Thermoelasticity

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MEEN 5330

• Thermoelasticity
• By
• Durga vara prasad vulisi

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Thermoelasticity

• Introduction and Basic Definition
• The study of the behavior of solids following
  certain elastic law is called elasticity,
  plastic law - plasticity, thermoelastic law -
  thermoelasticity.
• When the behavior of a linear elastic solid is
  influenced by thermal effects, the stressstrain
  relations depend upon the absolute temperature. A
  study of such coupled thermal and elastic effects
  is called thermoelasticity.

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Introduction Continued

• Changes in temperatures causes thermal effects on
  materials. Some of these thermal effects include
  thermal stress, strain, and deformation.
• Thermal deformation simply means that as the
  "thermal" energy of a material increases, so
  does the vibration of its atoms/molecules and
  this increased vibration results in what can be
  considered a stretching of the molecular bonds -
  which causes the material to expand.
• If the thermal energy of a material decreases,
  the material will shrink or contract.

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Relation With Fundamental Laws Of Continuum

• We consider only the linear theory based on the
  Lagrangian description.
• The Kinetic equation of state for a isotropic,
  homogeneous, thermoelastic solid is
• 
  
  (1)
• where Tabsolute temperature
• Reference temperature for
  the unstressed and unstrained state
• Coefficient of Linear
  thermal expansion
• Lames constants at the
  temperature

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Importance of Coefficient of Thermal Expansion

• Coefficient of thermal expansion is defined by
• For the accurate determination of internal stress
  distributions in advanced composite systems, the
  elastic and thermal properties of their
  constituents must be known.
• When Composites are manufactured at relatively
  high temperatures, large residual stresses can
  exist in the composites after manufacturing due
  to the mismatch in the thermal expansion
  coefficients .

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Lames Constants

• µ is normal shear constant , and the other
  constant is the bulk modulus less two thirds of
  the normal shear constant i.e.
  .
• These constants are related to more familiar
  constants E,? as follows
• The and vary with temperature but for
  the linear theory we assume them to be constant
  over a certain range of temperature the equation
  (1) is the familiar Hooke's law in linear
  elasticity with temperature term included.

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Elastic Constitutive Equation

• In general the linear relationship between
  components of the strain and stress tensors is
  where is a
  fourth order tensor.
• The isotropic constitutive law using index
  notation is given as
• 
  
• Which can be written as
• where E is young's modulus,? is Poisson
  ratio,dij is kronecka delta

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Sample problem

• Given the stress state of steel with elastic
  constants E207GPa ,?0.3
• find the total strain induced in the steel?
• Solution we know the elastic constitutive
  equation as
• so

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Solution continued

• Therefore
• Similarly calculating the other strain
  components we get the components of strain as

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Thermoelastic Constitutive Equation Derivation

• When thermal effects are taken into account
  (2) in which
  
  
  is the contribution from the stress field and
  is the contribution from the temperature field
  .
• is given by due to
  change from some reference temperature to T.
• Inserting the above equation together with
  Hooke's Law
• 
  
  (3)
• 
  
  
• we get
• 
  
  (4)
• 
  
  

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Duhamel -Neumannn Relations

• Equation (4) is known as Duhamel -Neumannn
  relations which may be inverted to get the thermo
  elastic constitutive equation (1)
• Heat conduction in an isotropic elastic solid is
  governed by the Fourier law of heat conduction
  where k is thermal conductivity of
  the body ,introducing specific heat at constant
  deformation we get
• 
  
  (5)
• 
  
  
  

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Coupled Heat Equation Maxwell's Relations

• The internal energy is assumed to be a function
  of the strain components and the temperature
  T, the energy equation may be expressed in the
  form
• 
  which is known as Coupled heat
  equation.
• Assuming the internal energy to be a function of
  the strains and temperature, the differential
  change of u due to
  changes of
  
• and T can be written as
• The above equation can be written as
  since du is perfect
  differential we conclude that
• 
  
  (6)

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Summary of the Basic Equations

• Equation (8) is known as Maxwell's relations.
• The basic equations in linear thermoelasticity
  are the equations of motion ,the stress
  -displacement relations, the kinetic equation of
  state and the coupled heat conduction equation
  they are as follows in the order stated
• 
  
  
• 
  
  
• 
  
  
• 
  
  
• 
  
  

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Sample problem

• Develop the strain energy density for a
  thermoelastic solid?
• Solution
• The simplest form of Strain energy function
  that leads to a linear stress strain relation is
  given by U1/2eijsij inserting the thermoelastic
  constitutive equation into the strain energy
  function gives
• U

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References

• 1 Chang.T.S,Frederick.D, "Continuum Mechanics",
  Scientific publishers
• 2 http//web.mit.edu/course/3/3.11/www/modules/c
  onst.pdf.
• 345George,E.Mase,?Continuum mechanics,
  Schaums Outlines,TMH
• 678910Chang.T.S,Frederick.D,"ContinuumMe
  chanics", Scientific publishers

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Homework problem

• Consider the stress state of steel with elastic
  constants E207GPa ,?0.3,k160GPa
• find the total thermal strain induced in
  the steel when its heated from temperature of
  100c to a temperature of 700c?

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