6 Austenitic stainless steels are commonly used as structural materials in high- eld superconducting magnet systems 7 because they retain high strength, ductility, and toughness at very low temperatures, and they are paramagnetic or 8 antiferromagnetic under the N eel temperature in their fully austenitic state. However, they are susceptible to strain- 9 induced martensitic transformation, especially at cryogenic temperatures, which modi es the material properties, induces 10 volume changes and additional strain hardening, and leads to ferromagnetic behavior. Thus, accurate predictions of the 11 structural performance of these materials at very low temperatures are of great interest for the conception and design 12 of these cryo-magnetic systems. In this paper, we propose an adequate constitutive model for the evolving bi-phase 13 material|austenite and martensite|based on a Hill-type incremental formulation. Two di erent versions of the model 14 are proposed based on the linear mean- eld homogenization scheme: Mori-Tanaka and Self-Consistent. Moreover, a rate- 15 independent nonlinear mixed kinematic-isotropic hardening law is used for each phase, and the martensitic transformation 16 is described by the nonlinear kinetic law proposed by Olson and Cohen (1975). The constitutive model is implemented 17 in ABAQUS/Standard through a UMAT user subroutine, for which a return mapping algorithm based on the implicit 18 backward Euler integration scheme is used and a closed-form expression of the consistent Jacobian tensor is provided. 19 The Mori-Tanaka and Self-Consistent approaches are evaluated in terms of their ability to describe the mechanical 20 behavior of the bi-phase aggregate by comparing the predictions of the homogenization schemes with unit-cell nite 21 element calculations with an explicit description of the martensite inclusions and the austenite matrix. The comparison 22 is carried out for di erent stress states with controlled triaxiality and Lode parameter under monotonic and cycling 23 loading, paying special attention to the evolution of the mechanical elds in each phase. The unit-cell calculations are 24 performed for both constant and evolving martensite volume fractions. In addition, numerical simulations of tensile tests 25 on samples subjected to di erent initial temperatures are carried out for the transforming bi-phase material and the 26 results are compared to experimental data for AISI 304L and AISI 316LN steels. 27 Keywords: 28 Homogenization, Multiscale modeling, Phase transformation, Cryogenic temperatures, Finite element analysis 29 Corresponding author. Tel. +34 916246015; Fax: +33 916249430. E-mail address:mariapilar.fernandez@uc3m.es Preprint submitted to Mechanics of Materials October 12, 2023 2 1. Introduction 30 Due to the high strength, ductility, and toughness that austenitic stainless steels exhibit at low temperatures, they are 31 the predominant structural material used for cryogenic applications. Since they are paramagnetic or antiferromagnetic 32 under the N eel temperature in their fully austenitic state, high- eld superconducting magnet systems are one of their 33 main application elds of interest (Sgobba, 2006; Barabash et al., 2007). The austenitic steel collars of the standard 34 Dipole magnet for the LHC at CERN, as well as those for the LHC's more recent upgrade (the High Luminosity LHC), 35 and the jacket of the ITER Cable-in-Conduit Conductors, are some examples of structural components made of austenitic 36 stainless steel (Bertinelli et al., 2006; Bordini et al., 2019; Devred et al., 2014). 37 Nevertheless, austenitic stainless steels are particularly prone to strain-induced martensitic transformation at cryo- 38 genic temperatures (Angel, 1954). The initially undeformed -austenite (face-centered cubic) may be transformed into 39 either-martensite (hexagonal close-packed) or 0-martensite (body-centered tetragonal) when submitted to plastic de- 40 formation. However, since the 0-martensitic transformation is generally much more extensive than the transformation 41 of-martensite (Mangonon and Thomas, 1970; Seetharaman, 1984; De et al., 2004), the partially transformed material 42 can be reasonably considered bi-phasic: 0-martensite sites dispersed in the -austenite matrix. The - 0transformation 43 may lead to a considerable evolution of material properties (e.g., strong hardening) and to ferromagnetic behavior of 44 the bi-phase material, which is problematic for the above-mentioned application (i.e., for superconducting magnet sys- 45 tems). Moreover, the transformation rate increases with decreasing temperature, so a nearly fully or fully transformed 46 0-martensitic material may be obtained at low temperatures and high levels of deformation (Fern andez-Pis on et al., 47 2021). For these reasons, it is necessary to formulate a speci c constitutive model for cryogenic applications that covers 48 the entire transformation range and resolves the inherent complexity of the evolving microstructure of austenitic steels. 49 There are only a few elastoplastic constitutive models in the open literature that have been expressly developed to 50 describe the mechanical behavior of austenitic steels at cryogenic temperatures. The majority of these works simply 51 modify the explicit functions of common plasticity models so they can adequately predict macroscopically the additional 52 strain hardening produced by the martensitic transformation at cryogenic temperatures, as in Lee et al. (2009), Ding et al. 53 (2019) and Homayounfard and Ganjiani (2022). More detailed approaches considering the evolving microstructure using 54 homogenization techniques for bi-phase materials are the Mori-Tanaka-based model proposed by Garion and Skocze n 55 (2002) and the Voigt-like formulation used by Kothari et al. (2019). The model proposed by Garion and Skocze n (2002) 56 is widely used in the literature, as demonstrated by Sitko et al. (2010), Ortwein et al. (2014), Kazemi et al. (2019) and 57 Schmidt et al. (2022), among others (some of these works additionally include speci c features to account for microdamage 58 and/or discontinuous plastic ow). 59 The original model proposed by Garion and Skocze n (2002) was developed at CERN for the LHC thin-walled cor- 60 3 rugated shells (called bellows) made of AISI 316L. Since these bellows undergo cycles of expansion and compression at 61 cryogenic temperatures, a distinctive feature of this constitutive framework was to include a mixed kinematic-isotropic 62 hardening for the bi-phase material valid at low temperatures. The authors considered the austenite as elasto-plastic 63 and the martensite as elastic, and they accounted for two contributions to the strain hardening of the bi-phase material: 64 the strain hardening of the austenite phase and the evolution of the composite's tangent sti ness. The austenite was 65 assumed to have a linear kinematic hardening, with the hardening coecient increasing linearly with the martensite 66 volume fraction. The bi-phase material's tangent sti ness tensor was computed via mean- eld homogenization using a 67 Hill-type incremental approach (Hill, 1965a) and the Mori-Tanaka scheme (Mori and Tanaka, 1973; Benveniste, 1987). A 68 new formulation was employed to reduce all anisotropic tangent sti ness operators to isotropic forms. The main simpli- 69 fying assumptions of the model of Garion and Skocze n (2002) were to consider that the strain increment of the bi-phase 70 material was the same as that of the austenite matrix and that this strain increment was only due to the plastic strains. 71 Moreover, the model introduced a simpli ed (but non-di erentiable) kinetic law of martensitic transformation based on a 72 linearization of the law proposed by Olson and Cohen (1975). The constitutive framework of Garion and Skocze n (2002) 73 is attractive because of its simplicity, but convergence between the experimental results and model predictions is limited 74 to strain values not exceeding 0 :2. 75 In this paper, we take the formulation of Garion and Skocze n (2002) as the starting point to develop an improved 76 constitutive model that is suitable for predicting the mechanical behavior of austenitic stainless steels at large strains 77 and cryogenic temperatures for any volume fraction of transformed martensite. Unlike the model of Garion and Skocze n 78 (2002), the constitutive framework proposed in this work is based on a pure mean- eld homogenization procedure: the 79 hardening of the bi-phase material is not directly provided, but the e ective material behavior is entirely determined by 80 homogenization and the constitutive behavior of the individual phases. Both individual phases, austenite and martensite, 81 are considered elastoplastic. A rate-independent nonlinear mixed kinematic-isotropic hardening model is used for each 82 of the phases: the kinematic contribution follows a two-term Chaboche's law (Chaboche et al., 1979; Lemaitre and 83 Chaboche, 1994), and the isotropic contribution uses the superposition of two exponential laws. The model adopts a Hill- 84 type incremental formulation (Hill, 1965a). Anisotropic algorithmic tangent operators are used everywhere except in the 85 Eshelby's tensor (Eshelby, 1957), which is computed with the isotropic form introduced by Bornert et al. (2001a,b). This 86 speci c selection of tangent sti ness operators and Eshelby's tensor is the suggested combination of Pierard and Doghri 87 (2006) for general loading conditions. In particular, Bornert's isotropization method has been selected over the spectral 88 decomposition method (Ponte-Casta~ neda, 1996; Chaboche and Kanout e, 2003) since the latter may lead to inaccurate 89 results for non-proportional loadings and does not provide an isotropic operator for a Chaboche nonlinear kinematic 90 hardening model (Doghri and Ouaar, 2003). Additionally, two di erent versions of the model are proposed based on the 91 4 linear homogenization approach: one using a Mori-Tanaka scheme (Mori and Tanaka, 1973; Benveniste, 1987) and another 92 using a Self-Consistent scheme (Hill, 1965b). The Mori-Tanaka scheme is simpler, but its validity is questioned when 93 the matrix-particle microtopology is lost for a high volume fraction of martensite, whereas the Self-Consistent implicit 94 scheme has no limitation on the volume fraction of transformed martensite (B ohm, 1998; Chatzigeorgiou et al., 2018). 