13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- kinetic energy, where ρ is the density of substance. Work of the external stresses βσ at the crack opening is equal to the work on the crack faces; its work on the bulk is zero because of symmetry. One can integrate the elementary work along faces and obtain the value fV βσ δ⋅ , where fVδ is an increment of the crack volume. Therefore, the potential energy of crack in the external stress field βσ can be written as 3 2 f 2 / VU V R G β β σ π σ =− =− ; negative sign here indicates that the micro-crack growth is energy-wise efficient in the tensile stress. Potential surface energy connected with two crack faces is equal to 2 2 SU Rπ γ = . Estimation of the dissipative function is more complex. The crack size changing is accompanied by generation of non-uniform fields of deformations and stresses in the crack vicinity, and the shear part of these stresses undergoes plastic relaxation. Due to the plastic strain the mechanical energy of the growing or collapsing crack is conversed in heat. On the base of the Orowan equation for dislocation plasticity, one can write the maximal plastic strain rate D t w b c ρ =& , where Dρ is the scalar density of dislocations in the substance, b is the modulus of the Burgers vector of dislocations, tc is the limiting speed of the dislocations movement. Using the ratio / / h R G βσ = as a characteristic value of deformation in the crack vicinity, we obtain a characteristic time of the plastic relaxation ( ) ( ) / / / D D t h R w Gb c β τ σ ρ = =& . This is a time interval sufficiently large for the plastic relaxation of excess shear stress in the crack vicinity if the form of crack is artificially frozen. Then the mechanical energy decrease rate can be estimated as / D K τ (external stresses and, respectively, VU are supposed to be fixed). Than the dissipative function is equal to ( ) ( ) ( )2 3 1/2 / D D t F K b c R R τ πρ ρ = = & . Substituting all obtained elements in the Lagrange equation, one can obtain the next equation for the crack radius: ( ) ( ) 2 2 2 3 4 ' 6 2 R R R R R G G β β σ σ ρ γ γ ⎛ ⎞ + =− + + ⎜ ⎟ ⎝ ⎠ & & , (1) where ( )( ) 2 ' /2 D t b c R R γ ρ ρ = & is irreversible energy (per unit area of crack faces) spending on plastic deformations; this energy dissipates both at the growth and at the collapse of the crack. It follows from the Eq. (1), that the crack grows when its radius exceeds a critical value ( ) ( ) 2 cr 2/3 R G βσ γ γ − ′ = + . Calculations show, that at high-rate deformation the value of γ′ is negligible with respect to γ for just formed cracks with critical radius. Therefore, initial crack growth occurs in a brittle mode, practically without energy dissipation on the plastic deformations. In particular, the next relation is valid with good accuracy ( ) 2 cr 2/3 R G β γ σ− = . But at the micro-crack growth, the irreversible energy γ′ increases and it can reach the value of the order of 2 : 1000 J /cm for large cracks, which exceeds the value of γ on the three orders of magnitude. 2.2. Nucleation of cracks. Weakened zones of the material Let us consider a micro-cracks ensemble. We denote the number of micro-cracks in unit volume of substance as n. To find the micro-cracks production rate we assume, that all they are generated due to thermal fluctuations. We assume that main contribution in / dn dt is provided by cracks with critical radius cr R , because cracks with cr R R< are being healed, and generation of cracks with
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