Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

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.     

含不完美界面的磁电复合材料倾斜裂纹问题

研究了含非完美界面的双层压电/压磁复合材料中压电相存在一个倾斜于界面的Ⅲ型裂纹问题。采用弹簧型耦合界面模型模拟非完美界面,运用Fourier积分变换法将裂纹面条件转化为奇异积分方程,并使用Lobatto-Chebyshev方法数值求解了裂纹尖端应力强度因子（SIF）。详细地研究了裂纹尖端SIF与界面参数、压电/压磁材料参数和材料的层厚、裂纹的倾斜角、裂纹与界面的距离等几何参数的关系。结果表明：力学不完美性可以独立地增大SIF,而磁学、电学不完美性只有与力学不完美性耦合时才会减小SIF;力学-电学、力学-磁学不完美性的耦合会减小SIF,而磁学-电学不完美性的耦合不会影响SIF;磁场作用下,增大压磁层弹性模量会减小SIF,而增大压电层压电系数,减小压电层弹性模量和介电常数,均会减小SIF;界面不完美性会影响SIF随裂纹倾斜角度或裂纹与界面之间距离的变化规律;在一定范围内增加压电层或压磁层厚度可以减小SIF。     

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.     

Arbitrarily oriented microcracks near the tip of an interface macrocrack in bonded dissimilar anisotropic materials

It is well known that microcracking in brittle materials results in a reduction of the stress intensity factor (SIF) and energy release rate (ERR). The reduced SIF or ERR represents crack tip shielding which is of significant interest to micromechanics and material science researchers. However, the effect of microcracking on the SIF and ERR is a complicated subject even for isotropic homogeneous materials, and becomes much more formidable in case of interface cracks in bonded dissimilar solids. To unravel the micromechanics of interface crack tip shielding in bonded dissimilar anisotropic solids, an interface crack interacting with arbitrarily oriented subinterface microcracks in bonded dissimilar anisotropic materials is studied. After deducing the fundamental solutions for a subinterface crack under concentrated normal and tangential tractions, the present interaction problem is reduced to a system of integral equations which is then solved numerically. A J‐integral analysis is then performed with special attention focused on the J₂‐integral in a local coordinate system attached to the microcracks. Theoretical and numerical results reassert the conservation law of the J‐integral derived for isotropic materials 1 , 2 also to be valid for bonded dissimilar anisotropic materials. It is further concluded that there is a wastage when the remote J‐integral transmits across the microcracking zone from infinity to the interface macrocrack tip. In order to highlight the influence of microstructure on the interfacial crack tip stress field, the crack tip SIF and ERR in several typical cases are presented. It is interesting to note that the Mode I SIF at the interface crack tip is quite different from the ERR in bonded dissimilar anisotropic materials.     

Interfacial fracture analysis of a piezoelectric–polythene composite cylindrical shell patch under axial shear

The present work studies the interfacial fracture in a piezoelectric cylindrical shell patch. The problem is solved by the methods of infinite trigonometric series and Cauchy singular integral equation, and the numerical results of the stress intensity factor (SIF) are obtained. The effects of the interfacial radius and crack’s location on the SIF are explained through the effects of the free surface, interfacial curvature, crack length, and interface end, respectively. An optimal stiffness matching relationship between the piezoelectric layer and dielectric substrate is suggested. The effects of the piezoelectric and dielectric coefficients are explained through the mechanism of piezoelectric stiffening.     

Determining the mixed mode stress intensity factors of surface cracks in functionally graded hollow cylinders

This work reports about an investigation on mixed mode stress intensity factors (SIFs) of three-dimensional (3D) surface cracks in hollow cylinders made-up of functionally graded material (FGM). A finite element implementation of the interaction energy integral in domain form is employed to extract the SIFs. In turn, surface cracks located at the inner and outer wall of the cylinders are considered, and the influence of exponentially varying Poisson’s ratio and Young’s modulus in radial direction on the SIFs is studied in detail. The computational results reported herein show that graded materials properties can significantly affect the magnitude and the distribution of SIFs along 3D crack fronts in FGM hollow cylinders.     

金属裂纹板复合材料修补结构的超奇异积分方程方法

根据应力强度因子在线弹性范围内具有可叠加性,将金属裂纹板复合材料修补结构进行简化,在表面裂纹线弹簧模型的基础上,建立了基于超奇异积分方程的Line-Spring模型。利用第二类Chebyshev多项式展开的方法,将超奇异积分方程转化为线性方程组,推导出以裂纹面位移表示的应力强度因子表达式,得到了裂纹尖端应力强度因子的数值解,并利用虚拟裂纹闭合法加以验证。参数分析确定了影响对称修补裂纹板应力强度因子的两个主要参数:胶层界面刚度和补片与金属板刚度比,为胶接修补结构的承载能力分析以及改进设计提供理论依据。     

Study of interface influence on crack growth: Application to Solid Oxide Fuel Cell like materials design

In the present work a mesh free method is used to study the effect of interfaces on the crack propagation path in multilayered like Solid Oxide Fuel Cells (SOFCs) materials. The partition of unity principle was introduced to model the crack tip singularity. The penalty method is used to model the materials discontinuity along the interfaces. The stress intensity factors (SIF) were computed using the interaction integral. The mesh free method is implemented and verified using results from the literature. A parametric study of crack propagation in mono-layer and bi-layer material is conducted. Finally an application to SOFC unit shows that crack initiated in the anode propagates towards the anode/electrolyte interface.     

