Stochastic analysis of a thermoelastic problem in functionally graded plates with uncertain material properties

The statistics (i.e., mean and variance) of temperature and thermal stress are analytically obtained in functionally graded material (FGM) plates with uncertainties in the thermal conductivity and coefficient of linear thermal expansion. These FGM plates are assumed to have arbitrary nonhomogeneous thermal and mechanical properties through the entire thickness of plate and are subjected to deterministic convective heating. The stochastic temperature and thermal stress fields are analysed by assuming the FGM plate is multilayered with distinct, random thermal conductivity and coefficient of linear thermal expansion in each layer. Vodicka’s method, which is a type of integral transform method, and a perturbation method are employed to obtain the analytical solutions for the statistics. The autocorrelation coefficients of each random property and cross-correlation coefficients between different random properties are expressed in exponential function forms as a non-homogeneous Markov random field of discrete space. Numerical calculations are performed for FGM plates composed of partially stabilised zirconia (PSZ) and austenitic stainless steel (SUS304), which have the largest dispersion of the random properties at the place where the volume fractions of the two constituent materials are both 0.5. The effects of the spatial change in material composition, thermal boundary condition and correlation coefficients on the standard deviations of the temperature and thermal stress are discussed.     

非均匀复合材料的动态热弹性断裂力学分析

对非均匀复合材料的动态热弹性断裂力学问题进行了研究,假设材料参数沿厚度方向为变化的,沿该方向将复合材料划分为许多单层,取每一单层材料参数为常数,应用Fourier变换法,在Laplace域内推导出了控制问题的奇异积分方程组,给出了热应力强度因子的表达式,然后利用Laplace数值反演,得出了裂纹尖端的动态应力强度因子．本文的方法具有以下特点：（1）多个垂直于厚度方向的裂纹,（2）材料可以为正交各向异性：（3）考虑了惯性效应．作为算例,研究了带有两个裂纹的功能梯度结构,分析了材料参数的变化对应力强度因子的影响．     

Transient thermal response of functionally graded piezoelectric laminates with an infinite row of parallel cracks normal to the bimaterial interface

In this paper, the problem of a functionally graded piezoelectric material strip(FGPM strip)containing an infinite row of parallel cracks perpendicular to the interface between the FGPM strip and a homogeneous layer is analyzed under transient thermal loading condition. The crack faces are supposed to be completely insulated. Material properties are assumed to be exponentially dependent on the distance from the interface. Using the Fourier transforms, the electro-thermoelastic problem is reduced to a singular integral equation, which is solved numerically. The stress intensity factors are computed and presented as a function of the normalized time, the nonhomogeneous and geometric parameters.     

An interaction energy integral method for nonhomogeneous materials with interfaces under thermal loading

A plane crack problem of nonhomogeneous materials with interfaces subjected to static thermal loading is investigated. A modified interaction energy integral method (IEIM) is developed to obtain the mixed-mode thermal stress intensity factors (TSIFs). Compared with the previous IEIM, the original point of this paper is: the domain-independence of the modified IEIM still stands in nonhomogeneous materials with interfaces under thermal loading. Therefore, the modified IEIM can still be applied to obtain the TSIFs of nonhomogeneous material even if the integral domain includes interfaces. The modified IEIM is combined with the extended finite element method (XFEM) to solve several thermal fracture problems of nonhomogeneous materials. Good agreement can be obtained compared with the analytic solutions and the domain-independence of the IEIM is verified. Therefore, the present method is effective to study the TSIFs of nonhomogeneous materials even when the materials contain interfaces. The influence of the discontinuity of the material properties (thermal expansion coefficient, thermal conductivity and Young’s modulus) on the TSIFs is investigated. The results show that the discontinuity of both thermal expansion coefficient and Young’s modulus affects the TSIFs greatly, while the discontinuity of thermal conductivity does not arouse obvious change of the TSIFs.     

Near-tip out of plane displacement fields for dynamic crack propagation in functionally graded materials

Asymptotic expansion for the out of plane displacement field around a crack propagating along the gradient in a functionally graded material is developed. The irregular behavior of one of the terms in the expansion at low crack speeds is further examined and a remedial solution, which is well behaved at low crack speeds, is proposed. The developed out of plane displacement field is used to estimate stress intensity factor from quasi-static finite element solution. The results indicate that inclusion of the proposed nonhomogeneity specific terms gives estimates of stress intensity factor, which are consistent with existing analytical predictions.     

Two mode III internal cracks located within two bonded functionally graded piezoelectric half planes respectively

This paper studies the mode III crack problem of two bonded functionally graded piezoelectric half planes which contain a crack respectively. These two cracks are located normal to the interface. All the material properties are assumed to vary along the direction of the crack line. A system of singular integral equations for electrically impermeable and permeable cracks is derived and solved numerically by using the Gauss–Chebyshev integration formula. The influence of the nonhomogeneous parameters and the dependence of the crack interactions on the stress and electric displacement intensity factors are investigated.     

