“Computational Civil Engineering - "Intersections" International Journal

124 S. Oller, Al.H. BarbatwhereG[ f ( σ )]f ( 0)0A 10f ⎜⎛σ− ⎟⎞f ( 0)01 e ⎝σ= −⎠ withf0( f ( ))A = ⎡gf / σ⎢⎣02( σ0)( σ )0− 0.5⎤⎥⎦−1f0max( σ ) = cis a parameter depending on the fractureg ( )maxenergy dissipationf[19]. The value f 0 σ 0 = c is obtained from theagreement with the first damage threshold, when the condition0max0G [ f ( σ )] − G[ c ] 0 is reached and f ( )max[ ] = G[ c ] ≡ 00 =G σ 0shows thedamage integration algorithm for each single point of the structure.0Box A1. Integration of the continuum damage equation at each structural point withexponential softening1. Compute the elastic prediction stress and the internal variable at current time“ t + ∆t ”, and equilibrium iteration “ i ”, ,2. Damage threshold checking:a. If: τ −τ max ≤ 0t+∆tt+∆t[ σ0] = C0:[ ε]i t+∆tit[ d] ; τ = [ G[ f ( σ0)]0 i 0 tτ = G[ f ( σ )]+∆t+∆t[ ]i t+∆tt+∆tThen ⎪⎧ [ σ] = [ σ ] ⎪⎫0⎨ i t+∆tmax ⎬[ d] ; τ = τ⎪ ⎭0and go to the EXIT⎪⎩b. If: τ − τ max > 0Then start with the damage constitutive integration3. Integration of the damage constitutive equation,maxτ = τiτ( 1−0 )⎡ 0i t+∆tτ A ⎤[ d ] = 1−τ⎢ e ⎥⎢⎣τ ⎥⎦4. Stress and tangent constitutive tensor actualization.i t+∆ti t+∆tt+∆tσ = (1−d ) σ5. EXITi[ ] [ ] [ ]iT t+∆t⎡ ∂Gf ( σ 0)[ C ] = (1 − d)C -⎢⎣00∂t+∆t[ ][ ( )] [ ] ⎡∂f( C0: ε)C0: ε ⊗f σ⎢∂ε0⎣⎤⎤⎥⎥⎦⎦t+∆t