Issue 47

R. Fincato et alii, Frattura ed Integrità Strutturale, 47 (2019) 231-246; DOI: 10.3221/IGF-ESIS.47.18 233 It is worth mentioning that the novel damage accumulation law can be applied to a general loading path, since the contribution of the inelastic strain depends on the orientation of the total stress rate. The Damage Subloading Surface model This subsection deals with the constitutive equations of the DSS. Starting from the definition of the additive decomposition of the total strain rate tensor, the expression of the co-rotational rate of the Cauchy stress is obtained. The theoretical framework and the main variables are here briefly introduced, for a complete discussion of the model please refer to [11]. The DSS model is a coupled elastoplastic and damage model that derives its formulation from a previous unconventional plasticity theory, named Extended subloading surface model (i.e. SS) [15]. The choice of an unconventional plasticity model [17] is mainly due to the ability of the SS theory to give a realistic description of the plastic strain accumulation and of the material ratcheting for cyclic mobility problems and for fatigue life investigations. In fact, this approach has been used successfully in different numerical analyses focused on predicting low- and high- cycle fatigue life [18,19], assessing welded structures [20, 21], and generally for analyzing the deformation behavior of metallic [22, 23] and granular [24–26] materials. a) b) Figure 1: a) Loading paths for proportional and non-proportional loading, b) sketch of the normal-yield and subloading surface. The main feature of the SS consists in the abolition of the neat distinction between the elastic and plastic domains, stating that irreversible deformations can be generated whenever the loading criterion is satisfied. In order to achieve this goal a second surface, named subloading surface, is generated by means of a similarity transformation from the conventional yield surface, here renamed normal yield-surface (see Fig. 1b). The subloading surface is defined to pass always through the current stress state, assuming the role of a loading surface that expands and contracts in the stress space depending on the loading or unloading of the material. The center of similarity s is not fixed in the stress space but it can move following the plastic strain rate (its speed is regulated by the material parameter c ). Some limitations were introduced by Hashiguchi [15] to prevent the protrusion of the similarity-center beyond the yield surface, which would lead to theoretical and numerical inconsistencies. In detail, a third surface, the similarity-center surface, is created as the locus of points for the similarity center and its size is regulated by a material parameter χ, see Fig. 1b. The present work assumes a finite elastoplasticity framework that adopts a hypoelastic base plasticity. In this context, the total strain rate tensor D is additively decomposed into an elastic part e D and a plastic part p D according to e p   D D D (1) The expression of the co-rotational rate of the Cauchy stress can be obtained from the previous Eqn. (1) through some mathematical manipulations. In the following lines, it will be shown how to re-write the elastic and plastic strain rates as functions of the stress rate. The elastic part of the stress rate is related to the co-rotational rate of the Cauchy stress by means of the fourth-order elasticity tensor pre-multiplied by the weakening function (1-D), following the approach proposed by Lemaitre [13].