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To2008-ConformingContact-PinLoadedStructure.pdf
Available online at www.sciencedirect.com
International Journal of Solids and Structures 45 (2008) 3935–3950
www.elsevier.com/locate/ijsolstr
On the conforming contact problem in a reinforced
pin-loaded structure with a non-zero second Dundurs’ constant
Q.D. To, Q.-C. He *
Université Paris-Est, Laboratoire de Modelisation et Simulation Multi Echelle, FRE 3160 CNRS, 5 Boulevard Descartes,
77454 Marne-la-Vallée Cedex 2, France
Received 7 December 2007; received in revised form 27 January 2008
Available online 10 March 2008
Abstract
Within the framework of two-dimensional linear elasticity, the unilateral frictionless contact between two conformal
cylindrical surfaces is governed by an integral equation in which the first and second Dundurs’ constants are involved.
In the case of elastic similarity characterized by the zero second Dundurs’ constant, the integral equation is considerably
simplified so as to lend itself to a closed-form solution. However, in the case of elastic dissimilarity defined by the non-zero
second Dundurs’ constant, the question of obtaining a closed-form solution to the integral equation is a much tougher one.
Starting from the integral equation established by To et al. [To, Q.D., He Q.-C., Cossavella, M., Morcant, K., Panait, A.,
2007. Closed-form solution for the contact problem of reinforced pin-loaded joints used in glass structures. Int. J. Solids
Struct. 44, 3887–3903] for the conformal contact problem originating from a reinforced pin-loaded joint used in tempered
glass structures, the present work proposes a new approximate analytical method to solve it in the case of elastic dissimilarity by minimizing an error function. The derived closed-form solution, valid not only for the conformal contact between
a pin and an infinite holed plate but also for the one between a pin and a finite holed plate, is shown to be in very good
agreement with available numerical results.
Ó 2008 Elsevier Ltd. All rights reserved.
Keywords: Conformal contact; Pin-loaded joints; Integral equation; Dundurs constants; Least-square method
1. Introduction
The contact between the border of a circular hole in a plate and the surface of a circular pin inserted into
the hole and subjected to a force is the prototype of a great number of unilateral conformal contact problems
encountered in civil, mechanical and aerospace engineering. The study of this prototype conformal problem
has a long history. A rather comprehensive list of relevant references can be found in a recent interesting paper
of Ciavarella et al. (2006).
*
Corresponding author. Tel.: +33 0 160 957 786; fax: +33 0 160 957 799.
E-mail address: [email protected] (Q.-C. He).
0020-7683/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijsolstr.2008.02.010
3936
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
When the friction between the pin surface and the hole border is neglected, the aforementioned conformal
contact problem formulated with the framework of two-dimensional (2D) elasticity is governed by an integral
equation involving both the first and second Dundurs’ constants (see, e.g., Barber, 2002) of the materials constituting the pin and plate. The contacting materials are said to be elastically similar or dissimilar according as
the second Dundurs’ constant is equal or not equal to zero. In the case of elastic similarity, the integral equation governing the conformal contact problem is considerably simplified so as to lend itself to a closed-form
solution. However, in the case of elastic dissimilarity, the problem of obtaining a closed-form solution to the
integral equation is a very tough one.
In his Ph.D. thesis, Persson (1964) was the first to give a closed-form solution when the contacting materials
are similar (see Johnson, 1985, section 5.3). Using a different method and limiting themselves also to the case
of elastic similarity and zero-clearance, Noble and Hussain (1969) derived another closed-form solution. In a
paper consisting of two parts, Ciavarella and Decuzzi (2001a,b) improved the results of Persson (1964) and
Noble and Hussain (1969) in the case of elastic similarity and proposed a method to deal with the case of elastic dissimilarity.
The present work is a continuation of a previous investigation by the authors (To et al., 2007). In the
latter, the problem of conformal contact between the border of a circular ring reinforcing a holed glass
plate of finite breadth and the surface of a circular pin in the ring and subjected to a force (Fig. 1)
was formulated as an integral equation and a closed-form solution was given for the case where the materials forming the ring and pin are elastically similar. This paper presents a new approximate analytical
method to derive closed-form solutions in the unsolved case of elastic dissimilarity. Our method is rather
different from the one proposed by Ciavarella and Decuzzi (2001b) to treat the case of elastic dissimilarity.
Indeed, on the basis of the numerical observation that the second Dundurs’ constant has little influence on
the contact pressure but significantly affects the relation between the contact area and the normalized
dimensionless loading parameter, Ciavarella and Decuzzi (2001b) adopted as an approximate one the contact pressure distribution corresponding to the case of elastic similarity and deduced a formula for the
contact area in the case of elastic dissimilarity. By contrast, in our method, the effects of the second Dundurs’ constant on the contact pressure and area are directly taken into account through approximating the
integral term involving the second Dundurs’ constant in the integral equation by an analytically tractable
function and by minimizing the resulting error. Furthermore, our results hold not only for a pin in a plate
of infinite breadth but also for a pin in a plate of finite breadth.
Fig. 1. Composition of a reinforced pin-loaded joint and conformal contact between the pin and ring.
