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.     

Antiplane crack at the interface of two bonded dissimilar graded piezoelectric materials

The problem of an antiplane crack situated in the interface of two bonded dissimilar graded piezoelectric half-spaces is considered under the permeable crack assumption. The mechanical and electrical properties of the half-spaces are considered for a class of functional forms for which the equilibrium equation has analytic solutions. By using an integral transform technique, the problem is reduced to dual integral equations which are transformed into a Fredholm integral equation by introducing an auxiliary function. The stress intensity factors are obtained in explicit form in terms of auxiliary functions. By solving the Fredholm integral equation numerically, the numerical results for stress intensity factors are obtained which have been displayed graphically to show the influence of the graded piezoelectric materials.     

Dynamic behavior of two collinear anti-plane shear cracks in a functionally graded layer bonded to dissimilar half planes

In recent years, the functionally graded materials (FGMs) have been widely applied in extremely high temperate environment. In this paper, the dynamic behavior of two collinear cracks in FGM layer bonded to dissimilar half planes under anti-plane shear waves is studied by the Schmidt method. By using the Fourier transform technique, the present problem can be solved with a dual integral equation. These equations are solved using the Schmidt method. The present method is used to illustrate the fundamental behavior of the interacting cracks in FGMs under dynamic loading. Furthermore, the effects of the geometry of the interacting cracks, the shear stress wave velocity of the materials and the frequency of the incident wave on the Dynamic Stress Intensity Factor are investigated.     

Interaction of three interfacial Griffith cracks between bonded dissimilar orthotropic half planes

The paper deals with the interaction of a pair of outer cracks on a central crack situated at the interface of two dissimilar orthotropic half-planes. The mixed boundary value problem is reduced to solving a pair of simultaneous singular integral equations which have finally been solved numerically by using Jacobi polynomials. The analytical expressions for stress intensity factors at the central crack tip and the expression of the strain energy release rate have been derived for general loading. Numerical values of the interaction effects of the outer cracks on the central crack have been calculated through stress magnification factors. It is seen that the interaction effects are either shielding or amplification depending on the size of the outer cracks and their spacing from the central crack.     

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.     

Angle cracks in two bonded functionally graded piezoelectric material under anti-plane shear

Consider two bonded functionally graded piezoelectric material (FGPM) with finite height. Each material contains an arbitrary oriented crack. The material properties are assumed in exponential forms in the direction normal to the interface. The crack surface condition is assumed to be electrically impermeable or permeable. Using the Fourier transform technique, the problem can be reduced to a system of singular integral equations, which are then solved numerically by applying the Gauss-Chebyshev integration formula to obtain the stress intensity factors at the crack tips. Numerical calculations are carried out to obtain the energy density factor S and the energy release rate G. In impermeable case, the energy release rate has been shown to be negative as the electric loads are applied. The positive definite characteristic of the energy density factor makes it possible for predicting the fracture behavior of the cracked structure. The influences of the non-homogeneous parameters and crack orientation on the energy density factors at the crack tips are discussed in detail. The results show that the energy density factor at the crack tip will be increased when the crack tip is located within the softer material.     

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.     

Screw dislocation interacting with twin interfacial edge cracks between two bonded dissimilar piezoelectric strips

This paper is concerned with the electroelastic potentials and the fracture parameters of a twin-edge-cracked piezoelectric bimaterial strip with a screw dislocation. By means of conformal mapping technique and the known dislocation solution, the antiplane displacement and inplane electric potentials are obtained in closed-form. The intensity factors and the energy release rate are extracted explicitly. In some limiting cases, the present solutions cover those in the literature.     

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.     

Dynamic behavior of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips

The dynamic interaction of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips subjected to the anti-plane shear harmonic stress waves was investigated. By using the Fourier transform, the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is jump of displacement across the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter, the circular frequency of the incident waves and the thickness of the strip upon stress, electric displacement and magnetic flux intensity factors of cracks.     

Mode III fracture problem of two arbitrarily oriented cracks located within two bonded functionally graded material strips

This paper shows the anti-plane crack problem of two bonded functionally graded material (FGM) strips. Each strip contains an arbitrarily oriented crack. The material properties of the strips are assumed in exponential forms varied in the direction normal to the interface. After employing the Fourier transforms, the unknowns are solved from the interface conditions, boundary conditions and the condition on the crack surfaces. The problem can then be reduced to a system of singular integral equations, which are solved numerically by applying the Gauss-Chebyshev integration formula to obtain the stress intensity factors at the crack tips. In the discussions, several degenerated problems are considered to demonstrate the influence of the non-homogeneous parameters, crack orientations, edge effects and the crack interactions on the normalized intensity factors. In general, the factors are larger when crack tips are located in stronger material. Also, the factors increase as the crack is oriented in the direction normal to the interface. The conclusions made in this research can be used to evaluate the safety of two bonded strips once the cracks exist inside the structure.     

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.     

Antiplane deformation of a partially bonded elliptical inclusion

A new method that introduces two holomorphic potential functions (the two-phase potentials) is applied to analyze the antiplane deformation of an elliptical inhomogeneity partially-bonded to an infinite matrix. Elastic fields are obtained when either the matrix is subject to a uniform longitudinal shear or the inhomogeneity undergoes a uniform shear transformation. The stress field possesses the square-root singularity of a Mode III interface crack, which, in the special case of a rigid line inhomogeneity, changes in order, as the crack tip approaches the inhomogeneity end. In the latter situation the crack-tip elastic fields are linear in two real stress intensity factors related to a strong and a weak singularity of the stress field.     

Thermal stresses around two parallel cracks in two bonded dissimilar elastic half-planes

Summary Thermal stresses around two parallel cracks in two bonded dissimilar elastic half-planes are determined. One of the cracks lies in the upper half-plane, while the other is in the lower half-plane. Uniform heat flow is assumed to be at right angles to the interface. Application of the Fourier transform technique reduces the problem to that of solving dual integral equations. To solve the equations, the difference of the crack surface temperature and those of the crack surface displacements are expanded in a series of functions which are automatically zero outside the cracks. The unknown coefficients in the series are solved by the Schmidt method. The stress intensity factors are calculated numerially for composite materials featuring a ceramic upper half-plane and a steel lower half-plane.

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Wärmespannungen um zwei parallele Risse in zwei verbundenen, verschiedenen, elastischen Halbunendlichplatten

Übersicht Es werden die Wärmespannungen um zwei parallele Risse in zwei verbundenen, verschiedenen, elastischen Halbunendlichplatten bestimmt. Einer der beiden Risse liegt in der oberen Halbunendlichplatte, der andere in der unteren. Es wird angenommen, daß ein gleichmäßiger Wärmefluß senkrecht zur Grenzfläche erfolgt. Die Anwendung der Fourier-Transformation reduziert das Problem auf die Lösung dualer Integralgleichungen. Zur Lösung der Gleichungen werden die Temperatur-sowie die Verschiebungsdifferenzen an der Rißoberfläche in eine Reihe von Funktionen entwickelt, die außerhalb der Risse automatisch zu Null werden. Die unbekannten Koeffizienten dieser Reihe werden dann über das Schmidt-Verfahren bestimmt. Anschließend werden für Verbundmaterialien, bei denen die obere Halbunendlichplatte aus Keramik und die untere aus Stahl besteht, die Spannungsintensitätsfaktoren numerisch berechnet.

     

Dynamic response of a crack in a functionally graded interface of two dissimilar piezoelectric half-planes

Summary  In this paper, the dynamic anti-plane crack problem of two dissimilar homogeneous piezoelectric materials bonded through a functionally graded interfacial region is considered. Integral transforms are employed to reduce the problem to Cauchy singular integral equations. Numerical results illustrate the effect of the loading combination parameter λ, material property distribution and crack configuration on the dynamic stress and electric displacement intensity factors. It is found that the presence of the dynamic electric field could impede of enhance the crack propagation depending on the time elapsed and the direction of applied electric impact. Received 4 December 2001; accepted for publication 9 July 2002 This work is supported by the National Natural Science Foundation of China through Grant No. 10132010.     

功能梯度夹层多个环形界面裂纹扭转冲击

研究位于功能梯度层和外部均匀材料之间多个环形界面裂纹的扭转冲击问题,功能梯度材料 (FGM)粘结在两种不同的弹性材料之间,功能梯度层和外部材料之间环形界面裂纹的数目是任意的．引进积分变换和位错密度函数将问题化为求解Laplace域里标准的Cauchy奇异积分方程,进而化为求解代数方程;应用Laplace数值反演技术,计算时域里的动应力强度因子(DSIF)．考查了结构几何尺度和材料特性对裂尖动态断裂特性的影响．数值结果表明,DSIF存在一个主峰,到达主峰后,在其相应的静态值附近波动并最终趋于稳定;增加FGM的梯度能减小DSIF的峰值．     

弹性功能梯度材料板条中周期裂纹的反平面问题

讨论了弹性功能梯度材料板条中裂纹的反平面问题. 用Fourier 变换方法得到了一个基本解. 这个基本解表示了实轴上一点作用有点位错时引起的影响. 利 用此基本解可得单裂纹和周期裂纹问题的奇异积分方程. 在周期裂纹求解时, 远处裂纹对于中央裂纹的影响作了有效的近似处理. 最后, 给出了数值结果, 它表示了材料性质对于裂纹端应力强度因子的影响.     

Anti-plane analysis of a functionally graded strip with multiple cracks

The stress fields are obtained for a functionally graded strip containing a Volterra screw dislocation. The elastic shear modulus of the medium is considered to vary exponentially. The stress components exhibit Cauchy as well as logarithmic singularities at the dislocation location. The dislocation solution is utilized to formulate integral equations for the strip weakened by multiple smooth cracks under anti-plane deformation. Several examples are solved and stress intensity factors are obtained.     

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.     

Nonlocal theory solution of two collinear cracks in the functionally graded materials

In this paper, the interaction of two collinear cracks in functionally graded materials subjected to a uniform anti-plane shear loading is investigated by means of nonlocal theory. The traditional concepts of the nonlocal theory are extended to solve the fracture problem of functionally graded materials. To make the analysis tractable, it is assumed that the shear modulus varies exponentially with the coordinate vertical to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near the crack tips. The nonlocal elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials. The magnitude of the finite stress field depends on the crack length, the distance between two cracks, the parameter describing the functionally graded materials and the lattice parameter of the materials.