PSI - Issue 12

Guido Violano et al. / Procedia Structural Integrity 12 (2018) 58–70 G. Violano et al. / Structural Integrity Procedia 00 (2018) 000–000

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Fig. 2. The normalized applied force F / (2 π ∆ γ R ) as a function of the dimensionless penetration δ/ε . Results are referred to: i) Bradley model (black); ii) Derjaguin model (cyan); iii) DMT thermodynamic approach (magenta); iv) improved DMT thermodynamic approach (blue); v) DMT Maugis model (green); vi) DMT force approach (red); vii) JKR theory (green dashed line); viii) numerical solution (orange line with circular markers). Calculations are performed at µ = 0 . 2.

thus neglecting the elastic coupling of the contact regions. In particular, denoting with ¯ d the distance between the plane of summits and the smooth plane and with ϕ ( z ) the heights distribution of asperities, the number of asperities in contact n c can be written as n c = n tot ∞ ¯ d ϕ ( z ) dz , (9) where n tot is the total number of the summits. The total real area and total load are obtained by summing the Hertzian relations A i = π R z i − ¯ d and F i = 4 E ∗ R 1 / 2 z i − ¯ d 3 / 2 / 3 obtained on the i th sphere, for all the asperities in contact A = n tot ∞ ¯ d A i ϕ ( z ) dz (10) F = n tot ∞ ¯ d F i ϕ ( z ) dz (11) In presence of adhesion and according to the Maugis’ approximation for contacts of the DMT type, the calculations of n c and A are not modified, but for F we have F = n tot ∞ ¯ d 4 E ∗ R 1 / 2 z i − ¯ d 3 / 2 / 3 − 2 π ∆ γ R ϕ ( z ) dz = 3 / 2

1 / 2 ∞

¯ d

z i − ¯ d

4 3

n tot E ∗ R

ϕ ( z ) dz − 2 π ∆ γ Rn c

(12)

Therefore, the total adhesive load F ad = 2 π ∆ γ Rn c is simply the sum of the constant adhesive contribution calculated for each asperity in contact.