EBB 220/3 MODEL FOR VISCO-ELASTICITY

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EBB 220/3MODEL FORVISCO-ELASTICITY

• DR AZURA A.RASHID
• Room 2.19
• School of Materials And Mineral Resources
  Engineering,
• Universiti Sains Malaysia, 14300 Nibong Tebal, P.
  Pinang
• Malaysia

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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 Hookes law
• Viscous component Newtons law
• Deformation of polymeric materials ? combination
  of Hookes law and Newtons law.

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Hookes law Newtons Law

• The behaviour of linear elastic were given by
  Hookes law

or

• The behaviour of linear viscous were given by
  Newtons 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
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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 Hookes
  law
• Viscous dashpots with viscosity h which follows
  Newtons law.
• The models are used to explain the phenomena
  creep and stress relaxation of polymers involved
  with different combination of this two basic
  elements.

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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
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• 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.

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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
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• 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
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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 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

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• .

• 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
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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 ee0 applied
  to the system given de/dt 0
• then
• Integration at t0 s s0 given

?
so earlier stress
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• 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

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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
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• 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
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• 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

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• 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
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• 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
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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.

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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

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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.

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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 its entire
  deformation history.
• This principle can be used to predict the overall
  creep and stress relaxation of polymeric
  materials

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• Botzmann proposed that
• The creep in a specimen is a function of its
  entire loading history
• Each loading step makes an independent
  contribution to the final deformation
• Overall deformation ? algebraic sum of each
  contribution

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Illustrating the Boltzman superposition principle
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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?

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Thank you