Engineering Analysis with Boundary ElementsBoundary element formulation of axisymmetric problems for an elastic halfspace M.F.F. Oliveiraa,n, N.A. Dumontb, A.P.S. Selvaduraic a Computer

Engineering Analysis with Boundary Elements 36 (2012) 1478–1492

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Boundary element formulation of axisymmetric problemsfor an elastic halfspace

M.F.F. Oliveira a,n, N.A. Dumont b, A.P.S. Selvadurai c

a Computer Graphics Technology Group, Department of Civil Engineering, Pontifical Catholic University of Rio de Janeiro, 22453 900 Rio de Janeiro, Brazilb Department of Civil Engineering, Pontifical Catholic University of Rio de Janeiro, 22453 900 Rio de Janeiro, Brazilc Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, Canada H3A 2K6

a r t i c l e i n f o

Article history:

Received 7 October 2011

Accepted 27 March 2012

Keywords:

Elastic halfspace

Boundary element method

Axisymmetric problems

Traction boundary value problems

97/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.enganabound.2012.03.015

esponding author. Tel.: þ55 21 2512 5984; f

ail addresses: [email protected],

[email protected] (M.F.F. Oliveira), dumont@pu

[email protected] (A.P.S. Selvadurai).

a b s t r a c t

Axisymmetric problems for an elastic halfspace are commonly analyzed by the boundary element (BE)

method by employing the axisymmetric fundamental solution for the fullspace. In such cases, the

discretization of the free surface is required, with its truncation at an appropriate location from the axis

of symmetry. This paper presents the BE implementation of the axisymmetric fundamental solution for an

elastic halfspace, given in terms of integrals of the Lipschitz–Hankel type, that satisfies in advance the

boundary condition of zero traction on the free surface and the decay of displacements in the far field.

Explicit equations for post-processing the results at internal points are provided, as well as adequate

numerical schemes to evaluate the boundary integrals arising in the method. This formulation can be easily

implemented in existing BE computational codes for axisymmetric fullspace problems, requiring only a few

modifications. Numerical results are provided to validate the proposed formulation.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The axisymmetric formulation in classical elasticity is usefulfor the analysis of problems in geomechanics [1,2], as well ascontact problems for cylinders, spheres and circular plates [3–8].Other applications involve the study of fracture mechanics phe-nomena and inclusions [5,9–11].

In particular, the BE method is advantageous for axisymmetricproblems, since it reduces the analysis of the three-dimensionaldomain to a one-dimensional mesh discretization requiring onlythe evaluation of linear integrals. However, the fundamentalsolutions involved are more complex, requiring special considera-tions on their manipulation and integration to correctly evaluatethe influence coefficients arising in the boundary integral equa-tions. Extensive surveys on the existing axisymmetric fundamen-tal solutions are given by Wang and Liao [12,13], Wang et al. [14]and Wideberg and Benitez [15].

The BE method for axisymmetric elasticity was first formu-lated by Cruse et al. [16], using the fullspace fundamentalsolution derived by Kermanidis [17]. Several contributions tothe formulation of the axisymmetric problem may be cited, suchas the expansion of non-symmetric boundary conditions byFourier series suggested by Mayr [18] and Rizzo and Shippy

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ax: þ55 21 3527 1848.

c-rio.br (N.A. Dumont),

[19,20], and the assessment of body forces by means of particularintegrals incorporated by Park [21]. Also, axisymmetricformulations have been developed for transverse isotropy [22],thermoelasticity [23], elastoplasticity [24] and viscoplasticity[25]. In elastodynamics, the works by Wang and Banerjee[26,27], Tsinopoulos et al. [28] and Yang and Zhou [29] in thefrequency domain should be mentioned. The method has alsobeen successfully applied to contact problems [30] and fracturemechanics [31].

For axisymmetric halfspace problems, the BE formulationemployed with the fullspace fundamental solution requiresthe discretization of the infinite free surface. In this case, thesurface must be truncated at a reasonable distance from theaxis of symmetry and the region of interest [32]. The disadvan-tage of such a scheme is that a large number of boundaryelements is needed to model the remote boundary satisfactorily,so that relative displacements in particular can be accuratelyevaluated.

An alternative way to deal with this problem is to use infiniteboundary elements, as suggested by Watson [33]. These infiniteelements, which simulate the decay of the displacements andstresses in the far field, are mapped onto a finite region in termsof an intrinsic coordinate system to facilitate the integration. Avariety of infinite elements can be found in the literature forthree-dimensional elasticity, depending on the mapping schemeused and the application [34–36]. However, such elements are notavailable for problems with axisymmetry, probably becausetreating the integration of the singular kernels over the mapped

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M.F.F. Oliveira et al. / Engineering Analysis with Boundary Elements 36 (2012) 1478–1492 1479

infinite elements is not straightforward for the fullspace funda-mental solution. Therefore, Kelvin’s fundamental solution isusually employed together with the available three-dimensionalsurface infinite elements for axisymmetric applications in thehalfspace [37–39], thus requiring the boundary surfaces to bediscretized.

Another way to treat this problem is to implement thefundamental solution that satisfies in advance the traction freeboundary condition on the free surface, which circumvents itsnumerical discretization. In elasticity, this approach was used byTelles and Brebbia [40] and Dumir and Mehta [41] to examineproblems for an isotropic and orthotropic halfplane, respectively.

This work presents a BE formulation for axisymmetric elasti-city problems for a halfspace [42] that makes use of the funda-mental solutions due to radial and axial ring loads embedded in ahalfspace derived by Hasegawa [43,44]. The resulting equationscould be manipulated by expressing the fundamental solutions interms of Lipschitz–Hankel integrals, as adopted by Selvadurai andRajapakse [5] using extensions to the solutions developed byMindlin [45] and Mindlin and Cheng [46]. Since the terms of thefullspace fundamental solution can be identified as constituentsof the halfspace fundamental solution, the proposed formulationcan be implemented by introducing only a few modifications inexisting axisymmetric computational codes. Explicit equationsare presented for expressing results at internal points as well asappropriate numerical schemes to accurately evaluate the inte-grals arising in the formulation. Problems related to torsionalloads, not addressed in this work, involve simpler fundamentalsolutions and can be examined in a similar manner.

Section 2 of this paper introduces the axisymmetric funda-mental solution for the elastic fullspace and an elastic halfspace.Section 3 presents the axisymmetric BE formulation, followed bySection 4 that deals with the numerical integration. Finally,Section 5 illustrates numerical examples that validate the pro-posed formulation.

Fig. 1. Ring loads in the elastic fullspace: (a) radial direction; (b) axial direction.

2. Axisymmetric fundamental solution

The axisymmetric fundamental solution for elasticity consistsof displacements un

ijðP,Q Þ and stresses sn

ijkðP,Q Þ due to ring loads inthe i-direction applied at Pðx,z0Þ and centered in the z-axis. Thecontinuum has shear modulus m and Poisson’s ratio n. Thesolutions are given in the cylindrical coordinate system (r,z).The indices j and k stand for the displacement and stresscomponents measured at Q ðr,zÞ.

For the fullspace, displacements due to ring loads were firstderived by Kermanidis [17], by applying Betti’s theorem to thePapkovich–Neuber solution [47] for an elastic medium of infiniteextent. Subsequently, Cruse et al. [16] and Bakr and Fenner [23]solved Navier’s equilibrium equations by expressing the displace-ments as Galerkin vectors [47] and considering ring loads as bodyforces. Also, Shippy et al. [48] integrated Kelvin’s solution [47] forthe three-dimensional infinite medium along a circular pathcentered on the axis of symmetry.

For the halfspace, Hasegawa [43,44] deduced displacementsand stresses from stress functions [49] obtained by means ofFourier and Hankel transforms and considering ring loads as bodyforces. Later, Selvadurai and Rajapakse [5] imposed boundaryconditions and continuity conditions to displacements and stres-ses expressed by Muki’s solution [50,51] and arrived at the samesolutions. These solutions were also obtained by Hanson andWang [52] as a particular case of the problem for the mediumwith transverse isotropy.

Both axisymmetric fundamental solutions for fullspace andhalfspace can be expressed by means of either integrals of the

Lipschitz–Hankel type involving products of Bessel functions [53],or complete elliptic integrals of the first and second types [54], orLegendre functions [54]. In this work, the approach presented bySelvadurai and Rajapakse [5] is adopted. Expressions are writtenin terms of integrals of the Lipschitz–Hankel type [53]

Ipqlðx,r; cÞ ¼

Z 10

JpðxtÞJqðrtÞe�cttl dt ð1Þ

in which p, q and l are integers, JpðxtÞ and JqðrtÞ are Besselfunctions of the first kind of order p and q, respectively. Theintegrals arising in the axisymmetric fundamental solutions areconvergent [53] and their closed form expressions in terms ofcomplete elliptic integrals of the first, second and third kinds [54]are listed in Appendix A.

