PSI - Issue 13

R.R. Yarullin et al. / Procedia Structural Integrity 13 (2018) 902–907 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

906

5

where  0 is angle determining position of initial point of the corner crack tip and  c is the angle corresponding to the last point of the crack tip. In the following representation of the numerical results, we will use the variable R in a range from 0 to 1. A value of R =0 indicates the exterior free surface of the compressor disk, while a value of R =1 indicates the slot surface (Fig. 3b). The distributions of the elastic and elastic-plastic constraint parameters along the crack tip in the compressor disk at elevated temperature are plotted in Fig. 5. It can be observed that all constraint parameters change in character along the crack tip from the free surface ( R =0) across the mid-plane (0

a) c) Fig. 5. Constraint parameter distributions along crack tip (1-initial, 2-middle, 3-final front) at elevated temperature. b)

As can be seen from Fig.5 all constraint parameters are very sensitive to crack tip shape. It should be noted that the aspect ratio for initial crack tip (indicated by the number one in Fig. 5) is equal to 1.0. In general, increasing the temperature from 23 °C to +300 °C resulted in increase of constraint parameters values at all levels of crack sizes, especially for crack front number one. 4.2. Elastic and plastic stress intensity factors Shlyannikov et al. (2016a) showed that stress-strain state in the rotating disk obtained from FE calculation was found to be biaxial, that is, the structure around the slot fillets of the key is subjected to both radial and tangential (or hoop) stresses. Stress biaxiality ratio     rr  is a function of the current value of the disk radius and varied from η= - 0.27 to η=+0.29. Moreover, looking at Fig. 1b and 3a, and taking the inclination crack angle into account, mixed mode crack behavior in compressor disk can be expected. To study the influence of the mixed mode loading conditions on material fracture resistance parameters, unlike pure mode I, it is necessary to calculate the two fracture parameters, namely, the mode I and II stress intensity factors K I and K 2 (SIF). Shlyannikov (2013) generalized the numerical method for calculation of the geometry dependent correction factors Y 1 and Y 2 for the SIF K I and K 2 under mixed mode fracture. The present study explores the direct use of FE solution results for calculating the SIF K I and K 2 ahead of the crack tip ( θ =0º): r K FEM    2 1  , r K FEM r    2 2  (3) where r and θ are polar coordinates centered at the crack tip, FEM i  are stresses obtained from the FE solution. For compressor disk, the plastic SIF K p in pure Mode I (or pure Mode II) can be expressed directly in terms of the corresponding elastic SIF K I and K 2 using Rice’s J -integral as was shown by Shlyannikov and Zakharov (2017):

1

 

  4 K K K K 2 2 2   2 1 1 2

   

   



n

1



,

3 4     ,  = 0.3

  4 1 1  

2

(4)

 



K



P

2

n I w

0 

where α and n are the hardening parameters, ν is Poisson’s ratio, w is characteristic size (in this case, the width of key), σ 0 is the yield stress and I n - the governing parameter of the elastic – plastic stress – strain fields in the form of I n -factor.