“Computational Civil Engineering - "Intersections" International Journal

122 S. Oller, Al.H. Barbat( σ 0q) = G[ f ( σ )] − G[ c( d )] ≤ 0, with q ≡ { d}F (A.9);0[ ]where G χderivative.is a monotonic scalar function, invertible and positive with positivesEvolution law for the internal damage variableThe evolution law for the internal damage variable can be written in the followinggeneral from:F( σ q)[ f ( σ )]0[ f (0)][ f ( )]0 ; Gd&∂∂= µ & ≡ µ &∂∂ σσ0(A.10)where µ is a non negative scalar value named damage consistency parameter,whose definition is close to the plastic consistency parameter λ . As in thePlasticity Theory, the evaluation of this parameter is made using the Ilyushin [18]consistency condition. From this condition, and from the properties of G[ χ], thefollowing function is obtained:∂G( ) [ ( )] [ ( )] [ f ( σ 0)]∂G ( ) ( )[ c( d )]F σ 0 ; q = 0 ⇒ G f σ 0 = G c d ⇒ f σ 0 = c d ⇒= (A.11)∂f( σ ) ∂c( d )and, from here, the permanency condition is deduced&∂G[ f ( σ 0)]G( )( )[ c( d )]f&∂F σc( d ) f&0; q = 0 ⇒σ 0 − & = 0 ⇒ ( σ 0) = c&( d ) (A.12)∂f( σ )∂c( d )0Observing the rate of the threshold damage function ∂G[ f ( σ )] ∂t= G&[ f ( )]0 / σ 0(Equation A.12) and comparing with the evolution law of the internal variable d &(Equation A.10), the following expression for the damage consistency parameter isobtained:d&∂G( )[ ( )][ f σ 0 ]f σ =∂f( σ 0)∂G[ f ( σ 0)]= µ &∂[ f ( σ )]G&0f&( σ )⎫⎪⎬⎪⎪⎭d&≡ G&[ f ( σ )] &0 ⇒ µ & ≡ f ( σ 0)∂f( σ ) ∂f( σ )∂σ00σ&0: 0 = : C 0∂σ00= c&( d0 0)⇒=: ε&=(A.13)Time integration over the rate of internal damage variable (Equation A.13) givesthe following explicit expression for the damage evaluation in each point of thesolid: