The universal algorithm based on complex hypersingular integral

ComputationalMechamcs18(1996)127-138 9 Springer-Verlag

1996

The universal algorithm based on complex hypersingular integral equation to solve plane elasticity problems S. G. Mogilevskaya

127 Abstract The effective numerical algorithm to solve a wide range of plane elasticity problems is presented. The method is based on the use of the complex hypersingular boundary integral equation (CHBIE) for blocky ~systems and bodies with cracks and holes. The BEM technique is employed to solve this equation. The unknown functions (displacement discontinuities (DD) or tractions) are approximated by Lagrange polynomials of the arbitrary degree. For the tip elements the asymptotics for the DD are taken into account. The boundaries of the blocks, cracks and holes are approximated by the arcs of the circles and the straight elements. In this case all the integrals (hypersingular, singular and regular) involved in this equation are evaluated in a closed form. Numerical results are given and compared either to the ones obtained by the other authors or to analytical solutions.

The early prejudice against HBIEs, when for the most part they have been avoided, seems to be overcome and by the efforts of a number of researchers [Kutt (1975); Ioakimidis (1982); Golberg (1983); Takakuda, Koizumi and Shibuya (1985); Linkov and Mogilevskaya (1986); Lin and Keer (1987); Kaya and Erdogan (1987); Budreck and Achenbach (1988); Cruse (1988); Gray, Martha and Ingraffea (1990); Krishnasamy, Schmerr, Rudolphi and Rizzo (1990); Chien, Rajiyah and Atluri (1990); Tsamasphyros and Dimou (1990); Wendland and Stephan (1990); Guiggiani, Krishnasamy, Rudolphi and Rizzo (1992); Cruse and Novati (1992), Lutz, Ingraffea and Gray (1992), etc.] HBIEs became an essential and a robust tool in boundary element analysis. Since a lot of papers concerning real HBIEs have been published it is impossible to cite all of them hefe. The different strategies of numerical treatment of these HBIEs were summarized and discussed in detail in the review article by 1 Tanaka, V. Sladek and J. Sladek (1994). Introduction The well-known advantages of the complex variables in 2-D The method of hypersingular integral equation (HBIE) tends elasticity problems stimulated the creation of the theory of to become a natural means for attacking problems involving displacement discontinuities. These are contact problems and complex hypersingular integrals and integral equations [Linkov and Mogilevskaya (1991, 1994, 1995); Ladopoulos (1992)]. The problems for bodies with cracks. HBIEs for these problems CHBIEs combine the merits of the two approaches into an contain as unknowns only the values (tractions and DD) that characterize the contact interaction. In the crack problems the effective way to solve plane problems with DD. This method was applied [Linkov and Mogilevskaya (1991), (1994)] to the use of HBIEs allows to avoid a well-known problem of problem of blocky systems with arbitrary shear moduli and degeneracy, when the upper and the lower crack surfaces Poisson's ratios and for bodies with cracks. The CHBIE for these coincide. problems allows to solve a wide range of plane elasticity problems by an universal way. The aim of this paper is to suggest an effective and universal Communicated by T. Cruse, 22 ]anuary 1996 algorithm of a numerical treatment of CHBIE. This algorithm is based on BEM. The basic idea is a possibility of the S. G. Mogilevskaya Department of Mathematics, Kuzbass Technical University, Vesennia smooth approximation of each smooth part of the boundary by a number of circular arcs and straight elements. So, only two 28, Kemerovo 650026, Russia types of the boundary elements are employed. If unknown Correspondence to: S. G. Mogilevskaya functions are approximated by the polynomials or the polynomials multiplied by a weight function (for the tip This work was supported by the International Science Foundation elements) all the integrals (hypersingular, singular and regular) (Grants R3X000, R3X300) funded by George Soros. The part of this work involved in this equation can be evaluated in a closed form. This was performed when the author was a visiting research fellow at the fact leads to improve accuracy of the calculations and to lower University of Minnesota. I am especially grateful to Prof. A. Linkov of the Institute of Problems of Mechanical Engineering (Russian computation costs. CHBIE then may be solved numerically by reduction to the system of linear algebraic equations and Academy of Science) for the fruitful discussion and Prof. T. Cruse of the University of Vanderbilt for much communication on by the use of collocation method. Nonconforming elements, the contents of this paper. when the collocation points are away from the edge points, are used. The choice of the collocation points, sizes of the boundary 1 The terminology refers to multiregions of interacting elastic bodies elements and the other important stages of the numerical with generalized interfaces ranging from fixed to having displacement implementation are discussed in detail. discontinuities. The terminology derives from Linkov (1983) and The method presented has been applied to the different was associated with the hypersingular formulation in Linkov and Mogilevskaya (199l). elasticity problems in finite and infinite geometry including

