Present study proposes a nonlinear formulation to study the large amplitude free vibration of composite laminated plates reinforced by graphene sheets. Volume fraction of graphene sheets as ...reinforcements may be different in the layers. Material properties of the constituents are assumed to be temperature dependent. Properties of the composite media are obtained using a refined Halpin–Tsai approach where the auxiliary parameters are included into the formulation to capture the size dependency of the nanocomposite media. With the aid of third order shear deformation plate theory and the von Kármán type of kinematic assumptions, the basic governing equations of the plate are established. Afterwards, a non-uniform rational B-spline (NURBS) based isogeometric finite element method is used to study the large amplitude free vibration response of the graphene reinforced composite plates in thermal environment. Large amplitude frequencies as a function of centre point deflection are provided for different functionally graded patterns, aspect ratios, side to thickness ratios and boundary conditions.
Present research deals with the postbuckling problem of carbon nanotube reinforced composite plates subjected to uniform temperature rise loading. Distribution of carbon nanotubes as reinforcements ...may be uniform or functionally graded. To account for the large deformations of the plate, von-Kármán type of geometrical nonlinearity is included into the formulation. The virtual displacements principle associated with the conventional Ritz formulation whose shape functions are selected as the Chebyshev polynomials is used to obtain the matrix representation of the nonlinear equilibrium equations. The solution method is general and may be used for arbitrary combination of boundary conditions. The postbuckling equilibrium path which is governed by a nonlinear eigenvalue problem is traced using a displacement control strategy. Results of this study are compared with the available data in the open literature for the cases of isotropic homogeneous plates and cross-ply laminated plates. Afterwards numerical results are given for FG-CNTRC plates. It is shown that, FG-X pattern results in higher buckling temperature and also decreases the postbuckling deflection of the plate. Furthermore, this type of composites are eager to exhibit the secondary instability which is designated with a snap-through phenomenon in the post-buckling equilibrium path.
A nonuniform rational B-spline isogeometric finite element formulation is presented in this research to analyze the thermal buckling behavior of composite laminated skew plates reinforced by graphene ...platelets. Formulation is based on the first-order shear deformation plate theory. It is assumed that each layer of the composite laminated plate may have different volume fraction of graphene platelets leading to a through-the-thickness piecewise functionally graded medium. The equivalent properties of the plate are obtained by means of the Halpin-Tsai rule. The developed solution method may be used for arbitrary combinations of boundary conditions. The accuracy of the developed formulation is depicted via comparison studies with respect to the available data in the open literature. Novel numerical results are also given to show the effects of volume fraction of graphene platelets, distributed patterns of graphene platelets, and geometric characteristics of the skew plate.
Shear buckling response of carbon nanotube reinforced composite (CNTRC) rectangular plates in thermal environment is investigated in this research. Distribution of CNTs across the plate thickness may ...be uniform or graded via a mid-plane symmetric pattern. Properties of the CNTRC plate are obtained using the modified rule of mixtures approach by introduction of efficiency parameters. First order plate theory is adopted to construct the basic equations of the plate. A two-dimensional Ritz formulation with Chebyshev basis polynomials is implemented to obtain the elastic and geometric stiffness matrices of the plate. The proposed solution method may be used for arbitrary in-plane and out-of-plane edge supports. After performing convergence and comparison studies to show the effectiveness and accuracy of the proposed method, parametric studies are conducted to examine the influences of boundary conditions, volume fraction of CNTs, graded pattern of CNTs, thermal environment and plate geometry. It is shown that, shear buckling capacity of the plate may be enhanced through functionally graded distribution of CNTs. Besides, enrichment of matrix with CNTs enhances the shear buckling loads of FG-CNTRC plates.
Free vibration characteristics of carbon nanotube reinforced composite spherical panels are studied in the present research. First order shear deformation shell theory and the Sanders kinematics are ...considered as the basic assumptions. Distribution of carbon nanotubes (CNTs) across the panel thickness may be uniform or functionally graded. Equivalent properties of the media are estimated according to a modified rule of mixtures approach which consists efficiency parameters to capture the size dependency of the properties. Using Hamilton’s principle and the conventional Ritz formulation, the matrix representation of the equations associated with the free vibration motion is obtained. Shape functions of the Ritz method are obtained according to the Gram-Schmidt process. The resulting eigenvalue problem is solved to obtain the frequencies as well as mode-shapes of the spherical panel reinforced with CNTs. Convergence and comparison studies are provided to assure the effectiveness and accuracy of the proposed method. Afterwards, parametric studies are given to explore the effects of volume fraction of CNTs, distribution pattern of CNT, boundary conditions and geometric characteristics of the panel. It is shown that, enrichment of the polymeric matrix with more CNT results in higher frequencies. Furthermore, graded pattern of CNT is an influential factor on frequencies.
Present study deals with the free vibration analysis of skew plates made from functionally graded carbon nanotube reinforced composites. Carbon nanotubes as reinforcements are distributed across the ...thickness of the plate. Distribution pattern may be uniform or functionally graded. The developed formulation from a Cartesian coordinate system is transformed to an oblique coordinate system to satisfy the boundary conditions. The virtual strain and kinetic energies of the plate are obtained using the first order shear deformation plate theory. Ritz method whose shape functions are developed according to the Gram–Schmidt process is implemented to construct an eigenvalue problem associated to the natural frequencies of the plate. The developed solution method is general and may be used for arbitrary boundary conditions of the plate. Results are compared for isotropic homogeneous and composite laminated plates in skew shape with the available data in the open literature. Afterwards numerical results are provided for skew plates reinforced with carbon nanotubes. It is shown that volume fraction of carbon nanotubes and their distribution pattern are both influential of natural frequencies of the carbon nanotube reinforced plates. Generally, the higher the volume fraction of carbon nanotubes, the higher the natural frequencies of the skew plate.
