Nonlinear Thermal Buckling Research Articles

Nonlinear thermal buckling analysis of temperature-dependent porous annular and circular microplates reinforced by graphene platelets by using isogeometric analysis method

In this research, by using isogeometric analysis (IGA) method, nonlinear thermal buckling analysis of porous circular and annular microplates reinforced by graphene platelets (GPLs) under uniform temperature rise is performed by taking both the temperature dependence of the material properties and the size effects into account. The closed-cell metal foam scheme combined with the Halpin-Tsai micromechanics model and the rule of mixture is utilized to identify the effective material properties of the porous nanocomposites for three GPL dispersion patterns and three porosity distributions. With the aid of the virtual work principle, the classical and the non-classical motion equilibrium formulation based on the higher-order shear deformation theory in conjunction with the modified strain gradient (MSG) theory for the nonlinear temperature-dependent buckling behaviour are established. The higher-order NURBS basis functions capable of accomplishing easily the smoothness of arbitrary continuity order are successfully utilized to solve the nonlinear governing equations by satisfying the C2-continuity of the displacement field required by the MSG theory. The thermal buckling behaviour is assessed in detail by using various parameters such as the thickness-to-radius ratio, the porosity coefficient, the GPL weight fraction and the material length scale as well as the porosity distribution scheme and the GPL dispersion pattern. The results show the significance of incorporating the temperature dependency of the material properties when analyzing the thermal buckling behaviour and present the combination scheme of porosity and GPL distributions for enhancing the thermal buckling resistance.

Nonlinear Thermal/Mechanical Buckling of Orthotropic Annular/Circular Nanoplate with the Nonlocal Strain Gradient Model.

This article presents the nonlinear investigation of the thermal and mechanical buckling of orthotropic annular/circular single-layer/bilayer nanoplate with the Pasternak and Winkler elastic foundations based on the nonlocal strain gradient theory. The stability equations of the graphene plate are derived using higher-order shear deformation theory (HSDT) and first-order shear deformation theory (FSDT) considering nonlinear von Karman strains. Furthermore, this paper analyses the nonlinear thermal and mechanical buckling of the orthotropic bilayer annular/circular nanoplate. HSDT provides an appropriate distribution for shear stress in the thickness direction, removes the limitation of the FSDT, and provides proper precision without using a shear correction coefficient. To solve the stability equations, the differential quadratic method (DQM) is employed. Additionally, for validation, the results are checked with available papers. The effects of strain gradient coefficient, nonlocal parameter, boundary conditions, elastic foundations, and geometric dimensions are studied on the results of the nondimensional buckling loads. Finally, an equation is proposed in which the thermal buckling results can be obtained from mechanical results (or vice versa).

Thermo-Mechanical Buckling and Non-Linear Free Oscillation of Functionally Graded Fiber-Reinforced Composite Laminated (FG-FRCL) Beams

We investigated the thermal buckling temperature and nonlinear free vibration of functionally graded fiber-reinforced composite laminated (FG-FRCL) beams. The governing nonlinear partial differential equations were derived from the Euler–Bernoulli beam theory, accounting for the von Kármán geometrical nonlinearity. Such equations were then reduced to a single equation by neglecting the axial inertia. Thus, the Galerkin method was applied to discretize the governing nonlinear partial differential equation in the form of a nonlinear ordinary differential equation, which was then solved analytically according to the He’s variational method. Three different boundary conditions were selected, namely simply, clamped and clamped-free supports. We also investigated the effect of power-index, lay-ups, and uniform temperature rise on the nonlinear natural frequency, phase trajectory and thermal buckling of FG-FRCL beams. The results showed that FG-FRCL beams featured the highest fundamental frequency, whereas composite laminated beams were characterized by the lowest fundamental frequency. Such nonlinear frequencies increase for an increased power index and a decreased temperature. Finally, it was found that FG-FRCL beams with [0/0/0] lay-ups featured the highest nonlinear natural frequency and the highest thermal buckling temperature, followed by [0/90/0] and [90/0/90] lay-ups, while a [90/90/90] lay-up featured the lowest nonlinear natural frequency and critical buckling temperature.

