Statics of FGM circular plate with magneto-electro-elastic coupling: axisymmetric solutions and their relations with those for corresponding rectangular beam
Abstract:This paper investigates the static behavior of a functionally graded circular plate made of magneto-electro-elastic (MEE) materials under tension and bending. The analysis is directly based on the three-dimensional governing equations for magnetoelectro-elasticity, with the boundary conditions on the upper and lower surfaces satisfied exactly and those on the cylindrical surface satisfied approximately (in the Saint Venant sense). The analytical solutions, derived with a direct displacement method, are valid f… Show more
“…where α is a constant associated with the radially varying parameters D and M and Poisson's ratio ν, β is a constant reflecting the comprehensive effect of the rotatory inertia and shear deformation, and C + ω and C − ω are two distinct roots of the quadratic equation (11) with respect to C ω in view of Eqs. (12) and (13).…”
Section: Exact Analytical Methodsmentioning
confidence: 99%
“…In practical applications, the thickness, modulus, and mass density of circular plates usually vary continuously or stepwisely, in one spatial dimension or more, to meet the requirement for optimizing the self-weight distribution, improving the local strength, or tailoring the dynamical properties. In recent years, it becomes very popular to use functionally graded structures [6][7][8][9][10][11] to avoid the interface mismatch on mechanical or thermal properties efficiently.…”
Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and flexural rigidity with the consideration of the effect of Poisson's ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.
“…where α is a constant associated with the radially varying parameters D and M and Poisson's ratio ν, β is a constant reflecting the comprehensive effect of the rotatory inertia and shear deformation, and C + ω and C − ω are two distinct roots of the quadratic equation (11) with respect to C ω in view of Eqs. (12) and (13).…”
Section: Exact Analytical Methodsmentioning
confidence: 99%
“…In practical applications, the thickness, modulus, and mass density of circular plates usually vary continuously or stepwisely, in one spatial dimension or more, to meet the requirement for optimizing the self-weight distribution, improving the local strength, or tailoring the dynamical properties. In recent years, it becomes very popular to use functionally graded structures [6][7][8][9][10][11] to avoid the interface mismatch on mechanical or thermal properties efficiently.…”
Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and flexural rigidity with the consideration of the effect of Poisson's ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.
“…A solution for the stresses and displacements of a FGMEE disk under tension and bending was presented by Wang et al. 32 Dai et al. 33 disclosed the elastic behavior of a rotating FGPM annular plate under hygrothermal loads.…”
Section: Introductionmentioning
confidence: 99%
“…Loghman et al 31 studied FGPM rotating disc and considered creep behavior. A solution for the stresses and displacements of a FGMEE disk under tension and bending was presented by Wang et al 32 Dai et al 33 disclosed the elastic behavior of a rotating FGPM annular plate under hygrothermal loads. The free vibration response of circular and annular MEE plates was investigated by Vinyas et al 34 Saadatfar 35 investigated the hygrothermal stress redistribution in an MEE rotating disk using an analytical method.…”
Effects of porosity, profile of thickness and angular deceleration on the stress and deformation of a fluid-saturated functionally graded porous magneto-electro-elastic rotating disc are investigated in this article. Since the angular velocity is taken to be variable, the disc is subjected to Lorentz force in two directions: radial and circumferential. It is assumed that material properties of the disc obey power-law function of radius. The disc is uniformly porous and its thickness varies as a function of radius. First, three coupled governing partial differential equations in terms of the displacement and electric potential are converted to ordinary differential equations employing the separation of variable method. Then, obtained equations are solved using the Runge–Kutta and shooting methods for the case of fixed–free boundary condition. The effect of variable angular velocity, thickness profile, inhomogeneity index, porosity, and magnetic field is studied and illustrated graphically. The results demonstrate that considering angular acceleration for the disc has a considerable effect on the Lorentz force resulted from the magnetic field. Besides, the angular velocity constant has a significant effect on the stresses and displacements in the presence of the magnetic field.
“…Li et al (2008) investigated the problem of a functionally graded, transversely isotropic, magnetoelectro-elastic circular plate acted on by a uniform load. Recently, static behavior of a MEE-FG circular plates under tension and bending is investigated by Wang et al (2015).…”
In this paper, vibration characteristics of magneto-electro-thermo-elastic functionally graded (METE-FG) nanobeams is investigated in the framework of third order shear deformation theory. Magneto-electro-thermo-elastic properties of FG nanobeam are supposed to vary smoothly and continuously along the thickness based on power-law form. To capture the small size effects, Eringen's nonlocal elasticity theory is adopted. By using the Hamilton's principle, the nonlocal governing equations are derived and then solved analytically to obtain the natural frequencies of METE-FG nanobeams. The reliability of proposed model and analytical method in predicting natural frequencies of METE-FG nanobeam is evaluated with comparison to some cases in the literature. Numerical results are provided indicating the influences of several parameters including magnetic potential, external electric voltage, temperature fields, power-law exponent, nonlocal parameter and slenderness ratio on the frequencies of METE-FG nanobeams. It is found that the vibrational behavior of METE-FG nanobeams is significantly impressed by these effects.
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