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Elastic circular inclusion in an infinite plane containing two cracks
Int. J. Engng Sci., 1973. Vol. 1 I. pp. 437-449.
Pergamon Press.
Printed in Great Britain
ELASTIC CIRCULAR INCLUSION IN AN INFINITE PLANE CONTAINING TWO CRACKS R. D. BHARGAVA
and RAJ RANI
BHARGAVA
Indian institute of Technology, Bombay, India Abstract-The elastic fields in an elastic circular inclusion and surrounding infinite matrix cont~ning two cracks symmetrically situated, are determined when the matrix is subjected to loads at infinity. In this problem, the elastic properties of inclusion could differ from those of the matrix. The Muskhelishvili’s technique is used. The solution depends upon two sets of suitable complexpotentials{@,,,(z),T,(Z)}, I@(Z), Uri(z)}for matrix and inclusion respectively, which solves the problem. 1. INTRODUCTION THE
INFINITE
plane containing an elastic circular inclusion is discussed in [ 11, [2] and 131. 133 also contains the case of elliptic insert. The crack problems are considered in 14-81 et al. The problem of two cracks under constant pressure in an infinite plane has been considered in [9]. The problem of ‘The interaction between a crack and an inclusion’ was considered by Atkinsion[6] and ‘The effect of a circular inclusion on the stresses around a line crack in sheet under tension’ by Tamate[7]. We extend the method given by Tamate to obtain the explicit solution to the problem of this paper. It may be seen that Atkinsion’s problem consists of one crack and a circular inclusion in an infinite plate, while the present problem consists of an inclusion and two symmetrically situated cracks along the horizontal diameter. 2. STATEMENT
OF
THE
PROBLEM
Consider the circular inclusion, of radius R and the centre origin, which is embedded in an infinite matrix containing two straight cracks L, and L, along x-axis (Fig. 1). It will be assumed that the matrix and the inclusion are perfectly welded along their common boundary L. The elastic properties of the inclusion may differ from those of matrix. The straight cracks L, and L, lie from f- b,O) to (-a, 0) and from (a, 0) to (b, 0) respectively. The matrix is subjected to prescribed loads at infinity. As shown by Muskhelishvili the stress components PxE, P,, and P,, and displacement components u,, U, can be expressed in terms of complex potentials cP(z), p (z) as follows:
pm+p,,= 2[(9(2)+ca)], P,,-P,,+2iP,,=
2[241’(z) +*(z)f.
(2.1)
---
2/.C(&,,+&,,,)
=
K@(Z)
--@(iI)
-Z@‘(Z)
-v’(Z).
(2.2)
The bar over the function represents the complex conjugate of the function. The comma denotes the differentiation with respect to subscript following it. As the inclusion boundary is a circle, it is convenient to introduce polar coordinates r, 0: z = x-t- iy = reie. The stress components P,,, PBB,Pro and derivatives of displacement 437
438
R. D.
BHARGAVA
and R. R. BHARGAVA
Fig. 1.
components
can be written in terms of tf, (2) , zl’ (z) as
pw-+-pfx3= 3@,(z) +rnl, P 08--P,,+2iP,,
= 2~z/~)~Z9p’(.z) -t*(z)].
(2.3)
2p(%efiU&B)
=iZ[K@‘(Z)-m+z~+(Z/z)wz)].
(2.4)
Here K = (3 - v)/( I + v) for generalized plane stress and K = 3 - 4v for plane strain case, v being the Poisson’s ratio and p is the shear modulus. It is convenient to introduce the complex potential R(z) in terms of Q,(z) and * (z) as follows: Q(z) = @(z) +z,G’(z) +@(z). Thus the stress and displacement @p(z) and 0(z) as:
components
(2.5)
may be written in terms of the functions
2[rD(z)+@‘z)l,
p,+p,,=
P ?#- iP,, = Q(z) +n(z)
+ (z-Z)&‘(Z)
and 2p(ur,3:+iuy.s)
=
K@(Z)
--n(z)
-
(z--+5’(z).
(2.7)
We now introduce the boundary conditions. (a) On the rims of the cracks, no external forces act. (b) There is no discontinuity in the stresses and displacements across the bounda~ of the inclusion. (c) The stresses at infinity are prescribed. These are the principal stresses IV, and iV2making angles cyand 7ri2 + CYwith x-axis.
