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.

Elastic-Plastic Analysis for Functionally Graded Thick-Walled Tube Subjected to Internal Pressure

2016 ◽  
Vol 8 (2) ◽  
pp. 331-352 ◽  
Author(s):  
Libiao Xin ◽  
Guansuo Dui ◽  
Shengyou Yang ◽  
Ying Liu
Keyword(s):  
Internal Pressure ◽  
Yield Criterion ◽  
Functionally Graded ◽  
Power Functions ◽  
Unknown Parameters ◽  
Volume Fractions ◽  
Elastic Plastic ◽  
Graded Material ◽  
Two Phases ◽  
Elastic Plastic Materials

AbstractThe elastic-plastic response of the functionally graded thick-walled tube subjected to internal pressure is investigated by using the relation of the volume average stresses of constituents and the macroscopic stress of composite material in micromechanics. The tube consists of two idealized isotropic elastic-plastic materials whose volume fractions are power functions of the radius. As the internal pressure increases, the deformations of one phase and two phases from elastic to plastic are analyzed. In order to simplify the calculations we assume both materials with the same Poisson's ratio. By using the assumption of a uniform strain field within the representative volume element and the Tresca yield criterion, the theoretical solutions are obtained for the case of two elastic phases and the case of two plastic phases, and the function of the radial displacement is presented for the case with both elastic and plastic phases. The yield criterion of functionally graded material is given in terms of the yield stresses and volume fractions of constituents rather than Young's modulus and yield stress with different unknown parameters of the whole material in the existing papers. Finally we also discuss the position where the plastic deformation first occurs and the conditions for which material first yields in the tube.

scholarly journals Calculation of Volume Fractions of In Situ TiB and Residual Stress Distributions in Functionally Graded Composite of Ti–TiB–TiB2

Materials ◽  
2019 ◽  
Vol 12 (23) ◽  
pp. 4006
Author(s):  
Youfeng Zhang ◽  
Shasha He ◽  
Wanwan Yang ◽  
Jiangwei Ren ◽  
Haijuan Kong
Keyword(s):  
Residual Stress ◽  
Matrix Composite ◽  
Target Volume ◽  
Functionally Graded ◽  
Gradient Field ◽  
Stress Distributions ◽  
Volume Fractions ◽  
Functionally Graded Composite ◽  
The Matrix ◽  
Graded Material

Ti matrix composite with a polylaminate structure was successfully fabricated via spark plasma sintering (SPS) process. A temperature gradient field (TGF) was obtained during the sintering to form functionally graded material (FGM) in a vacuum under 40 MPa for 5 min. The actual volume fractions of TiB in the matrix were calculated based on the X-ray diffraction pattern. The target volume fractions of TiB were 0%, 20%, 40%, 60%, 80% and 100%. The calculated TiB volume fractions were slightly higher than the target volume fractions in layers 2–4 and lower than the target volume fractions in layers 5–6 and the deviations in layers 4 and 5 were less than 5% of the target volume. Based on the elastic axial symmetry model, the residual stress distributions in the Ti matrix composite with a polylaminate structure were simulated, indicating a relatively low thermal residual stress in the FGM.

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.

TiN/Al2O3 Functionally Graded Material Fabricated by In Situ Reaction

2008 ◽  
Vol 368-372 ◽  
pp. 1835-1837
Author(s):  
Jian Hua Nie ◽  
Ya Wei Li ◽  
Hao Yan ◽  
Yong He Liang ◽  
Yuan Bing Li
Keyword(s):  
Phase Composition ◽  
Functionally Graded Material ◽  
Functionally Graded ◽  
Initial Reaction ◽  
Tio2 Powder ◽  
Functionally Graded Composite ◽  
Epma Analysis ◽  
Graded Material ◽  
Al Powder

TiN/Al2O3 functionally graded composite was fabricated by in-situ aluminothermic reduction of TiO2 in coke bed from mixtures of TiO2 powder and metal Al powder. The reaction process, phase composition, and microstructure of sample treated at 1500°C for 3h were analyzed by XRD, SEM and EPMA. The results indicated that the thermite reduction of TiO2 involves several transitional stages and its initial reaction temperature is lowered by prior reaction between Al and TiO2. EPMA analysis showed that the TiN/Al2O3 ratio in TiN/Al2O3 functionally graded material products changes gradually across the samples without distinct interface between the different layers. The microstructure of the composite changes gradually, and the size of TiN grains increases from the verge region of samples to the centre of samples. These results above were in agreement with thermodynamic analysis.

Circumferential waves in pre-stressed functionally graded cylindrical curved plates

2014 ◽  
Vol 21 (1) ◽  
pp. 87-97 ◽  
Author(s):  
Jiangong Yu ◽  
Chuanzeng Zhang ◽  
Xiaoming Zhang
Keyword(s):  
Initial Stress ◽  
Wave Equations ◽  
Axial Direction ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Initial Stresses ◽  
Polynomial Method ◽  
Coupled Wave Equations ◽  
Graded Material ◽  
Polynomial Series

AbstractInitial stress (pre-stress) in functionally graded material (FGM) structures is often inevitable because of the limitation of available manufacturing technology. On the basis of the “mechanics of incremental deformations”, the circumferential wave characteristics in FGM cylindrical curved plates under uniform initial stresses in the radial and axial directions are investigated. The Legendre polynomial series method is used to solve the coupled wave equations with variable coefficients. Through numerical examples, the convergence of the polynomial method is discussed. The influences of the initial stresses on the circumferential Lamb-like and the circumferential SH waves are investigated, respectively. Numerical results show that they are quite distinct. Moreover, the influences of the initial stress in the axial direction are very different from those in the radial direction, both on the dispersion curves and on the displacement and stress distributions.

scholarly journals Modeling of vibration for functionally graded beams

2016 ◽  
Vol 14 (1) ◽  
pp. 661-672 ◽  
Author(s):  
Gülsemay Yiğit ◽  
Ali Şahin ◽  
Mustafa Bayram
Keyword(s):  
Initial Conditions ◽  
Adomian Decomposition Method ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Fourier Series Expansion ◽  
Bernoulli Beam ◽  
Adomian Decomposition ◽  
Graded Material ◽  
Euler Bernoulli Beam

AbstractIn this study, a vibration problem of Euler-Bernoulli beam manufactured with Functionally Graded Material (FGM), which is modelled by fourth-order partial differential equations with variable coefficients, is examined by using the Adomian Decomposition Method (ADM).The method is one of the useful and powerful methods which can be easily applied to linear and nonlinear initial and boundary value problems. As to functionally graded materials, they are composites mixed by two or more materials at a certain rate. This mixture at a certain rate is expressed with an exponential function in order to try to minimize singularities from transition between different surfaces of materials as much as possible. According to the structure of the ADM in terms of initial conditions of the problem, a Fourier series expansion method is used along with the ADM for the solution of simply supported functionally graded Euler-Bernoulli beams. Finally, by choosing an appropriate mixture rate for the material, the results are shown in figures and compared with those of a standard (homogeneous) Euler-Bernoulli beam.

Dynamic analysis of functionally graded shell with piezoelectric layers based on elasticity

2009 ◽  
Vol 223 (9) ◽  
pp. 2039-2047 ◽  
Author(s):  
M Javanbakht ◽  
M Shakeri ◽  
S N Sadeghi
Keyword(s):  
Differential Equations ◽  
Piezoelectric Layer ◽  
External Surface ◽  
Internal Surface ◽  
Longitudinal Direction ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Galerkin Finite Element ◽  
Piezoelectric Layers ◽  
Functionally Graded Shell

A study on the elasticity solution of the functionally graded (FG) shell with two piezoelectric layers is presented. In this article, the structure is finitely long, simply supported, and FG with two piezoelectric layers under pressure and electrostatic excitation. The equations of equilibrium, which are coupled partial differential equations, are reduced to ordinary differential equations (o.d.e.) with variable coefficients by means of trigonometric function expansion in the longitudinal direction. The resulting o.d.e. are solved by the Galerkin finite-element method and the Newmark method. Numerical results are presented for a FG cylindrical shell with a piezoelectric layer as an actuator in the external surface and a piezoelectric layer as a sensor in the internal surface.

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