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Chapter 7: Mechanical Properties Chapter 7: Mechanical Properties Why mechanical properties? ISSUES TO ADDRESS... • Need to design that will withstand • and strain: Normalized and displacements. applied load and in-service uses for… Engineering : σ e = Fi / A0 εe = Δl / l0 Bridges for autos and people MEMS devices True : σT = Fi / Ai εT = ln(lf / l0 )

• Elastic behavior: When loads are small. Young 's Modulus : E [GPa] Canyon Bridge, Los Alamos, NM • behavior: and permanent skyscrapers Strength : σYS [MPa] (permanent deformation) Ulitmate Tensile Strength : σ [MPa] () TS • , , resilience, toughness, and : Define and how do we measure? Space exploration Space elevator? • Mechanical behavior of the various classes of materials.

1 2 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Stress and Strain Pure Tension Pure Compression F stress σ = normal e A Stress: Force per unit area arising from applied load. o l − lo strain εe = Tension, compression, shear, torsion or any combination. lo

σ = Eε Elastic e Stress = σ = force/area response

F Pure τ = shear e A Strain: physical deformation response of a o to stress, e.g., elongation. strain γ = tanθ τ = Gγ Elastic e response Pure Torsional Shear 3 4 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

1 Engineering Stress Common States of Stress

• Tensile stress, s: • Shear stress, t: • Simple tension: cable

F σ = Ao

Ski lift (photo courtesy P.M. Anderson) F • Simple shear: drive shaft σ = t Ao original area Fs = before loading Stress has units: N/m2 (or lb/in2 ) τ Ao Note: τ = M/AcR here. 5 6 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Common States of Stress Common States of Stress

• Simple compression: • Bi-axial tension: • Hydrostatic compression:

Ao

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

Pressurized tank Fish under water (photo courtesy Note: compressive (photo courtesy P.M. Anderson) Balanced Rock, Arches structure member (σ < 0). P.M. Anderson) National Park σθ > 0 (photo courtesy P.M. Anderson)

σz > 0 σ h < 0

7 8 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

2 Engineering Strain Elastic Deformation

• Tensile strain: • Lateral (width) strain: 1. Initial 2. Small load 3. Unload δ/2 bonds Lo wo stretch δ/2 return to δL/2 δL/2 initial • Shear strain: δ θ/2 F F Linear- γ = tan θ Strain is always elastic π/2 - θ dimensionless. Elastic means reversible! Non-Linear- elastic π/2 θ/2 δ 9 10 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Plastic Deformation of Strain Testing 1. Initial 2. Small load 3. Unload • Tensile specimen • Tensile test machine bonds planes stretch Often 12.8 mm x 60 mm & planes still sheared Adapted from Fig. 7.2, shear Callister & Rethwisch 3e. extensometer specimen

δ δelastic + plastic plastic gauge F length F

linear linear • Other types: Plastic means permanent! elastic elastic δ -compression: brittle materials (e.g., ) δplastic -torsion: cylindrical tubes, shafts. δelastic 11 12 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

3 Linear Example: Hooke’s Law

• Hooke's Law: σ = E ε (linear elastic behavior) • Modulus of Elasticity, E: Units: E [GPa] or [psi] (also known as Young's modulus) 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 For Cu, E = 110 GPa. σ Axial strain Axial strain

  Δl σ l0 E σ = Eε = E   ⇒ Δl =  l0  E ε (276MPa)(305mm) Δl = = 0.77mm Linear- Width strain 110x103 MPa Width strain

elastic

Hooke’s law involves axial (parallel to applied tensile load) elastic deformation.

13 14 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Elastic Deformation Mechanical Properties

1. Initial 2. Small load 3. Unload • Recall: Bonding vs distance plots

bonds stretch

return to initial δ F F Linear- elastic tension

Elastic means reversible! Non-Linear- compression elastic Adapted from Fig. 2.8 δ Callister & Rethwisch 3e.

15 16 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

4 Mechanical Properties Elasticity of

• Recall: Slope of stress strain plot (proportional to the E) • Elastic Behavior And Effects of Porosity 2 depends on bond strength of E= E0(1 - 1.9P + 0.9 P )

E larger E smaller Al2O3

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

Neither Glass or Alumina experience plastic deformation before fracture!

17 18 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Comparison of Elastic Moduli : Tangent and Secant Modulus • Tangent Modulus is experienced in service.

