In this paper, thermally induced vibration of annular sector plate made of functionally graded materials is analyzed. All of the thermomechanical properties of the FGM media are considered to be ...temperature dependent. Based on the uncoupled linear thermoelasticity theory, the one-dimensional transient Fourier type of heat conduction equation is established. The top and bottom surfaces of the plate are under various types of rapid heating boundary conditions. Due to the temperature dependency of the material properties, heat conduction equation becomes nonlinear. Therefore, a numerical method should be adopted. First, the generalized differential quadrature method (GDQM) is implemented to discretize the heat conduction equation across the plate thickness. Next, the governing system of time-dependent ordinary differential equations is solved using the successive Crank–Nicolson time marching technique. The obtained thermal force and thermal moment resultants at each time step from temperature profile are applied to the equations of motion. The equations of motion, based on the first-order shear deformation theory (FSDT), are derived with the aid of the Hamilton principle. Using the GDQM, two-dimensional domain of the sector plate and suitable boundary conditions are divided into a number of nodal points and differential equations are turned into a system of ordinary differential equations. To obtain the unknown displacement vector at any time, a direct integration method based on the Newmark time marching scheme is utilized. Comparison investigations are performed to validate the formulation and solution method of the present research. Various examples are demonstrated to discuss the influences of effective parameters such as power law index in the FGM formulation, thickness of the plate, temperature dependency, sector opening angle, values of the radius, in-plane boundary conditions, and type of rapid heating boundary conditions on thermally induced response of the FGM plate under thermal shock.
Considering the uncoupled thermoelasticity assumptions, a thermally induced vibration analysis for functionally graded material (FGM) conical shells is performed in this research. Thermo-mechanical ...properties of the conical shell are assumed to be temperature and position dependent. The conical shell is under rapid heating with various cases of thermal loads on the outer surface, whereas the opposite surface is kept at reference temperature or thermally insulated. Since the ratio of thickness to radius is much smaller than one, the transient heat conduction equation, for simplicity, may be established and solved for one-dimensional condition. Assuming temperature-dependent material properties, the heat conduction equation is nonlinear and should be solved using a numerical method. A hybrid generalized differential quadrature (GDQ) and Crank–Nicolson method is used to obtain the temperature distribution in thickness direction, respectively. Based on the first-order shear deformation theory and geometrically nonlinear assumptions, the equations of motion are obtained applying the Hamilton principle. Discretization of the equations of motion in the space domain and boundary conditions is performed by applying the GDQ method, and then, the system of highly nonlinear coupled ordinary differential equations is solved by the iterative Newmark time-marching scheme and well-known Newton–Raphson method. Since the thermally induced vibration of the conical shells is not reported in the literature, the results are compared with the case of a circular plate. Also, studies of the FGM conical shells for various types of boundary conditions, functionally graded patterns, and thermal loads are provided. The effects of temperature dependency, geometrical nonlinearity, semi-vertex angle, shell length, and shell thickness upon the deflections of the conical shells are investigated.
The current article deals with the nonlinear free vibration of nanocomposite circular plates reinforced with graphene platelets (GPL). The functionally graded (FG) plate is considered over a three ...parameter non-linear elastic foundation. In this research, three types of gradings are assumed for reinforcing the plate by GPLs. The Halpin-Tsai micromechanical rule is exploited to obtain the elastic modulus of the plate. The first order shear deformation plate theory associated with the nonlinear strain-displacement relations are applied to extract the governing motion equations. The generalized differential quadrature (GDQ) method is implemented to solve the equations of motion in the plate domain. Furthermore, an iterative displacement control technique associated with the weighted residual technique are used to linearize the present problem and obtain the linear and nonlinear frequencies. After examining the validation study, some parametric studies are tabulated and plotted to recognize the effects of the boundary condition, distribution types of GPL, GPL weight fraction, geometrical parameters, and elastic foundation parameters on the linear and non-linear free vibration behaviour of structure. It is shown that maximum frequencies of the plate belong to the FG-X case and the minimum ones are obtained in FG-O type.
