A penny-shaped crack in a functionally graded piezoelectric strip under thermal loading

The mixed-mode thermoelectromechanical fracture problem for a functionally graded piezoelectric material (FGPM) strip with a penny-shaped crack is considered. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. The thermal and electromechanical problems are reduced to singular integral equations and solved numerically. The stress and electric displacement intensity factors are presented for different crack size, crack position and material nonhomogeneity.     

Thermoelectroelastic response of a functionally graded piezoelectric strip with two parallel axisymmetric cracks

A mixed-mode thermoelectroelastic fracture problem of a functionally graded piezoelectric material strip containing two parallel axisymmetric cracks, such as penny-shaped or annular cracks, is considered in this study. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. Using integral transform techniques, the problem is reduced to that of solving two systems of singular integral equations. Systematic numerical calculations are carried out, and the variations of the stress and electric displacement intensity factors are plotted for various values of dimensionless parameters representing the crack size, the crack location and the material non-homogeneity.     

Thermal fracture analysis of a functionally graded orthotropic strip with a crack

This paper is concerned with the thermal fracture problem of a functionally graded orthotropic strip, where the crack is situated parallel to the free edges. All the material properties are assumed to be dependent only on the coordinate y (perpendicular to the crack surfaces). By using Fourier transform, the thermoelastic problem is reduced to those that involve a system of singular integral equations. Numerical results are presented to show the effects of the crack position and the material distribution on the thermal stress intensity factors.     

Transient response of a functionally graded piezoelectric strip with a penny-shaped crack under electric time-dependent loading

Summary The dynamic fracture problem for a functionally graded piezoelectric material (FGPM) strip containing a penny-shaped 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 time-dependent electric load. Integral transform techniques and dislocation density functions are employed to reduce the problem to the solutions of a system of singular integral equations. The stress and electric displacement intensity factors versus time are presented for various values of dimensionless parameters representing the crack size, the crack location and the material nonhomogeneity.     

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.     

Two parallel penny-shaped or annular cracks in a functionally graded piezoelectric strip under electric loading

In this paper, the mixed-mode fracture problem of a functionally graded piezoelectric material strip with two penny-shaped or annular cracks 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 electric loading. The problem is formulated in terms of a system of singular integral equations, which are solved numerically. Numerical calculations are carried out, and 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.     

Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a FGP substrate under static electromechanical load

In this paper, the Fourier integral transform–singular integral equation method is presented for the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a different functionally graded piezoelectric material. The properties of two materials, 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. The mixed boundary value problem is reduced to a singular integral equation over crack by applying the Fourier transform and the singular integral equation is solved numerically by using the Lobatto–Chebyshev integration technique. The analytic expressions of the stress intensity factors and the electric displacement intensity factors are derived. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

Crack in a bonded functionally graded magneto-electro-elastic strip

Investigated is the anti-plane problem of functionally graded magneto-electro-elastic strip sandwiched between two functionally graded strips. It is assumed that the material properties vary exponentially with the coordinate parallel to the crack. The crack is assumed to be either magneto-electrically impermeable or permeable. Fourier transforms are used to reduce the crack problems to a system of singular integral equations, which is solved numerically by application of the Gauss–Chebyshev integration formula. Numerical results show the effects of the material gradient parameter and crack configuration on the field intensity factors of the crack.     

Thermoelastic analysis of a partially insulated crack in a strip under thermal impact loading using the hyperbolic heat conduction theory

In this paper, the transient temperature and thermal stresses around a partially insulated crack in a thermoelastic strip under a temperature impact are obtained using the hyperbolic heat conduction theory. Fourier and Laplace transforms are applied and the thermal and mechanical problems are reduced to solving singular integral equations. Numerical results show that the hyperbolic heat conduction parameters, the thermal conductivity of crack faces, and the geometric size of the strip have significant influence on the dynamic temperature and stress field. The results based on hyperbolic heat conduction show much higher temperature and much more dynamic thermal stress concentrations in the very early stage of impact loading comparing to the Fourier heat conduction model. It is suggested that to design materials and structures against fracture under transient thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.     

Transient response of a center crack in a functionally graded piezoelectric strip under electromechanical impact

The dynamic fracture problem for a functionally graded piezoelectric strip containing a center crack parallel to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the medium vary continuously in the thickness direction, and that the strip is under in-plane mechanical and electric impact loadings. 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 material nonhomogeneity and the loading combination.     