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
3937
The paper is organized as follows. In the next section, we recall the setting of the problem and rewrite the
governing integral equation in a more tractable form. In Section 3, after approximating the integral related to
the second Dundurs’ constant by a finite series and minimizing the error in the sense of the least-square
method, a general closed-form solution to the governing integral equation is obtained, allowing the achievement of any desired degree of accuracy. In Section 4, the general solution derived in Section 4 is applied by
considering an infinite ring and a finite ring. The resulting approximate analytical results are compared with
the numerical and approximate analytical results provided by Ciavarella and Decuzzi (2001b) in the case of an
infinite plate and with the numerical results obtained by us with the aid of the finite element method in the case
of a plate of a finite breadth. These comparisons show that our approximate analytical results are very accurate. We conclude this paper by giving a few closing remarks in Section 5.
2. Setting of the problem and governing integral equation
Consider the problem of conformal contact between a bolt and a ring which is, by a resin layer, glued to the
border of a hole in a glass plate as in Fig. 1. Relative to a system of polar coordinates ðr; hÞ with the origin
coinciding with the hole center, the components of this reinforced pin-loaded joint are specified as follows
–
–
–
–
the
the
the
the
bolt: r < R0 ;
reinforcement ring: R1 < r < R2 ;
resin layer: R2 < r < R3 ;
glass plate: r > R3 .
The materials constituting the components are all taken to be linearly elastic, homogeneous and isotropic.
The conforming contact problem will be studied within the framework of plane elasticity. Thus, it is convenient to characterize material i ð¼ 0; 1; 2; 3Þ by the Kolosov constants li and ji which are expressed in terms
of the Young modulus Ei and the Poisson ratios mi by
(
3mi
ðplane stressÞ;
Ei
; ji ¼ 1þmi
li ¼
ð1Þ
2ð1 þ mi Þ
3 4mi ðplane strainÞ:
The integer subscript i taking the values 0, 1, 2, 3 denotes the bolt, the ring, the resin and the glass plate,
respectively. The frictionless unilateral contact between the bolt and the ring must verify the following conditions of Signorini type:
ur0 ðhÞ ur1 ðhÞ ¼ DR;
pðhÞ P 0;
if h 2 ½a; a;
ur0 ðhÞ ur1 ðhÞ < DR;
pðhÞ ¼ 0;
if h 62 ½a; a:
ð2Þ
Here, ur0 ðhÞ is the radial displacement of a point ðr; R0 Þ on the surface of the bolt, ur1 ðhÞ is the radial displacement of a point ðr; R1 Þ on the interior surface of the ring, the radius difference DR ¼ R1 R0 stands for the
clearance of the joint, pðhÞ represents the contact pressure which is taken to be positive, and a designates half
of the contact angle.
Under the condition that the shear modulus l2 of the resin layer is much smaller than the shear moduli l1
and l3 of the bolt and glass plate, i.e. l2 minðl1 ; l3 Þ, To et al. (2007) have shown that the integral equation
governing the foregoing frictionless contact problem takes the form
Z
0
a
1
X
pðnÞdn
h
sin nh c2 p
¼ ðb0 G0 þ b c2 G0 Þ
þ ðb1 2c2 ÞG1 þ
þ
bn Gn
cos h cos n
sin h
sin h sin h
n¼2
Z
h
pðnÞdn;
ð3Þ
0
which corresponds to Eq. (43) in To et al. (2007). Above, the coefficients bn ; Gn ; b and c2 are defined by
3938
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
Gn ¼
Z
a
pðnÞ cos nn dn for n ¼ 0; 1; 2; . . . ;
1
1 j0 2q þ j1 1
b0 ¼ ; b1 ¼ 2;
j1 þ1
j0 þ1
ðq 1Þl1
l0
þ
l1
l0
0
2½ðn2 þ nÞqnþ1 2ðn2 1Þqn þ ðn2 nÞqn1 2
bn ¼ l1
½q2n n2 qnþ1 þ 2ðn2 1Þqn n2 qn1 þ 1
1 þ jj01 þ1
þ1 l0
4pDR
;
b¼ j0 þ1
þ
R j1lþ1
l
1
0
c2 ¼
j1 1
l1
j1 þ1
l1
j0l1
0
þ j0lþ1
0
for n P 2;
ð4Þ
:
In these expressions, c2 is the well-known second Dundurs’ constant (Barber, 2002) and q is defined by
2
q ¼ ðR2 =R1 Þ ;
ð5Þ
which characterizes the size of the ring relative to that of the pin and whose value range is 1; þ1. Moreover,
we have an explicit relation for G1 due to the force equilibrium condition
Z a
G1 ¼
pðnÞ cos n dn ¼ F =2R:
ð6Þ
0
Normalizing the contact pressure pðnÞ as
RpðnÞ
F
qðnÞ ¼
ð7Þ
with R ¼ ðR0 þ R1 Þ=2, and introducing the first Dundurs’ constant
j1 þ1
l
j0lþ1
0
l1
þ j0lþ1
0
1
c1 ¼ j1 þ1
ð8Þ
in the integral equation (3), the latter can be written in the following more compact and more tractable form:
Z a
Z h
1
X
qðnÞdn
h
sin nh c2 p
¼ ðb0 H 0 kÞ
2ð1 þ c2 ÞH 1 þ
þ
bn H n
qðnÞdn
ð9Þ
sin h
sin h sin h 0
0 cos h cos n
n¼2
with
Hn ¼
Z
a
qðnÞ cos nn dn
for n ¼ 0; 2; 3; . . . ; H 1 ¼ 1=2;
0
k¼
bR
4pDR
;
¼ j1 þ1
F
þ j0 þ1
F
l1
l0
b0 ¼ qð1 þ c1 Þ
;
2ðq 1Þ
2
bn ¼
ð1 þ c1 Þ½ðn þ nÞqnþ1 2ðn2 1Þqn þ ðn2 nÞqn1 2
q2n n2 qnþ1 þ 2ðn2 1Þqn n2 qn1 þ 1
for n P 2:
ð10Þ
Owing to the previous reformulation, we clearly see that the normalized contact pressure qðnÞ depends only on
the four parameters, i.e. c1 ; c2 ; k and q. The first two ones c1 and c2 are the Dundurs’ constants; the third one
k represents a combination of the clearance DR, applied external force F and material elastic properties; the
fourth parameter q describes the relative size of the ring.
In Appendix A, it is shown that, by a variable change, Eq. (9) is equivalent to a generalized Prandtl integral
equation.
In the special case where the second Dundurs’ constant c2 ¼ 0, Eq. (9) reduces to
Z a
1
X
qðnÞdn
h
sin nh
¼ ðb0 H 0 kÞ
1þ
;
ð11Þ
bn H n
cos
h
cos
n
sin
h
sin h
0
n¼2
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
3939
which involves three parameters, i.e. c1 ; k and q. This simpler problem has already been solved in our previous
work (To et al., 2007). In what follows, we are concerned with treating the much more complicated situation
where c2 6¼ 0.
3. General approximate analytical solution
As the right-hand side terms of Eq. (9) contains an infinite series, it is analytically very difficult or even
impossible to find an exact analytical solution to it. However, approximate solutions can be obtained by making some physically
P1 based simplifications. It is known from To et al. (2007)
Pk that, if q is sufficiently large, the
infinite series n¼2 bn H n sin nh= sin h can be approximated by a finite one n¼2 bn H n sin nh= sin h so as to derive
a simple closed-form solution to Eq. (11).
When the material dissimilarity defined by c2 6¼ 0 takes place and when the ring becomes infinite, i.e.
q ¼ 1, Ciavarella and Decuzzi (2001b) proposed an approximate solution by retaining the solution for
c2 ¼ 0 while making the contact angle a dependent on c2 . At the same time, Iyer (2001), using the finite element method, found that the material dissimilarity has little effect on the contact pressure distribution in both
infinite and finite pin-loaded plates in the particular case where the pins and plates are all made of metallic
materials. Below, we propose a different and more efficient approach for obtaining approximate analytical
solutions in the general case when c2 6¼ 0 and q < 1. Remark that other approximation methods for solving
integral equations, such as the collocation method and the Bubnov–Galerkin method, can be found in the
handbook by Polyanin and Manzhirov (1998).
First, assuming that the value of the finiteness parameter q is large enough, the infinite series
P1
n¼2 bn H n sin nh= sin h in Eq. (9) can be replaced by a finite series as follows:
1
X
bn H n
n¼2
k
sin nh X
sin nh
’
;
bn H n
sin h
sin h
n¼2
ð12Þ
where the number k of initial terms involved in the finite series depends on the desired degree of accuracy.
Next, approximating the normalized pressure distribution qðnÞ by a finite series
qðnÞ ’ a0 þ
l
X
an cos nn;
ð13Þ
n¼1
the last term in the right-hand side of Eq. (9) involving the second Dundurs’ constant c2 has then the following
approximate expression:
"
#
Z h
l
X
c2 p
sin nh
h
an þ
a0 :
qðnÞdn ’ c2 p
ð14Þ
sin h 0
n sin h
sin h
n¼1
The main advantage of the previous two approximations is that they lead to expressing the right-hand side of
Eq. (9) in terms of the base functions h= sin h and sin nh= sin h with n P 1. Consequently, the technique elaborated in To et al. (2007) can be directly used to solve the resulting integral equation.
To determine the values of the coefficients an ðn ¼ 0; 1; 2; . . . ; lÞ, one simple and direct possibility is to consider the approximation (13) as a finite Fourier series with the coefficients determined by the usual formulae
for an even function:
a0 ¼
H0
;
p
an ¼
2H n
p
for n ¼ 1; 2; . . . ; l;
ð15Þ
where H n ðn ¼ 0; 1; 2; . . . ; lÞ are given by the first formula in (10). However, this way of specifying the values
of an does not provide the best approximation for the function qðnÞ with the finite series (13). To gain the best
one, we use the least-square method to calculate an . More precisely, the values of an ðn ¼ 0; 1; 2; . . . ; lÞ are
determined by minimizing the error function
3940
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
Eða0 ; a1 ; . . . ; ak Þ ¼
Z
a
"
qðnÞ a0 0
l
X
#2
an cos nn dn:
ð16Þ
n¼1
The necessary condition to achieving the minimum of this function leads to a system of linear equations
#
Z a"
l
X
a0 þ
an cos nh cos mh dh ¼ H m with m ¼ 0; 1; . . . ; l;
ð17Þ
0
n¼1
which allows the determination of an .
