2009

DOI: 10.3103/s1066369x09060036

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Solution of fifth-degree equations

E. N. Mikhalkin

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Cited by 3 publications

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“…There are some analytical approaches for solving a quintic equation, a polynomial of fifth degree, using the Hermit-Kronecker method and the Mellin method [30,31]. However, since one of the characteristic roots of equation ( 22) is 1 ¼ Àa 1 , we can factor this term from equation ( 22) and solve analytically the fourth degree algebraic equation [32].…”

Section: Solution Proceduresmentioning

confidence: 99%

Coupled thermopiezoelectric behaviour of a one-dimensional functionally graded piezoelectric medium based on C–T theory

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²

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³

2011

Proceedings of the Institution of Mechanical Engineers, Part C:

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In this article, the transient thermopiezoelectric behaviour of a one-dimensional (1D) functionally graded piezoelectric medium subjected to a moving heat source is investigated. The formulation is given based on the Chandrasekhariah and Tzou (C–T) generalized thermoelasticity theory to consider the details of energy transport in the material in comparison with the Lord–Shulman (L–S) generalized theory. All material properties are taken to vary exponentially along the length of the medium except for phase lags, the relaxation time, and the specific heat, which are taken to be constant. The governing partial differential equations are given in the three coupled fields of displacement, temperature, and electric potential based on the C–T theory. Using Laplace transform to eliminate the time dependency of the problem, an analytical method is presented to obtain the coupled fields in the Laplace domain. The solutions are then derived in time domain by employing the fast Laplace inversion technique. Numerical results are shown to display the effects of discontinuities on the temperature and stress distribution, non-homogeneity index and the phase-lag constants of heat flux and temperature gradient on the wave propagation of temperature and stress fields based on the dual-phase-lag model of the C–T. Furthermore, the results are compared between the C–T and L–S thermoelasticity theories. Finally, the results are validated with those reported in the literature.

“…There are some analytical approaches for solving a quintic equation, a polynomial of fifth degree, using the Hermit-Kronecker method and the Mellin method [30,31]. However, since one of the characteristic roots of equation ( 22) is 1 ¼ Àa 1 , we can factor this term from equation ( 22) and solve analytically the fourth degree algebraic equation [32].…”

Section: Solution Proceduresmentioning

confidence: 99%

Coupled thermopiezoelectric behaviour of a one-dimensional functionally graded piezoelectric medium based on C–T theory

¹

,

²

,

³

2011

Proceedings of the Institution of Mechanical Engineers, Part C:

View full text Add to dashboard Cite

In this article, the transient thermopiezoelectric behaviour of a one-dimensional (1D) functionally graded piezoelectric medium subjected to a moving heat source is investigated. The formulation is given based on the Chandrasekhariah and Tzou (C–T) generalized thermoelasticity theory to consider the details of energy transport in the material in comparison with the Lord–Shulman (L–S) generalized theory. All material properties are taken to vary exponentially along the length of the medium except for phase lags, the relaxation time, and the specific heat, which are taken to be constant. The governing partial differential equations are given in the three coupled fields of displacement, temperature, and electric potential based on the C–T theory. Using Laplace transform to eliminate the time dependency of the problem, an analytical method is presented to obtain the coupled fields in the Laplace domain. The solutions are then derived in time domain by employing the fast Laplace inversion technique. Numerical results are shown to display the effects of discontinuities on the temperature and stress distribution, non-homogeneity index and the phase-lag constants of heat flux and temperature gradient on the wave propagation of temperature and stress fields based on the dual-phase-lag model of the C–T. Furthermore, the results are compared between the C–T and L–S thermoelasticity theories. Finally, the results are validated with those reported in the literature.

Coupled Thermal Stresses in Advanced Smart Materials

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2019

Structural Integrity

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Thermopiezoelectric Analysis of a Functionally Graded Piezoelectric Medium

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2011

Int. J. Appl. Mechanics

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The thermopiezoelectrical behavior of a functionally graded piezoelectric medium (FGPM) is investigated in the present work. For the special case, the dynamic response of an FGPM rod excited by a moving heat source is studied. The material properties of the FGPM rod are assumed to vary exponentially through the length, except for specific heat and thermal relaxation time which are held constant for simplicity. The governing differential equations in terms of displacement, temperature, and electric potential are obtained in a general form that includes coupled and uncoupled thermoelasticity. The coupled formulation considers classical thermoelasticity as well as generalized thermoelasticity. Employing the Laplace transform and successive decoupling method, unknowns are given in the Laplace domain. Employing a numerical Laplace inversion method, the solutions are gained in the time domain. Numerical examples for the transient response of the FGPM rod are displayed to clarify the differences among the results of the generalized, coupled, and uncoupled theories for various nonhomogeneity indices. The results are verified with those reported in the literature.

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