Uneven Porosity Distributions Research Articles

Sound transmission loss of a porous heterogeneous cylindrical nanoshell employing nonlocal strain gradient and first-order shear deformation assumptions

The size-dependent sound transmission loss problem of an air-filled functionally graded material cylindrical shell subjected to a plane progressive sound wave with even and uneven porosity distributions was analytically studied using a nonlocal strain gradient and the first-order shear deformation theories. To simulate the heterogeneous material, the impressive material properties were supposed to be associated with the porosity volume fraction model, based on a power-law model. They were suggested to be constantly changeable along the thickness direction. The motion equations were extracted by Hamilton’s principle and then solved using the Fourier-Bessel series. The accuracy of the obtained formulation were strictly proved by comparing the data accessible in the literature. Parameter studies reveal the effects of material gradient index, size scale factors, porosity volume fraction, and incident angles on the variations in the amplitude of sound transmission loss through the nanoshell.

A refined inverse hyperbolic shear deformation theory for bending analysis of functionally graded porous plates

The modern engineering structures require the advanced engineering materials to resist the high temperatures and to provide high stiffness. In particular the functionally graded porous materials (FGPMs) introduced are expected to have these desired properties, consequently eliminating local stress concentration and de-lamination. In the present paper, a new shear strains shape function is chosen to research the bending analysis of functionally graded plates (FGPs) with uneven symmetrical, uneven asymmetrical and even distributions of porosity. The material properties of uneven porosity distributions along the thickness of the FGPs vary with cosine function. The present theory includes the influence of thickness stretching. This theory also fulfills the nullity of the shear stresses in the transverse direction on the top and bottom of the plate, thus avoids the use of a shear correction factor. The virtual displacement principle is employed to develop the equilibrium equations for porous FGPs. The Navier’s method is used to obtain the solutions of porous FGPs for simply supported (SS) conditions. The accuracy of the developed theory is established with numerical results of perfect and porous FGPs available in the open source. The transverse displacements and stress results have been reported and studied for evenly, unevenly symmetrical and unevenly asymmetrical distributions with different porosity volume fraction (PVF), thickness ratios and aspect ratios. From the numerical results it is concluded that the type of porosity distribution needs to be considered as a key factor in the optimal design of the porous FGPs.

Nonlinear free and forced vibrations of porous functionally graded pipes conveying fluid and resting on nonlinear elastic foundation

This paper studies the nonlinear free and forced vibrations of fluid-conveying pipes that are made of porous functionally graded materials and supported on nonlinear elastic foundation. A modified power-law function is adopted to model the porous functionally graded materials, with both even and uneven porosity distributions to represent different material imperfections. Equations of motion of the pipe system are then derived through Hamilton’s principle based on Euler–Bernoulli beam theory, with foundation and von Kármán nonlinearities and damping effect being considered. The nonlinear free vibration and the primary resonance of the pipe system are investigated analytically, by employing the variational iteration method and the direct method of multiple scales, respectively. Detailed parametric studies for the free and forced vibrations are then carried out numerically, considering different parameters including power-law index, porosity characteristics, fluid velocity, pipe geometries, foundation parameters and so on.

The thermal vibration characteristics of the functionally graded porous stepped cylindrical shell by using characteristic orthogonal polynomials

This paper presents a general approach for dealing with the thermal vibration characteristics of functionally graded porous stepped cylindrical shell (FGP-SCS) by employing characteristic orthogonal polynomials. The thermal vibration characteristics including free vibration, steady-state and transient response are investigated in detail. The governing equations are derived by Rayleigh-Ritz method and the displacement admissible functions are denoted by using characteristic orthogonal polynomials. The FGP-SCS is divided into N segments with different thickness and the coupling of adjacent segment is realized by connecting springs. The boundary conditions of FGP-SCS can be determined easily by selecting the proper stiffness values of boundary springs and the boundary conditions including classical and elastic boundary conditions. Three temperature distributions including uniform, linear and nonlinear distributions and two porosity distributions including even and uneven porosity distributions are taken into account. Meantime, the influences of some key parameters including power law index, porosity volume fraction, temperature change, geometric structure on the thermal vibration behaviors of FGP-SCS are investigated in detail.

