PSI - Issue 41

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Andrea Spagnoli et al. / Procedia Structural Integrity 41 (2022) 656–663 / Structural Integrity Procedia 00 (2022) 000–000 Spagnoli et al.

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1

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a 1 / a (theory) [-] a 1 / a (FEM) [-] K / K R (theory) [-] K / K R (FEM) [-]

.8 1 / a [-]

0.8

0.0 1

0.6

Norm. penetration force,

K / K R , a 1 / a

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0.2

0 0.2 0.4 0.6 0.8 1 Normalised needle radius , R / a [-] 0

Fig. 2. Theoretical and FE distribution of relative SIF and contact length versus relative radius of the needle.

0 0.2 1 Normalised contact area, a 1 / a [-] (3) 0.2 0.4 0.6

0 3 6 9 12 15 18 Norm. contact pressure, ( P/D ) /  a [-]

a)

b)

0.8 It has been shown in (Ståhle et al., 2017) that the crack surfaces detach from the needle profile at a coordinate x = a 1 , so that the cut is traction free for a 1 ≤ x < a (see Fig. 1). From the solution of a boundary value problem (Terzano et al., 2018), the position of the point of separation a 1 and the stress intensity factor at the crack tip K , are obtained. The results are plotted in Figure 2, against the normalised needle radius R / a . The SIF K is normalised with respect to K R , namely the stress intensity factor related to the needle filling the entire crack ( R / a = 1). This is defined as

)

√ π R

E ∗ 2

K R =

0 For a given radius of the needle, depending on the toughness of the material, a planar crack of semi-length a forms so that K = K c at the crack tips. In other words, the critical value of the relative radius, corresponding to the ratio between the radius and the critical crack semi-length, R / a c , is determined (along with the critical value of the contact relative semi-length a 1 / a c ) from the curve of R / a − K / K R in Figure 2 by posing K = K c . Note that, according to the present model, the needle penetration produces th crack development provided that R ≥ R c , where R c is a material-based length parameter, defined as 0 0.2 0.4 0.6 0.8 1 Normalised needle radius , R / a [-] The SIF K R turns out to be proportional to the square root of the radius of curvature at the crack tip (when R / a = 1, no detachment length a − a 1 develops).

.6 0.8 1

4 π

K c E ∗

2

R c =

(4)