scholarly journals Regularization of the Divergent Integrals. II. Application in Fracture Mechanics.

2007 ◽  
Vol 4 (2) ◽  
Author(s):  
Vladimir Zozulya
Keyword(s):  
Fracture Mechanics ◽  
Integral Equations ◽  
Convex Polygon ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Hypersingular Integrals ◽  
Divergent Integral ◽  
Contour Integrals ◽  
Weakly Singular ◽  
Arbitrary Convex

In this article the methodology for divergent integral regularization developed in [8] is applied for regularization of the weakly singular and hypersingular integrals, which arise when the boundary integral equations (BIE) methods are used to solve problems in fracture mechanics. The approach is based on the theory of distribution and the application of the Green theorem. The weakly singular and hypersingular integrals over arbitrary convex polygon have been transformed to the regular contour integrals that can be easily calculated analytically or numerically.

scholarly journals NONLINEAR PROBLEM OF FRACTURE MECHANICS OF AN INTERFACE CRACK SUBJECTED TO SHEAR WAVE

2021 ◽  
pp. 55-64
Author(s):  
Oleksandr Menshykov ◽  
Vasyl Menshykov ◽  
Olga Kladova
Keyword(s):  
Stress Intensity ◽  
Fracture Mechanics ◽  
Integral Equations ◽  
Shear Wave ◽  
Contact Zone ◽  
Interface Crack ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Contact Boundary ◽  
Crack Faces

Solution for the problem for an interface crack under the action of a harmonic shear wave in normal direction is presented. The contact of the crack faces is put into consideration. The problem is solved by the boundary integral equations method, the vector components in the boundary integral equations are presented by Fourier series. The unilateral Signorini boundary conditions are involved to prevent the interpenetration of the crack faces and the emergence of tensile forces in the contact zone. Amonton-Coulomb Friction Law included allows to put into consideration relative resting of the crack opposite faces or their relative motion when one crack face is slipping or sliding across another face. The contact boundary restrictions are implemented using the iterative correction algorithm. The mathematical model adequacy is checked by comparing with classical model solution for statics problems that takes into account the crack faces contact. Numerical researches of friction influence on the displacement and contact forces distribution, size of contact zone are carried out. Influence of the faces contact and value of the friction coefficient on the distribution of stress intensity coefficients of normal rupture and transverse shear, which are the parameters of the biomaterial fracture, are presented and analyzed. It is shown that the nature of change in the distribution of the stress intensity coefficients for the conditions of tensile and shear waves is fundamentally different. It is concluded that it is possible to extend the approach proposed to the problems of three-dimensional fracture mechanics for composites with interfacial cracks at arbitrary dynamic loading.

scholarly journals A General Algorithm for the Numerical Solution of Hypersingular Boundary Integral Equations

1992 ◽  
Vol 59 (3) ◽  
pp. 604-614 ◽  
Author(s):  
M. Guiggiani ◽  
G. Krishnasamy ◽  
T. J. Rudolphi ◽  
F. J. Rizzo
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Collocation Point ◽  
Three Dimensional ◽  
Careful Analysis ◽  
Boundary Integral ◽  
Hypersingular Integrals ◽  
Closed Surfaces ◽  
First Time ◽  
General Method

The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical algorithm. In the second part, a new general method for the evaluation of hypersingular surface integrals in the boundary element method (BEM) is presented. The proposed method can be systematically applied in any BEM analysis, either with open or closed surfaces, and with curved boundary elements of any kind and order (of course, provided the density function meets necessary regularity requirements at each collocation point). The algorithm operates in the parameter plane of intrinsic coordinates and allows any hypersingular integral in the BEM to be directly transformed into a sum of a double and a one-dimensional regular integrals. Since all singular integrations are performed analytically, standard quadrature formulae can be used. For the first time, numerical results are presented for hypersingular integrals on curved (distorted) elements for three-dimensional problems.

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