13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- The idea is to adapt this model in order to study the ballistic limit velocity and the residual velocity of the striker and with the hypothesis that the radius of the dimple is equal to the sheet radius. The elastic bending work is neglected. Thus, the total perforation work W is the sum of the bending plastic work Wb the stretching plastic work Ws, the petals’ bending work Wbf and the fracture work Wf (W = Wb + Ws + Wbf +Wf). The perforation work is the absorbed energy and so with the definition of the absorbed energy, the residual velocity Vr is given as a function of the initial velocity V0: 1 ( 0 − ) → √ 0 − √ 0 − ( + + + ) (2) To obtain the ballistic limit velocity, the initial velocity is taken equal to Vbl and the residual velocity Vr equal to 0 m/s. Thus, we obtain: √ √ ( + + + ) (3) where - the bending work Wb is given as a function of the yield stress , the initial thickness 0 the sheet’s radius 0 and the bending angle (calculated as a function 0 and the striker displacement z as ( ) 0 ). by: 0 0 (4) - the stretching work Ws of the sheet is given as a function of the yield stress , 0, 0, and the stretching strain by ( 0) 0 . The stretching strain can be defined thanks to the evolving surface area of the sheet and its original area ( (√ 0 − 0 )): ( 0) 0 (√ ) (5) - the bending work Wbf involved by each petals can be written as a function of the number of petals N, the striker radius r0, the thickness e, and γ the petals bending angle ( − − ) as (see Fig. 3): + 1+ (6) - the Fracture work Wf is due to crack propagation. It can be given as a function of the fracture toughness G, the thickness e and the length of fracture L by .The thickness e can be express with the stretching strain as 0 √ 0 . L is taken equal to the projection of the striker on the bending sheet ( √ 0 ). The parameter zb, is the ultimate bending striker displacement the parameter. Thus the fracture work is given by : 0 (√ )√ 0 (7) Here the angle of the conical striker is 60°, its mass is equal to 13 kg, and its diameter is 45 mm. The sheet thickness is 2 mm and its diameter is 148 mm. The static yield stress is 40 , the number of petals is taken equal to 4 and the ultimate bending striker displacement zb is 3 mm.
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