基于n阶GBT热-机载荷作用下FGM梁的振动特性和稳定性分析

研究了热-机载荷耦合作用下弹性地基FGM梁的振动特性与稳定性。考虑到材料的物性依赖于温度变化且组分沿梁厚按幂律分布。首先,基于一种扩展的n阶广义剪切变形梁理论(n-th GBT),应用Hamilton原理,统一建立了系统自由振动及屈曲问题力学模型的控制方程,采用一种改进型广义微分求积法(MGDQ)获得FGM梁静动态响应的数值解。其次,通过算例验证GBT的有效性并给出阶次n的理想取值,在丰富梁理论的同时,也可验证或改进其他各种剪切变形梁理论。最后,讨论并分析了升温、边界条件、初始轴向机械载荷、梯度指标、地基刚度、跨厚比等诸多参数对FGM梁振动特性和稳定性的影响。     

湿-热-机-弹耦合FGM梁的稳定性及振动特性

研究了初始轴向机械载荷作用下Winkler-Pasternak弹性地基上功能梯度材料（FGM）梁在湿-热环境中的稳定性及振动特性。假设温度和湿度沿梁厚度方向稳态分布,材料的物性依赖于温度且按Voigt混合幂律模型连续分布。首先,基于一种扩展的n阶广义梁理论,应用Hamilton原理,统一建立了以轴向位移、弯曲变形项挠度及剪切变形项挠度为基本未知函数FGM梁的屈曲及自由振动方程,采用Navier解法获得了FGM简支梁静动态响应的精确解。其次,通过算例验证并给出了该广义梁理论阶次n的理想取值,丰富梁理论的同时,可供验证或改进其他各种剪切变形梁理论。最后,着重探讨了3种湿-热分布下湿度与温度增加、初始轴向机械载荷、跨厚比、地基刚度、梯度指标等诸多参数对FGM梁稳定性和振动特性的影响。     

湿-热-机耦合作用下多孔功能梯度梁的振动及屈曲特性

采用一种拓展的n阶广义梁理论（GBT）,研究了轴向机械载荷作用下多孔功能梯度材料（FGM）梁在湿热环境中的振动及屈曲特性。考虑了材料的物性随温度变化,湿-热沿梁厚按三种不同类型分布,采用含孔隙率的修正Voigt混合率模型描述多孔功能梯度梁的材料属性,在宏-细观力学模型框架下应用Hamilton原理统一建立了系统的自由振动及屈曲方程,采用Navier法求解FGM简支梁的静动态响应。通过算例验证并讨论了GBT阶数n的理想取值,可用于丰富梁理论。探讨了湿热效应、湿-热-机耦合、孔隙率、材料梯度指标、跨厚比对FGM梁振动及屈曲特性的影响。结果表明:湿-热加剧降低了FGM梁的频率和临界载荷,且不同类型的湿热分布对其减小程度有显著差异;随着孔隙率增大,梁结构的整体刚度虽有所弱化,但在湿热环境中频率反而增大,稳定性增强;湿-热效应对多孔FGM细长梁频率和稳定性影响十分显著,但对短粗梁的影响比较有限。      

基于二元耦联性解耦下多孔FGM梁的热-力耦合振动与屈曲特性

采用一种改进型广义微分求积(MGDQ)法,数值研究了初始轴向机械力作用下含均匀孔隙的功能梯度材料(FGM)梁在热环境中的耦合振动及耦合屈曲特性。考虑了材料性质随温度的相关性,温度沿梁的厚度方向按不同类型稳态分布,采用含孔隙率修正的Voigt混合幂率模型来表征多孔FGM梁的材料属性。采用一种n阶广义梁理论(GBT),在Hamilton体系下统一建立描述该系统耦合振动及屈曲问题力学模型的控制方程。通过引入边界控制参数,可实施3种典型边界梁动态响应MGDQ法求解的MATLAB统一化编程。基于两种静动态力学行为之间的二元耦联性,编写循环子程序用来获得屈曲静态响应,该分析方法极大地简化了解耦过程并提高了计算效率。通过算例主要探究了梁理论、边界条件、温度分布、升温、初始轴向机械力、热-力耦合效应、孔隙率、梯度指标、跨厚比等诸多参数对多孔FGM梁振动及屈曲特性的影响,同时刻画并揭示了两种静动态力学行为之间的二元耦联性。     

Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory

This paper proposes a new higher-order shear deformation theory for buckling and free vibration analysis of isotropic and functionally graded (FG) sandwich beams. The present theory accounts a new hyperbolic distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion are derived from Lagrange's equations. Analytical solutions are presented for the isotropic and FG sandwich beams with various boundary conditions. Numerical results for natural frequencies and critical buckling loads obtained using the present theory are compared with those obtained using the higher and first-order shear deformation beam theories. Effects of the boundary conditions, power-law index, span-to-depth ratio and skin-core-skin thickness ratios on the critical buckling loads and natural frequencies of the FG beams are discussed.     

A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities

This article proposes a four-variable shear deformation refined beam theory for thermo-mechanical vibration characteristics of porous, functionally graded (FG) beams exposed to various kinds of thermal loadings by using an analytical method. Thermo-mechanical properties of functionally graded material (FGM) beams are supposed to vary through the thickness direction, and are estimated through the modified power-law rule in which the porosities with even and uneven types are approximated. The material properties of FGM beams are supposed to be temperature dependent. Porosities possibly occur inside FGMs during fabrication because of technical problems that lead to the creation of microvoids in these materials. The variation of pores along the thickness direction influences the mechanical properties. Thus, it is incumbent to predict the effect of porosities on the thermo-mechanical vibration behavior of FG beam in the present study. Four types of thermal loading, namely, uniform, linear, nonlinear, and sinusoidal temperature rises through the z-axis direction are discussed. The governing differential equations and boundary conditions of FG porous beams subjected to thermal loadings are formulated through Hamilton's principle, based on a four-variable refined theory that considers a constant transverse displacement and higher order variation of axial displacement through the depth of the beam without the need of any shear correction factors. An analytical solution procedure is used to achieve the natural frequencies of porous FG beams subjected to various temperature fields. The impact of several specific parameters such as power-law exponent, porosity volume fraction, different porosity distribution, and thermal effect on the vibration of the porous FG beams is perused and discussed in detail. It is deduced that these parameters play a notable role on the thermo-dynamic behavior of porous FG beams. Presented numerical results can serve as benchmarks for the future analyses of FG beams with porosity phases.     

基于n阶GBT初始轴向载荷影响下FGM梁的振动特性

研究了初始轴向载荷影响下弹性地基功能梯度材料(FGM)梁的振动特性。基于一种拓展的n阶广义剪切变形梁理论(n-GBT),以轴向位移、剪切变形挠度与弯曲变形挠度为基本未知函数,应用Hamilton原理,建立了该系统自由振动问题力学模型的控制方程。引入边界控制参数,采用一种改进型广义微分求积(MGDQ)法获得了FGM梁的静动态响应。通过算例验证并给出了GBT阶次n的理想取值,丰富梁理论的同时,可供验证或改进其它各种剪切变形梁理论;提供的数值分析方法切实可行,拓展了GDQ法的使用范围。最后,着重讨论并分析了初始轴向载荷、边界条件、梯度指标、地基刚度、跨厚比等参数对FGM梁振动特性的影响。     

