Page 73 - 《摩擦学学报》2020年第5期

628 摩 擦 学 学 报 第 40 卷

[26]
mation δ of asperity is given by . asperity in the state of elastic deformation can be given
2 (4−D) G (D−2) (lnγ) 1/2 ′(3−D)/2 as [23, 25]
(D−2)
1/2
′ (3−D)
δ = 2G (lnγ) (2r ) = a
π (3−D)/2 a ′
(2) a e = (6)
2
a a = πr ) is the nominal contact area of the
where ( ′ ′2 Substitution of Eq. (3) and Eq. (6) into Eq. (4) can
′
asperity, r is the nominal contact radius of the asperity, obtain the contact load ∆F e and contact pressure ∆P e of
′
in Fig. 1. the asperity in the state of elastic deformation, given by
[ (7−2D)/2 (D−2) 1/2 ( )]
R 2 G (lnγ) E 3π ′(4−D)/2
′
∆F e (a ) = 1− a (7)
Deformed π (3−D)/2 16
asperity r'
Rigid plane
r [ (9−2D)/2 (D−2) 1/2 ( )]
2 G (lnγ) E 3π
∆F e ′(2−D)/2
′
Deformation δ ∆P e (a )= = (3−D)/2 1− a
Original a e π 16
asperity (8)
The condition for initial yield of the asperity consi-
[26-27]
dering friction coefficient is given by Li et al
Nominal contact
area α' Real contact P m = 1.1k µ σ y (9)
area α
where P is critical contact pressure of the asperity; σ is
y
m
Fig. 1 The contact of a single asperity with rigid plane the yield strength of softer material; k is a frictional
μ
correction factor, it can be expressed as
The radius of curvature R of the asperity can be {
1−0.228µ 0 ⩽ µ ⩽ 0.3
expressed as k µ = (10)
0.932e −1.58(µ−0.3) 0.3 < µ ⩽ 0.9
a ′ a ′(D−1)/2
R ≈ = (3) where μ is friction coefficient.
2πδ 2 (5−D) (D−1)/2 G (D−2) (lnγ) 1/2
π
According to Eq. (8) and Eq. (9), the critical contact
[24]
Wang et al established a mechanical model consi-
area a c1 and the critical contact deformation δ c1 at the
′
dering asperity interaction based on elastic theory and
initial yield are given by
Hertz contact theory. After considering the interaction

 (9−2D) (2D−4) ( ) 2 1/(D−2)

between the asperities, the contact load ∆F of the a c1 =  2 G lnγ 1− 3π     (11)
′

2
2 2 (3−D)
  1.1 φ k π 16  
asperity in the state of elastic deformation can be exp- µ
ressed as   2
 
 
 
( )   1/2  
Ea 3/2 3π 1.1π k µ φ  


∆F = 1− (4) δ c1 =  ( )  R (12)




πR 16   3π  

 1−  
16
2
where a ( a = πr ) is the real contact area of the asperity,
where φ=σ /E denotes the characteristic parameter of
r is the real contact radius of the asperity; E is the y
material.
equivalent elastic modulus, it can be expressed as
Eq. (2) is divided by Eq. (12) to get the following
1 1−v 2 1 1−v 2 2
= + (5) expression
E E 1 E 2
( ) −2 (9−2D) (2D−4) (D−3) ( ) 2
δ 1.1k µ φ 2 G π lnγ 3π
where E and E are the elastic modulus of two rough = 1− (13)
1
2
δ c1 a ′(D−2) 16
surfaces respectively; v and v represent their Poisson’s
2
1
ratio respectively. Substitution of Eq. (11) into Eq. (13) can obtain
′ )(D−2)
2.2 Contact state analysis of the single asperity δ ( a c1
= (14)
In general, the deformation state of the asperity can δ c1 a ′
[28]
be divided into elastic deformation, elastic-plastic defor- According to the research of Kogut and Etsion ,
mation and full plastic deformation [16, 19-20] . According to the relationship between critical full plastic deformation
Hertz contact theory, the real contact area a of the δ of the asperity and critical elastic deformation δ of
c2
e
c1