scholarly journals COMPLETE ASYMPTOTIC EXPANSION M. WILLIAMS NEAR THE CRACK TIPS OF COLLINEAR CRACKS OF EQUAL LENGTHS IN AN INFINITE PLANE MEDIUM

2015 ◽  
pp. 188-225 ◽  
Author(s):  
Степанова Лариса ◽  
Росляков Павел
Keyword(s):  
Asymptotic Expansion ◽  
Infinite Plane ◽  
Collinear Cracks ◽  
Complete Asymptotic Expansion ◽  
Plane Medium ◽  
Crack Tips

Complete Williams Asymptotic Expansion near the Crack Tips of Collinear Cracks of Equal Lengths in an Infinite Plane

2016 ◽  
Vol 258 ◽  
pp. 209-212 ◽  
Author(s):  
Larisa Stepanova ◽  
Pavel Roslyakov ◽  
Tatjana Gerasimova
Keyword(s):  
Plane Elasticity ◽  
Analytical Determination ◽  
Geometrical Parameters ◽  
Infinite Plane ◽  
Variable Theory ◽  
Complex Variable Theory ◽  
Plane Medium ◽  
Crack Tips ◽  
Plane Elasticity Theory

The present study is aimed at analytical determination of coefficients in crack tip expansion for two collinear finite cracks of equal lengths in an infinite plane medium. The study is based on the solutions of the complex variable theory in plane elasticity theory. The analytical dependence of the coefficients on the geometrical parameters and the applied loads for two finite cracks in an infinite plane medium is given. It is shown that the effect of the higher order terms of the Williams series expansion becomes more considerable at large distances from the crack tips. The knowledge of more terms of the stress asymptotic expansions allows us to approximate the stress field near the crack tips with high accuracy.

scholarly journals MULTIPARAMETRIC ANALYSIS OF THE STRESS FIELD NEAR THE CRACK TIP

2017 ◽  
Vol 21 (10) ◽  
pp. 52-76
Author(s):  
L.V. Stepanova ◽  
P.S. Roslyakov
Keyword(s):  
Asymptotic Expansion ◽  
Stress Field ◽  
Mixed Mode ◽  
Infinite Plate ◽  
Higher Order ◽  
Analytical Determination ◽  
Mixed Mode Loading ◽  
Complete Asymptotic Expansion ◽  
Higher Order Terms ◽  
Crack Tips

The paper is devoted to analytical determination of coefficients of the Williams asymptotic expansion of the stress field in the neighborhood of two collinear crack tips in an infinite plate under mixed mode loading. On the basis of the Kolosof-Muskhelishvili approach the complete asymptotic expansion of the stress field in the vicinity of the crack tips of two collinear cracks of equal lengths under mixed mode loading is derived. The analysis of the higher order terms in the asymptotic expansion series is performed. It is clear that it is necessary to take into account the higher order terms.

The mutual effects between two unequal collinear cracks under compression

2016 ◽  
Vol 22 (5) ◽  
pp. 1205-1218 ◽  
Author(s):  
Yong Fan ◽  
Zheming Zhu ◽  
Jiming Kang ◽  
Yangcheng Fu
Keyword(s):  
Crack Surface ◽  
Analytical Formula ◽  
Surface Friction ◽  
Complex Stress ◽  
Infinite Plane ◽  
Collinear Cracks ◽  
Interval Distance ◽  
Crack Tips ◽  
Analytical Result ◽  
Abaqus Code

For two unequal collinear cracks under compression, which crack would propagate first, the longer one or the shorter one, and how they affected each other was studied. By using complex stress function theory and considering crack surface friction, the analytical formula of stress intensity factors (SIFs) for an infinite plane containing two unequal collinear cracks was obtained and the analytical results have been validated through numerical simulation by employing ABAQUS code, photoelastic experiments, and compressive tests. The results show that the numerical, photoelastic, and compressive test results agree well with the analytical result. Finally, the effects of crack length, crack surface friction, and crack interval distance between two crack tips on SIFs were analyzed, and the results show that the SIF values at the longer crack tips are always higher than those at the shorter crack tips; for each crack, the SIF value at the internal tip is always larger than that at the external tip.

scholarly journals EXPERIMENTAL DETERMINATION OF COEFFICIENTS OF A MULTIPARAMETER DECOMPOSITION OF FIELD OF CRACK TIP STRESSES: PHOTOELASTICITY METHOD

2017 ◽  
Vol 23 (1) ◽  
pp. 59-68 ◽  
Author(s):  
L. V. Stepanova ◽  
V. S. Dolgikh
Keyword(s):  
Asymptotic Expansion ◽  
Analytical Solution ◽  
Stress Field ◽  
Infinite Plate ◽  
Crack Tip ◽  
Digital Processing ◽  
Optical Methods ◽  
Collinear Cracks ◽  
Deformable Solid ◽  
Complete Asymptotic Expansion

The purpose of this study is multiparameter asymptotic analysis of the stress field in the immediate vicinity of the crack tip in a linearly elastic material and construction of complete asymptotic expansion of M. Williams stress field in the vicinity of the crack tip. Multiparametric analysis of the stress field is based on the polarization-optical methods of mechanics of a deformable solid (the method of photoelasticity). Digital processing of the results of optoelectronic measurements performed on a series of samples with cracks and notches is carried out. Different classes of samples from optically sensitive materials, in particular a sample with two collinear cracks under conditions of normal detachment, were considered. A set of programs has been prepared that makes it possible to determine the scale (amplitude) multipliers of complete asymptotic expansion of M.Villiams for the stress field at the crack tip. Using the basic law of photoelasticity, first five coefficients of complete asymptotic expansion of M. Williams are calculated. The results of the experiments are compared with the available analytical solution. It is shown that the results of processing optoelectronic measurements are in good agreement with the analytical solution obtained for an infinite plate with two collinear cracks.

scholarly journals Asymptotic expansions of Lambert series and related q-series

2017 ◽  
Vol 13 (08) ◽  
pp. 2097-2113 ◽  
Author(s):  
Shubho Banerjee ◽  
Blake Wilkerson
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Theta Functions ◽  
Lambert Series ◽  
Pochhammer Symbol ◽  
Complete Asymptotic Expansion ◽  
Polygamma Functions ◽  
Jacobi Theta Functions ◽  
Relative Errors ◽  
Asymptotic Forms

We study the Lambert series [Formula: see text], for all [Formula: see text]. We obtain the complete asymptotic expansion of [Formula: see text] near [Formula: see text]. Our analysis of the Lambert series yields the asymptotic forms for several related [Formula: see text]-series: the [Formula: see text]-gamma and [Formula: see text]-polygamma functions, the [Formula: see text]-Pochhammer symbol and the Jacobi theta functions. Some typical results include [Formula: see text] and [Formula: see text], with relative errors of order [Formula: see text] and [Formula: see text] respectively.

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