INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

Transcription

1 IC/8U/63 : "X7 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS S S V THE IMAGE FORCE ON THE DISLOCATION NEAE A FINITE LENGTH CRACK TIP C.W. Lung INTERNATIONAL ATOMIC ENERGY AGENCY L. Wang UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1984 Ml RAM ARE-TRIESTE

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3 IC/8V63 International Atomic Energy Agency United Nations Educational Scientific Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THE IMAGE FORCE ON THE DISLOCATION NEAR A FINITE LENGTH CRACK TIP C.W. Lung** International Centre for Theoretical Physics, Trieste, Italy L. Wang Insitute of Metal Research, Academia Siniea, Shenyang, China, ABSTRACT The image force on the dislocation near an elliptical hole was calculated by a conformal mapping technique. Allowing one axis of the ellipse to dwindle to zero,the finite length crack limit solution vas obtained. 1. Introduction Since the discovery of dislocation-free zone (DFZ) near a crack tip_ (Kobayashi Ohr 1980), efforts have been made to explain its formation theoretically. This DFZ may have some influences on the theory of elastic plastic fracture mechanics. Chang Ohr (l<)8l) have discussed the formation of the DFZ by applying the criterion of Rice Thomson (1973) which involves the image force due to the free surface of a crack tip. Lung Xiong (1933) calculated the dislocation distribution function in the plastic zone at a crack tip showed that there was a negative dislocation zone at the crack tip which may form DFZ after some processes of energy balancing. It seems worthwhile to analyze the problem of the image force on the dislocation near a crack tip. In their calculation, Rice Thomson (1973) Asaro (1975) pointed out that the image force on the dislocation near a crack tip is similar to that near a free plane. This conclusion has been quoted by many authors, however, we think that two questions should be clarified: (i) They did not consider the radius of curvature at the crack tip which may be very small. (li) The crack length they discussed was semi-infinite long which was true only in limiting cases. Generally, a crack is of finite length. 2. Conformal mapping function Using Eshelhy's expression (Eshelby, 1979) for the Image force on the dislocation near a circular hole, the Image force on the dislocation near an elliptical hole was calculated by conformal mapping technique. Allowing the axia of ellipse to dwindle to zero, the crack limit solution ia obtained. The transform function is By this transform, the elliptical contour In a physical plane {z-plane), MIRAMARE - TRIESTE June 1981* To be submitted for publication. Permanent address:institute of Metal Research,Acadetnia Siniea, Shenyang^ China. X i- -2- = 1 h 1 (2) can be mapped onto a circle in the C-plane c = R (R > l). R Is the radius of the circle. The relationships between the semi-long axis a, semi-short axis b in the z-plane the R, c of the circle In the -plane are

4 c R"), =:</(a.' t -b*j, b - (3) This reduces to Eghelby's expression for the image force on the dislocation' near a circular hole. 3. Let x=a+s s «y( f = b*/a, the radius of curvature of the ellipse at the long axis end), it can be proved that On the contrary, a circle In the x, -plane can be mapped onto an ellipse in the z-plane. The reversion of the transform function is 5= According to Eshelby, the change of potential energy on the real axis due to the image force on a screw dislocation with Burger's vector b s near a circular hole is E(J) in the ;;-plane Re E ($)=(/*tifarilnk$~ «V/ V whereat is the shear modulus. With conformal mapping, the change of potential energy due to the image force on the dislocation near the elliptic hole on the direction of real axis is E(z) = - E<$) (6) the same as where A = yu/2tf, This is^hice Thomson (1973) Asaro's (1975) expression for the dislocation image force anti (10) is the same as that due to a free plane. Then, we see that any curved plane seen by the dislocation at a distance much smaller than the radius ofcurvature is like a plane. That is^eice Thomson Asaro's expression would be correct if the distance were much smaller because the radius of curvature of the curved plane at the crack tip might be very small. Rice Thomson (1973) Asaro (1975) did not give the condition of how long the distance in comparison with the radius of curvature of a crack tip is to fit (10). 1*. Let b=0. This is the crack limit case. The image force near a finite length crack tip is Therefore, the image force in the z-plane ia _ de\d$\ _ c AS. (7) After some lengthy calculation, the Image force may be expressed as In our case, we choose plus sign in (U). 3. Image force on dislocation near circular, elliptical holes crack tip From (8), the following cases are analyzed: 1. If b/a=k{const.); then, (a+b)=a(l+k).that is, if both the long axis short axis decrease proportionally, the image force F would approach zero as a -» If a/b=l; then, -3- (8) (9) Taking an approximate expression, let s«; a ; then, F. 2; Ab /Its. That is, even in the ideal crack case, the image force is approximately half the value of the formula (10). In Rice Thomson (1973) Asaro's (1975) papers cricks were of infinite length.any finite length is a small quantity in comparison with infinity, furthermore, any small quantity in comparison with a finite length may be a smaller ijuantity of higher order. Therefore, they fit in with formula (10) k. Numerical results discussion In Fig. 3 we give the relationship between s/a. the following results appear: 2Tta F(s/a),/u.b s 1. Image force on dislocation decreases in the order of free plane, circular hole, elliptic hole crack; 2. At a distance s/a<0.005, the image force due to any curved plane is similar -to the free plane; when the distance s/a > 0.0O5, the differences among them (including crack, b = 0) will be quite large. -k-

5 3. In the general ease, cracks are of finite length blunted, we may use expressions (8) or (ll) instead of (10). t-plane ACKNOWLEDGMENTS One of the authors (C.W. Lung) would like to thank Prof. Abdus Salara, the International Atomic Energy Agency UNESCO for hospitality at the International Centre for Theoretical Physics, Trl«at«, *fe*n tfci-i paper vas written. The dislocation near a circular hole. REFERENCES Asaro, R.J. (1975) J. Physics F, 5, 22UQ. Chang, S.J. Ohr, S.M. O98l) J. Appl. Phys. 5g. 12, T17 1 * Eshelby, J.D. (19T9) in Dislocations in Solids. Vol.1, Ed. F.R.N. Narbarro, 1^7. Kobayashi, S. Ohr, S.M. (1980) Phil. Mag.Aji2, 763. Lung, C.W. Xiong, L.Y. (1983), Phjrs. Sfcat. Sol, (a) 77, Si. Bice, J.R. Thomson, R, (1973) Phil. Mag. 29, 73. z -plane Fig. 2 The dislocation near an elliptic hole. -5-

6 100 r BO 60 CM s /a Fig. 3 The relationship between the dislocation image force the distance from a hole. -7-