13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- 0 2 0 Γ 1 1 ( ) e e M M M M M dz z z z z . (45) With straight line element applied, the nearly singular integrals can be integrated analytically, and the separated parts have simple expressions. 5. Numerical examples 5.1. A central crack in finite plate For the first example, a central crack of length 2a in a homogeneous and linear piezoelectric plate is considered to verify the correctness of the two methods. As shown in Fig. 1, the plate with width 2L height 2H is under a pure uniform mechanical tensile loading of 0 1Mpa . Figure 1. A central crack in a piezoelectric plate Normalized stress intensity factor 0 / ( ) IK a is presented versus L/a of the finite plate, the results have been plotted and compared with the corresponding FEM results presented by Cao and Kuang [5] to test the accuracy of the present integration methods. Different heights are considered with H/a=0.368, 0.568, 0.968 and 4.618. The two methods coincide well with each other, and they also match well with the FEM results. It should be pointed out that the FEM results are obtained for the condition that the thickness b of the plate is 0.01a. It can be found from Fig.2 that the values of SIF are approaching to the stable values when the ratio of L/a is larger than 2.0, which means these values reach the case of a central crack in the corresponding strips with the height 2H. 1 2 3 4 5 0 2 4 6 8 10 Normalized K1 L/a 0.368(line) 0.568(line) 0.968(line) 4.618(line) 0.368(quadratic) 0.568(quadratic) 0.968(quadratic) 4.618(quadratic) 0.368(FEM) 0.568(FEM) 0.968(FEM) 4.168(FEM) Figure 2. Normalized stress intensity factors for different dimensions
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