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Research ArticleSolution of a Problem Linear Plane Elasticity with MixedBoundary Conditions by the Method of Boundary Integrals

Nahed S Hussein12

1 Department of Mathematics Faculty of Science Cairo University Giza Egypt2 Department of Mathematics Faculty of Science Taif University Saudi Arabia

Correspondence should be addressed to Nahed S Hussein husseinnahedyahoocom

Received 16 February 2014 Revised 6 April 2014 Accepted 6 April 2014 Published 8 May 2014

Academic Editor Kim M Liew

Copyright copy 2014 Nahed S Hussein This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A numerical boundary integral scheme is proposed for the solution to the system of eld equations of plane The stresses areprescribed on one-half of the circle while the displacements are givenThe considered problemwithmixed boundary conditions inthe circle is replaced by two problems with homogeneous boundary conditions one of each type having a common solution Theequations are reduced to a system of boundary integral equations which is then discretized in the usual way and the problem at thisstage is reduced to the solution to a rectangular linear system of algebraic equations The unknowns in this system of equations arethe boundary values of four harmonic functions which define the full elastic solution and the unknown boundary values of stressesor displacements on proper parts of the boundary On the basis of the obtained results it is inferred that a stress component has asingularity at each of the two separation points thought to be of logarithmic typeThe results are discussed and boundary plots aregivenWe have also calculated the unknown functions in the bulk directly from the given boundary conditions using the boundarycollocation method The obtained results in the bulk are discussed and three-dimensional plots are given A tentative form for thesingular solution is proposed and the corresponding singular stresses and displacements are plotted in the bulk The form of thesingular tangential stress is seen to be compatible with the boundary values obtained earlier The efficiency of the used numericalschemes is discussed

1 Introduction

The plane problem of the linear theory of elasticity hasreceived considerable attention long ago as being a simplifiedalternative to the more realistic three-dimensional problemsof practical interest A large class of two-dimensional prob-lems has been tackled using various analytical techniquesDue to the increasing mathematical difficulties encounteredin the theoretical studies of problems involving arbitraryboundary shapes or complicated boundary conditions manypurely numerical techniques have been developed in the pastfew decades which rely on finite difference or finite elementtechniques In both methods the natural boundary of thebody is usually replaced by an outer polygonal shape whichinvolves a multitude of corner points and necessarily addsor deletes parts to the region occupied by the body Thisin turn necessitates the application of boundary conditionson artificial boundaries a fact that introduces additional

inaccuracies into the solution Minimizing the error requireslarge computing times

Many of the disadvantages of the numerical techniquesare overcome by the use of alternative semianalytical treat-ments based on boundary integral formulations of the prob-lem Such approaches are usually classified under the generaltitle of boundary integral methods They have the advantageof reducing the volume of calculations by considering at onestage only the boundary values of the unknown functionsand then using them to find the complete solution inthe bulk In addition the procedure deals exclusively withthe real boundary of the medium (restricted though tocertain regularity conditions) and need not introduce arti-ficial boundaries An extensive account of integral equationmethods in potential theory and in elastostaticsmay be foundin [1 2] Natroshvili et al [3] give a brief review of boundaryintegral methods as applied to the theory of micropolarelasticity Constanda [4] investigates the use of integral

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 323178 11 pageshttpdxdoiorg1011552014323178

2 Mathematical Problems in Engineering

equations of the first kind in plane elasticity Atluri andZhu [5] present a meshless local Petrov-Galerkin approachfor solving problems of elastostatics Sladek et al [6] andRui et al [7] present meshless boundary integral methodsfor 2D elastodynamic problems Elliotis et al [8] present aboundary integral method for solving problems involving thebiharmonic equation with crack singularities Li et al [9]present a numerical solution for models of linear elastostaticsinvolving crack singularities

The solution to plane problems of elasticity for isotropicmedia with mixed boundary conditions is a difficult taskBoundary methods may be useful in providing such solu-tions especially when the geometry of the domain boundaryis not simple Several papers deal with such problemseither for Laplacersquos equation or for the biharmonic equationShmegera [10] finds exact solutions of nonstationary contactproblems of elastodynamics for a half-plane with frictioncondition in the contact zone in a closed form A newmethodof solution based on the use of Radon transform is usedSchiavone [11] presents integral solutions to mixed prob-lems in plane strain elasticity with microstructure Haller-Dintelmann et al [12] consider three-dimensional ellipticmodel problems for heterogeneous media including mixedboundary conditions Helsing [13] studies Laplacersquos equationunder mixed boundary conditions and their solution by anintegral equation method Problems of elasticity are alsoconsidered Lee et al [14 15] study singular solutions atcorners and cracks in linear elastostatics undermixed bound-ary conditions Explicit solutions are obtained Khuri [16]outlines a general method for finding well-posed boundary-value problems for linear equations of mixed elliptic andhyperbolic type which extends previous techniques Thismethod is then used to study a particular class of fullynonlinear mixed type equations