A new analytical approach of nonlinear thermal buckling of FG-GPLRC circular plates and shallow spherical caps using the FSDT and Galerkin method

A new analytical approach for nonlinear thermal buckling of Functionally Graded Graphene Platelet Reinforced Composite (FG-GPLRC) circular plates and shallow spherical caps using the first-order shear deformation theory (FSDT) is presented in this paper. The circular plates and shallow spherical caps are assumed to be subjected to uniformly distributed thermal loads. By applying the Galerkin method, the relations between thermal load–deflection are achieved to determine the postbuckling behavior and critical buckling loads of the considered structures. Special effects on the nonlinear thermal behavior of circular plates and shallow spherical caps with five different material distribution laws, different Graphene platelet (GPL) mass fractions, and geometrical dimensions are explored and discussed in numerical examples.

Nonlinear in-plane thermal buckling of rotationally restrained functionally graded carbon nanotube reinforced composite shallow arches under uniform radial loading

Nonlinear in-plane thermal buckling of rotationally restrained functionally graded carbon nanotube reinforced composite shallow arches under uniform radial loading

Nonlinear thermal buckling instability analysis of a rotating nanocomposite beam reinforced with graphene platelet via the Chebyshev–Ritz scheme

Resorting to the Chebyshev–Ritz scheme instability as well as post-instability response as a consequence of the buckling occurrence for a rotating nanocomposite beam in a uniform thermal environment is studied. The beam has been reinforced with graphene platelet. The Halpin–Tsai theory is utilized for the sake of specifying the elasticity modulus of the reinforced nanocomposite beam. The Chebyshev set of polynomial functions has been used to serve as a required independent set for the Ritz method. The static displacement field owing to the rotation of the beam and the thermal load is acquired by employing a potent algorithm. The outcomes indicate that for nanocomposite beams with the graphene platelet patterns in the model of O- and U-patterns rotating, respectively, below 9,326, and 7,420 rpm, the graphene reinforcing phase should not be used since it leads to a thermal buckling stiffness for the nanocomposite rotating beam smaller than the thermal buckling stiffness of the associated original rotating beam that just made of a host matrix. Moreover, developing the weight fraction of the graphene platelet for a U-type beam rotating below 9,628 rpm lessens the corresponding thermal buckling stiffness while an X-type beam always strengthens in a similar conditions.

On combination of all small-scale theories for nonlinear free vibrations and thermal buckling of nanobeams under thermal loading

Nonlinear free vibration characteristics of nanobeams exposed to thermal loading and surface effects are studied in this paper. Within the framework of the Euler–Bernoulli beam theory in conjunction with the nonlocal strain gradient theory, a size-dependent model for nanobeams subjected to thermal loadings has been developed with considering the von Kármán nonlinearity. In the current investigation, size dependencies are captured in the nonlinear free vibrations of nanobeams by using the strain gradient tensor and the curvature tensor. Moreover, the surface elasticity theory is employed to study the effects of surface elasticity on the dynamic behaviors of nanobeams. Governing equations of nonlocal strain gradient nanobeams are obtained using Hamilton’s principle. The influences of nonlocal parameter, length scale parameter resulting from the strain gradient theory, surface elasticity, and temperature on the nonlinear natural frequency of nanobeams are discussed in detail. Moreover, the hardening-softening behaviors of nanoscale beams are investigated in the presence of the curvature tensor. To validate the results of the proposed model for nanobeams, the results obtained in terms of natural frequencies are compared to those from available well-known references.

Study on Linear and Nonlinear Thermal Buckling Mode and Instability Characteristics for Engine Rotating Thin-Walled Blade

The thermal shock load has an important effect on the stability of thin-walled blades under high-speed operation of aircraft engines. According to the actual working conditions, the linear interpolation distribution of blade temperature is obtained by the numerical fitting method. A thermal buckling model is built to obtain the linear and nonlinear modal response of the blade through the finite element method. The results show that the blade stiffness changes under the influence of thermal buckling and the obvious torsional deformations are produced along the radial direction of the blade. Meanwhile, the largest deformation of about 1.3 mm and stress of 81 Mpa occurs on the blade tip for both the linear and nonlinear response. The buckling stress distribution and critical load factor of thermal buckling are also calculated, consistent with the rubbing part of blade. The changing radial length is the main reason for the distance reduction between the blade and casing, causing more probability of friction impact. Therefore, reasonable local thermal buckling technology is helpful to improve the design level of thermal-shock loaded blades.

