A special boundary-element formulation for multiple-circular-hole problems in an infinite plate

COMPUTER METHODS NORTH-HOLLAND

IN APPLIED

MECHANICS

AND ENGINEERING

50 (1985) 263-273

A SPECIAL BOUNDARY-ELEMENT FORMULATION FOR MULTIPLECIRCULAR-HOLE PROBLEMS IN AN INF~ITE PLATE* Y.Z. CHEN Division

of Engineering

Mechanics, Jiangsu Institute of Technology, P.O. Box 80, Zhenjiung, Jiangsu, 212003, The People’s Republic of China

Received 9 July 1984

A special boundary-element formulation is proposed to solve the multiple-circular-hoIe problems in an infinite plate. The main ideas of the proposed formulation are followed from two directions. The first formulation in which, all the unknowns are taken from the boundary tractions, is like the boundaryelement method. In the second, all the derivations of the solution are derived from the complex variable method of plane elasticity. Using the principle of superposition, a system of linear algebraic equations for the unknowns is established. Finally, several numerical examples are given.

1. ~ntr~uction The multiple-circular-bole problems in an infinite plate have got much attention by many investigators. Previously published papers in this field are by Savin [l] and Ling [2]. Clearly, the above-mentioned works represent the achievements in the old age in which the electronic computers were not available. Recently, the finite element method (FEM) and boundaryelement method (BEM) in the analysis of solid mechanics are investigated and developed on a large scale as computers are widely used. There are numerous references in the field of FEM and BEM. Obviously, the FEM and BEM were developed primarily for their generality and versatility. Now, if one examines the FEM and BEM in detail, one may find that the derivations and programs in the FEM for a rather simple problem are always very lengthy. For the derivations as well as the programs in the FEM have to be completed by manual labour, so sometimes the efficiency and economy of using FEM and BEM are challenged. We think that if the classical problems of solid mechanics are solved by the use of the classical technique and the electronic computer, sometimes, the efficiency and economy of solving the problems are satisfactory. A lot of our papers were written under guidance of the above-mentioned policy. In [3], the torsional rigidities of several kinds of bars were calculated by the use of 36 unknowns only, and in [4] the stress-concentration factors of notches in a finite plate were calculated by the use of 16 unknowns. Clearly, 36 unknowns are just the number of the unknowns in several elements of the FEM. Technically, the main ideas of this paper are followed from two directions. First, we take all the unknowns from the tractions on the circular boundaries, so the formulation proposed is *The author is grateful for support from the Science Fund of the Chinese Academy of Sciences. 00457825/85/$3.30

@ 1985, Elsevier Science Publishers B.V. (North-Holland)

Y.Z. Chen, Formulation for multiple-circular-hole problems

264

like the BEM. Second, all the derivations for solving the proposed problems are derived from the complex variable method of plane elasticity [5]. In other words, we have not used the current BEM, for example, as mentioned in [6].

2. Two particular

solutions

for two particular

cases

In order to solve the proposed problems, two particular solutions for two particular cases are presented. In the first case two concentrated forces, N and T, are applied at a prescribed point of the circular boundary, as show% Fig. 1. In the second case some distributed tractions as shown in Fig. 2. are applied along some arc element ajbj+,aj+, of the circular boundary, In the solution of the first problem, one has to solve the boundary-value problem of plane elasticity, as shown in Fig. 1. In Fig. 1, an infinite plate contains one circular hole with zero remote stresses, and two concentrated forces, N and T, are applied at the prescribed point of the circular boundary. It is well known [5] that the stress and displacement components in plane elasticity can be expressed as 6x

+

a,, =4Re@(z),

(1)

a,, + iu,, = Q(2) + Q(z) + Z@‘(z) + JqZ) ) -

2G(u

(2)

-

+ iu) = K(P(Z) - zq~‘(z) - I/J(Z),

(3)

where a(z) = P’(Z), P(Z), F(z) = 9’(z), $,(z > are the complex potentials

in plane elasticity (51, K = (3 - ~)/(l + V) (under the plane-stress condition), K = 3 - 4~ (under the plane-strain condition), and v is Poisson’s ratio. From [5, Sections 82, 82a], the boundary-value problem shown in Fig. 1 has the following solution f&(z)= p

(=>= 0

N+iT ~ 27rR (N-WR

a ~--( z-a -

K

I+KZ

a ) ’

1

a(2 - a) + (1 +

27r

1 ~)a.2

+(N+iT)R 27-r

i

1

1

(z-a)’

z2

K l+K

2a 2’

i ;j)

where a = Re’* .

