integral equation formulation for thin shells—revisited

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Engineering Analysis with Boundary Elements 31 (2007) 539–546

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Integral equation formulation for thin shells—revisited

Subrata Mukherjee�

Department of Theoretical and Applied Mechanics, Kimball Hall, Cornell University, Ithaca, NY 14853, USA

Received 22 February 2006; accepted 20 October 2006

Available online 18 December 2006

Abstract

While the matter of solving problems in fracture mechanics with boundary integral equations (BIEs) has received considerable

attention, the ‘‘conjugate’’ problem of thin shells has received less. The latter problem is revisited in this work. The attempt here is to

clarify certain issues that were not fully addressed in earlier publications. In particular, the issue of consistency of the relevant regularized

boundary integral equations, when collocated at corresponding points on two sides of a thin shell, is addressed carefully. Some comments

on the corresponding BIE formulation for fracture mechanics problems are made at the end of the paper in order to compare and

contrast these two problems.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Boundary integral equations; Cracks; Thin shells; Fracture mechanics

1. Introduction

The problem of a crack in an (infinite or finite) linearelastic medium has received considerable attention in theboundary integral equation (BIE) literature (please see [1]and the references therein). The failure of the standard BIEin the limit of an infinitely thin crack is well known; as isthe idea of employing the hypersingular BIE (HBIE) tosolve this problem. The conjugate problem of a thin shellhas been solved with an elasticity BIE approach by Liu [2].Certain issues related to this formulation have beendiscussed in [3] (see, also, [1]).

The formulation presented in [3] is revisited in this paper.Regularized displacement BIEs, collocated at points xþ (onthe upper) and x� (on the lower) surfaces of a thin shell,are carefully derived. It is shown that consistency of theseequations requires, as expected, that the displacement mustbe continuous across a thin shell. Similarly, one concludesfrom the consistency requirement of the correspondingregularized stress BIEs that the stress must also becontinuous across a shell. Of course, such continuity inthese fields is to be expected. It is interesting, however, to

e front matter r 2006 Elsevier Ltd. All rights reserved.

ganabound.2006.10.003

255 7143; fax: +1 607 255 2011.

ess: [email protected].

demonstrate that consistency of the relevant regularizedBIEs demands continuity of the relevant fields.Finally, it is shown that while these BIEs in the limit of a

thin crack are well posed and solvable (this is well known),those for thin shells are not! Fortunately, however, problemsinvolving thin shells of finite thickness have been successfullysolved numerically by Liu [2] using the displacement BIE inits standard form—i.e. by modeling the displacementsseparately on the two shell surfaces rather than just theirdifference, together with careful evaluation of the nearlysingular integrals that arise in his formulation.

2. Mathematical preliminaries

Formulae for integral of the traction kernel T over aboundary element, and for the solid angle subtended by aboundary element at a point, are very important for therest of this paper. Hence, these issues are addressed first.The standard BIEs for three-dimensional (3-D) linear

elasticity involve the traction kernel. This kernel has theform [1, Eq. (1.19)]:

Tik ¼ �1

8pð1� nÞr2fð1� 2nÞdik þ 3r;ir;kg

qr

qn

�

þ ð1� 2nÞðr;ink � r;kniÞ

�. ð1Þ

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x

y

r(x , y)+

+

+

n

s

L

+

Z axis

Ψ

Fig. 2. Surface element Sþwith bounding contour L

þ.

S. Mukherjee / Engineering Analysis with Boundary Elements 31 (2007) 539–546540

In the above rðx; yÞ ¼ y� x (with x a source and y a fieldpoint), r ¼ jrj, r;k ¼ qr=qyk ¼ ðyk � xkÞ=r, n is the unitoutward normal to the boundary at y, d is the Kroneckerdelta and n is the Poisson’s ratio of the elastic material.

