Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory

In this research, thermal buckling of circular plates compose of functionally graded material (FGM) is considered. Equilibrium and stability equations of a FGM circular plate under thermal loads are derived, based on the higher order shear deformation plate theory (3rd order plate theory). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. A buckling analysis of a functionally graded circular plate (FGCP) under various types of thermal loads is carried out and the result are given in closed-form solutions. The results are compared with the critical buckling temperature obtained for FGCP based on first order (1st order plate theory) and classical plate theory (0 order plate theory) given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of FGCP, whereas the first order and classical plate theory overestimates buckling temperature.     

Sound radiation of a functionally graded material cylindrical shell in water by mobility method

Based on the fundamental dynamic equations of functionally graded material (FGM) cylindrical shell, this paper investigates the sound radiation of vibrational FGM shell in water by mobility method. This model takes into account the exterior fluid loading due to the sound press radiated by the FGM shell. The FGM cylindrical shell was excited by a harmonic line radial force uniformly distributing along the generator. The FGM shell equations of motion, the Helmholtz equation in the exterior fluid medium and the continuity equation at fluid-shell interface are used in this vibroacoustic problem. The expressions of sound radiation efficiency and sound field of the FGM shell have been derived by mobility method. Radiation efficiency, modal mobility and the directivity pattern of the sound field are solved numerically. In particular, radiation efficiency and directivity pattern with various power law index are analyzed.     

Coupled thermoelasticity of functionally graded cylindrical shells

The coupled thermoelastic response of a functionally graded circular cylindrical shell is studied. The coupled thermoelastic and the energy equations are simultaneously solved for a functionally graded axisymmetric cylindrical shell subjected to thermal shock load. A second-order shear deformation shell theory that accounts for the transverse shear strains and rotations is considered. Including the thermo-mechanical coupling and rotary inertia, a Galerkin finite element formulation in space domain and the Laplace transform in time domain are used to formulate the problem. The inverse Laplace transform is obtained using a numerical algorithm. The shell is graded through the thickness assuming a volume fraction of metal and ceramic, using a power law distribution. The results are validated with the known data in the literature.     

Natural frequencies of rotating functionally graded cylindrical shells

Love’s first approximation theory is used to analyze the natural frequencies of rotating functionally graded cylindrical shells.To verify the validity of the present method,the natural frequencies of the simply supported non-rotating isotropic cylindrical shell and the functionally graded cylindrical shell are compared with available published results.Good agreement is obtained.The effects of the power law index,the wave numbers along the x-and θ-directions,and the thickness-to-radius ratio on the natural frequencies of the simply supported rotating functionally graded cylindrical shell are investigated by several numerical examples.It is found that the fundamental frequencies of the backward waves increase with the increasing rotating speed,the fundamental frequencies of the forward waves decrease with the increasing rotating speed,and the forward and backward waves frequencies increase with the increasing thickness-to-radius ratio.     

Stress analysis of functionally graded shells using a 7-parameter shell element

In this paper, finite element stress analysis of functionally graded structures using a high-order spectral/hp shell finite element is presented. The shell element is based on a seven-parameter first-order shear deformation theory in which the seventh parameter, in addition to the usual six degrees of freedom, is the thickness stretch. The continuum shell element is utilized for the numerical simulations of the fully geometrically nonlinear response of functionally graded elastic shell structures. Several nontrivial shell problems are considered to report deflections and stresses, the latter being the main focus of the current paper. It is found that the stresses computed in the current study agree only in some cases with those of ANSYS and/or ABAQUS and thus requires additional study to determine the cause of the disagreement.     

High-order shear theory for static analysis of functionally graded plates with porosities

The bending responses of porous functionally graded (FG) thick rectangular plates are investigated according to a high-order shear deformation theory. Both the effect of shear strain and normal deformation are included in the present theory and so it does not need any shear correction factor. The equilibrium equations according to the porous FG plates are derived. The solution to the problem is derived by using Navier's technique. Numerical results have been reported and compared with those available in the open literature for non-porous plates. The effects of the exponent graded and porosity factors are investigated.     

功能梯度圆柱壳-内声场耦合系统振声预测

针对功能梯度(FGM)圆柱壳与其内声场的振声特性分析需求,采用特征正交-里兹法研究圆柱壳厚度、材料梯度指数、边界支撑刚度、阻抗边界和简谐力位置等因素对振声系统的影响。圆柱壳的位移及内部声压变量通过特征正交多项式与周向傅里叶谐波函数构建,并以声压做功的方式考虑结构与声场之间耦合条件,进而结合里兹法得到耦合系统的振声模态和稳态响应。研究表明,由该模型所得到的数值解与文献及有限元结果具有较好的一致性。对参数的研究发现,壳体厚度、梯度指数、边界刚度、阻抗边界和激励位置等对振声系统的耦合特性均有较为明显影响,这为该类结构的低噪声设计及其内部声场优化提供了有益的参考。     

Analytical solution for the electromechanical behavior of a rotating functionally graded piezoelectric hollow shaft

In the present paper, the analytical solution for a radially piezoelectric functionally graded rotating hollow shaft is presented. The variation of material properties is assumed to follow a power law along the radial direction of the shaft. Two resulting fully coupled differential equations in terms of the displacement and electric potential are solved directly. Numerical results for different shaft geometries with different profiles of inhomogeneity are also graphically displayed.     

Torsional impact of transversely isotropic solid with functionally graded shear moduli and a penny-shaped crack

The torsional impact response of a penny-shaped crack in an unbounded transversely isotropic solid is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace transform and Hankel transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress fields are obtained. Investigated are the influence of material nonhomogeneity and orthotropy on the dynamic stress intensity factor. The peak value of the dynamic stress intensity factor can be suppressed by increasing the shear moduli's gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface.     

Exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder

We find closed-form solutions for axisymmetric plane strain deformations of a functionally graded circular cylinder comprised of an isotropic and incompressible second-order elastic material with moduli varying only in the radial direction. Cylinder's inner and outer surfaces are loaded by hydrostatic pressures. These solutions are specialized to cases where only one of the two surfaces is loaded. It is found that for a linear through-the-thickness variation of the elastic moduli, the hoop stress for the first-order solution (or in a cylinder comprised of a linear elastic material) is a constant but that for the second-order solution varies through the thickness. The radial displacement, the radial stress and the hoop stress do not depend upon the second-order elastic constant but the hydrostatic pressure and hence the axial stress depends upon it. When the two elastic moduli vary as the radius raised to the power two or four, the radial and the hoop stresses in an infinite space with a pressurized cylindrical cavity equal the pressure in the cavity. For an affine variation of the elastic moduli, the hoop stress in an internally loaded cylinder made of a linear elastic isotropic and incompressible material at the point is the same as that in a homogeneous cylinder. Here R_in and R_ou equal, respectively, the inner and the outer radius of the undeformed cylinder and R the radial coordinate of a point in the unstressed reference configuration.     

Nonlinear delamination buckling and expansion of functionally graded laminated piezoelectric composite shells

In this paper, an analytical method is presented to investigate the nonlinear buckling and expansion behaviors of local delaminations near the surface of functionally graded laminated piezoelectric composite shells subjected to the thermal, electrical and mechanical loads, where the mid-plane nonlinear geometrical relation of delaminations is considered. In examples, the effects of thermal loading, electric field strength, the stacking patterns of functionally graded laminated piezoelectric composite shells and the patterns of delaminations on the critical axial loading of locally delaminated buckling are described and discussed. Finally, the possible growth directions of local buckling for delaminated sub-shells are described by calculating the expanding forces along the length and short axis of the delaminated sub-shells.     

Nonlinear vibration of functionally graded graphene platelet-reinforced composite truncated conical shell using first-order shear deformation theory

In this study, the first-order shear deformation theory(FSDT) is used to establish a nonlinear dynamic model for a conical shell truncated by a functionally graded graphene platelet-reinforced composite(FG-GPLRC). The vibration analyses of the FG-GPLRC truncated conical shell are presented. Considering the graphene platelets(GPLs) of the FG-GPLRC truncated conical shell with three different distribution patterns, the modified Halpin-Tsai model is used to calculate the effective Young's modulus. Hamilton's principle, the FSDT, and the von-Karman type nonlinear geometric relationships are used to derive a system of partial differential governing equations of the FG-GPLRC truncated conical shell. The Galerkin method is used to obtain the ordinary differential equations of the truncated conical shell. Then, the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by the harmonic balance method. The effects of the weight fraction and distribution pattern of the GPLs, the ratio of the length to the radius as well as the ratio of the radius to the thickness of the FG-GPLRC truncated conical shell on the nonlinear natural frequency characteristics are discussed. This study culminates in the discovery of the periodic motion and chaotic motion of the FG-GPLRC truncated conical shell.     

Identifying the temperature effect on the vibrations of functionally graded cylindrical shells with porosities

The free thermal vibration of functionally graded material (FGM) cylindrical shells containing porosities is investigated. Both even distribution and uneven distribution are taken into account. In addition, three thermal load types, i.e., uniform temperature rise (UTR), nonlinear temperature rise (NLTR), and linear temperature rise (LTR), are researched to explore their effects on the vibration characteristics of porous FGM cylindrical shells. A modified power-law formulation is used to describe the material properties of FGM shells in the thickness direction. Love’s shell theory is used to formulate the strain-displacement equations, and the Rayleigh-Ritz method is utilized to calculate the natural frequencies of the system. The results show that the natural frequencies are affected by the porosity volume fraction, constituent volume fraction, and thermal load. Moreover, the natural frequencies obtained from the LTR have insignificant differences compared with those from the NLTR. Due to the calculation complexity of the NLTR, we propose that it is reasonable to replace it by its linear counterpart for the analysis of thin porous FGM cylindrical shells. The present results are verified in comparison with the published ones in the literature.     

Two-dimensional thermoelastic analysis of a functionally graded cylindrical panel due to nonuniform heat supply

This paper is concerned with the theoretical treatment of the steady-state thermoelastic problem of a functionally graded cylindrical panel due to nonuniform heat supply in the circumferential direction. The thermal and thermoelastic constants of the cylindrical panel are expressed as power functions of the radial coordinate. We obtain the exact solution for the two-dimensional temperature change in a steady state, and thermal stresses of a simple supported cylindrical panel under the state of plane strain. Some numerical results are shown in figures and tables. Furthermore, the influence of the nonhomogeneity of the material, the radius ratio and the span angle upon the temperature change, displacements and stresses is investigated.     

四边简支功能梯度圆锥壳的非线性振动分析

研究了四边简支条件下功能梯度圆锥壳的非线性自由振动。首先,通过Voigt模型和幂律分布模型描述了功能梯度材料的物理属性。然后,考虑von-Karman几何非线性建立了功能梯度圆锥壳的能量表达式,利用Hamilton原理推出圆锥壳的运动方程。在此基础上,采用Galerkin法,只考虑横向振动,功能梯度圆锥壳运动方程可简化为单自由度非线性振动微分方程。最后,通过改进的L-P法和Runge-Kutta法求解非线性振动方程,讨论功能梯度圆锥壳的非线性振动响应,分析几何参数和陶瓷体积分数指数对圆锥壳非线性频率响应的影响。结果表明,几何参数对非线性频率和响应的影响相较于陶瓷体积分数指数更明显;圆锥壳的几何参数和陶瓷体积分数指数通过改变非线性频率影响振动响应;功能梯度圆锥壳呈弹簧渐硬非线性振动特性。     

Non-linear analysis of functionally graded plates under transverse and in-plane loads

Large deflection and postbuckling responses of functionally graded rectangular plates under transverse and in-plane loads are investigated by using a semi-analytical approach. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The plate is assumed to be clamped on two opposite edges and the remaining two edges may be simply supported or clamped or may have elastic rotational edge constraints. The formulations are based on the classical plate theory, accounting for the plate-foundation interaction effects by a two-parameter model (Pasternak-type), from which Winkler elastic foundation can be treated as a limiting case. A perturbation technique in conjunction with one-dimensional differential quadrature approximation and Galerkin procedure are employed in the present analysis. The numerical illustrations concern the large deflection and postbuckling behavior of functional graded plates with two pairs of constituent materials. Effects played by volume fraction, the character of boundary conditions, plate aspect ratio, foundation stiffness, initial compressive stress as well as initial transverse pressure are studied.     

Interfacial fracture analysis of bonded dissimilar strips with a functionally graded interlayer under antiplane deformation

This paper provides the solution to the problem of dissimilar, homogeneous semi-infinite strips bonded through a functionally graded interlayer and weakened by an embedded or edge interfacial crack. The bonded system is assumed to be under antiplane deformation, subjected to either traction-free or clamped boundary conditions along its bounding planes. Based on the Fourier integral transform, the problem is formulated in terms of a singular integral equation which has a simple Cauchy kernel for the embedded crack and a generalized Cauchy kernel for the edge crack. In the numerical results, the effects of geometric and material parameters of the bonded system on the crack-tip stress intensity factors are presented in order to quantify the interfacial fracture behavior in the presence of the graded interlayer.     

Geometric nonlinear analyses of functionally graded beams using a tailored Lagrangian formulation

This paper presents a geometric nonlinear analysis formulation for beams of functionally graded cross-sections by means of a Total Lagrangian formulation. The influence of material gradation on the numerical response is investigated in detail. Two examples are given that illustrate the main features of the formulation, in which the behavior of beams of graded cross-sections is compared with homogeneous material beams. A motivation for this work is the potential development of functionally graded risers for the offshore oil exploration industry.     

A catastrophe model for the impact torsional buckling of elastic cylindrical shell

The catastrophe theory is used to study the impact buckling of elastic structures. A criterion for impact buckling is established based on the proposed catastrophe system, in whose bifurcation set the critical step load is located. By the present theory, the impact torsional buckling for a clamped cylindrical shell is studied and the critical step torques for different imperfections are given. Also, the static critical torque is given, and it is shown that the critical step torque is smaller than the critical static torque.The project is supported by National Natural Science Foundation of China     

金属圆柱壳受大质量低速冲击的屈曲变形

对钢质和铜质金属圆柱壳的轴向冲击动力响应进行了实验研究,记录了两种不同材料圆柱壳在大质量低速冲击下的冲击力时程曲线,得到其屈曲模态。采用高速摄像及模拟技术给出了钢质圆柱壳渐进屈曲的全过程,为理解钢质圆柱壳的屈曲机理提供了直观的结果。黄铜质圆柱壳在大质量低速冲击下, 出现整个壳面滿布屈曲波纹的塑性动力屈曲现象,说明高速冲击不是产生塑性动力屈曲的充要条件。像铜这样具有高密度的韧性材料,在大质量低速冲击下,会在轴向产生持续的压缩塑性流作用而出现塑性动力屈曲现象。