Mixed-mode J-integral formulation and implementation using graded elements for fracture analysis of nonhomogeneous orthotropic materials

The path-independent J_k^*-integral, in conjunction with the finite element method (FEM), is presented for mode I and mixed-mode crack problems in orthotropic functionally graded materials (FGMs) considering plane elasticity. A general procedure is presented where the crack is arbitrarily oriented, i.e. it does not need to be aligned with the principal orthotropy directions. Smooth spatial variations of the independent engineering material properties are incorporated into the element stiffness matrix using a “generalized isoparametric formulation”, which is natural to the FEM. Both exponential and linear variations of the material properties are considered. Stress intensity factors and energy release rates for pure mode I and mixed-mode boundary value problems are numerically evaluated by means of the equivalent domain integral especially tailored for orthotropic FGMs. Numerical results are discussed and validated against available theoretical and numerical solutions.     

XFEM fracture analysis of orthotropic functionally graded materials

In the present study, the extended finite element method (XFEM) has been used for fracture analysis of orthotropic functionally graded materials. Orthotropic crack tip enrichments have been used to reproduce the singular stress field near a crack tip. Moreover, the incompatible interaction integral method has been employed to extract the stress intensity factor components. Accuracy and convergence of the proposed method have been evaluated by numerical examples and quality results have been obtained by far fewer DOFs. Also, crack propagation in isotropic and orthotropic FGMs in the presence of crack tip enrichments has been investigated and various propagation criteria have been compared, and verified, if available, by experimental and numerical data in the literature. Application of XFEM in combination of the maximum circumferential tensile stress criterion for investigation of crack propagation in orthotropic FGM problems is performed for the first time.     

Determination of thermal stress intensity factors for an interface crack in a graded orthotropic coating-substrate structure

The thermal fracture problem of an interface crack between a graded orthotropic coating and the homogeneous substrate is investigated by two different approaches. For the case that most of the material properties in the graded orthotropic coating are assumed to vary as an exponential function, the integral transform and singular integral equation technique is used to obtain some analytical results. In order to analyze the case with more complex material distribution, an interaction integral is presented to evaluate the thermal stress intensity factors of cracked functionally graded materials (FGMs), and then the element-free Galerkin method (EFGM) is developed to obtain the final numerical results. The good agreement is obtained between the numerical results and the analytical ones. In addition, the influence of material gradient parameters and material distribution on the thermal fracture behavior is also presented.     

Interaction integrals for thermal fracture of functionally graded materials

This paper addresses finite element evaluation of the non-singular T-stress and mixed-mode stress intensity factors in functionally graded materials (FGMs) under steady-state thermal loads by means of interaction integral. Interaction integral provides an accurate and efficient numerical framework in evaluating these fracture parameters in FGMs under thermal as well as mechanical loads. We use a non-equilibrium formulation and the corresponding auxiliary (secondary) fields tailored for FGMs. Graded finite elements have been developed to account for the spatial gradation of thermomechanical properties. This paper presents various numerical examples in which the accuracy of the present method is verified.     

Interface crack problems in graded orthotropic media: Analytical and computational approaches

Interface crack problems in graded orthotropic media are considered using analytical and computational techniques. In the analytical formulation an interface crack between a graded orthotropic coating and a homogeneous orthotropic substrate is considered. The principal axes of orthotropy are assumed to be parallel and perpendicular to the crack plane. Mechanical properties of the medium are assumed to be continuous with discontinuous derivatives at the interface. The problem is formulated in terms of the averaged constants of plane orthotropic elasticity and reduced to a pair of singular integral equations which are solved numerically to compute the mixed mode stress intensity factors and the energy release rate. In the second part of the study, enriched finite elements are formulated and implemented for graded orthotropic materials. Comparisons of the finite element and analytical results show that enriched finite element technique is capable of producing highly accurate results for crack problems in graded orthotropic media. Finally, periodic interface cracking and the four point bending test for graded orthotropic solids are modeled using enriched finite elements and the results are briefly discussed.     

Stress intensity factor evaluations for a curved crack in orthotropic particulate composites using an interaction integral method

This paper develops a domain-independent interaction integral (DII-integral) for extracting mixed-mode stress intensity factors (SIFs) for orthotropic materials with complex interfaces. The DII-integral does not require material property gradients, and moreover its validity is not affected by material interfaces. Combined with the extended finite element method (XFEM), the DII-integral is employed to investigate a straight crack in an orthotropic functionally graded plate and a curved crack in orthotropic particulate composites.     

Thermal fracture analysis of a functionally graded orthotropic strip with a crack

This paper is concerned with the thermal fracture problem of a functionally graded orthotropic strip, where the crack is situated parallel to the free edges. All the material properties are assumed to be dependent only on the coordinate y (perpendicular to the crack surfaces). By using Fourier transform, the thermoelastic problem is reduced to those that involve a system of singular integral equations. Numerical results are presented to show the effects of the crack position and the material distribution on the thermal stress intensity factors.     

Finite element evaluation of mixed mode stress intensity factors in functionally graded materials

This paper is directed towards finite element computation of fracture parameters in functionally graded material (FGM) assemblages of arbitrary geometry with stationary cracks. Graded finite elements are developed where the elastic moduli are smooth functions of spatial co‐ordinates which are integrated into the element stiffness matrix. In particular, stress intensity factors for mode I and mixed‐mode two‐dimensional problems are evaluated and compared through three different approaches tailored for FGMs: path‐independent J^*_k‐integral, modified crack‐closure integral method, and displacement correlation technique. The accuracy of these methods is discussed based on comparison with available theoretical, experimental or numerical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

梯度复合材料裂纹扩展路径和起裂载荷的有限元分析

为了模拟功能梯度材料(FGM)在工程应用中可能会出现的断裂问题并计算相应的开裂载荷,通过编写用户自定义UEL子程序将梯度扩展单元嵌入到ABAQUS软件中模拟功能梯度材料的物理场,并编写交互能量积分后处理子程序计算裂纹尖端的混合模式应力强度因子(SIF),采用最大周向应力准则编写子程序计算裂纹的偏转角,并模拟了裂纹扩展路径,计算了裂纹的起裂载荷。讨论了材料梯度参数对裂纹扩展路径以及起裂载荷的影响规律。通过与均匀材料的对比,验证了功能梯度材料断裂性能的优越性。研究表明:外载平行于梯度方向时,垂直梯度方向的初始裂纹朝着等效弹性模量小的方向扩展,且偏转角在梯度指数线性时出现峰值,并随着组分弹性模量比的增加而变大;当外载和初始裂纹均平行于梯度方向时,材料等效弹性模量和断裂韧性的增加或者梯度指数的减小都导致起裂载荷变大。     

On the Moving Griffith Crack in a Nonhomogeneous Orthotropic Strip

A finite crack with constant length (Yoffe type crack) propagating in the functionally graded orthotropic strip under the plane loading is investigated by means of the Schmidt method. By using the Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of material properties, the thickness of the functionally graded orthotropic strip and the speed of the crack propagating upon the dynamic fracture behavior.     

K‐dominance of static crack tip in functionally gradient materials

K‐dominance of static crack tip in functionally gradient materials (FGMs) with a crack oriented along the direction of the elastic gradient is studied through coherent gradient sensing (CGS), digital speckle correlation method (DSCM) and finite element method (FEM). In the direction of crack propagation, the shear modulus has a linear variation with constant mass density and Poisson's ratio. First, the CGS and DSCM governing equations related to the measurements and the elastic solutions at mode I crack in FGMs are obtained in terms of the stress intensity factor, material constants and graded index. Secondly, two kinds of FGMs specimens and one homogenous specimen are prepared to observe the influences of the property variation on the K‐dominance. Then, CGS and DSCM experiments using three‐point‐bending of FGMs and homogenous beams are performed. Thirdly, based on the results of the experiments, the stress intensity factors of three kinds of specimens are calculated by CGS and DSCM. Meanwhile, the stress intensity factors are obtained by FEM. Finally, comparing the results from CGS, DSCM and FEM, the K‐dominance of mode‐I static crack tip in FGMs is discussed in detail.     

Crack analysis in unidirectionally and bidirectionally functionally graded materials

Mixed-mode crack analysis in unidirectionally and bidirectionally functionally graded materials is performed by using a boundary integral equation method. To make the analysis tractable, the Young's modulus of the functionally graded materials is assumed to be exponentially dependent on spatial variables, while the Poisson's ratio is assumed to be constant. The corresponding boundary value problem is formulated as a set of hypersingular traction boundary integral equations, which are solved numerically by using a Galerkin method. The present method is especially suited for straight cracks in infinite FGMs. Numerical results for the elastostatic stress intensity factors are presented and discussed. Special attention of the analysis is devoted to investigate the effects of the material gradients and the crack orientation on the elastostatic stress intensity factors.     

Developing new enrichment functions for crack simulation in orthotropic media by the extended finite element method

New enrichment functions are proposed for crack modelling in orthotropic media using the extended finite element method (XFEM). In this method, Heaviside and near‐tip functions are utilized in the framework of the partition of unity method for modelling discontinuities in the classical finite element method. In this procedure, by using meshless based ideas, elements containing a crack are not required to conform to crack edges. Therefore, mesh generation is directly performed ignoring the existence of any crack while the method remains capable of extending the crack without any remeshing requirement. Furthermore, the type of elements around the crack‐tip remains the same as other parts of the finite element model and the number of nodes and consequently degrees of freedom are reduced considerably in comparison to the classical finite element method. Mixed‐mode stress intensity factors (SIFs) are evaluated to determine the fracture properties of domain and to compare the proposed approach with other available methods. In this paper, the interaction integral (M‐integral) is adopted, which is considered as one of the most accurate numerical methods for calculating stress intensity factors. Copyright © 2006 John Wiley & Sons, Ltd.     

Thermo-mechanical behavior of a viscoelastic FGMs coating containing an interface crack

In this paper the plane thermo-mechanical behavior of a crack in a viscoelastic functionally graded materials (FGMs) coating with arbitrary material properties bonded to a homogeneous substrate is studied. In order to avoid the complex forms that describe the viscoelastic properties of FGMs, a multi-layered model for the FGMs coating is developed. The compliance and thermal conductivity in the multi-layered model linearly vary in each layer. In this mixed boundary value problem, the system is reduced to singular integral equations and solved numerically with the Lobatto-Chebyshev collocation technique. Using the correspondence principle and Laplace transform, the problem of an interface crack between a homogeneous substrate and a viscoelastic FGMs is solved. Some numerical examples are given to demonstrate the accuracy, efficiency and versatility of the multi-layered model. The numerical results confirm that the fracture toughness of materials can be greatly improved by the graded variation of material parameters. It is also confirmed that the specific variation of material parameters greatly influences the fracture behavior of viscoelastic FGMs coating.     

Dynamic stress intensity factors around two parallel cracks in a functionally graded layer bonded to dissimilar half-planes subjected to anti-plane incident harmonic stress waves

The time-harmonic problem of determining the stress field around two parallel cracks in functionally graded materials (FGMs) is studied. The Fourier transform technique is used to reduce the boundary conditions to four simultaneous integral equations which are then solved by expanding the differences of crack surface displacements in a series. The unknown coefficients in the series are obtained by the Schmidt method. Numerical calculations are carried out for dynamic stress intensity factors (DSIF) in FGMs.     

Residual stress and thermal properties of zirconia/metal (nickel, stainless steel 304) functionally graded materials fabricated by hot pressing

To analyse the residual stress and the thermal properties of functionally graded materials (FGMs), disc-type tetragonal zirconia polycrystal (TZP)/Ni- and TZP/stainless steel 304 (SUS304)-FGM were hot pressed, and compared with directly jointed materials (DJMs). The continuous interface and the microstructure of FGMs were characterized by electron probe microanalysis, wavelength dispersive spectrometry, optical microscopy and scanning electron microscopy. It has been verified that the defect-like cracks in FGMs are induced by the preferential shear stress and shown to cause fracture. These facts have corresponded well with the residual stress distribution analysed by the finite element method. The thermal diffusivity and the thermal conductivity of FGMs and DJMs were also measured by the laser flash technique. As a consequence, this work has described the interfacial stability, the residual stress release mechanism and the thermal protection characteristics of FGMs via a compositional gradient. This revised version was published online in November 2006 with corrections to the Cover Date.     

A thermomechanical fracture modeling and simulation for functionally graded solids using a residual-strain formulation

This paper addresses mixed-mode crack growth in two-dimensional functionally graded solids under thermomechanical loads, and investigates the effect of mechanical and thermal loads as well as the T-stress on their crack growth behavior. A novel residual strain-based formulation in the interaction integral method is developed and used for the accurate evaluation of mixed-mode stress intensity factors and/or the T-stress. Simulation of mixed-mode crack propagation in functionally graded materials including solid oxide fuel cells under thermomechanical loads is performed by means of the finite element method and the generalized interaction integrals in conjunction with a remeshing algorithm. An iterative procedure is used for crack growth simulation including the calculation of mixed-mode stress intensity factors and/or the T-stress by means of the generalized interaction integral method, determination of crack growth direction and crack initiation condition based on selected fracture criteria, and local automatic remeshing along the crack path. The present approach employs a user-defined crack increment at the beginning of the simulation. Crack trajectories and fracture parameters obtained by the present simulation for thermomechanical loads are assessed for some numerical examples in comparison with those for mechanical loads.     

Dynamic stress intensity factor of a functionally graded material under antiplane shear loading

Summary The dynamic response of a finite crack in an unbounded Functionally Graded Material (FGM) subjected to an antiplane shear loading is studied in this paper. The variation of the shear modulus of the functionally graded material is modeled by a quadratic increase along the direction perpendicular to the crack surface. The dynamic stress intensity factor is extracted from the asymptotic expansion of the stresses around the crack tip in the Laplace transform plane and obtained in the time domain by a numerical Laplace inversion technique. The influence of graded material property on the dynamic intensity factor is investigated. It is observed that the magnitude of dynamic stress intensity factor for a finite crack in such a functionally graded material is less than in the homogeneous material with a property identical to that of the FGM crack plane.     

功能梯度压电带中偏心裂纹对SH波的散射问题

功能梯度材料在机械、光电、核能、生物工程领域的应用非常广泛.但由于生产技术及工作环境等方面的原因,功能梯度材料内部常常产生各种形式的裂纹并最终导致材料破坏,这将会给材料所处的整个系统带来巨大损失.因此研究功能梯度材料的断裂问题对于该种材料的设计,制备和合理、安全的应用具有极大的促进作用.本文在压电材料线性宏观理论下,研究了功能梯度压电带中偏心裂纹对SH波的散射问题.借助于积分变换方法,在电非渗透型边界条件的情况下,将所考虑的问题转化为奇异积分方程,运用Gauss-Chebyshev数值积分方法对奇异积分方程进行了数值求解,进而得到了裂纹尖端的应力和电位移强度因子.     

Mixed-mode dynamic crack propagation in graded materials under thermo-mechanical loading

Mixed-mode dynamic crack growth behavior in functionally graded materials (FGMs) under thermo-mechanical loading is studied. Asymptotic analysis in conjunction with displacement potentials has been used to develop thermo-mechanical stress fields for a mixed mode propagating crack in a FGM. The shear modulus, mass density, thermal conductivity and coefficient of thermal expansion of the FGM are assumed to vary exponentially along the gradation direction. First, asymptotic temperature fields are derived for an exponential variation of thermal conductivity and later these temperature fields are used in deriving stress fields. Using asymptotic thermo-mechanical stress fields the variation of maximum shear stress, circumferential stress and strain-energy density as a function of temperature around the crack tip are generated. Finally, utilizing the minimum strain-energy density criterion and the maximum circumferential stress criterion, the crack growth direction for various crack-tip speeds, non-homogeneity coefficients and temperature fields are determined.