Large-amplitude nonlinear free vibrations of functionally graded plates with porous imperfection: A novel approach based on energy balance method

A novel approach, based on energy balance method, to deal with the large-amplitude nonlinear free vibration problem of the rectangular porous functionally graded plates is presented in this paper. Both even and uneven porosity distributions are considered. The governing equations of the plates are established via von Kármán theory and Lagrange equation. A new deformation assumption for the vibration problem of plates is proposed. This deformation assumption allows a quadratic relationship between the in-plane and out-plane displacements, thus is more advantageous than the conventional one to describe the displacement field of the plate during the vibration period. The nonlinear frequency of the plate is obtained by adopting the energy balance method cooperating with the Newton-Raphson iterative method. Numerical examples are provided to inspect the influences of the porosity distribution type, porosity volume fraction, geometrical parameters and vibration amplitude on the nonlinear free vibration characteristic of the plates. The results indicate that the nonlinear-to-linear frequency ratio is less sensitive to the porosity volume fraction than the linear frequency for the porous functionally graded plates, and the nonlinear-to-linear frequency ratio of plate with uneven porosity distribution is more sensitive to the porosity volume fraction than that with even porosity distribution.

Dynamic Stability of Bi-Directional Functionally Graded Porous Cylindrical Shells Embedded in an Elastic Foundation

This paper investigates the dynamic buckling of bi-directional (BD) functionally graded (FG) porous cylindrical shells for various boundary conditions, where the FG material is modeled by means of power law functions with even and uneven porosity distributions of ceramic and metal phases. The third-order shear deformation theory (TSDT) is adopted to derive the governing equations of the problem via the Hamilton’s principle. The generalized differential quadrature (GDQ) method is applied together with the Bolotin scheme as numerical strategy to solve the problem, and to draw the dynamic instability region (DIR) of the structure. A large parametric study examines the effect of different boundary conditions at the extremities of the cylindrical shell, as well as the sensitivity of the dynamic stability to different thickness-to-radius ratios, length-to-radius ratios, transverse and longitudinal power indexes, porosity volume fractions, and elastic foundation constants. Based on results, the dynamic stability of BD-FG cylindrical shells can be controlled efficiently by selecting appropriate power indexes along the desired directions. Furthermore, the DIR is highly sensitive to the porosity distribution and to the extent of transverse and longitudinal power indexes. The numerical results could be of great interest for many practical applications, as civil, mechanical or aerospace engineering, as well as for energy devices or biomedical systems.

Analysis of porosity effect on free vibration and buckling responses for sandwich sigmoid function based functionally graded material plate resting on Pasternak foundation using Galerkin Vlasov’s method

The objective of the present paper is to study the effect of porosity on vibration and buckling responses for sandwich plate supported with different boundary conditions at the edges. The formulation is performed for recently developed sandwich plate using new modified sigmoid function based functionally graded material plate of different symmetric and asymmetric configurations. Three different types of porosity are considered viz. even, uneven but symmetric, and uneven but non-symmetric. A new uneven non-symmetric porosity model is used in which micro-voids are varied in accordance with material property variation in the thickness direction in order to capture the accurate distribution of voids on the plate. The variational principle is employed to derive equilibrium equations. A closed form solution is obtained using the assumed solution with shape functions satisfying the edge boundary conditions using Galerkin Vlasov’s method. Free vibration and buckling analysis have been performed to study the effect of porosity, geometric configurations, inhomogeneity parameter, and buckling load parameter. These studies are performed on sandwich sigmoid function based functionally graded material plate for different boundary conditions and plate configurations. It is observed that in several cases with uneven symmetric porosity (P-2) distribution, the reduction in effective inertia appears to be more prominent in comparison to the reduction in effective stiffness of the plate and uneven non-symmetric porosity (P-3) distribution ascertains the realistic idealization for a plate with micro-voids. The higher modes of vibrations and nature of in-plane load are very sensitive to the type of porosity distribution considered. In addition, zero frequency parameter and zero critical buckling load in a load frequency curves correspond to critical buckling load and frequency parameter, respectively. The accuracy and validation of the present formulation are determined by considering different examples.