含裂纹功能梯度Euler-Bernoulli梁和Timoshenko梁的屈曲载荷计算与分析

含裂纹构件的屈曲载荷是结构是否安全的判定准则之一, 其计算与分析也是结构健康监测和安全评价中关注的重要问题。基于Euler-Bernoulli梁理论和Timoshenko梁理论, 建立了一种求解含裂纹功能梯度材料梁的屈曲载荷计算方法。首先裂纹导致的构件截面转角不连续性由转动弹簧模型进行模拟, 再根据功能梯度材料Euler-Bernoulli梁和Timoshenko梁的屈曲控制方程及其闭合解, 由传递矩阵法建立了求解含裂纹功能梯度材料梁在多种边界条件下屈曲载荷的循环递推公式和特征行列式, 使问题通过降阶的方法得到快速准确的解答。数值算例研究了剪切变形、 裂纹的不同数目及位置、 材料参数变化、 长细比和不同边界约束条件等对含裂纹功能梯度材料梁屈曲载荷的影响。结果表明该方法可以简单、 方便和准确地计算不同数目裂纹和任意边界条件下功能梯度材料梁的屈曲问题。      

A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading

Due to the variation in material properties through the thickness, bifurcation buckling cannot generally occur for plates or beams made of functionally graded materials (FGM) with simply supported edges. Further investigation in this paper indicates that FGM beams subjected to an in-plane thermal loading do exhibit some unique and interesting characteristics in both static and dynamic behaviors, particularly when effects of transverse shear deformation and the temperature-dependent material properties are simultaneously taken into account. In the analysis, based on the nonlinear first-order shear deformation beam theory (FBT) and the physical neutral surface concept, governing equations for both the static behavior and the dynamic response of FGM beams subjected to uniform in-plane thermal loading are derived. Then, a shooting method is employed to numerically solve the resulting equations. The material properties of the beams are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and to be temperature-dependent. The effects of material constants, transverse shear deformation, temperature-dependent material properties, in-plane loading and boundary conditions on the nonlinear behavior of FGM beams are discussed in detail.     

An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates

In this paper, an efficient and simple higher order shear and normal deformation theory is presented for functionally graded material (FGM) plates. By dividing the transverse displacement into bending, shear and thickness stretching parts, the number of unknowns and governing equations for the present theory is reduced, significantly facilitating engineering analysis. Indeed, the number of unknown functions involved in the present theory is only five, as opposed to six or even greater numbers in the case of other shear and normal deformation theories. The present theory accounts for both shear deformation and thickness stretching effects by a hyperbolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The obtained results are compared with 3-dimensional and quasi-3-dimensional solutions and those predicted by other plate theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of functionally graded plates.     

Bending,buckling and vibration of size-dependent functionally graded annular microplates

The bending, buckling and free vibration of annular microplates made of functionally graded materials (FGMs) are investigated in this paper based on the modified couple stress theory and Mindlin plate theory. This microplate model incorporates the material length scale parameter that can capture the size effect in FGMs. The material properties of the FGM microplates are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique. The higher-order governing equations and boundary conditions are derived by using Hamilton’s principle. The differential quadrature (DQ) method is employed to discretize the governing equations and to determine the deflection, critical buckling load and natural frequencies of FGM microplates. A parametric study is then conducted to investigate the influences of the length scale parameter, gradient index and inner-to-outer radius ratio on the bending, buckling and vibration characteristics of FGM microplates with hinged–hinged and clamped–clamped supports. The results show that the size effect on the bending, buckling and vibration characteristics is significant when the ratio of the microplate thickness to the material length scale parameter is smaller than 10.     

非均匀热载荷作用下功能梯度梁的非线性静态响应

由于功能梯度材料结构沿厚度方向的非均匀材料特性,使得夹紧和简支条件的功能梯度梁有着相当不同的行为特征。该文给出了热载荷作用下,功能梯度梁非线性静态响应的精确解。基于非线性经典梁理论和物理中面的概念导出了功能梯度梁的非线性控制方程。将两个方程化简为一个四阶积分-微分方程。对于两端夹紧的功能梯度梁,其方程和相应的边界条件构成微分特征值问题;但对于两端简支的功能梯度梁,由于非齐次边界条件,将不会得到一个特征值问题。导致了夹紧与简支的功能梯度梁有着完全不同的行为特征。直接求解该积分-微分方程,得到了梁过屈曲和弯曲变形的闭合形式解。利用这个解可以分析梁的屈曲、过屈曲和非线性弯曲等非线性变形现象。最后,利用数值结果研究了材料梯度性质和热载荷对功能梯度梁非线性静态响应的影响。     

