PSI - Issue 13

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

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3

was found (Figure 1b) (Terekhina, 2017). Results of the numerical simulation have shown that for the considered material under quasi-static tensile the half of the critical distance is equal to 0.85 mm. In further we will consider a specimen with a U-shaped stress concentrator with a notch radius of 1mm (Figure 1a). 3. Statistical model of defects evolution near stress concentrators Under deformation the structural evolution observed at all scale levels and leads to irreversible changes and fracture of materials. One of the possible descriptions of defect kinetics is the statistical model of defect ensemble. To describe the behavior of the volumes defects (microcrack) ensemble, it is necessary to determine parameters characterizing microcracks and having a meaning of the independent state variables. Such a variable can be introduced in terms of the dislocation theory (Naimark O.B. (2003)) Defect density tensor (structural-sensitive parameter) is defined by the averaging over the elementary volume of the symmetrical tensor s characterizing unit defect of shear-type: s n  p , (3) where n is volume concentration of defects. A thermomechanical process obeys the momentum balance equation and the first and second laws of thermodynamics. In the case of small deformation, these equations involve the following thermodynamic quantities: volumetric mass  , strain and stress tensors σ and ε , heat supply r , heat flux vector q , specific Helmholtz free energy F , and specific entropy  . The entropy production can be written as (Murakami (2012)): where  is a Nabla-Operator, the superposed dot stands for the material time derivative. We assume the following kinematical relationship for the material under study: ε ε ε p e p        , (5) where e ε and p ε are the elastic and plastic strain tensor, p - structural strain tensor, upper dot denotes time derivative. The free energy under isothermal condition is a function of two independent variables   ε p p , F F  , and thus equation (4) takes the form:   0 : 1 : :           p σ ε ε p p ε ε e p e e       F F , (6) By requiring that equation (6) should hold for any thermodynamic process, we have 0      T r       T q   , (4)

 F  ,

(7)

σ



e ε





 0

1 :  p σ ε p p   

p F



:

(8)

 

,





In (8) p  , p ε  are thermodynamic forces and σ , p   / F are thermodynamic fluxes. Under an assumption of local equilibrium thermodynamic forces and fluxes are linearly related (Glansdorff and Prigogine (1971)). Thus, we can obtain the following constitutive equations:

  

   ,

p F



ε p 

  σ

σ

(9)

Г Г





σ

pσ

