“Computational Civil Engineering - "Intersections" International Journal

“Computational Civil Engineering 2005”, International Symposium 123[ f ( σ )] dt = G[ f ( )]d = d& dt = G&∫ ∫ 0 σ 0(A.14)ttSubstituting Equation (A.14) in (A.5), the following expression for the rate of themechanical dissipation at each damaged point is established[ f ( σ 0)]f ( σ )( σ )& ∂G∂f0Ξ = Ψ G [ f ( σ )]: : ε&0 = Ψ00(A.15)∂ ∂σ0 C00The current value for the damage threshold c can be written, at timemax{ c ,max{ f ( )} ∀ ≤ s ≤ ts = t , asc = max σ00(A.16)Particular expression used for the damage threshold criterionThere are several ways to define the damage threshold criterion. In this work, theexponential of reference [19] for concrete structures is used. The scalar functionG χ (Equation A.11) is here defined in function of the unit normalized dissipation[ ]variable κ as [15]⎡r(σ 0 ) 1−r(σ 0 ) ⎤⎡ ⎤κ&= K ( σ 0 ) ⋅ Ξ = ⎢ + ⎥ ⋅ Ξ ⇒ 0 ≤ ⎢κ = ∫ κ&mmdt⎥≤ 1 (A.17)⎢⎣gfgC⎥⎦⎢⎣t ⎥⎦where d&33Ξ m = Ψ 0 is the damage dissipation and r ( σ)= σ σI=1 I I 1 I ascalar function to define the sign of the stress state at each point and at each timex = 0 ,5 x + x the McAully function. Theinstant of the damage process, being [ ]s∑ ∑ =variables g and g are the maximum values for the tension-compressionfcdissipation at each point, respectively [15]. By this way, the damage dissipationwill be always normalized to the maximum consumed energy during themechanical process.Using κ as an auxiliary variable, it is now possible to evaluate the damage functionG χ in the following form [19]:[ ]( κ )c( )( 1−κmax )maxc Acmaxd = G[ c( κ )]= 1−e with 0 ≤ c ≤ c(d) (A.18)cbut, under the damage condition f ( σ0) ≡ c( κ )as. This equation can be also written