The static bending behavior of porous functionally graded (PFG) micro-plate under the geometrically nonlinear analysis is studied in this article. A small-scale nonlinear solution is established using the Von-Kármán hypothesis and the modified couple stress theory (MCST). To obtain the deflection of the plate, the Reddy higher-order plate theory coupled with isogeometric analysis (IGA) is utilized. The distribution of porosities is assumed to be even and uneven across the plate’s thickness and the effective material properties of porous functionally graded micro-plate are calculated using the refined rule-of-mixture hypothesis. The influence of power index, porosity parameter and material length scale parameter on the nonlinear behaviors of static bending of porous FGM micro-plates are also investigated using several numerical examples.
相似文献The present work fills a gap on the postbuckling behavior of multilayer functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical and spherical shell panels resting on elastic foundations subjected to central pinching forces and pressure loadings. Based on a higher-order shear deformation theory and the von Kármán’s nonlinear strain–displacement relations, the governing equations of the FG-GPLRC cylindrical and spherical shell panels are established by the principle of virtual work. The non-uniform rational B-spline (NURBS) based isogeometric analysis (IGA), the modified arc-length method and the Newton’s iteration method are employed synthetically to obtain nonlinear load–deflection curves for the panels numerically. Several comparative examples are performed to test reliability and accuracy of IGA and arc-length method in present formulation and programming implementation. Parametric investigations are carried out to illustrate the effects of dispersion type of the graphene platelet (GPL), weight fraction of the GPL, thickness of the panel, radius of the panel and parameters of elastic foundation on the load–deflection curves of the FG-GPLRC shell panels. Some complex load–deflection curves of the FG-GPLRC cylindrical and spherical shell panels resting on elastic foundations may be useful for future references.
相似文献A nonlocal strain gradient model is developed in this research to analyse the nonlinear frequencies of functionally graded porous curved nanotubes. It is assumed that the curved nanotube is in contact with a two-parameter nonlinear elastic foundation and is also subjected to the uniform temperature rise. The non-classical theory presented for curved nanotubes contains a nonlocal parameter and a material length scale parameter which can capture the size effect. A power law distribution function is used to describe the graded properties through the thickness direction of curved nanotubes. The even dispersion pattern is used to model the porosities distribution. The high-order shear deformation theory and the von Kármán type of geometric non-linearity are utilized to obtain the nonlinear governing equations of the structure. The size-dependent equations of motion for the large amplitude vibrations of curved nanotubes are obtained by employing Hamilton’s principle. The analytical solutions are extracted for the curved nanotube with immovable hinged-hinged boundary conditions. Size-dependent frequencies of the curved nanotube exposed to thermal field are obtained using the two-step perturbation technique and Galerkin procedure. The effects of important parameters such as nonlocal and length scale parameters, temperature field, elastic foundation, porosity, power law index and geometrical parameters are studied in detail.
相似文献In this paper, the nonlinear dynamic response of functionally graded (FG) sandwich nanobeam associated with temperature-dependent material properties by considering the initial geometric imperfection is investigated. The size-dependent behavior of the FG sandwich nanobeam is simulated based on the nonlocal strain gradient theory, and Von Karman nonlinear hypothesis is used to model the geometrical nonlinearity. Moreover, the geometric imperfection is considered as a slight curvature satisfying the first mode shape, and four different FG sandwich patterns including two asymmetric configurations and two symmetric configurations are taken into account. The governing equation of the FG sandwich nanobeam subjected to thermal and harmonic external excitation loadings is derived on the basis of Hamilton’s principle. The numerical results are obtained by employing the multiple-scale method, which are also validated by comparison with two previous studies. Furthermore, comprehensive investigations into the influences of size-dependent parameters, external temperature variation, geometric imperfection amplitude, gradient index and sandwich configuration on the nonlinear characteristics of imperfect FG sandwich nanobeams are conducted through numerical results.
