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

304

3

where S - volume of the defect, n - unit normal to the crack plane. In case of the microshears tensor s is defined as:

1 2





s

 nl ln S ,

(2)



where S - intensity of the shear, n - unit normal to the shear plane, l - unit vector along the shear plane. Macroscopic tensor p characterizing volume concentration and orientation of defects (structural-sensitive parameter) can be determined by the averaging of the s over the elementary volume:

 p s n ,

(3)

where n - is concentration of defects. It is assumed that considered process of static deformation obeys the first and second thermodynamics laws. In case of small deformations these laws involve the following thermodynamic quantities: density ρ , strain and stress tensors σ and ε , absolute temperature T , heat supply r , heat flux vector q , specific Helmholtz free energy F , specific entropy ψ . The second law of thermodynamics has the form (Murakami (2012)):

         q r T T

,

(4)

ψ

0

where  - Nabla-Operator, the upper dot stands for the material time derivative. The media under consideration obeys the following kinematic relation:

   e p ε ε ε p ,

(5)

where e ε - elastic strain tensor, p ε - plastic strain tensor, p - structural strain tensor, upper dot denotes time derivative. It is assumed that under isothermal conditions the free energy is a function of elastic strain, structural strain   ,  e ε p F F and dissipation inequality (4) takes the form:

F

p F

1 ρ









: e ε

: p σ ε ε p  e

e

   p

:

0





.

(6)

ε





Let us assume that (6) holds for every thermodynamic process, then the following relations hold:

 ε F ρ ,  e

σ



(7)

p F

1 :







p σ ε p

p

:

0





 

.

(8)

ρ



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: