173
PART II: CONSOLIDATING EFFECT
CONTACT PHASE  POROELASTIC
Summary The lubricated clutch engagement behavior of two annular disks, one of which is covered with a layer of porous material, was analyzed at the consolidating contact phase based on the theory of poroelasticity. The porous friction material was treated as a porous elastic sponge saturated with a viscous fluid. Results are presented in terms of the relations of deflection versus time for various geometric and material parameters. Comparison made between analysis and experiment at squeeze film and consolidating contact phases indicates good qualitative agreement. The overall disk engagement behavior of the present model is also discussed.
Nomenclature a b c e xx eYY
ezz
Lf HP k G JO PO Q
r,z R Rf
t
u,v,w
u0
W X,YJ
yo
oli
(3
outside radius of annular disk inside radius of annular disk consolidation constant (h2p/k + l/kQ)’ lineal components of strain dilatation (= eXX + eyy + err) poroelastic constant porous facing thickness (= Q//J) shear modulus Bessel’s function of first kind, zeroth order load pressure on annular disk combined poroelastic constant (l/R, h/Hf) radial and axial coordinates (= c/b) poroelastic constant time displacement components along x, y, z direction Eigenfunction defined in eqn. (16) average axial displacement rectangular coordinates Bessel’s function of second kind, zeroth order ith eigenvalue [= (1  2 v)/2G(l v)]
saturation parameter [= p/(1 + h’p&)] [= 2(1 + V) G/3(1  2v) H,] viscosity Poisson’s ratio increment of liquid pressure in porous medium zcomponent normal stress permeability of porous medium Introduction In the previous paper (Part I), an annular disk lubricated clutch engagement model consisting of squeeze film phase, mixed asperity contact phase, and consolidating contact phase was introduced. The engagement behavior at the squeeze film phase has been analyzed considering the effects of surface roughness, the permeability and the elastic deformation of the porous, papertype friction material. An analysis of the engagement behavior at the consolidating contact phase is now presented. Together with the results obtained in Part I, a qualitative comparison of the theoretical predictions of film thickness and porous layer compression versus time relations is made with experimental results [l] . The overall disk engagement behavior is also discussed. The consolidating contact phase begins when physical contact between the disks has been made and the resilient porous facing starts to deform due to compression. The deformation continues as time elapses until a steady state is reached. This time dependent behavior has been observed experimentally in clutch plate squeeze film tests [l] . In this paper, this behavior was analyzed by considering a model based on the theory of poroelasticity [2  51. The model treats the papertype friction facing as a porous, elastic sponge which is saturated with a viscous fluid. As a load is applied, a gradual deformation will result which depends on the rate of fluid being “squeezed” out of the voids. Results are presented as the relations of normalized deflection versus dimensionless time for various parameters involving disk geometric dimensions and the permeability, elastic and poroelastic constants and the degree of fluid satiation of the porous material. Analysis Based on the modified Hooke’s law considering the liquid pressure effect in the pores of a poroelastic body and Darcy’s law characterizing the liquid flowing through the porous medium, Biot [ 31 derived the following four differential equations governing the stress distribution, liquid content and deformation as a function of time in a porous medium under given loading conditions. They are: G GV2u+
l2v
i3e

ax
175
ae
G
GVJ2v+ 
ay
lzv
G
GB2w +
az
l2v
and
(2)
au, _=O Q at
ae
kv20f&
(3)
1
_at
(4)
In these equations, u, v, w, uf are unknowns to be solved. G and v are the shear modulus and the Poisson’s ratio, respectively, of the elastic skeleton while k (= Qr/p ) is the coefficient of permeability of the poroelastic medium. X and Q are defined as (5)
and 1
x
1
(6)
Q =Rf6
where Rf and Hf are two additional physical constants. The inverse of Hf is a measure of the compressibility of the porous medium for a change in liquid pressure and the inverse of I+ is a measure of the change in liquid content for a given change in liquid pressure. INAL
SURFACE
OF CONTACT CE
POROUS
OF CONTACT
FACING .7
Fig. 1. Geometry and coordinate system.