cracks (curved, branched and kinked) and blocky systems problems. In all the problems considered an excellent agreement with the analytical (if they exist) or with numerical solutions obtained by the other authors was observed.

2 CHBIE for problems of blocky systems and bodies with cracks This equation was obtained by Linkov and Mogilevskaya (1991) by direct differentiating the corresponding complex singular equation obtalned by Linkov (1983). It can be written as 128

1 r Au('c) 9

TcN~~( z -a t)z

--

1

1

4- 2-"SL

--2--'~L

a 1 --

d r -}- 7 ~ . ~ ( a l - a3) o ( r ) $3 (z, t) d z ZTgl L

7Cl L

2 1 i ! ala(z)S4(z,t)dg-a-ffa(t) = f ( t ) , t e L

(1)

where L is the totality of external and internal boundaries; Au = u § -- u- is the DD; u + and u - are the limit values of the displacements ifwe approach the contour from its left or right side respectively; a = a9 + iG; G is a normal and G is a shear traction on L; the unit normal n is directed to the right of the direction of travel; the direction of travel is arbitrary for contact boundaries between blocks; it is counter clockwise for the external boundary; a bar over a symbol denotes complex conjugation; the origin of the coordinate system is assumed to be inside some block; i = x / l; 1 a 1 --

1

2/1 +

2#-'

I+K + a3

1+I( + a2 =

-

-

2/1 +

-

-

1Fy171 ,au(C) 2#~ 1[

4/2~

y

.~] ; 4-a~ + dtt(a)-Y-axx-2Zy d{

02 z- t &stln'?-~

o~

co

¦z

oy

z- t

dfdi 1 i dtdz(g-[)2-(z-t)

]

o -Audr

L -

~

dtdz'\

1

Audy

! a(q--e) -- al ~(r -- z) dz + (a 3- al). ~ eAu(f-Y 9 ,

-,

~ (r-z)

] J

(4) where #t, ~y are the elastic constants of the f-th block, z is the point inside this block. In the crack problem the stress intensity factors (SIFs) at the crack tips are important characteristics of fracture mechanics. To calculate SIFs for the curvilinear crack we will use the following formuli which can be obtained directly from the other ones suggested by Linkov (1975) nr Savruk (1981)

2 iK2)~

-

x / ~ /2iexp ( - icpl2) lim ( - ~ - I

dt ~ - t

~4 - ~ ~ ) ~ - 2 d t ( e - ~ ) ~

8 z--t

8ty

5y171

(K~ -

_ (

0 z--t dt 1 S3(z,t ) = ~-~ln z= ~--t dt~-~ S4(z't)-

where the point z 0 belongs to the interior region. Boundary and contact conditions may be different on different patts of L and may include prescribed tractions, or DD, or prescribed relationships between tractions and DD. Note that the Eq. (1) accounts for cracks if one puts p+ = # - = #; K+ = K- = K, for corresponding parts ofL (then a~ = a 3 = 0 for these parts). For the infinite elastic plane with p curvilinear cracks this equation was obtained by Linkov and Mogilevskaya (1990). In the p articular case of multiple straight cracks it was formulated also by Chen (1995). The CHBIE (1) have been proved to have zero index in the funcfional class of N. Muskhelishvili [see Linkov and Mogilevskaya (1995)]. It means that there is no need for additional conditions to be satisfied. The stresses in the {-th block inner points can be calculated by using the following formuli

aYz -- axx + 2iaxy - tc~ 4- 1 ~i

2/2-

~~+1[

(3)

2/2-

a~=' a~v' a| are the stresses at infinity (for a finite region these stresses are equal to zero); t = x 4-/y;

S2(z,t ) _

Re~~d'c =0, (r - z0)

;

~c= 3 -- 4v in plane strain; ~: = (3 -- v)/(1 + v) in plane stress; /2+,/2- are the shear moduli of the contacting blocks; v+,v are their Poisson's ratios; the sign " + " (" - " ) indicates the elastic constants for the region that is to the lefl (right) of the direction of travel on L;

St ( r , t ) -

dr = 0

L r--Zo

14-K-

2/2 +

f(t)-

Au(z)

y171

I+K-t-

We take 1/2#- = 0, u- = 0 for a finite blocky system; 1/2# + = 0, u + = 0 for a plane with hole. To exclude rigid body movement (if only tractions are given on the whole outer contour) the following additional conditions are introduced:

1

y

1 z-t dt z - t

dt(C-~)2"

(K 1 - iK2)z2 -

(2)

~

~ exp ( - i~“

lim~~z2 x/~-2-~z "

(5)

Here (p is the angle between the tangent in the left tip z, and axis Ox; ~ is the similar angle in the right tip z 2.

3

Numerical treatment Equation (1) can be solved by BEM. The basic idea of the algorithm presented is a possibility of the smooth representation of the main types of the plane curves by a number of circular arcs and straight segments. The different methods of such representation exist. Three parameters need to be known to define a circle. To provide a smooth representation of a curve one have to keep the tangent in the common point of the adjoining circles. We will use here the method suggested in the work (see Volkov (ed.), (1987)). The core point in this method is as follows. The points A~, Az..... A~ which belongs to some curve and the angle (p~ between the axis Ox and the tangent in the point A~ are known. At the first stage of the approximation orte can define the circle which links the points A z and A2 and has the angle O~ between the axis Ox and the tangent in the point A r In the case when the angle between the segment A~ A2 and axis Ox is equal to ~o~or % _+ ~, one can approximate the curve between A~ and A 2 by a straight segment. Then the angle (P2 between the axis Ox and the tangent of the circle (segment) in the point A 2 may be defined. At the second stage one has the same problem as at the first one but the points A 2, A 3 and the angle (P2 are known. Following this way, one can provide the smooth representation of the curve. In the work cited above (Volkov (ed.)) no algorithm have been suggested to choose the points A z, A2..... A~ if the equation of the curve are known. As a circle is a curve with the constant curvature one can choose the points in such a manner that the curvature between the neighboring points does not change dramatically. The approximations of the elliptic and parabolic arcs by using the method presented are shown in Fig. 1. So, the only two types of the boundary elements need to be employed to represent any smooth part of the boundary. We represent the contour L by a totality of such boundary elements Lj, j = 1..... n.

Because of the continuity constraints (C ~continuity need for Au) we will use the nonconforming elements. The tractions a and DD Au are approximated by complex Lagrange polynomials of the arbitrary degree m - 1 at the each element L r For the tip elements (in the crack problems) we will take the asymptotics for Au into account. Then for the element Lj we have

AuJ(~) ~ ~, Au~P~(~)

(6)

g=I m

#(z)~

~~Q~(v)

(7)

(=1

where "C - - l"~

i and

'Q~(z),

for ordinary elements

PJ(~)=, ~4 TG-(- ~a J ) ,J

for the left tip a j

,~G(~),

for the right tip bS;

Au~, cr) are the values of Au(r) and a(z) in the nodes z~(d = 1..... m) of j-th element. Later on we will discuss the choice of the nodes in detail. On substituting (6), (7) into (1), (3), we get the system of equations (last two of them are taken into account if the additional condifions (2) are used)

f [c%(t)Au~+ fio(t)A~. 4- 7o(t)a ~ + (o(t)a~]

'~

=1{'=1

- a2~ia (t) ~ 2~r/f(t) Y

Y,

Bf

J~z~21 A7

q,9 AG~ 0

2 ~ Re [Z0 (%) Au~87] ~ 0

As A s .