2.1. Ring loads in an elastic fullspace

The fundamental solution can be derived from Muki’s solution[50,51] of the Navier equilibrium equations for an elastic isotropicmedium,

ð1�2nÞ r2ur�ur

r2

� �þD,r ¼ 0 ð2Þ

ð1�2nÞr2uzþD,z ¼ 0 ð3Þ

where

D¼ ur,rþur

rþuz,z ð4Þ

Muki represented displacements by means of harmonic and bi-harmonic functions and used Hankel transforms and their corre-spondence to generalized Fourier–Bessel transforms to arrive at ageneral asymmetric solution. This solution can be specialized foraxisymmetry, leading to

ur ¼1

2

Z 10

dG

dzþ2H

� �½J1ðrtÞ�J�1ðrtÞ�t2 dt ð5Þ

uz ¼

Z 10ð1�2nÞd

2G

dz2�2ð1�nÞt2G

" #J0ðrtÞ dt ð6Þ

where

Gðt,zÞ ¼ ðAþBzÞeztþðCþDzÞe�zt ð7Þ

Hðt,zÞ ¼ EeztþFe�zt

ð8Þ

in which AðtÞ,BðtÞ, . . . ,FðtÞ are unknown functions.Consider a fullspace split into two parts, I and II, by a plane

normal to z at z¼ z0 as shown in Fig. 1. Applying Eqs. (5) and (6)and the regularity conditions for the displacements and stressesas z-71,

uI,IIi ðr,71Þ¼ 0, sI,II

ij ðr,71Þ¼ 0 ð9Þ

Fig. 2. Ring loads in the elastic halfspace: (a) radial direction; (b) axial direction.

M.F.F. Oliveira et al. / Engineering Analysis with Boundary Elements 36 (2012) 1478–14921480

the number of unknown functions can be reduced from 12 to 6.These functions can be determined using (i) the compatibilityconditions of displacements at the interface

uIiðr,z0Þ ¼ uII

i ðr,z0Þ ð10Þ

and (ii) the equilibrium conditions for radial and vertical unit ringloads applied at ðx,z0Þ

sIrrðr,z0Þ�sII

rrðr,z0Þ ¼1

2pxdðr�xÞ ¼1

2px

Z 10

J1ðxtÞJ1ðrtÞxt dt ð11Þ

sIrzðr,z0Þ�sII

rzðr,z0Þ ¼ 0 ð12Þ

and

sIzrðr,z0Þ�sII

zrðr,z0Þ ¼ 0 ð13Þ

sIzzðr,z0Þ�sII

zzðr,z0Þ ¼1

2px dðr�xÞ ¼1

2px

Z 10

J0ðxtÞJ0ðrtÞxt dt ð14Þ

where d is the Dirac delta function [55].The final expressions for displacements uI

iðr,zÞ and uIIi ðr,zÞ can

be combined, leading to the following equations for displace-ments unf

ij ðP,Q Þ:

unfrr ¼

1

16pmð1�nÞ fð3�4nÞI110�9z9 I111g ð15Þ

unfrz ¼

1

16pmð1�nÞzI101 ð16Þ

unfzr ¼�

1

16pmð1�nÞzI011 ð17Þ

unfzz ¼

1

16pmð1�nÞfð3�4nÞI000þ9z9I001g ð18Þ

where

z ¼ z0�z and Ipql ¼ Ipqlðx,r; c¼ 9z9Þ ð19Þ

and the superscript f stands for the fullspace fundamentalsolution.

Considering the constitutive equations in cylindrical coordi-nates

srr ¼ 2m ur,rþn

1�2nD� �

ð20Þ

srz ¼ mður,zþuz,rÞ ð21Þ

szz ¼ 2m uz,zþn

1�2nD

� �ð22Þ

the corresponding stresses snfijkðP,Q Þ are obtained as

snfrrr ¼

1

8rð1�nÞf�ð3�4nÞI110þ9z9I111þð3�2nÞrI101�r9z9I102g ð23Þ

snfrrz ¼

1

8ð1�nÞ fsignðzÞ2ð1�nÞI111�zI112g ð24Þ

snfrzz ¼

1

8ð1�nÞf�ð1�2nÞI101þ9z9I102g ð25Þ

snfzrr ¼

1

8rð1�nÞfzI011þr signðzÞ2nI001�rzI002g ð26Þ

snfzrz ¼

1

8ð1�nÞ f�ð1�2nÞI011�zI012g ð27Þ

snfzzz ¼

1

8ð1�nÞfsignðzÞ2ð1�nÞI001þzI002g ð28Þ

where, for a generic argument z,

signðzÞ ¼1 if zZ0

�1 if zo0

(ð29Þ

If the ring load is applied at the axis of symmetry, i.e., for x¼ 0,the load in the radial direction is naturally void and, as a

consequence, unfrj 9x ¼ 0 ¼ 0 and snf

rjk9x ¼ 0 ¼ 0. In such a case, the

fundamental solution for the vertical load simplifies to Kelvin’s

three-dimensional solution [47]. The expressions for unfzj 9x ¼ 0 and

snfzjk9x ¼ 0 can be derived by taking the limit as x-0 in Eqs.

(15)–(18) and Eqs. (23)–(28). Appendix A presents the limits of

the integrals of the Lipschitz–Hankel type as x-0.

2.2. Ring loads in an elastic halfspace

An analogous procedure can be carried out for the axisym-metric halfspace. Consider a plane normal to z at z¼ z0 and splitthe halfspace defined for zr0 into two parts, as depicted in Fig. 2.

Applying Eqs. (5) and (6) to each part of the halfspace leads to12 unknown functions, as in the fullspace problem. These func-tions can be evaluated by applying regularity conditions fordisplacements and stresses at z-�1 in part I,

uIiðr,�1Þ¼ 0, sI

ijðr,�1Þ¼ 0 ð30Þ

traction free boundary condition at the surface of part II,

sIIzjðr,0Þ ¼ 0 ð31Þ

as well as displacement compatibility conditions and equilibriumconditions for the radial and vertical ring loads expressed in Eqs.(10)–(14).

The expressions of displacements for parts I and II can becombined and a similar procedure can also be applied to thecomplementary halfspace zZ0. The final expressions of displace-ments unh

ij ðP,Q Þ and their corresponding stresses snhijkðP,Q Þ are

given by

unhij ¼ unf

ij þundij and snh

ijk ¼ snfijkþs

ndijk ð32Þ

in which unfij ðP,Q Þ and snf

ijkðP,Q Þ are the fullspace fundamentalsolutions given by Eqs. (15)–(18) and Eqs. (23)–(28). The index din the remaining terms und

ij ðP,Q Þ and sndijkðP,Q Þ refers to the

difference between the halfspace and fullspace fundamentalsolutions and are given by

undrr ¼

1

16pmð1�nÞ fð5�12nþ8n2ÞI110�ð3�4nÞ9z9I111þ2zz0 I112g ð33Þ

undrz ¼

1

16pmð1�nÞ f�4ð1�nÞð1�2nÞsignðzÞI100þð3�4nÞzI101

þ2zz0 signðzÞI102g ð34Þ

Fig. 3. Axisymmetric halfspace subjected to: (a) radial ring load; (b) vertical

ring load.


undzr ¼

1

16pmð1�nÞ f�4ð1�nÞð1�2nÞsignðzÞI010�ð3�4nÞzI011

þ2zz0 signðzÞI012g ð35Þ

undzz ¼

1

16pmð1�nÞfð5�12nþ8n2ÞI000þð3�4nÞ9z9I001þ2zz0 I002g

ð36Þ

and

sndrrr ¼

1

8rpð1�nÞ f�ð5�12nþ8n2ÞI110þð3�4nÞ9z9I111

�2zz0 I112þr½ð5�6nÞI101�signðzÞ½ð3�4nÞzþ3z0�I102

�2zz0 I112þ2zz0 I103�g ð37Þ

sndrrz ¼

1

8pð1�nÞf�signðzÞ2ð1�nÞI111þ½ð3�4nÞzþz0�I112

�signðzÞ2zz0 I113g ð38Þ

sndrzz ¼

1

8pð1�nÞ fð1�2nÞI101þsignðzÞ½ð3�4nÞz�z0�I102�2zz0 I103g

ð39Þ

sndzrr ¼

1

8rpð1�nÞfsignðzÞ4ð1�nÞð1�2nÞI010þð3�4nÞzI011

�signðzÞ2zz0 I012þr½�signðzÞð2�3nÞI001

þ½ð3�4nÞz�3z0�I002þsignðzÞ2zz0 I003�g ð40Þ

sndzrz ¼

1

8pð1�nÞ fð1�2nÞI011�signðzÞ½ð3�4nÞz�z0�I012�2zz0 I013g

ð41Þ

sndzzz ¼

1

8pð1�nÞ f�signðzÞ2ð1�nÞI001�½ð3�4nÞzþz0�I002

�signðzÞ2zz0 I003g ð42Þ

where

z ¼ z0 þz and I pql ¼ Ipqlðx,r; c¼ 9z9Þ ð43Þ

The above equations are valid for the halfspace defined either forzr0 or zZ0.