Inside approach: The following finite part integrals (FPintegrals in the sense of Mukherjee [4]) of the kernel T(see Fig. 1 below and (3.26) in [1]) are noted below:

Tikðx; yÞdSðyÞ ¼ �gðaÞik ðxÞ �

OðIÞðSþ;xÞ

4pdik, (2)

Tikðx; yÞdSðyÞ ¼ �gðcÞik ðxÞ �

OðIÞðS�;xÞ

4pdik, (3)

where

gðaÞik ðxÞ ¼

1� 2n8pð1� nÞ

�ik‘

I ðaÞLþ

1

rðx; yÞdz‘

þ�k‘m

8pð1� nÞ

I ðaÞLþ

r;iðx; yÞr;‘ðx; yÞ

rðx; yÞdzm. ð4Þ

In (2)–(4), x is a boundary point that can be xþ or x�

(see Fig. 1). The surface element Sþ� Sþ is a neighbor-

hood of xþ 2 Sþ. It is noted that, strictly speaking, theintegral on the left-hand side of (2) is strongly singular onlyfor x ¼ xþ and nearly strongly singular for x ¼ x�.Therefore, its FP designation is strictly valid only in thelimit of a very thin shell. (A similar comment also appliesto (3).)

Also, e is the alternating tensor, zk ¼ yk � xk, and theline integrals over L

þare evaluated in the anti-clockwise

sense when viewed from above. The expression for gðcÞik is

the same as that on the right-hand side of (4), except thatthe integral is now evaluated over L

�in the clockwise sense

when viewed from above. In the case of a very thin shell(shell thickness h! 0), gðaÞðxÞ � �gðcÞðxÞ.

The superscripts ðIÞ in (2) and (3) indicate approach toxþ from a point n inside the shell (see Fig. 1). The solidangle OðIÞ, subtended by S

þand S

�at xþ, are given by (see

Fig. 2 and [5])

OðIÞðSþ;xþÞ ¼

rðxþ; yÞ � nðyÞ

r3ðxþ; yÞdSðyÞ

¼

Z 2p

0

½1� cosðcðyÞÞ�dy, ð5Þ

x

x

S

S

S

S

SESE

+

++

n

n

+

--

--

^

^

Thin Shell

ξ

Fig. 1. Geometry of a thin shell. The unit normal point away from the

shell.

OðIÞðS�; x�Þ ¼

rðx�; yÞ � nðyÞ

r3ðx�; yÞdSðyÞ

¼

Z 2p

0

½1þ cosðcðyÞÞ�dy. ð6Þ

The solid angle O was first expressed in this way by Liu[2]. A Cartesian coordinate system OXYZ is chosen withthe origin at the source point (inside the shell) such that thepositive Z-axis intersects the appropriate surface (S

þfor

(5) and S�for (6)). In Fig. 2, c is the angle between the

positive Z-axis and rðxþ; yÞ with y 2 Lþ, and y is the angle

between the X -axis and the projection of rðxþ; yÞ in theX–Y plane (see Figs. 5 and 6 in [2]).It is noted here that for a very thin shell xþ � x�, so

that, one has OðIÞðSþ;xþÞ � OðIÞðS

þ; x�Þ and

OðIÞðS�; xþÞ � OðIÞðS

�;x�Þ.

Outside approach: The situation is very interesting if aboundary point x is approached by a point n outside theshell. The equations corresponding to (2) and (3) now are

Tikðx; yÞdSðyÞ ¼ �gðaÞik ðxÞ �

OðOÞðSþ;xÞ

4pdik, (7)

Tikðx; yÞdSðyÞ ¼ �gðcÞik ðxÞ �

OðOÞðS�;xÞ

4pdik. (8)

It is noted that the value of the tensor g is independent ofhow the point x is approached. The value of the solidangle, however, does depend on both the direction of thenormal to the surface as well as the direction of approach[5]. For a very thin shell, one has (see Fig. 2)

OðOÞðSþ; xþÞ ¼ �

Z 2p

0

½1þ cosðcðyÞÞ�dy ¼ �OðOÞðS�;xþÞ,

(9)

OðOÞðSþ; x�Þ ¼

Z 2p

0

½1� cosðcðyÞÞ�dy ¼ �OðOÞðS�;x�Þ.

