Issue 49

Yu. Bayandin et alii, Frattura ed Integrità Strutturale, 49 (2019) 243-256; DOI: 10.3221/IGF-ESIS.49.24

(microcracks and microshears) are introduced like dislocation density tensors and are described by symmetric strain tensors in the case of microcracks ik i k s sv v  and microshears   1 2 ik i k i k s s v l l v   . Here v is the unit vector to the base plane of the microcrack or slip plane of the microscopic shear, l is the unit vector in the direction of the shift, s is the volume of the microcrack or the microshear intensity. The averaging of the microscopic tensor ik s gives a macroscopic tensor of microcracks (microshears) density (defect density tensor)

s n 

p

(4)

ik

ik

which coincides with macroscopic strain induced by defects, where n is the concentration of defects. Statistical theory for mesoscopic defects has established various qualitative reactions of the material represented in the nonlinear form of out-of-equilibrium free energy of solid with defects F depending on defect density tensor ik p and the structural-scaling parameter c  characterizing the current susceptibility of the material to the nucleation and growth of defects. This parameter represents the ratio of the characteristic structural scales   3 0 ~ R r  , where R is the distance between the defects, 0 r is the average size of the defect nuclei. It was shown that in the intervals * 1, 1.3 c c           the responses of the material are characteristic for quasi-brittle, ductile and fine grain state, respectively [5, 30].

Figure 1: Characteristic material reactions on the growth of defects: (a) - for microshears, (b) - for microcracks

0 F p    for simple shear (  d

  , 

d p p ) and uni-axial for microcracks) is a

The curves in Fig. 1 demonstrate the solutions of the equation

strain ( 

s   , 

s p p )states. Metastability for stresses

(

for microshears,

ci   

cd   

cs   

consequence of the effect of the ordering (orientation transition) in the ensemble of defects. The value of stress  cs   for  c 

 defines the dynamic elastic limit (HEL) for quasi-brittle materials. The stress in the

   corresponds to the range of the elastic limit for materials with a plastic response (Fig. 1) [5].

metastable area

c 

* 

Figure 2: Self-similar solutions of the kinetic equation for strain induced by defects: periodic spatial structures ( *  

 ), autosoliton

) and localized dissipative structures (  c   )

c 

  

* 

waves (

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