Cruciform Crack Research Articles

Effect of T-stress on the fracture in an infinite two-dimensional decagonal quasicrystals of a cruciform crack with unequal arms

The T-stress near the tips of a cruciform crack with unequal arms in the two-dimensional (2D) decagonal quasicrystals (QCs) are analyzed. The associated boundary value problems can transform to a system of singular integral equations by using the integral transformation technique. According to the stress difference method, T-stresses of phonon field and phason field can be expressed as the sum of an integral involving crack opening displacements (CODs) and applied loading at infinity. CODs play a leading role in determining T-stresses of phonon field and phason field. The role of the T-stresses in brittle fracture for 2D decagonal QCs is emphasized. Moreover, the influence of the ratio of two crack-arm lengths b/a and the CODs on the T-stresses of a cruciform crack is examined.

Numerical solution of an integral equation arising in the problem of cruciform crack using Daubechies scale function

This paper is concerned with obtaining approximate numerical solution of a classical integral equation of some special type arising in the problem of cruciform crack. This integral equation has been solved earlier by various methods in the literature. Here, approximation in terms of Daubechies scale function is employed. The numerical results for stress intensity factor obtained by this method for a specific forcing term are compared to those obtained by various methods available in the literature, and the present method appears to be quite accurate.

Numerical solution of a singular integral equation arising in a cruciform crack problem

A singular integral equation arising in a cruciform crack problem is investigated in the present paper. Based on the convex technique, the piecewise Taylor-series expansion method is extended by introducing a weight parameter. An approximate solution of the singular integral equation is constructed and its convergence and error estimate are made. The variations of the approximate solutions associating with stress intensity factors are analyzed by considering internal pressures of power and sine functions, respectively. By comparing with the known methods, the observations reveal that a good approximation can be achieved using less derivative times, less discretization points, and a suitable weight parameter. The obtained results show that the crack growth is dependent on applied mechanical loadings.

Study of thermo‐elastic cruciform crack with unequal arms in an orthotropic elastic plane

The objective of this article is concerned with a study of a static cruciform crack problem in an infinite orthotropic elastic medium opened by prescribed normal pressure distributions subjected to steady state temperature field. The problem is reduced to Fredholm singular integral equations of first kind, which are solved using Chebyshev polynomials. Analytical expressions of the stress intensity factors and crack energies are found and the numerical computations for the orthotropic elastic material Boron‐epoxy composite for different particular cases are depicted through figures. The striking feature of the article is the graphical presentation of possibility of crack arrest and variations of crack energies with the variations of the arms of the cruciform crack.

The interaction between a screw dislocation and an interfacial cruciform crack and collinear linear cracks

The interaction between a screw dislocation and an interfacial cruciform crack and collinear linear cracks under loads at infinity was investigated. General solutions of complex potentials to this problem were derived by using complex potential theory. As illustrative examples, the closed form solution for a screw dislocation interacting with an interfacial cruciform crack and a linear crack is obtained. The stress intensity factor and critical stress intensity factor for dislocation emission are also calculated. The results show that the shielding effect increases with the increase of the shear modulus and the distance between the two cracks, but it decreases with the increase of dislocation azimuth and the distance between the dislocation and the cruciform crack tip. The critical loads at infinity for dislocation emission increase with the increment of the emission angle, the distance the two cracks and the vertical length of the cruciform crack.

Approximate solution to an integral equation with fixed singularity for a cruciform crack

A novel method for determining an approximate solution to an integral equation with fixed singularity is presented. This integral equation is encountered in solving a cruciform crack. On the basis of Taylor’s series for the unknown function, the integral equation can be transformed to a system of linear equations for the unknown and its derivatives when neglecting a sufficiently small quantity. Moreover, the n th-order approximation obtained is exact for a solution of a polynomial of degree less than or equal to n . The proposed method is simple, fast, and can be performed by symbolic computation using any personal computer. A test example is given to indicate the efficiency of the method. This method is also applicable to a variety of integral equations.

