Christine Poitou
email: christine.poitou-bernert@aphp.fr

Margot Denis

Zeina Chakhtoura

Catherine Uzan

Maryam Nikpayam

Jean-Michel Oppert

Just the tip of the iceberg: difficulties in assessing and managing

A 33-year-old woman with extreme obesity (weight 342 kg, body mass index = 100 kg/m²) was admitted to the emergency department with respiratory failure, limb edema and ascites. Her weight had increased over the last months with worsening of her general health status. As her corpulence limited physical examination and excluded standard imaging investigations, she was empirically treated with anticoagulants and antibiotics for a possible pulmonary embolism and pneumonia. She also received a loop diuretic and continuous positive airway pressure was initiated. After initial improvement of her respiratory status, she was transferred to a post-acute rehabilitation department. Ascites remained unchanged despite draining and diuretic treatment. After 5 months under medical supervision and following a controlled diet, she had lost a large amount of weight (-92 kg) but remained disabled, bedridden and needed assistance from 6 professionals for routine hygienic care.

She was referred to our obesity clinic for specialized nutritional care and etiological investigation of ascites. Ascites aspiration revealed a liquid exudate without bacterial pathogens. There was no evidence for hepatitis, cardiac or renal failure. Abdominal and pelvic intra-vaginal ultrasound performed at the patient's bed did not show any effusion in the Douglas pouch or behind the liver. However, we observed a very large abdominal mass of a multicystic nature. Because of technical difficulties, the ultrasonographer could not exclude or confirm an ovarianrelated disease. Although she had experienced substantial weight loss, the patient's body shape still prevented performing standard imaging, such as computed tomography (CT) or magnetic resonance imaging (MRI). A duodenal echo-endoscopy and a proctoscopy were performed and did not show any finding in favor of a gastrointestinal or pancreatic tumor. There were no sign of peritoneal carcinosis either. The marker CA 19-9 was increased (800 U/ml, N < 60) and CA125 was normal. Therefore, an ovarian tumor was the most probable diagnosis. Reviewing the case with the gynecological surgeon led to a decision to operate despite obvious surgical difficulties and a high-risk profile for anesthesia and surgical-related complications in this patient with extreme obesity.

Owing to further patient weight loss, a CT scan became feasible before surgery. It confirmed the presence of a large pelvic cystic mass measuring 40 × 24 × 40 cm, in contact with the left ovary (Fig. 1). The liver appeared normal. No peritoneal carcinosis was observed. Surgical removal of this mass was performed by laparotomy during a 6 h surgical procedure. Seventeen liters of gelatinous liquid were removed from the mass, which originated from the left ovary. The excised mass weighed 3 kg. Pathological examination concluded the mass was a mucinous borderline tumor without invasive implants. Borderline ovarian tumors (BOTs) account for 10-15% of epithelial tumors. Mucinous BOTs are the second most common tumor type (after serous) and account for 35 to 45% of BOTs. These tumors are usually large, unilateral, composed of multiple cystic spaces with variable diameter [START_REF] Morice | Borderline ovarian tumour: pathological diagnostic dilemma and risk factors for invasive or lethal recurrence[END_REF]. BOTs occur in women around the age of 40 years. The prognostic factors include characteristics of the tumor (stage, peritoneal implants), type of surgery, preoperative CA125, bilateral forms and age, but not obesity.

The major growth in size of that tumor was due, at least in part, to delayed care in an extreme form of obesity and was responsible for multiple severe complications. First, it was probably infected after multiple drains of ascites, and responsible for three episodes of meti-S Staphylococcus aureus sepsis. Second, the patient had several digestive symptoms likely due to mechanical compression from the tumor, which included esophagal reflux with erosive esophagitis, severe constipation with several sub-occlusive episodes and anorexia. These symptoms led to undernutrition with global edemas which required enteral nutrition over the course of a few months. The patient also exhibited severe anemia requiring several blood transfusions. Third, pelvic compression led to aggravation of lower limb venous insufficiency and lymphedema. This state of extreme obesity, associated with undernutrition and sarcopenia, was responsible for disability despite the young age of the patient.

After 7 months of hospitalization under specialized care, albuminemia was normalized and the patient's weight was 180 kg. She could sit alone and required only 2 people helping her with the toilet.

This case illustrates how extreme obesity may mask other comorbidities and highlights some of the difficulties encountered in assessment and management of severely obese patients. First, obese patients are vulnerable to multiple forms of weight bias and stigmatization, and this applies particularly to healthcare settings [START_REF] Phelan | Impact of weight bias and stigma on quality of care and outcomes for patients with obesity[END_REF]. Due to past negative experiences with health professionals, these patients may avoid or postpone medical care and present with more advanced and thus more difficult to treat conditions. They can also withdraw from full participation in proposed treatments with reduced adherence to treatment and counseling. In this context, even benign diseases can lead to severe complications and injuries. Second, corpulence exhibited during severe obesity poses a tremendous challenge to clinical diagnosis and treatment such as drug titration especially for anticoagulants and antibiotics but also for transportation and explorations including imaging [START_REF] Allison | Measurement challenges and other practical concerns when studying massively obese individuals[END_REF]. As such, conventional MRI and CT scan devices are limited by the gantry size and the table weight limits.

Given increasing obesity rates worldwide [START_REF]Trends in adult body-mass index in 200 countries from 1975 to 2014: a pooled analysis of 1698 population-based measurement studies with 19•2 million participants[END_REF], especially increasing rates of massively obese cases, and in order for these patients to receive adequate care in an equitable fashion [START_REF] Le | Obese patients and radiography literature: what do we know about a big issue?[END_REF], the healthcare delivery systems need to adapt and training/education in obesity management need to be developed for healthcare professionals. Adequate transportation and equipment, management and treatment need to be adapted to provide necessary resources at all levels of care to patients with morbid obesity.

Department of Breast and Gynecologic Surgery, AssistancePublique-Hôpitaux de Paris (AP-HP), Pitié-Salpêtrière Hospital, Paris, France

Inserm U938, Paris, France

Institut universitaire de cancérologie (IUC), Paris, France

Acknowledgements The authors would like to thank Dr T. Swartz for English editing. Fig. 1 CT scan of the pelvic region in a woman with extreme obesity. The white arrow indicates a large pelvic mass measuring 40 × 24 × 40 cm

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict of interest.


Takayuki Otsuka

Renald Brenner

Brigitte Bacroix

FFT-based modelling of

Keywords: Transformation Plasticity, Steel, FFT, Crystal Plasticity

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

During the thermomechanical treatments involved in steel processing, transformation plasticity or transformation induced plasticity (TRIP) occurs as a result of the variation of the volumic fraction of the phases during the solid state transformation. Two main mechanisms are classically invoked: (i) a displacive mechanism (i.e. Magee effect [START_REF] Magee | Transformation kinetics, Microplasticity and Aging of Martensite in Fe-31Ni[END_REF]) with a shape change during transformation and (ii) a diffusive mechanism (i.e. Greenwood-Johnson effect [START_REF] Greenwood | The Deformation of Metals under Small Stresses during Phase Transformations[END_REF]), implying nucleation and growth steps, with a volume change plastically accommodated [START_REF] Fischer | Transformation-induced plasticity(TRIP)[END_REF]. They both have been the subject of a number of experimental studies (see, among others [START_REF] Desalos | Comportement dilatométrique et mécanique des materiaux d'austénite métastable d'un l'acier A533[END_REF] [START_REF] Gautier | Plasticité de transformation durant la transformation perlitique d'un acier eutectoïde[END_REF][6] [START_REF] Petit-Grostabussiat | Experimental results on classical plasitcity of steels subjected to structural transformations[END_REF][8] [START_REF] Boudiaf | Experimental analysis of the correlation between martensitic transformation plasticity and the austenitic grain size in steels[END_REF] [START_REF] Holzweissig | In-situ characterization of transformation plasticity during an isothermal austenite-to-bainite phase transformation[END_REF]).

The phenomenon is known to play a central role during phase transformation in defining the final shape and residual stress state of heat treated materials. Phenomenological models have been developed and been used in the framework of finite element thermo-mechanical simulations [START_REF] Inoue | Simulation of Quenching Process of Carburized Steel Gear Wheel under Metallo-Thermo-Mechanical CouplingElsevier Science Publishers[END_REF] [START_REF] Ju | Modeling and experimental verification of martensitic transformation plastic behavior in carbon steel for quenching process[END_REF] [START_REF] Montalvo-Urquizo | Simulation of quenching involved in induction hardening including mechanical effects[END_REF].

