• 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.

• Advances in nano research
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• v.14 no.4
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• pp.303-312
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• 2023

In this study, the bifurcation analysis of functionally graded material is done using exponential volume fraction law. Shell theory of Love is used for vibration of shell. The Galerkin's method is applied for the formation of three equations in eigen value form. This eigen form gives the frequencies using the computer software MATLAB. The variations of natural frequencies (Hz) for Type-I and Type-II functionally graded cylindrical shells are plotted for exponential volume fraction law. The behavior of exponent of volume fraction law is seen for three different values. Moreover, the frequency variations of Type-I and -II clamped simply supported FG cylindrical shell with different positions of ring supports against the circumferential wave number are investigated. The procedure adopted here enables to study vibration for any boundary condition but for brevity, numerical results for a cylindrical shell with clamped simply supported edge condition are obtained and their analysis with regard various physical parameters is done.

• Structural Engineering and Mechanics
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• v.76 no.1
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• pp.129-139
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• 2020

In this piece of work, carbon nanotubes motion equations are framed by Kelvin's method. Employment of the Kelvin's method procedure gives birth to the tube frequency equation. It is also exhibited that the effect of frequencies is investigated by varying the different index of polynomial function. By using volume fraction for power law index, the fundamental natural frequency spectra for two forms of single-walled carbon nanotubes are calculated. The influence of frequencies against length-to-diameter ratios with varying power law index are investigated in detail for these tubes. Throughout the computation, it is observed that the frequency behavior for the boundary conditions follow as; clamped-clamped, simply supported-simply supported and these frequency curves are higher than that of clamped-free curves. Computer software MATLAB is utilized for the frequencies of single-walled carbon nanotubes.

• Advances in concrete construction
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• v.13 no.5
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• pp.367-376
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• 2022

Theoretical study of vibration distinctiveness of rotating cylindrical are examined for three volume fraction laws viz.: polynomial, exponential and trigonometric. These laws control functionally graded material composition in the shell radius direction. Functionally graded materials are controlled from two or more materials. In practice functionally graded material comprised of two constituent materials is used to form a cylindrical shell. For the current shell problem stainless steel and nickel are used for the shell structure. A functionally graded cylindrical shell is sanctioned into two types by interchanging order of constituent materials from inner and outer side for Type I and Type II cylindrical shell arrangement. Fabric composition of a functionally graded material in a shell thickness direction is controlled by volume fraction law. Variation of power law exponent brings change in frequency values. Influence of this physical change is investigated to evade future complications. This procedure is capable to cater any boundary condition by changing the axial wave number. But for simplicity, numerical results have been evaluated for clamped- simply supported rotating cylindrical shells. It has been observed from these results that shell frequency is bifurcated into two parts: one is related to the backward wave and other with forward wave. It is concluded that the value of backward frequency is some bit higher than that forward frequency. Influence of volume fraction laws have been examined on shell frequencies. Backward and forward frequency curves for a volume fraction law are upper than those related to two other volume fraction laws. The results generated furnish the evidence regarding applicability of present shell model and also verified by earlier published literature.

• Steel and Composite Structures
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• v.37 no.5
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• pp.621-632
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• 2020

This research deals with the study of vibrational behavior of armchair and zigzag single-walled carbon nanotubes invoking extended Love shell theory. The effects of different physical and material parameters on the fundamental frequencies are investigated. By using volume fraction for power law index, the fundamental natural frequency spectra for two forms of single-walled carbon nanotubes are calculated. The influence of frequencies against length-to-diameter ratios with varying power law index are investigated in detail for these tubes. To discretize the governing equation in eigen-value form, wave propagation approach is developed. Complex exponential functions have been used and the axial model depends on boundary condition that has been described at the edges of carbon nanotubes to calculate the axial modal dependence. Computer software MATLAB is utilized for the frequencies of single-walled carbon nanotubes and current results shows a good stability with comparison of other studies.