Fig. 1. Two concentrated

(6)

forces

N and T applied

at the prescribed

point of the circular

boundary

265

Y.Z. Chen, Formulation for multiple-circular-hole problems

Fig. 2. Some distributed

tractions

applied

along the arc element

of the circular

boundary.

Now we turn our attention to the second problem shown in Fig. 2. It is assumed that the arc in Fig. 2 is an element of the circular boundary, where a,Gaj+l aj = Reitam6),

bj+l =

Re’“,

aj+l = Rei(“+S) ,

and the angles (Yand S are shown in Fig. 2. Suppose, the boundary tractions along the element the following form (i.e. the interpolation equation) a, + r=

-

aibj+,kj+,

(7)

of the circular boundary

have

iu, = N(B)+iT(B)=A+B(~-l)=A+B(ei~B”‘-l), R,

a-s
(8)

In this case, if one takes A

= i

[(Nj+,

+ iT+l)-

(Nj +

iq)] tan 46 + $[(&+I + iT+1)+ (4 + i?;.)]

7

(94 Pb)

then the tractions at two ends/nodes

of the arc element have the following values

266

Y.Z. Chen. Formulation for multiple-circular-hole

the complex

Using (4) and (5) and the principle of superposition, the boundary-value problem shown in Fig. 2 become a

------“--“)(A

l a)

1

+

---

+ ( (Z-ff)2

Noting

the properties

(1+

1 Z2

potentials

corresponding

1)) R de,

+l?($-

~+KZ

z-a

problems

)(A+B(k-

to

W>

I))

K)az

-‘“)(A l+K

+ l?($-

Z’

l))]R

de.

(lib)

that Li = R2/a,

d0 = da/(k),

b,+, = R2/b,+,

from (6) and (7) after some manipulations

one gets

(124

+

c

_ (

Pb)

f2_f6@) __~t,+~R2(-~_~_~-~~)] z2

z2

where fi

=

aj+i

-

,,L-L

2Si,

f2=

aj,

a1

f4=

aj+l *

2aj,

a,+]

fs=~(-$$). I

f6(Z)= Log (=$)

>

f7(4=

I+’

&+-

(13)

Z - ffj ’

c = $[(Nj+, + i7;+,) - (Nj + iq)] cot S + $[(Nj+I •t iT+l) + (Nj + iq)] 7 1 D = (2R sin 6)i RN+1 +

The influences

of the tractions

i7;+,) - (Nj + iT)] e-‘” .

applied

along the arc element

(14)

a,Ga,+,

on the remote

point

Y.Z. Chen, Formulation for multiple-circular-hole problems

z = x + iy, which is located on the fictitious circle (Fig. 2), can be easily obtained influences are as follows iu,,,. = 2 Re a(z)-

a,,,, +

uo~e~+

e-“‘“‘[t@‘(z) + V(z)] ,

ia,<,. = 2 Re Q(z) + ezie’[Z@‘(z)+ q(z)] .

267

[5]. These

054 W-4

Nevertheless, the influences of the tractions along the arc element U,GkZj+, of the circle on the points, which are located on the circle itself, should be divided into two cases. In the first case the point bj+l is the midpoint of the arc element (Fig. 2). After some manipulations one easily gets the influences on the point bj+l (bj+l = R eip): a,, =

Re (A),

WW

0 d = Im (A),

(16b)

In the second case the point b is located outside the arc element O,G&+I (Fig. 2). After some manipulations one easily gets the influences on the point b (b = R eicu+‘), IpI > S) as a;r=O,

(174 VW

C&=0,

uee= fRe{(A-B)[-8-iLog~-2~~~‘e-i~] -6-iLogE

P2>

-2sinS-

(174

where 8, @, p1 and p2 are as shown in Fig. 2 and p1 = 2R sin @? + 6)

p2 = 2R sin f(P - 8).

v-9

So far the influences of the tractions applied along the arc element a,Qaj+, of the circle (1) on the points in the exterior of the circle are given by (15a) and (15b), (2) on the midpoint bj+l of the arc element (bj+l = R eia) by (16a), (16b) and (16c), and (3) on the point b (b = R eicu+@), 1PI > 8) outside th e arc element by (17a), (17b) and (17c), respectively. Clearly, the fundamentals of the proposed formulation rely on all the influences mentioned above.

3. A special boundary-element

formulation

In order to solve the proposed consider the multiple-circular-hole

for the proposed problem

problem, using the principle of superposition, we can problem (Fig. 3) as a superposition of several single-

268

Y.Z. Chen, Formulation for multiple-circular-hole

problems

Fig. 3. Multiple-circular-hole problem for an infinite plate with normal and tangential and r,(O) applied along the circular boundaries (only two circles are shown).

tractions

nl(O), t,(O), nz(61)

circular-hole problems with some undetermined tractions applied along the circular boundary. This idea was developed in [7]. From the viewpoint of BEM, the method of solution belongs to the indirect BEM [6]. Using this idea and dividing the circular boundary of each circle into several elements, the following special boundary-element formulation can be reached. (1) All the source points are located on the circular boundaries with equa1 space between them. The source points al, u2, . . . are shown in Fig. 3. In addition, at every source point the undetermined normal and tangential tractions are assumed (the IVl.; and T,, in the following equations (19a) and (19b)). (2) The interpolation equation of the normal and tangential tractions, as shown in (8), (9a) and (9b), is used. (3) All the observation points coincide with the midpoints of the neighbouring source points. The observation points bI, bz, . . . are shown in Fig. 3. Clearly, the normal and tangential tractions at the observation points are known and are obtained from the given boundary values (the nl.i and Q; in the following equations (19a) and (19b)). (4) The fundamentals of the proposed problem can be described as follows: One takes some suitable values of the normal and tangential tractions (Nt,i and & in (19a) and (19b)) at the source points a,, a2, . . . of each circle, so as to satisfy the boundary-value conditions of the normal and tangential tractions (n[,i and t/.i in (19a) and (19b)) at the observation points by, b 2, . . . of each circle. From the above special boundary-element formulation we get :

i:

/=I

i=l

5i

i=l

i=l

(fl,a,.n,k,h,,.NI,i+f

I.a,,t.k,b/,nTI.i)=nk,j,

k=l,T...,M

i=

tft.ai,n,k.b,,rNt.i+f

t,a,,r,k,*,,rTt,i)=fk,j

k=l,2,+,,.M,

~=L,2:**-J,

1,2,...,1,

(1%)

(19b)

Y.Z. Chen, Formulation for multiple-circular-hole problems

269

where M is the number of the circles, I is the number of the circular elements, and nk,j, are given boundary values of normal and tangential tractions at the discrete points of the circles. In above equations, the influence coefficients f have a definite meaning. For example, f I,ai.n,k, b,, f means that (i) the source is the normal traction (corresponding to am) applied at the point ai of the Ith circle; (ii) the influence is the tangential traction (corresponding to ati) applied at the point bj of the kth circle. Obviously, all these coefficients can be found from (Isa), (16a), (16b), (17a) and (17b). After the N,,i and T,,i (1~ 1,2, . . . , M, i = 1,2, . . . , I) are evaluated by solving (19a) and (19b), all the distributions of the stress and displacement components can be easily found from the second particular solution mentioned in the Section 2. Generally, the most interesting values to be investigated are the stresses u BBalong the circular boundaries. Similar to above derivations, the g,-values at the point bj of the kth circle can be expressed by tk,j

h4

I

(“@@hi = C C kL ai, /=I

n,k,

bjN/.i + g,,q,t,k. biTI,;)

3

(20)

i=l

k=l,2,...

,M,

j=l,2

,...,

I.

As before, the coefficients g in above equation can be easily found from (lSb), (16~) and (17~). It is important to point out that N1.i is the undetermined value placed on the node Ui, and 4i , ,+1a.r+l, so the influence coefficient f is the end-point of both arc element aizi and a,b. consists of two parts, one from the arc element Qisi and other from the arc element For simplification, the details of assembling the matrix in (19a) and (19b) are not &i+lai+l* cited here.

