Free vibrations of silica nanoparticle/short glass microfiber hybrid nanocomposite plates with various shapes resting on elastic foundation employing p-Ritz method

Modified strain gradient Reddy rectangular plate model for biaxial buckling and bending analysis of double-coupled piezoelectric polymeric nanocomposite reinforced by FG-SWNT

In this article, the surface stress effect on the biaxial critical buckling load of nonlocal polymeric nanocomposite rectangular plate reinforced by carbon nanotubes (CNTs) is presented. Various approaches such as Eshelby–Mori–Tanaka, the extended mixture rule, Halpin–Tsai, and micromechanical are used to determine the effective material properties of polymeric nanocomposite plate. The governing equations of equilibrium are obtained by using Hamilton’s principle. The Navier’s method is considered to obtain the biaxial critical buckling load of polymeric nanocomposite rectangular plate for simply supported boundary conditions. A detailed parametric study is conducted to explain the effects of aspect ratio, elastic foundation, surface stress and agglomeration on the biaxial buckling of nanocomposite plate. The results show that surface stress effect plays an important role at nanoscale. Also, the biaxial critical buckling load decreases with increasing the CNTs volume fraction in the inclusion (agglomeration effect). The results of this research can be used for micro-electro-mechanical and nano-electro-mechanical devices.

This paper considers the free vibration and flutter of carbon nanotube (CNT) reinforced nanocomposite plates subjected to supersonic flow. From the literature review, a great deal of research has been conducted on the free vibration and flutter response of high-volume CNT/nanocomposite structures; however, there is little research on the flutter instability of low-volume CNT/nanocomposite structures. In this study, free vibration and flutter analysis of classical CNT/nanocomposite thin plates with aligned and uniformly distributed reinforcement and low CNT volume fraction are performed. The geometry of the CNTs and the definition of the nanocomposite material properties are considered. The nanocomposite properties are estimated based on micromechanical modeling, while the governing relations of the nanocomposite plates are derived according to Kirchhoff’s plate theory with von Karman nonlinear strains. Identification of vibrational modes for nanocomposite thin plates and analytical/graphical evaluation of flutter are presented. The novel contribution of this work is the analysis of the eigenfrequencies and dynamic instabilities of nanocomposite plates with a low fraction of CNTs aligned and uniformly distributed in the polymer matrix. This article is helpful for a comprehensive understanding of the influence of a low-volume fraction and uniform distribution of CNTs and boundary conditions on the dynamic instabilities of nanocomposite plates.

In this article, stresses and free-vibration behaviors of annular circular piezoelectric nanocomposite plate reinforced by functionally graded single-walled boron nitride nanotubes (FG-SWBNNTs) embedded in an elastic foundation based on modified couple stress theory (MCST) are explored. The mechanical properties of FG-SWBNNT-reinforced nanocomposite plate are assumed to be graded in the direction of thickness and estimated through the micro-mechanical approach. The governing equations are obtained using the energy method. The natural frequencies and stresses of FG-SWBNNT-reinforced nanocomposite plate are computed using the differential quadrature method (DQM). An excellent agreement is observed between the obtained results and the results in the literature. Influences of the internal radius to the external radius, the thickness to the internal radius ratio, the material length scale parameter, the functionally graded parameter, temperature changes and elastic coefficients on the natural frequencies and stresses of the hollow circular nanocomposite plate are investigated. The results of this research show that the natural frequencies of the piezoelectric nanocomposite plate increase by increasing the material length scale parameter, the elastic foundation parameters, the ratio of the inner radius to the outer radius, the ratio of the thickness to the inner radius, and decreasing the power index and temperature change. The radial stress of the nanocomposite plate varies proportionally to its mode shape. The results can be employed to design smart structures in micro-electro-mechanical systems (MEMS).

Free vibration of viscoelastic double-bonded polymeric nanocomposite plates reinforced by FG-SWCNTs using MSGT, sinusoidal shear deformation theory and meshless method

