Odalric-Ambrym Maillard
email: odalric.maillard@inria.fr

Timothée Mathieu

Debabrota Basu

Farm-gym: A modular reinforcement learning platform for stochastic agronomic games

We introduce Farm-gym, an open-source farming environment written in Python, that models sequential decisionmaking in farms using Reinforcement Learning (RL). Farm-gym conceptualizes a farm as a dynamical system with many interacting entities. Leveraging a modular design, it enables us to instantiate from very simple to highly complicated environments. Contrasting many available gym environments, Farm-gym features intrinsically stochastic games, using stochastic growth models and weather data. Further, it enables to create farm games in a modular way, activating or not the entities (e.g. weeds, pests, pollinators), and yielding non-trivial coupled dynamics. Finally, every game can be customized with .yaml files for rewards, feasible actions, and initial/end-game conditions. We illustrate some interesting features on simple farms. We also showcase the challenges posed by Farm-gym to the deep RL algorithms, in order to stimulate studies in the RL community.

Introduction: motivation and challenges

Managing a farm during a growing season requires the farmer to take a sequence of decisions in a field to reach her objective. Farmers take these decisions under uncertainty (e.g. weather) while incorporating observations and various interactions between entities of the agronomic ecosystem (plant, weeds, pests, etc.) on-the-go. Developing novel management strategies is a time consuming and expensive process that often requires trial and error. Recently, an increasing attention has been drawn to improve farm management with data-driven methods, such as machine learning.

Reinforcement Learning (RL) [START_REF] Sutton | Reinforcement learning: An introduction[END_REF]) is a popular paradigm of machine learning that models the problem of decision making under uncertainty. Hence, while interacting with a dynamical system, called environment. Development of RL algorithms for different applications often commence with developing simulators or gamified platforms that mimic the challenges of the real-world complex problems in computationally controllable manner. For example, Atari games [START_REF] Bellemare | The arcade learning environment: An evaluation platform for general agents[END_REF]) provided a stimulating benchmark to develop RL algorithms in the context of computer games. Mujoco [START_REF] Todorov | MuJoCo: A physics engine for model-based control[END_REF] is proposed as a physics-driven engine for fostering RL algorithms for continuous control and robotics. Ope-nAI [START_REF] Brockman | Openai gym[END_REF] proposed a set of lightweight RL environments with a standardized Application Programming Interface (API), called OpenAI gym. Presently, gym API became a reference in the RL community to create standardized RL environments in order to compare performances of RL algorithms. Following gym, several platforms are developed to foster reproducible frameworks and improved algorithms. For example, the highway environment [START_REF] Leurent | An Environment for Autonomous Driving Decision-Making[END_REF] aims at safe planning in the context of traffic with multiple vehicles and unknown dynamics, and NLPgym (Ramamurthy, Sifa, and Bauckhage 2020) focuses of Natural Language Processing challenges. Often these gym frameworks have deterministic environments which are unknown and reveal themselves through noisy observations. Challenges. In contrast, growing plants in a farm is inherently stochastic as it involves external uncontrolled entities (e.g. weather, insects, weeds) and interaction between different entities (e.g. plants and soil). These interactions and impacts of the entities demonstrate a coupled dynamics, which is often hard to model. Since existing deep RL algorithms can handle stochasticity only in small environments, farming provides an interesting challenge to RL community.

Another key challenge is the ability to observe or measure a specific entity. In contrast to traditional gym environments, where everything can be measured or observed, each observation in agronomy incurs a cost that must be balanced with the need to act on the field (e.g. watering, harvesting). Hence, agronomic decision making invoke an observationaction trade-off, which is missing in RL literature. Hence, we develop Farm-gym to foster both the RL research, and interaction between agronomy and RL communities. Contributions. We develop Farm-gym 1 with the aim to create a modular environment that mimics the dynamics of an agricultural farm but still is simple enough to control and diagnose. Farm-gym takes a gamification approach in contrast to the high-precision simulators whose goal is to accurately model an individual entity. Each farm is constructed with multiple entities in a modular way, which can be seen as a game. We can choose which dynamics of the environment to simulate, and also to combine these dynamics to study specific coupling phenomena. Hence, Farm-gym proposes many variants of the farming problem, which can serve to model particular challenges, such as non-stationarity, coupling, long-term planning, active observation, etc. Specially, the Farm-gym platform is designed to host a number of environments built by combining different modules, cost-constraint and scores, such that each environment can be considered as a separate game of various and progressive difficulty. This provides even a non-expert an opportunity to play with and study parts of the complex farming problem with simple games.

