Issue 47

R. Fincato et alii, Frattura ed Integrità Strutturale, 47 (2019) 231-246; DOI: 10.3221/IGF-ESIS.47.18 232 [3]; Bao and Wierzbicki [4]; Brünig et al. [5]; Papasidero et al. [6]) pointed out that the stress triaxiality and the Lode angle are the main two factors that affect the ductility behavior in metals. In fact, monotonic loading tests, for different geometries and under different loading conditions (uniaxial extension, pure shear, plane strain, uniaxial compression), proved a different failure behavior for the same material. However, the literature still lacks a proper characterization of the damage evolution under non-proportional loading paths. In the recent years, Faleskog and Barsoum [7], Papasidero et al., Cortese et al. [8] and Algarni et al. [9] conducted a series of experiments aimed to identify the effect of the loading path on ductility. In detail, the work of Faleskog and Barsoum [7], Papasidero et al. [10] and Cortese et al. [8] consisted in the application of several non-proportional loading paths, as the result of the combination of tension-torsion or compression-torsion, on steel and aluminum tubular samples. On the other hand, Algarni et al. [9] tried to describe the crack formation on notched Iconel 718 bars in low cycle fatigue investigations. A common aspect that emerged from those previous works is that the deformation at fracture is higher when the load is proportional, suggesting that the damage accelerates whenever non- proportional loading conditions are triggered. The present paper aims to investigate the influence of the loading path on the ductile damage evolution. The constitutive model Damage Subloading Surface (DSS) [11] was modified to account for an additional inelastic contribution, named tangential plasticity, that arises during the non-proportionality of the load. In detail, the numerical analyses focus the attention on the structural response of a steel bridge column subjected to various loading conditions. A modified Mohr- Coulomb criterion [12] is adopted for the description of the damage behavior of SS400 steel pier, with a modification of the damage evolution law. The paper is organized as follows. A first section (i.e. theoretical framework ) introduces the definition of proportional and non- proportional loadings. Subsequently, the main features of the DSS model are briefly described. Finally, the ductile damage criterion and a novel ductile damage evolution law are presented. The second section, numerical analyses , is divided into two parts. A simple numerical example shows the different damage evolution under proportional and non-proportional loading paths, clarifying the role of the tangential plasticity presented in the first section. The second part of the numerical analyses deals with a steel column that undergoes a cyclic non-proportional loading. The results will show the necessity of considering the tangential plasticity for a more realistic description of the structural response. The last section reports the concluding remarks of the paper, summarizing the theoretical approach and the advantages of the model. T HEORETICAL FRAMEWORK he following section deals with the presentation of the main features of the elastoplastic and damage DSS model. A detailed description of this theory is available in a previous work of the authors [11], where the Lemaitre’s formulation [13] was adopted for the description of the ductile damage evolution. The present work considers a different approach, based on the modified Mohr-Coulomb criterion as presented by Bai and Wierzbicki [12]. Moreover, an additional modification of the ductile damage evolution law is here introduced to describe the difference in ductility under proportional and non-proportional loading paths. Therefore, the discussion is initially focused on the definition of proportional and non-proportional loads. According to the present literature [14,15], a load can be defined as proportional whenever the ratio among the principal stresses does not change during the loading. To this category belong, for example, uniaxial tension/compression, pure shear, or more complex conditions where the stress tensor components change ‘proportionally’. On the contrary, non- proportional loads are defined as all the types of loading that do not meet the previous condition. An example of proportional and non-proportional loading paths is schematically described in Fig. 1a. Hashiguchi and Tsutsumi [16] pointed out that in the traditional elastoplasticity, assuming an associated flow rule and a single smooth plastic potential, the generation of plastic deformation depends uniquely on the component of the stress rate along the normal to the plastic potential. This aspect becomes particularly relevant whenever the stress path deviates significantly from the proportional loading, as for example in instability phenomena with localization and/or bifurcation of the deformation. In this case, during the loading process, the stress rate deviates from the normal to the plastic potential due to a component of the stress rate directed along the tangent to the plastic potential, called tangential stress rate . Hashiguchi and Tsutsumi proposed a novel constitutive model to include the inelastic contribution (i.e. tangential plastic strain) generated by the tangential stress rate component to correct the overestimation of the material stiffness predicted by traditional models with a single smooth plastic potential. The fundamental concept of this paper is the inclusion of an additional inelastic stretching, the tangential plastic strain, to the ductile damage evolution law in order to explain the different failure mechanism under non-proportional loading paths. T