# modeling visco elasticity

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EBB 220/3MODEL FOR

VISCO-ELASTICITY

DR AZURA A.RASHIDRoom 2.19

School of Materials And Mineral Resources Engineering,

Universiti Sains Malaysia, 14300 Nibong Tebal, P. Pinang Malaysia

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.

INTRODUCTIONINTRODUCTION

• The behaviour of linear elastic were given by Hooke’s law:

Ee

E= Elastic modulus

= Stress

e=strain

de/dt = strain rate

d/dt = stress rate

= viscosity

ordt

deE

dt

d

• The behaviour of linear viscous were given by Newton’s Law:

dt

de

** This equation only applicable at low strain

Hooke’s law & Newton’s LawHooke’s law & Newton’s Law

• 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:

1. Elastic spring with modulus which follows Hooke’s law

2. Viscous dashpots with viscosity 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.

Mechanical ModelMechanical Model

CREEP STRESS RELAXATION

Constant strain is applied the stress relaxes as function of time

Constant stress is applied the strain relaxes as function of time

• The common mechanical model that use to explain the viscoelastic phenomena are:

1. Maxwell• Spring and dashpot align in series

2. Voigt• Spring and dashpot align in parallel

3. Standard linear solid• One Maxwell model and one spring align

in parallel.

• 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 springViscous dashpot

Maxwell ModelMaxwell Model

• Overall stress , 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:

ds eee

ds

anddt

deE

dt

d s

dt

ded

• 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 maka d/dt = 0. The equations can be written as:

• Maxwell model can predict the Newtonian behaviour the strain is predict to increased with time

dt

d

Edt

de 1

0

dt

de

.

• The behavior of Maxwell model during creep loading (constant stress, strain is predicted to increased linearly with time

This is not the viscoelastic behaviour of polymeric materials de/dt decreased with time

• 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 given

dt

d

Edt

de 1

dt

d

E

10

dtEd

Et

eksp0= earlier stress

• The term E is constant for Maxwell model and sometimes can be refered as time relaxation, 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

00

teksp

• 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.

Voigt ModelVoigt Model

Viscous dashpot

Elastic spring

• 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 din every elements are additive and the overall stress are

• Equation for spring and dahpot can be written as:

and

ds eee

ds

Ees dt

ded

• 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 so d/dt = 0. The equation can be written as:

• The simple differential equation given by:

Ee

dt

de

0

Ee

dt

de

Et

ekspE

e 10

• Constant ratio E can be replace with time relaxation,

• Changes in strain with time for Voigt model that having creep are given by:

0

0 1

teksp

Ee

Figure shows polymer behavior under creep deformation strain rate decreased with time

e and t=

• 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:

0Ee

or

0Ee

Behavior of Voigt model at different loading Stress relaxation

• 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.

Standard linear solidStandard linear solid

• 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

Viscous dashpot

Elastic spring

• 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.

SummarySummary

• 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

Boltzman superposition principle Boltzman superposition principle

• 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

Illustrating the Boltzman superposition principle

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?

Example of the exams questionExample of the exams question

Thank you