An analysis on thermal buckling of composite laminated annular sector plates reinforced with the graphene platelets is examined in this research. It is assumed that the graphene platelets fillers are ...randomly oriented and uniformly distributed in each ply of the composite media. Effective elasticity modulus of the nanocomposite media is extracted utilizing the modified Halpin-Tsai procedure which takes into account the size effects of the graphene fillers. Using the von Kármán type of geometrical nonlinearity and first order shear deformation plate theory, the governing equilibrium equations for the buckling of nanocomposite plates in sector shape under uniform temperature rise are established. Stability equations are obtained using the adjacent equilibrium criterion and solved by means of the generalized differential quadrature method. Numerical examples are given to study the effects of boundary conditions, weight fraction of the graphene platelets, and distribution pattern of the graphene platelets on critical temperature and the fundamental buckled shapes. Results represent that, with introduction of a small amount of graphene platelets into the isotropic matrix of the composite media, the critical buckling temperature of the plate may be enhanced.
•Thermal buckling and buckled shapes of GPLRC annular sector plates are obtained.•Increasing the weight fraction of GPLS may increase/decrease the critical buckling temperature depending of the functionally graded pattern.•FG-X pattern results in the maximum critical buckling temperature.•With introduction of even a low amount of GPLs, critical buckling temperature may be increased significantly.
In the current research, application of generalized differential quadrature element (GDQE) method is presented to analyze the free vibration of L-shaped plates. It is considered that the laminated ...composite plate is reinforced by graphene platelets (GPLs). Furthermore, the GPLs are randomly oriented and uniformly dispersed in each lamina. The GPL weight fraction changes from layer to layer based on the four functionally graded models. To develop the formulation, the effective Young modulus is calculated by means of the Halpin–Tsai micromechanical rule. GDQE method firstly divides total plate domain into three rectangular elements. First order shear deformation theory (FSDT) is employed to estimate the displacement components of each element. Moreover, the Hamilton principle is utilized to extract the motion equations of elements individually. The GDQ tool is applied to each element for separating these regions into sample grid points. Properly applying the compatibility conditions between the elements is a very important factor in the correctness and accuracy of the response. In order to present the validity and accuracy of the outcomes, results are compared with the existing information in the open literature. After that, novel results are demonstrated to examine the effects of GPL weight fraction and distribution, plate geometrical parameters and various boundary conditions on the natural frequencies and corresponding mode shapes of L-shaped plates.
•Generalized Differential Quadrature element method is applied to obtain the frequencies of an L-shape plate.•Plate is divided into three elements where the matching conditions are properly applied to the intersections.•Plate is reinforced with graphene platelets where the volume fraction of GPLs may vary through the thickness.•Arbitrary combinations of edge supports may be applied to the edges of the plate.
Based on the nonlinear dynamic analysis, thermally induced vibrations of the FGM shallow arches subjected to different sudden thermal loads are studied. Temperature and position dependence of the ...material properties are taken into account. Based on the uncoupled thermoelasticity assumptions, The non-linear one-dimensional transient heat conduction equation is solved numerically by a hybrid iterative GDQ method and Crank-Nicolson time marching scheme. A first order shear deformation arch theory (FSDT) is also combined with the von Kármán type of geometrical non-linearity and the Donnell kinematic assumption to obtain the equations of motion employing the Hamilton principle. Discretization of the highly coupled non-linear equations of motion is done by using the GDQ method in the arch domain. The solution of the system of the ordinary differential equations is established by means of a hybrid iterative Picard-Newmark scheme. Comparison is also made with the existing results for the case of isotropic homogeneous shallow arches, where good agreement is obtained. Also, parametric studies are proposed to show the effects of temperature dependency, geometrical non-linearity, arch thickness, power law index, and the type of thermal-mechanical boundary conditions upon the arch deflection.
Natural frequencies of circular deep arches made of functionally graded materials (FGMs) with general boundary conditions are obtained in this research based on the unconstrained higher-order shear ...deformation theory taking into account the depth change, complete effects of shear deformation, and rotary inertia. The material properties are assumed to vary continuously through the thickness direction of the arch. Displacement field within the arch is obtained through expansion up to an arbitrary order. Governing differential equations of the in-plane vibration are derived using Hamilton's principle. These equations are solved numerically utilizing the differential quadrature method (DQM) formulation. In order to illustrate the validity and accuracy of the presented results, results are compared with the available data in the open literature. Afterwards, novel numerical results are given for free vibration behaviour of the FGM deep arches with various boundary conditions.
