Issue 47

R. Fincato et alii, Frattura ed Integrità Strutturale, 47 (2019) 231-246; DOI: 10.3221/IGF-ESIS.47.18 235 o p p,D : = ; ; 1 M     N σ σ D N N N σ (8) The term M p,D in Eqn. (8) 1 is the elastoplastic modulus coupled with the damage. Its definition is derived by means of some mathematical manipulations from the consistency condition of Eqn. (3) 2 , as shown in [11].       , 1 1 2 1 2 ˆ ˆ ˆ : 1 1 1 3 3 p D p D dF M D U R c D dH F R R                                  α σ σ N σ s σ D     (9) Finally, combining the expressions of Eqs. (2) and (8) 1 into Eqn. (1) it is possible to obtain the co-rotational stress rate as a function of the total strain rate as follows , : : p D M           EN EN σ E D N E N  (10) The ductile damage evolution law for proportional and non-proportional loading paths Following a well-consolidated approach adopted by the continuum damage mechanics an arbitrary stress state can be described by using two dimensionless parameters: the stress triaxiality ST and the Lode angle parameter LA, defined as in Eqn. (11):   1.0 6 m Mises          (11) In particular, the Lode angle parameter field of existence covers all the possible loading conditions, from the uniaxial extension to the uniaxial compression passing by the pure shear (or plane strain) condition where it assumes null value. Therefore, every pair of variables , representing a unique stress state, can be used in the formulation of fracture envelope in the MC criterion. This criterion has been widely and successfully used in the prediction of the failure behavior of granular materials such as rock, soil and concrete. Recent applications [9,12,29], adopted successfully the MC criterion in investigations on metals ductility. The expression of the MC failure envelope can be found in [12], where Bai and Wierzbicki presented an analytical formula function of the two variables, ST and LA, and of eight material parameters. However, in case the effects of the pressure and of the LA on plasticity could be considered as negligible, a von Mises potential can be assumed and then the expression can be simplified in the following: 1 2 1 1 2 1 1 cos sin 3 6 3 6 N f c A c c                                              (12) where A , c 1 , c 2 and N are four user defined material parameters. Based on the previous Eqn. (12) the ductile damage evolution law can be written as     1 2 3 H , p f D H d      D  (13) where the constant d 1 is introduced to allow the damage to evolve after a certain amount of plastic deformations. The previous Eqn. (13) offers a good description of the damage evolution during proportional loading [30], however, it cannot explain the damage acceleration during the non-proportionality of the load. As recalled at the beginning of the theoretical framework section, the non-proportionality triggers a non-negligible component of the stress rate along the tangent to the plastic potential. Therefore, the main idea is to add to the ductile damage evolution an additional inelastic contribution, herein called tangential plastic strain rate , generated by the stress rate component tangential to the Mises potential. The tangential inelastic strain rate can be evaluated adopting the same assumptions proposed by Rudnick and Rice [31] and Hashiguchi and Tsutsumi in [16]. In particular:   ,   t D