Anti-plane fracture of a functionally graded material strip

This paper considers the anti-plane (or mode III) crack problem in a functionally graded material strip. The shear modulus of the strip is considered for a class of functional forms for which the equilibrium equation has an analytical solution. The problem is solved by means of singular integral equation technique. Both a single crack and a series of collinear cracks are studied. The results are tabulated and plotted to show the effect of the material nonhomogeneity and crack location on the stress intensity factors.     

Griffith crack moving in a piezoelectric strip

Summary  The dynamic problem of an impermeable crack of constant length 2a propagating along a piezoelectric ceramic strip is considered under the action of uniform anti-plane shear stress and uniform electric field. The integral transform technique is employed to reduce the mixed-boundary-value problem to a singular integral equation. For the case of a crack moving in the mid-plane, explicit analytic expressions for the electroelastic field and the field intensity factors are obtained, while for an eccentric crack moving along a piezoelectric strip, numerical results are determined via the Lobatto–Chebyshev collocation method for solving a resulting singular integral equation. The results reveal that the electric-displacement intensity factor is independent of the crack velocity, while other field intensity factors depend on the crack velocity when referred to the moving coordinate system. If the crack velocity vanishes, the present results reduce to those for a stationary crack in a piezoelectric strip. In contrast to the results for a stationary crack, applied stress gives rise to a singular electric field and applied electric field results in a singular stress for a moving crack in a piezoelectric strip. Received 14 August 2001; accepted for publication 24 September 2002 The author is indebted to the AAM Reviewers for their helpful suggestions for improving this paper. The work was supported by the National Natural Science Foundation of China under Grant 70272043.     

Eccentric crack in a piezoelectric strip bonded to half planes

We consider the problem of determining the singular stresses and electric fields in a piezoelectric ceramic strip containing an eccentric Griffith crack off the centre line bonded to two elastic half planes under anti-plane shear loading using the continuous crack-face condition. Fourier transforms are used to reduce the problem to the solution of two pairs of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and energy release rate are obtained.     

Stress intensity factor for orthotropic three layered material with crack under arbitrary anti-plane shear

A three-layered orthotropic material with a center crack is subjected to an arbitrary anti-plane shear loading. The problem is formulated as a mixed boundary value problem by using the Fourier integral transform method. This gives a Fredholm integral equation of the second kind. The integral equation is solved numerically and anti-plane shear stress intensity factors are analyzed in terms of the product of shear modulus, material orthotropy for each layer and the thickness of the outer layer and the order of the loading polynomial.     

A MOVING CRACK IN A NONHOMOGENEOUS MATERIAL STRIP

This paper considers an anti-plane moving crack in a nonhomogeneous material strip of finite thickness. The shear modulus and the mass density of the strip are considered for a class of functional forms for which the equilibrium equation has analytical solutions. The problem is solved by means of the singular integral equation technique. The stress field near the crack tip is obtained. The results are plotted to show the effect of the material non-homogeneity and crack moving velocity on the crack tip field. Crack bifurcation behaviour is also discussed. The paper points out that use of an appropriate fracture criterion is essential for studying the stability of a moving crack in nonhomogeneous materials. The prediction whether the unstable crack growth will be enhanced or retarded is strongly dependent on the type of the fracture criterion used. Based on the analysis, it seems that the maximum 'anti-plane shear' stress around the crack tip is a suitable failure criterion for moving cracks in nonhomogeneous materials.     

DYNAMIC BEHAVIOR OF SEVERAL CRACKS IN FUNCTIONALLY GRADED STRIP SUBJECTED TO ANTI-PLANE TIME-HARMONIC CONCENTRATED LOADS

This paper contains a theoretical formulations and solutions of multiple cracks sub- jected to an anti-plane time-harmonic point load in a functionally graded strip. The distributed dislocation technique is used to construct integral equations for a functionally graded material strip weakened by several cracks under anti-plane time-harmonic load. These equations are of Cauchy singular type at the location of dislocation, which are solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to evaluate the stress intensity factor and strain energy density factors （SEDFs） for multiple cracks with differ- ent configurations. Numerical calculations are presented to show the effects of material properties and the crack configuration on the dynamic stress intensity factors and SEDFs of the functionally graded strip with multiple curved cracks.     

