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Multiple crack problems of antiplane elasticity in an infinite body by using fredholm integral equation approach
Eng[neering Fracture Mechanics VoL 21, No. 3, pp. 473-478.1985 Printed ia the U,S.A.
0013-7944,'85 $3.00 + -00 9 1985 Pergamon Press Ltd.
MULTIPLE CRACK PROBLEMS OF ANTIPLANE ELASTICITY IN AN INFINITE BODY BY USING FREDHOLM INTEGRAL EQUATION APPROACH Y. Z. CHEN Division of Engineering Mechanics, Jiangsu Institute of Technology, Zhenjiang, Jiangsu, 212003 People's Republic of China Abstract--Two elementary solutions are present to solve the proposed problem. The first (second) elementary solution is defined as a solution that one pair of longitudinal concentrated forces is applied at a prescribed point of both edges of a single crack in infinite isotropic medium, with same magnitude and opposite direction (with same magnitude and same direction). Using the two elementary solutions and the principle of superposition, we found the proposed problem can be easily converted into a system of Fredholm integral equations. The system is solved numerically and SIF values at the crack tips can be easily calculated. Several numerical examples are given.
1. INTRODUCTION MANY investigators reduced the crack problem of fracture mechanics into a singular integral equation in mathematics. Refs.[l-3] are most famous in this field. As a counterpart of the traditional and classical approach (the singular integral equation (SIE) approach), recently we found the alternative (the Fredholm integral equation (FIE) approach) is more efficient [4, 5]. The particular feature of the FIE approach is that all the manipulations processed are elementary and straightforward. In this paper the multiple crack problems of antiplane elasticity in an infinite body are analyzed as follows. After deriving two elementary solutions and using the principle of superposition, the proposed problem can be easily converted into a system of Fredholm integral equations. Finally, the problems can be solved very easily9 There is no doubt of the particular advantage of the aforementioned approach. For example, a FORTRAN programme requires only about 100 cards and 5 minutes on FELIX C-256 computer to perform the calculations mentioned in the following sections. 2. TWO ELEMENTARY SOLUTIONS It is well known that, in antiplane elasticity the stresses, resultant force and displacement can be expressed by one complex potential ~(z) as follows[6]
(t)
,~, - i~,z = ~o'(z).
f=
(o~ dy - %.~dx) = (q~(z) - q~(z))/(2i) = Im[q~(z)].
(2) (3)
a w = (~(z) + f(z))/2 = Re[f(z)l.
where a~..and a~.~denote the stress component, w displacement in the longitudinal direction, G shear modulus, and f is the resultant force along some path from a fixed point zo to the moving point z. From (1) we have
a,~ = 89
(4)
- q,'(z)).
Suppose, there are some longitudinal tractions applied along the both edges of a single crack in an infinite body (Fig. 1). The boundary conditions can be written as
ay+=p(t)+q(t),
a~.~-=p(t)--q(t), 473
Itl
(5)
Y. Z. C H E N
474
C ~
I:
t
\o.i
~
p(t)-q(t)
Fig. I An infinite body containing a single crack.
If we let
r
= r
a(z)
=
(6)
(7)
-6(z).
After using (4), the conditions (5) become i[(b +(t) + fl - (t)]/2 = p ( t ) + q(t), i[q) - ( t ) + fl + (t)]/2 = p ( t ) -- q(t).
t[ < a
(8)
Itl < a
(9)
or
[alp(t) + fl(t)] + [~(t) + fl(t)]- = --4ip(t), [alP(t) - n(t)] + - [~(t) - ll(t)]- = --4iq(t). From above equations and the solution of the Hilbert problem[7], we get
1 f~ X(t)p(t)dt ~ X ( z ) _. t -- z
9 (z) = - - -
a(z) =
-
1 fa q__(t)dt ic rt .j _a t -- z + X(z----)'
I f~ X(t)p(t)dt+l I ~ q__(t)dt ic ~ rtY((z) J _ o t - z n d - , t - z + X(z)"
(lOa)
(lOb)
where c is a constant and
X ( z ) = (z 2 - a2) '/2.
(11)
Noting the properties that
X(z) = X(z),
and
X(t) = --X(t), H < a
(12)
and substituting (lOa) and (lOb) into (7), we get
Im (c) = O.