95 Based on the linear homogenization approach, the strain and stress elds in each individual phase are provided, which 96 are generally di erent for the two phases and for the bi-phase material. Moreover, the phase transformation is described 97 by the nonlinear kinetic law proposed by Olson and Cohen (1975). Furthermore, the constitutive model is implemented 98 in ABAQUS/Standard through a UMAT user subroutine that uses a return mapping algorithm based on the implicit 99 backward Euler integration scheme and provides a closed-form expression of the consistent Jacobian tensor. The material 100 parameters for the description of the mechanical behavior of the individual phases and of the kinetics of the martensitic 101 transformation are identi ed using the tensile experimental data for AISI 304L and AISI 316LN presented in Fern andez- 102 Pis on et al. (2021). Unit-cell nite element calculations with an explicit description of the martensite inclusions and the 103 austenite matrix are carried out for di erent stress states with controlled triaxiality and Lode parameter under monotonic 104 and cycling loading. Note that, since the pioneering work of Needleman (1972), nite element unit-cell computations 105 have been extensively used to provide fundamental understanding of the mechanical response of composite aggregates 106 (Christman et al., 1989; Shen et al., 1994; S rensen et al., 1995; Needleman et al., 2010). The unit-cell simulations 107 presented in this paper are performed for both constant and evolving martensite volume fractions. The comparison with 108 the predictions of the homogenization schemes allows us to evaluate the Mori-Tanaka and Self-Consistent approaches with 109 regard to their ability to describe the mechanical behavior of the bi-phase material. Additionally, numerical simulations 110 of tensile tests on samples subjected to di erent initial temperatures are carried out to demonstrate the capability of the 111 developed constitutive model to predict cryogenic martensitic transformation in austenitic stainless steels. 112 2. Constitutive framework 113 Two di erent mean- eld homogenization schemes have been used to model the mechanical behavior and marten- 114 sitic transformation of austenitic stainless steels subjected to cryogenic temperatures: Mori-Tanaka and Self-Consistent 115 schemes. Section 2.1 includes the constitutive description of the individual phases; Section 2.2 presents the kinetic law 116 used to model the martensitic transformation; and a detailed account of the bi-phase material and the homogenization 117 procedure can be found in Section 2.3. The -austenite plays the role of the matrix, while the 0-martensite takes the 118 form of inclusions embedded in the austenite matrix. Following Stringfellow et al. (1992), Papatrianta llou (2005) and 119 Msolli et al. (2016), among others, the martensite is represented by spherical inclusions isotropically dispersed in the 120 material. Moreover, perfect bonding between the two constituents is assumed, as for all mean- eld approaches (B ohm, 121 5 1998). 122 2.1. Individual phases 123 The constitutive description is taken to be the same for both -austenite and 0-martensite (hereinafter simply 124 referred to as austenite and martensite), so that the di erent mechanical behavior of the phases comes from the di erent 125 values of the material parameters (see Tables 1 and 2 in Section 4). 126 The total rate of deformation tensor dis decomposed into elastic deand plasticdpcomponents: 127 d=de+dp(1) where the elastic part of the rate of deformation tensor is related to the rate of stress by the incremental Hooke's law: 128 _r=Le:de=Le: (ddp) (2) where_()rdenotes an objective time derivative (Zaremba-Jaumann rate as in ABAQUS/Standard) of the Cauchy stress 129 tensor, andLeis the isotropic elastic sti ness tensor: 130 Le= 3Ivol+ 2Idev (3) withIvol=1 31 1andIdev=IIvolbeing the fourth-order volumetric and deviatoric projection tensors, respectively, 131 where 1is the second-order identity tensor and Iis the fourth-order symmetric identity tensor. Moreover, =E 3(12)132 and=E 2(1 +)are the elastic bulk and shear moduli, respectively, where Eis the Young's modulus and is the 133 Poisson's ratio. 134 The plastic part of the rate of deformation tensor is given by the associated ow rule: 135 dp=_@F @(4) with _being the rate of the plastic multiplier, where_() denotes di erentiation with respect to time, and Fis the von 136 Mises yield function: 137 F(;X;p) =r 3 2(sX) : (sX) y0+R(p) 0 (5) wheres=h1is the deviatoric part of the Cauchy stress tensor and h=1 3:1is the hydrostatic stress. Moreover, 138 Xis the backstress tensor for the kinematic hardening, y0is the initial yield stress, Ris the isotropic hardening function, 139 6 and p=Zt 0_p( )d is the accumulated plastic strain, where _p=r 2 3dp:dpis the accumulated plastic strain rate (which 140 corresponds to the rate of the plastic multiplier for von Mises plasticity). 141 A rate-independent nonlinear mixed kinematic-isotropic hardening model is assumed. The kinematic contribution is 142 implemented using Chaboche's law (Chaboche et al., 1979; Lemaitre and Chaboche, 1994), in which the backstress tensor 143 is de ned with two separate equations of the form proposed by Armstrong and Frederick (Armstrong et al., 1966): 144 X=2X j=1Xj; _Xr j= 2 3hjdp j_pXj (6) The isotropic hardening is modeled by the superposition of equivalent exponential hardening laws, so that the mixed 145 hardening model provides the same response for monotonic proportional loading independently of the contributions from 146 kinematic and isotropic hardening: 147 R=2X j=1Rj; Rj= (1 )hj j 1e jp (7) wherehjand jare the material parameters responsible for strain hardening and dynamic recovery, respectively. The 148 contribution of kinematic and isotropic hardening to the mixed model is controlled by the Bauschinger parameter , 149 which implies a purely kinematic model if = 1, a purely isotropic model if = 0, and a mixed model if 0 < < 1. 150 Moreover, following a Hill-type incremental formulation, the relationship between the rate of the stress tensor and 151 the total rate of deformation is: 152 _r=Ltan:d (8) whereLtanis the so-called reference modulus (see Section 6.1 in Doghri and Ouaar (2003)). In this work, the reference 153 modulus is taken as the instantaneous algorithmic tangent operator (see equation (A.42) in Appendix A.1). The reader 154 is referred to the works of Doghri and Ouaar (2003) and Pierard and Doghri (2006) to obtain insights into the di erences 155 between various reference moduli and their implications on homogenization methods. 156 2.2. Kinetics of martensitic transformation 157 The evolution of the martensite volume fraction f 0is modeled with the sigmoidal-type kinetic relationship proposed 158 by Olson and Cohen (1975): 159 f 0= 1e 1ep 4:5 (9) 7 whereis the rate of shear-band formation, is the probability that a shear-band intersection will result in the nucleation 160 of martensite, and  p is the accumulated plastic strain in the austenite. Note that the austenite volume fraction is readily 161 computed as f = 1f 0. The kinetic relation of Olson and Cohen (1975) is only valid under isothermal conditions. 162 Moreover, it does not account for the stress state and strain rate sensitivities of the martensitic transformation process. 163 However, especially for cryogenic temperatures, this kinetic law is very attractive due to the small number of parameters 164 that need to be identi ed and to the relatively simple mechanical tests that are needed for their identi cation (note that 165 tests at cryogenic temperatures are highly time- and resource-consuming). Indeed, although the original cryogenic model 166 of Garion and Skocze n (2002) was developed for cyclic loading, it uses a linearization of the Olson and Cohen (1975) 167 kinetic relation. The simpli ed kinetic relation of Garion and Skocze n (2002) has not been used in this work because it 168 leads to an unrealistic non-di erentiable stress-strain curve when the martensite content limit is reached. 