Spectral relationships for integral operators in ...

Applied Mathematics and Computation 118 (2001) 95±111 www.elsevier.com/locate/amc

Spectral relationships for integral operators in contact problem of impressing stamps M.A. Abdou Department of Mathematics, Alexandria University, Alexandria, Egypt

Abstract A method is given for solving the problem of mechanics of continuous media between a ®nite system of stamps varying width and an elastic half-space in a three dimensional formulation. The friction within the region of contact is neglected. The problem is solved in the Mathieu function form. Also, the problem of contact of impressing a system strip of stamps in an elastic half-space of a logarithmic series in the case of two symmetric intervals is solved. Some important relationships are investigated. Also the spectral relationships of the integral equations with Karlman kernel are obtained. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Contact problems; Mathieu function; Potential theory method; Integral equation of the ®rst kind; Logarithmic kernel

1. Introduction The mechanics mixed problems of continuous media have been studied by many authors (see [3,6,10,15]). Protsenko [14] used the potential theory method for solving the problem about the contact of a thin plate in the form of an in®nite strip lying on an elastic frictionless half-space in a three dimensional formulation. Popov [3] used the orthogonal polynomials method and its generalization, for the investigation of complex mixed problems of the mechanics of continuous media. In Ref. [2] Kovalenko studied the mixed problems of the mechanics of continuous media with boundary conditions speci®ed on a circle and he obtained a Fredholm integral equation of the ®rst kind with singular kernel. Aleksandrov and Pozharskii [12] studied the problems concerning the pressing of a thin linear inclusion in the form of a sti€ness rib into a plate having a wedge shape in planform. In Ref. [7] Abdou solved the three dimensional 0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 9 9 ) 0 0 1 6 7 - 8

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M.A. Abdou / Appl. Math. Comput. 118 (2001) 95±111

semi-symmetric Hertz contact problem of two rigid surfaces having two p di€erent elastic materials occupying the domain w ˆ fz ˆ 0; x2 ‡ y 2 < a2 g and the kernel takes the potential function form. Here, the method of orthogonal function applied for the investigation of complex mixed problems of the mechanics of continuous media, are based on the utilization of spectral relationships that invert the main part of the kernel of the integral equation of the problem under consideration. A suciently general approach to the derivation of spectral relationships that is based on potential theory is proposed. The kernel of the problem is represented in the Weber± Sonin integral formula. We prove that, the kernel of the consideration problem ful®lls a non-homogeneous wave equation. Also, here eigenfunctions are obtained in the problem of impressing a system strip stamps in an elastic halfspace of a logarithmic series in the case of two symmetric intervals. In the end, the spectral relationships of the integral equations with Karlman kernel are obtained. 2. Contact problem with potential kernel Consider a system of a three dimensional contact problem of frictionless impression of a system rigid stamps in the surface of an elastic half-space occupying the domain ÿ1 < x; y < 1; z > 0. The integral equation of such a problem [11] takes the form k Z Z X iˆ1



Xi

h

Pi …n; g†dn dg 2

…x ÿ n† ‡ …y ÿ g†

2

…x; y† 2 Xi ; h  G…1 ÿ m†

i1=2 ˆ 2phfi …x; y†

ÿ1



;

…2:1†

where fi …x; y† ˆ di ÿ ai x ‡ bi y ÿ fi …x; y†. Here, G is the displacement magnitude, m the Poisson's coecient, Pi …x; y† are the unknown normal stress under the stamps, Xi are the contact domain between the stamps and the half-plane surface. Also, fi …x; y† are the known functions describing the shape of the stamps base. While, di ÿ ai x ‡ bi y are the rigid displacements of the stamps under the action of forces Pi and moments Mix ; Miy . We valid Eq. (2.1) under the evident statics conditions Z Z Pi …x; y†dx dy ˆ Pi

…Pi constant†;

Xi

yPi …x; y†dx dy ˆ Mix ; Xi

…2:2†

Z Z

Z Z

Xi

xPi …x; y†dx dy ˆ Miy :