(10)

An interesting observation: In this discussion, x is aboundary point which can be either xþ or x� (see Fig. 1).Further, very thin shells are considered here.It is noted that (5), (6) and the last two lines of the

paragraph ‘‘Inside approach’’ in Section 2 imply that

OðIÞðSþ; xÞi� OðIÞðS

�;xÞ. (11)

ARTICLE IN PRESSS. Mukherjee / Engineering Analysis with Boundary Elements 31 (2007) 539–546 541

Also, from (9), (10),

OðOÞðSþ;xÞ � �OðOÞðS

�;xÞ (12)

and, for very thin shells,

gðaÞðxÞ � �gðcÞðxÞ. (13)

Therefore, from (2), (3), (11) and (13), one gets

Tikðx; yÞdSðyÞi� Tikðx; yÞdSðyÞ (14)

while, from (7), (8), (12) and (13),

Tikðx; yÞdSðyÞ � � Tikðx; yÞdSðyÞ. (15)

The integrands in (14) are approximately equal andopposite (see (1)), i.e.,

Tikðn; yþÞ � �Tikðn; y

�Þ; nayþ; nay� (16)

for any source point n inside, on or outside a very thinshell, with yþ and y� corresponding field points across athin shell. In view of (16), the result (14) appearsanomalous. The reason for this apparent anomaly is that(14) contains not regular but FP integrals (see [4,5]).

3. BIEs for thin linear elastic shells

BIEs for a thin shell (Fig. 1), composed of an isotropic,homogeneous, linear elastic material, are given below.

3.1. The regularized displacement BIE collocated on the

shell edge

The displacement BIE for a thin shell collocated at apoint x 2 SE (see Fig. 1), analogous to Eq. (3.20) in [1], hasthe form:

0 ¼

ZSE

½Uikðx; yÞtiðyÞ � Tikðx; yÞðuiðyÞ � uiðxÞÞ� dSðyÞ

þ

ZSþ½Uikðx; yÞqiðyÞ � Tikðx; yÞviðyÞ�dSðyÞ. ð17Þ

Eq. (17) is true for both inside and outside approach.In (17), u is the displacement and s is the traction. The

sum of the tractions, q, and the difference of thedisplacements, y, across the shell, are defined as

qiðyÞ ¼ tiðyþÞ þ tiðy

�Þ; viðyÞ ¼ uiðyþÞ þ uiðy

�Þ (18)

for twin points yþ and y� across a shell.The displacement kernels U is [1]:

Uik ¼1

16pð1� nÞGr½ð3� 4nÞdik þ r;ir;k�, (19)

where G is the shear modulus of the material.It is noted that the following equations have been used to

derive (17):

Tikðx; yþÞ ¼ �Tikðx; y

�Þ; x 2 SE ; yþ 2 Sþ; y� 2 S�,

(20)

ZqB

Tikðn; yÞdSðyÞ ¼�dik; n inside shell;

0; n outside shell;

((21)

where the shell boundary qB ¼ SE [ Sþ [ S�:

3.2. The displacement BIE collocated on the shell surface

3.2.1. Inside approach

In this section, the regularized BIEs are derived forcollocation at both xþ and x� in order to check that theyare consistent.

Collocation at xþ: The displacement BIE for a thin shellcollocated at xþ (see Fig. 1), analogous to Eq. (3.25) in [1],is

ukðxþÞ ¼

ZSE

½Uikðxþ; yÞtiðyÞ � Tikðx

þ; yÞuiðyÞ�dSðyÞ

þ

ZSþ

Uikðxþ; yÞqiðyÞdSðyÞ

�

ZSþ�S

þTikðx

þ; yÞviðyÞdSðyÞ

�

ZSþ

Tikðxþ; yÞ½viðyÞ � viðxÞ�dSðyÞ

� uiðxþÞ Tikðx

þ; yÞdSðyÞ

� uiðx�Þ Tikðx

þ; yÞdSðyÞ. ð22Þ

Regularized forms of the FP integrals in (22) are given in(2)–(6).It is noted that the following equation has been used to

obtain the fourth term on the right-hand side of (22):