A new method for suppressing nonlinear flutter and thermal buckling of composite lattice sandwich beams

A new method for suppressing nonlinear flutter and thermal buckling of composite lattice sandwich beams

Dynamic thermal buckling of spherical porous shells

The aim of present work is to address nonlinear dynamic thermal buckling of shallow spherical functionally graded porous shells subjected to transient thermal loading using the first order shear deformation theory (FSDT). A power-law distribution as well as cosine-type porosity distribution are used to model the variation of constituents through the shell thickness. Thermomechanical properties are assumed to be temperature dependent. Using Crank–Nicolson time marching scheme, an iterative procedure is employed to solve nonlinear transient heat conduction equation. For thermal boundary conditions, the outer surface of shells is kept at a reference temperature, while the inner surface experiences a sudden temperature rise. Geometrical type of nonlinearity in the sense of von-Karman is taken into account. The highly coupled nonlinear governing equations of motion are extracted by constructing the appropriate weak form and also using multi-term Ritz–Chebyshev method. The resulting ODEs are then reduced to a system of nonlinear algebraic equations by employing the well-known Newmark family of time integration schemes. The latter equations are solved by means of Newton–Raphson iteration procedure. Budiansky criterion is used to recognize critical parameters of dynamic instability of shells due to applied thermal shocks. Some comparison studies are conducted in order to verify the accuracy of results of the present work. Moreover, various parametric studies are performed to assess the influence of involved parameters.

Nonlinear thermal dynamic buckling and global optimization of smart sandwich plate with porous homogeneous core and carbon nanotube reinforced nanocomposite layers

The analytical investigation for the nonlinear thermal dynamic buckling of smart sandwich plate subjected to mechanical, thermal and electric loadings is presented. The sandwich plate is composed of a porous homogeneous core, two carbon nanotube reinforced composite (CNTRC) layers and two piezoelectric face sheets. Basic equations are derived based on the Reddy's higher order shear deformation plate theory and Hamilton's principle in which the initial imperfection and Pasternak-type elastic foundations are included. The external pressure is assumed to be uniformly distributed on the surface of the sandwich plate and depend on time according to the linear functions. The nonlinear dynamic response, the frequency – amplitude relation are obtained by using the Galerkin and Runge – Kutta methods and the critical dynamic buckling load is determined by using Budiansky – Roth criterion. Bees Algorithm is used to determine the maximum value of natural frequency of smart sandwich plate and the corresponding optimum values of geometrical and material parameters. The effects of geometrical parameters, CNT volume fraction, elastic foundations, temperature increment, initial imperfection and porosity coefficient on the nonlinear vibration and dynamic buckling of the smart sandwich plate are considered specifically. The numerical results are also compared with existing results using different theories.

Analyzing the Thermal Post-Buckling of Composite Plate Containing an Elliptical Cut-Out Using a Particle Semi-Energy Method

In this research, nonlinear thermal buckling and post-buckling behaviors of composite plate with the circular/elliptical cut-out are investigated using particle semi-energy (PSE) method. The semi-energy method is based on the solution of compatibility equation via an Airy force function and out-of-plane displacement function. The unknown parameters of these functions are determined by minimizing the potential energy. The integral of potential energy are replaced with summation of particles energy at perforated plate. The cut-out is modeled easily using these particles. The advantages of this method are easily cut-out modeling by particles and proposing just one of the displacement fields (i.e. out-of-plane). Based on the results, there is a good agreement (1.25%) between the post-buckling loads derived from PSE of this paper and experimental test of other literature. The accuracy of the finite element method (FEM) is 7.5% with respect to experimental test. The influences of temperature distribution, the cut-out size and elliptical cut-out rotating on post-buckling load and deflection of perforated plate are investigated.

Post-buckling analysis of imperfect nonlocal piezoelectric beams under magnetic field and thermal loading

An investigation of the nonlinear thermal buckling behavior of a nano-sized beam constructed from intelligent materials called piezo-magnetic materials has been presented in this article. The nano-sized beam geometry has been considered based on two assumptions: an ideal straight beam and an imperfect beam. For incorporating nano-size impacts, the nano-sized beam formulation has been presented according to nonlocal elasticity. After establishing the governing equations based on classic beam theory and nonlocal elasticity, the nonlinear buckling path has been obtained via Galerkin's method together with an analytical trend. The dependency of buckling path to piezo-magnetic material composition, electro-magnetic fields and geometry imperfectness has been studied in detail.

