Presentation on theme: "Polymer Viscoelasticity & Rheology"— Presentation transcript:
1 Polymer Viscoelasticity & Rheology Minshiya P.Assistant ProfessorM E S Keveeyam College Valanchery
2 Ideal or Newtonian fluids • Incompressible – the density is constant• Irrotational – the flow is smooth, no turbulence• Nonviscous – fluid has no internal friction ( η = 0)• Steady flow – the velocity of the fluid at each pointis constant in time.
3 Ideal or Newtonian fluids Under fixed and constant stress – strian or deformation increases continuoslyStrain is non recoverable (zero yield value)Strain, ε = f(σ, t)
5 Ideal or Newtonian fluids σs (T) α dv/drσs (T) = ηdv/dr = (1/φ) dv/dr (1)η = 1/φ = σs/dv/dr (2)η : coeff of viscosity or viscosity or internal friction of the liquidφ : fluidityeqn (1) & (2) : Newton’s law
6 d ε /dt = dv/dr ----- (3) d ε /dt :Strain rate Strain rate : rate of change in strain (deformation) of a material with respect to time(1) σs = η d ε /dt = (1/φ) d ε /dt (4)Shear stress α rate of shear strain
7 Non-Newtonian fluidA non-Newtonian fluid is a fluid whose flow properties differ in any way from those of Newtonian fluids.Viscosity dependent on shear rate or shear rate historyEg:- Many salt solutions, molten polymers &many commonly found substances such asketchup, custard, toothpaste, starchsuspensions, paint, blood, and shampoo
8 Newtonian V/s Non –Newtonian Non -Newtonian fluidRelation between the shear stress and theshear rate is differentCan even be time-dependent.Coefficient of viscosity cannot be defined.Newtonian fluidRelation between shear stress andshear rate is linear, passing throughthe originThe constant of proportionalitybeing the coefficient of viscosityInadequate to describe non-NewtonianfluidsThe concept of viscosity is used tocharacterize the shear propertiesof a fluid
10 Ideal or Elastic solids Hooke's lawForce needed to extend or compress a spring by some amount is proportional to that amountis a constant characteristic of the spring, stiffness.
11 Ideal or Elastic solids Under stress, deformed instantaneously and experiences an internal force that oppose the deformation and restore it to its original state if the external force (stress) is no longer appliedStrain α Applied stressF/A = k ∆L/ L0σ = Eε E : Youngs modulus
12 Ideal or Elastic solids Under a constant stress, an ideal elastic solid deform immediately to a fixed & constant level in eqm with applied stress and will not deform or strain further with time.
13 Stress–strain curve1. Ultimate strength 2. Yield strength – corresponds toyield point 3. Rupture 4. Strain hardening region 5. Necking region A: Engineering stress (F/A0) B: True stress (F/A)
14 1. Ultimate tensile strength (UTS) Also termed as tensile strength (TS) or ultimate strength : maximum stress that a material can withstand while being stretched or pulled before failing or breaking.
15 2. Yield (engineering)Yield strength or yield point : The stress at which a material begins to deform plasticallyPrior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed.Once the yield point is passed, some fraction of the deformation will be permanent and non-reversible.
16 3. Fracture* Separation of an object or material into two, or more, piecesunder the action of stress* The fracture of a solid almost always occurs due to thedevelopment of certain displacement discontinuity surfaceswithin the solid.* Displacement develops in this case perpendicular to the surfaceof displacement, it is called a normal tensile crack or simply acrack* Displacement develops tangentially to the surface ofdisplacement, it is called a shear crack, slip band, or dislocation.
17 4. Strain hardening Also known as Work hardening (cold working) Strengthening of a metal by plastic deformation.This occurs because of dislocation movements anddislocation generation within the crystal structure ofthe material.Most non-brittle metals (with high melting point)& several polymers can be strengthened
18 Necking: a mode of tensile deformation Relatively large amounts of strain localize disproportionately in a small region of the materialThe resulting prominent decrease in local cross-sectional area provides the basis for the name "neck"A PE sample witha stable neck
19 Viscoelasticity & Rheology Pure Newtonian viscosity & pure elastic behaviour are idealAll motions of real bodies have flow and elastisityElastoviscous liquids or viscoelastic solidsPolymers are viscoelastic
21 OVERVIEW POLYMERS TREATED AS SOLIDS Strength Stiffness Toughness POLYMERS TREATED AS FLUIDS Viscosity of polymer meltsElastic properties of polymer meltsVISCOELASTIC PROPERTIES CreepStress relaxation
22 • Stress Relaxation and Creep – Chemical versus Physical Processes – Analysis with Springs and Dashpots – Relaxation and Retardation Times – Physical Aging of Glassy Polymers
23 Creep Deformation under a constant load as a function of time 3 kgTimeDeformation under a constant load as a function of timeCreep or Creep strain may be either fully recoverable with time or involve permanant deformation with partial recovery
24 Stress Relaxation Constant deformation experiment 3 kg4 kg5 kg6 kgTimeConstant deformationexperimentStress decreases with timeSlowly drops to zero as the material undergo permanant set to the stretched length