If the ring load is applied at the axis of axisymmetry (i.e. x¼ 0),

unhrj 9x ¼ 0 ¼ 0 and snh

rjk9x ¼ 0 ¼ 0. In the case of a vertical load, unhzj 9x ¼ 0

and snhzjk9x ¼ 0 can be derived by taking the limit as x-0 in Eqs.

(33)–(36) and Eqs. (37)–(42). The terms undij ðP,Q Þ and snd

ijkðP,Q Þ are

singular only at z¼0. Notice that the implementation of thehalfspace fundamental solution requires only a few changes to acode where the fullspace solution is already implemented.

3. Boundary element formulation

3.1. Boundary integral equation

In the absence of body forces, the displacements uiðPÞ in thedomain O of a halfspace can be expressed in terms of displace-ments uiðQ Þ and traction forces tiðQ Þ ¼ sijZj along the boundary Gby Somiglianas’s identity for axisymmetric problems [16]

uiðPÞ ¼�2pZG

tnhij ðP,Q ÞujðQ Þr dGþ2p

ZG

unhij ðP,Q ÞtjðQ Þr dG ð44Þ

where Gðr,zÞ ¼Gi [ Gs [ G0 is the boundary of the meridian plane,shown in Fig. 3, and Zi is the outward unity normal to G. In thisfigure, Gi, Gs and G0 represent the internal boundary, the loadedportion of the boundary at z¼0 and the traction free extent of theboundary at z¼0, respectively.

The fundamental solutions unhij and tnh

ij ¼ snhijknk are displace-

ments and traction forces in the halfspace that satisfy in advancethe boundary conditions at z¼0. Since, by definition, there are notractions tiðQ Þ on G0, Eq. (44) simplifies to

uiðPÞ ¼�2pZGi


ZG


where G ¼Gi [Gs. Evaluating the above equation at the boundaryleads to the following integral equation:

cijujðPÞ ¼ �2p�ZGi


ZG


in which the first integral should be considered in the sense ofCauchy’s principal value and

cij ¼dij if Pðx,z0ÞAGs

c ij if Pðx,z0ÞAGi

(ð47Þ

The constants cij correspond to the discontinuous part of thefirst integral of Eq. (45) when Pðx,z0ÞAGi and their evaluation ispresented in Section 3.2. Since the term tnd

ij of the fundamentalsolution in Eq. (32) has no singularities, only tnf

ij needs to beconsidered and the resulting constants cij are the same ones as forthe implementation of the fullspace fundamental solution [16].These constants can also be evaluated indirectly, by applyingknown analytical solutions, such as hydrostatic stress, planestress and plane strain, to the final system of equations [56].

By approximating displacements and tractions along theboundary, which is discretized with nn nodes, we obtain theconventional equation

Hpquq ¼ Gpqtq, p,q¼ 1: :2nn ð48Þ

where Hpq and Gpq are the influence matrices, and uq and tq arenodal displacements and tractions. Solutions can be obtained byapplying boundary conditions and rearranging the above equa-tion. In this work, traction discontinuities were represented byduplicating the corresponding nodal degree of freedom [57].

3.2. Evaluation of the constants cij

The constants cij correspond to the discontinuous part of thefirst integral of Eq. (45) and can be expressed as

cij ¼ dijþ2p limE-0

ZGE

tnfij ðP,Q Þr dG¼ 2p lim

E-0

ZGE

tnfij ðP,Q Þr dG ð49Þ

Fig. 4. Integration of constants cij for (a) xa0 and (b) x¼ 0.


where GE and GE are portions of the circumference of radius E, asdepicted in Fig. 4.

For the traction forces tnfij in the fullspace, as the distance r

between Pðx,z0Þ and Q ðr,zÞ tends to zero, the modulus m of thecomplete elliptic integrals tends to unity, and accordinglyEðmÞ-1 and KðmÞ-1 in the integrals Ipql. The integral inEq. (49) can be simplified by expanding K(m) as an infinite seriesfor mo1 as [54]

KðmÞ ¼1

2p1þ

1

2

� �2

mþ1 � 3

2 � 4

� �2

m2þ1 � 3 � 5

2 � 4 � 6

� �2

m3þ � � �

" #ð50Þ

and using the following geometric relations:

r¼ xþE cos y, z¼ z0 þE sin y

nr ¼� cos y, nz ¼�sin y, dGE ¼�E dy ð51Þ

When m and n refer to the same node, the following expres-sions are obtained:

crr ¼1

4pð1�nÞsin 2y1�sin 2y2

2þ2ð1�nÞDy

� �

crz ¼ czr ¼1

4pð1�nÞ½sin2 y1�sin2 y2�

czz ¼1

4pð1�nÞ�sin 2y1�sin 2y2

2þ2ð1�nÞDy

� �ð52Þ

for xZ0, where y2 ¼ y1�Dy and Dy is the internal angle betweeny1 and y2, and

crr ¼ 1

crz ¼ 0

czr ¼1

4pð1�nÞ½�cos3 y1þcos3 y2�

czz ¼1

4pð1�nÞ½sin y1½2ð1�nÞ�cos2 y1��sin y2½2ð1�nÞ�cos2 y2�

ð53Þ

for x¼ 0. On the other hand, when m and n do not refer to thesame node, cmn ¼ 0.

The expressions given by Eq. (52) were obtained by Cruse et al.[16] and coincide with the constants for plane strain elasticity[58]. Correspondingly, Eq. (53) can be derived by integrating theconstants for the three-dimensional elasticity [58] over the axis ofsymmetry. For x¼ 0, only czz is required for a computationalimplementation since other constants are either multiplied byzero values of ur or correspond to ur values that a priori are knownto be zero at the axis z, as remarked by Graciani et al. [59].

3.3. Displacements and stresses in the domain

From the solutions uiðQ Þ and tiðQ Þ along the boundary,displacements at a point Pðx,z0Þ in the domain can be obtainedby Somigliana’s identity, expressed in Eq. (45). Stresses in thedomain can be evaluated by applying Somigliana’s identity to theconstitutive relations given by Eqs. (20)–(22), leading to

sijðPÞ ¼ 2pZG

tnhijkðP,Q ÞukðQ Þr dGþ2p

ZG

unhijkðP,Q ÞtkðQ Þr dG ð54Þ

Similar to the decomposition given by Eq. (32), functions unhijk

and tnhijk can also be expressed as

unhijk ¼ unf

ijkþundijk and t

nhijk ¼ t

nfijkþt

ndijk ð55Þ

where unfijk and t

nfijk are derived from the fullspace fundamental

solutions and were tabulated by Tan [60,61] in terms of completeelliptic integrals. The evaluation of unh

ijk and tnhijkl is a cumbersome

task since it involves the derivatives of the fundamental solutionsunh

ij and tnhij . However, they can be written in a more compact form

in terms of integrals of the Lipschitz–Hankel type, as listed inAppendix B. This procedure is also valid for the evaluation ofdisplacements and stresses on the non-discretized boundary G0.

3.4. Displacements and stresses along the boundary

Stresses at a point Pðx,z0Þ of the boundary can be obtained bysubstituting for ui in the constitutive relations for axisymmetry,Eqs. (20)–(22), with Eq. (45). As a result, the integral equationbecomes hypersingular. This integral was first presented forfullspace problems by Lacerda and Wrobel [62], with contribu-tions by Mukherjee [63] regarding its numerical integration.

Because of the complexity in evaluating these hypersingularintegrals, this work adopts the approach of interpolating thenodal results along each boundary element in a local coordinatesystem [64].

4. Numerical integration

As only the meridian of the axisymmetric body needs to bediscretized, the integrals can be evaluated along the boundaryGðr,zÞ, for successive sub-boundaries Gt between two consecutivenodes of an element. Owing to the singularity of the fundamentalsolution, adequate numerical schemes must be adopted to eval-uate the integrals

Gtia ¼ 2p

ZGt

unhij ðP,Q ÞNaðQ ÞrðQ Þ dGðr,zÞ ð56Þ

bHt

ia ¼ 2pZGt

tnhij ðP,Q ÞNaðQ ÞrðQ Þ dGðr,zÞ ð57Þ

where the index t identifies the part of the boundary beingintegrated and Na is the interpolation function for a given nodea in the element. The various singularities occur only at theextremities of the integration intervals.