Nonlinear thermal buckling and postbuckling analysis of bidirectional functionally graded tapered microbeams based on Reddy beam theory

In the present study, the nonlinear thermal buckling, postbuckling, and snap-through phenomenon of higher-order shear deformable tapered bidirectional functionally graded (BDFG) microbeams are comprehensively investigated under different types of thermal loading, for the first time. The thermomechanical properties of the BDFG microbeam are assumed to be functions of temperature, axial, and thickness directions. Reddy’s parabolic shear deformation beam theory with the von Karman nonlinearity is employed to derive the variable coefficient governing nonlinear differential equations on the basis the physical neutral surface concept. Size-dependent effect is captured in the formulation employing the modified couple stress theory. The generalized differential quadrature method (GDQM) is used to discretize the motion equations considering different boundary conditions. The resulting system of nonlinear algebraic equations is solved iteratively using Newton’s method. Theoretical analysis and numerical results indicate that, depending on the shear deformation beam theory, boundary conditions, and the type of thermal load, the response of the BDFG microbeam may be of the bifurcation or snap-through buckling type of instability. Numerical parametric studies are conducted to explore the influences of thermal load type, material property gradient indexes, boundary conditions, material temperature dependency, taperness ratio, and microstructural length scale on critical thermal buckling load, thermal postbuckling, and equilibrium paths of the BDFG microbeam.

Nonlinear static and dynamic thermal buckling analysis of imperfect multilayer FG cylindrical shells with an FG porous core resting on nonlinear elastic foundation

In this article, a semi-analytical and analytical method for the nonlinear static and dynamic thermal buckling behavior of imperfect multilayer FG cylindrical shells with an FG porous core in the thermal environment is investigated. The structure is embedded within a generalized nonlinear elastic foundation which is composed of a two-parameter Winkler-Pasternak foundation augmented by a nonlinear cubic stiffness. Two types of multilayer FG cylindrical shells with an FG porosity core distribution consist of a uniform porosity distribution and a non-uniform porosity distribution core are considered. Using the Galerkin method with regard to the von Kármán equations, the discretized motion equation is obtained. The fourth-order Runge-Kutta method is utilized to obtain the nonlinear dynamic thermal buckling responses. The effects of various geometrical characteristics, material parameters, and elastic foundation coefficients are investigated on the nonlinear static thermal buckling and dynamic thermal buckling behavior of the multilayer FG system with an FG porous core. The results show that the various types of porosity core, imperfection and the elastic foundation parameters have strongly effect on the thermal buckling behaviors of the multilayer FG cylindrical shells with an FG porous core.

A new $${\varvec{n}}$$th-order shear deformation theory for isogeometric thermal buckling analysis of FGM plates with temperature-dependent material properties

This study presents an isogeometric analysis (IGA) for investigating the buckling behavior of functionally graded material (FGM) plates in thermal environments. The material properties of the FGM plate are considered to be graded across the thickness, and temperature dependency of the material properties is taken into account. A new nth-order shear deformation theory with the von Karman type of geometric nonlinearity, in which the optimum order number to best approximate the thermal buckling problem can be chosen, is developed. The principle of virtual work is used to derive the governing equations for the nonlinear thermal buckling analysis. The nth-order shear deformation theory is incorporated into the non-uniform rational B-spline-based IGA which fulfills the $$C^{1}$$ -continuity requirement of the proposed higher-order plate theory. The discrete nonlinear system equations are solved by utilizing the modified Newton–Raphson iterative technique. Parametric studies on the buckling behavior of FGM plates subjected to diverse through-thickness temperature variations are performed, and the influence of temperature dependency of the material properties is examined. Results validate the performance accuracy and effectiveness of the proposed IGA based on the nth-order shear deformation theory, and they demonstrate that temperature-dependent material properties should be included in the thermal buckling analysis.