To apply the above poroelasticity theory to the analysis of disk engagement behavior at the consolidating contact phase, the coordinate system shown in Fig. 1 is used. Let the annular deformable porous facing be firmly attached on a bottom impermeable plate. The initial contact surface is at z = 0. Wis the average amount of porous facing deflection and is a function of time. The steady state value of @ is the maximum of the porous facing deflection. Let the annular porous facing support a load p. = CT,, and assume that contacts at the top and bottom with the impermeable disks are perfect such that no liquid can escape through these contact surfaces; the liquid is free to escape laterally. Because of the axial loading and the low Poisson’s ratio of paper type porous material, the only component of displacement which has to be considered is the axial displacement component w. This w,
176
as well as the liquid pressure uf, will be functions of the coordinate r and z, and the time t. Also since the thickness of the porous facing is small compared with the disk radial dimension, eqn. (3) and eqn. (4), expressed in cylindrical coordinates, become 1 a2w p az2
,!$=O
a20f
+1
ar2
r
where P=
aa,
(7) +a2Qf
az2
ar
x a2w =_+k a2at
1
ao,
(Qk)
at
1  2v
(8)
(9)
2G(1_y)
Integrating eqn. (7) with respect to z and partial differentiating with t yields _$=fi$f
(10)
Substituting eqn. (10) into eqn. (8) one obtains a20f 1 auf +_+=ar2 r ar
a20f az2
1 aof c
at
(11)
where 1 1 =k(hZp+i) C
Equation (11) is seen to have the same form as a heat conduction or diffusion equation. By solving af from eqn. (11) considering appropriate initial and boundary conditions and substituting it into eqn. (lo), the deflection versus time relation for the porous facing then can be determined by solving eqn. (10). The boundary conditions of eqn. (11) for the present disk engagement problem are : (i)
$
=0
(ii)
uf = 0
at
2 = O,H,
at
r =a,b
(12)
The first condition indicates that no liquid may escape through the top and the bottom of the porous facing because of perfect contact there. The second condition means that the liquid is allowed to escape freely through the inner and outer disk lateral surfaces. The initial condition is Of
=h
,14,:,,
fort=0
(13)
177
This condition was derived based on the consideration that the change of liquid content in the porous medium is zero when the load is applied [3]. It can also be written as ot=;(l;)p,
fort=0
(14)
where pi=
p
1 + A2pQ
pi is zero if the porous medium is completely saturated, i.e., the volume change of the medium is equal to the amount of liquid squeezed out. If the porous medium contains air bubbles, /Iithen will be a positive value larger than zero. pi, therefore, is a measure of the degree of liquid saturation in a porous medium. The solution of eqn. (11) with boundary conditions (12) and the initial condition (14) is obtained as Jo
ca;t
(ajb)
Jo(ajb)
+ Jo(cu/a) u”(air)ee
where Uo (&jr) = Yo (aja) Jo (air)  Jo (ala) Yo (air)
(16)
and aj is the ith eigenvalue satisfying Ye(aje) Jo(ajb)
 Jo(aja)
Yo(ajb)
=0
(17)
Jo and Y. are zeroth order Bessel functions of first and second kind, respectively. The deflection w may be found by integrating eqn. (10) twice and the result is w(z,r,t)=ponP
[
1 F
1
(zHP)
m
C
Jo(ajb)
j= 1 Jo (ajb)
uo(ajr)
(18)
PPo(zlJ) The average of the deflection of the surface (z = 0) becomes
s a
b w(O,r,t)rdr
6 (090 _ PPoH,
PPoH,
Ja
rdr
=I;;;i;
e_ca;t
+ Jo (aja)
::(ri)
178
CtE,? /b
2
e
where &i = ojb and R = a/b. Results Figures 2 through 4 show how the average surface deflections of a loaded annular porous medium progress with time for various geometric, elastic and poroelastic parameters. The deflections are negativeexponentially decayed with respect to time. Generally, the time of retardation becomes long if the annular porous medium has a large a/b ratio, low permeable material structure, no air bubble content and the viscosity of the liquid is high. w PP,HP
0002
0004
0006
0006
0010
0012
0014
0016
00
 E
hHp
05
IO
Fig. 2. Deflection
01
02
us. time curves 
04
03
geometric
size and air bubble content
007
0
Fig. 3. Deflection
effects.
I
0 05
0 IO
us. time curves 
0 15
fully saturated
0 20
porous
medium.
In Fig. 2, two graphs are shown to compare the effect of the a/b ratio. Note the large difference between the two different time scales. The effect of the existence of the air bubbles in the medium has also been shown. The completely saturated porous medium (pi = 0) needs more time to reach the
179
ct b’
Fig. 4. Deflection VS.time curves  partially saturated porous medium.
same amount of compression compared with a partially saturated medium because air is compressible and can be driven out of the medium quickly. Also, for a fully saturated porous medium, the instantaneous deflection at the moment of load application (t = O+) is very small. This instantaneous deflection increases when the medium becomes partially saturated, and is almost linearly dependent on the degree of air content. Similar deflection uersus time curves are shown in Figs. 3 and 4 for given pi/p values with a/b ratio increasing by an increment of 0.2. The effects of the size of the annular porous medium and its degree of liquid saturation on time retardation again are demonstrated. General discussion Overall disk engagement
behavior
The disk engagement behavior was investigated analytically by considering an engagement model consisting of the squeeze film phase, the mixed asperity contact phase and the consolidating contact phase. Solutions at the squeeze film phase (Part I) and consolidating contact phase (Part II) have been obtained. Their quantitative comparison with the experiments in the form of film thickness and porous facing deflection uersus time cannot be made presently due to the lack of sufficient experimental data as well as the complete information on testing conditions and the porous facing material constants. However, by examining the film thicknesstime relations shown in Part I (Fig. 12) for the squeeze film phase and the deflection curves given in Figs. 2  4 (Part II) for the consolidating contact phase, qualitative agreement with the experimental results reported in ref. 1 is seen to be good. If information on the friction material’s permeability, its elastic and poroelastic constants, as well as accurate roughness measurements, are available, good quantitative comparisons between the theory and experiment as shown in Fig. 5 and Fig. 6 (Part II) are expected. It is also expected that a small discrepancy will occur in the neighborhood where the two solutions intersect since this region corresponds to the area of roughness interaction. The solutions considering the mixed asperity

I
Experimental    Squeeze Film Phase Conrolidoling Contocl Phose
.Theory
.
0.1 SEC
TIME
Fig. 5. Comparison of squeeze film and paper compression traces for new ground paper plate.

Exoerimentof SqkexeFilmPhose    Consolidoiii
Conioct Phese )Theory
TIME
Fig. 6. Comparison of squeeze film and paper compression traces for ground pressed plate.
contact phase at that region are currently not available. However, for nonrotating disks, this region is occupied only for a relatively short time period, compared with the other two phases, hence its effect on the total engagement behavior is not expected to be significant. This mixed phase will become one of the important stages in the whole engagement process when relative sliding speed occurs between the disks. Under this situation, the mixed phase will cover the transition period from the domination of squeeze and hydrodynamic actions (relative sliding speed is maximum) at the beginning of asperity contact to the domination of adhesive friction (zero relative sliding speed  disks rotate at equal speed) at the completion of the eng~ement. As mentioned in Part I, the exact behavior of this phase is difficult to analyze completely due to involvement of many physical and chemical interface factors. However, a first approximation analysis is still possible. Results obtained should be able to show how the density, permeability, compliance, surface roughness structure and the adhesive friction properties of the wet friction material, as well as the transmission fluid properties, will effect the engagement behavior. The terminology of “consolidating contact” was borrowed from soil mechanics in dealing with the settlement phenomenon of water saturated
181
soil under load. The wording also indicates the deformation of porous medium associated with the liquid being squeezed out from its voids. This is similar to the wet clutch friction material deformation under contact load so the terminology was adopted: The consolidating contact phase can be considered as progressing simultaneously with the mixed asperity contact phase, but, it occurs inside the body of the porous facing instead of at the contact surface. The deflection versus time relations presented in Part II do not include the disk sliding speed effect. However, it will not be difficult to extend the current analysis taking into account the sliding speed. The load can be assumed to move on the surface with given velocity, and on the surface, in addition to normal load, a shear stress component due to friction force should be considered too. These solutions will be much more interesting for interpreting the engagement behavior than the present stationary solutions since, in reality, the disk does rotate. In ref. 1, a simple concept of wet clutch engagement was introduced. The postulated engagement model employs three phases, the squeeze film phase, squash phase and the adhesive contact phase. In the present two papers, the engagement model is considered to consist of a squeeze film phase, a mixed asperity contact phase and a consolidating contact phase. The analysis of the squeeze film phase has been extended by considering the additional effects due to friction material elastic deformation and its surface roughness. The mixed asperity contact phase should be able to cover both squash and adhesive phases since the adhesive contact occurs continuously but with an increasing degree of domination as the engagement progresses toward completion. The consolidating phase which deals with the total porous facing surface deformation under contact load is clearly demonstrated in squeeze film testing results given in ref. 1 corresponding to the later stage of the squeeze film traces. This phenomenon was briefly discussed in ref. 1 but was not considered as a significant phase in its engagement model. Thus, it may be seen, that the present engagement model has extended and refined the modeling concept proposed in ref. 1. However, further investigations in the mixed asperity contact and consolidating contact phases are needed, especially with the consideration of disk speed effect, in order to establish a complete and satisfactory clutch engagement theory useful in understanding its intriguing behavior and helpful in its research and diagnostic studies. Conclusion The engagement behavior of wet clutches used in automotive transmissions was analyzed by considering an engagement model consisting of three phases: the squeeze film phase, the mixed asperity contact phase and the consolidating contact phase. Solutions have been obtained for the squeeze film phase and consolidating contact phase. Investigation of the mixed asperity contact phase is not completed yet.
182
In the squeeze film phase, it was found that increases of the permeability, resilience and surface roughness of the porous paper facing friction material will contribute to the decrease of film pressure distribution and load capacity, and reduction of squeeze time. In the consolidating contact phase, the deflection of the contact surface decays negativeexponentially with respect to time. The time of retardation increases as the porous facing has larger contact area, lower permeability, less air bubble content, and higher liquid viscosity. The analytical results at these two phases agree qualitatively with experimental data. Acknowledgements The author wishes to express his appreciation to Mr. A. E. Anderson of Scientific Research Staff, Ford Motor Company, for his suggestions and comments during the course of this study. References 1 A. E. Anderson, Friction and wear of paper type wet friction elements, SAE paper 720521. 2 A. Bedford and J. D. Ingram, A continuum theory of fluid saturated porous media, J. Appl. Mech., (1971) 1. 3 M. A. Biot, General theory of threedimensional consolidation, J. Appl. Phys., 12 (1941) 155. 4 G. DeJosselin DeJong, Application of stress functions to consolidation problems, Proc. 4th Int. Conf. on Soil Mechanics and Foundation Engrg., I (3a/13) (1957) 320. 5 J. Mandel, Consolidation des couches d’argiles, Proc. 4th Int. Conf. on Soil Mechanics and Foundation Engrg., I (3a/21) (1957) 360.