<<

Normal effects (Weissenberg et al. 1947)

4: non-linear

Main goals Ranges of viscoelasticity: the Deborah number

- To develop constitutive models that can describe non-linear phenomena such as rod climbing

- To use the equations in practical applications such as polymer processing and soft tissue mechanics 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 Shear thinning

- Data for a LDPE melt - Lines from the Kaye-Bernstein, Kearsley, Zapas (K-BKZ) model

For small shear rates:

Normal stress coefficients Shear thinning & time dependent Interrelations between shear functions

For small shear rates: Cox-Merz rule

Gleissle mirror rule

Lodge-Meissner; after step shear:

LDPE Steady shear (solid line) Cox Merz (open symbols) Gleissle mirro rule (solid points)

Extensional thickening Extensional thickening

Time-dependent, uniaxial uniaxial extensional viscosity extensional viscosity versus shear viscosity

Non-linear behavior in uniaxial extensional viscosity is more sensitive to molecular archtecture than non-linear behavior in shear viscosity PS, HDPE: no branches 2 LDPE’s: branched, tree-like Stressing Second order fluid

Simplest constitutive model that predicts a first normal stress difference:

Upper convective derivative (definition): m = ½ uniaxial extension m = 1 biaxial extension Finger tensor: m =0 planar extension with the substantial or material derivative: Extremely rare results

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 Upper Convected Maxwell model: shear flow

Start-up flow

1-D Maxwell model (linear viscoelasticity): Homogenous flow = 0

- Non-linear viscoelastic model (product of and ) Symmetry:

- 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 Upper Convected Maxwell model: shear flow

Start-up flow Stress growth, non-zero components:

Steady state results (time derivatives are zero)

Viscosity and first normal stress coefficient are constant Upper Convected Maxwell model:extension Upper Convected Maxwell model

Steady state uniaxial extension Integral form

How to use integral models ?

Different shear histories . Extremely extension thickening, viscosity rises to infinity when : → : strain accumulated between t an t’

Upper Convected Maxwell model Upper Convected Maxwell model

Step shear strain Start-up uniaxial extension

Start-up steady shear Upper Convected Maxwell model Upper Convected Maxwell model

Start-up uniaxial extension Start-up uniaxial extension

Upper Convected Maxwell model Upper Convected Maxwell model

Start-up uniaxial extension Integral form; multi mode

Corresponding differential forms Upper Convected Maxwell model Upper Convected Maxwell model

Summarizing: Oldroyd-B (1950) Including the (viscous) solvent contribution 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 HWM polyisobutylene in Con’s poly(1-butene) / kerosene - No second normal stress difference - No shear rate dependence of viscosity and first normal stress difference __ UCM equation (i.e no shear thinning) --- Oldroyd-B - extensional thickening is too severe

Works for very dilute solutions (< 0.5% concentration) and dilute solutions with very high solvent viscosities (Boger fluids)

More accurate constitutive models More accurate constitutive models Integral constitutive models Integral constitutive models

Lodge: How to obtain the right energy function:

Lodge, step strain Two invariants IB, IIB and time t: - lots of experiments required (problem!) - or guidance from molecular theory

Use general elastic solid:

Time-dependent elastic First, restrict tot simple shear flow energy function

Special case: Lodge: More accurate constitutive models More accurate constitutive models

Integral constitutive models: simple shear Integral constitutive models: simple shear / step strain

The function can be obtained by taking the tome derivative of the relaxing after step shear strain

With the expression: one can obtain and

More accurate constitutive models More accurate constitutive models

Time-strain factorabilibty Simple shear / step strain

- 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

Time-strain factorability!! (works also for other than shear) More accurate constitutive models More accurate constitutive models

Stress growth Simple shear / step strain /damping function h(γ) With Gi, λi and h(γ) known predictions for various shear flows can be made

Steady state viscosity & first Wagner (1976) (--) normal stress coefficient

Laun (1978) (__)

Khan & Larson (1987) Stress relaxation

Notice that the Lodge-Meisner relation is obeyed ( ) A single damping function captures a wealth of non-linear shear data for many polymer melts

More accurate constitutive models More accurate constitutive models

Factorization doesn’t work always Concentrated polystyrene solution More general expression:

Also problems with strain reversal (complete failure)

Doesn’’t work for other type of flows (extensional)

Based on a molecular theory, limited applicability More accurate constitutive models Maxwell-type differential equations

Recent integral models: - Wagner stress function - Pom-Pom model (Larson-McLeish)

Modifies the rate of stress build up

Modifies the rate of stress decay

Multi-modes required to describe experimental data

Larson model Phan Thien-Tanner model

Step shear

Uniaxial extension

Step biaxal extension

Maxwell-type differential equations Notice: mostly only one Maxwell-type differential equations non-linear parameter!! Maxwell-type differential equations Maxwell-type differential equations

Recent differential models Example: viscosity and normal stress - Pom-Pom differential approximation coefficients for Johson-Segelman model - eXtended Pom-Pom (XPP) - Rolie-Poly Steady state:

Some algebra →

Summary