1

Sci.Rep.Fukushima Univ.No.30 (1980)

A Mixed

83

Problem for an Infinite Elastic Medium Containing a Circular Inclusion and a Circular Hole*

Katsuo SUGIURA

Facult1l of E:du cat on, Fukushima Universitll , Fukushima, J apan .

(11eei?ed 10 S eptember, 1980)

ABSTRACT

This paper presents the mixed boundary valae .problem for an inf inite elastic medium

which contains a circular hole and a circular inclusion with defferent elastic properties from the matrix.The inclusion is perfectly bonded to the matrix along the circumference,

the circular hole is it1 contact with two rigid stamps.The effect of a circular inclusion on the stress state along the rim of the circular hole

is discussed.

The conformal mapping and the complex variable technique are used to obtain an ana_ lytical solution for the problem.

Numerical calculations are carried out for the stress distribution along the eireumfer_

once of the circular hole due to the configuration and elastic properties of the matrixand the inclusion.

Introduction

Mixed boundary value problems for an inf inite medium with two circular holes have been discussed by several authors [1- 5]. However,the particular mixed problem for a double-connected rigion, i.e. the contact problem for an infinite elastic medium containing two circular boundaries has not been worked out_ In recent years, the author [6] has been discussed the mixed boandary value problem for the infinite re_glen bounded by two nonintersecting circular holes,one of which subjected to the uniform pressure and the other is in contact with two r igid stamps.

This paper continues the previous work and deals with the mixed boundary value problem for an infinite elastic medium (matrix) which contains a circular hole and a circular inclusion of different elastic material. The circular inclusion is bonded to the matrix along the circumference,the hole is in contact with two rigid stamps. On the basis of the two-dimensional theory of elasticity,the effect of an inclusion on the stress state along the rim of the circular hole is discussed.The conformal mapping and the complex variable approach are used throughout to convert this mixed boundary value problem te a Hilbert problem [7].

To clarify the effect of the configuration and the elastic properties of the matrix *) Contributed to Confererice on Japan Soc. Meeh. Eng. November 1978,Sendai

2

84 K Sugiura :Mixed Problem for an Elastic Matrix

and the inclusion on the stress distribution along the circumference of the circular hole, numerical calculations are carried out and the results are shown graphically.

Statement of the problem and basic equations

We employ a rectangular coordinate system, as shown in Fig.1. We consider an in_ finite elastic matrix containing two holes whose boundaries are non_intersecting cjr_ oles L1 and L2, of radii R and R2,respectively, and whose centres are distant d (> R,十R2) apart.A circular disk of the different elastic material from the matrix is inserted into the circular hote L, in the matrix and bonded round. It will be as_ sumed that this infinite elastic medium is compressed under diametrally opposite forces P applied through two rigid stamps of equal width which are perfectely bonded to the matrix along the contact arcs L2 and L;zof .L . Let the matrix and the inclusion be homogeneous and isotropic.

In the z-plane (z=?十i?), we denote the region inside the circle L2by S2 and theremaing part of the plane by S;. Moreover, in the region S2, let S-'be the interior of a circle L,and S? the exterior (Fig.2a)_As a very powerful method of solv-

ing the problem, we use conformal mapping.By [8],the function

(1)

Fig 1 maps the plane region 21=χ十ill eon_ formally on to the region j = el十i e2(Fig.2), where

a = -d十R,γδ= 一dγ十Rl,

γ= (1十A)/(1-A),

A2= 1(d- R,)2- Rj21/1(d十R1)2- .R21.

(a) 2-plarie (b) ?-plane

Fig.2

The circles .1,and L2 in the z-plane are mapped onto the concentric circles Γ1(l?1= 1)

3

Sci.Rep. Fukushima Univ.No.30 (1980) 85

and Γ2(l? l = r-_ - (3十R,γ)/(a十R2)> 1) in the ?-plane,.respectively. The regions S; and S2 correspond to the resions Σ+inside and the region Σ2 outside the circle 「2, respectively. Consequently,the region S' is mapped on the interor Σ:l ot the circle r,,the infinite region S?,consisting of the points outside two given circles L and L2, is mapped on the ring Σ22 bounded by the two concentric circles Γl and 12.