The micromechanical modelling of transformation plasticity dates back to the pioneering theoretical work of Leblond et al. [START_REF] Leblond | A Theoretical and Numerical Approach to the Plastic Behaviour of Steels During Phase Transformations-I. Derivation of General Relations[END_REF] based on a rigorous homogenisation procedure. Few years later, an approximate analytical model has been derived from [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels -I: Case of deal-plastic phases[END_REF] [START_REF] Leblond | Mathematicl Modelling of Transformation Plasticity in Steels II: Coupling with Strain Hardening Phenomena[END_REF] to describe transformation plasticity due to the Greenwood-Johnson mechanism. Since then, developments have been suggested based on this approach [START_REF] Taleb | A micromechanical modeling of the Greenwood-Johnson mechanism in transformation induced plasticity[END_REF]. Besides, it is worth mentioning that a variety of mean-field models have been developed; see, for instance, [START_REF] Diani | Mircromechanical modelling of the transformation induced plasticity (TRIP) phenomenon in steels[END_REF][19] [START_REF] Fischlschweiger | A mean-field model for transformation induced plasticity including backstress effects for non-proportional loadings[END_REF]. Apart from these works, numerical micromechanical modelling has also been performed, making use of the finite-element method (FEM) to study diffusional transformations. With increasing complexity, numerical investigations have first considered the case of a two-phase material with plasticity, and various nucleation rules, 𝐽 2 [START_REF] Leblond | A Theoretical and Numerical Approach to the Plastic Behaviour of Steels During Phase Transformations-I. Derivation of General Relations[END_REF] [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels -I: Case of deal-plastic phases[END_REF][21] [START_REF] Barbe | Numerical modelling of the plasticity induced during diffusive transformation. Case of a cubic array of nuclei[END_REF] [START_REF] Barbe | Numerical modelling of the plasticity induced during diffusive transformation. An ensemble averaging approach for the case of random arrays of nuclei[END_REF], and more recently, the case of polycrystalline materials with crystalline plasticity at the slip system level, together with a microstructure described by a Poisson-Voronoi tessellation [START_REF] Barbe | A numerical modelling of 3D polycrystal-to-polycarystal diffusive phase transformations involving crystal plasticity[END_REF].

In the context of classical plasticity (i.e. without solid phase transformation), an efficient numerical scheme based on fast Fourier transform (FFT) [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF] has been successfully applied to a variety of problems and constitutive relations [START_REF] Lebensohn | Orientation image-based micromechanical modellilng of subgrain texture evolution in polycrystalline copper[END_REF][27] [START_REF] Lee | Modeling the viscoplastic micromechanical response of two-phase materials using Fast Fourier Transfroms[END_REF][29] [START_REF] Lebensohn | An elasto-viscoplastic formulation based on fast Fourier transforms for the prediction of micromechanical fields in polycrystalline materials[END_REF]. This alternative approach to FEM allows to consider large polycrystalline aggregates with reasonable CPU time and memory allocation. Its accuracy has been discussed by confronting with FEM simulation results [START_REF] Eisenlohr | A spectral method solution to crystal elasto-viscoplasticity at finite strains[END_REF]. Besides, meshing of the microstructure is not necessary as the computation is directly made on the digital image of the material (regular grid of pixels in 2D or voxels in 3D). These features are especially convenient to consider experimental microstructural data obtained by fine-scale EBSD or X-ray diffraction contrast tomography [START_REF] Ludwig | X-ray diffraction contrast tomography: a novel technique for three-dimensional grain mapping of polycrystals. i. direct beam case[END_REF][33] [START_REF] Grennerat | Experimental characterization of the intragranular strain field in columnar ice during during transient creep[END_REF]. By contrast to FEM, the FFT method is limited to a periodic homogenisation scheme which makes it less general.

In the present work, we first investigate the application of the FFT method in the context of plasticity induced by diffusional transformation. Following previous investigation, an enhanced numerical scheme is employed incorporating the pre-hardening effect, which induces the back stress and thus the anisotropic effect on transformation plasticity. Then, use is made of FFT reference results on representative polycrystalline aggregates for a critical analysis of two existing analytical micromechanical models [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels -I: Case of deal-plastic phases[END_REF] [START_REF] Taleb | A micromechanical modeling of the Greenwood-Johnson mechanism in transformation induced plasticity[END_REF]. These analytical models are compared for varying material data (transformation expansion coefficient).

Formulation of FFT-based numerical scheme

The FFT numerical scheme proposed by Moulinec and Souquet [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF] relies on the Green functions method to solve a periodic boundary value problem for heterogeneous media. For completeness, its formulation is first recalled. It offers a straightforward framework to consider stress-free strains related to the solid-state transformation process. Then, the constitutive local equations for crystalline plasticity and transformation kinetic are detailed.

2.1. Lippmann-Schwinger equation for periodic media 2.1. 1. Formulation In the case of a periodic boundary problem, the local displacement can be divided into fluctuation and average terms such that:

𝒖(𝒙) = 𝒖 ' (𝒙) + 𝜀𝒙 (1)
where, is a periodic displacement. The strain field reads: 𝑢 ' (𝑥) 𝜺(𝒖(𝒙)) = 𝜺(𝒖 ' (𝒙)) + 𝜀

with . 〈𝜺(𝒖'(𝒙))〉 = 0 For a heterogeneous elasto-plastic material, the rate-form of constitutive relation reads:

𝝈(𝒙) = 𝑪(𝒙):𝜺 𝑒 (𝒙) = 𝑪(𝒙):(𝜺(𝒙) -𝜺 𝑝 (𝒙) -𝜺 𝑚 (𝒙))

where is the elastic tensor at local position and , , and are elastic, total, plastic and 𝑪(𝑥) 𝑥 𝜺 𝑒 𝜺 𝜺 𝑝 𝜺 𝑚 transformation strain tensor respectively. Equivalently, the constitutive law can be rewitten:

𝝈(𝑥) = 𝑪(𝒙):𝜺(𝒙) + 𝝉(𝒙) (4)
By introducing homogeneous reference media with elasticity , equation ( 4) can be rewritten as: 𝑪 0 𝝈(𝒙) = 𝑪 0 :𝜺(𝒙) + (𝑪(𝒙) -𝑪 0 ):𝜺(𝒙) + 𝑪(𝒙):(𝜺 𝑝 (𝒙) + 𝜺 𝑚 (𝒙)) = 𝑪 0 :𝜺(𝒙) + 𝝉(𝒙) , , , ∀𝑥 ∈ 𝑉 div𝝈 = 0 ∀𝑥 ∈ 𝑉 𝑢 ' # 𝝈 ⋅ 𝒏 -# [START_REF] Gautier | Plasticité de transformation durant la transformation perlitique d'un acier eutectoïde[END_REF] where is a polarisation tensor which comprises the contributions from the elastic heterogeneity, 𝝉(𝒙) the plastic strain and transformation strains (eigenstrains). Solution of the problem reads:

. 𝜺 ( 𝒙 ) = 𝜀 -𝜞 𝟎 * 𝝉

FFT-based iterative algorithms have been proposed to solve equation [START_REF] Taleb | Experimental analysis of transformation plasticity[END_REF]. In the following, use has been made of the fixed-point method initially proposed by [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF].

