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VoL 27, No. I, pp. I09..-116, 2W0
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ANALYSIS OF A DISSIMILAR HNITE WEDGE UNDER ANTIPLANE DEFORMATION
M.H.Kargernovin School o f Mechanical Engineering, Sherif University of Technology, Tehran, Iran S.J.Fariborz Department of Mechanical Engineering Amirkabir University of Technology, Tehran, Iran
(Received 18 May 1999; acceptedfor print 9 November 1999)
Introduction
The stress analysis in a composite infinite wedge has been the s~ject of numerous i n v ~ o m . Probably, the major interest m the wedge geometry can be attributed to the fact that half-plane and edge crack are special cases of wedge problem~ Stress analysis in bi-material wedge was done by BOI~D'[1], Hein and Erdogen [2],Dempsey and S ~ [3], and Dwpsey [4]. ~ o t i c stress anmy~ m the neighbornooo oz me common apex ot a regime__ comiatiag otimtropic wedges with differmt mater;ads is accomplished by Blinova!and Linko~, [5]. In all above works the wedge was under in-plane deformation. The antiplane shear deformation of an isotropic wedge under static and dynamic loading is solved by Morozov and Naflmt [6]. Ma and Hour [7], studied the anfiplane shear deformation of dissimilar anisotropic wedges. In the present article the ~ti'plane.defo..r~tion of a wedge consisting of two isotropic and perfectly bonded wedg.es with a .finite radius m analyzed. Two problems regarding the type of boundary data on the circular pomon of the boundary are studied. The traction f l ~ Problem A, and fixed displacement, Problem B, conditions are imposed on the circular arc. Various boundary data are considered on the radial edges. The solution is accomplished by employing the finite Mellin transforms. In contrast to the isotropic wedge under identical boundary conditions, ..Kar~arnovin et al. [8], it is not possible to determine explicitly the power of leading term of displacement field at the wedge apex; the solUtion of transceaden~ equations is ~ . These equations are identical with those obtained in [7], for infinite wedges. The dominant term of solution for displacement is obtained in the vicinity of the wedge apex.
,Formulation ~md PrQblem Solution
The dissimilar wedge is composed of two isotropic wedges having different material prop~!'es. These wedges are perfectly bonded together tkmg the common edge OB and extended infinitely in the direction perpendicular to the ..w~.g.e.plgne, Fig.0 ). Tharadius of wedges is equal to a and the apex angles are c~ and an. The dissimilar Wedge m subjected to anUplane shear deformation. Therefore, the only existing displacement component is the out of plane component W(r, 0), and the n o n - v ~ stress components are xn(r~0) and T~(r,O). The constitutive relationships for isotropic materials under foregoing conditions are 0W _p0W Zrzftt"~-', Z0z r a0 (1) 109
110
M.H. KARGARNOVIN and S. J. FARIBORZ
C
O
Figure 1- Schematic view of a finite wedge with radius a comprising of two different materials with wedge angles oti and otn. where I-t designates, shear modulus of the material. The Navier's equation is 02W l a W 1 02W &~ + ~ - 0 (2) r & r 2 305 In problem A, i.e., cases A1, A2 and A3, the circular segment of the wedge circumference is free from traction. Therefore, ~= (a, 0) = 0 (3) In problem B, i.e., cases BI, B2 and B3, the wedge is fixed on the circular segment of the W(a,o) = 0 boundary. Thus, The finite Mellin transform of the first and second kinds are represented, respectively, as
* a 2S
(4)
Mt [W(r, 0), S] = W, (S, O) = ~(-g7" rS~)W(r, e)dr 0 r * a a2S M2 [W(r, 0), S] = W~ (S, 0) = ~(--~- + r~~)W(r, 0)dr
0 r
(S)
where S is a complex transform parameter. The inversions of these transforms are as follows M]~[w~(S, 0),r] = W(r, 0) - ( - 1 ) J c +]~r S W j ("S , 0)dS (j = 1,2 ) (6) 2hi c-i~o The application of the Mellin transform of the first kind in conjunction with integration by parts on (2) yields
