
85Fundamentals of Friction
==ε+V
dR
Vi
(4.40)
Using the force components,
=+PF
j
(4.41)
one can write similarly to the derivation of Equations (4.37) and (4.38):
=⋅ =ε +SP
V
FV WV T
/
(4.42)
∫
== ⋅
=
∂
∂
⋅
=ε
∂
∂
+
∂
∂
=
−
XS
P
T
Vdt
x
P
T
dR
S
V
i
S
V
j
P
T
1
(4.43)
And, nally, the phenomenological relationships and Equation (3.28):
=ε +
−
V
LF
1
x
1
11 12
(4.44)
V
LF
1
y 12 22
(4.45)
Further resolving Equations (4.44) and (4.45) for the force components:
=ε −F
TV
L
W
x
12
(4.46)
=−=
−
+ε
−
W
TV
L
FV
TL
V
TL
yy
12
11
2
12
2
(4.47)
The limit of ε → 0 with any nite value of V
x
yields
=−F
L
W
12
(4.48)
=
−
WV
TL
y
11 ...