Let complex potentials 0*(z),V*(z) be denfined in the z-plane and ?-plane are connected by the function (1), the potentials 0*(z),V*(2) can be considered as the function of 「.New notation is used by designating Φ*[a'(0] as Φ(i ), and so on. Putting ?=ρe''1,the lines ρ= const. and f1= const. form, in the z-pti ne,an orthogonal curvilinear network. Hence, when no body forces are present, the stress components o'p,c;1, てp,f in the curvilinear coordinate system (p,'1) and the displacemente compo- nents u ,u. in the rectangular coordinate can be expressed in terms of the potentials Φ(? ,V(? of the complex variable ?= ρe''7 :

2)

(3)

(4)

where ｵi s the shear modulus 'c=3- 4y for plane deformation and 'c=(3-v)/(1十y) for general plane stress,y is Poisson's ratio in addition, u:= 3u /a'l and u,,= au /a'fl Let 02*(z),??(z)be defined and holomorphic at every points of the region S; except for possible singularities in S+.Then,the corresponding functions Φ2(a ,Φ(0 are defined for the region Σ;of j-plane and have singular ities in the region Σ+

We now extende the definition of Φ2(「)to the region Σ by putting

co'(? 02(? = 一ω(?? 2(r'/? 十(r'/? )ω(? ???r'/? 十(r'/? )io (「2/?? (? ,?∈Σ2(5)

(6)

This fomula expresses V2(j) for j in Σ;terms of 02(a which has been extended by (5). Hence, the components of stress and displacement in the matrix filling the re- gion Sj may be expressed in trems of the single function Φ2(j),

(7)

(8)

4

86 K.Sugiura :Mixed Problem for an Elastic Matrix.

-(e(?)/1?ω'(? l]?2(-? - i??o'(f o'(? [?/1r(,;(r'/-? l- 1/1?ω'(? l]?2(-? ,

i∈Σ22 (9)

where ｵ2and lt2denote the elastic constants of the matrix and ? (? is given by (6). We suppose that the contact arcs L,2 and L22of L2 subtend the angle 200 at the origin respectively (see Fig.1).The union of the arcs .L' Lu will be denote by L:,,so that L,2= L?十? ,and the remaining part of L2by Li .By (1), the arcs L,and .L2 corres_ pond to the arcs 「;?= I 2十「22 and 「2=「2十「? of the circle 「2,respectively.Let 2ωk be the central angle subtended by the arc Γ2,, then the angle 2ω and the angle 200k are connected by the formula

ω2,=l'an-'[j学(γ'-1)sinω、/{('-??(?-?

±(2γ与 一号)号c。sa,-(与)γ}]l (10)

where the upper and lower signs apply to k= 1 and k= 2,respectively.Let the complex potentials for the inclusion be Φ,?z),V,?z), then the correspond_

ing functions Φ,(0 ,Vl(0 have to be holomorphic in Σ,.We denote by [00'(t)02(t)]'and [ω'(t)02(t)]-the boundary values of ω'(? 02(? as ?

approaches at the point t ot 「2 fromΣ'and Σ.2 ,respectively. By (8),(9),the bound_ ary conditions of this problem can be written as

(a) On the ar e .L2,u.十jug= const.;

に2[a''(t)Φ(t)]'十[ω(t)02(t)]-= 0, t∈Γ2. (11)

(b) On the arc L2,op十iてp,1= 0;

[Co'(t)Φ2(t)]+- [(,'(t)Φ2(t)]-=0,t∈Γ2. (12)

(c) On the common circle L,, ie,on the circle 「1;

02(e)十? 2(一σ)- Ii[tO(σ)一Φ2・(一σ)十一a;(if)-? 2(? ]/1σ(l;(0)l

= Φ1(σ)十? ,(ii;)- 一σ[to(e)? 1(一σ)十一ce(lf)-V,(i )]/1eω(of)l, (13)

(ｵ1/ｵ2)[に2σω,(of)一σω(?)? 2(i )十一σtaj(σ)? (一σ)十一ω・(-e「)-V2(i )l]

= に,σa;(o・)0 1(t;)一σω'(e)? 1(? 十一o?ω(σ)? l(-o')十i3 (if)-V,(ii:)l, (14)

where σ=e'l and ｵl, に,are eleastic constants of the inclusion. In addition,(d) Since the external forces applied te L2 have the resultant vector

(-P,0), we get

5

Sci.Rep. Fukushima Univ. No.30 (1980) 87

i Sr:[ta'(t)Φ2(t)l'- 1ω(t)02(t)「]dt= - P. (15)

Futher, the condition of single-valuedness of displacements corresponding to these potentials must be satisfied.

Complex Potentials

The problem is reduced to construct the potential functions Φ2(a , V2(i), Φ,(0 and 当(j), which satisfy the boundary conditions (11) to (14)and the condition of single valuedness of displacements_

Condition (12) indicates that ω(?)02(? is continued analytically through the bound-ary arc 「2. Condition (11) expresses the Hilbelt boundary value problem for the func- tion (l;'(j;')02(?). In order to solve this Hilbert problem, we investigate the singulari- ties (o'(?Φ2(?).From (4) we are that ω'(? has a pole of the second ober at ?-- 一γ(1< γくr),corresponding fe z=co. Since the stress and rotation vanish at infinity in the matrix,Φ2(e ls e[(j 十γ)] near i = 一γ_ Thus ω'(aΦ2(0 = 0(1)near the point ?= 一γ, in other words,?= 一γwill not be a pole of ω'(i:')Φ2(? . Further,(,'(?Φ2(i:') may have the poles at the points ?= 0 and ?= oc, because the points of 2-plane corresponding to 「= 0 and j = °°lie in the regions S2 and S'l occupying the hole and the inclusion,respectively.Consequently,the general solution of (11) is given by

(o'(?02(?)= X?? Σb.(j:/r)'', (l 6)

where bn are unknown coefficients and X(? is a particular solution problem (11), is given by

x (? = [(?_ re-,a')(?+ re-,'。)]-i -'β[(?_ re'。 )(?十re 。,)]-i + β,

of the Hilbert

(17)

where β一(Inに2)/(2π).Here X(0 1s understood as the branch,holomorphic in the plane cut along Γ2, such that 「X(a →1 as l i t→°°.

The complex potentials 01(0 ,当(0 for the elastic inclusion are to be holomorphic in the region Σ2,, So that we can represent in the form,

ω'(? o (?)= -',i g ?n' r2n_e n ' }「∈Σ+,, (18)

c。(? v ,(?)= 1,i hn??r n o

where g , h.are unknown coefficients.The coefficients b,,,gn, and h・have to be de- termined so that they satisfy the conditions (13),(14) and (15) and the condition of single_valuedness of displacements.

Since the function (,,(? 02(?) is holomorphic in the region Σ',except at the points j = Oand j = 00,respectively,we have expansions

6

88 K.Sugiura :Mixed Problem for an Elastic Matrix

(19)

(20)

where M =? P,b_ N = k◆oQb_ ., (21)

P.,Q. in (21) are the coeff icients in the power series expansions of Xて? in Σ;and Σ respectively,

where

(22)

(23)

(m=1,2,_)_

here γe-'-i?J=(2J- 1)/2::i:iβ(io=(to= 1,and the respective components p,q and (jmay be obtained by replacing the ω21 in (16) by ω22, - (o22 in turn.

