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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1. Introduction Plastic deformations of materials depend on the rate of loading. In case of quasi-static loading conditions, rate independent elastoplastic models can give an accurate description of the material behavior, however, whenever the rate of application of the load exceeds the quasi-static conditions a different description of the material rheology needs to be considered. Materials such as metals, plastic polymers, rubbers, geomaterials present a time-dependent mechanical behavior under practical loading conditions. The necessity to describe this phenomenon led to the development of rate-dependent constitutive models which can be grouped into two categories: the overstress model (Bingham, 2018; Perzyna, 2016; Prager, 1962) and the creep model (Norton, 1929; Odqvist, 1966). The present work adopts the overstress framework for the description of the viscoplastic deformations of metals since the creep model cannot be reduced to the elastoplastic constitutive model in a quasi-static deformation Hashiguchi (2017). In detail, the present work aims to upgrade the formulation of the overstress subloading model (i.e. OSS hereafter) introduced by Hashiguchi (2017). In Hashiguchi (2017) the OSS model is formulated without the similarity center internal variable and therefore is not suitable for investigating cyclic mobility problems. Moreover, here, an internal isotropic scalar damage variable is strongly coupled with the plastic deformation in the framework of the continuum damage mechanics (CDM). A previous work of the authors (Fincato and Tsutsumi, 2017c) presented a coupled elastoplastic and damage constitutive model (i.e. DSS hereafter) where the damage rate did not affect the elastic strain rate but merely the elastic state, this drawback is corrected in the present formulation of the algorithm for rate-dependent materials. The aim of this work is to show the main features of the damage overstress subloading surface model DOSS, pointing out its advantages compared with other conventional models based on the overstress concept. The paper is organized as follows. Section 2 presents an overview of the OSS model and its constitutive equations. Section 3 deals with the numerical implementation of the constitutive equation in a fully-implicit integration scheme (i.e. closest-point projection method). Section 4 introduces the numerical analyses, comparing the model response for two different cyclic loading rate conditions. Finally, section 5 presents the concluding remarks. Nomenclature D : strain rate tensor D e : elastic strain rate tensor D vp : viscoplastic strain rate tensor 0 E : fourth-order elasticity tensor of undamaged material   0

1 D   E E : fourth-order elasticity tensor of damaged material , σ σ  : Cauchy stress tensor, and Cauchy stress corotational rate , α α  : back stress tensor, back stress corotational rate , s s  : similarity centre tensor, similarity centre corotational rate D : scalar isotropic damage variable (0 ≤ D ≤ 1) F : isotropic hardening function F 0 : initial size of the normal-yield surface (i.e., yield stress) H: cumulative plastic strain variable R : subloading surface similarity ratio  1 R   R d : dynamic loading surface similarity ratio  0 R m : limit for the dynamic loading surface  m R  R e : constant defining the size of the elastic sub-domain vp  : viscoplastic multiplier  : parameter limiting movement of s c : constant influencing s translation speed K, h 1 , h 2 : material parameters for the isotropic hardening s 1 , s 2 : material parameters for the damage  0  d m R R    1