..

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Poisson's ratio, ν

Young’s Modulus, E Metals Alloys 1200 1000 800 600 400

E(GPa)

200 100 80 60 40

109 Pa

Graphite Composites Ceramics Polymers /fibers Semicond

• Poisson's ratio, ν:

Units: ν dimensionless

Diamond

Tungsten Molybdenum Steel, Ni Tantalum Platinum Cu alloys Zinc, Ti Silver, Gold Aluminum

Si carbide Al oxide Si nitride

ν =−

Carbon fibers only

CFRE(|| fibers)*

<111>

Si crystal

width strain Δw / w ε =− =− L axial strain Δl / l ε

Aramid fibers only

<100>

AFRE(|| fibers)*

Glass-soda

Glass fibers only

Magnesium, Tin

GFRE(|| fibers)*

Axial strain

Concrete GFRE*

20

CFRE* GFRE( fibers)*

Graphite

10 8 6 4

εL

Polyester PET PS PC

2

CFRE( fibers)* AFRE( fibers)*

Epoxy only

PP HDPE

1 0.8 0.6 0.4

PTFE

Wood(

ε Based on data in Table B2, Callister 6e. Composite data based on reinforced epoxy with 60 vol% of aligned carbon (CFRE), aramid (AFRE), or glass (GFRE) fibers.

metals: ν ~ 0.33 ceramics: ν ~ 0.25 polymers: ν ~ 0.40

Width strain

grain)

-ν

Why does ν have minus sign?

LDPE

0.2

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Limits of the Poisson Ratio

Poisson Ratio: materials specific

• Poisson Ratio has a range –1 ≤ ν ≤ 1/2

Metals:

Look at extremes • No change in aspect ratio:

Ir 0.26

Solid Argon:

0.25

ν =−

Δw / w = −1 Δl / l

Covalent Solids:

• Volume (V = AL) remains constant: Hence, ΔV = (L ΔA+A ΔL) = 0.

ΔV =0. So,

ΔA / A = −ΔL /L

In terms of width, A = w 2, then ΔA/A = 2 w Δw/w2 = 2Δw/w = –ΔL/L. Hence,

ν =−

Δw / w (− Δl / l) =− € = 1/ 2 Δl / l Δl / l 1 2

W 0.29

Ni 0.31

Cu 0.34

Al 0.34

Ag 0.38

Incompressible solid. Water (almost).

Si 0.27

Ionic Solids:

MgO

Silica Glass:

0.20

Ge 0.28

Al2O3 0.23

TiC 0.19

generic value ~ 1/4

0.19

Polymers:

Network (Bakelite) 0.49

Chain (PE) 0.40 ~generic value

Elastomer:

Hard Rubber (Ebonite) 0.39

(Natural) 0.49

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Au 0.42

generic value ~ 1/3

Δw /w = Δl /l

€

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Other Elastic Properties

Example: Poisson Effect Tensile stress is applied along cylindrical brass rod (10 mm diameter). Poisson ratio is ν = 0.34 and E = 97 GPa. • Determine load needed for 2.5x10 –3 mm change in diameter if the deformation is entirely elastic?

M

τ

• Elastic Shear modulus, G:

G 1

τ=Gγ

simple Torsion test

γ

M

Width strain: (note reduction in diameter)

Axial strain:

mm)/(10 mm) =

P

• Elastic Bulk modulus, K:

–2.5x10 –4

P

Given Poisson ratio

P Pressure test: Init. vol = Vo. Vol chg. = ΔV

εz = –εx/ν = –(–2.5x10 –4)/0.34 = +7.35x10 –4 Axial Stress:

• Special relations for isotropic materials: E E So, only 2 independent elastic G= K= 2(1+ ν) 3(1− 2ν) constants for isotropic media

σz = Eεz = (97x10 3 MPa)(7.35x10 –4) = 71.3 MPa.

Required Load: F = σzA 0 = (71.3 MPa)π(5 mm)2 = 5600 N. 25 MatSE 280: Introduction to Engineering Materials

δ = FL o δw = −ν Fw o EA o EA o F Ao

wo

• For linear elastic, isotropic case, use “linear superposition”.

