13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- Parameter ( ) a t defines the radius of the contact area and its value depends on the contact time. Values ( )t rσ and ( )t θσ correspond to maximum values of main stresses in cylindrical coordinates. Therefore, the radial component of stresses ( )t rσ is considered, since usually tensile stresses lead to fracture. Below ( )t rσ is denoted by ( )tσ . The expressions (3)-(5) can be used to determine the expression for the threshold energy of the spherical particle. By threshold energy we mean the minimum quantity of energy, which is necessary to spend on initiation of the threshold fracture pulse during impact intersection. In this work the incubation time criterion is applied for fracture prediction [4]: ( ) ∫ − = t t c t RV s ds τ σ σ τ , , max 1 . (6) The static strength of the material cσ can be experimentally measured. Parameter τ corresponds to incubation time of the fracture. It characterizes the time period for preparing the media to fracture or phase transformation. The incubation time is a material strength constant and its value can be measured experimentally or derived by computational and experimental methods. In the work [5] different interpretations of the incubation time are shown for various problems. The application of the incubation time criterion caused by this criterion takes into account the process dynamics. The dynamic strength properties of materials appear when the loading duration has a similar value as the incubation time [6,7]. When the time of loading significantly exceeds the incubation time, the media resistance is specified by static strength properties. After the substitution of the expressions (1), (3) and (5) into (4), the fracture criterion becomes: ∫ − ⎟ = ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − t t c t ds t s h R k τ τσ π π ν 0 0 sin max 2 1 2 . (7) The integral in the expressions (7) takes the maximum value at the time ( ) 2 0 τ = + t t . The impact duration, radius and velocity of the particle should be introduced in dimensionless form: τ τ λ 0 = , τp d c R R = , p d c V V = . From now on ( ) ( )( )ν ν ρ ν 1 1 2 1 + − = − m pc E is the propagation velocity of the dilatational wave in the elastic media with density mρ . Then, the accepted criterion (7) gives the following expression for calculation of the threshold velocity in dimensionless form: ( ) ( ) [ ] ∫ + − ⎟ − − = ⎠ ⎞ ⎜ ⎝ ⎛ 2 1 2 1 2 5 1 sin λ λ λ λ π α ds H s H s s Vd d , (8) where ( ) ( ) ( ) ( ) ( ) ( )1 5 2 4 5 1 2 5 4 1 5 5 3 1 2 4 31 2 π ν ν πσ ρ α − − = − cr d pc E is dimensionless parameter, ρ is material density of the particle, ( ) H s is the Heaviside function. The value of the threshold radius can be calculated from following expression for the impact duration: ( ) 1 5 2 2 5 0 1 1 3.2 V R E t ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ≈ ν πρ τ τ λ .
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