![Chapter 6 - 11
Linear Elastic Properties
• Hooke's Law:
σ = E ε
σ
Linear-
elastic
• Modulus of Elasticity, E:
(also known as Young's modulus)
E
ε
• Elastic deformation is nonpermanent and reversible!
– generally valid at small deformations
– linear stress strain curve
compression
tension
Units:
E: [GPa] or [psi]
1 GPa = 109 Pa](https://image.slidesharecdn.com/01-ch06-250520202613-59827d17/85/01-ch0666666666-pdf-classiterrrrdvdvfvbffh-11-320.jpg)
01-ch0666666666 pdf classiterrrrdvdvfvbffh
- 1. Chapter 6 - 1 ISSUES TO ADDRESS... • When a metal is exposed to mechanical forces, what parameters are used to express force magnitude and degree of deformation? • What is the distinction between elastic and plastic deformations? • How are the following mechanical characteristics of metals measured? (a) Stiffness (b) Strength (c) Ductility (d) Hardness • What parameters are used to quantify these properties? Chapter 6: Mechanical Properties of Metals
- 2. Chapter 6 - 2 Common States of Stress • Simple tension: cable Ski lift (photo courtesy P.M. Anderson) σ = F Ao = cross-sectional area of cable (with no load) F = force F Tensile stress = σ A0
- 3. Chapter 6 - 3 Common States of Stress (cont.) • Torsion (a form of shear): drive shaft Ski lift (photo courtesy P.M. Anderson) s τ = F A M M = moment 2R Ac As F M AcR = AC = cross-sectional area of drive shaft (with no load)
- 4. Chapter 6 - 4 (photo courtesy P.M. Anderson) Canyon Bridge, Los Alamos, NM o σ = F A • Simple compression: Note: structure members are under compression (F < 0 and σ < 0). (photo courtesy P.M. Anderson) OTHER COMMON STRESS STATES (i) Ao Balanced Rock, Arches National Park
- 5. Chapter 6 - 5 • Bi-axial tension: • Hydrostatic compression: Pressurized tank σ < 0 h (photo courtesy P.M. Anderson) (photo courtesy P.M. Anderson) OTHER COMMON STRESS STATES (ii) Fish under water σz > 0 σθ > 0
- 6. Chapter 6 - 6 Stress-Strain Testing • Typical tensile test machine Fig. 6.3, Callister & Rethwisch 10e. (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.) specimen extensometer • Typical tensile specimen Fig. 6.2, Callister & Rethwisch 10e.
- 7. Chapter 6 - 7 Units for stress: MPa = 106 Pa = 106 N/m2 or lbf /in2 Engineering Stress • Shear stress, τ: Area, Ao F F τ = F Ao • Tensile stress, σ: original cross-sectional area before loading σ = F Ao Area, Ao F F
- 8. Chapter 6 - 8 Engineering Strain • Tensile strain (εz): • Lateral strain (εx): Both tensile and shear strain are dimensionless • Shear strain (γ): θ y x γ = Δx/y = tan θ εz = Δl lo Δl/2 lo do - Δd εx = d0 Δd/2
- 9. Chapter 6 - 9 • Simple tension: Δl = Flo EAo Δd = - ν Fdo EAo • Deflection is dependent on material, geometric, and loading parameters. • Materials with large elastic moduli deform less Useful Linear Elastic Relationships Ao
- 10. Chapter 6 - 10 • Simple torsion Useful Linear Elastic Relationships (cont.) α = 32Mlo πdo 4 G M = moment α = angle of twist do lo
