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

Maxwell-type differential equations

Recent differential models- Pom-Pom differential approximation- eXtended Pom-Pom (XPP)- Rolie-Poly

Maxwell-type differential equations

Example: viscosity and normal stress coefficients for Johson-Segelman model

Steady state:

Some algebra

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Summary