169 2.3. Bi-phase material (homogenization) 170 The constitutive equations for the bi-phase material and the homogenization procedure are presented next, using 171 the incremental tangent linearization method (also called Hill-type incremental formulation (Hill, 1965a)) adapted to 172 the case of interest for which the volume fraction of the individual phases evolves. In what follows, subscripts and 0173 are used to refer to austenite and martensite phases, respectively. Moreover, the subscripts + 0andHM denote the 174 bi-phase material and the homogenized elds derived from the homogenization average theorems, respectively. 175 The total rate of deformation tensor of the bi-phase material d + 0is decomposed into elastoplastic dHMand 176 transformation dTRcomponents: 177 d + 0=dHM+dTR (10) where the elastoplastic component corresponds to the homogenized strain rate eld (recall that the total rate of deforma- 178 tion in the individual phases is composed of the corresponding elastic and plastic components, as shown in equation (1)). 179 The transformation component is an additional inelastic term associated with the phase transformation, so it does not 180 directly contribute to the strain-hardening of the individual phases (Stringfellow et al., 1992). This term is typically de- 181 composed into volumetric and deviatoric parts, dTR(vol)anddTR(dev), respectively. The volumetric term is related to the 182 volume change and the deviatoric term to the shape change, both resulting from the transformation process. Following 183 Garion and Skocze n (2002) and Homayounfard and Ganjiani (2022), among others, we only consider here the volumetric 184 part: 185 dTR=dTR(vol)=1 3_f 0v1 (11) 8 where v= 0:05 is the relative volume change during the phase transformation (Garion et al., 2006). 186 The rate of the stress tensor of the bi-phase material corresponds to the homogenized stress rate eld: 187 _r + 0=_r HM (12) The homogenized strain and stress elds are determined using the strain and stress average theorems: HM = 188 (1f 0) +f 0 0andHM= (1f 0) +f 0 0, respectively (see equation (2.5) in B ohm (1998)). In this work, the 189 de nition of the strain tensor provided by ABAQUS/Standard is adopted, _r=d, see Nguyen and Waas (2016). The 190 aforementioned average theorems lead to the following homogenized strain rate and stress rate elds: 191 dHM=dHM(noTR )+dHM(TR) (13) _r HM=_r HM(noTR )+_r HM(TR) (14) where the term denoted by the subscript HM(noTR ) considers a constant volume fraction of the phases and the term 192 denoted by the subscript HM(TR) accounts for the rate of martensite volume fraction: 193 dHM(noTR )= (1f 0)d +f 0d 0;dHM(TR)=_f 0( 0 ) (15) _r HM(noTR )= (1f 0)_r +f 0_r 0; _r HM(TR)=_f 0( 0 ) (16) The deformation rate in each individual phase (see Section 2.1) is related to the total deformation rate in the bi-phase 194 material as follows: 195 dp=Atan p:d + 0 (17) where the subscript prefers to either the austenite or the martensite phase ( p= orp= 0, respectively), and Atan pis 196 the instantaneous strain concentration tensor. This fourth-order tensor is given by: 197 Atan p=Atan p(noTR ): I+ ( 0 ) +1 3v1 1 (18) withAtan p(noTR )being the instantaneous strain concentration tensor for the auxiliary problem that considers a constant 198 volume fraction of the phases: 199 9 dp=Atan p(noTR ):dHM(noTR ) (19) where the explicit form of Atan p(noTR )depends on the selected homogenization scheme (the expressions corresponding to 200 the Mori-Tanaka and Self-Consistent approaches are provided later). The second factor in equation (18) results from the 201 relationship between the total deformation rate in the bi-phase material d + 0and the term of the homogenized strain 202 rate eld that considers a constant volume fraction of the phases dHM(noTR )(see equations (A.78) in Appendix A), 203 where the second-order tensor is de ned such that _f 0= :dHM(noTR ). 204 Mean- eld homogenization schemes di er from each other by the selection of the concentration tensors. Namely, 205 Mori-Tanaka and Self-Consistent schemes' strain concentration tensors rely on the dilute concentration tensor, which 206 is the practical outcome of Eshelby's equivalent inclusion theory (Eshelby, 1957). This fourth-order tensor provides a 207 relationship between the homogeneous uniform strain in an inclusion and the far- eld strain in the parent medium 208 in which the inclusion is embedded. Following the incremental tangent linearization method, the instantaneous dilute 209 concentration tensor is given by: 210 di=Atan dil(i;0):d0;Atan dil(i;0)= I+Stan-iso 0;spher : Ltan 01: Ltan iLtan 01 (20) where the subscripts iand 0 refer to the inclusion and the parent medium, respectively, and Stan-iso 0;spheris the fourth-order 211 instantaneous Eshelby's tensor, which depends on the material properties of the parent medium and on the shape and 212 orientation of the inclusion. The general expression of the Eshelby's tensor is rather complex (see, for example, equation 213 (2.22) in B ohm (1998)), so in this paper we use the closed-form expression for spherical inclusions in an isotropic parent 214 medium: 215 Stan-iso 0;spher =1 31 +tan 0 1tan 0Ivol+2 1545tan 0 1tan 0Idev (21) wheretan 0=3tan 02tan 0 2(3tan 0+tan 0)is the tangent Poisson's ratio and tan 0andtan 0are the tangent bulk and tangent shear moduli 216 of the isotropic parent medium, respectively. Because the algorithmic tangent operator of the parent medium (austenite 217 or homogenized material depending on the homogenization scheme) is anisotropic in general, the isotropic estimate 218 proposed by Bornert et al. (2001a,b) is used (as denoted by the superscript tan-isoin Eshelby's tensor), resulting in 219 tan 0=1 3Ivol::Ltan 0andtan 0=1 10Idev::Ltan 0. Recall that the isotropic estimates are only used to compute the Eshelby's 220 tensor, while anisotropic algorithmic tangent operators are used everywhere else. 221 The instantaneous strain concentration tensors of the Mori-Tanaka and Self-Consistent schemes are presented next, 222 using the incremental tangent linearization method for the auxiliary problem that considers a constant volume fraction 223 10 of the phases: 224 Mori-Tanaka scheme . This explicit approach assumes the matrix-type phase (austenite) as the parent medium in 225 which the particle-type phase (martensite) is embedded. According to Eshelby's theory, the two phases are related 226 as follows:d 0=Atan dil( 0; ):d (see the equivalence with equation (20) 1). Using the previous equation and the rate 227 form of the strain average theorem given by equation (15) 1, the instantaneous strain concentration tensor for each 228 phase, austenite and martensite, is given by: 229 d =Atan (noTR ):dHM(noTR );Atan (noTR )= (1f 0)I+f 0Atan dil( 0; )1 (22) d 0=Atan 0(noTR ):dHM(noTR );Atan 0(noTR )=Atan dil( 0; ): (1f 0)I+f 0Atan dil( 0; )1 (23) Self-Consistent scheme . This implicit approach assumes the homogenized material as the parent medium in 230 which each individual phase is treated as embedded. According to Eshelby's theory, for each phase (austenite and 231 martensite), the instantaneous strain concentration tensor corresponds to the instantaneous dilute concentration 232 tensor (see the equivalence with equation (20) 1): 233 d =Atan (noTR ):dHM(noTR );Atan (noTR )=Atan dil( ;HM (noTR )) (24) d 0=Atan 0(noTR ):dHM(noTR );Atan 0(noTR )=Atan dil( 0;HM (noTR )) (25) where the instantaneous strain concentration tensor for each phase is dependent on the tensor Ltan HM(noTR )(see equa- 234 tion (20) 2), which relates the homogenized strain rate and stress rate elds _r HM(noTR )=Ltan HM(noTR ):dHM(noTR )235 and is itself a function of the instantaneous strain concentration tensors, as shown by the following expression (valid 236 for any mean- eld homogenization approach): 237 Ltan HM(noTR )= (1f 0)Ltan :Atan (noTR )+f 0Ltan 0:Atan 0(noTR ) (26) which is derived from the rate form of the stress average theorem (equation (16) 1), in which the rate of the stress 238 tensor in each phase is expressed as a function of the corresponding rate of deformation (equation (8)), and the 239 11 latter is related to the homogenized strain rate eld by the corresponding instantaneous strain concentration tensors 240 (equation (19)). 241 3. Numerical implementation 242 The constitutive model has been implemented at integration point level within the nite element solver ABAQUS/ 243 Standard using a UMAT user material subroutine. The UMAT code is included as Supplementary Material . Note that 244 the subroutine is speci cally designed to handle the kinetics of martensitic transformation as described in Section 2.2 245 (Olson and Cohen (1975) model). 246 A sketch of the implementation procedure is shown in Fig. 1. At each time step, the total strain increment of the 247 bi-phase material,  (n+1) + 0, together with the mechanical elds of the previous step, (n) + 0and(n) + 0, are provided 248 by ABAQUS to update the stress components, (n+1) + 0, and to provide the Jacobian tensor ( DDSDDE as denoted in 249 ABAQUS). State variables of the problem ( SDV as denoted in ABAQUS) are the two terms of the isotropic hardening 250 function in the two phases: R1 ,R2 ,R1 0, andR2 0; the accumulated plastic strain in the two phases:  p and p 0; the 251 volume fraction of martensite: f 0; the back stresses in the two phases: X1 ,X2 ,X1 0, andX2 0; the mechanical elds in 252 the two phases:  , , 0, and 0; the plastic strain in the two phases: p andp 0; and the algorithmic tangent operators 253 for the two phases: Ltan andLtan 0. The update of the internal tensorial variables at each time step during a nite strain 254 analysis takes into account the incremental rotation of the material basis system. Thus, the second-order tensorial internal 255 variables are rotated to the current con guration using the prede ned ABAQUS function ROTSIG , which performs the 256 following calculation:  R(n+1) (n)()(n)R(n+1) (n)T, where  R(n+1) (n)=R(n+1)R(n)Tis the incremental rotation ( DROT as 257 denoted in ABAQUS). The objective increments are hence expressed as ( )(n+1)= ()(n+1)R(n+1) (n)()(n)R(n+1) (n)T. 258 For the fourth-order tensorial internal variables, Ltan andLtan 0, the operation that requires their rotations to the current 259 con guration is performed as shown in equation (A.44) in Appendix A.1. 260 Firstly, from the total strain increment in the bi-phase material corresponding to the current time step ( n+ 1), 261 the contribution of each individual phase is assumed for the rst ( k= 1) iterative step of the global scheme:  (n+1) (k=1)262 and (n+1) 0(k=1). For the rst time step ( n= 1) it is assumed that  (n=1) (k=1)= (1f(n=1) 0(k=1))(n=1) + 0and (n=1) 0(k=1)= 263 f(n=1) 0(k=1)(n=1) + 0, while for the rest of the time steps  (n+1) (k=1)= (n) q (n+1) + 0: (n+1) + 0 q (n) + 0: (n) + 0 and 264 (n+1) 0(k=1)= (n) 0q (n+1) + 0: (n+1) + 0 q (n) + 0: (n) + 0 . This is performed in the Initialization box represented in 265 the gure. Subsequently, the constitutive equations for each individual phase (Section 2.1) are solved using the numerical 266 implementation explained in Appendix A.1. In the gure, this step is represented by the Austenite phase andMartensite 267 phase boxes (notice the loop, which represents the sub-iterative process required to solve the problem). Each box uses 268 the strain increment of the speci c k-th iteration of the global iterative scheme as input, along with the internal variables 269 12 of the corresponding individual phase. Once the current value of the accumulated plastic strain in the austenite phase for 270 thek-th iteration of the global iterative scheme is known, the kinetics of martensitic transformation (Section 2.2) can be 271 solved. This step is represented in the gure by the Kinetics of transformation box, and the corresponding implementation 272 is described in Appendix A.2. Afterwards, the constitutive equations for the bi-phase material (Section 2.3) are solved, 273 and the bi-phase material's mechanical behavior is calculated for the k-th iteration of the global iterative scheme. 274 In the gure, this step is represented by the Bi-phase material (homogenization) box, which uses as inputs the state 275 variables together with the outputs of the Austenite phase ,Martensite phase , and Kinetics of transformation boxes. The 276 numerical implementation, which depends on the homogenization scheme (Mori-Tanaka or Self-Consistent), is described 277 in Appendix A.3. The required iterative scheme for the implicit Self-Consistent approach is represented in the gure 278 by a dashed loop, indicating that it is only needed if the Self-Consistent approach is selected. The outcome of the Bi- 279 phase material (homogenization) box is the corrected strain increment for each individual phase. Then, the di erence 280 between the strain increment assumed at the beginning of the global iterative scheme and the corrected strain increment 281 is calculated. If the di erence is larger than a prede ned tolerance, a new iteration is performed using the corrected value 282 of the strain increment for each phase as input in the new iterative step of the global scheme:  (n+1) (k+1)and (n+1) 0(k+1). 283 The whole process, which includes the Austenite phase ,Martensite phase ,Kinetics of transformation , and Bi-phase 284 material (homogenization) boxes, is recomputed until the convergence condition is met. When convergence is ful lled, 285 the converged stress components (n+1) + 0and state variables are given to ABAQUS/Standard. The Jacobian tensor also 286 has to be given to ABAQUS/Standard at the end of each time step. The solver uses this tensor in its global Newton 287 scheme to achieve an accurate assessment of the incremental kinematics. To ensure computational eciency with fast 288 convergence, the consistent Jacobian tensor is used; its numerical implementation is given in Appendix A.4. 289 Note that the same numerical implementation, based on the integration scheme explained in Appendix A.1, is 290 utilized for each of the individual phases. In contrast, in the works of Stringfellow et al. (1992) and Papatrianta llou 291 et al. (2006), two separate integration schemes are employed for the martensite phase, each addressing one of the two 292 contributions that these works de ne in the hardening process of the martensite. This strategy is adopted to explicitly 293 account for the assumption that a unit of martensite that forms at a given austenite plastic strain should inherit the strain- 294 hardened dislocation structure of its parent austenite (Stringfellow et al., 1992). Thus, the hardening contribution from 295 the fraction of new transformed martensite is solved separately from the contribution of the already existing martensite 296 at the beginning of the current time step (each fraction of martensite has a di erent accumulated plastic strain associated 297 with it). Note that this strategy is not fully consistent with a pure mean- eld homogenization procedure, where only 298 the average elds in the individual phases are de ned; moreover, it might lead to inconsistencies with the constitutive 299 equations. For this reason, an explicit implementation of this feature is neglected in the present model. 300 13 Kinetics of transformationMartensite phaseAustenite phase no convergence convergenceBi-phase material (homogenization)Initialization SOLVERUMAT Figure 1: Sketch of the implementation of the homogenized constitutive model (Mori-Tanaka or Self-Consistent scheme) in ABAQUS/Standard. 4. Identi cation of material parameters 301 Two sets of material constants are to be identi ed: the parameters involved in the mechanical behavior of the 302 individual phases|austenite and martensite|and the parameters involved in the kinetics of martensitic transformation. 303 The rst set of parameters consists of the Young's modulus Eand the Poisson's ratio (equation (3)), the initial 304 yield stress y0(equation (5)), and the parameters describing the mixed kinematic-isotropic hardening: ,h1,h2, 1, 305 and 2(equations (6) and (7)), all of which are speci c to each individual phase. On the other hand, the parameters 306 involved in the expression for the kinetics of martensitic transformation proposed by Olson and Cohen (1975) are and 307 (equation (9)). 308 The constitutive model is calibrated to describe the mechanical behavior of two di erent austenitic steel grades that 309 are widely used in superconducting magnet system applications: AISI 304L and AISI 316LN. The experiments performed 310 by Fern andez-Pis on et al. (2021) are the basis for the identi cation of the constitutive parameters. The data set includes 311 uniaxial tensile stress{strain curves and the corresponding evolution of the volume fraction of martensite with the strain 312 for the two aforementioned grades at room temperature (300 K), liquid nitrogen temperature (77 K), and liquid helium 313 temperature (4 K); see Figs. 4, 13, and 14 therein, where the stress and strain are referred to the bi-phase material. 314 Exceptionally, the values of the Poisson's ratio and the Bauschinger parameter, = 0:3 and = 0:45, respectively, 315 14 are taken from Garion et al. (2006) for the same family of steels (AISI 304 and AISI 316L) at cryogenic temperatures. 316 For these two material parameters, identical values for the two individual phases are assumed for the two grades over 317 the whole temperature range analyzed in this work. For the rest of the parameters, an explicit temperature-evolution 318 function is considered, and the coecients of the function are determined by best- t operation with the aforementioned 319 experimental data presented in Fern andez-Pis on et al. (2021). The idea is to cover the entire analyzed temperature 320 range, unlike conventional approaches that independently determine the material parameters at a certain temperature. 321 The temperature-evolution function selected for the material parameters involved in the mechanical behavior of the 322 individual phases is shown in equation (27), which is similar to the one used by Hosseini et al. (2015) for a Chaboche 323 kinematic hardening model. Moreover, the same functions proposed by Fern andez-Pis on et al. (2021), which are based 324 on the experimental data provided by Olson and Cohen (1975), are adopted here with regard to the material parameters 325 involved in the kinetics of martensitic transformation, as shown in equation (28). 