Abou-Dina and Ghaleb [17 18] proposed a method todeal with the static plane problems of elasticity in stressesfor homogeneous isotropic media occupying simply con-nected regions The method relies on the representation ofthe biharmonic stress function in terms of two harmonicfunctions and on the well-known integral representation ofharmonic functions expressed in real variablesThese authorsapplied theirmethod to a number of examples with boundaryconditions of the first or of the second type only but didnot consider mixed conditions Constanda [19] discussesKupradzersquos method of approximate solution in linear elas-ticity The same author [20] explains the advantages andconvenience of the use of real variables due to its generalityin dealing with the different forms of the boundary unlikethe approach based on the use of complex variables ldquowherethe essential ingredients of the solution must be constructedin full for every individual situationrdquo Abd-Alla et al [21]investigated the effect of the nonhomogeneity on the com-posite infinite cylinder of isotropic material Abd-Alla andFarhan [22] investigated the effect of the nonhomogeneity onthe composite infinite cylinder of orthotropicmaterials Abd-Alla et al [23] investigated the effect of magnetic field andnonhomogeneity on the radial vibrations in hollow rotatingelastic cylinder Abd-Alla et al [24] studied the propagationof Rayleigh waves in a rotating orthotropic material elastic

half-space under initial stress and gravity The extensiveliterature on the topic is now available and we can onlymention a few recent interesting investigations in [25ndash28]

In the present paper we propose a numerical schemefor the solution to a mixed boundary-value problem ofplane linear elasticity for homogeneous isotropic elasticbodies occupying a circular domain Part of the boundaryis subjected to a given pressure and the remaining partof the boundary is fixed The initial problem with mixedboundary conditions is replaced by two subproblems withhomogeneous boundary conditions one of each type havinga common solution Following the scheme presented in [17]the equations for each of these two subproblems are reducedto a system of boundary integral equations which are thendiscretized in the usual way and the problem at this stageis reduced to the solution to a linear system of algebraicequations The obtained results are thoroughly discussedand graphs are given In particular we put in evidencethe singular behavior of the tangential stress componentat the two separation boundary points Three-dimensionalplots for the stress function the stress components and thedisplacement components in the whole domain obtainedby the boundary collocation method are also provided Asingular solution is proposed the corresponding singularstresses and displacements are plotted The ensuing form ofthe singular tangential stress is seen to be compatible with theboundary values obtained earlier The efficiency of the usednumerical schemes is discussed All figures were producedusing Mathematica 70 software

2 Problem Formulation and Basic Equations

Let us consider 119863 as the circular region occupied by theisotropic elastic medium Its contour 119862 has parametricrepresentation

119909 = 119886 cos (120579) 119910 = 119886 sin (120579) 0 le 120579 lt 2120587 (1)

where (119909 119910) denote a pair of orthogonal Cartesian coordi-nates in the plane of 119863 with origin 119874 at the center of thecircle and 120579 is the polar angle measured from the 119909-axis Forapplication we will take 119886 = 1 Let n be the unit outwardsnormal to119862 and 120591 the unit vector tangent to119862 at any arbitrarypoint in the positive sense associated with 119862 One has

120591 =

120596i +

119910

120596j n =

119910

120596i minus 120596j (2)

where the dot over a symbol denotes differentiation withrespect to 120579 and

120596 = radic2 + 1199102 = 119886 = 1 (3)

The general equations of the linear theory of elasticity fora homogeneous and isotropic material are well establishedand may found in standard references In what followswe will quote these equations as presented in [17] withoutproof to be used throughout the paper In the absence ofbody forces the equations of equilibrium are automatically

Mathematical Problems in Engineering 3

satisfied if the identically nonvanishing stress components aredefined through the stress function 119880 by the relations

120590119909119909=1205972119880

1205971199102 120590

119910119910=1205972119880

1205971199092 120590

119909119910=1205972119880

120597119909120597119910 (4)

With respect to polar coordinates the stress componentsare

120590119903119903=1

119903

120597119880

120597119903+1

1199032

1205972119880

1205971205792 120590

120579120579=1205972119880

1205971199032

120590119903120579=1

1199032

120597119880

120597120579minus1

119903

1205972119880

120597119903120597120579

(5)

It is assumed that the stress function 119880 isin 1198624(119863) and thatits second order partial derivatives are univalued functions inthe whole region119863 Hookersquos law reads

120590119909119909=

V119864(1 + V) (1 minus 2V)

[120597119906

120597119909+120597V120597119910] +

119864

(1 + V)120597119906

120597119909

120590119910119910=

V119864(1 + V) (1 minus 2V)

[120597119906

120597119909+120597V120597119910] +

119864

(1 + V)120597119906

120597119910

120590119909119910=

119864

2 (1 + V)[120597119906

120597119910+120597V120597119909]

(6)

where 119864 and V are Youngrsquos modulus and Poissonrsquos ratiorespectively for the considered elastic medium

The compatibility condition for the solution to (6) forthe displacement components leads to the following homo-geneous biharmonic equation for the stress function 119880

Δ2119880 = 0 (7)

The stress function 119880 solving (7) is

119880 = 119909Φ + 119910Φ119888+ Ψ (8)

where Φ and Ψ are two harmonic functions the superscript1198881015840 denotes the harmonic conjugate and 119863 is the closureof 119863 Since the boundary integral representation is to beused it seems adequate to suppose from the outset thatthe functions Φ and Ψ and their conjugates belong to theclass of functions 1198622(119863) The following representation forthe mechanical displacement components may be easilydeduced

