• Steel and Composite Structures
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• v.30 no.6
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• pp.603-615
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• 2019

In the present work, free vibration of beams made of imperfect functionally graded materials (FGMs) including porosities is investigated. Because of faults during process of manufacture, micro voids or porosities may arise in the FGMs, and this situation causes imperfection in the structure. Therefore, material properties of the beams are assumed to vary continuously through the thickness direction according to the volume fraction of constituents described with the modified rule of mixture including porosity volume fraction which covers two types of porosity distribution over the cross section, i.e., even and uneven distributions. The governing equations of power law FGM (P-FGM) and sigmoid law FGM (S-FGM) beams are derived within the frame works of classical beam theory (CBT) and first order shear deformation beam theory (FSDBT). The resulting equations are solved using separation of variables technique and assuming FG beams are simply supported at both ends. To validate the results numerous comparisons are carried out with available results of open literature. The effects of types of volume fraction function, beam theory and porosity volume fraction, as well as the variations of volume fraction index, span to depth ratio and porosity volume fraction, on the first three non-dimensional frequencies are examined in detail.

• Journal of the Korean Ceramic Society
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• v.35 no.9
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• pp.988-994
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• 1998

In the manufacturing of vitreous bonded CBN tool the porosity change associated with various processing conditions, I. e. the sintering temperature and the size and the amount of abrasive grits was observed. In the case of sintering of vitreous bond material only the specimen density reached the maximum at 950$^{\circ}C$ and then the total porosity was increased slightly with the temperature above 950$^{\circ}C$. In the sintering of a-brasive grits and the vitreous bond material together a marked increase in the total porosity was found with the temperature above 950$^{\circ}C$ Reducing the grit size at the constant volume fraction of abrasive grits showed an increase in the total porosity at whole sintering temperature. On the contrary. it was observed that increasing the volume fraction of abrasive grits with a same size showed the increased open porosity simultaneously with decreased closed porosity at whole sintering temperature.

• Structural Engineering and Mechanics
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• v.72 no.1
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• pp.61-70
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• 2019

The functionally graded materials (FGM) used in plates contain probably a porosity volume fraction which needs taking into account this aspect of imperfection in the mechanical bahavior of such structures. The present work aims to study the effect of the distribution forms of porosity on the bending of simply supported FG plate reposed on the Winkler-Pasternak foundation. A refined theory of shear deformation is developed to study the effect of the distribution shape of porosity on static behavior of FG plates. It was found that the distribution form of porosity significantly influence the mechanical behavior of FG plates, in terms of deflection, normal and shear stress. It can be concluded that the proposed theory is simple and precise for the resolution of the behavior of flexural FGM plates resting on elastic foundations while taking into account the shape of distribution of the porosity.

• Nuclear Engineering and Technology
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• v.34 no.5
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• pp.433-435
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• 2002

The relation between sintered densities and porosities in UO$_2$ pellets is investigated. The open porosity decreases linearly up to about 95% T.D.,(theoretical density) as the sintered density increases whereas, above 96% T.D., sintered UO$_2$ pellets do not have any open pores. The fraction of open porosity to the total porosity also decreases linearly as the sintered density increases, though the slope is lower than that of open porosity and, above 95% T.D., the fraction decreases rapidly to approach a zero.

• Structural Engineering and Mechanics
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• v.82 no.2
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• pp.151-161
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• 2022

In this study, the effect of porosity distribution pattern on the free vibration analysis of porous FG plates with various boundary conditions is studied. The material properties of the plate and the porosities within the plate are considered to vary continuously through the thickness direction according to the volume fraction of constituents defined by the modified rule of the mixture, this includes porosity volume fraction with four different types of porosity distribution over the cross-section. The governing partial differential equation of motion for the free vibration analysis is obtained using hyperbolic shear deformation theory. An analytical solution is presented for the governing PDEs for various boundary conditions. Results of the presented solution are compared and validated by the available results in the literature. Moreover, the effects of material and porosity distribution and geometrical parameters on vibrational properties are investigated.

