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Elastic stress analysis of jointed half-planes
Elastw Stress A nalysh of Jointed Half-Planes /9y K. T. SUNDARA RAJA IYENGAR Department of Civil Engineering Indian Institute of Science, Bangalore, India a n d R. s. ALWAR Department of Applied Mechanics Indian Institute of Technology, Madras, India
ABSTRACT: Using a Fourier-integral approach, the problem of stress analysis in a composite plane consisting of two half-planes of different elastic properties rigidly joined along their boundaries has been solved. The analysis is done for a force acting in one of the half-planes for both cases when the force acts parallel and perpendicular to the interface. As a particular case, the interface stresses are evaluated when the interface is smooth. Some properties of the normal stress at the interface are discussed both for plane stress and plane strain conditions. Introduction
The stress distribution in the case of a force acting at a point within one of the two semi-infinite elastic solids joined along the plane boundary was investigated by Rongved (1) using Papkowitch functions. Later Frasier and Rongved (2) used the above three-dimensional solution to derive the solution for a plane-strain problem of two jointed semi-infinite plates subjected to a force at a point within one of the two plates (Figs. 1 and 2). By suitably
if Q
X
!
FIG. 1.
267
Y
In(¢rfoc¢
K. T. Sundara Raja Iyengar and R. S. A lwar
changing the elastic constants , their solution may also be used for a plane stress problem. Recently Dundurs (3) treated the plane stress problem of two elastic hMf-planes joined along their boundaries with a concentrated force inside one of them. He derived the solution using the complex-variable approach to plane problems in elasticity, and assumed that the interface is smooth. In this paper, a direct solution is given to the problem of two elastic halfplanes rigidly joined at the interface with a concentrated load in one of them. The solution is obtained using the Fourier-Integral approach. The case of the smooth interface solved in a different way by Dundurs becomes a particular case of the problem treated here. The solution is obtained assuming plane stress conditions. While discussing these reslflts, results are also given for plane strain conditions.
S t a t e m e n t of the Problem The problem has been considered in two parts: one in which the force is acting parallel to the interface, and the other in which the force is acting perpendicular to the interface. These and the orientation of the x and y axes are shown in Figs. 1 and 2. C a s e / - - T h e problem is shown in Fig. 1. E~, tL~ and E2, tL2 are the modulus of elasticity and Poisson's ratio of the two plates 1 and 2, respectively. The force P acts parallel to the interface at a depth 'a' below it. Since the problem is assumed to belong to the plane stress case, it can be formulated in terms of Airy's stress function. Hence, we have to determine a stress function for each half-plane such that each stress function satisfies the biharmonic equation; thus, if ~j is the stress function for the j~h layer (j = 1,2)
v4ei
=
0.
(1)
The stress components in the jth layer are ( ~ ) j = ~Oy " ~2 '
(~y)~ = ~"~; Ox2 '
(r~); =
~; ~x~y"
(2)
By assuming normal and shear stresses at the interface as Fourier-Integrals with unknown functions, the two half-planes can be treated separately. Such a procedure helps to set up the equations for the required unknown functions. This procedure has been used (4) in treating composite infinite strips. Let the unknown normal and shear stresses at the interface in the problem of Fig. 1 be (z~)~ = (r~,) ~ =
// //
K~ sin c~yda
(3)
L, cos ~ydo~.
(4)
Consider the two half-planes separately.
268
Journal of
The Franklin Institute
Elastic Stress Analysis of Jointed Half-Planes For half-plane 1: the boundary conditions are at x -- 0 K . sin c~ydc~
(cr.)1 = .
.
(5)
(r~v)l = f0 L. cosaydaJ In addition to the above, there is a singularity due to the force P at a point distant 'a' from the interface.
For half-plane 2: at x = 0 K . sin aycl~
(~)2 =
~
(r.v)2 =
--
.
(6)
/0 L~
In Eqs. 5 and 6 subscripts 1, 2 indicate the layers to which the stresses refer. Across the interface, the stresses ax and r ~ and the displacement components u and v must be eontinuous. Again putting subscripts 1 and 2 for the two half-planes, we have at x = 0 (*x) l =
(~)~
(**~)~ (u)~
-
(r,~)= L
_ (~)~ [.
(~)~
(~)~
(7)
J
F r o m Eqs. 3 and 4 it can be seen t h a t the first two relationships of Eq. 7 are automatically satisfied. Solution for the first layer: The solution for this half-plane having a concentrated load at a distance 'a' from the interface and having the interface stresses given by Eq. 5 can be built up of two stress functions. T h e first function which can be considered as a 'singular part' m a y be taken as the solution for a full plane with a concentrated load (5). The second function can thus be chosen that, when combined with the first one, it satisfies the required b o u n d a r y conditions of the problem. Hence ~l =
~i
I
+ ~oltl
where ~f =
fo ~ -~ 1 [A,
+
a(a -- x)B.]e -"(a-~) sin ayda,
(a -- x) /> 0
(8)
where
Vol. 278, No. 4, October 1964
A, =
1-~__~lP}
B,=
? 1-t-#1PJ4
(9)
269
K. T. Sundara Raja Iyengar and R. S. Alwar
and ~t" =
~1
-~ [C~ + ~ x D r ] e -"~ sin ayd%
x ) O.
