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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

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

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

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)

Potential Applications of FGMs

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

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 TheTemperature 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.

Governing Equations(Timoshenko Beam Theory)

The components of displacements in Timoshenko’s Theory:

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:

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:

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:

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 :

Solution Procedure(Boundary Conditions and Initial Values)

Mechanical B.C. and I.V:

Thermal B.C. and I.V:

Solution Procedure(Finite Fourier Transformation)

Based on Fourier series theory, the inverse transformation can be expressed by:

Solution Procedure(Applying Finite Fourier Transformation)

The system of dimensionless coupled equations :

Applying finite Fourier transformation :

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 :

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.

Results (verification)

- Aluminum Beam considerations:
- Length:0.25m Height: 0.0022m
- Simply supported boundary condition
- Ambient temperature: T0=293K
- Upper surface: Step function heat flux
- Lower surface: thermally insulated

Results (verification)

Deflection history of an aluminum beam at midpoint with the coupled thermoelasticity assumption.

Deflection history at the midpoint of the beam for n=0 and h/l=0.003125.

Results (verification)

- FGM Beam considerations:
- Length:0.8m Height: 0.0025m
- Simply supported boundary condition
- Ambient temperature: T0=293K
- Upper surface: Step function heat flux
- Lower surface: convection to the surrounding ambient with coefficient of hc=10000 W/m2K

Temperature change history at the midpoint of the beam at the upper side for n=0 and h/l=0.003125.

Deflection history at the midpoint of the beam for n=0 and h/l=0.001.

Maximum midplane axial displacement history of the beam for different power law indices and h/l=0.003125.

Deflection history at the midpoint of the beam for different power law indices and h/l=0.003125.

Deflection history at the midpoint of the beam for different power law indices and h/l=0.003125.

Temperature change history at the midpoint of the beam at the upper side for different power law indices and h/l=0.003125.

Temperature change distribution at the midpoint of the beam across the thickness direction at t=3 for different power law indices .

Normal stress history at the midpoint of the beam at the upper side for different power law indices and h/l=0.003125.

Maximum shear stress history of the beam for different power law indices and h/l=0.003125.

Results (slenderness ratio effect)

Deflection history at the midpoint of the beam for n=0 for different slenderness ratios.

Results (slenderness ratio effect)

Temperature change history at the midpoint of the beam at the upper side for n=0 and for different slenderness ratios.

Coupling factors for different power low indices

Deflection history at the midpoint of the beam for n=0 and h/l=0.001 with the uncoupled and coupled thermoelasticity assumptions.

Deflection history at the midpoint of the beam for n=0 and h/l=0.001 with coupled thermoelasticity assumptions.

Temperature change history at the midpoint of the beam at the upper side for n=0 and h/l=0.001 with the uncoupled and coupled assumptions.

.

Deflection history at the midpoint of the beam for n=20 and h/l=0.003125 with the uncoupled and coupled thermoelasticity assumptions.

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.

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.

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 .

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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