PSI - Issue 26

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347 M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

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zone, where the conditions of plane stress are completely recovered at su ffi cient distance ( ∼ 2 h ); this hypothesis can be later verified by analyzing the results.

Fig. 3: (a) Representation of the various regions with the correspondent stress state near a crack / notch tip. (b) 3D finite element model of a sharp V-notch surrounded by a cylindrical region. The local Cartesian coordinate system ( x , y , z ) is centered at the notch tip on the mid-plane, as reported specifically in Fig. 2a.

As shown in Fig. 3b, a 3D FE model of a cylindrical portion of a sharply-V-notched plate of thickness (2 h ) 20 mm and radius R = h was built by using the commercial FE software Abaqus / Standard, through Python language. The model consisted of 15-node wedge elements (C3D15 in Abaqus) around the V-notch tip, surrounded by 20-node hexahedral (brick) elements (C3D20 in Abaqus) along the radial direction, whose biggest element size was 2.1 mm on the outer boundary of the cylinder, progressively reduced towards the notch tip down to 0.0001 mm (the nearest node at 0.00005 mm from the notch tip). The supplementary V-notch angle γ (Fig. 2b) was divided in 16 parts, whereas the plate thickness was meshed with weighted divisions from the mid-plane, whose first element size was h / 10, i.e. 1 mm, toward the external surface, with the smallest element size being 0.01 mm. This mesh resulted in ∼ 120000 elements and ∼ 500000 nodes, slightly varying with the angle 2 α (Fig. 2b). Due to the symmetry of the problem with respect to the mid-plane under Mode 1, only one half of the cylinder was modeled, as evident in Fig. 3b. Consequently, symmetric displacement boundary conditions were applied on the nodes of the mid-plane ( z = 0), constraining the displacements along z (coordinate system in Fig. 2a). Moreover, assuming that the conditions of plane stress are attained at distance equal to half of the thickness from the notch tip, a plane stress far-field displacement (Eq. (1a)-(1b)) under mode 1 was imposed to the outer periphery of the cylinder. This displacement field, i.e. the components u r and u θ (coordinate system Fig. 2a), can be obtained as a function of the asymptotic plane stress NSIF ( K ∞ 1 ) under mode 1 by simply integrating for separation of variables the constitutive equation (Lamè Equation) in terms of displacements, and knowing the expression of the first singular term of the asymptotic expansion of the stress field around a V-notch (see Lazzarin and Tovo (1998)), as follows:

f rr ( θ, λ 1 , χ 1 ) − ν f θθ ( θ, λ 1 , χ 1 ) λ 1

r λ 1 K ∞ 1

1 E √ 2 π 1 E √ 2 π

u r ( r , θ ) =

(1a)

(1 + λ 1 ) + χ 1 (1 − λ 1 ) r λ 1 K ∞ 1 (1 + λ 1 ) + χ 1 (1 − λ 1 ) 1 + ν

f θθ ( θ, λ 1 , χ 1 ) d θ − ν +

f rr ( θ, λ 1 , χ 1 ) d θ , (1b)

λ 1

1 λ 1

θ

θ

u θ ( r , θ ) =

0

0