230 Computational Mechanics of Composite Materials probabilistic characteristics)of Ao.Let us assume that the crack in a weld is growing according to the Paris-Erdogan law,cf.(A5.26),described by the equation =cyo元"a是 (5.10) dN and that y≠Y(a).Then a-jclYso)"dN (5.11) Q which gives by integration that 1 (5.12) -罗+ a+1=Cyo元)N+D,Der Taking for N=0 the initial condition a=a,it is obtained that (5.13) a for K=學-1，B=CYo√元m (5.14) Therefore,the number of cycles to failure is given by (5.15) The following equation is used to determine the probabilistic moments of the number of cycles for a crack to grow from the initial length a;to its final length af AN=j- 1 -da 4C(△K (5.16) Substituting for AK one obtains △W=19 1da Corπ号y"a (5.17)230 Computational Mechanics of Composite Materials probabilistic characteristics) of ∆σ. Let us assume that the crack in a weld is growing according to the Paris-Erdogan law, cf. (A5.26), described by the equation ( ) 2 m C Y a dN da m = ∆σ π (5.10) and that Y≠Y(a). Then ( ) ∫ = ∫C Y∆ dN a da m m σ π 2 (5.11) which gives by integration that a C( ) Y N D m m m = ∆ + − + − + σ π 1 2 2 1 1 , D ∈ℜ (5.12) Taking for N=0 the initial condition a=ai, it is obtained that a N a k i − β = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ 1 1 (5.13) for 1 2 κ = − m , ( ) m β κai C Y σ π κ = ∆ (5.14) Therefore, the number of cycles to failure is given by β 1 N f = (5.15) The following equation is used to determine the probabilistic moments of the number of cycles for a crack to grow from the initial length ai to its final length af: ( ) ∫ ∆ ∆ = f i a a m da C K N 1 (5.16) Substituting for ∆K one obtains ( ) ∫ ∆ ∆ = f i m m a m a m da C Y a N 2 2 1 1 σ π (5.17)