Weight function for an elliptic crack in an infinite medium-II. – Shear loading

In a recent work (Roy and Saha, 2000) analytical expression for weight function has been deduced for an elliptic crack under normal loading. The integral equation method developed earlier was used in conjunction with Jacobi's polynomials in solving the problem. In the present work the same method is used to solve the title problem. New results of weight function have been given for the first time for the case of an elliptic crack under shear loading. The analytical expressions for mode II and mode III weight functions given, can be used to evaluate mode II and mode III stress intensity factors for an elliptic crack under polynomial as well as non-polynomial loadings. Some examples have been cited as illustrations.     

Determination of stress intensity factor solutions for cracks in finite-width functionally graded materials

A generalized method to determine the stress intensity factor equations for cracks in finite-width specimens of functionally graded materials (FGMs), based on force balance in regions ahead of the crack tip is provided. The method uses the Westergaard's stress distribution ahead of the crack in an infinite plate and is based on the requirement of isostrain deformation of layers of varying moduli ahead of the crack tip. It is shown that the modified Westergaard equation describes the normal stress distribution and the singular stress state ahead of the crack tip in a reasonably accurate manner. Based on this, closed-form analytical equations for the stress intensity factors of cracks in finite-width center cracked specimens were derived. Comparisons of the K values from the analytical equations with that obtained from FEM simulations indicate that the derived stress intensity factor equations for FGMs are reasonably accurate. For the finite-width center-cracked-tension (CCT) specimen, the errors are less than 10% for most of the crack lengths for materials with the outer layer modulus ratios varying from 0.2 to 5. The stress intensity factors were found to be sensitive to the absolute values of moduli of the layers, the modulus ratio of the outer layers as well as the nature of gradation including the increasing and the decreasing functional forms. The stress intensity factor equations are convenient for engineering estimates of stress intensity factors as well as in the experimental determinations of fracture toughness of FGMs.     

含任意方向裂纹功能梯度材料的应力分析研究

功能梯度材料是在航空航天领域的需求背景下发展起来的,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,因此研究含任意方向裂纹功能梯度材料的断裂问题具有重要意义。以含有任意方向裂纹的功能梯度材料为对象,运用积分变换方法,给出了相应材料平面问题的位移场的形式解。通过引入辅助函数并利用相关条件,可将问题转化为求解一组带有Cauchy核的奇异积分方程,继而采用Lobatto-Chebyshev方法对奇异积分方程进行数值求解。最后分析了裂纹方向、材料非均匀指数、载荷条件对混合型应力强度因子的影响。      

Calculation of mixed mode stress intensity factors for an elliptical subsurface crack under arbitrary normal loading

Normal loading causes mixed fracture modes in an elliptical subsurface crack because of the nonsymmetrical geometry with respect to the crack face. In this paper, mixed mode weight functions (MMWFs) for elliptical subsurface cracks in an elastic semi‐infinite space under normal loading are derived. Reference mixed mode stress intensity factors (MMSIFs), calculated by finite element analysis, under uniform normal loading are used to derive MMWFs. The cracks have aspect ratios and crack depth to crack length ratios of 0.2–1.0 and 0.05 to infinity, respectively. MMWFs are used to calculate MMSIFs for any point of the crack front under linear and nonlinear two‐dimensional (2D) loadings. So, in order to evaluate the fatigue crack growth phenomenon under complicated 2D stress distributions, MMWFs can be easily used. The comparison between the MMSIFs obtained from the MMWFs and finite element analysis indicates high accuracy.     

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.     

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.     

Fracture mechanics analysis of geometrically nonlinear shear deformable plates

This paper presents boundary integral equations for fracture mechanics analysis of geometrically nonlinear shear deformable plates. A radial basis function and dual reciprocity method are utilized to evaluate the derivative terms and the domain integrals that appear in the formulations, respectively. Numerical examples of the clamped and simply supported plates containing a center crack subjected to uniform transversal loadings are presented. Displacement extrapolation technique is used to compute the stress intensity factors (SIFs). Stress intensity factors of mode I for plate bending and membrane problems are presented. The normalized stress intensity factors in membrane significantly increase after few increments of the load while the normalized stress intensity factors in bending decrease. Less displacement and rotational constraints in cracked plates under uniform transversal loadings will raise the stress intensity factors. The bending stress intensity factors of a central crack in clamped square plate were found to be the highest values compared to those for clamped non-square plates.     