(10)
The final stress components are ( ~ ) I = --
/;
[ A ~ + a ( a -- x)Br?e -~(~-~) sin ayda -
(~)~ =
/;
[-Cr + axD~?e - ~ sin c~yda
[-(A~ -- 2Br) + a(a -- x)B~-]e -"C"-~) sin aydc~
+ ( r ~ ) l = --
2
/;
(11) [(C. -- 2D.) + c~xD.~e - ~ sin ~yda
[ ( A . - Be) + a(a - x)B~e-"(~-~) cos ~yd~ +
F
[(Ca - Dr) + ~xD.~e -"~ cos ayda
The corresponding displacements are (u,) = - E 1
c~{ (l + # ~ ) [ A . + c ~ ( a - x ) B . ~ + +-E~
l f f l l~
(1-u~)B~}e -"("-~) sin ~yda
(l+m)(C.+c~xDr)+(1-u~)D.}e
-~" sin c~yda
(12)
(v)x=~-~l f f a l { 2 B " - ( l + m ) ~ A . + a ( a - - x ) B . 3 } e -~("-~) cos ayda E,
{2D,-- (l+~t)(Cr+axD,)}e-"" cos ayda
Putting in the boundary conditions of Eq. 5 into Eq. 11 we get C. = - K . - (A~ + aaB~)e -~" } Dr = C. - L~ - r_(A~ - B . ) + aaU~]e_. ~ ."
(13)
Solution for the second layer: This is a half-plane with the boundary stresses
given by Eq. 6. Hence the solution fo ~
1
~F~ + a x G ~ e -r~ sin ayd~,
x >/ 0
(14)
where
G.
270
= -- K~, Lr -- K. "
(15)
Journal of The Franklin Institute
Elastic Stress .! nalysis of Jointed Half-Planes
The stress components are (q~)2 =
(F~ + axG,~)e-" sin ayda
-- f0 ~
O-y) 2 ~---
(16)
o"~ [ (F,~ -- 2G,~) + axG.']e -"~ sin ayda fo ~ [. (F. -- G.) + axG.Te -"~ cos ayda
Txy) 2 ~-
It can be seen from Eqs. 15 and 16 that the boundary conditions specified by Eq. 6 are satisfied. The corresponding displacements are (u)2 --- ~
a {(1 + g2)(F. -~ axG,,) + (1 - ~2)G~}e-"~sinayda •
(~')"
1 ~o~1 { 2 G . -
E-~
(17)
(1 + ~,.,)(F~ + c~xG,,)}e-"~cosayda
Continuity conditions: As previously stated, the last two relations of Eq. 7
are to be satisfied. Hence from Eqs. 12 and 17 we arrive at the following relationships : [-(1 -q- ~1)(A,~ + aaB,~) -~ (1 - ~l)B~Te -"~ - [(I+ul)C.+
(1-ul)D.7-E~[(1
+u2)F~+
(1 - u 2 ) G ~ ] = 0
(18)
and [-(1 + gl)(A. + aaB,~) - 2B,~e -"~ +[(1
E1
+#~)C,-
+t~)F~--2G~]
2D~--ff2[(1
= 0.
(19)
Using Eqs. 9, 13 and 15 we can write Eqs. 18 and 19 as two equations in the two unknowns K~ and L~. Hence
-
where PI=
-- 2
P~K~ -- Q~L~ = I~
(20)
Q , K ~ + PIL~ = I2
(21)
( E,) 1+-~2
Ox = (1 - ~1) - F .E1 ~ (1 - ,.,) I1
--
-
P
-
(22) [(1
-
.1)
-
.a(l
+
,1)]e
-"~
12 = P r(1 + #t)aa -- 2~e -"a
Vol. 278, No. 4, October 1964
271
K. T. Sundara Raja Iyengar and R. S. Alwar L~, and substituting them in 3 and 4, we
Solving Eqs. 20 and 21 for K~ and have stresses at the interface, (~,)~ =
fo P,x + Q,I
px 2 _ Qx2 sin
ayd~
l
fo ~ P~I~ + QII~ COS ayda / "
(r~),
P12
Qt2
(23)
;
After simplification these will be i (a~) =
P 1 { ~-p2Q,~.[(1
a
-,~)P~+2Q,]y2+a s 2a~y
-- (1 + ,~)(Px + Q~) (y, + a,)2
} -.
(-r,~)' =
PTrP~ 1
(24)
Q12 { [2P1 + (1 -- .x)Q~] y2 +a a'-' -- (1 + u ~ ) ( P ~ + Q ~ )
~_~7a2~ I a(a2 -- Y~) ]
Eqs. 11, 12, 16, 17, 22 and 23 constitute the rigorous solution for the problem of Fig. 1. Frietionless interface: If the interface is smooth, the shear stress there is zero, i.e., L~ = 0 and the fourth relationship in Eq. 7 is not valid. Hence from Eq. 20 we have I1
K~ = ~ .
