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. References Barber, J.R., 2002. Elasticity. Kluwer Academic Publisher. Ciavarella, M., Strozzi, A., Baldini, A., Giacopini, M., 2007. Normalization of load and clearance eﬀects in ball-in-socket-like replacements. Proc. I. Mech. Eng. H: J. Eng. Med. 221, 601–611. Ciavarella, M., Baldini, A., Barber, J.R., Strozzi, A., 2006. 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