(-a--~=+ s )w~ (s,o) + 2 s , s w ( < o) = 0
Similarly employing the Mellin transform of the second kind on (2), leads to
2
92
2
*
(7)
(307 + S2) W; (S, 0) + 2 a s+~0W(a,0r0) = 0
(8)
Applying the boundary condition (4) on (7), and the boundary data (3) with the aid of the first of(I) on (8) leads to the following equation for both problems A and B d2W] +$2 Wj' =0 (j =1,2) (9) d 05 where j=2, and j=l, for problem A and B, respectively. The solution to (9) is k W ; ( S , 0 ) = kAj(S)sin(S0) + k Bj (S ) cos (S 0) (j,k=l,2) (10) where k designates the wedge number. Employing the condition of continuity of displacement and stress component x~, along the interface, 0 = 0, two unknown coefficients of each solution in (10) are calculated in terms of the other coefficients. The Mellin transforms of displacement in two
FINITE WEDGE DEFORMATIONS regions appear as
111
'w~. (s, 0) = & (s) sin (s 0) + Bj (s) cos (s 0) 2W~(S,O) =(1~/1~) Ai(S)sin(SO) + Bi (S ) cos (S 0) ( j = ] , 2 ) (11) In the followin~ two sections, various boundary data on the radial edges are enforced to compute the unknowns m (11).
Problem A
Depending upon the prescribed boundary data on segments OA and OC in Fig. (1), three different cases of traction-displacement, displacement-displacement and traction-traction may be recognized in each problem. The analyses of these cases for Problem A being taken up separately in this section.
Case AI - Tmction-Disvlacement
Let the wedge be fixed on the boundary OA and subjected to a concentrated antiplane shear traction with intensity P, on the edge OC. Therefore, the boundary conditions may be written as X'%z(r,ctl) = PS(r- h) 0
n, the residues of(21) at the simple pole, S = - n should be used to determine the leading terms of displacement fields. The results lead to 1W (r, 0) = [ (Ix ii / ~tl ) cos (n an) sin(n O) + sin(n an) cos (n 0)] r ° / D2 (n)
(26)
2W(r, 0)= sin[n(0 +aii)] r"/D2(n) where D2(n) should be calculated from (22). It is interesting to note that for nSi (i=1,2,3,...), and carrying out the contour integration (14), (21) and (32), respectively, by determining the residues at these poles. Moreover, depending upon the values of wedge angles and shear modulus ratio, the aforementioned transcendental equations may have several roots which are less than unity. The stress fields corresponding to these roots are singular at the wedge apex. The plot of first two roots of(16) which produce singular stress fields for a wedge with an= 30 °, shear modulus ratio ga//.t~ = 0.1, where oq changes from 15° to 330 ° is shown in Fig.(2).
1.1 1 0.9 0.8 0.7
0.6 0.5
0.4 0.3 0.2 0.1
I ,,
I
I
I
I
I
0
1
2
3 ALPHA1 (tad)
4
5
6
Figure 2- Plot of the first two solutions of transcendental equation for (7.11=x/6 and~li/p.l = O. l.
(19) vs. cx~
Problem B
In essence the analysis of problem B parallels that of problem A. Consequently, a brief account of the analysis is given. The Mellin transform of second kind is utilized in problem A, whereas in problem B the solution method is based on the employment of finite Mellin transform of the first
FINITE WEDGE DEFORMATIONS
115
kind. In the sequel, as treated in problem A, different boundary data on the radial edges of the wedge is considered.
Case BI - Traetion-Disvlaeement
The boundary conditions are represented by (12). The finite Mellin transform of the first kind of boundary data (12) with the aid of the second (1), yields
d
dO
~ W ; ( S , ~ x ) = ( P / , I ) (ae~/b~-h ~ )
(37)
2W~ (S, -- (~II) = 0
Applying (37) to (11) withj=l, utilizing the inversion formula (6), and carrying out the eomour integration analogous to case A1, results in
tW(r,0) = (-P/la t EtSt)(a-2S'hS'-h -s' )[eos(Stau) sin(st0)+ (I.tt IV.n) sin(stcq0 eos(St0)]r s' 2W(r,0) = (-P/ p.n EtSI)( a-as' hs*-h-s' ) sin[st(0+ctH)] rs* (38)
where E~ is given in (18) and $1 is the smallest positive root of (16).