- 11. Chapter 6 - 11 Linear Elastic Properties • Hooke's Law: σ = E ε σ Linear- elastic • Modulus of Elasticity, E: (also known as Young's modulus) E ε • Elastic deformation is nonpermanent and reversible! – generally valid at small deformations – linear stress strain curve compression tension Units: E: [GPa] or [psi] 1 GPa = 109 Pa
- 12. Chapter 6 - 12 Metals Alloys Graphite Ceramics Semicond Polymers Composites /fibers E(GPa) Based on data in Table B.2, Callister & Rethwisch 10e. Composite data based on reinforced epoxy with 60 vol% of aligned carbon (CFRE), aramid (AFRE), or glass (GFRE) fibers. Elastic Modulus – Comparison of Material Types 0.2 8 0.6 1 Magnesium, Aluminum Platinum Silver, Gold Tantalum Zinc, Ti Steel, Ni Molybdenum Graphite Si crystal Glass -soda Concrete Si nitride Al oxide PC Wood( grain) AFRE( fibers) * CFRE* GFRE* Glass fibers only Carbon fibers only Aramid fibers only Epoxy only 0.4 0.8 2 4 6 10 20 40 60 80 100 200 600 800 1000 1200 400 Tin Cu alloys Tungsten <100> <111> Si carbide Diamond PTFE HDPE LDPE PP Polyester PS PET CFRE( fibers) * GFRE( fibers)* GFRE(|| fibers)* AFRE(|| fibers)* CFRE(|| fibers)*
- 13. Chapter 6 - 13 Elastic deformation is nonpermanent and reversible! Elastic Deformation 2. Small load Force, F Δl bonds stretch 1. Initial 3. Unload return to initial F Δl Linear- elastic Non-Linear- elastic Atomic configurations—before, during, after load (force) application = metal atom
- 14. Chapter 6 - 14 Influence of Bonding Forces • Elastic modulus depends on interatomic bonding forces • Modulus proportional to slope of interatomic force- interatomic separation curve Fig. 6.7, Callister & Rethwisch 10e. Interatomic Separation r Interatomic Force F Stongly bonded – larger E Weakly bonded – smaller E dF dr æ è ç ö ø ÷ ro
- 15. Chapter 6 - 15 Poisson's ratio • Poisson's ratio, ν: Units: ν: dimensionless For most metals, ceramics and polymers: 0.15 < ν ≤ 0.50 metals: ν ~ 0.33 ceramics: ν ~ 0.25 polymers: ν ~ 0.40 εz εx -ν ε ν = - z εx compression tension
- 16. Chapter 6 - 16 • Elastic Shear modulus, G: τ G γ τ = G γ Other Elastic Properties simple torsion test M M • Elastic constant relationships for isotropic materials: 2(1+ν) E G = 3(1-2ν) E K = = moment 0 • Elastic Bulk modulus, K: Pressure test: Init. vol. = Vo Vol. chg. = ΔV P = P P P = -K ΔV Vo P -ΔV K Vo hydrostatic pressure 0
- 17. Chapter 6 - 17 Plastic deformation is permanent and nonrecoverable. Plastic Deformation (Metals) F Δl linear elastic linear elastic 3. Unload atoms remain displaced Δlplastic 1. Initial = metal atom 2. Apply load F Δlelastic + bonds stretch & atoms displaced Δlplastic Δlplastic
- 18. Chapter 6 - 18 Plastic Deformation • Stress-strain plot for simple tension test: Adapted from Fig. 6.10 (a), Callister & Rethwisch 10e. • Plastic Deformation is permanent and nonrecoverable stress, σ strain, ε Stressed into Plastic Region, Elastic + Plastic εp plastic strain Elastic Deformation Stress Removed, Plastic Deformation Remains