• Secant Modulus is effective modulus at 2% strain. - grey cast is also an example • Modulus of 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) initial E Glass (pyrex) 70 Stress (MPa) SiC (fused or sintered) 207-483 secant E Graphite (molded) ~12 tangent E High modulus C- 400 Nanotubes ~1000 Normalize by density, 20x wire. strength normalized by density is 56x wire. %strain 1 2 3 4 5 ….. 19 20 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

5 Young’s Modulus, E Poisson's ratio, ν

Graphite Metals Composites Ceramics Polymers Alloys / Semicond • Poisson's ratio, ν: Units: ν dimensionless 1200 1000 800 Diamond 600 Si carbide width strain Δw / w εL 400 Al oxide Carbon fibers only Molybdenum Si nitride ν = − = − = − Steel, Ni CFRE(|| fibers)* E(GPa) 200 Tantalum <111> axial strain Δ / ε ε Platinum Si crystal l l L Cu alloys <100> Aramid fibers only 100 Zinc, Ti 80 , Gold Glass-soda AFRE(|| fibers)* 60 Aluminum Glass fibers only Magnesium, 40 GFRE(|| fibers)* Concrete Axial strain 9 20 GFRE* 10 Pa CFRE* GFRE( fibers)* 10 Graphite ε 8 CFRE( fibers)* 6 AFRE( fibers)* Based on data in Table B2, Callister 6e. Polyester 4 PET Composite data based on metals: ν ~ 0.33 PS reinforced epoxy with 60 vol% - Epoxy only ν 2 PC of aligned carbon (CFRE), ceramics: ν ~ 0.25 PP aramid (AFRE), or glass (GFRE) fibers. 1 HDPE 0.8 Width strain polymers: ν ~ 0.40 0.6 ( grain) PTFE 0.4 Why does ν have minus sign? 0.2 LDPE 21 22 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Limits of the Poisson Ratio Poisson Ratio: materials specific

• Poisson Ratio has a range –1 ≤ ν ≤ 1/2 Metals: Ir W Ni Cu Al Ag Au 0.26 0.29 0.31 0.34 0.34 0.38 0.42 Look at extremes generic value ~ 1/3 • No change in aspect ratio: Δ w /w = Δl /l Argon: 0.25 Δw / w ν = − = −1 Covalent : Si Ge Al2O3 TiC Δl / l 0.27 0.28 0.23 0.19 generic value ~ 1/4 € • (V = AL) remains constant: ΔV =0. Ionic Solids: MgO 0.19

Hence, ΔV = (L ΔA+A ΔL) = 0. So, ΔA / A = −ΔL /L Silica Glass: 0.20

In terms of width, A = w2, then ΔA/A = 2 w Δw/w2 = 2Δw/w = –ΔL/L. Polymers: Network (Bakelite) 0.49 Chain (PE) 0.40 ~generic value

w / w ( 1 / ) Elastomer: Hard Rubber (Ebonite) 0.39 (Natural) 0.49 Hence, Δ − 2 Δl l ν = − = − € = 1/ 2 Incompressible solid. Δl / l Δl / l Water (almost). 23 24 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

6 Example: Poisson Effect Other Elastic Properties Tensile stress is applied along cylindrical brass rod (10 mm M diameter). Poisson ratio is ν = 0.34 and E = 97 GPa. • Elastic Shear τ G simple –3 modulus, G: • Determine load needed for 2.5x10 mm change in diameter if 1 γ Torsion test the deformation is entirely elastic? τ = G γ

Width strain: (note reduction in diameter) M • Elastic Bulk ε = Δd/d = –(2.5x10–3 mm)/(10 mm) = –2.5x10–4 P x modulus, K: Axial strain: Given Poisson ratio P P test: ε = –ε /ν = –(–2.5x10–4)/0.34 = +7.35x10–4 z x Init. vol = Vo. Vol chg. = ΔV

3 –4 • Special relations for isotropic materials: Axial Stress: σz = Eεz = (97x10 MPa)(7.35x10 ) = 71.3 MPa. E E So, only 2 independent elastic Required Load: F = σ A = (71.3 MPa)π(5 mm)2 = 5600 N. G = K = z 0 2(1+ ν) 3(1− 2ν) constants for isotropic media 25 26 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Useful Linear Elastic Relationships Complex States of Stress in 3D • There are 3 principal components of stress and (small) strain. • Simple tension: • Simple torsion: • For linear elastic, isotropic case, use “linear superposition”. 2ML FLo Fw o o • Strain || to load by Hooke’s Law: = /E, i=1,2,3 (maybe x,y,z). δ = δw = −ν α = εi σi r 4 EAo EA o π o G • Strain ⊥ to load governed by Poisson effect: εwidth = –νεaxial. F M = moment α stress δ/2 = angle of twist σ1 σ2 σ3 Total Strain Ao strain /E - /E - /E L ε1 σ1 νσ2 νσ3 in x o Lo wo ε2 -νσ1/E σ2/E -νσ3/E in y ε3 -νσ1/E -νσ2/E σ3/E in z 2r Poisson δ o w /2 In x-direction, total linear strain is: 1 ε1 = {σ 1 − ν(σ 2 +σ 3 )} E εz , σz • Material, geometric, and loading parameters all contribute to . 1 • Larger elastic moduli minimize elastic deflection. or = {(1+ ν)σ 1 − ν(σ 1 +σ 2 +σ 3 )} E 27 28 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

