Issue 38

R. Fincato et alii, Frattura ed Integrità Strutturale, 38 (2016) 231-236; DOI: 10.3221/IGF-ESIS.38.31 232 that is also a function of the isotropic and kinematic hardening and the temperature. The damage affects the yield function and reduces the stiffness though the definition of the effective stress, first introduced by Kachanov [9]. The main difference between these two approaches is that the void growth model neglects the effect of damage on the elastic behaviour, limiting the softening behaviour during material loading. The present paper uses continuum damage mechanics to describe the damage as an internal variable and aims to couple the ductile damage constitutive equations with those of the subloading surface model, which is an unconventional [10] plasticity theory initially proposed by Hashiguchi [11, 12]. These models combine the advantages of the plasticity model, which describes the accumulation of irreversible contributions during a generic deformation process (i.e., monotonic, non-proportional, cyclic), with the degradation of the mechanical properties because of the large plastic strains. C ONSTITUTIVE EQUATIONS Subloading surface model he subloading surface model is regarded as an unconventional plastic model because inelastic contributions can be calculated for every change in the stress state during material loading. This is achieved by removing the separation of elastic and plastic domains, stating that the material always behaves non-linearly. A subloading surface is introduced by a similarity transformation from the conventional plastic potential (i.e., normal-yield surface, Fig. 1). The subloading surface functions as a loading surface always passing through the current stress state and expanding or contracting in the stress space, depending on the loading or unloading of the sample. The analytical expressions for these two surfaces are f F H f RF H ˆ ˆ ( ) ( ), normal-yield surface ( ) ( ) subloading surface     σ σ σ α σ (1) Here, σ is the Cauchy stress, α is the back-stress, F is the isotropic hardening function (defined later), H is the isotropic hardening variable, R is the similarity transformation ratio, and σ and α are the conjugate Cauchy stress and conjugate back-stress for the subloading surface, respectively, which are expressed as R ˆ ˆ , , ,         σ σ α α s s σ σ s s s α  (2) The model is extended by the introduction of a mobile similarity centre, s , which moves freely in the stress space following the development of plastic strain. However, some limits are necessary to avoid the similarity centre crossing the plastic potential. This would lead to both theoretical and numerical inconsistencies such as the subloading surface being impossible to define. Therefore, the similarity-centre surface, defined as the locus of points for the similarity centre and its expansion within the normal-yield surface, is limited by a user-defined parameter,  ( 0 1    ). Figure 1 : Sketch of the subloading and normal-yield surface. T