Lecture12 Notes

Lecture12 Notes - Lecture 12: Networks and Gels Origin of...

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Unformatted text preview: Lecture 12: Networks and Gels Origin of Rubbery Elasticity November 9, 2010 1 Monday, November 8, 2010 Networks and Gels, Rubber Elasticity and Viscoelasticity • • • • • • Polymers have unique physical/mechanical properties Overview of mechanical properties and elastic properties of materials Many of the unique properties are the result of cross-linking The entropy of the chain drives rubber elasticity Polymers can be crosslinked in different ways Water swollen crosslinked networks are use in biomedical applications 2 Monday, November 8, 2010 MATERIALS UNDER LOAD: DEFORMATION 3 Monday, November 8, 2010 Stress and Strain Compression and Tension: stress (σ) : applied force per unit area strain (ε) : deformation (as a result of stress) Typically expressed in MPa σ= F A0 ε= li − l0 Δl = l0 l0 Dimensionless θ Shear: stress (τ) : applied force per unit area strain (γ) : deformation (as a result of stress) Δx Fs γ= = tan θ τ= y A0 F⊥ l0 A0 Tension Monday, November 8, 2010 l y F|| F⊥ A0 l Δx A0 l0 compression shear 4 Common engineering loads torsion bending hydrostatic 5 Monday, November 8, 2010 Poisson ratio Poisson’s ratio (ν) is the ratio of lateral and axial strains ⎛ Δlx ⎞ ⎜⎟ −εx −εy ⎝ l0 x ⎠ υ= = = εz εz ⎛ Δlz ⎞ ⎜⎟ ⎜l ⎟ ⎝ 0z ⎠ ν = -1.0 to 0.5 ν ~ 0.0 cork ν ~ 0.25 ceramics ν ~ 0.33 metals σz Δlx/2 Δlz/2 Δl0y Δl0z ν ~ 0.40 polymers ν ~ 0.5 rubber ν > 0.5 impossible (at a certain strain a material would reach negative volume); instead voids form auxetic: negative ν, gets thicker when stretched Monday, November 8, 2010 Δly/2 Δl0x σz 6 Biomedical applications • • • • Drug delivery Filters Cushions ePTFE (υ < -12) Expanded PTFE sliding rotation Evans & Caddock. (1989) J. Phys. D: Appl. Phys. 22 1883-1887 7 Monday, November 8, 2010 Elastic deformation • Hooke’s Law F = k Δx F = resisting force; k = spring constant Δx = displacement 8 Monday, November 8, 2010 Stress, Strain and Modulus P = −B σ = Eε σ= F A0 ε= ! ΔV ΔV0 li − l0 Δl = l0 l0 τ= " Monday, November 8, 2010 Fs A0 γ= Δx = tan θ y $ ! E τ = Gγ σ = −Kε -K G " K is called “B” in your text # 9 A quick note about compressive modulus… Compression in 2D, expansion F⊥ in 3rd direc0on F⊥ A0 l l0 l F|| Compression in 3D A0 l0 A0 This experiment measures the elas0c modulus in compression, or the compressive modulus of elas0city (Young’s modulus, E) Common experiment for polymers This experiment measures the bulk modulus, or the compressive modulus (B or K) An ideal rubber is incompressible 10 Monday, November 8, 2010 Poisson’s Ratio For isotropic materials: E K E = 3(1 − 2ν )K E Young’s (Elastic) Modulus K Bulk (Compressive) Modulus K= E 3(1 − 2ν ) G Shear Modulus G= E 2(1 + ν ) K is called “B” in your text Monday, November 8, 2010 G E = 2G(1 + ν ) K= G= 2G(1 + ν ) 3(1 − 2ν ) 3(1 − 2ν )K 2(1 + ν ) 11 Elastic Properties of Materials • • • For a very simple elastic material, the stress/strain curve is very informative This is an idealized version, and real materials, especially polymers have more complex behavior In order to understand the complex behavior, we first need to understand an important component, the elastic behavior of polymers 12 Monday, November 8, 2010 Elastic Deformation Hooke’s Law: σ = Eε, for elastic deformation in compression and tension E is also called Young’s modulus The greater E, the more resistant the material is to deformation; the material is stiffer. elastic Compliance: ε1 D= = σE plastic ! # " Elastic deformation is recoverable; plastic deformation is permanent 13 Monday, November 8, 2010 Hooke Elastic Behavior • • • • Highly crystalline materials may exhibit Hookean elastic behavior at very low strains (<0.2%) This elasticity is energy-driven; when force is applied, the bond lengths deviate from the (minimum energy) equilibrium The displacement of lattice atoms accompanying the stress increases the potential energy, PE, with approximately no change in entropy. The elastic force f causes a deformation/displacement Δl of atoms from their equilibrium states, which is determined from the slope of f vs. r (l) plot k2 PE = k" x = x 2 121 PE = kx + k ' x 3 + … 2 3 f = kx + k ' x 2 + … 2 f = kΔl + k ' Δl 2 + … Essen;ally, Hooke’s law at low strains 14 Monday, November 8, 2010 Modulus is related to bonding for non-polymeric materials Strongly bonded ⎛ ∂F ⎞ ⎜ ⎟ ⎝ ∂r ⎠ r0 Energy to separate Separation r Weakly bonded ⎛ ∂F ⎞ Modulus ∝ ⎜ ⎟ ⎝ ∂r ⎠ r0 Modulus proportional