13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- cr R R> is the low-probability process. Formation work for critical crack is ( ) ( ) cr cr cr V S A U R U R = + (it is reversible process without plastic dissipation), and we obtain ( ) 2 cr cr 2/3 A R π γ = ⋅ ⋅ . The probability of such fluctuations is ( ) ( ) cr exp / B P A k T = − [16], where T is the substance temperature, Bk is the Boltzmann constant. Concentration of the fluctuating centers can be estimated as ( ) 3 g cr 1/ 8 n R = . Fluctuation frequency f can be estimated as the frequency of transverse phonon with wavelength equal to the critical crack diameter ( ) cr / 2 t f c R = . As a result, the cracks nucleation rate is 2 cr g 4 cr 2 exp 16 3 t B c R dn P n f dt R k T π γ ⎛ ⎞ ⋅ ⋅ = ⋅ ⋅ = ⎜ − ⎟ ⎝ ⎠ . (2) The Eq. (2) determines the nucleation rate in the pure, defect-free material. In the real solids there are structural defects, such as dislocations, grain boundaries in polycrystals, inclusions in alloys etc. Micro-crack formation in the defective (weakened) regions requires smaller work in comparison with the undefective one, since these regions already possess raised energy relative to the defect-free crystal material (atoms in it are weaker bounded with each other). Primary origination of voids near the defects is observed in MD simulations [7,10]. We have to consider an influence of defects on the micro-cracks formation. Faultiness of current region of the material can be characterized by a parameter *γ in such a way, that difference ( )* γ γ − defines the crack surface formation energy per unit area in this defective region, *γ γ < . Then the formation work of the critical crack is equal to ( ) ( ) ( ) * * 2 cr cr 2/3 A R γ π γ γ = − , where cr R is determined by γ because the crack must grow further in the defect-free regions of crystal, hence, it should be stable at this value of surface tension. We suppose that the weakened zones are exponentially distributed on *γ : ( ) ( ) * * 0 exp / n n γ γ γ = ⋅ − Δ where γΔ is a distribution parameter. The product ( ) * * n d γ γ is the number of micro-cracks nucleation centers in unit volume of substance with the faultiness parameter belonging to * d γ interval near the *γ . The constant 0n is determined by the normalization condition: ( ) ( ) ( ) * * 0 0 1 exp / gn n d n γ γ γ γ γ γ = = ⋅Δ ⋅ − − Δ ∫ . Than the crack generation rate is defined by the following expression: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 cr 4 2 cr cr exp 2 / 3 exp / 16 1 2 / 3 1 exp / B t B R k T c dn dt R R k T πγ γ γ π γ γ γ ⎡ ⎤ − − − Δ ⎣ ⎦ = ⎡ ⎤ − Δ ⋅⎡⎣ − − Δ ⎤⎦ ⎣ ⎦ . (3) At 0 γΔ = (homogeneous material) this equation turns back to the Eq. (2). Thus, the proposed model of fracture contains two empirical parameters: γ and γΔ . The first of them, γ, is of the order of surface tension, and the second one, γΔ , is defined by the degree of material faultiness. 2.3. Continuous formulation Here and further we operate with macroscopic length scale, which is much larger than the micro-crack size and the distance between the micro-cracks. It means that each physically small volume of substance contains a set of micro-cracks. Therefore, we use macroscopic fields of substance density, velocity, stresses, deformations, et alia, which are averaged over such physically small volumes. Let us assume that in considered substance element all micro-cracks have identical
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