DEM study of the porosity distribution of pellet sandpile formed by ternary size particles

Porosity is an important parameter to reflect the internal structure of the heap, and it is related to the permeability of a packed bed. Thus, understanding and further adjusting the porosity distribution will help to control the gas-solid reaction in a chemical reactor. In this work, sandpiles formed by ternary size particles with variable mass ratio were researched by discrete element method. The porosity increases with the increase of the mass fraction of small particles. Different dump methods (Drop Test and Chute Test) both cause four distinct porosity regions (Zone A, B, C and D), but the locations are not the same. Changing the drop height in Drop Test and chute properties in Chute Test both have influence on the porosity zone, especially for the upper porosity. Furthermore, size segregation will lead to uneven porosity distribution, but there is no evident relationship between porosity and size segregation in sandpile.

Thermo-Mechanical Vibration Analysis of Imperfect Inhomogeneous Beams Based on a Four-Variable Refined Shear Deformation Beam Theory Considering Neutral Surface Position

In this disquisition, an exact solution method is developed for analyzing the vibration characteristics of porous functionally graded (FG) beams by considering neutral surface position and different thermal loadings via a four-variable shear deformation refined beam theory. Four types of environmental conditions through the z-axis direction are supposed as: uniform (UTR), linear (LTR), nonlinear (NLTR) and sinusoidal (STR) temperature rises. Mechanical properties of porous FG beams are supposed to vary through the thickness direction and are modeled via the modified power-law. The modified power-law is formulated using the concept of even and uneven porosity distributions. Since the variation of pores along the thickness direction influences the mechanical properties, porosity plays a key role in the mechanical response of FG structures. The governing differential equations and related boundary conditions of porous FG beams are subjected to temperature field that is derived by Hamilton's principle based on a four-variable refined theory which verifies shear deformation regardless of any shear correction factor. The Navier-type solution procedure is used to achieve the natural frequencies of porous-FG beams supposed to various thermal loadings which satisfies the simply-simply boundary condition. A parametric study is led to carry out the effects of material graduation exponent, porosity volume fraction, different porosity distribution, and thermal effect on dimensionless frequencies of porous FG beams. It is concluded that these parameters play noticeable roles in the vibration behavior of imperfect FG beams. Presented numerical results can be applied as benchmarks for future designs of imperfect FG structures with porosity phases.

Vibration and sound radiation analysis of temperature-dependent porous functionally graded material plates with general boundary conditions

This paper investigates the vibration and sound radiation of porous functionally graded material (FGM) plates with temperature gradient along the thickness. The even and uneven porosity distributions are taken into consideration in the present study. The formulation of the FGM plate is based on the first-order shear deformation theory (FSDT) with the temperature dependence of material properties considered. Each displacement component of the FGM plate is expanded as the Fourier cosine series combined with auxiliary functions in order to satisfy the general boundary conditions. A substantial number of numerical examples concerning the vibration and sound radiation responses of the heated plate are carried out to validate the convergence and accuracy of the proposed method. Finally, the effects of key parameters including the boundary condition, volume fraction index, thermal environment, temperature dependence of material properties and porosity volume fraction on the structural and acoustic responses of the FGM plate are discussed in detail.

Multiparametric Analytical Solution for the Eigenvalue Problem of FGM Porous Circular Plates

Free vibration analysis of the porous functionally graded circular plates has been presented on the basis of classical plate theory. The three defined coupled equations of motion of the porous functionally graded circular/annular plate were decoupled to one differential equation of free transverse vibrations of plate. The one universal general solution was obtained as a linear combination of the multiparametric special functions for the functionally graded circular and annular plates with even and uneven porosity distributions. The multiparametric frequency equations of functionally graded porous circular plate with diverse boundary conditions were obtained in the exact closed-form. The influences of the even and uneven distributions of porosity, power-law index, diverse boundary conditions and the neglected effect of the coupling in-plane and transverse displacements on the dimensionless frequencies of the circular plate were comprehensively studied for the first time. The formulated boundary value problem, the exact method of solution and the numerical results for the perfect and imperfect functionally graded circular plates have not yet been reported.

Analyzing post-buckling behavior of continuously graded FG nanobeams with geometrical imperfections

This research is concerned with post-buckling investigation of nano-scaled beams constructed from porous functionally graded (FG) materials taking into account geometrical imperfection shape. Hence, two types of nanobeams which are perfect and imperfect have been studied. Porous FG materials are classified based on even or uneven porosity distributions. A higher order nonlinear refined beam theory is used in the present research. Both perfect and imperfect nanobeams are formulated based on this refined theory. A detailed study is provided to understand the effects of geometric imperfection, pore distribution, material distribution and small scale effects on buckling of FG nanobeams.

Nonlocal strain gradient theory for the magneto-electro-elastic vibration response of a porous FG-core sandwich nanoplate with piezomagnetic face sheets resting on an elastic foundation

The free vibration analysis of a nonlocal strain gradient elastic sandwich nanoplate with porous graded core and piezomagnetic face sheets is presented in this paper. The rectangular elastic sandwich nanoplate is resting on Pasternak's foundation. Porosities are distributed evenly and unevenly through the thickness of the core. The gradation of material properties having porosities is described using a modified power-law function. A nonlocal parameter and a strain gradient parameter are employed to describe both stiffness reduction and stiffness enhancement of nanoplates. The governing equations of the motion are derived from Hamilton’s principle based on the first order shear deformation theory. In addition, Eringen’s nonlocal strain gradient piezo-magneto-elasticity theory is used to consider nanoscale effects. The analytical solution is presented to solve seven governing equations of motion using Navier’s solution. Eventually, the natural frequency is surveyed for different side length ratios, nonlocal coefficient, porosity volume fraction, and parameters of foundation numerically with even and uneven porosity distributions.

Vibration analysis of porous FG nanoshells with even and uneven porosity distributions using nonlocal strain gradient elasticity

This paper studies the free vibrational behavior of porous functionally graded nanoshells using nonlocal strain gradient theory. A nonlocal parameter and a strain gradient parameter are employed to describe both stiffness reduction and stiffness enhancement of nanoshells. Porosities are evenly and unevenly distributed thorough the thickness of the nanoshell. The gradation of material properties having porosities is described using a modified power-law function. The nanoshell is modeled via first-order shear deformation theory, and Galerkin’s method is implemented to obtain vibration frequencies. Shape functions which satisfy available classical and nonclassical boundary conditions in nonlocal strain gradient theory are proposed. It is shown that the vibrational behavior of the nanoshell is influenced by the porosity volume fraction, porosity distribution, nonlocal coefficient, strain gradient coefficient, boundary conditions and radius-to-thickness ratio.

Dynamic response of porous functionally graded material nanobeams subjected to moving nanoparticle based on nonlocal strain gradient theory

Up to now, nonlocal strain gradient theory (NSGT) is broadly applied to examine free vibration, static bending and buckling of nanobeams. This theory captures nonlocal stress field effects together with the microstructure-dependent strain gradient effects. In this study, forced vibrations of NSGT nanobeams on elastic substrate subjected to moving loads are examined. The nanobeam is made of functionally graded material (FGM) with even and uneven porosity distributions inside the material structure. The graded material properties with porosities are described by a modified power-law model. Dynamic deflection of the nanobeam is obtained via Galerkin and inverse Laplace transform methods. The importance of nonlocal parameter, strain gradient parameter, moving load velocity, porosity volume fraction, type of porosity distribution and elastic foundation on forced vibration behavior of nanobeams are discussed.

Aero-hygro-thermal stability analysis of higher-order refined supersonic FGM panels with even and uneven porosity distributions

In this paper, porosity-dependent aero-hygro-thermal instability of functionally graded (FG) panels is investigated by developing a higher-order refined shear deformable theory for the first time. Porosities are randomly distributed around the cross section of FG panel. Hygro-thermo-elastic material properties of porous FG panel are described using a modified power-law function accounting for even and uneven porosity distributions. Based on the presented refined shear deformation theory, it is possible to examine instability regions of thicker FG panels without using a shear correction factor. Employing extended Hamilton’s principle, the governing equations of FG panel under supersonic airflow simulated via first-order piston theory are obtained. It is concluded that porosity volume fraction, type of porosities, hygro-thermal environment, material gradation and slenderness ratio have major roles on the prediction of divergence and flutter boundaries of porous FG panels.

Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method

Flexural vibration analysis of beams made of functionally graded materials (FGMs) with various boundary conditions is considered in this paper. Due to technical problems during FGM fabrication, porosities and micro-voids can be created inside FGM samples which may lead to the reduction in density and strength of materials. In this investigation, the FGM beams are assumed to have even and uneven distributions of porosities over the beam cross-section. The modified rule of mixture is used to approximate material properties of the FGM beams including the porosity volume fraction. In order to cover the effects of shear deformation, axial and rotary inertia, the Timoshenko beam theory is used to form the coupled equations of motion for describing dynamic behavior of the beams. To solve such a problem, Chebyshev collocation method is employed to find natural frequencies of the beams supported by different end conditions. Based on numerical results, it is revealed that FGM beams with even distribution of porosities have more significant impact on natural frequencies than FGM beams with uneven porosity distribution.

1H‐NMR Microscopy of Tablets

A1H‐nuclear magnetic resonance (NMR) microscopy method was utilized for the first time to determine the porosity distribution of physically intact tablets. The main advantage of this newly developed method was that porosity cross sections through whole tablets or specific locations could be obtained without mechanically destroying the tested tablet. This was achieved by filling tablet cavities with silicone oil under vacuum. The amount of silicone oil locally within the tablet was then determined by1H‐NMR microscopy, revealing the inverse inner structure. To reduce the measuring time, a paramagnetic gadolinium complex was added to the silicone oil. The cross sectional signals produced by1H‐NMR microscopy through the tablet were transformed into a color image by a specially designed computer graphic program. To improve the signal‐noise ratio an algorithm of 3D‐filtering was introduced. The maximal spatial resolution achieved with this method was about 95 μm for a cube's edge length corresponding to some 380 000 positions in a 9‐mm‐diameter compression‐coated tablet. Uneven porosity distributions within tablets, cracks, or cavities could be visualized with this newly developed method. Different compaction mechanisms were observed with plastic‐ or brittle‐type tablets. The different states of densification during compaction of powders could be detected. The integrity of compression coatings was determined to be dependent on the pressure load and the location of the core within the coat.

Effect of uneven porosity distribution in cement paste and mortar on reinforcing steel corrosion

The influence of heterogeneity of cementitious matrices, e.g. uneven porosity distribution in repair cement paste or mortar, on surface corrosion of the reinforcing steel was investigated using a low frequency impedance technique. Rebar embedded in lower porosity matrices had lower surface impedance and phase angle values in the frequency range 1Hz to 10mHz. The corrosion reactions appeared to preferentially occur on the surface of rebar embedded in denser matrices. This observation was valid in cases with and without admixed chloride ions. A possible explanation of the surface corrosion mechanism of reinforcing steel is given. The phenomenon is explained by the depletion of oxygen concentration at the steel-cement paste interface resulting in the formation of an “anodic area”. The chloride ions accelerate the corrosion reaction by destroying the iron oxide film and reducing the pH values at the steel-cement paste surface.