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The relationship between Pi ; Mix ; Miy and di ; ai ; bi can be determined from expressing the equilibrium of the stamps on the half-space, when ai ˆ bi ˆ 0 we set Mix ˆ Miy ˆ 0. Using the potential theory method [11], we introduce the simple layers potential of densities Pi …x; y† distributed over the domain Xi . k Z Z X Pi …n; g†dn dg q : …2:3† Vi …x; y; z† ˆ 2 2 iˆ1 …x ÿ n† ‡ …y ÿ g† ‡ z2 Xi The functions Vi …x; y; z† are harmonic everywhere except on a plane slit in the domain Xi and its vanish at in®nity as Pi Rÿ1 …Pi ˆ constant;  p 2 2 2 R ˆ x ‡ y ‡ z †. Moreover the functions Vi …x; y; z† are continuous in all space including the domain Xi and its normal derivatives take the form    oVi 2pPi …x; y†; …x; y† 2 Xi ; …2:4† ˆ 0; …x; y† 62 Xi : oz z!0 On the slits, we set the following boundary condition: Vi …x; y; 0† ˆ 2phfi …x; y†

……x; y† 2 Xi †:

…2:5†

The solution of the integral equation (2.1) is equivalent to the problem (2.3)± (2.5) of determining the harmonic functions Vi …x; y; z†. The contact problem concerning the frictionless impression of a system stamps of strip planform in an elastic half-space representing by (2.1), with the aid of Ref. [10], reduces to the integral equation k Z ai X /i …ki ; t†K0 …ki jt ÿ xj†dt ˆ phfi …ki ; x†; …2:6† iˆ1

ÿai

where K0 …t† are the Macdonald functions and ki are arbitrary positive number. The solution of the integral equation (2.6), reduces to the following boundary value problem (see (7.1)±(7.5) of [10, pp. 161])

and

o2 Vi o2 Vi ‡ 2 ÿ ki Vi ˆ 0 …x 62 Xi ; z 6ˆ 0†; ox2 oz  1 oVi …ki ; x; 0†  ‡ ph /i …ki ; x†; jxj < ai ; ˆ 0; jxj > ai oz Vi …ki ; x; 0† ˆ fi …ki ; x†; Vi …ki ; x; z† ! 0

jxj 6 ai ;

……x2 ‡ z2 † ! 1†;

where we assumed Vi …x; y† ˆ fi …ki ; x† coski y;

Pi …x; y† ˆ /i …ki ; x† cos ki y

…2:7† …2:8†

…2:9†

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and we introduced the potential functions  q k Z ai 1 X K0 ki …x ÿ t†2 ‡ z2 /i …ki ; t†dt: Vi …ki ; x; y† ˆ 2ph iˆ1 ÿai

…2:10†

Let us introduce the local Cartesian coordinates xi hi zi , placing their origins in the middle of the segments ‰ÿai ; ai Š …i ˆ 1; 2; . . . ; k† and let us perform the variable substitution xi ai xi and zi ai zi . Passing now in each local coordinate system xi ; hi ; zi to the elliptic cylinder coordinates according to the formulae xi ˆ ai cosh ni cosgi ;

zi ˆ ai sinh ni sin gi

…0 6 gi 6 2p; 0 6 ni < 1†: …2:11†

Using Eq. (2.11) in (2.6)±(2.9), we have o2 Vi o2 Vi ai k2i … cosh2ni ÿ cos 2gi †; ‡ 2 ÿ 2 ogi on2i

Vi ˆ 0

…2:12†

Vi …ki ; 1; gi † ˆ 0;

…2:13†

and Vi …ki ; 0; gi † ˆ fi …ki ; ai cosgi †; while Eq. (2.8) becomes /i …ki ; ai cos gi † ˆ

h oVi ai j sin gi j on nˆ0

…for all values of gi †:

…2:14†

Assuming the functions fi …ki ; x† to be such that it can be expanded into a uniformly convergent series of periodic Mathieu functions (see Ref. [9]) in the intervals ai 6 xi 6 ai   1 X k2i a2i fi …ki ; ai cosgi † ˆ cin Cen …gi ; ÿqi † qi ˆ …2:15† 4 nˆ0 we will seek the solution of (2.12) in the form Vi …ki ; ni ; gi † ˆ Ui …ni †Wi …gi †:

…2:16†

Using (2.16) in (2.12), we arrive to study the Mathieu equations Wi 00 ‡ …ai ‡ 2qi cos2gi † Ui00 ÿ …ai ‡ 2qi cosh 2ni † Ui …0†Wi …gi † ˆ fi …ki ; ai cosgi †;

Wi ˆ 0; Ui ˆ 0 …ai are constants†; Ui …1† ˆ 0:

…2:17†

Furthermore, following the general theory of Mathieu function [9], we write the general solution (2.16) of the boundary value problem (2.12) formulated to satisfy the condition of (2.17) in the form Vi …ki ; n; g† ˆ