For RL researchers, the agronomy-inspired Farm-gym offers a bouquet of stimulating challenges to study and experiment with, such as intrinsically stochastic dynamics, interaction and coupled dynamics, the observation-action trade-off, and user-defined score function2 . We experimentally demonstrate the impact of the coupled dynamics (Section 3), and the challenges encountered by classical deep RL algorithms (Section 4). Experimental results indicate that designing learning strategies capable of simultaneously solving many Farm-gym games will help pave the way towards real-life applicable RL.

While we provide a few initial games and entity modules, the platform allows easy addition of new games and modules (e.g. fungus, slugs). Hence, Farm-gym also offers the possibility to study games involving more real-like complex and refined interactions in future. Related works. [START_REF] Garcia | Use of reinforcement learning and simulation to optimize wheat crop technical management[END_REF] first proposed an RL agent interacting with a crop simulator to learn to maximize wheat yield while restricting nitrogen fertilization. Recently, [START_REF] Sun | Reinforcement learning control for water-efficient agricultural irrigation[END_REF][START_REF] Chen | A reinforcement learning approach to irrigation decision-making for rice using weather forecasts[END_REF][START_REF] Yang | Deep reinforcement learning-based irrigation scheduling[END_REF]) use different RL algorithms to maximize the yield while learning the best irrigation schedules. [START_REF] Trépos | Apprentissage par renforcement pour l'optimisation de la conduite de culture du colza[END_REF]) learns a technical itinerary that considers the date of sowing and the potential weather conditions. Unfortunately, none of these works provides an open source and standardized RL environment.

In this context, a few gym environments are proposed that model various decision-making problems while trying to optimize the yield of a given crop. For example, CropGym (Overweg, Berghuijs, and Athanasiadis 2021) focuses on fertilization strategies. [START_REF] Kemanian | The Cycles Agroecosystem Model: Fundamentals, Testing, and Applications[END_REF] introduces CyclesGym that simulates multi-year crop strategies (e.g. crop-rotation strategies). But often the models considered here are deterministic like traditional gym environments. Recently, [START_REF] Gautron | gym-DSSAT: a crop model turned into a Reinforcement Learning environment[END_REF]) introduces gym-DSSAT, an environment with maize fertilization and irrigation problems. It is backed by the DSSAT crop models, allowing accurate simulation of a wide range of realworld growing conditions. However, apart from the weather, all other processes are deterministic.

The existing works consider specific settings of interest and focus on optimizing simple strategies (e.g. date of sowing or fertilization) for maximizing yield output with accurate models of plants. They do not consider either interactions between different entities (e.g. weeds, insects and plants), or complete stochasticity as seen in real-world. In contrast, Farm-gym provides a holistic platform to study 2 Farm-Gym: An RL platform for solving gamified agronomy challenges Modeling an agronomic system such as a farm, is a challenging task for several reasons. As mentioned before, a farm is an intrinsically stochastic system, e.g. subject to the variation of weather or pest attacks, with interaction of many entities at spatial (plant, crop, etc.) and temporal scales (day/night, week, month, year). This creates a coupled dynamics (predator-prey dynamics). Additionally, observations are costly and should be actively acquired in order to optimize a risk-sensitive and personalized objective functions. Here, we elaborate the components of Farm-gym.

Interacting entities. A Farm-gym environment consists of a set of fields. Each field has a pre-defined size and contains various entities. Hence, we conceptualize a farm as a big dynamical system, with states given by the concatenation of all statevariables of all entities. Due to the high number of observation, a subset of the observations can be selected, either to guide the learner or to model partially observed phenomena. Since a learner only accesses a subset of all internal variables, a Farm-gym problem is a Partially Observable Markov Decision Process (POMDP). This is similar in spirit to the real problems faced by farmers with unobserved factors. Unlike classical RL environments that are static POMDPs, here the system evolves even when no action is taken. A learner, while not completely knowing the dynamics, must find when to use each action depending on the task. While for classical MDP and POMDPs models, the states or observations are given for free to the learner, Farm-gym promotes the case of actively-observed MDPs.

