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Y Z Chen - One of the best experts on this subject based on the ideXlab platform.

  • evaluation of the t stress for multiple cracks in an elastic half plane using Singular Integral Equation and green s function method
    Applied Mathematics and Computation, 2014
    Co-Authors: Y Z Chen
    Abstract:

    This paper studies the T-stress problem for multiple cracks in an elastic half-plane using the Singular Integral Equation and Green's function method. In the relevant perturbation field, the boundary of the elastic half-plane is traction free. In the solution, the complex potentials are decomposed into two parts, the principal part and the complementary part. The principal part of the complex potentials is derived from dislocation distributions along the crack faces. The role of the complementary part is to eliminate the traction along the boundary of half-plane caused by the principal part. Finally, a Singular Integral Equation is formulated in which the dislocation distributions are unknown function. The explicit formulae for evaluating the stress intensity factors and the T-stresses at the crack tips are provided. Several numerical examples are presented.

  • numerical solution of Singular Integral Equation for multiple curved branch cracks
    Structural Engineering and Mechanics, 2010
    Co-Authors: Y Z Chen, X Y Lin
    Abstract:

    In this paper, numerical solution of the Singular Integral Equation for the multiple curved branch-cracks is investigated. If some quadrature rule is used, one difficult point in the problem is to balance the number of unknowns and Equations in the solution. This difficult point was overcome by taking the following steps: (a) to place a point dislocation at the intersecting point of branches, (b) to use the curve length method to covert the Integral on the curve to an Integral on the real axis, (c) to use the semi-open quadrature rule in the integration. After taking these steps, the number of the unknowns is equal to the number of the resulting algebraic Equations. This is a particular advantage of the suggested method. In addition, accurate results for the stress intensity factors (SIFs) at crack tips have been found in a numerical example. Finally, several numerical examples are given to illustrate the efficiency of the method presented.

  • a semi analytic solution for multiple curved cracks emanating from circular hole using Singular Integral Equation
    Applied Mathematics and Computation, 2009
    Co-Authors: Y Z Chen, X Y Lin, Z X Wang
    Abstract:

    This paper provides an elastic solution for an infinite plate containing multiple curved edge cracks emanating from a circular hole. A fundamental solution is suggested, which represents a particular solution for a concentrated dislocation in an infinite plate with the traction free hole. The generalized image method and the concept of the modified complex potentials are used in the derivation of the fundamental solution. After using the fundamental solution and placing the distributed dislocations at the prospective sites of cracks, a Singular Integral Equation is formulated. The Singular Integral Equation is solved by using the curve length method in conjunction with the semi-opening quadrature rule. By taking an additional point dislocation at the hole center, the number of the unknowns is equal to the number of the resulting algebraic Equations. This is a particular advantage of the suggested method. Finally, several numerical examples are given to illustrate the efficiency of the method presented. Numerical examinations are carried out and sufficient accurate results have been found.

  • perturbation method for the solution of a zener stroh crack with a slightly curved configuration
    Acta Mechanica, 2009
    Co-Authors: Y Z Chen, X Y Lin, Z X Wang
    Abstract:

    This paper investigates the Zener–Stroh crack with curved configuration in plane elasticity. A Singular Integral Equation is suggested to solve the problem. Formulae for evaluating the SIFs and T-stress at the crack tip are suggested. If the curve configuration is a product of a small parameter and a quadratic function, a perturbation method based on the Singular Integral Equation is suggested. In the method, the Singular Integral Equation can be expanded into a series with respect to the small parameter. Therefore, many Singular Integral Equations can be separated from the same power order for the small parameter. These Singular Integral Equations can be solved successively. The solution of the successive Singular Integral Equations will provide results for stress intensity factors and T-stress at the crack tip. It is found that the behaviors for the solution of SIFs and T-stress in the Zener–Stroh crack and the Griffith crack are quite different. This can be seen from the presented comparison results.

  • Singular Integral Equation method for the solution of multiple curved crack problems
    International Journal of Solids and Structures, 2004
    Co-Authors: Y Z Chen
    Abstract:

    Abstract A method based on complex potentials for distributions of dislocations along curved cracks is used to solve multiple curved crack problems in plane elasticity. The method allows evaluation of the interaction between curved cracks. A crack problem is reduced to a system of Singular Integral Equations and the crack curve length is taken as the coordinate in the associated Integral Equations. A crack is then mapped on the real axis in an interval (− a , a ), where 2 a is the length of crack and the original Singular Integral Equations are transformed accordingly. The method allows cracks with a general curvature and is not restricted to slightly curved crack configurations. The resulting Singular Integral Equation system is solved through Gauss quadrature. A few numerical examples of problems with two cracks are given and crack interaction, i.e., the shielding effect of a curved crack surrounding by another, is also studied.

