Stress relaxation

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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

τ(t) Rubber (permanent network)

time

VII. Stress-Strain relationship

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Boltzmann superposition principle deformation

Stress relaxation Characteristic relaxation time: λ

stress λ

t

deformation

stress

VII. Stress-Strain relationship

t0

t1

t2

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t

4

Boltzmann superposition principle

deformation

dγ(ti) stress

t

If small deformations are imposed in the past, they have additive effects on the stress at time t.

t-ti If at time t, a deformation dγ(ti) is imposed, the stress at time t coming from this deformation is:

(Linear regime) VII. Stress-Strain relationship

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VII.2 Creep (fluage):

t

t

Constant stress:

t

Solid elastic: J=1/G Newtonian fluid: J=t/η

Creep compliance VII. Stress-Strain relationship

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Creep recovery test

τ

Solid elastic: J=1/G Newtonian fluid: J=t/η

τ = τ0

γ/τ0

γr(t,t0)/τ0

γ(t,t0)/τ0 Jo, elastic component 0 VII. Stress-Strain relationship

t0 MECA 2141

t 7

Definitions of mechanical quantities

•  Creep compliance •  Creep recovery compliance •  Magnitude of complex compliance •  Storage compliance •  Loss compliance VII. Stress-Strain relationship

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VII.3. Maxwell model: Basic empirical modelisation

•  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

,

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Determination of the stress relaxation after cessation of steady-shear flow:

0

t

(!: integration limits) if

,

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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

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t 22

Combining Viscous and elastic response

Viscoelastic solid

γ

τ τ0

t

0

γ

τ τ0

0 VII. Stress-Strain relationship

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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

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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

VII. Stress-Strain relationship

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VII. Stress-Strain relationship

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One-mode Maxwell model:

If ωλ