A laminate composite with an infinite row of parallel cracks normal to the interfaces

Under the assumption of plane strain, the distribution of stresses in a laminate composite due to the presence of an infinite row parallel cracks situated normal to the bond lines is obtained. Using an integral transform technique, the problem is reduced to that of solving a Fredholm integral equation of the second kind. The singular stress field near the tip of a crack is then derived. The effects on the stress intensity factor due to the presence of many cracks in a laminate-composite are illustrated graphically.     

A [90/0]s composite laminate with part-through matrix cracks in adjacent plies

A [90/0]_s orthotropic composite laminate with part-through matrix cracks is considered. Stress intensity factors are determined for the cracks using a linear-elastic analysis. These matrix cracks run along the fiber direction of the individual plies. The crack-geometry considered here is one where the matrix cracks in adjacent plies form a cross-like pattern in the plan view of the laminate. The plies are assumed bonded by thin resin-rich adhesive layers. These adhesive layers are modeled as distributed shear springs. Each ply of the laminate is modeled as a thin elastic orthotropic layer under plane stress. The laminate is subject to both tensile and shear loading. The mathematical model for the stresses and displacements in the layers reduces to a pair of Fredholm integral equations which are solved numerically. The stress intensity factors show a strong dependence on crack-sizes and nature of loading. In particular, the magnitudes of the stress intensity factors for the matrix crack in the 0° layer are increased significantly by the crack in the adjacent 90° layer.     

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.     

The elastostatic problem of an infinite strip containing a periodic row of line cracks

The two dimensional problem of an infinite strip containing a periodic row of line cracks, each subjected to arbitrary but identical pressure distribution, is solved by superposing two solutions: the problem of a row of line cracks in an infinite medium and the solution of an infinite strip loaded at the edges. This procedure leads to a system of simultaneous integral equations. The solution is obtained by reduction of these equations into algebraic equations with the help of Fourier expansion of involved functions. An analytic expression for the stress-intensity factor is also derived. Boundary conditions are later modified to get the solution of the problem of an infinite sheet containing doubly periodic array of line cracks.     

Tension of a cylindrical bar having an infinite row of circumferential cracks

In this paper, the crack problems in the case of a cylindrical bar having a circumferential crack and a cylindrical bar having an infinite row of circumferential cracks under tension are analyzed by the body force method. The stress field for a periodic array of ring forces in an infinite body is used to solve the problems. The solution is obtained by superposing the stress fields of ring forces in order to satisfy a given boundary condition. The stress intensity factors are calculated for various geometrical conditions. The obtained values of stress intensity factor of a single circumferential crack are considered to be more reliable than the results of other paper's. As the crack becomes very shallow, the stress intensity factor of a row of circumferential cracks approaches the value corresponding to that of a row of edge cracks in a semi-infinite plate under tension. As the crack becomes very deep, it approaches the values corresponding to that of a single deep circumferential crack.     

Dynamic stress intensity factors around three cracks in an infinite elastic plane subjected to time-harmonic stress waves

The time-harmonic problem for an infinite elastic plane weakened by three parallel cracks has been solved. In this problem, two cracks are situated symmetrically on either side of a central crack and incident stresses impinge perpendicular to the cracks. Using the Fourier transform technique, the boundary conditions are reduced to four simultaneous integral equations. To solve the equations, the differences of displacements inside the cracks are expanded in a series. The unknown coefficients in the series are solved by the Schmidt method. The dynamic stress intensity factors are calculated numerically for several crack configurations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Analysis of elliptical cracks perpendicular to the interface of two joined transversely isotropic solids

This paper presents a boundary element analysis of elliptical cracks in two joined transversely isotropic solids. The boundary element method is developed by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of stress intensity factors (SIFs) are obtained by using crack opening displacements. The results of the proposed method compare well with the existing exact solutions for an elliptical crack parallel to the isotropic plane of a transversely isotropic solid of infinite extent. Elliptical cracks perpendicular to the interface of transversely isotropic bi-material solids of either infinite extent or occupying a cubic region are further examined in detail. The crack surfaces are subject to the uniform normal tractions. The stress intensity factor values of the elliptical cracks of the two types are analyzed and compared. Numerical results have shown that the stress intensity factors are strongly affected by the anisotropy and the combination of the two joined solids.     

Two coplanar Griffith cracks in an infinite elastic layer under torsion

Summary We consider the problem of determining the stress distribution in an infinitely long isotropic homogeneous elastic layer containing two coplanar Griffith cracks which are opened by internal shear stress acting along the lengths of the cracks. The faces of the layer are assumed to be stress free. The cracks are located in the middle plane of the layer parallel to its faces. By using Fourier transforms, we reduce the problem to the solution of a set of triple integral equations with a cosine kernel and a weight function. These equations are solved exactly by using finite Hilbert transform techniques. Finally we derive the closed form expressions for the stress intensity factors and the crack energy. Solutions to the following problems are derived as particular cases: (i) a single crack in an infinite layer under torsion, (ii) two coplanar cracks in an infinite space under torsion, (iii) a single crack in an infinite space under torsion.     

On the solution of doubly periodic array of cracks

We present a correct procedure for the determination of the stress intensity factor at the tip of a crack in a doubly periodic array contained in an infinite elastic solid. It is based on the use of weight functions for an infinite row of periodic cracks, thus avoiding the use of divergent double infinite summations.     

Mutual influence of two parallel coaxial cracks in a composite material with initial stresses

By using the approaches of three-dimensional linearized mechanics of deformed bodies, we study the axisymmetric problem of two parallel coaxial circular cracks in an infinite composite material with initial stresses acting in the planes of the cracks. The resolving system of Fredholm integral equations of the second kind is deduced and the representations of the stress intensity factors in the vicinity of the crack tips are obtained. The dependences of the stress intensity factors on the initial stresses and the distance between the cracks are determined. For two types of composite materials, namely, for a layered composite with isotropic layers and a composite with stochastic reinforcement by short ellipsoidal fibers, the stress intensity factors are computed and their dependences on the initial stresses, physicomechanical characteristics of the composites, and geometric parameters of the problem are investigated. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 4, pp. 58–67, July–August, 2008.     

Anti-plane transient analysis of planes with multiple cracks

In this paper, the transient dynamic stress intensity factor is determined for multiple curved cracks under impact loading. The dislocation method has rarely been applied to the problems involving dynamic loading. The transient response of Volterra-type dislocation in a plane is obtained by means of the Cagniard-de Hoop method. The distributed dislocation technique is used to construct integral equations for an infinite isotropic plane weakened by cracks. These equations are of Cauchy singular type at the location of dislocation which are solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to determine stress intensity factors for multiple smooth cracks. Numerical results are obtained to validate the formulation and illustrate its capabilities.     

Dynamic stress intensity factors around two parallel cracks in an infinite elastic plate

Summary Dynamic stresses around two parallel cracks in an infinite elastic plate are obtained. An incoming shock stress wave impinges on the cracks at right angles to their faces. The Fourier-Laplace transform technique is utilized to reduce the problem to dual integral equations. To solve these equations, the differences in the crack surface displacements are expanded in a series of functions which are zero outside the cracks. The unknown coefficients occurring in those series are solved using the Schmidt method. The stress intensity factors defined in the Laplace transform domain are inverted numerically, in the physical space.     

Theoretical stress analysis for a non-isotropic body of cylindrical configuration containing a row of cracks

A problem of stress analysis for a long circular cylinder has been dealt with in this paper by analytical methods. The cylinder is assumed to be made of an elastic material which is not isotropic but the elastic properties are considered to be similar in directions perpendicular to the axis of the cylinder. The body under consideration is supposed to contain an infinite row of penny-shaped cracks which are parallel to each other and located periodically along the cylinder-axis. All the cracks are assumed to be opened by the same distribution of internal pressure on their surfaces. By choosing appropriate potential functions the problem is treated mathematically through the use of integral equation approach. Numerical results for the stress-intensity factor, the strain energy and the critical pressure, obtained on the basis of the analysis are also given.     

Scattering of antiplane shear waves by a finite crack in piezoelectric laminates

Summary Following the dynamic theory of linear piezoelectricity, we consider the scattering of horizontally polarized shear waves by a finite crack in a composite laminate containing a piezoelectric layer. The piezoelectric layer is bonded between two half-spaces of a different elastic solid. The crack is normal to the interfaces and is placed at an equal distance away from them. Both cases of a partially broken layer and a completely broken layer are studied. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a singular integral equation. The propagation of symmetric first mode is studied numerically, and the dynamic stress intensity factor and the dynamic energy release rate are obtained for some piezoelectric laminates.     

Three co-planar moving Griffith cracks in an infinite elastic medium

The dynamic in-plane problem of determining the stress and displacement due to three co-planar Griffith cracks moving steadily at a subsonic speed in a fixed direction in an infinite, isotropic, homogeneous medium under normal stress has been treated. The static problem of determining the stress and displacement around three co-planar Griffith cracks in an infinite isotropic elastic medium has also been considered. In both the cases, employing Fourier integral transform, the problems have been reduced to solving a set of four integral equations. These integral equations have been solved using finite Hilbert transform technique and Cook's result [16] to obtain the exact form of crack opening displacement and stress intensity factors which are presented in the form of graphs.     

The stress intensity factors of a star-shaped array of cracks in an infinite elastic solid

The problem of determining the stress intensity factors and crack formation energy of a radial system of line cracks in an infinite elastic solid is reduced to the solution of a singular integral equation. The equation is solved numerically for the special case in which the cracks are opened by a constant pressure.     

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.     

Four co-planar Griffith cracks in an infinite elastic medium

The dynamic in-plane problem of determining the stress and displacement due to four co-planar Griffith cracks moving steadily at a subsonic speed in a fixed direction in an infinite, isotropic, homogeneous medium under normal stress has been treated. The static problem of determining the stress and displacement in an infinite isotropic elastic medium has also been considered. In both cases, employing the Fourier integral transform, the problems have been reduced to solving a set of five integral equations. These integral equations have been solved using the finite Hilbert transform technique to obtain the exact form of crack opening displacement and stress intensity factors which are presented in the form of graphs.     

Transient analysis of stress waves around two coplanar griffith cracks under impact load

The problem of determining the transient stress distribution in an infinite elastic medium weakened by two coplanar Griffith cracks is considered. To the surfaces of the cracks, an internal pressure is applied suddenly. The problem is reduced to that of solving dual integral equations in the Laplace transform domain and those are solved by a series-expansion method. The dynamic stress intensity factors are computed numerically.     

On elastodynamical problem of interfacial Griffith cracks in composite media

The plane strain problem of determining stress intensity factors for two equal and parallel moving interfacial Griffith cracks in composite media consisting of an orthotropic layer bonded two dissimilar orthotropic half planes is considered. The problem is reduced to solution of two pair of simultaneous singular integral equations containing Cauchy kernels. Expressions for stress intensity factor are obtained for the case of a general loading distribution.