α = 2ML o πr o4 G

• Strain || to load by Hooke’s Law: εi=σi/E,

i=1,2,3 (maybe x,y,z).

• Strain ⊥ to load governed by Poisson effect: εwidth = –νεaxial .

M = moment α = angle of twist

Lo

δw /2

• There are 3 principal components of stress and (small) strain.

• Simple torsion:

δ/2

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Complex States of Stress in 3D

Useful Linear Elastic Relationships • Simple tension:

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stress strain

ε1 ε2 ε3

Lo 2ro

σ1

σ2

σ3

σ1/E -νσ1/E -νσ1/E

-νσ2/E σ2/E -νσ2/E

-νσ3/E -νσ3/E σ3/E

In x-direction, total linear strain is:

• Material, geometric, and loading parameters all contribute to deflection. • Larger elastic moduli minimize elastic deflection.

Total Strain in x in y in z

1 {σ − ν (σ 2 + σ 3 )} E 1 1 = {(1+ ν )σ 1 − ν (σ 1 + σ 2 + σ 3 )} E

ε1 = or

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Poisson

εx = Δd/d =

–(2.5x10 –3

εz , σz

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Complex State of Stress and Strain in 3-D Solid

Plastic (Permanent) Deformation

• Hooke’s Law and Poisson effect gives total linear strain: ε1 =

1 {σ − ν (σ 2 + σ 3 )} E 1

(at lower temperatures, i.e. T < Tmelt/3)

1 {(1+ ν )σ 1 − ν (σ 1 + σ 2 + σ 3 )} E

or

• For uniaxial tension test σ1= σ2 =0,

• Simple tension test:

so ε3= σ3/E and ε1=ε2= –νε3.

• Hydrostatic Pressure:

P = σ Hyd =

σ 1 + σ 2 + σ 3 Tr σ = 3 3

ε1 =

Elastic+Plastic at larger stress

engineering stress, σ

Elastic initially

1 {(1+ ν )σ 1 − 3ν P} E

permanent (plastic) after load is removed

• For volume (V=l1l2l3) strain, ΔV/V = ε1+ ε 2+ ε3 = (1-2ν)σ3/E

εp

ΔV P = 3(1− 2ν ) V E

Bulk Modulus, B or K:

P = –K ΔV/V so K = E/3(1-2ν)

engineering strain, ε plastic strain

(sec. 7.5)

Adapted from Fig. 7.10 (a), Callister & Rethwisch 3e.

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Yield Stress, σY

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Yield Points and σYS • Yield-point phenomenon occurs when elastic plastic transition is abrupt.

• Stress where noticeable plastic deformation occurs. when εp = 0.002

For metals agreed upon 0.2%

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No offset method required.

tensile stress, σ

• P is the proportional limit where deviation from linear behavior occurs.

• In steels, this effect is seen when dislocations start to move and unbind for interstitial solute.

σy P

Elastic recovery

Strain off-set method for Yield Stress • Start at 0.2% strain (for most metals). • Draw line parallel to elastic curve (slope of E). • σY is value of stress where dotted line crosses stress-strain curve (dashed line).

Eng. strain, ε εp = 0.002

Note: for 2 in. sample

• Lower yield point taken as σY . For steels, take the avg. stress of lower yield point since less sensitive to testing methods.

• Jagged curve at lower yield point occurs when solute binds dislocation and dislocation unbinding again, until work-hardening begins to occur.

ε = 0.002 = Δz/z ∴ Δz = 0.004 in 31

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Compare Yield Stress, σYS

Stress-Strain in Polymers • 3 different types of behavior

σy(ceramics) >>σy(metals) >> σy(polymers)

For plastic polymers: • YS at maximum stress just after elastic region. • TS is stress at fracture!

Brittle

plastic

Room T values Highly elastic

• Highly elastic polymers: • Elongate to as much as 1000% (e.g. silly putty). • 7 MPa < E < 4 GPa 3 order of magnitude! • TS(max) ~ 100 MPa some metal alloys up to 4 GPa MatSE 280: Introduction to Engineering Materials

Based on data in Table B4, Callister 6e . a = annealed hr = hot rolled ag = aged cd = cold drawn cw = cold worked qt = quenched & tempered

33

(Ultimate) Tensile Strength, σTS

For Metals: max. stress in tension when necking starts, which is the metals work-hardening tendencies vis-à-vis those that initiate instabilities.

