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.     

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.     

Anti-plane stress intensity, energy release and energy density at crack tips in a functionally graded strip with linearly varying properties

The fracture problem of a crack in a functionally graded strip with its properties varying in a linear form along the strip thickness under an anti-plane load is considered. The embedded anti-plane crack is located in the middle of strip half way through the thickness. The third mode stress intensity factor is derived using two different methods. In the first method, by employing Fourier integral transforms, the governing equation is converted to a singular integral equation, which is subsequently solved numerically by the collocation method based on Chebyshev polynomials. Then, the problem is solved by means of finite element method in which quadrilateral 8-node singular elements around each crack tip are used. After inspecting the validity of the solution technique, effects of crack geometry and non-homogeneous material parameter on the stress intensity, energy release and energy density are studied and the results of analytical and FEM solutions are compared.     

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.     

Functionally graded piezoelectric strip with a penny-shaped crack under electromechanical loadings

In this paper, the mixed-mode penny-shaped crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in-plane electromechanical loadings. The problem is formulated in terms of a system of singular integral equations. The stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.     

A finite crack in a semi-infinite strip of a graded piezoelectric material under electric loading

In this paper, the mixed-mode crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in-plane electric loading. The problem is formulated in terms of a system of singular integral equations. The stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.     

Impact failure prediction of Mode III crack in orthotropic functionally graded strip

Mode III impact of a crack in an orthotropic functionally graded strip is investigated. The shear moduli in two directions of the material are assumed to vary proportionately with gradient. Laplace transform and Fourier cosine transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Energy density factor criterion is applied to obtain the maximum of minimum energy density and direction of crack initiation. Numerical results are given graphically. The effects of orthotropy, nonhomogeneity and height of the strip on the energy density factor are discussed.     

Electromechanical impact of an impermeable parallel crack in a functionally graded piezoelectric strip

The dynamic fracture problem for a functionally graded piezoelectric material (FGPM) strip containing a crack parallel to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the strip vary continuously along the thickness direction of the strip, and that the strip is under the in-plane mechanical and electric impact. Integral transform techniques and dislocation density functions are employed to reduce the problem to the solutions of a system of singular integral equations. The dynamic stress and electric displacement intensity factors versus time are presented for various values of dimensionless parameters representing the crack size, the crack location, the material nonhomogeneity and the loading combination.     

Transient thermal response of functionally graded piezoelectric laminates with an infinite row of parallel cracks normal to the bimaterial interface

In this paper, the problem of a functionally graded piezoelectric material strip(FGPM strip)containing an infinite row of parallel cracks perpendicular to the interface between the FGPM strip and a homogeneous layer is analyzed under transient thermal loading condition. The crack faces are supposed to be completely insulated. Material properties are assumed to be exponentially dependent on the distance from the interface. Using the Fourier transforms, the electro-thermoelastic problem is reduced to a singular integral equation, which is solved numerically. The stress intensity factors are computed and presented as a function of the normalized time, the nonhomogeneous and geometric parameters.     

THE PERIODIC CRACK PROBLEM IN BONDED PIEZOELECTRIC MATERIALS

The problem of a periodic array of parallel cracks in a homogeneous piezoelectric strip bonded to a functionally graded piezoelectric material is investigated for inhomogeneous continuum.It is assumed that the material inhomogeneity is represented as the spatial varia- tion of the shear modulus in the form of an exponential function along the direction of cracks. The mixed boundary value problem is reduced to a singular integral equation by applying the Fourier transform,and the singular integral equation is solved numerically by using the Gauss- Chebyshev integration technique.Numerical results are obtained to illustrate the variations of the stress intensity factors as a function of the crack periodicity for different values of the material inhomogeneity.     

Mode III crack problem in a functionally graded magneto-electro-elastic strip

Considering the material properties to be one-dimensionally dependent, this paper studied an anti-plane problem for an embedded crack and edge crack perpendicular to the boundary of a functionally graded magneto-electro-elastic strip. The crack is assumed to be either magneto-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. Numerical results show the effects of the loading combination parameter, material gradient parameter and crack configuration on the field intensity factors and the energy release rates of the functionally graded magneto-electro-elastic strip.     

ANTI-PLANE FRACTURE ANALYSIS OF FUNCTIONALLY GRADIENT MATERIAL INFINITE STRIP WITH FINITE WIDTH

The special case of a crack under mode III conditions was treated, lying parallel to the edges of an infinite strip with finite width and with the shear modulus varying exponentially perpendicular to the edges. By using Fourier transforms the problem was formulated in terms of a singular integral equation. It was numerically solved by representing the unknown dislocation density by a truncated series of Chebyshev polynomials leading to a linear system of equations. The stress intensity factor (SIF) results were discussed with respect to the influences of different geometric parameters and the strength of the non-homogeneity. It was indicated that the SIF increases with the increase of the crack length and decreases with the increase of the rigidity of the material in the vicinity of crack. The SIF of narrow strip is very sensitive to the change of the non-homogeneity parameter and its variation is complicated. With the increase of the non-homogeneity parameter, the stress intensity factor may increase, decrease or keep constant, which is mainly determined by the strip width and the relative crack location. If the crack is located at the midline of the strip or if the strip is wide, the stress intensity factor is not sensitive to the material non-homogeneity parameter.     

