PSI - Issue 42

4

Shihao Bian/ Structural Integrity Procedia 00 (2019) 000 – 000

4

Shihao Bian/ Structural Integrity Procedia 00 (2019) 000 – 000

Shihao Bian et al. / Procedia Structural Integrity 42 (2022) 172–179

175

ℎ = ( − ) ̵ Ͳ̵ ̈́ ̈́Ψ

(5) (5)

where is the initial temperature, and is the identity tensor. 2.3. Mechanical behavior Isotropic thermo-elasticity is considered, based on Young modulus and Poisson ratio . Isotropic hardening is assumed, based on a Johnson-Cook law (G. R. Johnson and Cook, 1983) = ( + ) (1 − ( − − ) ) (6) where represents the yield stress and the equivalent plastic strain. and represent respectively the room and melting temperatures (assumed to be equal to 300 and 1356 K respectively). Last, , , and are material parameters. 2.4. Material parameters All parameters are extracted from literature. and can be found in (Penzhorn et al., 2012). One trap has been considered, with a trapping energy equal to 0.7 eV (Guillermain, 2016), and a density such that log = 25.26 − 2.33 −5.5 site/m 3 (adapted from (Kumnick and H. H. Johnson, 1980) to account for an initial dislocation density equal to 10 12 -10 14 m/m 3 ). Heat transfer parameters have been taken from (Umbrello et al., 2007), with an temperature independent expansion coefficient (Sayman et al., 2009). Last, thermomechanical parameters have been extracted from (Chandrasekaran et al., 2005; Umbrello et al., 2007). where ̵ Ͳ̵ ̈́ is the initial temperature, and is the identity tensor. 2.3. Mechanical behavior Isotropic thermo-elasticity is considered, based on Young modulus and Poisson ratio . Isotropic hardening is assumed, based on a Johnson-Cook law {Johnson:1983tr} ͳ ͓Ͳ ʹ͵͵Ͷ ͷ͸͹̵ ͵Ͳ ʹ͵͵Ͷ Ͷ (6) where ͳ represents the yield stress and ͓ the equivalent plastic strain. ʹ͵͵Ͷ and ͷ͸͹̵ ͵Ͳ represents respectively the room and melting temperatures (assumed to be equal to 300 and 1356 K respectively). Last, , , and are material parameters. 2.4. Material parameters All parameters are extracted from literature. Ǩ a Ǩ ca be found in {Penzhorn:2012ds}. Two traps have been consider d, with the same trapping energy (0.7 eV {Guillermain:yvOpv0eM}), but different densities: ̶ ǡͺ ͺͻ site/m {Penzhorn:2012ds} and ̶ ǡȀ ǣ Ǣ ǤǢ ൌ Ǩ site/m 3 {Kumnick:1980dc}. Heat transfer parameters have been take from {U brello:2007hv}, with an temperature-independant exp nsion coefficient {Sayman:2009cf}. Last, thermomechanical parameters have been extracted from {Umbrello:2007hv, Chandrasek ran:2005vk}. Shihao Bian/ Structural Integrity Procedia 00 (2019) 000 – 000 • a UEXPAN subroutine, which can de ermine the th rma expansio , furthermor , th use of this subrout ne mak s it possible not to wo ry about the th rm l contribution t the d formations transmitted by Abaqus to the UMAT subroutine. 3: a User ELement (UEL) subrout ine: to add a degree of freedom at each nodes of the Finit e Element model, and t o link this degree of fre d m t o a t ransient heat t ransfer problem. 4: a User EXPANsion (UEXPAN) subrout ine: t o comput e t he thermal expansion at each point . The deformat ion considered in t he UMAT is t he purely mechanical one ( ✏ − ✏ t h ). The fl owchart of this implement at ion is presented on fi gure 1. 5

Heat Transfert

Thermo-Mechanically assisted hydrogen flux (diffusion / trapping)

Thermo-Mechanical behavior

Fig. 2 Implementation scheme of a strongly coupled thermo-chemo-mechanical problem in Abaqus software. 3. ’Ž‡‡–ƒ–‹‘ The chemo-thermo-mechanical system described in the three paragraphs is resolved in a fully coupled way (strong coupling) using the Abaqus Finite Element software (Simulia, 2011), based on the 'coupled temp-displacement' procedure and user subroutines (see (Charles et al., 2017; Vasikaran et al., 2020) for details): Fig. 2. Implementation scheme of a strongly c l t ermo-chemo-mechanical problem in Abaqus software. 3. The chemo-thermo-mechanical system described in the three paragraphs is resolved in a fully coupled way (strong coupling) using the Abaqus Finite Element software {Simulia:2011ux}, based on the 'coupled temp-displacement' procedure and user subroutines (see {Charles:2017ci, Vasikaran:2020bi} for details): 3.1. Material parameters The parameters used are globally the same as those used in (Benannoune et al., 2020), the thermomechanical parameters are taken from (Umbrello et al., 2007), with an expansion coefficient considered to be independent of temperature(Sayman et al., 2009). Finally, the data for hydrogen diffusion are taken from(Penzhorn et al., 2012; 2010). Two types of traps are con idere here, with the sam nergy (equal to 0.7 eV (Pe zhorn et al., 2012)0) but different densiti s: Ƭ ̶ ǡ͸ ൌ ͺǤʹ × ͳͲ ͸͹ site/mm 3 {Penzhorn:2012ds} and • – Ƭ ̶ ǡǦ ൌ ʹ͵Ǥʹ͸ − ʹǤ͵͵’ ͺ ͻǤͻǢ Ǩ where ȗ ͓ is the eq ivalent plas c strain(McNabb and Foster, 1963). The trapping kinetics parameters are the same as in (Penzhorn et al., 2012)0. 3.2. Configuration of problem The geometry of DFW (in the form of a CAD file provided by the ITER Organization) is presented in Figure 2(a). For the sake of simplicity, only a part of this geometry will be modelled (indicated by a red rectangle). Each cylindrical Figure 1: User Subrout ines developed to add chemo-thermo-mechanical features to Abaqus software 2 DFW Reference case In t he following, the r ference con fi gurat ion is present ed 2.1 Geomet r y & M esh The global geomet ry of the DFW have been provided by ITER (in the form of a CAD fi le), and is presented on Figure 2a. For the sake of simplicity, only a small sect ion of this geomet ry is considered (indicat ed by a red rect angle). This sect ion (Figure 2b) is modeled in Abaqus i 2D, and is meshed wit h around 4000 fully integr ted linear ele nt s. Themesh has been opt imized based on the result s