PSI - Issue 13

Yaroslav Dubyk et al. / Procedia Structural Integrity 13 (2018) 1502–1507 Dubyk at al./ Structural Integrity Procedia 00 (2018) 000–000

1503

2

Nomenclature r  ,   , r   , r u , u 

Stresses and displacements in a polar coordinates

unknown real coefficients

1 n  , 2 n  , 1 n  , 2 n  , n  , n 

for plane stress Poisson ratio

3 1      



Williams' functions in a polar coordinates Williams' functions in a Cartesian coordinates

 ,  ,

  ,

 

n r W n x W

n W n W

n r W

y  ,



xy

n W

1. Describe asymptotic behavior and distribution of stresses and displacements in the vicinity of the crack tip. In such a way, a clear understanding of the idea of the coefficient of stress intensity factor (SIF) can be demonstrated. Furthermore, other authors employed Williams functions to study cracks located in the boundary between two media (Malíková, Seitl, 2017). The possibility of two media being joint at a certain angle has been taken into account as well (Dolgov, Soroka, 2004). 2. These functions can also be used to determine the area of the body around crack where SIF based stress dominates. Furthermore, employing these functions one is able to demonstrate that second largest stress by magnitude is uniform, acting parallel to crack and does not depend on r . This stress is called T-stress is extensively used to modify fracture criteria (Bouledroua, Meliani, Pluvinage, 2016) nowadays. 3. These functions represent a simple enough mathematical tool used to derive solutions of boundary value problems with cracks. Initially, the Williams-functions-based method was called “Boundary Collocation Method” and was quite a popular in the beginning of practical fracture mechanics in the 60s of last century The clear advantage of this method was that Williams functions are the exact analytical solution inside the body and that the deviation of the calculated parameters (stresses or displacements) on the boundary from the predefined ones can be clear measure of the solution accuracy. In the well-known book on fracture mechanics by Savruk (1988) a considerable number of solution derived using this method is presented. Further modification of the method lies in the approximate satisfaction of the boundary conditions on the whole boundary of the body. This was achieved by introduction of the integral quadratic error and minimization of this error by means of the method of least squares (Pei-Qing, 1985; Fett 1997). Furthermore, in Pei-Qing (1985) a rather pessimistic opinion was expressed regarding the practical application of the method because of the convergence issues. It was pointed out that in some unpredictable cases SIF values were diverging, meaning that while increasing the number of collocation points the results do not converge to a specific value. Even though, sometimes results do converge to a specific value, this value turns out to be wrong. If a large number of Williams functions are selected, then the calculated stresses on the boundary turn out to consist of highly fluctuated functions which cannot be reduced by error minimization. Neither minimization in the given points (interpolation approximation) nor minimization at all points of the boundary performed by means of integration on elementary regions of the boundary (averaged interpolation) cannot be used in this case. In the case of an interpolation method, alteration by the number of points on the boundary and by the number of Williams series terms may lead to a considerable divergence of the results. In case of analytical integration, the increase in elementary regions number boosted convergence while the increase in the number of Williams series terms increased divergence. Such an unpredictable behavior of the solutions brought the Pei-Qing (1985) to a paradoxical conclusion that the Williams functions do not form a complete set of functions, so such functions cannot guarantee the correct solution of the plane body with arbitrary geometry and lead to limitations in the method of collocations. The actual assessment of capabilities of the Williams functions and collocation method are of the major concern for the current study. We aim to identify the actual advantages and disadvantages as well as the limits of employability of the Williams functions. Two preliminary considerations give insight on the main idea of work: 1. There are two types of Eri functions in polar coordinates. The first type are functions having a physical meaning in the vicinity of the center of coordinates 0 r  and contain polar radius in positive power n r , 0 n  . The second type comprises functions converging at infinity and containing terms n r  . The first type is applied to study continuous restricted circles while the second type is used to study infinite bodies with the circular hole. Applied together, these functions accurately describe the behavior of a ring. Obviously, the effectiveness of these functions