1 X rin Fe Kn …n; ÿqi † Cen …g ÿ qi †: nˆ0

…2:18†

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The coecients rin are found from the set of in®nite system ri` ‡

n X 1 X rsn T`n …s; i† ˆ s6ˆi nˆ0

ci` ; Fe Kn …n; ÿqi †

Qn` …s; i†Ce` …0; ÿqi † ; T`n …s; i† ˆ Fe K` …0; ÿqi †

…2:19†

where cin are the coecients of expansion of the function of (2.15). Di€erentiating (2.18) and using the result in (2.14), we obtain /i …ki ai cos g† ˆ

1 X h Fe Kn0 …0; qi † cin Cen …g; ÿqi † ai j sin gj nˆ0 Fe Kn …0; ÿqi †

…2:20†

Using (2.20) in (2.6), we arrive at the following spectral relationship for the Mathieu functions: k Z ai Ce ‰ cos ÿ1 n ÿ q Š X n i ai q  K0 …ki jn ÿ xj†dn 2 2 iˆ0 ÿai ai ÿ n   pFe Kn …0; ÿqi † ÿ1 x Ce cos ; ÿ q …2:21† ˆ n i : Fe Kn0 …0; qi † ai Equation (2.21) represents the spectral relationships for a system of Fredholm integral equation of the ®rst kind with Macdonald kernel. 3. Weber±Sonin integral formula Using the polar coordinates Eq. (2.1) takes the form k Z ai Z p X Pi …q; /†q dq d/ p ˆ 2phfi …r; h†: 2 ‡ q2 ÿ 2rq cos…h ÿ /† r ÿp ÿa i iˆ1 To separate the variables, one assumes   cosmh; cosmh; fi …r; h† ˆ fim …r† Pi …q; h† ˆ Pim …r† sin mh; sin mh: Using (3.2) in (3.1), we have X Z ai Lm …r; q†Pim …q†q dq ˆ 2phgim …r†; iˆ0

ÿai

…3:1†

…3:2†

…3:3†

where Z Lm …r; q† ˆ

p

cos m/ d/ p : 2 r ‡ q2 ÿ 2rq cos / ÿp

…3:4†

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M.A. Abdou / Appl. Math. Comput. 118 (2001) 95±111

To write Eq. (3.4) in the Weber±Sonin form, ®rstly we use the following relation (3(a)): Z 2p cosm/ d/ 2p…a†m zm F …a; m ‡ a; m ‡ 1; z2 † ˆ …3:5† 2 †a m! …1 ÿ 2z cos / ‡ z 0 and   
4z ÿ2a 1 1 2 F a; a ‡ 2 ÿ b; b ‡ 2 ; z ˆ …1 ‡ z† F a; b; 2b; 1 ‡ z2   …3:6† C…m ‡ a† : jzj < 1 Re a > 0; …a†m ˆ C…a† Hence, the kernel (3.4) takes the form ! p 2 pC…m ‡ 1=2†…rq†m 4rq 1 1 F m ‡ 2 ; m ‡ 2 ; 2m ‡ 1; ; Lm …r; q† ˆ 2m‡1 2 m…r ‡ q† …r ‡ q† …3:7† where F …a; b; c; z† is the Gauss hypergeometric function and C…x† is the Gamma function. Secondly, using the famous relation (3(b)) Z 1 aa ba 2ÿb F …a ‡ …1 ÿ b†=2† Ja …ax†Ja …bx†xÿb dx ˆ 2aÿb‡1 C…a ‡ 1†C……1 ‡ b†=2† …a ‡ b† 0   …3:8†  F a ‡ …1 ÿ b†=2; a ‡ 1=2; 2a ‡ 1; …4ab=…a ‡ b†2 † : Eq. (3.7) takes the form Z 1 Jm …tq†Jm …tr†dt Lm …r; q† ˆ 2p 0

…Jm …x† is the Bessel function†:

Using the following notations p r q u ˆ ; v ˆ ; wi …r† ˆ rPim …r†; ai ai

p hi …r† ˆ h rgim …r†;

we can write Eq. (3.3), with the aid of (4.9), in the form k Z 1 X K…u; v†wi …u†du ˆ hi …v†; iˆ1

ÿ1

where k…u; v† ˆ

p uv

Z 0

1

Jm …tu†Jm …tv†dt:

…3:9†

…3:10†

…3:11†

…3:12†

Eq. (3.11) represents a system of Fredholm integral equation of the ®rst kind with kernel (3.12) in the form of Weber±Sonin integral formula.

