In this paper, nonlinear vibration analysis of Functionally graded piezoelectric (FGP) beam with porous materials is investigated based on the Timoshenko beam theory. Material properties of FG porous beam are described according to the rule of mixture which is modified to approximate material properties with porosity phases. Ritz method is used to obtain the governing equation which is then solved by a direct iterative method to determine the nonlinear vibration frequencies of FGP porous beam subjected to different boundary conditions. The effects of external electric voltage, material distribution profile, porosity volume fraction, slenderness ratios and boundary conditions on the nonlinear vibration characteristics of the FGP porous beam are discussed in detail. The results indicate that piezoelectric layers have a significant effect on the nonlinear frequencies. Also, it is found that porosity has a considerable influence on the nonlinear frequency, especially when the electric voltage is applied.

This study presents critical buckling of Functionally graded soft ferromagnetic porous (FGFP) rectangular plates, under magnetic field with simply supported boundary condition. Equilibrium and stability equations of a porous rectangular plate in transverse magnetic field are derived. The geometrical nonlinearities are considered in the Love- Kirchhoff hypothesis sense. The formulations are compared to those of homogeneous isotropic plates were given in the literature. In this paper the effect of pore pressure on critical magnetic field of plate and the effect of important parameters of poroelastic material on buckling capacity are investigated. Also the compressibility of fluid and porosity on the buckling strength are studied.

This investigation presents material nonlinear analysis of a cantilever bar element made of Functionally graded material with porosity properties. The material properties of bar element are considered as changing though axial direction based on the Power-Law distribution and uniform porosity distribution. The stress-strain relation of the material is considered as a nonlinear property according to a Power-Law function. The cantilever bar element is subjected to a point load at the free end. In order to obtain more realistic solution for the nonlinear problem and axially material distribution, nonlinear finite element method is used. In the obtaining of finite element equations, the virtual work principle is used and, after linearization step, the tangent stiffness matrix and residual vector are obtained. In the nonlinear solution process, the incremental force method is implemented and, each load step, the nonlinear equations are solved by using the Newton-Raphson iteration method. In the numerical results, effects of material nonlinearity parameters, porosity coefficients, material distribution parameter and aspect ratios on nonlinear static deflections of the bar are presented and discussed. The obtained results show that the material nonlinear behaviour of the bar element is considerably affected with porosity and material graduation.

The spacecraft and space shuttles demand novel engineering materials to meet the required properties. This can be accomplished by altering the material properties in more than one direction. The introduction of inplane bidirectional Functionally graded materials with porosity are expected to exhibit these properties. This paper presents the buckling analses of inplane bidirectional (2-D) Functionally graded porous plates (IBFGPPs) considering uniform porosity distribution in uni-axial and bi-axial compression. The effective modulus of elasticity of the material is varied in in x-and y-axes by employing the rule of mixtures. The higherorder theory used for the study of buckling response meets the nullity requirements at plate’s upper and lower surface and derived the equations of motion thru Lagrange equations. The displacement functions are formulated in simple algebraic polynomials, incorporating admissible functions to satisfy the simply supported conditions in both axial and transverse directions. The components of admissible functions are derived by Pascal’s triangle. Accurateness of this theory is judged by comparing it to existing numerical data in the literature. The effect of thickness ratio’s (a/h), aspect ratio’s (b/a), exponents (ζ_1and ζ_2) in η_1 and η_2-direction, and the porosity on the buckling response of IBFGPPs are examined comprehensively. The numerical findings provided here serve as reference solutions for evaluating diverse plate theories and for comparing them against results obtained through alternative analytical and finite element techniques. From the obtained results, it can be inferred that the proposed theory facilitates the assessing of buckling tendencies of in-plane bi-directional porous FG plates produced through sintering process and could be deemed as a pivotal in the process of optimizing the design of the IBFGPPs.

This paper presents a novel investigation into the free vibration of porous folded plates using the differential transformation method (DTM). The porosity is Functionally graded (FG) along the thickness of the plate, resulting in material properties that vary with the z-coordinate. The motion equations for each plate segment are derived based on classical plate theory (CPT), with simply-supported boundary conditions applied at the front edges, allowing the transformation of partial differential equations into ordinary differential equations. The differential transformation method is then employed to discretize the motion equations in the x-direction. By applying boundary conditions at the remaining edges and ensuring continuity at the joints, the eigenvalue problem is formulated, leading to the calculation of natural frequencies and mode shapes of the folded plate. The mathematical model is validated through comparisons with finite element method (FEM) results and existing literature. Results indicate that Type C porosity distributions exhibit the highest stiffness and resonant frequency compared to other porosity types. While frequency behavior is consistent across mode numbers regardless of porosity distribution and plate length, the impact of the porosity parameter on the frequency of Type C plates is demonstrably less significant than on other porosity types.