�

ZSþ

Tikðxþ; yÞ½uiðyÞ � uiðx

þÞ�dSðyÞ

�

ZS�

Tikðxþ; yÞ½uiðyÞ � uiðx

�Þ�dSðyÞ

¼ �

ZSþ

Tikðxþ; yÞ½viðyÞ � viðxÞ�dSðyÞ. ð23Þ

Collocation at x�: Eq. (22), collocated at x�, has theform:

ukðx�Þ ¼

ZSE

½Uikðx�; yÞtiðyÞ � Tikðx

�; yÞuiðyÞ�dSðyÞ

þ

ZS�

Uikðx�; yÞqiðyÞdSðyÞ

þ

ZS��S

�Tikðx

�; yÞviðyÞdSðyÞ

þ

ZS�

Tikðx�; yÞ½viðyÞ � viðxÞ�dSðyÞ


�; yÞdSðyÞ

� uiðx�Þ Tikðx

�; yÞdSðyÞ. ð24Þ

With Uikðx; yþÞ ¼ Uikðx; y�Þ; Tikðx; yþÞ ¼ �Tikðx; y�Þ;the first four terms on the right-hand side of (24) are easily

ARTICLE IN PRESSS. Mukherjee / Engineering Analysis with Boundary Elements 31 (2007) 539–546542

shown to be equal to the corresponding ones on the right-hand side of (22). The last two terms above must again beevaluated using (2)–(6).

Regularized versions: In general, equations like (22) and(24), containing finite parts of integrals, are not compu-table; and corresponding computable regularized versionsbecome necessary. These regularized versions of (22) and(24) are derived below.

Use of (2)–(6) makes the last two terms on the right-handside of (22):

gðaÞik ðx

þÞviðxÞ þukðx

þÞ þ ukðx�Þ

2�

vkðxÞ

4p

Z 2p

0

cosðcðyÞÞdy

(25)

while use of (2)–(6) makes the last two terms on the right-hand side of (24):

gðaÞik ðx

�ÞviðxÞ þukðx

þÞ þ ukðx�Þ

2�

vkðxÞ

4p

Z 2p

0

cosðcðyÞÞdy.

(26)

The relationship gðcÞik ðxÞ ¼ �g

ðaÞik ðxÞ has been used to

derive (25)–(26).Moving the second term in (25) over to the left-hand side

yields ð12ÞvkðxÞ on the left-hand side of (22). Moving the

second term in (26) over to the left-hand side yields�ð12ÞvkðxÞ on the left-hand side of (24). Finally, theregularized version of (22) can be written as

ð12ÞvkðxÞ ¼

ZSE



þ

ZSþ


�

ZSþ�S

þTikðx


�

ZSþ


þ gðaÞik ðx

þÞviðxÞ �vkðxÞ

4p

Z 2p

0

cosðcðyÞÞdy ð27Þ

while that of (24) has the same right-hand side as that of(22) but with the left-hand side equal to �ð1

2ÞvkðxÞ. In view

of this fact, one must have, for a very thin shell, viðxÞ ¼

ð12Þðuiðx

þÞ � uiðx�ÞÞ ¼ 0!

3.2.2. Outside approach

It is interesting to consider the situation when the pointxþ in Fig. 1 is approached from outside the shell. Thedisplacement BIE, in this case, has the form:

0 ¼

ZSE



þ

ZSþ


�

ZSþ�S

þTikðx


�

ZSþ



þ; yÞdSðyÞ � uiðx�Þ Tikðx

þ; yÞdSðyÞ.

ð28Þ

Using (7)–(9), the last two terms on the right-hand sideof (28) become

gðaÞik ðx

þÞviðxÞ �vkðxÞ

2�

vkðxÞ

4p

Z 2p

0

cosðcðyÞÞdy. (29)

Moving the second term in (29) over to the left-hand sideyields vkðxÞ=2 and one finally gets exactly the sameequation as (27).Following a very similar procedure as above, it can also

be shown that one gets the same equation whether thepoint x� in Fig. 1 is approached from inside or fromoutside the shell.

3.3. The stress BIE collocated on the shell surface—inside

approach

3.3.1. Collocation at xþ

The stress BIE for a thin shell collocated at xþ (seeFig. 1), analogous to Eq. (3.30) in [1], is

sijðxþÞ ¼

ZSE

½Dijkðxþ; yÞtkðyÞ � Sijkðx

þ; yÞukðyÞ�dSðyÞ

þ

ZSþ�S

þ½Dijkðx

þ; yÞqkðyÞ � Sijkðxþ; yÞvkðyÞ�dSðyÞ

þ

ZSþ

Dijkðxþ; yÞ½sk‘ðyÞ � sk‘ðxÞ�n‘ðyÞdSðyÞ

�

ZSþ

Sijkðxþ; yÞ½vkðyÞ � vkðxÞ

� dknðxÞðyn � xþn Þ�dSðyÞ

� vkðxÞ Sijkðxþ; yÞdSðyÞ

þ um;nðxþÞ ½Ek‘mnDijkðx

þ; yÞn‘ðyÞ

� Sijmðxþ; yÞðyn � xþn Þ�dSðyÞ

þ um;nðx�Þ ½Ek‘mnDijkðx

þ; yÞn‘ðyÞ

� Sijmðxþ; yÞðyn � x�n Þ�dSðyÞ. ð30Þ

In (30), one has

sijðxÞ ¼ sijðxþÞ � sijðx

�Þ, (31)

dijðxÞ ¼ ui;jðxþÞ � ui;jðx

�Þ. (32)

Also, E is the elasticity tensor (withsij ¼ Eijk‘uk;‘; sij ¼ Eijk‘dk‘) and the new kernels D and S

are

Dijk ¼ Eijmn

qUkm

qxn

¼ lqUkm

qxm

dij

þ mqUki

qxj

þqUkj

qxi

� �¼ �Sijk, ð33Þ

ARTICLE IN PRESSS. Mukherjee / Engineering Analysis with Boundary Elements 31 (2007) 539–546 543

Sijk ¼ Eijmn

qSkpm

qxn

np ¼ lqSkpm

qxm

npdij

þ mqSkpi

qxj

þqSkpj

qxi

� �np

¼G

4pð1� nÞr33qr

qn½ð1� 2nÞdijr;k þ nðdikr;j þ djkr;iÞ

�

�5r;ir;jr;k��þ

G

4pð1� nÞr3½3nðnir;jr;k þ njr;ir;kÞ

þ ð1� 2nÞð3nkr;ir;j þ njdik þ nidjkÞ � ð1� 4nÞnkdij �

ð34Þ

with

Sijk ¼ Eijmn

qUkm

qyn

¼ �1

8pð1� nÞr2½ð1� 2nÞðr;idjk þ r;jdik � r;kdijÞ

þ 3r;ir;jr;k�. ð35Þ

The first FP integral on the right-hand side of (30) above(see (3.36) in [1]) can be written as

Sijkðxþ; yÞdSðyÞ ¼ �Eijmn

I ðaÞLþSk‘mðx

þ; yÞ�‘nt dzt, (36)

where the integral over Lþ

is evaluated in the anti-clockwise sense when viewed from above.

Referring to (3.37), (3.38) and (3.40) in [1], the second FPintegral on the right-hand side of (30) becomes

JðIÞijmnðS

þÞ � ½Ek‘mnDijkðx

þ; yÞn‘ðyÞ

� Sijmðxþ; yÞðyn � xþn Þ�dSðyÞ

¼ Eijpq½Ið1ÞmnpqðS

þÞ þ I ð2ÞmnpqðS

þÞ� ð37Þ

with, from (3.44) and (3.45) in [1] and (2):

I ð1ÞmnpqðSþÞ � � hðaÞmnpqðx

þÞ ¼ �

I ðaÞLþ½Ek‘mnUpkðx

þ; yÞ

� Sm‘pðxþ; yÞðyn � xþn Þ��‘qt dzt, ð38Þ

I ð2ÞmnpqðSþÞ ¼ �

Z ðIÞSþ

Tmpðxþ; yÞdnq dsðyÞ

¼ gðaÞmpðxþÞdnq þ

OðIÞðSþ;xþÞ

4pdmpdnq. ð39Þ

Following a similar procedure, the last FP integral onthe right-hand side of (30) can be written as

JðIÞijmnðS

�Þ � ½Ek‘mnDijkðx

þ; yÞn‘ðyÞ

� Sijmðxþ; yÞðyn � x�n Þ�dSðyÞ

¼ � EijpqhðcÞmnpqðxþÞ

þ Eijpq gðcÞmpðxþÞdnq þ

OðIÞðS�; xþÞ

4pdmpdnq

" #. ð40Þ

As for the tensor g, a superscript ðaÞ on the fourth ranktensor h denotes anti-clockwise integration over L