The singularities arising in the halfspace fundamental solu-tions unh

ij ðP,Q Þ and tnhij ðP,Q Þ depend on the singularities of their

individual terms, given in Eq. (32). Table 1 summarizes thesesingularities, where r is the distance between the points Pðx,z0Þand Q ðr,zÞ. For the fullspace terms, the singularity type dependson the point Pðx,z0Þ at which the ring loads are applied. If the ringloads are placed on the axis of symmetry, i.e. x¼ 0, the halfspacefundamental solutions coincide with the three-dimensionalKelvin’s solutions [47] with their corresponding singularities.The remaining terms und

ij and tndij present no singularities for

Fig. 5. Two illustrations of case 1: Pðx,z0ÞAG and Pðx,z0Þ=2Gt for: (a) Pðx,z0ÞAGi and

Q ðr,zÞAGs; (b) Pðx,z0ÞAGs and Q ðr,zÞAGi .

Fig. 6. Case 2.1: Pðx,z0ÞAGi and Pðx,z0ÞAGt for xa0: (a) rðZÞ9Z ¼ �1¼0;

(b) rðZÞ9Z ¼ 1¼0.

Table 1Singularities of the halfspace fundamental solutions.

Ring load location

Singularity type

For all z For z¼0

unfij tnf

ij undij

xa0 lnðrÞ lnðrÞ and 1=r lnðrÞx¼ 0 1=r 1=r2 1=r


za0. On the other hand, for z¼0 the traction term tndij vanishes

and only undij presents singularity.

In this section, the integration cases identified for Gtia and bH t

iaare grouped according to the position of Pðx,z0Þ in relation to thepart of the boundary along which the integration is carried out aswell as to the axis of symmetry. The numerical schemes employedin this work to evaluate regular integrals, weakly singular integralsof logarithmic terms and the finite part of singular integrals oforder 1=r are briefly presented in Appendix C.

4.1. Case 1: Pðx,z0Þ=2Gt

If the point Pðx,z0Þ does not belong to the portion of theboundary being integrated, as illustrated in Fig. 5, then r40and accordingly both Gt

ia and bH t

ia expressed in Eqs. (56) and (57)are regular and can be evaluated by the Gauss–Legendre quad-rature rule [65]. For each portion of the boundary, these integralscan be rewritten in terms of the natural coordinate ZA ½�1;1� as

Gtia ¼ 2p

Z 1

�1unh

ij NaðZÞrðZÞJðZÞ dZ ð58Þ

bH t

ia ¼ 2pZ 1

�1tnh

ij NaðZÞrðZÞJðZÞ dZ ð59Þ

where JðZÞ is the Jacobian of the coordinate transformation. Incomparison to the fullspace implementation, it suffices to add tothe existing integration scheme the terms represented by und

ij and tndij .

4.2. Case 2: Pðx,z0ÞAGt , Gt �Gi

If Q ðr,zÞAGi and Pðx,z0Þ belongs to the portion of the boundaryalong which the integration is carried out, the singularities in the

fundamental solutions are entirely due to the fullspace terms unfij

and tnfij , for either case x40 or x¼ 0. The integration of Gt

ia and bH t

ia

for these singular terms is discussed below in Cases 2.1 and 2.2.

The terms undij and tnd

ij present no singularities for za0 and

therefore, it is sufficient to add to existing fullspace codes the

integration of these terms in Gtia and bH t

ia by using the Gauss–

Legendre quadrature scheme.Case 2.1: xa0. If the point Pðx,z0Þ coincides with one of the

nodes of the portion of the element being integrated and xa0,then rðZÞ9Z ¼ Z0 ¼0 for either Z0 ¼ �1 or Z0 ¼ 1, as depicted in Fig. 6.In this case, because of the complete elliptic integrals, both unf

ij

and tnfij present singularity of order lnðrÞ. Moreover, tnf

ij alsopresents a singularity of order 1=r. As a consequence, Gt

ia isweakly singular and bHt

ia presents both weak and strong singula-rities, the latter ones to be evaluated in terms of finite part.

The singular terms in Gtia and bHt

ia can be isolated by decom-

posing displacements and traction forces as

unfij ðP,Q Þ ¼ unf

KijKðmÞþunf

EijEðmÞ ð60Þ

tnfij ðP,Q Þ ¼ tnf

KijKðmÞþtnf

EijEðmÞ ð61Þ

where unfKij

, unfEij

and tnfKij

are regular functions and tnfEij

has singularity

of order 1=r. Also, the complete elliptic integrals K(m) and E(m)may be approximated as [54]

KðmÞ ¼ K1ðmÞ�K2ðmÞln m ð62Þ

EðmÞ ¼ E1ðmÞ�E2ðmÞln m ð63Þ

where

m¼4xr

ðxþrÞ2þz2, m ¼

r2

ðxþrÞ2þz2ð64Þ

are the modulus and the complementary modulus of the com-plete elliptic integrals of the fundamental solutions. The poly-nomials K1ðmÞ, K2ðmÞ, E1ðmÞ and E2ðmÞ are listed in Appendix C.

Using Eqs. (60)–(63) in Eq. (56), Gtia can be evaluated by the

numerical scheme suggested by Bialecki et al. [66], summarized inAppendix C. This approach splits the original integral into a regularintegral and a weakly singular logarithmic integral, yielding

Gtia ¼ 2p

Z 1

�1unf

KijK1þK2 ln

ð1�Z0ZÞ2

4m

" #(

þunfEij

E1þE2 lnð1�Z0ZÞ2

4m

" #)NaðZÞrðZÞJðZÞ dZ

�8pZ 1

0½unf

KijK2þunf

EijE2�ln ~Z Nað ~ZÞrð ~ZÞJð ~ZÞ d ~Z ð65Þ

where ZA ½�1;1� and ~ZA ½0;1� are the natural coordinate vari-ables. The first integral is regular and can be evaluated by the


Gauss–Legendre quadrature rule while the second integral haslogarithmic singularity and should be evaluated by the weightedlogarithmic Gauss quadrature rule [65]. Although the completeelliptic integral E(m) presents no singularity, its nonsingularapproximation also includes logarithmic terms which are alsoisolated to enhance numerical convergence.

Owing to Eqs. (60)–(63), bHt

ia as given by Eq. (57) also includeslogarithmic terms in tnf

KijKðmÞ and tnf

EijE2ðmÞln m. Thus, their inte-

gration can be carried out by means of a regular integral and aweakly logarithmic singular integral, similar to the procedureproposed for Gt

ia. On the other hand, the integral of tnfEij

E1ðmÞ existsonly in terms of the finite part, to be numerically evaluated by thescheme proposed by Dumont and Souza [67] for singular integralsin terms of order 1=r over a curved boundary. This procedureemploys the Gauss–Legendre quadrature and an additional cor-rection term, as summarized in Appendix C. Hence, bH t

ia can begiven by

bHt

ia ¼ 2pZ 1

�1tnf

KijK1þK2 ln

ð1�Z0ZÞ2

4m

" #(

þtnfEij


4m

" #)NaðZÞrðZÞJðZÞ dZ

�8pZ 1

0½tnf

KijK2þtnf

EijE2�ln ~Z Nað ~ZÞrð ~ZÞJð ~ZÞ d ~Z

�2pZ0½tnfEijrNaðZÞrðZÞ�Z ¼ Z0 ln92J9Z ¼ Z0�

Xng

m ¼ 1

wgm

1þZgm

( )ð66Þ

where Z0 is the value of Z at the singularity point (which is either

�1 or 1) and Zgm and wm

g are the abscissae and weights of theGauss–Legendre quadrature scheme for ng points. In the above

equation, the limit of tnfEijrNaðZÞrðZÞ for Z-n0 is difficult to obtain

analytically and has been evaluated numerically by extrapolation.Alternatively, the elements of the matrix bHt

ia that require theevaluation of singular integrals can be obtained indirectly byapplying to Eq. (48) known analytical solutions for simple stressstates, as suggested by Bakr and Fenner [56].

Case 2.2: x¼ 0. If x¼ 0, as depicted in Fig. 7, despite the 1=rsingularity of the displacement fundamental solutions, Gt

iabecomes regular due to the presence of r multiplying its integrandand can be integrated as presented in Eq. (58) for Case 1.

Similarly, the presence of r in the integrand of bH t

ia makes its

singularity of order 1=r, even though the singularity of tnfij is

actually of order 1=r2. Thus, the integration scheme suggested byDumont and Souza [67] can be applied as well, leading to

bHt

ia ¼ 2pZ 1

�1tnf

ij NaðZÞrðZÞJðZÞ dZ

Fig. 7. Case 2.2: Pðx,z0ÞAGi and Pðx,z0ÞAGt for x¼ 0: (a) rðZÞ9Z ¼ �1¼0; (b)

rðZÞ9Z ¼ 1¼0.

�2pZ0½tnfij rNaðZÞrðZÞ�Z ¼ Z0 ln92J9Z ¼ Z0�

Xng

m ¼ 1

wgm

1�Zgm

( )ð67Þ

4.3. Case 3: Pðx,z0ÞAGt , Gt �Gs

If Q ðr,zÞAGs, i.e. z¼0, and Pðx,z0Þ belongs to the portion of theboundary along which the integration is carried out, singularities

arise in the fundamental solutions unhij for either case x40 or x¼ 0.

