13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- micro-cracks nucleation and growth and by the equations for determination of the plastic distortion ikw through the dislocations movement (see, for example, [17]). For determination of the pressure and temperature, the wide-range equations of state [18,19] were used in the forms of dependences ( ) , P P Uρ = and ( ) , T T Uρ = . All equations have been integrated in time by the explicit Euler scheme with variable time step, which has been selected from the stability condition { } min 0.1 / , 0.001/ zz t R R ε Δ ≤ & & . Parameters γ and γ Δ had been chosen by fitting with the experimental data [1-4,20-26] and with the of molecular-dynamics (MD) simulation results [7,9,10] for the strain rate dependences of the material strength; the obtained values of these parameters are summarized in the Table 1. Table 1. Parameters of the fracture model. Abbreviations: m/c – monocrystalline, p/c – polycrystalline. Cu Al Ti Fe Zn Mo Substance m/c p/c m/c p/c p/c p/c m/c m/c p/c γ, J/m2 0.95 0.60 0.57 0.47 1 1 0.8 1.1 0.7 γ Δ , J/m2 0.029 0.029 0.016 0.016 0.027 0.036 0.029 0.034 0.034 3. Results and discussion Strain dependences of tensile stress have been obtained in calculations; the typical dependences are shown in Fig. 2(a). Stress grows up with strain at the initial stage. Then the tension reaches some critical level, an intensive nucleation and growth of the cracks begins. It leads to relaxation of tensile stresses; the increase of stresses is changed on the decrease. Non-zero level of tensile stresses is held till the complete destruction of substance. The moment of complete destruction corresponds to the slump of stresses down to zero. The maximal obtained value of tensile stresses has to be treated as the dynamic (spall) strength of the material spσ . Dynamic strength increases with the strain rate, because the growth rate of the total volume of cracks has to be proportional to the strain rate for effective relaxation of the tensile stresses. This total volume is determined by the number and size of cracks; therefore, nucleation and growth rates have to be increased for relaxation at the increased strain rate, these rates are obtained at higher level of the tensile stresses. Fig. 2(b) shows the strain dependences of the average radius R, the critical radius cr R and the concentration n of the micro-cracks. Nucleation of cracks starts at the earliest stages of deformation, but its concentration is insignificant initially. At the low acting stresses, voids are nucleated only in weakened zones with a very high value of faultiness *γ γ ≈ , concentration of which is negligibly small. The critical radius decreases with the increase of strains and of the tensile stresses; zones with lower faultiness begin to contribute in nucleation of cracks, and the concentration of cracks grows fast. When the concentration reaches some value, the relaxation on micro-cracks becomes a dominant process, and the tensile stresses begin to decrease (see Fig. 2 (a)). It initiates an increase of the critical radius and slump of the nucleation rate of new cracks, only the growth of the existing micro-cracks takes place. This growth continues up to the reaching of the complete destruction, after than, R means a typical size of fragments in the destructed material. It is 30 μm and 7 μm for the strain rates 106 s-1 and 107 s-1 correspondently. The calculated strain rate dependencies of the dynamic strength for monocrystalline and polycrystalline copper and aluminum are presented in Fig. 3 in comparison with the experimental data and MD simulation results. The model parameters γ and γΔ (Table 1) have been chosen for
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