Formally, the actions are interventions defined with entities, and observations that request the current value of a specific state variable. The observations can incur some cost depending on the score function. Finally, a reward can be computed when a game stops. Terminal (or end-game) conditions are defined as Boolean functions computed on the state variables. Typical end-game conditions include reaching a certain day of the year, or when all plants are dead or harvested. But more complex conditions can be defined. The full user-guide is available on the github page3 . Modular environment. Modular design of Farm-gym allows customizing agronomy games with co-interacting entities. For example, Figure 2 demonstrates evolution of a single Plant (bean) and Weeds.

In Farm-gym, each instantiated farm generates configurable (YAML) files to specify the subset of allowed actions, the cost of each action/observation, the rewards, and the initial conditions. For instance, beyond the classical yield, Farm-gym allows to include the soil health index, a biodiversity index in the reward, etc.

Interestingly, Farm-gym also allows generating games with several entities of the same type, such as different plants for a poly-culture agrosystem (e.g. carrots and beans), different types of fertilizers, etc. This is to facilitate modelling poly-culture or crop-rotation games. Specifically, each entity comes with parameters, that can be instantiated in a YAML file for various instances. Currently, Farm-gym has three plants (corn, bean, tomato), three type of soils (clay, loam, sand), two cides (insecticide, pesticide), and two fertilizers (slow, fast diffusion). Each of them are different instances of the same entity with different parameters. We are also working on incorporating more instances. Highly stochastic environment. In Farm-gym, the weather can be loaded from historical data to which stochastic noise is added.

Further, as many entities depend on some "favorable conditions" to transit from one state to another, we find it con-venient to introduce a generic function in order to model this principle for different entities. Formally, we introduce an exponential general linear function

E(β, y, Y) = exp   -β 0 - j∈J β j d(y j , Y j )   ,
where β = (β 0 , (β j ) j ) is a vector of weight parameters, y = (y j ) j is a vector of real-values, Y = (Y j ) j a set of intervals in R, and d(y, Y) denotes the distance from y to the set Y. Each entity transition (e.g. from seed to sprout, etc.) is modeled using a probability distribution that follows an aforementioned exponential model. This allows building an intrinsically stochastic environment. It should be noted that these plant models are inspired by characteristics of real plants. However, the goal of Farm-gym is not to reproduce accurate high-precision dynamics of actual plants from an agronomy perspective. Rather, the goal is to offer a gamified and approximate version to create flexible models and algorithms that can be combined and extended. Challenges. Using stochastic processes in the dynamics of each entity (e.g. stochastic growth models) challenges the learning algorithms. Indeed, today's algorithms (e.g. GOexplore [START_REF] Ecoffet | Go-explore: a new approach for hardexploration problems[END_REF])) designed to explore and solve complex environments such as Montezuma [START_REF] Bellemare | The arcade learning environment: An evaluation platform for general agents[END_REF]) are based on function approximation using deep learning and are not designed to handle a stochastic dynamic. In a stochastic environment, one may not obtain the same trajectory twice when applying the same policy. As a result, algorithms struggle to explore and successfully solve a far-gym environment, as demonstrated in Section 4. On the other hand, strategies inspired from multi-armed bandit (Lattimore and Szepesvári 2019) are designed to handle noise, however they struggle with large state-action space dynamics. Hence, we expect Farm-gym to foster interesting crossbreeding between deep RL community focusing on scalable algorithms and deterministic environments, and RL theory community focusing on model-based RL and multiarmed bandits with stochasticity. Large and structured action space. Each entity comes with its own set of actions, which can be parametrized. For instance, the action watering from entity soil requires the amount of water to be specified. Further Farm-gym supports asking for a list of observations and interventions each day. This blows up the action space to A k , when allowing k many actions a day, in comparison to daily action space A. Hence, a farm game can feature a large number of actions. These actions are mixed, i.e. they take both continuous and discrete values. On the other hand, the action space is very structured, and some actions (e.g. removing weeds) directly affect only part of the system. But a learner has to learn this structure to optimize the actions. We believe this fosters research on studying reinforcement learning in the context of a large and structured action space. Observation-action trade-off. In Farm-gym, the agent must perform a specific action (observation-step) with an associated non-negative cost in order to retrieve the relevant information, e.g. the amount of water in the soil, the average number of insects in the field, etc. before intervening. The agent must learn to identify when these additional observations are crucial to improve the management of the field, i.e. it must learn when to act for observing. To some extent this requires the agent to evaluate the state of its knowledge about the system and update its values by observing as a farmer would. This creates a non-trivial problem of activelyobserved MDPs, which is hardly studied in RL community.

