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Derivation of a new analytical solution for a general two-dimensional finite-part integral applicable in fracture mechanics. (English) Zbl 0772.73062

The study of two-dimensional and three-dimensional crack problems for a variety of bodies and crack geometries leads to boundary value problems which can be reduced to a governing hypersingular integral equation of the form \[ \begin{split} {\rlap{\;=}\int\kern-.5em\rlap{\;=}\int_ \Omega r^{-3}} W(x_ 0,y_ 0)dx_ 0 dy_ 0+\int\int_ \Omega W(x_ 0,y_ 0) K(x_ 0,y_ 0;x,y)dx_ 0 dy_ 0=\\ -4\pi\mu^{-1}(1- \nu)p(x,y), \qquad x,y\in\Omega\end{split} \tag{1} \] where \(\Omega\) denotes the crack domain with boundary \(\partial\Omega\), \(W(x_ 0,y_ 0)\) the unknown crack opening displacement, \(r\) the distance defined by \(r^ 2=(x_ 0-x)^ 2+(y_ 0-y)^ 2\), \(p(x,y)\) is arbitrary pressure distribution on the crack faces, \(\nu\) is the Poisson ratio and \(\mu\) the shear modulus of the isotropic elastic medium. \(K(x_ 0,y_ 0;x,y)\) is a bounded regular kernel which represents free boundary effects. If the crack lies in an infinite elastic medium, \(K\) vanishes.
The numerical solution of the problems of the above type were obtained by finite element method and boundary element method. A. C. Kaya [“Applications of integral equations with strong singularities in fracture mechanics”, Ph. D. Thesis, Lehigh Univ. (1984)] used the expression \(W(x_ 0,y_ 0)=g(x_ 0,y_ 0)\sqrt{Z(x_ 0,y_ 0)}\), where \(Z(x_ 0,y_ 0)\) represents the geometric description of the crack boundary \(\partial\Omega\), the unknown function \(g(x_ 0,y_ 0)\) defines a two-dimensional surface and can be approximated in terms of a finite double power series: \[ g(x_ 0,y_ 0)=\sum_{i_ =0}^{N_ 1} \sum_{j=0}^{N_ 2} a_{ij} x_ 0^ i y_ 0^ j.\tag{2} \] Inserting this in (1) leads to the following linear algebraic equation system \[ \sum_{i=0}^{N_ 1} \sum_{j=0}^{N_ 2} a_{ij} [EW_{ij}(x_ k,y_ k)+H_{ij} (x_ k,y_ k)]=-4\pi\mu^{-1} (1- \nu)p(x_ k,y_ k), \quad k=1,\dots,M, \tag{3} \] for \((N_ 1+1)(N_ 2+1)<M\) unknown coefficients \(a_{ij}\) by selecting appropriate collocation points \((x_ k,y_ k)\in\Omega\). Here \(H_{ij}(x,y)\) denotes a regular integral and \(EW_{ij}(x,y)\) the following finite-part integral: \[ EW_{ij}(x,y)=\rlap{\;=}\int\kern-.5em\rlap{\;=}\int_ \Omega r^{-3} x_ 0^ iy_ 0^ j \sqrt{Z(x_ 0,y_ 0)} dx_ 0 dy_ 0. \tag{4} \] The selection of \(N_ 1\) and \(N_ 2\) in (2) should be sufficiently high for a proper approximation of \(W(x_ 0(,y_ 0)\) and requires the solution of (4) for all combinations of \(i\) and \(j\) which appear in (3). Therefore for enhanced accuracy, it is desirable to have closed form solution of the singular integral in (4).
With this objective, the authors study analytically and extend Kaya’s concept to an inclined elliptical domain \(\Omega\). By using a suitable transformation, the initial two-dimensional singular integral in (4) is reduced to a one-dimensional finite-part integral, and the closed form solution is obtained. The exact expressions derived for the integrals with arbitrary integers \(i\) and \(j\) increase the accuracy of the numerical results and simultaneously lead to quicker numerical results. The finite-part integral is expressed in closed form as function of complete elliptical integrals or Gauss hypergeometric functions, respectively. The correctness of the analytical solutions obtained is checked numerically by using expansion and quadrature methods. Formulae for special cases and some values \(i\), \(j\) and their numerical verifications are given in the appendices.

MSC:

74R99 Fracture and damage
74S30 Other numerical methods in solid mechanics (MSC2010)
33C75 Elliptic integrals as hypergeometric functions
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