þ, and a

superscript ðcÞ on it denotes clockwise integration over L�

(when viewed from above). For a very thin shell,hðaÞðxÞ � �hðcÞðxÞ.Putting everything together, the last two FP integrals on

the right-hand side of (30) become

� EijpqhðaÞmnpqðxþÞdmnðxÞ þ EijpngðaÞmpðx

þÞdmnðxÞ

þ Eijmnum;nðxþÞ

OðIÞðSþ;xþÞ

4p

þ Eijmnum;nðx�Þ

OðIÞðS�;xþÞ

4p. ð41Þ

Now, using (5)–(6), the last two terms in (41) become

ð12ÞðsijðxþÞ þ sijðx

�ÞÞ �sijðxÞ

4p

Z 2p

0

cosðcðyÞÞdy. (42)

Finally, the regularized form of (30) is

ð12ÞsijðxÞ

¼

ZSE



þ

ZSþ�S

þ½Dijkðx


þ

ZSþ


�

ZSþ

Sijkðxþ; yÞ½vkðyÞ � vkðxÞ

� dknðxÞðyn � xþn Þ�dSðyÞ

þ EijmnvkðxÞ

I ðaÞLþSk‘mðx

þ; yÞ�‘nt dzt

� EijpqdmnðxÞhðaÞmnpqðx

þÞ þ EijpndmnðxÞgðaÞmpðx

þÞ

�sijðxÞ

4p

Z 2p

0

cosðcðyÞÞdy. ð43Þ

3.3.2. Collocation at x�

The stress BIE for a thin shell collocated at x� (seeFig. 1), is

sijðx�Þ ¼

ZSE

½Dijkðx�; yÞtkðyÞ � Sijkðx

�; yÞukðyÞ�dSðyÞ

þ

ZS��S

�½Dijkðx

�; yÞqkðyÞ þ Sijkðx�; yÞvkðyÞ�dSðyÞ

�

ZS�

Dijkðx�; yÞ½sk‘ðyÞ � sk‘ðxÞ�n‘ðyÞdSðyÞ

þ

ZS�

Sijkðx�; yÞ½vkðyÞ � vkðxÞ

� dknðxÞðyn � x�n Þ�dSðyÞ

þ vkðxÞ Sijkðx�; yÞdSðyÞ

þ um;nðxþÞ ½Ek‘mnDijkðx

�; yÞn‘ðyÞ

� Sijmðx�; yÞðyn � xþn Þ�dSðyÞ

ARTICLE IN PRESS

β3

β2β1 α2

α1

α3

SE

SE

S-

S+

Fig. 3. Local coordinates on shell surfaces.

S. Mukherjee / Engineering Analysis with Boundary Elements 31 (2007) 539–546544

þ um;nðx�Þ ½Ek‘mnDijkðx

�; yÞn‘ðyÞ

� Sijmðx�; yÞðyn � x�n Þ�dSðyÞ. ð44Þ

In the limit of a thin shell, with Dijkðx; yþÞ ¼ Dijkðx; y�Þ,Sijkðx; yþÞ ¼ �Sijkðx; y�Þ, n‘ðy

þÞ ¼ �n‘ðy�Þ, the first four

terms on the right-hand side of (44) are easily shown to beequal to the corresponding ones on the right-hand side of(30). Using (36), it is seen that the fifth terms are also equal.Also, the last two terms of (44) can be shown to be equal tothe corresponding ones in (30).

Finally, the regularized version of (44) has �ð12ÞsijðxÞ on

its left-hand side, while its right-hand side equals that of(43). Hence, for a very thin shell, one must have [1,3]:

sijðxÞ ¼ sijðxþÞ � sijðx

�Þ ¼ 0. (45)

Eq. (45) implies that

qiðxÞ ¼ tiðxþÞ þ tiðx

�Þ ¼ 0. (46)

4. Determination of gradients and stresses from

displacements and tractions for moderately thin shells

It is assumed here that tractions are prescribed on the shellsurfaces Sþ and S� and displacements are prescribed on theshell edge SE . (A common situation is that SE is completelyclamped with u ¼ 0 for x 2 SE). Other permissible boundaryconditions, of course, can be modeled as well.

