Multiple crack problems of antiplane elasticity in an infinite body

Twe elementary solutions are presented for case of a pair of normal or tangential concentrated unit forces acting at a point of both edges of a single crack in an infinite plane isotropic elastic medium. Using these two elementary solutions and the principle of superposition, we found that the multiple crack problems can be easily converted into a system of Fredholm integral equations. Finally, the system obtained is solved numerically and the values of the stress intensity factors at the crack tips can be easily calculated. Two numerical examples are given in this paper. A system of Fredholm integral equations is complex form is also presented. We found that the system of Fredholm integral equations can be easily reduced from the system of singular integral equations given by Panasyuk[1]     

Singular integral equation method for multiple Zener–Stroh crack problems in antiplane elasticity

In this paper, the multiple Zener–Stroh crack problems in anti-plane elasticity are studied. The crack faces are assumed to be traction free, and dislocation distributions on the cracks are chosen as the unknown functions in the solution. The singular integral equations for the problem are obtained. The constraint equations are also derived from the condition of the accumulation of dislocation on the cracks. After solving the integral equations, the stress intensity factors at crack tips can be evaluated immediately. Numerical examples are given. It is found that interactions between the Zener–Stroh cracks are quite different from those for the Griffith cracks, in qualitative and quantitative aspects.     

New Fredholm integral equation for multiple crack problem in plane elasticity and antiplane elasticity

A singular integral equation for the multiple crack problem of plane elasticity is formulated in this paper. In the formulation we choose the crack opening displacement (COD) as unknown function and the resultant force as the right hand term of the equation. After using Vekua's regularization procedure or making inversion of the Cauchy singular integral in the equation, a new Fredholm integral equation is obtainable. The obtained Fredholm integral equation is compact in form and easy for computation. After solving the equation, the CODs of the cracks and the stress intensity factors (SIFs) at the crack tips can be derived immediately. Similar formulation for the multiple crack problem of antiplane elasticity is also presented. Finally, numerical examples are given to demonstrate the use of the proposed integral equation approach.     

Dual boundary integral equation formulation in antiplane elasticity using complex variable

This paper investigates the dual boundary integral equation formulation in antiplane elasticity using complex variable. Four kinds of boundary integral equation (BIE) are studied, and they are the first complex variable BIE for the interior region, the second complex variable BIE for the interior region, the first complex variable BIE for the exterior region, and the second complex variable BIE for the exterior region. The first BIE for the interior region is derived from the Somigliana identity, or the Betti’s reciprocal theorem in elasticity. A displacement versus traction operator is suggested. After using this operator, the second BIE for the interior region is derived. Similar derivations are performed for the first and second BIEs for the exterior region. In the case of the exterior boundary, two degenerate boundary cases are studied. One is the curved crack case, and other is the case of a deformable line. All kernels in the suggested BIEs are expressed in terms of complex variable.     

Hypersingular integral equation approach for the multiple crack problem in an infinite plate

Summary In this paper, a hypersingular integral equation for the multiple crack problem in an infinite plate is formulated. The unknown functions involved in the equation are the crack opening displacements (CODs) while the right hand terms are the tractions applied on the crack faces. Some particular hypersingular integrals are quadratured in a closed form. After the CODs are approximated by a weight function multiplied by a polynomial, the hypersingular integrals in the equation can be evaluated in a closed form, and the regular integrals can be integrated numerically. Numerical examples with the calculated stress intensity factors (SIFs) at the crack tips are given.     

Solution of periodic group crack problems by using the Fredholm integral equation approach

Summary Periodic group cracks composed of infinitely many groups numbered from j = -∞,...-2,-1,0,1,2,...to j = ∞ placed periodically in an infinite plate are studied. The same loading condition and the same geometry are assumed for cracks in all groups. The Fredholm integral equation is formulated for the cracks of the central group (or the 0-th group) collecting the influences from the infinite neighboring groups. The influences from many neighboring groups on the central group are evaluated exactly, and those from remote groups approximately summed up into one term. The stress intensity factors can be directly evaluated from the solution of the Fredholm integral equation. Numerical examples show that the suggested technique provides very accurate results. Finally, several numerical examples are presented, and the interaction between the groups is addressed.     

A new singular integral equation for the classical crack problem in plane and antiplane elasticity

A new singular integral equation (with a kernel with a logarithmic singularity) is proposed for the crack problem inside an elastic medium under plane or antiplane conditions. In this equation the integral is considered in the sense of a finite-part integral of Hadamard because the unknown function presents singularities of order ?3/2 at the crack tips. The Galerkin and the collocation methods are proposed for the numerical solution of this equation and the determination of the values of the stress intensity factors at the crack tips and numerical results are presented. Finally, the advantages of this equation are also considered.     