Free vibration analysis of simply supported sandwich beams with lattice truss core

Free vibration of AISI 304 stainless steel sandwich beams with pyramidal truss core is investigated in the present paper. The lattice truss core is transformed to a continuous homogeneous material. Considering the deformation characteristics of the sandwich beam, the following assumptions are made: (1) the thickness of the sandwich beam remains constant during deformation; (2) for the thin face sheets, only bending deformation is considered, neglecting the effect of transverse shear deformation; (3) for the core, only shear deformation is considered as the core is too weak to provide a significant contribution to the bending stiffness of the sandwich beam. The shear stress is assumed to be constant along the thickness of the core. The governing equation of free vibration is derived from Hamilton's principle, and the natural frequencies are calculated under simply supported boundary conditions. Finally, numerical simulation is carried out to get the mode shapes and natural frequencies. Our results show that the theoretical solutions agree well with the numerical results. It indicates the present method would be useful for free vibration analysis of sandwich beams with lattice truss core.     

Thermo-electrical buckling of piezoelectric functionally graded material Timoshenko beams

In this article, buckling analysis of functionally graded material (FGM) beams with or without surface-bonded piezoelectric layers subjected to both thermal loading and constant voltage is studied. Thermal and mechanical properties of FGM layer is assumed to follow the power law distribution in thickness direction, except Poisson’s ratio which is considered constant. The Timoshenko beam theory and nonlinear strain-displacement relations are used to obtain the governing equations of piezoelectric FGM beam. Beam is assumed under three types of thermal loading and various types of boundary conditions. For each case of boundary conditions, existence of bifurcation-type buckling is examined and for each case of thermal loading and boundary conditions, closed-form solutions are obtained which are easily usable for engineers and designers. The effects of the applied actuator voltage, beam geometry, boundary conditions, and power law index of FGM beam on critical buckling temperature difference are examined.     

Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory

Investigated herein is the free vibration characteristics of microbeams made of functionally graded materials (FGMs) based on the strain gradient Timoshenko beam theory. The material properties of the functionally graded beams are assumed to be graded in the thickness direction according to the Mori–Tanaka scheme. Using Hamilton’s principle, the equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FGM microbeams including size effect. A detailed parametric study is performed to indicate the influences of beam thickness, dimensionless length scale parameter, and slenderness ratio on the natural frequencies of FGM microbeams. Moreover, a comparison between the various beam models on the basis of the classical theory (CT), modified couple stress theory (MCST), and strain gradient theory (SGT) is presented for different values of material property gradient index. It is observed that the value of gradient index play an important role in the vibrational response of the microbeams of lower slenderness ratios. It is further observed that by increasing the length-to-thickness ratio of the microbeam, the value of dimensionless natural frequency tends to decrease for all amounts of the gradient index.     

Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory

Investigated herein is the free vibration characteristics of microbeams made of functionally graded materials (FGMs) based on the strain gradient Timoshenko beam theory. The material properties of the functionally graded beams are assumed to be graded in the thickness direction according to the Mori–Tanaka scheme. Using Hamilton’s principle, the equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FGM microbeams including size effect. A detailed parametric study is performed to indicate the influences of beam thickness, dimensionless length scale parameter, and slenderness ratio on the natural frequencies of FGM microbeams. Moreover, a comparison between the various beam models on the basis of the classical theory (CT), modified couple stress theory (MCST), and strain gradient theory (SGT) is presented for different values of material property gradient index. It is observed that the value of gradient index play an important role in the vibrational response of the microbeams of lower slenderness ratios. It is further observed that by increasing the length-to-thickness ratio of the microbeam, the value of dimensionless natural frequency tends to decrease for all amounts of the gradient index.     

Free vibration analysis of pre-twisted rotating FGM beams

The natural frequencies of vibration of a rotating pre-twisted functionally graded cantilever beam are investigated. Rotating cantilever beam with pre-twist made of a functionally gradient material (FGM) consisting of metal and ceramic is considered for the study. The material properties of the FGM beam symmetrically vary continuously in thickness direction from core at mid section to the outer surfaces according to a power-law form. Equations of motion for free vibration are derived using Lagrange’s equation and the natural frequencies are determined using Rayleigh–Ritz method. The effect of parameters such as the pre-twist angle, power law index, hub radius and rotational speed on the natural frequencies of rotating functionally graded pre-twisted cantilever beams are examined through numerical studies and comparison is made with the numerical results obtained using other methods reported in literature. The effect of coupling between chordwise and flapwise bending modes on the natural frequencies has also been investigated.     

Effect of three-parameter viscoelastic medium on vibration behavior of temperature-dependent non-homogeneous viscoelastic nanobeams in a hygro-thermal environment

In this article, free vibration of functionally graded (FG) viscoelastic nanobeams resting on viscoelastic foundation subjected to hygrothermal loading is investigated employing a higher order refined beam theory which captures shear deformation influences needless of any shear correction factor. The three-parameter viscoelastic medium consists of parallel springs and dashpots as well as a shear layer. Temperature-dependent material properties of FGM beam are graded across the thickness via the power-law model. Employing non-local elasticity theory of Eringen and Hamilton's principle, non-local governing equations of a size-dependent viscoelastic nanobeam are obtained and solved analytically for various boundary conditions. To verify the reliability of the developed model, the results of the current work are compared with those available in literature. The effects of viscoelastic foundation parameters, internal damping coefficient, hygrothermal loading, non-local parameter, gradient index, mode number, and slenderness ratio on the vibrational characteristics of nanoscale viscoelastic FG beams are explored.     

An assessment of several displacement-based theories for the vibration and stability analysis of laminated composite and sandwich beams

Several displacement-based theories are assessed by analyzing the free vibration and the buckling behaviors of laminated beams with arbitrary layouts as well as soft-core sandwich beams. The equations governing the dynamic response of laminated structures are derived by using Hamilton’s principle. However, equations of equilibrium for buckling problems are given by employing the principle of virtual displacements. Moreover, using Navier’s technique and solving the eigenvalue equations, analytical solutions based on the global–local higher-order theory used in this paper are first presented in present study. At the same time, the effect of the order number of higher-order shear deformation as well as interlaminar continuity of transverse shear stress on the global response of both laminated beams and soft-core sandwiches has been also studied. Numerical results show that by increasing the order number of in-plane and transverse displacement components, the global higher-order theories can reasonably predict the natural frequencies and the critical loads of laminated beams with arbitrary layouts and soft-core sandwich beams whereas these global higher-order theories are still less accurate compared to the global–local higher-order theory and the zig-zag theory used herein.     

Thermoelasticity analysis of functionally graded beam with integrated surface piezoelectric layers

This paper presents analytical solution for functionally graded material (FGM) beams integrated with piezoelectric actuator and sensor under an applied electric field and thermo-mechanical load. In FGM host properties is assumed to vary exponentially in thickness direction and the Poisson’s ratio is held constant. The hybrid beam is in a state of plane stress and the piezoelectric is composed of orthotropic materials. The beam is simply supported with the bottom surface traction free and zero temperature. By using of state-space method in thickness direction and Fourier series in longitudinal direction, the solution can be made. To verify the accuracy of the present formulation, numerical results for the simple case is compared with results obtained in the published literature. Finally, effects of FGM index, electromechanical coupling, thickness ratio and thermo-mechanical surface boundary condition on the bending behaviour of beam are investigated.