PSI - Issue 13

M. Hrstka et al. / Procedia Structural Integrity 13 (2018) 1123–1128 Hrstka, M., Zˇ a´k, S., Vojtek, T. / Structural Integrity Procedia 00 (2018) 000–000

1124

2

tension bending torsion

1.0

0.8

pure shear, mode III position pure shear, mode II position

σ/σ max and τ/τ max

0.6

0.4

0.2

0.0

0

1

2

3

4

5

6

r [mm]

Fig. 1: Comparison of normalized stress distribution along the radial direction for the specimen in Fig. 2. The position r = 0 mm corresponds to the centre of the specimen and r = 6 mm is the notch tip surface.

explained as follows. The stress concentration factor (SCF) of notches under shear-mode loading is smaller than under tensile loading due to di ff erent stress distribution. A lower SCF is always associated with a low stress gradient in radial direction so that the static equilibrium is kept. Under mode III loading, the SCF is smaller in torsion than in tension and bending. Neubers simple solutions of SCFs (denoted k N , Neuber (1958)) for shallow elliptical notches under tension (equal to the solution of Inglis (1913) for elliptical holes loaded in tension), bending and torsion are: k N , tension = k N , bending = 1 + 2 a ρ , and k N , torsion = 1 + a ρ (1) where a is the elliptical notch depth and ρ is the notch root radius Therefore, there is a principal reason for smaller SCFs under torsion loading. For the real notch of the investigated cylindrical specimen in this work (see Fig. 2), the SCFs were determined by FEM under tension, bending and torsion as follows: k tension = 6 . 56, k bending = 5 . 05, k torsion = 3 . 09. Under mode II loading, the stress distribution is completely di ff erent. In case of a cylindrical bar or a symmetrical double-edge notch plate loaded by simple shear, the original stress field without the notch has the maximum value in the middle of the cross section and the shear stress is zero at the free surface. In presence of a notch, the shear stress quickly increases to the maximum value closely below the surface and then decreases with a very low stress gradient. Therefore, under mode II loading the influence of the notch is much deeper than in mode III. The shear stress distribution along the radial direction normalised to the maximum value for various types of loading are presented in Fig. 1. Such distribution has two important consequences: • Under shear-mode loading, the solution of stress intensity factor (SIF) for cracks emanating from a notch is possible to replace by the solution for cracks in a smooth bar after the crack is much longer than under tensile loading. There is a certain critical crack length, for which the crack gets out of the influence of the notch and these two solutions merge together. Lefort (1978) published a simple formula for estimation of this critical length a 0 for mode I loading in dependence on the notch root depth d and radius ρ : a 0 ρ = 0 . 5 d ρ 1 / 3 (2) Critical lengths for mode II and mode III loading are subject of current investigation. Preliminary results re ported in Horn´ıkova´ et al. (2018) showed that the critical crack length is much larger under mode II loading than under mode I and mode III loading. • Plastic zones are larger under shear-mode loading and the SSY condition is harder to fulfil even for relatively low loading levels.