Elastic circular inclusion in an infinite plane containing two cracks
439
The condition (a) may be written as P& - iP$, = 0,
(2.8)
the subscripts + and - indicate the limiting values as a point on the crack is approached from the positive (y > 0) or negative (y < 0) part of x,y plane. When represented in terms of complex potentials Q(z) and R(z) by (2.6) these conditions (2.8) give dual homogeneous Hilbert problem which may be written as: Q,‘(t) +0-(t) The superscripts
= 0; W(t) +sz+(t) = 0.
(2.9)
have the same meaning as defined above; and t is a point on the crack. 3. SOLUTION
FOR
MATRIX
The solution of equation (2.9) may be written directly with the help of complex variable technique. Taking into account that the functions Cp(z) and n(z) can have poles of various orders at origin and using the conditions (c), these can be written as:
a&(z)
=-;rt+;g4n(;)n++,(z)[(2r+r’)($;$) 2
(3.1)
(3.2) where X,(z) =
a2(z-a)-“*(~-~)-1/*(z+a)-l,2(Z+~)_1,2~
In the above equations the subscript m indicates that the potential functions a(z) refer to the matrix; and as is well known r=f(N1+N2);
r’=-t(Nl-ZV2)e-2i”;
(3.3) a,(z),
(3.4)
and also F = F(7r/2, d(b*-a*)/b) is elliptic integral of first kind and E = E(7r/2, d/(b*- a*)/b) is the elliptic integral of second kind. The constants A,‘s and B,‘s are unknowns, and are to be determined. In the region 0 < lzl < a, the functions Qm(z) and qIr, (z) may be expanded by Laurent series as follows: (3.5)
440
R. D.
BHARGAVA
and R. R. BHARGAVA
(3.6)
where S,,, is Kronecker
delta, and (3.8)
4. SOLUTION
FOR
INCLUSION
We now discuss the case of inclusion. The complex potentiafs for inclusion ]zl d R may be written as: (4.1) The subscript i denotes the case for inclusion and G,, H, are the unknown constants to be determined. 5. DETERMINATION
OF
CONSTANTS
The constants A,, B,t, G,, and H, are determined by the condition (6) which impties that on the common boundary of circular inclusion i.e. at r = R
(5.1) In terms of complex potential functions, at, ( Reie) f Cp, ( Rei@)- Re+P~ = 4[>+ ( fi”e@)+ cDi(j&@) - ~e-~e~)
( ReiB) - e+*, - e-2@-)
( Re@) *
(5.2)
Elastic circular inclusion in an infinite plane containing two cracks
441
and ff
( Reie) -
[K,&~
am( Reie) i-
Re-ie@im( Reie) + e-2’e*m(Reie) ]
= KicDi(Re”‘) - @i(Reie) -f-Re+e@8f(j%?) i- em2jeqi (ReiB).
(5.3)
The subscripts m and i have the same meaning as above; JL~,,pli are shear moduli of matrix and inclusion respectively. Substituting the values of Qmr V\Ir,from (3.5) and $, ‘Pi from (4.1) in (5.2) and (5.3) and then comparing the various powers of ele the following equations are obtained:
= (n-
_.!sKm Pm
i
’a
“D,-/”
Pm
>
1)&-l-H,_,(l-SI,,).
n = 1,2,. . .
-BB_,+CLiE_n_2 Pm
(rtt I) t 0
=
KiG,-G&,,.
(5.4)
n=O,l,...
(5.5)
The values of G, and H, may be written in terms of D,, D_,, E, and E_,, which are the linear functions ofA,‘s and B,‘s, as follows: c
=
h’Gn/Pm)+ 1 E RD,_ t&/&d - 1
A
0a
1+Ki
+
(/&l/&n)-’ l+Ki
H,=
n=o,
0
:
%,-
1+
Ki
& i
f2 >
(n+2)2
i
4
-%_,_,+ >
R -n-4_ (n+ 1) k-4, 0a
I,...
(5.6)
Substituting the values of G, and H, from (5.5) and (5.4) and solving, the following relations are obtained: D_, = 0,
(Pi’Pm)-l [(~-1)(~)2’ti,+(~~-‘E,-,]. D-n =- (/&Km/&n) +1
n=2,3,...
(5.7)
E_, = 0,
(5.8)
R. D.
442
BHARGAVA
and R. R. BHARGAVA
Comparing the values of D_, obtained from (3.6), and (5.7), the following equation is obtained:
4, =
-~~&-h-2 0
(5.9) n=2,3,...
Equating the two values of E_, obtained from (3.7), and (5.8) the equations for determining Bk’s are obtained: (5.10) and
(5.11)
n=o,
l,...
