The influence of higher order terms of Williams series on a more accurate description of stress fields around the crack tip

Most of fracture analyses often require an accurate knowledge of the stress/displacement field over the investigated body. However, this can be sometimes problematic when only one (singular) term of the Williams expansion is considered. Therefore, also other terms should be taken into account. Such an approach, referred to as multi-parameter fracture mechanics is used and investigated in this paper. Its importance for short/long cracks and the influence of different boundary conditions are studied. It has been found out that higher-order terms of the Williams expansion can contribute to more precise description of the stress distribution near the crack tip especially for long cracks. Unfortunately, the dependences obtained from the analyses presented are not unambiguous and it cannot be strictly derived how many of the higher-order terms are sufficient.

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Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part I

PNRPU Mechanics Bulletin ◽

10.15593/perm.mech/2020.4.20 ◽

2020 ◽

pp. 237-249

Author(s):

L. V Stepanova

Keyword(s):

Series Expansion ◽

Digital Image ◽

Power Series Expansion ◽

Crack Tip ◽

Higher Order ◽

Accurate Estimation ◽

Principal Stresses ◽

Isochromatic Fringe ◽

Higher Order Terms ◽

Fringe Patterns

This study aims at obtaining coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending by the use of the digital photoelasticity method. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give a more accurate estimation of the near-crack-tip stress, strain and displacement fields and extend the domain of validity for the Williams power series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. By means of the developed tool, the fringe patterns that contain the whole field stress information in terms of the difference in principal stresses (isochromatics) are captured as a digital image, which is processed for quantitative evaluations. The developed tool allows us to find points that belong to isochromatic fringes with the minimal light intensity. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of the stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion. The iterative procedure of the over-deterministic method is utilized to find the higher order terms of the Williams series expansion. The procedure is based on the consistent correction of the coefficients of the Williams series expansion. The first fifteen coefficients are obtained. The experimentally obtained coefficients are used for the reconstruction of the isochromatic fringe pattern in the vicinity of the crack tip. The comparison of the theoretically reconstructed and experimental isochromatic fringe patterns shows that the coefficients of the Williams series expansion have a good match.

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Application of the Williams Expansion near a Bi-Material Interface

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.754.206 ◽

2017 ◽

Vol 754 ◽

pp. 206-209 ◽

Cited By ~ 2

Author(s):

Lucie Malíková ◽

Stanislav Seitl

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Crack Tip ◽

Higher Order ◽

Simplified Model ◽

The Finite Element Method ◽

Material Interface ◽

Higher Order Terms ◽

Element Method ◽

Crack Tip Stress

A simplified model of a crack approaching a bi-material interface is modelled by means of the finite element method in order to investigate the significance of the higher-order terms of the Williams expansion for the proper approximation of the opening crack-tip stress near the bi-material interface. The discussion on results is presented and the importance of the higher-order terms proved.

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Higher-order crack-tip creep stress fields in materials under biaxial loading

Strength of Materials ◽

10.1007/s11223-008-9079-y ◽

2008 ◽

Vol 40 (6) ◽

pp. 615-628 ◽

Cited By ~ 2

Author(s):

V. N. Shlyannikov ◽

N. V. Boichenko

Keyword(s):

Biaxial Loading ◽

Crack Tip ◽

Higher Order ◽

Stress Fields ◽

Creep Stress

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Analytical Solutions for Crack-Tip Higher Order Stress Fields in Thermal Barrier Coating

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.47-50.1023 ◽

2008 ◽

Vol 47-50 ◽

pp. 1023-1026

Author(s):

Yao Dai ◽

Chang Qing Sun ◽

Sun Qi ◽

Wei Tan

Keyword(s):

Crack Tip ◽

Higher Order ◽

Functionally Graded ◽

Stress Fields ◽

Thermal Barrier ◽

Barrier Coating ◽

Graded Material ◽

Stress Functions ◽

Order Stress ◽

Analytical Expressions

Analytical expressions for crack-tip higher order stress functions for a plane crack in a special functionally graded material (FGM), which has an variation of elastic modulus in 1 2 power form along the gradient direction, are obtained through an asymptotic analysis. The Poisson’s ratio of the FGM is assumed to be constant in the analysis. The higher order fields in the asymptotic expansion display the influence of non-homogeneity on the structure of crack-tip fields obviously. Furthermore, it can be seen from expressions of higher order stress fields that at least three terms must be considered in the case of FGMs in order to explicitly account for non-homogeneity effects on the crack- tip stress fields. These results provide the basis for fracture analysis and engineering applications of this FGM.

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