A Functionally Graded Plane With a Circular Inclusion Under Uniform Antiplane Eigenstrain

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
X. Wang ◽  
E. Pan ◽  
A. K. Roy
Keyword(s):  
General Solution ◽  
Shear Modulus ◽  
Helmholtz Equation ◽  
Series Expansion ◽  
Original Problem ◽  
Circular Inclusion ◽  
Functionally Graded ◽  
Numerical Results ◽  
Stress Fields ◽  
Homogeneous Material

The problem of a functionally graded plane with a circular inclusion under a uniform antiplane eigenstrain is investigated, where the shear modulus varies exponentially along the x direction. By introducing a new function which satisfies the Helmholtz equation, the general solution to the original problem is derived in terms of series expansion. Numerical results are then presented which demonstrate clearly that for a functionally graded plane, the strain and stress fields inside the circular inclusion under uniform antiplane eigenstrains are intrinsically nonuniform. This phenomenon differs from the corresponding homogeneous material case where both the strain and stress fields are uniform inside the circular inclusion.

Analytical Solutions of Stresses in Functionally Graded Circular Hollow Cylinder with Finite Length

2004 ◽  
Vol 261-263 ◽  
pp. 651-656 ◽  
Author(s):  
Z.S. Shao ◽  
L.F. Fan ◽  
Tie Jun Wang
Keyword(s):  
Young’S Modulus ◽  
Hollow Cylinder ◽  
Finite Length ◽  
Analytical Solutions ◽  
Radial Direction ◽  
Young's Modulus ◽  
Functionally Graded ◽  
Numerical Results ◽  
Stress Fields ◽  
Simply Supported

Analytical solutions of stress fields in functionally graded circular hollow cylinder with finite length subjected to axisymmetric pressure loadings on inner and outer surfaces are presented in this paper. The cylinder is simply supported at its two ends. Young's modulus of the material is assumed to vary continuously in radial direction of the cylinder. Moreover, numerical results of stresses in functionally graded circular hollow cylinder are appeared.

scholarly journals An Analytical Closed-Form Solution for Free Vibration of Functionally Graded Circular Plates with Even and Uneven Porosity Distributions

2018 ◽  
Author(s):  
Krzysztof Kamil Żur
Keyword(s):  
General Solution ◽  
Boundary Conditions ◽  
Circular Plate ◽  
Volume Fraction ◽  
Functionally Graded ◽  
Numerical Results ◽  
Axisymmetric Vibration ◽  
Gradient Index ◽  
Circular Plates ◽  
Coupled Equations

Free axisymmetric and non-axisymmetric vibration analysis of the unsaturated porous functionally graded circular plates has been presented on the basis of classical plate theory. The defined coupled equations of motion for the porous functionally graded circular plate were decoupled based on the properties of the physical neutral surface. The one general solution of the decoupled equation of motion was obtained as linear combinations of the multiparametric special Bessel functions for the functionally graded circular plate with even and uneven porosity distributions. The influences of the even and uneven distributions of porosity, gradient index, diverse boundary conditions and the negligible effect of the coupling in-plane and transverse displacements on the dimensionless frequencies of the circular plate were comprehensively studied. The obtained numerical results show the differences and significant effect of considered types of distributions of porosities, values of the gradient index and the porosity volume fraction on the distribution of eigenfrequencies of the circular plates. Additionally, the obtained multiparametric general solution of the defined differential equation will allow to study the influences of diverse additional complicating effects such as stepped thickness, cracks, additional mounted elements expressed by only additional boundary conditions on the dynamic behavior of the porous functionally graded circular/annular plates. The formulated boundary value problem, the method of solution and the obtained numerical results for the perfect and imperfect functionally graded circular plates have not yet been reported. The present paper fills this void in the literature.

scholarly journals Nonlinear multiscale simulation of instabilities due to growth of an elastic film on a microstructured substrate

