On the singularity of temperature gradient near an inclined crack terminating at bimaterial interface

Transcription

1 International Journal of Fracture 58: , Kluwer Academic Publishers. Printed in the Netherlands. 319 On the singularity of temperature gradient near an inclined crack terminating at bimaterial interface WEN-HWA CHEN 1 and CHIN-CHENG HUANG 2 adepartment of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan, Republic of China 21nstitute of Nuclear Energy Research, Lung-Tan, Taiwan, Republic of China Received 20 June 1991; accepted 25 March 1992 Abstract. This paper deals with the singularity of temperature gradient near an inclined crack terminating at a bimaterial interface. The temperature field is solved by considering the continuity of temperature and heat flux at the interface and appropriate thermal boundary conditions on crack surfaces. The singularity of temperature gradient around the crack tip is then studied for the cases for which the temperature on crack surfaces is prescribed or crack surfaces are insulated. It is found that, unlike the oscillatory singularity of the stress field, no oscillatory character near the crack tip is observed for these problems. The dependence of the singularity of temperature gradient on the inclined angle of crack and thermal conductivity ratio of two dissimilar media is also shown. 1. Introduction The study of heat conduction in dissimilar media with cracks under given thermal loadings is becoming increasingly important and has attracted considerable attention in the design of various structures such as electronic circuit board, ceramic composite and pipeline with cladding, etc. As is known, there are unavoidable cracks or crack like defects arising in the vicinities of bonding interfaces during the manufacturing of structures. The local thermal stresses at the regions near the imperfections are elevated by the intensified temperature gradient. This may initiate crack propagation or breakdown of the structures even under normal thermal conditions. Hence, to predict the failure behavior, an accurate analysis of the intensification of thermal gradient in dissimilar media with cracks is required. In the literature [1-5], the heat conduction for two-dimensional homogeneous media with cracks subjected to various types of heat transfer conditions on crack surfaces has been discussed. The r- 1/2 type singularity of temperature gradient around the crack tip was observed. r denotes the radial distance from the crack tip. For dissimilar media with cracks, however, little work has been done. Recently, Kuo [6] studied a crack situated at the interface between two semi-infinite dissimilar media subjected to a uniform heat flow. The temperature distribution of the problem was obtained by superimposing the temperature field for a perfectly bonded composite solid and a series of distributed thermal dipoles at the crack location. However, the singularity of temperature gradient around the crack tip was not noted. The objective of this work is thus devoted to the study of the singularity of temperature gradient near an inclined crack terminating at the bimaterial interface. The inclined angle of crack is between 0 and n. Using the method of separation of variables, the general solution of the temperature field is obtained by considering the continuity of temperature and heat flux at the perfectly bonded interface. Two types of thermal boundary conditions on crack surfaces are treated: (i) the temperature is prescribed and

2 320 W.-H. Chen and C.-C. Huang (ii) the crack surfaces are insulated. It is found that, unlike the oscillatory singularity of the stress field (except for the crack terminating at the interface perpendicularly) [7-9], no oscillatory character of the temperature gradient near the crack tip is observed in this problem. The dependence of the singularity of temperature gradient on the inclined angle of crack and thermal conductivities of dissimilar media is also drawn, respectively. 2. Temperature solution As shown in Fig. 1, consider a bimaterial plate with an inclined crack terminating at its interface. Two dissimilar media I and II of the plate are assumed to be homogeneous, isotropic and perfectly bonded along the interface. (r, 0) is the polar coordinate with the origin selected at the crack tip. fl is the inclined angle between the crack and the interface and 0 ~< fl ~< ~. fll and ~3 denote the regions above and below the crack in medium I, respectively, f12 represents the region in medium II. The steady state heat conduction equation for the temperature field T(r, O) in the plate can be written in terms of polar coordinates as c~2t 1 c3t 1 c32t ~r~ = O. (1) r ~-r r By the continuity of temperature and heat flux at the bimaterial interface, the following boundary conditions can be found as (a) T1 = T2 at O=fl, (2) (b) T2 = T3 at 0= -(n-fl), (3). ~3T1 3Ta (c) kl~-=k2~ at0=/? (4) # f ~edi2 I ~ f ) z O Interface Medium II Fig. 1. An inclined crack terminating at the bimaterial interface.

3 Singularity of temperature gradient 321 and,~t~ ~T3 (d) kz-~=kl-~ at0=-(n-fl). (5) Where T1, T2 and T3 denote the temperatures in regions f~l, ~r~ 2 and f~3, respectively, k I and k2 are the thermal conductivities of media I and II. To solve the temperature field near the crack terminating at the bimaterial interface, two kinds of thermal boundary conditions on the crack surfaces are considered: (1) the temperature is prescribed and (ii) the crack surfaces are insulated. (i) The temperature is prescribed Since the crack is situated in medium I, the additional thermal boundary conditions on crack surfaces are shown as Tx=T at0=n (6) and T 3 = T at 0 = --zc, (7) where T is the prescribed temperature. By the method of separation of variables, the general solution of T(r, O) can be obtained from (1) as T(r, O) = 7"+ ~ rx"(c"l cos C~ sin 2.0), (8) tl=l where 2, is an eigenvalue governing the singularity of temperature gradient, C] and C~ (n = 1, 2... ~) are the real constants to be determined by appropriate thermal boundary conditions. Considering the continuity conditions as stated in (2) to (5) and boundary conditions (6) and (7), the necessary and sufficient condition for nontrivial solution of T(r, O) can be stated as cos 2. fl sin 2.fl - cos 2.fl - sin 2. fl cos 2.(rr - fl) sin 2.(re - fl) cos 2.(zt -/~) - sin 2.(n - fl) co sin 2. fl - o9 cos 2. fl - sin 2. fl cos 2. fl sin 2.(n - fl) cos 2.(n - fl) -co sin 2.(n - fl) -~o cos 2.(n - fl) cos 2. n sin 2. rc cos 2.n -sin 2.n =0, where ~o = kl/k 2 is the thermal conductivity ratio of two dissimilar media. The characteristic equation for determining the eigenvalue 2, can be thus found as (1 -- o2)[sin 22.0r -- fl) + sin 22.fl-1 -- (1 + o9) 2 sin 22.1r = O. (9)

4 322 W.-H. Chen and C.-C. Huang (ii) The crack surfaces are insulated When the crack surfaces are insulated, the thermal boundary conditions on crack surfaces are = 0 at 0 = ~ (10) and ~T =0 at0=-n. (11) Following similar procedures as mentioned above, the general solution of T(r, O) for this case can be obtained from (1) as T(r, O) = ~ rx"(c] cos C~ sin 2.0). (12) n=l Considering the continuity conditions (2) to (5) and the boundary conditions on crack surfaces as mentioned in (10) and (11), the condition for nontrivial solution of T(r, O) can be written as cos 2.fl sin 2.fl -cos 2.fl -sin 2.fl cos 2.(r~ - fl) sin 2.(n - fl) cos 2.(n - fl) -sin 2.(rt - fl) o sin 2.fl -ocos 2.fl -sin 2,fl cos 2.fl sin 2.0r - fl) cos 2.(n - fl) - 09 sin 2.(z~ - fl) - ~o cos 2.(n - fl) - sin 2. zt cos 2.re sin 2.n cos 2.7z =0. Again, the characteristic equation which determines 2. is (1 - o2)[sin 22.(n - fl) + sin 22.fl] + (1 + ~o) z sin 22.n = 0. (13) It is interesting to note that (13) is nothing but (9) if 09 in (13) is replaced by 1/o. For the special case of an interfacial crack, i.e. fl = 0 or n, the characteristic equation for both cases can be further reduced as sin 22.1z = 0. (14) That is, the eigenvalue 2, is independent of the thermal conductivities of two dissimilar media and is exactly the same as that for homogeneous and isotropic cracked plates [1-5].

5 Singularity of temperature gradient Singularity of temperature gradient near crack tip The singularity of temperature gradient near the crack tip is governed by the term obtained from (8) and (12) as ~T ~-~ ~ O(r -~) 0 < (5 ~< 1, (15) where 6(= 1-21) represents the order of singularity of temperature gradient for which 0 < 21 ~< 1.21 can be evaluated from (9) and (13) for n = 1. The values of 6 as defined in (15) for an inclined crack terminating at the bimaterial interface can be computed by solving (9) and (13) numerically. As shown in Fig. 2, the dependence of (5 on the inclined angle ~ and thermal conductivity ratio o9 for the case in which the temperature is prescribed in the crack surfaces is observed. Since (5 is symmetric with respect to fl = 90, only the results between fl = 0 and fl = 90 are shown. For a specified value of r, (5 becomes larger as o9 decreases. That means, for a given medium I, the singularity of temperature gradient becomes stronger as the thermal conductivity k2 increases. Further, larger (5 is observed as ~ increases for o9 < 1 (i.e. kl < k2). But, as o9 > l(kt > k2), the variation of (5 is reversed. For the case of insulated crack surfaces, since (13) has exactly the same form as (9) except for replacing o9 by 1/o9, the dependence of (5 on fl and o9 can also be shown in Fig. 3, and opposite dependence of (5 on and o9 is concluded. In addition, for homogeneous and isotropic cracked plate (o9 = 1), as stated in an earlier section, (5 is found to be 0.5 for any ft. This also agrees with that reported in the literature [1-5] = ~ ~ I I I! I O" " ~" 75" 9~ Fig. 2. Singularity of temperature gradient versus crack inclined angle and thermal conductivity ratio as the temperature on crack surfaces is prescribed.

6 324 W.-H. Chen and C.-C ~ Huang D.B, " I I I 16" 30",5" 8'0",o" a Fig. 3. Singularity of temperature gradient versus crack inclined angle and thermal conductivity ratio for insulated crack. 4. Conclusion The general solution of the temperature field near an inclined crack terminating at a bimaterial interface has been examined and the singularity of the temperature gradient around a crack tip studied for two cases for which the temperature on the crack surfaces is prescribed or the crack surfaces are insulated. It is found that, unlike the oscillatory singularity of the stress field (except for the crack terminating at the interface perpendicularly), no oscillatory character of temperature gradient near the crack tip is observed. Opposite dependence of the singularity of temperature gradient on the crack inclined angle and thermal conductivity ratio for these two cases is also noted. The present work would be helpful for the heat conduction analysis of bimaterial cracked plates. References 1. G.C. Sih, ASME Journal of Heat Transfer 87 (1965) A.F.Emery, P.K. Neighbors, A.S. Kobayashi, and W.J. Love, ASME Journal of Pressure Vessel Technology 100 (1977) W.H. Chen and K. Ting, Nuclear Engineering and Design 90 (1985) K. Ting and W.H. Chert, AIAA Journal 26 (1988) W.H. Chen and K. Ting, Computational Mechanics 4 (1989) A.Y. Kuo, ASME Journal of Applied Mechanics 57 (1990) F. Erdogan, ASME Journal of Applied Mechanics 32 (1965) D.B. Bogy, ASME Journal of Applied Mechanics 38 (1971) D.N. Fenner, International Journal of Fracture 12 (1976)