Mode III fracture of an arbitrary oriented crack in two dimensional functionally graded material

In this paper, a two dimensional functionally graded material (2D-FGM) under an anti-plane load with an internal crack is considered. The crack is oriented in an arbitrary direction. The material properties are assumed to vary exponentially in two planar directions. The problem is analyzed and solved by two different methods namely Fourier integral transforms with singular integral equation technique, and then by the finite element method. The effects of crack orientation, material non-homogeneity, and other parameters on the value of stress intensity factor (SIF) are studied. Finally, the obtained results for Mode III stress intensity factor of different methods are compared.     

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.     

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.     

Closed-form solution for two collinear mode-III cracks in an orthotropic elastic strip of finite width

The problem of an orthotropic strip containing two collinear cracks normal to the strip boundaries is considered. The Fourier series method is used to reduce the associated boundary value problem to triple series equations, then to a singular integral equation, which can be solved analytically. Under remote uniform antiplane shear loading, the stress field and the crack sliding displacement are determined analytically and stress intensity factors are also given in a closed form.     

功能梯度压电板条中电绝缘型运动裂纹的电弹性场

基于三维弹性理论和压电理论，对材料系数按指数函数规律分布的功能梯度压电板条中的反平面运动裂纹问题进行了求解。利用Fourier积分变换方法将电绝缘型运动裂纹问题化为对偶积分方程，并进一步归结为易于求解的第二类Fredholm积分方程。通过渐近分析，获得了裂纹尖端应力、应变、电位移和电场的解析解，给出了裂纹尖端场各个变量的角分布函数，并求得了裂纹尖端场的强度因子，分析了压电材料物性梯度参数、几何尺寸及裂纹运动速度对它们的影响。结果表明，对于电绝缘型裂纹，功能梯度压电板条中运动裂纹尖端附近的各个场变量都具有-1／2阶的奇异性；当裂纹运动速度增大时，裂纹扩展的方向会偏离裂纹面。     

FRACTURE ANALYSIS OF A FUNCTIONALLY GRADED STRIP UNDER PLANE DEFORMATION

In this paper the plane elasticity problem for a functionally graded strip containing a crack is considered. It is assumed that the reciprocal of the shear modulus is a linear function of the thickness-coordinate, while the Possion's ratio keeps constant. By utilizing the Fourier transformation technique and the transfer matrix method, the mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. The influences of the geometric parameters and the graded parameter on the stress intensity factors and the strain energy release rate are investigated. The numerical results show that the graded parameters, the thickness of the strip and the crack size have significant effects on the stress intensity factors and the strain energy release rate.     

Fracture behavior of a bonded magneto-electro-elastic rectangular plate with an interface crack

In this paper the anti-plane problem for an interface crack between two dissimilar magneto-electro-elastic plates subjected to anti-plane mechanical and in-plane magneto-electrical loads is investigated. The interface crack is assumed to be either magneto-electrically impermeable or permeable, and the position of the interface crack is arbitrary. The finite Fourier transform method is employed to reduce the mixed boundary-value problem to triple trigonometric series equations. The dislocation density functions and proper replacement of the variables are introduced to reduce these series equations to a standard Cauchy singular integral equation of the first kind. The resulting integral equation together with the corresponding single-valued condition is approximated as a system of linear algebra equations which can be easily solved. Field intensity factors and energy release rates are determined numerically and discussed in detail. Numerical results show the effects of crack configuration and loading combination parameters on the fracture behaviors of crack tips according to energy release rate criterion. The study of this problem is expected to have applications to the investigation of dynamic fracture properties of magneto-electro-elastic materials with cracks.     

Dynamic stress intensity factors of two collinear mode-III cracks perpendicular to and on the two sides of a bi-FGM weak-discontinuous interface

