1. EBB 220/3 MODEL FOR VISCO-ELASTICITY
DR AZURA A.RASHID
Room 2.19
School of Materials And Mineral Resources Engineering,
Universiti Sains Malaysia, Nibong Tebal, P. Pinang
Malaysia

2. INTRODUCTION
It is difficult to predict the creep and stress relaxation for polymeric materials.
It is easier to predict the behaviour of polymeric materials with the assumption  it behaves as linear viscoelastic behaviour.
Deformation of polymeric materials can be divided to two components:
Elastic component – Hooke’s law
Viscous component – Newton’s law
Deformation of polymeric materials  combination of Hooke’s law and Newton’s law.

3. Hooke’s law & Newton’s Law
The behaviour of linear elastic were given by Hooke’s law: or
The behaviour of linear viscous were given by Newton’s Law:
E= Elastic modulus
s = Stress
e= strain
de/dt = strain rate
ds/dt = stress rate
h= viscosity
** This equation only applicable at low strain

4. Mechanical Model
Methods that used to predict the behaviour of visco-elasticity.
They consist of a combination of between elastic behaviour and viscous behaviour.
Two basic elements that been used in this model:
Elastic spring with modulus which follows Hooke’s law
Viscous dashpots with viscosity h which follows Newton’s law.
The models are used to explain the phenomena creep and stress relaxation of polymers involved with different combination of this two basic elements.

5. STRESS RELAXATION
CREEP
Constant strain is applied  the stress relaxes as function of time
Constant stress is applied  the strain relaxes as function of time

6. The common mechanical model that use to explain the viscoelastic phenomena are:
Maxwell
Spring and dashpot  align in series
Voigt
Spring and dashpot  align in parallel
Standard linear solid
One Maxwell model and one spring  align in parallel.

7. Maxwell Model
Maxwell model consist of spring and dashpot in series and was developed to explain the mechanical behaviour on tar.
On the application of stress, the strain in each elements are additive.
The total strain is the sum of strain in spring & dashpot. The stress each elements endures is the same.
Elastic spring
Viscous dashpot

8. Overall stress s, overall strain e in the system is given by:
es = strain in spring and ed = strain in dashpot dashpot
Because the elements were in series  the stress is the same for all elements,
Equations for spring and dashpot can be written as:
and

9. For Maxwell model, the strain rate is given as
The accuracy of prediction the mechanical behaviour of Maxwell model can be confirm.
In creep case, the stress at s = s0 maka ds/dt = 0. The equations can be written as:
Maxwell model can predict the Newtonian behaviour  the strain is predict to increased with time

10. .
The behavior of Maxwell model during creep loading (constant stress, s0 strain is predicted to increased linearly with time
This is not the viscoelastic behaviour of polymeric materials  de/dt decreased with time

11. Integration at t=0 s= s0 given 
May be this model is useful to predict the behaviour of polymeric materials during stress relaxation.
In this case, the strain is constant e=e0 applied to the system given de/dt =0
then
Integration at t=0 s= s0 given 
so= earlier stress

12. The term h/E is constant for Maxwell model and sometimes can be refered as time relaxation, t0 written as:
The exponential decreased in stress can be predicted  give a better representation of polymeric materials behaviour.
Stress were predicted completely relaxed with time period  it is not the normal case for polymer

13. Voigt Model Can also known as the Kelvin model.
It consists of a spring and dashpot in parallel.
In application of strain, the stress of each element is additive, and the strain in each element is the same.
Elastic spring
Viscous dashpot

14. The parallel arrangement of spring and dashpot gives the strain e are the same for the system given by:
es = strain in spring and ed = strain in dashpot
Because the elements in parallel  stress s din every elements are additive and the overall stress are
Equation for spring and dahpot can be written as:
and

15. For Voigt model, the strain rate are
The accuracy of prediction the mechanical behaviour of Voigt model can be confirm.
In creep case, stress is s = so so ds/dt = 0. The equation can be written as:
The simple differential equation given by:

16. Constant ratio h/E can be replace with time relaxation, t0.
Changes in strain with time for Voigt model that having creep are given by:
Figure shows polymer behavior under creep deformation strain rate decreased with time
e so /.E and t=

17. Voigt model fails to predict the stress relaxation behaviour of polymer
When the strain is constant at e0 and dan de/dt = 0 the equation shows:
 The linear response is shown in the figure: or
Behavior of Voigt model at different loading  Stress relaxation

18. Standard linear solid As shown:
Maxwell model can accurately predict the phenomenon stress relaxation to a first approximation.
Voigt Model can accurately predict the phenomenon creep to a first approximation.
Standard linear solid model was developed to combined the Maxwell and Voigt model  to describe both creep & stress relaxation to a first approximation.

19. In consist  one Maxwell elements in parallel with a spring.
Elastic spring
Viscous dashpot
In consist  one Maxwell elements in parallel with a spring.
The presence on this second spring will stop the tendency of Maxwell element undergoing viscous flow during creep loading  but will still allow the stress relaxation to occur

20. Summary
There were a lots of attempts to discover more complex model that can give a good approximation to predict viscoelastic behaviour of polymeric materials.
When the elements used is increased  mathematical can be more complex.
It can be emphasis that mechanical models can only gives mathematical representations for mechanical behaviour only  it not much help to predict the behaviour of viscoelasticity at molecular level.

21. Boltzman superposition principle
Linear viscoelastic theory is Boltzman superposition principle.
It is the first mathematical statement of linear viscoelastic behaviour that allows the state of stress or strain in a viscoelastic body to determine  from a knowledge of it’s entire deformation history.
This principle can be used to predict the overall creep and stress relaxation of polymeric materials

22. Botzmann proposed that:
The creep in a specimen is a function of it’s entire loading history
Each loading step makes an independent contribution to the final deformation
Overall deformation  algebraic sum of each contribution

23. Illustrating the Boltzman superposition principle

24. Example of the exams question
What is the purpose of mechanical model in visco-elasticity theories?
Gives a brief description how the chosen mechanical model can be used to estimate the creep or stress relaxation behavior for polymeric materials?

25. Thank you