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4. Partial di€erential equation To prove that the kernel (3.12) ful®ls a non-homogeneous wave equation, di€erentiate it with respect to u and v, respectively, we obtain     o o 1 1 1 ÿ K…u; v† ˆ ÿ K…u; v† ou ov 2 u v Z 1   p t Jm0 …tu†Jm …tv† ÿ Jm …tu†Jm0 …tv† dt: …4:1† ‡ uv 0

The second derivatives take the form    2    o o2 1 o 1o 3 1 1 ÿ K…u; v† ÿ ÿ K…u; v† ˆ ÿ K…u; v† ou2 ov2 u ou v ov 4 u2 v 2 Z  p 1 2  00 t Jm …tu†Jm …tv† ÿ Jm …tu†Jm00 …tv† dt: ‡ uv 0

…4:2† Using the relations (3(b)) n n Jn0 …z† ˆ Jnÿ1 …z† ÿ Jn …z† Jn0 …z† ˆ Jn …z† ÿ Jn‡1 …z†; z z in (4.1) and (4.2) we have  2  o o2 ÿ K…u; v† ˆ …h…u† ÿ h…v††K…u; v†; ou2 ov2 where

ÿ
h…x† ˆ m2 ÿ 14 xÿ2


…m 6ˆ  12 :

…4:3†

…4:4†

Eq. (4.4) represents a non-homogeneous wave equation.

5. Contact problem with logarithmic kernel Consider the system of the integral equation k Z X iˆ1

ai ÿai

k…t† ˆ

1 2

k

x ÿ t /i …t†dt ˆ pfi …x† k

Z

1 ÿ1

tanh u jut e du u

under the static conditions Z ai /i …t†dt ˆ P < 1: ÿai

…j ˆ

k 2 …0; 1†; p ÿ1†;

…5:1† …5:2†

…5:3†

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The boundary value problem (5.1)±(5.3) represents the contact problem of a strip occupying the region 0 6 y 6 h, made of material satis®es Hook's law (see Ref. [10]). The strip, in the absence of mass force, lies without fraction on a rigid support, a system of rectangular stamps is impressed into the boundary of a strip y ˆ h. Assume the frictional forces in the contact area between the stamps and the strip are small, so it can be neglected. Also, assume the width of the area of contact is independent of the magnitude of the force applied. As in Ref. [3, pp. 32], we can write the kernel in the form Z  1 1 tanh u jut pt  xÿy e du ˆ ÿ ln tanh ; k 2 …0; 1† : tˆ k…t† ˆ 2 ÿ1 u 4 k …5:4† If k ! 1 and …x ÿ y† is very small, so that it satis®es the condition tanh a ' a, then we have   pt 4k d ˆ ln : …5:5† ln th ˆ ln jtj ÿ d 4 p Here the kernel (5.2) takes the form k…t† ˆ ‰ÿ ln jx ÿ yj ‡ dŠ: So (5.1) becomes  k Z ai  X 1 ln ‡ d /i …t†dt ˆ pfi …x†; jx ÿ tj iˆ1 ÿai under the condition (5.3). We introduce the logarithmic potential function 2 3 Z k a i X 1 6 7 Ui …x; y† ˆ 4 log q ‡ d 5/i …t†dt: 2 ÿa 2 i iˆ1 …x ÿ t† ‡ y

…5:6†

…5:7†

Eqs. (5.7) and (5.3) reduce to the Dirichlet boundary value problem o2 o2 ‡ ……x; y† 62 …ÿai ; ai ††; ox2 oy 2 Ui …x; y†jyˆ0 ˆ pfi …x† …x 2 …ÿai ; ai ††;    p 1 r ˆ x2 ‡ y 2 ; Ui …x; y† ' P ln ‡ d r   1 P ln ‡ d ! finite term …as r ! 1†: r DUi …x; y† ˆ 0;

Dˆ

…5:8†

The solution of the integral equation (5.6) is equivalent to the solution of the Dirichlet problem (5.8). After the function Ui …x; y† in (5.8) has been constructed, the density of the potential /i …x† will be determined from the formula

M.A. Abdou / Appl. Math. Comput. 118 (2001) 95±111

/i …x† ˆ ÿ

1 oUi …x; y† lim sgn y
p y!0 oy

…x 2 …ÿai ; ai ††:

Assume the density source function   1 Wi …x; y† ˆ Ui …x; y† ÿ P ln ‡ d : r

103

…5:9†

…5:10†

So, Eq. (5.8) can be written as DWi …x; y† ˆ 0 ……x; y† 62 … ÿ ai ; ai ††; Wi …x; y†jyˆ0 ˆ pf …x† ÿ P … ln jxj ÿ d† … x 2 … ÿ ai ; ai ††; Wi …x; y† ! 0