As we have summarised the functioning and challenges posed by Farm-gym, we experimentally instantiate them in the following sections.

Illustrating coupled dynamics in a farm

In this section, we illustrate basic results on a few farms featuring various entities: Soil, Plant, Pollinators, Weeds, Pests. They highlight the non-trivial dynamics that emerges, when multiple entities interact in Farm-gym. For simplicity of exposure, we use simple hand-crafted policies, all farms are 1 × 1, and the agent has access to all observations.

In Figure 3, for clay-type Soil, we illustrate the effect of having pollinators in the field. Different plants react differently to the pollinators. For instance, corn does not depend much on insect pollination, while beans depend a lot on it.

In Figure 4, we illustrate another coupling phenomenon between weeds and pests. While weeds compete with the plant in terms of nutrients and water, some can also protect the main plant from pests. In that case, removing the weeds may not be the best option. We consider the learner spreads a tiny amount of herbicide at different frequencies. At low frequency, the effect is to slow-down the developments of weeds. In contrast, while used at high frequency, herbicide kills the weeds, causing the pests to attack the plant only.

Challenges for training a deep RL agent

To investigate the challenges present when learning on farmgym environments, we train a popular deep RL algorithm, PPO (Proximal Policy Optimization) [START_REF] Schulman | Proximal policy optimization algorithms[END_REF], on a basic farm. The farm consists of a 1x1 plot of land, the crop are beans, and the weather is from the north of France. We include bugs and weeds. The farmer can use water, herbicide, pesticide, and fertilizer, or do nothing. There are 9 possible actions with discrete values, and 17 observations which are, for simplification, in a box space. Further details are given in Appendix B. All the observations are free and the algorithm takes one action at each time step. In Figure 5, we present the cumulative reward observed while the training the PPO agent. We observe that the raw plot (represented in light blue in Figure 5) is highly variable. Thus, we use smoothing, similar to Tensorboard's 4 smoothing, in order to visualize the increase of the cumulative reward. This variability of the reward is a big challenge for PPO, and it is linked to the stochastic nature of the environment. The results indicate that PPO might not be the optimal choice for this environment as it does not take into account the timedependent dimension of the problem. However, it gives a benchmark and shows that a classical RL algorithm manage to learn something on this environment. On the other hand, we design an expert agent tailored to the same farm. This agent uses expert knowledge, such as beans should be planted after April, and there should be a small amount of pesticide and herbicide used before planting to clean the ground. The mean episodic reward, computed on 128 evaluation episodes and 15 seeds, is 433 ± 50 for the expert agent against 75 ± 79 for PPO. This shows that classical deep RL agents are still far from expert agent on this task, and poses a challenge to be solved in future.

A Additional experiments on farm dynamics

Like many agronomy platforms, Farm-gym models the weather and soil conditions. To illustrate the effect of soil on the plant cycle, we explore a simple deterministic policy that waters the same amount each day and harvests when the plant reaches a ripe stage. In this experiment, the weather is fixed to an (artificial) dry weather. Thus, the water input only comes from watering. Farm-gym enables us to easily customize the weather with .csv files.

Figure 6 illustrates a clear difference between the type of soils. For example, with less that 2L of water input in the sand soil, the plant does not grow, while it grows in clay.

B Additional Experiments on learning with an RL-agent

The farm considered in Section 4 consists of an 1x1 farm, with beans in a loam ground in the north of France that has rainy and moderately cold weather. There are pests, weeds and pollinators. The farmer have 9 possible actions each day: doing nothing, watering 1L or 5L of water, sowing some seeds, harvesting, scatter some pesticides, scatter some herbicide, scatter some fertilizer, removing weeds by hand. There are 17 observations given freely each day: day (from 1 to 365), temperature (mean, max and min in °C), rain amount (categorical), sun-exposure (from 1 to 5), consecutive dry days, stage of growth of the plant (categorical), number of fruits (int), size of the plant (in cm), soil wet surface (m2.day-1), fertilizer amount (kg), pollinators occurrence (proportion) , weeds grow (int), weeds flowers (int), weight of fruits (in g), micro-life health index (in percent).

An episode begins with nothing in the soil, and it ends once either the plant has grown and has been harvested, or the plant is dead, or a year ends. The reward consists of the amount of fruit harvested (in g, given only once at the end of the episode), and a negative reward of -2 given each day for which the micro-life index went below 10%.

