Viscoelasticity

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Viscoelasticity

• By
• Vikram chetkuri
• Varsha maddala

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Abstract

• Viscoelasticity is time dependent material. In
  this report we study about the introduction of
  viscoelasticity, examples of viscoelasticity,
  various Phenomena, Creep, Relaxation, Recovery,
  and Viscoelastic models like Maxwell Model,
  Kelvin Model, Three-parameter viscous models,
  Burgers Model and Generalized Maxwell and Kelvin
  Models.

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Viscoelasticity

• Time dependent material behavior where the
  stress response of that material depends upon
  both the strain applied and strain rate at which
  it was applied.1
• Examples
• Silly Putty
• Synthetic polymers,wood human tissue and metals
  at high temperature

They are called as time dependant materials
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Introduction

• The time dependent behavior of materials can be
  studied by3
• Creep
• Stress relaxation
• Constant rate Straining

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Creep

• The slow and continuous deformation of a
  material under constant stress is called creep
• 
  One dimension
• Creep is generally described in 3 stage
• Primary creep- Concave is facing down
• Secondary creep-Near constant rate
• Tertiary creep-It increases and fractures

s (t) s0H (t)
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Creep Response
Creep response If the load is removed , a
reverse elastic strain followed by recovery of a
portion of the creep strain will occur at a
continuously decreasing rate.2
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Stress relaxation

• The gradual decrease of stress when the material
  is held at constant strain is called the stress
  relaxation.3
• One
  dimension

e (t) e0H (t)
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Viscoelastic models

• These models can be used to describe viscoelastic
  materials and to establish their differential
  equations
• Used to compare stress-strain-time relation of
  viscoelastic materials.4

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Basic elements

• The two basic elements were
• Helical spring
• Hooks law(sEe)
• Dash pot
• Piston moving in a cylinder with perforated
  bottom

s? (de/dt)
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Maxwell model

• It consists of linear spring and linear dashpot
  connected in series
• sE e1 (spring) s ? (de2 /dt) (Dash pot)
• The total strain and strain rate is given by
• e e1e2 e e1e2

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Maxwell cont

• Strain- time relaxation can be obtained by
• e(t)ss0/Esd0/?t

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Generalized Maxwell Model
A sequence of Maxwell units in series is called a
Generalized Maxwell model
Total strain in generalized Maxwell series model
is
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Maxwell units in parallel
Total strain
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Kelvin/Voigt Model
Stress strain relationship
e (E/?)e s / ?
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Generalized Kelvin model
Kelvin units arranged in series is called a
generalized Kelvin model Total strain for this
model is
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Standard Linear Solid

• A three-parameter model constructed from two
  springs and one dashpot is known as the standard
  linear model.

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Standard Linear solid cont

• For the unit one the linear elastic strain is
  given by,
• sE1 e 1 (For spring)
• There fore the strain rate can be written as,
  
  e1 s /E1
• The differential equation for the standard linear
  solid from the stress-strain relation would be
  given by

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Three parameter Viscous Model

• A Three parameter viscous model is constructed by
  two dashpots and one spring
• Strain, ee 1e 2
• Strain rate,
• e e1 e2

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Three parameter Viscous Model Cont..

• The linear elastic strain is given by,
• s?1 e 1(For spring )
• e 2 (s 2 / E2 ) (s /?2 )
• The Total strain rate in Three-parameter viscous
  model is given by

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Example Of Kelvin Model

• Problem
• The deformation response of a certain polymer can
  be described by Voigt model. If E200 Mpa and ?
  2 1012 Mpa-s. Compute the relaxation time with
  a steady stress of 10 Mpa. Plot strain rate curve
  with an increment of one second in time?

Solution
Relaxation time can be calculated by
Stress strain relation for the Kelvin model can
be estimated by the expression
e s/E(1- e Et/?)
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Results of Kelvin Model example

• The table shows the value of strain for a
  increment of 1 second in time
• The values of strain are tabulated by
  substituting the values of different times t
  in Stress strain relation for the Kelvin model

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Graph
From the graph we can see Strain increases for
certain time and remains constant. Initially
stress increases and remains constant at certain
point.
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Homework problem

• Derive the constitutive relation for the
  viscoelastic model shown below.

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References

• References-
• 1)http//www.emba.uvm.edu/iatridis/me301/viscoela
  sticity_intro.pdf
• 2) William N.Findley, James S.Lai and Kasif
  Onaran.., Creep and Relaxation of Nonlinear
  Viscoelastic Materials.Dover Publications, INC.,
  New York.
• 3) http//silver.neep.wisc.edu/lakes/VEnotes.html
  
• 4) Wilhelm Flugge., Viscoelasticity.Springer-Ver
  lag New York.Heidelberg 1975.
• 5)George E.Mase Schaums Outline Series
  Continuum Mechanics.McGraw-Hill.
• 6) http//ocw.mit.edu/OcwWeb/index.htm
• 7)Daniel Frederick, Tien Sun Chang Continuum
  Mechanics Allyn and Bacon Series.

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Thank you
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