- 19. Chapter 6 - 19 • Yield strength = stress at which noticeable plastic deformation has occurred Yield Strength Adapted from Fig. 6.10 (a), Callister & Rethwisch 10e. • Transition from elastic to plastic deformation is gradual σy = yield strength Note: for 5 cm sample ε = 0.002 = Δz/z Δz = 0.01 cm when εp = 0.002 σ (stress) ε (strain) σy εp = 0.002
- 20. Chapter 6 - 20 Room temperature values Based on data in Table B.4, Callister & Rethwisch 10e. a = annealed hr = hot rolled ag = aged cd = cold drawn cw = cold worked qt = quenched & tempered Yield Strength – Comparison of Material Types Graphite/ Ceramics/ Semicond Metals/ Alloys Composites/ fibers Polymers Yield strength, σ y (MPa) PVC Hard to measure , since in tension, fracture usually occurs before yield. Nylon 6,6 LDPE 70 20 40 60 50 100 10 30 200 300 400 500 600 700 1000 2000 Tin (pure) Al (6061)a Al (6061)ag Cu (71500)hr Ta (pure) Ti (pure)a Steel (1020)hr Steel (1020)cd Steel (4140)a Steel (4140)qt Ti (5Al-2.5Sn) a W (pure) Mo (pure) Cu (71500)cw Hard to measure, in ceramic matrix and epoxy matrix composites, since in tension, fracture usually occurs before yield. HDPE PP humid dry PC PET ¨
- 21. Chapter 6 - VMSE: Virtual Tensile Testing 21
- 22. Chapter 6 - 22 Tensile Strength • Metals: Maximum on stress-strain curve appears at the onset of noticeable necking Adapted from Fig. 6.11, Callister & Rethwisch 10e. y strain Typical response of a metal Fracture strength Neck – acts as stress concentrator engineering TS stress engineering strain • Tensile strength (TS) = maximum stress on engineering stress-strain curve.
- 23. Chapter 6 - 23 Tensile Strength: Comparison of Material Types Si crystal <100> Graphite/ Ceramics/ Semicond Metals/ Alloys Composites/ fibers Polymers Tensile strength, TS (MPa) PVC Nylon 6,6 10 100 200 300 1000 Al (6061)a Al (6061)ag Cu (71500)hr Ta (pure) Ti (pure)a Steel (1020) Steel (4140)a Steel (4140)qt Ti (5Al-2.5Sn)a W (pure) Cu (71500)cw LDPE PP PC PET 20 30 40 2000 3000 5000 Graphite Al oxide Concrete Diamond Glass-soda Si nitride HDPE wood( fiber) wood(|| fiber) 1 GFRE(|| fiber) GFRE( fiber) CFRE(|| fiber) CFRE( fiber) AFRE(|| fiber) AFRE( fiber) E-glass fib C fibers Aramid fib Based on data in Table B4, Callister & Rethwisch 10e. 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. Room temperature values
- 24. Chapter 6 - 24 • Ductility = amount of plastic deformation at failure: • Specification of ductility -- Percent elongation: -- Percent reduction in area: Ductility lf Ao Af lo Adapted from Fig. 6.13, Callister & Rethwisch 10e. tensile strain, ε tensile stress, σ low ductility high ductility
- 25. Chapter 6 - = s de 0 ey ò 25 Resilience • Resilience—ability of a material to absorb energy during elastic deformation • Energy recovered when load released • Resilience specified by modulus of resilience, Ur Ur = Area under stress-strain curve to yielding If assume a linear stress-strain curve this simplifies to y y r 2 1 U ε σ ≅ εy Fig. 6.15, Callister & Rethwisch 10e.