The values of B,‘s are determined from (5.10) and (5.11) by substituting the values for PI/I..&,,&, t%,, T, f”, Rla, Ha. The values of An’s are determined with the help of expression (5.9) using the values of B,‘s determined earlier. Therefore, the potentials am(z) and ?v, (z) are known. The constants G, and H, are determined from (5.6) using the values ofA,‘s and B,‘s already obtained from (5.9), (5.10) and (5.11). Hence the functions @(i(z) and Vi(z) are also known. The stress fields in the matrix and inclusion can now be determined by the use of (2.1) using the values of {am, Vr,} and {@i, Yr,} determined above. Thus, the problem is completely solved as all the constants have been determined, and the functions {Q(z), 9 (z) } satisfy all the conditions. 6. CONCLUSION
As is obvious, to obtain the solution one will have to solve infinite number of linear simultaneous equations. Fortunately the convergence is very fast. Some particular cases of interest were solved numerically and the results have been given in the
Elastic circular inciusion in an infinite plane containing two cracks
443
form of graphs. In each case, the load acts in the y-direction at inanity, the diameter of inclusion is 2, and the crack length (b - a) = 2. In Fig. 2, the stress intensity (&f/V) = l$t (~P,,(x, 0))) near the crack tip (a, 0) is taken along y-axis and is plotted v/s a (the distance of crack tip from the o~gin) along x-axis for various values of ~~~~~. The value of stress P,, at infinity is N. The values that were taken are: ~/~~ = 0, Q, 1 and 3. The case ~~~~ = 0 refers to that of the hole, and ~*~~~ = 1, when the elastic properties of matrix and inclusion are the same and hence in practice, there is no inclusion. This is the case of a matrix with
ot
I
I.0
Fig. 2. Stress intensityl~
I.5
I
I
2.0 a
2.5
for various values of&(tl,,,, vs the distance of crack tip from origin.
two cracks only. The analytical results for this case agree with those obtained in (9). Note that the vatues of a have been taken to vary from 1.05 to 2.75. As is clear from figure, (i) for any fixed value of a, the stress intensity decreases for increasing values of ~~/~~, (ii) for the range of values of ~~/~~ from 0 to 1, the stress intensity diminishes with the crack tip moving further. In the case when ~~~~ > 1, the stress intensity increases when a changes from l-05 to approximately 2.0 and then stabihses. The trends of the curves indicate that for (~~~~~) > 1, (&IN) + 0 as n + 1, and for (~*/~~) < 1, (~~~~ + 03as a -+ 1. For (~i~~~~ = 1, it may directly be proved from the results of this paper or from (9) that (~~~) = (1/~2~(b~-~~))[6~(EfF) -a”]. Nume~cally, the curve comes as a straight line, which is verified by the above theoretical result. For Figs. 3 and 4, the same numerical data have been taken, namely the diameter of the inciusion is 2 and the crack length (b - a) is 2, also the stresses at infinity is in the y-direction. In Fig. 3(i) the hoop stress P*~/~ on the boundary of inclusion (along y-axis) is plotted v/s the angle 8 made with x-axis, for three values of ~1~~ = 3, 1,3 for the case a = 1.5. it is interesting to note that hoop stress is positive and maximum at the x-axis (0 = 01, and increases as the ratio of shear moduli bilks increases. The same graphs for hoop stresses in inclusion vls the angular orientation of the points on the inclusion boundary from x-axis as in Fig. 3(i) are plotted for other values
444
R. D.
BHARGAVA
and R. R. BHARGAVA
of a = 2.0 and 2.5 in Fig. 3(ii, iii). The figures indicate that for larger values of a, the curves smoothen out. In Fig. 4(i) the graphs for hoop stress in matrix v/s the angle 13,is again repeated. This shows the behaviour of hoop stress. The graphs are drawn for the same data described already as in Fig. 3. As expected the behaviour of the matrix (in Fig. 4(i)) is opposite to that of the inclusion at the interface. As pi/pm decreases, the hoop stress at 0 = 0 increases.
Fig. 3(a).
Fig. 3(b).
Elastic circuiar inclusion in an infinite plane containing two cracks
445
It may be seen that for the case when ki/pm = 1, the curves for hoop stresses for inclusion and matrix across the circular boundary are the same. The Fig. 4(ii, iii) give the hoop stresses for matrix, along y-axis v/s 0 for a = 2-O and 2.5. The values of the hoop stresses are approximately the same near about 58” for a = l-5.2-0 and 2.5.
Fig. 3(c).
Fig. 3(d).
LIES Vol. 1I. No. 4-D
446
R. D. BHARGAVA
and R. R. BHARGAVA
0.
