Research Papers: Design and Analysis

Solution for Equation of Two-Dimensional Transient Heat Conduction in Functionally Graded Material Hollow Sphere With Piezoelectric Internal and External Layers

[+] Author and Article Information
M. Jabbari, S. M. Mousavi, M. A. Kiani

Mechanical Engineering Department,
Islamic Azad University,
South Tehran Branch,
Tehran, Iran

Contributed by the Pressure Vessel and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received August 30, 2015; final manuscript received May 20, 2016; published online August 5, 2016. Assoc. Editor: Albert E. Segall.

J. Pressure Vessel Technol 139(1), 011201 (Aug 05, 2016) (6 pages) Paper No: PVT-15-1206; doi: 10.1115/1.4033702 History: Received August 30, 2015; Revised May 20, 2016

In this paper, an exact solution for the equation of two-dimensional transient heat conduction in a hollow sphere made of functionally graded material (FGM) and piezoelectric layers is developed. Transient temperature distribution, as a function of radial and circumferential directions and time with general thermal boundary conditions on the inside and outside surfaces, is analytically obtained for different layers, using the method of separation of variables and Legendre series. The results are the sum of transient and steady-state solutions that depend upon the initial condition for temperature and heat source, respectively. The FGM properties are assumed to depend on the variable r and they are expressed as power functions of r.

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Suresh, S. , and Mortensen, A. , 1998, Fundamentals of Functionally Graded Materials: Processing and Thermomechanical Behavior of Graded Metals and Metal–Ceramic Composites, IOM Communications, London.
Miyamoto, Y. , Kaysser, W. A. , Rabin, B. H. , Kawasaki, A. , and Ford, R. G. , 1999, Functionally Graded Materials: Design, Processing and Applications, Springer, New York.
Tiersten, H. F. , 1969, Linear Piezoelectric Plate Vibrations, Plenum Press, New York.
Ootao, Y. , Tanigawa, Y. , and Nakanishi, N. , 1991, “ Transient Thermal Stress Analysis of a Nonhomogeneous Hollow Sphere Due to Axisymmetric Heat Supply,” Trans. Jpn. Soc. Mech. Eng. Ser. A, 57(1991), pp. 1581–1587. [CrossRef]
Ootao, Y. , and Tanigawa, Y. , 1994, “ Three-Dimensional Transient Thermal Stress Analysis of Nonhomogeneous Hollow Sphere With Respect to Rotating Heat Source,” Trans. Jpn. Soc. Mech. Eng., 60A(578), pp. 2273–2279. [CrossRef]
Ootao, Y. , Akai, T. , and Tanigawa, Y. , 1995, “ Three-Dimensional Transient Thermal Stress Analysis of a Nonhomogeneous Hollow Circular Cylinder Due to a Moving Heat Source in the Axial Direction,” J. Therm. Stresses, 18(5), pp. 497–512. [CrossRef]
Jabbari, M. , Sohrabpour, S. , and Eslami, M. R. , 2003, “ General Solution for Mechanical and Thermal Stresses in a Functionally Graded Hollow Cylinder Due to Nonaxisymmetric Steady-State Loads,” ASME J. Appl. Mech., 70(1), pp. 111–118. [CrossRef]
Poultangari, R. , Jabbari, M. , and Eslami, M. R. , 2008, “ Functionally Graded Hollow Spheres Under Non-Axisymmetric Thermo-Mechanical Loads,” Int. J. Pressure Vessels Piping, 85(5), pp. 295–305. [CrossRef]
He, X. Q. , Ng, T. Y. , Sivashanker, S. , and Liew, K. M. , 2001, “ Active Control of FGM Plates With Integrated Piezoelectric Sensors and Actuators,” Int. J. Solid Struct., 38(9), pp. 1641–1655. [CrossRef]
Jabbari, M. , Mohazzab, A. H. , and Bahtui, A. , 2009, “ One-Dimensional Moving Heat Source in a Hollow FGM Cylinder,” ASME J. Appl. Mech., 131(2), p. 021202.
Eslami, M. R. , Babai, M. H. , and Poultangari, R. , 2005, “ Thermal and Mechanical Stresses in a Functionally Graded Thick Sphere,” Int. J. Pressure Vessels Piping, 82(7), pp. 522–527. [CrossRef]
Lutz, M. P. , and Zimmerman, R. W. , 1996, “ Thermal Stresses and Effective Thermal Expansion Coefficient of Functionally Graded Sphere,” J. Therm. Stresses, 19(1), pp. 39–54. [CrossRef]
Jafari Fesharaki, J. , Jafari Fesharaki, V. , Yazdipoor, M. , and Razavian, B. , 2012, “ Two-Dimensional Solution for Electro-Mechanical Behavior of Functionally Graded Piezoelectric Hollow Cylinder,” Appl. Math. Model., 36(11), pp. 5521–5533. [CrossRef]
Alibeigloo, A. , and Nouri, V. , 2010, “ Static Analysis of Functionally Graded Cylindrical Shell With Piezoelectric Layers Using Differential Quadrature Method,” Compos. Struct., 92(8), pp. 1775–1785. [CrossRef]
Li, X. F. , Peng, X. L. , and Lee, K. Y. , 2010, “ Radially Polarized Functionally Graded Piezoelectric Hollow Cylinders as Sensors and Actuators,” Eur. J. Mech. A/Solids, 29(4), pp. 704–713. [CrossRef]


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Fig. 1

Geometry of the laminated shell

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Fig. 2

Transient temperature distribution in the outer piezolayer hollow cylinder

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Fig. 3

Transient temperature distribution in FGM hollow cylinder

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Fig. 4

Transient temperature distribution in the inner piezo layer hollow cylinder

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Fig. 5

Transient temperature with various power law indices at  θ=π/4 and t = 5 s

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Fig. 6

Temperature distribution in the outer piezo layer at θ=−π

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Fig. 7

Temperature distribution in FGM layer at θ=−π

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Fig. 8

Temperature distribution in the inner piezo layer at θ=−π



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