Equations [START_REF] Gautier | Plasticité de transformation durant la transformation perlitique d'un acier eutectoïde[END_REF] in Fourier space are written as follows:

, 𝝈 ( 𝝃 ) = i𝑪 0 :(𝒖 ' (𝝃)⨂𝝃) + 𝝉(𝝃) i𝝈 ( 𝝃 ) ⋅ 𝝃 = 0 [START_REF] Petit-Grostabussiat | Experimental results on classical plasitcity of steels subjected to structural transformations[END_REF] where, is frequency, and the non-italic character represents imaginary number. By eliminating 𝝃 i 𝝈 from equation [START_REF] Petit-Grostabussiat | Experimental results on classical plasitcity of steels subjected to structural transformations[END_REF], we obtain

𝒖'(𝝃) = i 2 (𝑵 0 ⨂𝝃 + 𝝃⨂𝑵 0 )𝝉(𝝃) (8)
where,

, 𝑵 0 (𝝃) = 𝑲 0 (𝝃) -1 𝑲 0 (𝝃) = 𝑪 0 :(𝝃⨂𝝃) (9)
𝜺(𝝃) = i 2 (𝝃⨂𝒖 ' (𝝃) + 𝒖 ' (𝝃)⨂𝝃) =-𝜞 0 (𝝃):𝝉(𝝃) (10)
𝜞 0 (𝝃) = 1 4 (𝑁 0 𝑙𝑖 𝜉 𝑗 𝜉 𝑘 + 𝑁 0 𝑘𝑖 𝜉 𝑗 𝜉 𝑙 + 𝑁 0 𝑙𝑗 𝜉 𝑖 𝜉 𝑘 + 𝑁 0 𝑘𝑗 𝜉 𝑖 𝜉 𝑙 ) (11)
is a periodic Green's operator. Inverse Fourier transformation of equation [START_REF] Holzweissig | In-situ characterization of transformation plasticity during an isothermal austenite-to-bainite phase transformation[END_REF] gives the strain 𝜞 0 field within the heterogeneous medium.

The iterative algorithm to solve this problem is described below.

𝑒 𝑛 = 〈‖div(𝝈 𝑛 )‖ 2 〉 = 〈‖𝝃 ⋅ 𝝈 𝑛 (𝝃)‖ 2 〉 < 𝜖 otherwise 𝑒 𝑛 = 〈‖div(𝝈 𝑛 )‖ 2 〉 ‖〈𝝈 𝑛 〉‖ = 〈‖𝝃 ⋅ 𝝈 𝑛 (𝝃)‖ 2 〉 ‖𝝈 𝑛 (𝟎)‖ < 𝜖 (c) 𝜺 𝑛 + 1 (𝝃) = 𝜺 𝑛 (𝜉) -𝜞 0 (𝝃):𝝈 𝑛 (𝝃) ∀𝜉 ≠ 0 and 𝜺 𝑛 + 1 (0) = 𝜀 (d) 𝜺 𝑛 + 1 = 𝐹𝐹𝑇 -1 (𝜺 𝑛 + 1 ) (e) 𝝈 𝑛 + 1 (𝒙) = 𝑔(𝜺 𝑛 + 1 (𝒙)) ∀𝑥 ∈ 𝑉
where, is a constitutive equation which relates strain and stress (equation ( 3), in the present case). 𝑔

In the case where the macroscopic stress is imposed, the procedure (c) is modified as:

(c') .

𝜺 𝑛 + 1 (𝝃) = 𝜺 𝑛 (𝜉) -𝜞 0 (𝝃):𝝈 𝑛 (𝝃) ∀𝜉 ≠ 0 and 𝜺 𝑛 + 1 (0) = 〈𝜺 𝑛 〉 + 𝑪 𝟎 -1 :(𝝈 𝑛 -〈𝝈 𝑛 -1 〉) Plastic strain is the result of dislocation glide. The slip occurs along certain directions on certain planes. Such combinations of directions and planes are called slip systems. Let be the slip rate on 𝛾 𝛼 the slip system. The plastic strain rate is the determined by the sum of slip rates on all slip 𝛼 systems:

𝜺 𝑝 = ∑ 𝛼 𝒑 𝛼 𝛾 𝛼 (12)
where is the Schmid tensor of slip system. 𝒑 𝛼 𝛼

𝒑 𝛼 = 1 2 (𝒔 𝛼 ⨂𝒎 𝛼 + 𝒎 𝛼 ⨂𝒔 𝛼 ) (13)
and are respectively the slip direction and slip plane normal of the slip system. According 𝒔 𝛼 𝒎 𝛼 𝛼 to the Schmid law, the slip system becomes active when the resolved shear stress becomes equal 𝜏 𝛼 to the Critical Resolved Shear Stress (CRSS) :

𝑔 𝛼 |𝜏 𝛼 | = |𝒑 𝛼 :𝝈| = 𝑔 𝛼 (14)
During elasto-plastic deformation, the equation ( 14) is satisfied. In other words, stress increases according to the hardening of each slip system.

For such a rate-independent problem, Hutchinson [START_REF] Hutchinson | Elastic-plastic behaviour of poly crystalline metals and composites[END_REF] proposed the following solution for determining the slip rate of each slip system by using the consistence condition:

𝜏 𝛼 = sgn(𝜏 𝛼 )𝑔 𝛼 (15)
For the small deformation problems, equation ( 15) can be rewritten as:

𝝈:𝒑 𝛼 = ∑ 𝛽 ℎ 𝛼𝛽 𝛾 𝛽 (16)
and the general form of the hardening law is:

𝑔 𝛼 = ∑ 𝛽 ℎ 𝛼𝛽 |𝛾 𝛽 | (17)
with the hardening coefficients. They can be expressed phenomenologically with an additional ℎ 𝛼𝛽 evolution law. We adopt the following form [START_REF] Peirce | An analysis of non-uniform and localized deformation in ductile single crystals[END_REF]:

ℎ 𝛼𝛼 = ℎ = 𝐻 0 sech 2 [ 𝐻 0 ∑ 𝛽 𝛾 𝛽 𝜏 𝑠 -𝜏 0 ] (18)
ℎ 𝛼𝛽 = 𝑞ℎ + (1 -𝑞)ℎ𝛿 𝛼𝛽 (19)
where is a coefficient parameter, is the initial yield stress value and is the saturated stress 𝐻 0 𝜏 0 𝜏 𝑠 value. The parameter , which describes latent hardening, takes values from 1.0 to 1.4 depending on 𝑞 the material.

By using the constitutive laws, one obtains.

𝝈:𝒑 𝛼 = 𝑪:(𝜺 -𝜺 𝑝 -𝜺 𝑡ℎ -𝜺 𝑚 ):𝒑 𝛼 = 𝑪:(𝜺 -𝜺 𝑡ℎ -𝜺 𝑚 ):𝒑 𝛼 -∑ 𝛽 𝒑 𝛼 :𝑪:𝒑 𝛽 𝛾 𝛽 = ∑ 𝛽 ℎ 𝛼𝛽 𝛾 𝛽 (20)
and the slip rates are expressible as:

with

𝛾 𝛼 = 𝒇 𝛼 :(𝜺 -𝜺 𝑡ℎ -𝜺 𝑚 ) 𝒇 𝛼 = ∑ 𝛽 𝑌 𝛼𝛽 𝑪:𝒑 𝛽 (21)
with and

𝑌 𝛼𝛽 = (𝑋 𝛼𝛽 ) -1 𝑋 𝛼𝛽 = ℎ 𝛼𝛽 + 𝒑 𝛼 :𝑪:𝒑 𝛽 (22)
The matrix has to be non-singular. For that reason, the maximum number of active slip systems, 𝑋 𝛼𝛽 i.e. rank of the matrix , is limited to 5. From equation [START_REF] Ganghoffer | Finite element calculation of the micromecanics of a diffusional transformation[END_REF], we can find the value of the shear 𝑋 𝛼𝛽 strain rates, which in turn depends on the prescribed strain rate or stress rate.

For the elasto-plastic transition problem, initial values of and are generally different. The 𝜏 𝛼 𝑔 𝛼 incremental form of equation ( 15) during a finite time increment can be written as follows:

𝜏 𝛼 + Δ𝜏 𝛼 = sgn(𝜏 𝛼 )(𝑔 𝛼 + Δ𝑔 𝛼 ) (23)
Note that and are the values evaluated at the previous step. represents one increment during 𝜏 𝛼 𝑔 𝛼 Δ one time step.

Consequently, equation ( 16) takes the form:

(𝝈 + Δ𝝈):𝒑 𝛼 = sgn(𝜏 𝛼 )𝑔 𝛼 + ∑ 𝛽 ℎ 𝛼𝛽 Δ𝛾 𝛽 (24)
Following the previous discussion, the plastic slip increments can be expressed as (see Appendix A):

Δ𝛾 𝛼 = ∑ 𝛽 𝑌 𝛼𝛽 (𝝈:𝒑 𝛽 + 𝒑 𝛽 :𝑪:(Δ𝜺 -Δ𝜺 𝑡ℎ -Δ𝜺 𝑚 ) -sgn(𝜏 𝛽 )𝑔 𝛽 ) (25)
If the resolved stress equates CRSS at the previous step, then the equation ( 25) becomes exactly equivalent to as equation ( 21).

Microstructures

A cubic area volume with regularly allocated calculation points (voxels) is divided into cells (grains).

This procedure is realised by using the Voronoi tessellation method. First, grain seeds are spread inside the cube. The number of the grains is thus fixed at the beginning of the calculation. Then, the surfaces which have same distances from two neighbouring grain seeds are considered to be flat grain boundaries between these two grains. All voxels which are included into one grain have the same crystallographic orientation. Hereafter, initial grain seeds and initial crystallographic orientations of each grain are randomly distributed.

The heterogeneity of the polycrystal then derives solely from the crystalline orientation distributions together with the local anisotropy of the single crystal behaviour.

Phase transformation model

Phase transformation is considered here to be diffusive, isotropic and isothermal. As a consequence, there are no thermal strains within the polycrystal. This simplifies the problem and provides the basic results of transformation plasticity. In accordance with the numerical homogenisation scheme with periodic displacement boundaries, the periodicity of the phase transformation has to be imposed.

One example is shown in Fig. 1 (b); the 100 initial grains and 100 growing new grains (red coloured circles).

A finite intermediate zone between parent and daughter phases, where the volume fraction of new phase takes value of , is defined at the circumference of the daughter phase particles. 0 ≤ 𝜉 ≤ 1 Within this intermediate zone, the volume fraction of the daughter phase obeys KJMA (Kolmogolov-Johnson-Mehl-Avrami) type equation [START_REF] Arimoto | Heat treatment simulations and their validation[END_REF]:

𝜉 = 1 -exp ( -𝑏𝑡 𝑛 ) (26)
where and are temperature dependent material parameters defining the intermediate zone 𝑏 𝑛 thickness. When the frontier of a new grain reaches a calculation point, the value of time in 𝑡 equation ( 26) is set to be zero and the transformation begins according to the equation.

When the volume fraction exceeds 50%, the constitutive behaviour of the daughter phase is considered to be transformed and being a new phase. At the same time, the crystallographic orientation is changed according to Kurdjumow-Sachs (K-S) relation [START_REF] Cabus | Etude et modélisation des textures de transformations de phases dans les aciers destinés à l'emboutissage[END_REF]. There are 24 possible variants for the K-S relation; in the present calculation, one of these 24 variants is randomly selected for the new grain.

The calculations start with 100 % parent grains which are subject to a constant uniaxial stress loading along the z-axis. The constant applied stress value takes values between -90 MPa and 90

MPa is held until phase transformation is complete, i.e. the calculation reaches 100 % of daughter phase throughout the cubic region. with the exact theoretical result of Leblond [START_REF] Leblond | A Theoretical and Numerical Approach to the Plastic Behaviour of Steels During Phase Transformations-I. Derivation of General Relations[END_REF] for an isotropic polycrystalline aggregate undergoing phase transformation without macroscopic applied stress.

In this particular case, it has been shown that the overall plastic strain is necessarily null. As a consequence, the exact macroscopic strain of the polycrystal reads

𝜀 = 𝜀 𝑚 = 𝛽𝜉 (27)
with , the macroscopic volume fraction of daughter phase. The results obtained for an austenitic 𝜉 polycrystal composed of 100 grains with random orientations are shown in Fig. 2. Note that this case uses the same elastic constants for each phase. The numerical response agrees with the exact result.

The chosen numerical microstructure can thus be considered isotropic. It is noted that there is no macroscopic plastic strain despite the plastic deformation occurs at the crystal (microscopic) level. Evolution of the macroscopic strain with the volume fraction of transformed phase (no applied stress, isotropic polycrystal).

Calculation of transformation plasticity

In the present study, the mechanical behaviour of carbon steel undergoing austenite-pearlite phase transformation will be investigated. The set of plasticity parameters for this steel grade is shown in Table 1. The macroscopic stress/strain relation of austenite phase and pearlite phase are calculated by using FFT model and are depicted in Fig. 3. The selected material parameters have been adjusted

in order to get a good fit between FFT calculations and experimental data (the experimental data corresponding to the austenite phase are also shown in Fig. 3) In order to analyse thoroughly each model, two cases of transformation expansion coefficients will be considered ( and ). The new phase grain seeds are spread

𝛽 = 1.5 × 10 -3 𝛽 = 4.4 × 10 -3
randomly and consequently, subsequent isotropic grain growth starts.

Case of small transformation expansion ( )

𝛽 = 1.5 × 10 -3
The stress-strain calculation during phase transformation is performed in a reference volume element with periodic boundaries. In this procedure, the equivalent plastic strain accumulation is pronounced at the vicinity of the daughter phase as demonstrated in Fig. 4.

The evolution of the average volume fraction of the new phase under stress free condition is shown in Fig. 5 (a). Within the period of phase transformation, the strain values change according to the magnitude of applied stress (transformation plastic strain). This additional strain effect (transformation plasticity) is depicted in Fig. 5 (b). As the applied stress value increases, the resulting total strain value also increases. As seen below, transformation plasticity is naturally obtained at the macroscopic scale during phase transformation as a result of the scale-transition numerical approach. The transformation plastic strain after phase transformation is plotted in Fig. 6 as a function of the applied stress. One can observe in the Fig. 6 that there is indeed a linear relation between applied stress and transformation plastic strain when applied stress is less than 60 MPa. This relation agrees with the predictions of the models by Greenwood-Johnson [START_REF] Greenwood | The Deformation of Metals under Small Stresses during Phase Transformations[END_REF], Leblond [START_REF] Leblond | A Theoretical and Numerical Approach to the Plastic Behaviour of Steels During Phase Transformations-I. Derivation of General Relations[END_REF] or Inoue [START_REF] Inoue | Simulation of Quenching Process of Carburized Steel Gear Wheel under Metallo-Thermo-Mechanical CouplingElsevier Science Publishers[END_REF]. The factors of proportionality (transformation plasticity coefficient) are equal to 𝐾 + 𝑃 = 2.20 × 10 -5 (MPa -1 ) and (MPa -1 ), where the superscripts "+" and "-" denote transformation 𝐾 - 𝑃 = 2.15 × 10 -5 plasticity under tensile applied stress and compressive applied stress respectively. Thus, there is no significant difference between tensile and compressive applied stresses in the present case. To make a quantitative comparison with Leblond's model, the transformation plasticity coefficient by Leblond's model is (MPa -1 ); a good agreement between the two approaches. We 𝐾 𝑃 = 2.17 × 10 -5 can then conclude that the FFT model is able to predict correctly the transformation plasticity due to a diffusive Greenwood-Johnson process.

In addition to the proportional relationship between applied stress and transformation plastic strain, the results by FFT predict satisfactorily the experimental evidence of non linearity between applied stress and transformation plastic strain when applied stress is relatively large.

Case of large transformation expansion (

)

𝛽 = 4.4 × 10 -3
The large transformation strain induced plasticity is analysed. All the conditions except the transformation strain are the same as those used for the small transformation expansion case. Fig. 7 is the transformation plastic strain calculated by FFT numerical model depending on the applied constant stresses. From the obtained data, the transformation plastic coefficient is (MPa -1 ) for 𝐾 + 𝑃 = 8.44 × 10 -5 tensile and (MPa -1 ) for compressive applied stress. There is a slight discrepancy 𝐾 - 𝑃 = 9.38 × 10 -5 between transformation plastic coefficients under tensile and compressive stresses but the difference is below 10 % even in the large transformation expansion case. The nonlinearity behaviour for large applied stress is found to be more pronounced than in the small transformation expansion case.