• Advances in nano research
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• v.9 no.4
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• pp.251-261
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• 2020

In this article, vibration attributes of single walled carbon nanotubes based on Galerkin's method have been investigated. The influence of power law index subjected to different end supports has been overtly examined. Application of the Hamilton's variational principal leads to the formation of partial differential equations. The effects of different physical and material parameters on the fundamental frequencies are investigated for armchair and zigzag carbon nanotubes with clamped-clamped, simply supported and clamped-free boundary conditions. By using volume fraction for power law index, the fundamental natural frequency spectra for two forms of Single-Walled Carbon Nanotubes (SWCNTs) are calculated. The influence of frequencies against length-to-diameter ratios with varying power law index are investigated in detail for these tubes. MATLAB software package has been utilized for extracting tube frequency spectra. The obtained results are confirmed by comparing with available literature.

• Computers and Concrete
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• v.29 no.2
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• pp.117-126
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• 2022

The vibrational characteristic of three-layered cylindrical shell (CS) submerged in fluid with the ring support has been studied. The inner and outer layer is supposed to construct by isotropic layer. The composition of central layer is of functionally graded material type. Acoustic Wave condition has been utilized to present the impact of fluid. The central layer of cylindrical shell (CS) varies by volume fraction law that has been expressed in terms of polynomial. The main shell frequency equation has been obtained by theory of Love's shell and Rayleigh-Ritz technique. The oscillation of natural frequency has been examined under a variety of end conditions. The dependence of axial model has been executed with the help of characteristic beam function. The natural frequencies (NFs) of functionally graded material (FGM) shell have been observed of cylindrical shell along the shell axial direction. Different physical parameters has been used to examine the vibration characteristics due to the effect of volume fraction law. MATLAB software has been used to get result.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1997.04a
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• pp.777-781
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• 1997

In this study, the effect of strain-rate and inital solid fraction on the deformation behaviour of semi-solid material is investigated, when semi-solid forging is performed by the process of closed-die compression using A356 alloy of which the above results can be practically applied for industrical purpose. In order to simulate densification in the deformation of semi-solid material, the semi-solid material is assumed to be composed of solid region following rigid visco-plastic material, the liquid region following Darcy's law for the liquid flow saturated in the interstitial space. Simulation results of closed-die compression and simple upsetting under different strain-rate and initial solid fraction are compared.

• Proceedings of the Korean Society For Composite Materials Conference
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• 2005.11a
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• pp.66-69
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• 2005

Mechanical strength of functionally graded composite plates that composed of ceramic, functionally graded material and metal layers is investigated using 3-D finite element method. In FGM layer, material properties are assumed to be varied continuously in the thickness direction according to a simple power law distribution in terms of the volume fraction of a ceramic and metal. The 3-D finite element model is adopted by using an IS-node solid element to analyze more accurately the variation of material properties in the thickness direction. Numerical results are compared with those of the previous works. In addition, the displacements, the tensile stresses and the compressive stresses are analyzed for the variation of FGM thickness ratio and volume fraction distribution.Mechanical strength of functionally graded composite plates that composed of ceramic, functionally graded material and metal layers is investigated using 3-D finite element method. In FGM layer, material properties are assumed to be varied continuously in the thickness direction according to a simple power law distribution in terms of the volume fraction of a ceramic and metal. The 3-D finite element model is adopted by using an IS-node solid element to analyze more accurately the variation of material properties in the thickness direction. Numerical results are compared with those of the previous works. In addition, the displacements, the tensile stresses and the compressive stresses are analyzed for the variation of FGM thickness ratio and volume fraction distribution.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.20 no.12
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• pp.3715-3727
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• 1996

Compression behavior of semi-solid aluminum alloys with controlled solid fractions was investigated in the present study. The stress and strain relationships were obtained from the compression test. Variations of the solid fraction distribution and the material behaviour were investigated for various friction coeffieiants and die speedsd. For a finite element analysis, the semi-solid material was described by a compressible regid viscoplastic model for the solid region and darcy's law for the liquid region. The computed results were compared with experimental data for the validity of the yield criteria.