PSI - Issue 39

D. Fedotova et al. / Procedia Structural Integrity 39 (2022) 419–431 Author name / Structural Integrity Procedia 00 (2019) 000–000

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9

5. Elastic and plastic stress intensity factors An important and one of the main elements of the structure of stress, strain, and displacement fields at the crack tip are scale factors in the form of amplitude coefficients or stress intensity factors (SIFs). These amplitude coefficients have the physical meaning of fracture resistance parameters of the material and are generally dependent on the applied load, the cracked body configuration, and the elastic-plastic material properties. The elastic SIFs for the employed CTS specimen for mode I ( K I ) and mode II ( K II ) loading conditions have the form:

2     , r

2     r

K

K

(8)

I

yy

II

xy

To describe the mixed-mode crack growth along the curvilinear crack path for the elastic plane problem, it is necessary to express the equivalent SIF, which is a function of mode I K I and mode II K II SIFs. Chang et al. (2006) proposed an equation for the elastic equivalent SIF as follows           * * 2 2 * * 2 * 2 1 1 cos 1 cos 4 sin 5 3cos 2 2 eqv I I II II III K K K K K K                                    (9)

where the branching fracture angle is determined from the equality as





    

    

   

   

1

*

*

*





  

  

  

  

3 2

3 2

3 2

2 

2 

2 

2

*

*

2 II

*

2

(10)

sin sin 

4

cos

3sin

5sin

sin

0

K

K K

K

K







 











I

I II

III

2

There are limited studies in the literature on fracture resistance parameters for strain gradient plasticity, and these studies are limited to a few particular cases and do not have sufficient generalization. Recently, Shlyannikov et al. (2021) presented numerical and analytical formulations for the plastic SIFs, which are fracture resistance parameters and are applicable in the domain of validity of CMSG plasticity. From Shlyannikov et al. (2021), we adopt the following numerical formulation for the amplitude   , FEM P A r 

FEM P K factors for CMSG plasticity, which are given by

and plastic stress intensity

  ,

  , r r         , ˆ FEM FEM

FEM A r

(11)

P

ij

ij

FEM FEM P P K A r  

(12)

where r r l  is the nondimensional distance to the crack tip and  is the power of the stress singularity. In Eq. (14), the angular distributions of the stress component   ˆ , FEM ij r   are normalised, such that FEM FEM ij ij Y     . Fig. 5 shows the numerical distributions of the equivalent elastic (Fig. 5a and 5b)), plastic SIFs according to the classical HRR solutions (Fig. 5c and 5d)) and new plastic SIFs for SGP (Fig. 5e and 5f) for each of the experimental crack paths in the CTS specimens for all tested materials. The left row in Fig. 5 shows the SIF distributions as a function of the normalized crack length Σ a/w for pure mode I, while the right row corresponds to the initial pure mode II with subsequent cyclic mixed-mode fracture. In the distributions of the SIFs shown in Fig. 5, the following features can be mentioned. For steel 34X and Ti-6Al-4V and aluminum alloys, the values of the plastic classic HRR SIFs are inhomogeneous along the curvilinear crack paths, Fig. 5d. In the ductile steel P2M, the distributions of the plastic classic HRR SIF monotonically increase along the crack path. Fig. 5a and 5с clearly illustrate the influence of the plastic properties of materials on the dependence of the elastic and plastic SIFs on the crack length using the example of mode I. Thus, for