…as r ! 1†:

Consequently Eq. (5.9) is transferred to   1 oWi …x; y† ÿ pP d…x† ; /i …x† ˆ ÿ lim sgn y
p y!0 oy

…5:11†

…5:12†

where d…x† is the Dirac-delta function. We construct the solution of the boundary value problem (5.11) by the method of conformal mapping (see Ref. [1]), that transforms a given complicated region into a simpler one. To this end, we note that the mapping function p ai ai …5:13† z ˆ W …n† ˆ …n ‡ nÿ1 † …n ˆ qe jh ; z ˆ x ‡ jy; j ˆ ÿ1† 2 2 maps the region in (x, y) plane into the region outside the unit circle c, such that w0 …n† does not vanish or becomes in®nite outside the unit circle c. The mapping function (5.13) maps the upper and the lower half-plane ……x; y† 2 …ÿai ; ai †† into the lower and the upper of the semi-circle q ˆ 1, respectively. Moreover the point z ˆ 1 will be mapped onto the point n ˆ 0. Using the parametric equation of (5.13) and under the condition (5.2), we can rewrite the density source of the logarithmic potential of Eq. (5.10) in the form   2q ‡d Wi …q; h† ˆ Ui0 …q; h† ÿ P ai       …5:14† ai 1 ai 1 q‡ qÿ coshi ; sin hi Ui0 …q; h† ˆ Ui 2 2 q q In view of Eq. (5.14) the boundary value problem of (5.11) is transformed to o2 Wi 1 oWi 1 o2 Wi ‡ ‡ ˆ 0 …q 6 1; ÿ p < h < p†; oq2 q oq q2 oh2

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Wi …0; hi † ˆ 0;



 2 Wi …1; hi † ˆ f0i …hi † ÿ P ln ‡ d ; a

f0i …hi † ˆ fi …ai cos hi †:

…5:15†

Consequently, after using the chain rule, Eq. (5.12) is transformed to   oWi : …5:16† /i …ai cos hi † ˆ …pai j sin hi j†ÿ1 P ‡ oq qˆ1 6. Fourier series method To solve the Dirichlet problem of (5.15), we use the Fourier series method (see Ref. [13]) Wi …q; hi † ˆ

1 X nˆ0

ain ˆ

1 p

Z

ain qn cos nhi ; p

ÿp

ÿp 6 hi 6 p;

fi0 …h† cosnh dh;

a0 ˆ

1 2p

Z

p

ÿp

fi0 …h†dh:

…6:1†

Substituting (6.1) in (5.15), then using the di€erentiating result in (5.16) (see Ref. [8]), we obtain  ÿ1 p coshi …pai sin hi † ; n ˆ 1; 2; . . . ; …6:2† /i …ai cos hi † ˆ ÿ1 P …pai sin hi † ; nˆ0 and Pˆ

  ÿ1 Z p 2 fi0 …h†dh: 2p ln ‡ d ai ÿp

Finally, substitute (6.2) in (5.1), we have the following relationship a   l Zi  X 1 Tn …t=ai † p… log 2=ai ‡ d†; n ˆ 0; ln ‡ d p dt ˆ p 2 T …x=ai †; n P 1; 2 jx ÿ tj ai ÿ t n n iˆ1 ÿai

where Tn …x† is the Chebyshev polynomial of the ®rst kind. We can derive some important relationships from Eq. (6.4): (i) If, n ˆ 2m, xi sin ni =2 ; ˆ ai sin ai =2 and if n ˆ 2m + 1,

ti sin gi =2 ˆ sin ai =2 ai

…6:3†

…6:4†

M.A. Abdou / Appl. Math. Comput. 118 (2001) 95±111

xi tan ni =2 ; ˆ ai tan ai =2

105

ti tan gi =2 ˆ ai tan ai =2

… ÿ ai < ni ; gi < ai ; ai ˆ p; m ˆ 0; 1; 2; . . . ; i ˆ 1; 2; . . . ; k†; the integral operator (6.4) becomes ! k Z ai X 1 ln ‡ d Pi …g†dg: KP ˆ j 2j sin nÿg iˆ1 ÿai 2

…6:5†

…6:6†

Hence, with the aid (6.5), the spectral relations (6.4) take the following form   " # sin g=2   k Z ai T cos…g=2†dg X 2m sin ai =2 1 sin n=2 p ˆ l2m T2m ln ‡d sin ai =2 j 2j sin nÿg 2… cos g ÿ cosai † iˆ1 ÿai 2 …6:7† and n Z X

