PSI - Issue 40

V. Kibitkin et al. / Procedia Structural Integrity 40 (2022) 223–230 V. Kibitkin et al/ Structural Integrity Procedia 00 (2022) 000 – 000

227

5

4. Average deformation and its development Another approach is based on the fact that each vector field can be quantitatively associated with one or two characteristics for better understanding and more accurate separation of the stages of destruction. Under conditions of active compression of the material at a constant velocity, the displacement field is generally homogeneous, if a foreign object – a crack or a local vortex – does not interfere with this typical scenario. If the deformation process proceeds uniformly, then the deformation at each point of the sample should be approximately the same. This requirement is satisfied by the conditions that the dependences of the longitudinal and transverse displacements are linear functions of coordinates.

x x u (x, y) a x b y c      , x x

y y u (x, y) a x b y c      , y y

(1)

4

xx yy   





2

2

2 ) (     ( x y a b b a x y

2

(2)

(

)

)





xy

2 1 / ( ) n t t     .

(3)

Here , , , , , x x x y y y a b c a b c – constants determined from experimental data, n  – average specific deformation. The approximation was performed by the least square’s method. To take into account the degree of deviation from the corresponding plane, the standard deviation of the displacements  u was also calculated:

/ x y D D M N  ,

/ D t    ,

  

u

n

1 1 j  M N i  

1 1 j  M N i  

, i j , x i j (( ) ( ) ) u u  a x

a

, y i j (( ) ( ) ) u u  , y i j

x D

D

,

.

(4)





2

2

y

M(N) – the number of rows (columns) of x u (x, y) , y u (x, y) arrays, presented in discrete form. Here the variance of the displacements  n is normalized to the number of points in the field. It is an important characteristic that also allows you to identify the main stages of destruction. Based on Fig. 5 and taking into account the above data, it is possible to identify roughly the boundaries of the fracture stages. At the first stage ( 42 100 s t   ), the “ flow ” significantly differs from the average one, since here the decisive role is played by a crack, the edges of which are opened according to the normal fracture mode (Fig. 3, b). The second stage ( 100 180 s t   ) is associated with the formation of numerous deformation bands. The third stage ( 180 s t  ) is associated with the gradual homogenization of the inelastic “ flow ” and the uniform distribution of deformation over the entire volume of the sample. These data are approximate, since it is quite difficult to distinguish clearly the stage boundaries, especially taking into account the statistical scatter of the physical parameters of ceramics during manufacturing.