25 Reasons for Stress Relaxation and Creep 1. Chain Scission:Oxidation, Hydrolysis.
26 Reasons for Stress Relaxation and Creep (Contd.) 2. Bond Interchange:Polyester, PolysiloxanePolyamides, Polyethers
27 Reasons for Stress Relaxation and Creep (Contd.) 3. Segmental Relaxations:Thirion Relaxation
28 Viscous Flow: slippage of chains past one another, fast reptation All physical reasons for stress relaxation and creep are related to molecular motion (strain) induced by stress, therefore, must be related to conformational changes
29 Mechanical models of viscoelsatic materials Maxwell & Kelvin (Voigt) ElementUsing Springs (purely elastic) and Dashpots (purely viscous)For a spring σ = E ε (Hooke’s Law)For a dashpot σ = η dε/dt (Newton’s Law)
30 Store energy and respond instantaneously. SpringDashpotUnder stress piston moves through the fluidRate α stressNo recovery on removingDissipate energy in the form of heat.Springs Stretched instantaneously under stress & holding that stress indefinitelyStore energy and respond instantaneously.
32 Creep Experiment • Maxwell Model Spring & Dashpot are in series η• Maxwell ModelSpring & Dashpot are in seriesDeveloped to explain the behavior of pitch and tarMaxwell assumed that such materials can undergo viscous flow and also respond elasticallyCombine Hooke’s and Newton’s laws
33 Creep Experiment - Maxwell Model (contd) ηStrains are additive , ie; ε = εelast + εviscε(t) = σ/G + σt/η1st term -- instantaneous elastic response2nd term -- the viscous retarded response.Both elements feel the same stressResidual stress at time t, σ = σ0 exp ( -t/λ)λ : relaxation time , σ0 : initial stressε = σ /E
34 Creep Experiment Kelvin (Voigt) Element Spring & dashpot are in parallelCombine Hooke’s and Newton’s lawsBoth elements feel the same strainStress is additive , ie; σ = σelast + σviscσ = ε Ε + η dε/dt
35 Creep Experiment - Kelvin (Voigt) Element Both spring & dashpot undergo concerted motionDashpot slowly responds to the stress & viscous flow decreases asymptoticallyStrain rate at Dashpot, d ε /dt = σ/ ηSpring bears all the stress – spring & dashpot stop deforming together- creep stopsOverall stress is constantε(t) = σ0 [1 - exp(-t /η)]This is a Creep Experiment.The strain reaches a limiting value at long times.
36 Stress Relaxation Experiments ηMaxwell Element Apply an instantaneous deformation, ε0 andkeep it constant, record the decaying stress:ε = εelast + εviscdε/dt = (1/G) dσ/dt + σ/η
37 Stress Relaxation Experiments (contd.) Since the shear strain is kept constantdσ/dt = -Gσ/ησ(t) = σ0 exp(-Gt/η)Stress relaxation modulus G(t) = σ(t) / ε0This is a Stress Relaxation Experiment.The quantity τ1 = η/G has units of time and is called the Relaxation Time
38 Stress Relaxation Stress-Strain Behaviors Maxwel Model Kelvin-Voigt Models
39 Stress Relaxation and Creep The Four Element Model Combination of Maxwell and Kelvin elements in seriesUsed to describe the viscoelastic deformation of polymers in simple terms
40 The Four Element ModelOA: Instantaneous extension (Maxwell element, E1)AB: Creep or retarded deformation (Kelvin element )BC1: Instantaneous & partial recovery (Maxwell element E1)C1D:Time dependant recovery (Creep recovery) – t1 to t2 (Kelvin element)DE: Permanant deformationNote that the recovery is not complete.ABC1DC2C3Et1t2F
41 Relaxation timeMaxwell ModelSpring & Dashpot are in seriesε = εelast + εviscdε/dt = 1/E dσ /dt + σ /ηdσ /dt = E dε/dt - E σ /η= E dε/dt - σ /λ : λ = η/E – Relaxation time
42 Relaxation time: Order of magnitude of time required for a certain proportion of polymer chain to relax (ie; to responds to the external stress )Time required for a chemical reaction to takes placeBond interchange, degradation, hydrolysis & oxidation
43 Retardation time Kelvin (Voigt) Element Spring & dashpot are in parallelσ = σelast + σviscσ = ε Ε + η dε/dtε= σ/E - η/E dε/dtUnder cont stress the eqn integrated toε= σ/E (1-e - (E/ η)t )ε= σ/E (1-e- t/τ) : τ = η/E Retardation time
44 For four element model under cont stress ε = ε1+ ε2 + ε3ε= σ/E1 + σ/E2 (1-e- t/τ) + (σ /η3) telastic term Viscoelastic effect Viscous effectRetardation time: time required for E2 & η2 in kelvin element to deform 1-1/e or % of the total expected creep
45 Time-Temperature Superposition Principle – Method of Superposition– WLF Equation and Application
46 Time-Temperature Superposition The Stress relaxation modulus depends on temperature and time.
47 First: Choose an arbitrary reference temperature T0. Second: Shift, along the time axis, the stress relaxation modulus curve recorded at T just above or below T0, so that these two curves superpose partially.Third: Repeat the procedure until all curves have beenshifted to be partially superposed with the previous ones.Four: Keep track of aT, the amount a curve recorded at T is shifted. No shift for curve recorded at T0.