It is first noted that displacement gradients and stressesfor x 2 SE , if desired, can be obtained by employing themethod suggested by Lutz et al. [6]. For 3-D problems,local coordinates ak ðk ¼ 1; 2; 3Þ, are chosen at x 2 SE suchthat the a3-axis is normal and the a1 and a2 axes aretangential to SE at x. Now, tangential differentiation of thedisplacement interpolation functions provides the quanti-ties uk;d; k ¼ 1; 2; 3; d ¼ 1; 2: The remaining displacementgradients at x are obtained from the formulae:

u1;3 ¼t1G� u3;1;

u2;3 ¼t2G� u3;2;

u3;3 ¼ð1� 2nÞt32ð1� nÞG

�n

1� nðu1;1 þ u2;2Þ;

9>>>>>>=>>>>>>;

(47)

where G is the shear modulus of the elastic shell. Finally,sij ¼ Eijk‘uk;‘:

Determination of dk‘ðxÞ with x 2 Sþ, in terms of qk

and vk require application of the above idea on both surfacesof a shell (Fig. 3). First, in the ak coordinate frame, one has

dkg ¼ vk;g; k ¼ 1; 2; 3; g ¼ 1; 2. (48)

The remaining components of d are obtained as follows:

On Sþ; in the ak frame: u1;3ðxþÞ ¼ tþ1 =G � ðuþ3 Þ;1;

On S�; in the bk frame: u1;3ðx�Þ ¼ t�1 =G � ðu�3 Þ;1;

On S�; in the ak frame: �u1;3ðx�Þ ¼ t�1 =G þ ðu�3 Þ;1:

9>=>;(49)

Adding the first and last of (49), one has

At xþ in the ak frame: d13 ¼q1

G� d31. (50)

Similarly,

d23 ¼q2

G� d32, (51)

d33 ¼ð1� 2nÞq3

2ð1� nÞG�

n1� n

ðd11 þ d22Þ. (52)

It should be noted that in the above uþk � ukðxþÞ,

tþk � tkðxþÞ, and similarly at x�. Finally, s is related to d

by the equation sij ¼ Eijk‘dk‘.

5. Some comments on the exterior (crack) problem

The exterior (crack) problem is discussed in many papersand books. An example of use of the present formulation isavailable in Mukherjee and Mukherjee [1]. The geometry isshown in Fig. 4. It is noted that the unit normals now pointaway from the solid body.The results presented below have been published before.

This short section is presented here in the interest ofcompleteness of the present paper. It allows one tocompare and contrast two classes of problems with thinfeatures—thin bodies and thin gaps (see Section 6).


outer surface of the body

From [1], this equation, for x 2 qB0 is

0 ¼

ZqB0

½Uikðx; yÞtiðyÞ � Tikðx; yÞðuiðyÞ � uiðxÞ�dSðyÞ

þ

ZSþ½Uikðx; yÞqiðyÞ � Tikðx; yÞviðyÞdSðyÞ. ð53Þ


crack surface

The displacement BIE, collocated at a point on the cracksurface (see Fig. 4), is derived carefully. The insideapproach is considered to start inside the crack while theoutside approach starts from outside the crack, but frominside the body. The directions of the normals on Sþ andS� are reversed in Fig. 4 compared to those in Fig. 1.

ARTICLE IN PRESS

∂B0

∂B0

Crackx

x

x S+

S-

S-

+

-

^

^

ξ

n+

n−

S+^

Fig. 4. Geometry of a solid body with a crack. The unit normals point

away from the body.

S. Mukherjee / Engineering Analysis with Boundary Elements 31 (2007) 539–546 545

Therefore, the leading signs in the solid angle formulae (5),(6), (9) and (10) must now be reversed (e.g. nowOðIÞðS

þ;xþÞ ¼ �

R 2p0 ½1� cosðcðyÞÞ�dy). For the same rea-

son, application of Stokes’ theorem now changes anappropriate integral on S

þto a clockwise integral on its

boundary Lþ.

For the fracture problem, four distinct cases—approach

xþ or x�, either from inside or from outside the crack—all

lead to the same regularized BIE. That is,

ð12Þðukðx

þÞ þ ukðx�ÞÞ ¼

ZqB0

½Uikðxþ; yÞtiðyÞ

� Tikðxþ; yÞuiðyÞ�dSðyÞ

þ

ZSþ


�

ZSþ�S

þTikðx


�

ZSþ


� gðaÞik ðx

þÞviðxÞ

þvkðxÞ

4p

Z 2p

0


It is noted that (54) is identical to (3.28) in [1]. (Pleasenote that the integral in g is clockwise in (3.28) and theangle c in Fig. 3.2 of [1] is redefined as p� c in Fig. 2 ofthe present paper.)

