Mechanical Properties ISSUES TO ADDRESS... • Stress and strain: What are they and why are they used instead of load and deformation? • Elastic behavior: When loads are small, how much deformation occurs? What materials deform least? • Plastic behavior: At what point does permanent deformation occur? What materials are most resistant to permanent deformation? • Toughness and ductility: What are they and how do we measure them?
Engineering Stress • Tensile stress, s: • Shear stress, t: F F F t t Area, A F s Area, A F s F t F F t s F t = F lb N t s = A = f or o 2 2 in m A o original area before loading Stress has units: N/m2 or lbf/in2
Stress and Strain Stress: Force per unit area arising from applied load. Tension, compression, shear, torsion or any combination. Stress = σ = force/area Strain: ε – physical deformation response of a material to stress, e.g., elongation.
Common States of Stress F F A = cross sectional o area (when unloaded) F = s s s A o M F s A o A c F s = t A o M 2R • Simple tension: cable Ski lift(photo courtesy P.M. Anderson) • Torsion (a form of shear): drive shaft
OTHER COMMON STRESS STATES (1) A o Canyon Bridge, Los Alamos, NM (photo courtesy P.M. Anderson) F = s Balanced Rock, Arches A National Park o • Simple compression: Note: compressive structure member (s < 0 here). (photo courtesy P.M. Anderson)
OTHER COMMON STRESS STATES (2) Fish under water s > 0 q s< 0 s > 0 z h • Bi-axial tension: • Hydrostatic compression: Pressurized tank (photo courtesy P.M. Anderson) (photo courtesy P.M. Anderson)
Engineering Strain d /2 d L e = e = L L w L o w o o o d /2 L g = Dx/y= tan q • Tensile strain: • Lateral strain: - d • Shear strain: q x y 90º - q Strain is always dimensionless. 90º Adapted from Fig. 6.1 (a) and (c), Callister 7e.
Elastic Deformation 1. Initial 2. Small load 3. Unload bonds stretch return to initial d F F Linear- elastic Non-Linear- elastic d Elastic means reversible!
Plastic Deformation (Metals) 1. Initial 2. Small load 3. Unload bonds p lanes stretch still & planes sheared shear d plastic d elastic + plastic F F linear linear elastic elastic d d plastic Plastic means permanent!
1. Elastic Materials Return to the their original shape when the applied load is removed. P Unloading Loading d
2. Plastic Materials No deformation is observed up to a certain limit. Once the load passes this limit, permanent deformartions are observed. P Limit Unloading Loading δ Plastic deformation
• Typical tensile specimen Adapted from Fig. 6.2, Callister 7e. specimen extensometer gauge length Stress-Strain Testing • Typical tensile test machine Adapted from Fig. 6.3, Callister 7e. (Fig. 6.3 is taken from H.W. Hayden, W.G. Moffatt, and J. Wulff, The Structure and Properties of Materials, Vol. III, Mechanical Behavior, p. 2, John Wiley and Sons, New York, 1965.)
F • Hooke's Law: s = Ee s E F e simple Linear- tension test elastic Linear Elastic Properties • Modulus of Elasticity, E: (also known as Young's modulus)
Axial strain Width strain Example: Hooke’s Law • Hooke's Law: σ = E ε (linear elastic behavior) Copper sample (305 mm long) is pulled in tension with stress of 276 MPa. If deformation is elastic, what is elongation? For Cu (polycrystalline), E = 110 GPa. Hooke’s law involves axial (parallel to applied tensile load) elastic deformation.
Elastic Deformation F d 1. Initial 2. Small load 3. Unload bonds stretch return to initial d F Linear- elastic Elastic means reversible! Non-Linear- elastic
eL e e n - L n = - e Poisson's ratio, n • Poisson's ratio, n: metals: n ~ 0.33ceramics: n~ 0.25polymers: n ~ 0.40 Units: E: [GPa] or [psi] n: dimensionless • > 0.50 density increases • < 0.50 density decreases (voids form)
Mechanical Properties • Slope of stress strain plot (which is proportional to the elastic modulus) depends on bond strength of metal Adapted from Fig. 6.7, Callister 7e.
