Analytical Solution for the Thermopiezoelastic Behavior of a Smart Functionally Graded Material Hollow Sphere Under Radially Symmetric Loadings

2015 ◽  
Vol 137 (6) ◽  
Author(s):  
M. Jabbari ◽  
A. R. Barati
Keyword(s):  
Differential Equations ◽  
Functionally Graded Material ◽  
Hollow Sphere ◽  
Material Properties ◽  
Functionally Graded ◽  
Feedback Gain ◽  
Modern Engineering ◽  
Graded Material ◽  
Power Law Function ◽  
Fgm Layer

An analytical study of the piezothermoelastic behavior of a functionally graded material (FGM) hollow sphere with integrated piezoelectric layers as a sensor and actuator under the effect of radially symmetric thermo-electro-mechanical loading is carried out. The material properties of the FGM layer are assumed to be graded in the radial direction according to a power law function. Governing differential equations are developed in terms of the components of the displacement field, the electric potential and the temperature of each layer of the smart FGM hollow sphere. The resulting differential equations are solved analytically. Numerical examples are given and discussed to show the significant influence of grading index of material properties and feedback gain on the mechanical–electrical responses. This will be useful for modern engineering design.

scholarly journals Exponential matrix method for the solution of exact 3D equilibrium equations for free vibrations of functionally graded plates and shells

2017 ◽  
Vol 21 (1) ◽  
pp. 77-114 ◽  
Author(s):  
Salvatore Brischetto
Keyword(s):  
Differential Equations ◽  
Functionally Graded Material ◽  
Material Properties ◽  
Matrix Method ◽  
Three Dimensional ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Equilibrium Equations ◽  
Graded Material ◽  
Exponential Matrix

The present paper analyzes the convergence of the exponential matrix method in the solution of three-dimensional equilibrium equations for the free vibration analysis of functionally graded material structures. The three-dimensional equilibrium equations are written in general orthogonal curvilinear coordinates for one-layered and sandwich plates and shells embedding functionally graded material layers. The resulting system of second-order differential equations is reduced to a system of first-order differential equations redoubling the variables. This system is exactly solved using the exponential matrix method and harmonic displacement components. In the case of functionally graded material plates, the differential equations have variable coefficients because of the material properties which depend on the thickness coordinate z. For functionally graded material shells, the differential equations have variable coefficients because of both changing material properties and curvature terms. Several mathematical layers M can be introduced to approximate the curvature terms and the variable functionally graded material properties to obtain differential equations with constant coefficients. The exponential matrix is applied to solve the resulting system of partial differential equations with constant coefficients, where the used expansion has a very fast convergence ratio. The present work investigates the convergence of the proposed method related to the order N used for the expansion of the exponential matrix and to the number of mathematical layers M used for the approximation of curvature shell terms and variable functionally graded material properties. Both N and M values are analyzed for different geometries, thickness ratios, materials, functionally graded material laws, lamination sequences, imposed half-wave numbers, frequency orders, and vibration modes.

Stability of the Size-Dependent and Functionally Graded Curvilinear Timoshenko Beams

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
J. Awrejcewicz ◽  
A. V. Krysko ◽  
S. P. Pavlov ◽  
M. V. Zhigalov ◽  
V. A. Krysko
Keyword(s):  
Differential Equations ◽  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
Stability Loss ◽  
Functionally Graded ◽  
Beam Deflection ◽  
Timoshenko Beams ◽  
Size Dependent ◽  
Rigid Layer ◽  
Graded Material

The size-dependent model is studied based on the modified couple stress theory for the geometrically nonlinear curvilinear Timoshenko beam made from a functionally graded material having its properties changed along the beam thickness. The influence of the size-dependent coefficient and the material grading on the stability of the curvilinear beams is investigated with the use of the setup method. The second-order accuracy finite difference method is used to solve the problem of nonlinear partial differential equations (PDEs) by reducing it to the Cauchy problem. The obtained set of nonlinear ordinary differential equations (ODEs) is then solved by the fourth-order Runge–Kutta method. The relaxation method is employed to solve numerous static problems based on the dynamic approach. Eight different combinations of size-dependent coefficients and the functionally graded material coefficient are used to study the stress-strain responses of Timoshenko beams. Stability loss of the curvilinear Timoshenko beams is investigated using the Lyapunov criterion based on the estimation of the Lyapunov exponents. Beams with/without the size-dependent behavior, homogeneous beams, and functionally graded beams having the same stiffness are investigated. It is shown that in straight-line beams, the size-dependent effect decreases the beam deflection. The same is observed if the most rigid layer is located on the top of the beam. In the curvilinear Timoshenko beam, such a location of the most rigid layer essentially improves the beam strength against stability loss. The observed transition of the largest Lyapunov exponent from a negative to positive value corresponds to the transition from a precritical to postcritical beam state.

Geometrically nonlinear dynamic response and vibration of shear deformable eccentrically stiffened functionally graded material cylindrical panels subjected to thermal, mechanical, and blast loads

2018 ◽  
Vol 22 (3) ◽  
pp. 658-688 ◽  
Author(s):  
Nguyen Dinh Duc ◽  
Ngo Duc Tuan ◽  
Pham Hong Cong ◽  
Ngo Dinh Dat ◽  
Nguyen Dinh Khoa
Keyword(s):  
Dynamic Response ◽  
Functionally Graded Material ◽  
Material Properties ◽  
Nonlinear Dynamic ◽  
Functionally Graded ◽  
Blast Loads ◽  
Temperature Dependent ◽  
Nonlinear Dynamic Response ◽  
Cylindrical Panels ◽  
Graded Material

Based on the first order shear deformation shell theory, this paper presents an analysis of the nonlinear dynamic response and vibration of imperfect eccentrically stiffened functionally graded material (ES-FGM) cylindrical panels subjected to mechanical, thermal, and blast loads resting on elastic foundations. The material properties are assumed to be temperature-dependent and graded in the thickness direction according to simple power-law distribution in terms of the volume fractions of the constituents. Both functionally graded material cylindrical panels and stiffeners having temperature-dependent properties are deformed under temperature, simultaneously. Numerical results for the dynamic response of the imperfect ES-FGM cylindrical panels with two cases of boundary conditions are obtained by the Galerkin method and fourth-order Runge–Kutta method. The results show the effects of geometrical parameters, material properties, imperfections, mechanical and blast loads, temperature, elastic foundations and boundary conditions on the nonlinear dynamic response of the imperfect ES-FGM cylindrical panels. The obtained numerical results are validated by comparing with other results reported in the open literature.

Finite Element Analysis of Human Tibia Modeled as a Functionally Graded Material

2019 ◽  
Vol 2 (3) ◽  
Author(s):  
Ashish Tiwari ◽  
Pankaj Wahi ◽  
Niraj Sinha
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Stress Distribution ◽  
Functionally Graded Material ◽  
Material Properties ◽  
Functionally Graded ◽  
Element Analysis ◽  
Cross Sectional ◽  
Human Tibia ◽  
Graded Material

Human tibia, the second largest bone in human body, is made of complex biological material having inhomogeneity and anisotropy in such a manner that makes it a functionally graded material. While analyses of human tibia assuming it to be made of different material regions have been attempted in past, functionally graded nature of the bone in the mechanical analysis has not been considered. This study highlights the importance of functional grading of material properties in capturing the correct stress distribution from the finite element analysis (FEA) of human tibia under static loading. Isotropic and orthotropic material properties of different regions of human tibia have been graded functionally in three different manners and assigned to the tibia model. The nonfunctionally graded and functionally graded models of tibia have been compared with each other. It was observed that the model in which functional grading was not performed, uneven distribution and unrealistic spikes of stresses occurred at the interfaces of different material regions. On the contrary, the models with functional grading were free from this potential artifact. Hence, our analysis suggests that functional grading is essential for predicting the actual distribution of stresses in the entire bone, which is important for biomechanical analysis. We find that orthotropic nature of the bone tends to increase the maximum von Mises stress in the entire tibia, while inclusion of cross-sectional inhomogeneity typically increases the stresses across normal cross section. Accordingly, our analysis suggests that both orthotropy as well as cross-sectional inhomogeneity should be included to correctly capture the stress distribution in the bone.

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