(13)
For the displacement must be single-valued, so from (3) and (6) we have
Re(~b(z)
d z ) = O.
(14)
Multiple crack problems of antiplane elasticity
475
Substituting (10a) into above equation and processing some manipulations, we also have
Re(c)=O
(15)
Finally, we get the solution as follows
q~(z)-------l'2(z)=
l fa X(t..)p(t_)d t
rcX(z) J_a
t-z
1 f" q ( t ) d t
(16)
:~j_~ t - - z
First elementary solution If we let p(t)=f(t--s),
q(t) = 0.
I/[
(17)
where 6 means Dirac function, then we easily get X(s) ~ ( z ) = ~ X ( z ) ( z --
(18)
s)
and the stress intensity factors at the crack tips A and B as follows
K3.a
\Tra(a + s)]
K3.B
\ ~ ( a -- s)]
(19)
Now the stress field given by ,~(z) shown in (18) is called the first elementary solution:
Second elementary solution If we let p(t) = 0,
q(t)=cS(t--s)
[tl
(20)
where ~ means the Dirac function, then we easily get qb(Z) =
1
~(z
-
s)
(21)
and the stress intensity factors at the tips A and B (Fig. 1) as follows
K3~ = K3.n = 0.
(22)
The stress field given by ~/,(z) shown in (21) is called the second elementary solution. Using the complex variable formulation of antiplane elasticity, one can get the traction ay, at the point z = x + iy located on the dashed line CD (Fig. 1), which has an inclined angle cz with respect to the ox axis, as o'r,z = o-rz cos a - o-x~sin a = -hn(ep(z) e~").
(23)
It is important to point out that the stress distribution on the whole region exterior to the crackcontour can be easily found from the above derivations. Obviously, these distributions are continuous in the region exterior to the crack contour. In addition, we shall see later, the two elementary solutions play an important role in our approach although they are very simple in the form.
476
Y . Z . CHEN
3. A SYSTEM OF F R E D H O L M INTEGRAL E Q U A T I O N S F O R T H E M U L T I P L E CRACK PROBLEMS Suppose, we have an infinite body with N cracks and the remote stresses are equal to zero. The applied longitudinal tractions on both edges of each crack are shown in Fig. 2(a). For example, the applied longitudinal forces on the upper edge of Kth crack is pk(s~) + qk(sk), and on the lower edge is pk(sk) -- q~(sk). NOW the problem, shown in Fig. 2(a), can be considered as a superposition of N single crack problems, shown in Fig. 2(b), with some undetermined forces on both edges on this crack. For example, in Fig. 2(b) the longitudinal forces on the upper and lower edges of Kth crack are Pk(sk) + qk(sk) and Pk(sk) -- qk(sk) respectively. As mentioned above, for any forces applied along the both edges of a single crack always cause a continuous distribution of stress in the region exterior to this crack, so the function qk(sk) is known and only Pk(sk) needs to be determined (Fig. 2b). Furthermore, using the two elementary solutions obtained and the principle of superposition, we get the following system of Fredholm integral equations N
Pk(sk) + ~ ' I= I
[.at
I
N
fu(sk, st)P,(st) dst = pk(sk) -- ~ '
J'-at
f
l
j_
I=I
gk~(Sk, s,)q,(st) ds~. k = 1,2 .... N
(24)
at
Here, the symbol Z" means that the term corresponding to l = k must be excluded in the summation. Using (23), the kernels f u and gu in the above equations can be found from (18) and (21) respectively. Obviously, the kernels in (24) are regular. Note that, the kernelsfu(sk, st) and gk~Sk, S~) have a definite meaning. For example, the fu(sk, st) (gu(sk, sl)) means that: (i) the influence source is a pair of longitudinal unit forces with opposite direction (with same direction) and is located at the point (st, 0) of the lth crack, (ii) the kind of the influence is a pair of distributed longitudinal forces acting at the point (sk, 0) of the kth crack (Fig. 2). After the system of integral eqns (24) is solved, or in another wording, the undetermined functions Pk(Sk) (k = 1,2 . . . . N) are obtained, the stress intensity factors at the crack tips can be obtained from K3a,k = ~
ak -- 1
f_'~
\ a k + s~ ,"
ds, (k = 1,2 ..... N)
(25a)
( k - - 1,2 ..... N)
(25b)
. S \ 1/2
where A or B represents the left or right crack tip respectively. So far the theoretical basis of proposed approach is established.