326 E(T) =x1 1x2e(T=x3) ; y0(T) =x4 1x5e(T=x6) ; h 1(T) =x7 1x8e(T=x9) ; h2(T) =x10 1x11e(T=x12) ; 1(T) =x13 1x14e(T=x15) ; 2(T) =x16 1x17e(T=x18) (27) (T) =2 1 +e3(T1)2 2;  (T) =2 1 +e3(T1)(28) The identi cation procedure is split into three parts: (i) identi cation of the coecients corresponding to the material 327 parameters of the austenite phase; (ii) identi cation of the coecients corresponding to the material parameters of the 328 martensite phase; and (iii) identi cation of the coecients corresponding to the material parameters of the kinetics of 329 martensite transformation. The determination of the coecients is described as follows: 330 (i) We take as reference the aforementioned experimental stress{strain curves, but we only use the data with a strain 331 level corresponding to a volume fraction of martensite lower than 10% (so the e ect of the martensite phase 332 on the selected data is considered to be negligible). For each grade, the selected data is represented as a x 333 pTsurface, with xbeing the axial stress component corresponding to the tensile loading direction, and the 334 surface is then tted to the function x=y0(T) +h1(T) 1(T) 1e 1(T)p +h2(T) 2(T) 1e 2(T)p . This function 335 is adapted from equation (5), where a pure isotropic hardening is used because it is sucient for determining 336 the parameters (monotonic loading test). The best- tting condition using the least squares method provides the 337 coecients corresponding to the functions E(T),y0(T),h1(T),h2(T), 1(T), and 2(T) for the austenite phase 338 (see equation (27)). 339 15 AISI 304L x1(MPa) x2x3(K)x4(MPa) x5x6(K) 2:061051:521072:291016:671024:771011:00103 Austenite x7(MPa) x8x9(K)x10(MPa) x11x12(K) phase 1 :301056:841011:001032:141031:611011:00103 x13 x14x15(K) x16 x17x18(K) 1:011021:4210101:001033:211005:801021:27102 x1(MPa) x2x3(K)x4(MPa) x5x6(K) 2:061051:521072:291013:271036:641011:00103 Martensite x7(MPa) x8x9(K)x10(MPa) x11x12(K) phase 2 :911034:051011:001031:711045:851011:00103 x13 x14x15(K) x16 x17x18(K) 1:601015:881026:671024:611002:371036:09101 Kinetic 1(K) 23(K1)1(K) 23(K1) relation 3 :361021:311014:691022:661022:101008:80102 AISI 316LN x1(MPa) x2x3(K)x4(MPa) x5x6(K) 2:061051:521072:291012:651036:631019:89102 Austenite x7(MPa) x8x9(K)x10(MPa) x11x12(K) phase 6 :031046:661019:381025:871033:141014:53102 x13 x14x15(K) x16 x17x18(K) 1:101022:401011:001032:001006:941059:99102 x1(MPa) x2x3(K)x4(MPa) x5x6(K) 2:061051:521072:291013:271036:641011:00103 Martensite x7(MPa) x8x9(K)x10(MPa) x11x12(K) phase 2 :911034:051011:001031:711045:851011:00103 x13 x14x15(K) x16 x17x18(K) 1:601015:881026:671024:611002:371036:09101 Kinetic 1(K) 23(K1)1(K) 23(K1) relation 4 :621025:891004:291039:001014:841004:94102 Table 1: Identi cation of the coecients of the temperature-evolution parameters corresponding to the austenite and martensite phases and to the kinetic relation of Olson and Cohen (1975), see equations (27) and (28). (ii) Unit-cell nite element calculations with a homogenized depiction of the bi-phase aggregate are performed under 340 monotonic uniaxial tensile loading (see Section 5.1, with triaxility X + 0= 1=3 and Lode parameter L + 0=1). 341 The Self-Consistent scheme version of the UMAT user subroutine described in Section 3 is employed, but the 342 subroutine includes a simpli ed form of the kinetic law of martensitic transformation: f 0= 1e 1e + 0x4:5 , 343 where + 0xis the axial strain component of the bi-phase material that corresponds to the tensile loading direction 344 16 (see Fig. 3). This simpli ed law reduces the coupling between the kinetics of transformation and the homogenization 345 approach, thereby facilitating the parameter identi cation operation (note that in the original law, equation (9), 346 the driving force for the transformation is the accumulated plastic strain in the austenite phase). The nite element 347 calculations are performed for the AISI 304L at three di erent constant temperatures (300 K, 77 K, and 4 K). The 348 calculations use the material parameters for the austenite phase identi ed as described in the previous paragraph 349 as well as the material parameters for the kinetic relation proposed by Fern andez-Pis on et al. (2021) (see Table 2 350 therein). Note that the material parameters for the kinetic relation are taken directly from Fern andez-Pis on et al. 351 (2021) in this step of the calibration because this former study assumed the aforementioned simpli ed kinetic law 352 of martensitic transformation. Moreover, in accordance with the experiments, the bi-phase material is initially 353 considered 100% austenite ( f(n=1) 0 = 0). The coecients corresponding to the functions y0(T),h1(T),h2(T), 354 1(T), and 2(T) for the martensite phase are identi ed for the best correlation between the numerical results and 355 experimental tensile stress{strain curves in the bi-phase material, now covering the full range of strain shown in the 356 source gure (Fig. 4 in Fern andez-Pis on et al. (2021)). The coecients corresponding to the function E(T) are taken 357 to be identical to those obtained for the austenite phase in accordance with Msolli et al. (2016) and Perdahco glu 358 and Geijselaers (2011a,b), among others, who make the simplifying assumption that the elastic properties of both 359 individual phases are equal. 360 (iii) Unit-cell nite element calculations with a homogenized depiction of the bi-phase aggregate are performed under 361 monotonic uniaxial tensile loading. The precise Self-Consistent scheme version of the UMAT user subroutine, 362 as described in Section 3, is employed (including the implementation of the original kinetic law of martensitic 363 transformation shown in equation (9)). The nite element calculations are performed for the two grades, AISI 304L 364 and AISI 316LN, at the three di erent constant temperatures. The calculations use the material parameters for the 365 austenite and martensite phases identi ed as described in the previous paragraphs. The coecients corresponding 366 to the functions (T) and(T) (see equation (28)) are identi ed for the best correlation between the numerical 367 results and the experimental data of the evolution of the volume fraction of martensite with the strain in the 368 bi-phase material reported in Fern andez-Pis on et al. (2021). 369 Table 1 provides the nal values of the coecients for the temperature-evolution functions (equations (27) and (28)). 370 Recall that these functions and associated coecients are meant to be valid for the three analyzed temperatures|300 K, 371 77 K, and 4 K|as well as for any other temperature Tin the range under consideration. Particularizing these functions 372 for the analyzed temperatures provides the material parameters given in Table 2. Note that the nal material parameters 373 for the kinetic relation ( and) are di erent from the ones proposed by Fern andez-Pis on et al. (2021), which have also 374 17 AISI 304L AISI 316LN T= 300 KT= 77 KT= 4 KT= 300 KT= 77 KT= 4 K E(GPa) 191 206 206 191 206 206 y0(MPa) 238 323 348 272 753 888 Austenite h1(MPa) 10020 33922 40657 5040 16721 19980 phaseh2(MPa) 1676 1769 1795 2301 3686 4011 1 101 101 101 74 81 83 2 1:23 2 :87 3:02 2 :00 2 :00 2:00 E(GPa) 191 206 206 191 206 206 y0(MPa) 338 925 1090 338 925 1090 Martensite h1(MPa) 1319 1638 1727 1319 1638 1727 phaseh2(MPa) 3581 6275 7034 3581 6275 7034 1 14:49 14 :90 15:01 14 :49 14 :90 15:01 2 3:10 4 :57 4:59 3 :10 4 :57 4:59 Kinetic  4:47 6 :53 6:53 0 :98 2 :00 2:22 relation  0:10 2 :10 2:10 0 :00 3 :17 4:77 Table 2: Identi cation of the material parameters corresponding to the austenite and martensite phases and to the kinetic relation of Olson and Cohen (1975), see equations (27) and (28), for the three analyzed temperatures: 300 K, 77 K, and 4 K. been used here in the second step of the calibration. It is due to the fact that in the third step of the calibration, where 375 those parameters are determined, the original kinetic law of martensitic transformation is used (equation 9). Moreover, 376 the material parameters for the martensite phase of the AISI 316LN are the same as those of the AISI 304L. This is 377 because, unlike for AISI 304L, the experimental stress-strain curves for AISI 316LN steel do not show any signi cant 378 change as martensite forms. Therefore, only the experimental stress-strain curves for AISI 304L are used to calibrate 379 the martensite phase (as previously mentioned), and it is assumed that the ow stress and strain hardening material 380 parameters for martensite are the same for an AISI 316LN steel grade. 381 Section 6.2 includes a comparison between tensile test calculations using the calibrated model and experimental 382 data for both steel grades at the three analyzed temperatures (see Figs. 19 and 20). Since the material is initially 383 fully austenitic, this comparison gives information on the accuracy of the calibration performed for the austenite phase. 384 Regarding the martensite phase, the result of an additional unit-cell nite element simulation considering a non-evolving 385 martensitic material ( f 0= 100% and _f 0= 0) is compared with experimental stress-strain curves for martensite at 386 T= 300 K taken from the literature (the authors are not aware of any experimental martensite's stress{strain curves at 387 18 cryogenic temperatures). The comparison is shown in Fig. 2, where the experimental data taken from Zaera et al. (2012) 388 corresponds to the mechanical behavior of the single martensite phase on AISI 304 samples tensile tested by Rodr guez- 389 Mart nez et al. (2011), and the experimental data supplied by the manufacturer KVA STAINLESS corresponds to a 390 martensitic stainless steel AISI 410. A good correlation between the numerical results and the experimental data is 391 obtained, which indicates the careful calibration developed for the martensite phase. 