Pengpeng Shi - One of the best experts on this subject based on the ideXlab platform.

  • Singular Integral Equation method for 2d fracture analysis of orthotropic solids containing doubly periodic strip like cracks on rectangular lattice arrays under longitudinal shear loading
    Applied Mathematical Modelling, 2020
    Co-Authors: Pengpeng Shi
    Abstract:

    Abstract The doubly periodic arrays of cracks represent an important mesoscopic model for analysis of the damage and fracture mechanics behaviors of materials. Here, in the framework of a continuously distributed dislocation model and Singular Integral Equation approach, a highly accurate solution is proposed to describe the fracture behavior of orthotropic solids weakened by doubly periodic strip-like cracks on rectangular lattice arrays under a far-field longitudinal shear load. By fully comparing the current numerical results with known analytical and boundary element solutions, the high precision of the proposed solution is verified. Furthermore, the effects of periodic parameters and orthotropic parameter ratio on the stress intensity factor, crack tearing displacement, and effective shear modulus are studied, and an analytically polynomial estimation for the equivalent shear modulus is proposed in a certain range. The interaction distances among the vertical and horizontal periodic cracks are quite different, and their effects vary with the orthotropic parameter ratio. In addition, the dynamic problem is discussed briefly in the case where the material is subjected to harmonic longitudinal shear stress waves. Further work will continue the in-depth study of the dynamics problem of the doubly periodic arrays of cracks.

Z K Eshkuvatov - One of the best experts on this subject based on the ideXlab platform.

S A Plaksa - One of the best experts on this subject based on the ideXlab platform.

Khairullina L. - One of the best experts on this subject based on the ideXlab platform.

  • Uniform Wavelet-Approximation of Singular Integral Equation Solutions
    2020
    Co-Authors: Khairullina L., Ozhegova A.
    Abstract:

    © 2018, Pleiades Publishing, Ltd. In this article we consider a Singular Integral Equation of the first kind with a Cauchy kernel on a segment of the real axis, which is a mathematical model of many applied problems. It is known that such an Equation is exactly solved only in rare cases, therefore, the problem of its approximate solution with obtaining uniform error estimates is very actual. This Equation is considered on a pair of weighted spaces that are constrictions of the space of continuous functions. The correctness of the problem of solving this Equation on a chosen pair of spaces of the desired elements and right-hand sides gives the possibility of its approximate solution with a theoretical justification. The numerical method proposed in this article is based on the approximation of the unknown function by Chebyshev wavelets of the second kind. Uniform error estimates are established depending on the structural properties of the initial data. The numerical experiment in the Wolfram Mathematica package showed a good convergence rate of the approximate solution to the exact one

  • Convergence of approximate solutions to two-dimensional Singular Integral Equation with Cauchy kernel in the Integral metrics
    2020
    Co-Authors: Khairullina L.
    Abstract:

    © 2016,International Journal of Pharmacy and Technology. All rights reserved.The article is devoted to approximate solution to two-dimensional Singular Integral Equation with the Cauchy kernel by the moment method. It is known from the theory of Singular Equations that finding an exact solution to such closed form Equation is possible only in certain cases,and regular and Singular Integrals with complex densities have to be calculated to obtain a numerical result. Therefore,developments of approximate solution methods followed by theoretical justification are important for the theory and,in particular,for applications. In this study the convergence of the approximate solution method with regard to a Singular Integral Equation is set forth in the broadest function space,namely,in the space of square summable [-1,1]×[-1,1] functions L = L [-1,1]2. For the correct statement of the problem,a pair of weighted spaces of the required elements and right parts are introduced,which are restrictions of the space of summable functions. Correctness of the considered Equation is proved. Solution to the characteristic Equation,as well as norms of a Singular operator and an operator opposite to it are given. Computational scheme of the moment method is formed. Theorem about unique solvability of the resulting system of linear algebraic Equations in the Integral metric is proven

  • On the uniform convergence of the method of waveletquadratures for the solution of a Singular Integral Equation of the first kind
    2020
    Co-Authors: Khairullina L., Ozhegova A.
    Abstract:

    © 2019 IOP Publishing Ltd. We propose a method of wavelet-quadratures for the solution of a Singular Integral Equation of the first kind with a Cauchy kernel on a segment of the real axis, which is a mathematical model of many applied problems. To solve this Equation, a computational scheme is constructed, based on the approximation of the unknown function by Chebyshev wavelets of the second kind and using the quadrature Gauss formula. Uniform estimates of the error of approximate solutions are obtained, which take into account the structural properties of the initial data. A numerical experiment was carried out using the Wolfram Mathematica package