Dynamic stability analysis of porous functionally graded beams under hygro-thermal loading using nonlocal strain gradient integral model

We present a study on the dynamic stability of porous functionally graded(PFG)beams under hygro-thermal loading.The variations of the properties of the beams across the beam thicknesses are described by the power-law model.Unlike most studies on this topic,we consider both the bending deformation of the beams and the hygro-thermal load as size-dependent,simultaneously,by adopting the equivalent differential forms of the well-posed nonlocal strain gradient integral theory(NSGIT)which are strictly equipped with a set of constitutive boundary conditions(CBCs),and through which both the stiffness-hardening and stiffness-softening effects of the structures can be observed with the length-scale parameters changed.All the variables presented in the differential problem formulation are discretized.The numerical solution of the dynamic instability region(DIR)of various bounded beams is then developed via the generalized differential quadrature method(GDQM).After verifying the present formulation and results,we examine the effects of different parameters such as the nonlocal/gradient length-scale parameters,the static force factor,the functionally graded(FG)parameter,and the porosity parameter on the DIR.Furthermore,the influence of considering the size-dependent hygro-thermal load is also presented.

Complete solutions for elastic fields induced by point load vector in functionally graded material model with transverse isotropy

The paper develops and examines the complete solutions for the elastic field induced by the point load vector in a general functionally graded material(FGM)model with transverse isotropy.The FGMs are approximated with n-layered materials.Each of the n-layered materials is homogeneous and transversely isotropic.The complete solutions of the displacement and stress fields are explicitly expressed in the forms of fifteen classical Hankel transform integrals with ten kernel functions.The ten kernel functions are explicitly expressed in the forms of backward transfer matrices and have clear mathematical properties.The singular terms of the complete solutions are analytically isolated and expressed in exact closed forms in terms of elementary harmonic functions.Numerical results show that the computation of the complete solutions can be achieved with high accuracy and efficiency.

Cracked elastic substrate strip with functionally graded coating under thermal-mechanical loading

This paper investigates the functionally graded coating bonded to an elastic strip with a crack under thermal- mechanical loading. Considering some new boundary conditions, it is assumed that the temperature drop across the crack surface is the result of the thermal conductivity index which controls heat conduction through the crack region. By the Fourier transforms, the thermal-elastic mixed boundary value problems are reduced to a system of singular integral equations which can be approximately solved by applying the Chebyshev polynomials. The numerical computation methods for the temperature, the displacement field and the thermal stress intensity factors （TSIFs） are presented. The normal temperature distributions （NTD） with different parameters along the crack surface are analyzed by numerical examples. The influence of the crack position and the thermal-elastic non- homogeneous parameters on the TSIFs of modes I and 11 at the crack tip is presented. Results show that the variation of the thickness of the graded coating has a significant effect on the temperature jump across the crack surfaces when keeping the thickness of the substrate constant, and the thickness of functionally graded material （FGM） coating has a significant effect on the crack in the substrate. The results can be expected to be used for the purpose of gaining better understanding of the thermal-mechanical behavior of graded coatings.

Analytical solution for functionally graded anisotropic cantilever beam subjected to linearly distributed load

The bending problem of a functionally graded anisotropic cantilever beam subjected to a linearly distributed load is investigated. The analysis is based on the exact elasticity equations for the plane stress problem. The stress function is introduced and assumed in the form of a polynomial of the longitudinal coordinate. The expressions for stress components are then educed from the stress function by simple differentiation. The stress function is determined from the compatibility equation as well as the boundary conditions by a skilful deduction. The analytical solution is compared with FEM calculation, indicating a good agreement.

Nonlinear analysis on dynamic buckling of eccentrically stiffened functionally graded material toroidal shell segment surrounded by elastic foundations in thermal environment and under time-dependent torsional loads

The nonlinear analysis with an analytical approach on dynamic torsional buckling of stiffened functionally graded thin toroidal shell segments is investigated. The shell is reinforced by inside stiffeners and surrounded by elastic foundations in a thermal environment and under a time-dependent torsional load. The governing equations are derived based on the Donnell shell theory with the yon Karman geometrical nonlinearity, the Stein and McElman assumption, the smeared stiffeners technique, and the Galerkin method. A deflection function with three terms is chosen. The thermal parameters of the uniform temperature rise and nonlinear temperature conduction law are found in an explicit form. A closed-form expression for determining the static critical torsional load is obtained. A critical dynamic torsional load is found by the fourth-order Runge-Kutta method and the Budiansky-Roth criterion. The effects of stiffeners, foundations, material, and dimensional parameters on dynamic responses of shells are considered.

A Theoretical Analysis of Functionally Graded Beam under Thermal Loading

Modeling of the behavior for Functionally Graded Beam (FGB) under thermal loading is introduced in the present work. The material properties are assumed to vary according to power function along the thickness of the beam. The effects of several parameters such as thermal expansion parameter, thermal conductivity and modulus of elasticity on the resultant axial stress of the FG beam have been studied. For thermal loading the steady state of heat conduction with power and exponentially variations through the thickness of FGB, is considered. The results obtained show that temperature distribution plays very important parameter controlling thermal resultant distribution of stresses and strains.

Crack Propagation Path in Two-Directionally Graded Composites Subjected to Mixed-Mode Ⅰ+Ⅱ Loading

Crack propagation path in two-directionally graded composites was investigated by the finite element method.A graded extended finite element method(XFEM)was employed to calculate displacement and stress fields in cracked graded structures.And a post-processing subroutine of interaction energy integral was implemented to extract the mixed-mode stress intensity factors(SIFs).The maximum hoop stress(MHS)criterion was adopted to predict crack growth direction based on the assumption of local homogenization of asymptotic crack-tip fields in graded materials.Effects of material nonhomogeneous parameters on crack propagation paths were also discussed in detail.It is shown that the present method can provide relatively accurate predictions of crack paths in two-directionally graded composites.Crack propagates in the decreasing direction of effective Young′s modulus.The shape and steepness of property gradient perpendicular to the crack surface have great influences on crack paths.Through redesigning material property reasonably,crack growth in graded material can be changed to improve mechanical behaviours of cracked structures.

High heat loading performance of actively cooled W/Cu FGM-based components

A functionally graded material-based actively water-cooled tungsten-copper mockup with a dimension of 30 mm×30 mm×25 mm was designed and fabricated by infiltration-brazing method.The thicknesses of the pure W layer and W/Cu graded layer were 2 and 3 mm,respectively.High heat flux tests were performed on the mockup using an e-beam device.There is no damage occurring on the joint after heat loading at 5 MW/m2.The temperature on the pure W surface is less than 500°C after irradiation for 100 s at 5 MW/m2,while the temperature on the brazing seam/copper surface is around 200°C.

Geometrically Nonlinear Bending Analysis of Metal-Ceramic Composite Beams under Thermomechanical Loading

A new method is developed to derive equilibrium equations of Metal-Ceramic beams based on first order shear deformation plate theory which is named first order shear deformation beam theory2(FSDBT2). Equilibrium equations obtained from conventional method (FSDBT1) is compared with FSDBT2 and the case of cylindrical bending of Metal-Ceramic composite plates for non-linear thermomechanical deformations and various loadings and boundary conditions. These equations are solved by using three different methods (analytical, perturbation technique and finite element solution). The through-thickness variation of the volume fraction of the ceramic phase in a Metal-Ceramic beam is assumed to be given by a power-law type function. The non-linear strain-displacement relations in the von-Kármán sense are used to study the effect of geometric non-linearity. Also, four other representative averaging estimation methods, the linear rule, Mori-Tanaka, Self-Consistent and Wakashima-Tsukamoto schemes, by comparing with the power-law type function are also investigated. Temperature distribution through the thickness of the beams in thermal loadings is obtained by solving the one-dimensional heat transfer equation. Finally it is concluded that for Metal-Ceramic composites, these two theories result in identical static responses. Also the displacement field and equilibrium equations in the case of cylindrical bending of Metal-Ceramic plates are the same as those supposed in FSDBT2.

INFLUENCE OF TEMPERATURE-DEPENDENT PROPERTIES ON TEMPERATURE RESPONSE AND OPTIMUM DESIGN OF C/M FGMS UNDER THERMAL CYCLIC LOADING

The influence of temperature-dependent properties on temperature response and optimum design of newly developed ceramic-metal functionally graded materials under cyclic thermal loading and high temperature gradient environment is studied. The thermal conductivity of the material is considered to be dependent on the temperature. In this paper, the temperature response of the material is calculated using a nonlinear finite element method. Emphasis is placed on the influence of temperatue-dependent properties on the thermal response and insulation property of the material render the different graded compositional distributions and different heat flux magnitudes. Through the analysis, it is suggested that the influence of temperature-dependent properties can not be neglected in the temperature response analysis and the optimum design process of the material must be based on the temperature-dependent temperature analysis theory.

INFLUENCE OF TEMPERATURE-DEPENDENT PROPERTIES ON THERMAL STRESSES RESPONSE AND OPTIMUM DESIGN OF C/M FGMS UNDER THERMAL CYCLIC LOADING

The influence of temperature-dependent properties on thermal stresses response and optimum design of newly developed ceramic-metal functionally graded materials under cyclic thermal loaning and high temperature gradient environment is studied. The thermal conductivity of material is considered to be dependent on the temperature. In this paper, the thermal stresses response of the material is calculated rising a nonlinear finite element method. Emphasis is placed on the influence of temperature-dependent properties on the thermal stresses response characteristics, the thermal stresses relaxation property and the thermal stresses history under the different graded compositional distributions and different heat flux magnitudes. Through tile analysis. it is suggested that the influence of temperature-dependent properties can not be neglected In the thermal stresses response analysis and the optimum design process of the material must be based on the temperature-dependent thermo-elastic-plastic theory.

THERMO-ELASTIC-PLASTIC RESPONSE AND OPTIMUM DESIGN OF CERAMIC-METAL FGMS-CYCLIC THERML LOADING PROBLEM

The materials are made with a graded composition and microstructure in the thickness direction from the ceramic side to the metal side. The cyclic thermal loading and high temperasure gradient environment are simulated by heating the ceramic surface with a cyclic hear flux input and cooling the metal surface with a flowing liquid niterogen. The thermal and themo-elastic-plastic response of the materials is calculated using the isotropic hardening model and kinetic hardening model. Emphasis is placed on the response analysis under the different graded compositional distributions. Through the response analysis, the optimum design process of the graded composition under the dynamic case is established, which is bused on a unified viewpoint of the heat insulation property, thermal stress relaxation property and stress history feature.

ANALYSIS OF SHAKEDOWN OF FG BREE PLATE SUBJECTED TO COUPLED THERMAL-MECHANICAL LOADINGS

The static and kinematic shakedown of a functionally graded （FG） Bree plate is analyzed. The plate is subjected to coupled constant mechanical load and cyclically varying temperature. The material is assumed linearly elastic and nonlinear isotropic hardening with elastic modulus,yield strength and the thermal expansion coeffcient varying exponentially through the thickness of the plate. The boundaries between the shakedown area and the areas of elasticity,incremental collapse and reversed plasticity are determined,respectively. The shakedown of the counterpart made of homogeneous material with average material properties is also analyzed. The comparison between the results obtained in the two cases exhibits distinct qualitative and quantitative difference,indicating the importance of shakedown analysis for FG structures. Since FG structures are usually used in the cases where severe coupled cyclic thermal and mechanical loadings are applied,the approach developed and the results obtained are significant for the analysis and design of such kind of structures.

Buckling analysis of functionally graded plates partially resting on elastic foundation using the differential quadrature element method

We extend the differential quadrature element method (DQEM) to the buckling analysis of uniformly in-plane loaded functionally graded (FG) plates fully or partially resting on the Pasternak model of elastic support. Material properties of the FG plate are graded in the thickness direction and assumed to obey a power law distribution of the volume fraction of the constituents. To set up the global eigenvalue equation, the plate is divided into sub-domains or elements and the generalized differential quadrature procedure is applied to discretize the governing, boundary and compatibility equations. By assembling discrete equations at all nodal points, the weighting coefficient and force matrices are derived. To validate the accuracy of this method, the results are compared with those of the exact solution and the finite element method. At the end, the effects of different variables and local elastic support arrangements on the buckling load factor are investigated.

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 straindisplacement 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.

Thermal-induced snap-through buckling of simply-supported functionally graded beams

The instability of functionally graded material(FGM)structures is one of the major threats to their service safety in engineering applications.This paper aims to clarify a long-standing controversy on the thermal instability type of simply-supported FGM beams.First,based on the Euler-Bernoulli beam theory and von K′arm′an geometric nonlinearity,a nonlinear governing equation of simply-supported FGM beams under uniform thermal loads by Zhang’s two-variable method is formulated.Second,an approximate analytic solution to the nonlinear integro-differential boundary value problem under a thermal-induced inhomogeneous force boundary condition is obtained by using a semiinverse method when the coordinate axis is relocated to the bending axis(physical neutral plane),and then the analytical predictions are verified by the differential quadrature method(DQM).Finally,based on the free energy theorem,it is revealed that the symmetry breaking caused by the material inhomogeneity can make the simply-supported FGM beam under uniform thermal loads occur snap-through postbuckling only in odd modes;furthermore,the nonlinear critical load of thermal buckling varies non-monotonically with the functional gradient index due to the stretching-bending coupling effect.These results are expected to provide new ideas and references for the design and regulation of FGM structures.

Study and Commercial Application of Graded Optimization of Catalysts for Fixed Bed Residue Upgrading Unit

The SHIFT-G technology of inverse catalyst loading is used to optimize the catalyst grading in the residue hydrotreating unit. The results, taken from pilot tests and commercial units, have showed that the optimized catalyst grading system can reasonably distribute the reaction load, effectively improve the prop- erties of hydrotreated products, prolong the operating cycle and promote economic benefits.

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.

On asymmetric bending of functionally graded solid circular plates

Based on the three-dimensional elasticity equations, this paper studies the elastic bending response of a transversely isotropic functionally graded solid circular plate subject to transverse biharmonic forces applied on its top surface. The material prop- erties can continuously and arbitrarily vary along the thickness direction. By virtue of the generalized England＇s method, the problem can be solved by determining the expres- sions of four analytic functions. Expanding the transverse loarl in Fourier series along the circumferential direction eases the theoretical construction of the four analytic functions for a series of important biharmonic loads. Certain boundary conditions are then used to determine the unknown constants that are involved in the four constructed analytic functions. Numerical examples are presented to validate the proposed method. Then, we scrutinize the asymmetric bending behavior of a transversely isotropic functionally graded solid circular plate with different geometric and material parameters.

Buckling analysis of functionally graded nanobeams under non-uniform temperature using stress-driven nonlocal elasticity

In this work,the size-dependent buckling of functionally graded(FG)Bernoulli-Euler beams under non-uniform temperature is analyzed based on the stressdriven nonlocal elasticity and nonlocal heat conduction.By utilizing the variational principle of virtual work,the governing equations and the associated standard boundary conditions are systematically extracted,and the thermal effect,equivalent to the induced thermal load,is explicitly assessed by using the nonlocal heat conduction law.The stressdriven constitutive integral equation is equivalently transformed into a differential form with two non-standard constitutive boundary conditions.By employing the eigenvalue method,the critical buckling loads of the beams with different boundary conditions are obtained.The numerically predicted results reveal that the growth of the nonlocal parameter leads to a consistently strengthening effect on the dimensionless critical buckling loads for all boundary cases.Additionally,the effects of the influential factors pertinent to the nonlocal heat conduction on the buckling behavior are carefully examined.