Equilibrium of an internal transverse crack in a semiinfinite elastic body with thin coating

Static elasticity problems for a half-plane and a strip weakened by a rectilinear transverse crack are studied. In each case, the upper boundary of the body is reinforced by a flexible patch. Various versions of conditions on the lower boundary are considered in the case of the strip. The crack is maintained in the open state by distributed normal forces. The method of generalized integral transforms reduces solving the problem for the equations of equilibriumto solving a singular integral equation of the first kind with the Cauchy kernel with respect to the derivative of the crack opening function. The solutions of the integral equation are constructed by the small parameter and collocation methods for various combinations of the geometric and physical parameters of the problem, and the structure of the solutions is analyzed. The values of the stress intensity factor (SIF) near the crack vertex are obtained.     

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.     

ANALYSIS OF A CRACK IN A FUNCTIONALLY GRADED STRIP WITH A POWER FORM SHEAR MODULUS

The plane strain problem of a crack in a functionally graded strip with a power form shear modulus is studied. The governing equation in terms of Airy＇s stress function is solved exactly by means of Fourier transform. The mixed boundary problem is then reduced to a system of singular integral equations and is solved numerically to obtain the stress intensity factor at crack-tip. The maximum circumferential stress criterion and the strain energy density criterion are both employed to predict the direction of crack initiation. Numerical examples are given to show the influence of the material gradation models and the crack sizes on the mode-I and mode-II stress intensity factors. The dependence of the critical kink-angle on the crack size is examined and it is found that the crack kink-angle decreases with the increase of the normalized crack length, indicating that a longer crack tends to follow the original crack-line while it is much easier for a shorter crack to deviate from the original crack-line.     

Representation of plasticity at an interface crack by inclined strip yield superdislocation model

The basic equations of the representation of plasticity at an interface crack by an inclined strip yield superdislocation model are derived. With a special combination of the material properties and small-scale yielding case, the problem is reduced to an algebraic equation in an unknown, the ratio of the plastic zone size. Some discussions on the limitations of this model are presented.supported by Japan Society for the Promotion of Science.     

Crack nucleation in a strip of varying thickness under force loading

A mathematical model of crack nucleation in a strip of varying thickness under force loading is constructed. It is assumed that the loading of the strip by external forces gives rise to prefracture zones, which are modeled as regions of weakened interparticle bonds in the material. Solution of the problem of the equilibrium of an isotropic strip of varying thickness, with a crack nucleus reduces to solving a system of nonlinear singular integral equations with a Cauchy type kernel to determine the forces in the region of crack nucleation. The condition of crack nucleation in a strip of varying thickness is formulated using the criterion of the limiting stretching of material bonds.     

The collinear crack problem for an orthotropic functionally graded coating–substrate structure

A theoretical treatment of antiplane crack problem of two collinear cracks on the two sides of and perpendicular to the interface between a functionally graded orthotropic strip bonded to an orthotropic homogeneous substrate is put forward. Various internal cracks and crack terminating at the interface and crack crossing the interface configurations are investigated, respectively. The problem is formulated in terms of a singular integral equation with the crack face displacement as the unknown variable. The asymptotic stress field near the tip of a crack crossing the interface is examined, and it is shown that, unlike the corresponding stress field in piecewise homogeneous materials, in this case, the “kink” in material property at the interface does not introduce any singularity. Numerical calculations are carried out, and the influences of the orthotropy and nonhomogeneous parameters and crack interactions on the mode III stress intensity factors are investigated.     

Strip yield zones around ring-shaped crack in transversely isotropic thick layer

A ring-shaped crack under uniform load in an infinitely long elastic–perfectly plastic thick layer is considered. The problem is formulated for a transversely isotropic material by using integral transform technique. Due to the geometry of the configuration, Hankel integral transform technique was chosen and the problem was reduced to a singular integral equation which is solved numerically by using Gaussian Quadrature Formulae and the values were evaluated at discrete points. The plastic zone widths were obtained by using the plastic strip model after stress intensity factors were obtained. Numerical results are plotted for various ring-shaped crack sizes and transversely isotropic materials. It was found that the width of the plastic zone at the inner edge of the crack was greater than the outer one.     

The dynamic response of an edge crack in a functionally graded orthotropic strip

The dynamic response of a functionally graded orthotropic strip with an edge crack perpendicular to the boundaries is studied. The material properties are assumed to vary continuously along the thickness direction. Laplace and Fourier transforms are applied to reduce the problem to a singular integral equation. Numerical results are presented to illustrate the influences of parameters such as the nonhomogeneity constant and geometry parameters on the dynamic stress intensity factors (SIFs).