Thermal fracture analysis of orthotropic functionally graded materials using an equivalent domain integral approach

A new computational method based on the equivalent domain integral (EDI) is developed for mode I fracture analysis of orthotropic functionally graded materials (FGMs) subjected to thermal stresses. By using the constitutive relations of plane orthotropic thermoelasticity, generalized definition of the J-integral is converted to an equivalent domain integral to calculate the thermal stress intensity factor. In the formulation of the EDI approach, all the required thermomechanical properties are assumed to have continuous spatial variations through the functionally graded medium. Developed methodology is integrated into a fracture mechanics research finite element code FRAC2D using graded finite elements that possess cubic interpolation. Steady-state and transient temperature distribution profiles in orthotropic FGMs are computed using the finite elements based heat transfer analysis software HEAT2D. EDI method is validated and domain independence is demonstrated by comparing the numerical results obtained using EDI to those calculated by an enriched finite element method and to those available in the literature. Single and periodic edge crack problems in orthotropic FGMs are examined in order to study the influences of principal thermal expansion coefficient and thermal conductivity components, relative crack length and crack periodicity on the thermal stress intensity factors. Numerical results show that among the three principal thermal expansion coefficient components, the in-plane component perpendicular to the crack axis has the most significant influence on the mode I stress intensity factor. Gradation profile of the thermal expansion coefficient parallel to the crack axis is shown to have no effect on the outcome of the fracture analysis.     

A periodic array of cracks in functionally graded materials subjected to transient loading

A periodic array of cracks in an infinite functionally graded material under transient mechanical loading is investigated. In-plane normal (mode I) and shear (mode II) loading conditions are considered. For each individual loading mode, a singular integral equation is derived, in which the crack surface displacements are unknown functions. Numerical results are obtained to illustrate the variation of the stress intensity factors as a function of the crack periodicity for different values of material inhomogeneity, either at the transient state or steady state. The material inhomogeneity can increase or decrease the mode I and mode II stress intensity factors. Compared with the single crack solution, it is also shown that multiple cracking may decrease the mode I stress intensity factors, but enhance the mode II stress intensity factors significantly.     

Finite Element Weight Function Application for a Cracked Disk

A practical application of the weight function method is used in this paper in order to find values of the stress intensity factor for a cracked disk subjected to different loadings. The finite element method is used in order to obtain discrete values of the crack face displacement in a reference loading case, namely inertia forces due to uniform rotation. These values are interpolated and a general expression of the displacements is obtained, which is further used to determine the stress intensity factor in this case. With the weight function equation, stress intensity factors for other loadings are obtained and the results are compared with those reported by other authors. Very good agreement was obtained, showing thus the reliability of this approach.     

Determination of approximate point load weight functions for embedded elliptical cracks

This paper presents the application of weight function method for the calculation of stress intensity factors in embedded elliptical cracks under complex two-dimensional loading conditions. A new general mathematical form of point load weight function is proposed based on the properties of weight functions and the available weight functions for two-dimensional cracks. The existence of this general weight function form has simplified the determination of point load weight functions significantly. For an embedded elliptical crack of any aspect ratio, the unknown parameters in the general form can be determined from one reference stress intensity factor solution. This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body. The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to fluctuating non-linear stress fields resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

A semi-infinite-crack model for determining mode I stress intensity factors using crack surface displacements

A semi-infinite-crack model is used to supplement the conic section simulation method for determining stress intensity factors of finite cracked bodies under mode I loadings. The actual displaced crack surface profile is found by finite element analysis. For each crack surface segment between two neighbouring nodes, a set of model parameters is found by using the displacements of these two nodes. A stress intensity factor estimate is then calculated from the closed-form formula associated with the model. It is found that near-tip crack surface displacements produce model parameters that are sufficient for quantifying the stress intensity factor. The semi-infinite-crack model can be used either as a stand alone model or in conjunction with the ellipse simulation procedure to form a systematic approach. It is shown that this model can be applied to different geometries and loadings with excellent accuracy.     

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.     

Direct adjustment procedure for weight functions of graded materials

The weight function procedure simplifies the determination of stress intensity factors. If the weight function is known for a crack in a component of homogeneous material, the stress intensity factor can be obtained by multiplying this function by the stress distribution and integrating it over the crack length. In the case of graded materials, weight functions are seldom available in the literature. The main point of this paper is to demonstrate whether or not the approximate direct adjustment procedure for the derivation of weight functions is also valid in the case of graded materials. In this study it has been found that the procedure is applicable and no changes are necessary compared with the procedure for homogeneous materials.     

A new weight function for one‐dimensional subsurface cracks under general loading

In the present study, weight functions (WFs) of a subsurface crack were derived by proposing a new general form for approximate one‐dimensional WF. The WFs coefficients were considered as a function of crack length to depth ratio and were obtained using reference stress intensity factors (SIFs) of 16 cracks under uniform, linear, and parabolic normal and shearing loadings. The verification was performed by comparison of the straight and coupled SIFs calculated by WF and finite element modelling under some complicated loadings. In conclusion, the WFs can be simply and effectively employed for evaluating the cracks under any complex stress distributions.     

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.     

Precise computation and error control of stress intensity factors and certain integral characteristics in anisotropic inhomogeneous materials

The numerical simulation of quasi-static crack propagation is closely related to the computation of characteristics such as stress intensity factors or energy release rates. In this work, ideas are proposed, how such quantities can be calculated precisely in linear elastic, anisotropic and inhomogeneous plane structures. Stress intensity factors and other local characteristics can be evaluated in terms of functionals depending on solutions of certain elasticity problems. The approach used here to calculate these functional values precisely with Galerkin finite elements is a dual-weighted-residual method for adaptive mesh refinement and a posteriori error control. Especially in structures under mixed-mode loadings contact of crack faces can occur. A numerical realization of mutual non-penetration conditions of inequality type for the crack faces and the effect of such constraints on stress intensity factors is shown. Numerical results are presented for anisotropic and functionally graded materials.     

A weight function method for mixed modes hole‐edge cracks

A weight function approach is proposed to calculate the stress intensity factor and crack opening displacement for cracks emanating from a circular hole in an infinite sheet subjected to mixed modes load. The weight function for a pure mode II hole‐edge crack is given in this paper. The stress intensity factors for a mixed modes hole‐edge crack are obtained by using the present mode II weight function and existing mode I Green (weight) function for a hole‐edge crack. Without complex derivation, the weight functions for a single hole‐edge crack and a centre crack in infinite sheets are used to study 2 unequal‐length hole‐edge cracks. The stress intensity factor and crack opening displacement obtained from the present weight function method are compared well with available results from literature and finite element analysis. Compared with the alternative methods, the present weight function approach is simple, accurate, efficient, and versatile in calculating the stress intensity factor and crack opening displacement.     

The fracture analysis of a graded coating/substrate system of finite thickness with arbitrary spatial variations of coating properties: Plane deformation

A new multi-layered model for functionally graded materials (FGMs) with continuously varying elastic properties is developed. The model divides the FGM into multiple layers. In each layer the material properties vary linearly and are continuous on the sub-interfaces. With this new multi-layered model, we solve the crack problems of an FGM coated strip under the in-plane deformation. The method employs the Fourier integral transform technique and singular integral equation theory. The stress intensity factors are calculated. Comparisons between the present model and other existing models show some advantages of the new model: (i) it involves no discontinuities of the material properties at the sub-interfaces; and (ii) it can be used to analyze the crack problems of FGMs with properties of arbitrary variations.