Fig. 4(a).
40 t
I
PO-
Fig. 4(b)
Elastic circular inclusion in an infinite plane containing two cracks
Fig. 4(c).
Fig. 4(d).
R. D. BHARGAVA and R. R. BHARGAVA
448
In Fig. 5, two curves for stress intensity at the crack tip (a, 0) are drawn, for two cases when infinite plate contains (i) a circular inclusion and a straight crack and (ii) a circular inclusion
inclusion with two symmetrically situated cracks. For, one crack and an the curve is drawn with dotted line and is the same as given by Atkinsion in
[6]. The other curve with the continuous
line is for two cracks and an inclusion. In
Y
A !I -
I
I
I
\
4-
15 -
17I I
I
I
I
14
16
16
2
8”
Distance from inclusion
Fig. 5.
each case the radius of the inclusion has been taken as unity. The tension at infinity is in the y-direction. The two curves for stress intensity show the remarkable difference in the stress intensity, but as expected the pattern is the same in the two cases. Acknowledgment-Authors
are extremely grateful to the referee for his comments which have improved
the paper.
REFERENCES J. DUNDURS and M. HET’ENYI,J. uppl. Mech.(Trans. A.S.M.E., series E) 28 103 (1966). 111 [21 R. D. BHARGAVA and 0. P. KAPOOR, Proc. Camb.phil. Sot., 62 113 (1966). [31 N. I. MUSKH E LISHVILI, Some Basic Problems of fhe Mathematical Theory of Elasticity. Noordhoff(1953).
[41 G. C. SIH and J. R. RICE, J. appl. Mech. 31477 (1964). [51 F. ERDOGAN, J. appl. Mech. 30,232 (1963). 161 C. ATKINSION, Inr. J. Engng Sci. 10 127 (1972). [71 0. TAMATE, Int. J. Engng Sci. 4 257 (1969). @I T. J. WI L LMORE, Q. J. Mech. uppl. Math. 53 (1949). 191 M. LOWENGRU B and K. N. SHRIVASTVA, Int. J. EngngSci. 6 359 (1968). (Received
26 January
1972)
Elastic circular inclusion
in an infinite plane containing two cracks
449
Resume-Les
champs Blastiques dans une inclusion circulaire elastique et une matrice intinie l’entourant et contenant deux fissures disposees symetriquement, sont determines lorsque la matrice est soumise a des charges a l’infini. Dans ce probleme, les proprit%es elastiques d’une inclusion pourraient btre differentes de celles de la matrice. La technique de Muskhelishvili est utilisee. La solution repose sur deux ensembles de potentiels complexes convenables {Q,,,(z), TV,(z)} , {Qi (z) , Pi(z)} respectivement pour la mathce et l’inelusion, qui r solvent le probltme.
Zusammenfassung-Es werden die elastischen Felder in einer elastischen kreisformigen Einschliessung und umgebenden unendlichen zwei symmetrisch gelagerte Risse enthaltenden Matrize bestimmt, wenn die Matrize Lasten im Unendlichen unterliegt. In diesem Problem konnen die elastischen Eigenschaften der Einschliessung von denen der Matrize verschieden sein. Muskhelishvili’s Methode wird verwendet. Die Losung hlngt von zwei Satzen geeigneter komplexer Potentiale {Q,,,(z), qIrm(z)}, {Qi(z), T,(z)} fiir Matrize bzw. Einschliessung ab, die das Problem l&en. Sommarlocontenente
Si determinano i campi elastici in un’inclusione circolare elastica e la matrice infinita circostante
due incrinature sistemate simmetricamente, quando la matrice i sottoposta a carichi all’ inhnito. In quest0 problems, le proprieta elastiche dell’inclusione potrebbero essere diverse da quelle della matrice. Si fa uso della tecnica di Muskhelishvili. La soluzione dipende da due serie di potenziale complessi idonei {~,n(z),\v,“(z)},{~i(z),~i(z)}
perlamatriceeI’inclusionerispettivamente,ilcherisolveilproblems.
