Issue 47

R. Fincato et alii, Frattura ed Integrità Strutturale, 47 (2019) 231-246; DOI: 10.3221/IGF-ESIS.47.18 236 • Only the deviatoric part of the stress is responsible for the inelastic stretch, which allows us to conclude that the tangential inelastic strain is triggered by the deviatoric tangential component of the stress rate. • The tangential stretching , as well as the plastic stretching , smoothly develop with the similarity ratio R (i.e. is a function of the similarity ratio). • The tangential stretching does not contribute to the material hardening [15]. • the tangential stretching is linearly related to the co-rotational stress rate. In detail, the last hypothesis was introduced to obtain a simple constitutive model that could be used in general loading paths. In fact, more complex corner theories limit the application to simple loading cases. The tangential plastic strain rate can be written as 2 2 1 1 1 1 ˆ ; ; 2 3 2 1 T t t t T T R A A G G T R                         D σ σ I σ I I I N N σ     (14) where A is a scalar variable, function of the shear modulus G pre-multiplied by the (1-D) factor (i.e.   1 G G D   ), the similarity ratio R and two material parameters   1 1 0 1 T T   and   2 2 0 T T  . In particular, if T 1 is equal to zero no tangential plastic strain rate is generated, on the other hand, if T 1 tends to unity and assuming a fully plasticized state (i.e. R = 1) the tangential plastic strain rate is maximized. The term in Eqn. (14) 2 is the fourth-order deviatoric tangential (or projection) tensor [15] and it is function of the fourth-order identity tensor and the second-order identity tensor I . In conclusion, the ductile damage evolution law of Eqn. (13), can be rewritten to consider the additional inelastic contribution of Eqn. (14) 1 as       3 1 3 2 3 H ; 2 3 , p t i p p D T f T D H d H T H H             D D D D  (15) The constant   3 3 0 1 T T   has been introduced to regulate the amount of the contribution of the tangential stretching to the damage. The second of Eqn. (15) reports the cumulative inelastic strain variable H i , as the sum of the cumulative plastic strain H D and the cumulative tangential plastic strain H T . The advantage of the law in Eqn. (15) 1 is that the damage is subjected to an additional contribution uniquely when the stress rate deviates from the normal to the plastic potential. Therefore, it is suitable to describe the failure behaviour under both proportional and non-proportional loading paths. N UMERICAL ANALYSES he numerical analyses in this paper were carried out by implementing the previous set of constitutive equations via user subroutine for the commercial code Abaqus (ver. 6.14-5). The section is subdivided in two sub-sections. The first example deals with a two steps loading of a plate element with the scope of showing the features of the ductile damage law in Eqn. (15) 1 . The second numerical example shows an application of the DSS model to describe the structural response of a thin wall bridge pier under cyclic non-proportional loading, reproducing the experimental results carried out by Nishikawa et al . [32]. Two step loading example This numerical example consists in subjecting a plate element, under the assumption of plane strain, to a sequence of two different loading conditions. During the first step (i.e. step 1) the plate is pulled, under uniaxial boundary conditions, up to 10% of the nominal axial strain. Subsequently, a simple shear condition is applied to generate a 20% angular distortion (i.e. step 2). Please refer to the sketch of Fig. 2a. The numerical test was designed this way to maximize the effect of the tangential plastic strain. In fact, after the uniaxial extension, the stress state belongs to a fully plasticized state, where the similarity ratio R equal to unity gives the maximum contribution according to Eqn. (14) 3 . Moreover, the sudden change of direction between the loading step 1 and the loading step 2 triggers a non-negligible component of the stress rate along the tangent to the plastic potential. T t D p D t D ˆ  I I