Solutions of multiple crack problems of a circular plate or an infinite plate containing a circular hole by using Fredholm integral equation approach

Presented is an elementary solution, which is a particular solution of the circular plate containing one crack. The solution consists of two parts and satisfies the following conditions: (i) the first part corresponds to a pair of normal and tangential concentrated forces acting at a prescribed point on both edges of a single crack; (ii) the second part corresponds to some distributed tractions along both edges of the crack; (iii) the obtained elementary solution, i.e. the sum of the first and second parts, satisfies a traction free condition on the circular boundary. Using this elementary solution and taking some undetermined density of the elementary solution along each crack, a system of Fredholm integral equations of multiple crack problems can always be obtained. The multiple crack problems of an infinite plate containing a circular hole can be solved in a similar way. Several numerical examples are given in this paper.     

A variational boundary integral equation method for an elastodynamic antiplane crack

This paper investigates the transient wave scattering by a crack by means of the Boundary Integral Equation Method (BIEM). The author has developed a new formulation to solve the BIE for the Crack Opening Displacement (COD). The resolution is done directly in the time domain. The solution is represented by means of a retarded double layer potential, and the resulting BIE, with the COD as unknown, has a hypersingular kernel. The corresponding difficulty is overcome by using a variational method. We present the application of this method to an antiplane crack, describe the approximate problem and finally give some numerical results.     

A new boundary integral equation for notch problem of antiplane elasticity

In this paper the notch problem of antiplane elasticity is discussed and a new boundary integral equation is formulated. In the problem, the distributed dislocation density is taken to be the unknown function. Unlike the usual choice, the resultant force function is taken as the right hand term of the integral equation; therefore, a new boundary integral equation for the notch problem of antiplane elasticity with a weaker singular kernel (logarithmic) is obtained. After introducing a particular fundamental solution of antiplane elasticity, the notch problem for the half-plane is discussed and the relevant boundary integral equation is formulated. The integral equations derived are compact in form and convenient for computation. Numerical examples demonstrated that high accuracy can be achieved by using the new boundary equation.     

Numerical solutions of a hypersingular integral equation for antiplane elastic curved crack problems of circular regions

Summary. In this paper, a hypersingular integral equation for the antiplane elasticity curved crack problems of circular regions is suggested. The original complex potential is formulated on a distribution of the density function along a curve, where the density function is the COD (crack opening displacement). The modified complex potential can also be established, provided the circular boundary is traction free or fixed. Using the proposed modified complex potential and the boundary condition, the hypersingular integral equation is obtained. The curve length method is suggested to solve the integral equation numerically. By using this method, the usual integration rule on the real axis can be used to the curved crack problems. In order to prove that the suggested method can be used to solve more complicated cases of the curved cracks, several numerical examples are given.     

A hypersingular boundary integral equation for antiplane crack problems for a class of inhomogeneous anisotropic elastic materials

An antiplane multiple crack problem is considered for a class of inhomogeneous anisotropic elastic materials. The problem is reduced to a boundary integral equation involving hypersingular integrals. The boundary integral equation may be solved numerically using standard procedures. Some crack problems for a particular inhomogeneous material is considered in detail and the stress intensity factors are obtained in order to assess the effect of the anisotropy and inhomogeneity on the stress field near the crack tips.     

Boundary element analysis of an integral equation for three-dmensional crack problems

In another paper, the authors proposed an integral equation for arbitrary shaped three-dimensional cracks. In the present paper, a discretization of this equation using a tensor formalism is formulated. This approach has the advantage of providing the displacement discontinuity vector in the local basis which varies as a function of the point of the crack surface. This also facilitates the computation of the stress intensity factors along the crack edge. Numerical examples reported for a circular crack and a semi-elliptical surface crack in a cylindrical bar show that one can obtain good results, using few Gaussian points and no singular elements.     

A direct traction boundary integral equation method for three-dimension crack problems in infinite and finite domains

This paper presents a direct traction boundary integral equation method (TBIEM) for three-dimensional crack problems. The TBIEM is based on the traction boundary integral equation (TBIE). The TBIE is collocated on both the external boundary and one of the crack surfaces. The displacements and tractions are used as unknowns on the external boundary and the relative crack opening displacements (CODs) are introduced as unknowns on the crack surface. In our implementation, all the surfaces of the considered structure are discretized into discontinuous elements to satisfy the continuity requirement for the existence of finite-part integrals, and special crack-front elements are constructed to capture the crack-tip behavior. To calculate the finite-part integrals, an adaptive singular integral technique is proposed. The stress intensity factors (SIFs) are computed through a modified COD extrapolation method. Numerical examples of SIFs computation are presented to demonstrate the accuracy and efficiency of our method.