On the shear stress distribution between a functionally graded piezoelectric actuator and an elastic substrate and the reduction of its concentration

Recent advances in material processing technologies allow the production of piezoelectric materials with functionally graded material properties. We investigate the implications of functionally graded piezoelectric materials when used as actuators for structural control by examining the distribution of the actuating shear stress under a piezoelectric actuator of a functionally graded material (FGM) on an isotropic elastic half-space. It is shown that FGM materials can be used to adjust the shear stress distribution. In particular, the concentration near the edges of a conventional homogeneous piezoelectric actuator can be significantly reduced in an FGM actuator.     

Analytical elastic solutions for pressurized hollow cylinders with internal functionally graded coatings

The main objective of this work is to obtain analytical solutions for thick-walled cylinders subjected to internal and external pressure in which the entire wall is made of functionally graded material or of only a thin functionally graded coating present on the internal homogeneous wall. We assume that the materials are isotropic with constant Poisson’s ratio; as far as the Young modulus is concerned, we consider a power and an exponential. The proposed analytical solutions show the effects of the different profiles describing the graded properties of the materials on the stress and displacement fields; in addition, comparisons between graded coating and conventional homogeneous coating highlight the advantage of the graded material on the interface stress reduction. Furthermore, we show how even a thin graded coating can be useful to satisfy the requirements of a specific application without having to make an entire wall with graded properties. This investigation permits us to optimize the elastic response of cylinders under pressure by tailoring the thickness variation of the elastic properties and to reduce manufacturing costs given by the technological limitations that occur to produce entire functionally graded walls.     

Non-destructive Detection of a Circular Cavity in a Finite Functionally Graded Material Layer Using Anti-plane Shear Waves

A semi-analytical method is proposed to investigate the non-destructive detection of a circular cavity buried in a functionally graded material layer bonded to homogeneous materials, and the multiple scattering effect of shear waves is described accurately. The image method is used to satisfy the traction free boundary condition at the edge of the functionally graded material layer. The analytical solutions of wave fields are expressed by employing wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions at the edge and around the cavity. The analytical and numerical solutions of dynamic stress concentration factors around the cavity are presented. The effects of the position of the cavity in the material layer, the incident wave number, and the properties of the two phases of materials on the dynamic stress concentration factors are analyzed. Analyses show that when the buried depth of the cavity and the thickness of the layer are relatively small, the properties of the two phases of materials have great effect on the distribution of dynamic stress around the cavity. In the region of higher frequency, the effects of the position of the cavity and the properties of the two phases of materials on the maximum dynamic stress are greater.     

Exact solutions for characterization of electro-elastically graded materials

The present paper deals with a class of functionally graded materials (FGM), called active FGM that has electro-elastically graded material phases. An active FGM system leads to minimization of stress concentration that arises due to mismatch in the electrical and elastic properties of the constituent phases. This work focuses on the characterization of the through thickness stresses of an active FGM subjected to electrical excitation. The structure is comprised of a substrate, an electro-elastically graded layer and an active layer. A formulation for exact solutions of the system based on Euler–Bernoulli theory is presented. Power-law variation of the composition of the two phases in the graded layer is considered. Performance of linearly gradient FGM for a range of stiffness and electrical property ratios of the active and substrate materials have been studied. It is observed that the electrical strain component and the compositional gradation significantly influence the stress characteristics of the active FGM.     

Path planning of functionally graded material objects for layered manufacturing

Path planning is a fundamental process-planning task in layered manufacturing. However, currently, most researches focus on path planning of homogeneous objects and few approaches for path planning of functionally graded material objects have been reported in the literature. Unlike homogeneous objects, functionally graded materials objects contain continuously varying material information as well as geometry information, which makes their path planning more complex and difficult than those of homogeneous objects. This paper presents an approach for path planning of functionally graded material objects. First, the continuous material distribution in each layer is changed into discrete step-wise gradings by subdividing the functionally graded materials slice into multimaterial subregions, which can be regarded as homogeneous material regions so that the functionally graded material layer can be built by mixing materials into desired volume ratio. Then the tool paths and process parameters for each multimaterial subregion in the slice are generated separately. An algorithm that summarized the procedure is described and an example is also presented.     

Dynamic stress from a cylindrical inclusion buried in a functionally graded piezoelectric material layer under electro-elastic waves

This paper presents a theoretical method to investigate the multiple scattering of electro-elastic waves and dynamic stress around a subsurface cylindrical inclusion in a functionally graded piezoelectric material layer bonded to homogeneous piezoelectric materials. The analytical solutions of wave fields are expressed by employing wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions around the inclusion. The image method is used to satisfy the mechanical and electrically short conditions at the free surface of the structure. Through the numerical solutions of dynamic stress concentration factors around the inclusion, it is found that when the cylindrical inclusion possesses higher rigidity and greater piezoelectric constant than the two phases of functionally graded materials, the dynamic stress around the inclusion increases greatly. When the distance between the surface of the structure and the inclusion is smaller, the effect of the properties of the inclusion becomes greater. When the cylindrical inclusion possesses lower rigidity and smaller piezoelectric constant than the two phases of functionally graded materials, the maximum dynamic stress shows little difference; however, the variation of the distribution of the dynamic stress around the inclusion is greater. The effect of the properties of the inclusion on the dynamic stress around the inclusion is greater than that on the electric field. The effects of wave frequency on the dynamic stress and electric field are also examined.     

The effect of notch depth on J-integral and critical fracture load in plates made of functionally graded aluminum–silicone carbide composite with U-notches under bending

The present paper deals with the effect of notch depth on J-integral and critical fracture load in a plate made of functionally graded aluminum–silicone carbide composite (Al–SiC) with U-notch under bending. The weight fraction of SiC particles varies from 0% to 20% through the specimen width. Using three criteria namely mean stress (MS), point stress (PS), and averaged strain-energy density (ASED), the critical fracture load has been predicted and its variation with respect to the notch depth has been studied. A comparison of the J-integral between functionally graded and homogeneous Al–SiC composite were made, where the notch tip in the functionally graded material is situated in a layer with same mechanical properties as the homogeneous composite. The results indicated that in the case where the notch scene is toward brittleness increment the critical J-integral in functionally graded material (FGM) is larger than that of in homogeneous material with the same mechanical properties at the notch tip. Therefore, FGM is more convenient than homogeneous material against fracture.     

Fatigue crack growth simulations of interfacial cracks in bi-layered FGMs using XFEM

An investigation of fatigue crack growth of interfacial cracks in bi-layered materials using the extended finite element method is presented. The bi-material consists of two layers of dissimilar materials. The bottom layer is made of aluminium alloy while the upper one is made of functionally graded material (FGM). The FGM layer consists of 100 % aluminium alloy on the left side and 100 % ceramic (alumina) on the right side. The gradation in material property of the FGM layer is assumed to be exponential from the alloy side to the ceramic side. The domain based interaction integral approach is extended to obtain the stress intensity factors for an interfacial crack under thermo-mechanical load. The edge and centre cracks are taken at the interface of bi-layered material. The fatigue life of the interface crack plate is obtained using the Paris law of fatigue crack growth under cyclic mode-I, mixed-mode and thermal loads. This study reveals that the crack propagates into the FGM layer under all types of loads.     

Finite Element Analysis of Mono- and Bi-Adhesively Bonded Functionally Graded Adherend

The stress concentration at the end of bonded lap joints is a major concern in the design and application of adhesive joints, and, therefore, many research works have been carried out to reduce the stress level in the bond line. Most of the proposed methods focus on changing adhesive geometries or properties to achieve an optimized model. In this paper, the stress and strain distribution for adherend with functionally graded properties was analyzed to investigate the effect of the adherend material properties and the type of joint on the stress distribution within bond line. The effect of ceramic volume fraction of the functionally graded materials (FGMs) on the stress concentration has been studied. Also, bi-adhesive joint is used as an alternative stress reduction technique for the joint. Results show that using bi-adhesively joint technique together with high-ceramic volume fraction FGMs can significantly reduce the shear and peel stress in the lap joint.     

Elastic-plastic stress analysis in a long functionally graded solid cylinder with fixed ends subjected to uniform heat generation

Elastic-plastic deformation of a solid cylinder with fixed ends, made of functionally graded material (FGM) with uniform internal heat generation is investigated, based on Tresca’s yield criterion and its associated flow rule, considering four of the material properties to vary radially according to a parabolic form. These four material properties are yield strength, modulus of elasticity, coefficients of thermal conduction and thermal expansion, assumed to be independent of temperature as Poisson’s ratio which is taken as constant. The materials which compose the functionally graded cylinder are supposed to be elastic-perfectly plastic materials. Expressions for the distributions of stress, strain and radial displacement are found analytically in terms of unknown interface radii. After determining these radii numerically by means of Mathematica 5.2, the distributions are plotted versus dimensionless radius, increasing heat generation, to compare the FGM cylinder with the homogeneous one. The numerical values used in this work for material parameters are arbitrarily chosen to point out the effect of the non-homogeneity on the stress distribution. The results obtained show that the stress distribution, as well as the development of plastic region radii, is influenced substantially by the material non-homogeneity.     

Fracture investigation of U‐notch made of tungsten–copper functionally graded materials by means of strain energy density

The averaged strain energy density over a well‐defined control volume was employed to assess the fracture of U‐notched specimens made of tungsten–copper functionally graded materials under prevalent mode II loading. The boundary of control volume was evaluated by using a numerical method. Power law function was employed to describe the mechanical properties (elasticity modulus, Poisson's ratio, fracture toughness and ultimate tensile stress) through the specimen width. The effect of notch tip radius and notch depth on notch stress intensity factors and mode mixity parameter χ were assessed. In addition, a comparison based on fracture load between functionally graded and homogeneous W–Cu was made. Furthermore, in this research, it was shown that the mean value of the strain energy density over the control volume can be accurately determined using coarse meshes for functionally graded materials.     

Theoretical analysis of heat conduction problems of nonhomogeneous functionally graded materials for a layer sandwiched between two half-planes

Functionally graded materials are the materials whose material properties are smoothly varying along one axis, and they are used as buffer layers to connect two dissimilar materials. By choosing proper functionally graded parameters, the material properties at the interface can be identical to prevent the interfacial fracture problem. This study analyzes the heat conduction problem of nonhomogeneous functionally graded materials for a layer sandwiched between two half-planes. From the Fourier transform method, the full-field solutions of temperature and heat flux are obtained in explicit forms. Numerical calculations based on the analytical solutions are performed and are discussed in detail. The continuous characteristics of the temperature and heat flux along the interface are emphasized, and some interesting phenomena are presented in this study. It is noted that the temperature and heat flux fields along the interface for nonhomogeneous functionally graded materials are continuous if the conductivities are identical at the interface. Furthermore, the temperature and heat flux q _y have the identical contour slopes across the interface.     

具有中等组分不同网状结构功能梯度构件的三维动力特性分析

功能梯度材料具有复杂的细部结构,其内部构造远比匀质材料复杂,因此其构件动力分析很难求得其解析解。该文建议一种新颖的功能梯度构件动力分析的细观元法。细观力学研究的目的在于建立材料的宏观性能同其组分材料性能及细观结构之间的定量关系,它可揭示不同的材料组合具有不同的宏观性能的内在机制。此法可实现材料细观结构到构件宏观响应的直接过渡分析,而计算单元与自由度又等同一般常规有限元,却使得组成功能梯度材料构件的各种材料细观构造得到反映。通过细观元技术,对具有中等组分不同网状结构功能梯度构件进行三维动力特性分析,并给出其三维固有频率及振型的三维分布,特别是给出了不同网格结构功能梯度板件应力振型的平面等值线图差异。结果表明:不同细观网格结构对功能梯度材料结构三维动力响应有较明显影响。     

Numerical simulation of time-dependent fracture of graded bimaterial metallic interfaces

Stationary cracks along and near interfaces between two time-dependent materials are simulated using the finite element method (FEM) to examine crack tip fields and candidate driving force parameters for crack growth. Plane strain conditions are assumed. In some cases, a thin transition layer is included between the two materials. This transition layer features a functionally graded blend of properties of the two materials. An example of such a system is that of weld metal (WM) and base metal (BM) in a weldment, with the transition layer corresponding to the heat-affected zone (HAZ). Numerical solutions for the stress and strain fields of homogeneous and heterogeneous Compact Tension (C(T)-type) specimens are presented. The equivalent domain integral technique is employed to compute the J-integral for elastic-plastic cases as well as the C(t)-integral and transition times for creep behavior. Results from parametric studies are curve-fit in terms of transition layer thickness and crack position, inelastic property mismatches, and other independent model parameters. Results indicate that the incorporation of functionally graded transition layer regions leads to less concentrated stress and strain components along interfaces ahead of the crack tip. It is also shown that the computed fracture parameters are influenced by the transition layer properties.     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.     

Stress analysis for bonded materials with a graded interfacial layer under a rigid punch

It is well known that functionally graded materials can be used to eliminate the stress discontinuity that is often encountered in multilayer composites. In this article, the stress analysis for the coating-functionally graded interfacial layer-substrate structure under a rigid spherical punch is investigated. A linear multi-layer model is used to model the graded interfacial layer with arbitrary varying materials properties along the thickness direction. The spherical indentation problem is formulated in terms of a singular integrate equation with the method of transfer matrix and Hankel integral transform technique. The stress components in the coating-graded interfacial layer-substrate structure are calculated by solving the equation numerically. The results show that stiffens ratio and the gradient index of the graded interfacial layer has a significant effect on the distribution of stress components.     

STRESS INTENSITY FACTORS FOR A FUNCTIONALLY GRADED MATERIAL CYLINDER WITH AN EXTERNAL CIRCUMFERENTIAL CRACK

This paper presents mode I stress intensity factors for external circumferentially cracked hollow cylinders, which are assumed to be made of functionally graded materials and subjected to remote uniform tension. The conventional finite element method is improved by introducing isoparametric transformation for simulating the gradient variations of material properties in the finite elements. This improved finite element method is verified to be effective and efficient. Various types of functionally graded materials and different gradient compositions for each type are investigated. The results show that the material property distribution has a quite considerable influence on the stress intensity factors.     

功能梯度材料的断裂与屈曲驱动断裂的简化分析

作为一类先进的复合材料,功能梯度材料(FGM)能综合利用多种材料的物理性能,同时材料性质的连续变化也使其具有许多优越的力学性能。本文对功能梯度材料中平行于界面的裂纹的断裂参数进行了计算,并分析了梯度变化的薄膜在压应力作用下的屈曲驱动扩展。研究结果表明:功能梯度材料能有效地减小界面中的应力集中及它对材料中缺陷的作用,从而不同程度地提高了材料的强度和韧性。     

间隙波在功能梯度压电板和压电半空间结构中的传播

研究了间隙波在功能梯度压电板和压电半空间结构中的传播性质.功能梯度压电板的材料性能沿x2方向呈指数变化,首先推导了间隙波传播时的解析解,利用界面条件得到了间隙波的频散方程,基于推导的频散方程,结合数值算例分析了功能梯度压电材料的梯度、压电层厚度以及材料性能对间隙波相速度的影响,研究结果对功能梯度压电材料的覆层结构在声波器件中的应用具有重要的参考价值.     

Elastic solution of a two-dimensional functionally graded thick truncated cone with finite length under hydrostatic combined loads

In this paper, a thick truncated hollow cone with finite length made of two-dimensional functionally graded materials (2D-FGM) subjected to combined loads as internal, external and axial pressure is considered. The volume fraction distribution of materials and geometry are assumed to be axisymmetric but not uniform along the axial direction. The Finite Element Method based on the Rayliegh-Ritz energy formulation is applied to obtain the elastic behavior of the functionally graded thick truncated cone. By using this method, the effects of semi-vertex angle of the cone and the power law exponents on the distribution of different types of displacements and stresses are considered. The results show that using 2D-FGM leads to a more flexible design so that both the maximum stresses and stress distribution can be controlled by the material distribution.