T-stress near the tips of a cruciform crack with unequal arms

The T-stress near the tips of a crack of cross shape embedded in an isotropic elastic solid is analyzed. The integral transform technique is employed to convert the associated boundary value problem to a system of singular integral equations. According to the stress difference method, T-stresses can be expressed as a sum of an integral involving crack opening displacement (COD) and applied loading at infinity. Obtained results indicate that, in addition to applied loading, T-stresses at the horizontal (vertical) crack tips depend on the COD of the vertical (horizontal) crack surface. COD plays a leading role in determining T-stresses, in particular for a cruciform crack of two crack-arm lengths of the same order. Moreover, T-stresses for a single-crack limiting case can be recovered from the present results as the length of one arm approaches zero. For a biaxial tension of the same magnitude, T-stresses are present for a cruciform crack, but absent for a single crack. Finally, for several cases of interest, T-stresses for a cruciform crack are evaluated and compared with those for a single crack, and the influence of the ratio of two crack-arm lengths b/ a and the COD on the T-stress of a cruciform crack is examined.

Study of a static cruciform crack problem in an orthotropic elastic plane

This paper deals with the plane strain problem of determining stress intensity factors and crack energies of a static cruciform crack in an infinite orthotropic elastic medium. The problem is reduced to the solution of two simultaneous singular integral equations with Cauchy-type singularities. Expressions for stress intensity factors (SIF) and crack energies for the case general loading distribution are obtained. Numerical results for some special cases are presented for the Boron-Epoxy composite material.

An algorithm for the approximate solution of integral equations of Mellin type

The cruciform crack problem of elasticity gives rise to an integral equation of the second kind on [0,1] whose kernel has a fixed singularity at (0,0). We introduce a transformation of [0,1] onto itself such that an arbitrary number of derivatives vanish at the end points 0 and 1. If the transformed kernel is dominated near the origin by a Mellin kernel then we have given conditions under which the use of a modified Euler-Maclaurin quadrature rule and the Nystrom method gives an approximate solution which converges to the exact solution of the original equation. The method is illustrated with a numerical example.

Analysis of stress intensity factors for three-dimensional cruciform cracks

The stress intensity factors for three-dimensional cruciform surface cracks in a semi-infinite body are numerically calculated by the body force method. Mindlin's point force solution is used for the derivation of basic equations to express the influence coefficient of triangular elements, into which the crack is divided. The interactions between crossed crack planes as well as contact between crack surfaces are considered in the iterative manner. Stress intensity factors for a cruciform median crack and a cruciform semicircular crack under a point force on the surface of a semi-infinite solid are analyzed. The possibility of growth of a median crack toward the free surface of the semi-infinite solid is discussed. A cruciform semicircular surface crack under remote uniaxial tension, or under combined tension and compression is also analyzed. The effect of contact of crack surfaces on stress intensity factors is discussed.

Nyström interpolants based on the zeros of Legendre polynomials for a non-compact integral operator equation

We consider two different Nystrom interpolants for the numerical solution of the following singular integral equation (formule...), 0 < x ≤ 1, arising from a problem of determining the distribution of stress in a thin elastic plate in the vicinity of a cruciform crack. These interpolants originate from the discretization of the integral by two different quadrature formulas of interpolatory type based on the zeros of Legendre orthogonal polynomials. The first quadrature is of product type and integrates exactly the kernel; the second one is the well-known Gauss-Legendre formula. First we derive uniform convergence estimates for the two basic quadrature rules

Numerical solution for a cruciform crack problem

Numerical solution for a cruciform crack problem

Thermoelastic problem of an orthotropic elastic plane containing a cruciform crack

The problems of determining the thermal stress and displacement fields in an infinite orthotropic plane containing a cruciform crack when (I) the shape of the crack is prescribed and (II) the cracks are opened by given normal pressures, are reduced to mixed boundary value problems for the quarter plane. An integral transform technique, based upon displacement potentials, is employed for the case of a steady-state temperature field. Closed form solution is obtained for problem I, whereas the solution of problem II has been reduced to solving a Fredholm integral equation of second kind with non-singular kernel. The expressions for the stress intensity factor and the energy of crack formation are derived. In problem II, numerical results are presented for the case of linear loading functions.

Cruciform crack in an orthotropic elastic plane

The problems of determining the stress and displacement fields in an infinite orthotropic plane containing a cruciform crack 387-1, y=0 and 387-2, x=0 when (I) the shape of the crack is prescribed and (II) the cracks are opened by given normal pressures, are reduced to mixed boundary value problems for the quarter plane. Using integral transform techniques, a closed form solution is obtained for problem I, whereas the solution of problem II has been reduced to solving a Fredholm integral equation of second kind with non-singular kernel. Numerical calculation of the stress intensity factor and crack energy in the case of a linear loading function for various crack lengths are presented for problem II, using the values of material constants for a Boron-Epoxy composite.

The general solution to an infinite orthotropic plate with a cruciform crack under arbitrary anti-plane shear stresses

In accordance with the basic theorem of the Mellin transform, the general solution of a cruciform crack in an orthotropic infinite plate subjected to an arbitrary longitudinal shear stress is found in this study. I think the result of the stress intensity factor of this problem will certainly interest engineers because it is not only formulated in simply closed form but also independent of material constants.

On the numerical treatment of an integral equation arising from a cruciform crack problem

AbstractWe consider the following two integral equations magnified image the latter subject to the condition ∫v(y)dy = 0, arising from the same problem of determining the distribution of stress in a thin elastic plate in the vicinity of a cruciform crack. In particular, we prove existence and uniqueness of the solutions of the two equations above in C[0, 1]. We also analyse the behaviour of ϕ(x) near the end‐point x = 0.The therotical result obtained in the first part are then used in the last section to derive optimal rates of convergence for numerical methods, such as Galerkin, collocation and Nyström, applied to equation (1).

Cruciform crack propagation in plane strain

The problem of an asymmetric cruciform crack emanating from a point defect and rapidly expanding in a linear elastic space, under plane strain homogeneous polynomial-form loading, is analysed by means of the Busemann-Chaplygin technique and analytic function theory. The problem is reduced to a singular integral equation of the Carleman type which can be solved by well known numerical methods. The result is compared with that of the single-crack limiting case.

The general solution to an infinite plate with a cruciform crack under arbitrary normal stresses

With the aid of the basic theorem of the Mellin transform, the exact solution to an infinite sheet weakened by a cruciform crack subjected to arbitrary normal stresses is obtained in this paper. I think the new result of the stress intensity factor of this problem will certainly interest designers because it is formulated in simply closed form. Especially, we can also find that the new method recommended in this article will be employed to solve the more complex problems which cannot be treated by methods given in previous research.

The cruciform crack running in a biaxial tension field

Under a biaxial tension field, failure in a small region can give rise to simultaneous rapid crack extension along multiple paths. To gain insight into this phenomenon, the dynamic analysis of a cruciform crack growing from a point in a biaxial tension field is considered. An analytic solution is at present not available. However, by treating the cruciform arms as dislocation arrays, the problem is reduced to a pair of coupled integral equations. By following a scheme for plane crack analysis, these equations are uncoupled, and solved numerically by a standard technique for singular integral equations. Examination of the solution indicates that a plane crack is more likely to occur than a cruciform crack in a uniform biaxial tension field, and the crack openings will be larger.

Dynamic studies of running half-plane and cruciform cracks

The dynamic problems of a crack running perpendicularly into a half-plane surface and a cruciform crack running in an unbounded solid under the action of moving point forces are analyzed. The cracks are treated as dislocations distributed w.r.t. Speed, so that the problems reduce to singular integral equations with Dirac functions as non-homogeneous terms. By extracting physically significant limit cases with analytical solutions, the terms are removed, and the resulting equations solved numerically by a standard technique. Dynamic stress intensity factor and crack opening data are presented. For the cruciform crack, this data is compared with that for the plane crack limit case.