This work deals with the nonlinear buckling and post-buckling of stiffened nanocomposite plates reinforced functionally graded carbon nanotubes (FG CNTRC) resting on elastic foundation in thermal environment. Obtained results showed that the properties of the nanocomposited plates embedded with single-walled carbon nanotubes are dependent on temperature and altered according to linear functions of the thickness. The governing equations are derived by the third-order shear deformation plate theory taking into account von Kármàn geometrical nonlinearity and solved by both the Airy’s stress function and Galerkin method. In numerical results, the influences of various types of distribution and volume fractions of carbon nanotubes, geometrical parameters, elastic foundations on the nonlinear buckling and post-buckling behaviour of stiffened FG-CNTC plates subjected mechanical, thermal loading and both are demonstrated.
 Keywords: Stiffened FG CNTRC plates; Buckling and Post buckling analysis; Third-order shear deformation theory; Thermal environment; Galerkin method
 This work deals with the nonlinear buckling and post-buckling of stiffened nanocomposite plates reinforced functionally graded carbon nanotubes (FG CNTRC) resting on elastic foundation in thermal environment. Obtained results showed that the properties of the nanocomposited plates embedded with single-walled carbon nanotubes are dependent on temperature and altered according to linear functions of the thickness. The governing equations are derived by the third-order shear deformation plate theory taking into account von Kármàn geometrical nonlinearity and solved by both the Airy’s stress function and Galerkin method. In numerical results, the influences of various types of distribution and volume fractions of carbon nanotubes, geometrical parameters, elastic foundations on the nonlinear buckling and post-buckling behaviour of stiffened FG-CNTC plates subjected mechanical, thermal loading and both are demonstrated.

In this paper, thermal- buckling behavior of the functionally graded (FG) nanocomposite plates reinforced with graphene oxide powder (GOP) is studied under three types of thermal loading once the plate is supposed to be rested on a two-parameter elastic foundation. The effective material properties of the nanocomposite plate are considered to be graded continuously through the thickness according to the Halpin-Tsai micromechanical scheme. Four types of GOPs\' distribution namely uniform (U), X, V and O, are considered in a comparative way in order to find out the most efficient model of GOPs\' distribution for the purpose of improving the stability limit of the structure. The governing equations of the plate have been derived based on a refined higher-order shear deformation plate theory incorporated with Hamilton\'s principle and solved analytically via Navier\'s solution for a simply supported GOP reinforced (GOPR) nanocomposite plate. Some new results are obtained by applying different thermal loadings to the plate according to the GOPs\' negative coefficient of thermal expansion and considering both Winkler-type and Pasternak-type foundation models. Besides, detailed parametric studies have been carried out to reveal the influences of the different types of thermal loading, weight fraction of GOP, aspect and length-to-thickness ratios, distribution type, elastic foundation constants and so on, on the critical buckling load of nanocomposite plates. Moreover, the effects of thermal loadings with various types of temperature rise are investigated comparatively according to the graphical results. It is explicitly shown that the buckling behavior of an FG nanocomposite plate is significantly influenced by these effects.

The hygrothermal deformation of nanocomposite piezoelectric plates containing internal pores lying on elastic foundations is illustrated in this paper by utilizing a novel quasi-3D plate theory (Q3DT). This nanocomposite plate has been strengthened by functionally graded graphene platelets (FG GPLs). For the purpose of identifying the FG porous materials, four alternative patterns of porosity distribution are employed, with the first pattern having a uniform distribution and the others having an uneven one. The material properties of the reinforced plate are estimated based on the Halpin–Tsai model. From the proposed theory and the virtual work principle, the basic differential equations are derived. The Levy method is used to convert the deduced partial differential equations to ordinary ones. The differential quadrature method (DQM) as a fast-converging method is utilized to solve these equations for various boundary conditions. The minimal number of grid points needed to obtain the converging solution is found by introducing a convergence study. After validating the obtained results with the studies of other researchers, this study’s findings are provided tabularly and graphically with numerous comprehensive discussions to examine the impact of the various factors of the proposed responding system.

Free vibration analysis of size‐dependent carbon nanotube‐reinforced composite (CNTRC) nanoplates, resting on the visco‐Pasternak foundation is studied. The Mori–Tanaka approach, considering carbon nanotube (CNT) agglomeration, is employed to derive the material properties of the composite nanoplates. The analysis is carried out for the uniform distribution (UD) and randomly oriented (RO) distribution of single‐walled carbon nanotubes (SWCNTs) while the nanoplate and the foundation are considered to be temperature‐dependent. Using Hamilton's principle, the governing differential equations are derived based on the Eringen's nonlocal elasticity theory and sinusoidal shear deformation theory (SSDT). The natural frequency of the nanoplates is then obtained by the Navier's analytical solution. To verify the method presented, the results are compared with those in the literature. Detailed parametric studies are performed to discuss the influences of CNTs agglomeration, nonlocal parameter, temperature, foundation parameters, the volume fraction of CNTs, plate length‐to‐thickness, and aspect ratios on the free vibration of the CNTRC nanoplates. POLYM. COMPOS., 40:E1479–E1494, 2019. © 2018 Society of Plastics Engineers

3-D free vibration analysis of annular plates on Pasternak elastic foundation via p-Ritz method