!

ai

1 ‡d ln nÿg 2 sin iˆ1 ÿai 2   tan n=2 ; ˆ l2m‡1 T2m‡1 tan ai =2

where



l2m ˆ

  tan g=2 T2m‡1 tan cos…g=2†dg ai =2 p 2… cosg ÿ cosai †

ÿ1

p…2m† ; ÿp ln … sin ai =2 ‡ d †;

…6:8†

m P 1; mˆ0

and ÿ1

l2m‡1 ˆ p…2m ‡ 1† ;

m P 1:

…6:9†

(ii) Di€erentiating (6.4) with respect to x, we have k 1X p iˆ1 k Z X iˆ1

Z

X Tn …t=ai † dt p ˆ aÿ1 i Unÿ1 …x=ai † 2 2 t ÿ x ÿ t a ÿai iˆ0 i ai

aj

ÿaj

…n P 1†;

dt p ˆ 0 …jxj < ai †; …t ÿ x† a2i ÿ t2

where Un …x=a† is the Chebyshev polynomial of the second kind. Using (6.7), (6.8) in (6.10), we have k 1X 2 iˆ1

  tan g=2 g ÿ n Tn tan ai =2 cos …g=2† p dg cot 2 2… cosg ÿ cosai † ÿai

Z

ai

…6:10†

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M.A. Abdou / Appl. Math. Comput. 118 (2001) 95±111

8 0; n ˆ 0; jnj < ai ; > >   > < tan n=2 ˆ cosec…ai =2†U2mÿ1 tan ai =2 ; n ˆ 2m; m ˆ 1; 2; . . . ; >   >   > : cosec…ai =2†U2mÿ1 tan n=2 ‡ …ÿ1†m sin ai tan ai 2mÿ2 ; n ˆ 2m ÿ 1 tan ai =2 1‡ cos ai =2 4 …6:11† and   tan g=2 cot gÿn Tn tan sec g=2 2 ai =2 p dg 2… cos g ÿ cos ai † ÿai   8 ÿ
tan n=2 < cosec…ai =2† sec2 n2 Unÿ1 tan …n P 1†; ai =2 ˆ ÿ
: sec a2i tan n2 ; n ˆ 0; jnj < ai :

k 1X 2 iˆ1

Z

ai

…6:12†

The reader must know that, the previous relationships can be derived with the aid of the following relations: dt cos…n=2† cos …n=2† sin …a=2† sec…g=2† p ˆ
p dn † a sin …gÿn 2… cos g ÿ cosa† …t ÿ x† a2 ÿ t2 2 and

…6:13†

    g n nÿg g g cos ˆ cos ‡ cos ; 2 2 2 2 2       g n n gÿn n cos ˆ cos cos ‡ : cos 2 2 2 2 2 cos

7. Contact problem with Karlman kernel Consider a system, in a half-space, of contact problem of frictionless impression of a system rigid stamps in the surface of elastic whose modulus changes according to a power low. Such problem [10] can reduce to the following integral equations n Z ai X /i …t†dt …0 6 l < 1†; …7:1† l ˆ fi …x† jx ÿ tj iˆ1 ÿai under the conditions Z ai /i …y†dy ˆ P < 1; ÿai

…7:2†

M.A. Abdou / Appl. Math. Comput. 118 (2001) 95±111

we introduce, the general Karlman potential function k Z ai X /i …t†dt h il=2 ˆ Ui …x; y†: iˆ1 ÿai …x ÿ t†2 ‡ y 2

107

…7:3†

The solution of Eq. (7.1) with (7.2) reduces to the boundary value problems   l oUi o2 o2 ˆ0 …x; y† 62 … ÿ ai ; ai †; D ˆ 2 ‡ 2 ; DUi ‡ ox oy y oy Ui …x; y†yˆ0 ˆ fi …x†; Ui …x; y† ' Prÿl

…r ! 1; r ˆ

p x2 ‡ y 2 †

…7:4†

The complete solution of (7.1) is given by l

/i …x† ˆ k lim sin yjyj y!0

oUi ; oy

C…l=2† x 2 …ÿai ; ai †; k ˆ p 2 pC……1 ‡ l†=2†

…7:5†

where C(t) is the Gamma function To obtain the solution of the boundary value problems (7.4), ®rstly, we assume Ui …x; y† ˆ jyj

ÿl=2

Vi …x; y†;