Micromechanics-based thermoelastic model for functionally graded particulate materials with particle interactions

Thermoelastic behavior of functionally graded particulate materials is investigated with a micromechanical approach. Based on a special representative volume element constructed to represent the graded microstructure of a macroscopic material point, the relation between the averaged strains of the particle and matrix phases is derived with pair-wise particle interactions, and a set of governing equations for the thermoelastic behavior of functionally graded materials is presented. The effective coefficient of thermal expansion at a material point is solved through the overall averaged strain of two phases induced by temperature change under the stress-free condition, and is shown to exhibit a weak anisotropy due to the particle interactions within the graded microstructure. When the material gradient is eliminated, the proposed model predicts the effective coefficient of thermal expansion for uniform composites as expected. If the particle interactions are disregarded, the proposed model recovers the Kerner model. The proposed semi-analytical scheme is consistent and general, and can handle any thermal loading variation. As examples, the thermal stress distributions of graded thermal barrier coatings are solved for two types of thermal loading: uniform temperature change and steady-state heat conduction in the gradation direction.     

Antiplane internal crack normal to the edge of a functionally graded piezoelectric/piezomagnetic half plane

This paper deals with the antiplane magnetoelectroelastic problem of an internal crack normal to the edge of a functionally graded piezoelectric/piezomagnetic half plane. The properties of the material such as elastic modulus, piezoelectric constant, dielectric constant, piezomagnetic coefficient, magnetoelectric coefficient and magnetic permeability are assumed in exponential forms and vary along the crack direction. Fourier transforms are used to reduce the impermeable and permeable crack problems to a system of singular integral equations, which is solved numerically by using the Gauss-Chebyshev integration technique. The stress, electric displacement and magnetic induction intensity factors at the crack tips are determined numerically. The energy density theory is applied to study the effects of nonhomogeneous material parameter β, edge conditions, location of the crack and load ratios on the fracture behavior of the internal crack.     

Modeling method for a crack problem of functionally graded materials with arbitrary properties—piecewise-exponential model

A new model, piecewise-exponential model (PE model), is developed to investigate the crack problem of the functionally graded materials (FGMs) with arbitrary properties. In the PE model, the functionally graded material is divided into some nonhomogeneous layers along the gradient direction of the properties, with each layer’s properties varying exponentially. By this way, the real material properties can be approached by a series of exponential functions. Since the real material properties are used on both surfaces of each nonhomogeneous layer, the nature of continuously varying properties of FGMs can be approached accurately. The influences of the local nonhomogeneity on the crack-tip fields can be fully considered. By using the new model, the fracture problem of a functionally graded strip with arbitrary properties and a crack vertical to the free surfaces is studied. The integral transform method, the theory of residues and the theory of singular integral equation are applied. Some representative samples with different kinds of nonhomogeneous properties are analyzed and the corresponding stress intensity factors (SIFs) are presented. It is shown that the PE mode is effective for investigating the crack problems of the FGMs with arbitrary properties.     

Mode III fracture problem of an arbitrarily oriented crack in a FGPM strip bonded to a FGPM half plane

This paper studies the mode III electro-elastic field of a cracked functionally graded piezoelectric strip bonded to a functionally graded piezoelectric half plane. The crack is oriented in arbitrary direction. The material properties along x-axis vary in exponential form. By using the Fourier transform, the problem can be formulated into a system of singular integral equations and solved by applying the Gauss–Chebyshev integration formula. The effects come from the edge, crack orientation and the nonhomogeneous material parameters on intensity factors are discussed graphically.     

平行于功能梯度材料夹层的币型裂纹起裂条件

分析了功能梯度材料中币型裂纹的扩展问题．裂纹平行于无限域中功能梯度材料夹层,受有与裂纹面成任意角度的拉应力．假定功能梯度材料夹层与两个半无限域均匀介质完全粘合,其弹性模量沿厚度方向变化．采用基于层状材料广义Kelvin基本解的边界元方法分析裂纹问题,给出了均布正应力和剪应力作用下裂纹的应力强度因子、将应力强度因子耦合于应变能密度断裂判据,讨论了裂纹体在拉伸应力作用下的起裂条件．     

Anti-plane shear of an arbitrary oriented crack in a functionally graded strip bonded with two dissimilar half-planes

An internal crack located within a functionally graded material (FGM) strip bonded with two dissimilar half-planes and under an anti-plane load is considered. The crack is oriented in an arbitrary direction. The material properties of strip are assumed to vary exponentially in the thickness direction and two half-planes are assumed to be isotropic. Governing differential equations are derived and to reduce the difficulty of the problem dealing with solution of a system of singular integral equations Fourier integral transform is employed. Semi closed form solution for the stress distribution in the medium is obtained and mode III stress intensity factor (SIF), at the crack tip is calculated and its validity was verified. Finally, the effects of nonhomogeneous material parameter and crack orientation on the stress intensity factor are studied.     