392 Experiments: Numerical simulation Figure 2: Comparison of the stress-strain curve for the martensite phase resulting from the calibration performed in this work for AISI 304L and experimental data at T= 300 K. The results of the simulations are shown with a black solid line. The experimental data from Rodr guez- Mart nez et al. (2011) and Zaera et al. (2012), red markers, corresponds to the mechanical behavior of the single martensite phase on AISI 304 samples. The experimental data supplied by the manufacturer KVA STAINLESS, green markers, corresponds to a martensitic stainless steel AISI 410 (Data source: https://www.kvastainless.com). 5. Finite element models 393 This section describes the main features of the nite element models created in ABAQUS/Standard to assess the Mori- 394 Tanaka and Self-Consistent homogenization schemes with regard to their ability to describe the mechanical behavior and 395 martensitic transformation of AISI 304L and AISI 316LN austenitic steels. Two di erent models have been developed: 396 (i) a cubic unit-cell subjected to controlled macroscopic triaxiality and Lode parameter; and (ii) a at tensile sample 397 subjected to prescribed displacement. 398 5.1. Unit-cell 399 The reference con guration of the unit-cell is de ned by the domains 0 xL0, 0yL0, and 0zL0, where 400 (x;y;z ) is a Cartesian coordinate system with origin located at the bottom right corner of the cell and L0= 1 mm (see 401 Fig. 3(a)). The unit-cell plots included in Figs. 8 and 17 are referred to this coordinate system. We have performed two 402 di erent types of unit-cell calculations related to the representation of the material microstructure: 403 19 Figure 3: Cubic unit-cell nite element model: (a) dimensions and Cartesian coordinate system ( x;y;z ), (b) loading directions for the case of monotonic loading simulations with  + 0I,  + 0II, and  + 0IIIbeing the principal values of the macroscopic stress tensor of the bi-phase material. Explicit depiction of the martensite inclusions and the austenite matrix . The mechanical behavior of the 404 individual phases is modeled separately using the constitutive equations of Section 2.1. The material is considered 405 to initially contain martensite inclusions that are represented as spherical particles which are perfectly bonded to 406 the austenite matrix (consistent with the assumptions of the homogenization schemes, see Section 2). Di erent 407 initial volume fractions of martensite have been investigated: 25%, 45%, and 65%. We have performed calculations 408 with a single central inclusion, and with multiple inclusions distributed in the unit-cell, see Figs. 4(a) and 4(b)-(e), 409 respectively. Note that two distinct cases with FCC-symmetric (face-centered cubic) distributions of inclusions have 410 been examined: one involves a representative volume element with particles intersecting the unit-cell boundaries, 411 see Fig. 4(b), while the other features a unit-cell with multiple inclusions and a boundary layer devoid of particles, 412 see Fig. 4(c). We utilize the latter case for comparison with the con gurations shown in Figs. 4(d)-(e), which 413 include random spatial distribution of multiple inclusions and a boundary layer devoid of particles (note that 414 models 4(b)-(c) essentially produce identical results for the same volume fraction of particles, see Section 6.1.1). 415 The simulations with random spatial distribution of the particles have been carried out with inclusions of the 416 same size and also with inclusions of di erent sizes, see Fig. 4(d)-(e). The particles with the same size have radii 417 of 0:04475 mm, 0 :054436 mm, and 0 :061534 mm for the martensite volume fractions of 25%, 45%, and 65%, 418 respectively (note that these inclusion sizes are used both in microstructures with random and FCC-symmetric 419 spatial distributions of particles and a boundary layer devoid of inclusions, see Figs. 4(c)-(d)). For the particles 420 with di erent sizes, see Fig. 4(e), following Giompliakis (2014), we use three di erent families of particles with 421 radiifRi;Rii;Riiig=f9 4Rmin;7 4Rmin;Rmingand particle concentrations fFi;Fii;Fiiig=f0:5F 0;0:25F 0;0:25F 0g, 422 20 whereF 0is the total volume fraction of martensite in the unit-cell and Rminis the radius used in the monodisperse 423 microstructure such that Rmin= 0:04475 mm;0:054436 mm, and 0 :061534 mm for F 0= 25%;45%, and 65%, 424 respectively. The unit-cell has been discretized with ten-node quadratic tetrahedral elements (C3D10 in ABAQUS 425 notation), see Fig. 3(b). A very ne mesh near the spherical particles is necessary to capture the initial geometry 426 and shape evolution of the martensite inclusions during loading. The number of elements increases with the number 427 of inclusions and the martensite volume fraction. For the calculations with a single central particle and a martensite 428 volume fraction of 25%, the mesh is discretized with 34000 elements, while the mesh of the unit-cell with a FCC- 429 symmetric spatial distribution of the particles, a boundary layer devoid of particles, and 65% martensite consists 430 of900000 elements. A mesh sensitivity analysis has been carried out to con rm that the quantitative results 431 and general trends provided in this paper are largely independent of the discretization (the results are not shown 432 for the sake of brevity). A workstation AMD Milan 7453 @ 2 :75 GHz with 56 cores has been used to perform 433 the calculations. The computational cost of each simulation varied from 2 hours to 4 weeks, depending on the 434 microstructure considered, using simultaneously all the cores of the workstation. 435 Figure 4: Unit-cell nite element model with explicit depiction of the martensite inclusions colored in red and the austenite matrix colored in blue: (a) single central inclusion, (b) representative volume element of a FCC-symmetric spatial distribution of inclusions with the same size and particles intersecting the boundaries, (c) FCC-symmetric spatial distribution of inclusions with the same size and a boundary layer devoid of particles, (d) random spatial distribution of inclusions with the same size and a boundary layer devoid of particles, and (e) random spatial distribution of inclusions with di erent sizes and a boundary layer devoid of particles. The nite element models correspond to the case of F 0= 25%. Homogenized depiction of the bi-phasic aggregate . The e ective mechanical behavior of the material is 436 modeled using the constitutive equations of Section 2.3. Simulations are performed using both versions of the 437 UMAT user subroutine (Section 3) corresponding to the Mori-Tanaka and Self-Consistent approaches. The unit- 438 cell has been meshed with 12 ten-node quadratic tetrahedral elements (this is the minimum number of C3D10 439 elements required to model a cubic geometry). Note that the number of elements is 3000 times less than in the 440 simulations performed with a single central martensite inclusion and 75000 times less than in the calculations with a 441 FCC-symmetric spatial distribution of the particles, a boundary layer devoid of particles, and 65% martensite. The 442 calculations have been performed using a single core of a laptop computer with an Intel(R) Core(TM) i5 8350U 443 21 CPU @ 1:70 GHz. The computational cost of a simulation varied between 5 and 40 minutes, depending on the 444 homogenization scheme considered, i.e., these calculations are signi cantly faster than the simulations with an 445 explicit depiction of the martensite inclusions and the austenite matrix. 