(25)
Consequently the interface normal stress is
(~),=
P 1 [ ( 1 - ~ 1 ) y2 y
+ a2
~" P1
2 ~(y2_++ _~)a~y ] a~)2 j"
(26)
Case 2 - - T h e problem is shown in Fig. 2. In this case the force P is acting in a direction perpendicular to the interface. As in the previous case, the unknown normal and shear stresses at the interface are assumed to be in the form of Fourier-Integrals, i.e., (a~) i --
K , cos
aydc~ .
(27)
(r,~)' = fo L, sin ayd~ J Solution for layer 1: As worked out in the previous case, the stress function for this half-plane is made up of two parts, i.e., @I =
272
~I l +
~PzII
Journal of The Franklin Institute
Elastic Stress Analysis of Jointed Half-Planes
where ¢11
= ~ ~1 (A. + ayB.)e-"~ cos axda, Y>/ 0 J0
-~ (C~ + axD.)e -"~ cos ayda,
x ) O.
(28) (29)
The stresses and displacements are (~)1 =
1-(A. - 2B,) + ayB.~e -"u cos axda -
(~)1 = --
/:
/;
(C. + axD~)e -"~ cos ayda
(A~ + ayB.)e - ~ cos axd~ + ~o 1-(C~ - 2D~) d- a x D ~ e -"~ cos aycla
(r~)l = --
/;
l (30)
[-(A. -- B.) + a y B ~ e - ~ sin axda - f ~ [-(C~ -- D.) -I- axD.Te - ~ sin aycla ..Io
and (U) I
lf0~l~ { ( 1
= E1
+VII1 f o ~ l
+ ~1)(A. + ayB.) -- 2B.}e-"~ sin axda
{(1 d-~1)(C, d - a x D . ) d- (1 - ~ l ) D . } e - " x c o s a y d a (31)
(v)~ = ~1 ~0~1~ {(1 + ~ ) ( A ~ + ayB.) + (1 -- #l)B~}e -~y cos axd~
1 ~o~1~ { (1 -4- ~1) (C. -4- axD~) -- 2D.}e - ~ sin (~yda
d- ~
In order to satisfy the boundary conditions on x = 0 given by Eq. 27 and to have the singularity at x --- a, the following relationships are valid P A~ = ~ ( 1 B.=
--~l) sinaa
- 2 -P~ (1 + gl) s i n a a
(32)
C~ = - K . -4- _2 1-2 d- (1 d- ~l)aa~e -~" 7r
D.=C.+L.
Vol. 218, No. 4, October 1954
~73
K. T. Sundara Raja Iyengar and R. S. Alwar
X
I
"
Inlcrfar.¢
]El t~l It
(I
I, !
FIo. 2.
Solution for layer 2: This being similar to Case 1, we h a v e ~"- =
f(i
"27,2(F. + axG~)e -"~ cos ayda,
x ~/ O,
(33)
the stresses are (a.)~ = (%) o =
/0
(r~)2 = --
(F. + axG.)e - ~ cos ayda ["( F . -- 2G.) + axG.;e -"~ cos ayda
.,
(34)
[ (F~ -- G.) + axG.?e -"~ sin ayda
the d i s p l a c e m e n t s are (u)., = N
~ [(1 + ~2) (F. + axG.) + (1 - g2)G.3e -"~ cos ayda
(35) 1 f o * 1 [(1 + g,,) (F~ + axG.) - 2 G ~ e -"~ sin ayda T o satisfy conditions at t h e interface given b y Eq. 27 we h a v e
G.
K.-L."
(36)
Continuity conditions: T h e s e are at x = 0
(~)' = - (~)~1 (v),= (v)~ l"
274
(37)
Journal of The Franklin Institute
Elastic Stress Analysis of Jointed Half-Planes Using Eq. 31 and 35 and satisfying Eq. 37, and after simplification by using Eqs. 32 and 36, we get the following two linear equations to determine K~ and L..
PIK. + Q1L~ = - 2In ) Q,K. ~- P , L . 14 -1- (1 - #,)I8I where
P1
(38)
and Q1 are given by Eq. 22 and p
Ia = ~
1-(1 ~- ~ ) a a -t- 2-]e -"~
(39) P (1 -1- #l)2aae -an
14
J
Solving Eqs. 38 and using Eq. 27, we derive expressions for the interface stresses
P
131-2 , -}- (1 -- #I)Q1-] ~__I4Q1 COS ayda
f
(~*)'
- "J 0
=
. . . .