In this case, it is preferable to manipulate and simplify the entireexpressions rather than treat the singularities in the fullspace and the

remaining terms separately. The integral bH t

ia vanishes since tnhij ¼ 0

satisfies the traction free boundary condition. The integration of Gtia

for these singular terms are discussed below in Cases 3.1 and 3.2.Case3.1: xa0. If Q ðr,zÞAGs, xa0 and Pðx,z0Þ belongs to the

portion of the element along which the integration is carried out,as depicted in Fig. 8, the term und

ij presents logarithmic singularityand the evaluation of Gt

ia involves an improper integral.For z¼0 and xa0, the fundamental solution unh

ij may bedecomposed as

unhij ðP,Q Þ ¼ unh

KijKðmÞþunh

EijEðmÞþunh

Lijð68Þ

The remaining functions are given by

unhKrr¼ð1�nÞðx2

þr2Þ

2p2mðxþrÞxr, unh

Krz¼ 0

unhKzr¼ 0, unh

Kzz¼ð1�nÞ

p2mðxþrÞð69Þ

unhErr¼�ð1�nÞðxþrÞ

2p2mxr, unh

Erz¼ 0, unh

Ezr¼ 0, unh

Ezz¼ 0 ð70Þ

unhLrr¼ 0, unh

Lrz¼

�ð1�2nÞhsign

4pmx if rox

�ð1�2nÞhsign

8pmxifr¼ x

0 if r4x

8>>>>><>>>>>:

unhLzr¼

0 if rox,

�ð1�2nÞhsign

8pmrifr¼ x,

�ð1�2nÞhsign

4pmrif r4x,

8>>>>><>>>>>:unh

Lzz¼ 0 ð71Þ

where hsign is either 1 or �1 for the halfspace defined for zr0 orzZ0, respectively.

Fig. 8. Case 3.1: Pðx,z0ÞAGs , Pðx,z0ÞAGt and xa0 for: (a) rðZÞ9Z ¼ �1 ¼ 0; (b)

rðZÞ9Z ¼ 1 ¼ 0.

Fig. 9. Case 3.2: Pðx,z0ÞAGs , Pðx,z0Þ=2Gt , x¼ 0 and rðZÞ9Z ¼ 1¼0.

Fig. 10. Boundary element model of a halfspace subjected to a uniform pressure

p on a circular surface of radius R¼5 m.


The complete elliptic integrals can be approximated as in Eqs.(62) and (63). Similar to the procedure presented for Case 2.1, onearrives at

Gtia ¼ 2p

Z 1

�1unh

KijK1þK2 ln

ð1�Z0ZÞ2

4m

" #(

þunhEij


4m

" #þunh

Lij

)NaðZÞrðZÞJðZÞ dZ

�8pZ 1

0½unh

KijK2þunh

EijE2�ln ~ZNað ~ZÞrð ~ZÞJ ~Z ð ~ZÞ d ~Z ð72Þ

in which the polynomials K1, K2, E1 and E2 are listed in Appendix C.Case 3.2: x¼ 0. If Q ðr,zÞAGs, x¼ 0 and Pðx,z0Þ belongs to the

portion of the element along which the integration is carried out,as depicted in Fig. 9, the term und

ij also has singularity of order 1=r.However, similar to Case 2.2, the presence of r multiplying theintegrand of Gt

ia makes the integral regular and the Gauss–Legendre quadrature rule can be applied.

Only a few modifications are needed in the integration schemewhen comparing the fullspace and halfspace computationalcodes. In the boundary element implementation, only the caseof Gt

ia for the integration carried out along part of Gs should becoded separately. In all other cases, the additional terms in thehalfspace fundamental solution can be handled using the Gauss–Legendre quadrature rule.

5. Numerical examples

In the following, some numerical examples are presented inorder to validate the proposed formulation. The BE method foraxisymmetric problems was programmed in FORTRAN 90/95, forlinear and quadratic elements. Six and eight integration pointswere used in the Gauss–Legendre and the logarithmic-weightedGauss quadratures, respectively.

5.1. Example 1: circular load on a halfspace

Fig. 10 illustrates the halfspace zr0 with shear modulus m andPoisson’s ratio n¼ 0:25, subjected to a uniform compressivenormal stress p over a circular region of radius R¼5 m. Theanalytical expressions for vertical displacement and axial stresscan be found in Selvadurai [68] and Milovic [69] as

uz ¼pR

2m ½2ð1�nÞI10�1ðR,r; zÞþzI100ðR,r; zÞ� ð73Þ

szz ¼ p A�n

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1þð1þtÞ2

q n2þt2�1

n2þð1�tÞ2EðkÞ

"8><>: þ1�t

1þtP0ðk,qÞ

�9>=>;ð74Þ

in which

n¼z

R, t¼

r

R, k2

¼4t

n2þðtþ1Þ2, q¼

4t

ðtþ1Þ2ð75Þ

and

A¼

1 if roR

1=2 if r¼ R

0 if r4R

8><>: ð76Þ

and the z-axis is positive in the direction shown in the figure.This problem was modeled with 11 nodes and five quadratic

elements, as depicted in Fig. 10. Results at internal points wereevaluated using a square grid of 25 m spaced by 0.5 m. Fig. 11shows good agreement between analytical and numerical resultsfor vertical displacement uz and axial stress szz. Displacementsand stresses on the surface z¼0 for r4R are evaluated as resultsat internal points, because of the halfspace formulation. In thiscase, as the free surface G0 does not require any discretization, theevaluation of results in its nearby region is exempted from thetreatment of the quasi-singular integrals arising in the fullspaceformulation due to the boundary effect.

5.2. Example 2: halfspace medium with an irregularly shaped cavity

subjected to a stress field

Consider a localized ring load pn ¼ ð1;1Þ MN applied to thecoordinates P¼ ð3 m,�5 mÞ of an elastic halfspace with shearmodulus m¼ 10 MPa and Poisson’s ratio n¼ 0:3. As proposed, thisring load has unit projections in both radial (pn

r ) and axial (pnz )

directions, according to Fig. 2. The displacement and the stressfields produced at any point Q ¼ ðr,zÞ can be evaluated by directlyapplying the halfspace fundamental solution of Eq. (32)

ui ¼ unhrj pn

r þunhzj pn

z ð77Þ

sij ¼ snhrjkpn

r þunhzjkpn

z ð78Þ

Let an irregular, axisymmetric patch, given by the segmentABCDEF shown in Fig. 12, be drawn in this elastic medium,generating a cavity by removal of the enclosed material. Sincethe ring load is inside the cavity, as shown in the figure, thecorresponding displacement and stress fields given by the aboveequations remain unaltered in the halfspace domain of interest ifthe traction forces corresponding to Eq. (78) are consequently

Fig. 12. Boundary element model of a halfspace medium with an irregularly

shaped cavity subjected to a stress field (dimensions in meters).

Fig. 13. Results along the segments A0B0 and C0D: (a) radial and vertical displace-

ments; (b) radial, vertical and shear stress components.

0

-5

-10

-25

-20

-25

z (m

)

0 5 10 15 20 25

r (m)

AnalyticalBEM

0.0

-0.5

-1.5

-2.5

-3.5

-2.0

-3.0

-1.0

-4.0u (m)

0

-2

-3

-1

-40.0 0.5 1.0 1.5 2.0

r/R

z/R

-1.0

0.0

-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

-0.9AnalyticalBEM

Fig. 11. Results of a vertical circular load p on a halfspace: (a) vertical displace-

ments uzðr,zÞ (m); (b) vertical stresses szzðr,zÞ=p.


applied to the cavity’s surface. Observe that no stress singularitiesactually occur in the vicinity of the reentrant corners given bynodes B, C, E and F, for the applied ring load, although a high

stress gradient is expected in the vicinity of the whole cavity. Avariety of numerical problems with mixed boundary conditionscan be formulated for the proposed example, as displacements orboundary tractions caused by the ring load can be complementa-rily prescribed along different parts of the cavity’s boundary,according to Eqs. (77) and (78).

The following numerical results are obtained for tractionforces given by Eq. (78) applied to the cavity – Neumannboundary conditions – for a BE model with 43 nodes and 21quadratic elements, as depicted in Fig. 12. The nodal displacementresults along the boundary ABCDEF are shown in Fig. 14, ascompared with the analytical solution given by Eq. (77). Displace-ments and stresses evaluated at 20 internal points along thesegments A0B0 and C0D0 of Fig. 12 are presented in Fig. 13. Table 2presents the global relative errors for the displacements andstresses evaluated along the boundary and in the domain usingthe norm

Error¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPðxai�xniÞ2P

x2ai

sð79Þ

Table 2Global errors of displacements and stresses along the boundary and in the domain.