446 The macroscopic stress tensor in the unit-cell (bi-phase material) is de ned as the volumetric averaging of the 447 microscopic (local, i.e., de ned at each integration point) Cauchy stress tensor  + 0=1 VcellR VcelldV(Vadillo et al., 448 2016; Hosseini et al., 2022), with Vcell=R VcelldVbeing the total volume of the unit-cell. Moreover, the macroscopic 449 strain tensor of the bi-phase material is de ned as the volumetric averaging of the microscopic strain tensor " + 0= 450 1 VcellR VcelldV, where the de nition of the microscopic strain tensor provided by ABAQUS/Standard has been adopted, 451 _r=d, see Nguyen and Waas (2016). Note that for the simulations with the homogenized depiction of the bi-phasic 452 aggregate, the so-called microscopic stress and strain tensors, and, are referred to the bi-phase material,  + 0and 453  + 0, respectively, see Fig. 1. Furthermore, the formulations of the macroscopic stress and strain tensors of the bi-phase 454 material are equivalent to the stress and strain average theorems applied in homogenization procedures, see equation (4.1) 455 in Chatzigeorgiou et al. (2018) and equation (1.4) in B ohm (1998). Similarly, the macroscopic stress and strain tensors 456 for each individual phase are de ned as p=1 VpR VcellpdVpand"p=1 VpR VcellpdVp, withVp=R VcelldVpbeing the total 457 volume of each individual phase, where the subscript prefers to either the austenite or the martensite phase ( p= or 458 p= 0, respectively). In the simulations with explicit depiction of the martensite inclusions and the austenite matrix, 459 the macroscopic elds of each individual phase only account for the elements whose material corresponds to this speci c 460 phase (Srivastava et al., 2015). It means that, for the martensite phase, ( )pdVp= ()dVfor elements within the particles 461 and ()pdVp= 0 for elements within the matrix, and the inverse for the austenite phase. On the other hand, for the 462 simulations with the homogenized depiction of the bi-phasic aggregate, the macroscopic elds are calculated accounting 463 for all the elements in the unit-cell using ( )pdVp= ()pfpdV, withfpbeing the microscopic volume fraction of the 464 considered phase (see Section 2.2). Note that the formulations of the macroscopic stress and strain tensors for each 465 individual phase are equivalent to the phase-wise uniform stress and strain elds employed in mean- eld homogenization 466 approaches, see equation (1.16) in B ohm (1998). Furthermore, the macroscopic volume fraction of martensite is de ned as 467 F 0=1 VcellR VcelldV 0where, in simulations with explicit depiction of the martensite inclusions and the austenite matrix, 468 dV 0=dVfor elements within the particles and dV 0= 0 for elements within the matrix; while, for the simulations with 469 homogenized depiction of the bi-phasic aggregate, dV 0=f 0dVfor all the elements. Note that the terms microscopic 470 and macroscopic adopt the conventional de nitions used in unit-cell nite element simulations: microscopic refers to the 471 integration point level, and macroscopic refers to a volumetric average in the unit-cell. Notice the di erence with literature 472 dealing exclusively with homogenization notions, where the term macroscopic is reserved for the bi-phase material. 473 The stress state in the unit-cell (bi-phase material) and in the individual phases is described by the corresponding 474 22 macroscopic stress triaxiality X=h and macroscopic Lode parameter L=2IIIIII IIII. Note that  h=I+II+III 3475 and =q 3 20:0are the macroscopic hydrostatic stress and the macroscopic e ective stress, respectively, with  I, 476 II, and III(IIIIII) being the principal values of the macroscopic stress tensor and 0=h1being the 477 macroscopic deviatoric stress tensor. Moreover, for representation purposes, we also de ne a macroscopic e ective strain 478 as"=q 2 3"0:"0, where"0=""h1is the macroscopic deviatoric strain tensor and "h="I+"II+"III 3is the macroscopic 479 volumetric strain, with "I,"II, and"III("I"II"III) being the principal values of the macroscopic strain tensor. 480 Note that for the previous de nitions, the subscripts ( + 0, , or 0, denoting the bi-phase material, austenite phase, and 481 martensite phase, respectively) have been omitted because these de nitions can be applied separately to any of them by 482 using the corresponding macroscopic stress and strain tensors that were previously introduced. Hereinafter, when these 483 variables appear without a subscript, it means that they refer to the three of them separately: the bi-phase material, the 484 austenite phase, and the martensite phase. 485 In the simulations, periodic boundary conditions have been applied on the unit-cell using the equations for the nodal 486 displacements reported in Appendix A of Dakshinamurthy et al. (2021). The multi-point constraint subroutine developed 487 by Dakshinamurthy et al. (2021) has been employed to impose controlled values of macroscopic stress triaxiality and 488 macroscopic Lode parameter in the unit-cell (bi-phase material), X + 0andL + 0, respectively|see also Hosseini 489 et al. (2022) and Vishnu et al. (2023). The loading directions are parallel to the principal directions of the macroscopic 490 stress tensor of the bi-phase material, such that the major loading direction corresponds to the principal stress direction 491 associated with  + 0Iand the minor loading direction corresponds to the principal stress direction associated with 492  + 0III; see Fig. 3(b) for the case of the monotonic loading simulations (for the cyclic loading simulations, the direction 493 of  + 0Iand  + 0IIIswitches, see Fig. 12). 494 The calculations with an explicit description of the microstructure are to be compared with the simulations performed 495 with the Mori-Tanaka and Self-Consistent approaches to evaluate the ability of the homogenization schemes to describe 496 the e ective behavior of the bi-phase material and the individual phases. Section 6.1 includes calculations performed under 497 monotonic and cyclic loading for di erent values of macroscopic triaxiality X + 0=0:5, 0:5, 1:5, and 3 and macroscopic 498 Lode parameter L + 0=1, 0, and 1 for constant and evolving martensite volume fraction. The calculations are 499 performed using the material parameters corresponding to AISI 304L austenitic stainless steel at the constant temperature 500 ofT= 77 K (see Table 2). Additional calculations using the material parameters for AISI 304L at T= 4 K and for 501 AISI 316LN at T= 77 K are presented in Appendix B and Appendix C, respectively. Note that the material parameters 502 were identi ed against experiments based on the homogenized model predictions for the evolving bi-phase material under 503 tensile loading (see Section 4). The unit-cell simulations presented in Section 6.1 do not involve any prior calibration of 504 the homogenized model (no further calibrations were performed for constant martensite volume fractions or for di erent 505 23 loading conditions). Therefore, the comparison between the unit-cell simulations using the homogenization schemes 506 and the calculations with an explicit description of the microstructure (Section 6.1) is independent of the calibration 507 procedure. The idea is to avoid using the same set of data for both parameter identi cation and model evaluation. 508 5.2. Tensile test 509 The geometry of the tensile sample is based on a \non-proportional test piece" according to the standard ISO-6892-4 510 (2015), see Fig. 5. This is the sample used by Fern andez-Pis on et al. (2021) to perform the tensile experiments that are 511 employed in Section 4 to calibrate the constitutive model. Material points are referred to using a Cartesian coordinate 512 system with positions in the reference con guration denoted as ( x;y;z ), the origin of coordinates is located in the center 513 of mass of the sample, see Fig. 5. 514 (a) (b) 81.15 417 R6 R254.210 10 22 xy z ClampedPrescribed displacement Figure 5: Tensile sample nite element model: (a) geometry and dimensions in millimeters (thickness is 4 mm), (b) mesh and boundary conditions. The boundary conditions used in the nite element model are such that the right-side surface of the sample is clamped: 515 Ux(40:575;y;z;t ) =Uy(40:575;y;z;t ) =Uz(40:575;y;z;t ) = 0 URx(40:575;y;z;t ) =URy(40:575;y;z;t ) =URz(40:575;y;z;t ) = 0(29) and a uniform displacement is prescribed in the axial direction on the left-side surface of the sample: 516 Ux(40:575;y;z;t ) =U0 Uy(40:575;y;z;t ) =Uz(40:575;y;z;t ) = 0 URx(40:575;y;z;t ) =URy(40:575;y;z;t ) =URz(40:575;y;z;t ) = 0(30) 24 whereUandURare the displacement and rotation along a given direction (indicated by the subscript x,y, orz, see 517 Fig. 5) and U0is the magnitude of the prescribed displacement, which is taken to be between 15 mm and 20 mm, 518 depending on the sample's material (AISI 304L or AISI 316LN) and the testing temperature, in order to ensure that the 519 material's ultimate tensile strength is attained. 520 Moreover, a constant temperature is imposed in the nite element simulations: 521 T(x;y;z;t ) =T0 (31) whereT0is 300 K, 77 K, or 4 K. The analysis is isothermal, the sample's temperature is uniform in space and remains 522 constant over time. This assumption is based on the observation that under cryogenic conditions and low strain rates, the 523 heat released during the test (generated by the deformation or phase transformation) is rapidly removed by the cryogenic 524 coolant. The reader is referred to the work of Tabin et al. (2016, 2019), which shows local temperature rises in the form 525 of-Dirac function (temperature spikes) on tensile samples during testing at 4 K. 526 The e ective mechanical behavior of the material is modeled using the constitutive equations of Section 2.3 (i.e., the 527 homogenized depiction of the bi-phasic aggregate). Simulations are performed using the UMAT user subroutine described 528 in Section 3. The material is considered to be fully austenitic in the undeformed con guration. The martensitic transfor- 529 mation occurs as the loading proceeds. The sample has been meshed with 450 eight-node tri-linear brick elements (C3D8 530 in ABAQUS notation). A mesh convergence study has been performed in which the time evolution of di erent critical 531 output variables, namely stress, strain, volume fraction of martensite, and necking inception, were compared against 532 di erent mesh sizes. While we have found some mesh sensitivity in the numerical results, we have checked that it does 533 not signi cantly a ect our results, neither qualitatively nor quantitatively. The calculations have been carried out using 534 a single core of a laptop computer with an Intel(R) Core(TM) i5 8350U CPU @ 1 :70 GHz. The computational cost of a 535 simulation varied between 10 and 45 minutes, depending on the rate of the martensitic transformation (the calculations 536 are faster for cases showing a slow or nonexistent transformation) and on the homogenization scheme considered (the 537 Mori-Tanaka scheme was 10% faster than the Self-Consistent for the case with the maximum time di erence between 538 the two). Note that the computational cost is not signi cantly higher than in the unit-cell calculations using the same 539 homogenized model because the unit-cell calculations include cyclic loading simulations where the accumulated macro- 540 scopic e ective strain of the bi-phase material is more than three times higher than in the tensile simulations. The tensile 541 test calculations are to be compared with the experiments of Fern andez-Pis on et al. (2021) to check the ability of the 542 developed homogenized model to predict the martensitic transformation in AISI 304L and AISI 316LN steels subjected 543 to cryogenic temperatures (see Figs. 19 and 20). 