Green's functions for the stress intensity factor evolution in finite cracks in orthotropic materials

The elastodynamic response of an infinite orthotropic material with finite crack under concentrated loads is examined. Solution for the stress intensity factor history around the crack tips is found. Laplace and Fourier transforms are employed to solve the equations of motion leading to a Fredholm integral equation on the Laplace transform domain. The dynamic stress intensity factor history can be computed by numerical Laplace transform inversion of the solution of the Fredholm equation. Numerical values of the dynamic stress intensity factor history for some example materials are obtained. This solution can be used as a Green's function to solve dynamic problems involving fini te cracks.     

Dynamic Stress Intensity Factor of Interfacial Finite Cracks in Orthotropic Materials

The elastodynamic response of an infinite non-homogeneous orthotropic material with an interfacial finite crack under distributed normal and shear impact loads is examined. Solution for the stress intensity factor history around the crack tips is found. Laplace and Fourier transforms are employed to solve the equations of motion leading to a Fredholm integral equation on the Laplace transform domain. The dynamic stress intensity factor history can be computed by numerical Laplace transform inversion of the solution of the Fredholm equation. Numerical values of the dynamic stress intensity factor history for some materials are obtained. Interfacial cracks between two different materials and between two pieces of the same material but different fiber orientation are considered. Bimaterial formulation of a crack problem is shown to converge to the mono-material formulation, derived independently, in the limiting case when both materials are the same.     

Impact response of a strip composite with a finite crack

The dynamic response of a central crack in a strip composite under normal impact is analyzed. The crack is oriented normally to the interfaces. Laplace and Fourier transform techniques are used to reduce the elastodynamic problem to a pair of dual integral equations. The integral equations are solved by using an integral transform technique and the result is expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factor is determined and its dependence on time, the material properties and the geometrical parameters is discussed.     

Impact behavior of an inclined edge crack in a layered medium with a graded nonhomogeneous interfacial zone: antiplane deformation

Summary The impact response of an inclined edge crack in a layered medium with a functionally graded interfacial zone is investigated under the state of antiplane deformation. The interfacial zone is modeled by a nonhomogeneous interlayer having the power-law variations of shear modulus and mass density between the coating and the substrate of dissimilar homogeneous properties. Based on the Laplace and Fourier integral transform technique and the coordinate transformations of basic field variables, the transient crack problem is reduced to the solution of a singular integral equation with a generalized Cauchy kernel in the Laplace transform domain. The crack-tip response in the physical domain is recovered through the inverse Laplace transform to evaluate the dynamic mode III stress intensity factors as functions of time. The peak values of the dynamic stress intensity factors are further obtained versus the crack orientation angle, addressing the effects of crack obliquity on the overshoot characteristics of the transient crack-tip behavior for various combinations of material and geometric parameters of the layered medium.     

Transient analysis of a piezoelectric strip with a permeable crack under anti-plane impact loads

The problem of a through permeable crack situated in the mid-plane of a piezoelectric strip is considered under anti-plane impact loads for two cases. The first is that the strip boundaries are free of stresses and of electric displacements, and the second is that the strip boundaries are clamped rigid electrodes. The method adopted is to reduce the mixed initial-boundary value problem, by using integral transform techniques, to dual integral equations, which are further transformed into a Fredholm integral equation of the second kind by introducing an auxiliary function. The dynamic stress intensity factor and energy release rate in the Laplace transform domain are obtained in explicit form in terms of the auxiliary function. Some numerical results for the dynamic stress intensity factor are presented graphically in the physical space by using numerical techniques for solving the resulting Fredholm integral equation and inverting Laplace transform.     

Impact response of a finite crack in an orthotropic strip

Summary The elastodynamic response of a finite crack in an infinite orthotropic strip under normal impact is investigated in this study. The crack is situated symmetrically and oriented in a direction normal to the edges of the strip. Laplace and Hankel transforms are used to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution to the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor for some fiber-reinforced composite materials are obtained and the results are graphed to display the influence of the material orthotropy.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

Fracture analysis of a functionally graded coating-substrate structure with a crack perpendicular to the interface - Part II: Transient problem

In Part I of this paper, the static problem of a functionally graded coating-substrate system with an internal or edge crack perpendicular to the interface has been studied. In this part, the transient response of the structure is considered under an in-plane impact. Laplace and Fourier transforms are applied to reduce the mixed boundary value problem to a singular integral equation which is solved in the Laplace domain numerically. The dynamic stress intensity factors (DSIFs) are obtained by numerical Laplace inversion technique. The influences of material constants and geometry parameters on the dynamic stress intensity factors are studied. It is found that the DSIFs for an internal crack rise rapidly to a peak and then tend to the steady value without obvious oscillations, but the DSIFs for an edge crack have more obvious oscillations after rising to a peak with the increasing of the nonhomogeneity constant.     

Dynamic stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid

This paper considers the transient stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid. The crack surfaces are impermeable. By virtue of the integral transform methods, the poroelastodynamic mixed boundary value problems is formulated as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind in the Laplace transform domain. Time domain solutions are obtained by inverting Laplace domain solutions using a numerical scheme. A parametric study is presented to illustrate the influence of poroelastic material parameters on the transient stress intensity. The results obtained reveal that the dynamic stress intensity factor of poroelastic medium is smaller than that of elastic medium and the poroelastic medium with a small value of the potential of diffusivity shows higher value of the dynamic stress intensity factor.     

Transient analysis of stress waves around two rectangular cracks under impact load

The three-dimensional response of two rectangular cracks in an infinite elastic medium to impact load is investigated in this paper. Fourier and Laplace transforms are applied and the problem is reduced to that of solving dual integral equations in the Laplace transform domain. To solve these equations, the crack surface displacement is expanded in a double series of functions which are zero outside of the cracks. The unknown coefficients accompanied in that series are solved with the aid of the Schmidt method. The dynamic stress intensity factors are computed numerically.     

Impact response of a crack in a semi-infinite body with a surface layer under longitudinal shear

Summary The impact response of a crack in a semi-infinite body with a surface layer which is subjected to antiplane shear deformation is considered in this study. The semi-infinite body contains a crack near an interface. Using Laplace and Fourier transforms, the case of a crack perpendicular to the interface is reduced to a set of triple integral equations in the Laplace transform plane. The solution to the triple integral equations is then expressed in terms of a singular integral equation of the first kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factors at the crack tips are obtained for several values of time, material constants, and geometrical parameters.With 8 Figures     

Dynamic singular stresses in glassfiber reinforced plastics with a broken layer at low temperatures

The plane strain dynamic singular stress problem for glassfiber reinforced plastics with a broken layer at low temperatures is considered. With the order of stress singularity around the tip of the crack which is normal to and ends at the interface between orthotropic elastic materials, Laplace and Fourier transforms are used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gauss-Jacobi integration formula. Numerical calculations are carried out, and the dynamic stress intensity factors at different temperatures are shown graphically.     

Axisymmetric elastodynamic response of a flat annular crack to normal impact waves

The axisymmetric response of a flat annular crack in an infinite medium subjected to normal impact load is investigated in this study. A step stress is applied to the crack surface. The singular solution is equivalent to solutions of the problem of diffraction of normally incident tension wave by a flat annular crack, and the problem of the sudden appearance of a flat annular crack in a uniform tensile stress field. Laplace and Hankel transforms are used to reduce the problem to the solution of a set of triple integral equations in the Laplace transform domain. These equations are solved by using a integral transform technique and the result is expressed in terms of a singular integral equation of the first kind with the kernel which is improved by means of a contour integration on the Riemann surface. A numerical Laplace inversion routine is used to recover the time dependence of the solution. Numerical results of the dynamic stress intensity factor are obtained to show the influence of inertia, the ratio of the inner radius to the outer one and Poisson's ratio on the load transmission to the crack tip.     

Dynamic mode II stress intensity factors of an infinite cracked medium subjected to transient concentrated forces

In this article the determination of dynamic mode II stress intensity factors of a crack embedded in an infinite medium subjected to transient concentrated line forces is investigated. The concentrated line forces act at an arbitrary distance away from the crack, including the special case when the forces act precisely on the crack surfaces. Laplace and Fourier transforms are used to reduce the mixed boundary value problem to a standard Fredholm integral equation of the second kind in Laplace transform domain, which is solved numerically. Via the numerical inversion of Laplace transform, the dynamic mode II stress intensity factors at the crack tips are obtained and presented in graphical form for various geometry parameters. It is found that the point of application of the concentrated forces, which induce the maximum value of the dynamic mode II stress intensity factors, is precisely on the crack surface for horizontal concentrated forces, whereas for vertical forces, it is at some distance away from the crack.     

Impact response of a finite crack in an orthotropic piezoelectric ceramic

Summary The transient dynamic stress intensity factor and dynamic energy release rate were determined for a cracked piezoelectric ceramic under normal impact in this study. A plane step pulse strikes the crack and stress wave diffraction takes place. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion technique is used to compute the values of the dynamic stress intensity factor and the dynamic energy release rate for some piezoelectric ceramics, and the results are graphed to display the electroelastic interactions.     

Torsional impact response of a penny-shaped crack lying on a bimaterial interface

The torsional impact response of a penny-shaped crack lying on a bimaterial interface is considered in this study. Laplace and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution to the dual integral equations is expressed in terms of a Fredholm integral equation of the second kind with a finite integral kernel. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factor is determined and its dependence on time and material constants is discussed.     

Displacement Discontinuity Method for Fracture Mechanics Analysis of Reissner Plates: Static and Dynamic

This paper is concerned with the displacement discontinuity method applied to the shear deformable plates (Reissner’s and Mindlin’s theories) with cracks subjected to static and dynamic loads. Fundamental solutions of dislocation are derived using the Fourier transform method and the Laplace transformation technique. Boundary integral equations are presented in terms of rotations/displacement on the crack surfaces. The Chebyshev polynomials of the second kind are used to evaluate the integral equations with hypersingular kernels on the crack boundaries and determine the stress intensity factors at the crack tips. Comparisons are made with other numerical solutions to demonstrate the proposed method is accurate both for static and dynamic problems.     

Weight Function for a Crack in a Two-dimensional Orthotropic Medium Under Impact Shear Loading

The paper examines the elastodynamic response of an infinite two-dimensional orthotr- opic medium containing a central crack under impact shear loading. Laplace and Fourier integral transforms are employed to reduce the problem to a pair of dual integral equations in the Laplace transformed plane. These equations are reduced to integral differential equations, which have been solved in the low frequency domain by iterations. To determine time dependence, these equations are inverted to yield the dynamic stress intensity factor (SIF) for shear point force loading that corresponds to the weight function for the crack under shear loading. It is used to derive SIF for polynomial loading.     

Evaluation of the stress intensity factors and the T-stress in periodic crack problem

This paper investigates the T-stress at crack tips in the periodic crack problem. Remote tension in the y-direction is applied to cracks with an arbitrary inclined angle. The original stress field can be considered a superposition of a uniform stress field and a perturbation stress field. The problem of evaluating the stresses in the perturbation field can be considered a superposition of many single crack problems. A Fredholm integral equation is suggested for the solution of the perturbation stress field. In the equation, the loading on the crack face is chosen as unknown quantity. Once the integral equation is solved, the stress intensity factors and the T-stress at the crack tip can be evaluated immediately. For solving the integral equation and evaluating stresses in the perturbation field, the remainder estimation technique is suggested for evaluating the influences on the central crack from infinite cracks. The technique can considerably improve convergence in computation. Many results for the stress intensity factors and the T-stresses in periodic cracks are presented. It is shown that the interaction is significant for the closer cracks.     

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.