After choosing the values of k and l and after calculating all the coefficients an , the final equation approximating Eq. (9) can be written in the general compact form
Z a
maxðk;lÞ
X
h
qðnÞdn
¼ b0 H 0 k þ c2 pa0
ð1 þ c2 c2 pa1 Þ þ
½bn H n signðk nÞ
sin h
0 cos h cos n
n¼2
þ c2 pan signðl nÞ
sin nh
;
n sin h
ð18Þ
where signðxÞ is the step function defined by signðxÞ ¼ 0 for x < 0 and signðxÞ ¼ 1 for x P 0.
As in To et al. (2007), by a suitable change of variables, the integral equation (18) governing the contact
pressure distribution can be transformed into a Cauchy singular integral equation whose solution is given
for example by Peters (1963). To avoid repetition, no details are given here and only the solution for the contact pressure is provided:
pffiffiffi
Z 1
Z r
H 0 sin n
sin n d
dr
uðxÞ xdx
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffi
pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi ;
qðnÞ ¼
ð19Þ
rx
rt 0
2mp tð1 tÞ p t dt t
where
t ¼ sin2 ðn=2Þ=m; m ¼ sin2 ða=2Þ; x ¼ sin2 ðh=2Þ=m;
h
ð1 þ c2 c2 pa1 Þ
u½xðhÞ ¼ b0 H 0 k þ c2 pa0
sin h
maxðk;lÞ
X
sin nh
:
ðbn H n signðk nÞ þ c2 pan signðl nÞÞ
þ
n
sin h
n¼2
ð20Þ
After carrying out the integrations in the expression (19) of qðnÞ, we can write (see To et al., 2007 for more
details)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cosðnÞ þ m sin ðn=2Þ
QðmÞ cosðn=2Þ
þ cosðn=2ÞR½sin2 ðn=2Þ m sin2 ðn=2Þ; ð21Þ
pffiffiffiffiffiffiffiffiffiffiffiffi
qðnÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ C ln 1m
m sin2 ðn=2Þ
where QðmÞ is a function of m; C is a constant and R½sin2 ðn=2Þ is a polynomial function of sin2 ðn=2Þ whose
coefficients depend on H n ðn ¼ 0; 1; . . . ; maxðk; lÞÞ and m. The coefficients H n and m are determined by the
system of equations:
8
>
1 ¼ 1=2;
<H
Ra
ð22Þ
qðnÞ cos nn dn ¼ H n with n ¼ 0; 1; . . . ; maxðk; lÞ;
0
>
:
QðmÞ ¼ 0:
The last condition QðmÞ ¼ 0 in (22) is derived from the condition ensuring non singularity of qðnÞ at n ¼ a
determined by Eq. (21).
To illustrate the above general results, we now consider two important particular cases and present the corresponding results.
Case 1: k ¼ 2 and l ¼ 0
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
3941
From the system (17), we deduce a0 ¼ H 0 =a. The normalized contact pressure admits the closed-form
solution
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos
n=2
þ
m sin2 n=2 8b2 H 2
k þ a0 c2 pÞ þ
pffiffiffiffiffiffiffiffiffiffiffiffi
ln cos n=2 sin2 n=2
qðnÞ ¼ p
p
1
m
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 cos n=2
m sin2 n=2 m sin2 n=2½2ð1 mÞb2 H 2 1 c2 ;
p
2ðb0 H 0
ð23Þ
where the contact angle a expressed in terms of m and the coefficients H 0 and H 2 are specified upon solving the
three nonlinear equations
8
mðb0 H 0 k þ c2 pa0 Þ þ 12 mð2 mÞð1 þ c2 Þ
>
>
>
>
2
>
>
2mð1 mÞ b2 H 2 ¼ 12 ;
>
>
>
< 1
2
2 mð2 3mÞðb0 H 0 k þ c2 pa0 Þ þ mð1 mÞ ð1 þ c2 Þ
>
>
12 mð4 14m þ 20m2 9m3 Þb2 H 2 ¼ H 2 ;
>
>
>
>
>
H 0 ðb0 H 0 k þ a0 c2 pÞ lnð1 mÞ ð1 þ c2 Þm
>
>
:
b2 H 2 mð3m 2Þ ¼ 0:
ð24Þ
Case 2: k ¼ 2 and l ¼ 1
In this case, from (17) we obtain
a0 ¼
ða þ sin a cos aÞH 0 sin a
;
a sin a cos a þ a2 2 sin2 a
a1 ¼
2 sin aH 0 a
:
a sin a cos a þ a2 2 sin2 a
Concerning the contact pressure, we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2ðb0 H 0 k þ a0 c2 pÞ cos n=2 þ m sin n=2 8b2 H 2
2
pffiffiffiffiffiffiffiffiffiffiffiffi
ln qðnÞ ¼ þ p cos n=2 sin n=2
p
1m
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 cos n=2
2
m sin n=2 m sin2 n=2½2ð1 mÞb2 H 2 1 c2 þ c2 pa1 :
p
The system of equations allowing us to compute a; H 0 ; H 2 is provided by
8
mðb0 H 0 k þ c2 pa0 Þ þ 12 mð2 mÞð1 þ c2 c2 pa1 Þ
>
>
>
>
>
>
2mð1 mÞ2 b2 H 2 ¼ 12 ;
>
>
>
< 1
2
2 mð2 3mÞðb0 H 0 k þ c2 pa0 Þ þ mð1 mÞ ð1 þ c2 c2 pa1 Þ
1
2
3
>
>
> 2 mð4 14m þ 20m 9m Þb2 H 2 ¼ H 2 ;
>
>
>
> H 0 ðb0 H 0 k þ a0 c2 pÞ lnð1 mÞ ð1 þ c2 c2 pa1 Þm
>
>
:
b2 H 2 mð3m 2Þ ¼ 0:
ð25Þ
ð26Þ
ð27Þ
Up to now, we have elaborated a general approach to obtain an approximate analytical solution to the
integral equation governing the conformal contact problem in a reinforced pin-loaded joint with a nonzero second Dundurs’ constant c2 6¼ 0. In particular, by setting c2 ¼ 0, we recover the solution provided
by our previous paper (To et al., 2007). In the next section, we apply our foregoing general solution
to the important special case of an infinite two-dimensional body by posing q ¼ 1 and compare the corresponding results with the relevant ones existing in the literature; to validate the general solution for a
two-dimensional body of finite breadth, it is confronted with the finite element solution obtained for a
real reinforced pin-loaded joint.