OnpeAeJIeHbI ynpyrHe nOJIll BO YIIpyrbiM, KpyrOBbIM BKJIMYIeHBeM A OTKpy~aIomeii 6eCKOHe’IHOkMaTpFWe, COAep~aLUefiABe CUMMeTpAKHOpaCnOAOx(eHHbIeTpeIUUHblB CJQ’Yae,KOrAa MaTpIiUa IIOABepxeHa Harpy3KaM B 6eCKOHeYHOCTH. B 3T02i npo6neMe B03MOmHOpa3naYue ynpyrux CBOiCTBBKIIK)‘IeHnIIOT TeX MaTpIiUbI. npnMeHeH MeTOA Mycxenumsunu. PemeHIie 3aBUCnT OT AByX CHCTeMnpHrOAHbIX KOMIIAeKCHbIX IIOTeHUUanOB{4,,,(Z), I/.,(Z)}, {e&(Z), I&(Z)} AAR MaTpnUbI a BKAIO’IeHIiRCOOTBeTCTBeHHO, ‘ITO pemri~ npo6nebfy. A6CTPaKT
Pergamon Press.
Printed in Great Britain
ELASTIC CIRCULAR INCLUSION IN AN INFINITE PLANE CONTAINING TWO CRACKS R. D. BHARGAVA
and RAJ RANI
BHARGAVA
Indian institute of Technology, Bombay, India Abstract-The elastic fields in an elastic circular inclusion and surrounding infinite matrix cont~ning two cracks symmetrically situated, are determined when the matrix is subjected to loads at infinity. In this problem, the elastic properties of inclusion could differ from those of the matrix. The Muskhelishvili’s technique is used. The solution depends upon two sets of suitable complexpotentials{@,,,(z),T,(Z)}, I@(Z), Uri(z)}for matrix and inclusion respectively, which solves the problem. 1. INTRODUCTION THE
INFINITE
plane containing an elastic circular inclusion is discussed in [ 11, [2] and 131. 133 also contains the case of elliptic insert. The crack problems are considered in 14-81 et al. The problem of two cracks under constant pressure in an infinite plane has been considered in [9]. The problem of ‘The interaction between a crack and an inclusion’ was considered by Atkinsion[6] and ‘The effect of a circular inclusion on the stresses around a line crack in sheet under tension’ by Tamate[7]. We extend the method given by Tamate to obtain the explicit solution to the problem of this paper. It may be seen that Atkinsion’s problem consists of one crack and a circular inclusion in an infinite plate, while the present problem consists of an inclusion and two symmetrically situated cracks along the horizontal diameter. 2. STATEMENT
OF
THE
PROBLEM
Consider the circular inclusion, of radius R and the centre origin, which is embedded in an infinite matrix containing two straight cracks L, and L, along x-axis (Fig. 1). It will be assumed that the matrix and the inclusion are perfectly welded along their common boundary L. The elastic properties of the inclusion may differ from those of matrix. The straight cracks L, and L, lie from f- b,O) to (-a, 0) and from (a, 0) to (b, 0) respectively. The matrix is subjected to prescribed loads at infinity. As shown by Muskhelishvili the stress components PxE, P,, and P,, and displacement components u,, U, can be expressed in terms of complex potentials cP(z), p (z) as follows:
pm+p,,= 2[(9(2)+ca)], P,,-P,,+2iP,,=
2[241’(z) +*(z)f.
(2.1)
---
2/.C(&,,+&,,,)
=
K@(Z)
--@(iI)
-Z@‘(Z)
-v’(Z).
(2.2)
The bar over the function represents the complex conjugate of the function. The comma denotes the differentiation with respect to subscript following it. As the inclusion boundary is a circle, it is convenient to introduce polar coordinates r, 0: z = x-t- iy = reie. The stress components P,,, PBB,Pro and derivatives of displacement 437
438
R. D.
BHARGAVA
and R. R. BHARGAVA
Fig. 1.
components
can be written in terms of tf, (2) , zl’ (z) as
pw-+-pfx3= 3@,(z) +rnl, P 08--P,,+2iP,,
= 2~z/~)~Z9p’(.z) -t*(z)].
(2.3)
2p(%efiU&B)
=iZ[K@‘(Z)-m+z~+(Z/z)wz)].
(2.4)
Here K = (3 - v)/( I + v) for generalized plane stress and K = 3 - 4v for plane strain case, v being the Poisson’s ratio and p is the shear modulus. It is convenient to introduce the complex potential R(z) in terms of Q,(z) and * (z) as follows: Q(z) = @(z) +z,G’(z) +@(z). Thus the stress and displacement @p(z) and 0(z) as:
components
(2.5)
may be written in terms of the functions
2[rD(z)+@‘z)l,
p,+p,,=
P ?#- iP,, = Q(z) +n(z)
+ (z-Z)&‘(Z)
and 2p(ur,3:+iuy.s)
=
K@(Z)
--n(z)
-
(z--+5’(z).
