Creep recovery compliance. ⢠Magnitude of complex compliance. ⢠Storage .... phi=0.34. Pom-pom polymer diluted at di
Outlines: Part I: phenomenology – introduction to rheology I.
Introduction
III. Glass, thermosetting and thermoplastic materials IV. Solutions and melts V.
Flow properties
VI. Concentrated linear polymers: Diffusion and relaxation times VII. Time – Temperature Superposition (TTS) VIII. Viscoelastic Fluid – stress-strain relationship (linear regime)
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VII. Viscoelastic Fluid – stress-strain relationship (linear regime) Entangled polymer melt or highly concentrated (thermoplastic) Deformations: shear or elongation, linear (non-linear: see later)
Stress relaxation:
Creep (fluage):
deformation
stress
time
time
Constant deformation
Constant stress
VII. Stress-Strain relationship
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VII.1. Stress relaxation γ
τ(t)
Temporary network
γ0 time
τrel
time
The molecules move in order to recover their stable coil-shape structure
• Spring – Elastic behavior – No time dependence – Response can be described with
Spring
• Dashpot – Viscous behavior – Time dependence – Response can be described with
VII. Stress-Strain relationship
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Dash pot
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Relaxation behavior of an ideally elastic body τ
τ2 τ1
t
γ
Spring
γ2 γ1
t VII. Stress-Strain relationship
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Viscoelastic materials: • Maxwell element – Used for relaxation processes – Viscoelasticity by spring and dashpot in series
• Kelvin-Voigt element – Used for description of creep processes – Viscoelasticity by spring and dashpot (amortisseur) in parallel VII. Stress-Strain relationship
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Maxwell model - Used for describing time-dependent phenomena (relaxation effects) characterized by very small displacement gradients - First step in modeling memory fluids For shear deformation: Hookean solid: Newtonian fluid: Maxwell model (1867):
VII. Stress-Strain relationship
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G
η
γ1
γ2
Strain: γ = γ 1 + γ 2 Stress:
Shear viscosity/shear modulus = λ=relaxation time (s) If η = η0 (zero-shear viscosity), λ= η0 /G = λ1 Steady-state response: Newtonian Sudden change in stress: Hookean VII. Stress-Strain relationship
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(used for small deformations)
G(t-t’)
Sudden deformation: or With boundary condition:
VII. Stress-Strain relationship
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(Sudden deformation):
Stress relaxation t
t
VII. Stress-Strain relationship
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Determination of the stress growth for steady-shear flow:
0
t
(!: integration limits) if
,
t MECA 2141
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Determination of the stress relaxation after cessation of steady-shear flow:
0
t
(!: integration limits) if
,
t MECA 2141
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Effect of a sudden constant stress: At t=0,
If
reversible
VII. Stress-Strain relationship
irreversible
t
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t
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(Effect of a sudden stress:)
reversible
irreversible
- The strain evolution is linear with time. - The irreversible part increase with time: model for describing liquid. Maxwell model: for liquid-like response
VII. Stress-Strain relationship
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Solid-like response: The Voigt-Kelvin model
G
γ1
η γ2
Retardation time
For a fixed (sudden) stress:
Reversible deformation (but not sudden): = constant VII. Stress-Strain relationship
t
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t
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After a constant stress:
t
t VII. Stress-Strain relationship
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Combining Viscous and elastic response
Viscoelastic liquid
γ
τ τ0
t
0
τ
γ
t
0
τ0
0 VII. Stress-Strain relationship
t0
t
0 MECA 2141
t0
t 22
Combining Viscous and elastic response
Viscoelastic solid
γ
τ τ0
t
0
γ
τ τ0
0 VII. Stress-Strain relationship
t0
t MECA 2141
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Start-up behavior of a viscoelastic system
t0
0 τ(t)
t
τ(t)
Viscoelastic solid
Elastic t τ(t)
t τ(t) Viscoelastic liquid
Viscous 0
t0
VII. Stress-Strain relationship
t
0 MECA 2141
t0
t 24
Generalization of Maxwell: If
Convention:
VII. Stress-Strain relationship
Spectrum of relaxation times
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Spectrum of relaxation times:
G(t) Expressed in log-scale:
Continuous relaxation spectrum
VII. Stress-Strain relationship
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Very useful for the data analysis: description of G(t) by only few modes
VII. Stress-Strain relationship
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Generalized model: retardation spectrum
For a fixed (sudden) stress:
Retardation function: (Solid: Jk=1/Gk) VII. Stress-Strain relationship
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VII.4. Oscillatory shear – Maxwell model and Kelvin-Voigt