L Ma/Intemational Journal of Solids and Structures 47(2010)3214-3220 y Pre-switching Post-switc Fig. 6. A 90 switching elementary domain with an original orientation a near the crack ti et al 1983). the shear stress criterion(Evans and Cannon, 1986), and D(xs, ys)=1. Inserting Eq (6.1)into Eq (4.7)we get the transformation strain energy criterion(see, e.g. Lambropoulos, 1986), the energy switching criterion(Hwang et al, 1995)and 4 (s)=-4Eok1 other criteria for transformation toughening, which are reviewed 2π(1+K in the reference (Hannink et al, 2000). According to different mate- Substituting Eq (6.2) into Eq (5.1), we get AK of the crack tip due to be used to evaluate the transformed zone and assess the degree of transformation toughening. While the problem of transforma- AK=AK +iAK ∫4(x,y)D(x,ydA tion criterion remains a controversial issue. its further discussion lies outside the scope of this pape In this section, our aim is to verify the validity of the obtained fundamental solutions and to demonstrate the efficiency of the for- (1+k)ls√」 mulations obtained above. Three simple but representative trans- formation toughening examples will be investigated for given transformation zones. In order to attack the inverse problem of identifying the most appropriate transformation criterion, different This result is completely consistent with the results obtained by transformation zones can be considered and compared using the IcMeeking and Evans(1982). explicit solutions given below. 6.2. Example 2: a semi-infinite crack enclosed by a transformed wake 6.1. Example 1: interaction between an infinite crack and a transformed spot Consider a semi-infinite crack enclosed by a transformed wake as shown in Fig. 8. Suppose the radius of the transformed circular his example has been studied by McMeeking and Evans area in front of crack tip is R, the infinite transformed area is A, through an Eshelby-type approach. So the existing solutions transformation is a dilatational transformation problem can be regarded as a standard solution to verify the D(r, 0)=1. The influence function is identical (6.2)which y of the obtained fundamental solutions in this paper. can be rewritten as R<r, interacts with an infinite crack, as shown in Fig. 7. Suppose f(r, 0)=-46oA-rtcos30-isin2e a transformed circular region of radius R and center at(r, 0). the transformation strain within the region is dilatational (1+K) ExD= Eyo= Eo (6.1) Substituting Eq (6. 4)into Eq. (5. 1). after some straightforward :∠ Fig. 7. Interaction between an infinite crack and a transformed spot(McMeeking Fig 8. A crack is enclosed by a transformed wakeet al., 1983), the shear stress criterion (Evans and Cannon, 1986), the transformation strain energy criterion (see, e.g. Lambropoulos, 1986), the energy switching criterion (Hwang et al., 1995) and other criteria for transformation toughening, which are reviewed in the reference (Hannink et al., 2000). According to different mate￾rial transformation mechanisms, the corresponding criterion can be used to evaluate the transformed zone and assess the degree of transformation toughening. While the problem of transforma￾tion criterion remains a controversial issue, its further discussion lies outside the scope of this paper. In this section, our aim is to verify the validity of the obtained fundamental solutions and to demonstrate the efficiency of the for￾mulations obtained above. Three simple but representative trans￾formation toughening examples will be investigated for given transformation zones. In order to attack the inverse problem of identifying the most appropriate transformation criterion, different transformation zones can be considered and compared using the explicit solutions given below. 6.1. Example 1: interaction between an infinite crack and a transformed spot This example has been studied by McMeeking and Evans (1982) through an Eshelby-type approach. So the existing solutions of the problem can be regarded as a standard solution to verify the valid￾ity of the obtained fundamental solutions in this paper. A transformed circular region of radius R and center at (r,h), R  r, interacts with an infinite crack, as shown in Fig. 7. Suppose the transformation strain within the region is dilatational, ex0 ¼ ey0 ¼ e0 ð6:1Þ and D(xs,ys) = 1. Inserting Eq. (6.1) into Eq. (4.7) we get f4ðsÞ ¼ 4e0l ffiffiffiffiffiffi 2p p ð1 þ jÞ 1 s ffiffi s p ð6:2Þ Substituting Eq. (6.2) into Eq. (5.1), we get DK of the crack tip due to presence of the transformed spot as DK ¼ DKI þ iDKII ¼ Z Z A f4ðxs; ysÞDðxs; ysÞdA ¼ 4e0lpR2 ffiffiffiffiffiffi 2p p ð1 þ jÞ 1 s ffiffi s p   ¼ 4e0lpR2 ffiffiffiffiffiffi 2p p ð1 þ jÞ r3 2 cos 3 2 h i sin 3 2 h   ð6:3Þ This result is completely consistent with the results obtained by McMeeking and Evans (1982). 6.2. Example 2: a semi-infinite crack enclosed by a transformed wake Consider a semi-infinite crack enclosed by a transformed wake as shown in Fig. 8. Suppose the radius of the transformed circular area in front of crack tip is R, the infinite transformed area is A, transformation is a dilatational transformation (ex0 = ey0 = e0), and D(r,h) = 1. The influence function is identical to Eq. (6.2) which can be rewritten as f4ðr; hÞ ¼ 4e0l ffiffiffiffiffiffi 2p p ð1 þ jÞ r3 2 cos 3 2 h i sin 3 2 h   ð6:4Þ Substituting Eq. (6.4) into Eq. (5.1), after some straightforward manipulation, we get Fig. 6. A 90 switching elementary domain with an original orientation w near the crack tip. Fig. 7. Interaction between an infinite crack and a transformed spot (McMeeking and Evans, 1982). Fig. 8. A crack is enclosed by a transformed wake. L. Ma / International Journal of Solids and Structures 47 (2010) 3214–3220 3219