Exact solution of the exterior fundamental mixed problem

V.A,Babeshko

494

2. 3, 4. 5.

6, 7. 8.

9.

and

V.V.Kalinchuk

I. N., On the proof of some uniqueness theorems encountered in the theory of steady-state vibrations. Dokl. Akad. Nauk SSSR, Vol. 80, Np3, 1951. Kupradze, V. D., Boundary value problems of the theory of steady-state elastic vibrations. Uspekhi Matem. Nauk, Vol. 88, NQ3, 1953. On the theory of dynamic contact problems. Dokl.Akad. Babeshko, V. A., Nauk SSSR, Vol.201. Np3. 1971. Babeshko, V. A., On an effective method of solution of certain integral equations of the theory of elasticity and mathematical physics. PMM Vol. 31, Ngl, 1967. Shtaerman, A. I., Contact Problem of Elasticity Theory. Gostekhizdat, MoscowLeningrad. 1949. Aleksandrov, V. M. and Kucherov, V. A., Some problems on the effect of two stamps on an elastic strip. Ineh. Zh. , Mekh, Tverd. Tela, Np4, 196 8. Zabreiko, P. P., Koshelev, A. I., Krasnosel’skii. M. A., Mikhlin, V. Ia., Integral EquaRakovshchik, L. S. and Stetsenko, S.G., tions. “Nat&a”, Moscow, 1968. Dispersion of flexural waves in an elastic circular cylinder. Yih-Hsing Pao, Trans. ASME, Ser, E., J. Appl, Me&., Vol.29, Npl, 1962, Vekua.

Translated

by M.D. F. UDC 539.3

EXACT SOLUTION OF THE EXTERIORFUNDAMENTAL MIXED PROBLEM PMM Vol.38, Np3, 1974, pp. 534-538 V.I. FABR~ANT (Ul’ianovsk) (Received January 10, 1973) An exact solution of the f~damental exterior mixed problem with a circular line of separation of the boundary conditions for a transversely isotropic halfspace is proposed. The interior fundamental mixed problem for an isotropic half-space has been examined in Cl, 23, 1 Let us consider a transversely isotropic half-space z > 0 whose planes of isotropy are parallel to the boundary. We understand the problem with the following conditions on the boundary z = 0: l

0, = @(PY cp)l %I2 = %/z (P,

cp)

w = w (PI OP), u Y = =% (PI CF)

%x = %r (P, cp) (P <

a)

% = n, (P, 0) (P >

4

to be the exterior fundamental mixed problem. We introduce the complex tangential displacements u = u, + iu, and shear stres= If the requirement of decomposability in Fourses Z L= Tzx + iz*, 7 = z,, - it,. ier series in the angular coordinate is imposed on the given and desired functions, then

Exact

solution of the exterior

fundamental

mixed problem

495

by using the method in [3] we reduce the problem of determining the stresses outside the circle P > a to the infinite system of integral equations (1.1)

*s x

m

2pn-1

s c

P

dX

JGq

2nHup”-1

(4 + Ga[ (2n - 1) 2%GI~%_,,+~

@--

2npsl

i/a-sr

’ 5-n (2) x-“+‘d~

s

= F-tat, (PI

c;l,l

(0)

Eas f

042, i)

P

I;-n+f.(p>= a

fka+1

(P) c2

G&_,+~ (s) + G,[(2n- 1).+- 245,+1 (4

8-n Jf/s'L

dS

Re {aS, (p) eing} = zu, (p) einp + W-n (PI eviw an (s) eino + a_* p-1

Cl = B i- Y1YaH7

(s) emin’ ds _

j/3=3

c, = B -

YlYd

The constants a, p, yl, yz, II are determined by the elastic characteristics of the material of the half-space @I]. The complex functions F, and Qtk are known from the condition of the problem. The quantities U,, crk, ak are coefficients of the Fourier

496

V.I.Fabrikant

series expansions of the corresponding functions in the angular coordinate. The axisymmetric exterior problem corresponds to n = 0. Its detailed solution is presented in [4& 2, Without limiting the generality, the first two equations in (1.1) can be considered homogeneous. Indeed, because of the linearity of the equations, this can be achieved by making the third equation complicated. The solution of the system (1.1) can then be sought as

Here f,, is the desired complex stress function, C and I), are constants to be determined. Substituting (2.1) in the first two equations of (1.1) satisfies them identically if the following conditions are satisfied

(2.2)

m

2n;Hcm2~+~5kf* (x) a

Then we use the following of equations (1.1)

(2)

,T,(~, tJ =

If*(t) dt]2-2-2

(x2 - is)+* = 0

a

integral

$

'j k=O

which originates

I' (n +'/ar (n +

1 -

upon substituting

4 1‘P/2+ 4 L'O +

(2.1) into the third

k) (+)"" k)

Here Qn is an even ~lynomial in z, t, whose coefficients are determined by known recursion relationships [S]. Let us divide both sides of the third equation in (1.1) by pn, let us differentiate with respect to p, multiply the result by pz” / v/p2 - P and integrate with respect to p between t and 00. Use of the known properties of hy~rgeome~ic functions [Et] permits separating the integral equation into singular and degenerate parts. The governing equa-

Exact solution of the exterior fundamental mixed problem

tion to determine

fn is

Here 03

xn(r)z-&S

& (P,2, F($

497

t> =

c1

Pnap-a -@*64 -

~ JI’-~

dp

*--l r (‘/a +

n -

c,

kEO

pn

k)

t

21(

)

i -i;-

r@--++)

-+--tz+k;

1-n-j-k;

(2.4)

x

f

1

Since n > k, then R, is an even polynomial, and the integrals corresponding to the degenerate part of the kernel are elementary functions for any n. The exact solution of (2.3) is [4]

fn (4 = qp

t [XnC(t) Y, (0 + K18(4 Y, @)I+ AaY, (4

(2.5)

Here

where A, is an arbitrary constant which corresponds to the homogeneous solution of(2.3). The mag~tudes of the constants corresponding to the degenerate part of the kernel, as well as A, and D,, are determined from the system of linear algebraic equations to which the two conditions (2.2) are added. The formulated problem is solved in general form. 3, Let us consider an example. Within the circle p < a a normal concentrated force P is applied at a distance b from the center. We assume the exterior of the circle to be rigidly clamped. Let us determine the stresses in the framing. We have a problem with boundary conditions on the boundary z = 0 CI, =

u=w=Q

PS (p -

b) 6 (9 -

01, z = 0

(p <

a)

(3.1)

(P > a)

For this case we determine F,,,

(P) = PfIo@‘p+1,

F_,+1 = 0

(3.2)

498

V.I.Fabrlkant

Further computations n = 0 we obtain

are carried out separately

for each harmonic.

For example,

for (3*3)

@>a)

1 The results of calculating

the first harmonic

are (3.5)

The constants tions

f),,

A,,

B,

are determined “p

fl T&

from the linear system of algebraic

equa-

(1)

a

a

a s

ds-

Pb=O

prin-

It should be noted that the system of stresses in the clamped part is such that its cipal vector equals P exactly. This latter is easily shown by direct integration. If b = 0, then the problem becomes axisymmetric and the unique nonzero stress function is

For CC= 0 the solution

is expressed in elementary

functions

499

Exact solution of the exterior fundamental mixed problem

In the case of an isotropic body this latter holds for a Poisson’s ratio of one-half. REFERENCES 1.

2. 3. 4. 5.

Mossakovskii. V. I. , The fundamental mixed problem of the theory of elasticity for a half-space with a circular line of separation of the boundary conditions. PMM Vol. 18, No2, 2954. Ufliand, Ia. S, , Integral Transforms in Elasticity Theory Problems. Akad. Nauk SSSR Press, Moscow-Leningrad, 1963. Effect of shearing force and tilting moment on a cylindrical Fabrikant, V. I., punch attached to a transversely isotropic half-space. PMM Vol. 35. NpX,1971. Exterior axisymmetric mixed problem for a transversely isoFabrikant, V.I., tropic halflspace, PMM Vol. 36, No5, 1972. Gradshtein, I.S. and Ryzhik. I. M., TablesofIntegrals,Sums.Series and Products. Fizmatgiz. Moscow, ,1963. Translated by M.D. F. UDC 539.3

PMM Vol. 38, NJ3. 1974, pp. 539-550 0. V, SOTKILAVA and G. P. CHEREPANOV (Moscow) (Received June 29, 1973) Within the framework of the plane problem of the theory of elasticity, we. consider the eq~librium of an elastic plane with thin different elastic inclusions, situated along a straight line. We give the formulation of the boundary value problems on the basis of the approach adopted from the theory of a thin airfoil, We present an effective method for obtaining the exact solution ofa general class of problems of the above indicated type. We analyze theeffect of the inclusions on the strength and we formulate criteria for the initiation of brittle fracture. I, Forrnui~t~oR of the boundary value problem, In many materials which represent a practical interest, we frequently encounter thin elastic inclusions of a different material. Such are, for example, layers of graphite in cast iron, areas of oxidined metal in alloys, layers of low strength clay or sand in tectonic faults, welds, etc. The inclusions in the basic material lead to stress ~ncentrations which affect essentially the strength properties of the material as a whole. We consider the deformation of an unbounded, elastic, homogeneous, isotropic space with an arbitrary number of thin cylindrical inclusions of a different elastic material. Let the plane XII be some cross section of these cylinders. We assume that each of these inclusions has in the plane zy an axis of symmetry which coincides with the s -