FI~--Q--Y--
J0f~ I31-(1 -- #~)P~W2Q ~~p1_ Q~2
(r~)i
+ I4P1 sin ayda
l
(40)
J"
After simplification these are
P
O'x) i
2
/
a
P12 -- QI: t ]-2PI ~- (1 -- #I)QI~ y~ _1_ aS
a(a 2 _ y2)
+ (1 + #l)(p~ + QI) -~ -+-a~ (Tzy) ~
P 7
1 p~2 _
[
Q1z .
(41)
y
[-(1 - - ] z l ) P i nt-
}
2Ql~ y2 _}_ a s
2a2y
-t- (1 -]- ~ ) ( P 1 -b Q1) (y~ -b aS)i
Frictionless interface: In this case, by putting (rxu)~ -- 0 in Eq. 27 and using only the first condition of Eq. 37, we get 213 P1
(42)
P 1 [ 2a (l "k ~l)a(a2 -- Y2) ] -~ P1 y2 q_ as -b (-~-~-~)~ .
(43)
K. =
and the interface normal stress is
(cr~)~ --
Plane strain conditions: Until now, solutions for the two cases have been derived by assuming plane stress conditions• The same solutions also hold valid for
Vol. m , No. 4, Octobe, 1964
9,75
K. T. Sundara Raja 7yengar and R. S. Alwar
c~ 11
I
II
f~
r~
%
v
v
II
c)~ll
~g
276
II
~o~ ~g
Journal of The Franklin Institute
Elastic Stress Ana ysis of Jointed Half-Planes
I
I
t
I
II
el
+
~ +
v
+
+
I
II
+
I
I
I II
c~
+
+ I
%
II
+ %
I [
~
+
"1
+ +
+
~2 I
I
I
v
:L
I
I
%
r.r.l
+
i ~
+
+
% I
I
~1~
v
I
i
v
I
II
v
o~v
II
0
~ , ~ .~
VoW.27s, so. 4, October z96,~
°~
277
K. T. Sundara Raja Iyengar and R. S. Alwar
E plane strain conditions if the elastic constants E and g are replaced by 1 - g2 and ~
1 -
t~'
respectively.
Particular case: In both cases, if wc put in solutions, E1 = E~ and /~, = gx2,
we arrive at the solution for a full plane with a load P in the y and x directions respectively. It can easily be verified that the solutions in these cases reduce to those well-known of the infinite plane.
Discussion of Results B y working with smoothly jointed elastic half-planes, Dundurs has deduced some interesting properties of the normal stress transmitted across the interface. To deduce the same from the solutions derived here, the interface stresses (normal and shear) are given in Tables 1 and 2 for plane stress and strain conditions for both cases of loading, and for both rough and smooth interfaces. In the same tables, stresses prevailing in the full plane are also given. F r o m Tables I and I I it can be seen that normal stress at the smooth interface m a y be written as 4 (a,)',mooth = -- p~ ax* (44) where ~** is the stress at the plane of interface in a homogeneous full plane with the same loading as in the composite plane. Equation 44 is true in b o t h Cases 1 and 2 and for plane stress and plane strain conditions. Hence, for identical materials, i.e., E i = E2 and ~ l - ~2, P~ = - 4 , Eq. 44 indicates t h a t the normal stress across a 'lubricated cut' in a full plane remains unchanged. B y comparing the interface normal stresses (ax) in both problems of rough and smooth interface given in the two tables, we can arrive at the following conclusions : (1) In the ease of plane stress, these two interface normal stresses are equal if Q~ = 0. Then, the elastic constants have the following relationships, E._A1= 1 -- ttl E2 1 - tt:"
(45)
(2) In the :case of plane strain, the two normal stresses at the interface are equal if Q~ = 0. This gives two sets of relationships among the elastic constants,
(i) ~ 1 = ~ - = ½ or
(ii)
E1 1 + _t____A= 2 i - 2g_____A E~ 1 -4- it1 1 -- 2#z
(46)
i.e., G I - - ~
G2
1
-- 2~i
1 - 2~
Xi or
--
h2
~i ~
--
#2
where G arm ), are the rigidity modulus and Lame's constants, respectively.
278
Journal of The Franklin Institute
Elastic Stress Ana'ysis of Jointed Half-Planes Therefore, in the case of plane stress, when the interface is smooth, stress distribution in the composite plane is independent of Poisson's ratio of the plane which is free from internal force. B u t in the case of rough interface, stress distribution is dependent on the ~'s of b o t h planes. References (1) L. Rongved, "Forces in the Interior of One of Two Joined Semi-infinite Solids," Proc. Sec. Mid-west. Conf. on Solid Mech., Lafayette, Ind. 1955. (2) Frasier and L. Rongved, "Forces in the Plane of Two Joined Semi-infinite Plates," ,four. Appl. Mech., Vol. 24, p. 582, 1957. (3) J. Dundurs, "Force in Smoothly Joined Elastic Half-planes," Jour. Eng. Mech. Div. of ASCE., Vol. 88, p. 25, 1962. (4) K. T. Sundara Raja Iyengar, "Stress Distribution in Composite Infinite Strips," Bulletin of Polish Academy of Science, Vol. 9, No. 12, p. 659, 1961. (5) K. Girkmann, "Fli~chentragwerke," Vienna, Springer Verlag, 1959.