119864

1 + V119906 = minus

120597119880

120597119909+ 4 (1 minus V) Φ

119864

1 + V120592 = minus

120597119880

120597119910+ 4 (1 minus V) Φ119888

(9)

In terms of the harmonic functions Φ Φc and Ψ thestress and the displacement components are expressed asfollows

120590119909119909= 1199091205972Φ

1205971199102+ 2120597Φ119888

120597119910+ 1199101205972Φ119888

1205971199102+1205972Ψ

1205971199102 (10)

120590119910119910= 1199091205972Φ

1205971199092+ 2120597Φ

120597119909+ 1199101205972Φ119888

1205971199092+1205972Ψ

1205971199092 (11)

120590119909119910= minus 119909

1205972Φ

120597119909120597119910minus 1199101205972Φ119888

120597119909120597119910minus1205972Ψ

120597119909120597119910 (12)

119864

1 + V119906 = (3 minus 4V) Φ minus 119909

120597Φ

120597119909minus 119910120597Φ119888

120597119909minus120597Ψ

120597119909 (13)

119864

1 + V120592 = (3 minus 4V) Φ119888 minus 119909

120597Φ

120597119910minus 119910120597Φ119888

120597119910minus120597Ψ

120597119910 (14)

3 Boundary Integral Representation ofthe Basic Equations

In what follows we present the boundary integral represen-tation of the basic equations and boundary conditions to beused in the sequel We closely follow the guidelines of [17]

31 Boundary Integral Representation of Harmonic FunctionsLet us consider that 119891 isin 1198622(119863) is harmonic in 119863 We usethe well-known integral representation for 119891 at an arbitraryfield point (119909 119910) in 119863 in terms of the boundary values of thefunction 119891 and its complex conjugate 119891119888 in the form

119891 (119909 119910) =1

2120587∮119904

[119891 ( 119904)120597

120597 119899ln119877 + 119891119888 ( 119904) 120597

120597 119904ln119877]119889 119904 (15)

where 119877 is the distance between the point (119909 119910) in 119863and the current integration point (119909( 119904) 119910( 119904)) on 119878 Therepresentation of the conjugate function is given by

119891119888(119909 119910) =

1

2120587∮119904

[119891119888( 119904)120597

120597 119899ln119877 minus 119891 ( 119904) 120597

120597 119904ln119877]119889 119904 (16)

The integral representations (15) and (16) for the har-monic functions119891 and119891119888 replace the usual Cauchy-Riemannconditions

120597119891

120597119909=120597119891119888

120597119910

120597119891

120597119910= minus120597119891119888

120597119909 (17)

When the point (119909 119910) tends to a boundary point (119909(119904) 119910(119904))relation (15) yields

119891 (119904) =1

120587∮119904

[119891 ( 119904)120597

120597 119899ln119877 + 119891119888 ( 119904) 120597

120597 119904ln119877]119889 119904 (18)

Replacing (120597120597 119899) ln119877 by (120597120597 119904)Θ in (15) (16) and theirboundary version (18) where Θ is the complex conjugate ofln119877 it is readily seen that these integral relations are invariantunder the transformation of parameter from the arc length119904 to any other suitable parameter This property makes themethod more flexible

4 Mathematical Problems in Engineering

32 Conditions for the Uniqueness of the Solution Beforedealing with each of the two above-mentioned fundamentalproblems we first turn to the conditions to be satisfied inorder to determine the unknown harmonic functions inan unambiguous manner This is of primordial importancefor any numerical treatment of the problem for a properuse of the solving algorithm We will require the followingsupplementary conditions to be satisfied at the point 119876

0(119904 =

0) of the boundary in order to determine the totality ofthe arbitrary integration constants appearing throughoutthe solution process These additional conditions have nophysical implications on throughout the problem

(i) The vanishing of the function 119880 and its first orderpartial derivatives at 119876

0

119880 =120597119880

120597119909=120597119880

120597119910= 0 (19)

Or equivalently

119880 =120597119880

120597119904=120597119880

120597119899= 0 (20)

which in terms of the boundary values of the unknownharmonic functions give

119909 (0)Φ (0) + 119910 (0)Φ119888(0) + Ψ (0) = 0

119909 (0)Φ(0) + 119910 (0)Φ

119888

(0) + Ψ(0)

+ 119909(0)Φ (0) + 119910

(0)Φ119888(0) = 0

119909 (0)Φ119888

(0) minus 119910 (0)Φ(0) + Ψ

119888

(0)

+ 119910(0)Φ (0) minus 119909

(0)Φ119888(0) = 0

(21)

(ii) The vanishing of the expression

119909 (0)Φ119888(0) minus 119910 (0)Φ (0) + Ψ

119888(0) = 0 (22)

This last additional condition amounts to determining thevalue of Ψ119888 at 119876

0and this is chosen to simplify the formule

33 Boundary Condition for the First Fundamental Problem ofElasticity In the first fundamental problem we are given theforce distribution on the boundary 119878 of the domain119863

Let

f = 119891119909i + 119891119910j = 119891120591120591 + 119891119899n (23)

denote the external force per unit length of the boundaryThen at a general boundary point 119876 the stress vector

120590119899= f (24)

or in components

120590119909119909119899119909+ 120590119909119910119899119910= 119891119909 120590

