Download  3.Model Viscoelasticity.ppt

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EBB 220/3
MODEL FORVISCOELASTICITY
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 viscoelasticity.
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
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 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
model during creep loading
(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
at different loading
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
during creep loading but will still allow the stress
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 viscoelasticity 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