Error (%)

Along the boundary In the domain

ur uz ur uz srr srz szz

1.17 0.35 0.44 0.06 1.06 0.45 0.48

Fig. 14. Radial and vertical displacements along the boundary ABCDEF.


where xa are the analytical results given by Eqs. (77) and (78), xn

are the numerical results and the summation index i refers to thenumber of results evaluated. All numerical results present goodagreement with the analytical results. This example has also beenanalyzed for mixed boundary conditions, with displacementsgiven by Eq. (77) considered as prescribed at some of the nodalpoints: the numerical results turned out to be visually indistin-guishable from the ones of Figs. 13 and 14.

6. Concluding remarks

The expression of the fundamental solutions by means ofintegrals of the Lipschitz–Hankel type with products of Besselfunctions was shown advantageous for the case of axisymmetricproblems. For both fullspace and halfspace problems, each term ofthe fundamental solutions was investigated, allowing the order ofthe singularities to be identified and isolated. Moreover, allexpressions that appear in the boundary element formulationcould be written in a more compact manner than given in thetechnical literature, providing more concise equations to beimplemented computationally and making it easier to find theirlimiting expressions in the cases of ring loads applied on the axisof symmetry. This is evident in the expressions for evaluatingdisplacements and stresses at domain points in terms of Somigli-ana’s identity.

Finally, this more compact representation has made explicitthat the halfspace fundamental solution incorporates the full-space fundamental solution. As the difference terms betweenthese two fundamental solutions present singularities only on thesurface of the halfspace, the implementation of the halfspace

formulation turned out to require only a few modifications toexisting codes for examining the fullspace problem.

Acknowledgments

Most of this work was developed by the first author during asandwich doctoral stage spent at the Department of Civil Engi-neering and Applied Mechanics, McGill University, and supportedby the Brazilian agency CAPES/PDEE. The authors also gratefullyacknowledge the support of the Brazilian agencies CNPq andFAPERJ.

Appendix A. Integrals of the Lipschitz–Hankel type involvingproducts of Bessel functions

The integrals of the Lipschitz–Hankel type involving productsof Bessel functions can be represented by

Ipqlðx,r; cÞ ¼

Z 10

JpðxtÞJqðrtÞe�cttl dt ðA:1Þ

where p, q and l are integers; and JpðxtÞ and JqðrtÞ are Besselfunctions of the first kind of order p and q, respectively. Theconvergent integrals of this type were tabulated by Eason et al.[53] and the expressions used in this formulation are

I000 ¼2k

pA1KðmÞ ðA:2Þ

I110 ¼�2ðk2�2Þ

pkA1KðmÞ�

4

pkA1EðmÞ ðA:3Þ

I100 ¼

�kc

2pxffiffiffiffiffixr

p KðmÞ�L0ðn,mÞ

2xþ

1

xif x4r

�kc

2px2KðmÞþ

1

2xif x¼ r

�kc

2pxffiffiffiffiffixr

p KðmÞþL0ðn,mÞ

2xif xor

8>>>>>>>>><>>>>>>>>>:ðA:4Þ

I001 ¼2ck3

pk2A3

1

EðmÞ ðA:5Þ

I111 ¼�4ck

pA31

KðmÞ�2ckðk2

�2Þ

pk2A3

1

EðmÞ ðA:6Þ

I101 ¼k

pxA1KðmÞþ

k3A2

pxk2A3

1

EðmÞ ðA:7Þ

I002 ¼�2c2k5

pk2A5

1

KðmÞ�2k3

pk2A3

1

1þ2c2k2

ðk2�2Þ

k2A2

1

24 35EðmÞ ðA:8Þ

I112 ¼2k

pA31

2þc2k2ðk2�2Þ

k2A2

1

24 35KðmÞþ2k

pk2A3

1

k2�2þ

2c2k2ðk4þk

2Þ

k2A2

1

24 35EðmÞ

ðA:9Þ

I102 ¼�ck5A2

pxk2A5

1

KðmÞþck3

pxk2A3

1

3�2k2A2ðk

2�2Þ

k2A2

1

24 35EðmÞ ðA:10Þ

I003 ¼2ck5

pk2A5

1

3þ4c2k2

ðk2�2Þ

k2A2

1

24 35KðmÞ


�2ck5

pk4A5

1

�6ðk2�2Þ�

c2k2

A21

8k4

k2þ23

!" #EðmÞ ðA:11Þ

I113 ¼�2ck3

pk2A5

1

3ðk2�2Þþ

2c2k2

A21

2k4

k2þ3

!" #KðmÞ

�2ck3

pk4A5

1

6ðk4þk

2Þþ

c2k2ðk2�2Þð3k

2þ8k4

Þ

k2A2

1

24 35EðmÞ ðA:12Þ

I103 ¼k5

pxk2A5

1

A2�5c2þ4k2ðk2�2Þc2A2

k2A2

1

24 35KðmÞ

þk3

pxk2A3

1

�3þ2k2ðk2�2ÞðA2�5c2Þ

k2A2

1

þc2k4A2

k2A4

1

8k4

k2þ23

!24 35EðmÞ

ðA:13Þ

I004 ¼�2k5

pk2A5

1

3þ24c2k2

ðk2�2Þ

k2A2

1

þc4k4ð24k4

þ41k2Þ

k4A4

1

24 35KðmÞ

�4k5

pk4A5

1

3ðk2�2Þþ

3c2k2ð8k4þ23k

2Þ

k2A2

1

þ4c4k4

ðk2�2Þð6k4

þ11k2Þ

k4A4

1

24 35EðmÞ

ðA:14Þ

I114 ¼6k3

pk2A5

1

k2�2þ

4c2k2ð2k4þ3k

2Þ

k2A2

1

þc4k4ðk2�2Þð8k4

þ5k2Þ

k4A4

1

24 35KðmÞ

þ12k3

pk4A5

1

k2þk

4þ

c2k2ðk2�2Þð3k

2þ8k4

Þ

k2A2

1

24þ

c4k4ð8�4k

2�3k

4�4k

6þ8k

8Þ

k4A4

1

35EðmÞ ðA:15Þ

I104 ¼�ck5

pxk2A5

1

�15þ4k2ðk2�2Þð3A2�7c2Þ

k2A2

1

þc2k4A2ð24k4

þ71k2Þ

k4A4

1

24 35KðmÞ

�ck5

pxk4A5

1

�30ðk2�2Þþ

k2ð8k4þ23k

2Þð3A2�7c2Þ

k2A2

1

24þ

8c3k4A2ðk2�2Þð6k4

þ11k2Þ

k4A4

1

35EðmÞ ðA:16Þ

in which Ipqlðx,r; cÞ ¼ Iqplðr,x; cÞ and

A1 ¼ 2ffiffiffiffiffixr

p, A2 ¼ x2

�r2�c2, A3 ¼�x2þr2�c2 ðA:17Þ

In the above expressions, K(m) and E(m) are the complete ellipticintegrals of the first and second kinds, respectively,

KðmÞ ¼

Z p=2

0ð1�m sin y2

Þ�1=2 dy ðA:18Þ

EðmÞ ¼

Z p=2

0ð1�m sin y2

Þ1=2 dy ðA:19Þ

The modulus k, the complementary modulus k and the parameterm are given by

k¼2

ffiffiffiffiffiffix r

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxþrÞ2þc2

q , k ¼

ffiffiffiffiffiffiffiffiffiffiffiffi1�k2

qand m¼ k2

ðA:20Þ

In Eq. (A.4), L0ðn,mÞ is the Heuman complete elliptic integralexpressed as

L0ðn,mÞ ¼2

pffiffiffiffiffiffiffiffiffiffi1�np

ffiffiffiffiffiffiffiffiffiffiffiffi1�

m

n

rPðn,mÞ

� �ðA:21Þ

where Pðn,mÞ is the complete elliptic integral of the third kinddefined as

Pðn,mÞ ¼

Z p=2

0ð1�n sin y2

Þ�1ð1�m sin y2

Þ�1=2 dy ðA:22Þ

and n is the characteristic number

n¼A2

1

ðxþrÞ2ðA:23Þ

Note that all Lipschitz–Hankel integrals Ipqlðx,r; cÞ listed above arewritten in terms of K(m), E(m) and PðmÞ, which can be numeri-cally evaluated by duplication as proposed by Carlson [70,71].