Vol. 278, No. 4, October 1964
279
ABSTRACT: Using a Fourier-integral approach, the problem of stress analysis in a composite plane consisting of two half-planes of different elastic properties rigidly joined along their boundaries has been solved. The analysis is done for a force acting in one of the half-planes for both cases when the force acts parallel and perpendicular to the interface. As a particular case, the interface stresses are evaluated when the interface is smooth. Some properties of the normal stress at the interface are discussed both for plane stress and plane strain conditions. Introduction
The stress distribution in the case of a force acting at a point within one of the two semi-infinite elastic solids joined along the plane boundary was investigated by Rongved (1) using Papkowitch functions. Later Frasier and Rongved (2) used the above three-dimensional solution to derive the solution for a plane-strain problem of two jointed semi-infinite plates subjected to a force at a point within one of the two plates (Figs. 1 and 2). By suitably
if Q
X
!
FIG. 1.
267
Y
In(¢rfoc¢
K. T. Sundara Raja Iyengar and R. S. A lwar
changing the elastic constants , their solution may also be used for a plane stress problem. Recently Dundurs (3) treated the plane stress problem of two elastic hMf-planes joined along their boundaries with a concentrated force inside one of them. He derived the solution using the complex-variable approach to plane problems in elasticity, and assumed that the interface is smooth. In this paper, a direct solution is given to the problem of two elastic halfplanes rigidly joined at the interface with a concentrated load in one of them. The solution is obtained using the Fourier-Integral approach. The case of the smooth interface solved in a different way by Dundurs becomes a particular case of the problem treated here. The solution is obtained assuming plane stress conditions. While discussing these reslflts, results are also given for plane strain conditions.
S t a t e m e n t of the Problem The problem has been considered in two parts: one in which the force is acting parallel to the interface, and the other in which the force is acting perpendicular to the interface. These and the orientation of the x and y axes are shown in Figs. 1 and 2. C a s e / - - T h e problem is shown in Fig. 1. E~, tL~ and E2, tL2 are the modulus of elasticity and Poisson's ratio of the two plates 1 and 2, respectively. The force P acts parallel to the interface at a depth 'a' below it. Since the problem is assumed to belong to the plane stress case, it can be formulated in terms of Airy's stress function. Hence, we have to determine a stress function for each half-plane such that each stress function satisfies the biharmonic equation; thus, if ~j is the stress function for the j~h layer (j = 1,2)
v4ei
=
0.
(1)
The stress components in the jth layer are ( ~ ) j = ~Oy " ~2 '
(~y)~ = ~"~; Ox2 '
(r~); =
~; ~x~y"
(2)
By assuming normal and shear stresses at the interface as Fourier-Integrals with unknown functions, the two half-planes can be treated separately. Such a procedure helps to set up the equations for the required unknown functions. This procedure has been used (4) in treating composite infinite strips. Let the unknown normal and shear stresses at the interface in the problem of Fig. 1 be (z~)~ = (r~,) ~ =
// //
K~ sin c~yda
(3)
L, cos ~ydo~.
(4)
Consider the two half-planes separately.
268
Journal of
The Franklin Institute
Elastic Stress Analysis of Jointed Half-Planes For half-plane 1: the boundary conditions are at x -- 0 K . sin c~ydc~
(cr.)1 = .
.
(5)
(r~v)l = f0 L. cosaydaJ In addition to the above, there is a singularity due to the force P at a point distant 'a' from the interface.
For half-plane 2: at x = 0 K . sin aycl~
(~)2 =
~
(r.v)2 =
--
.
(6)
/0 L~
In Eqs. 5 and 6 subscripts 1, 2 indicate the layers to which the stresses refer. Across the interface, the stresses ax and r ~ and the displacement components u and v must be eontinuous. Again putting subscripts 1 and 2 for the two half-planes, we have at x = 0 (*x) l =
(~)~
(**~)~ (u)~
-
(r,~)= L
_ (~)~ [.
(~)~
(~)~
(7)
J
F r o m Eqs. 3 and 4 it can be seen t h a t the first two relationships of Eq. 7 are automatically satisfied. Solution for the first layer: The solution for this half-plane having a concentrated load at a distance 'a' from the interface and having the interface stresses given by Eq. 5 can be built up of two stress functions. T h e first function which can be considered as a 'singular part' m a y be taken as the solution for a full plane with a concentrated load (5). The second function can thus be chosen that, when combined with the first one, it satisfies the required b o u n d a r y conditions of the problem. Hence ~l =
~i
I
+ ~oltl
where ~f =
fo ~ -~ 1 [A,
+
a(a -- x)B.]e -"(a-~) sin ayda,
(a -- x) /> 0
(8)
where
Vol. 278, No. 4, October 1964
A, =
1-~__~lP}
B,=
? 1-t-#1PJ4
(9)
269
K. T. Sundara Raja Iyengar and R. S. Alwar
and ~t" =
~1
-~ [C~ + ~ x D r ] e -"~ sin ayd%
x ) O.
