Chapter 7: Mechanical Properties Chapter 7: Mechanical Properties Why mechanical properties? ISSUES TO ADDRESS... • Need to design materials that will withstand • Stress and strain: Normalized force 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 • Plastic behavior: dislocations and permanent deformation skyscrapers Yield Strength : σYS [MPa] (permanent deformation) Ulitmate Tensile Strength : σ [MPa] (fracture) TS • Toughness, ductility, resilience, toughness, and hardness: 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 Shear stress τ = shear e A Strain: physical deformation response of a o material to stress, e.g., elongation. strain γ = tanθ Elastic τ = Gγ 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 Metals 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., concrete) 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 Elasticity 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) 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 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 Energy 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 Ceramics • Recall: Slope of stress strain plot (proportional to the E) • Elastic Behavior And Effects of Porosity 2 depends on bond strength of metal 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 Polymers: Tangent and Secant Modulus • 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) initial E Glass (pyrex) 70 Stress (MPa) SiC (fused or sintered) 207-483 secant E Graphite (molded) ~12 tangent E High modulus C-fiber 400 Carbon Nanotubes ~1000 Normalize by density, 20x steel 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 /fibers Semicond • Poisson's ratio, ν: Units: ν dimensionless 1200 1000 800 Diamond 600 Si carbide width strain Δw / w εL 400 Tungsten 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 Silver, Gold Glass-soda AFRE(|| fibers)* 60 Aluminum Glass fibers only Magnesium, 40 Tin 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 Wood( 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 Solid Argon: 0.25 Δw / w ν = − = −1 Covalent Solids: Si Ge Al2O3 TiC Δl / l 0.27 0.28 0.23 0.19 generic value ~ 1/4 € • Volume (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 Δw / w (− 1 Δ / ) Elastomer: Hard Rubber (Ebonite) 0.39 (Natural) 0.49 Hence, ν = − = − 2 l l = 1/ 2 Incompressible solid. Δ / €Δ / Water (almost). l l l l 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 Pressure 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 ) 3(1 2 ) constants for isotropic media + ν − ν 25 26 MatSE 280: Introduction to Engineering Materials ©D.D.