Dynamic responses of a functionally graded pyroelectric hollow sphere for spherically symmetric problems

The spherically symmetric thermoelastic problem of a functionally graded pyroelectric hollow sphere is considered. The functionally graded material is restricted to the variations of density and other material constants as a certain power of the radial distance. By virtue of the introduction of a dependent variable as well as the separation of variables technique, a Volterra integral equation of the second kind related to a function with respect to time is derived, which is solved successfully by using an interpolation polynomial to approximate the unknown function. The corresponding recursive formulae are derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently and is suitable for a functionally graded pyroelectric hollow sphere with an arbitrary thickness and subjected to arbitrary spherically symmetric thermal loads. Numerical results are finally presented.     

Nonlinear analysis of functionally graded laminates considering piezoelectric effect

In this paper, static bending analysis of functionally graded plates with piezoelectric layers has been carried out considering geometrical nonlinearity in different sets of mechanical and electrical loadings. Only the geometrical nonlinearity has been taken into account. The governing equations are obtained using potential energy and Hamilton??s principle. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect by using higher order elements. The present finite element used displacement and electric potential as nodal degrees of freedom. Results are presented for two constituent FGM plate under different mechanical boundary conditions. Numerical results for FGM plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response of the plate are also studied.     

Dynamic response of an anti-plane shear crack in a functionally graded piezoelectric strip

The dynamic response of a cracked functionally graded piezoelectric material (FGPM) under transient anti-plane shear mechanical and in-plane electrical loads is investigated in the present paper. It is assumed that the electroelastic material properties of the FGPM vary smoothly in the form of an exponential function along the thickness of the strip. The analysis is conducted on the basis of the unified (or natural) crack boundary condition which is related to the ellipsoidal crack parameters. By using the Laplace and Fourier transforms, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, the electric field, FGPM gradation, crack length, and electromechanical coupling coefficient.     

Analysis of nonlinear vibration response of a functionally graded truncated conical shell with piezoelectric layers

The nonlinear vibration response of a functionally graded materials (FGMs) truncated conical shell with piezoelectric layers is analyzed. The vibration amplitude is suppressed by the positive and inverse piezoelectric effects. And the bifurcation phenomenon is described to reveal the motion state of the conical shell. Firstly, a truncated conical shell composed of three layers is described. And the effective material properties of the FG layer are defined by the Voigt model and the power law distribution. Next, the electric potentials of piezoelectric layers are defined as cosine distribution along the thickness direction. Meanwhile, the constant gain negative velocity feedback algorithm is used to suppress the vibration amplitude by the electric potential produced by the sensor layer. Thereafter, considering the first-order shear deformation theory and the von Karman nonlinearity, the relationship between the strain and displacement is defined. And the corresponding energy of the conical shell is calculated. After that, the motion equations of the conical shell are derived based on the Hamilton principle. Again, the nonlinear single degree of freedom equation is derived by the Galerkin method and the static condensation method. In the end, the nonlinear vibration response of FGMs truncated conical shell with piezoelectric layers under the external excitation is analyzed via using the harmonic balance method and the Runge-Kutta method. The effects of various parameters, such as ceramic volume fraction exponent, external excitation’s amplitude, control gain and geometric parameters on the nonlinear vibration response of the system are evaluated by case studies. Results indicate that the control gain plays an important role on the suppression of the vibration amplitude. The ceramic volume fraction exponents are not sensitive to the nonlinear vibration response compared with other parameters. The bifurcation behavior is observed under different parameters. The FGMs truncated conical shell with piezoelectric layers has three types of motion state, such as periodic motion, multi-periodic motion, and chaos motion.

     

Anti-plane moving crack in a functionally graded piezoelectric layer between two dissimilar piezoelectric strips

The dynamic propagation of a crack in a functionally graded piezoelectric material (FGPM) interface layer between two dissimilar piezoelectric layers under anti-plane shear is analyzed using integral transform approaches. The properties of the FGPM layers vary continuously along the thickness. The FGPM layer and two homogeneous piezoelectric layers are connected weak-discontinuously. A constant velocity Yoffe-type moving crack is considered. The Fourier transform is used to reduce the problem to two sets of dual integral equations, which are then expressed to the Fredholm integral equations of the second kind. Numerical values on the dynamic energy release rate (DERR) are presented for the FGPM to show the effects on electric loading, gradient of the material properties, crack moving velocity, and thickness of the layers. The following are helpful to increase resistance to crack propagation in the FGPM interface layer: (a) certain direction and magnitude of the electric loading, (b) increasing the thickness of the FGPM interface layer, and (c) increasing the thickness of the homogeneous piezoelectric layer to have larger material properties than those of the crack plane in the FGPM interface layer. The DERR always increases with the increase of crack moving velocity and the gradient of the material properties.     

