Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

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     

Impact response of a cracked orthotropic medium-revisited

Elastodynamics response of an infinite orthotropic medium containing a central crack under impact loading has been investigated. Laplace and Fourier transforms have been employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform domain which has finally been solved by the method of iteration in the low frequency case. Analytic expressions for the stress intensity factors and crack opening displacement are also obtained for low frequency.     

Stress Intensity Factor of an Edge Crack in Bonded Orthotropic Materials

The article presents the problem of an edge crack under normal point loading terminating perpendicular to the surface of an orthotropic strip of finite thickness which is bonded to another orthotropic half plane. Expressing the displacements and stresses in plane strain condition in terms of harmonic functions, the problem is reduced to a pair of simultaneous integral equations with Cauchy type singularities, which are finally been solved by the Hilbert transform technique. The analytical expression of stress intensity factor (SIF) at the crack tip for large thickness of the strip is calculated, which corresponds to the weight function of a crack under normal loading. The influences of elastic constants of two different orthotropic materials, distinct arbitrary locations of normal point loading on the crack surface and length of the crack on the dynamic SIF are depicted through graphs.     

An interface crack in a functionally graded piezoelectric bi-layer under anti-plane shear impact

The transient response of an interface crack between two dissimilar functionally graded piezoelectric material (FGPM) layers under anti-plane shear impact loading is analyzed using the integral transform method. The properties of the FGPM layers vary continuously along the thickness, and the two layers are connected weak-discontinuously. Laplace transform and Fourier transform are used to reduce the problem to two sets of dual integral equations, which are then expressed to the Fredholm integral equations of the second kind. Numerical values on the dynamic energy release rate are presented for the FGPM to show the effects on the electric loading, variation and gradient of material properties, and thickness of layers. Following things are helpful to increase the resistance of transient fracture of interface crack in FGPMs: (a) increase of the material properties from the interface to the upper or lower free surface; (b) decrease of weak discontinuity at the interface; (c) increase of the gradient of material properties; (d) certain direction and magnitude of the electric loading; and (e) increase of the thickness of the FGPM layer.     

Impact response of an anti-plane crack in a FGPM bonded to a homogeneous piezoelectric strip

A finite crack under transient anti-plane shear loads in a functionally graded piezoelectric material (FGPM) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPM vary continuously according to exponential functions along the thickness of the strip, and that the two layered strips is under combined anti-plane shear mechanical and in-plane electrical impact loads. The analysis is conducted on the electrically unified crack boundary condition. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Fredholm integral equations of the second kind in the Laplace transform domain. Then, a numerical Laplace inversion is performed and the dynamic intensities are obtained as functions of time and geometric parameters, which are displayed graphically.     

Transient thermoelectroelastic response of a functionally graded piezoelectric strip with a penny-shaped crack

Transient response of a penny-shaped crack in a plate of a functionally graded piezoelectric material (FGPM) is studied under thermal shock loading conditions. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the crack faces are completely insulated. By using both the Laplace and Hankel transforms, the thermal and electromechanical problems are reduced to a singular integral equation and a system of singular integral equations which are solved numerically. The intensity factors vs. time for various crack size, crack position and material nonhomogeneity are obtained.     

Transient analysis of multiple parallel cracks under anti-plane dynamic loading

The problem of a homogeneous linear elastic body containing multiple non-collinear cracks under anti-plane dynamic loading is considered in this work. The cracks are simulated by distributions of dislocations and an integral equation relating tractions on the crack planes and the dislocation densities is derived. The integral equation in the Laplace transform domain is solved by the Gaussian–Chebyshev integration quadrature. The dynamic stress intensity factor associated with each crack tip is calculated by a numerical inverse Laplace scheme. Numerical results are given for one crack and two or three parallel cracks under normal incidence of a plane horizontally shear stress wave.     

Anti-plane shear problem for an edge crack in a finite orthotropic plate

The problem of an edge crack in a finite orthotropic plate under anti-plane shear is considered. The boundary collocation method is used to calculate the mode III stress intensity factor (SIF). For the case in which the material is isotropic, the present results agree very well with those obtained by using the integral equation method. Furthermore, the method can be extended readily for general cases with arbitrary geometrical and boundary loading conditions and material properties.     

Finite length crack moving in a viscoelastic strip under impact—I. Theory

A dynamic crack problem of the transient type was considered within the context of the linear theory of viscoelasticity. This involves a finite-length crack moving in a strip-like viscoelastic body under impact loading. Laplace and Fourier transforms were utilized and the resulting dual integral equations were reduced to a Fredholm integral equation of the second kind.     

