Mechanical Properties Elastic deformation Plastic deformation Fracture
I. Elastic Deformation S s u s y e u e T I II III e For a typical ductile metal: I. Elastic deformation II. Stable plastic deformation III. Unstable deformation IV. Fracture Stress-strain relation is linear (Hooke s Law) s = Ee Strain is recoverable relaxation along the load line to zero stress S e
Importance of Elastic Behavior Engineering design Most structures are designed to remain below yield For example, s < s y /3 (boiler and pressure vessel code) s << s y (turbine blades, springs) Elastic failure Utlimate strength Buckling Flutter Bridges, other structures Aircraft
The Tensile Moduli: Young s Modulus and Poisson s Ratio L 0 (1+e z ) D 0 (1+e y ) Assume isotropic material Need two elastic moduli, E and ν Young s modulus: s z = Ee z Poisson s ratio: e x = e y = νe z Volume change: (E = Young s modulus) (ν = Poisson s ratio) ΔV V 0 = (1+ e x )(1+ e y )(1+ e z ) [ ] 1 e x + e y + e z [ ] ΔV V 0 = e z (1 2ν)
Multiaxial Deformation uniaxial tension hydrostatic pressure simple shear balanced shear Linear elastic stresses are additive s x = e x = 1 [ E s x ν(s y + s z )] E (1 2ν)(1+ ν) (1 ν)e x + ν(e y + e z ) [ ] Get e y, e z, s y, s z by interchanging x, y, z.
The Physical Moduli: Bulk and Shear Moduli uniaxial tension hydrostatic pressure simple shear balanced shear Fundamental properties are reflected in the response to Change of volume (pressure) Change of shape (shear) E = E 0 (V ) + E conf (E conf from atom arrangement at given V) volume shear bulk modulus (β) shear modulus (G)
Physical Basis of Elastic Moduli compression E = ν = shear 9βG (G + 3β) 3β 2G 2(G + 3β) Elastic moduli balance β = resistance to volume change G = resistance to shape change Four cases: β >> G incompressible solid E ~ 3G ν ~ 0.5 β ~ 3G normal metal E ~ 2.7G ν ~ 0.35 β ~ G strong directional covalent E ~ 2.25G ν ~ 0.125 β << G bond angles preserved E ~ 9 β ν ~ -1 (No natural solid has ν < 0)
Engineering the Elastic Modulus Elastic properties reflect atomic bonding Microstructure manipulation has little effect Small composition changes cause ΔE c(e 2 -E 1 ) One exception is Li in Al Adding Li to Al E, ρ E significantly Al-(1-2.5)Li alloys of interest for aircraft Composite materials High-modulus materials are usually brittle Add high-modulus fiber or particle to ductile matrix Combine high modulus with useful toughness Examples: Fiberglass (glass-epoxy) Graphite-epoxy SiC-Al
Composite Materials Fiber: E = E 1 f 1 + E 2 f 2 - stiffer element dominates Particulate: 1 E = f 1 E 1 + f 2 E 2 - softer element dominates Fiber composites: ~ uniform strain (e 1 ~ e 2 = e) s = P A = s 1 A 1 A A + s 2 2 = (E A 1 f 1 + E 2 f 2 )e Particulate composites: ~ uniform stress (s 1 ~ s 2 = s) E = E 1 f 1 + E 2 f 2 e = ΔL = ΔL 1 L 1 + ΔL 2 L 2 = e 1 f 1 + e 2 f 2 = f 1 + f 2 L 0 L 1 L 0 L 2 L 0 E 1 1 E = f 1 E 1 + f 2 E 2 E 2 s
Composite Materials Fiber: E = E 1 f 1 + E 2 f 2 - stiffer element dominates Particulate: 1 E = f 1 E 1 + f 2 E 2 - softer element dominates Fiber composites: High modulus but directional properties Use in applications where loading is uniaxial Or, use 2-d or 3-d configurations Less directionality, but lower modulus Particulate composites: Lower modulus effect, but More isotropic Relatively tough and formable Used for multiaxial loads
Elastic Failures Lattice instability The ultimate strength of a solid (nanoindentation) Failures in tension and shear ( inherent ductile-brittle transition) Buckling Beams and buildings Sheets (dents and crushing in automobiles) Vibration and flutter Bridge failures (flutter - Tocoma Narrows) Aircraft (flutter, whirl mode - the Lockheed Electra) Hovercraft (flagellation)
Buckling Failure Elastic response is geometrically unable to support load Catastrophic failure at hinge point Common failures: Beams in earthquakes Dents and crumpling in automobiles Folds in impacts and crashes 200A Fall, MSE 2008 290M Fall, 2006
Buckling Failure Buckled heicopter main rotor after impact with ground MSE 290M Fall, 2006
Vibrational Instability (Flutter) Natural frequencies depend on Geometry Constraint Density and stiffness Structural damping Damps normal motions Drive with natural frequency Flutter failure Bridges Whirlmode in prop aircraft 200A Fall, MSE 2008 290M Fall, 2006
The Tacoma Narrows Bridge (1940) MSE 290M Fall, 2006