PSI - Issue 13

Mauro Ricotta et al. / Procedia Structural Integrity 13 (2018) 1560–1565 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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discovered for first time that local stresses can possess an oscillatory nature. Williams analysis was later extended to bending loads by Sih and Rice (1964), and to any opening angle in the presence of in-plane loadings by Bogy (1971), Demsey and Sinclair (1981), amongst the others. In Bogy (1971) and Demsey and Sinclair (1981) it is proved that the near tip stress fields are governed by a singular term r s , where s  depends on the V-notch opening angle and the elastic properties of the dissimilar materials, whereas r is the distance from the point of singularity. Differently, the intensity of the local stress fields can be quantified by a Generalized Stress Intensity Factor, H, thought of as an extension of Gross and Mendelson parameter (Gross and Mandelson 1972) to bi-material problems (Lazzarin et al (2002)). Therefore, it can be adopted as a fracture parameter controlling the local failure in mechanical components made of dissimilar materials (see for example Lazzarin et al (2002) and references reported therein). However, the numerical evaluation of H and of the associate strength of singularity requires very accurate meshes and large computational efforts, thus commonly hampering the adoption of this criterion in the engineering practice. The main aim of the present work is to overcome this limitation by extending to isotropic bi-material corners the Peak Stress Method (PSM), which was proposed by Meneghetti and Lazzarin (2007) to estimate the stress intensity factor at the tip of a geometrical singular point with relatively coarse mesh patterns. 2. Theoretical background Consider a bi-material corner between two elastic materials (Fig. 1a). Generally speaking, the stress distributions in the very close neighborhood of the corner can be given as a one term asymptotic expansion according to Eq. (1) (Lazzarin et al 2002): s ij ij (r, ) H r f ( )      (1) In Eq. (1), ij f ( )  are the stress angular functions, H is the Generalised Stress Intensity Factor associated to the leading order term of the stress distribution and -s (with s<0) is the singularity degree of the stress field, depending on the notch angle, the plane hypothesis used (plane stress or plane strain) and Dundurs’ parameters (Dundurs 1969), defined as (under plane strain):

1 2 G (k 1) (k 1)G G (k 1) (k 1)G       1 1 2 1

1 2 G (k 1) (k 1)G G (k 1) (k 1)G       1 1 2 1

(2)

 

 

2

2

2

2

where subscript i = 1 or 2 specifies the material, G i is the shear elastic modulus and, finally, k i is equal to 3 – 4  i . The numerical evaluation of H and of the associate strength of singularity requires very accurate meshes and large computational efforts, thus commonly hampering the adoption of this criterion in the engineering practice.

Fig. 1. (a) schematic view of the singular zone showing the Cartesian and polar coordinate system and (b) a pointed V-notch. The intensity of mode I asymptotic stress field in the close neighborhood of a pointed V-notch can be quantified by the Notch Stress Intensity Factor (NSIF), defined as (Gross and Mendelson 1972): 1 1 1 0 r 0 K 2 lim ( ) r         (3)