Issue 41

A. Mardaliazad et alii, Frattura ed Integrità Strutturale, 41 (2017) 504-523; DOI: 10.3221/IGF-ESIS.41.62 510 model within 1994 to 2000 [26-29], known as the “Karagozian and Case Concrete (KCC or K&C)” model, by applying several significant enhancements to the material model 16. The Karagozian and Case concrete model, implemented in LS-DYNA, consists of three-invariant failure surface formulation, i.e. 1 2 3 ( , , ) f f I J J  , where 1 I is the first principal invariant of the Cauchy stress, 2 J and 3 J are the second and third principal invariants of the deviatoric part of the Cauchy stress, respectively. It is more convenient to use a cylindrical set of coordinate system for the cohesive frictional material since the principal stresses can be immediately recast into this format as Eq. (4). The formulation of KCC model is explained by means of this cylindrical coordinate system as a clearer way of guidance for readers. 1 2 3 cos 1 2 cos( 2 3) 3 3 cos( 2 3)                                                (4) where, the  ,  and  are the Haigh-Westergaard coordinates defined as Eq. (5).   1 2 3 1 3 3 3 3 2 2 1 3 3 2 2 3 [ ] 3 3 1 cos 3 [ 3 ] 2 2 eq eq I p J q J r r J q J                                   (5) The failure function of the KCC material model can be derived by Eq. (6) in terms of the Haigh-Westergaard stress invariants. ˆ ( , , ) ( , ) [ ( ), ] c f f f r p           (6) where, ( , ) c f   corresponds with the failure surface in the triaxial compression state of stress, when the Lode angle θ is equal to 60  , and the non-dimensional function ˆ[ ( ), ] r p   is the ratio between the current radius of the failure surface and the compressive meridian. The Eq. (6) is discussed in detail below. The Karagozian and Case concrete model takes advantage of the three-fixed independent failure surfaces in the compressive meridian (    plane), which correspond to the yield, the ultimate and the residual strength of the material. Hence, the function ( , ) c f   is defined for each one of these failure surfaces separately by Eqs. (7), (8) and (9). 0 1 2 2 0.55 ( ) 2 3 [ ( ) ;( ) 0.45( ) ] 3 3 2 y m m y y y y a a a                  (7) 0 1 2 2 ( ) 2 3 3 3 m a a a       (8) 1 2 2 ( ) 3 r f f a a      (9) The stress invariants for an axisymmetric loading condition, when 1 a    is the axial stress and 2 3 l      are the lateral stresses, can be re-written as Eq. (10).