7 Complex State of Stress and Strain in 3-D Solid Plastic (Permanent) Deformation • Hooke’s Law and Poisson effect gives total linear strain: (at lower , i.e. T < Tmelt/3) 1 1 ε1 = {σ 1 − ν(σ 2 +σ 3 )} or {(1+ ν)σ 1 − ν(σ 1 +σ 2 +σ 3 )} E E • Simple tension test: Elastic+Plastic • For uniaxial tension test σ = σ =0, so ε = σ /E and ε =ε = –νε . 1 2 3 3 1 2 3 engineering stress, σ at larger stress

• Hydrostatic Pressure: Elastic σ +σ +σ Tr 1 1 2 3 σ initially P = σ Hyd = = ε1 = {(1+ ν)σ 1 − 3νP} 3 3 E permanent (plastic) after load is removed

• For volume (V=l1l2l3) strain, ΔV/V = ε1+ ε2+ ε3 = (1-2ν)σ3/E ΔV P ε engineering strain, ε = 3(1− 2ν) p V E plastic strain Adapted from Fig. 7.10 (a), , B or K: P = –K ΔV/V so K = E/3(1-2ν) (sec. 7.5) Callister & Rethwisch 3e. 29 30 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Yield Stress, σY Yield Points and σYS

• Stress where noticeable plastic deformation occurs. • Yield-point phenomenon occurs when elastic plastic transition is abrupt. For metals agreed upon 0.2% when εp = 0.002 No offset method required. tensile stress, σ • P is the proportional limit where deviation from linear behavior occurs. • In , this effect is seen when σy dislocations start to move and unbind Strain off-set method for Yield Stress for interstitial solute. P • Start at 0.2% strain (for most metals).

• Draw line parallel to elastic curve (slope of E). • Lower yield point taken as σY. • σY is value of stress where dotted line crosses stress-strain curve (dashed line). • Jagged curve at lower yield point Elastic For steels, take the avg. occurs when solute binds recovery stress of lower yield point since less sensitive to and dislocation unbinding again, until testing methods. Note: for 2 in. sample work- begins to occur. Eng. strain, ε ε = 0.002 = Δz/z ε p = 0.002 ∴ Δz = 0.004 in 31 32 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

8 Stress-Strain in Polymers Compare Yield Stress, σYS

• 3 different types of behavior For plastic polymers: σy(ceramics) • YS at maximum stress just Brittle after elastic region. >>σy(metals) • TS is stress at fracture! >> σy(polymers)

plastic Room T values

Highly elastic Based on data in Table B4, Callister 6e. a = annealed hr = hot rolled ag = aged cd = cold drawn cw = cold worked • Highly elastic polymers: qt = quenched & tempered • 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 33 34 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

(Ultimate) Tensile Strength, σTS Metals: Tensile Strength, σTS

• Maximum possible engineering stress in tension. For Metals: max. stress in tension when necking starts, which is the metals work-hardening tendencies vis-à-vis TS those that initiate instabilities. F = fracture or σy dF = 0 Maximum eng. Stress (at necking) ultimate strength dσ dA T = − i dF = 0 = σT dAi + AidσT σT Ai stress Typical response of a metal Neck – acts Fractional fractional Increase in decrease engineering as stress decreased force due to Increased force due to Flow stress in load- concentrator decrease in gage diameter increase in applied stress strain bearing engineering strain area • Metals: occurs when necking starts. At the point where these two competing changes dσ dA d dσ T = − i = l i ≡ dε ⇒ T = σ • Ceramics: occurs when crack propagation starts. in force equal, there is permanent neck. T T σT Ai l i dεT • Polymers: occurs when polymer backbones are If σ = K(ε )n, then n = ε Determined by slope of “true stress” - “true strain” curve T T T n = strain-hardening coefficient aligned and about to break. 35 36 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