TS F = fracture or ultimate strength

engineering stress

Typical response of a metal

strain engineering strain

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Metals: Tensile Strength, σTS

• Maximum possible engineering stress in tension.

σy

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Neck – acts as stress concentrator

dF = 0

Maximum eng. Stress (at necking)

dσ T dA =− i σT Ai

dF = 0 = σ T dAi + Ai dσ T decreased force due to decrease in gage diameter

Increased force due to increase in applied stress

At the point where these two competing changes in force equal, there is permanent neck.

• Metals: occurs when necking starts. • Ceramics: occurs when crack propagation starts. • Polymers: occurs when polymer backbones are aligned and about to break.

Determined by slope of “true stress” - “true strain” curve

Fractional Increase in Flow stress

dσ T dA dl = − i = i ≡ dεT σT Ai li

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dσ T = σT dεT

⇒

If σ T = K(εT )n , then n = εT n = strain-hardening coefficient

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fractional decrease in loadbearing area

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Compare Tensile Strength, σTS Metals/ Alloys

Tensile strength, TS (MPa)

5000 3000 2000 1000

300 200 100 40 30 20

Graphite/ Ceramics/ Semicond

Polymers

Composites/ fibers C fibers Aramid fib E-glass fib

Steel (4140) qt W (pure) Ti (5Al-2.5Sn) a Steel (4140)cwa Cu (71500) Cu (71500) hr Steel (1020) Al (6061) ag Ti (pure) a Ta (pure) Al (6061) a

A FRE(|| fiber) GFRE(|| fiber) CFRE(|| fiber)

Diamond Si nitride Al oxide

Si crystal <100>

Glass-soda Concrete Graphite

Nylon 6,6 PC PET PVC PP HDPE

Example for Metals: Determine E, YS, and TS

Room T values

Stress-Strain for Brass

• Young’s Modulus, E

(bond stretch)

σ − σ 1 (150 − 0)MPa E= 2 = = 93.8GPa ε2 − ε1 0.0016 − 0

TS(ceram) ~TS(met) ~ TS(comp) >> TS(poly)

• 0ffset Yield-Stress, YS

€

(plastic deformation)

YS = 250 MPa

wood(|| fiber)

• Max. Load from Tensile Strength TS

GFRE( fiber) CFRE( fiber) A FRE( fiber)

€

d 2 Fmax = σTS A0 = σTS π  0  2

LDPE

Based on data in Table B4, Callister & Rethwisch 3e.

10

wood (

fiber)

• Gage is 250 mm (10 in) in length and 12.8 mm (0.505 in) in diameter. • Subject to tensile stress of 345 MPa (50 ksi)

 2 12.8x 10−3 m  = 450MPa π = 57,900N   2  

• Change in length at Point A, Δl = εl0

€

1

Δl = εl0 = (0.06)250 mm = 15 mm

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Ductility (%EL and %RA)

Temperature matters (see Failure) Most metals are ductile at RT and above, but can become brittle at low T

• Plastic tensile strain at failure:

%EL =

Lf − Lo

bcc Fe

Lo

x100

Adapted from Fig. 7.13, Callister & Rethwisch 3e.

• Another ductility measure:

cup-and-cone fracture in Al

%RA =

Ao − Af Ao

• Note: %RA and %EL are often comparable. - Reason: crystal slip does not change material volume. - %RA > %EL possible if internal voids form in neck.

brittle fracture in mild steel 39

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x100

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Toughness

Resilience, Ur

• Energy to break a unit volume of material, or absorb energy to fracture. • Approximate as area under the stress-strain curve.