"~
, ~ "k~
I ~'r
x Fig. 2. Utilization of the principle of superposition for the solution of the multiple crack problems (a) Infinite body containing N cracks, (b) Infinite body containing a single crack.
Multiple crack problems of antiplane elasticity
477
1.8 --F38
-- -- --F~
1.6
0.60
1.4
--
0,50
--
0,40
i%
|
!
1.2
r
1.0
.
S---I
!_
.
X,~-~-~B ~ec
0,30
i
0.8
~---d
~D
1-2 .1
0.20
IF~= I
I
0.6
0.'I0
0.4
0~00
0
0.2
0.4
0.6
0.8
1.0
2a/d
Fig. 3. A pair of equal cracks inclined to each other (see (28))9
0
0.2
0.4
0.6
0.8
1.0
2a/d
Fig. 4. The values of stress intensity factors at four crack tips in the case of a uniform longitudinal traction applied on the upper edge of the crack AB (see (28)).
1,4
F3D
--F98
1.3
1". 2
1.1
1.0
0;9 0
I
I
0.2
0;4
q---0.6
0.8
1.0
2a/d
Fig. 5. The values of stress intensity factors at two crack tips of two parallel cracks, where B(D) represents the right crack tip of the lower (upper) crack (see (28))9
4. NUMERICAL EXAMPLES Three numerical examples are carried out in this paper. For all the derivations and computations are very simple, so we only explain the brief outline of computations. It is well known that the one possible way to solve the Fredholm integral equation in our occasion numerically and approximately, is that an integral equation is converted into an algebraic equation by using some quadrature rule. In our computations we use Chebyshev quadrature rule f~
an" ( 2 m - 1)rr ( 2 m - 1)rr a G(t) dt -~ -'~ ~= ~ G(t,,) sin 2M ' t m = a cos 2M m
(26)
then the system of Fredholm integral eqns (24) are converted into a system of algebraic equations9
478
Y.Z. CHEN
In this way, from (24) the values o f Pk(sk) at M discrete points can be easily obtained. W e use (25a,b) and the following Chebyshev rule ff
F(t) dt
a
~r M
(a ~ -- I'll/2 ~ ?"~ Z -
,~ =
F(tm),
trn = a COS
t
(2m 27,1
1)rr
(27)
The S I F values at all the crack tips can be also obtained. In the condition o f M = 9, the calculated numerical results are plotted in Figs. 3, 4 and 5. In the three figures, we express the S I F values as follows
K3~, = -- F3AT (na )'~, K3.B = F~B~(na )'a,
K,,c = --1%~ (~a )'~, K3.D = F3ox (na ) '~.
(28)
where A, B, C and D represent the four crack tips o f two cracks, 9 is the remote stress applied, and a is the half length o f crack. T h e results shown in the Fig. 3 and Fig. 4 were first o b t a i n e d in this paper. In addition, the results for inner crack tip shown in the Fig. 5 coincided with those cited in Ref. [6], so the a c c u r a c y o f o u r a p p r o a c h is examined by this numerical example. REFERENCES [1] v. v . Panasyuk, P. Savruk and A. P. Datsyshyn, A general method of solution of two-dimensional problems in the theory
of cracks, Engng Fracture Mech., 9, 481--497(1977). [2] F. Erdogan, Mixed boundary-value problems in mechanics, Mechanics Today 4, Pergamon Press, New York, 1978. [3] P. S. Theocaris, and N. I. loakimidis, Numerical integration methods for the solution of singular integral equations, Quartly AppL Maths. 35, 173-183 (1977). [4] Y. Z. Chen, A Fredholm integral equation approach for multiple crack problems in an infinite plate, Engng Fracture Mech. (In press) (1983). [5] Y. Z. Chert Reducing crack problem of a circular plate or an infinite plate containing a circular hole into Fredholm integral equation, Int. J. Fracture 23, R 101-104 (1983). [6] M. Isida, Method of Laurant series expansion for internal crack problems, Mechanics of Fracture I (Methods of Analysis and Solutions of Crack Problems), (Edited by G. C. Sih,), Noordhoff Int, New York (1973). [7] N. I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity. Groningen, Netherlands, 1953. (Receired 21 December 1983)
0013-7944,'85 $3.00 + -00 9 1985 Pergamon Press Ltd.