Other Elastic Properties M t G simple torsion test g M • Special relation for isotropic materials: E = G 2(1+n) • Elastic Shear modulus, G: t = Gg
Young’s Moduli: Comparison 1200 10 00 Diamond 8 00 6 00 Si carbide 4 00 Tungsten Carbon fibers only Al oxide Molybdenum Si nitride Steel, Ni C FRE(|| fibers)* 2 00 Tantalum <111> Si crystal Platinum A ramid fibers only Cu alloys <100> 10 0 Zinc, Ti 8 0 Silver, Gold A FRE(|| fibers)* Glass - soda Aluminum 6 0 Glass fibers only Magnesium, G FRE(|| fibers)* 4 0 Tin Concrete GFRE* 2 0 CFRE * G FRE( fibers)* G raphite 10 8 C FRE( fibers) * 6 AFRE( fibers) * Polyester 4 PET PS Epoxy only PC 2 PP HDP E 1 0.8 0.6 Wood( grain) PTF E 0.4 LDPE 0.2 Graphite Ceramics Semicond Metals Alloys Composites /fibers Polymers E(GPa) Based on data in Table B2, Callister 7e. Composite data based on reinforced epoxy with 60 vol% of aligned carbon (CFRE), aramid (AFRE), or glass (GFRE) fibers. 109 Pa
Plastic (Permanent) Deformation (at lower temperatures, i.e. T < Tmelt/3) • Simple tension test: Elastic+Plastic at larger stress engineering stress, s Elastic initially permanent (plastic) after load is removed ep engineering strain, e Adapted from Fig. 6.10 (a), Callister 7e. plastic strain
s tensile stress, P Elastic recovery Eng. strain, e = 0.002 ep Yield Stress, σY • Stress where noticeableplastic deformation occurs. When εp = 0.002 For metals agreed upon 0.2% • P is the proportional limit where deviation from linear behavior occurs. σY • 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). Note: for 2 in. sample ε = 0.002 = Δz/z Δz = 0.004 in Adapted from Fig. 6.10 (a), Callister 7e.
For steels, take the avg. stress of lower yield point since less sensitive to testing methods. Yield Points and σYS • Yield-point phenomenon occurs when elastic plastic transition is abrupt. No offset method required. • In steels, this effect is seen when dislocations start to move and unbind for interstitial solute. • Lower yield point taken as σY. • Jagged curve at lower yield point occurs when solute binds dislocation and dislocation unbinding again, until work-hardening begins to occur.
Yield Strength : Comparison Graphite/ Metals/ Composites/ Ceramics/ Polymers Alloys fibers Semicond 2 0 00 qt Steel (4140) 10 00 a Ti (5Al-2.5Sn) W (pure) (MPa) 7 00 6 00 cw Cu (71500) 5 00 Mo (pure) a Steel (4140) 4 00 cd y Steel (1020) s , 3 00 ag Al (6061) hr Steel (1020) 2 00 ¨ a Ti (pure) Ta (pure) Hard to measure, hr Cu (71500) Hard to measure in tension, fracture usually occurs before yield. in ceramic matrix and epoxy matrix composites, since since in tension, fracture usually occurs before yield. 100 dry Yield strength, 70 PC 60 Nylon 6,6 a Al (6061) PET 50 humid PVC 40 PP 30 H DPE 20 LDPE Tin (pure) 10 Room T values Based on data in Table B4, Callister 7e. a = annealed hr = hot rolled ag = aged cd = cold drawn cw = cold worked qt = quenched & tempered
TS F = fracture or ultimate strength Neck – acts as stress concentrator y stress engineering Typical response of a metal strain engineering strain (Ultimate) Tensile Strength, TS • Maximum stress on engineering stress-strain curve. Adapted from Fig. 6.11, Callister 7e. • Metals: occurs when noticeable necking starts. • Polymers: occurs when polymer backbonechains are aligned and about to break.
Graphite/ Metals/ Composites/ Ceramics/ Polymers Alloys fibers Semicond 5000 C fibers Aramid fib 3000 E-glass fib 2000 qt Steel (4140) (MPa) A FRE (|| fiber) 1000 Diamond W (pure) GFRE (|| fiber) a Ti (5Al-2.5Sn) C FRE (|| fiber) a Steel (4140) cw Si nitride Cu (71500) hr Cu (71500) Al oxide Steel (1020) 300 ag Al (6061) a Ti (pure) 200 Ta (pure) a Al (6061) Si crystal wood(|| fiber) 100 <100> strength, TS Nylon 6,6 Glass-soda PET PC PVC GFRE ( fiber) 40 Concrete PP C FRE ( fiber) 30 A FRE( fiber) H DPE Graphite 20 L DPE 10 Tensile wood ( fiber) 1 Tensile Strength : Comparison Room Temp. values Based on data in Table B4, Callister 7e. a = annealed hr = hot rolled ag = aged cd = cold drawn cw = cold worked qt = quenched & tempered AFRE, GFRE, & CFRE = aramid, glass, & carbon fiber-reinforced epoxy composites, with 60 vol% fibers.
Ductility (%EL and %RA) • Plastic tensile strain at failure: Adapted from Fig. 6.13, Callister 7e. • Another ductility measure: • Note: %RA and %EL are often comparable. - Reason: crystal slip does not change material volume. - %RA > %EL possible if internal voids form in neck.
Toughness small toughness (ceramics) E ngineering tensile large toughness (metals) s stress, very small toughness Adapted from Fig. 6.13, Callister 7e. (unreinforced polymers) e Engineering tensile strain, • Energy to break a unit volume of material • Approximate by the area under the stress-strain curve. Brittle fracture: elastic energyDuctile fracture: elastic + plastic energy
1 @ s e U r y y 2 Resilience, Ur • Ability of a material to store energy • Energy stored best in elastic region If we assume a linear stress-strain curve this simplifies to Adapted from Fig. 6.15, Callister 7e.
Elastic Strain Recovery • • Unloading in step 2 allows elastic strain to be recovered from bonds. • Reloading leads to higher YS, due to work-hardening already done Adapted from Fig. 6.17, Callister 7e.
apply known force measure size e.g., of indent after 10 mm sphere removing load Smaller indents d D mean larger hardness. most brasses easy to machine cutting nitrided plastics Al alloys steels file hard tools steels diamond increasing hardness Hardness • Resistance to permanently indenting the surface. • Large hardness means: --resistance to plastic deformation or cracking in compression. --better wear properties. Adapted from Fig. 7.18.
Hardness: Measurement • Rockwell • No major sample damage • Each scale runs to 130 but only 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
Hardness: Measurement Table 6.5
True Stress & Strain Note: S.A. changes when sample stretched • True stress • True Strain Adapted from Fig. 6.16, Callister 7e.
s s large hardening y 1 s y small hardening 0 e hardening exponent: ( ) n n = 0.15 (some steels) s = e K T T to n = 0.5 (some coppers) “true” strain: ln(L/Lo) “true” stress (F/A) Hardening • An increase in sy due to plastic deformation. • Curve fit to the stress-strain response:
Variability in Material Properties • Elastic modulus is material property • Critical properties depend largely on sample flaws (defects, etc.). Large sample to sample variability. • Statistics • Mean • Standard Deviation where n is the number of data points
Design or Safety Factors d 1045 plain carbon steel: L o s = 310 MPa y 5 TS = 565 MPa F = 220,000N • Design uncertainties mean we do not push the limit. • Factor of safety, N Often N is between 1.2 and 4 • Example: Calculate a diameter, d, to ensure that yield does not occur in the 1045 carbon steel rod below. Use a factor of safety of 5. d = 0.067 m = 6.7 cm
Summary • 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 sy. • Toughness: The energy needed to break a unit volume of material. • Ductility: The plastic strain at failure.