3.model visco-elasticity.ppt

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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

    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 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