The mechanical model was established for the anti-plane dynamic fracture problem for two collinear cracks on the two sides of and perpendicular to a weak-discontinuous interface between two materials with smoothly graded elastic properties, as opposed to a sharp interface with discontinuously changing elastic properties. The problem was reduced as a system of Cauchy singular integral equations of the first kind by Laplace and Fourier integral transforms. The integral equations were solved by Erdogan's collocation method and the dynamic stress intensity factors in the time domain were obtained through Laplace numerical inversion proposed by Miller and Guy. The influences of geometrical and physical parameters on the dynamic stress intensity factors were illustrated and discussed, based on which some conclusions were drawn: (a) to increase the thickness of the FGM strip on either side of the interface will be beneficial to reducing the DSIF of a crack perpendicular to a bi-FGM interface and embedded at the center of one of the FGM strips; (b) To increase the rigidity of the FGM strip where the crack is located will increase the DSIF. However, when the material in one side of the interface is more rigid, the DSIF of the interface-perpendicular embedded crack in the other side will be reduced; (c) To decrease the weak-discontinuity of a bi-FGM interface will not necessarily reduce the stress intensity factor of a crack perpendicular to it, which is different from the case of interfacial crack; (d) For two collinear cracks with equal half-length, when the distance between the two inner tips is less than about three times of the half-length, the interaction of them is intensified, however, when the distance is greater than this the interaction becomes weak.     

Layered piezoelectric medium with interface crack under anti-plane shear

The theory of linear piezoelectricity is applied to solve the anti-plane shear problem of a piezoelectric layer sandwiched by two dissimilar homogeneous materials with a crack at the interface. Both mechanical and electrical loads are applied to the piezoelectric laminate. By the use of Fourier transforms, the mixed boundary value problem is reduced to a singular integral equation which is solved numerically to determine the stress intensity factors for several layered piezoelectric media, and the results are presented in graphical form.     

Transient response of a piezoelectric ceramic strip with an eccentric crack under electromechanical impacts

The transient response of a piezoelectric strip with an eccentric crack normal to the strip boundaries under applied electromechanical impacts is considered. By using the Laplace transform, the mixed initial-boundary-value problem is reduced to triple series equations, then to a singular integral equation of the first kind by introducing an auxiliary function. The Lobatto–Chebyshev collocation technique is adopted to solve numerically the resulting singular integral equation. Dynamic field intensity factors and energy release rate are obtained for both a permeable crack and an impermeable crack. The effects of the crack position and the material properties on the dynamic stress intensity factor are examined and numerical results are presented graphically.     

A numerical approach for a crack problem by Gauss–Chebyshev quadrature

In this paper, a problem of a crack in an orthotropic strip is studied under plane strain conditions. It is assumed that normal displacements and shear stresses do not act on neither of the boundaries of the strip. Cauchy-type singular integral equation for the crack problem is derived by using the theory of plane elasticity and the Fourier transformation technique. A quadrature collocation approach is adopted for the numerical solutions of the singular integral equation. The effect of relative thickness and mechanical properties of strip on Mode I stress intensity factors (SIFs) are examined under different loading conditions. Some sample results are given for SIFs; also, material orthotropy and geometrical effects are discussed in detail.     

Closed-form solution for two collinear cracks in a piezoelectric strip

Under the permeable electric boundary condition, the problem of two collinear anti-plane shear cracks lying at the mid-plane of a piezoelectric strip is investigated. By using the Fourier transform, the associated problem is reduced to a singular integral equation. Solving the resulting equation analytically, the electro-elastic field intensity factors and energy release rates at the inner and outer crack tips can be determined in explicit form. Numerical results for PZT-5H piezoelectric ceramic are also presented graphically. The results reveal that the effect of electric field on crack growth in piezoelectric materials is dependent on applied elastic displacement.     

Dynamic stress intensity factors of multiple cracks in a functionally graded orthotropic half-plane

In the present paper dynamic stress intensity factor and strain energy density factor of multiple cracks in the functionally graded orthotropic half-plane under time-harmonic loading are investigated. By utilizing the Fourier transformation technique the stress fields are obtained for a functionally graded orthotropic half-plane containing a Volterra screw dislocation. The variations of the material properties are assumed to be exponential forms which the equilibrium has an analytical solution. The dislocation solution is utilized to formulate integral equation for the half-plane weakened by multiple smooth cracks under anti-plane deformation. The integral equations are of Cauchy singular type at the location of dislocation which are solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to determined stress intensity factor and strain energy density factors (SEDFs) for multiple smooth cracks under anti-plane deformation. Numerical examples are provided to show the effects of material properties and the crack configuration on the dynamic stress intensity factors and SEDFs of the functionally graded orthotropic half-plane with multiple curved cracks.     

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.