In this paper, an aerothermoelastic analysis of Functionally graded plate containing porosities in yawed hypersonic flows is investigated. Due to some incorrect manufacturing processes, two different types of porosity, namely, even and uneven distributions are taken into account. The third-order piston theory is utilized to estimate the unsteady aerodynamic pressure induced by the hypersonic airflow. The material properties of a plate are assumed to vary across the thickness direction according to a simple power law. Based on classical plate theory, the motion equations are developed with geometric nonlinearity taking into consideration of von Karman strains. The formulations are established based on Hamilton’s principle while the generalized differential quadrature method is employed to solve the nonlinear aerothermoelastic equations. Due to lower computational efforts‎, the method of generalized differential quadrature is used to obtain accurate results. Moreover, the assumed mode method along with the Runge-Kutta integration algorithm is used as a solution method. The reliability and precision of the obtained results are validated by published literature. Then, the influence of porosity distribution, porosity coefficient, and yawed flow angle are discussed in detail. In general, this paper shows that even porosity distribution would have a more destabilizing effect compared with the ‎uneven porous model. And also, for both porosity distributions, the chaotic behavior appears in higher top surface temperature but even porosity distribution has a profound effect on chaotic motion.

Free vibrations of the cylindrical-hemispherical shell made of Functionally graded porous materials with variable thickness are investigated in the cylindrical coordinate system. In this study, two boundary conditions is examined. The first B.C is assumed free-free and second B.C is considered clamped-free in cylindrical and hemispherical, respectively. The present analysis is based on three-dimensional elasticity relations and the Ritz method, using orthogonal polynomials such as Legendre as admissible functions. Convergence of results for natural frequencies is shown. The results were compared with the previous results of the finite element method and analytical studies. After confirming the accuracy of the results by adding the porosity effect, its effect on the frequency of the hemispherical-cylindrical shell was investigated. The results show that the shell frequency decreased with an increase in the porosity coefficient. Natural frequencies were obtained with different geometric parameters. Finally, the results show that the porosity coefficient and geometric parameters have a significant effect on the natural frequency of the porous complex shells. The results show that torsional modes (n=0^T) and axisymmetric modes (n=0^A) play a more important role in clamped-free B.C compared to free-free B.C. Decreasing the thickness generally decreases the frequencies in all modes. Also, it can be seen that with the increase of the porosity coefficient in the cylindrical-hemispherical shell due to the decrease in stiffness, the frequencies decrease in all modes.

The analysis of the bending behavior of rotating porous disks with exponential thickness variation consisting of viscoelastic Functionally graded material is illustrated. The study of bending in the porous disk was done using the first-order shear deformation theory. The porous disk is under the effect of a combination of mechanical stresses and thermal distribution. All material factors for the porous disk change across the thickness as a power law of radius. To solve the mathematical structure by using the semi-analytical technique for displacements in the porous disk, and then to treat the structure model with viscoelastic material by the correspondence principle and Illyushin’s approximation manner. Numerical outcomes including the effect of porosity parameter, inhomogeneity factor, and relaxation time are presented with three different sets of boundary conditions for the solid and hollow disks. A comparison between porous and perfect disk with numerous values of porosity parameters and different inhomogeneity factors have been shown to emphasize the importance of complex mathematical structure in modern engineering mechanical designs.

A novel two-dimensional elasticity solution is presented in this paper, specifically designed for studying the vibration of Functionally graded porous (FGP) beams. The kinetics of the beam are defined by two-dimensional elasticity theory, and Lagrange’s equations are used to derive the governing equations of motion. The Ritz method devises the expansion of displacement variables in polynomial and trigonometric series in the thickness and axial directions. Furthermore, microvoids can emerge as a result of technical issues during the manufacture of Functionally graded materials (FGMs), leading to the development of porosities. The porosity distribution functions, one for three porosity distributions: uniform porosity (UP), non-uniform porosity-I (NUP-I), and non-uniform porosity-II (NUP-II), are considered in the problem. This study investigates the impact of the gradation exponents (p) in the z-direction, the slenderness ratio (L/h), the distribution of porosity, the porosity coefficient (e), and various boundary conditions on the natural frequencies. A comparison with the findings from higher-order shear deformation theory (HSDT) validated the accuracy and effectiveness of the proposed methodology.