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Or
r2 1 1
R,-RZ 2
z2 -z,
= 2
R2
R,
(5)
2R,R2 r
If the displacements of the points
m
and
n
at the t-
axis direction are, respectively, wl and w2, then the
shorten quantity between any two points at the z-axis,
4, can be written as following,
8=z2-z,+w1 +w2
Substituting
Eq.(5) into (6), we have,
w, +w2 =,5 -,8 r2
R, - R2
Q-
(6)
(7)
where 2R,R2 . This is the displacement
equation or geometric relationship of the ball-socket
elastic contact modelling at the hip joint.
2.2 Physical
Relationships and Equilibrium
Equations on the Modelling
It is assume that the displacement at any point M
on the contact area is w due to the contact pressures. If
the element area is taken as
sdrpds ,
here the s is the
distance from the point M to the element area, while
the
cp
is the angle between the secant line through the
point M and the oy axis (see Figure 2 ). Then, the
contact pressure acting on the element area can be
written as following,
gsdrpds
235
Fig 2
.
The coordinate system for calculating
pressures on the contact area.
where the A is the elastic contact area.
Substituting Eq.(9) into (7), we have,
(k,+k2)Jjgdsdcp=(5 -fire
(10)
A
z z
where
k,
_
= 1^ E' , kz 2
= 1)T E
2
It is
because of the fact that the elastic contact area
between the hip head and acetabulum is the circular
one, therefore, at the boundary of the contact area, the
following relationship must be held,
w, + w2 =0 (11)
Thus, from Eq.(7) or (10), the radii of the elastic
contact area can be determined as following,
rc = S/,fl
(12)
The integral
equation
(
10) can be
solved by the
inverse
method. The results
are given as following:
where the q is the contact pressures on the contact
area.
rc
[3)rP(k, +k2)R,RZ
3
Thus, according to the Boussinesq Problem's
4(R, - R2 )
solutions [30], the displacement at any point M due to
3P
the pressures
gsdcpds
can be written as following,
qo =
27rr
2
c
(13)
2
1-
'
=
dr
ds (8)
2
1
2
W
j
jg
p
3P r
2
z 2
7r
E A
q(,;, 77) =
2
z (1
_
^ r - ^ +q
If the two elastic bodies contact each other, then
we have,
r.
27rr
37rP(k, + k2 )
4rc
W, - 1- u' ffq drp ds
7zE A
Where the r,
is radii of the contact area
;
the
q0
is
(9)
the maximum contact stress; the q is the stress
distributions of the contact area
; the P is
the resultant
W2-1-uz
JJgdrpds
7r E2 A
force acting on the hip joint.