13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- The yield function takes the form of the Drucker and Prager yield criterion to account for different evolution of strength under tension and compression. The evolution of the yield surface is controlled by the two hardening variables !!" and !!". In terms of effective stresses, the yield function takes the form [22]: = 1 1− −3 + !!" !"# − !"# − ! !!" ≤0 (1) where, , and are dimensionless constants, !"# is the algebraic maximum principal stress, the hydrostatic pressure, which is a function of the first stress invariant ! (defined as = ! /3), and the Von Mises equivalent stress, described as = 2 ! (where ! is the second deviatoric stress invariant). , and are given as: = !! !! −1 2 !! !! −1 (2) = !! !! −1 −(1+ ) (3) =3(1− !) 2 ! −1 (4) where, !! and !! are the compressive and compressive biaxial elastic limit respectively, !! the tensile elastic limit, and ! is the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian at the elastic limit for any given value of the pressure invariant such that the maximum principal stress is negative (the default value is 2/3) [22]. To compute the non-elastic stress-strain behaviour, the plastic-damage model assumes non-associated potential flow as [22]: !" = ( ) (5) where the flow rule G is the Drucker-Prager hyperbolic function [22]: = !! !! ! + ! − (6) where ϵcc is the eccentricity, which defines the rate at which the plastic potential function approaches the asymptote. ψ is the dilation angle (measured in the - plane at a high confining pressure. The post-yield behaviour in tension is defined according to the energy criterion Gf. Quasi-brittle behaviour is characterised by a stress-displacement response and the fracture energy is invoked by specifying the post-yield stress as a function of cracking displacement. The tensile damage is converted to a cracking displacement value using a relationship where the specimen length l is assumed to be one unit length in FE implementation [22]. The post-yield behaviour in compression is defined as a displacement-hardening/softening behaviour. A cracking displacement !!" is defined as the total displacement minus the elastic displacement corresponding to the undamaged material. This requires the softening data in terms of the cracking displacement [22]: !!" = !!" − 1 1− ! ! ! (7)
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