(2.7)
We now introduce the boundary conditions. (a) On the rims of the cracks, no external forces act. (b) There is no discontinuity in the stresses and displacements across the bounda~ of the inclusion. (c) The stresses at infinity are prescribed. These are the principal stresses IV, and iV2making angles cyand 7ri2 + CYwith x-axis.
Elastic circular inclusion in an infinite plane containing two cracks
439
The condition (a) may be written as P& - iP$, = 0,
(2.8)
the subscripts + and - indicate the limiting values as a point on the crack is approached from the positive (y > 0) or negative (y < 0) part of x,y plane. When represented in terms of complex potentials Q(z) and R(z) by (2.6) these conditions (2.8) give dual homogeneous Hilbert problem which may be written as: Q,‘(t) +0-(t) The superscripts
= 0; W(t) +sz+(t) = 0.
(2.9)
have the same meaning as defined above; and t is a point on the crack. 3. SOLUTION
FOR
MATRIX
The solution of equation (2.9) may be written directly with the help of complex variable technique. Taking into account that the functions Cp(z) and n(z) can have poles of various orders at origin and using the conditions (c), these can be written as:
a&(z)
=-;rt+;g4n(;)n++,(z)[(2r+r’)($;$) 2
(3.1)
(3.2) where X,(z) =
a2(z-a)-“*(~-~)-1/*(z+a)-l,2(Z+~)_1,2~
In the above equations the subscript m indicates that the potential functions a(z) refer to the matrix; and as is well known r=f(N1+N2);
r’=-t(Nl-ZV2)e-2i”;
(3.3) a,(z),
(3.4)
and also F = F(7r/2, d(b*-a*)/b) is elliptic integral of first kind and E = E(7r/2, d/(b*- a*)/b) is the elliptic integral of second kind. The constants A,‘s and B,‘s are unknowns, and are to be determined. In the region 0 < lzl < a, the functions Qm(z) and qIr, (z) may be expanded by Laurent series as follows: (3.5)
440
R. D.
BHARGAVA
and R. R. BHARGAVA
(3.6)
where S,,, is Kronecker
delta, and (3.8)
4. SOLUTION
FOR
INCLUSION
We now discuss the case of inclusion. The complex potentiafs for inclusion ]zl d R may be written as: (4.1) The subscript i denotes the case for inclusion and G,, H, are the unknown constants to be determined. 5. DETERMINATION
OF
CONSTANTS
The constants A,, B,t, G,, and H, are determined by the condition (6) which impties that on the common boundary of circular inclusion i.e. at r = R
(5.1) In terms of complex potential functions, at, ( Reie) f Cp, ( Rei@)- Re+P~ = 4[>+ ( fi”e@)+ cDi(j&@) - ~e-~e~)
( ReiB) - e+*, - e-2@-)
( Re@) *
(5.2)
Elastic circular inclusion in an infinite plane containing two cracks
441
and ff
( Reie) -
[K,&~
am( Reie) i-
Re-ie@im( Reie) + e-2’e*m(Reie) ]
= KicDi(Re”‘) - @i(Reie) -f-Re+e@8f(j%?) i- em2jeqi (ReiB).
(5.3)
The subscripts m and i have the same meaning as above; JL~,,pli are shear moduli of matrix and inclusion respectively. Substituting the values of Qmr V\Ir,from (3.5) and $, ‘Pi from (4.1) in (5.2) and (5.3) and then comparing the various powers of ele the following equations are obtained:
= (n-
_.!sKm Pm
i
’a
“D,-/”
Pm
>
1)&-l-H,_,(l-SI,,).
n = 1,2,. . .
-BB_,+CLiE_n_2 Pm
(rtt I) t 0
=
KiG,-G&,,.
(5.4)
n=O,l,...
(5.5)
The values of G, and H, may be written in terms of D,, D_,, E, and E_,, which are the linear functions ofA,‘s and B,‘s, as follows: c
=
h’Gn/Pm)+ 1 E RD,_ t&/&d - 1
A
0a
1+Ki
+
(/&l/&n)-’ l+Ki
H,=
n=o,
0
:
%,-
1+
Ki
& i
f2 >
(n+2)2
i
4
-%_,_,+ >
R -n-4_ (n+ 1) k-4, 0a
I,...
(5.6)
Substituting the values of G, and H, from (5.5) and (5.4) and solving, the following relations are obtained: D_, = 0,
(Pi’Pm)-l [(~-1)(~)2’ti,+(~~-‘E,-,]. D-n =- (/&Km/&n) +1
n=2,3,...