Pre-deformation effect on transformation strains

Heat treatment process in steels is often accompanied with pre-deformation, such as hot rolling and hot forging. Hence, the effect of pre-deformation is inherently linked to the precise shape and residual stress analyses. As a basic analysis, let us conduct a uniaxial tension simulation followed by phase transformation introducing back stress effect on each slip system for the following simulations, such that

𝜏 * (𝛼) = 𝜏 (𝛼) -𝑎 (𝛼) (28)
where is effective resolved shear stress and is back stress value on the slip system. Now 𝜏 * (𝛼) 𝑎 (𝛼) 𝛼 let be Armstrong-Frederick type back stress model [START_REF] Frederick | A mathematical representation of the multiaxial Bauschinger effect[END_REF], such that:

𝑎 (𝛼) 𝑎 (𝛼) = 𝐶 1 (𝐶 2 𝛾 (𝛼) -𝑎 (𝛼) |𝛾 (𝛼) |) (29)
where and are material constants. 𝐶 1 𝐶 2

A uniaxial tensile calculation up to 10 % strain of polycrystalline aggregates is conducted followed by phase transformation calculation under applied constant stress. This sequence is depicted in Fig. 8.

At 1000 step (10 % strain), the unloading process to a prescribed stress is calculated and then the phase transformation calculation is initiated. The obtained transformation strains under free applied stress are demonstrated in Fig. 9. A remarkable diverge of (pre-tensile direction) from the other strain components is observed. The 𝜀 𝑧𝑧 strain takes minus value at the beginning of transformation even though the transformation is an 𝜀 𝑧𝑧 expanding process. The other two components take the same larger value. Obviously, this diverge is controlled by the classical plastic strain and thus the total transformation strain (volume change) keeps constant from the input value. In fact, 𝜀 𝑚 𝑖𝑖 = 𝜀 𝑚 11 ( = 0.002146) + 𝜀 𝑚 22 ( = 0.002153) + 𝜀 𝑚

33

. The mechanism of this remarkable phenomenon ( = 0.000296) = 0.004595≅3 × 𝛽( = 0.001536) can be given as follows.

During pre-tension, the yield surface moves, and thus material gets hardened for the loading direction. By contrast, the material is softer after deformation for the opposite direction exhibiting anisotropy. To accommodate the volume difference between parent and daughter phases, as we have seen in the previous section, plastic strain occurs at the vicinity of the daughter phase. If the material is isotropic, this plastic strain gives no apparent macroscopic strain. However, it is not applicable for the anisotropic materials. The present case, the anisotropy in the RVE is induced by the pre-tensile calculation. Therefore, it is hard to deform in the tensile direction resulting in the small (even negative because of the deformation in the other directions) plastic strain yielding large plastic strain in the other directions (because the incompressive plastic deformation). This phenomenon can also be found in the experimental works by Taleb et al. [START_REF] Holzweissig | In-situ characterization of transformation plasticity during an isothermal austenite-to-bainite phase transformation[END_REF] in bainitic phase transformation in 16MND5 steel. The experimental results appear to be very similar to the calculations. Above explanation is also valid for the pre-compressed case. To confirm this, -10 % pre-compressed followed by unloading and phase transformation calculations are carried out. The calculated results of free dilatation are shown in Fig. 10. It is observed that the total strain in z direction is larger than those of the other directions even though the local transformation strain is always isotropic. Again, the total volume change remains the same value with that of non-hardening condition. In analogy with the previous calculations, following pre-hardening, strains under several applied stresses during phase transformation are calculated. In this case, the transformation plastic strain depending on the applied stress is plotted in Fig. 11. For small applied stresses (smaller than 60 MPa), the proportional factors (transformation plastic coefficients) of each case are given by follows.

Pre-tension : MPa -1 , MPa -1 𝐾 + 𝑃 = 2.12 × 10 -5 𝐾 - 𝑃 = 1.17 × 10 -5 Pre-compression:

MPa -1 , MPa -1 𝐾 + 𝑃 = 1.36 × 10 -5 𝐾 - 𝑃 = 2.01 × 10 -5 The superscript of "+" represents positive applied constant applied stress during phase transformation, and "-" represents negative applied stress. It is found that the linear relation between applied stress and transformation plastic strain is kinked because of the pre-hardening effect. Leblond has developed an analytical model which allows to calculate the plastic strain state of the parent phase which surrounds a spherical inclusion of the daughter phase [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels -I: Case of deal-plastic phases[END_REF]. According to the Greenwood-Johnson mechanism, parent phase crust expansion by phase transformation induces some plastic strain in the surrounding parent phase. Leblond expressed the transformation plastic strain rate as a function of the volume fraction change, neglecting plastic strain in the daughter phase, such that,

Assessment of existing analytical models

𝜀 𝑡𝑝 = (1 -𝜉) 〈 𝛿𝜀 𝑝 1 𝛿𝜉 〉 𝑉 1 𝜉 (30)
where is equivalent plastic strain rate caused by transformation plasticity and 𝜀 𝑡𝑝 〈𝛿𝜀 𝑝 1 /𝛿𝜉〉 𝑉 1 represents the average value within the parent phase of the ratio of plastic strain increment due to volume fraction change. Starting with the equation [START_REF] Lebensohn | An elasto-viscoplastic formulation based on fast Fourier transforms for the prediction of micromechanical fields in polycrystalline materials[END_REF], it is assumed that for small or moderately high applied stresses, the parent phase is entirely elastoplastic whereas the daughter phase remains elastic (or its plastic strain rate remains always much smaller than that of the parent phase).

Using Prandtl-Reuss relation, the equation ( 30) can be rewritten as:

𝜀 𝑡𝑝 = 3(1 -𝜉) 2𝜎 𝑦 1 〈 𝛿𝜀 𝑒𝑞 1 𝛿𝜉 〉 𝑉 1 𝑆𝜉 (31)
where is the applied macroscopic deviatoric stress value. The total equivalent strain increment due 𝑆 to volume change can be obtained by neglecting external stress effects

〈 𝛿𝜀 𝑒𝑞 1 𝛿𝜉 〉 𝑉 1 =-2𝛽 1 -𝜉 ln (𝜉) (32)
Finally we obtain

𝜺 𝑡𝑝 = { 𝟎 if 𝜉 ≤ 0.03 ( -2𝛽 𝜎 1 𝑦 ) ln (𝜉)𝜉 3 2 𝒔 if 𝜉 > 0.03 (33)
The total value of transformation plastic strain under constant applied stress is

𝜺 𝑡𝑝 = 1 ∫ 0.03 ( -2𝛽 𝜎 1 𝑦 ) ln (𝜉)𝜉 3 2 𝒔𝑑𝜉≅0.8648 ( -2𝛽 𝜎 1 𝑦 ) 3 2 𝒔 ( 34
)
This model has been shown by Leblond to estimate correctly the transformation plastic strain in comparison with the experimental results. An extended model to strain hardening materials is suggested [START_REF] Leblond | Mathematicl Modelling of Transformation Plasticity in Steels II: Coupling with Strain Hardening Phenomena[END_REF], namely,

𝜺 𝑡𝑝 = { 𝟎 if 𝜉 ≤ 0.03 ( -2𝛽 𝜎 1 𝑦 ) ln (𝜉)𝜉 3 2 𝒔 * if 𝜉 > 0.03 (35)
where is a deviatoric effective stress; the deviatoric back stress is extracted from the deviatoric 𝒔 * stress.

As we have seen above, transformation plasticity models are often expressed as a linear relationship between applied stress and transformation plastic strain. This is confirmed by many experimental works [START_REF] Inoue | Simulation of Quenching Process of Carburized Steel Gear Wheel under Metallo-Thermo-Mechanical CouplingElsevier Science Publishers[END_REF][40] [START_REF] Otsuka | Effect of Volume Expansion on Transformation Plasticity during Ferrite and Martensite Transformation of Steel[END_REF]. However, when the applied stress exceeds a certain threshold value, the linear relation does not hold anymore. The threshold value is often set equal to be a half of the value of yield stress value of weaker phase [START_REF] Greenwood | The Deformation of Metals under Small Stresses during Phase Transformations[END_REF] [START_REF] Videau | Experimental study of the transformation induced plasticity in a Cr-Ni-Mo-Al-Ti steel[END_REF]. To the authors' knowledge, no theoretical or numerical explanation of this threshold value has been provided yet. However, a phenomenological modelling of the transformation plasticity including non-linear regime has been given by Leblond [START_REF] Leblond | Mathematicl Modelling of Transformation Plasticity in Steels II: Coupling with Strain Hardening Phenomena[END_REF]:

𝜺 𝑡𝑝 = { 0 if 𝜉 ≤ 0.03 ( -3𝛽 𝜎 1 𝑦 ) ℎ ( 𝜎 𝑒𝑞 𝜎 𝑢 ) ⋅ ln (𝜉)𝜉𝒔 if 𝜉 > 0.03 (36)
with