MULTIPLE CRACK PROBLEMS OF ANTIPLANE ELASTICITY IN AN INFINITE BODY BY USING FREDHOLM INTEGRAL EQUATION APPROACH Y. Z. CHEN Division of Engineering Mechanics, Jiangsu Institute of Technology, Zhenjiang, Jiangsu, 212003 People's Republic of China Abstract--Two elementary solutions are present to solve the proposed problem. The first (second) elementary solution is defined as a solution that one pair of longitudinal concentrated forces is applied at a prescribed point of both edges of a single crack in infinite isotropic medium, with same magnitude and opposite direction (with same magnitude and same direction). Using the two elementary solutions and the principle of superposition, we found the proposed problem can be easily converted into a system of Fredholm integral equations. The system is solved numerically and SIF values at the crack tips can be easily calculated. Several numerical examples are given.
1. INTRODUCTION MANY investigators reduced the crack problem of fracture mechanics into a singular integral equation in mathematics. Refs.[l-3] are most famous in this field. As a counterpart of the traditional and classical approach (the singular integral equation (SIE) approach), recently we found the alternative (the Fredholm integral equation (FIE) approach) is more efficient [4, 5]. The particular feature of the FIE approach is that all the manipulations processed are elementary and straightforward. In this paper the multiple crack problems of antiplane elasticity in an infinite body are analyzed as follows. After deriving two elementary solutions and using the principle of superposition, the proposed problem can be easily converted into a system of Fredholm integral equations. Finally, the problems can be solved very easily9 There is no doubt of the particular advantage of the aforementioned approach. For example, a FORTRAN programme requires only about 100 cards and 5 minutes on FELIX C-256 computer to perform the calculations mentioned in the following sections. 2. TWO ELEMENTARY SOLUTIONS It is well known that, in antiplane elasticity the stresses, resultant force and displacement can be expressed by one complex potential ~(z) as follows[6]
(t)
,~, - i~,z = ~o'(z).
f=
(o~ dy - %.~dx) = (q~(z) - q~(z))/(2i) = Im[q~(z)].
(2) (3)
a w = (~(z) + f(z))/2 = Re[f(z)l.
where a~..and a~.~denote the stress component, w displacement in the longitudinal direction, G shear modulus, and f is the resultant force along some path from a fixed point zo to the moving point z. From (1) we have
a,~ = 89
(4)
- q,'(z)).
Suppose, there are some longitudinal tractions applied along the both edges of a single crack in an infinite body (Fig. 1). The boundary conditions can be written as
ay+=p(t)+q(t),
a~.~-=p(t)--q(t), 473
Itl
(5)
Y. Z. C H E N
474
C ~
I:
t
\o.i
~
p(t)-q(t)
Fig. I An infinite body containing a single crack.
If we let
r
= r
a(z)
=
(6)
(7)
-6(z).
After using (4), the conditions (5) become i[(b +(t) + fl - (t)]/2 = p ( t ) + q(t), i[q) - ( t ) + fl + (t)]/2 = p ( t ) -- q(t).
t[ < a
(8)
Itl < a
(9)
or
[alp(t) + fl(t)] + [~(t) + fl(t)]- = --4ip(t), [alP(t) - n(t)] + - [~(t) - ll(t)]- = --4iq(t). From above equations and the solution of the Hilbert problem[7], we get
1 f~ X(t)p(t)dt ~ X ( z ) _. t -- z
9 (z) = - - -
a(z) =
-
1 fa q__(t)dt ic rt .j _a t -- z + X(z----)'
I f~ X(t)p(t)dt+l I ~ q__(t)dt ic ~ rtY((z) J _ o t - z n d - , t - z + X(z)"
(lOa)
(lOb)
where c is a constant and
X ( z ) = (z 2 - a2) '/2.
(11)
Noting the properties that
X(z) = X(z),
and
X(t) = --X(t), H < a
(12)
and substituting (lOa) and (lOb) into (7), we get
Im (c) = O.
(13)
For the displacement must be single-valued, so from (3) and (6) we have
Re(~b(z)
d z ) = O.
