13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- equation for its radius. Macro-level is described by continuum mechanics equations with additional terms, which takes the micro-cracks ensemble into account. 2.1. Voids growth equation Let β r to be a unit vector directed along the maximal tensile stress. Tensile stress ik i k βσ σ β β = can initiate formation and growth of the opening mode cracks perpendicular to β r (see Fig. 1). Let us consider such a micro-crack, which is supposed to be isolated and axially-symmetrical. Radius of crack is R and its half-thickness is h; its volume fV can be estimated as a volume of cylinder 2 f 2 V R h π = ⋅ . Opening of crack creates non-uniform field of displacements in its vicinity. Created by a localized disturbance, the displacements have to decay with distance with on the scale R. Maximal displacements in β r direction corresponds to the crack faces, and it is h+ and h− for opposite faces. Therefore, the ratio / h R can be used as an estimation of the strain value in the crack vicinity. Corresponding stress can be estimated as ( ) / G h R . In the loaded medium this internal stress is superposed on the external macroscopic tensile stress βσ . And the total stress has zero normal components on the crack faces to maintain the constant form at the fixed radius R. Hence, we obtain an estimation / h R G βσ = for the crack half-thickness; than the crack volume is 3 f 2 / V R G β π σ = . Usage of the last two formulas assumes that the crack thickness instantly adapts to variations of R or βσ (a quasi-static approximation), it is valid when the typical time of variations τ ( / R R τ≈ & or 1/ τ ε ≈ &, where ε& is the strain rate) is much longer than the transient period / t R c . Therefore, conditions for the quasi-static approximation are the next: t R c << & and / tc R ε<< & ; these inequalities usually take place. Figure 1. A separate micro-crack; the real crack shape is approximated by cylinder. The crack grows due to separation of atoms along the plane of crack. The growth rate is restricted by inertia of the surrounding material. It is possible to formulate a Lagrange equation [14] for the micro-crack growth, using the radius R as a generalized coordinate. In this approach system is characterized by the Lagrange function S V L K U U = − − , where K is the kinetic energy of the substance movement due to the growth of micro-crack; SU is the surface energy of the crack faces; VU is the potential energy of crack in the field of external stresses βσ , and by the dissipative function F, which is equal to the one half of the mechanical energy decrease rate due to plastic deformation in the micro-crack vicinity. Expression 2 f K V R ρ = & is used as an estimation of the
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