3942
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
4. Application and validation of the approximate analytical solution
4.1. Infinite ring q ¼ 1
The results derived in the previous section hold for any value q 21; þ1. The special case q ¼ 1 associated to an infinite ring is of important interest, since all the analytical results reported in the literature about
the conformal contact problem in a pin-loaded joint with c2 6¼ 0 are limited to this case and can be used for
comparison.
Setting q ¼ 1, the governing integral equation (9) is simplified enormously, because
b0 ¼ 1 þ c1
;
2
bn ¼ 0
for n P 2:
ð28Þ
In particular, accounting for these conditions in (23)–(27), we obtain the explicit results detailed below.
Case 1: k ¼ 2 and l ¼ 0
Recalling a0 ¼ H 0 =a, using (28) in (23) and (24) and noting that the second equation in (24) is redundant for
the case under consideration, we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðb0 H 0 k þ H 0 c2 p=aÞ cos n=2 þ m sin n=2 2ð1 þ c2 Þ
pffiffiffiffiffiffiffiffiffiffiffiffi
ln cos
n=2
m sin2 n=2; ð29Þ
qðnÞ ¼ p
p
1m
(
mðb0 H 0 k þ c2 pH 0 =aÞ þ mð2mÞ
ð1 þ c2 Þ ¼ 12 ;
2
H 0 ðb0 H 0 k þ H 0 c2 p=aÞ lnð1 mÞ mð1 þ c2 Þ ¼ 0:
Combining the equations in (30) leads to the nonlinear equation characterizing a:
"
#
ð1 mÞ2 1 þ c1 ð1 mÞ2
þ
lnð1 mÞ m
kþ
2m
2
2m
"
#
2
ð2 mÞpc2 ð1 mÞ p ð1 þ c1 Þð2 mÞ
pc2 p 1 þ c1 2 m
þ c2 lnð1 mÞ
þ þ c2 m
¼0
2ma
4
a
2m
2a
a
2
ð30Þ
ð31Þ
with m ¼ sin2 a=2. Note that the first three terms in the left-hand member of this equation do not involve the
second Dundurs’ constant c2 .
Case 2: k ¼ 2 and l ¼ 1
In this case, it is convenient to rewrite (25) in the form
a0 ¼ b0 H 0 þ c 0 ;
a1 ¼ b1 H 0 þ c 1
ð32Þ
with
a þ sin a cos a
;
a sin a cos a þ a2 2 sin2 a
2 sin a
b1 ¼
;
a sin a cos a þ a2 2 sin2 a
b0 ¼
sin a
;
a sin a cos a þ a2 2 sin2 a
a
c1 ¼ :
a sin a cos a þ a2 2 sin2 a
c0 ¼ ð33Þ
Then, introducing (28) into (26) and (27) and observing that the second equation in (27) is not useful for the
case under consideration, it follows that
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2ðb0 H 0 k þ a0 c2 pÞ cos n=2 þ m sin n=2 2ð1 þ c2 a1 c2 pÞ
pffiffiffiffiffiffiffiffiffiffiffiffi
ln cos n=2
qðnÞ ¼ p
p
1m
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m sin2 n=2;
ð34Þ
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
(
mðb0 H 0 k þ c2 pa0 Þ þ mð2mÞ
ð1 þ c2 c2 pa1 Þ ¼ 12 ;
2
H 0 ðb0 H 0 k þ a0 c2 pÞ lnð1 mÞ mð1 þ c2 c2 pa1 Þ ¼ 0:
3943
ð35Þ
Combining (32) and (35) yields the following nonlinear equation for a:
"
#
ð1 mÞ2 1 þ c1 ð1 mÞ2
kþ
þ
lnð1 mÞ m
2m
2
2m
þ c2 ½D1 D5 þ D2 D3 D7 D4 D6 þ c2 ðD2 D5 D4 D7 Þ ¼ 0
ð36Þ
with
2
ðm 1Þ
ð2 mÞ
ð1 pc1 Þ;
k; D2 ¼ pc0 2
2m
1 þ c1
ð2 mÞ
pb1 ;
; D4 ¼ pb0 þ
D3 ¼ b0 ¼ 2
2
ð2 mÞ
lnð1 mÞ þ m pb1 ;
D5 ¼
2
D1 ¼
2
ð1 mÞ
lnð1 mÞ m;
2m
ð2 mÞ
lnð1 mÞ þ m ðpc1 1Þ:
D7 ¼
2
D6 ¼
ð37Þ
Observe that the first three terms in the left-hand side of (36) are identical to those in the left-hand side of (31)
and do not include c2 .