119909119910119899119909+ 120590119910119910119899119910= 119891119910 (25)

The stress function 119880 at the boundary point 119876

120597119880

120597119904(119904) = minus (119904) 119884 (119904) + 119910 (119904)119883 (119904)

120597119880

120597119899(119904) = minus 119910 (119904) 119884 (119904) minus (119904)119883 (119904)

(26)

or in terms of the unknown harmonic functions

119909 (119904) Φ (119904) + 119910 (119904)Φ119888

(119904) + Ψ (119904) + (119904) Φ (119904) + 119910 (119904)Φ119888(119904)

= minus (119904) 119884 (119904) + 119910 (119904)119883 (119904)

119909 (119904) Φ119888(119904) minus 119910 (119904) Φ (119904) + Ψ

119888(119904) + 119910 (119904)Φ (119904) minus (119904) Φ

119888(119904)

= minus 119910 (119904) 119884 (119904) minus (119904)119883 (119904)

(27)

34 Boundary Condition for the Second Fundamental Problemof Elasticity In this problem we are given the displacementvector on the boundary 119878 of the domain119863 Let this vector bedenoted

d = 119889119909i + 119889119910j = 119889120591120591 + 119889119899n (28)

Multiplying the restriction of expression (13) to theboundary 119878 by (119904) and that of expression (14) by 119910(119904) andadding one gets

(3 minus 4V) ( (119904) 120601 (119904) + 119910 (119904) 120601119888(119904))

minus 119909 (119904) 120601 (119904) minus 119910 (119904) 120601119888(119904) minus (119904)

=119864

1 + V( (119904) 119889

119909(119904) + 119910 (119904) 119889

119910(119904)) 120596

(29)

Similarly if one multiplies the restriction of expression(13) to the boundary 119878 by 119910(119904) and that of expression (14) by(119904) and subtracts one obtains

(3 minus 4V) ( 119910 (119904) 120601 (119904) minus (119904) 120601119888 (119904))

minus 119909 (119904) 120601119888(119904) + 119910 (119904) 120601 (119904) minus

119888(119904)

=119864

1 + V( 119910 (119904) 119889

119909(119904) minus (119904) 119889

119910(119904)) 120596

(30)

These last two relations may be conveniently rewritten as

(3 minus 4V) ( (119904) 120601 (119904) + 119910 (119904) 120601119888(119904)) minus 119909 (119904) 120601 (119904)

minus 119910 (119904) 120601119888(119904) minus (119904) =

119864

1 + V119889120591(119904) 120596

(3 minus 4V) ( 119910 (119904) 120601 (119904) minus (119904) 120601119888 (119904)) minus 119909 (119904) 120601119888 (119904)

+ 119910 (119904) 120601 (119904) minus 119888(119904) =

119864

1 + V119889119899120596

(31)

35 Boundary Conditions for the Third Fundamental Problemof Elasticity This is a problem with mixed boundary condi-tions For definiteness we will restrict further considerations

Mathematical Problems in Engineering 5

to the case where one-half of the boundary has a prescribedpressure on it while the other half of the boundary is fixedThis problem will be replaced by two subproblems each withhomogeneous boundary condition The first subproblem isof the first kind It involves the given known pressure onthe same half of the boundary as the initial problem and anunknown stress on the other half This stress is expressedthrough its normal and tangential components respectivelydenoted

119899 120591 The second subproblem is of the second type

It involves zero displacement on the same half of the bound-ary as the initial problem and an unknown displacement onthe other half This displacement is expressed through itsnormal and tangential components respectively denoted

119899

120591In what follows we will apply this idea to solve three

problems for the ellipse the nearly circular boundary andthe Cassini oval In choosing these boundaries we have triedto keep away from boundaries involving singular pointsas our main task is to deal with the mixed boundaryconditions which on its own includes two boundary pointsof separation which need special attention

36 Calculation of the Harmonic Functions at Internal PointsHaving determined the boundary values of the harmonicfunctions formulae (15) and (16) may now be used tocalculate the values of these functions at any point (119909 119910)inside the domain For this we write

119877 = radic(119909 minus 119909 (1199041015840))2

+ (119910 minus 119910 (1199041015840))2

120597 ln (119877)120597119899

= n sdot nabla (ln119877)

120597 ln (119877)120597119904

= 120591 sdot nabla (ln119877)

(32)

We can also proceed otherwise In fact if we write downexpansions of the four harmonic functions involved in thecalculations in terms of some adequately chosen basis wecan then determine the expansion coefficients using thewell-known boundary collocation method (BCM) This isin fact the method we have used to calculate the unknownfunctions in the circular domain The expansions of the fourbasic harmonic functions in terms of polar harmonics are asfollows

Φ =

119873

sum

119899=1

119903119899(119860119899cos 119899120579 + 119861

119899sin 119899120579)

Φ119888=

119873

sum

119899=1

119903119899(119860119899sin 119899120579 minus 119861

119899cos 119899120579)

Ψ =

119873

sum

119899=1

119903119899(119864119899cos 119899120579 + 119863

119899sin 119899120579)

Ψ119888=

119873

sum

119899=1

119903119899(119864119899sin 119899120579 minus 119863

119899cos 119899120579)

