PLANE PROBLEM OF THERMAL CONDUCTIVITY AND ELASTICITY FOR TWO JOINED DISSIMILAR HALF-PLANES WITH CURVED INCLUSIONS AND CRACKS UDC 539.375

M. P. Savruk and V. M. Zelenyak

Problems of determination of the temperature fields and stresses in an infinite plane with inclusions and cracks have been reduced to integral equations with respect to closed (lines of separation of the material) and open (incisions) contours. Below is considered the case of a piecewise uniform plane consisting of two joined dissimilar half-planes containing inclusions and cracks. The system of integral equations does not contain an unknown function for the line of the joint of the half-planes. Problem of Thermal Conductivity. Let an infinite plane consist of a matrix (two joined half-planes S+ and S- with the line of separation L 0) weakened by'N-M curved incisions Lk(k = M + i, N), and inclusions S k with the smooth closed boundaries Lk(k = I, M), Let us assume that the contours Lk(k = 0, N) do not have common points. Each contour L-Lk k = i, N) is linked to an XkOkY k local system of coordinates, the OkX k axis of which forms the angle =k with the Ox axis, which coincides with the contour L0, and the points O k are determined in the xOy system bY the complex coordinates z~. We will assume the positive direction of bypass of the contours Lk(k = 0, M) to be thas with which the area of the inclusion or the upper half-plane S+ remains on the left (Fig. i). Let us assume that on the closed contours Lk(k = i, M) and the contour L0 there is ideal thermal contact t E L ~, k-----0,M;

(1)

X" k0Tz/c)n=%(tk) +-F~(tk), tk ~ Lk, k = M + I,N,

(2)

k~OT+/Oa=XOT-/On,

T+ = T -.

and on the edges of the incisions the heat flows

satisfying the condition - - o,

(3)

k~M+l

Lo

Fig. i.

S§ $"

z 0

Infinite piecewise-uniform plane with incisions.

G. V. Karpenko Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, Lvov. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 24, No. 2, pp. 23-28, March-April, 1988. Original article submitted April 3, 1987. 124

0038-5565/88/2402-0124512.50 9 1988 Plenum Publishing Corporation

are known, that is, the total quantity of heat exiting through the contour L = ULk(k = M + i, N) is equal to zero. Here n is the external normal to the closed contour L k (k = I, M) or to the left edge of the incision L k (k = M + i, N); X(% k) is the coefficient of thermal conductivity of the matrix (inclusion Sk), T(x, y) is temperature, tk are the affixes of the points on the contour L k in the local system of eoordinates, s k is the arc abscissa of the point tk; A~ = X, if the incision L k belongs to the matrix, ~ = lv if L k belongs to the inclusion Sv (v = I,----M); ~ = ~+(~-) for the upper (lower) half-plane, ~o =l+. and the superscripts "+" and "-" indicate the limiting values of the corresponding functions from the left and from the right of the contour L k. Let us use integral representation of the complex potential of temperature for an infinite plane with inclusions and incisions F(z) = f'(z) [i]

1 ;Ho(t)dt ~ 1 ~ ! Hk(tk)dt. +FI(z) '

F(z)=;7 where

"r

Hk(t~)= 7k'(tk)+i~(tk)e•

(4)

-z

y~'(t~)=0 (k = l,~);Tk=t~exp(iak)+zg;

Ok i s t h e a n g l e

composed o f t h e p o s i t i v e t a n g e n t t o t h e c o n t o u r Lk a t t h e p o i n t t k w i t h t h e OkXk a x i s , and F~(z) d e t e r m i n e s t h e t e m p e r a t u r e in a c o n t i n u o u s u n i f o r m p l a n e w i t h o u t i n c l u s i o n s f o r which the characteristics o f t h e u p p e r and l o w e r h a l f - p l a n e s a r e t h e same. H a v i n g now s a t i s f i e d c o n t o u r L0 we o b t a i n

with the use of potential

2.-5,_,

(4) the boundary condition

t r,--,

+

( 1 ) on t h e

J'

ao= (x+ - ~-)/(x+ + x-).

(5)

Having substituted solution (5) in Eq. (4) we find

1 ~ [ H.(t.)dlk+Fo(z), rk-- z

F (z) = ~

(6)

where

eo (z)

a0' ~ nk (t,) at~ + F~ (z), ~0' ~t k-I

Tk--z

[--ao, z ~ S +, T ~ S + .

Having now satisfied with the use of potential (6) the boundary conditions (i) and (2) on each of the contours L k (k = i, N), we arrive at a system of N integral equations relative to Pk(tk) (k = I-~-M) and y~(tk) (k = M + i, N):

Lk

,~(8.--1).%(tn)/'h*n@anlm{Fo(Tn')exPIi(On'+O~.)]}, n=l,N~ H e r e 6n = 1 w i t h n

= 1, M; 6 n = 0

T,/--~t,," exp (i a . ) + z ~ ; The s o l u t i o n

with

n-----~l+l,N;

(7)

Knk(th, tn') ~-A~ exp[i(O,/+an)]/[i(T~--Tn')];

An-----[ (X.--~) /(X~4-~) ] 6 . + I

of system (7) must satisfy

the conditions

f Tn"(t=)dtn=O, n---,44+ l,N, Ln

125

providing continuity of the temperature in by-passing of the contours of the cracks. Problem of Thermal Elasticity. Let us consider an infinite plane consisting of the two joined half-planes Se and S- and inclusions with shear moduluses of G+, G-, and G n and also with the Poisson's ratios of B+, ~-, a~d ~n" The body is weakened by N-M curved incisions (Fig. I). Let us assume that the plane is under the action of the steady temperature field T(x, y). On the lines of the junction of the inclusions and the matrix and on the contour L 0 the stresses are continuous, the displacements undergo the break

INn (&) + irn (tn)] + - [~n (&) +it. (G)l:, (Un+~Vn)+--(Un+tVn)--= (1--~no)Z~(tn),

(8)

tn 6 Zn, nf0,M;

the crack edges are not in contact, and on them is specified the self-balanced load

[gn(tn)+Wn(tn)] +-=p:(&),

tn G t.,

n=M+

1,N.

(9)

Here Nn(tn) and Tn(t n) are the normal and tangential components of the forces, un and v n are the components of the displacements, and 6n0 is the Kronecker symbol. Let us use the complex potentials of stresses for an infinite piecewise uniform plane with incisions [i]

0~(z)

= ~ ~

t -----7-

+ O (z);

(Odt +,r(,). ,oo (t - z)~ I

'i[

(lO)

Here

1 ~, ;

r

Qk(tk)exp(i=k)dtk"

, kffi!

9 (z)=~=,

r~-z

Tk--z

(rk--z) ~

s

Qk(tk) = I &k'(td'

t~ G L~, k = i,'M;

I g; (t~) + i~./(1 + ~,). [/+ (td - / ~ , = 8 [ a n d