Experimental Investigations on Deformation and Fracture Behavior of Glass Sphere Filled Epoxy Functionally Graded Materials

In this paper, deformation and fracture behavior of glass sphere filled epoxy functionally graded materials (FGM) are numerically evaluated and experimentally studied. The fabrication of the FGM is described in detail, and the spatial gradation of elastic modulus and the microscopic structure in FGM are measured and analyzed. The deformation and fracture characterization of the FGM specimen with a crack oriented along the direction of the elastic gradient under three point bend are studied by the experimental and the finite element method. The influences of crack location at both the stiff and the compliant sides of the FGM specimen on crack initiation, deformation field and stress intensity factor are analyzed. The results are: (a) The neutral-axis in the FGM specimen under three-point-bending will shift toward the stiffer side; (b) The initial fracture load increases with the increase of elastic modulus at the crack tip; (c) The elastic gradients shield a crack on the compliant side and lower the stress intensity factor when compared to the one with crack on the stiff side. These results will be useful for better design and reliable evaluation of FGM.     

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.     

Stress intensity factor analysis for an interface crack between dissimilar isotropic materials under thermal stress

Thermal stresses, one of the main causes of interfacial failure between dissimilar materials, arise from different coefficients of linear thermal expansion. Two efficient numerical procedures in conjunction with the finite element method (FEM) for the stress intensity factor (SIF) analysis of interface cracks under thermal stresses are presented. The virtual crack extension method and the crack closure integral method are modified using the superposition method. The SIF analyses of some interface crack problems under mechanical and thermal loads are demonstrated. Very accurate mode separated SIFs are obtained using these methods.     

On the influence of a bimaterial interface on dynamic stress intensity factors

Summary The method of dynamic Green's function and the integral transforms are applied to investigate the elasto-dynamic stress intensity factor of a crack straddling an interface of a bimaterial composite. The crack which extends to infinity on one side is assumed to extend an arbitrary distancea on the other side of the interface. Anti-plane line loads are suddenly applied at timet=0 on either side of the crack surface at arbitrary distancesl ₁ andl ₂ from the interface. The effect of the interface on the dynamic stress field near the crack tip is studied. It is found that the transmitted wave through the interface and reflected wave from the interface serve to increase or decrease the stress field in the vicinity of the crack tip depending on the elastic properties of the two materials.     

Stress intensity factors for large aspect ratio surface and corner cracks at a semi-circular notch in a tension specimen

Stress intensity factor solutions for semi-elliptic surface and quarter-elliptic corner cracks emanating from a semi-circular notch in a tension specimen are presented. A threedimensional finite-element analysis in conjunction with the equivalent domain integral was used to calculate stress intensity factors (SIF). SIF solutions for surface or corner crack (crack length to depth ratio of 2) at a notch are presented for a wide range of crack sizes and notch radii. Results showed that the SIF are larger for larger crack lengths and for larger notch radii. The SIF are nearly constant all along the crack front for deep surface cracks and for all corner cracks analysed.     

A new weight function approach using indirect boundary integral method

A new weight function approach to determine SIF (stress intensity factor) using the indirect boundary integral method has been presented. The crack opening displacement field was represented by one boundary integral in the form of a single-layer potential whose kernel was modified from the fundamental solution. The proposed method enables the calculation of SIF using only one SIF formula without any modification of the crack geometries symmetric in a two-dimensional plane, e.g. a center crack in a plate with or without an internal hole, double edge cracks, circumferential crack or radial cracks in a pipe. The application procedure for this variety of crack geometries is very simple and straghtforward with only one SIF formula. The necessary information in the analysis is two reference SIFs. The analysis results, using several examples, verified that the present closed-form solution was in good agreement with those of the literature and applicable to various crack geometries.     

Cracking in coating-substrate composites with multi-layered and FGM coatings

This investigation evaluates, by finite element method, the stress intensity factors (SIF) of cracked multi-layered and functionally graded material (FGM) coatings of a coating-substrate composite, due to the action of uniform normal stress on the crack surfaces. The substrate is assumed to be homogeneous material, while the coating consists of multi-layered media or sigmoid FGMs. The sigmoid FGM is a kind of FGM in which the material properties of the coating are governed by two power-law functions of volume fractions such that the functions of the material property represent sigmoid distributions in the thickness direction, simply called S-FGM in this paper. For the multi-layered coatings, one, two, and four-layered homogeneous coatings with stepwise changing volume fractions are considered. The primary problem addressed herein is the appearance of a crack in the coating surface and its expansion into the substrate along the direction perpendicular to the interface between the coating and the substrate. The results show that if the coating is stiffer than the substrate, a crack in a one-layered coating is much more susceptible to propagation into the substrate than a crack in the two- or four-layered coating. But crack growth can be effectively averted by using an S-FGM coating. However, if the coating is softer than the substrate, the S-FGM coating behaves like a bridge to connect the soft coating and the stiff substrate, and facilitates the expansion of the crack expanding into the substrate. Whereas the one-layered coating can more effectively prevent the crack from propagating into the substrate than can the two- or four-layered coating. The investigation also indicates that the material gradations of S-FGMs influence SIFs obviously only when the crack tip is inside the coating that is stiffer than the substrate. As the crack extends through the coating and into the substrate, the material gradation of the S-FGM coating and the material mismatch of the multi-layered coating slightly bear on the values of SIF.     

Effect of vertex singularities on stress intensities near plate free surfaces

The stress intensity factor (SIF) distribution along the front of a through‐the‐thickness crack is significantly influenced by the presence of the 3D corner (vertex) singularities. All past 3D finite element studies indicated that for mode I, SIF rapidly decreases near the free surface and for mode II, it sharply increases. From the previous numerical simulations, it is unclear what the limiting values of SIF near the surface are and whether these values are infinite or bounded at the vertex point. In this paper, we conduct a careful finite element study and propose a theoretical equation, which describes the SIF behaviour near the vertex. We demonstrate that the asymptotic behaviour of SIF near the surface is governed by the difference in the strength of the corner and edge singularities. Furthermore, we validate our numerical approach and calculations by utilising the invariant properties of J‐integral.     

A model for interface cracks in layered orthotropic solids: Convergence of modal decomposition using the interaction integral method

The mode‐partitioning problem for bimaterial interfaces is still not resolved by the classical fracture mechanics approach in a satisfactory manner. Stress oscillations and overlapping crack faces are a direct consequence of the rigorous solution of the elastic boundary value problem, if the constitutive law changes discontinuously across the interface. Conversely, continuously varying material properties, also referred to as functionally graded materials (FGM), avoid these physically not admissible drawbacks. In this case the crack tip fields are of the same nature as those known from homogeneous materials. Therefore, the well‐established stress intensity factor concept can be used without any changes. Following this motivation an FGM‐interface model for delaminated composite beam structures was developed and its characteristics with respect to the modal decomposition of the crack tip fields were investigated. The considered beam structures consisted of two orthotropic layers, each of a different material. The spatial variation of the material properties in the interface region was modeled by a tanh ‐function introducing one transition parameter that controlled the FGM‐gradient. Four load cases were analyzed for each structural configuration: either a unit normal force or a unit bending moment was imposed on each end of the split beam. Thus, any load case can be simply reconstructed from the presented results by means of superposition. The stress intensity factors for modes I and II were then evaluated using an interaction integral method along with the finite element method. The corresponding results are given depending on the mesh density of the interface region, the integration domain and the transition parameter. In this manner, the influence of the transition parameter on the mode ratio and on the convergence behavior of the modal decomposition scheme with respect to its integration domain was identified. Finally, the ability of the FGM‐interface model to represent bimaterial interfaces while still maintaining the advantages of crack analysis in homogeneous materials was highlighted. Copyright © 2008 John Wiley & Sons, Ltd.     

Interaction between an interface and cracks

The elastic field of edge dislocation embedded in one of two well bonded blocks is easily given by combining the edge dislocation solution in infinite plane and the method of analytic function. Using the elementary field and the principle of superposition, the problem of interaction between multi-cracks and interface is converted into a group of singular integral equations which can be numerically solved with the aid of the Chebyshev polynomial. The analytical results show that the connection between the stress intensity factors (SIF) and the Dunders' parameters depends on the bimaterial system. Furthermore, the depth beneath the interface is presented where the effect of the interface on SIF of cracks is reduced to a specified small scale. The authors point out that the depth is largely dependent on load phase angle and the distribution of cracks, but a little dependent on the orientation of the crack when there exists only one.     

A new weight function approach using indirect boundary integral method

A new weight function approach to determine SIFs (stress intensity factors) using the indirect boundary integral method has been presented. The crack opening displacement field was represented by one boundary integral term in the form of a single-layer potential whose kernel was modified from the fundamental solution. The proposed method enables the calculation of SIFs using only one SIF solution, without any modification for the crack geometries symmetric in the two-dimensional plane, e.g. a center crack in a plate with or without an internal hole, double edge cracks, circumferential cracks or radial cracks in a pipe. The application procedure for this variety of crack geometries is very simple and straightforward with only one SIF solution. The necessary information in the analysis is two reference SIFs. The analysis results using several examples verified that the present closed-form solution was in good agreement with those of the literature and applicable to various crack geometries.