(33)

while the stress function is

119880 = 119909Φ + 119910Φ119888+ Ψ (34)

and the quantities of practical interest are

2120583119906 =

119873

sum

119899=1

119903119899(119860119899cos 119899120579 + 119861

119899sin 119899120579)

minus

119873

sum

119899=1

119899119903119899(119860119899cos (119899 minus 2) 120579 + 119861

119899sin (119899 minus 2) 120579)

minus

119873

sum

119899=1

119899119903119899minus1(119864119899cos (119899 minus 1) 120579 + 119863

119899sin (119899 minus 1) 120579)

2120583120592 =

119873

sum

119899=1

119903119899(119860119899sin 119899120579 minus 119861

119899cos 119899120579)

minus

119873

sum

119899=1

119899119903119899(minus119860119899sin (119899 minus 2) 120579 + 119861

119899cos (119899 minus 2) 120579)

minus

119873

sum

119899=1

119899119903119899minus1(minus119864119899sin (119899 minus 1) 120579 + 119863

119899cos (119899 minus 1) 120579)

(35)

The equations for the normal and the tangential stresseson any given element of area with unit normal (119899

119909 119899119910) inside

the body or on its boundary are given by the followingformulae

120590119899119899=

119873

sum

119899=1

119903119899minus1119860119899(cos (119899 minus 1) 120579 ((3119899 minus 1198992) 1198992

119903+ (1198992+ 119899) 119899

2

120579)

+ 2 (1198992minus 119899) 119899

119903119899120579sin (119899 minus 1) 120579)

+

119873

sum

119899=1

119903119899minus1119861119899(sin (119899 minus 1) 120579 ((3119899 minus 1198992) 1198992

119903+ (1198992+ 119899) 119899

2

120579)

+ 2 (1198992minus 119899) 119899

119903119899120579cos (119899 minus 1) 120579)

+

119873

sum

119899=1

119903119899minus2119862119899(cos 119899120579 ((119899 minus 1198992) 1198992

119903+ (1198992minus 119899) 119899 minus 120579

2)

+ 2 (1198992minus 119899) 119899

119903119899120579sin 119899120579)

+

119873

sum

119899=1

119903119899minus2119863119899(sin 119899120579 ((119899 minus 1198992) 1198992

119903+ (1198992minus 119899) 119899

2

120579)

+2 (119899 minus 1198992) 119899119903119899120579cos 119899120579)

120590119899120591=

119873

sum

119899=1

119903119899minus1119860119899(cos (119899 minus 1) 120579 ((minus2119899 + 21198992) 119899

119903119899120579)

+ (1198992minus 119899) (119899

2

119903minus 1198992

120579) sin (119899 minus 1) 120579)

+

119873

sum

119899=1

119903119899minus1119861119899(sin (119899 minus 1) 120579 ((minus2119899 + 21198992) 119899

119903119899120579)

6 Mathematical Problems in Engineering

030

025

020

015

010

005

minus005

1 2 3 4 5 6

Φ(120579)

120579

(a)

Φc(120579)

030

025

020

015

010

005

minus005

1 2 3 4 5 6

120579

(b)

1 2 3 4 5 6

Ψ(120579)

02

minus02

minus04

minus06

120579

(c)

1 2 3 4 5 6

Ψc(120579) minus02

minus04

minus06

120579

(d)

Figure 1 Boundary values of the basic harmonic functions

+ (119899 minus 1198992) (1198992

119903minus 1198992

120579) cos (119899 minus 1) 120579)

+

119873

sum

119899=1

119903119899minus2119862119899(cos 119899120579 (minus2119899 + 21198992) 119899

119903119899120579+ (1198992minus 119899) 119899

2

120579

+ 2 (1198992minus 119899) (119899

2

119903minus 1198992

120579) sin 119899120579)

+

119873

sum

119899=1

119903119899minus2119863119899(sin 119899120579 (minus2119899 minus 21198992) 119899

119903119899120579

+ (119899 minus 1198992) (1198992

119903minus 1198992

120579) cos 119899120579)

(36)

The relevant boundary relations are discretized in theusual way by considering a partition of the boundary As aresult the actual boundary is replaced by a contour formedby broken lines The differential and integral equations thusreduce to a rectangular system of linear algebraic equationswhich are solved by the least squares method The conver-gence of the solution to the discretized system of equationsto the solution to the initial problem was discussed elsewhere[20] Here we only notice the existence of removable singu-larities in the formulae of integral representation of harmonicfunctions These are dealt with in the manner explainedin [18] Also the tangential derivatives of the unknownharmonic functions have to be evaluated carefully We havecalculated these derivatives using 15 points

4 Numerical Results and Discussion

The force acting on one-half of the boundary is a pressure ofintensity 119891 given by

119891 = minus1199010(sin 120579)6 120587 lt 120579 le 2120587 (37)

The other half of the boundary is completely fixed

119906 = 120592 = 0 0 lt 120579 le 120587 (38)

For definiteness we have taken 1199010= 1 The motivation

for the above choice of the pressure on half the boundary is tomake the pressure distribution tend to zero smoothly enoughat both ends of its interval of definition

The above boundary integral equations are solved numer-ically from which we have obtained the boundary valuesof the harmonic functions 120601 120601119888 120595 120595119888

119899 120591 119899 and

120591

and accordingly of the stress function 119880 The boundarywas discretized by placing a number of nodal points on itas explained 240 boundary nodes were needed in order toget the present results The nodal points were distributeduniformly on the boundary The results are shown below

Figure 1 gives the boundary values of the four basicharmonic functions

All the four figures show a weak discontinuity at the point120579 = 120587 It goes without saying that the same takes place for120579 = 0 from symmetry considerations The emplacements ofthese discontinuities are referred to by arrows on the figures

Mathematical Problems in Engineering 7

120579

02

01

minus01

35 40 45 50 55 60u120591(120579)

(a)

minus02

minus04

minus06

minus08

minus10

120579

35 40 45 50 55 60

un(120579)

(b)

minus02

minus04

02

04

120579

05 10 15 20 25 30120590120591(120579)

(c)

120579

05 10 15 20 25 30minus005

minus010

minus015

minus020

minus025

minus030

minus035

120590n(120579)

(d)

Figure 2 Boundary values of the normal and tangential stress and displacement components

Figure 2 shows the normal and the tangential compo-nents of the unknown displacement and stress on the relevantparts of the boundary As may be noticed three of figuressome difficulties were encountered when performing thecalculations in the vicinity of the singular points at 120579 = 0 andat 120579 = 120587 It was not possible to increase the number of nodalpoints beyond 240 for stability reasons in order to improvethe results

The normal displacement component reached zero valueat both separation points as should be while the tangentialdisplacement component failed to do so but the resultsimproved as the number of nodes was augmented up to acertain limit Unwanted oscillations appeared on the curvesfor stress near these two points Curve fitting techniques bypolynomial functions were used to improve the curves Addi-tionally two logarithmic functions based on the singularitieswere used for fitting only the function

120591 These are the

smooth curves on the figures based on these observationsOne suspects the presence of a logarithmic behavior of thefunction

120591at the singular points

Figure 3 shows the boundary values of the stress func-tion The curve is skewed towards the second half of theboundary but thus asymmetry should not raise any concernsas it depends on the additional conditions imposed on thisfunction

minus02

minus04

minus06

minus08

1 2 3 4 5 6

120579

U(120579)

Figure 3 Boundary values of the stress function

We have used the well-known boundary collocationmethod to directly compute the unknown functions 120590

119899119899 120590119899120591

119906 and 120592 on concentric circles centered at the origin inside thedomain using (33)ndash(36) together with the given boundaryconditionsThus there is no ambiguity in the meaning of thenormal unit vector appearing in the equations A maximumnumber of 120 nodes were usedThe results are shown on thefollowing three-dimensional plots where we have also shownon each of them the circular region in which the unknownfunctions are plotted (Figure 4)

8 Mathematical Problems in Engineering

minus10 minus05 00 05 10

minus10minus05

000510

02

01

00

minus01

minus02

(a) 119906

minus10 minus05 00 05 10

minus10

minus05

00

05

10

02

04

06

08

00

(b) V

minus10 minus05 00 05 10

minus10

minus05

00

05

10

00

minus05

minus10

(c) 120590119899119899

minus10 minus05 00 05 10

minus10

minus05000510

02

00

minus02

minus04

(d) 120590119899120591Figure 4 Displacement and stress components inside the circle by BCM

The surfaces for the displacement components are in con-formity with our expectations The normal stress componentis regular while the tangential stress component shows thesingular behavior mentioned above (Figure 5)

Here again we notice a singular behavior of the stresscomponents 120590

119910119910and 120590119909119910at the two singular boundary points

In the other cases the comparison ismadewith the correctionof the analytical results obtained in [29] as a special caseThe numerically obtained results are compared with thoseobtained analytically in [30]

5 On the Singular Solution

Wepropose to add a function120595(119904)with boundary singularities

to the function 120595 in order to get the required logarithmicbehavior of the function 120590

119899120591at the singular points (plusmn119886 0)

This function was proposed by Abou-Dina and Ghaleb [31]in connection with the solution to some boundary-valueproblems for Laplacersquos equation in rectangular domains hereit is used in a special setting

Figure 6 shows the emplacements of the singularities offunction 120595

(119904) as well as the variables 120579

1 1205792 1205881 and 120588

2

As to the function 120595s is of the form

120595119904=1

2120587(1205882

1(sin 2120579

1ln 1205881+ 1205791cos 2120579

1))

+ (1205882

2(sin 2120579

2ln 1205882+ 1205792cos 2120579

2))

(39)

where

1205881= radic1199032 minus 2119886119903 cos 120579 + 1198862

1205882= radic1199032 + 2119886119903 cos 120579 + 1198862

1205791= tanminus1 119886 minus 119903 cos 120579

minus119903 sin 120579

1205792= tanminus1 119903 cos 120579 + 119886

minus119903 sin 120579

(40)

Figure 7 shows the singular stress and displacementcomponents obtained from the singular function 120595

119904 These

functions are also labeled ldquo119904rdquo The singular behavior offunction 120590

119899120591is clear It is recalled that the normal and the

tangential components of stress are calculated on concentriccircles centered at the origin inside the domain thus themeaning of the unit normal vector is clear

The details of the calculations and the final results forthe stresses and displacements when the singular function isadded will be considered in a separate publication for othertypes of boundaries

6 Conclusions

The following conclusions are due(1) We have considered a boundary-value problem of

the plane theory of elasticity with mixed boundary

Mathematical Problems in Engineering 9

minus10 minus05 00 05 10

minus10minus05

000510

minus10

minus05

00

(a) 120590119909119909

minus10 minus05 00 05 10

minus10minus05

000510

minus10

minus15

minus05

00

(b) 120590119910119910

minus10 minus05 00 05 10

minus10minus05

000510

minus10

0

10

(c) 120590119909119910

Figure 5 Stress components inside the circle by BCM

12057911205792

12058811205882

Figure 6 Singular points

conditions in a circle Half of the boundary is sub-jected to pressure while the other half is completelyfixed The shape of the boundary was chosen as thesimplest in order to focus on problems raised by theboundary singularities We have also chosen smoothboundary pressure that decreases smoothly enough tozero towards the points of separation

(2) The correct calculation of this type of problemsrequires a large number of boundary points at whichthe unknowns are to be calculated For the presentcase 240 points could be reached without obtainingsatisfactory results in the whole boundaryThe reason

for this is the presence of singular boundary points atthe separation points of the boundary conditions

(3) To get the solution on the boundary we have replacedour problem by two subproblems each with homo-geneous boundary condition of one type having acommon solution

(4) The calculations on the boundary were performedusing a known boundary integral technique involvingharmonic functions only including regularizationand a careful calculation of the tangential derivativesof functions using 15 points

(5) The boundary calculations indicated a logarithmicbehavior of the tangential stress component on thefixed part of the boundary

(6) The solution inside the domain was obtained bythe collocation method directly using the prescribedboundary conditions A maximum number of 120uniformly distributed nodes were used

(7) In solving the arising systems of linear algebraic equa-tions we have used least squares andQR-factorizationtechniques both yielded the same results Each time alinear system of equations was solved we verified thatthe obtained results satisfy the systemThe errors didnot exceed 1

(8) We have proposed a solution having a logarithmicboundary singularity to improve the solution Theabsolute errors in satisfying the boundary conditionson an interval and including the separation pointscould thus be reduced from nearly 3

10 Mathematical Problems in Engineering

minus10

minus05

00

0510

minus10

minus05

00

05

10

minus10 minus05 00 05 10

(a) 119906(119904)

minus10 minus05 00 05 10

minus10

minus05

00

05

10

minus10

minus05

00

05

10

(b) V(119904)

minus10 minus05 00 05 10

minus10minus05

000510

minus1

0

1

(c) 120590(119904)119899119899

minus10 minus05 00 05 10

minus10 00 10

minus1

0

1

(d) 120590(119904)119899120591

Figure 7 Singular displacement and stress components inside the circle

(9) The obtained results show stress concentration andthus indicate the need to introduce domains of pos-sible plastic behavior of the material around the twoboundary separation points

(10) The method extends to other geometries of theboundary In the presence of corner points a smooth-ing process must be applied Numerical experi-ments have clearly indicated that the best results areobtained for boundaries of smoothness of the fourthorder

(11) Future work will involve more complicated shapes ofthe boundary and other types of boundary condi-tions

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V D Kupradze Methods of Potential in Theory of ElasticityFizmatgiz Moscow Russia 1963

[2] M A Jaswon and G T Symm Integral Equation Methods inPotential Theory and Elastostatics Academic Press LondonUK 1977

[3] D Natroshvili I G Stratis and S Zazashvili ldquoBoundary inte-gral equation methods in the theory of elasticity of hemitropic

materials a brief reviewrdquo Journal of Computational and AppliedMathematics vol 234 no 6 pp 1622ndash1630 2010

[4] C Constanda ldquoIntegral equations of the first kind in planeelasticityrdquo Quarterly of Applied Mathematics vol 53 no 4 pp783ndash793 1995

[5] S N Atluri and T-L Zhu ldquoMeshless local Petrov-Galerkin(MLPG) approach for solving problems in elasto-staticsrdquo Com-putational Mechanics vol 25 no 2 pp 169ndash179 2000

[6] J Sladek V Sladek and R Van Keer ldquoMeshless local boundaryintegral equation method for 2D elastodynamic problemsrdquoInternational Journal for Numerical Methods in Engineering vol57 no 2 pp 235ndash249 2003

[7] Z Rui H Jin and L Tao ldquoMechanical quadrature methodsand their splitting extrapolations for solving boundary integralequations of axisymmetric Laplace mixed boundary valueproblemsrdquo Engineering Analysis with Boundary Elements vol30 no 5 pp 391ndash398 2006

[8] M Elliotis G Georgiou and C Xenophontos ldquoThe singularfunction boundary integral method for biharmonic problemswith crack singularitiesrdquo Engineering Analysis with BoundaryElements vol 31 no 3 pp 209ndash215 2007

[9] Z-C Li P-C Chu L-J Young and M-G Lee ldquoModelsof corner and crack singularity of linear elastostatics andtheir numerical solutionsrdquo Engineering Analysis with BoundaryElements vol 34 no 6 pp 533ndash548 2010

[10] S V Shmegera ldquoThe initial boundary-value mixed problemsfor elastic half-plane with the conditions of contact frictionrdquoInternational Journal of Solids and Structures vol 37 no 43 pp6277ndash6296 2000

