4: non-linear viscoelasticity - materials technologypeters/4k400/rheol_chap04.pdf · 4: non-linear...
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4: non-linear viscoelasticity
Normal stress effects (Weissenberg et al. 1947)
Main goals
- To develop constitutive models that can describe non-linearphenomena such as rod climbing
- To use the equations in practical applications such as polymer processing and soft tissue mechanics
Ranges of viscoelasticity: the Deborah number
A dissappointing point of view …… Three major topics:
- Non-linear phenomena (time dependent)- Normal stress difference- Shear thinning- Extensional thickening
- Most simple non-linear models
- More accurate models
Normal stress differences in shear
Normal stress coefficients
For small shear rates:
Shear thinning
- Data for a LDPE melt- Lines from the Kaye-Bernstein, Kearsley, Zapas (K-BKZ) model
Shear thinning & time dependent viscosity Interrelations between shear functions
Cox-Merz ruleFor small shear rates:
Gleissle mirror rule
LDPESteady shear (solid line)Cox Merz (open symbols) Gleissle mirro rule (solid points)
Lodge-Meissner; after step shear:
Extensional thickening
Time-dependent, uniaxial extensional viscosity
PS, HDPE: no branches2 LDPE’s: branched, tree-like
Extensional thickening
uniaxial extensional viscosityversus shear viscosity
Non-linear behavior in uniaxialextensional viscosity is moresensitive to molecular archtecturethan non-linear behavior in shearviscosity
m = ½ uniaxial extensionm = 1 biaxial extensionm =0 planar extension
Extremely rare results
Stressing viscosities Second order fluid
Simplest constitutive model that predictsa first normal stress difference:
Upper convective derivative (definition):
with the substantial or material derivative:
Finger tensor:
Second order fluid in simple shear Second order fluid in uniaxial extension
For: extensional thickening for low extension rates
Useful for non-uniform complex flow
More complex model (Criminale-Ericksen-Filby):
Can not predict neither stress Growth or stress relaxation
Upper Convected Maxwell model
1-D Maxwell model (linear viscoelasticity):
- Non-linear viscoelastic model (product of and )
- Small strain: non-linear terms dissapear, material derivative
- Steady flow, small strain rate: Newtonian flow
- Transient flow, high strain rates: Neo-Hookean.
Upper Convected Maxwell model: shear flow
Start-up flow
Homogenous flow = 0
Symmetry:
Upper Convected Maxwell model: shear flow
Start-up flow
Steady state results (time derivatives are zero)
Viscosity and first normalstress coefficient are constant
Upper Convected Maxwell model: shear flow
Stress growth, non-zero components:
Upper Convected Maxwell model:extension
Steady state uniaxial extension
Extremely extension thickening, viscosity rises to infinity when :
.
→
Upper Convected Maxwell model
Integral form
How to use integral models ?
Different shear histories
: strain accumulated between t an t’
Upper Convected Maxwell model
Step shear strain
Start-up steady shear
Upper Convected Maxwell model
Start-up uniaxial extension
Upper Convected Maxwell model
Start-up uniaxial extension
Upper Convected Maxwell model
Start-up uniaxial extension
Upper Convected Maxwell model
Start-up uniaxial extension
Upper Convected Maxwell model
Integral form; multi mode
Corresponding differential forms
Upper Convected Maxwell model
Summarizing:
Pro’s- Recaptures all of the linear viscoelastic modelling- Newtonian / Neo-Hookean behavior for the limiting case (slow / fast flow)- Predicts first normal stress difference and extensional thickening
Con’s- No second normal stress difference- No shear rate dependence of viscosity and first normal stress difference
(i.e no shear thinning)- extensional thickening is too severe
Works for very dilute solutions (< 0.5% concentration) and dilute solutionswith very high solvent viscosities (Boger fluids)
Upper Convected Maxwell model
Including the (viscous) solvent contribution
Oldroyd-B constitutive equation (1950)
HWM polyisobutylene in poly(1-butene) / kerosene
__ UCM equation--- Oldroyd-B
More accurate constitutive modelsIntegral constitutive models
Lodge:
Lodge, step strain
Use general elastic solid:
Time-dependent elasticenergy function
Special case: Lodge:
More accurate constitutive models
How to obtain the right energy function:
Two invariants IB, IIB and time t:- lots of experiments required (problem!)- or guidance from molecular theory
First, restrict tot simple shear flow
Integral constitutive models
More accurate constitutive models
Integral constitutive models: simple shear
With the expression: one can obtain and
More accurate constitutive models
Integral constitutive models: simple shear / step strain
The function can be obtained by taking the tome derivativeof the relaxing shear stress after step shear strain
More accurate constitutive models
Simple shear / step strain
Time-strain factorability!!(works also for other than shear)
More accurate constitutive models
Time-strain factorabilibty
- M(t-t’) from linear viscoelastic measurements- Non-linear measurements for U(IB,II B)- For τ12 and N1 the so-called damping function h(γ) needs to be measured
More accurate constitutive models
Simple shear / step strain /damping function h(γ)
Wagner (1976) (--)
Laun (1978) (__)
Khan & Larson (1987)
Notice that the Lodge-Meisnerrelation is obeyed ( )
More accurate constitutive models
With Gi, λi and h(γ) known predictions forvarious shear flows can be made
Steady state viscosity & firstnormal stress coefficient
Stress growth
Stress relaxation
A single damping function capturesa wealth of non-linear shear data formany polymer melts
More accurate constitutive models
Factorization doesn’t work always Concentrated polystyrene solution
Also problems with strainreversal (complete failure)
Doesn’’t work for other type of flows (extensional)
More accurate constitutive models
More general expression:
Based on a molecular theory, limited applicability
More accurate constitutive models
Recent integral models: - Wagner stress function- Pom-Pom model (Larson-McLeish)
Maxwell-type differential equations
Modifies the rate of stress build up
Modifies the rate of stress decay
Multi-modes required to describe experimental data
Max
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Notice: mostly only onenon-linear parameter!!
Max
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Step shear
Step biaxal extension
Uniaxial extension
Larson model Phan Thien-Tanner model
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