PSI - Issue 5

O. Plekhov et al. / Procedia Structural Integrity 5 (2017) 438–445 A. Vshivkov et al. / Structural Integrity Procedia 00 (2017) 000 – 000

444

7

K

, I II

, I II yy

f

,





( , )2 I II

r 

2

K

, I II

, I II xy

f

,

  we can write the for multiaxial loading as follow:    2 2 1 1 II I II I e bf af bf af f    

( , )3 I II

r 

2

  2



  3    af bf

 . 2

3 (7) Substitution of equation (7) into equation (6) gives us an estimation of the evolution of plastic zone caused by application of shear stress. The structure of plastic zone at crack tip under monotonic loading is presented in figure 7b. As a result, we can analyze the plastic deformation at crack tip under multiaxial loading based on equation (5). The biaxial coefficient can be taken into account using equation (7). For cyclic loading we have to consider energy dissipation in cyclic plastic zone at crack tip: . mon p cyc p tot p U U U   To explain the experimental fact reported in previous paragraph we have to calculate the total energy increment - 3 1 1 1 I 1 2 II I II II I af   bf af bf

dl dU cyc p

tot p U . For loading condition considered in our experiment it can be shown that

0 

and plastic energy at

crack tip   tot

p U can be written as follow:

  2

2 dN U W A W dl tot p    1

(8)

,

where  A – stress amplitude. Based on the equation (8) we can conclude that for small crack rate the plastic work and, as a consequence, energy dissipation at crack tip is proportional to the applied stress amplitude and can decrease during experiments with constant stress intensity factor. The calculation of analytical relations for W 1 , W 2 can be carried out similarly to the monotonic loading. The biaxial coefficient changes the function f e but keeps the constant the structure of the equation (8). It allows us to predict the existence of peculiarities of energy dissipation at crack tip reported in Iziumova (2016) for multiaxial loading. In this work the experimental and theoretical study of dissipated energy were carried out during fatigue crack propagation. Based on a contact heat flux sensor, an experimental technique for the application of the method of infrared thermography for measuring the energy dissipation during fatigue test has been developed. The device allows us to measure heat dissipation under uniaxial and multiaxial loadings. Decrease in the power of the dissipated energy was shown during the fatigue crack propagation with a constant stress intensity factor. In order to explain this effect, the theoretical analysis of the energy dissipation zone formation at the crack tip was carried out. Using the coupling between Young’s modulus and the secant plasticity modulus, the analysis of the plastic zone shape and dissipated energy value at the crack tip under uniaxial and multiaxial loadings has been made. A good agreement between theoretical and experimental results illustrates the possibility of the application of the method to description of the energy dissipation under multiaxial loading. 5. Conclusion

Acknowledgements

This work was supported by the grant of the President of Russian Federation for support of young Russian scientists and leading scientific schools [MK-1236.2017.1] and the Russian Foundation for Basic research [grant number 16-31-00130]. The authors would like to thank Prof. Jürgen Bär for experimental support of the work.