ℎ ( 𝜎 𝑒𝑞 𝜎 𝑢 ) = { 1 if 𝜎 𝑒𝑞 𝜎 𝑢 ≤ 1 2 1 + 3.5 ( 𝜎 𝑒𝑞 𝜎 𝑢 - 1 2 ) if 𝜎 𝑒𝑞 𝜎 𝑢 > 1 2 (37)
where

𝜺 𝑡𝑝 = { ( -2𝛽 𝜎 1 𝑦 ) ln (𝜉 𝑙 )𝜉 3 2 𝒔 if 𝜉 ≤ 𝜉 𝑙 ( -2𝛽 𝜎 1 𝑦 ) ln (𝜉)𝜉 3 2 𝒔 if 𝜉 > 𝜉 𝑙 (38)
with

𝜉 𝑙 = 𝜎 𝑦 1 2𝛽 4𝜇 + 3𝐾 9𝐾𝜇 (39)
where and are bulk and shear elastic moduli respectively. Thus, Taleb's model does not restrict 𝐾 𝜇 the parent phase to be entirely elastoplastic until the volume fraction of daughter phase reaches the value indicated by equation [START_REF] Frederick | A mathematical representation of the multiaxial Bauschinger effect[END_REF]. Consequently, Taleb's model is expected to be more accurate during the beginning of phase transformation. After the volume fraction of daughter phase becomes , this model follows Leblond's model. 𝜉 𝑙

Comparison of the analytical models with FFT numerical results

First, transformation plastic strain for the small transformation expansion case (

) by 𝛽 = 1.5 × 10 -3 analytical solutions under several applied stresses is plotted and confronted with FFT numerical results in Fig. 12. Note that the line "Leblond" signifies Leblond's linear equation [START_REF] Belkhabbaz | Prediction of the overall behavior of a 3D microstructure of austenitic steel by using FFT numerical scheme[END_REF] and "Leblond-Mod." is non-linear equation [START_REF] Hutchinson | Elastic-plastic behaviour of poly crystalline metals and composites[END_REF]. "Leblond-Mod." is an enhanced "Leblond" model comprises the nonlinearity under high external stress. All these models predict approximately the same transformation plastic strain values when applied stress is relatively small, whilst Leblond's modified model overestimates transformation plastic strain at large applied stresses. The mechanism of this nonlinearity will be discussed in the later section. Next discussion is on the kinematic evolution of strain during phase transformation under applied stress. Fig. 13 shows the comparison of the evolution curves of the transformation plastic strain predicted by FFT, Leblond's model [START_REF] Belkhabbaz | Prediction of the overall behavior of a 3D microstructure of austenitic steel by using FFT numerical scheme[END_REF] and Taleb model [START_REF] Cabus | Etude et modélisation des textures de transformations de phases dans les aciers destinés à l'emboutissage[END_REF]. According to Fig. 13, Taleb's model and FFT results agree at the beginning of phase transformation process. By contrast, Leblond's model overestimates the transformation plastic strain at this stage especially under small applied stresses. This agreement between Taleb's model and FFT results at the beginning of transformation can be explained by the Taleb's modification which considers that the parent phase is not entirely elastoplastic at the beginning of phase transformation. The modification is rather important when transformation expansion is relatively small. Both Leblond and Taleb's models underestimate the transformation plastic strain during the second half of phase transformation. This seems to be caused by plastic deformation of the daughter phase which is neglected by the analytical models. Thus, it is counterbalanced by an underestimation of the transformation plastic strain during the final stage for Leblond's model. On the other hand, for Taleb's model, it underestimates whole transformation plastic strain because of the underestimation at the final stage (and due to precise estimation at its initial stage) caused by the inappropriate assumption of elastic daughter phase as seen below.

Effect of plastic deformation in daughter phase

To enlighten the each contribution to transformation plasticity by parent and daughter phases, equivalent plastic strain of each phase is separately shown in Fig. 14 The equivalent plastic strain in parent phase takes maximum value when approximately 50 % transformed. After 50 %, the equivalent plastic strain in parent phase decreases due to the decrease in volume fraction. The non-negligible strain in daughter phase accumulates in conjunction with the strain decrease in parent phase. Therefore, neglect of the strain in daughter phase is obviously a major agent for the underestimation at the end of the phase transformation.

These discussions lead a conclusion that the underestimation of Leblond's and Taleb's models in the latter half of phase transformation derives from the assumption of elastic daughter phase. To confirm this analysis, FFT numerical simulations with elastic daughter phase (infinite yield stress) are carried out. The comparison among FFT, Leblond's equation ( 33) and Taleb's equation [START_REF] Cabus | Etude et modélisation des textures de transformations de phases dans les aciers destinés à l'emboutissage[END_REF] under several applied stresses is shown in Fig. 15. again there is no significant difference between tensile and compressive stress conditions.

Comparing to the daughter phase elastoplastic case, the transformation plasticity coefficient value is about 20 % smaller for daughter phase elastic case. This appears also to be in quantitative agreement with the discrepancy in total transformation plastic strain between FFT numerical model and Taleb's model. This proved that plasticity in daughter phase should be incorporated especially when the daughter phase is relatively soft as ferrite phase. Relation between applied stress and transformation plastic strain (pearlite: elastic case).

Mechanism that causes nonlinearity under high applied stress

When the applied stress is large, both Leblond's and Taleb's model underestimate transformation plastic strain as shown in Fig. 13 (c) and (d). In these cases, the linear relation between transformation plastic strain and applied stress is never satisfied (Fig. 6). The reason of this nonlinearity is investigated here by FFT computations.

Many models, such as Greenwood-Johnson [START_REF] Greenwood | The Deformation of Metals under Small Stresses during Phase Transformations[END_REF] or Inoue [START_REF] Inoue | Unified Transformation-Thermoplasticity and the Application (in Japanese)[END_REF], predict a linear relationship between applied stress and transformation plastic strain. Therefore, they are valid only for small applied stress conditions. As previously discussed, experimental results show that when the applied stress exceeds half the yield stress of the parent phase, nonlinearity occurs.

Leblond has analytically modelled transformation plasticity according to the Greenwood-Johnson mechanism [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels -I: Case of deal-plastic phases[END_REF]. This model contains 5 main hypotheses which are recalled here.

H1. The microscopic elastic compliance tensor may be equated to the macroscopic overall elastic compliance tensor.

H2. For small or moderately high applied stresses, the austenitic phase is entirely plastic, but the 𝛼 phase remains elastic or its plastic strain rate remains always much smaller than that of the 𝛾 phase.

H3. Both phases are ideal-plastic and obey the von Mises criterion and the Prandtl-Reuss flow rule.

H4. Correlations between and can be neglected.

𝛿𝜀 𝑒𝑞 1 /𝛿𝑧 𝑠 1
H5. For small applied stresses, the average stress deviator in phase 1 is almost equal to the overall average stress deviator.

We now intend to check which assumptions remain valid, and which are not, for the high applied stress regime.

Hypothesis 1 is fulfilled because local elastic anisotropy does not significantly affect transformation plasticity.

Concerning hypothesis 2, Fig. 17 shows a probability density distribution of equivalent plastic strain when 50 % transformed state calculated by FFT (64 3 voxels, 100 grains). The results show that most voxels remain elastic, which is against the hypothesis. The larger the applied stress, the more voxels deform plastically. It means that even though there are many parts which remain elastic, this hypothesis cannot be the reason for the nonlinearity under high applied stress. (c) 70MPa (d) 90MPa

Fig. 18 Effect of correlation between (1) and ( 2) at several applied stress.

