PSI - Issue 18

Riccardo Fincato et al. / Procedia Structural Integrity 18 (2019) 75–85 Author name / Structural Integrity Procedia 00 (2019) 000–000

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The advantages of the OSS are the ability to give a better description of the smooth stress-strain response for quasi static loading conditions and to model the viscoplastic response in case of an impact load. The smooth transition between purely elastic and viscoplastic deformations is obtained by abolishing the neat distinction between the elastic and the fully viscoplastic domain, assuming that the material always behaves in a viscoplastic manner for all the stress states that satisfy the loading criterion. In order to do that two surfaces are introduced in addition to the conventional plastic potential (here renamed normal-yield surface ): the subloading surface and the dynamic loading surface . The first has an identical definition as in the rate-independent model (Hashiguchi, 1989), and it always passes through the current elastoplastic stress state ss σ , expanding and contracting in the stress space depending on the loading or unloading of the material. The dynamic loading surface was introduced as a loading surface that always passes through the current Cauchy stress state σ , in order to model the overstress effect in case of dynamic loading conditions. Both the subloading and the dynamic loading surfaces are obtained by means of a similarity transformation from the normal yield surface. In detail, the size ratio between the subloading and the dynamic loading surfaces against the normal yield surface is defined by the variables R and R d , respectively. The subloading surface shrinks to a point for a null stress state ( R = 0) and it can expand until it coincides with the normal-yield surface (fully developed viscoplastic state, R = 1). The dynamic loading surface also shrinks to a point in a null stress state condition ( R d = 0), however, it can expand beyond the normal-yield surface to model the overstress effect ( R d > 1). The range of variation of R and R d , together with the nomenclature used to define the different stress states is reported in the following Eq. (1). Fig. 1a reports a sketch of the aforementioned surfaces. In Hashiguchi (2009) a limitation on the upper limit of the dynamic loading ratio is introduced to correct the drawback of the conventional overstress theory that predicts a purely elastic response in case of an impact load, see Fig. 1b. The size of the dynamic loading surface is fixed by a material parameter to be R m times the size of the normal-yield surface, allowing the development of inelastic strains even for high rate loading conditions Fig. 1c. 0 1 0 1 sub-yield stress state 1 0 1 mix stress state 1 1 fully viscoplastic stress state d d d R and R R and R R and R          (1). The following work adopts a finite elastoplasticity framework under the assumption of a hypoelastic base plasticity. The total strain rate D can be additionally decomposed into an elastic part D e and a viscoplastic part D vp (see Eq, (2) 1 ). The analytical expression of the dynamic loading surface is given as follows: Where f is a generic function of the stress expressing the plastic potential (herein, the von Mises criterion is assumed), D (0 ≤ D ≤ 1) is the isotropic scalar damage variable, F is the isotropic hardening function and H is the isotropic hardening variable (i.e. cumulative viscoplastic strain) and the conjugated Cauchy stress σ is defined as :   ˆ 1 , ˆ if 0 D R          σ σ α s s α α s s     (3). It has to be noted that the previous Eq. (2) 2 the dynamic loading surface has been written in the damaged configuration, following the approach proposed by Chaboche (1999) and followed by other authors (i.e. Grammenoudis et al. , 2009, Badreddine et al. , 2017; Badreddine and Saanouni, 2017). Consequently, the conjugated Cauchy stress has been also written in the damaged configuration. The expressions of the back stress α  and similarity center s  are instead left in the undamaged configuration, since this allows an easier implementation of the numerical scheme. The expression of the isotropic hardening function F of Eq. (2) and the evolution law of for the back stress are given as follows: 1 1 d d d R R      α α         σ  1   D D D e     vp d f D R F H   (2).