13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Then, the threshold radius is determined in dimensionless form by: 1 5 d d dR V β λ = , (9) where ( ) ( ) 2 2 2 5 3.2 1 c E p d ρ ν π β − = is dimensionless parameter. If the threshold velocity (8) and radius (9) are known, then the threshold energy of the particle can be calculated as: 3 2 3 2 W R V πρ = . (10) In dimensionless form, (10) is given by the formula: 3 2 d d d R V W W = = ω , (11) where the parameter 3 2 3 5 pc πρτ ω= has the dimension of energy and is determined by material constants. Thus, the quantity dW determines the minimum dimensionless value of particle energy required for half-space fracture. Figure 1 presents the graphs of dependence of the energy (11) on the impact duration (Fig. 1a) and radius (Fig. 1b), where the half-space material is zinc and parameters the value 3 / 3200kg m =ρ . Figure 1. Dependence of the threshold energy of the spherical particle on: a – the impact duration, b – particle radius These graphs show that the value of the threshold energy has the marked minimum distinct from equal zero. Hence, it is possible to decrease the energy costs for fracture by controlling such parameters as radius and velocity of particle. Threshold Energy of a Cylindrical Particle The similar analysis can be provided for cylindrical particles with a constant circle section of radius R. The cylinder height is equal to 4 3 H R = , that follows from the assumption that the cylinder mass is equal to the ball mass. In the cylindrical case the contact force ( ) P t and approach ( ) h t are related as follows: ( ) ( ) P t kh t = , (12) where ( ) 2 2 1 ν− = k RE . а 0 0.6 1.2 1.8 2.4 3 2 10 11 − × 4 10 11 − × 6 10 11 − × 8 10 11 − × 1 10 10 − × dW τ λ / 0t= 0 0.03 0.06 0.09 0.12 0.15 2 10 11 − × 4 10 11 − × 6 10 11 − × 8 10 11 − × 1 10 10 − × dW dR b
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