Linear Viscoelasticity (Its All a Matter of Time)

Linear Viscoelasticity (its all a matter of time)

Gareth H. McKinley Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139

Including material from: Randy Ewoldt, University of Illinois Gerry Fuller, Stanford University Tim Lodge, U. Minnesota

Introduction to Viscoelasticity | Göteborg, Sweden | Aug. 21 2019LVE-1

Key References

• Rheology: Principles, Measurements and Applications – C.W. Macosko (Wiley, 1994)

• Viscoelastic Properties of Polymers – J.D. Ferry (3rd Edition, 1980)

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The Goal of Today… (in honor of 20th anniversary) VISCOELASTICITY • “I’m going to learn Jiu-Jitsu VISCOELASTICITY?

Neo(Hookean) How to Learn Viscoelasticity

Before… After… want some more???

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So What is a Complex Fluid? Isaac Newton, FRS 1672 • Complex fluids possess an underlying microstructure that can be affected by (and in turn then affect) a flow field – Gives rise to Visco-Elasticity • Examples include: – Polymer solutions, polymer melts, liquid crystals – Foams, gels, bubbly-liquids, – Suspensions, emulsions, slurries, mud.. – Food stuffs, paints, adhesives and other consumer products Non- • Basically everything except air, oil, water! Newtonian • These fluids violate Newton’s law of viscosity : Robert Hooke, FRS 1663 ∂vx t τ yx = µ τ = µ{∇v + ∇v } Neo-Hookean!!! ∂y •Rheology: study of the material properties of complex fluids in specified/known flow fields •Non-Newtonian Fluid Dynamics: self-consistent solutions of conservation of mass, momentum PLUS a constitutive model (rheological equation of state) For example: “Neo-Hookean Dumbbell Model” *Posthumous Portrait by Rita Greer (Wikipedia)

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Four Key Rheological Phenomena

shear Impact time

force ( ) shear thinning G(t) : viscoelasticity η γ! torque extension shear

N ( ) 1 γ! shear normal stresses ηE (ε!) strain hardening Image from Chris Macosko’s book 5

Rheology = study of deformation of complex materials

EXAMPLES: Dilute Polymer Solution (oil additive) Entangled Polymers (polyethylene melt)

Surfactant Solution (body wash)

Emulsion Suspension Gel (mayonnaise) (latex paint) (gelatin) 6

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Probing the Viscoelastic Response of Polymeric Materials

Natural Time Scale of Complex Fluids

• Natural time scale λ ≡ τ material ≈ 10 sec. • The Deborah number is a dimensionless measure compared with the time scale of the deformation...

http://www.sillyputty.com

Images courtesy of Cambridge Polymer Group

De < 1 Viscous Liquid

De ~ 1 Viscoplastic ‘Solid’ De ~ 10 Elastic Solid

(Relaxation time of the material) τ material De ~ = (The timescale of the process) T process Brittle Solid De ~1000! ©MIT; Harold.Edgerton Strobe Lab.

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Origin of the Deborah number see M. Reiner, Physics Today, January 1964, p. 62

Judges 5:5

A creep test:

A creep test (constant load) on silly putty

Another creep test (silicone gum!!)

From Youtube:

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USGS Image Library “Byrd Glacier Flow”. Photo: Dr Bill Servais (http://www.rosssea.info/glaciers.html) 11

10mm 0˚ 30˚ 60˚ 90˚

Elastic

increasing amplitude (of applied stress and/or deformation) Viscous “Linear viscoelasticity” is the limit of small loading amplitude (does not disrupt microstructure)

ideal elastic : ideal viscous : viscoelastic d = G ! = ⌘ = ⌘˙ dt

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ideal elastic : ideal viscous : d = G viscoelastic = ⌘ = ⌘˙ stress strain dt (old literature: elastico-viscous)! strain-rate “shear rate”

Basic tests: apply measure calculate γ Step strain: γ 0 s G modulus

t t t

s s0 γ J Creep: compliance t t t γ s G’ Dynamic: G” complex wt wt moduli w

Stress Relaxation Modulus, G(t)

Mathematically: g Impose: step strain g 0 = constant Heaviside Step Function (How? Use a rheometer) time

(t, 0) Observe : stress (t, 0) Material function : G(t, 0)= 0

log G st

Linear response: g small enough so that G(t) is independent of g

t LVElog-14 t

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Stress Relaxation Modulus, G(t), Limiting Cases

τs Hookean (t, 0) G(t, 0)= 0

Experimental difficulty: True “step” displacement impossible (instrument inertia at short times)

time

τ sτ s Newtonian viscoelastic solid

viscoelastic liquid

time time LVE-15

Maxwell Viscoelastic Model

• Let’s motivate these linear tests using a simple model

• Elastic solid: e = G spring

• Viscous liquid: v = ⌘˙ dashpot

• The Maxwell model is a superposition of these two:

total strain : = e + v

Go ηo stress is equal : = e = v

⌘0 d d d d + = ⌘0 + = ⌘0 G0 dt dt ! dt dt = ⌘ /G : relaxation time 0 0 16

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Example: Maxwell Model in Step Strain

• In a step-strain experiment, the material is subjected to an instantaneous strain of magnitude go • Following the step, the strain is held fixed at this constant value and the stress is monitored in time.

• The initial stress is then G0go • Solving the Maxwell model, we have 0 d d + = ⌘ dt 0 dt t/ t/ = (t = 0)e = G00e • Dividing the stress by the strain, we get the “relaxation modulus”

t/ • No strain dependence G(t)=G0e • Single relaxation time

t/ G e γ γ0 G(t) 0

t t 17

Comparing with Measurements of G(t)

1. Impose: step strain g 0 = constant g

2. Observe : stress (t, 0)

(t, 0) time 3. Material function : G(t, 0)= 0 t/ 4. Choose a model and fit to data; e.g. Maxwell G(t)=G0e Modulus vs. time: AP Example: Wormlike micellar solution (e.g. hair Modulusshampoo) vs. time: AP

G 0 G G0 0 G0

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There are many other possible models… Relaxation and creep function of (A) Maxwell, (B) Voigt, and (C) Kelvin model (standard linear solid).

Eiji Tanaka, and Theo van Eijden CROBM 2003;14:138-150

Generalized Maxwell Model

G(t) G exp( t / ) = ∑ k − λk k

A spectrum of relaxation times {λk} “Prony Series” Extract spectrum by inversion (tricky) or Fit to model e.g. for polymers, Rouse, Zimm, reptation… G(t) usually follows a kind of “equipartition”: ρ N G(t) av k T exp( t / ) = B ∑ − λk M k

other models: G(t) = At −n power-law spectra, near gel point LVE-20

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Fitting to a sum of exponentials

A spectrum of relaxation modes {λk} “Prony Series” “Mechanical Spectroscopy”

Gi k =1 k =1 G(t) k =2

k =3

k =4

k =N ...

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Creep Compliance, J(t)

s s0 Impose : step stress 0 = constant

time

(t, 0) Observe : strain (t, 0) Material function : J(t, 0)= 0

J viscoelastic liquid ϕ = 1/η ˙ (t, 0) dJ Fluidity : (t, 0)= = 0 dt viscoelastic solid

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