Geometry effects of substrate and coating layers on the thermal stress response of FGM structure

Summary Due to transient temperature change, the plane strain elastic-plastic problem for a functionally graded material (FGM) bonded to a homogeneous coating layer and a metal substrate is considered by the use of the finite element method (FEM). The substrate and the coating are assumed to be aluminum and partially stabilized zirconia, respectively. The FGM layer is a particulate composite of aluminum and partially stabilized zirconia with volume fractions continuously varying through the thickness. Generally in high temperature applications, the FGM system is sandwiched between a substrate layer and a coating layer. The coating layer increases the protection from heat but decreases the thermal shock resistance while the substrate layer increases the rigidity of the structure and decreases strength-related properties at high temperature. In order to compromise the thickness of both the coating and substrate layers, different values of the substrate and coating thickness are studied in order to evaluate their effects on the thermal stress response of the FGM structure. Since the main objective of the FGMs is using them in different applications with severe thermal loading conditions, the thermal stresses may be so high that some reinforcements may be fractured and/or debonded from the matrix giving a weakening effect instead of a reinforcing one. Hence, the behaviors of the reinforcements and the matrix are essential to be studied. In this regard, microscopic constitutive equations along with the temperature-dependent properties of the constituent materials are considered to enable us obtaining more realistic results of thermal stresses. Since the FGM structures are fabricated at high temperatures, thermal residual stresses are produced. In order to find out the importance of the consideration of the residual stresses arising from the fabrication process, the FGM structure with stress-free conditions is heated to the operating temperature, and its thermal stress response is compared with that one where the residual stresses are taken into account. Also, several functional forms of gradation of the constituents in the FGM layer are examined to reach the optimum profile giving the minimum stress level for the FGM structure under thermo-elasto-plastic behavior.     

含FGM的涂层结构中热残余应力的分析与优化

本文利用有限元方法和优化理论,对含ＦＧＭ（Ｆｕｎｃｔｉｏｎａｌｌｙ　Ｇｒａｄｅｄ　Ｍａｔｅｒｉａｌｓ）层的热喷涂构件中的残余应力进行了数值分析,并获得了ＦＧＭ内各组份体积含量分布的最优化形式和参数ｐ。同时,我们也研究了喷涂构件的几何形状、涂层及基底的材料性能对于ｐ的影响规 律。在本文的分析中,考虑了基底材料和ＦＧＭ的塑性变形以及其性能对于温度的依赖。本文 的工作将有利于含ＦＧＭ层的热喷涂构件的设计与生产。     

Thermo-elastic Stresses in a Functional Graded Material Under Thermal Loading,Pure Bending and Thermo-mechanical Coupling

Analytical expressions have been derived for the through thickness stresses of a Functional graded materials (FGMs) thin plate subjected to thermal loading, pure bending and thermo-mechanical coupling, respectively. The structure is comprised of a metallic layer, a ceramic layer and a functional graded layer. Continuous gradation of the volume fraction in the FGM layer is modeled in the form of an "m" power polynomial of the coordinate axis in thickness direction of the plate. Numerical scheme of discretizing the continuous FGM layer with different graded distributions such as linear (m=1), quadratic (m=2) and square root (m=0.5) has been developed by the averaging technique of composites. Solutions for the stress distributions have been derived for the system under thermal loading, pure bending and thermo-mechanical coupling, respectively.     

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.     

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.     

Evaluation of R-curve behavior of ceramic-metal functionally graded materials by stable crack growth

This paper deals with the fracture toughness and R-curve behavior of ceramic-metal functionally graded materials (FGMs). A possibility of stable crack growth in a three-point-bending specimen is examined based on the driving force and resistance for crack growth in FGMs, and the distribution of fracture toughness or R-curve behavior is evaluated on FGMs fabricated by powder metallurgy using partially stabilized zirconia (PSZ) and stainless steel (SUS 304). The materials have a functionally graded surface layer (FGM layer) with a thickness of 1 mm or 2 mm on a SUS 304 substrate. Three-point-bending tests are carried out on a rectangular specimen with a very short crack in the ceramics surface. On the three-point-bending test, a crack is initiated from a short pre-crack in unstable manner, and then it propagates in stable manner through the FGM layer with an increase in the applied load. From the relationship between applied load and crack length during the stable crack growth in the FGM layer, the fracture toughness is evaluated. The fracture toughness increases with an increase in a volume fraction of SUS 304 phase.     

Dynamic stress from a subsurface cylindrical inclusion in a functionally graded material layer under anti-plane shear waves

The multiple scattering of shear waves and dynamic stress resulting from a subsurface cylindrical inclusion in a functionally graded material (FGM) layer bonded to homogeneous materials are investigated, and the analytical solution of this problem is derived. Image method is used to satisfy the traction free boundary condition of the FGM layer. The analytical solutions of wave fields around the actual and image inclusions are expressed by employing wave functions expansion method, and the expanded mode coefficients are determined by satisfying the continuous boundary conditions around the inclusions. Through the numerical solutions of dynamic stress concentration factors (DSCFs) around the inclusion, the effects of the position of the inclusion in the material layer, the properties of the inclusion, and the properties of the two phases of composites on the DSCFs are analyzed. Analyses show that when the cylindrical inclusion is stiffer than the two phases of FGMs, 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 is softer than the two phases of FGMs, the maximum dynamic stress shows little difference; however, the variation of the distribution of the dynamic stress around the inclusion is greater.     

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。     

Three-dimensional analysis of a functionally graded coating/substrate system of finite thickness

The concept of functionally graded material (FGM) is currently actively explored in coating design for the purpose of eliminating the mismatch of thermomechanical properties at the interfaces and thus increasing the resistance of coatings to functional failure. In the present paper, three-dimensional elastic deformation of a functionally graded coating/substrate system of finite thickness subjected to mechanical loading is investigated. A comparative study of FGM versus homogeneous coating is conducted to examine the effect of the coating type on stress and displacement fields in the system.     

燃烧合成TiC-Al_2O_3/Fe梯度材料及其抗热震行为

采用自蔓延高温合成结合准热等静压（ＳＨＳ／ＰＨＩＰ）制备出了具有对称结构的 ＴｉＣ－Ａｌ２Ｏ３／Ｆｅ梯度材料,并对其抗热冲击及抗热疲劳行为进行了测试和分析·结果表明, ＳＨＳ／ＰＨＩＰ制备的 ＴｉＣ－Ａｌ２Ｏ３／Ｆｅ梯度材料具有预期的梯度式组成,在热冲击和热疲劳实验过程中均无梯度层间横向贯穿裂缝,克服了传统陶瓷／金属直接接合界面的热应力剥落．     

Interface cracking between functionally graded coatings and a substrate under antiplane shear

Coating technology plays a significant role in a number of applications such as high temperatures, corrosion, oxidation, wear, and interface. In this paper, we investigate the interface cracking between ceramic and/or functionally graded coatings (FGM coatings) and a substrate under antiplane shear. Four coating models are considered, namely single layered homogeneous coating, double layered piece-wise homogeneous coating, single layered FGM coating and double layered coating with an FGM bottom coat. Mode III stress intensity factors (SIFs) are calculated for the different coating models. In the case of μ_L > μ₀ where μ₀ is the shear modulus of the substrate and μ_L the shear modulus of the material at the surface of the coating, it is found that the single layered FGM coating reduces SIF slightly, whereas the coating system with a top homogeneous layer and a thin FGM bottom layer reduces SIF significantly. In the case of μ_L < μ₀ the SIF is found to be larger for the FGM coatings than for the homogeneous coatings. The FGM coating, however, may still be superior to homogeneous coatings in this case as FGM coatings usually provide better bonding strength between the coating and substrate. Finally, the applicability of the SIF concept in the fracture of FGM coatings is discussed. Large modulus gradients in thin coatings may seriously restrict the application of SIFs as the SIF-dominant zone may fall into the crack tip nonlinear deformation and damage zone. The same argument exists for some interphase models in interface crack solutions.     

燃烧合成TiC-A12O3/Fe梯度材料及其抗热震行为

采用自蔓延高温合成结合准热等静压(SHS/PHIP)制备出了具有对称结构的TiC-A12O3/Fe梯度材料,并对其抗热冲击及抗热疲劳行为进行了测试和分析。结果表明,SHS/PHIP制备的TiC-A12O3/Fe梯度材料具有预期的梯度式组成,在热冲击和热疲劳实验过程中均无梯度层间横向贯穿裂缝,克服了传统陶瓷/金属直接接合界面的热应力剥落。     

Functionally graded semiconductor layers for devices application

We report on various aspects of the application of functionally graded materials (FGM) for devices fabrication. The broad spectrum of problems ranging from design and technology of graded structures to its characterization is discussed. The main attention is focused on the application of FGM as an active area of a new photo-detector. The influence of various profile distributions of AIIIBV grading layers composition and layers configuration on the photo-detector spectral characteristics are discussed.     

燃烧合成TiC-Al2O3/Fe梯度材料及其抗热震行为

采用自蔓延高温合成结合准热等静压（SHS／PHIP）制备出了具有对称结构的TiC-Al2O3/Fe梯度材料,并对其抗热冲击及抗热疲劳行为进行了测试和分析,结果表明,SHS／PHIP制备的TiC-Al2O3/Fe梯度材料具有预期的梯度式组成,在热冲击和热疲劳实验过程中均无梯度层间横向贯穿裂缝,克服了传统陶瓷／金属直接接合界面的热应力剥落。     

Material selection in the design of the tibia tray component of cemented artificial knee using finite element method

Aseptic loosening of the tibial component; which may be caused by mechanical stress shielding in the bone and may require revision surgery; is the primary concern of total knee replacement (TKR). The stiffness of the implant material had a marked influence on the stresses developed in the constituents and surrounding bones of the artificial knee and then will affect the bone stress shielding. Therefore, the functionally graded materials had been developed as a potential tibia tray material of TKR due to its improved capability of stress distribution. In the current investigation two dimensional finite element models have been developed to study bone and interface stresses for six different tibial prothesises (titanium, CoCrMo and four functional graded materials “FGM” models). The utilization of FGM tibia tray with elastic modulus changing gradually in vertical direction downwardly showed a favorable stress distribution outcome. Furthermore, the results has revealed that the FGM tibia tray will reduce the stress shielding in the surrounding bones of the artificial knee which will increase the life of the total knee prosthesis.     

HA-Ti生物功能梯度材料微观组织及热应力缓和特性

本文用粉末冶金法制备了ＨＡ－Ｔｉ系生物ＦＧＭ,并测定了ＨＡ－Ｔｉ复合体材料的弹性模量和热膨胀系数．应用经典叠层板理论和热弹性力学理论分析了ＨＡ－Ｔｉ系ＮＦＧＮ双层板和ＨＡ－Ｔｉ系ＦＧＭ的制备残余热应力和热应变．结果表明,ＨＡ－Ｔｉ系生物ＦＧＭ呈现出宏观不均匀性与微观连续性的组织特征．ＨＡ－Ｔｉ系复合体材料的弹性模量在Ｔｉ－ＨＡ８０达到谷值,并受到气孔率的影响．其热膨胀系数随着ＨＡ含量和测试温度的升高而增大．残余热应力和残余热应变强烈依赖于组成分布,ＦＧＭ由于组成梯度化减小了成分变化幅度,其最大残余拉应力只有ＨＡ／Ｔｉ直接叠合体（ＮＦＧＭ）的１／３,具有显著缓和热应力的功能.     

An alternative model for elastic thermal stresses in two materials joined by a graded layer

The elastic stress distribution resulting from thermal mismatch in a film/graded layer/substrate system is a critical issue. Recently by a three-variable method Hsueh and Lee formulated a closed-form analytical solution [Hsueh CH, Lee S. Modeling of elastic thermal stresses in two materials joined by a graded layer. Composites: Part B 2003;34:747–52] in order to overcome the complexity of the traditional analytical models. The paper is devoted to developing an alternative analytical model for graded beams by Zhang’s two-variable method for multilayered beams. To illustrate applications of the present solution, specific results are calculated for the GaAs/graded GaAs–Si/Si systems. The transition phenomenon between tensile stress and compressive stress in the graded beam is investigated in detail for different graded parameters or different thickness. The present results agree well with existing analytical results. The differences between the present two-variable model and Hsueh’s three-variable model are also discussed.     

Anti-plane problem of periodic interface cracks in a functionally graded coating-substrate structure

In this paper, the interface cracking between a functionally graded material (FGM) and an elastic substrate is analyzed under antiplane shear loads. Two crack configurations are considered, namely a FGM bonded to an elastic substrate containing a single crack and a periodic array of interface cracks, respectively. Standard integral-transform techniques are employed to reduce the single crack problem to the solution of an integral equation with a Cauchy-type singular kernel. However, for the periodic cracks problem, application of finite Fourier transform techniques reduces the solution of the mixed-boundary value problem for a typical strip to triple series equations, then to a singular integral equation with a Hilbert-type singular kernel. The resulting singular integral equation is solved numerically. The results for the cases of single crack and periodic cracks are presented and compared. Effects of crack spacing, material properties and FGM nonhomogeneity on stress intensity factors are investigated in detail.     

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.