PSI - Issue 13

Jan Klusák et al. / Procedia Structural Integrity 13 (2018) 1261–1266 Jan Klusák & Ond ř ej Krepl/ Structural Integrity Procedia 00 (2018) 000–000

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2

and bi-material notches are singular stress concentrators (theoretical stress goes to infinity at for the tip of the notches). They are defined by their geometry and material properties of the two parts; see Fig. 1. Note that for the same material characteristics of both parts, bi-material notch becomes to be sharp notch in homogeneous material. The material properties are given by the elastic constants of Young’s moduli and Poisson’s ratios. The solution mostly presupposes the approximation of plane strain or plane stress. For the case of a bi-material notch a perfect bonding (displacement and traction continuity) is assumed at the interface. The material characteristics, therefore, change by step at the interface. Furthermore, the notch surfaces are traction-free.

Fig. 1. The sharp bi-material notch with a Cartesian coordinate system at its tip

The stress distribution (Klusák et al. (2010), Ayatollahi and Dehghany (2010)) is described by the relation:

n

1 k   



p k

  

(1)



1 H r

f

ij

ijk

where the indices ( i , j )  ( r ,  ) are polar coordinates. The symbol H k stands for the Generalized Stress Intensity Factor (GSIF) with the unit of [ H k ] = MPa  m p k . The f ijk (  ) is the angular eigenfunction, which is dimensionless. The stress singularity exponents are given by p k = 1- λ k , where λ k is the k -th eigenvalue of the problem, which is real or complex number. In most of the geometrical and material configurations of V-notches and bi-material notches there are two real exponents p 1 and p 2 in the interval (0 , 1) corresponding to the singular terms of the series. Higher order eigenvalues can be either real or complex numbers. Since their real parts are greater than one, these eigenvalues correspond to the non-singular higher order terms. Note that the series above (1) is for simplicity written for real eigenvalues and real GSIFs. Such form of the series provides satisfactory description for most of the cases. In majority of the fracture mechanics analyses of sharp and bi-material notches only the singular terms are used for description of stress field (Williams (1952), Seweryn et al. (1996)) and following determination of crack onset conditions. In Kim et al. (2009) and Ayatollahi et al. (2011) the effect of first non-singular stress term on stress distribution in case of sharp V-notch is studied. The effect of the first non-singular term in case of bi-material notches is studied in Ayatollahi et al. (2010) and Mirsayar et al. (2014). In Klusák et al. (2014) they have shown the significance of consideration of the first non-singular term in the case of sharp bi-material orthotropic plate. A study which has shown the effect of non-singular terms in the cases of sharp material inclusion has been conducted in Krepl and Klusák (2017). 2. Stability criteria of sharp and bi-material notches A sharp and bi-material notches as well as a crack in homogeneous material are all the singular stress concentrators. Thus we suppose that the mechanism of crack initiation in the notches is the same as the mechanism of crack propagation in single material. The criteria of the direction of crack initiation at a sharp notch or bi-material notch tip and the criteria of the stability of the notches are derived in analogy to the approaches of a crack in homogeneous