…7:6†

to eliminate the term oUi =oy. Secondly we use the transformations mapping (5.13), to obtain the boundary value problem in polar coordinates in the form " # 1 1 ‡ 2 Vi0 …q < 1† DVi0 …q; h† ‡ l…2 ÿ l† 2 4q sin 2 hi …q2 ÿ 1† ÿl=2    ai 1 qÿ sin hi Vi0 …q; hi †jqˆ1 ˆ fi …ai cos hi † 2 q Vi0 …q; h†jqˆ0 ˆ 0 where



Vi0 …q; hi † ˆ Vi

…ÿp < hi 6 p†;

…7:7†

     ai 1 ai 1 q‡ qÿ cos hi ; sin hi ˆ Vi …x; y† q q 2 2

and Dˆ

o2 1 o 1 o2 ‡ ‡ : oq2 q oq q2 oh2

…7:8†

The solution of (7.7) can be obtained by separating the variables, for this aim, we assume Vi0 …q; h† ˆ R…q†wi …h†:

…7:9†

108

M.A. Abdou / Appl. Math. Comput. 118 (2001) 95±111

Hence, we have

" # d2 Ri dRi q2 2 ‡ l…2 ÿ l† ‡q ÿ a i Ri ˆ 0 q dq do2 …q2 ÿ 1†2 2

…0 6 q 6 1†

…7:10†

and   d2 wi l…2 ÿ l† 2 ‡ ai ‡ wi ˆ 0 dh2 4 sin 2 h

…ÿp < hi 6 p†;

…7:11†

where a2i are the constants of separation. As in Ref. [9] the general solution of (7.10) and (7.11), respectively take the form Ri …q† ˆ qn‡l=2 …1 ÿ q2 †

l=2

F …l=2; n ‡ l; n ‡ 1 ‡ l=2; q2 †;

…R…0† ˆ 0; 0 6 q < 1; n ˆ 0; 1; 2; . . . ; n ˆ ai ÿ l=2†

…7:12†

and wi …h† ˆ jai sin hi j

l=2

Cnl=2 … cos hi † …ÿp < hi 6 p; n ˆ 0; 1; 2 . . .†:

…7:13†

Cnl=2 …t†

is a Gegenbauer where F …a; b; c; z† is the hypergeometric function, while polynomial. Using (7.12) and (7.13) in (7.6), with the aid (7.9), we have Ui0 …q; hi † ˆ qn‡l F …l=2; n ‡ l; n ‡ 1 ‡ l=2; q2 †Cnl=2 … coshi †;       1 1 1 1 q‡ qÿ coshi ; sin hi ˆ Ui …x; y†; Ui0 …q; h† ˆ U 2 q 2 q …0 6 q < 1; ÿp < hi < p; n ˆ 0; 1; 2; . . .†

…7:14†

The complete solution of the problem, can be obtained as ®rstly writing Eq. (7.5) in polar coordinates /…ai cos hi † ˆ

C…l=2†…ai sin hi †lÿ1 l oUi0 ; p l‡1  1‡l  lim…1 ÿ q2 † q!1 oq p2 C 2

0 < hi < p:

…7:15†

Secondly using (7.14) after di€erentiating with respect q in (7.15), ®nally, we have C…l†C…n ‡ 1 ‡ l=2† lÿ1 …ai sin hi † Cnl=2 … cos hi †; /i …ai cos hi † ˆ p l p2 C‰…1 ‡ l†=2ŠC…n ‡ l† 0 < hi < p:

…7:16†

Hence, inserting (7.16) in (7.1), after using the second equations in (7.4), we obtain the following spectral relationships:

M.A. Abdou / Appl. Math. Comput. 118 (2001) 95±111 k Z X iˆ1

ai ÿai

Cnl=2 …t=ai †dt l

…1ÿl†=2

jx ÿ tj …a2i ÿ t2 †

ˆ kn Cnl=2 …x=ai †;

kn ˆ pC…n ‡ l†‰n!C…l† cos …pl=2†Šÿ1

109

jxj < ai ;

…n P 0†:

…7:17†

8. Conclusions From the above results and discussions, the following may be concluded: (1) For the contact problems, when a system of impressing stamps lay on a strip of an elastic material and under certain conditions, most of its reduce to a system of integral equations of the ®rst kind with di€erent kernel k Z ai X K…u; v†/i …v†dv ˆ fi …u†; K/ ˆ iˆ1

ÿai

K : L2 …ÿai ; ai † ! L2 …ÿai ; ai †;

…8:1†

where K…u; v† 2 C …‰ ÿ ai ; ai Š  ‰ ÿ ai ; ai Š†;

i ˆ 1; . . . ; k

and the discontinuous kernel of (8.1) must satisfy the relation Z ai Z ai k 2 …u; v†du dv ˆ m2 < 1: ÿai