(10)
The final stress components are ( ~ ) I = --
/;
[ A ~ + a ( a -- x)Br?e -~(~-~) sin ayda -
(~)~ =
/;
[-Cr + axD~?e - ~ sin c~yda
[-(A~ -- 2Br) + a(a -- x)B~-]e -"C"-~) sin aydc~
+ ( r ~ ) l = --
2
/;
(11) [(C. -- 2D.) + c~xD.~e - ~ sin ~yda
[ ( A . - Be) + a(a - x)B~e-"(~-~) cos ~yd~ +
F
[(Ca - Dr) + ~xD.~e -"~ cos ayda
The corresponding displacements are (u,) = - E 1
c~{ (l + # ~ ) [ A . + c ~ ( a - x ) B . ~ + +-E~
l f f l l~
(1-u~)B~}e -"("-~) sin ~yda
(l+m)(C.+c~xDr)+(1-u~)D.}e
-~" sin c~yda
(12)
(v)x=~-~l f f a l { 2 B " - ( l + m ) ~ A . + a ( a - - x ) B . 3 } e -~("-~) cos ayda E,
{2D,-- (l+~t)(Cr+axD,)}e-"" cos ayda
Putting in the boundary conditions of Eq. 5 into Eq. 11 we get C. = - K . - (A~ + aaB~)e -~" } Dr = C. - L~ - r_(A~ - B . ) + aaU~]e_. ~ ."
(13)
Solution for the second layer: This is a half-plane with the boundary stresses
given by Eq. 6. Hence the solution fo ~
1
~F~ + a x G ~ e -r~ sin ayd~,
x >/ 0
(14)
where
G.
270
= -- K~, Lr -- K. "
(15)
Journal of The Franklin Institute
Elastic Stress .! nalysis of Jointed Half-Planes
The stress components are (q~)2 =
(F~ + axG,~)e-" sin ayda
-- f0 ~
O-y) 2 ~---
(16)
o"~ [ (F,~ -- 2G,~) + axG.']e -"~ sin ayda fo ~ [. (F. -- G.) + axG.Te -"~ cos ayda
Txy) 2 ~-
It can be seen from Eqs. 15 and 16 that the boundary conditions specified by Eq. 6 are satisfied. The corresponding displacements are (u)2 --- ~
a {(1 + g2)(F. -~ axG,,) + (1 - ~2)G~}e-"~sinayda •
(~')"
1 ~o~1 { 2 G . -
E-~
(17)
(1 + ~,.,)(F~ + c~xG,,)}e-"~cosayda
Continuity conditions: As previously stated, the last two relations of Eq. 7
are to be satisfied. Hence from Eqs. 12 and 17 we arrive at the following relationships : [-(1 -q- ~1)(A,~ + aaB,~) -~ (1 - ~l)B~Te -"~ - [(I+ul)C.+
(1-ul)D.7-E~[(1
+u2)F~+
(1 - u 2 ) G ~ ] = 0
(18)
and [-(1 + gl)(A. + aaB,~) - 2B,~e -"~ +[(1
E1
+#~)C,-
+t~)F~--2G~]
2D~--ff2[(1
= 0.
(19)
Using Eqs. 9, 13 and 15 we can write Eqs. 18 and 19 as two equations in the two unknowns K~ and L~. Hence
-
where PI=
-- 2
P~K~ -- Q~L~ = I~
(20)
Q , K ~ + PIL~ = I2
(21)
( E,) 1+-~2
Ox = (1 - ~1) - F .E1 ~ (1 - ,.,) I1
--
-
P
-
(22) [(1
-
.1)
-
.a(l
+
,1)]e
-"~
12 = P r(1 + #t)aa -- 2~e -"a
Vol. 278, No. 4, October 1964
271
K. T. Sundara Raja Iyengar and R. S. Alwar L~, and substituting them in 3 and 4, we
Solving Eqs. 20 and 21 for K~ and have stresses at the interface, (~,)~ =
fo P,x + Q,I
px 2 _ Qx2 sin
ayd~
l
fo ~ P~I~ + QII~ COS ayda / "
(r~),
P12
Qt2
(23)
;
After simplification these will be i (a~) =
P 1 { ~-p2Q,~.[(1
a
-,~)P~+2Q,]y2+a s 2a~y
-- (1 + ,~)(Px + Q~) (y, + a,)2
} -.
(-r,~)' =
PTrP~ 1
(24)
Q12 { [2P1 + (1 -- .x)Q~] y2 +a a'-' -- (1 + u ~ ) ( P ~ + Q ~ )
~_~7a2~ I a(a2 -- Y~) ]
Eqs. 11, 12, 16, 17, 22 and 23 constitute the rigorous solution for the problem of Fig. 1. Frietionless interface: If the interface is smooth, the shear stress there is zero, i.e., L~ = 0 and the fourth relationship in Eq. 7 is not valid. Hence from Eq. 20 we have I1
K~ = ~ .
