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EBB 220/3

MODEL FORVISCO-ELASTICITY

DR AZURA A.RASHIDRoom 2.19

School of Materials And Mineral Resources Engineering,

Universiti Sains Malaysia, 14300 Nibong Tebal, P. Pinang

Malaysia

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It is difficult to predict the creep and stress relaxationfor polymeric materials.

It is easier to predict the behaviour of polymericmaterials with the assumption it behaves as linear

viscoelastic behaviour.

Deformation of polymeric materials can be divided totwo components:

Elastic component Hookeslaw

Viscous component Newtonslaw

Deformation of polymeric materials combination ofHookeslaw and Newtonslaw.

INTRODUCTION

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The behaviour of linear elastic were given byHookes

law:

Ee

E= Elastic modulus

= Stresse=strain

de/dt= strain rated/dt= stress rateh= viscosity

ordt

deE

dt

d

The behaviour of linearviscous were given by

NewtonsLaw:

dtdeh

** This equation only applicable at low strain

Hookes law & Newtons Law

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Methods that used to predict the behaviour of visco-elasticity.

They consist of a combination of between elasticbehaviour and viscous behaviour.

Two basic elements that been used in this model:

1. Elastic spring with modulus which follows Hookeslaw

2. Viscous dashpots with viscosity h which followsNewtonslaw.

The models are used to explain the phenomena creepand stress relaxation of polymers involved with different

combination of this two basic elements.

Mechanical Model

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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

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The common mechanical model that use to

explain the viscoelastic phenomena are:

1. Maxwell

Springand dashpotalign in series

2. Voigt

Spring and dashpotalign in parallel

3. Standard linear solid

One Maxwellmodel and one springalignin parallel.

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Maxwell model consist of spring and dashpot in seriesand was developed to explain the mechanical

behaviour on tar.

On the application of stress, the strain in each elements

The total strain is the sum of strain in spring & dashpot.

The stress each elements endures is the same.

Elastic springViscous dashpot

Maxwell Model

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Overall stress , overall strain e in the system is givenby:

es= strain in spring ande

d= 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

and

dt

deE

dt

d s

dt

dedh

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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 0maka d/dt= 0. Theequations can be written as:

Maxwell model can predict the Newtonian behaviour

the strain is predict to increased with time

h

dtd

Edtde

1

h

0

dt

de

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.

The behavior of Maxwell

(constant stress, 0 strainis predicted to increased

linearly with time

This is not the viscoelasticbehaviour of polymeric

materialsde/dtdecreased

with time

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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 thesystem given de/dt=0

then

Integration at t=0 0 given

h

dt

d

Edt

de 1

h

dt

d

E

10

dtEd

h

h

Et

eksp0o= earlier stress

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The term h/E is constant for Maxwell model andsometimes can be refered as time relaxation, t0 writtenas:

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

0

0t

teksp

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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 Model

Viscous dashpot

Elastic spring

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For Voigt model, the strain rate are

The accuracy of prediction the mechanical behaviour ofVoigt model can be confirm.

In creep case, stress is oso d/dt= 0. Theequationcan be written as:

The simple differential equation given by:

hh

Ee

dt

de

h

h

0Ee

dt

de

h

Eteksp

E

e 10

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Constant ratio h/E can be replace with timerelaxation, t0

Changes in strain with time for Voigt model thathaving creep are given by:

0

01

t

teksp

Ee

Figure shows polymer

behavior under creep

deformation strain ratedecreased with time

eo /E and t=

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Voigt model fails to predict the stress relaxation

behaviour of polymer

When the strain is constant at e0and dan de/dt= 0 theequation shows:

The linear response is shown in the figure:hh

0Eeor

0Ee

Behavior of Voigt model

Stress relaxation

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As shown:

Maxwell modelcan accurately predict thephenomenon stress relaxation to a firstapproximation.

Voigt Model can accurately predict thephenomenon creep to a firstapproximation.

Standard linear solid model wasdeveloped to combined the Maxwell andVoigt model to describe both creep &stress relaxationto a first approximation.

Standard linear solid

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In consist one Maxwell elements inparallel with a

spring.

The presence on this second spring will stop the

tendency of Maxwell element undergoing viscous flow

relaxation to occur

Viscous dashpot

Elastic spring

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There were a lots of attempts to discover morecomplex model that can give a goodapproximation to predict viscoelastic behaviourof polymeric materials.

When the elements used is increased mathematicalcan bemore complex.

It can be emphasis that mechanical models canonly gives mathematical representations formechanical behaviour only it not muchhelp to predict the behaviour of viscoelasticity atmolecular level.

Summary

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Linear viscoelastic theory is Boltzmansuperposition principle.

It is the first mathematical statement of linear

viscoelastic behaviour that allows the state ofstress or strain in a viscoelastic body todetermine from a knowledge of its entiredeformation history.

This principle can be used to predict theoverall creep and stress relaxation ofpolymeric materials

Boltzman superposition principle

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Illustrating the Boltzman superposition principle

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What is the purpose of mechanical model

in visco-elasticity theories?

Gives a brief description how the chosenmechanical model can be used to estimate

the creep or stress relaxation behavior for

polymeric materials?

Example of the exams question

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hank you