In the following, some useful limits are given, forr0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2þc2p

:

limx-0

I000 ¼1

r0

limx-0

I102

x¼�

3cð3r2�2c2Þ

2r70

limx-0

I110

x¼

r

2r3limx-0

I012 ¼3rc

r50

limx-0

I100

x¼

c

2r30

limx-0

I003 ¼�3cð3r2�2c2Þ

r70

limx-0

I010 ¼c�r0

rr0

limx-0

I113

x¼�

15rcð3r2�4c2Þ

2r90

limx-0

I001 ¼c

r30

limx-0

I103

x¼

3ð3r4�24r2c2þ8c4Þ

2r90

limx-0

I111

x¼

3rc

2r50

limx-0

I013 ¼�3rðr2�4c2Þ

r70

limx-0

I111

x¼

3rc

2r50

limx-0

I013 ¼�3rðr2�4c2Þ

r70

limx-0

I101

x¼�

r2�2c2

2r50

limx-0

I004 ¼3ð3r4�24r2c2þ8c4Þ

r90

limx-0

I011 ¼r

r30

limx-0

I114

x¼

45rðr4�12r2c2þ8c4Þ

2r110

limx-0

I002 ¼�r2�2c2

r50

limx-0

I104

x¼

15cð15r4�40r2c2þ8c4Þ

2r110

limx-0

I112

x¼�

3rðr2�4c2Þ

2r70

limx-0

I014 ¼�15crð3r2�4c2Þ

r90

Appendix B. Expressions for evaluating stresses in the domain

As presented in Section 3.3, stresses in the domain can berecovered by integrating the terms unh

ijkðP,Q Þ ¼ unfijkþund

ijk andtnhijkðP,Q Þ ¼ t

nfijkþt

ndijk along the boundary, where

tnðÞ

ijk ¼ snðÞ

ijklZl ¼ snðÞ

ijlkZl ðB:1Þ

For xa0, the terms above can be written in terms of theLipschitz–Hankel integrals as

unfrrr ¼

1

8pð1�nÞ1

x½�P5I110þ9z9I111�þP4I001�9z9I012

ðB:2Þ

unfrrz ¼

1

8pð1�nÞ �zI101

x�signðzÞ2nI001þzI002

( )ðB:3Þ

unfrzr ¼

1

8pð1�nÞ f�signðzÞ2P1I111þzI112g ðB:4Þ

unfrzz ¼

1

8pð1�nÞf�P2I101�9z9I102g ðB:5Þ


unfzzr ¼

1

8pð1�nÞ f�P2I011þ9z9I012g ðB:6Þ

unfzzz ¼

1

8pð1�nÞ f�signðzÞ2P1I001�zI002g ðB:7Þ

undrrr ¼

1

8pð1�nÞ1

x½�P7 I110þP59z9I111�2zz0 I112�þP6 I011

�signðzÞðP5z0 þ3zÞI012þ2zz0 I013

oðB:8Þ

undrrz ¼

1

8pð1�nÞ1

x½signðzÞ4P1P2 I100�P5zI101

�signðzÞ2zz0 I102��signðzÞ2P3 I001þðP5z0�3zÞI002

þsignðzÞ2zz0 I003

oðB:9Þ

undrzr ¼

1

8pð1�nÞ f�signðzÞ2P1 I111þðP5z0 þzÞI112�signðzÞ2zz0 I113g

ðB:10Þ

undrzz ¼

1

8pð1�nÞ fP2 I101�signðzÞðP5z0�zÞI102�2zz0 I103g ðB:11Þ

undzzr ¼

1

8pð1�nÞfP2 I011þsignðzÞðP5z0�zÞI012�2zz0 I013g ðB:12Þ

undzzz ¼

1

8pð1�nÞ f�signðzÞ2P1 I001�ðP5z0 þzÞI002

�2signðzÞzz0 I003g ðB:13Þ

snfrrrr ¼

m4pð1�nÞ

1

x½P4I101�9z9I102�þ

1

xr½�P5I110þ9z9I111�

þ

1

r½P4I011�9z9I012��3I002�9z9I003

ðB:14Þ

snfrrrz ¼

m4pð1�nÞ

1

x½signðzÞ2P1I111�zI112��signðzÞ2I012

þzI013

�ðB:15Þ

snfrrzz ¼

m4pð1�nÞ

1

x½�P2I101þ9z9I102�þ I002�9z9I003

ðB:16Þ

snfrzrr ¼

m4pð1�nÞ

1

r½�signðzÞ2P1I111þzI112�þsignðzÞ2I102

�zI103

�ðB:17Þ

snfrzrz ¼

m4pð1�nÞ fI112�9z9I113g ðB:18Þ

snfrzzz ¼

mzI103

4pð1�nÞðB:19Þ

snfzzrr ¼

m4pð1�nÞ

1

r½�P2I011þ9z9I012�þ I002�9z9I003

ðB:20Þ

snfzzrz ¼�

mzI013

4pð1�nÞðB:21Þ

snfzzzz ¼

m4pð1�nÞ

fI002þ9z9I003g ðB:22Þ

sndrrrr ¼

m4pð1�nÞ

1

x½P6 I101�signðzÞðP5zþ3z0ÞI102

þ2zz0 I103�þ

1

xr½�P7 I110�2zz0 I112þP59z9I111�

þ1

r½P6 I011�signðzÞðP5z0 þ3zÞI012þ2zz0 I013�

�5I002þ39z9I003�2zz0 I004

oðB:23Þ

sndrrrz ¼

m4pð1�nÞ

1

x½�signðzÞ2P1 I111þðP5zþz0ÞI112

�signðzÞ2zz0 I113�þsignðzÞ2I012�ð3zþz0ÞI013

�signðzÞ2zz0 I014

oðB:24Þ

sndrrzz ¼

m4pð1�nÞ

1

x½P2 I101þsignðzÞðP5z�z0ÞI102�2zz0 I103�

�I002þsignðzÞð�3zþz0ÞI003þ2zz0 I004

oðB:25Þ

sndrzrr ¼

m4pð1�nÞ

1

r½�signðzÞ2P1 I111þðP3z0 þzÞI112

�signðzÞ2zz0 I113�þsignðzÞ2I102�ðzþ3z0ÞI103

þsignðzÞ2zz0 I104

oðB:26Þ

sndrzrz ¼

m4pð1�nÞ

f�I112þ9z9I113�2zz0 I114g ðB:27Þ

sndrzzz ¼

m4pð1�nÞ f�zI103�signðzÞ2zz0 I104g ðB:28Þ

sndzzrr ¼

m4pð1�nÞ

1

r½P2 I011þsignðzÞðP5z0�zÞI012�2zz0 I013�

�I002þsignðzÞðz�3z0ÞI003þ2zz0 I004

oðB:29Þ

sndzzrz ¼

m4pð1�nÞ fzI013�signðzÞ2zz0 I014g ðB:30Þ

sndzzzz ¼

m4pð1�nÞ

f�I002�9z9I003�2zz0 I004g ðB:31Þ

where

P1 ¼ 1�n, P2 ¼ 1�2n, P3 ¼ 2�3n

P4 ¼ 3�2n, P5 ¼ 3�4n, P6 ¼ 5�6n, P7 ¼ 5�12nþ8n2 ðB:32Þ

For x¼ 0, these functions can be obtained in terms of limits

only. The required expressions for limx-0Ipql and limx-0 I pql can

be found in Appendix A. The terms unfijk and t

nfijk were also

presented by Tan [60,61] in terms of elliptic integrals.

Appendix C. Numerical integration schemes

This appendix presents the numerical schemes used to eval-uate the integrals arising in the boundary element formulationsfor axisymmetric problems. As only the meridian of the axisym-metric boundary needs to be discretized, these integrals areevaluated along the boundary Gðr,zÞ, for each portion betweenconsecutive nodes of an element.

C.1. Regular integral

Let f ðr,zÞ be a regular function on G, in the sense that it can beapproximated by a polynomial of a not too high degree in thedomain of interest. Then, its integral can be expressed in a naturalcoordinate system Z in the interval ½�1;1� and approximated bythe Gauss–Legendre quadrature rule [65], arriving atZG

f ðr,zÞ dG¼Z 1

�1f ðZÞJðZÞ dZffi

Xng

m ¼ 1

½f ðZÞJðZÞ�Z ¼ Zgm

wgm ðC:1Þ


where

JðZÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidr

dZ

� �2

þdz

dZ

� �2s

ðC:2Þ

is the Jacobian transformation between the global and naturalcoordinate systems. The coefficients Zg

m and wmg are the abscissas

and weights of the Gauss–Legendre quadrature rule for ng pointswithin the interval ð�1;1Þ, which suffice to exactly evaluate theintegral of a polynomial of order 2ng�1.