- 26. Chapter 6 - 26 • Toughness of a material is expressed in several contexts • For this chapter, toughness = amount of energy absorbed before fracture • Approximate by area under the stress-strain curve—units of energy per unit volume Toughness Brittle fracture: small toughness Ductile fracture: large toughness very small toughness (unreinforced polymers) tensile strain, ε tensile stress, σ small toughness (ceramics) large toughness (metals)
- 27. Chapter 6 - sT = F Ai eT = ln ℓ i ℓ o ( ) sT = s 1+e ( ) eT = ln 1+e ( ) 27 True Stress & Strain • True stress • True strain Adapted from Fig. 6.16, Callister & Rethwisch 10e. where Ai = instantaneous cross-sectional area Conversion Equations: valid only to the onset of necking
- 28. Chapter 6 - 28 True Stress-True Strain Relationship • Most alloys, between point of yielding and onset of necking −− n and K values depend on alloy and treatment −− n = strain-hardening exponent −− n < 1.0 • σT vs. εT -- influence of n. σT = K εT ( )n σT εT larger n small n
- 29. Chapter 6 - 29 Elastic Strain Recovery Fig. 6.17, Callister & Rethwisch 10e. Stress Strain 3. Reapply load 2. Unload D Elastic strain recovery 1. Load initial yield strength = σyo yield strength for 2nd deformation = σyi
- 30. Chapter 6 - 30 Hardness • Measure of resistance to surface plastic deformation— dent or scratch. • Large hardness means: -- high resistance to deformation from compressive loads. -- better wear properties. one indenter type- 10 mm sphere apply known force measure size of indent after removing load d D Smaller indents mean larger hardness. increasing hardness most plastics brasses Al alloys easy to machine steels file hard cutting tools nitrided steels diamond
- 31. Chapter 6 - 31 Measurement of Hardness • Examples: – Rockwell A Scale – 60 kg load/diamond indenter – Superficial Rockwell 15T Scale – 15 kg load/ 1/16 in. indenter • Rockwell hardness designation: (hardness reading) HR • Examples: 57 HRA; 63 HR15T • Hardness range for each scale: 0−130 HR; useful range: 20−100 HR Rockwell Hardness • Several scales—combination of load magnitude, indenter size
- 32. Chapter 6 - 32 Measurement of Hardness (cont.) • Single scale • Brinell hardness designation: (hardness reading) HB Brinell Hardness – P = load (kg) – 500 kg P 3000 kg (500 kg increments) • Relationships—Brinell hardness & tensile strength – TS (psia) = 500 x HB – TS (MPa) = 3.45 x HB
- 33. Chapter 6 - s = S xi - x ( ) 2 n -1 é ë ê ê ê ù û ú ú ú 1 2 x = Sxi n 33 Variability of Material Properties • Measured material properties—always scatter in values for same material • Statistical treatments • Typical value—take average value, for some parameter x: • Degree of scatter—use standard deviation, s n = number of measurements xi = specific measured value i = 1 n i = 1 n
- 34. Chapter 6 - 34 • Because of design uncertainties allowances must be made to protect against unanticipated failure • For structural applications, to protect against possibility of failure—use working stress, σw, and a factor of safety, N Depending on application, N is between 1.2 and 4 Design/Safety Factors yield strength sw = sy N
- 35. Chapter 6 - 35 Example Problem: A cylindrical rod, to be constructed from a steel that has a yield strength of 310 MPa, is to withstand a load of 220,000 N without yielding. Assuming a value of 4 for N, specify a suitable bar diameter. Design/Safety Factors (cont.) 220,000 N p d 2 æ è ç ö ø ÷ 2 4 sw = sy N Steel rod: σy = 310 MPa F = 220,000 N d d = 0.060 m = 60 mm Solving for the rod diameter d yields
- 36. Chapter 6 - 36 • Applied mechanical force—normalized to stress • Elastic deformation: −−non-permanent; occurs at low levels of stress −−stress-strain behavior is linear Summary • Plastic deformation −−permanent; occurs at higher levels of stress −−stress-strain behavior is nonlinear • Degree of deformation—normalized to strain • Stiffness—a material's resistance to elastic deformation −−elastic (or Young's) modulus
- 37. Chapter 6 - 37 • Strength—a material's resistance to plastic deformation −−yield and tensile strengths • Ductility—amount of plastic deformation at failure −−percents elongation, reduction in area Summary (cont.) • Hardness—resistance to localized surface deformation & compressive stresses −−Rockwell, Brinell hardnesses
- 38. Chapter 6 - 38 Core Problems: Self-help Problems: ANNOUNCEMENTS Reading: