任意梯度分布功能梯度涂层平面裂纹分析

提出可以分析任意梯度功能梯度材料的分层模型,并采用该模型研究功能梯度涂层平面裂纹问题.采用Fourier变换和传递矩阵法将该混合边值问题化为奇异积分方程组,通过数值求解获得应力强度因子.考察了分层模型的有效性,以及材料梯度变化形式、结构几何尺寸和材料梯度参数对裂纹应力强度因子的影响,发现结构几何尺寸、材料梯度变化形式、...     

Fracture analysis of a functionally graded strip with arbitrary distributed material properties

In this paper, the plane elasticity problem for a crack in a functionally graded strip with material properties varying arbitrarily is studied. The governing equation in terms of Airy stress function is formulated and exact solutions are obtained for several special variations of material properties in Fourier transformation domain. A multi-layered model is employed to model arbitrary variations of material properties based on two linear-distributed material softness parameters. The mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. Comparisons with other two existing multi-layered models have been made. Some numerical examples are given to demonstrate the accuracy, efficiency and versatility of the model. Numerical results show that fracture toughness of materials can be greatly improved by graded variation of elastic modulus and the influence of the specific form of elastic modulus on the fracture behavior of FGM is limited.     

The Anti-plane Dynamic Fracture Analysis of Functionally Graded Materials with Arbitrary Spatial Variations of Material Properties

Anti-plane dynamic fracture analysis is presented for functionally graded materials (FGM) with arbitrary spatial variations of material properties. The FGM with the material properties varying continuously in an arbitrary manner is modeled as a multi-layered medium with the elastic modulus and mass density varying linearly in each sub-layer and continuous at the interfaces between two adjacent sub-layers. With this linearly inhomogeneous multi-layered model, the problem of a crack in a graded interfacial zone bonded to two homogeneous half-spaces or in a coating bonded to a homogeneous half-space subjected to the anti-plane shear impact load is investigated. Laplace and Fourier transforms and transfer matrix are applied to reduce the associated mixed boundary value problem to a Cauchy singular integral equation which is solved numerically in the Laplace transformed domain. The dynamic stress intensity factors (DSIF) are obtained by using the numerical technique of Laplace inversion.     

Study of a propagating finite crack in functionally graded piezoelectric materials considering dielectric medium effect

This paper provides a study of the problem of a propagating finite crack under in-plane loading in functionally graded piezoelectric materials (FGPMs). The analytical formulations are developed by Fourier transforms and the resulting singular integral equations are solved by using Chebyshev polynomials. By using a dielectric crack model with deformation-dependent electric boundary condition, numerical simulations are made to show the effects of the dielectric medium, the gradient of material properties and the speed of crack propagation on the fracture parameters, such as the stress, electric displacement and crack opening displacement intensity factors. A critical state for the electromechanical loading applied to the FGPMs is observed, which determines whether the traditionally impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model. The validity of this dielectric crack model is also examined by comparing the results of different existing crack models.     

Three-dimensional mixed-mode stress intensity factors for cracks in functionally graded materials using enriched finite elements

Three-dimensional enriched finite elements are used to compute mixed-mode stress intensity factors (SIFs) for three-dimensional cracks in elastic functionally graded materials (FGMs) that are subject to general mixed-mode loading and constraint conditions. The method, which advantageously does not require special mesh configuration/modifications and post-processing of finite element results, is an enhancement of previous developments applied so far on isotropic homogeneous and isotropic interface cracks. The spatial variation of FGM material properties is taken into account at the level of element integration points. To validate the developed method, two- and three-dimensional mixed-mode fracture problems are selected from the literature for comparison. Two-dimensional cases include: inclined central crack in a large FGM medium under uniform tensile strain and stress loadings, a slanted crack in a finite-size FGM plate under exponentially varying tensile stress loading and an edge crack in a finite-size plate under shear traction load. The three-dimensional example models a deflected surface crack in a finite-size FGM plate under uniform tensile stress loading. Comparisons between current results and those from analytical and other numerical methods yield good agreement. Thus, it is concluded that the developed three-dimensional enriched finite elements are capable of accurately computing mixed-mode fracture parameters for cracks in FGMs.     

功能梯度材料涂层平面裂纹分析

研究粘接于均质基底材料上功能梯度涂层平面裂纹问题. 假设功能梯度材料剪切模量的倒数为坐标的线性函数，而泊松比为常数. 采用Fourier变换和传递矩阵法将该混合边值问题化为奇异积分方程组，通过数值求解获得 应力强度因子. 考察了材料梯度变化形式、结构几何尺寸和材料梯度参数对裂纹应力强度因子的影响，发现 功能梯度材料涂层尺寸、裂纹长度以及材料梯度参数均对应力强度因子有显著影响.     

Interface crack problem in layered orthotropic materials under thermo-mechanical loading

In the present paper, the behavior of an interface crack for a homogeneous orthotropic strip sandwiched between two different functionally graded orthotropic materials subjected to thermal and mechanical loading is considered. It is assumed that interface crack is partly insulated, and the temperature drop across the crack surfaces is the result of the thermal resistance due to the heat conduction through the crack region. The elastic properties of the material are assumed to vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the crack orientation. The complicated mixed boundary problems of equations of heat conduction and elasticity are converted analytically into singular integral equations, which are solved numerically. The main objective of the paper is to study the effects of material nonhomogeneity parameters and the dimensionless thermal resistance on the thermal stress intensity factors for the purpose of gaining better understanding of the thermal behavior of graded layer.     

Mode III fracture of an arbitrary oriented crack in two dimensional functionally graded material

In this paper, a two dimensional functionally graded material (2D-FGM) under an anti-plane load with an internal crack is considered. The crack is oriented in an arbitrary direction. The material properties are assumed to vary exponentially in two planar directions. The problem is analyzed and solved by two different methods namely Fourier integral transforms with singular integral equation technique, and then by the finite element method. The effects of crack orientation, material non-homogeneity, and other parameters on the value of stress intensity factor (SIF) are studied. Finally, the obtained results for Mode III stress intensity factor of different methods are compared.     

Mode III crack problems of two bonded functionally graded strips with internal cracks

This paper deals with the anti-plane problem of two bonded functionally graded finite strips. Each strip contains an internal crack normal to the interface. The material properties of two strips are assumed to vary along the direction of the crack lines. A system of singular integral equations is derived and then solved numerically by using Gauss–Chebyshev integration formula. The influences of nonhomogeneous parameters, crack interactions and two edge conditions on the mode III stress intensity factors are investigated.     

Three-dimensional thermoelastic analysis of finite length laminated cylindrical panels with functionally graded layers

Based on the 3D thermoelasticity theory, the thermoelastic analysis of laminated cylindrical panels with finite length and functionally graded (FG) layers subjected to three-dimensional (3D) thermal loading are presented. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The variations of the field variables across the panel thickness are accurately modeled by using a layerwise differential quadrature (DQ) approach. After validating the approach, as an important application, two common types of FG sandwich cylindrical panels, namely, the sandwich panels with FG face sheets and homogeneous core and the sandwich panels with homogeneous face sheets and FG core are analyzed. The effect of micromechanical modeling of the material properties on the thermoelastic behavior of the panels is studied by comparing the results obtained using the rule of mixture and Mori–Tanaka scheme. The comparison studies reveal that the difference between the results of the two micromechanical models is very small and can be neglected. Then, the effects of different geometrical parameters, material graded index and also the temperature dependence of the material properties on the thermoelastic behavior of the FG sandwich cylindrical panels are carried out.     

Periodic cracks in a functionally graded piezoelectric layer bonded to a piezoelectric half-plane

Studied is the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material. The properties of the functionally graded piezoelectric strip, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. Integral transform and dislocation density functions are employed to reduce the problem to the solution of a system of singular integral equations. The effects of the periodic crack spacing, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

正交各向异性功能梯度材料反平面裂纹尖端应力场

采用积分变换-对偶积分方程方法,研究了正交各向异性功能梯度材料反平面裂纹问题,文中假定材料沿两个主轴方向的剪切模量成比例按双参数梯度模型变化,通过求解对偶积分程并考虑变形Bessel函数的渐特性,推导出了裂纹尖端应力场,最后考察了材料非均匀性及正交性对应力强度因子的影响。     

Mixed-mode fracture analysis of orthotropic functionally graded materials under mechanical and thermal loads

Mixed-mode fracture problems of orthotropic functionally graded materials (FGMs) are examined under mechanical and thermal loading conditions. In the case of mechanical loading, an embedded crack in an orthotropic FGM layer is considered. The crack is assumed to be loaded by arbitrary normal and shear tractions that are applied to its surfaces. An analytical solution based on the singular integral equations and a numerical approach based on the enriched finite elements are developed to evaluate the mixed-mode stress intensity factors and the energy release rate under the given mechanical loading conditions. The use of this dual approach methodology allowed the verifications of both methods leading to a highly accurate numerical predictive capability to assess the effects of material orthotropy and nonhomogeneity constants on the crack tip parameters. In the case of thermal loading, the response of periodic cracks in an orthotropic FGM layer subjected to transient thermal stresses is examined by means of the developed enriched finite element method. The results presented for the thermally loaded layer illustrate the influences of the material property gradation profiles and crack periodicity on the transient fracture mechanics parameters.     

黏弹性功能梯度材料裂纹问题的有限元方法

针对组分材料体积含量任意分布的黏弹性功能梯度材料裂纹问题建立有限元分析途径. 通过Laplace变换,将黏弹性问题转化到象空间中求解,基于反映材料非均匀的梯度单元和裂纹尖端奇异特性的奇异单元计算象空间中的位移、应力和应变场,应用虚拟裂纹闭合方法得到应变能释放率,分别由应力和应变能释放率确定应力强度因子. 给出这些断裂参量在物理空间和象空间之间的对应关系,由数值逆变换求出其在物理空间的相应值. 文中分析两端均匀受拉的黏弹性边裂纹板条,首先针对松弛模量表示为空间函数和时间函数乘积的特殊梯度材料进行计算,结合对应原理验证方法的有效性. 然后分析组分材料体积含量具有任意梯度分布的情形,由Mori-Tanaka方法预测象空间中的等效松弛模量. 计算结果表明,蠕变加载条件下,应变能释放率随时间增加,其增大程度与黏弹性组分材料体积含量相关. 由于梯度材料的非均匀黏弹性性质,产生应力重新分布,导致应力强度因子随时间变化,其变化范围与组分材料的体积含量分布方式有关.     

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.     

The interaction of two parallel Mode-I limited-permeable cracks in a functionally graded piezoelectric material

In this paper, the interaction of two parallel Mode-I limited-permeable cracks in a functionally graded piezoelectric material was investigated by using the generalized Almansi's theorem. In the analysis, the electric permittivity of the air inside the crack was considered. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surface. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses and the singular electric displacements at the crack tips in functionally graded piezoelectric materials carry the same forms as those in homogeneous piezoelectric materials; however, the magnitudes of intensity factors depend on the electric permittivity of the air inside the crack and the gradient parameter of functionally graded piezoelectric material properties. It was also revealed that the crack shielding effect is also present in functionally graded piezoelectric materials.     

Mode-3 spontaneous crack propagation in unsymmetric functionally graded materials

The effects of the inhomogeneous material property variation on the mode-3 crack propagation characteristics have been analyzed. The spectral form of the elastodynamic boundary integral equations are derived for a functionally graded material, which is used as the numerical tool in our analysis. The material property gradient is assumed to vary unsymmetrically in the direction normal to the fracture plane. A parametric study has been performed by systematically varying the inhomogeneity length scale. In comparison with a homogeneous material, an unsymmetric functionally graded material offers lesser fracture resistance. The fracture resistance progressively decreases with increase in inhomogeneity, quantified by the increase in crack sliding displacement jumps, crack tip velocities and accelerations. The material property inhomogeneity affects only the transient crack propagation velocities, while the quasi-steady-state velocity remains unaltered.     

AXISYMMETRIC BENDING OF TWO-DIRECTIONAL FUNCTIONALLY GRADED CIRCULAR AND ANNULAR PLATES

Assuming the material properties varying with an exponential law both in the thick- ness and radial directions,axisymmetric bending of two-directional functionally graded circular and annular plates is studied using the semi-analytical numerical method in this paper.The de- flections and stresses of the plates are presented.Numerical results show the well accuracy and convergence of the method.Compared with the finite element method,the semi-analytical nu- merical method is with great advantage in the computational efficiency.Moreover,study on ax- isymmetric bending of two-directional functionally graded annular plate shows that such plates have better performance than those made of isotropic homogeneous materials or one-directional functionally graded materials.Two-directional functionally graded material is a potential alter- native to the one-directional functionally graded material.And the integrated design of materials and structures can really be achieved in two-directional functionally graded materials.     

Buckling of an orthotropic graded coating with an embedded crack bonded to a homogeneous substrate

To simulate buckling of nonuniform coatings, we consider the problem of an embedded crack in a graded orthotropic coating bonded to a homogeneous substrate subjected to a compressive loading. The coating is graded in the thickness direction and the material gradient is orthogonal to the crack direction which is parallel with the free surface. The elastic properties of the material are assumed to vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the crack orientation. The loading consists of a uniform compressive strain applied away from the crack region. The graded coating is modeled as a nonhomogeneous medium with an orthotropic stress–strain law. Using a nonlinear continuum theory and a suitable perturbation technique, the plane strain problem is reduced to an eigenvalue problem describing the onset of buckling. Using integral transforms, the resulting plane elasticity equations are converted analytically into singular integral equations which are solved numerically to yield the critical buckling strain. The Finite Element Method was additionally used to model the crack problem. The main objective of the paper is to study the influence of material nonhomogeneity on the buckling resistance of the graded layer for various crack positions, coating thicknesses and different orthotropic FGMs.