13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- { } b b b Δ = Δ u L q , on b ∂Ω ; { } c c c Δ = Δ u L q , on c ∂Ω ; { } 1 1 1 m m m Δ = Δ u L q , on 1m ∂Ω ; (6) { } 2 2 2 m m m Δ = Δ u L q , on 2m ∂Ω ; { } 3 3 3 m m m Δ = Δ u L q , on 3m ∂Ω where e Δq , b Δq , 1m Δq , 2m Δq , 3m Δq and c Δq are generalized displacement increment vectors, and L, bL , 1mL , 2mL , 3mL and cL are interpolation matrices. Substituting Eq. (4), (5) and (6) into the energy function (1), and setting the first variations of mc element Π with respect to the stress coefficients m Δβ and c Δβ , respectively to zero, yields following two weak forms of the kinematic relations. Let { }i d β corresponds to the correction to βΔ ’s in the ith iteration, the weak form of the kinematic relations may be expressed as: 1 1 2 3 2 3 e i b i m i m m i e mb mm mm mm c i c i cb cc m i m i d d d d c d d d d ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ⎤ − ⎧ ⎫ ⎡ ⎤⎪ ⎪ = ⎢ ⎥ ⎨ ⎬ ⎨ ⎬ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎩ ⎭ ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ q q q H 0 G G G 0 G G β q 0 G 0 G 0 0 β 0 H q q (7) where T m m m m m d Ω = Ω ∫ H P S P (8) T c c c c c d Ω = Ω ∫ H P S P (9) m T e T e e d ∂Ω = ∂Ω ∫ G P n L (10) 1 1 1 m m T m T m mm d ∂Ω = ∂Ω ∫ G P n L (11) 2 2 2 m T m T m mm m d ∂Ω = ∂Ω ∫ G P n L (12) 3 3 3 m T m T m mm m d ∂Ω = ∂Ω ∫ G P n L (13) c T c T c cc c d ∂Ω = ∂Ω ∫ G P n L (14) m T b T b mb b d ∂Ω = ∂Ω ∫ G P n L (15) c T b T b cb b d ∂Ω = ∂Ω ∫ G P n L (16) or in a condensed form 1 i i d d − =H G q β (17) Setting the first variation of the total energy functional (1) with respect to e Δq , b Δq , m Δq and c Δq to zero, results in the weak form of the traction reciprocity conditions as
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