It is known that the distribution of stresses in a rectangular plate is the same as the applied stresses on the boundaries when the loading is uniform or linearly varying. For other types of ...compressive loads, for instance parabolic compressive loading, the distribution of stresses in the plate is different from the applied loads at the boundaries of the plate. For such conditions, to obtain the buckling loads of the plate, an accurate prebuckling analysis should be performed. The present research aims to obtain the buckling loads and buckling pattern of composite plates reinforced with carbon nanotubes with uniform or functionally graded distribution across the plate thickness. The properties of the composite media are obtained based on a modified rule of mixtures approach with the introduction of efficiency parameters. First-order shear deformation plate theory is used to approximate the plate kinematics. The plate is subjected to uniaxial compressive loads which vary as parabolic functions across the width of the plate. At first, using the Ritz method and Airy stress function formulation, the distribution of stress resultants in the plate domain is obtained as a two-dimensional elasticity formulation. Afterwards, by means of the Chebyshev polynomials as the basic functions of the Ritz solution method, an eigenvalue problem is established to obtain the buckling load and buckling shape of the plate. Comparison studies are provided to assure the accuracy of the presented formulation for isotropic homogeneous and cross-ply laminated plates. Afterwards, parametric studies are performed for composite plates reinforced with carbon nanotubes.
Chebyshev polynomial functions are used in the Lagrangian multipliers method to study the free vibration characteristics of rectangular moderately thick composite plates reinforced with carbon ...nanotubes (CNTs). Plate is resting on point supports. Distribution of CNTs across the plate thickness is considered to be either uniform or functionally graded. Properties of the plate are obtained using a refined rule of mixtures approach which includes the efficiency parameters to capture the size dependent characteristics of the composite plate. Using a Ritz solution method, an eigenvalue problem is established which results in natural frequencies and mode shapes of the plate. Based on the developed solution method, number and position of point supports are arbitrary and also various boundary conditions may be assumed for the four edges of the plate. After performing comparison studies for isotropic homogeneous plates on point supports, parametric studies are provided to explore the vibration characteristics of the carbon nanotube reinforced composite plates on point supports. It is shown that, frequencies of the plate increase as the volume fraction of CNTs increases.
Thermal postbuckling analysis and the axisymmetric static and dynamic snap-through phenomena due to static/sudden uniform lateral pressure in a thermally postbuckled functionally graded material ...circular plate are performed in this research. Plate is formulated using the first order shear deformation plate theory. Thermo-mechanical properties of the plate are assumed to be temperature dependent where dependency is described according to the higher order Touloukian representation. Two types of temperature loading are considered. Uniform temperature rise and heat conduction across the thickness direction. The one dimensional heat conduction equation in the thickness direction is obtained and discreted via the central finite difference method. The obtained system of equations is nonlinear since the thermal conductivity itself is a function of the unknown nodal temperatures. Using the von-Kármán assumptions, the governing equations of the plate are obtained in a matrix representation with the aid of the conventional Ritz method whose shape functions are developed using the Gram-Schmidt process. At first thermal postbuckling analysis is performed which is a nonlinear problem with respect to both temperature and displacements. Afterwards, response of the bulged thermally postbuckled plate is obtained under the static and dynamic uniform pressure. Snap-through phenomenon may be observed in both static and dynamic loading cases, due to the immovability of the edge of the plate and the initial deflection caused by postbuckling deflection. To capture the snapping phenomenon and trace the path beyond the limit loads, cylindrical arch-length technique is used. In dynamic snap-through analysis, the effect of structural damping is also included. Numerical results of this study reveal that the structure is sensitive to the initial deflection caused by thermal postbuckling load. Increasing the temperature prior to mechanical loads enhances the snap-through intensity and also increases both the upper and lower limit loads. As shown, dynamic snap-through loads are lower than the static ones, however dynamic snap-through intensity is more than the static snap-though intensity. Furthermore, structural damping enhances the dynamic buckling loads of the plate and decreases the dynamic postbuckling deflection of the plate.
•Thermal Post-buckling of FGM circular plates in analyzed.•The possibility of snap-through phenomenon due to uniform static or dynamic pressures is investigated.•Influence of structural damping on dynamic snap-through is analyzed.•For antipathetic loading, snap-through may take place in both static and dynamic senses.
In this study, free vibration characteristics of composite plates reinforced with single walled carbon nanotubes is investigated. Distribution of the carbon nanotubes through the thickness of the ...panel may be uniform or functionally graded. Properties of the composite media are obtained according to a refined rule of mixtures approach which contains the efficiency parameters. First order shear deformation shell theory and Donnell-type kinematic assumptions are used. To establish the eigenvalue problem of the system, the energy based Ritz method with Chebyshev polynomials as the basis functions is implemented. The resulting eigenvalue problem is solved to obtain the natural frequencies of the system as well as the associated mode shapes. After performing comparison studies for the simpler cases, numerical results are given for vibration characteristics of carbon nanotube reinforced cylindrical panels. Numerical results reveal that, frequencies of the panel are dependent to both, volume fraction of carbon nanotubes and their distribution pattern across the thickness. Increasing the volume fraction of carbon nanotubes increases the frequencies of the panel.