9 Compare Tensile Strength, σTS Example for Metals: Determine E, YS, and TS Graphite/ Metals/ Composites/ Ceramics/ Polymers Alloys fibers Semicond Stress-Strain for Brass • Young’s Modulus, E (bond stretch) 5000 C fibers Room T values Aramid fib 3000 E-glass fib σ −σ (150 − 0)MPa ) E = 2 1 = = 93.8GPa 2000 TS(ceram) Steel (4140) qt ε2 −ε1 0.0016 − 0 AFRE(|| fiber) 1000 W (pure) Diamond GFRE(|| fiber) ~TS Ti (5Al-2.5Sn) a (met) MPa CFRE(|| fiber) Steel (4140) a ( Cu (71500) cw Si nitride hr ~ TS • 0ffset Yield-Stress, YS (plastic deformation) Cu (71500) (comp) € Steel (1020) Al oxide 300 Al (6061) ag Ti (pure) a >> TS YS = 250 MPa TS 200 Ta (pure) (poly) Al (6061) a 100 Si crystal wood(|| fiber) <100> Nylon 6,6 Glass-soda PC PET • Max. Load from Tensile Strength TS 40 PVC GFRE( fiber) Concrete PP 30 CFRE( fiber) € 2 AFRE( fiber) d 0 HDPE Fmax = σTS A0 = σTS π  20 Graphite  2 

strength, LDPE  2 Based on data in Table B4, 12.8x10−3m 10 = 450MPa  π = 57,900N Callister & Rethwisch 3e.  2  • Gage is 250 mm (10 in) in length and 12.8 mm   (0.505 in) in diameter. wood ( fiber) • Subject to tensile stress of 345 MPa (50 ksi) • Change in length at Point A, Δl = εl0

Tensile € Δl = εl0 = (0.06)250 mm = 15 mm 1 37 38 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Temperature matters (see Failure) Ductility (%EL and %RA)

Most metals are ductile at RT and above, but can become brittle at low T • Plastic tensile strain at failure: L − L %EL = f o x100 L o bcc Fe

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

A − A • Another ductility measure: %RA = o f x100 A o • Note: %RA and %EL are often comparable. - Reason: crystal slip does not change material volume. cup-and-cone fracture in Al brittle fracture in mild steel - %RA > %EL possible if internal voids form in neck. 39 40 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

10 Toughness Resilience, Ur

• Energy to break a unit volume of material, • Resilience is capacity to absorb energy when deformed elastically 2 or absorb energy to fracture. and recover all energy when unloaded (=σ YS/2E). • Approximate as area under the stress-strain curve. • Approximate as area under the elastic stress-strain curve.

Engineering small toughness (ceramics) tensile ε large toughness (metals) U = ∫ Y σ dε stress, σ r o ε ε 2 σ ε σ 2 ε very small toughness = ∫ Y Eε dε ~ E Y = Y Y = Y U = f σ dε (unreinforced polymers) o 2 2 2E T ∫o Area up to 0.2% strain

Engineering tensile strain, ε If linear elastic

Brittle fracture: Ductile fracture: elastic + plastic energy 41 42 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Elastic Strain Recovery 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. materials are more brittle than metals. Why? D σyi • Consider mechanism of deformation

σyo – In crystalline materials, by dislocation motion 2. Unload – In highly ionic solids, dislocation motion is difficult • few slip systems

Stress • resistance to motion of ions of like charge (e.g., anions) past one another.

1. Load 3. Reapply load Strain

Adapted from Fig. 7.17, Elastic strain Callister & Rethwisch 3e. recovery 43 44 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

11 Strength of Ceramics - Strength of Ceramics - Flexural Strength • RT behavior is usually elastic with brittle failure. • 3-point bend test employed for RT Flexural strength. • 3-point bend test employed (tensile test not best for brittle materials).

Al O cross section 2 3 d R b δ = midpoint rect. circ. deflection

• Determine elastic modulus according to: Rectangular cross-section d 3F L F 3 f • Typical values: F L σ fs = 2 x E (rect. cross section) b 2bd Material σfs (MPa) E(GPa) = 3 F δ 4bd Si nitride 250-1000 304 slope = Circular cross-section 3 L= length between Si carbide 100-820 345 δ F L 8F L load pts R f Al oxide 275-700 393 E = (circ. cross section) σ = b = width δ 4 fs 3 δ 12πR πd d = height or glass (soda-lime) 69 69 linear-elastic behavior diameter Data from Table 7.2, Callister & Rethwisch 3e. 45 46 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Stress-Strain in Polymers Influence of T and on Thermoplastics

