Coupled Thermoelasticity for FGM Timoshenko Beam

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Outline

• The Introduction of Functionally Graded Materials
• The Introduction of Coupled Thermoelasticity of
  Beams
• Coupled Thermoelasticity of Functionally Graded
  Timoshenko Beams
• Governing Equations
• Solution Procedure
• Results
• Conclusions

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Functionally Graded Materials Concept
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Functionally Graded Materials History

• 1984 Sendai Group proposed a concept of FGM
  (Nino, Koizumi and Hirai)
• 1985 Establishing the concept of FGM
• 1986 Investigation and research conducted for FGM
  (with Special Coordination Funds for Promoting
  Science and Technology)
• 1987 Launching a National Project called FGM Part
  I (with Special Coordination Funds for Promoting
  Science and Technology) (to be ended in March,
  1991) Title" Research on the Generic Technology
  of FGM Development for Thermal Stress Relaxation
  The 1st FGM symposium
• 1990 The 1st International Symposium on
  Functionally Graded Materials (FGM 1990) in
  Sendai, Japan

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A Typical Metal-Ceramic FGM
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FGMs Attributes

• High thermal resistance
• High abrasion resistance (ceramic face)
• Corrosion resistance
• High impact resistance
• Weldable/boltable to metal supports
• Biological compatibility

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Potential Applications of FGMs

• Aerospace and Aeronautics
• Advanced aircraft and aerospace engines
• Rocket heat shields
• Thermal barrier coatings
• Power plants
• Thermal barrier coatings
• Heat exchanger tubes
• Manufacturing
• Machine tolls
• Forming and cutting tools
• Metal casting, forging processes
• Smart Structures
• Functionally Graded Piezoelectric Materials
  (FGPMs)
• Shape memory alloys

• MEMS and sensors
• Electronics and Optoelectronics
• Optical fibers for wave high-speed transmission
  (Asahi Glass Company)
• Computer circuit boards
• Cellular phone
• Biomaterials
• Artificial bones, joints
• Teeth
• Cancer prevention
• Others
• Baseball Cleats (MIZUNO Inc.)
• Razor Blades (Matsushita Electric Works, Ltd.)
• Titanium Watches (CITIZEN commercialized ASPEC)

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Fabrication Methods of FGMs

• Constructive Processes
• Solid State Powder Consolidation
• Plasma Spray Forming
• Laser Cladding
• Transport-Based Processes
• Settling and Centrifugal Separation
• Infiltration Process

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Mathematical Models

• Exponential Formulation

• Power Law Formulation

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Coupled Thermoelasticity of beams (Review)

• Shieh , 1979 ,Eigensolutions for Coupled
  Thermoelastic Vibrations of Timoshenko Beams.
• Massalas and Kalpakidis, 1983-1984, Coupled
  Thermoelastic Vibrations of Euler-Bernoulli and
  Timoshenko Beams, Analytical solution by using
  finite Fourier Transformation.
• Maruthi Rao and Sinha, 1997, Finite Element
  Coupled Thermostructural Analysis of Composite
  Beams, Neglecting The Temperature change across
  the thickness direction.
• Manoach and Ribeiro, 2004, Coupled Thermoelastic
  Large Amplitude Vibrations of Timoshenko Beams,
  Using finite difference approximation and modal
  coordinate transformation.
• Sankar and Tzeng, 2002, Thermal Stress in
  Functionally Graded Beams.
• Babaei et al, 2007, Coupled Thermoelasticity of
  Functionally Graded Beams, Galerkin finite
  element solution with C1-contious shape function.

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Governing Equations(Timoshenko Beam Theory)
The components of displacements in Timoshenkos
Theory
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Governing Equations(Stress-Displacement
equations)
The strain-displacement equations
The stress- strain equations
The stress-displacement equations for FGM
Timoshenko beam
The Temperature change across the thickness
direction is assumed to be linear
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Governing Equations(Equation of Motion)
The equation of motion of a beam based on
Timoshenko theory
Where
According to Stress-displacement equations and
assumed Temperature function
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Governing Equations(Energy Equation)
Classical Coupled Thermoelasticity Assumption
Based on the first law of Thermodynamics
Energy equation for FGM Timoshenko beam in the
coupled form
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Governing Equations(Energy Equation)
For more accuracy in this analysis, the energy
equation is considered to residue (Res) equation
and it may be made orthogonal with respect to dz
and zdz as
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Solution Procedure(Dimensionless Values)
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Solution Procedure(Dimensionless Equations)
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Solution Procedure(Boundary Conditions and
Initial Values)
Mechanical B.C. and I.V
Thermal B.C. and I.V
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Solution Procedure(Finite Fourier Transformation)
Based on Fourier series theory, the inverse
transformation can be expressed by
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Solution Procedure(Applying Finite Fourier
Transformation)
The system of dimensionless coupled equations
Applying finite Fourier transformation
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Solution Procedure(Laplace Transform)
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Solution Procedure(Analytical Laplace Inverse)
The solution of unknown variables can be obtained
in Laplace domain as
Mellin's inverse formula, An integral formula for
the inverse Laplace transform
In practice, computing the complex integral can
be done by using the Cauchy residue theorem
The solution of unknown variables can be obtained
in time domain as
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Results (verification)

• Comparison with previous aluminum beam solutions
• Analytical solution reported by Massalas and
  Kalpakidis for Euler-Bernoulli beam.
• Finite element solution presented by Babaei et
  al. for Euler-Bernoulli beam.
• Comparison with recent FGM beam solution
• Finite element solution presented by Babaei et
  al. for functionally graded Euler-Bernoulli beam.