• Resilience is capacity to absorb energy when deformed elastically and recover all energy when unloaded (=σ2YS/2E). • Approximate as area under the elastic stress-strain curve.

small toughness (ceramics)

Engineering tensile stress, σ

large toughness (metals) very small toughness (unreinforced polymers)

ε UT = ∫of σ dε

ε Ur = ∫oY σ dε ε2 σ ε σ2 ε = ∫oY Eε dε ~ E Y = Y Y = Y 2 2 2E Area up to 0.2% strain If linear elastic

Engineering tensile strain, ε Brittle fracture: elastic energy Ductile fracture: elastic + plastic energy 41 MatSE 280: Introduction to Engineering Materials

Elastic Strain Recovery

D

Stress

Ceramic materials are more brittle than metals. Why? • Consider mechanism of deformation – In crystalline materials, by dislocation motion – In highly ionic solids, dislocation motion is difficult • few slip systems • resistance to motion of ions of like charge (e.g., anions) past one another.

2. Unload

1. Load

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Ceramics Mechanical Properties

• Unloading in step 2 allows elastic strain to be recovered from bonds. • Reloading leads to higher YS, due to work-hardening already done.

σyi σyo

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3. Reapply load

Strain Adapted from Fig. 7.17, Callister & Rethwisch 3e.

Elastic strain recovery 43

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Strength of Ceramics - Elastic Modulus

Strength of Ceramics - Flexural Strength

• RT behavior is usually elastic with brittle failure. • 3-point bend test employed (tensile test not best for brittle materials).

• 3-point bend test employed for RT Flexural strength. Al2O3

cross section

d b

rect.

R δ = midpoint deflection

circ.

• Determine elastic modulus according to:

F

x

F slope = δ

δ

linear-elastic behavior

E=

F L3 δ 4bd 3

Rectangular cross-section

d

(rect. cross section)

b

σ fs =

3Ff L

• Typical values:

2bd 2

Material

Si nitride 250-1000 304 Si carbide 100-820 345 Al oxide 275-700 393 glass (soda-lime) 69 69

Circular cross-section

F L3 (circ. cross section) E= δ 12πR 4

R

σ fs =

8Ff L

πd 3

L= length between load pts b = width d = height or diameter

σ fs (MPa) E(GPa)

Data from Table 7.2, Callister & Rethwisch 3e.

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Stress-Strain in Polymers

Influence of T and Strain Rate on Thermoplastics

brittle polymer

• Decreasing T... -- increases E -- increases TS -- decreases %EL

plastic elastomer elastic moduli – less than for metals

©D.D. Johnson 2004/2006-2008

Adapted from Fig. 7.22, Callister & Rethwisch 3e.

• Increasing strain rate... -- same effects as decreasing T.

σ(MPa) 80 4°C 60

Plots for semicrystalline PMMA (Plexiglas)

20°C

40

40°C

20 0

60°C 0

0.1

0.2

ε

to 1.3 0.3

Adapted from Fig. 7.24, Callister & Rethwisch 3e. (Fig. 7.24 is from T.S. Carswell and J.K. Nason, 'Effect of Environmental Conditions on the Mechanical Properties of Organic Plastics", Symposium on Plastics, American Society for Testing and Materials, Philadelphia, PA, 1944.)

• Fracture strengths of polymers ~ 10% of those for metals. • Deformation strains for polymers > 1000%. – for most metals, deformation strains < 10%. 47 MatSE 280: Introduction to Engineering Materials

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Stress-Strain in Polymers • Necking appears along entire sample after YS!

Time-dependent deformation in Polymers

• Mechanism unlike metals, necking due to alignment of crystallites. Load vertical

• Stress relaxation test: -strain in tension to εο and hold. - observe decrease in stress with time.

• Large decrease in Er for T > Tg. 105 Er (10 s) 3 in MPa 10

10 -1

• Representative Tg values (°C):

time

See Chpt 8

E r (t ) =

Fig. 7.28, Callister & Rethwisch 3e. (Fig. 7.28 from A.V. Tobolsky, Properties and Structures of Polymers, Wiley and Sons, Inc., 1960.)

