9 Measurement of Crack Tip Displacement Field in Desiccating Paste 69 where fIn .r; / D rn=2 2G n cos n 2 n 2 cos n 2 2 Cn n 2 C. 1/nocos n 2 o; fIIn .r; / D rn=2 2G n sin n 2 n 2 sin n 2 2 Cn n 2 . 1/nosin n 2 o; gIn .r; / D rn=2 2G n sin n 2 C n 2 sin n 2 2 n n 2 C. 1/nosin n 2 o; gIIn .r; / D rn=2 2G n cos n 2 n 2 cos n 2 2 Cn n 2 . 1/nocos n 2 o: (9.2) In Eqs. 9.1 and 9.2, u and v are the displacement component in x-direction and y-direction, respectively; Tx and Ty represent the rigid body translations along the x-direction and y-direction, respectively; Ris rigid body rotation; r and are polar coordinate at a crack tip. The unknown coordinate of the crack tip is represented by the polar coordinates as follows: r Dq.x x0/ 2 C.y y0/ 2; Dtan 1 y y0 x x0 ; (9.3) where x0 and y0 represent the position of a crack tip with respect to the arbitrary Cartesian coordinate system in the image. is substituted with (3 )/(1C ) for plane stress and 3 for plane strain where is Poison’s ratio. The coefficients of the first terms, AI1 andAII1 relate to stress intensity factors, KI andKII, through the relations of AI1 DKI= p2 ; AII1 D KII= p2 : (9.4) AIn, AIIn, Tx, Ty, R, x0 andy0 are unknown parameters and are determined by an iterative procedure based on the NewtonRaphson method [14, 15]. Therefore, the stress intensity factors can be evaluated if the measured crack tip displacement field can be approximated by Eqs. 9.1 and 9.2. Furthermore, the crack tip position (x0, y0) is also determined exactly. Equation 9.2 is the theoretical solution for a semi-infinite crack in elastic material. The boundary effects of finite-size specimen and the interaction between cracks appear in higher-order terms [14, 15]. 9.4 Experimental Result A whole image example of a completely desiccated paste is shown in Fig. 9.2a. It is found that a crack nucleates from the stress concentration point and propagates straightly between the glass plates because of restricting the shrinkage direction of the paste with the glass plates. This fact means that the present method can control the crack nucleation position. Fig. 9.2 An example of the desiccation crack that is controlled the nucleation position between the glass plates. (a) Whole image taken after desiccation and (b) magnified image during crack propagation
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