The three deformation models elastic, plastic, or elastic–plastic give relation-ships for three important geometric parameters of the joint: the relative real contact areaA r /A a, the c
Trang 11 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
[341],(81)
Lines: 3311 to 3334
———
0.60109pt PgVar
———
Normal Page
* PgEnds: Eject
[341],(81)
CLA roughness= L1
L
0
rms roughness=
1
L
L
0
y2(x) dx (m) (4.230)
wherey(x) is the distance ofpoints in the surface from the mean plane (Fig 4.22)
andL is the length ofa trace that contains a sufficient number ofasperities For
Gaussian asperity heights with respect to the mean plane, these two measures of surface roughness are related (Mikic and Rohsenow, 1966):
σ =
π
2 · CLA
A second very important surface roughness parameter is the absolute mean asperity slope, which is defined as (Cooper et al., 1969; Mikic and Rohsenow, 1966; and DeVaal et al 1987)
y x
y x
1
2
m2=dy dx2/ 2
m dy dx= /
m1=dy dx1/ 1
Y
Y
Figure 4.22 Typical joint between conforming rough surfaces (From Hegazy, 1985.)
Trang 21 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
[342], (82)
Lines: 3334 to 3384
———
3.76213pt PgVar
———
Normal Page PgEnds: TEX [342], (82)
m = L1
L
0
dy(x) dx dx (rad) (4.231) The effective rms surface roughness and the effective absolute mean asperity slope for a typical joint formed by two conforming rough surfaces are defined as (Cooper
et al., 1969; Mikic, 1974; Yovanovich, 1982)
σ = σ2
1+ σ2
2 and m = m2
1+ m2
Antonetti et al (1991) reported approximate relationships form as a function of σ for
several metal surfaces that were bead-blasted
The three deformation models (elastic, plastic, or elastic–plastic) give relation-ships for three important geometric parameters of the joint: the relative real contact areaA r /A a, the contact spot densityn, and the mean contact spot radius a in terms
ofthe relative mean plane separation defined asλ = Y/σ The mean plane separation
Y and the effective surface roughness are illustrated in Fig 4.22 for the joint formed
by the mechanical contact oftwo nominally flat rough surfaces
The models differ in the mode of deformation of the contacting asperities The three modes of deformation are plastic deformation of the softer contacting asperities, elastic deformation of all contacting asperities, and elastic–plastic deformation of the softer contacting asperities For the three deformation models there is one thermal contact conductance model, given as (Cooper et al., 1969; Yovanovich, 1982)
h c=2nak s (W/m2· K) (4.233)
wheren is the contact spot density, a is the mean contact spot radius, and the effective
thermal conductivity ofthe joint is
k s = 2k1k2
k1+ k2
and the spreading/constriction parameter ψ, based on isothermal contact spots, is
approximated by
where the relative contact spot size is √
A r /A a The geometric parametersn, a
andA r /A aare related to the relative mean plane separationλ = Y/σ.
4.16.1 Plastic Contact Model
The original plastic deformation model of Cooper et al (1969) has undergone signif-icant modifications during the past 30 years First, a new, more accurate correlation equation was developed by Yovanovich (1982) Then Yovanovich et al (1982a) and
Trang 31 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
[343],(83)
Lines: 3384 to 3393
———
3.097pt PgVar
———
Normal Page PgEnds: TEX [343],(83)
Figure 4.23 Vickers microhardness versus indentation diagonal for four metal types (From Hegazy, 1985.)
Hegazy (1985) introduced the microhardness layer which appears in most worked
metals Figures 4.23 and 4.24 show plots ofmeasured microhardness and macrohard-ness versus the penetration deptht or the Vickers diagonal d V These two measures of
indenter penetration are related:d v /t = 7 Figure 4.23 shows the measured Vickers
microhardness versus indentation diagonal for four metal types (Ni 200, stainless steel 304, Zr-4 and Zr-2.5 wt % Nb) The four sets of data show the same trends:
that as the load on the indenter increases, the indentation diagonal increases and the Vickers microhardness decreases with increasing diagonal (load) The indentation diagonal was between 8 and 70µm
Figure 4.24 shows the Vickers microhardness measurements and the Brinell and Rockwell macrohardness measurements versus indentation depth The Brinell and Rockwell macrohardness values are very close because they correspond to large in-dentations, and therefore, they are a measure of the bulk hardness, which does not change with load According to Fig 4.24, the penetration depths for the Vickers mi-crohardness measurements are between 1 and 10µm, whereas the larger penetration
Trang 41 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
[344], (84)
Lines: 3393 to 3406
———
3.73402pt PgVar
———
Normal Page PgEnds: TEX [344], (84)
Figure 4.24 Vickers, Brinell, and Rockwell hardness versus indentation depth for four metal types (From Hegazy, 1985.)
depths for the Brinell and Rockwell macrohardness measurements lie between ap-proximately 100 and 1000µm
The microhardness layer may be defined by means ofthe Vickers microhardness measurements, which relate the Vickers microhardnessH V to the Vickers average indentation diagonald V (Yovanovich et al., 1982a; Hegazy, 1985):
H V = c1
d
V
d0
c2
whered0 represents some convenient reference value for the average diagonal, and
c1andc2are the correlation coefficients It is conventional to setd0= 1 µm Hegazy
(1985) found thatc1is closely related to the metal bulk hardness, such as the Brinell hardness, denoted asH B.
The original mechanical contact model (Yovanovich et al., 1982a; Hegazy, 1985) required an iterative procedure to calculate the appropriate microhardness for a given surface roughnessσ and m, given the apparent contact pressure P and the coefficients
c1andc2 Song and Yovanovich (1988) developed an explicit relationship for the micro-hardnessH p, which is presented below Recently, Sridhar and Yovanovich (1996b) developed correlation equations between the Vickers correlation coefficientsc1 and
Trang 51 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
[345],(85)
Lines: 3406 to 3454
———
-0.74191pt PgVar
———
Normal Page
* PgEnds: Eject
[345],(85)
c2and Brinell hardnessH Bover a wide range ofmetal types These relationships are also presented below
con-tacting asperities, the contact geometric parameters are obtained from the following relationships (Cooper et al., 1969; Yovanovich, 1982):
A r
A a =
1
2erfc
√
2
(4.237)
n = 1
16
m
σ
2 exp(−λ2)
erfc(λ/√2) (4.238)
a =
8
π
σ
mexp
λ2 2
erfc
λ
√
2
(4.239)
na = 1
4√
2π
m
σ exp
−λ2
2
(4.240)
Correlation of Geometric Parameters
A r
A a = exp(−0.8141 − 0.61778λ − 0.42476λ2− 0.004353λ3)
n = m
σ
2 exp(−2.6516 + 0.6178λ − 0.5752λ2+ 0.004353λ3)
a = mσ(1.156 − 0.4526λ + 0.08269λ2− 0.005736λ3)
and for the relative mean plane separation
λ = 0.2591−0.5446
ln P
H p
−0.02320
ln P
H p
2
−0.0005308
ln P
H p
3
(4.241) The relative mean plane separation for plastic deformation is given by
λ =√2 erfc−1
2P
H p
(4.242) whereH pis the microhardness ofthe softer contacting asperities
from the relative contact pressureP /H p For plastic deformation of the contacting asperities, the explicit relationship is (Song and Yovanovich, 1988)
P
H p =
P
c1(1.62σ/m) c2
1/(1+0.071c2)
(4.243)
Trang 61 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
[346], (86)
Lines: 3454 to 3509
———
2.71118pt PgVar
———
Long Page PgEnds: TEX [346], (86)
where the coefficientsc1andc2are obtained from Vickers microhardness tests The Vickers microhardness coefficients are related to the Brinell hardness for a wide range ofmetal types
co-efficientsc1andc2are obtained from Vickers microhardness measurements Sridhar and Yovanovich (1996b) developed correlation equations for the Vickers coefficients:
c1
3178 = 4.0 − 5.77H∗
B + 4.0 H∗
B 2− 0.61 H∗
B 3 (4.244)
c2= −0.370 + 0.442
H
B
c1
(4.245) whereH Bis the Brinell hardness (Johnson, 1985; Tabor, 1951) andH∗
B = H B /3178.
The correlation equations are valid for the Brinell hardness range 1300 to 7600 MPa
The correlation equations above were developed for a range of metal types (e.g., Ni200, SS304, Zr alloys, Ti alloys, and tool steel) Sridhar and Yovanovich (1996b) also reported a correlation equation that relates the Brinell hardness number to the Rockwell C hardness number:
BHN= 43.7 + 10.92 HRC − HRC2
5.18 +
HRC3
340.26 (4.246)
for the range 20≤ HRC ≤ 65
dimen-sionless contact conductanceC cis
C c≡ h cσ
k s m =
1
2√
2π
exp(−λ2/2)
1− 1
2erfc(λ/√2)
1.5 (4.247)
The correlation equation ofthe dimensionless contact conductance obtained from theoretical values for a wide range ofλ and P /H p is (Yovanovich, 1982)
C c≡h c
k s
σ
m = 1.25
P
H p
0.95
(4.248)
which agrees with the theoretical values to within±1.5% in the range 2 ≤ λ ≤ 4.75.
It has been demonstrated that the plastic contact conductance model ofeq (4.248) predicts accurate values ofh cfor a range of surface roughnessσ/m, a range ofmetal
types (e.g., Ni 200, SS 304, Zr alloys, etc.), and a range ofthe relative contact pres-sureP /H p (Antonetti, 1983; Hegazy, 1985; Sridhar, 1994; Sridhar and Yovanovich,
1994, 1996a) The very good agreement between the contact conductance models and experiments is shown in Fig 4.25
In Fig 4.25 the dimensionless contact conductance model and the vacuum data for different metal types and a range of surface roughnesses are compared over two
Trang 71 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
[347],(87)
Lines: 3509 to 3531
———
2.397pt PgVar
———
Long Page PgEnds: TEX [347],(87)
Figure 4.25 Comparison ofa plastic contact conductance model and vacuum data (From Antonetti, 1983; Hegazy, 1985.)
decades ofthe relative contact pressure defined asP /H e, whereH ewas called the
effective microhardness ofthe joint The agreement between the theoretical model
developed for conforming rough surfaces that undergo plastic deformation of the con-tacting asperities is very good over the entire range ofdimensionless contact pressure
Because ofthe relatively high contact pressures and high thermal conductivity ofthe metals, the effect of radiation heat transfer across the gaps was found to be negligible for all tests
4.16.2 Radiation Resistance and Conductance for Conforming Rough Surfaces
The radiation heat transfer across gaps formed by conforming rough solids and filled with a transparent substance (or its in a vacuum) is complex because the geometry of
Trang 81 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
[348], (88)
Lines: 3531 to 3570
———
3.24652pt PgVar
———
Normal Page
* PgEnds: Eject
[348], (88)
TABLE 4.18 Radiative Conductances for Black Surfaces
the microgaps is very difficult to characterize and the temperatures of the bounding solids vary in some complex manner because they are coupled to heat transfer by conduction through the microcontacts
The radiative resistance and the conductance can be estimated by modeling the heat transfer across the microgaps as equivalent to radiative heat transfer between two gray infinite isothermal smooth plates The radiative heat transfer is given by
Q r = σA a F12
T4
j1 − T4
whereσ = 5.67×10−8W/(m2· K4) is the Stefan–Boltzmann constant andT j1andT j2
are the absolute joint temperatures ofthe bounding solid surfaces These temperatures are obtained by extrapolation ofthe temperature distributions within the bounding solids The radiative parameter is given by
1
F12
1
2
where 1and 2are the emissivities ofthe bounding surfaces The radiative resistance
is given by
R r =T j1 − T j2
Q r = T j1 − T j2
σA a F12
T4
j1 − T4
j2
(K/W) (4.251) and the radiative conductance by
h r =A Q r
a (T j1 − T j2 )=
σF12
T4
j1 − T4
j2
T j1 − T j2 (W/m2· K) (4.252)
The radiative conductance is seen to be a complex parameter which depends on the emissivities 1and 2and the joint temperaturesT j1andT j2 For many interface prob-lems the following approximation can be used to calculate the radiative conductance:
T4
j1 − T4
j2
T j1 − T j2 ≈ 4(T j )3
where the mean joint temperature is defined as
Trang 91 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
[349],(89)
Lines: 3570 to 3612
———
-0.25409pt PgVar
———
Normal Page
* PgEnds: Eject
[349],(89)
T j =1
2(T j1 + T j2 ) (K) Ifwe assume blackbody radiation across the gap, 1 2 = 1 givesF12 = 1
This assumption gives the upper bound on the radiation conductance across gaps formed by conforming rough surfaces If one further assumes that T j2 = 300 K
andT j1 = T j2 + ∆T j, one can calculate the radiation conductance for a range of
values of∆T j andT j The values ofh r for black surfaces represent the maximum
radiative heat transfer across the microgaps For microgaps formed by real surfaces, the radiative heat transfer rates may be smaller Table 4.18 shows that when the joint temperature is T j = 800 K and ∆T j = 1000 K, the maximum radiation
conductance is approximately 161.5 W/m2· K This value is much smaller than the
contact and gap conductances for most applications whereT j < 600 K and ∆T j <
200 K The radiation conductance becomes relatively important when the interface is formed by two very rough, very hard low-conductivity solids under very light contact pressures Therefore, for many practical applications, the radiative conductance can
be neglected, but not forgotten
4.16.3 Elastic Contact Model
The conforming rough surface model proposed by Mikic (1974) for elastic deforma-tion ofthe contacting asperities is summarized below (Sridhar and Yovanovich, 1994, 1996a)
param-eters are (Mikic, 1974)
A r
A a =
1
4erfc
√
2
(4.253)
n = 1
16
m
σ
2 exp(−λ2)
erfc(λ/√2) (4.254)
a = √2 π
σ
mexp
λ2 2
erfc
λ
√
2
(4.255)
na = 1
8√ π
m
σ exp
−λ2
2
(4.256) The relative mean plane separation is given by
λ =√2erfc−1
4P
H e
(4.257) The equivalent elastic microhardness according to Mikic (1974) is defined as
Trang 101 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
[350], (90)
Lines: 3612 to 3662
———
5.38135pt PgVar
———
Short Page
* PgEnds: Eject
[350], (90)
where the effective Young’s modulus of the contacting asperities is
1
E = 1− ν21
E1
+1− ν22
E2
(m2/N) (4.259)
Greenwood and Williamson (1966), Greenwood (1967), and Greenwood and Tripp (1970) developed a more complex elastic contact model that gives a dimensionless elastic microhardnessH e /mEthat depends on the surface roughness bandwidthα
and the separation between the mean planes ofthe asperity “summits,” denoted as
λs For a typical range ofvalues ofα and λs (McWaid and Marschall, 1992a), the
value ofMikic (1974) (i.e.,H e /mE = 0.7071) lies in the range obtained with the
Greenwood and Williamson (1966) model There is, at present, no simple correlation for the model of Greenwood and Williamson (1966)
conduc-tance for conforming rough surfaces whose contacting asperities undergo elastic de-formation is (Mikic, 1974; Sridhar and Yovanovich, 1994)
h cσ
k s m=
1
4√ π
exp
− λ2/2
1− 1
4erfc(λ/√2)
1.5 (4.260)
The power law correlation equation based on calculated values obtained from the theoretical relationship is (Sridhar and Yovanovich, 1994)
h cσ
k s m = 1.54
P
H e
0.94
(4.261)
has an uncertainty ofabout ±2% for the relative contact pressure range 10−5 ≤
P /H e ≤ 0.2.
forA r /A a , n, and a for the relative contact pressure range 10−6≤ P /H e ≤ 0.2 are
A r
A a = 12exp(−0.8141 − 0.61778λ − 0.42476λ2− 0.004353λ3)
n = m
σ
2 exp(−2.6516 + 0.6178λ − 0.5752λ2+ 0.004353λ3)
a = √1
2
σ
m (1.156 − 0.4526λ + 0.08269λ2− 0.005736λ3)
and the relative mean planes separation
... radiation heat transfer across the gaps was found to be negligible for all tests4.16.2 Radiation Resistance and Conductance for Conforming Rough Surfaces
The radiation heat transfer. .. because they are coupled to heat transfer by conduction through the microcontacts
The radiative resistance and the conductance can be estimated by modeling the heat transfer across the microgaps... transfer across the microgaps as equivalent to radiative heat transfer between two gray infinite isothermal smooth plates The radiative heat transfer is given by
Q r = σA