In this article authors obtained solutions to determine the ground curved sections free vibrations of thin-walled large-diameter pipelines with a liquid flow, based on a geometrically nonlinear version of the semi-momentless toroidal shells medium bending theory by V.Z. Vlasov and V.V. Novozhilov. The pipeline is a toroidal shell, the design scheme is presented in the form of a half torus. Angle β = 1800. The shell is laid on an elastic foundation and makes contact with the ground along a narrow strip. The problem of the soil pressure influence on the shell along a narrow strip is solved as a contact problem, using Fourier's series and an impulse function. The shell is exposed to the cooperative effect of the internal operating pressure, the liquid flowing pressure, the elastic soil foundation, and changes in the geometric characteristics. Motion equations of the middle shell surface are obtained taking into account the geometric and mechanical characteristics, and all the components of the shell material inertial forces. The hinged fastening of the shell ends is taken as the limiting condition. Using the semi-momentless shell theory assumptions, displacements in the longitudinal and circumferential directions are obtained. The solution to the problem of determining the free vibrations comes down to solving the problem of determining AB matrix values. The solutions obtained make it possible to determine the free vibrations frequencies at various wavenumbers values in the longitudinal and circumferential directions, and also make it possible to determine the internal operating pressure contribution, the soil bed coefficient, and geometric characteristics to the pipeline free vibrations frequencies.

This paper investigates the size dependent effect on the vibration analysis of a porous nanocomposite viscoelastic plate reinforced by functionally graded-single walled carbon nanotubes (FG-SWCNTs) by considering nonlocal strain gradient theory. Therefore, using energy method and Hamilton\'s principle, the equations of motion are derived. In this article, the effects of nonlocal parameter, aspect ratio, strain gradient parameter, volume fraction of carbon nanotubes (CNTs), damping coefficient, porosity coefficient, and temperature change on the natural frequency are perused. The innovation of this paper is to compare the effectiveness of each mentioned parameters individually on the free vibrations of this plate and to represent the appropriate value for each parameter to achieve an ideal nanocomposite plate that minimizes vibration. The results are verified with those referenced in the paper. The results illustrate that the effect of damping coefficient on the increase of natural frequency is significantly higher than the other parameters effect, and the effects of the strain gradient parameter and nonlocal parameter on the natural frequency increase are less than damping coefficient effect, respectively. Furthermore, the results indicate that the natural frequency decreases with a rise in the nonlocal parameter, aspect ratio and temperature change. Also, the natural frequency increases with a rise in the strain gradient parameter and CNTs volume fraction. This study can be used for optimizing the industrial and medical designs, such as automotive industry, aerospace engineering and water purification system, by considering ideal properties for the nanocomposite plate.

In this article, the buckling and vibration analysis of a double-bonded nanocomposite piezoelectric plate reinforced by a boron nitride nanotube based on the Eshelby-Mori-Tanaka approach is developed using modified couple stress theory under electro-thermo-mechanical loadings surrounded by an elastic foundation. Using Hamilton's principle, the governing equations of motion are obtained by applying a modified couple stress theory and the Eshelby-Mori-Tanaka approach for piezoelectric material and Kirchhoff plate. These equations are coupled for the double-layer plate using the Pasternak foundation and solved using Navier’s type solution. Then the dimensionless frequencies and critical buckling load for simply-supported boundary conditions are obtained. The effects of material length scale parameter, elastic foundation coefficients, aspect ratio (a/b), length to thickness ratio (a/h), transverse and longitudinal wave numbers on the dimensionless natural are investigated. The dimensionless frequency of a double-bonded nanocomposite piezoelectric plate increases with increasing length to thickness ratio and decreases with increasing aspect ratio. In addition, the effect of the elastic foundation on the dimensionless frequency of double-bonded nanocomposite piezoelectric plates is more considerable for higher elastic medium parameters. The critical buckling load also decreases with an increase in the dimensionless material length scale parameter.

A finite element model using four-unknown shear deformation theory integrated with the nonlocal theory is proposed for the bending and free vibration analysis of functionally graded (FG) nanoplates resting on elastic foundations. The present study developed the four-node quadrilateral element using Lagrangian and Hermitian interpolation functions for analysis of the membrane and bending displacement fields of FG nanoplates. Such a finite element formulation is suitable to investigate for the FG nanoplates resting on the elastic medium foundation with the stiffness matrices, the mass matrices and the load vectors using the second derivatives. The material properties of FG nanoplates are assumed to vary through the thickness direction by a power rule distribution of volume-fractions of the constituents. The equation of motion for FG nanoplates resting on the elastic foundation is obtained through Hamilton’s principle. Several numerical results are presented to demonstrate the accuracy and reliability of the present approach in comparison with other existing methods. In addition, the effects of geometrical parameters, material parameters, nonlocal parameters on the static bending and the free vibration responses of the nanoplates is also investigated in detail.

Nonlinear dynamic analysis of damaged Reddy–Bickford beams supported on an elastic Pasternak foundation