In Figure 7, we present the number of times each action is used during the evaluation of a trained PPO agent over 128 episodes. It shows that the agent can learn a sparse set of frequent actions to take. It also learns actions related to the micro-life index maintenance, such as not frequently using herbicides and pesticides, in addition to the ones leading to higher yields, such as watering and scattering fertilizers. This result shows an opportunity to learn lessons about farming using an off-the shelf RL agent and Farm-gym. In Figure 8, we present the cumulative reward observed while the training the DQN agent. Similarly to PPO, the raw plot is highly variable, hence we display its smooth version. As for PPO, the results indicate that DQN might not be the optimal choice for this environment.

Figure 1 :

 1 Figure 1: Interactions between Farm-gym entities.

Figure 2 :

 2 Figure 2: Single field, 3 × 4 bean farm at 4 different days.

Figure 3 :

 3 Figure 3: Effect of watering with or without pollinators. All plots are built using 100 runs. Reward is the yield.

Figure 4 :

 4 Figure 4: Effect of spreading low herbicide on weeds, with or without pest insects. Reward is the yield.

Figure 5 :

 5 Figure 5: Training of one PPO agent on a basic farm. Reward is the yield penalized by low health index (details in App B).

Figure 6 :

 6 Figure 6: Effect of constant watering in different soils.

Figure 7 :

 7 Figure 7: Actions of a given trained PPO algorithm taken during 128 evaluation episodes

Figure 8 :

 8 Figure 8: Training of one DQN agent on a basic farm.

Code is available at: https://github.com/farm-gym/farm-gym

The score function is defined as the reward of the user minus the cost due to observation and intervention.

https://github.com/farm-gym/farm-gym/blob/main/ FarmGym.pdf

Acknowledgments

We acknowledge H.D Carvajal and J. Teigny for their support and discussions. O-A. Maillard acknowledges the support of the Métropole Européenne de Lille (MEL), ANR, Inria, Université de Lille, I-SITE ULNE through the AI chair Apprenf number R-PILOTE-19-004-APPRENF. T. Mathieu acknowledges the support of the A.Ex Inria SR4SG.


Eyram Tsekpuia

Adrien Guery

Nathalie Gey

Stéphane Berbenni
email: stephane.berbenni@univ-lorraine.fr

A microstructure-based three-scale homogenization model for predicting the elasto-viscoplastic behavior of duplex stainless steels

Keywords: A. Microstructures, EBSD, Homogenization, Crystal plasticity, Duplex steels, Elastoviscoplastic material

come

A microstructure-based three-scale homogenization model for predicting the elasto-viscoplastic behavior of duplex stainless steels

Introduction

To understand and predict the mechanical behaviors of multiphase and polycrystalline materials with complex microstructures, advanced micromechanical models need to be developed not only considering macroscopic fields (stresses and strains) but also fields in the different phases.

