Chapter 7. Mechanical Properties

Transcription

1 Chapter 7 Mechanical Properties

2 Chapter 7 Plastic Deformation, Ductility and Toughness

3 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 do dislocations cause permanent deformation? What materials are most resistant to permanent deformation? Toughness and ductility: What are they and how do we measure them? Ceramic Materials: What special provisions/tests are made for ceramic materials?

4 Loading Forms

5 Common States of Stress Simple tension: cable F F Ao = cross sectional area (when unloaded) F A o Torsion (a form of shear): drive shaft Ski lift A c M 2R M Fs A o F s A o Note: = M/A c R here.

6 Other Common Stress States_1 Simple compression: A o Canyon Bridge, Los Alamos, NM (photo courtesy P.M. Anderson) Balanced Rock, Arches National Park (photo courtesy P.M. Anderson) F A o Note: compressive structure member (s < 0 here).

7 Other Common Stress States_2 Bi-axial tension: Hydrostatic compression: Pressurized tank (photo courtesy P.M. Anderson) Fish under water (photo courtesy P.M. Anderson) > 0 z > 0 s< 0 h

8 Elastic Deformation 1. Initial 2. Small load 3. Unload bonds stretch return to initial F Elastic means reversible! F Cf. Anelasticity Linearelastic Non-Linearelastic

9 Engineering Stress F t F t F Area, A Area, A Fs F t = F t lb A 2 f N = or 2 o in m = F s A o Fs F F t original area before loading Stress has units: N/m 2 or lb f /in 2

10 Engineering Strain Tensile strain: L o w o /2 L o Lateral strain: L L w o Shear strain: x L /2 = x/y = tan y 90º 90º - Strain is always dimensionless.

11 Engineering Stress & Strain (1) Tensile stress σ = F A o + tensile - compressive psi = MPa 1 MPa = 1 MN/m 2 (2) Tensile strain ε = li - l l 0 0 = Δl l 0 (3) Shear stress τ = F A o (4) Shear strain γ = x h = tan θ

12 Geometric Considerations of Stress State σ' = σ cos θ = 1+ cos 2 σ 2 2 θ 90- τ' = σ sin θ cosθ = σ sin 2θ 2 Area : A/cos

13 Stress-Strain Testing Typical tensile test machine Typical tensile specimen extensometer specimen gauge length

14 Stress & Strain Curves Region I Region II Region III

15 Yield Point Phenomena

16 Linear Elastic Properties Modulus of Elasticity, E: (also known as Young's modulus) Hooke's Law: = E F E Linearelastic F simple tension test

17 Elastic Deformation_1 Slope of stress strain plot (which is proportional to the elastic modulus) depends on bond strength of metal

18 Elastic Deformation_2 The stress is linearly proportional to strain σ = Eε The slope of curve : Modulus of elasticity Young's modulus ( Typical metals ~ 10 9 psi ) (1) Deformation is recoverable. (2) Deformation is linear. ( Hooke's law ) (3) Modulus determined by atomic bonding. (atom type and crystal structure) τ = Gγ ( G : shear modulus )

19 Young s Moduli: Comparison E(GPa) 10 9 Pa Metals Alloys Tungsten Molybdenum Steel, Ni Tantalum Platinum Cu alloys Zinc, Ti Silver, Gold Aluminum Magnesium, Tin Graphite Ceramics Semicond Diamond Si carbide Al oxide Si nitride <111> Si crystal <100> Glass-soda Concrete Graphite Polymers Polyester PET PS PC Composites /fibers Carbon fibers only CFRE( fibers)* Aramid fibers only AFRE( fibers)* Glass fibers only GFRE( fibers)* GFRE* CFRE* GFRE( fibers)* CFRE( fibers)* AFRE( fibers)* Epoxy only Eceramics > Emetals >> Epolymers 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 PP HDPE PTFE Wood( grain) 0.2 LDPE

20 Non-elastic Behaviors

21 Poisson s Ratio_1 Poisson's ratio, : ε v = - ε L L metals: ~ 0.33 ceramics: ~ 0.25 polymers: ~ 0.40 : dimensionless -

22 Poisson s Ratio_2 elongation compression ν = - lateral strain axial strain σ Z = Eε Z ε x = ε y = - νε Z ν= - ε ε x z = - ε ε y z Poisson's ratio is determined by two opposing factor : (1) Conservation of volume (2) Conservation of bond angles rigid body ν= 0 diamond ν= 0.2 soft body ν= 0.5 natural rubber ν= 0.49 most structural materials ν 1/3

23 Other Elastic Properties Elastic Shear modulus, G: = G G M simple torsion test M Elastic Bulk modulus, K: P = -K ΔV V o Special relations for isotropic materials: P K V V o P P P Pressure test: Vol. init = V o Vol. chg = V G = E 2(1 +ν) K = E 3(1-2ν)

24 Useful Linear Elastic Relationships Simple tension: Simple torsion: Ao FL o EA o F w o L Fw o EA o /2 L o 2ML o r o 4 G M = moment = angle of twist L o 2r L /2 o Material, geometric, and loading parameters all contribute to deflection. Larger elastic moduli minimize elastic deflection.

25 Anelasticity - Time dependent elastic deformation - Elastic deformation continue after the stress application, and finite time is required for complete recovery after unloading. - Viscoelastic behavior of polymers, mud.