C.2. Weakly singular integral of logarithmic terms

Let f ðr,zÞ be a regular function and rðr,zÞ the distance betweenthe points Pðx,z0Þ and Q ðr,zÞ on the boundary Gðr,zÞ. It is necessaryto evaluate the following weakly singular integral:ZG

f ðr,zÞ ln rðr,zÞ dG¼Z 1

�1f ðZÞln rðZÞJðZÞ dZ ðC:3Þ

for the case rð�1Þ ¼ 0 or rð1Þ ¼ 0. A unified treatment of bothcases may be obtained by expressing

rðZÞ ¼ rðZ,Z0Þð1�Z0ZÞ ðC:4Þ

where Z0 is equal to either �1 or 1 and rðZ,Z0Þ is the non-vanishing part of rðZÞ for ZAð�1;1Þ. Then, the integral of Eq. (C.3)may be decomposed as [72]ZG

f ðr,zÞ ln rðr,zÞ dG¼Z 1

�1f ðZÞln½2rðZÞ�JðZÞ dZ

þ2

Z 1

0f ð ~ZÞln ~ZJð ~ZÞ d ~Z ðC:5Þ

in which the transformation to the natural coordinate system~ZA ½0;1� is given by

~Z ¼ 12ð1�Z

0ZÞ ðC:6Þ

The resulting integrals can be approximated by the Gauss–Legendre and logarithmic weighted Gauss quadratures rules[65], leading toZG

f ðr,zÞ ln rðr,zÞ dGffiXng

m ¼ 1

½f ðZÞln½2rðZÞ�JðZÞ�Z ¼ Zgm

wgm

þXnl

m ¼ 1

½f ð ~ZÞln ~ZJð ~ZÞ� ~Z ¼ Zlm

wlm ðC:7Þ

The coefficients Zlm and wm

l are the abscissas and weights of thelogarithmic weighted Gauss quadrature rule for nl points withinthe interval ð0;1Þ, which suffice to exactly evaluate the integral ofa polynomial of order 2nl�1.

The above integration scheme is obtained from a transforma-tion of variables and the use of Gauss–Legendre and logarithmicweighted Gauss quadrature rules. Other approaches can also beemployed [67,72].

C.2.1. Weakly singular integral of terms with the complete elliptic

integral of the first order

Let f ðr,zÞ be a regular function and K(m) the complete ellipticintegral of the first order with modulus

m¼4xr

ðxþrÞ2þðz0�zÞ2ðC:8Þ

given in terms of the coordinates of points Pðx,z0Þ and Q ðr,zÞ on theboundary Gðr,zÞ. The following weakly singular integralZG

f ðr,zÞKðmÞ dG¼Z 1

�1f ðZÞKðmÞJðZÞ dZ ðC:9Þ

needs to be evaluated, which actually encompasses two singula-rities in the case of KðmÞ-1 since m¼1 for either Z¼�1 or

Z¼ 1. The integration scheme presented was proposed by Bialeckiet al. [66].

The complete elliptic integral K(m) can be approximated, for0rmo1 and within an error Eo2� 10�8, by the expression [54]

KðmÞ ¼ K1ðmÞ�K2ðmÞln m ðC:10Þ

where

m ¼r2

ðxþrÞ2þðz0�zÞ2ðC:11Þ

is the complementary modulus of the complete elliptic integraland

K1ðmÞ ¼ a0þa1mþ � � � þa4m4

K2ðmÞ ¼ b0þb1mþ � � � þb4m4ðC:12Þ

are polynomials whose coefficients are given by

a0 ¼ 1:38629436112, b0 ¼ 0:5

a1 ¼ 0:09666344259, b1 ¼ 0:12498593597

a2 ¼ 0:03590092383, b2 ¼ 0:06880248576

a3 ¼ 0:03742563713, b3 ¼ 0:03328355346

a4 ¼ 0:01451196212, b4 ¼ 0:00441787012 ðC:13Þ

Substituting the approximation given by Eq. (C.10) in the weaklysingular integral in Eq. (C.9), the singular term can be isolated toobtainZG

f ðr,zÞKðmÞ dG¼ZG

f ðr,zÞ K1ðmÞþK2ðmÞlnrðr,zÞ2

m

" #dG

�2

ZG

f ðr,zÞK2ðmÞ ln rðr,zÞ dG ðC:14Þ

Applying the schema for regular and weakly singular integrals,presented in the previous sections, to the first and secondintegrals of the above equation, respectively, leads toZG

f ðr,zÞKðmÞ dG

ffiXng

m ¼ 1

f ðZÞ K1ðmÞþ2K2ðmÞ ln1�Z0Z2ffiffiffiffiffimp

� �JðZÞ

Z ¼ Zg

m

wgm ðC:15Þ

�4Xnl

m ¼ 1

½f ð ~ZÞK2ðmÞln ~ZJð ~ZÞ� ~Z ¼ Zlm

wlm ðC:16Þ

where ~Z is given in Eq. (C.6).

C.2.2. Weakly singular integral of terms with the complete elliptic

integral of the second order

Let f ðr,zÞ be a regular function and E(m) the complete ellipticintegral of the second order with modulus m, given in terms of thecoordinates of points Pðx,z0Þ and Q ðr,zÞ on the boundary Gðr,zÞ. Thefollowing weakly singular integralZG

f ðr,zÞEðmÞ dG¼Z 1

�1f ðZÞEðmÞJðZÞ dZ ðC:17Þ

needs to be evaluated for the case of m¼1 for either Z¼�1 orZ¼ 1. Although EðmÞa1 for this case, the quasi-singular termscan be isolated to enhance the convergence of the numericalintegration.

The complete elliptic integral E(m) can be approximated, for0rmo1 and within an error Eo2� 10�8, by the expression [54]

EðmÞ ¼ E1ðmÞ�E2ðmÞln m ðC:18Þ


where m is given by Eq. (C.8) and

E1ðmÞ ¼ 1þa1mþ � � � þa4m4

E2ðmÞ ¼ b1mþ � � � þb4m4ðC:19Þ

are the polynomials whose coefficients are given by

a1 ¼ 0:44325141463, b1 ¼ 0:24998368310

a2 ¼ 0:06260601220, b2 ¼ 0:09200180037

a3 ¼ 0:04757383546, b3 ¼ 0:04069697526

a4 ¼ 0:01736506451, b4 ¼ 0:00526449639 ðC:20Þ

The polynomial approximation of E(m) presents no singularity,since E2ðmÞ has no free coefficients, according to Eq. (C.18).However, the presence of ln m causes the integrand of Eq.(C.17) to be non-analytical, which requires a special numericaltreatment.

In a manner similar to that used in the previous section, thefollowing expression can be obtained for the numerical evalua-tion of the weakly singular integral given by Eq. (C.17)ZG

f ðr,zÞEðmÞ dG

ffiXng

m ¼ 1

f ðZÞ E1ðmÞþ2E2ðmÞ ln1�Z0Z2ffiffiffiffiffimp

� �JðZÞ

Z ¼ Zg

m

wgm ðC:21Þ

�4Xnl

m ¼ 1

f ð ~ZÞE2ðmÞln ~ZJð ~ZÞ�

~Z ¼ Zlm

wlm ðC:22Þ

for ~Z given by Eq. (C.6).

C.3. Cauchy principal value of the singular integral of order 1=r

Let f ðr,zÞ be a regular function and rðr,zÞ the distance betweenthe points Pðx,z0Þ and Q ðr,zÞ on the boundary Gðr,zÞ. The stronglysingular integralZG

f ðr,zÞ

rðr,zÞdG ðC:23Þ

has to be evaluated for the case rð�1Þ ¼ 0 or rð1Þ ¼ 0. This integralmay be obtained as a sum of a Cauchy principal value and adiscontinuous term asZG

f ðr,zÞ

rðr,zÞdG¼ PV

ZG

f ðr,zÞ

rðr,zÞdGþc ðC:24Þ

The evaluation of the discontinuous term c of the stronglysingular integrals appearing in the boundary element formula-tions is addressed in Section 3.2.

The Cauchy principal value is best evaluated in terms of twofinite-part integrals, denoted by �

R, for the boundary segments

adjacent to the singularity point rðr,zÞ ¼ 0.In what follows, the integration scheme proposed by Dumont

and Souza [67] is used. Using the notation of Eq. (C.6), the regularfunction can be expanded as a Taylor series to obtain thefollowing normalized integral of Eq. (C.23) over the curvedboundary G

�ZG

f ðr,zÞ

rðr,zÞdG¼�Z0½f ðZÞln9r9�Z ¼ Z0 þ �

Z 1

�1

f ðZÞrðZÞ JðZÞ dZ ðC:25Þ

The resulting quadrature rule for evaluating Cauchy’s principalvalue of the strongly singular integral of (C.23) is given by

�ZG

f ðr,zÞ

rðr,zÞdGffi

Xng

m ¼ 1

f ðZÞrðZÞ

JðZÞ� �

Z ¼ Zgm

wgm

�Z0½f ðZÞ�Z ¼ Z0 ½ln92r9�Z ¼ Z0�Xng

m ¼ 1

wgm

1�Zgm

( )ðC:26Þ

where

½rðZÞ�Z ¼ Z0 ¼ ½JðZÞ�Z ¼ Z0 ðC:27Þ

The above scheme, that employs the Gauss–Legendre quadraturerule and an additional correction term, evaluates exactly thisintegral for a polynomial function of order 2ng . Other numericalintegration schemes for the strongly singular integral can be used[72,73].

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Documents

Engineering Analysis with Boundary ElementsBoundary element formulation of axisymmetric problems for an elastic halfspace M.F.F. Oliveiraa,n, N.A. Dumontb, A.P.S. Selvaduraic a Computer