13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- 5. A circular plate of finite dimensions The above model can be extended to the case of a circular plate of finite dimensions. In this case the dimensionless function of the energy of crack w(n, l), similar to the function Eq. (4), takes the form: for a clamped plate ( ) ( ) ( ) ( ) m l l l l C l l n n n w n l nl B 2 2 3 2 2 2 3 10 3 3 2ln 6 ln 1 / sin / cos 1 ( , ) − − − + + − + π κ π = + for a freely supported plate ( ) ( ) ( ) ( ) ( )( ) ( ) [ ] − + ++ −μ− +μ+ +−μ + π κ π = + − 21 1 ln 1 6ln 4 2ln 4 31 / sin / cos 10 ( , ) 2 2 2 2 3 2 l l l C l l l l n n n w n l nl B m for a clamped annular plate ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] r l r l r l C r l r l r l l n n n w nl B i i i i i i m + + − − + + + + − + + + π κ π = + − 2ln 3 3 ln 6 1 2 / sin / cos 10 2 3 2 2 2 3 2 where this time γ = R h W w n l * ( , ) l = L/R* 2 2 *3 2 2 4 ) 3 (1 2 Ic R K E h B −μ π ≡ 13 1 4 2 ≥ −μ π κ= 2 2 3 (1 ) (1 ) 1 ( 1)ln ( , ) 2 l l l l C −μ + +μ − μ+ μ = ( ) ( ) ( ) ( )( ) 2 2 2 2 1 1 ln 21 1 3 l l l l C −μ − +μ + −μ − +μ = , R* is the radius of the annular plate, ri = Ri/L – dimensionless inner radius of the annular plate, m – the parameter that determines the magnitude of the critical plate deflection or elastic energy stored in the plate at the crash moment. 6. Discussion of the Obtained Results and Accepted Assumptions First, we note that at the first glance it seems rather strange that the obtained “optimal” values of the number of sectors (or cracks) are independent (except for the wedge opening angle Φ) of any geometrical and physical parameters of the model: the plate thickness, its rigidity, and fracture viscosity. To understand this fact, we recall that, for a given wedge (with angle Φ at the vertex), it is required to find a system of cracks of number and length such that the energy necessary to create such a system (this energy is the sum of the energies of formation of new surfaces and the energy of
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