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.

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.

scholarly journals Nondestructive Evaluation of Functionally Graded Subsurface Damage on Cylinders in Nuclear Installations Based on Circumferential SH Waves

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Zhen Qu ◽  
Xiaoqin Shen ◽  
Xiaoshan Cao
Keyword(s):  
Mechanical Properties ◽  
Phase Velocity ◽  
Nondestructive Evaluation ◽  
Surface Damage ◽  
Subsurface Damage ◽  
Functionally Graded ◽  
Horizontal Shear ◽  
Steel Cylinder ◽  
Sh Waves ◽  
Subsurface Layers

Subsurface damage could affect the service life of structures. In nuclear engineering, nondestructive evaluation and detection of the evaluation of the subsurface damage region are of great importance to ensure the safety of nuclear installations. In this paper, we propose the use of circumferential horizontal shear (SH) waves to detect mechanical properties of subsurface regions of damage on cylindrical structures. The regions of surface damage are considered to be functionally graded material (FGM) and the cylinder is considered to be a layered structure. The Bessel functions and the power series technique are employed to solve the governing equations. By analyzing the SH waves in the 12Cr-ODS ferritic steel cylinder, which is frequently applied in the nuclear installations, we discuss the relationship between the phase velocities of SH waves in the cylinder with subsurface layers of damage and the mechanical properties of the subsurface damaged regions. The results show that the subsurface damage could lead to decrease of the SH waves’ phase velocity. The gradient parameters, which represent the degree of subsurface damage, can be evaluated by the variation of the SH waves’ phase velocity. Research results of this study can provide theoretical guidance in nondestructive evaluation for use in the analysis of the reliability and durability of nuclear installations.

scholarly journals Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Bochao Chen ◽  
Li Qin ◽  
Fei Xu ◽  
Jian Zu
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Variable Coefficients ◽  
Series Solutions ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Analytical Series ◽  
Residual Power

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.

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 Lamb Waves in a Functionally Graded Composite Plate with Nonintegral Power Function Volume Fractions

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Xiaoshan Cao ◽  
Zhen Qu ◽  
Junping Shi ◽  
Yan Ru
Keyword(s):  
Differential Equations ◽  
Lamb Waves ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Power Functions ◽  
Volume Fractions ◽  
Functionally Graded Composite ◽  
Propagation Behavior ◽  
Graded Material ◽  
Thermal Stress Relaxation

An analytical modelling is carried out to determine the Lamb wave’s propagation behavior in a thermal stress relaxation type functionally graded material (FGM) plate, which is a composite of two kinds of materials. The mechanical parameters depend on the volume fractions, which are nonintegral power functions, and the gradient coefficient is the power value. Based on the theory of elastodynamics, differential equations with variable coefficients are established. We employ variable substitution for theoretical derivations to solve the ordinary differential equations with variable coefficients using the Taylor series. The numerical results reveal that the dispersion properties in some regions are changed by the graded property, the phase velocity varies in a nonlinear manner with the gradient coefficient, nondispersion frequency exists in the first mode, and the set of cutoff frequencies is a union of two series of approximate arithmetic progressions. These results provide theoretical guidance not only for the experimental measurement of material properties but also for their nondestructive testing.

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.

scholarly journals Properties of Love Waves in Functional Graded Saturated Material

Materials ◽  
2018 ◽  
Vol 11 (11) ◽  
pp. 2165 ◽  
Author(s):  
Zhen Qu ◽  
Xiaoshan Cao ◽  
Xiaoqin Shen
Keyword(s):  
Power Series ◽  
Transversely Isotropic ◽  
Variable Coefficients ◽  
Love Waves ◽  
Series Method ◽  
Power Series Method ◽  
Material Parameters ◽  
Non Destructive Evaluation ◽  
Saturated Layer ◽  
Non Destructive

In the present study, the propagation of Love waves is investigated in a layered structure with two different homogeneity saturated materials based on Biot’s theory. The upper layer is a transversely isotropic functional graded saturated layer, and the substrate is a saturated semi-space. The inhomogeneity of the functional graded layer is taken into account. Furthermore, the gradient coefficient is employed as the representation of the relation with the layer thickness and the material parameters, and the power series method is applied to solve the variable coefficients governing the equations. In this regard, the influence of the gradient coefficients of saturated material on the dispersion relations, and the attenuation of Love waves in this structure are explored, and the results of the present study can provide theoretical guidance for the non-destructive evaluation of functional graded saturated material.

scholarly journals Symmetry Analysis and Exact Solutions to the Space-Dependent Coefficient PDEs in Finance

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Hanze Liu
Keyword(s):  
Power Series ◽  
Exact Solutions ◽  
Variable Coefficients ◽  
Similarity Solutions ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Series Method ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Generalized Power Series

The variable-coefficients partial differential equations (vc-PDEs) in finance are investigated by Lie symmetry analysis and the generalized power series method. All of the geometric vector fields of the equations are obtained; the symmetry reductions and exact solutions to the equations are presented, including the exponentiated solutions and the similarity solutions. Furthermore, the exact analytic solutions are provided by the transformation technique and generalized power series method, which has shown that the combination of Lie symmetry analysis and the generalized power series method is a feasible approach to dealing with exact solutions to the variable-coefficients PDEs.

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