•Displacement of the arch is estimated by means of a higher order theory up to an arbitrary order.•Different types of boundary conditions are covered in this research.•GDQ method is used to solve the governing equations.•Properties are distributed across the depth of the arch using a power law function.
A theoretical study of photoionization from the outermost orbitals of Be, Mg and Ca atoms endohedrally confined in C
60
is reported. The fullerene ion core, comprised of sixty C
4+
, is smudged into ...a continuous jellium charge distribution, while the delocalized cloud of carbon valence electrons
plus
the encaged atom are treated in the time-dependent local density approximation (TDLDA). Systematic evolution of the mixing of outer atomic level with states of the C
60
valence band is found along the sequence. This is found to influence the plasmonic enhancement of atomic photoionization at low energies and the geometry-revealing confinement oscillations at high energies in significantly different ways: (a) the extent of enhancement is mainly determined by the strength of atomic ionization, giving the strongest enhancement for Be even though Ca suffers the largest mixing. But (b) strongest collateral oscillations are uncovered for Ca, since, relative to Be and Mg, the mixing causes the highest photoelectron production at confining boundaries of Ca. The study paints the first comparative picture of the atomic valence photospectra for alkaline earth metallofullerenes in a dynamical many-electron framework.
Present study deals with the dynamic snap-through phenomenon of an isotropic shallow spherical cap under transient type of thermal loading. The inner surface of the shell is subjected to sudden ...temperature elevation, whereas the outer surface is kept at reference temperature. Transient thermal shock is applied uniformly and shell thickness is assumed to be thin enough. Therefore, transient heat conduction equation is analytically solved across the thickness direction (one dimensional). Immovable simply-supported boundary conditions are assumed for the shell. Since the boundary conditions and the applied thermal shock are axisymmetric, the governing motion equations of the shell are restricted to the case of axisymmetric. First order shear deformation theory of shells is utilized to approximate the displacement field. The von Kármán type of geometrical non-linearity is used in strain-displacement relations. With the establishment of the associated Hamilton principle, differential equations of motion are extracted. Motion equations are discretized within the shell domain by means of the harmonic differential quadrature method (HDQM). The Newmark time marching scheme based on the constant average acceleration method is applied to turn the motion differential equations into a system of algebraic equations at each time step. Highly coupled equations are solved implementing the well-known Newton-Raphson iterative technique. By means of the Budiansky criterion, critical thermal shock parameters are distinguished. Critical dynamic snap-through temperatures are also verified using the phase-plane presentation. Comparison study is performed to validate the formulation and solution method of the present research with simple cases. Also, parametric investigations are performed to demonstrate the geometrical effects on the dynamic critical buckling temperatures of the shallow spherical shells subjected to thermal shock.
•Symmetric, nonlinear coupled motion equations of spherical shell under thermal shock are solved.•Dynamic snap-through phenomenon is investigated for shallow spherical shells.•Budiansky criterion and phase plane approaches are used to detect the critical dynamic buckling temperatures.•The importance of inertia effects which results in thermally induced vibration phenomenon is highlighted.
By proposing the unifier factors, a unified formulation for the generalized coupled thermo-visco-elasticity response of hollow spheres based on the Lord–Shulman, Green–Lindsay, and Green–Naghdi ...models is developed. Kelvin–Voigt model is used in the current research to consider the viscoelastic response of the sphere. This formulation is used to capture the thermoviscoelastic response of finite isotropic hollow sphere under thermal shock. After transforming the one-dimensional radial equations into the non-dimensional form, finite element method (FEM) based on the Galerkin weak formulation is implemented. The governing equations are discretized through the radial domain utilizing FEM. The determined system of ordinary differential equations is traced using the average acceleration method as a type of Newmark time marching scheme. After validation, novel numerical results are demonstrated to analyze the influences of viscoelastic damping on the response of hollow sphere under rapid surface heating based on various types of generalized theories.
•Interaction, propagation and reflection of thermal and mechanical waves are investigated.•The Galerkin version of FEM method is used.•A viscoelastic sphere is considered by assuming the Kelvin–Voigt model.•Three different theories, namely Green–Naghdi, Green–Lindsay and Lord–Shulman are used.
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