4. Numerical examples For the purpose of explaining the use of above-mentioned investigation, the boundary-value problem for two circular holes shown in Fig. 4 is considered. The five cases considered are as follows Case Case Case Case Case

A. B.

a;,, = 0,

a> =p, u;, = 0,

C.

muxx - p,

UTY= p, G

D. E.

uYX= 0, -UXX- 0,

u&=0, UTY= 0

= P,

p1 = 0, p1= 0, PI = 0, p1=p, p1 = 0,

p2=0.

(214

p*=o.

(21’4 WC)

p2=0. p2=0.

(214

p2=p.

(214

In all five cases, we have taken I = 28 in (19a), (19b) and (20). From the symmetric conditions of loading and geometry the number of the unknowns is 56 (=4x (28/2)), so the scale of computation is rather small. The uJp values obtained at several points of the two circular boundaries for the five particular cases are listed in Tables l-5, respectively. Several numerical results obtained by Ling [2] are also listed in Tables 2 and 3. Comparison proves that the accuracy of presented numerical results is excellent.

Y.Z. Chen, Formulation for multiple-circular-hole problems

270

B

Fig. 4. Boundary

Table I u&p values at several b

a

e

1 1

1

0.5

I 1 1

1.0

I 1 I 1 1 1 1 1

1 1 2 3 4 5

1.5 2.0 2.5 3.0 1.0 1.0 1.0 1.0

points

of the two circular

for two circular

boundaries

point E

point D

point C

-0.9111 -0.9110 -0.9156 -0.9233 -0.9319 -0.9402 -0.7388 -0.6365 -0.6101 -0.6369

2.596 2.636 2.676 2.716 2.753 2.786 1.878 1.278 0.8933 0.6657

-0.3900 -0.3724 -0.4968 -0.6126 -0.7005 -0.7652 -0.2318 -0.4028 -0.6581 -0.9143

-0.5979 -0.7324 -0.8105 -0.8745

points

of two circular

b

a

e

point F

point E

1 1

1 1

0.5 1.0

-0.9233 -0.8974

1 1

1 1

1.5 2.0

1 1 1 1 1 1

1 1 2 3 4 5

2.5 3.0 1.0 1.0 1.0 1.0

3.272 3.167 (3.151) 3.113 3.082 (3.066) 3.062 3.050 3.407 3.695 4.008 4.328

in the parentheses

-0.8908 -0.8948 -0.9036 -0.9139 -0.6358 -0.5111 -0.5285 -0.6526 are obtained

boundaries point

D

4.070 3.271 (3.264) 3.084 3.029 (3.020) 3.011 3.006 4.090 4.951 5.715 6.342 by Ling [2].

holes in an infinite

plate.

in Case A (see (21a) and Fig. 4)

point F

Table 2 u&p values at several

a Values

conditions

point B

point A

2.935 3,001 3.048 3.099

-0.9886 - 1.018 - 1.070 -- 1,137

in Case B (see (21b) and Fig. 4) point C

point B

point A

3.230 3.113 3.056 3.105

- 1.043 -1.156 - 1.349 - 1.568

3.138 3.251 3512 3.819

271

Y.Z. Chen, Formulation for multiple-circular-hole problems Table 3 u&p values at several points of two circular boundaries in Case C (see (21~) and Fig. 4) b

a

e

point F

point E

point D

point C

point B

point A

1 1

1 I

0.5 1.0

1.674 1.739

1 1

1 1

1.5 2.0

1 1 I 1 1

1 1 2 3 4

1

5

2.5 3.0 1.0 1.0 1.0 1.0

2.361 2.256 (2.255)” 2.197 2.158 (2.158) 2.131 2.110 2.668 3.059 3.398 3.692

3.680 2.899 (2.887) 2.587 2.416 (2.411) 2.311 2.241 3.858 4.549 5.057 5.428

2.632 2.381 2.246 2.230

1.892 1.845 1.699 1.530

2.149 2.232 2.442 2.682

1.785 1.821 1.849 1.872 1,242 0.7671 0.3648 0.0131

a Values in the parentheses are obtained by Ling [Z]. Table 4 gw/p values at several points of two circular boundaries in Case D (see (21d) and Fig. 4) b

a

e

point F

point E

point D

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 2 3 4 5

OS 1.0 1.5 2.0 2.5 3.0 1.0 1.0 1.0 1.0

0.3264 0.2615 0.2030 0.1630 0.1340 0.1123 0.6885 1.091 1.440 1.746

-0.4624 -0.3537 -0.2716 -0.2153 -0.1748 -0.1447 -0.9718 -1.534 -2.001 -2.418

2.642 1.145 0.6850 0.4634 0.3363 0.2557 2.177 2.898 3.431 3.807

point C 0.0384 0.7535 0.9020 0.9527 0.9743 0.9848 0.3309 -0.0385 -0.2199 -0.2124

point B

point A

1.136 1.093 1.057 1.036 1.024 1.016 1.004 0.8074 0.5157 0.2134

1.034 0.9945 0.9941 0.9954 0.9965 0.9974 1.062 1.306 1.706 2.129

Table 5 u&p values at several points of two circular boundaries in Case E (see (21e) and Fig. 4) b

a

e

point F

point E

point D

point C

point B

point A

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 2 3 4 5

0.5 1.0 1.5 2.0 2.5 3.0 1.0 1.0 1.0 1.0

1.034 0.9945 0.9941 0.9954 0.9965 0.9974 0.9794 0.9678 0.9578 0.9460

1.136 1.093 1.057 1.036 1.024 1.016 1.214 1.301 1.366 1.431

0.0384 0.7535 0.9020 0.9527 0.9743 0.9848 0.6808 0.6503 0.6262 0.6205

2.642 1.145 0.6850 0.4634 0.3363 0.2557 1.301 1.419 1.465 1.443

-0.4624 -0.3537 -0.2716 -0.2153 -0.1748 -0.1447 -0.1127 0.0376 0.1837 0.3168

0.3624 0.2615 0.2030 0.1630 0.1340 0.1123 0.0872 -0.0739 -0.2638 -0.4448

272

Y.Z. Chen, ~~rrnula~~~r~for rnui~~p~e-circ~iar-h#le problems

There are many outputs in the cum~utation, so it is di~cu~t tu present all of them here. Now two particular distributions of gee/p along two circular boundaries are shown in Fig. 5. In the computation, the pfane-stress condition and v = 0.3 are assumed.

P

6 Fig. 5, Particular distributions Fig. 4).

of w&p along two circular boundaries

in Case B; b = 1. a = 2, e = 1 (see (21h) and

Y.Z. Chen, Formulation for multiple-circular-hole problems

273

5. Conclusions From our practice and the above derivations, several conclusions can be reached. (1) The efficiency of above-mentioned formulation is out of doubt. For example, using the FORTRAN program on the FELIX C-256 computer (the average number of computations 200000 per second) only about 350 cards were needed to perform the calculation described above, and 60 seconds of CPU time to perform the calculation for one set of a prescribed geometry of the circular holes. (2) From the idea mentioned above we can also reduce the proposed problem to a solution of the Fredholm integral equations, as we have done in [7]. The interpolation equation (18) can reflect the differences (i.e. the term ((ZVj+r+ iT+,) - (Nj + iT,)) in (9a) and (9b)) of the tractions between two end-points of the arc element, and all the influences can be expressed in an explicit form, so the proposed approach is more effective than the Fredholm integralequation approach. (3) Clearly, the proposed approach can be easily extended to the cases of multiple-elliptichole problems and some similar problems of antiplane elasticity.

References [l] G.N. Savin, Stress Concentration around Holes (Pergamon Press, Oxford, 1961). [2] C.B. Ling, On the stresses in a plate containing two circular holes, J. Appl. Phys. 19 (1948) 77-82. [3] Y.Z. Chen and Y.H. Chen, Solutions of the torsion problem for bars with [-, [-, +- and T-cross-section by a harmonic function continuation technique, Internat. J. Engrg. Sci. 19 (1981) 791-804. [4] Y.Z. Chen, Solution of plane notch problems for a finite plate by the generalized variational method, Comput. Meths. Appl. Mech. Engrg. 42 (1984) 57-70. [5] N.I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity (Noordhoff, Leyden, 1953). [6] C.A. Brebbia, Progress in Boundary Element Methods (Pentech Press, London, 1981). [7] Y.Z. Chen, A Fredholm integral equation approach for multiple crack problems in an infinite plate, Engrg. Fract. Mech. 20 (1984) 767-776.