(5.7)
E_, = 0,
(5.8)
R. D.
442
BHARGAVA
and R. R. BHARGAVA
Comparing the values of D_, obtained from (3.6), and (5.7), the following equation is obtained:
4, =
-~~&-h-2 0
(5.9) n=2,3,...
Equating the two values of E_, obtained from (3.7), and (5.8) the equations for determining Bk’s are obtained: (5.10) and
(5.11)
n=o,
l,...
The values of B,‘s are determined from (5.10) and (5.11) by substituting the values for PI/I..&,,&, t%,, T, f”, Rla, Ha. The values of An’s are determined with the help of expression (5.9) using the values of B,‘s determined earlier. Therefore, the potentials am(z) and ?v, (z) are known. The constants G, and H, are determined from (5.6) using the values ofA,‘s and B,‘s already obtained from (5.9), (5.10) and (5.11). Hence the functions @(i(z) and Vi(z) are also known. The stress fields in the matrix and inclusion can now be determined by the use of (2.1) using the values of {am, Vr,} and {@i, Yr,} determined above. Thus, the problem is completely solved as all the constants have been determined, and the functions {Q(z), 9 (z) } satisfy all the conditions. 6. CONCLUSION
As is obvious, to obtain the solution one will have to solve infinite number of linear simultaneous equations. Fortunately the convergence is very fast. Some particular cases of interest were solved numerically and the results have been given in the
Elastic circular inciusion in an infinite plane containing two cracks
443
form of graphs. In each case, the load acts in the y-direction at inanity, the diameter of inclusion is 2, and the crack length (b - a) = 2. In Fig. 2, the stress intensity (&f/V) = l$t (~P,,(x, 0))) near the crack tip (a, 0) is taken along y-axis and is plotted v/s a (the distance of crack tip from the o~gin) along x-axis for various values of ~~~~~. The value of stress P,, at infinity is N. The values that were taken are: ~/~~ = 0, Q, 1 and 3. The case ~~~~ = 0 refers to that of the hole, and ~*~~~ = 1, when the elastic properties of matrix and inclusion are the same and hence in practice, there is no inclusion. This is the case of a matrix with
ot
I
I.0
Fig. 2. Stress intensityl~
I.5
I
I
2.0 a
2.5
for various values of&(tl,,,, vs the distance of crack tip from origin.
two cracks only. The analytical results for this case agree with those obtained in (9). Note that the vatues of a have been taken to vary from 1.05 to 2.75. As is clear from figure, (i) for any fixed value of a, the stress intensity decreases for increasing values of ~~/~~, (ii) for the range of values of ~~/~~ from 0 to 1, the stress intensity diminishes with the crack tip moving further. In the case when ~~~~ > 1, the stress intensity increases when a changes from l-05 to approximately 2.0 and then stabihses. The trends of the curves indicate that for (~~~~~) > 1, (&IN) + 0 as n + 1, and for (~*/~~) < 1, (~~~~ + 03as a -+ 1. For (~i~~~~ = 1, it may directly be proved from the results of this paper or from (9) that (~~~) = (1/~2~(b~-~~))[6~(EfF) -a”]. Nume~cally, the curve comes as a straight line, which is verified by the above theoretical result. For Figs. 3 and 4, the same numerical data have been taken, namely the diameter of the inciusion is 2 and the crack length (b - a) is 2, also the stresses at infinity is in the y-direction. In Fig. 3(i) the hoop stress P*~/~ on the boundary of inclusion (along y-axis) is plotted v/s the angle 8 made with x-axis, for three values of ~1~~ = 3, 1,3 for the case a = 1.5. it is interesting to note that hoop stress is positive and maximum at the x-axis (0 = 01, and increases as the ratio of shear moduli bilks increases. The same graphs for hoop stresses in inclusion vls the angular orientation of the points on the inclusion boundary from x-axis as in Fig. 3(i) are plotted for other values
444
R. D.
BHARGAVA
and R. R. BHARGAVA
of a = 2.0 and 2.5 in Fig. 3(ii, iii). The figures indicate that for larger values of a, the curves smoothen out. In Fig. 4(i) the graphs for hoop stress in matrix v/s the angle 13,is again repeated. This shows the behaviour of hoop stress. The graphs are drawn for the same data described already as in Fig. 3. As expected the behaviour of the matrix (in Fig. 4(i)) is opposite to that of the inclusion at the interface. As pi/pm decreases, the hoop stress at 0 = 0 increases.
Fig. 3(a).
Fig. 3(b).