(ii2(i) is represented within Σ+by the Laurent series

Φ(i)=- l t .(i/r)', i ∈Σ+, (24)

wher e An=k? o? d,,_k, (25)

and d,,= r2bn_2十2rγbn_1十γbn. (26)

Hence, substituting (20) and (25) into (9),the expression of (u'(?)? ,(? in Laurentseries is given by

ω'(?)V2(? =? i nB.(?/r)n, (27)

where B = N._,,,?21- (n十1)ΣekA,,_1_k (28)e .l 1

(? )kt (k=1 2 _ )

Substituting (18),(19),(20),(24)and (27) into the continuity condition in (13) and (14),.and comparing terms which contain the same power of o', we arrive at the rota_ tions

7

Sci.Rep.Fukushima Univ.No.30 (1980)

2Co- D- Eo= 2A,- G- H?, }

(,c1- 1)C十D?十Eo= ｵ,2[('e2- 1)Ao十G 十rio],

C,,=[- G_n Hn A_n)], ?

fel o n= ｵ,2[?2?nA ,;一λ''(G _,,十H _n- A _f,)] ,

(n=1,2 Cn- D - En= [A-A_n一λn(Gn十H- An)],

en- D n- E = 一ｵ,2?にzλ-nA_,,十λn(Gn十H - An)j ,

where λ一1/r, ｵ,2= ｵ1/ｵ2,

and C.= yg 十(1一δo,n)2γg._,十(1一δo,,,一δ1,;,)g _2

Dn= hn十(1一δo,n)2γhn_,十(1一δon一δ,,.n)h _

E.= e1(n十1)C._,十(1- 1;o..)ε2nC.十(1一δo,,,一δ1,,)e.(n- 1)on_,,

Gn= e1(n 十1)?A .1十e2nA.十e3(n - 1)?-jAn_l,

Hn= γ2λ一'B._2十2γλ一1B._l十Bn,

a αγ十δ γδ

he「e ε1=fit(γ__l ),ε2--R1(γ 1),e= R,1(γ2_ 1)・

Moreover, from (15) we get

°° r 、/ ;t o βl ? ,+ °' 1

ΣS h.= - P,n=_。。 2(1十;c2)

where cos(β0,十n )d Cos t7十COSω22)(COSf1- COSω21

here δ,=:in[{cos η一2:')22 sinω212- 17 }/ j cos t7「 sin学 }]

Also the condition of single-valuedness of displacements gives

M_,=0, N_,=0.

89

(29)

(30)

(31)

2

3

4

3

3

3

(35)

Equations (29),(30),(32),and (35) are simultaneous equations for determining the coefficients b., g and h .

By eliminating gn and h from (29)and (30), we obtain

(に一ｵ1K;2)XnA -?K:十ｵ1)λ(G_ 十Hn- A_n)= 0, 、(n=1,2,_), (36)

(ｵl2- 1)λ(G 十H - A )十(1十ｵ12に2)λA_.= 0,

Hence,the epuations (32),(35)and (36) are the required epuations to determine the coefficients bnof (l'2(? .

8

90 K.Sugiura :Mixed Pr.oblem for an Elastic Matrix

Numerical Examples

In order to clar ify the effect of inclusion on the stress in the infinite matrix, numerical calculations were carr ied out for the following values1

'c= に2=2.0,2ω= 60 ,R?/d= 0.3,

R,/d=0.1 and 0.5,

ｵ2=ｵ1/ｵ2=0.2,0.5,2.0.In every case, the set of linear equations in (32), (35) and (36) is solved by the method of iteration and by taking the first 61 equations involving the first 61 unknowns (bo,_ b?,b_,,_b_3,)_ The computations carried out revealed that the conver_ gence of iteration is fair ly good.

From (2)and (3),we obtain the stress on the circular L2.The stress along the contact arc L? (_L 十L22)are given by

R1(

R (

- 1)r

- 1)r

(1十に le o s 1?十 COS (l;- Co s t?- Co s ec211

( 1 十 ;C2) e 一β1'l ' 'i 十・ 1

leos 17 COS ω2211 COS 17- COS ω11

Σd coslPii,十(n- 1),11,

Σdsin1βδ,十(n- 1)171, 7) 3

where the upper sign is chosen for the stress on L ,while the lower sign corres_ ponds to the stress on L22, and d.and δ,are determined by (26)and (34) respectively.