2020 ◽  
Vol 90 (11) ◽  
pp. 2397-2412
Author(s):  
Iman Valizadeh ◽  
Oliver Weeger
Keyword(s):  
Deformation Gradient ◽  
Multiscale Simulation ◽  
Numerical Results ◽  
Unit Cell ◽  
Homogeneous Material ◽  
Two Phase ◽  
Biological Phenomena ◽  
Elastic Film ◽  
Living Tissues ◽  
Analytical Approaches

Abstract The objective of this contribution is the numerical investigation of growth-induced instabilities of an elastic film on a microstructured soft substrate. A nonlinear multiscale simulation framework is developed based on the FE2 method, and numerical results are compared against simplified analytical approaches, which are also derived. Living tissues like brain, skin, and airways are often bilayered structures, consisting of a growing film on a substrate. Their modeling is of particular interest in understanding biological phenomena such as brain development and dysfunction. While in similar studies the substrate is assumed as a homogeneous material, this contribution considers the heterogeneity of the substrate and studies the effect of microstructure on the instabilities of a growing film. The computational approach is based on the mechanical modeling of finite deformation growth using a multiplicative decomposition of the deformation gradient into elastic and growth parts. Within the nonlinear, concurrent multiscale finite element framework, on the macroscale a nonlinear eigenvalue analysis is utilized to capture the occurrence of instabilities and corresponding folding patterns. The microstructure of the substrate is considered within the large deformation regime, and various unit cell topologies and parameters are studied to investigate the influence of the microstructure of the substrate on the macroscopic instabilities. Furthermore, an analytical approach is developed based on Airy’s stress function and Hashin–Shtrikman bounds. The wavelengths and critical growth factors from the analytical solution are compared with numerical results. In addition, the folding patterns are examined for two-phase microstructures and the influence of the parameters of the unit cell on the folding pattern is studied.

Forces on a Circular Hole Located in Functionally Graded Material

2011 ◽  
Vol 704-705 ◽  
pp. 631-635
Author(s):  
Xian Feng Wang ◽  
Feng Xing ◽  
Norio Hasebe
Keyword(s):  
Circular Hole ◽  
Constant Term ◽  
Complex Stress ◽  
Functionally Graded ◽  
Homogeneous Material ◽  
Dimensional Problem ◽  
Stress Components ◽  
Graded Material ◽  
Stress Functions ◽  
Stress Function Method

The complex stress function method is used in this study to formulate the 2-dimensional problem for nonhomogeneous materials. The Young’s modulus E varies linearly with the coordinate x and the Poisson’s ratio of the material is assumed constant and. The stress components and the boundary conditions are expressed in terms of two complex stress functions in explicit forms. It is noted that the constant term in stress functions has an influence on the stress components, which is different from the homogeneous material case. Subsequently, the problem of a nonhomogeneous plane containing a circular hole subjected to a uniform internal pressure is studied.

Love Wave in an Isotropic Half-Space with a Graded Layer

2013 ◽  
Vol 325-326 ◽  
pp. 252-255
Author(s):  
Li Gang Zhang ◽  
Hong Zhu ◽  
Hong Biao Xie ◽  
Jian Wang
Keyword(s):  
Shear Modulus ◽  
Half Space ◽  
Analytical Solutions ◽  
Love Wave ◽  
Elastic Half Space ◽  
Functionally Graded ◽  
Thickness Direction ◽  
Vibration Model ◽  
General Dispersion ◽  
Functionally Graded Layer

This work addresses the dispersion of Love wave in an isotropic homogeneous elastic half-space covered with a functionally graded layer. First, the general dispersion equations are given. Then, the approximation analytical solutions of displacement, stress and the general dispersion relations of Love wave in both media are derived by the WKBJ approximation method. The solutions are checked against numerical calculations taking an example of functionally graded layer with exponentially varying shear modulus and density along the thickness direction. The dispersion curves obtained show that a cut-off frequency arises in the lowest order vibration model.