26 Plastic Deformation Deformation is plastic by dislocation movements which are not recoverable. Plastic deformation P : proportional limit σ y : yield strength ultimate tensile strength : 50MPa ~3,000MPa work hardening ( iron wire once bended )

27 Elastic Strain Recovery

28 Plastic (permanent) Deformation (at lower temperatures, i.e. T < T melt /3) Simple tension test: engineering stress, Elastic+Plastic at larger stress Elastic initially permanent (plastic) after load is removed p engineering strain, plastic strain

29 Yield Strength, σy Stress at which noticeable plastic deformation has occurred. when p = (0.2%) tensile stress, y = yield strength y Note: for 2 inch sample = = z/z z = in p = engineering strain,

30 Yield Strength: Comparison 2000 Metals/ Alloys Steel (4140) qt Graphite/ Ceramics/ Semicond Polymers Composites/ fibers Yield strength, y (MPa) Ti (5Al-2.5Sn) W (pure) a Cu (71500) Mo (pure) cw Steel (4140) a Steel (1020) cd Al (6061) ag Steel (1020) hr Ti (pure) Ta (pure) a Cu (71500) hr Al (6061) a Hard to measure, since in tension, fracture usually occurs before yield. dry PC Nylon 6,6 PET humid PVC PP HDPE Hard to measure, in ceramic matrix and epoxy matrix composites, since in tension, fracture usually occurs before yield. Room T values a = annealed hr = hot rolled ag = aged cd = cold drawn cw = cold worked qt = quenched & tempered 10 Tin (pure) LDPE

31 Tensile Strength, TS Maximum stress on engineering stress-strain curve. TS y engineering stress Typical response of a metal strain engineering strain Metals: occurs when noticeable necking starts. Polymers: occurs when polymer backbone chains are aligned and about to break. * Cermics:?? F = fracture or ultimate strength Neck acts as stress concentrator

32 Tensile Strength: Comparison Tensile strength, TS (MPa) Metals/ Alloys Steel (4140)qt W (pure) Ti (5Al-2.5Sn)a Steel (4140)a Cu (71500)cw Cu (71500)hr Steel (1020) Al (6061)ag Ti (pure)a Ta (pure) Al (6061)a Graphite/ Ceramics/ Semicond Diamond Si nitride Al oxide Si crystal <100> Glass-soda Concrete Graphite Polymers Nylon 6,6 PC PET PVC PP LDPE HDPE Composites/ fibers C fibers Aramid fib E-glass fib TS(ceram) AFRE( fiber) GFRE( fiber) CFRE( fiber) wood( fiber) GFRE( fiber) CFRE( fiber) AFRE( fiber) wood( fiber) ~TS(met) ~ TS(comp) >> TS(poly) Room T values Based on data in Table B4, Callister 6e. 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.

33 Ductility, %EL Plastic tensile strain at failure: %EL L L f o L o x 100 Engineering tensile stress, smaller %EL larger %EL (ductile If %EL>5%) L o A o A f L f Engineering tensile strain, Another ductility measure: f = x 100 %RA A o - A o A

34 Brittle and Ductile

35 Resilience, U r Ability of a material to store energy Energy stored best in elastic region U r = ε y 0 σdε If we assume a linear stressstrain curve this simplifies to U r 1 2 σ y ε y Adapted from Fig. 6.15, Callister 7e.

36 Toughness Energy to break a unit volume of material Approximate by the area under the stress-strain curve Engineering tensile stress, s small toughness (ceramics) large toughness (metals) very small toughness (unreinforced polymers) Engineering tensile strain, Brittle fracture: elastic energy Ductile fracture: elastic + plastic energy

37 True Stress & Strain Note: S.A. changes when sample stretched True stress True Strain σ T= F A i ε T = ln ( ) l i l o σ ε T T = ln ( ) ( 1+ε) =σ 1+ε ε T = l l 0 dl l =ln( if A σ T l l 0 i li=a0 =σ(1+ε) l )=ln( l, 0 ε 0 T + Δl l 0 = ln ) =ln( l+ε) (1+ε) A i : instantaneous cross-sectional area

38 Hardening An increase in y due to plastic deformation. y1 y0 large hardening small hardening Curve fit to the stress-strain response: T = K ( T ) n hardening exponent: n= 0.15 (some steels) to n = 0.5 (some coppers) true stress (F/A) true strain: ln(l/l o )

39 Measuring Elastic Modulus Room T behavior is usually elastic, with brittle failure. 3-Point Bend Testing often used. --tensile tests are difficult for brittle materials. cross section d R b rect. circ. F L/2 L/2 Location of max tension = midpoint deflect ion

40 Measuring Strength Determine elastic modulus according to: F x slope = F linear-elastic behavior E F L 3 4bd 3 F rect. cross section L 3 12 R 4 circ. cross section Flexural strength: fail 1.5F fs m max L bd 2 F rect. x Fmax F maxl R 3 Typ. values: Material fs(mpa) E(GPa) Si nitride Si carbide Al oxide glass (soda) max

41 Hardness_1 Measure of a materials resistance to localized plastic deformation (1) simple and inexpensive (2) nondestructive Depend on type of indenter - Rockwell Hardness - Brinell Hardness - Knoop and Vickers Hardness

42 Hardness_2 Resistance to permanently indenting the surface. Large hardness means: --resistance to plastic deformation or cracking in compression. --better wear properties. e.g., 10 mm sphere apply known force measure size of indent after removing load D d Smaller indents mean larger hardness most plastics brasses Al alloys easy to machine steels file hard cutting tools nitrided steels diamond increasing hardness

43 Hardness Measurement_1 Rockwell No major sample damage Each scale runs to 130 but only useful in range 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

44 Hardness Measurement_2

45 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

46 Design or Safety Factors Design uncertainties mean we do not push the limit. Factor of safety, N Often N is σ y between 1.2 and 4 σ working = N 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 σ 220,000N ( 2 π d / 4) working 5 = σ y N d = m = 6.7 cm 1045 plain carbon steel: = 310 MPa TS = 565 MPa F = 220,000N L o

47 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.