(14)
Multiple crack problems of antiplane elasticity
475
Substituting (10a) into above equation and processing some manipulations, we also have
Re(c)=O
(15)
Finally, we get the solution as follows
q~(z)-------l'2(z)=
l fa X(t..)p(t_)d t
rcX(z) J_a
t-z
1 f" q ( t ) d t
(16)
:~j_~ t - - z
First elementary solution If we let p(t)=f(t--s),
q(t) = 0.
I/[
(17)
where 6 means Dirac function, then we easily get X(s) ~ ( z ) = ~ X ( z ) ( z --
(18)
s)
and the stress intensity factors at the crack tips A and B as follows
K3.a
\Tra(a + s)]
K3.B
\ ~ ( a -- s)]
(19)
Now the stress field given by ,~(z) shown in (18) is called the first elementary solution:
Second elementary solution If we let p(t) = 0,
q(t)=cS(t--s)
[tl
(20)
where ~ means the Dirac function, then we easily get qb(Z) =
1
~(z
-
s)
(21)
and the stress intensity factors at the tips A and B (Fig. 1) as follows
K3~ = K3.n = 0.
(22)
The stress field given by ~/,(z) shown in (21) is called the second elementary solution. Using the complex variable formulation of antiplane elasticity, one can get the traction ay, at the point z = x + iy located on the dashed line CD (Fig. 1), which has an inclined angle cz with respect to the ox axis, as o'r,z = o-rz cos a - o-x~sin a = -hn(ep(z) e~").
(23)
It is important to point out that the stress distribution on the whole region exterior to the crackcontour can be easily found from the above derivations. Obviously, these distributions are continuous in the region exterior to the crack contour. In addition, we shall see later, the two elementary solutions play an important role in our approach although they are very simple in the form.
476
Y . Z . CHEN
3. A SYSTEM OF F R E D H O L M INTEGRAL E Q U A T I O N S F O R T H E M U L T I P L E CRACK PROBLEMS Suppose, we have an infinite body with N cracks and the remote stresses are equal to zero. The applied longitudinal tractions on both edges of each crack are shown in Fig. 2(a). For example, the applied longitudinal forces on the upper edge of Kth crack is pk(s~) + qk(sk), and on the lower edge is pk(sk) -- q~(sk). NOW the problem, shown in Fig. 2(a), can be considered as a superposition of N single crack problems, shown in Fig. 2(b), with some undetermined forces on both edges on this crack. For example, in Fig. 2(b) the longitudinal forces on the upper and lower edges of Kth crack are Pk(sk) + qk(sk) and Pk(sk) -- qk(sk) respectively. As mentioned above, for any forces applied along the both edges of a single crack always cause a continuous distribution of stress in the region exterior to this crack, so the function qk(sk) is known and only Pk(sk) needs to be determined (Fig. 2b). Furthermore, using the two elementary solutions obtained and the principle of superposition, we get the following system of Fredholm integral equations N
Pk(sk) + ~ ' I= I
[.at
I
N
fu(sk, st)P,(st) dst = pk(sk) -- ~ '
J'-at
f
l
j_
I=I
gk~(Sk, s,)q,(st) ds~. k = 1,2 .... N
(24)
at
Here, the symbol Z" means that the term corresponding to l = k must be excluded in the summation. Using (23), the kernels f u and gu in the above equations can be found from (18) and (21) respectively. Obviously, the kernels in (24) are regular. Note that, the kernelsfu(sk, st) and gk~Sk, S~) have a definite meaning. For example, the fu(sk, st) (gu(sk, sl)) means that: (i) the influence source is a pair of longitudinal unit forces with opposite direction (with same direction) and is located at the point (st, 0) of the lth crack, (ii) the kind of the influence is a pair of distributed longitudinal forces acting at the point (sk, 0) of the kth crack (Fig. 2). After the system of integral eqns (24) is solved, or in another wording, the undetermined functions Pk(Sk) (k = 1,2 . . . . N) are obtained, the stress intensity factors at the crack tips can be obtained from K3a,k = ~
ak -- 1
f_'~
\ a k + s~ ,"
ds, (k = 1,2 ..... N)
(25a)
( k - - 1,2 ..... N)
(25b)
. S \ 1/2
where A or B represents the left or right crack tip respectively. So far the theoretical basis of proposed approach is established.
"~
, ~ "k~
I ~'r
x Fig. 2. Utilization of the principle of superposition for the solution of the multiple crack problems (a) Infinite body containing N cracks, (b) Infinite body containing a single crack.
Multiple crack problems of antiplane elasticity
477
1.8 --F38
-- -- --F~
1.6
0.60
1.4
--
0,50
--
0,40
i%
|
!
1.2
r
1.0
.
S---I
!_
.
X,~-~-~B ~ec
0,30
i
0.8
~---d
~D
1-2 .1
0.20
IF~= I
I
0.6
0.'I0
0.4
0~00
0
0.2
0.4
0.6
0.8
1.0
2a/d
Fig. 3. A pair of equal cracks inclined to each other (see (28))9
0
0.2
0.4
0.6
0.8
1.0
2a/d
Fig. 4. The values of stress intensity factors at four crack tips in the case of a uniform longitudinal traction applied on the upper edge of the crack AB (see (28)).
1,4
F3D
--F98
1.3
1". 2
1.1
1.0
0;9 0
I
I
0.2
0;4
q---0.6
0.8
1.0
2a/d
Fig. 5. The values of stress intensity factors at two crack tips of two parallel cracks, where B(D) represents the right crack tip of the lower (upper) crack (see (28))9
4. NUMERICAL EXAMPLES Three numerical examples are carried out in this paper. For all the derivations and computations are very simple, so we only explain the brief outline of computations. It is well known that the one possible way to solve the Fredholm integral equation in our occasion numerically and approximately, is that an integral equation is converted into an algebraic equation by using some quadrature rule. In our computations we use Chebyshev quadrature rule f~
an" ( 2 m - 1)rr ( 2 m - 1)rr a G(t) dt -~ -'~ ~= ~ G(t,,) sin 2M ' t m = a cos 2M m
(26)
then the system of Fredholm integral eqns (24) are converted into a system of algebraic equations9
478
Y.Z. CHEN
In this way, from (24) the values o f Pk(sk) at M discrete points can be easily obtained. W e use (25a,b) and the following Chebyshev rule ff
F(t) dt
a
~r M
(a ~ -- I'll/2 ~ ?"~ Z -
,~ =
F(tm),
trn = a COS
t
(2m 27,1
1)rr
(27)
The S I F values at all the crack tips can be also obtained. In the condition o f M = 9, the calculated numerical results are plotted in Figs. 3, 4 and 5. In the three figures, we express the S I F values as follows
K3~, = -- F3AT (na )'~, K3.B = F~B~(na )'a,
K,,c = --1%~ (~a )'~, K3.D = F3ox (na ) '~.
(28)
where A, B, C and D represent the four crack tips o f two cracks, 9 is the remote stress applied, and a is the half length o f crack. T h e results shown in the Fig. 3 and Fig. 4 were first o b t a i n e d in this paper. In addition, the results for inner crack tip shown in the Fig. 5 coincided with those cited in Ref. [6], so the a c c u r a c y o f o u r a p p r o a c h is examined by this numerical example. REFERENCES [1] v. v . Panasyuk, P. Savruk and A. P. Datsyshyn, A general method of solution of two-dimensional problems in the theory
of cracks, Engng Fracture Mech., 9, 481--497(1977). [2] F. Erdogan, Mixed boundary-value problems in mechanics, Mechanics Today 4, Pergamon Press, New York, 1978. [3] P. S. Theocaris, and N. I. loakimidis, Numerical integration methods for the solution of singular integral equations, Quartly AppL Maths. 35, 173-183 (1977). [4] Y. Z. Chen, A Fredholm integral equation approach for multiple crack problems in an infinite plate, Engng Fracture Mech. (In press) (1983). [5] Y. Z. Chert Reducing crack problem of a circular plate or an infinite plate containing a circular hole into Fredholm integral equation, Int. J. Fracture 23, R 101-104 (1983). [6] M. Isida, Method of Laurant series expansion for internal crack problems, Mechanics of Fracture I (Methods of Analysis and Solutions of Crack Problems), (Edited by G. C. Sih,), Noordhoff Int, New York (1973). [7] N. I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity. Groningen, Netherlands, 1953. (Receired 21 December 1983)