Issue 28

M.Malnati, Frattura ed Integrità Strutturale, 28 (2014) 12-18; DOI: 10.3221/IGF-ESIS.28.02 13 Similarly, themethodology proposed in the present paper has its foundation on a plasticity approach, since usage is made of “yield surfaces” created in the stress space during the advancement of the stress history. But on the other side, the description remains here merely in the stress space: no explicit use of the mesoscopic plastic strain is made, but the equivalent fatigue stress is calculated directly on each created stress cycle. The yield surfaces used in the present approach just represent a way to describe at the macroscopic level the underlying phenomenon related to the plasticity arising at a mesoscopic level. From amore general point of view, the present work follows the same scope of the initial work of Jabbado [3] to extend the endurance criterion of Dang Van [9] to a finite life assessment using a continuous damage evaluation. Nevertheless, differently from [3] a notion of stress cycle is used here, even though the cycles are not counted and extracted from the global sequence - as it is done in a rainflowmethod (see e.g. [1]) - but they are simply created and updated in a continuous way while the stress history advances. This continuous damage approach has the same viewpoint of the cited work of Jabbado [3] or for example of the methodology proposed by Stefanov [10]. On the other hand, the principle to adapt a multi-surface concept to fatigue assessment by creating the stress cycles during the stress history advancement has some similaritieswith thework ofHerbland [11]. C ONSTRUCTIONOFTHESTRESSCYCLES he basic ingredient of the present approach is the geometrical creation of closed surfaces in the stress space while the stress point is moving on its path, in a way identical to the classical plasticity theory (see e.g. [12, 13]). Each created yield surface is afterward associated to an amount of fatigue damage, evaluated using the basic fatigue material properties. This usage of the yield surfaces is analogous towhat is done for the stress cycles after their extraction by a classical cycle-counting, like the rainflowmethod. For this reason, the terms “yield surface” and “stress cycle”will be used hereafter as synonyms. The rules describedhere below resume the stress cycles creation.  The equation defining a yield surface in the stress space has the form f P (  – X C ) =  y (1) where f P is a scalar function that in accordance with the hypothesis used by Jabbado [3] and by many classical multiaxial criteria (see [1, 4]) takes theVonMises equivalent stress: f P (  ) =       2 2 2 1 2 [ ] I II II III III I            (2) where  I ,  II ,  III are the principal stresses of  . Let us note explicitly that f P is function of the only deviatoric part s of the stress tensor  : f P (  ) = f P ( s ) (3) s = dev  =  - p H I where p H = tr(  ) / 3 (4) Hence each yield surface is a VonMises hypersphere fully described by its centre X C (back-stress tensor) and its size  y .  In the same way of a classical plasticity criterion, when the current stress state moves along the stress history a yield surface is hardened if the stress lies on the surface itself and is moving outwards. These conditions will be integrated in the Eqs. (5.a), (5.b) and (12.a), (12.b) written below. As a consequence the stress point will be always inside - or exactly on – a yield surface, but never outside.  A multi-surface model is used, inspired to the one proposed by Mroz and described e.g. in [12]. In such a model, more than one yield surface can exist at the same time. However a significant dissimilaritywith the concept ofMroz is that intersections between surfaces are allowed in the present approach. As a consequence the existing surfaces are not necessarily all nested inside each other: this issue in the frame of multi-surface plasticity models is discussed for instance in ref. [14].  The following hardening rules are used. When the deviatoric part s of the stress tensor  has an increment d s , among the surfaces that are hardened at a given instant the one havingmaximum  y is hardenedby the following superposed isotropic / kinematic hardening law: T