Surface contact behavior of functionally graded thermoelectric materials indented by a conducting punch

The contact problem for thermoelectric materials with functionally graded properties is considered. The material properties, such as the electric conductivity, the thermal conductivity, the shear modulus, and the thermal expansion coefficient, vary in an exponential function. Using the Fourier transform technique, the electro-thermoelastic problems are transformed into three sets of singular integral equations which are solved numerically in terms of the unknown normal electric current density, the normal energy flux, and the contact pressure. Meanwhile, the complex homogeneous solutions of the displacement fields caused by the gradient parameters are simplified with the help of Euler's formula. After addressing the non-linearity excited by thermoelectric effects,the particular solutions of the displacement fields can be assessed. The effects of various combinations of material gradient parameters and thermoelectric loads on the contact behaviors of thermoelectric materials are presented. The results give a deep insight into the contact damage mechanism of functionally graded thermoelectric materials(FGTEMs).     

Stochastic thermoelastic problem of a functionally graded plate under random temperature load

This study attempts to derive the statistics of temperature and thermal stress in functionally graded material (FGM) plates exposed to random external temperatures. The thermomechanical properties of the FGM plates are assumed to vary arbitrarily only in the plate thickness direction. The external temperatures are expressed as random functions with respect to time. The transient temperature field in the FGM plate is determined by solving a nonhomogeneous heat conduction problem for a multilayered plate with linear nonhomogeneous thermal conductivity and different homogeneous heat capacity in each layer. The autocorrelations and power spectrum densities (PSDs) of temperature and thermal stress are derived analytically. These statistics for FGM plates composed of partially stabilised zirconia (PSZ) and austenitic stainless steel (SUS304) are computed under the condition that the fluctuation in the external temperature can be considered as white noise or a stationary Markov process.     

功能梯度材料的黏弹性断裂问题

功能梯度材料(FGM)是一种不同于传统复合材料的新型工程复合材料 [1], 国内外关于FGM的断裂力学方面的研究发展非常迅速. 关于FGM静态裂纹问题,学者们研究了不同类型裂纹尖端场的应力强度因子 [2-5], 探讨了有限长裂纹在不用载荷作用下的传播等问题. 而关于动态裂纹问题,也已经取得很大成就 [6-9]. FGM一个很重要的应用是高温结构材料,在强大的热环境中,很多材料都呈现出黏弹性. 因此,研究FGM的黏弹性断裂力学非常具有实际价值.对此,众多研究 [10-14]提出不同的分析模型,并在不同受载条件,通过理论计算,分析了黏弹性裂纹尖端场的力学 行为.本文考查了功能梯度材料板条中界面裂纹垂直于梯度方向时的黏弹性断裂问题,首先利用有限元法求解线弹性功能梯度材料板条的裂纹尖端场,然后根据黏弹性的对应性原理,求解出黏弹性功能梯度材料板条裂纹问题的应力场强度因子.      

DYNAMIC STRESS FIELD AROUND THE MODE Ⅲ CRACK TIP IN AN ORTHOTROPIC FUNCTIONALLY GRADED MATERIAL

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Thermoelastic contact mechanics for a flat punch sliding over a graded coating/substrate system with frictional heat generation

The problem of thermoelastic contact mechanics for the coating/substrate system with functionally graded properties is investigated, where the rigid flat punch is assumed to slide over the surface of the coating involving frictional heat generation. With the coefficient of friction being constant, the inertia effects are neglected and the solution is obtained within the framework of steady-state plane thermoelasticity. The graded material exists as a nonhomogeneous interlayer between dissimilar, homogeneous phases of the coating/substrate system or as a nonhomogeneous coating deposited on the substrate. The material nonhomogeneity is represented by spatially varying thermoelastic moduli expressed in terms of exponential functions. The Fourier integral transform method is employed and the formulation of the current thermoelastic contact problem is reduced to a Cauchy-type singular integral equation of the second kind for the unknown contact pressure. Numerical results include the distributions of the contact pressure and the in-plane component of the surface stress under the prescribed thermoelastic environment for various combinations of geometric, loading, and material parameters of the coated medium. Moreover, in order to quantify and characterize the singular behavior of contact pressure distributions at the edges of the flat punch, the stress intensity factors are defined and evaluated in terms of the solution to the governing integral equation.     

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。     

Quasi-static growth and dynamic extension of anti-plane shear crack in periodically nonhomogenous elastic material

The quasi-static growth of an anti-plane shear crack in a periodically nonhomogeneous elastic material is considered. An effective velocity dependent energy release rate is found for a semi-infinite crack. Approximate expressions of stress intensity factors for in-plane cracks in periodically nonhomogeneous material are also given.     

Torsional impact response of a penny-shaped crack in a functional graded strip

The torsional impact response of a penny-shaped crack in a nonhomogeneous strip is considered. The shear modulus is assumed to be functionally graded such that the mathematics is tractable. Laplace and Hankel transforms were used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Explicit expressions of both the dynamic stress intensity factor and the energy density factor were derived. And it is shown that, as crack driving force, they are equivalent for the present crack problem. Investigated are the effects of material nonhomogeneity and strip‘s highness on the dynamic fracture behavior.Numerical results reveal that the peak of the dynamic stress intensity factor can be suppressed by increasing the nonhomogeneity parameter of the shear modulus, and that the dynamic behavior varies little with the adjusting of the strip‘ s highness.