The collinear cracks in an inhomogeneous medium subjected to transient load

Summary A laminate model is employed to solve the elastodynamic problem of a collinear crack in an inhomogeneous material. The inhomogeneous material is treated as a series of thinner layer. The Laplace and Fourier transforms are used to reduce the problem to a set of singular integral equations that is solved numerically. Numerical results of two collinear cracks in a functionally graded material strip are obtained to show the influence of material inhomogeneity and crack position on crack tip field intensities.     

Indentation of a cantilever beam or plate

In this paper the general plane problem for a semi-infinite strip fixed at its short end, containing a crack perpendicular to its voundaries is considered. The strip is under the effect of a stamp. By extending the crack to the surfaces, one can reduce the problem to that of c cantilever beam or plate. Integral transform technique is used to provide an exact formulation of this problem, in terms of a system of four singular integral equations one of them being second kind. Stress singularities at the corners of the fixed-end, at the crack tips and at the end points of the contact region undermeath the stamp are obtained from the singular integral equations which are then solved numerically.     

Transient internal crack problem for a nonhomogeneous orthotropic strip (Mode I)

Supposing the material properties to be one-dimensionally dependent, this paper studied the transient internal crack problem for a functionally graded orthotropic strip. Integral transforms and dislocation density functions are employed to reduce the problem to singular integral equations. Numerical results show the effects of the material parameter βh, the crack configuration and the isotropic or orthotropic property on the dynamic stress intensity factors.     

Crack tip opening displacement of a Dugdale crack in a three-phase cylindrical model composite material

The analytical investigation of the plastic zone size of a crack in three-phase cylindrical model composite material was carried out. The physical problem is simulated as a crack near a circular inclusion (a single fiber) in the composite matrix, while the three-phase cylindrical composite model is used to represent the composite matrix. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small scale yielding, a thin strip of yielded plastic zone is introduced at each crack tip. Using the solution for a three-phase model with a single dislocation in the matrix phase as the Green’s function, the physical problem is formulated into a set of singular integral equations. By employing Erdogan and Gupta’s method, as well as iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacements.     

Pull-off test for a cracked layer bonded to a rigid foundation

This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semiinfinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.     

Stress intensities in a strip reinforced by stiffeners at the edges

The problem of a cracked or perforated strip, reinforced by stiffeners, is considered. Since each defect (crack or hole) is simulated by a continuous distribution of dislocations, the problem of a single dislocation, lying in the reinforced strip, has been solved via Fourier transform. Consequently, a singular integral equation is set up for any crack or hole in the strip. Numerical results, regarding stress intensities at straight crack tips, are presented.     

Pull-off test for a cracked layer bonded to a rigid foundation

This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semi-infinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.     

Scale effects in the initiation of cracking of a scarf joint

The singular stress field at the interface-corner of a bi-material scarf joint is analysed for a strip of finite width, w, under remote tension and bending. The two substrates are taken as linear elastic and isotopic. The intensity of the singular stress field is calculated using a domain integral method, and is plotted as a function of joint geometry and material mismatch parameters. It is envisaged that the intensity of singularity can serve as a valid fracture criterion provided the zone of nonlinearity is fully embedded within the singular elastic field. It is assumed that fracture initiates when the magnitude of the corner singularity attains a critical value; consequently, the fracture strength of the joint depends upon the size of the structure. In addition, the interfacial stress intensity factor and the associated T-stress are determined for an edge interfacial crack. When the crack is short with respect to the width of the strip, the stress intensity factor is dominated by the presence of the corner singularity; a boundary layer formulation is used to determine the coupling between the crack tip field and the interface-corner field. The solution suggests that an interfacial crack grows unstably with a rapidly increasing energy release rate, but with only a small change in mode mix. This revised version was published online in July 2006 with corrections to the Cover Date.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Thermoelastic interaction of two offset interfacial cracks in bonded dissimilar half-planes with a functionally graded interlayer

This paper deals with the thermoelasticity problem of bonded dissimilar half-planes with a functionally graded interlayer, weakened by a pair of two offset interfacial cracks. The material nonhomogeneity in the graded interlayer is represented by spatially varying thermoelastic moduli expressed in terms of exponential functions. The cracks are assumed to be thermally insulated disturbing a steady-state uniform heat flow, and the solution is obtained within the framework of linear plane thermoelasticity. The Fourier integral transform method is employed, and the formulation of the current nonisothermal crack problem is reduced to two sets of Cauchy-type singular integral equations for temperature and thermal stress fields in the bonded system. In the numerical results, parametric studies are conducted so that the variations in mixed-mode thermal stress intensity factors are presented as a function of offset crack distance for various geometric and material combinations of the dissimilar homogeneous media bonded through the thermoelastically graded interlayer, elaborating thermally induced singular interaction of the two neighboring interfacial cracks.