Investigation of anti-plane shear behavior of a Griffith crack in a piezoelectric material by using the non-local theory

In this paper, the behavior of a Griffith crack in a piezoelectric material under anti-plane shear loading is investigated by using the non-local theory for impermeable crack surface conditions. By using the Fourier transform, the problem can be solved with two pairs of dual integral equations. These equations are solved using Schmidt method. Numerical examples are provided. Contrary to the previous results, it is found that no stress and electric displacement singularity is present at the crack tip.     

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.     

Elastodynamics of a crack on the bimaterial interface

The paper is an application of boundary integral equations to the problem of a crack located on the bimaterial interface under time-harmonic loading. A system of linear algebraic equations is derived for solving the problem numerically. The distributions of the displacements and tractions at the bimaterial interface are obtained and analysed for the case of a penny-shaped crack under normal tension-compression wave. The dynamic stress intensity factors (normal and shear modes) are also computed. The results are compared with those obtained for the static case.     

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.     

Stress intensity factors in anisotropic sandwich plate with a part-through crack under mixed mode deformation

An adhesively bonded anisotropic plate containing a part-through crack under mixed mode deformation is investigated. The problem is reduced to a pair of Fredholm integral equations of the second kind by mathematical analysis. By solving these equations numerically shear stresses of the adhesive and stress intensity factors in the cracked plate are obtained. Numerical results are presented for tension loading and shear loading, respectively, for various fiber orientations of the laminated composite. The results indicate a fairly strong dependence of the stress intensity factor on the fiber orientation of plate.     

The time-dependent interaction between inclusions and a cracked matrix subjected to antiplane shear

Summary. The time-dependent interaction between multiple circular inclusions and a cracked matrix in the antiplane viscoelastic problem is discussed in this paper. The fundamental elastic solution is obtained as a rapidly convergent series in terms of complex potentials via successive iterations of Möbius transformation in order to satisfy continuity conditions on multiple interfaces. Based on the correspondence principle, the Laplace transformed viscoelastic solution is then directly determined from the corresponding elastic one. In association with the singular integral technique, the time-dependent mode-III stress intensity factor of the crack tip can be solved numerically in a straightforward manner. Finally, some typical examples of an arbitrary crack lying in a matrix with various material properties under various loading types are also discussed. The results show that, depending on the relative locations and material properties of inclusions, the evolution of the stress intensity factor (SIF) may increase or decrease with time.     

Two general equations for deriving SIF expressions of some axisymmetric finite domain problems

Using the previous analytical method (Wang QZ, The crack-line (plane) stress field method for estimating SIFs—a review. Engineering Fracture Mechanics 1996; 55(4): 593–603.) and Green's function approach, two general equations are formulated for deriving approximate stress intensity factor (SIF) expressions for two categories of finite domain problems: (1) a finite-width strip with a center crack; (2) a circular cylinder with a concentric penny-shaped crack, both under various axisymmetric tensile loading at the crack faces. Examples with concentrated and distributed (up to quadratic variation) loading conditions are given to show the efficiency of these two general equations. As compared with the previous method, now the necessity of finding out the exact crack-line (plane) stress solution for the counterpart infinite problem is eventually waived. Another merit is that some SIF results for concentrated loading cases derived by using the general equations may have better accuracy than those given by the previous method. These two general equations are almost identical in form except for a small difference. Examples also show that the dimensionless SIF expressions for some problems in category (1) are identical with those in category (2), and there exists a regular correspondence between their loading conditions. Such identities in the dimensionless SIF expressions are useful in applications. Several example solutions given in this paper fill in the vacancy of missing solutions in present SIF handbooks, while other solutions are much simpler than the corresponding solutions in SIF handbooks.     

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.     

Boundary integral equation method to solve embedded planar crack problems under shear loading

The solution of three-dimensional planar cracks under shear loading are investigated by the boundary integral equation method. A system of two hypersingular integral equations of a three-dimensional elastic solid with an embedded planar crack are given. The solution of the boundary integral equations is succeeded taking into consideration an appropriate Gauss quadrature rule for finite part integrals which is suitable for the numerical treatment of any plane crack without a polygonal contour shape and permit the fast convergence for the results. The stress intensity factors at the crack front are calculated in the case of a circular and an elliptic crack and are compared with the analytical solution.     

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.