𝛿𝜀 𝑒𝑞 1 /𝛿𝑧 𝑠 1
According to Fig. 18, it is clear that correlation between and can be neglected. Thus, the 𝛿𝜀 𝑒𝑞 1 /𝛿𝑧 𝑠 1 hypothesis 4 is well verified. In fact, as shown in Fig. 19, the resultant transformation plastic strains calculated with H.4 and without H.4 do not show substantive difference. This verification has also been performed by Leblond et al. [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels -I: Case of deal-plastic phases[END_REF] by using FEM with J 2 plasticity. The maximum difference between average stress of phase 1 and overall stress until 50 % phase transformation is less than 7 %. Hence, the hypothesis 5 is likely to be valid. To confirm this, the transformation plastic strain values with several applied stress are shown in Fig. 21; 〈𝛿𝜀 𝑒𝑞 1 /𝛿𝑧〉 𝑉 1 • without H.5 and with H.5. In this case, is calculated by using

〈𝑠 1 〉 𝑉 1 〈𝛿𝜀 𝑒𝑞 1 /𝛿𝑧〉 𝑉 1 • 𝑆 〈𝛿𝜀 𝑒𝑞 1 /𝛿𝑧〉 𝑉 1
analytical solution given by equation [START_REF] Ludwig | X-ray diffraction contrast tomography: a novel technique for three-dimensional grain mapping of polycrystals. i. direct beam case[END_REF]. As shown in Fig. 21, the hypothesis 5 is verified. In contrast to Fig. 19, where one can observe the nonlinearity between applied stress and transformation plastic strain, Fig. 21 shows almost linear relation between them; the difference seems to arise from whether the term is calculated by FFT or by analytic solution given

〈𝛿𝜀 𝑒𝑞 1 /𝛿𝑧〉 𝑉 1
by equation [START_REF] Ludwig | X-ray diffraction contrast tomography: a novel technique for three-dimensional grain mapping of polycrystals. i. direct beam case[END_REF].

Fig. 22 shows the transformation plastic strains calculated by equation [START_REF] Eisenlohr | A spectral method solution to crystal elasto-viscoplasticity at finite strains[END_REF] in which the term .is given by analytical solution using equation [START_REF] Ludwig | X-ray diffraction contrast tomography: a novel technique for three-dimensional grain mapping of polycrystals. i. direct beam case[END_REF] (1. circle symbol) and FFT solution

〈𝛿𝜀 𝑒𝑞 1 /𝛿𝑧〉 𝑉 1
(2. cross symbol). An important discrepancy is observed. Fig. 22 Difference between analytical solution (circle) and FFT (cross) for .

〈𝛿𝜀 𝑒𝑞

1 /𝛿𝜉〉 𝑉 1

From the discussion above, it can be concluded that the estimation of term is 〈𝛿𝜀 𝑒𝑞 1 /𝛿𝑧〉 𝑉 1 responsible for the nonlinearity under high applied stress condition. In addition, it shows that the analytical solution given by equation ( 32) is accurate for small applied stress but it is not valid under high applied stress value.

The reason why equation ( 32) is invalid under high applied stress is that the equivalent strain increment is supposed to be initiated only by transformation expansion. The assumption is 𝛿𝜀 𝑒𝑞 (𝑥) valid under small applied stress but applied stress effect on equivalent strain increment is no longer negligible when applied stress is large.

3.3. Application of Leblond's model for strain hardening materials [START_REF] Leblond | Mathematicl Modelling of Transformation Plasticity in Steels II: Coupling with Strain Hardening Phenomena[END_REF] to pre-hardening case Finally, modified Leblond's for strain hardening materials (equation ( 35)) are compared with the FFT numerical results. In this calculation, back stress for 10 % initial straining is estimated using equation [START_REF] Suquet | Multi-scale modeling of the mechanical behavior of polycrystalline ice under transient creep[END_REF]. The obtained results for pre-tension case followed by stress-free transformation and 50 MPa applied stress are shown in Fig. 23. The kinematic change in strain during phase transformation is surprisingly in good agreement between Leblond's model and numerical model especially for the stress free condition. The initial curves for 50 MPa applied stress case are slightly different each other but the final strain values are very close. Another example is for pre-compression case. The back stress is again estimated by equation [START_REF] Suquet | Multi-scale modeling of the mechanical behavior of polycrystalline ice under transient creep[END_REF].

The strain evolutions under stress absent and 50 MPa applied stress cases are demonstrated in Fig. 24. The Leblond's analytical model for straining materials revealed that it is applicable to estimate transformation strain and transformation plasticity for both tensile and compressive pre-straining materials.

Conclusion

An elasto-plastic model for the prediction of the behaviour of a polycrystalline material subjected to diffusive phase transformation effect (transformation expansion) has been developed by using the discrete fast Fourier transform (FFT) numerical scheme. This alternative approach to the finite element method, based on Green functions, offers a simple framework to take into account the effect of the local volume expansion on the plastic response of the material. The model confirms that Greenwood-Johnson effect is one of the main mechanisms that cause transformation plasticity.

Then, the numerical model is applied to a more complicated phenomenon with pre-hardening effect.

The pre-hardening leads to a distinct transformation strain evolution, namely pre-tension cause the small transformation dilatation (for the tensile direction) remaining total volume being same. The pre-compression case is totally opposite to the pre-tension case. This effect can be explained by the back stress (kinematic hardening of the austenite phase) and the anisotropy. The calculation results are in good agreement with experimental results by Taleb et al..

Finally, Leblond's and Taleb's models are compared with the FFT results shedding light on the quantitative analysis on the transformation plasticity. For the case of a small transformation expansion, the results by FFT are in quantitative agreement with the Leblond's model in terms of the final transformation plastic strain value. However, Leblond's model overestimates the transformation plastic strain at the beginning of phase transformation and underestimates it at the end. On the other hand, the Taleb's model shows a good agreement with the FFT numerical solution at the beginning of transformation but underestimates the transformation plastic strain during the latter half of phase transformation, which causes an underestimation of transformation plasticity. As

Taleb has mentioned in his paper, the assumption that "the austenite phase is entirely plastic during phase transformation" is responsible for the overestimation of Leblond's model at the beginning of phase transformation. In addition, it is now clear that the underestimation of Leblond's model at the end of phase transformation can be explained by the plastic deformation in daughter phase. By using FFT results, it has been confirmed that the average ratio of the equivalent plastic strain increment over volume fraction increment, namely , is responsible for the nonlinearity.

〈𝛿𝜀 𝑒𝑞

1 /𝛿𝜉〉 𝑉 1

For large transformation expansion case, both Leblond and Taleb models underestimate the transformation plastic strain. In the case of large transformation expansion, the period during which the parent phase is not entirely elastoplastic is short and thus the differences between models by Leblond and Taleb are small. Both models underestimate transformation plastic strain since they neglect the effect of an external stress on plastic strain of parent phase as well as the plastic strain in the daughter phase. These models are considered to be valid when the parent phase is much harder, i.e. bainitic or martensitic.

𝜎 𝑖𝑗 = 𝐶 𝑖𝑗𝑘𝑙 𝜀 𝑖𝑗 [START_REF] Otsuka | A simple identification of transformation plastic behaviour and some data for heat treating materials[END_REF] because total strain rate derives only from elastic strain.

When resolved shear stress exceeds the value , the material satisfies the yield condition 𝜏 𝛼 = 𝜎 𝑖𝑗 𝑝 𝛼 𝑖𝑗 𝑔 𝛼 and the problem becomes elastoplastic. The values can be found according to the initial hardness 𝑔 𝛼 of each slip system and their hardening law.

First of all, let us introduce Hutchinson method which allows one to find the magnitude of slip on each slip system. Suppose that the work hardening and resolved stress increment of slip system 𝛼 It can also be expressed as following equations.

𝛥𝜏 𝛼 = sgn(𝜏 Thus, the slip increment is calculated by using matrix , the inverse matrix of :

𝑌 𝛼𝛽 𝑋 𝛼𝛽 , 𝑌 𝛼𝛽 = (𝑋 𝛼𝛽 ) -1 (A- 6)
such that;

Δ𝛾 𝛼 = ∑ 𝛽 𝑌 𝛼𝛽 𝐶 𝑖𝑗𝑘𝑙 Δ𝜀 𝑘𝑙 𝑝 𝛽 𝑖𝑗 (A- 7)
This result rigorously applies only for slip systems which are in the elastoplastic regime during the whole strain increment.

Second, let us consider the situation where the state of slip system of previous increment was 𝛼 elastic and presently elastoplastic. Here, the trajectory of one step can be divided in two paths as depicted in Fig. A-2; one is elastic and the other is elastoplastic.