Adaptive complex frequency with V-cycle GMRES for preconditioning 3D Helmholtz equation

Geophysics ◽  
2021 ◽  
pp. 1-40
Author(s):  
Wenhao Xu ◽  
Yang Zhong ◽  
Bangyu Wu ◽  
Jinghuai Gao ◽  
Qing Huo Liu
Keyword(s):  
Linear System ◽  
Helmholtz Equation ◽  
3D Models ◽  
Numerical Results ◽  
Spatial Sampling ◽  
Gmres Method ◽  
Complex Frequency ◽  
Iterative Solver ◽  
Helmholtz Equations ◽  
Sampling Density

Solving the Helmholtz equation has important applications in various areas, such as acoustics and electromagnetics. Using an iterative solver together with a proper preconditioner is key for solving large 3D Helmholtz equations. The performance of existing Helmholtz preconditioners usually deteriorates when the minimum spatial sampling density is small (approximately four points per wavelength [PPW]). To improve the efficiency of the Helmholtz preconditioner at a small minimum spatial sampling density, we have adopted a new preconditioner. In our scheme, the preconditioning matrix is constructed based on an adaptive complex frequency that varies with the minimum spatial sampling density in terms of the number of PPWs. Furthermore, the multigrid V-cycle with a GMRES smoother is adopted to effectively solve the corresponding preconditioning linear system. The adaptive complex frequency together with a GMRES smoother can work stably and efficiently at different minimum spatial sampling densities. Numerical results of three typical 3D models show that our scheme is more efficient than the multilevel GMRES method and shifted Laplacian with multigrid full V-cycle and a symmetric Gauss-Seidel smoother for preconditioning the 3D Helmholtz linear system, especially when the minimum spatial sampling density is large (approximately 120 PPW) or small (approximately 4 PPW).

Displacement and Stress Fields in a Functionally Graded Fiber-Reinforced Rotating Disk With Nonuniform Thickness and Variable Angular Velocity

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
Y. Zheng ◽  
H. Bahaloo ◽  
D. Mousanezhad ◽  
A. Vaziri ◽  
H. Nayeb-Hashemi
Keyword(s):  
Radial Direction ◽  
Volume Fraction ◽  
Rotating Disk ◽  
Outer Radius ◽  
Functionally Graded ◽  
Circumferential Direction ◽  
Stress Fields ◽  
Fiber Reinforced ◽  
Inner Radius ◽  
Nonuniform Thickness

Displacement and stress fields in a functionally graded (FG) fiber-reinforced rotating disk of nonuniform thickness subjected to angular deceleration are obtained. The disk has a central hole, which is assumed to be mounted on a rotating shaft. Unidirectional fibers are considered to be circumferentially distributed within the disk with a variable volume fraction along the radius. The governing equations for displacement and stress fields are derived and solved using finite difference method. The results show that for disks with fiber rich at the outer radius, the displacement field is lower in radial direction but higher in circumferential direction compared to the disk with the fiber rich at the inner radius. The circumferential stress value at the outer radius is substantially higher for disk with fiber rich at the outer radius compared to the disk with the fiber rich at the inner radius. It is also observed a considerable amount of compressive stress developed in the radial direction in a region close to the outer radius. These compressive stresses may prevent any crack growth in the circumferential direction of such disks. For disks with fiber rich at the inner radius, the presence of fibers results in minimal changes in the displacement and stress fields when compared to a homogenous disk made from the matrix material. In addition, we concluded that disk deceleration has no effect on the radial and hoop stresses. However, deceleration will affect the shear stress. Tsai–Wu failure criterion is evaluated for decelerating disks. For disks with fiber rich at the inner radius, the failure is initiated between inner and outer radii. However, for disks with fiber rich at the outer radius, the failure location depends on the fiber distribution.

The Design Tradeoffs of Linear Functionally Graded Piezoceramic Actuators

Aerospace ◽  
2003 ◽  
Author(s):  
Paul W. Alexander ◽  
Diann Brei
Keyword(s):  
Material Properties ◽  
High Stress ◽  
Functionally Graded ◽  
Homogeneous Material ◽  
Stress Concentrations ◽  
Practical Applications ◽  
Complex Electric Field ◽  
Design Tradeoffs ◽  
A Minor ◽  
The Cost

It is common practice to reduce the voltage level within piezoelectric actuators by utilizing multiple layers, typically bonded together. Unfortunately, this has a tendency to result in device failure due to delamination. For example, with benders the typical lifetime is 105 to 106 cycles, limiting its use in practical applications. This poses an interesting design tradeoff: the stroke is increased due to sharper gradients between material layers; however, the higher gradients lead to high stress concentrations at those interfaces. One approach to reducing these stresses is to grade the material properties through a monolithic piece of piezoceramic so that no interfaces or bonding elements exist, but this comes at the cost of stroke. This paper explores the design tradeoff inherent to monolithic functionally graded piezoelectrics. An analytical free-displacement model for a monolithic piezoceramic beam with a generic gradient is derived. Key to this is the inclusion of the complex electric field distribution which rises from the non-homogeneous material properties. This model is used along with finite element models to examine the effect of continuous linear and stepwise material gradients on the displacement performance as well as the stress levels. The study shows that using monolithic functionally graded piezocermics can significantly reduce the stresses with only a minor impact on the device stroke.

Distortion of Polymer Matrix Composite Panels Under Transverse Thermal Gradients

2007 ◽  
Author(s):  
Pei Gu ◽  
R. J. Asaro
Keyword(s):  
General Solution ◽  
Polymer Matrix ◽  
Thermal Loading ◽  
Polymer Matrix Composites ◽  
Functionally Graded ◽  
Thermal Distortion ◽  
Matrix Composites ◽  
Graded Materials ◽  
Internal Forces ◽  
Analytical Expressions

This paper discusses the distortion of panels made by fiber reinforced polymer matrix composites under transverse thermal-loading conditions. We formulate thermal distortion from the bending theory of functionally graded materials. General solution for thermal distortion is derived in terms of material variation and temperature profile along the thickness of the panels. Using the general solution, analytical expressions of thermal distortion and associated internal forces for commonly used end supports are obtained. From these solutions, we discuss the failure mechanism induced by thermal distortion and design criteria that can be placed to prevent such failure. The roles of geometry, temperature and temperature gradients, and the competition among them in the process of structural failure are addressed.

Contact Analysis of Functionally Graded Materials Using Smoothed Finite Element Methods

2019 ◽  
Vol 17 (05) ◽  
pp. 1940012 ◽  
Author(s):  
Y. F. Zhang ◽  
J. H. Yue ◽  
M. Li ◽  
R. P. Niu
Keyword(s):  
Finite Element ◽  
Functionally Graded Materials ◽  
Contact Point ◽  
Linear Complementarity Problems ◽  
Contact Problems ◽  
Contact Analysis ◽  
Functionally Graded ◽  
Numerical Results ◽  
Graded Materials ◽  
Stiffness Matrices

In the paper, the smoothed finite element method (S-FEM) based on linear triangular elements is used to solve 2D solid contact problems for functionally graded materials. Both conforming and nonconforming contacts algorithms are developed using modified Coulomb friction contact models including tangential strength and normal adhesion. Based on the smoothed Galerkin weak form, the system stiffness matrices are created using the formulation procedures of node-based S-FEM (NS-FEM) and edge-based S-FEM (ES-FEM), and the contact interface equations are discretized by contact point-pairs. Then these discretized system equations are converted into a form of linear complementarity problems (LCPs), which can be further solved efficiently using the Lemke method. The singular value decomposition method is used to deal with the singularity of the stiffness matrices in the procedure constructing the standard LCP, which can greatly improve the stability and accuracy of the numerical results. Numerical examples are presented to investigate the effects of the various parameters of functionally graded materials and comparisons have been made with reference solutions and the standard FEM. The numerical results demonstrate that the strain energy solutions of ES-FEM have higher convergence rate and accuracy compared with that of NS-FEM and FEM for functionally graded materials through the present contact analysis approach.

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