Mathematical Problems in Engineering 11

[11] P Schiavone ldquoIntegral solutions of mixed problems in a theoryof plane strain elasticity with microstructurerdquo InternationalJournal of Engineering Science vol 39 no 10 pp 1091ndash1100 2001

[12] R Haller-Dintelmann H-C Kaiser and J Rehberg ldquoEllipticmodel problems including mixed boundary conditions andmaterial heterogeneitiesrdquo Journal de Mathematiques Pures etAppliquees Neuvieme Serie vol 89 no 1 pp 25ndash48 2008

[13] J Helsing ldquoIntegral equationmethods for elliptic problemswithboundary conditions of mixed typerdquo Journal of ComputationalPhysics vol 228 no 23 pp 8892ndash8907 2009

[14] M-G Lee L-J Young Z-C Li and P-C Chu ldquoCombinedTrefftz methods of particular and fundamental solutions forcorner and crack singularity of linear elastostaticsrdquo EngineeringAnalysis with Boundary Elements vol 34 no 7 pp 632ndash6542010

[15] M-G Lee L-J Young Z-C Li and P-C Chu ldquoMixed typesof boundary conditions at corners of linear elastostatics andtheir numerical solutionsrdquo Engineering Analysis with BoundaryElements vol 35 no 12 pp 1265ndash1278 2011

[16] M A Khuri ldquoBoundary value problems for mixed type equa-tions and applicationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 17 pp 6405ndash6415 2011

[17] M S Abou-Dina and A F Ghaleb ldquoOn the boundary integralformulation of the plane theory of elasticity with applications(analytical aspects)rdquo Journal of Computational and AppliedMathematics vol 106 no 1 pp 55ndash70 1999

[18] M S Abou-Dina and A F Ghaleb ldquoOn the boundary integralformulation of the plane theory of elasticity (computationalaspects)rdquo Journal of Computational and Applied Mathematicsvol 159 no 2 pp 285ndash317 2003

[19] C Constanda ldquoOn Kupradzersquos method of approximete solutionin linear elasticityrdquo Bulletin of the Polish Academy of SciencesMathematics vol 39 pp 201ndash204 1991

[20] C Constanda ldquoThe boundary integral equation method inplane elasticityrdquo Proceedings of the American MathematicalSociety vol 123 no 11 pp 3385ndash3396 1995

[21] A M Abd-Alla T A Nofal and A M Farhan ldquoEffect of thenon-homogenity on the composite infinite cylinder of isotropicmaterialrdquo Physics Letters A General Atomic and Solid StatePhysics vol 372 no 29 pp 4861ndash4864 2008

[22] A M Abd-Alla and A M Farhan ldquoEffect of the non-homogenity on the composite infinite cylinder of orthotropicmaterialrdquo Physics Letters A General Atomic and Solid StatePhysics vol 372 no 6 pp 756ndash760 2008

[23] A M Abd-Alla G A Yahya and S R Mahmoud ldquoEffect ofmagnetic field and non-homogeneity on the radial vibrationsin hollow rotating elastic cylinderrdquoMeccanica vol 48 no 3 pp555ndash566 2013

[24] AM Abd-Alla SM Abo-Dahab and T A Al-Thamali ldquoProp-agation of Rayleigh waves in a rotating orthotropic materialelastic half-space under initial stress and gravityrdquo Journal ofMechanical Science and Technology vol 26 pp 2815ndash2823 2012

[25] L W Zhang Z X Lei K M Liew and J L Yu ldquoStatic anddynamic of car- bon nanotube reinforced functioally gradedcylinderical panelsrdquo Composite Structures vol 111 pp 205ndash2122014

[26] L W Zhang P Zhu and K M Liew ldquoThermal buckling offunction- ally graded plates using a local kriging meshlessmethodrdquo Composite Sturctures vol 108 pp 472ndash492 2014

[27] P Zhu L W Zhang and K M Liew ldquoGeometrically nonlinearthermo- mechanical analysis of moderately thick functionally

graded plates us- ing local petror-Galerkin approach withmoving kriging interpolationrdquo Composite Structures vol 107pp 298ndash314 2014

[28] K M Liew Z X Lei J L Yu and L W Zhang ldquoPostbucklingof carbon nanotube-reinforced functionally graded cylindricalpanels under axial compression using a meshless approachrdquoComputer Methods in Applied Mechanics and Engineering vol268 pp 1ndash17 2014

[29] A S Gjam H A Abdusalam and A F Ghaleb ldquoSolutionfor a problem of linear plane elasticity with mixed boundaryconditions on an ellipse by the method of boundary integralsrdquoJournal of the Egyptian Mathematical Society vol 21 no 3 pp361ndash369 2013

[30] M S Abou-Dina and A F Ghaleb ldquoOn the boundary integralformulation of the plane theory of thermoelasticity (analyticalaspects)rdquo Journal of Thermal Stresses vol 25 no 1 pp 1ndash292002

[31] M S Abou-Dina andA F Ghaleb ldquoA variant of Trefftzrsquosmethodby boundary Fourier expansion for solving regular and singularplane boundary-value problemsrdquo Journal of Computational andApplied Mathematics vol 167 no 2 pp 363ndash387 2004

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Mathematical Problems in Engineering

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