(25)
Consequently the interface normal stress is
(~),=
P 1 [ ( 1 - ~ 1 ) y2 y
+ a2
~" P1
2 ~(y2_++ _~)a~y ] a~)2 j"
(26)
Case 2 - - T h e problem is shown in Fig. 2. In this case the force P is acting in a direction perpendicular to the interface. As in the previous case, the unknown normal and shear stresses at the interface are assumed to be in the form of Fourier-Integrals, i.e., (a~) i --
K , cos
aydc~ .
(27)
(r,~)' = fo L, sin ayd~ J Solution for layer 1: As worked out in the previous case, the stress function for this half-plane is made up of two parts, i.e., @I =
272
~I l +
~PzII
Journal of The Franklin Institute
Elastic Stress Analysis of Jointed Half-Planes
where ¢11
= ~ ~1 (A. + ayB.)e-"~ cos axda, Y>/ 0 J0
-~ (C~ + axD.)e -"~ cos ayda,
x ) O.
(28) (29)
The stresses and displacements are (~)1 =
1-(A. - 2B,) + ayB.~e -"u cos axda -
(~)1 = --
/:
/;
(C. + axD~)e -"~ cos ayda
(A~ + ayB.)e - ~ cos axd~ + ~o 1-(C~ - 2D~) d- a x D ~ e -"~ cos aycla
(r~)l = --
/;
l (30)
[-(A. -- B.) + a y B ~ e - ~ sin axda - f ~ [-(C~ -- D.) -I- axD.Te - ~ sin aycla ..Io
and (U) I
lf0~l~ { ( 1
= E1
+VII1 f o ~ l
+ ~1)(A. + ayB.) -- 2B.}e-"~ sin axda
{(1 d-~1)(C, d - a x D . ) d- (1 - ~ l ) D . } e - " x c o s a y d a (31)
(v)~ = ~1 ~0~1~ {(1 + ~ ) ( A ~ + ayB.) + (1 -- #l)B~}e -~y cos axd~
1 ~o~1~ { (1 -4- ~1) (C. -4- axD~) -- 2D.}e - ~ sin (~yda
d- ~
In order to satisfy the boundary conditions on x = 0 given by Eq. 27 and to have the singularity at x --- a, the following relationships are valid P A~ = ~ ( 1 B.=
--~l) sinaa
- 2 -P~ (1 + gl) s i n a a
(32)
C~ = - K . -4- _2 1-2 d- (1 d- ~l)aa~e -~" 7r
D.=C.+L.
Vol. 218, No. 4, October 1954
~73
K. T. Sundara Raja Iyengar and R. S. Alwar
X
I
"
Inlcrfar.¢
]El t~l It
(I
I, !
FIo. 2.
Solution for layer 2: This being similar to Case 1, we h a v e ~"- =
f(i
"27,2(F. + axG~)e -"~ cos ayda,
x ~/ O,
(33)
the stresses are (a.)~ = (%) o =
/0
(r~)2 = --
(F. + axG.)e - ~ cos ayda ["( F . -- 2G.) + axG.;e -"~ cos ayda
.,
(34)
[ (F~ -- G.) + axG.?e -"~ sin ayda
the d i s p l a c e m e n t s are (u)., = N
~ [(1 + ~2) (F. + axG.) + (1 - g2)G.3e -"~ cos ayda
(35) 1 f o * 1 [(1 + g,,) (F~ + axG.) - 2 G ~ e -"~ sin ayda T o satisfy conditions at t h e interface given b y Eq. 27 we h a v e
G.
K.-L."
(36)
Continuity conditions: T h e s e are at x = 0
(~)' = - (~)~1 (v),= (v)~ l"
274
(37)
Journal of The Franklin Institute
Elastic Stress Analysis of Jointed Half-Planes Using Eq. 31 and 35 and satisfying Eq. 37, and after simplification by using Eqs. 32 and 36, we get the following two linear equations to determine K~ and L..
PIK. + Q1L~ = - 2In ) Q,K. ~- P , L . 14 -1- (1 - #,)I8I where
P1
(38)
and Q1 are given by Eq. 22 and p
Ia = ~
1-(1 ~- ~ ) a a -t- 2-]e -"~
(39) P (1 -1- #l)2aae -an
14
J
Solving Eqs. 38 and using Eq. 27, we derive expressions for the interface stresses
P
131-2 , -}- (1 -- #I)Q1-] ~__I4Q1 COS ayda
f
(~*)'
- "J 0
=
. . . .
FI~--Q--Y--
J0f~ I31-(1 -- #~)P~W2Q ~~p1_ Q~2
(r~)i
+ I4P1 sin ayda
l
(40)
J"
After simplification these are
P
O'x) i
2
/
a
P12 -- QI: t ]-2PI ~- (1 -- #I)QI~ y~ _1_ aS
a(a 2 _ y2)
+ (1 + #l)(p~ + QI) -~ -+-a~ (Tzy) ~
P 7
1 p~2 _
[
Q1z .
