PSI - Issue 8

Matteo Loffredo / Procedia Structural Integrity 8 (2018) 265–275 M. Lo ff redo / Structural Integrity Procedia 00 (2017) 000–000

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For a good prediction of the final RS state, it is fundamental to account properly for the unloading phase occurring downstream a tensile plastic strain. Indeed the unloading behavior can be significantly a ff ected by Bauschinger e ff ect (Parker et al. (2003b)), that is the material loss of yielding performance when load is reversed after having experi enced a plastic strain in tension. This reduction is function of the level of plastic strain reached in tension. Moreover a significant hardening modulus follows the reverse yielding, that is as well function of the tensile plastic strain. This globally leads, for the case of autofrettage of cylindrical pressure vessels, to a reverese yielding occurring at the bore after unloading, and then, to compressive RS lower than the one that would have been predicted using a model not accounting for the Bauschinger e ff ect (Chen (1985); Livieri and Lazzarin (2001); Parker et al. (2003a); Parker; Parker et al., 1998); Gibson et al. (2005, 2012); Troiano et al. (2006, 2012); ASME (1997)). Several models are available in literature for modelling the Bauschinger e ff ect. The simplest was an analytical model proposed by Chen (Chen (1985)) that assumes a bilinear unloading hardening law and Tresca’s yield criterion. Nu merical or Finite Element (FE) models have been also developed, limited to the case of an autofrettaged cylinder, with various degrees of complexity and accuracy (see de Swardt (2006)), either based on the update of material proper ties based on accumulated plastic strain (Troiano et al. (2006, 2012)) or on an update of elastic constants (Jahed and Dubey). The aim of this paper is building a constitutive model to accurately reproduce the uniaxial unloading behavior (therefore using non linear unloading hardening laws) based on the evolution of a Von-Mises yield locus. This result is achieved by mixing non-linear kinematic hardening and non-linear isotropic softening. Moreover a procedure for tuning it using experimental data coming from uniaxial tension-compression tests is proposed. The Bauschinger e ff ect is treated from an experimental and numerical standpoint for a material of common use, the AISI 4140. Firstly the set-up and the results of an experimental campaign of uniaxial tension-compression on AISI 4140 steel are presented and discussed. Secondly the constitutive model is formulated and guidelines for program ming it in a Finite Element (FE) commercial code are provided. Then a tuning procedure, using data gathered during experimental phase, is presented. In the end, the model has been programmed in ANSYS commercial software as an User Programmable Feature (UPF) and has been implemented into the FE model of an autofrettaged hollow cylinder. The qualitative trend of resulting RS profile is reported and discussed.

Nomenclature

a b L p

Inner radius of the hollow cylinder. Outer radius of the hollow cylinder. Axial length of the hollow cylinder.

Hollow cylinder autofrettage pressure. σ θθ Residual hoop stress in the hollow cylinder. E Young’s Modulus. ν Poisson’s ratio. σ (L) y Tensile yield point. σ (U)

y Compressive yield point. σ y Radius of the yield locus. C (L) 1 First typical constant of the loading phase kinematic hardening. C (L) 2 Second typical constant of the loading phase kinematic hardening. H (L) 1 First typical constant of the loading phase isotropic softening. H (L) 2 Second typical constant of the loading phase isotropic softening. Ω Counter parameter. C (U) 1 First typical constant of the unloading phase kinematic hardening. C (U) 2 Second typical constant of the unloading phase kinematic hardening. α Backstress vector. β Memory backstress vector. n Plastic flow vector. p Plastic strain vector.