Page 74 - 《摩擦学学报》2020年第5期

第 5 期 孙献光, 等: 考虑摩擦系数和微凸体相互作用的粗糙表面接触热导分形模型 629

the asperity is expressed as where H=λσ denotes the hardness of softer material and
y
(15) λ is a constant value.
δ c2 = 110δ c1
Substitution of Eq. (15) into Eq. (14) to obtain the 2.3 Total real contact area and contact load
′
critical full plastic deformation area a c2 of the asperity, According to the study of Yan and Komvopou-
[23] ′
given by los , the area distribution function of the asperities n(a )
′ can be written as
a c1
′
a c2 = (16)
110 1/(D−2) D−1
′
a
n(a ) = a ′(D−1)/2 ′−(D+1)/2 (28)
L
When δ ≤δ<δ c2 or a c2 < a ⩽ a c1, the asperity is 2
′
′
′
c1
′
in the state of elastic-plastic deformation. The real cont- where a L is the largest nominal contact area of the
act area a , contact load ∆F , contact pressure ∆P of asperities.
ep
ep
ep
the asperity can be expressed as According to the above analysis, the total elastic
′
a (1+ f(δ)) contact area A , the total elastic-plastic contact area A ,
ep
e
a ep = (17)
2 and the total fully plastic contact area A can be
p
 ( ) 1/2 
′ δ expressed as
a (1+ f(δ))    
∆F ep =  1.1k µ σ y (1− f(δ))+ H f(δ)  

2 δ c1 ∫ a L ′ D−1   ( ′ ) (3−D)/2  
′
′

′

(18) A e = a e n(a )da = a L × 1− a c1  (29)




′
a ′ c1 6−2D a L
( ) 1/2
δ ∫ ′
∆P ep = 1.1k µ σ y (1− f(δ))+ H f(δ) (19) a c1 D−1
′
A ep = a ep n(a )da = a L ×
′
′
δ c1
6−2D
a ′ c2
where f(δ) is sample function [29] and it can be expressed
′
 ∫ a c1 
  ′(1−D)/2 ′ ′  
as  ( ) (3−D)/2 ( ) (3−D)/2 a f(a )da  
  ′ ′ 3− D  
 a c1 a c2 a ′ 
 c2 
( ) 2    − +   
2(δ−δ c1 ) δ−δ c1  a L a L 2 a ′ (3−D)/2  
′
′

f(δ)= − (20)    L   
110δ c1 −δ c1 110δ c1 −δ c1
(30)
According to Eq. (14), Eq. (20) can be rewritten as
∫ ′ ( ) (3−D)/2 ( ) (3−D)/2 
a c2 ′ ′
[ ′ )(D−2) ] [ ′ )(D−2) ] 2 D−1  a c2 a s  

′
′
′
2 ( a c1 1 ( a c1 A p = a p n(a )da= a L ×   −   
′
′
f(a )= −1 − −1 (21) a ′ 3− D a L a L
′
109 a ′ 109 2 a ′ s (31)
Then, Eqs. (17~19) can be adapted to where a s is the smallest nominal contact area of the
′
a (1+ f(a )) asperities, a s = π(L s /2) .
2
′
′
′
a ep = (22)
2
Through the above analysis, the total real contact
a (1+ f(a ))
′
′
∆F ep (a ) = area A of the whole contact surfaces can be expressed as
′
r
2
(23)  ′ ′
[ ] A e + A ep + A p a c1 ⩽ a L
( ′ )(D−2)/2 

a c1 
(1− f(a ))+ H f(a ) 

′
′
1.1k µ σ y ′ ′ ′ (32)
a ′ A r =  A ep +A p a c2 ⩽ a L < a c1




′ ′
A p 0 < a L < a c2
( ′ )(D−2)/2
a c1
′ ′ ′
∆P ep (a ) = 1.1k µ σ y (1− f(a ))+ H f(a ) (24) The total elastic contact load F , the total elastic-
a ′ e
plastic contact load F , and the total full plastic contact
When δ ≤δ or a ⩽ a c2 , the asperity is in the state ep
′
′
c2
load F can be expressed as
of full plastic deformation. The real contact area a , p
p
contact load ∆F , contact pressure ∆P of the asperity When D≠2.5,
p
p
∫ a L
′
can be expressed as D−1 ′(4−D)/2
′
′
′
F e = ∆F e (a )n(a )da = a L ×
a p = a ′ (25) a ′ c1 5−2D
[ (7−2D)/2 (D−2) 1/2 ( )]  ( ′ ) (5−2D)/2 
′
∆F p (a ) = Ha ′ (26) 2 G (lnγ) E 3π   a c1  

1− × 1−   

π (3−D)/2 16 a L
′
∆P p = H (27) (33)