Nonlinear Flutter Suppression and Thermal Buckling Elimination for Composite Lattice Sandwich Panels

A mechanism is proposed to suppress the nonlinear flutter and thermal buckling of composite lattice sandwich panels with tetrahedral truss core in supersonic airflow by means of the Winkler–Pasternak elastic foundation. The strain-displacement relations are evaluated according to the von Kármán large deflection and first-order shear deformation theories. The aerodynamic pressure acting on the sandwich panel is formulated by the supersonic piston theory. The nonlinear equations of motion of the structure are established by Hamilton’s principle in conjunction with the assumed mode method. By using the Winkler–Pasternak elastic foundation, an effective method for eliminating the nonlinear thermal buckling of the structure is presented. The natural frequencies of the sandwich panels are validated through comparing with the published results. The parametric investigations on the shearing layer and Winkler parameters are conducted. The influences of several parameters, including ply angle of laminated face sheets, and radius and material properties of the trusses on the nonlinear flutter, and thermal buckling behaviors are investigated. The numerical results attest that the nonlinear thermal buckling effect can be completely eliminated, and the nonlinear flutter of the composite lattice sandwich panels can be effectively suppressed by adjusting the elastic foundation parameters.

Nonlinear bending and thermal post-buckling behavior of functionally graded piezoelectric nanosize beams using a refined model

We in this paper analyze the problem of the nonlinear bending, thermal buckling and post-buckling of functionally graded piezoelectric material nanobeams based on Eringen’s nonlocal elasticity theory. The beams with immovable clamped ends are exposed to the external electric voltages, a uniform transverse load as well as uniform temperature change. A refined beam model which can transform the Reddy beam model into the Timoshenko beam model is introduced into this study so that we can distinguish the role of transverse shear deformation in complicated stress field. The governing equations are induced by using the generalized variation principle, and then the approximate analytical solution of the FGP material nanobeams for nonlinear, thermal buckling, post-buckling are obtained by using a two-step perturbation method. Subsequently, detailed parametric studies are carried out to get an insight into the effects of different physical parameters, including the slenderness ratio, small scale parameter, volume fraction index and external electric voltages.

Isogeometric analysis for nonlinear buckling of FGM plates under various types of thermal gradients

This study attempts at analyzing buckling responses of functionally graded material (FGM) plates under diverse types of thermal loadings. An accurate and effective numerical approach based on isogeometric analysis (IGA) to predict the nonlinear thermal buckling behavior is developed. A refined higher-order shear deformation theory (HSDT) which accounts for the geometric nonlinearity in the von Kármán sense is presented and used to derive the equilibrium and governing equations for FGM plate in thermal environments. Two different types of transverse shear functions are considered in the refined HSDT. IGA uses non-uniform rational B-spline (NURBS) basis functions which enable to accomplish easily the smoothness of arbitrary continuity order. Thus, the present method satisfies the C1-continuity of the displacement field required by the proposed HSDT. Several numerical examples involving buckling behavior of various kinds of FGM plates subjected to uniform, linear and nonlinear temperature distributions across the thickness are simulated, and the results are compared to the analytical solutions for the verification purpose. Parametric studies are also carried out on FGM plates under the different through-thickness temperature variations to scrutinize the thermal buckling features. Temperature gradient through the thickness by the one-dimensional (1-D) heat conduction and thermal profile developed by the three-dimensional (3-D) heat conduction are also taken into account. Results demonstrate that the proposed IGA method can be used as an accurate and effective numerical tool for analyzing the thermal buckling responses of FGM plates, and the 3-D heat conduction needs to be considered in the buckling analysis of FGM plates subjected to thermal conduction.

Nonlinear Static and Dynamic Thermal Buckling Analysis of Spiral Stiffened Functionally Graded Cylindrical Shells with Elastic Foundation

In this paper, an analytical method is used to study the nonlinear static and dynamic thermal buckling analysis of imperfect spiral stiffened functionally graded (SSFG) cylindrical shells. The SSFG cylindrical shell is surrounded by a linear and nonlinear elastic foundation. The proposed linear model is based on the two-parameter elastic foundation (Winkler and Pasternak). A three-parameter elastic foundation with hardening/softening cubic nonlinearity is used for nonlinear model. The material properties are temperature dependent and assumed to be continuously graded in the thickness direction. Also, for thermal buckling analysis, the uniform and linear temperature distribution in thickness direction is considered. The SSFG cylindrical shells are considered with various angles for spiral stiffeners. The strain–displacement relations are obtained based on the von Kármán nonlinear equations and the classical plate theory of shells. The smeared stiffener technique and the Galerkin method are applied to solve the nonlinear problem. In order to find the nonlinear dynamic thermal buckling responses, the fourth-order Runge–Kutta method is used. To validate the results, comparisons are made with those available in literature and good agreements are shown. The effects of various geometrical and material parameters are investigated on the nonlinear static and dynamic thermal buckling response of SSFG cylindrical shells.