PSI - Issue 41

Us

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

661

6

0 0.2 0.4 0.6 0.8 1 Normalised needle radius , R / a [-] 0 0.5 1 1.5 2 2.5 3 3.5 4 Norm. strain energy, (d U s / d D ) /  a 2 [-] Theory (uniform pressure) Theory (Hertzian pressure) FEM

1

0.8

0.6

Fig. 3. Theoretical and FE distribution of normalised strain energy versus relative radius of the needle.

4. Results in terms of penetration force

0.4

As it is shown in Figure 3, the contact pressure is well described by the Hertzian distribution. Hence, by inserting Eq. 5 in Eq. 8, we get (the shear modulus µ is equal to E / 3 and to E ∗ / 4 for a linear elastic incompressible material)

0.2

dU s dD =

0

πµ a 2 1

(9)

0

Considering that dU G / dD = 2 aG c , at the onset of deep penetration (the frictional contribution is not considered), the penetration force F p is equal to

1 2 3 4 5 6 7 strain energy, (d U s / d D ) /  a 2 [-] a c R

3 6 9 12 15 18 contact pressure, ( P/D ) /  a [-] (10) (11)

a)

F p = πµ a 2

1 + 2 aG c

Ogden (  = 9) Neo-Hookean (  = 2) Linear elastic

This expression can be conveniently written in a dimensionless form as (Figure 4)

a 1 a c

= π

c R

G c µ R

2 a

F p µ R 2

2

+ 2