相似文献The problem of the nonlinear thermal buckling and post-buckling of magneto-electro-thermo-elastic functionally graded porous nanobeams is analyzed based on Eringen’s nonlocal elasticity theory and by using a refined beam model. The beams with immovable clamped ends are exposed to the external electric voltages, magnetic potentials, a uniform transverse load and uniform temperature change. For the first time, the four types of porosity distribution in the nanobeam are considered and compared in complex electric–magnetic fields. Besides, the new formula of the effective material properties is proposed in this paper to simultaneously estimate the material distribution and porosity distribution in the thickness direction. The generalized variation principle is used to induce the governing equations, then the approximate analytical solution of the METE-FG nanobeams based on physical neutral surface is obtained by using a two-step perturbation technique. Finally, detailed parametric analyses are performed to get an insight into the effects of different physical parameters, including the slenderness ratio, small-scale parameter, volume fraction index, external electric voltages, magnetic potentials, porosity coefficient and different porosity distributions, for providing an effective way to improve post-buckling strength of porous beams.
相似文献In this paper, to improve the vibrational response of microstructures, the impact of the nonlinear modal analysis of axially functionally graded (AFG) truncated conical micro-scale tube including the thermal loading for the different type of cross sections such as uniform section, linear tapered section, convex section, the exponential section are studied that are applicable for various application, for example, the micro-thermal fins, macro-/micro-fluid-flow diffuser, fluid-flow nozzle, fluid-flow throat, micro-sensor, etc. The nonlinear equations are obtained applying Hamilton’s principles based on the modified couple stress to determine the size effect and Euler–Bernoulli beam theory considering the von-Kármán’s nonlinear strain. The material combination varies along the tube’s length, denouncing the AFG tube made by metal and ceramic phases. The nonlinear equations are solved by applying a couple of homotopy perturbation methods (HPM) to calculating the nonlinear results and the generalized differential quadrature method (GDQM) to providing the initial conditions. The linear and nonlinear results presented the effect of various cross sections and other parameters on the micro-tube frequency that are valuable to design and manufacture the micro-electro-mechanical systems (MEMS).
相似文献In this article, the free vibration response of sandwich plates with porous electro-magneto-elastic functionally graded (MEE-FG) materials as face sheets and functionally graded carbon nanotube-reinforced composites (FG-CNTRC) as core is investigated. To this end, four-variable shear deformation refined plate theory is exploited. The properties of functionally graded material plate are assumed to vary along the thickness direction of face sheets according to modified power-law expression. Furthermore, properties of FG-CNTRC layer are proposed via a mixture rule. Hamilton’s principle with a four-variable tangential–exponential refined theory is used to obtain the governing equations and boundary conditions of plate. An analytical solution approach is utilized to get the natural frequencies of embedded porous FG plate with FG-CNTRC core subjected to magneto-electrical field. A parametric study is led to fulfill the effects of porosity parameter, external magnetic potential, external electric voltage, types of FG-CNTRC, and different boundary conditions on dimensionless frequencies of porous MEE-FG sandwich plate. It is noteworthy that the numerical consequences can serve as benchmarks for future investigations for this type of structures with porous mediums.
相似文献In this paper, an analytical method is used to study the nonlinear primary resonance of imperfect spiral stiffened functionally graded (SSFG) cylindrical shells with internal stiffeners. The SSFG cylindrical shell is surrounded by linear and nonlinear elastic foundation and the effect of structural damping on the system response is also considered. The material properties of the shell and stiffeners are assumed to be continuously graded in the thickness direction. Three-parameter nonlinear elastic foundation model is consists of two-parameter linear elastic foundation (Winkler and Pasternak) and one hardening/softening cubic nonlinearity parameter. Based on the von Kármán nonlinear equations and the classical plate theory of shells, the strain–displacement relations are derived. The smeared stiffener technique is used to the model of the internal stiffeners. Using the Galerkin method, the partial differential equations of motion are discretized. The nonlinear primary resonance is analyzed by means of the multiple scales method. The effects of various geometrical characteristics, material parameters and elastic foundation coefficients are investigated on the nonlinear primary resonance.
