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 bamboo structure can be generally viewed as a Functionally graded composite material constituted by long and aligned cellulose fibres embedded in a lignin matrix. Analysing the transversal section of a bamboo culm, one can observe that the fibre distribution is variable through its thickness. The non-uniform distribution of fibres prevents the direct application of equations used to model the behaviour of composite materials, as the rule of mixtures equations for strength and modulus of elasticity. These equations assume, besides the perfect bonding between fibre and matrix, uniform distribution of the fibres in the matrix. In bamboo, the fibre distribution follows an organized pattern with a higher concentration of fibres on the outer surface of the culm. Establishing how this variation occurs, the basic equations from the composite materials approach can be modified in order to model the mechanical behaviour of bamboo. This paper presents the meso-structure analyses of bamboo culms through Digital Image Analysis. The variation of the volume fraction of the cellulose fibres across the transversal section of the bamboo is established. The developed methodology is successfully applied to study the volume fraction variation of fibres in two different samples of bamboo species Phyllostachys heterocycla pubenscens, commonly know as "Moso".      

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.

In this paper, using homotopy analysis method, an analytical solution for the nonlinear free vibrations of the Functionally graded porous micropipes conveying fluid flow is presented. The equations of motion are obtained based on Euler-Bernoulli beam theory and modified couple stress theory with consideration of geometric nonlinearity. It is assumed that the micropipe is porous and the porosity distribution is in three forms; uniform, non-uniform symmetric, and non-uniform asymmetric distributions. The Hamilton principle is used to obtain the governing equations of motion. Also, the Galerkin method is used to convert partial differential equations to ordinary differential equations. Finally, by considering immoveable simply-supported boundary conditions and using the homotopy analysis method, the analytical solution for the governing equations is performed. The results obtained from this method has been verified by the Runge-Kutta numerical method which shows that the homotopy analysis method has good accuracy by considering two terms of the Taylor series. The results showed that between the proposed porosity distribution schemes in the micropipe, the non-uniform asymmetric distribution pattern is the most suitable, because the microtube becomes unstable at a higher fluid velocity.

In this work, a new mathematical model of thermoelasticity theory has been considered in the context of a new consideration of heat conduction with fractional order theory. A Functionally graded isotropic unbounded medium is considered subjected to a periodically varying heat source in the context of space-time non-local generalization of three-phase-lag thermoelastic model and Green-Naghdi models, in which the thermophysical properties are temperature dependent. The governing equations are expressed in Laplace-Fourier double transform domain and solved in that domain. Then the inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand are obtained numerically in complex domain by using Laguerre’s method and the inversion of Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the thermal displacement, temperature and thermal stress are obtained for a hypothetical material. Finally, the obtained results are presented graphically to show the effect of non-local fractional parameter on thermal displacement, temperature and thermal stress. A comparison of the results for different theories (three-phase-lag model, GN model II, GN model III) is presented and the effect of non-homogeneity is also shown. The results, corresponding to the cases, when the material properties are temperature independent, agree with the results of the existing literature.

In the present paper, the buckling problem of rectangular Functionally graded (FG) plate with arbitrary edge supports is investigated. The present analysis is based on the classical plate theory (CPT) and large deformation is assumed for deriving stability equations. The plate is subjected to bi-axial compression loading. Mechanical properties of FG plate are assumed to vary continuously along the thickness of the plate according to different volume of fraction functions of constituents. These functions are assumed to have power law distributions. The displacement function is assumed to have the form of double Fourier series, of which derivatives are legitimized using Stokes’ transformation method. The advantage of using this method is the capability of considering effect of any possible combination of boundary conditions on the buckling loads. The out-plane displacement distribution is assumed using Fourier Sinus Series. This results in a general eigenvalue problem which can be used for evaluating the buckling load under different edge conditions, plate aspect ratios and various volume fraction functions. For generality of problem, plate is elastically restrained using some rotational and translational springs at four edges. Some numerical examples are presented and compared the to numerical results of finite element method using ABAQUS and other researchers’ results to validate the proposed method. It has been shown that there is good agreement between them.

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.

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.

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.