(41)
y
[-(1 - - ] z l ) P i nt-
}
2Ql~ y2 _}_ a s
2a2y
-t- (1 -]- ~ ) ( P 1 -b Q1) (y~ -b aS)i
Frictionless interface: In this case, by putting (rxu)~ -- 0 in Eq. 27 and using only the first condition of Eq. 37, we get 213 P1
(42)
P 1 [ 2a (l "k ~l)a(a2 -- Y2) ] -~ P1 y2 q_ as -b (-~-~-~)~ .
(43)
K. =
and the interface normal stress is
(cr~)~ --
Plane strain conditions: Until now, solutions for the two cases have been derived by assuming plane stress conditions• The same solutions also hold valid for
Vol. m , No. 4, Octobe, 1964
9,75
K. T. Sundara Raja 7yengar and R. S. Alwar
c~ 11
I
II
f~
r~
%
v
v
II
c)~ll
~g
276
II
~o~ ~g
Journal of The Franklin Institute
Elastic Stress Ana ysis of Jointed Half-Planes
I
I
t
I
II
el
+
~ +
v
+
+
I
II
+
I
I
I II
c~
+
+ I
%
II
+ %
I [
~
+
"1
+ +
+
~2 I
I
I
v
:L
I
I
%
r.r.l
+
i ~
+
+
% I
I
~1~
v
I
i
v
I
II
v
o~v
II
0
~ , ~ .~
VoW.27s, so. 4, October z96,~
°~
277
K. T. Sundara Raja Iyengar and R. S. Alwar
E plane strain conditions if the elastic constants E and g are replaced by 1 - g2 and ~
1 -
t~'
respectively.
Particular case: In both cases, if wc put in solutions, E1 = E~ and /~, = gx2,
we arrive at the solution for a full plane with a load P in the y and x directions respectively. It can easily be verified that the solutions in these cases reduce to those well-known of the infinite plane.
Discussion of Results B y working with smoothly jointed elastic half-planes, Dundurs has deduced some interesting properties of the normal stress transmitted across the interface. To deduce the same from the solutions derived here, the interface stresses (normal and shear) are given in Tables 1 and 2 for plane stress and strain conditions for both cases of loading, and for both rough and smooth interfaces. In the same tables, stresses prevailing in the full plane are also given. F r o m Tables I and I I it can be seen that normal stress at the smooth interface m a y be written as 4 (a,)',mooth = -- p~ ax* (44) where ~** is the stress at the plane of interface in a homogeneous full plane with the same loading as in the composite plane. Equation 44 is true in b o t h Cases 1 and 2 and for plane stress and plane strain conditions. Hence, for identical materials, i.e., E i = E2 and ~ l - ~2, P~ = - 4 , Eq. 44 indicates t h a t the normal stress across a 'lubricated cut' in a full plane remains unchanged. B y comparing the interface normal stresses (ax) in both problems of rough and smooth interface given in the two tables, we can arrive at the following conclusions : (1) In the ease of plane stress, these two interface normal stresses are equal if Q~ = 0. Then, the elastic constants have the following relationships, E._A1= 1 -- ttl E2 1 - tt:"
(45)
(2) In the :case of plane strain, the two normal stresses at the interface are equal if Q~ = 0. This gives two sets of relationships among the elastic constants,
(i) ~ 1 = ~ - = ½ or
(ii)
E1 1 + _t____A= 2 i - 2g_____A E~ 1 -4- it1 1 -- 2#z
(46)
i.e., G I - - ~
G2
1
-- 2~i
1 - 2~
Xi or
--
h2
~i ~
--
#2
where G arm ), are the rigidity modulus and Lame's constants, respectively.
278
Journal of The Franklin Institute
Elastic Stress Ana'ysis of Jointed Half-Planes Therefore, in the case of plane stress, when the interface is smooth, stress distribution in the composite plane is independent of Poisson's ratio of the plane which is free from internal force. B u t in the case of rough interface, stress distribution is dependent on the ~'s of b o t h planes. References (1) L. Rongved, "Forces in the Interior of One of Two Joined Semi-infinite Solids," Proc. Sec. Mid-west. Conf. on Solid Mech., Lafayette, Ind. 1955. (2) Frasier and L. Rongved, "Forces in the Plane of Two Joined Semi-infinite Plates," ,four. Appl. Mech., Vol. 24, p. 582, 1957. (3) J. Dundurs, "Force in Smoothly Joined Elastic Half-planes," Jour. Eng. Mech. Div. of ASCE., Vol. 88, p. 25, 1962. (4) K. T. Sundara Raja Iyengar, "Stress Distribution in Composite Infinite Strips," Bulletin of Polish Academy of Science, Vol. 9, No. 12, p. 659, 1961. (5) K. Girkmann, "Fli~chentragwerke," Vienna, Springer Verlag, 1959.
Vol. 278, No. 4, October 1964
279