PSI - Issue 5

Anastasiia Kostina et al. / Procedia Structural Integrity 5 (2017) 302–309 Anastasiia Kostina et al. / Structural Integrity Procedia 00 (2017) 000 – 000

305

4

  

     p F

p ε

σ + - ρ   σ



,

(9)

σ

pσ

 

     p - F ρ +

σ

p σ  , σ

p

  

(10)

p

where

σ  ,

p σ  , p  are kinetic coefficients. We suppose that processes of plastic deformation and defect 0   ). In case of a quasi-brittle fracture plastic deformation has small effect on 0   ) and considered process is described by the following constitutive equations:

accumulation are independent ( p σ

the deformation process ( σ

tr( ) 2  e e σ ε E ε λ + μ ,

(11)

 

     p - F ρ +

σ

p σ  , σ

p

  

(12)

p

where λ , μ - Lame constants, E - second-order identity tensor. In order to close system of equations it is necessary to involve approximation of function   σ p - F ρ 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 max ( )                 σ σ p p p p p β s - F ρ a q k σ  , (13) where max σ - maximum value of the stress tensor component near the concentrator, β , 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   β s allows definition of characteristic length scale by the following expression (Samarskii et al. (1995)): Equations (11)-(14) will be used further for the explanation of the fracture mechanisms near stress concentrators of the quasi-brittle materials under static loading. 3. Application of the linear elastic critical distance theory to the static strength assessment of the quasi-brittle material According to the critical distance theory, static failure can be predicted with the use of the linear elastic solution for the stress field near the stress concentrator with an error not exceeding 20% (Susmel and Taylor (2008)). In this work the point method of the critical distance theory was applied to the static strength assessment of the notched specimen under tensile loading:   11 c 0 / 2, 0    σ r L θ σ , (15) c π 2 1   k q L s s . (14)