PSI - Issue 28

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R.M. Zhabbarov et al. / Procedia Structural Integrity 28 (2020) 1768–1773 Author name / Structural Integrity Procedia 00 (2019) 000–000

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Malikova and Vesely (2014) employed the multi-parameter description of the crack-tip fields either. They introduced the Williams expansion for approximation of the stress and displacement distribution. The over-deterministic method is used for calculation of the coefficients of the power series terms because it requires only a conventional FE analysis. Particularly, the plastic zone extent in a cracked specimen subjected to I+II mixed mode of loading is investigated using Rankine criterion. Results determined via finite element analysis are compared to those calculated by means of the stress distribution expressed via the Williams expansion under consideration of various numbers of initial terms of the series. The size of the plastic zone is calculated for various values of tensile strength of the material modeled in the numerical simulations. It is shown that the higher-order terms of the Williams expansion can be significant when the extent of the nonlinear zone is large enough in comparison to the typical structural dimensions. The present study is aimed at numerical determination of the higher order coefficients in Williams’ series expansion in the classical specimen for linear fracture mechanics – a plate with central crack using finite element method. This type of experimental specimen allows us to elucidate the influence of the higher order terms of the Williams series expansion on the stress field description. Asymptotic expression for the stress field in a plane medium with a traction free crack submitted to mode I and mode II load have been derived:   2 /2 1 ( ) , 1 ( , ) m k m k k ij k m ij m k r a r f            (1) with index associated to the fracture mode; coefficients related to the geometric configuration, load and mode. Angular functions depending on stress components and mode. Analytical expressions for circumferential eigenfunctions are available (Karihaloo and Xiao (2001), Hello et al. (2012)). The coefficients m k a are the unknown mode I parameters. The SIFs can be computed from the coefficients as 1 1 2 I K a   and 2 1 2 II K a    . 1 2 a is related to T-stress as 1 1 2 4 . o a    It should be noted that the coefficients are found theoretically for some crack geometries. Thus, Hello et al. (2012) obtained the coefficients of the WE for an infinite plate with the central crack:   1 1 3 1/2 2 1/2 1 1 2 1 22 2 22 2 ( 1) (2 )! / 2 ( !) (2 1)a , / 4, 0 n n n n k a n n n a a               (2) for mode II crack problem. 2. Numerical experiments When the Williams series expansion (WE) is used it is necessary to determine the coefficients of the WE terms m k a . They can be obtained experimentally and numerically by the over-deterministic method. As it is pointed out by Malikova and Vesely (2014) this procedure is advantageous because of its low demands on the finite element software. The over-deterministic method needs to know only the stress field (and/or displacement field) around the crack. The plate with the central crack was modeled in SIMULIA Abaqus with the following dimensions: height was equal to 40 cm, width was equal to20 cm, the crack length was equal to1 cm. These dimensions were chosen to simulate an infinite plate with small crack to compare the algorithm developed here with the analytical solution (4), (5). The typical mesh around the crack tips is shown in fig. 1. The results of FEM analysis are shown in figs. 2- 7. Thus, one can compare the theoretical results (2), (3) with the FEM analysis. for mode I crack problem;   2 1 ( 1) (2 )! / 2 ( !) (2 1)a , n n n n n n          3 1/2 2 1/2 2 2 1 n 12 2 0 k a a  (3)