Futher, - (,'2,< 17< a) f or L2 , π一(,)22< 17< π十ω22 for .L22

The stress on the traction-free part L2 (L,十La) are given by

4 e 一β'? ・- ? ,1 _ 。。_ _ _

where

and

R1( - 1)r COS 11十 Co s a;2 COS 01121- Co s t? ? d.si n113δ2十(n- 1)fi t,

?2= In[{cos '?二2(';望一sin 7 2°2' 1/f cos 11子 s in'7子 1]

} (38)

(39)

Let the point on .L2 denote R.2e'8,the value of 17 in (34),(37)and (38) is given dy

_ , _,l R、R (1一γ')sine 1 f ,?. ,/- l;an i - (aγ十 R 2COS 十 2γ

Fig.3 and 4 show respectively the distribution of o・pand rp,,1 along the contact arc L Fig.5 shows the distribution of cl? on L?, where (j=Pi (2R ω). In the Fig.3 and

9

Sci.Rep.Fukushima Univ.No.30 (1980) 91

4, (σp)21 indicates the stress (fp on L2,, and so on.In the case of R i d=0.1 the stress distr ibution along ? are not affected execs_

sively by the r igidity modulus of inclusion. But for the ease R/d=0。5 the shear modulus of inclusion has a great influence on the stress distribution around L2. Further, we see that in the case of ｵ,2=ｵl/ｵ2= 0.5 the stress distribution is about the same as the result given in [6],

0

2

4

6

8

0

2

〇

0

0

0

-

-

J-

一

一

一

一

一

o'/

b

0 02 04 0.6 0.8 10c i t,; ,l l 80'-e )/l・'

6

4

2

0

2

4

6

0

0

0

〇

0

0

4-

一

一

一

0「

/

.l

0

2

4

6

8

0

0

0

0

0

1

0」一

一

一

一

一

〇'

/

-0

0.4 0.6 0。8 l.0e ,ω,(,go of),ω

(a1.R,/d= 0_1 (b)R,/d=0_5

Fig_3 Distridutions of o,along contact part of L2

0 0 2 0.4 0.6 0_8 108 /ω ,(l ge e )/(.1

6

4

2

0

2

4

6

0

0

0

0

0

0

4

J

-

-

一

〇'

/

.i

0 .4 0 .6 0 .8 l .0

e /ω .(i80 8 1/u

(a) R?/d=0 1 (b)R id=0.5

Fig.4 Distr ibution of rp? along contact part of L2。

10

92

3.5

3.0

2.5

2.0

56

K. Sugiura :Mixed Problem for an Elastic Matrix

00

25

20

l.0

90° l20 -60° 90°e

(a)R,/d= 0.1 (b)R,/d=0.5

Fig_5 Distribut on of c;1along tracton-free part of L

l20

Acknowledgment

The author wishes to express his hearty thanks to Professor Emeritus 0.Tamate of Tohoku University for his kind advice and many helpful discussions. Numerical calculations were performed on the NEAC 2200 at the computing Center of Tohoku Univef sity.

REFERENCES

[1]Weine1,Z.,Z.angew.Math.Mech.,175 (1937_10),276.[2]Ling,C.B.,J.App1.phys.,19_1 (1948),77[3]Miyao,K.,Trans.Japan Soe.Mech.Eng.,22_123(1956_11),77[4]Haddon,R.A.W.,Quart.J.Mech.Appl.Math.,20 3 (1967),277.[5] Sioya,S.,Trans.Japan Soc.Mech.Eng.,36_286 (1970-6),886.[6] Sugiura,K.,Trans. Japan Soe.Mech.Eng.,43-372 (1977-8),2852.[7] Muskhelishvilli,N.I.,Some Basic Problem of Mathematical Theory of Elasticity,

Noordhoff,Groningen,1953.[8] Von Koppenfels,W,and Stallmann,F.,Praxis der Konformen Abbildung,

Springer-Verlag,1959.