Among these materials, cast duplex stainless steels are widely used in nuclear power plants in the primary loop of the cooling system of pressurized reactors (PWR) due to their good corrosion resistance and their superior mechanical properties (yield stress, ductility) [START_REF] Solomon | Duplex stainless steels: a tale of two phases[END_REF][START_REF] Bonnet | Relationship between evolution of mechanical properties of various cast duplex stainless steels and metallurgical and aging parameters: outline of current EDF programmes[END_REF][START_REF] Bethmont | Properties of cast austenitic materials for light water reactors[END_REF][START_REF] Charles | Corrosion Resistance Properties[END_REF]. These two-phase polycrystalline steels are constituted of a -phase (ferrite) with volume fractions generally varying from 5% to 25% and a -phase (austenite). They exhibit a large yield stress due to the presence of bodycentered cubic ferrite (b.c.c. crystal structure, the "hard phase"), while ductility is ensured thanks to the presence of face-centered cubic austenite (f.c.c. crystal structure, the "soft phase"). However, during in service behavior at about 300°C and aging during long periods of time (e.g. 10000h), while the mechanical behavior of austenite is not affected, ferrite may exhibit embrittlement and material's fracture toughness decreases [START_REF] Bonnet | Relationship between evolution of mechanical properties of various cast duplex stainless steels and metallurgical and aging parameters: outline of current EDF programmes[END_REF][START_REF] Devillers-Guerville | Rupture d'aciers inoxydables austéno-ferritiques moulés, fragilisés par vieillissement à 350-400[END_REF][START_REF] Magnin | Mechanical twinning in ferritic stainless steels[END_REF][START_REF] Trautwein | Influence of Long-Time Aging of CF8 and CF8M Cast Steel at Temperatures Between 300 and 500°C on Impact Toughness and Structural Properties[END_REF][START_REF] Delliou | Integrity and Life Assessment of Cast Duplex Stainless Steel Elbows of EDF PWR Main Coolant Piping[END_REF]. Indeed, a spinodal decomposition of the ferrite -phase (Fe-Cr solid solution) into -phase (Cr-rich phase) andphase (Fe-rich phase) occurs during aging besides a G-phase precipitation [START_REF] Auger | Atom probe and transmission electron microscopy study of aging of cast duplex stainless steels[END_REF][START_REF] Pareige | Kinetics of G-phase precipitation and spinodal decomposition in very long aged ferrite of a Mo-free duplex stainless steel[END_REF][START_REF] Badyka | Quantification of hardening contribution of G-Phase precipitation and spinodal decomposition in aged duplex stainless steel: APT analysis and micro-hardness measurements[END_REF]. Therefore, the damage mode of ferrite is modified and changes from ductile to brittle fracture due to cleavage cracking. The cleavage mode was observed to be controlled by tensile stresses [START_REF] Bugat | Comportement et endommagement des aciers austéno-ferritiques vieillis : une approche micromécanique[END_REF][START_REF] Bugat | Microstructure and damage initiation in duplex stainless steels[END_REF][START_REF] M'cirdi | Comportement et eendommagement sous sollicitation mécanique d'un acier austénoferritique moulé vieilli[END_REF][START_REF] M'cirdi | Experimental approach of a crystallographic cleavage criterion in a cast aged duplex stainless steel[END_REF]. Therefore, good estimates of these stresses in both phases and especially in the ferritic phase are needed for as received and aged materials. Furthermore, the stress/strain responses of cast duplex stainless steels are strain rate sensitive as shown by the experimental cyclic responses [START_REF] Bugat | Comportement et endommagement des aciers austéno-ferritiques vieillis : une approche micromécanique[END_REF], which needs the development of a multiscale elasto-viscoplastic model.

Internal stresses in these materials were studied by [START_REF] Verhaeghe | Internal stresses in an austenoferritic duplex stainless steel[END_REF] identifying their sources directly from Bauschinger tests for aged and non-aged materials. Damage and fracture mechanisms were studied using TEM by [START_REF] Verhaeghe | Damage and rupture mechanisms in an austenoferritic duplex steel[END_REF], X-Ray diffraction methods by [START_REF] M'cirdi | Experimental approach of a crystallographic cleavage criterion in a cast aged duplex stainless steel[END_REF] and SEM with field measurements by [START_REF] Bugat | Microstructure and damage initiation in duplex stainless steels[END_REF]. In a cast aged duplex stainless steel, a crystallographic cleavage criterion, based on a stress normal to a {100} plane in ferrite was reported by M' Cirdi et al. (2001). More recently, in situ synchrotron and neutron diffractions were used to study the evolution of lattice strains in duplex steels during a tensile test and the occurrence of damage in the ferrite [START_REF] Baczmanski | Elastoplastic deformation and damage process in duplex stainless steels studied using synchrotron and neutron diffractions in comparison with a self-consistent model[END_REF]).

New experimental in-situ DIC and EBSD-based methods (Mollens, 2022a,b,c) reveal the heterogeneous deformation and micro-texture complexities, which motivates the development of new EBSD-informed microstructure-based multiscale homogenization model for predicting the elasto-viscoplastic behavior of duplex stainless steels. The microstructure heterogeneities are related to microstructure genesis and local correlations arise due to orientation relationships (OR).

For duplex stainless steels, different orientation relationships were reported in the literature [START_REF] Haghdadi | Effect of ferriteto-austenite phase transformation path on the interface crystallographic character distributions in a duplex stainless steel[END_REF]Mollens et al., 2022b). Most important observed ones are the Kurdjumov-Sachs (KS), Pitsch (P) and Nishiwama-Wassermann (NW). This microstructure genesis can also be simulated by advanced phase field models, considering the KS-OR from ferrite to austenite [START_REF] Song | Phase-field simulation of austenite growth behavior: Insights into the austenite memory phenomenon[END_REF]. From the mechanical ground, full field crystal plasticity models based on Finite Element modeling (CPFEM) are good candidates to predict internal stress and strain fields at the microstructure level of complex microstructures (see e.g. [START_REF] