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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For contacts of the DMT type, adhesion can be introduced in the Persson’s theory by computing the adhesive force according to eq. (14) (see Persson & Scaraggi (2014)). In this work, about the gap probability distribution P ( u ), we have used a more accurate expression, as given in A ff errante et al.(2018), with respect to the formulation proposed by Almqvist et al. (2011). Therefore, denoting with ζ the magnification, P ( u ) writes as

1 A 0

( dA ( ζ ) / d ζ )

d ζ −

P ( u ) ≈

1 / 2 ×

(2 π h 2

rms e ff ( ζ ))

exp −

rms e ff ( ζ )

+ exp −

rms e ff ( ζ ) rms ( ζ ) = q ,

( u − u 1 ( ζ )) 2 2 h 2

( u + u 1 ( ζ )) 2 2 h 2

(16)

( ζ ) = h − 2

1 ( ζ )

− 1 / 2

where h rms e ff 1 ( ζ ) is the average separation in the surface area that moves out of contact when the magnification is increased of an infinitesimal quantity d ζ , and can be calculated as (see Yang & Persson (2008)) u 1 ( ζ ) = ¯ u ( ζ ) + ( d ¯ u ( ζ ) / d ζ ) A ( ζ ) / ( dA ( ζ ) / d ζ ) (17) where ¯ u ( ζ ) is the mean interfacial separation, which is given by (see Yang & Persson (2008)) rms ( ζ ) + u − 2 , being h 2 > q 0 ζ d 2 q C ( q ). Moreover u

γ + 3 (1 − γ ) erf 2

E ∗

σ

1 2 √ π D ( ζ )

d 2 q qC ( q ) w ( q )

∞

E ∗

e −

w ( q ) σ

d σ

2

w ( q ) σ

¯ u ( ζ ) =

(18)

σ 0

where D ( ζ ) = { q ∈ R 2 | q

L ≤ | q | ≤ ζ q L } , and

w ( q ) =   1

2 D q

− 1 / 2

d 2 q q 2 C q  

(19)

2 | q

being D q = { q ∈ R

L ≤ | q | ≤ q } .

3.4. Discussion

Calculations with the ICHA model and Persson’s theory are performed on self-a ffi ne fractal surfaces with PSD described by a power law

if q L ≤ | q | ≤ q 0 if q 0 ≤ | q | ≤ q 1

C 0 C 0 q −

C ( q ) =

,

(20)

2( H + 1)

where H is the Hurst exponent and q is the wave vector, being q = | q | . The quantities q L and q 1 are the lower and upper cut-o ff wave vectors, while q 0 is the crossover wave vector from the power law C 0 q − 2( H + 1) for q > q 0 to the constant C 0 for q < q 0 (also known as roll-o ff wave vector). Specifically, the fractal surface are numerically generated by exploiting the spectral method developed in Putignano et al. (2012) with a fixed root mean square (rms) roughness amplitude h rms = 0 . 52 nm. Moreover, the short cut-o ff spatial frequency q L is fixed to 2 . 5 × 10 − 5 m − 1 , and q 0 = 4 q L .