PSI - Issue 37

Yulia Pirogova et al. / Procedia Structural Integrity 37 (2022) 1049–1056 Yulia Pirogova / Structural Integrity Procedia 00 (2021) 000 – 000

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3

3). In that case, the value of the volume fraction was p =0.5, the minimum radius of the spheres

, the

0.25

min r =

maximum radius m ах r = 2.

Fig. 2 Representative volume with the shape of inclusions sphere а =10, min r 0.25 = , m ах r 0.75, 1, 1.5 = .

Fig. 3 Representative volume with the shape of inclusions sphere

0.25 min r = ,

а =5, 10, 15, 20, 25 и 30.

2;

m ах r =

3. Morphological characterization Multipoint correlation functions are used to evaluate the influence of the spatial position of inclusions. The probabilistic meaning of the correlation function is that the closer the modulus of this function is to 1, the stronger the linear relationship between the points, and conversely, the closer the modulus of this function is to 0, the weaker the linear relationship between the sections. A random indicator function can be used to formalize information about the geometry of heterogeneous two-phase structures ( ) r  : ( ) 1, 0, I M r V r r V   =   ò ò , (1) where I V is the volume occupied by the inclusion, M V is the volume occupied by the matrix, r is radius-vector with components 1 2 3 ( , , ) x x x , ( ) ' r  is fluctuation of a random indicator function, which has the form ( ) ( ) ( ) ( ) ' r r r r p     = − = − , where p is volume fraction of inclusions. The n -th order correlation function for heterogeneous structures can be represented in the following form: ( ) ( ) ( ) ( ) 1 1 2 1 2 ( , ,..., ) ( ) ( )... ( ) ( ) ( ) ... ( ) C n n n n K r r r r r r r p r p r p           = = − − − (2) To find the values of the correlation function for each step, the product of indicator function fluctuation at the point and at the point taken at the distance of a given step should be calculated. This work employs the approach