Analytical investigation on axisymmetric free vibrations of moderately thick circular functionally graded plate integrated with piezoelectric layers

In this paper, a free vibration analysis of moderately thick circular functionally graded (FG) plate integrated with two thin piezoelectric (PZT4) layers is presented based on Mindlin plate theory. The material properties of the FG core plate are assumed to be graded in the thickness direction, while the distribution of electric potential field along the thickness of piezoelectric layers is simulated by sinusoidal function. The differential equations of motion are solved analytically for two boundary conditions of the plate: clamped edge and simply supported edge. The analytical solution is validated by comparing the obtained resonant frequencies with those of an isotropic host plate. The emphasis is placed on investigating the effect of varying the gradient index of FG plate on the free vibration characteristics of the structure. Good agreement between the results of this paper and those of the finite element analyses validated the presented approach.     

Effect of material inhomogeneity on the rotating functionally of a graded orthotropic hollow cylinder

The effect of the material inhomogeneity on the stress field of a rotating orthotropic infinite hollow cylinder made of functionally graded materials is investigated. The original functionally graded hollow cylinder is approximated by the laminate model, of which the solution will gradually approach the exact one with the increase in number of fictitious layers. The analysis is performed by means of the state space method. The validity of the solution is verified by utilizing the exact solution of a special non-homogeneous rotating hollow cylinder and earlier results. The distributions of the radial and tangential stresses of the internally pressurized rotating functionally graded hollow cylinder for various material inhomogeneity parameters are depicted graphically. The degree of the orthotropy on the stress fields is also discussed.     

Thermo-electro-mechanical vibration and buckling analysis of a functionally graded piezoelectric porous cylindrical microshell

Journal of Mechanical Science and Technology - In this paper, the vibration and buckling behavior of a functionally graded piezoelectric porous cylindrical microshell under...     

Dynamic propagation of a finite eccentric crack in a functionally graded piezoelectric ceramic strip

The dynamic propagation of an eccentric Griffith crack in a functionally graded piezoelectric ceramic strip under anti-plane shear is analyzed using the integral transform method. A constant velocity Yoffe-type moving crack is considered. Fourier transform is used to reduce the problem to a pair of dual integral equations, which is then expressed in a Fredholm integral equation of the second kind. We assume that the properties of the functionally graded piezoelectric material vary continuously along the thickness. The impermeable crack boundary condition is adopted. Numerical values on the dynamic stress intensity factors are presented for the functionally graded piezoelectric material to show the dependence of the gradient of material properties, crack moving velocity, and eccentricity. The dynamic stress intensity factors of a moving crack in functionally graded piezoelectric material increases when the crack moving velocity, eccentricity of crack location, material property gradient, and crack length increase. This paper was recommended for publication in revised form by Associate Editor Hyeon Gyu Beom Jeong Woo Shin received a B.S. and M.S. degree in Mechanical Engineering from Yonsei University in Seoul, Korea in 1998 and 2000, respectively. A major field of Mr. Shin is fracture mechanics. He is currently working on the KARI (Korea Aerospace Research Institute) as a senior researcher. He conducted load analysis of fixed wing aircraft and full scale airframe static test at the KARI. He is now developing landing gear in the KHP (Korea Helicopter Program) as a performance engineer.     

Dynamic characteristics of an eccentric crack in a functionally graded piezoelectric ceramic strip

The dynamic response of an eccentric Griffith crack in functionally graded piezoelectric ceramic strip under anti-plane shear impact loading is analysed using integral transform method. Laplace transform and Fourier transform are used to reduce the problem to two pairs of dual integral equations, which are then expressed to Fredholm integral equations of the second kind. We assume that the properties of the functionally graded piezoelectric material vary continuously along the thickness. The impermeable crack boundary condition is adopted. Numerical values on the dynamic stress intensity factors are presented for the functionally graded piezoelectric material to show the dependence of the gradient of material properties and electric loadings.     

