scholarly journals Analytical Solution for Bending Analysis of Axially Functionally Graded Angle-Ply Flat Panels

2018 ◽  
Vol 2018 ◽  
pp. 1-17 ◽  
Author(s):  
Agyapal Singh ◽  
Poonam Kumari ◽  
Rupam Hazarika
Keyword(s):  
Analytical Solution ◽  
Numerical Solutions ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Variation Principle ◽  
Kantorovich Method ◽  
Dimensional Solution ◽  
Power Series Method ◽  
Arbitrary Boundary ◽  
Property Variation

In this paper, the analytical solution is presented for axially functionally graded (AFG) angle-ply flat panels subjected to arbitrary boundary condition. Material properties of AFG panels are assumed to vary linearly along x-direction. Reissner-type variation principle is used to derive the governing equations in mixed form. By employing extended Kantorovich method (EKM), a set of nonhomogeneous ordinary differential equations (ODEs) are obtained along the in-plane (x) and thickness (z) direction. The system of ODEs along the z-direction has constant coefficients, solved analytically. However, the system of ODEs along x-direction has variable coefficients, solved using modified power series method. The influence of property variation on the deflection and stresses is studied and discussed comprehensively for different sets of boundary conditions. Numerical results are validated through comparison with 3D FE. The presented analytical solution can serve as a benchmark for assessing the accuracy of the two-dimensional solution or 3D numerical solutions.

A boundary integral investigation for unsteady modified Helmholtz problems of some other classes of anisotropic functionally graded materials

2021 ◽  
Vol 10 (01) ◽  
pp. 2150005
Author(s):  
Moh. Ivan Azis
Keyword(s):  
Functionally Graded Materials ◽  
Numerical Solutions ◽  
Free Variable ◽  
Variable Coefficients ◽  
Boundary Integral ◽  
Functionally Graded ◽  
Time Variable ◽  
Graded Materials ◽  
The Laplace Transform ◽  
Constant Coefficients

Numerical solutions for a class of unsteady modified Helmholtz problems of anisotropic functionally graded materials are sought. The governing equation which is a variable coefficients equation is transformed to a constant coefficients equation. The time variable is transformed using the Laplace transform. The resulted partial differential equation of constant coefficients and time free variable is then converted to a boundary integral equation, from which boundary element solutions can be obtained. Some examples are considered to verify the accuracy, convergence and consistency of the numerical solutions. The results show that the numerical solutions are accurate, convergent and consistent.

scholarly journals An analytical solution for a fractional heat-like equation with variable coefficients

2017 ◽  
Vol 21 (4) ◽  
pp. 1759-1764 ◽  
Author(s):  
Run-qing Cui ◽  
Tao Ban
Keyword(s):  
Analytical Solution ◽  
Power Series ◽  
Solution Process ◽  
Fractional Power ◽  
Variable Coefficients ◽  
Series Method ◽  
Power Series Method ◽  
Fractional Power Series

The fractional power series method is used to solve a fractional heat-like equations with variable coefficients. The solution process is elucidated, and the results show that the method is simple but effective.

scholarly journals An Extended Kantorovich Method Used for Static Solution of an Arbitrarily Edge Supported Sandwich Cylindrical Shell Panel

2019 ◽  
Vol 8 (6) ◽  
pp. 4184-4193
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Laminated Composite ◽  
Static Solution ◽  
Radial Component ◽  
Kantorovich Method ◽  
Power Series Method ◽  
Arbitrary Boundary ◽  
Shell Panel

Exact solution of complex problems like composite shells with arbitrarily supported boundary conditions through analytical three-dimensional (3-D) approach is mathematically challenging. In the present work an analytical 3-D elasticity solution for the static bending problem of a laminated composite cylindrical shell panel having any arbitrary boundary conditions is proposed. The governing Partial Differential Equations (PDE) problems are obtained by the application of the Ressiner-type mixed variational principle in cylindrical coordinate system. The extended Kantrovich method [10] is applied to solve these equations by reducing them to Ordinary Differential Equations (ODE). Further, the set of ODEs corresponding to the radial component & the circumferential components are solved utilizing modified power series method & Pagano’s approach respectively. Through numerical studies of sandwich shell panels it is shown that this method accurately predicts the deflections, stresses, boundary effects and interfacial disruptions being generated of laminate scheme, material property variations and configuration of the shell panel. Crucially, this is achieved with just two or three terms and few iterations, hence attributes faster computation as compared to other numerical techniques.

scholarly journals Asymptotic Analytical Solution on Lamb Waves in Functionally Graded Nano Copper Layered Wafer

2021 ◽  
Vol 11 (10) ◽  
pp. 4442
Author(s):  
Yifeng Hu ◽  
Xiaoshan Cao ◽  
Yi Niu ◽  
Yan Ru ◽  
Junping Shi
Keyword(s):  
Power Series ◽  
Phase Velocity ◽  
Nondestructive Evaluation ◽  
Lamb Waves ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Velocity Variation ◽  
Multilayered Structures ◽  
Power Series Method ◽  
Copper Wafers

In this study, the feasibility of using Lamb waves in functionally graded (FG) nano copper layered wafers in nondestructive evaluation is evaluated. The elastic parameters and mass densities of these wafers vary with thickness due to the variation in grain size. The power series technique is used to solve the governing equations with variable coefficients. To analyze multilayered structures, of which the material parameters are continuous but underivable, a modified transfer matrix method is proposed and combined with the power series method. Results show that multiple modes of Lamb waves exist in FG nano copper wafers. Moreover, the gradient property leads to a decrease in phase velocity, and the absolute value of the phase velocity variation is positively correlated with the gradient coefficient. The phase velocity variation and variation rate in Mode 2 are smaller than those in other modes. The findings indicate that Mode 4 is recommended for nondestructive evaluation. However, if the number of layers is greater than four, the dispersion curves of the Lamb waves in the multilayer structures tend to coincide with those in the equivalent uniform structures. The results of this study provide theoretical guidance for the nondestructive evaluation of FG nanomaterial layered structures.

Two-Dimensional Free Vibration Analysis of Axially Functionally Graded Beams Integrated with Piezoelectric Layers: An Piezoelasticity Approach

2020 ◽  
Vol 12 (04) ◽  
pp. 2050037
Author(s):  
Agyapal Singh ◽  
Poonam Kumari
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Numerical Results ◽  
Two Dimensional ◽  
Kantorovich Method ◽  
Piezoelectric Layers

For the first time, a two-dimensional (2D) piezoelasticity-based analytical solution is developed for free vibration analysis of axially functionally graded (AFG) beams integrated with piezoelectric layers and subjected to arbitrary supported boundary conditions. The material properties of the elastic layers are considered to vary linearly along the axial ([Formula: see text]) direction of the beam. Modified Hamiltons principle is applied to derive the weak form of coupled governing equations in which, stresses, displacements and electric field variables acting as primary variables. Further, the extended Kantorovich method is employed to reduce the governing equation into sets of ordinary differential equations (ODEs) along the axial ([Formula: see text]) and thickness ([Formula: see text]) directions. The ODEs along the [Formula: see text]-direction have constant coefficients, where the ODEs along [Formula: see text]-direction have variable coefficients. These sets of ODEs are solved analytically, which ensures the same order of accuracy for all the variables by satisfying the boundary and continuity conditions in exact pointwise manner. New benchmark numerical results are presented for a single layer AFG beam and AFG beams integrated with piezoelectric layers. The influence of the axial gradation, aspect ratio and boundary conditions on the natural frequencies of the beam are also investigated. These numerical results can be used for assessing 1D beam theories and numerical techniques.

Analytical Solution for Centrifugal Force Effect in Functionally Graded Hollow Sphere

2011 ◽  
Vol 110-116 ◽  
pp. 2829-2837
Author(s):  
Mohsen Jabbari ◽  
Amir Hossein Mohazzab
Keyword(s):  
Analytical Solution ◽  
Power Law ◽  
Hollow Sphere ◽  
Radial Stress ◽  
Radial Displacement ◽  
Direct Method ◽  
Functionally Graded ◽  
Power Series Method ◽  
Centrifugal Load ◽  
Material Stiffness

This paper presents the effect of centrifugal load in functionally graded hollow sphere. Analytical solution for stresses is determined using the direct method and the power series method. The material stiffness varies continuously across the thickness direction according to the power law functions of radial directions. Increasing the angular velocity results in increasing the all above quantities. With increasing the power law indices the radial displacement, the shear and circumferential stresses due to centrifugal load all are decreased and the radial stress due to centrifugal load increased.

scholarly journals Asymptotic Solution and Numerical Simulation of Lamb Waves in Functionally Graded Viscoelastic Film

Materials ◽  
2019 ◽  
Vol 12 (2) ◽  
pp. 268 ◽  
Author(s):  
Xiaoshan Cao ◽  
Haining Jiang ◽  
Yan Ru ◽  
Junping Shi
Keyword(s):  
Power Series ◽  
Asymptotic Solution ◽  
Viscoelastic Material ◽  
Lamb Waves ◽  
Exact Analytical Solution ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Dispersion Property ◽  
Power Series Solution ◽  
Power Series Method

To investigate Lamb waves in thin films made of functionally graded viscoelastic material, we deduce the governing equation with respect to the displacement component and solve these partial differential equations with complex variable coefficients based on a power series method. To solve the transcendental equations in the form of a series with complex coefficients, we propose and optimize the minimum module approximation (MMA) method. The power series solution agrees well with the exact analytical solution when the material varies along its thickness following the same exponential function. When material parameters vary with thickness with the same function, the effect of the gradient properties on the wave velocity is limited and that on the wave structure is obvious. The influence of the gradient parameter on the dispersion property and the damping coefficient are discussed. The results should provide nondestructive evaluation for viscoelastic material and the MMA method is suggested for obtaining numerical results of the asymptotic solution for attenuated waves, including waves in viscoelastic structures, piezoelectric semiconductor structures, and so on.

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