相似文献The nonlinear resonance responses of functionally graded (FG) cylindrical microshells with the elastic medium is investigated by considering thermal and scale effects. First, using the modified couple stress theory, the nonlinear dynamics model for FG microshell are established. Then the reduced nonlinear differential equations are derived by Galerkin’s method and static condensation. Finally, subharmonic, superharmonic and primary resonances of FG cylindrical microshells are analyzed by a perturbation method. In addition, the bifurcation characteristics of the nonlinear dynamic responses are investigated by some numerical examples. The effects of key parameters (modal damping, excitation frequency, foundation medium, scale parameter and thermal effect) on the nonlinear resonance responses are also discussed by numerical simulation.
相似文献This paper deals with dynamic stability of functionally graded (FG) nanocomposite annular plate reinforced with graphene platelets (GPLs) subjected to a periodic radial compressive load in thermal environment. On the basis of modified Halpin–Tsai micromechanics model, the effective elastic modulus of structure is estimated. Taking into consideration of the first-order shear deformation theory, the governing motion equations are obtained via the Hamilton’s principle. Afterward, the partial differential motion equations are discretized into a system of Mathieu–Hill equations by utilizing generalized differential quadrature method. Furthermore, the Bolotin’s technique is applied to determine the principle unstable zone of functionally graded graphene platelet reinforced composite (FG-GPLRC) annular plate. The accuracy and validity of current study are examined by comparing the fundamental natural frequencies of structure with those published in available literature. Eventually, to investigate the influences of number of layers, GPLs patterns and their geometric, boundary conditions, geometrical parameters of structure, static load factor, and temperature change on the DIRs of FG-GPLRC multilayer annular plate, different parametric studies are performed.
相似文献In the current work, the dynamic behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plate with negative Poisson’s ratio (NPR) is investigated by combining higher-order shear deformation theory and large deflection theory. First, explicit solutions are proposed to predict the effective Poisson’s ratio (EPR) of the laminates. Taking carbon nanotube-reinforced composite (CNTRC) as an example, the maximum NPR is obtained for \(\left( { \pm \theta } \right)_{{3{\text{T}}}}\) laminate as well. Results show that the EPR (\(v_{13}^{\text{e}}\),\(v_{23}^{\text{e}}\)) can range from a positive value of 0.311 to a negative value of 0.63. For the dynamic response problem, the asymptotic solutions with a two-step perturbation approach are derived for FG-CNTRC plates to capture the relationship between the center deflection and time. Several key factors such as functionally graded distribution, variations in the elastic foundation, and thermal stress produced by changing the temperature field are considered in the subsequent analysis. Numerical simulations are carried out to examine the corresponding dynamic behavior of FG-CNTRC plates when these factors are taken into account.
相似文献To impart desirable material properties, functionally graded (FG) porous silicon has been produced in which the porosity changes gradually across the material volume. The prime objective of this work is to predict the influence of the surface free energy on the nonlinear secondary resonance of FG porous silicon nanobeams under external hard excitations. On the basis of the closed-cell Gaussian-random field scheme, the mechanical properties of the FG porous material are achieved corresponding to the uniform and three different FG patterns of porosity dispersion. The Gurtin–Murdoch theory of elasticity is implemented into the classical beam theory to construct a surface elastic beam model. Thereafter, with the aid of the method of multiple time-scales together with the Galerkin technique, the size-dependent nonlinear differential equations of motion are solved corresponding to various immovable boundary conditions and porosity dispersion patterns. The frequency response and amplitude response associated with the both subharmonic and superharmonic hard excitations are obtained including multiple vibration modes and interactions between them. It is found that for the subharmonic excitation, the nanobeam is excited within a specific range of the excitation amplitude, and this range shifts to higher excitation amplitude by incorporating the surface free energy effects. For the superharmonic excitation, by taking surface stress effect into account, the excitation amplitude associated with the peak of the vibration amplitude enhances. Moreover, in the subharmonic case, it is demonstrated that by increasing the porosity coefficient, the value of the excitation frequency at the joint point of the two branches of the frequency-response curve reduces. In the superharmonic case, it is revealed that an increment in the value of porosity coefficient leads to decrease the peak of the oscillation amplitude and the associated excitation frequency.
相似文献This article presents a comprehensive analysis to investigate the static buckling stability and static deflection of axially single-walled (SW) functionally graded (FG) carbon nanotubes reinforced composite (CNTRC) plates with temperature-dependent material properties and graded by different functions, for the first time. The distribution of the carbon nanotubes is described by two functions, one for the x-direction CNTs distribution (CNTRC-x plate) and another for z-direction CNTs distribution (CNTRC-z plate). The graduation functions of CNTs are unidirectional (UD CNTRC), FG-X CNTRC, FG-O CNTRC, and FG-V CNTRC. The extended rule of mixture and the molecular dynamics simulations are exploited to evaluate the equivalent mechanical properties of FG-CNTRC plate. Equilibrium equations are formulated using principal of Hamilton and solved analytically using Galerkin method to cover various boundary conditions. New higher order shear deformation theory is proposed. The numerical results gained by the proposed solution are verified by comparing with those of published ones. Numerical results present influences of gradation function, inhomogeneity parameters, aspect ratio, thickness ratio, boundary conditions and temperature on the static buckling and deflection of FG-CNTRC plate using modified higher order shear theories.
相似文献In this research, thermal buckling and forced vibration characteristics of the imperfect composite cylindrical nanoshell reinforced with graphene nanoplatelets (GNP) in thermal environments are presented. Halpin–Tsai nanomechanical model is used to determine the material properties of each layer. The size-dependent effects of GNPRC nanoshell is analyzed using modified couple stress theory. For the first time, in the present study, porous functionally graded multilayer couple stress (FMCS) parameter which changes along the thickness is considered. The novelty of the current study is to consider the effects of porosity, GNPRC, FMCS and thermal environment on the resonance frequencies, thermal buckling and dynamic deflections of a nanoshell using FMCS parameter. The governing equations and boundary conditions are developed using Hamilton’s principle and solved by an analytical method. The results show that, porosity, GNP distribution pattern, modified couple stress parameter, length to radius ratio, mode number and the effect of thermal environment have an important role on the resonance frequencies, relative frequency change, thermal buckling, and dynamic deflections of the porous GNPRC cylindrical nanoshell using FMCS parameter. The results of current study can be useful in the field of materials science, micro-electro-mechanical systems and nano electromechanical systems such as microactuators and microsensors.
相似文献This study investigates the effects of fluid–structure and soil–structure interaction on the free vibration response of functionally graded sandwich plates. To this aim, an exemplary problem is analyzed, whereas a metal/ceramic sandwich plate is placed at the bottom of a tank filled in with fluid. Two cases are considered: (i) soft core, i.e., a sandwich plate with metal core and ceramic skins, and (ii) hard core, i.e., a sandwich plate with ceramic core and metal skins. In both cases, the skins are modelled as suitable functionally graded materials (FGMs). The soil is modelled as a Pasternak foundation. The free vibration analysis is carried out according to the extended higher order sandwich plate theory (EHSAPT). The fluid is assumed to be inviscid, incompressible, and irrotational. Hamilton’s principle is exploited to deduce the governing equations and the corresponding boundary conditions. The Rayleigh–Ritz method with two-variable orthogonal polynomials is used to compute the natural frequencies of the sandwich plate. The adopted approach is first validated through comparison with results published in the literature. Then, the effects are studied of several parameters on the dynamic response of the system.
相似文献The paper presents a novel nonlocal strain gradient isogeometric model for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates. To observe the length scale and size-dependency effects of nanostructures, the nonlocal strain gradient theory (NSGT) is considered. The present model is efficient to capture both nonlocal effects and strain gradient effects in nanoplate structures. In addition, the material properties of the FG-CNTRC are assumed to be graded in the plate thickness direction. Based on the higher order shear deformation theory (HSDT), the weak form of the governing equations of motion of the nanoplates is presented using the principle of virtual work. Afterward, the natural frequency and deflection of the nanoplates are made out of isogeometric analysis (IGA). Thanks to higher order derivatives and continuity of NURBS basic function, IGA is suitable for the weak form of NSGT which requires at least the third-order derivatives in approximate formulations. Effects of nonlocal parameter, strain gradient parameter, carbon nanotube (CNT) volume fraction, distributions of CNTs and length-to-thickness ratios on deflection and natural frequency of the nanoplates are examined and discussed in detail. Numerical results are developed to show the phenomenon of stiffness-softening and stiffness-hardening mechanisms of the present model.
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