Free vibration analysis of functionally graded coupled circular plate with piezoelectric layers

Based on classical plate theory (CLPT), free vibration analysis of a circular plate composed of functionally graded material (FGM) with its upper and lower surfaces bounded by two piezoelectric layers was performed. Assuming that the material properties vary in a power law manner within the thickness of the plate the governing differential equations are derived. The distribution of electric potential along the thickness direction in piezoelectric layers is considered to vary quadratically such that the Maxwell static electricity equation is satisfied. Then these equations are solved analytically for two different boundary conditions, namely clamped and simply supported edges. The validity of our analytical solution was checked by comparing the obtained resonant frequencies with those of an isotropic host plate. Furthermore, for both FGM plate and FGM plate with piezoelectric layers, natural frequencies were obtained by finite element method. Very good agreement was observed between the results of finite element method and the method presented in this paper. Then for the two aforementioned types of boundary conditions, the values of power index were changed and its effect on the resonant frequencies was studied. Also, the effect of piezoelectric thickness layers on the natural frequencies of FGM piezoelectric plate was investigated. This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim Saeed Jafari Mehrabadi received his B.S. in mechanical Engineering from Azad University, Arak, Iran, in 1992. He then received his M.S. from Azad University, Tehran, Iran in 1995. Now he is a faculty member of the department of mechanical engineering in Azad university of Arak, Iran and PhD student of Azad University, Science and Research Campus, Pounak, Tehran, Iran. His interests include computational methods and solid mechanics such as vibration, buckling.     

Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.     

Stochastic hybrid numerical method for transient analysis of stress field in functionally graded thick hollow cylinders subjected to shock loading

This investigation aims to study the random stresses in a functionally graded (FG) thick hollow cylinder with uncertain material properties subjected to mechanical shock loading using a hybrid numerical method. The mechanical properties are considered to vary across thickness of FG cylinder as a nonlinear power function of radius. The stresses are obtained by solving Navier equation and using Galerkin finite element and Newmark finite difference methods. The Monte Carlo simulation is used to generate the random mechanical properties for the problem. The failure probabilities and time history analysis of stresses are determined for various coefficient of variation considering various grading patterns of mechanical properties. The presented hybrid numerical method is effective, with high capability for stochastic analysis of dynamic and transient analysis of FG structures with various boundary conditions.     

Experimental investigation of surface roughness effect on a free-flight sphere in a Ludwieg tube

Understanding the aerodynamic coefficients of meteoroid fragments, deorbiting space debris, or launch vehicle stages through atmospheric reentry is essential for ground risk assessments. In high enthalpy flow, surface roughness is a crucial factor affecting the aerodynamic coefficient. In this work, the effect of surface roughness on drag coefficient is investigated experimentally within a Ludwieg tube at a Mach 4 test condition. The test model is a sphere with a 5 mm diameter. Three different types of surface roughness are considered using a pre-heating process. Shadowgraph technique was used to visualize the flow features and model behavior. Based on the acquired high-speed images, the drag coefficients were obtained using an image tracking technique. Results show that the drag coefficient decreases with corresponding increases in surface roughness for the given flow condition, implying the importance of surface roughness effect in ground risk assessments.

     

Nonlinear dynamic response of a simply supported rectangular functionally graded material plate under the time-dependent thermalmechanical loads

An analysis on nonlinear dynamic characteristics of a simply supported functionally graded materials (FGMs) rectangular plate subjected to the transversal and in-plane excitations is presented in the time dependent thermal environment. Here we look the FGM Plates as isotropic materials which is assumed to be temperature dependent and graded in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents. The geometrical nonlinearity using Von Karman’s assumption is introduced. The formulation also includes in-plane and rotary inertia effects. In the framework of Reddy’s third-order shear deformation plate theory, the governing equations of motion for the FGM plate are derived by the Hamilton’s principle. Then the equations of motion with two-degree-of-freedom under combined the time-dependent thermomechanical loads can be obtained by using Galerkin’s method. Using numerical method, the control equations are analyzed to obtain the response curves. Under certain conditions the periodic and chaotic motions of the FGM plate are found. It is found that because of the existence of the temperature which relate to the time the motions of the FGM plate show the great difference. A period motion can be changed into the chaotic motions which are affected by the time dependent temperature.     

Characteristics of a transiently propagating crack in functionally graded materials

When a crack propagates with acceleration, deceleration and time rates of change of stress intensity factors, it is very important for us to understand the effects of acceleration, deceleration and time rates of change of stress intensity factors on the individual stresses and displacements at the crack tip. Therefore, the crack tip stress and displacement fields for a transiently propagating crack along gradient in functionally graded materials (FGMs) with an exponential variation of shear modulus and density are developed and the characteristics of a transiently propagating crack from the fields are analyzed. The effects of the rate of change of the stress intensity factor and the crack tip acceleration on the individual stresses at the crack tip are opposite each other. Specially, the isochromatics (constant maximum shear stress) of Mode I tilt backward around the crack tip with an increase of crack tip acceleration, and tilt forward around the crack tip with an increase of the rate of change of the dynamic mode I stress intensity factor. This paper was recommended for publication in revised form by Associate Editor Chongdu Cho Kwang-Ho Lee received a Ph.D. degree in Yeungnam University in 1993. Dr. Lee is currently a professor at the School of Mechanical and Automotive Engineering at Kyungpook National University in Korea. He also had worked in KOMSCO as an engineer and researcher (1982.3–1996.2). He is interested in the fields of fracture and stress analysis on the composite, interface, nano and functionally graded materials by theoretical and experimental mechanics. Specially, his major interest is analysis of dynamic crack tip fields. Young-Jae Lee received his B.S degree in Agricultural Civil Engineering from Gyeongsang National University (GNU) in 1982. He then received his M.S. and Ph.D. degrees from GNU in 1984 and 1995, respectively. Dr. Lee is currently a professor at the department of Civil Engineering at Kyungpook National University in Korea. From 2005 to 2006, he had served as an editor of Korea Institute for Structure Maintenance and Inspection. His research interests are in the area of evaluation, diagnosis and optimum design of structure. Sang-Bong Cho received a Ph. D. degree from Tokyo University in 1989. Dr. Cho is currently a professor at the division of Mechanical and Automation Engineering at Kyungnam University in Korea. His research interests are in the area of fracture mechanics, FEM stress analysis and fretting fatigue.     

The effect of scanning strategy on laser fusion of functionally graded H13/Cu materials

In recent years, research has been undertaken on manufacturing of functionally graded materials (FGM) using layered manufacturing technologies (LMT), also commonly known as rapid prototyping (RP), solid free-form fabrication (SFF) etc. The use of LMT to build complex FGM parts could meet optimum engineering design for various applications such as high performance die casting tools. In this context, H13 tool steel is considered a suitable material because of the high resistance to thermal fatigue and dimensional stability. Nevertheless, H13 with a low heat conduction coefficient is not thermally efficient for certain part geometries. With this in mind, the use of functionally graded techniques to disperse copper to specific regions/volumes of a H13 mould could lead to higher performance of the tool. The laser fusion of different proportions of Cu (0, 25 and 50% by weight) powder dispersed in H13 were analysed in this work. Additionally, different laser strategies were used to statistically analyse effects with respect to the composition of Cu. It was found that the refill strategy produces better results compared to all other scanning methods. The H13 with 25% Cu mix produced a homogeneous structure but cracks were observed along the cross section of the 25% Cu specimens. The pure H13 had a lower porosity with fine dendrite structures. The H13 with 50% Cu produced a non-homogeneous structure. This paper also discusses the microhardness tests results with respect to Cu composition and scanning strategy. Also, in order to identify the effect of powder composition layer on the subsequent layers and the cooling rate effect, samples were produced and analysed. One started with 100% H13 and ended with H13+50% Cu while the other started with H13+50% Cu and ended with 100% H13.     

Analysis of the modal frequency of a functionally graded cylindrical shell

Journal of Mechanical Science and Technology - The modal frequencies of a functionally graded material (FGM) cylindrical shell with different geometry dimensions and material constituents are...     

Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments

Nonlinear bending analysis is presented for a simply supported, functionally graded rectangular plate subjected to a transverse uniform or sinusoidal load and in thermal environments. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded plate are based on Reddy's higher-order shear deformation plate theory that includes thermal effects. Two cases of the in-plane boundary conditions are considered. A mixed Galerkin-perturbation technique is employed to determine the load-deflection and load-bending moment curves. The numerical illustrations concern nonlinear bending response of functional graded rectangular plates with two constituent materials. The influences played by temperature rise, the character of in-plane boundary conditions, transverse shear deformation, plate aspect ratio and volume fraction distributions are studied.