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Results (verification)

• Aluminum Beam considerations
• Length0.25m Height 0.0022m
• Simply supported boundary condition
• Ambient temperature T0293K
• Upper surface Step function heat flux
• Lower surface thermally insulated

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Results (verification)
Deflection history of an aluminum beam at
midpoint with the coupled thermoelasticity
assumption.
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Results (verification)
Deflection history at the midpoint of the beam
for n0 and h/l0.003125.
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Results (verification)

• FGM Beam considerations
• Length0.8m Height 0.0025m
• Simply supported boundary condition
• Ambient temperature T0293K
• Upper surface Step function heat flux
• Lower surface convection to the surrounding
  ambient with coefficient of hc10000 W/m2K

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Results (verification)
Temperature change history at the midpoint of the
beam at the upper side for n0 and h/l0.003125.
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Results (verification)
Deflection history at the midpoint of the beam
for n0 and h/l0.001.
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Results (FGM effect)
Maximum midplane axial displacement history of
the beam for different power law indices and
h/l0.003125.
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Results (FGM effect)
Deflection history at the midpoint of the beam
for different power law indices and h/l0.003125.
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Results (FGM effect)
Deflection history at the midpoint of the beam
for different power law indices and h/l0.003125.
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Results (FGM effect)
Temperature change history at the midpoint of the
beam at the upper side for different power law
indices and h/l0.003125.
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Results (FGM effect)
Temperature change distribution at the midpoint
of the beam across the thickness direction at t3
for different power law indices .
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Results (FGM effect)
Normal stress history at the midpoint of the beam
at the upper side for different power law indices
and h/l0.003125.
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Results (FGM effect)
Maximum shear stress history of the beam for
different power law indices and h/l0.003125.
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Results (slenderness ratio effect)
Deflection history at the midpoint of the beam
for n0 for different slenderness ratios.
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Results (slenderness ratio effect)
Temperature change history at the midpoint of the
beam at the upper side for n0 and for different
slenderness ratios.
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Results (coupled effect)
Coupling factors for different power low indices
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Results (coupled effect)
Deflection history at the midpoint of the beam
for n0 and h/l0.001 with the uncoupled and
coupled thermoelasticity assumptions.
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Results (coupled effect)
Deflection history at the midpoint of the beam
for n0 and h/l0.001 with coupled
thermoelasticity assumptions.
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Results (coupled effect)
Temperature change history at the midpoint of the
beam at the upper side for n0 and h/l0.001 with
the uncoupled and coupled assumptions. .
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Results (coupled effect)
Deflection history at the midpoint of the beam
for n20 and h/l0.003125 with the uncoupled and
coupled thermoelasticity assumptions.
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Results (mathematical functions)
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Conclusions

• In this work, proper dimensionless values
  decrease numerical difficulties accompanied with
  coupled thermoelasticity problems.
• Analytical Laplace inverse method does not have
  time marching and numerical Laplace inverse
  problems.
• Results show that for larger values of power law
  indices which provide most metal-rich FGM, the
  lateral deflection of an FGM beam does not
  decrease constantly due to applied thermal shock.
  There is an optimum value for FGM parameter in
  which this value is minimum.
• For most metal-rich FGM beam The temperature
  distribution decreases in value due to higher
  thermal conductivity.

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Conclusions

• In general, due to the applied thermal shock, the
  frequency of the FGM beam vibration is decreased
  when the beam constituent materials change from
  the ceramic-rich to the metal-rich. But the
  amplitude is increased.
• Results show that the behavior of Timoshenko
  beam under thermal shock is not usually
  distinguishable with the Euler-Bernoulli beam
  specially for less slenderness ratios.
• Results show that the real coupled solution is
  not usually identifiable respect to the uncoupled
  case. However, the effect of coupling is like
  damping which can be recognized when coupling
  factors are magnified.

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Suggestions for future researches

• Using higher-order polynomial functions to
  approximate temperature change across the
  thickness.
• Assuming internal damping effect in equation of
  motion for more accurate analysis.
• Using the two-dimensional classical
  thermoelasticity equations to analyze an FGM
  beam.
• Classical coupled thermoelasticity of
  functionally graded plate based on the
  first-shear deformation theory and higher-shear
  deformation theory .

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Some Important References

• Babaei M. H. , Eslami M.R., Coupled
  Thermoelasticity Analysis for FGM Beams, A
  Thesis Submitted for The Degree of Master of
  Science, Mechanical Engineering Department,
  Amirkabir University of Technology, 2006.
• Massalas C. V. and Kalpakidis V. K., "Coupled
  Thermoelastic Vibrations of a Simply Supported
  Beam", J. Sound and Vibration, Vol. 88, No. 3,
  pp. 425-429, 1983.
• Massalas C. V. and Kalpakidis V. K., "Coupled
  Thermoelastic Vibrations of a Timoshenko Beam ,
  Lett. Appl. Engng. Sci., Vol. 22, No. 4, pp.
  459-465, 1984.
• Babaei M. H., Abbasi M. and Eslami M. R.,
  "Coupled Thermoelasticity of Functionally Graded
  Beams , J. Thermal Stress, Submitted for review.
• Krylov V. I. and Skoblya N. S., A Handbook of
  Methods of Approximate Fourier Transformation and
  Inversion of the Laplace Transformation, Moscow
  Mir Publishers, 1977.

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