ÿai

Also the unknown potential functions /i are continuous in the domain of integration and satisfy the normal condition. Moreover the unknown potential functions satisfy the Lipschitz condition with respect to the second argument. (2) The integral operator (8.1) is a positive compact and self-adjoint operator, so we may write K/ ˆ kn /n where kn and /n are the eigenvalues and the eigen functions of the integral operator, respectively. (3) A system of a three dimensional contact problem of frictionless impression of rigid stamps in the surface of an elastic (G, m) half-space occupying the domain ÿ1 < x; y < 1; z ˆ 0 reduces to an integral equation of the ®rst kind with symmetric kernel. (4) The potential function kernel reduces to the Weber±Sonin integral formula Z p 1 Jm …tu†Jm …tv†dt; K…u; v† ˆ uv 0

which represents a non-homogeneous wave equation (see Eq. (4.4)). (5) The value of the kernel (4.12) can be represented in the Legender polynomials as follows:

110

M.A. Abdou / Appl. Math. Comput. 118 (2001) 95±111

K…u; v† ˆ

1 …uv†m‡1=2 X C…n ‡ m ‡ 1=2†Pnm …u†Pnm …v† ÿ1 2 2 nˆ0 C …1 ‡ m ‡ n†…m ‡ 2n ‡ 1=2†

…Pnm …x† is Legender polynomial;

n ˆ 0; 1; 2; . . .†

…8:2†

(6) The contact problem of the zero harmonic symmetric kernel of the potential function is included as a special case when m ˆ 0. Also, the contact problem of the ®rst and higher order …m ˆ 1; 2; . . .† harmonic is included as a special case. (7) A system of plane contact problem of the impression of two symmetrically arranged stamps in an elastic half-space with logarithmic kernel, or with Karlman kernel, represents a Fredholm integral equation of the ®rst kind and its kernel takes the form Z p 1 m t J1=2 …tu†J1=2 …tv†dt; 0 6 m < 1; …8:3† K…u; v† ˆ uv 0

where m ˆ 0 for logarithmic kernel and 0 6 m < 1 for Karlman kernel, also 1=2 for symmetric and skew-symmetric, respectively. (8) This paper can be considered as a generalization of the work of contact problem in continuous media which is discussed in Refs. [8,10,11]. 9. Unlinked references [4,5] References [1] A.G. Sveshnikov, A.N. Tkhonov, The Theory of Functions of Complex Variable, Mir, Moscow, 1982. [2] E.V. Kovalenko, Some approximate method of solving integral equations of mixed problems, Prkil. Math. Mech. 53 (1) (1989) 85±92. [3] G.Ya. Popov, Contact Problems for a Linearly Deformable Base, Kiev-Odessa, 1982. [4] H. Bateman, A. Erdely, Higher Transcendental Functions, vol. 1, Nauka, Moscow, 1963. [5] H. Bateman, A. Erdely, Higher Transcendental Functions, vol. 2, Nauka, Moscow, 1973. [6] I.Ya. Shtearman, Contact Problem of Elasticity Theory, Moscow, 1949. [7] M.A. Abdou, Fredholm integral equation of the second kind with potential kernel, J. Comp. Appl. Math. 72 (1996) 161±167. [8] M.A. Abdou, S.A. Hassan, Boundary value of a contact problem, PV. M.A. 5 (3) (1994) 311± 316. [9] N.V. Machalan, Theory and Application of Mathieu Functions, Mir, Moscow, 1954. [10] V.M. Aleksandrov, E.V. Kovalenko, Problems in The Mechanics of Continuous Media With Mixed Boundary Conditions, Nauka, Moscow, 1986. [11] V.M. Aleksandrov, Development of The Theory of Contact Problems in The USSR, Nauka, Moscow, 1976.

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[12] V.M. Aleksandrov, D.A. Pozharskii, On contact problems for wedge-shaped plates, J. Appl. Math. Mech. 55 (1) (1991) 114±119. [13] V.S. Vladimirov, Equations of Mathematical Physics, Mir, Moscow, 1984. [14] V.S. Prostenko, V.G. Protsenko, Contact problem with several stamps, Dokl. Akad. Nauk. Ukrssr Ser. A, 10, 1972. [15] V.S. Prostenko, V.L. Rvachev, Plate in the form of an in®nite strip on an elastic half-space, Prkil. Math. Mech. 40 (2) (1976) pp. 298±305.