5.3. The regularized stress BIE collocated on the crack

surface

This time, the result is

ð12Þðsijðx

þÞ þ sijðx�ÞÞ

¼

ZqB0



þ

ZSþ�S

þ½Dijkðx


þ

ZSþ


�

ZSþ

Sijkðxþ; yÞ½vkðyÞ � vkðxÞ � dknðxÞðyn � xþn Þ�dSðyÞ

� EijmnvkðxÞ

I ðaÞLþSk‘mðx

þ; yÞ�‘nt dzt

þ EijpqdmnðxÞhðaÞmnpqðx

þÞ

� EijpndmnðxÞgðaÞmpðx

þÞ þsijðxÞ

4p

Z 2p

0


Again, (55) is identical to (3.46) in [1], once thedifferences in the directions of the line integrals, and thedefinitions of the angle c, in the two publications, are takeninto account.

6. Solution strategies for crack and for shell problems

First, it is assumed that tractions are prescribed on Sþ

and on S� in either problem, with appropriate displace-ments and/or tractions prescribed on the outer boundary ofthe body for the crack problem, and on the shell edge forthe shell problem, respectively.The strategy for solving the crack problem is well known

(see, e.g. [1,3]). A dot product of (55) is first taken withnðxþÞ, making its left-hand side equal tiðx

þÞ � tiðx�Þ.

Eqs. (53) and the new equation (called (550)) can now besolved for the displacement difference vðxÞ and for theunspecified components of the tractions and displacementson qB0. (Note that d and s can be obtained in terms of vand q—see Section 3.2.2 of [1].) Finally, (54) can be used,as a post-processing step, to find uðxþÞ and uðx�Þ.Unfortunately, however, the situation is quite different

for moderately thin shells. Eq. (27) only contains v and q ofthe shell surface—neither the separate displacements northe tractions! The same, in effect, is true for (43) dottedwith nðxþÞ (with d and s obtained from v and q—seeSection 4). On physical grounds, the displacement differ-ence v on a moderately thin shell does depend on theseparate tractions sðxþÞ and sðx�Þ, and not just on thetraction sum qðxÞ (e.g. q ¼ 0 can mean either both zero orequal and opposite non-zero applied tractions on Sþ andS�, respectively). Hence, (27) cannot be solved for v given q

(it has to be singular) and the same is true for (43) (dottedwith nðxþÞ)! Fortunately, however, problems with very thinshells have been solved by Liu (see, e.g. [2]). For shells withsmall but finite thickness, Liu successfully employs thedisplacement BIE in its standard form (with both uðxþÞ anduðx�Þ separately rather than v), together with carefulevaluation of nearly singular integrals, in his work.

Acknowledgment

This research has been supported by Grant #CMS-0508466 of the National Science Foundation to CornellUniversity.

References

[1] Mukherjee S, Mukherjee YX. Boundary methods—elements, contours

and nodes. Boca Raton, FL: CRC Press, Taylor and Francis Group;

2005.

ARTICLE IN PRESSS. Mukherjee / Engineering Analysis with Boundary Elements 31 (2007) 539–546546

[2] Liu YJ. Analysis of shell-like structures by the boundary element

method based on 3-D elasticity: formulation and verification. Int J

Numer Methods Eng 1998;41:541–58.

[3] Mukherjee S. On boundary integral equations for cracked and for thin

bodies. Math Mech Solids 2001;6:47–64.

[4] Mukherjee S. Finite parts of singular and hypersingular integrals with

irregular boundary source points. Eng Anal Boundary Elem

2000;24:767–76.

[5] Mukherjee S. CPV and HFP integrals and their applications

in the boundary element method. Int J Solids Struct 2000;37:

6623–34.

[6] Lutz ED, Ingraffea AR, Gray LJ. Use of ‘simple solutions’ for

boundary integral methods in elasticity and fracture analysis. Int J

Numer Methods Eng 1992;35:1737–51.

integral equation formulation for thin shells—revisited

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