Antiplane Crack Problem Research Articles

The eigen-functions of anti-plane crack problems in non-homogeneous materials

For the basic elastic fracture mechanics problems,we discussed the reasonability of introducing the Williams' solution of homogeneous materials to non-homogeneous materials in this paper.Taking the anti-plane crack problem as an example,we give its governing equations for arbitrary order eigen-function,and provide the Williams' solution for anti plane crack problem in non-homogeneous materials.This eigen-function theoretically reveals the effect of non-homogeneity on the structure of crack tip field.

Mode III fracture problem of an FGPM strip bonded to an FGPM half-plane with embedded arbitrarily oriented cracks

This article studies the anti-plane crack problem of a functionally graded piezoelectric material (FGPM) cracked strip bonded to an FGPM-cracked half-plate. Each crack is arbitrarily oriented. The material properties are assumed to be exponential forms varied in the direction normal to the interface. Using Fourier transforms, the problem can be reduced to a system of singular integral equations, which are solved numerically by applying the Gauss–Chebyshev integration formula. Numerical calculations are carried out to obtain the crack driving forces, such as the stress intensity factors (K), energy density factors (S), and the energy release rates (G), at the crack tips. The influences of the non-homogeneous parameter, crack orientation, electrical displacement, and the crack length on the normalized factors and rates are graphically discussed.

On an antiplane crack problem for functionally graded elastic materials

This paper examines an antiplane crack problem for a functionally graded anisotropic elastic material in which the elastic moduli vary quadratically with the spatial coordinates. A solution to the crack problem is obtained in terms of a pair of integral equations. An iterative solution to the integral equations is used to examine the effect of the anisotropy and varying elastic moduli on the crack tip stress intensity factors and the crack displacement. doi:10.1017/S1446181111000551

On a Displacement Based Solution to an Antiplane Crack Problem for Inhomogeneous Anisotropic Elastic Materials

This paper employs a displacement based method to examine an antiplane crack problem for an inhomogeneous elastic material in which the elastic moduli vary continuously with the spatial coordinates. Expressions for the crack tip stress intensity factors and the crack displacement are obtained in terms of Chebyshev polynomials. Numerical results are obtained for some particular inhomogeneous elastic materials.

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.

ON AN ANTIPLANE CRACK PROBLEM FOR FUNCTIONALLY GRADED ELASTIC MATERIALS

AbstractThis paper examines an antiplane crack problem for a functionally graded anisotropic elastic material in which the elastic moduli vary quadratically with the spatial coordinates. A solution to the crack problem is obtained in terms of a pair of integral equations. An iterative solution to the integral equations is used to examine the effect of the anisotropy and varying elastic moduli on the crack tip stress intensity factors and the crack displacement.

Stress field near circular-arc interface crack tip based on electric saturation concept

Within the framework of nonlinear electroelasticity, the anti-plane problem of a circular-arc interfacial crack between a circular piezoelectric inhomogeneity and an infinite piezoelectric matrix subjected to a far-field uniform loading is investigated by an electrical strip saturation model, the complex variable method, and the method of analytical continuation. Explicit closed form expressions for the complex potentials in both the matrix and the inclusion, and the stress intensity factor at the crack tip are presented. Comparison with some related solutions based on the linear electroelastic theory shows the validity of the present solutions

A path-independent integral for the characterization of solute concentration and flux at biofilm detachments

A path-independent (conservation) integral is developed for the characterization of solute concentration and flux in a biofilm in the vicinity of a detachment or other flux limiting boundary condition. Steady state conditions of solute diffusion are considered and biofilm kinetics are described by an uptake term which can be expressed in terms of a potential (Michaelis–Menten kinetics). An asymptotic solution for solute concentration at the tip of the detachment is obtained and shown to be analogous to that of antiplane crack problems in linear elasticity. It is shown that the amplitude of the asymptotic solution can be calculated by evaluating a path-independent integral. The special case of a semi-infinite detachment in an infinite strip is considered and the amplitude of the asymptotic field is related to the boundary conditions and problem parameters in closed form for zeroth and first order kinetics and numerically for Michaelis–Menten kinetics.

Transient anti-plane crack problem for two bonded functionally graded piezoelectric materials

In this paper, the dynamic anti-plane crack problem for two bonded functionally graded piezoelectric materials is considered. The crack is perpendicular to the interface and assumed to be electrically impermeable or permeable. Integral transforms are employed to reduce the problem to Cauchy singular equations that can be solved numerically. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor and the energy density factor are studied. It is found that for the impermeable case, the normalized dynamic stress intensity factor may increase or decrease in different time domains determined by the sign and magnitude of λ.

A meshless local boundary integral equation method for dynamic anti-plane shear crack problem in functionally graded materials

This paper presents a meshless local boundary integral equation method (LBIEM) for dynamic analysis of an anti-plane crack in functionally graded materials (FGMs). Local boundary integral equations (LBIEs) are formulated in the Laplace-transform domain. The static fundamental solution for homogeneous elastic solids is used to derive the local boundary–domain integral equations, which are applied to small sub-domains covering the analyzed domain. For the sub-domains a circular shape is chosen, and their centers, the nodal points, correspond to the collocation points. The local boundary–domain integral equations are solved numerically in the Laplace-transform domain by a meshless method based on the moving least–squares (MLS) scheme. Time-domain solutions are obtained by using the Stehfest's inversion algorithm. Numerical examples are given to show the accuracy of the proposed meshless LBIEM.

A Griffith interface crack in a micro-periodic reinforced elastic composite

This paper is devoted to the anti-plane static problem of an interface Griffith crack in a micro-periodic composite solid. The repeated unit cell of the body under consideration consists of four different elastic materials taking the shape of the fibers of infinite length with rectangular cross-sections. An approximate analysis is carried out within the homogenized model with microlocal parameters. The analytical results obtained may be used for the interface crack problems in micro-periodic fiber-reinforced composites as well as in solids with layered or chessboard periodic microstructures.

Out-of-plane interface cracks in dissimilar piezoelectric materials

This paper investigates the problem of an anti-plane interfacial crack between two dissimilar piezoelectric material layers. A single crack is first considered. The effect of interaction of two collinear cracks in the medium on the field intensity factors is investigated. The solutions of several particular cases, including an infinite piezoelectric bi-material and a piezoelectric material bonded to an elastic medium, are given. The bi-material constants governing the behavior of the crack tip fields are identified. By considering the crack as a notch of finite thickness, it is shown that the thickness of the notch has a pronounced influence on the crack tip field. The results for the assumption of a permeable crack represent the limit case where the notch thickness is reduced to zero.

Transient Anti-Plane Crack Problem for a Functionally Graded Piezoelectric Coating Bonded to an Elastic Layer

Transient Anti-Plane Crack Problem for a Functionally Graded Piezoelectric Coating Bonded to an Elastic Layer

Transient anti-plane crack problem of a functionally graded piezoelectric strip bonded to elastic layers

This paper examined the dynamic electromechanical behavior of a crack in a functionally graded piezoelectric layer bonded between two elastic layers under the combined anti-plane mechanical shear and in-plane electric impacts. Fourier cosine transforms are used to reduce the problem to the solution of a set of singular integral equations. It is found that the impermeable crack condition is more reasonable than the permeable crack condition to analyze the influence of electric loading, and the energy density factor is more acceptable than the energy release rate to be used as the fracture criterion. In addition, numerical results are also presented to show the influences of the crack position, electromechanical combination factor and material gradient parameter on the fracture behavior.

Transient Anti-Plane Crack Problem in a Functionally Graded Smart Structure

The dynamic anti-plane crack problem in a functionally graded smart structure is considered. Integral transforms are employed to reduce the problem to the solution of singular integral equations. Numerical results are presented to show the nfluences of electrical crack condition, crack position, electromechanical combination factor and material gradient parameter on the fracture behavior.

Fracture mechanics analysis models for functionally graded materials with arbitrarily distributed properties (Modes II and III problems)

In Wang et al. (2002), the mode I crack problem has been analyzed for a functionally graded material strip with arbitrarily distributed properties. For the completeness of the research, this paper provides an analysis method for the mode II in plane and mode III anti-plane elastic crack problems for a functionally graded material (FGM) strip containing a crack along the gradient direction. In the analysis, the gradient region is treated as a number of homogeneous layers stacked along the gradient direction. The analytical models developed in this paper for the modes II and III problems, together with the model developed in Wang et al. (2002) for the mode I crack problem complete the analysis of the mixed-mode crack problem for functionally graded materials with a crack parallel the gradient direction. Those analytical models allow the material properties be any continuous functions along the thickness direction of the strip.

Green?s functions for anti-plane deformations of a circular arc-crack at the interface of piezoelectric materials

The anti-plane deformation problem of an interfacial debounding crack between a circular piezoelectric inclusion and a piezoelectric matrix is investigated by means of the complex variables method. For a line load applied within the matrix or inside the inclusion, Green’s functions are presented for the complex potentials, intensity factors and electric fields on the crack faces, respectively, in closed and explicit form. The solutions are valid for both permeable and impermeable crack models. It is shown that, in the general case of permeable cracks, the electric field singularity is always proportional to the stress singularity.

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.

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

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.

Antiplane Crack Problem in Functionally Graded Piezoelectric Materials

In this paper the problem of a finite crack in a strip of functionally graded piezoelectric material (FGPM) is studied. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric permitivity of the FGPM vary continuously along the thickness of the strip, and that the strip is under an antiplane mechanical loading and in-plane electric loading. By using the Fourier transform, the problem is first reduced to two pairs of dual integral equations and then into Fredholm integral equations of the second kind. The near-tip singular stress and electric fields are obtained from the asymptotic expansion of the stresses and electric fields around the crack tip. It is found that the singular stresses and electric displacements at the tip of the crack in the functionally graded piezoelectric material carry the same forms as those in a homogeneous piezoelectric material but that the magnitudes of the intensity factors are dependent upon the gradient of the FGPM properties. The investigation on the influences of the FGPM graded properties shows that an increase in the gradient of the material properties can reduce the magnitude of the stress intensity factor.