.

(8)

ASA4 A,

j=ld=l

~/~//@~xA,"~

In this system


O

(16)

a

and to the following integrals and expressions for crack tip elements (only to evaluate ao, flo ) (11)

G:=,o (Tz_Ff where a, bare respectively the beginning and the end points of the element L. Both of these integrals can be evaluated analytically for any curvilinear element [Linkov and Mogilevskaya (1991), Linkov, Zubkov and Mogilevskaya (1994)]. We have ~Py

1 ~

=--~dy

(12)

~x/~-c)z~dz G;=~ -~~~-f2

~ z-t - 2 ! (~-~_t)87x/k(z - y

, s>O (17)

In the cases of the straight segment and circular arc all of the integrals (16), (17) are evaluated analytically. If te[a,b] the calculation of the rest integrals in (9) is eren simpler (if the same types of elements are employed). Consider now both of these cases of the boundary elements.

where

4(1) =

(-1)m-lfi%;...;d~(m-2)= (1/2) rr

4.1 The integrals for the straight element

~ z/c,;

The equation of the straight segment

r~dvap b-a

4(m--1)=--~%;

~ - a = g T a ( y a)

4(m)=l;

[a,b] has

the form

(18)

r=~d

a) te[a,b] In this case Eq. (18) can be written as

for ordinary elements (13) for tip elements;

z-t f -~

b-a b-d

(19)

Taking imo account (2) and (I9), we find S~(z, t) = S2(r,t) = S3(z,t) = $4 ('L, t ) = 0

(20)

and the corresponding integrals in (9) are equal to zero, b) tr

4.2 Integrals for the circular arc The equation of the circular arc ab can be written in the form (,c -- Z~) ('c -- Zc) = R2

b]

(24)

where z~ is a center of the circle and R is its radius.

Using Eq. (18) and the following relafions

b-a ~~=~~~;

a) teab

b-a ,/~7-c)=~,/~-c),

(21)

In this case using Eq. (24), we have 131

"c--t_,c--zy171

we get the following expressions for all the basic integrals (16), (17)

R2+(zy171171

zc--t

Then from the formuli (2) it follows

}

zSd,c

b-

b a+~s228228

df

s,(z,t)=0;

= i C:(a b-a_'~'(b-a~ '-'+' ,:o

,. - ~ - - ; ~ " ) t g-~7)

~_ "cdy

b - 8 t ,cd"c

~~ - J 7 m 7 - ~

s;-'+'(t)

S4(z,t ) -

b-- ,i

a+~--~(f-a)

!~ - ~ ' Py

(f-t)'df

s+l

b - - a ( b - - - ~-)s+2 __ ( a - - { ) s + 2

4b-a

(22)

s+2

Zc-t; (25)

c

iQ(z)dr

~-~-z~ ;

1

~ dt(s) b*-a '

~ Q(z) dz = ~,~=1= }k(~~-c),c'df

i Ql(r)dz

need to be evaluated in (9). The first two of them can be evaluated by using the formuli (12)- (15) and taking the coniunction, if one puts t = zy The third integral is equal to b

and

G4=

1

s87

and only the integrals

a

= a+~_ 8

1

1 ~--Z

{~---7 - Dm7 ai

G;= z ( / - t ) ' d f =

s2(,c,t) (y

s

(26)

where Ay is defined from the first formula (13). b) t~ab

b-a Using Eq. (24), we have

} ,cSdz

87

z ' ( r - zc)dz

1

zSdz_ R2L (27)

Gs~ 5

('C

where the integral

2!~Wk(z-c)cdz

5) 2

b

b - I;+l(t)-2J"

a

_ ~:>~~,,_2i~_t+~y

a--t+6_d(f-~)

b rsdr L f JaR=+ ( < - f)(~-z0 • can be evaluated by the following recurrence relation

(~-F) ~

~2~,

where/~(t) is defined by the correspondence formuli (14) (by the first one in (22) and by the second one in (23).