544 25 6. Results 545 The calculations performed with the unit-cell model and the tensile sample are reported in Sections 6.1 and 6.2, 546 respectively. 547 6.1. Unit-cell calculations 548 The results obtained with the explicit representation of the martensite inclusions and the austenite matrix are 549 systematically compared to the calculations carried out using Mori-Tanaka and Self-Consistent homogenization schemes. 550 The calculations are performed with parameters corresponding to AISI 304L steel tested at T= 77 K, see Table 2. 551 Section 6.1.1 shows results for di erent values of stress triaxiality X + 0=0:5, 0:5, 1:5, and 3 and Lode parameter 552 L + 0=1, 0, and 1, for monotonic and cyclic loading, and for a constant volume fraction of martensite of F 0= 45%. 553 Section 6.1.2 includes calculations performed with two additional values of constant martensite volume fraction, F 0= 554 25% and 65%, for stress triaxiality X + 0= 0:5, and Lode parameter L + 0=1. Simulations with evolving volume 555 fraction of martensite for X + 0= 0:5 andL + 0=1 are shown in Section 6.1.3. Note that the calculations presented 556 in Sections 6.1.1 and 6.1.2 assess the predictive capacity of both homogenization models to capture the mechanical 557 response of unit-cells with an explicit representation of the martensitic microstructure. The mechanical behavior of the 558 individual phases in both the homogenization models and the simulations with microstructure representation is the same. 559 It is important to emphasize that there has not been any prior calibration of the homogenized models to align with the 560 mechanical response of the unit-cells featuring microstructure representation. 561 6.1.1. Salient results 562 Fig. 6 shows the evolution of the normalized macroscopic e ective stress =y0 with the macroscopic e ective strain 563 "for calculations with di erent microstructures with an explicit representation of the martensite inclusions: (a) single 564 central particle, (b) representative volume element of a FCC-symmetric spatial distribution of inclusions with the same 565 size and particles intersecting the boundaries, (c) FCC-symmetric spatial distribution of inclusions with the same size 566 and a boundary layer devoid of particles and (d) random spatial distribution of particles with di erent sizes. The 567 volume fraction of martensite is F 0= 45%. The choice of this high martensite volume fraction is motivated by the 568 fact that the model has been speci cally developed for cryogenic temperatures, during which certain grades show large 569 amount of transformed martensite. Note also that a microstructure with a random distribution of particles of the same 570 size (similar to the one shown in Fig. 4(d) for F 0= 25%) is not possible for a martensite content of F 0= 45%. 571 The results with explicit depiction of the microstructure are compared with the predictions of the Mori-Tanaka and 572 Self-Consistent homogenization schemes for the bi-phase material, the austenite phase, and the martensite phase (for 573 the homogenized model, a constant f 0= 45% is imposed). The idea is to assess the capability of the homogenized 574 26 models to describe the mechanical response of unit-cell simulations with explicit microstructure representation. The 575 calculations correspond to monotonic loading, with the stress triaxiality and Lode parameter being X + 0= 0:5 and 576 L + 0=1 (axisymmetric tension), respectively. The macroscopic e ective strain in the bi-phase material is imposed 577 to reach " + 0= 0:4, and the colored markers indicate the corresponding value of the macroscopic e ective strain in 578 the individual phases. Recall that the behavior of the individual phases is the same in both homogenization schemes 579 and in the simulations with microstructure representation, and that the behavior of the phases was determined based 580 on the tensile experiments presented in Fern andez-Pis on et al. (2021). Consequently, the comparison performed between 581 Mori-Tanaka and Self-Consistent predictions with unit-cell simulations including explicit representation of the austenite 582 and martensite microstructure does not entail any a priori calibration of the homogenization schemes. 583 As can be seen in Fig. 6, the predictions of the homogenization schemes for the e ective behavior of the bi-phase 584 material are very close to the results obtained with the four microstructures considered (single central particle, represen- 585 tative volume element of a FCC-symmetric spatial distribution of inclusions with the same size and particles intersecting 586 the boundaries, FCC-symmetric spatial distribution of inclusions with the same size and a boundary layer devoid of 587 particles, and random spatial distribution of particles); notably, the Self-Consistent model virtually overlaps with the 588 unit-cell calculations with explicit representation of the martensite inclusions. However, the partitioning of the strain 589 between the two phases calculated with the homogenization schemes shows important di erences with the simulations 590 with explicit representation of the microstructure since both Self-Consistent and Mori-Tanaka underestimate the strain 591 in the martensite and overestimate the strain in the austenite. The largest di erences are obtained with the Mori-Tanaka 592 model. For instance, for the calculation with a single central particle (see Fig. 6(a)), for a macroscopic e ective strain 593 in the bi-phase material of  " + 0= 0:4, the Self-Consistent and Mori-Tanaka models underestimate the strain in the 594 martensite by 20% and 45%, respectively, and overestimate the strain in the austenite by 14% and 27%. Note that, 595 since the martensite is the hard phase, the e ective strain is signi cantly smaller than in the austenite (as expected). In 596 fact, the larger strain in the martensite predicted by the Self-Consistent scheme explains the slightly higher stress in the 597 bi-phase material obtained by this scheme compared to the Mori-Tanaka scheme (the Self-Consistent  + 0=y0 " + 0 598 curve lies above the Mori-Tanaka curve). Moreover, note that, unlike in the original model of Garion and Skocze n (2002), 599 the strain increment in the bi-phase material does not match the strain increment in the austenite matrix. Note also that 600 the results shown in Figs. 6(b) and 6(c) for the representative volume element of an FCC-symmetric spatial distribution 601 of inclusions with the same size and particles intersecting the boundaries (Fig. 4(b)), and for the FCC-symmetric spatial 602 distribution of inclusions with the same size and a boundary layer devoid of particles (Fig. 4(c)), respectively, are virtually 603 the same. These two representations of the FCC-symmetric spatial distribution of inclusions are essentially equivalent. 604 Fig. 7 shows the evolution of the macroscopic e ective strain in the austenite  " and the martensite  " 0with the 605 27 (a) (b) (c) (d) Figure 6: Unit-cell nite element simulations. Monotonic loading. Material 304L, temperature T= 77 K, and volume fraction of martensite F 0= 45%. Comparison of Mori-Tanaka and Self-Consistent homogenization schemes with calculations corresponding to di erent microstructures: (a) single central particle, Fig. 4(a), (b) FCC-symmetric spatial distribution of inclusions with the same size and particles intersecting the boundaries, Fig. 4(b), (c) FCC-symmetric spatial distribution of inclusions with the same size and a boundary layer devoid of particles, Fig. 4(c), and (d) random spatial distribution of particles with di erent sizes, Fig. 4(e). Evolution of the normalized macroscopic e ective stress =y0 with the macroscopic e ective strain  ". Results corresponding to the bi-phase material, the austenite phase, and the martensite phase for a stress triaxiality and Lode parameter in the unit-cell equal to X + 0= 0:5 andL + 0=1, respectively. The colored markers indicate the value of the macroscopic e ective strain in the individual phases for a macroscopic e ective strain in the bi-phase material of  " + 0= 0:4. (For interpretation of the