• Steel and Composite Structures
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• v.49 no.6
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• pp.713-728
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• 2023

This paper aims to investigate the free vibration analysis of FG plates, taking into account the effects of even and uneven porosity. The study employs the Hellinger-Reisner functional and obtains the element's bending stress and membrane stress fields from the analytical solution of the governing equations of the thick plate and plane problem, respectively. The displacement field serves as the second independent field. While few articles on free vibration analysis of circular plates exist, this paper investigates the free vibration of both rectangular and circular plates. After validating the proposed element, the paper investigates the effects of porosity distributions on the natural frequency of the FG porous plate. The study calculates the natural frequency of thin and thick bending plates with different aspect ratios and support conditions for various porosity and volume fraction index values. The study uses three types of porosity distributions, X, V, and O, for the uneven porosity distribution case. For O and V porosity distribution modes, porosity has a minor effect on the natural frequency for both circular and rectangular plates. However, in the case of even porosity distribution or X porosity distribution, the effect of porosity on the natural frequency of circular and rectangular plates increases with an increase in the volume fraction index.

• Smart Structures and Systems
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• v.26 no.2
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• pp.253-262
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• 2020

In this paper, a new higher order shear deformation model is developed for static analysis of functionally graded beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. The model account for higher-order variation of transverse shear strain through the depth of the beam and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The present work aims to study the effect of the distribution forms of porosity on the bending of simply supported FG beam. Based on the present higher-order shear deformation model, the equations of motion are derived by the principle of virtual works. Navier type solution method was used to obtain displacement and stresses, and the numerical results are compared with those available in the literature. A comprehensive parametric study is carried out to assess the effects of volume fraction index, porosity fraction index, and geometry on the bending of imperfect FG beams. It can be concluded that the proposed model is simple and precise for the resolution of the behavior of flexural FGM beams while taking into account the shape of distribution of the porosity.

• Steel and Composite Structures
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• v.42 no.5
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• pp.711-722
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• 2022

In this paper, hyperbolic shear deformation plate theory is developed for thermal buckling of functionally graded plates with porosity by dividing transverse displacement into bending and shear parts. The present theory is variationally consistent, and accounts for a quadratic variation of the transverse shearstrains across the thickness and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Three different patterns of porosity distributions (including even and uneven distribution patterns, and the logarithmic-uneven pattern) are considered. The logarithmic-uneven porosities for first time is mentioned. Equilibrium and stability equations are derived based on the present theory. The non-linear governing equations are solved for plates subjected to simply supported boundary conditions. The thermal loads are assumed to be uniform, linear and non-linear distribution through-the-thickness. A comprehensive parametric study is carried out to assess the effects of volume fraction index, porosity fraction index, aspect ratio and side-to-thickness ratio on the buckling temperature difference of imperfect FG plates.

• Coupled systems mechanics
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• v.9 no.6
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• pp.499-519
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• 2020

The effect of porosity on the thermo-mechanical behavior of simply supported functionally graded plate reposed on the Winkler-Pasternak foundation is investigated analytically in the present paper using new refined hyperbolic shear deformation plate theory. Both even and uneven distribution of porosity are taken into account and the effective properties of FG plates with porosity are defined by theoretical formula with an additional term of porosity. The present formulation is based on a refined higher order shear deformation theory, which is based on four variables and it still accounts for parabolic distribution of the transverse shearing strains and stresses through the thickness of the FG plate and takes into account the various distribution shape of porosity. The elastic foundation is described by the Winkler-Pasternak model. Anew modified power-law formulation is used to describe the material properties of FGM plates in the thickness direction. The closed form solutions are obtained by using Navier technique. The present results are verified in comparison with the published ones in the literature. The results show that the dimensionless and stresses are affected by the porosity volume fraction, constituent volume fraction, and thermal load.

• Earthquakes and Structures
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• v.10 no.6
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• pp.1429-1449
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• 2016

The effect of porosity on bending and free vibration behavior of simply supported functionally graded plate reposed on the Winkler-Pasternak foundation is investigated analytically in the present paper. The modified rule of mixture covering porosity phases is used to describe and approximate material properties of the FGM plates with porosity phases. The effect due to transverse shear is included by using a new refined shear deformation theory. The number of unknown functions involved in the present theory is only four as against five or more in case of other shear deformation theories. The Poisson ratio is held constant. Based on the sinusoidal shear deformation theory, the position of neutral surface is determined and the equation of motion for FG rectangular plates resting on elastic foundation based on neutral surface is obtained through the minimum total potential energy and Hamilton's principle. The convergence of the method is demonstrated and to validate the results, comparisons are made with the available solutions for both isotropic and functionally graded material (FGM). The effect of porosity volume fraction on Al/Al2O3 and Ti-6Al-4V/Aluminum oxide plates are presented in graphical forms. The roles played by the constituent volume fraction index, the foundation stiffness parameters and the geometry of the plate is also studied.