Elastic circuiar inclusion in an infinite plane containing two cracks
445
It may be seen that for the case when ki/pm = 1, the curves for hoop stresses for inclusion and matrix across the circular boundary are the same. The Fig. 4(ii, iii) give the hoop stresses for matrix, along y-axis v/s 0 for a = 2-O and 2.5. The values of the hoop stresses are approximately the same near about 58” for a = l-5.2-0 and 2.5.
Fig. 3(c).
Fig. 3(d).
LIES Vol. 1I. No. 4-D
446
R. D. BHARGAVA
and R. R. BHARGAVA
0.
Fig. 4(a).
40 t
I
PO-
Fig. 4(b)
Elastic circular inclusion in an infinite plane containing two cracks
Fig. 4(c).
Fig. 4(d).
R. D. BHARGAVA and R. R. BHARGAVA
448
In Fig. 5, two curves for stress intensity at the crack tip (a, 0) are drawn, for two cases when infinite plate contains (i) a circular inclusion and a straight crack and (ii) a circular inclusion
inclusion with two symmetrically situated cracks. For, one crack and an the curve is drawn with dotted line and is the same as given by Atkinsion in
[6]. The other curve with the continuous
line is for two cracks and an inclusion. In
Y
A !I -
I
I
I
\
4-
15 -
17I I
I
I
I
14
16
16
2
8”
Distance from inclusion
Fig. 5.
each case the radius of the inclusion has been taken as unity. The tension at infinity is in the y-direction. The two curves for stress intensity show the remarkable difference in the stress intensity, but as expected the pattern is the same in the two cases. Acknowledgment-Authors
are extremely grateful to the referee for his comments which have improved
the paper.
REFERENCES J. DUNDURS and M. HET’ENYI,J. uppl. Mech.(Trans. A.S.M.E., series E) 28 103 (1966). 111 [21 R. D. BHARGAVA and 0. P. KAPOOR, Proc. Camb.phil. Sot., 62 113 (1966). [31 N. I. MUSKH E LISHVILI, Some Basic Problems of fhe Mathematical Theory of Elasticity. Noordhoff(1953).
[41 G. C. SIH and J. R. RICE, J. appl. Mech. 31477 (1964). [51 F. ERDOGAN, J. appl. Mech. 30,232 (1963). 161 C. ATKINSION, Inr. J. Engng Sci. 10 127 (1972). [71 0. TAMATE, Int. J. Engng Sci. 4 257 (1969). @I T. J. WI L LMORE, Q. J. Mech. uppl. Math. 53 (1949). 191 M. LOWENGRU B and K. N. SHRIVASTVA, Int. J. EngngSci. 6 359 (1968). (Received
26 January
1972)
Elastic circular inclusion
in an infinite plane containing two cracks
449
Resume-Les
champs Blastiques dans une inclusion circulaire elastique et une matrice intinie l’entourant et contenant deux fissures disposees symetriquement, sont determines lorsque la matrice est soumise a des charges a l’infini. Dans ce probleme, les proprit%es elastiques d’une inclusion pourraient btre differentes de celles de la matrice. La technique de Muskhelishvili est utilisee. La solution repose sur deux ensembles de potentiels complexes convenables {Q,,,(z), TV,(z)} , {Qi (z) , Pi(z)} respectivement pour la mathce et l’inelusion, qui r solvent le probltme.
Zusammenfassung-Es werden die elastischen Felder in einer elastischen kreisformigen Einschliessung und umgebenden unendlichen zwei symmetrisch gelagerte Risse enthaltenden Matrize bestimmt, wenn die Matrize Lasten im Unendlichen unterliegt. In diesem Problem konnen die elastischen Eigenschaften der Einschliessung von denen der Matrize verschieden sein. Muskhelishvili’s Methode wird verwendet. Die Losung hlngt von zwei Satzen geeigneter komplexer Potentiale {Q,,,(z), qIrm(z)}, {Qi(z), T,(z)} fiir Matrize bzw. Einschliessung ab, die das Problem l&en. Sommarlocontenente
Si determinano i campi elastici in un’inclusione circolare elastica e la matrice infinita circostante
due incrinature sistemate simmetricamente, quando la matrice i sottoposta a carichi all’ inhnito. In quest0 problems, le proprieta elastiche dell’inclusione potrebbero essere diverse da quelle della matrice. Si fa uso della tecnica di Muskhelishvili. La soluzione dipende da due serie di potenziale complessi idonei {~,n(z),\v,“(z)},{~i(z),~i(z)}
perlamatriceeI’inclusionerispettivamente,ilcherisolveilproblems.
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