material replacing only the stresses with the effective ones and assigning a state equation for D. Bonora [1] proposed the following expression for the damage dissipation potential, ( ) n n cr D p D D D S S Y f + - - × ú ú û ù ê ê ë é - ÷÷ × ø ö çç è æ = - 2 1 0 2 0 1 2 1 a a (3) where, Dcr is the critical value of the damage variable for which ductile failure occurs, S0 is a material constant and n is the material hardening exponent. a is the damage exponent that determines the shape of the damage evolution curve and is related to the nature of the bound between brittle inclusions and the ductile matrix. Thus, the constitutive equation set for isotropic hardening material is given by: strain decomposition p ij e ij T ij e e e & & & = + (4) elastic strain rate & & & e n s n s d ij e ij kk ij E D E D = + - - - 1 1 1 (5) plastic strain rate eq ij ij p p ij D s f s l ¶s ¶ e l 1 21 3 - = = & & & & (6) plastic multiplier (1 ) p D R f r p =- = = - & & & & l ¶ ¶ l (7) kinetic law of damage evolution Y f D D ¶ ¶ l& & =- = ( ) ( ) p p D D D D cr eq H f th cr f & × × - ÷ ÷ ø ö ç ç è æ × - × - a a a s s e e a 1 1 0 / ) ln( (8) Detailed description on the derivation of these Equations can be found elsewhere, [1]. In Equation (9) stress triaxiality effects are accounted by the function f(sH/seq) defined as, ( ) ( ) 2 3 1 2 1 3 2 ÷ ÷ ø ö ç ç è æ × = + + × - ÷ ÷ ø ö ç ç è æ eq H eq H f s s n n s s (9) that is derived assuming that ductile damage mechanism is governed by the total elastic strain energy, Lemaitre [5]. Here, sH = skk/3 is the hydrostatic part of the stress tensor and n is the Poisson’s ratio. The model requires five material parameters in order to be applied. The strain threshold (in uniaxial monotonic loading) eth, at which damage processes are activated; the theoretical failure strain ef, at which ductile failure under completely uniaxial state of stress conditions occurs; the initial amount of damage present in the material, D0; the critical damage, Dcr, at which failure occurs and the damage exponent, a, that control the shape of damage evolution with plastic strain. Experimental procedure for damage parameters identification can be found elsewhere, [6]. EXTENSION TO REVERSAL PLASTIC FLOW Ductile damage formulations available in the literature always address tensile loading configuration, since it is well known that positive stress triaxiality enlarges nucleating voids in the material microstructure. The possible effect on damage variable due to compressive loading is usually neglected in the theoretical formulations. The major consequence of this limitation is that the damage variable, D, has to be associated to the total effective accumulated plastic strain, usually indicated with p, that plays the role of the associated internal variable. In the literature, little attention is given to the effects on the
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