]'

=

l

~-[ 1

I~ < - g

b ' - a'

s

( t- z c

R2") Zc_t 2

In ( b - - 5) (b - z-!/= lo

0;8 09

5 equctt e[-s

F,,

20

0.95 0.98 0.99 6 equat et-s

¦

16

x/o

0.96 0.98 0.99 "~-\/= x/Q

07 0¦ 09iS0 -I

Fig. 2. Different schemes of representation of the surface of the straight crack

f

6

8

10

14

15

F1

1.000814

1.00048

1.000252 1.000071 1.000004 0.999947 0.999885

Table 1. DependenceofnormalizedSIF on the collocation points

133

(1981)): + Me (IM + I ~ ) t a n ~ - - T - i(t M - - t ~ )

F+ -- iF+ = ~ “

t[nM~2-

3I M - I~

where the lower (upper) index corresponds to the left (right) tip. The values of the integrals I~ are expressed through the elementary functions and the elliptic integrals K(k), 17,(k) if M = 1, 2, 3, 4. Hefe we consider two cases M = 2 and M = 4. For these cases we have

134

I + = 2E (sina) -- K ( s i n y 1

Fig. 3. Geometry and loading for the branched crack

I4+ --

Table 3. Normalized SIFs for branched crack

I4 - 2 c o s o K ( c ),

n

[(1 -- 4COS200K(C) + 4COS2Or (c)]

1

Kitagava and Yuuki Isita

CHBIE

2 3 4

1.0 0.9415 0.8636

5

0.7972

6 8 10 12

0.6592 0.5979 0.5511

1.000004 0,940965 0.863475 0,79725 0.742762 0.659574 0.59889 0.55245

1.0 0.94152 0.86354 0.79717 0.74255 0.65899 0.59794 0.55106

2cosc~

I Z = K(sinc0

c=tany

Suppose y = n/4 for M = 2 a n d ~ = n/8 for M = 4. To calculate SIFs each arc is a p p r o x i m a t e d b y 4 circular elements. The n o r m a l i z e d SIFs at the crack tips are shown in Table 4.

5.3 Elliptic and parabolic cracks

5.2 The system of M circular cracks This problem is shown in Fig. 4. In this case the normalized SIFs take the form (FI -- iF2 = (Kl -- iK2)/a V / ~ ) (see Savruk

ff

t

Consider the elliptic and parabolic cracks subjected to remote uniform tension (a~ = ax~ = 1, a~v = 0). The approximation of these cracks by the circular arcs is shown in Fig. 1. Due to the symmetry (Oy is line of symmetry) just half of the arcs are shown. Each circular arc is represented by one boundary element. The normalized SIFs (F 1 - iF2 = (K 1 -- i K 2 ) / a ~ ) are shown in Table 5 and compared with the ones obtained by Savruk (1981). He used the parameterizations to represent ellipse and parabola and solved a singular integral equation by using Gauss-Ghebyshev quadrature for a segment [ - 1,1].

5.4 V-shaped crack, finite plate This example was considered and compared to FEM results by Zang and Gudmundson (1988). The geometry and loading

6,,

I

~

---"~ ff Table 4. Normalized SIFs for M circular crack

ff

M

Analytical solution

CHBIE

2 4

0.57287 • 0,213565i 0.522670•

0.572925 • 0,213548i 0.522768•

Fig. 4. Geometry and loading for the system ofM circular cracks

elliptic crack

Table 5. Normalized SlFs for elliptic and parabolic cracks

parabolic crack

Savruk

CHBIE

Savruk

CHBIE

0.3576 _ 0,4610i

0.3576__ 0,4603i

0.8158• 0,3266i

0.81566• 0,32632i

Yl,

0'n=1.;fr,t-=0

co 0'3

0'n=l.; 0'.c=0

Fig. 7. Geometry and loading for the system of 4 square blocks --36--

p

_1

Fig. 5. Geometry and loading for the V-shaped crack

are shown in Fig. 5. They used 72 linear boundary elements to discretize outer boundary and used 25 Gauss-points for each straight part of the crack. For the integral along outer boundary, the standard BEM technique was utilized. For the integral along crack Gauss-Chebyshev quadrature was used. To calculate the same scheme by using CHBIE only 11 boundary elements were employed to represent outer boundary and crack (8 for outer boundary and 3 for crack). The results (Au=, Auf, where Au= and Au~ are normal and tangential DD) are plotted in Fig. 6. 5.5

Blocky system Problem for four square blocks are shown in Fig. 7. The interaction coefficient being k 1, we have - A u 9 = k1 a=; - Au~ = k 1y171 at each contact. To provide analytical solution (y = 1, y = 0, ~7=y= 0) we suppose 2p = 1, v = 0. We used only one element for each of the squares sides. The numerical results by using CHBIE provided values which were correct to 6 digits.

1.8 O

135

1.2

:D

6 Condusions The following merits of the CHBIE in crack and contact problems: (i) they coincide only those values that characterize contact interaction and allow to avoid the problem of degeneracy in the crack problems; (ii) they have a zero index (no additional conditions need to be satisfied); (iii) they combine the merits of two approaches (HBIE and complex variables) impel us to develop the algorithm of their numerical solution. The possibility to use the arcs of the circles for the smooth approximation of the plane curves leads to the fact that the only two types of the boundary elements (straight segments and circular arcs) need to be employed to solve CHBIE by BEM. All the integrals (hypersingular, singular and regular) involved in the CHBIE are evaluated analytically if unknown functions are approximated by a polynomials or a polynomials multiplied by a weight function. It simplifies the most difficult stage of BEM and leads to accuracy improvement and lowers the computation cost. The simple formuli allow to control accuracy in calculation of regular integrals. The given recommendations concerning the choice of the collocation points, the boundary elements sizes provide a possiblity to obtain the most precious results if one uses the nonconforming elements. The further way to improve accuracy is to use the conforming elements. This work is in progress. The numerical results for the systems with cracks and blocks allow us to condude that CHBIE is a very efficient tool for solving problems with DD.

0.6 7

Appendix

0 0

3

6

9

12

15

18 X

7.1 Evaluation of the integral I'q (u) Consider the first expression from the formula (14)

0.8

0 :D . O Then we have

(i) The case of ordinary element. Using the integration by parts, we have

i P~(z) dy

= k [H~_~ (u) + (u - C)Hr(U)]

+~ ~-~

~-t 1 m

(3S)

/"

s-

i

where (b

K(u)

1 m + A-jS=~I d d ( s ) ( s - -

dz

1)G; -2

and To evaluate H~ the following recurrence relation will be used

b

T--t

(z--r) H~(u)-(r_l)(c_u)L(e_u)~

9

k

= k,A-~ª

["

1~- r--~ H~_l(u) , r > l

(e -- t) 2

1 ~-T-;xVS-~(b--t) Ay

~/k(u-c)-x/b-a +,/b a -irc8"1

Ho(u ) = 2 b x / ~ 8a

Pf(T) (e -- t ) 2 a -- 8

L t

t

--

a'-'(a--t) 1

/

m

(39)

L} +(s)(,-1)!~=~ 1

where e = b for the left tip, e = a for the right tip.

m

by

y

(43)

(il) r = - ( q + l + i - s ) = > 0 Following a similar process as in (38), we have

The last integral in (43) can be written as b

b 8 (Il) : ~ (T -- u ) r % / k ( T -- C)dT : k [Br+ 1 (u) ~- (u -- c) Br (/.,/)] a

M, =,o ( ~ _ f)2

i~0

~

(44)

where

By using (24), we have r

G2, b

if

a

b

8

a

b-F

b

i = s-- 1

+ G~

G~-2 ~

if

i= s

if

i