PSI - Issue 33

Toru Yagi et al. / Procedia Structural Integrity 33 (2021) 1225–1234 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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3. Analysis 3.1. FEM analysis

To find out where the stress wave reflects and see the difference of the loading mode of bending and tensile test, the finite element method, FEM, analysis has been performed. The condition of the FEM analysis is shown in Fig. 11. This is a three-dimensional, elastic, dynamic-implicit analysis with a nodal release method. The minimum element size is 0.2 mm. The shape of the crack front is straight two-dimensional shape. The dynamic stress intensity factor d is used for the evaluation of this analysis. d value shows the driving force of the propagating crack. d is normalized by velocity dependent function k ( v ) which is approximate expression for exact solution by Rose (1976), shown in equation (1)-(3). Since d value is dependent on the crack velocity, it is normalized to compare the gradient of d value as the crack propagates.

Fig. 11 Mesh division of FEM model d ( ) = ( ) d (0) ( ) ≈ (1 − ) √1 − ℎ ℎ ≈ 2 1 ( 2 ) 2 [1 − ( 2 1 )] 2

(1) (2)

(3) The results of this FEM analysis are shown in Fig. 12. (a) is the result of the bending model. The trend of d is decreasing as the crack propagates. When the crack velocity 400 m/s or more, d decreases more rapidly than the static solution. (b) is that of the tensile model. The trend of d is unchanged or slightly increasing on the tensile mode. This is against the experimental results showing the facts of arrest even in tensile mode specimens. Based on the previous study done by Willoughby, d of low crack velocity should decrease more rapidly than that of high crack velocity, but the results of the analysis are completely opposite. The traditional theory of reflected stress waves cannot describe the pop-in phenomenon.