Case B 2
- Displacemem-Displacement
The boundary conditions (19) are considered in this case. We should mention that the boundary condition on the edge OC must be consistent with (4), at point C. For the sake of brevity we have considered only one term of the Taylor series representation in the neighborhood of the wedge apex, of the actual boundary condition. Analysis analogous to what follows should be performed for other terms of the series. The Meilin transform of boundary data in view of the first of (5), yields, Wt (S, ct~ ) = 2S a ~÷s/(n 2 - S2 ) Applying the above conditions to (11) and making use of the inversion formula (6), we have
1W(r) 0)
"S 21Wl,( ,-- C(,II) = 0
(39)
-1
,ti
c~i® Sa~+S[(rtn/pa)eos(Sctn) sin(S0)+ sin(Sc~n) cos(S0)]
c-i~ J (n~-S~) t~(S)
r-SdS
(40)
2W(r,0 ) = - 1 ni
c~i® Sa~+S sin[S(0 +an)]r_Sd s
c-i® ( n 2 - S ~) I ~ (S)
In the event of Sl n, and Si = n, the displacement fields are (26) and (27), respectively.
Case B3- Traction- Traction The boundary data are those of case A3. The Mellin transform of the first kind of (29) becomes,
116
M.H. KARGARNOVIN and S. J. FARIBORZ
[ 2 wt* (s,- a.)] = ( P / ~ , . ) (a 2"/ h l - hl)
(42)
[ Wt (S, o~i)] = (P / ~.~i) (a 2S/ h~ - h~)
Utilizing these conditions, the unknowns in (11) with j=l may he calculated. Employing the inversion formula (6), and carrying out the contour integration leads to t W (r, 0) = { (a-2S' h~' - h~s' ) cos [ St ( ctt - 0 )] + ( a -2s, h s, _ h s, ) [ (~t ix / ~t l) sin (SI aii) sin(sl 0) - cos(S1 ctii) cos (Sl 0)] } r s, (43) W (r, 0) = ( - P / P-t S~ E4 ) { ( a -2s, h s, _ h2s, ) [ cos (St ctu ) cos(St 0) + (~t i / ~tn) sin(S1 at) sin (St 0)]- ( a -2sl h s, _ hi-S, ) cos [ St (cqi + 0 )]} r s' where E4 is given in (36).
Conclusion
The dissimilar wedge with finite radius under antiptane shear deformation has been analyzed in this article. The finite Mellin transform is employed to solve the governing differential equation. The boundary data on the circular arc in problem A is traction free and in problem B is the zero displacement components. The boundary data on radial segments are traction-displacement, displacement-displacement and traction-traction. The dominant solution for displacement field near the wedge apex is obtained. In the particular case of infinite wedge the solution may be obtained by simply letting the wedge radius approaches infinity.
Acknowledgment
The authors wish to express their gratitude to the Office of Research Affairs of Sharif University of Technology for the financial support to conduct this research.
References
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2. 3. 4. 5. 6. 7. 8. 9.
D.B. Bogy, J. Appl. Meek 38, p. 377, (1971). V.L. Hein, and F. Erdogan, Int. J. Fracture Mech.,7, p. 317, (1971). J.P. Dempsey and G.B. Sinclair, J. of Elasticity, 11, p. 317, (1981). J.P. Dempsey, J. of Adhesion Sci. Technol., 9, p. 253, (1995). V.T. Blinova and A. M. Linkov, J. Appl. Maths Mechs, 59, p. 187, (1995). N.F. Morozov and MA. Narbut, J. Appl. Maths Mechs, 59, p. 307, (1995). C.C. Ma and B.L. Hour, Int. J. Solids Structure, 25, p. 1295, (1989). M H . Kargarnovin, A.R. Shahani and S.J. Fariborz, Int. J. Solids Structures, 34, p. 113, (1996). J.P. Dempsey and G.B. Sinclair, J. of Elasticity, 9, p. 373, (1979).