brittle polymer σ • Decreasing T... (MPa) -- increases E 80 4°C Plots for -- increases TS semicrystalline -- decreases %EL 60 PMMA (Plexiglas) plastic 20°C elastomer • Increasing 40 40°C elastic moduli strain rate... – less than for metals Adapted from Fig. 7.22, Callister & Rethwisch 3e. -- same effects 20 to 1.3 as decreasing T. 60°C 0 0 0.1 0.2 ε 0.3 Adapted from Fig. 7.24, Callister & Rethwisch 3e. (Fig. 7.24 is from T.S. • Fracture strengths of polymers ~ 10% of those for metals. Carswell and J.K. Nason, 'Effect of Environmental Conditions on the Mechanical Properties of Organic ", Symposium on Plastics, • Deformation strains for polymers > 1000%. American Society for Testing and Materials, Philadelphia, PA, 1944.) – for most metals, deformation strains < 10%. 47 48 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

12 Stress-Strain in Polymers Time-dependent deformation in Polymers

• Stress relaxation test: • Large decrease in Er for T > Tg. • Necking appears along • Mechanism unlike metals, necking -strain in tension to (amorphous entire sample after YS! due to alignment of crystallites. εο 5 10 rigid solid polystyrene) and hold. E (10 s) (small relax) Fig. 7.28, Callister & r 3 Load vertical in MPa10 Rethwisch 3e. - observe decrease in transition (Fig. 7.28 from A.V. 101 region Tobolsky, Properties stress with time. and Structures of -1 Polymers, Wiley and 10 Sons, Inc., 1960.) tensile test viscous -3 (large relax) εo strain 10 60 100 140 180 T(°C) σ(t) Tg time • Representative Tg values (°C): • Relaxation modulus: PE (low density) - 110 •Align crystalline sections •After YS, necking PE (high density) - 90 by straightening chains in proceeds by σ(t) PVC + 87 the amorphous sections Er (t) = unraveling; hence, ε PS +100 neck propagates, o PC +150 See Chpt 8 unlike in metals! Selected values from Table 11.3, Callister & Rethwisch 3e. 49 50 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

True Stress and Strain Why use True Strain?

F • Up to YS, there is volume change due to Poisson Effect! Engineering stress σ = Initial area always Relation before necking A0 • In a metal, from YS and TS, there is plastic deformation, as F σT = σ 1+ε True stress σt = instantaneous area ( ) dislocations move atoms by slip, but ΔV=0 (volume is constant). Ai A A ε = ln 1+ε 0l0 = i l i € li T ( ) True strain ε = ln Relative change − + t ε = ln l i →ln l i l0 l0 = ln(1+ε) l0 Necking: 3D state of stress! t l0 l0 € Test length Eng. Eng. Eng. 0-1-2-3 0-3 € 0 2.00 Strain 1 2.20 0.1 2 2.42 0.1 Sum of incremental strain 3 2.662 0.1 0.662/2.0 TOTAL 0.3 0.331 does NOT equal total strain!

True 2.2 2.42 2.662 2.662 Sum of incremental strain ln ln ln ln Strain εt = + + = 2.0 2.20 2.42 2.00 does equal total strain. 51 52 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

13 Hardening Using Work-Hardening

• An increase in σy due to plastic deformation. Influence of “cold working” on low-. σ large hardening 2nd drawn σy 1 1st drawn 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 54 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Hardness Hardness: Measurement • Resistance to permanently indenting the surface. • Rockwell • Large hardness means: --resistance to plastic deformation or cracking in compression. – No major sample damage --better wear properties. – Each scale runs to 130 (useful in range 20-100). – Minor load 10 kg – Major load 60 (A), 100 (B) & 150 (C) kg • 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. 55 56 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

14 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 x – Mean x = n n 1 n  2 2 x x – Standard Deviation Σ( i − )  s =    n −1    where n is the number of data points 57 58 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

Design Safety Factors Summary

• Design uncertainties mean we do not push the limit. • Factor of safety, N (sometime given as S) • Stress and strain: These are size-independent Often N is between 1.2 and 4 measures of load and , respectively. σy σ = • Elastic behavior: This reversible behavior often working N shows a linear relation between stress and strain. To minimize deformation, select a material with a • Ex: Calculate diameter, d, to ensure that no yielding occurs large elastic modulus (E or G). in the 1045 carbon steel rod. Use safety factor of 5. • Plastic behavior: This permanent deformation behavior occurs when the tensile (or compressive) σ y uniaxial stress reaches sy. σworking = N • Toughness: The energy needed to break a unit 220,000N volume of material.   5 • Ductility: The plastic strain at failure. π d2 / 4   d = 0.067 m = 6.7 cm 59 60 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008

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