to slope of interatomic force-separation curve at equilibrium spacing (r = ro) ∴ depend on atomic bond type Covalent (ceramics) are stiffer than metallic (metals) 15 Monday, November 8, 2010 Other properties • • • • Stiffness of a material (ceramic, metal) is related to the stiffness of the chemical bonds between atoms Strength is related to the cohesive strength of the bond between atoms (depth of the PE well) In both cases, the stress strain curves are obviously more complex than these simple explanations Real materials are only elastic over a very small strain range (2% or less) 16 Monday, November 8, 2010 Stress/strain for polymeric materials • • • In contrast to small molecule materials, polymers exhibit elasticity over a very broad range (500% or more, sometimes!) Remember, for polymers, typically the chains are either folded into lamellae (crystalline phase), or coiled up (amorphous phase). We don’t really see fully extended chains. For polymers, we are not stretching the bonds between atoms, but simply stretching out the chains, or unwinding the chains 17 Monday, November 8, 2010 Polymers under tension 18 Monday, November 8, 2010 Young’s Moduli: Comparison Metals Alloys 1200 10 00 800 600 400 E(GPa) 200 10 0 80 60 40 109 Pa Graphite Composites Ceramics Polymers /fibers Semicond Diamond Tungsten Molybdenum Steel, Ni Tantalum Platinum Cu alloys Zinc, Ti Silver, Gold Aluminum Magnesium, Tin Si carbide Al oxide Si nitride Carbon fibers only C FRE(|| fibers)* <111> Si crystal Aramid fibers only <100> A FRE(|| fibers)* Glass -soda Glass fibers only G FRE(|| fibers)* Concrete GFRE* 20 10 8 6 4 2 1 0.8 0.6 0.4 0.2 Monday, November 8, 2010 CFRE * G FRE( fibers)* G raphite Polyester PET PS PC C FRE( fibers) * AFRE( fibers) * 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. Epoxy only PP HDP E PTF E LDPE Wood( grain) 19 Plastic Deformation and Yield upper yield point lower yield point Yield is the stress at which a material begins to behave plastically Ultimate strength is the stress at the maximum of the stress-strain curve. 20 Monday, November 8, 2010 Tensile Response: Brittle & Plastic Near Failure σ(MPa) fibrillar structure x briFle failure onset of necking near failure plas0c failure x Ini0al unload/reload ε aligned, networked cross ­ case linked case crystalline regions slide semi ­ crystalline case amorphous regions elongate crystalline regions align Stress ­strain curves adapted from Fig. 15.1, Callister 7e. Inset figures along plas0c response curve adapted from Figs. 15.12 & 15.13, Callister 7e. (Figs. 15.12 & 15.13 are from J.M. Schultz, Polymer Materials Science, Pren0ce ­Hall, Inc., 1974, pp. 500 ­501.) Monday, November 8, 2010 21 Tensile Response: Elastomer Case σ (MPa) x briFle failure x plas0c failure x elastomer ε ini0al: amorphous chains are kinked, cross ­linked. final: chains are straight, s0ll cross ­linked Stress ­strain curves adapted from Fig. 15.1, Callister 7e. Inset figures along elastomer curve (green) adapted from Fig. 15.15, Callister 7e. (Fig. 15.15 is from Z.D. Jastrzebski, The Nature and Proper;es of Engineering Materials, 3rd ed., John Wiley and Sons, 1987.) Deforma0on is reversible! • Compare to responses of other polymers:  ­ ­ briFle response (aligned, crosslinked & networked polymer)  ­ ­ plas0c response (semi ­crystalline polymers) 22 Monday, November 8, 2010 Yield Strength : Comparison Metals/ Alloys 300 200 Ti (5Al-2.5Sn) a W (pure) Cu (71500) cw Mo (pure) Steel (4140) a Steel (1020) cd Al (6061) ag Steel (1020) hr Ti (pure) a Ta (pure) Cu (71500) hr 100 70 60 50 40 Al (6061) a 30 20 10 Tin (pure) Monday, November 8, 2010 ¨ dry PC Nylon 6,6 PET PVC humid PP HDPE LDPE in ceramic matrix and epoxy matrix composites, since in tension, fracture usually occurs before yield. 700 600 500 400 Composites/ fibers Hard to measure, σ y (MPa) Yield strength, 10 00 Polymers Steel (4140) qt Hard to measure , since in tension, fracture usually occurs before yield. 20 00 Graphite/ Ceramics/ Semicond The amount of stress required to produce a specific amount of plastic strain (often 0.002) 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 23 Mechanical Properties of Natural Materials 24 Monday, November 8, 2010 Mechanical Properties of Natural Materials 25 Monday, November 8, 2010 Mechanical Properties of Natural Materials 26 Monday, November 8, 2010 Mechanical Properties of Natural Materials 27 Monday, November 8, 2010 Next time… • Entropy driven elasticity 28 Monday, November 8, 2010 ...
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