𝜏 𝛼 𝑛 = 𝑔 𝛼 𝑛 sgn(𝜏 𝛼 𝑛 )

𝑔 𝛼 𝑛 -1 sgn(𝜏 𝛼 𝑛 -1 ) In this case, (A-3) can be rewritten as:

𝜏 𝛼 𝑛 -1 𝛥𝑔 𝛼 sgn(𝜏 𝛼 𝑛 -1 ) 𝛥𝜏 𝛼
𝜏 𝛼 𝑛 -1 + 𝛥𝜏 𝛼 =
(A- 9)
Uniaxial tensile numerical tests using the FFT numerical scheme with the two proposed method (Results A-7 and A-9)) have been carried out. Relation between applied stress and transformation plastic strain (pearlite: elastic case).

Fig. 1 Sample

 1 Fig. 1 Sample Voronoi tessellation and phase transformation; 100 grains in 64 3 cubic space.

2. 3 . Applications 2 . 3 . 1 .

 3231 Validation of the numerical modelIn order to validate the FFT-based model, the numerical results have been assessed by comparison

  Fig.2Evolution of the macroscopic strain with the volume fraction of transformed

Fig. 3

 3 Fig. 3 Macroscopic stress/strain curves of Austenite and Pearlite phase (calculations and experiment at 670 o C).

Fig. 4 Fig. 5 Fig. 6

 456 Fig. 4 Evolution of volume fraction of new phase and macroscopic total strain.

Fig. 7

 7 Fig. 7 Relation between applied stress and transformation plastic strain.

Fig. 8 Fig. 9

 89 Fig. 8 Calculation sequence for pre-tension transformation plasticity.

Fig. 10

 10 Fig. 10 Transformation strain with pre-compression.

  Fig. 11 Relation between transformation plastic strain and applied stress.

3. 1 .

 1 Descriptions of selected existing analytical models 3.1.1. Leblond's model

Fig. 12

 12 Fig.12Relation between applied stress and transformation plastic strain.

Fig. 13

 13 Fig. 13 Comparison of evolution of transformation plastic strain.

Fig. 14

 14 Fig. 14 Strain evolutions of parent and daughter phases during phase transformation.

  Fig. 15Comparison of evolution of transformation plastic strain (daughter: elastic case).

Fig. 15

 15 Fig.15shows that Taleb's equation[START_REF] Cabus | Etude et modélisation des textures de transformations de phases dans les aciers destinés à l'emboutissage[END_REF] closely agrees with FFT calculation results. Leblond's model overestimates the transformation plasticity, while Taleb's model agrees well with FFT during the whole transformation process when the daughter phase is elastic. Fig.16represents the relation between applied stress and transformation plastic strain for daughter phase elastic condition. The transformation plasticity coefficient is (MPa -1 ) and 𝐾 𝑃 = 1.68 × 10 -5

Fig. 16

 16 Fig. 16Relation between applied stress and transformation plastic strain (pearlite: elastic

Fig. 17 1 ( 1 /𝛿𝜉〉 𝑉 1 •

 17111 Fig. 17 Equivalent plastic strain distribution at . 𝜉 = 0.5

𝑠 1 Hypothesis 5 Fig. 20

 1520 Fig.[START_REF] Cherkaoui | Couplings between plasticity and martensitic phase transformation: overall behavior of polycrystalline TRIP steels[END_REF] Effect of correlation between and .𝛿𝜀 𝑒𝑞1 /𝛿𝑧 𝑠 1

Fig. 21

 21 Fig. 21 Difference between average stress value of phase 1 and overall stress.

Fig. 23

 23 Fig. 23 Relation between FFT and Leblond model results of pre-tension case.

Fig. 24

 24 Fig. 24 Relation between FFT and Leblond model results of pre-compression case.

1 Fig. A- 1

 11 Fig. A-1 Work hardening and resolved stress increment during plastic deformation.

Fig. A- 2

 2 Fig. A-2 Division of trajectory in one step into two paths (elastic and elastoplastic).

Fig. A- 3

 3 Fig. A-3 Calculated stress-strain curves with different strain increment par step.

Fig. A- 4 Fig. 2 Evolution

 42 Fig. A-4 Calculated stress-strain curves with the two proposed methods for the determination

Fig. 3 Fig. 4 Fig. 5

 345 Fig. 3 Macroscopic stress/strain curves of Austenite and Pearlite phase

Fig. 6

 6 Fig. 6 Relation between applied stress and transformation plastic strain.

Fig. 7 Fig. 8 Fig. 9 Fig. 10 Fig. 11

 7891011 Fig. 7 Relation between applied stress and transformation plastic strain.

Fig. 12

 12 Fig. 12 Relation between applied stress and transformation plastic strain.

Fig. 13 Fig. 14

 1314 Fig. 13 Comparison of evolution of transformation plastic strain.

Fig. 16

 16 Fig.16

Fig. 20

 20 Fig. 17 Equivalent plastic strain distribution at . 𝜉 = 0.5

Fig. 21

 21 Fig. 21 Difference between average stress value of phase 1 and overall stress.

Fig. 22 Fig. 23 Fig. 24 1 Fig. A- 1 1 𝛥𝑔

 222324111 Fig. 22 Difference between analytical solution (circle) and FFT (cross) for .〈𝛿𝜀 𝑒𝑞 1 /𝛿𝜉〉 𝑉 1

Fig. A- 3

 3 Fig. A-3 Calculated stress-strain curves with different strain increment par step.

Fig. A- 4

 4 Fig. A-4

Table

Stress <σ zz >(MPa) Strain <ε zz >

  Parameters for crystal plasticity calculation.

		300
		250
		200
		150
		100		Austenite
		50		Pearlite
				Austenite(Exp.)
		0
		0.000	0.003	0.006	0.009
			Austenite		Pearlite
	Bulk modulus (MPa)	135833		150000
	Shear modulus (MPa)	62692		69231
	𝐻 0	(MPa)	10		550
	𝜏 0	(MPa)	30		100
	𝜏 𝑠	(MPa)	40		130
		𝑞	1.0		1.0
	Number of grains	100		100

  .

	Equivalent plastic strain ε p, -	0.0005 0.0010 0.0015 0.0020	0MPa 50MPa 80MPa		10MPa 60MPa 90MPa	30MPa 70MPa	, -Equivalent plastic strain ε p	0.0005 0.0010 0.0015 0.0020	0MPa 50MPa 80MPa		10MPa 60MPa 90MPa	30MPa 70MPa
		0.0000						0.0000
		0	2	4	6	8	10	0	2	4	6	8	10
					Time t, s						Time t, s

Table 1

 1 Parameters for crystal plasticity calculation.

			Austenite	Pearlite
	Bulk modulus (MPa)	135833	150000
	Shear modulus (MPa)	62692	69231
	𝐻 0	(MPa)	10	550
	𝜏 0	(MPa)	30	100
	𝜏 𝑠	(MPa)	40	130
		𝑞	1.0	1.0
	Number of grains	100	100

based modelling of transformation plasticity in polycrystalline materials during diffusive phase transformation

  Development Bureau, Nippon Steel and Sumitomo Metal Corporation, 20-1 Shintomi Futtsu Chiba 293-8511 Japan otsuka.6gx.takayuki@jp.nssmc.com Renald BRENNER Institut Jean le Rond d'Alembert, Université Pierre et Marie Curie, CNRS, UMR7190, 4 place Jussieu 75005 Paris Cedex 5 France Brigitte BACROIX Laboratoire des Sciences des Procédés et des Matériaux, Université Paris 13, CNRS, UPR 3407, 99 av. J-B Clément 93430 Villetaneuse France

	by
	Takayuki OTSUKA
	Corresponding author
	Technical Research &

Modified rate-independent model for elasto-plastic transient problems

In this appendix, the computation of the plastic slip increments during a strain loading step is described. A modification of the Hutchinson's method [START_REF] Hutchinson | Elastic-plastic behaviour of poly crystalline metals and composites[END_REF] is proposed for the elastoplastic transition. It allows to consider larger strain increment without affecting the overall response.

The plastic strain rate is given by summation of each slip rate on the slip systems, such that

where, is slip system is a Schemid tensor. So, the stress rate of elastoplastic problem can be 𝛼 𝑝 𝛼 𝑖𝑗 found as:

where, is total strain rate and is elastic stiffness tensor. On the other hand, the 𝜀 𝑘𝑙 𝐶 𝑖𝑗𝑘𝑙 (𝑥) constitutive equation of elastic problem is given simply as: