1. Polymer Viscoelasticity & Rheology
Minshiya P.
Assistant Professor
M E S Keveeyam College Valanchery

2. Ideal or Newtonian fluids
• Incompressible – the density is constant
• Irrotational – the flow is smooth, no turbulence
• Nonviscous – fluid has no internal friction ( η = 0)
• Steady flow – the velocity of the fluid at each point
is constant in time.

3. Ideal or Newtonian fluids
Under fixed and constant stress – strian or deformation increases continuosly
Strain is non recoverable (zero yield value)
Strain, ε = f(σ, t)

4. Ideal or Newtonian fluids

5. Ideal or Newtonian fluids
σs (T) α dv/dr
σs (T) = ηdv/dr = (1/φ) dv/dr (1)
η = 1/φ = σs/dv/dr (2)
η : coeff of viscosity or viscosity or internal friction of the liquid
φ : fluidity
eqn (1) & (2) : Newton’s law

6. d ε /dt = dv/dr ----- (3) d ε /dt :Strain rate
Strain rate : rate of change in strain (deformation) of a material with respect to time
(1) σs = η d ε /dt = (1/φ) d ε /dt (4)
Shear stress α rate of shear strain

7. Non-Newtonian fluid
A non-Newtonian fluid is a fluid whose flow properties differ in any way from those of Newtonian fluids.
Viscosity dependent on shear rate or shear rate history
Eg:- Many salt solutions, molten polymers &
many commonly found substances such as
ketchup, custard, toothpaste, starch
suspensions, paint, blood, and shampoo

8. Newtonian V/s Non –Newtonian
Non -Newtonian fluid
Relation between the shear stress and the
shear rate is different
Can even be time-dependent.
Coefficient of viscosity cannot be defined.
Newtonian fluid
Relation between shear stress and
shear rate is linear, passing through
the origin
The constant of proportionality
being the coefficient of viscosity
Inadequate to describe non-Newtonian
fluids
The concept of viscosity is used to
characterize the shear properties
of a fluid

9. Types of non-Newtonian behaviour

10. Ideal or Elastic solids
Hooke's law
Force needed to extend or compress a spring by some amount is proportional to that amount
is a constant characteristic of the spring, stiffness.

11. Ideal or Elastic solids
Under stress, deformed instantaneously and experiences an internal force that oppose the deformation and restore it to its original state if the external force (stress) is no longer applied
Strain α Applied stress
F/A = k ∆L/ L0
σ = Eε E : Youngs modulus

12. Ideal or Elastic solids
Under a constant stress, an ideal elastic solid deform immediately to a fixed & constant level in eqm with applied stress and will not deform or strain further with time.

13.                                                                                       
Stress–strain curve
1. Ultimate strength 2. Yield strength – corresponds to
yield point 3. Rupture 4. Strain hardening region 5. Necking region A: Engineering stress (F/A0) B: True stress (F/A)

14. 1. Ultimate tensile strength (UTS)
Also termed as tensile strength (TS) or ultimate strength : maximum stress that a material can withstand while being stretched or pulled before failing or breaking.

15. 2. Yield (engineering)
Yield strength or yield point : The stress at which a material begins to deform plastically
Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed.
Once the yield point is passed, some fraction of the deformation will be permanent and non-reversible.

16. 3. Fracture
* Separation of an object or material into two, or more, pieces
under the action of stress
* The fracture of a solid almost always occurs due to the
development of certain displacement discontinuity surfaces
within the solid.
* Displacement develops in this case perpendicular to the surface
of displacement, it is called a normal tensile crack or simply a
crack
* Displacement develops tangentially to the surface of
displacement, it is called a shear crack, slip band, or dislocation.

17. 4. Strain hardening Also known as Work hardening (cold working)
Strengthening of a metal by plastic deformation.
This occurs because of dislocation movements and
dislocation generation within the crystal structure of
the material.
Most non-brittle metals (with high melting point)
& several polymers can be strengthened

18. Necking: a mode of tensile deformation
Relatively large amounts of strain localize disproportionately in a small region of the material
The resulting prominent decrease in local cross-sectional area provides the basis for the name "neck"
A PE sample with
a stable neck

19. Viscoelasticity & Rheology
Pure Newtonian viscosity & pure elastic behaviour are ideal
All motions of real bodies have flow and elastisity
Elastoviscous liquids or viscoelastic solids
Polymers are viscoelastic

20. Polymer viscoelasticity
Inter-relationship among elasticity, flow & molecular motion
Time dependent flow behaviour
Creep & Stress relaxation

21. OVERVIEW POLYMERS TREATED AS SOLIDS Strength Stiffness Toughness
POLYMERS TREATED AS FLUIDS Viscosity of polymer melts
Elastic properties of polymer melts
VISCOELASTIC PROPERTIES Creep
Stress relaxation

22. • Stress Relaxation and Creep – Chemical versus Physical Processes – Analysis with Springs and Dashpots – Relaxation and Retardation Times – Physical Aging of Glassy Polymers

23. Creep Deformation under a constant load as a function of time
3 kg
Time
Deformation under a constant load as a function of time
Creep or Creep strain may be either fully recoverable with time or involve permanant deformation with partial recovery

24. Stress Relaxation Constant deformation experiment
3 kg
4 kg
5 kg
6 kg
Time
Constant deformation
experiment
Stress decreases with time
Slowly drops to zero as the material undergo permanant set to the stretched length

25. Reasons for Stress Relaxation and Creep
1. Chain Scission:
Oxidation, Hydrolysis.

26. Reasons for Stress Relaxation and Creep (Contd.)
2. Bond Interchange:
Polyester, Polysiloxane
Polyamides, Polyethers

27. Reasons for Stress Relaxation and Creep (Contd.)
3. Segmental Relaxations:
Thirion Relaxation

28. Viscous Flow: slippage of chains past one another, fast reptation
All physical reasons for stress relaxation and creep are related to molecular motion (strain) induced by stress, therefore, must be related to conformational changes

29. Mechanical models of viscoelsatic materials
Maxwell & Kelvin (Voigt) Element
Using Springs (purely elastic) and Dashpots (purely viscous)
For a spring σ = E ε (Hooke’s Law)
For a dashpot σ = η dε/dt (Newton’s Law)

30. Store energy and respond instantaneously.
Spring
Dashpot
Under stress piston moves through the fluid
Rate α stress
No recovery on removing
Dissipate energy in the form of heat.
Springs Stretched instantaneously under stress & holding that stress indefinitely
Store energy and respond instantaneously.

31. Maxwell & Kelvin (Voigt) Element

32. Creep Experiment • Maxwell Model Spring & Dashpot are in seriesη
• Maxwell Model
Spring & Dashpot are in series
Developed to explain the behavior of pitch and tar
Maxwell assumed that such materials can undergo viscous flow and also respond elastically
Combine Hooke’s and Newton’s laws

33. Creep Experiment - Maxwell Model (contd)η
Strains are additive , ie; ε = εelast + εvisc
ε(t) = σ/G + σt/η
1st term -- instantaneous elastic response
2nd term -- the viscous retarded response.
Both elements feel the same stress
Residual stress at time t, σ = σ0 exp ( -t/λ)
λ : relaxation time , σ0 : initial stress
ε = σ /E

34. Creep Experiment Kelvin (Voigt) Element
Spring & dashpot are in parallel
Combine Hooke’s and Newton’s laws
Both elements feel the same strain
Stress is additive , ie; σ = σelast + σvisc
σ = ε Ε + η dε/dt

35. Creep Experiment - Kelvin (Voigt) Element
Both spring & dashpot undergo concerted motion
Dashpot slowly responds to the stress & viscous flow decreases asymptotically
Strain rate at Dashpot, d ε /dt = σ/ η
Spring bears all the stress – spring & dashpot stop deforming together- creep stops
Overall stress is constant
ε(t) = σ0 [1 - exp(-t /η)]
This is a Creep Experiment.
The strain reaches a limiting value at long times.

36. Stress Relaxation Experimentsη
Maxwell Element Apply an instantaneous deformation, ε0 and
keep it constant, record the decaying stress:
ε = εelast + εvisc
dε/dt = (1/G) dσ/dt + σ/η

37. Stress Relaxation Experiments (contd.)
Since the shear strain is kept constant
dσ/dt = -Gσ/η
σ(t) = σ0 exp(-Gt/η)
Stress relaxation modulus G(t) = σ(t) / ε0
This is a Stress Relaxation Experiment.
The quantity τ1 = η/G has units of time and is called the Relaxation Time

38. Stress Relaxation Stress-Strain Behaviors Maxwel Model
Kelvin-Voigt Models

39. Stress Relaxation and Creep The Four Element Model
Combination of Maxwell and Kelvin elements in series
Used to describe the viscoelastic deformation of polymers in simple terms

40. The Four Element Model
OA: Instantaneous extension (Maxwell element, E1)
AB: Creep or retarded deformation (Kelvin element )
BC1: Instantaneous & partial recovery (Maxwell element E1)
C1D:Time dependant recovery (Creep recovery) – t1 to t2 (Kelvin element)
DE: Permanant deformation
Note that the recovery is not complete. A B C1 D C2 C3 E t1 t2 F

41. Relaxation time
Maxwell Model
Spring & Dashpot are in series
ε = εelast + εvisc
dε/dt = 1/E dσ /dt + σ /η
dσ /dt = E dε/dt - E σ /η
= E dε/dt - σ /λ : λ = η/E – Relaxation time

42. Relaxation time: Order of magnitude of time required for a certain proportion of polymer chain to relax (ie; to responds to the external stress )
Time required for a chemical reaction to takes place
Bond interchange, degradation, hydrolysis & oxidation

43. Retardation time Kelvin (Voigt) Element
Spring & dashpot are in parallel
σ = σelast + σvisc
σ = ε Ε + η dε/dt
ε= σ/E - η/E dε/dt
Under cont stress the eqn integrated to
ε= σ/E (1-e - (E/ η)t )
ε= σ/E (1-e- t/τ) : τ = η/E Retardation time

44. For four element model under cont stress
ε = ε1+ ε2 + ε3
ε= σ/E1 + σ/E2 (1-e- t/τ) + (σ /η3) t
elastic term Viscoelastic effect Viscous effect
Retardation time: time required for E2 & η2 in kelvin element to deform 1-1/e or % of the total expected creep

45. Time-Temperature Superposition Principle
– Method of Superposition
– WLF Equation and Application

46. Time-Temperature Superposition
The Stress relaxation modulus depends on temperature and time.

47. First: Choose an arbitrary reference temperature T0.
Second: Shift, along the time axis, the stress relaxation modulus curve recorded at T just above or below T0, so that these two curves superpose partially.
Third: Repeat the procedure until all curves have been
shifted to be partially superposed with the previous ones.
Four: Keep track of aT, the amount a curve recorded at T is shifted. No shift for curve recorded at T0.