Impact response of a functionally graded piezoelectric plate with a vertical crack

The dynamic fracture problem for a functionally graded piezoelectric plate containing a crack perpendicular to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the medium vary continuously in the thickness direction. Integral transform techniques and dislocation density function are employed to reduce the problem to the solution of a singular integral equation. Mode I dynamic energy density factors are presented for an internal crack as well as an edge crack for various values of dimensionless parameters representing the size and location of the crack and the material nonhomogeneity.     

The dynamic response of an edge crack in a functionally graded orthotropic strip

The dynamic response of a functionally graded orthotropic strip with an edge crack perpendicular to the boundaries is studied. The material properties are assumed to vary continuously along the thickness direction. Laplace and Fourier transforms are applied to reduce the problem to a singular integral equation. Numerical results are presented to illustrate the influences of parameters such as the nonhomogeneity constant and geometry parameters on the dynamic stress intensity factors (SIFs).     

Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation

This paper investigates the nonlinear flexural dynamic behavior of a clamped Timoshenko beam made of functionally graded materials (FGMs) with an open edge crack under an axial parametric excitation which is a combination of a static compressive force and a harmonic excitation force. Theoretical formulations are based on Timoshenko shear deformable beam theory, von Karman type geometric nonlinearity, and rotational spring model. Hamilton’s principle is used to derive the nonlinear partial differential equations which are transformed into nonlinear ordinary differential equation by using the Least Squares method and Galerkin technique. The nonlinear natural frequencies, steady state response, and excitation frequency-amplitude response curves are obtained by employing the Runge–Kutta method and multiple scale method, respectively. A parametric study is conducted to study the effects of material property distribution, crack depth, crack location, excitation frequency, and slenderness ratio on the nonlinear dynamic characteristics of parametrically excited, cracked FGM Timoshenko beams.     

Higher-order crack tip fields for functionally graded material plate with transverse shear deformation

The crack tip fields are investigated for a cracked functionally graded material (FGM) plate by Reissner’s linear plate theory with the consideration of the transverse shear deformation generated by bending. The elastic modulus and Poisson’s ratio of the functionally graded plates are assumed to vary continuously through the coordinate y, according to a linear law and a constant, respectively. The governing equations, i.e., the 6th-order partial differential equations with variable coefficients, are derived in the polar coordinate system based on Reissner’s plate theory. Furthermore, the generalized displacements are treated in a separation-of-variable form, and the higher-order crack tip fields of the cracked FGM plate are obtained by the eigen-expansion method. It is found that the analytic solutions degenerate to the corresponding fields of the isotropic homogeneous plate with Reissner’s effect when the in-homogeneity parameter approaches zero.     

Nonlinear dynamic instability of edge-cracked functionally graded graphene-reinforced composite beams

Nonlinear Dynamics - This work investigates nonlinear dynamic instability of edge-cracked functionally graded (FG) graphene nanoplatelet (GNP)-reinforced composite (GNPRC) beams consist of...     

Aeroelastic tailoring of a plate wing with functionally graded materials

A functionally graded material (FGM) provides a spatial blend of material properties throughout a structure. This paper studies the efficacy of FGM for the aeroelastic tailoring of a metallic cantilever plate-like wing, wherein a genetic algorithm provides Pareto trade-off curves between static and dynamic aeroelastic metrics. A key comparison is between the effectiveness of material grading, geometric grading (i.e. plate thickness variations), and using both simultaneously. The introduction of material grading does, in some cases, improve the aeroelastic performance. This improvement, and the physical mechanism upon which it is based, depends on numerous factors: the two sets of metallic material parameters used for grading; the sweep of the plate; the aspect ratio of the plate; and whether the material is graded continuously or discretely.     

Dynamic response of a crack in a functionally graded interface of two dissimilar piezoelectric half-planes

Summary  In this paper, the dynamic anti-plane crack problem of two dissimilar homogeneous piezoelectric materials bonded through a functionally graded interfacial region is considered. Integral transforms are employed to reduce the problem to Cauchy singular integral equations. Numerical results illustrate the effect of the loading combination parameter λ, material property distribution and crack configuration on the dynamic stress and electric displacement intensity factors. It is found that the presence of the dynamic electric field could impede of enhance the crack propagation depending on the time elapsed and the direction of applied electric impact. Received 4 December 2001; accepted for publication 9 July 2002 This work is supported by the National Natural Science Foundation of China through Grant No. 10132010.     

Dielectric crack problem for a magnetoelectroelastic strip with functionally graded properties

The paper presents a fracture analysis for an electromagnetically dielectric crack in a functionally graded magnetoelectroelastic strip. It is considered that the material properties are varying exponentially along the width direction. Under the assumption of the in-plane magneto-electro-mechanical loadings, the dielectric crack is simulated by using the semi-permeable crack-face boundary conditions. The Fourier transform technique is applied to solve the boundary-value problem and four coupling singular integral equations are determined. A nonlinear system of algebraic equations is further derived and solved numerically to determine the electromagnetic field inside the crack. Then the field intensity factors of stress, electric displacement, and magnetic induction are given. Through the numerical computations, the effects of the material non-homogeneity and the permeability of crack interior on the electric displacement and the magnetic induction at the crack faces are studied. The variations of the intensity factors of stress, electric displacement, and magnetic induction versus the geometry of the crack, the strip width, and the material non-homogeneity are presented in graphics respectively.     

Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings

Based on the classical nonlinear von Karman plate theory, axisymmetric large deflection bending of a functionally graded circular plate is investigated under mechanical, thermal and combined thermal–mechanical loadings, respectively, and axisymmetric thermal post-buckling behavior of a functionally graded circular plate is also investigated. The mechanical and thermal properties of functionally graded material (FGM) are assumed to vary continuously through the thickness of the plate, and obey a simple power law of the volume fraction of the constituents. Governing equations for the problem are derived, and then a shooting method is employed to numerically solve the equations. Effects of material constant n and boundary conditions on the temperature distribution, nonlinear bending, critical buckling temperature and thermal post-buckling behavior of the FGM plate are discussed in details.     

Nonlinear vibration analysis of isotropic plate with inclined part-through surface crack

The four modes of vibration of an isotropic rectangular plate with an inclined crack are investigated. It is assumed that the crack remains continuous and its center is located at the center of the plate. The governing nonlinear equation of the transverse vibration of the plate with the plate boundary conditions being simply-supported on all edges is developed. The multiple scale perturbation method is utilized as the solution procedure to find the steady-state frequency response equations for all the four modes of vibration. The equations for the free and forced vibrations are derived and their frequency responses are presented. A special case of large-scale excitation force has also been considered. The parameter sensitivity analysis for the angle of crack, length of crack and the position of the external applied excitation force is performed. It has been shown that according to the aspect ratio of the plate, the vibration modes can have either nonlinear hardening effect or nonlinear softening behavior.     

A dielectric crack in a functionally graded piezoelectric layer

Considering the dielectric effects inside a crack, the problem of an electrically dielectric crack in a functionally graded piezoelectric layer is addressed in this paper. The energetically consistent crack-face boundary conditions are utilized to analyze the effects of a dielectric of crack interior. Applying the Fourier transform technique, the boundary-value problem is reduced to solving three coupling singular equations. Then a system of non-linear algebraic equations is obtained and the field intensity factors along with the energy release rate are given. Numerical results show the differences of the electric displacement inside a crack, the stress and electric displacement intensity factors and the energy release rate using the permeable, impermeable, semi-permeable and energetically consistent boundary conditions respectively. The effects of the material non-homogeneity, the applied electric field and the discharge field of crack interior on the electrostatic traction acting on the crack faces and the energy release rate are further studied through the energetically consistent boundary conditions.     

Effects of SH waves in a functionally graded plate

A computational method is presented to investigate SH waves in functionally graded material (FGM) plates. The FGM plate is first divided into quadratic layer elements (QLEs), in which the material properties are assumed as a quadratic function in the thickness direction. A general solution for the equation of motion governing the QLE has been derived. The general solution is then used together with the boundary and continuity conditions to obtain the displacement and stress in the wave number domain for an arbitrary FGM plate. The displacements and stresses in the frequency domain and time domain are obtained using inverse Fourier integration. Furthermore, a simple integral technique is also proposed for evaluating modified Bessel functions with complex valued order. Numerical examples are presented to demonstrate this numerical technique for SH waves propagating in FGM plates.     

Nonlinear nonplanar vibration of a functionally graded box beam

A functionally graded material (FGM) is a type of material designed to change continuously within the solid. It can be designed for specific applications such as thermal barrier coatings, corrosion protection, biomedical materials, space/aerospace industries, automotive applications, compliant mechanisms etc. In these applications, many primary and secondary structural elements can be idealized as beams. So, the aim of the present work is to study the nonlinear nonplanar vibration of a clamped-free slender box beam made of a FGM. More specifically, the cross section consisting of two isotropic materials, connected by a FG layer, is considered. To correctly describe the dynamic characteristics of the system, the nonlinear integro-differential equations used in this work, which consider the flexural–flexural–torsional couplings that occur in the nonplanar motions of the beam, include both geometric and inertial nonlinearities. In addition, the Galerkin method is applied to obtain a set of discretized equations of motion, which are in turn solved by numerical integration using the Runge–Kutta method. A detailed parametric analysis using several tools of nonlinear dynamics, unveils the complex dynamics of the FG beam in the main resonance region. The FG beam displays a complex nonlinear dynamic behavior with several coexisting planar and nonplanar solutions, leading to an intricate bifurcation scenario. Special attention is given to the symmetry breaking of beam dynamics and its influence on the bifurcations and instabilities. The results show that even small variations in cross section and material gradation have profound influence on the bifurcation diagrams and the dynamic behavior of the structure.     

Dynamic internal crack problem of a functionally graded magneto-electro-elastic strip

In this paper the dynamic anti-plane problem for a functionally graded magneto-electro-elastic strip containing an internal crack perpendicular to the boundary is investigated. The crack is assumed to be either magneto-electrically impermeable or permeable. Integral transforms and dislocation density functions are employed to reduce the problem to Cauchy singular integral equations. Numerical results show the effects of loading combination parameter, material gradient parameter and crack configuration on the dynamic response. With the magneto-electrically permeable assumption, both the magnetical and electrical impacts have no contribution to the crack tip field singularity. However, with the impermeable assumption, both the applied magnetical loads and electrical loads play a dominant role in the dynamic fracture behavior of crack tips. And for the two kinds of crack surface conditions, increasing the graded index can all retard the crack extension.     

Thermal buckling analysis of functionally graded beam with longitudinal crack

The thermal buckling problem of functionally graded beam with longitudinal crack is presented in the paper. The whole beam is divided into four sub-beams and each one is modeled as a Timoshenko beam. The buckling governing equation of each sub-beam in thermal environment is established by using Hamilton Principle. Combining with the boundary conditions, the continuous conditions of the displacements and the forces, the buckling governing equations are solved by both the analytical and numerical methods. The buckling modes and critical buckling temperatures are obtained, and the effects of the functionally graded index, crack length, crack depth, and crack longitudinal location on the buckling characteristics of beams are discussed in numerical examples.     

Electromechanical impact of a crack in a functionally graded piezoelectric medium

The Mode-I transient response of a functionally graded piezoelectric medium is solved for a through crack under the in-plane mechanical and electric impact. Integral transforms and dislocation density functions are employed to reduce the problem to singular integral equations. Numerical results display the effects of the loading combination parameter λ and the material parameter βa on the dynamic stress intensity factor and electric displacement intensity factor. The energy density factor criterion is applied to obtain the maximum of the minimum energy density factor and the direction of crack initiation.     

Near-tip out of plane displacement fields for dynamic crack propagation in functionally graded materials

Asymptotic expansion for the out of plane displacement field around a crack propagating along the gradient in a functionally graded material is developed. The irregular behavior of one of the terms in the expansion at low crack speeds is further examined and a remedial solution, which is well behaved at low crack speeds, is proposed. The developed out of plane displacement field is used to estimate stress intensity factor from quasi-static finite element solution. The results indicate that inclusion of the proposed nonhomogeneity specific terms gives estimates of stress intensity factor, which are consistent with existing analytical predictions.     

Dynamic stress analysis of a functionally graded material plate with a circular hole

This paper is to study the two-dimensional dynamic stress of a functionally graded material (FGM) plate with a circular hole under plane compressional waves at infinity. With using the method of piece-wise homogeneous layers, the dynamic stress distribution of the FGM plate having radial arbitrary material parameters is derived based on the complex variable method. As examples, numerical results are presented for the FGM plate having given radial shear modulus, density and Poisson’s ratio. It is found that the dynamic stress around the circular hole in the FGM plate can be effectively reduced by choosing the proper change ways of the radial material parameters for different frequency incident wave.     

Mode III crack problem in a functionally graded magneto-electro-elastic strip

Considering the material properties to be one-dimensionally dependent, this paper studied an anti-plane problem for an embedded crack and edge crack perpendicular to the boundary of a functionally graded magneto-electro-elastic strip. The crack is assumed to be either magneto-electrically impermeable or permeable. Integral transform and dislocation density functions are employed to reduce the problem to the solution of a system of singular integral equations. Numerical results show the effects of the loading combination parameter, material gradient parameter and crack configuration on the field intensity factors and the energy release rates of the functionally graded magneto-electro-elastic strip.