13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- ∆ =σ ∆ = ∆ µ ∆ λ ϑϑ λ u I I 2 Ic 2 I 2 11 I I ( ) * * f K K K (10) The system (10) is readily solved, yielding the crack advancement ∆c and the critical value KIc * of the mode I GSIF KI * under pure mode I loading, i.e. the generalized fracture toughness [5]: I I I 1 u ch 2 1 u ) 2(1 Ic Ic ( ) ( ) −λ −λ λ − σ =ξ ω σ =ξ ω l K K * , with I I I 1 2 11 2 1 I ( 0, ) 2(2 ) ( ) −λ λ − λ µ ϑ= ω π ξ ω =λ (11) where lch = (KIc / σu) 2 is the Irwin length. Note that, since ξ is equal to unity for ω equal to 0 or π, the generalized fracture toughness equals the fracture toughness for a cracked geometry and the tensile strength for a flat edge. In the case of mixed mode loading, the critical values of the GSIFs can be obtained by substituting eqns (6) and (8) into the system (5): ξ δ + δ ψ δ = ψ+µ δ ψ ξ µ δ +µ δ δ = λ ϑϑ λ ϑϑ λ λ +λ λ ) ( ) ( tan tan tan II I II II I I II I Ic I 2 2 22 12 2 11 2 2 Ic I f f K K K K * * * * (12) where the mode I GSIF has been normalized with respect to the generalized fracture toughness KIc * , the finite crack advancement with respect to lch (δ = ∆ / lch) and ψ is the mode mixity, defined as: * I ch * II I II arctan K K l λ −λ ψ= (13) The system (12) can be recast as: = ψ+µ δ ψ µ δ +µ δ δ + δ ψ δ− ξ δ + δ ψ δ = λ λ +λ λ λ ϑϑ λ ϑϑ λ ϑϑ λ ϑϑ 0 tan tan tan tan 2 2 22 12 2 11 2 II I II I Ic I II II I I II I II I ) ( ) ( f f f f K K * * (14) The technique of Lagrange multipliers can now be exploited to solve eqn (14). In fact, eqn (14) can be interpreted as a constrained minimization problem, since, once the geometry, material and loading are fixed (i.e. ω and ψ are given), the actual crack advancement ∆c and crack orientation ϑc are the ones that minimize the first equation, i.e. the dimensionless failure load KIf * / KIc *, under the constraint represented by the second equation. Once the critical value KIf * of the mode I GSIF is determined, the corresponding critical value KIIf * of the mode II GSIF is provided by eqn (13).
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