Normal stress effects (Weissenberg et al. 1947)
4: non-linear viscoelasticity
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 viscosity 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 viscosities 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 constitutive equation (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 shear stress 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