60 100 140 180 T(°C) Tg

PE (low density) PE (high density) PVC PS PC

• Relaxation modulus: •After YS, necking proceeds by unraveling; hence, neck propagates, unlike in metals!

viscous liquid

(amorphous polystyrene)

10 -3 (large relax)

strain σ(t)

•Align crystalline sections by straightening chains in the amorphous sections

transition region

101

tensile test

εo

rigid solid (small relax)

σ(t ) εo

- 110 - 90 + 87 +100 +150

Selected values from Table 11.3, Callister & Rethwisch 3e.

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True Stress and Strain Engineering stress σ =

€

True strain

Why use True Strain?

F A0

Initial area always

Relation before necking

F Ai

instantaneous area

σ T = σ (1+ ε )

True stress σt =

l εt = ln i l0

Relative change

€

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• Up to YS, there is volume change due to Poisson Effect! • In a metal, from YS and TS, there is plastic deformation, as dislocations move atoms by slip, but ΔV=0 (volume is constant).

εT = ln (1+ ε )

l l − l0 + l0 ε t = ln i   → ln i = ln(1+ ε ) l0 l0

Necking: 3D state of stress!

Eng. Strain

€

Test

length

0 1 2 3 TOTAL

2.00 2.20 2.42 2.662

Eng. 0-1-2-3

Eng. 0-3

0.1 0.1 0.1 0.3

0.662/2.0 0.331

True 2.2 2.42 2.662 2.662 Strain ε t = ln 2.0 + ln 2.20 + ln 2.42 = ln 2.00

A0 l 0 = Ai l i

Sum of incremental strain does NOT equal total strain! Sum of incremental strain does equal total strain.

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Hardening

Using Work-Hardening Influence of “cold working” on low-carbon steel.

• An increase in σy due to plastic deformation.

σ 2nd drawn 1st drawn

large hardening

σy 1 σy

small hardening

0

Undrawn wire

ε • Curve fit to the stress-strain response after YS: Processing: Forging, Rolling, Extrusion, Drawing,… • Each draw of the wire decreases ductility, increases YS. • Use drawing to strengthen and thin “aluminum” soda can. 53 MatSE 280: Introduction to Engineering Materials

54 MatSE 280: Introduction to Engineering Materials

©D.D. Johnson 2004/2006-2008

Hardness

©D.D. Johnson 2004/2006-2008

Hardness: Measurement

• Resistance to permanently indenting the surface. • Large hardness means:

• Rockwell – No major sample damage – Each scale runs to 130 (useful in range 20-100). – Minor load 10 kg – Major load 60 (A), 100 (B) & 150 (C) kg

--resistance to plastic deformation or cracking in compression. --better wear properties.

• A = diamond, B = 1/16 in. ball, C = diamond

• HB = Brinell Hardness – TS (psia) = 500 x HB – TS (MPa) = 3.45 x HB Adapted from Fig. 7.18.

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Hardness: Measurement

Account for Variability in Material Properties • Elastic modulus is material property • Critical properties depend largely on sample flaws (defects, etc.). Large sample to sample variability. • Statistics n

Σ xn x= n

– Mean

– Standard Deviation

n  Σ xi − x s =  n −1 

(

)

1 2 2

    

where n is the number of data points 57 MatSE 280: Introduction to Engineering Materials

• Design uncertainties mean we do not push the limit. • Factor of safety, N (sometime given as S) Often N is between 1.2 and 4 σy σ working = N • Ex: Calculate diameter, d, to ensure that no yielding occurs in the 1045 carbon steel rod. Use safety factor of 5.

220,000N   π  d2 / 4   

©D.D. Johnson 2004/2006-2008

Summary

Design Safety Factors

σ working =

58 MatSE 280: Introduction to Engineering Materials

©D.D. Johnson 2004/2006-2008

σy N

5

• Stress and strain: These are size-independent measures of load and displacement, respectively. • Elastic behavior: This reversible behavior often shows a linear relation between stress and strain. To minimize deformation, select a material with a large elastic modulus (E or G). • Plastic behavior: This permanent deformation behavior occurs when the tensile (or compressive) uniaxial stress reaches s y. • Toughness: The energy needed to break a unit volume of material. • Ductility: The plastic strain at failure.

d = 0.067 m = 6.7 cm 59

MatSE 280: Introduction to Engineering Materials

©D.D. Johnson 2004/2006-2008

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