PSI - Issue 13

A. Vedernikova et al. / Procedia Structural Integrity 13 (2018) 1165–1170 Author name / Structural Integrity Procedia 00 (2018) 000–000

1168

4

  

   

p F

p σ Г     

σ

(10)

Г

,

p

pσ



where σ Г , pσ Г , p Г are kinetic coefficients. The processes of plastic deformation and defect accumulation are independent ( 0 pσ  Г ). Under quasi-brittle fracture conditions the plastic deformation has small effect on the deformation process ( 0 σ  Г ). The considered process can be described by the following constitutive equations:   e e σ tr ε E ε   2   , (11)

  

   

p F

p σ Г     

σ

(12)

Г

,

p

pσ



where  ,  - Lame constants, E - second-order identity tensor. In order to close system of equations it is necessary to involve approximation of function p σ    F  which determines equilibrium state of material with defects. Large stress gradients lead to the nonlocal effects in the defect ensembles, which can be described by the following relation (Lifshitz and Pitaevskii (1981)):     p p σ p p p σ             s k a q F Г    p , (13) where max  - maximum value of the stress tensor component near the concentrator,  , s - degree of polynomials, q , k , a - material parameters. Evolution equation for the dissipative defect structure describes its turning into infinity for a finite time on a characteristic length scale c L . It has been shown that condition 1   s  allows definition of characteristic length scale by the following expression (Samarskii et al. (1995)):      max Equations (11)-(14) will be used for the explanation of the fracture mechanisms near stress concentrators of the quasi-brittle materials under static loading. 4. Application of the proposed model to the static strength assessment To explain physical meaning of the obtained critical distance let us to consider quasi-static tension of the specimen with a U-shaped stress concentrator with a notch radius of 1mm using the proposed model. Finite-element mesh of the specimen i s presented in figure 2a. Figure 2b presents distribution of the stress component in the tensile direction. Figure 3c presents evolution of the 11 p spatial distribution over the cut plane along the loading direction. It can be seen that 11 p becomes more and more localized with the loading time. The more pronounced heterogeneity of the 11 p can be seen in the cross-sectional area of the notch perpendicular to the loading direction. Figure 3a shows simulation results of the defect density along the line characterizing distance to the notch in the plane with the maximum normal stress. It can be seen that there is no equilibrium defect concentration and the dissipative structure is localized on the spatial scale which is equal to the half of the critical distance obtained earlier. To get this scale equalled to the 0.085 mm, we have used the dimensionless material parameters included in equation (12) : 7 4.7 10   k , 50.9  q , 1  s , 2   , 10.1  a . Initial uniform distribution of 11 p is replaced by the heterogeneous 11 p with the localization zones where we can observe sharp increase in the defect density (figure 3b and 3c). Simulation results have shown that blow-up regime of the defect kinetics can be obtained when 11  is greater than critical value c  q k s s L c 1 2    . (14)