In Ciavarella and Decuzzi (2001b), the equation characterizing the contact angle a takes the form
E0 DR 2 ðc1 þ 1Þ½lnðb2 þ 1Þ þ 2b4 4c2
¼
:
þ
F
pð1 c1 Þ
pð1 c1 Þðb2 þ 1Þb2
ð38Þ
Above, b ¼ tanða=2Þ; E0 ¼ E0 =ð1 m20 Þ for plane strain and E0 ¼ E0 for plane stress. It is convenient to express E0 in terms of l0 and j0 as
E0 ¼
8l0
;
j0 þ 1
ð39Þ
which is valid both for the plane strain and stress cases. In addition, the left-hand member of (38) is related to
the quantity k in Eq. (10) by
DRE0
4k
¼
:
ð1 c1 Þp
F
Remark that b2 ¼ tan2 ða=2Þ can be also expressed in terms of m ¼ sin2 ða=2Þ by
m
:
b2 ¼
1m
Finally, the formula (38) of Ciavarella and Decuzzi (2001b) can be recast into
"
#
ð1 mÞ2 1 þ c1 ð1 mÞ2
k þ
þ
lnð1 mÞ m þ c2 ¼ 0:
2m
2
2m
ð40Þ
ð41Þ
ð42Þ
It is interesting to note that this equation is different from (31) or (36) only in the terms involving c2 . In other
words, when c2 ¼ 0, (31), (36) and (42) reduce to the exact nonlinear equation for a with k and c1 as the two
parameters, i.e.
"
#
ð1 mÞ2 1 þ c1 ð1 mÞ2
k þ
þ
lnð1 mÞ m ¼ 0;
ð43Þ
2m
2
2m
3944
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
when c2 6¼ 0, they are different and have k, c1 and c2 as the three parameters.
To demonstrate the robustness of our approximate analytical solution, we compare it with the numerical
solution and approximate analytical solution given by Ciavarella and Decuzzi (2001b). In Table 1, the contact
angle a ¼ 22:62 is fixed in advance but k and q0 ¼ qð0Þ are computed. From Table 1, it is seen that our solution is very accurate in comparison with the numerical results of Table 2 in Ciavarella and Decuzzi (2001b), in
particular when l ¼ 1. In Table 2, k ¼ 0 (neat fit contact) is considered and the contact angle a is calculated. It
is remarked that the results obtained by the present method with l ¼ 1 are closer to the numerical results than
those provided by the approximate method of Ciavarella and Decuzzi (2001b). In this sense, our approximate
analytical method improves the one of Ciavarella and Decuzzi (2001b).
The variation of the contact angle a with the normalized loading parameter k is illustrated in Fig. 3 with one
extreme value c1 ¼ 1 of c1 and in Fig. 4 with another extreme value c1 ¼ 1 of c1 . In each of these two figures,
the dependence of a on c2 is shown by taking c2 ¼ 0 and c2 ¼ 0:5. In addition to the curves corresponding to
the solutions of our Eqs. (31) and (36), to the solution of Eq. (42) of Ciavarella and Decuzzi (2001b) and to the
solution of (43), the Hertzian contact curve is also plotted in Figs. 3 and 4 according to the formula
rffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
2
F
j1 þ 1 j0 þ 1
¼
a¼
þ
:
ð44Þ
k
2pDR
l1
l0
Table 1
Comparison between the present solution and the numerical solution given in Table 2 of Ciavarella and Decuzzi (2001b)
c1
0.5
1/3
0
1/3
0.5
c2
a
0.175
0.117
0
0.117
0.175
knum
22.62
22.62
22.62
22.62
22.62
12.32
12.14
11.77
11.44
11.26
q0num
1.639
1.641
1.645
1.649
1.651
l¼0
l¼1
kðl¼0Þ
q0ðl¼0Þ
kðl¼1Þ
q0ðl¼1Þ
12.18
12.04
11.76
11.49
11.35
1.623
1.631
1.645
1.660
1.667
12.23
12.08
11.76
11.45
11.29
1.638
1.641
1.645
1.650
1.652
Table 2
Comparison between the present solution and the numerical solution and approximative solution given in Table 3 of Ciavarella and
Decuzzi (2001b) (C–D)
c1
c2
k
anum
1
0
0
1
0.50
0.25
0.25
0.50
0
0
0
0
87.76
98.02
75.73
80.96
C–D
Present
aCD
aðl¼0Þ
aðl¼1Þ
94.85
103.9
74.14
76.32
83.94
94.34
77.25
85.67
87.69
97.98
75.75
81.01
Fig. 2. A quarter of the pin-loaded joint used in the numerical analysis.
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
3945
100
Contact angle α [°]
80
C-D, gamma2=0.5
Present l=0, gamma2=0.5
Present l=1, gamma2=0.5
Analytic gamma2=0
Hertzian contact
60
40
20
0
1
2
3
4
5
6
7
8
9
10
Coefficient λ
Fig. 3. Relation between coefficient k and contact angle a (case c1 ¼ 1; c2 ¼ 0 and 0.5).
100
Contact angle α [°]
80
C-D, gamma2=-0.5
Present l=0, gamma2=-0.5
Present l=1, gamma2=-0.5
Analytic gamma2=0
Hertzian contact
60
40
20
0
1
2
3
4
5
6
7
8
9
10
Coefficient λ
Fig. 4. Relation between coefficient k and contact angle a (case c1 ¼ 1; c2 ¼ 0 and 0.5).
The derivation of this formula can be found in Johnson (1985), Hills et al. (1993) or Ciavarella and Decuzzi
(2001a).
It is seen from Figs. 3 and 4 that, when k is relatively large, the contact angle is small and all the curves are
very close to each other, so that the Hertzian contact regime prevails as expected. Physically, this situation can
be achieved by a small force F or/and a large clearance DR. However, when k has a small value which occurs
for a large force F or/and a small clearance DR, the curve associated to the Hertzian contact is quite far from
the other curves. According to the Hertz formula (44), the contact angle tends to infinity as k goes to 0, which
is physically inadmissible.
From Figs. 3 and 4, we also see that: (i) the effect of the second Dundurs’ constant c2 on the contact angle a
is negligible when k is large or equivalently when a is small; (ii) the effect becomes very important when k is
small or equivalently when a is large. This conclusion refines the relevant one made by Ciavarella and Decuzzi
(2001b). The three curves corresponding to our Eqs. (31) and (36) and to Eq. (42) of Ciavarella and Decuzzi
(2001b) are close to each other, though our approach is different from theirs. Our approach, based on a direct
3946
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
approximate solution of the governing integral equation, appears to be more accurate (see Table 1). However,
our expressions for the contact pressure and angle are more complicated than those given by Ciavarella and
Decuzzi (2001b). At the same time, our approach has the definitive advantage of being applicable to the case
of a finite joint with c2 6¼ 0, which has not been treated up to now.
4.2. Finite ring q < 1
In this case, bn 6¼ 0 for n P 2. The accuracy of the solution depends on the choice of both k and l. To evaluate the performance of the approximate formula, we consider a realistic case where the bolt and the ring are
made of stainless steel (Steel) and aluminium (Al) respectively, and vice versa (see Table 3). In regard to finiteness parameter q, two values are considered: q ¼ 4 when R2 ¼ 2R1 and q ¼ 9 when R2 ¼ 3R1 . The analytic
solution is first obtained by the general method presented in the previous section and then compared with
the result by the finite element method. The parameters used by the analytical solution are presented in Table
4 (case l ¼ 0) and Table 5 (case l ¼ 1).
Because the contact pressure does not depend on the geometry of the glass plate and the soft layer as long
as the rigidity of the latter is sufficiently soft, we consider a case of a square glass plate of dimension
200 200 19 mm as an example. The following geometrical and mechanical parameters are used:
– Geometric parameters: R0 ¼ 15 mm; R1 ¼ 15 mm; R2 ¼ 30 mm or 45 mm, R3 ¼ 60 mm; L ¼ 200 mm
(width and length of the glass plate), e ¼ 19 mm (thickness of the glass plate);
– Soft layer ðR2 6 r 6 R3 Þ: E2 ¼ 0:5 GPa; m2 ¼ 0:2;
– Glass plate (R3 6 r and jxj 6 L=2 and jyj 6 L=2Þ: R3 ¼ 60 mm, E3 ¼ 70 GPa, m3 ¼ 0:2;
– Total force applied at the center of the bolt: P ¼ P x ¼ 19 kN;
Table 3
Recapitulation of the mechanical properties involved in the governing equation
E0
Al–steel
70
Steel–Al
200
m0
E1
m1
c1
c2
0.33
200
0.3
0.481
0.157
0.3
70
0.33
0.481
0.157
Table 4
Coefficients calculated for the case l ¼ 0
c1
c2
q
b0
b2
a
H0
H2
a0
Al–steel
0.481
0.157
9
4
0.292
0.345
0.494
1.882
89.31
90.76
0.680
0.701
0.144
0.113
0.436
0.443
Steel–Al
0.481
0.157
9
4
0.833
0.987
1.412
5.377
84.99
87.16
0.668
0.704
0.157
0.100
0.450
0.423
Table 5
Coefficients calculated for the case l ¼ 1
c1
c2
q
b0
b2
a
H0
H2
a0
a1
Al–steel
0.481
0.157
9
4
0.292
0.345
0.494
1.882
88.09
90.08
0.671
0.695
0.157
0.121
0.131
0.195
0.470
0.388
Steel–Al
0.481
0.157
9
4
0.833
0.987
1.412
5.377
85.56
87.34
0.672
0.706
0.150
0.097
0.166
0.275
0.426
0.286
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
3947
– Force per thickness is equal F ¼ P =e ¼ 1 (kN/mm);
– Boundary conditions: ux ðx ¼ L=2; yÞ ¼ 0; uy ðx; y ¼ 0Þ ¼ 0.
The glass structure is analyzed under the plane stress assumption and using its geometric symmetry. The
model and its boundary conditions are shown in Fig. 2 and is identical to the one depicted by Fig. 6 in To
et al. (2007).
The comparison between the analytic solution and the numerical solution given by FEM is presented in
Figs 5–8. This comparison leads to the following two comments:
– The approximate analytical solution is in good agreement with the numerical one for q ¼ 4 and q ¼ 9.
– As q decreases and c1 increases, the difference between the analytical and numerical solutions
becomes nonP1
negligible (see Fig. 8). This can be explained by the fact the coefficient bn in the series n¼2 bn H n sin nh= sin h
is function of both ð1 þ c1 Þ and q.PTo improve the accuracy in such a case, the number k should be
increased, i.e. more initial terms in 1
n¼2 bn H n sin nh= sin h have to be used.
Normalized contact pressure
0.6
0.5
q=pR/F
0.4
0.3
FEM
Analytic l=1
Analytic l=0
0.2
0.1
90
81
72
63
54
45
36
27
18
9
0
0.0
Angle (°)
Fig. 5. Contact pressure distribution for the case of aluminium pin, steel ring and R2 ¼ 45 mm.
Normalized contact pressure
0.6
0.5
q=pR/F
0.4
0.3
FEM
Analytic l=1
Analytic l=0
0.2
0.1
90
81
72
63
54
45
36
27
18
9
0
0.0
Angle (°)
Fig. 6. Contact pressure distribution for the case of steel pin, aluminium ring and R2 ¼ 45 mm.
3948
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
Normalized contact pressure
0.6
0.5
pR/F
0.4
0.3
FEM
Analytic l=1
Analytic l=0
0.2
0.1
90
81
72
63
54
45
36
27
18
9
0
0.0
Angle (°)
Fig. 7. Contact pressure distribution for the case of aluminium pin, steel ring and R2 ¼ 30 mm.
5. Closing remarks
In this paper, an approximate analytical method has been proposed to derive closed-form solutions to the
conformal contact problem in a reinforced pin-loaded joint in the case of elastic dissimilarity. This method,
quite different from that used by Ciavarella and Decuzzi (2001b) to treat the latter, is based on the approximation of the terms involving the second Dundurs’ material constant c2 by a finite series and on the minimization of the error function and has led to very accurate analytical results in comparison with the available
numerical ones. The work presented above has satisfactorily completed the previous work (To et al., 2007).
However, as pointed out in the latter, we believe that taking into account the friction between the pin and ring
could be only done numerically (see e.g Renaud and Feng, 2003; Iyer, 2001; Lin and Lin, 1999; Hyer and
Klang, 1985).
Normalized contact pressure
0.6
0.5
pR/F
0.4
0.3
FEM
Analytic l=1
Analytic l=0
0.2
0.1
90
81
72
63
54
45
36
27
18
9
0
0.0
Angle (°)
Fig. 8. Contact pressure distribution for the case of steel pin, aluminium ring and R2 ¼ 30 mm.
Q.D. To, Q.-C. He / International Journal of Solids and Structures 45 (2008) 3935–3950
3949
The elastic conforming contact problem in a ball-loaded structure is the three-dimensional counterpart of
the one in a pin-loaded structure. This problem is of both theoretical and practical importance (see, e.g., Ciavarella et al., 2007). Solving this three-dimensional problem in a closed-form way would constitute a challenge.
Appendix A
Let us show that Eq. (9) can be recast into a generalized Prandtl equation (see Gori et al., 1998). First, we
make the following variable change:
x ¼ sin2 ðh=2Þ=m;
t ¼ sin2 ðn=2Þ=m;
which implies that
h ¼ cos1 ð1 2xmÞ;
n ¼ cos1 ð1 2tmÞ;
2mdt
cos h cos n ¼ 2mðt xÞ; dn ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
2
1 ð1 2tmÞ
With the help of the above variable change, Eq. (9) becomes
Z x
Z 1
q1 ðtÞdt
þ gðxÞ
q1 ðtÞdt ¼ f ðxÞ;
tx
0
0
where
qðcos1 ð1 2tmÞÞ
q1 ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
1 ð1 2tmÞ2
f ðxÞ ¼ ðb0 H 0 kÞ
gðxÞ ¼ 2mc2 p
;
sin hðxÞ
1
X
hðxÞ
sin nhðxÞ
þ 2ð1 þ c2 ÞH 1 þ
:
bn H n
sin hðxÞ
sin hðxÞ
n¼2
Then, posing
Z x
q1 ðtÞ ¼ yðxÞ;
0
Eq. (9) takes the final form
Z 1 0
y ðtÞdt
þ gðxÞyðxÞ ¼ f ðxÞ;
tx
0
which is a generalized Prandtl equation.
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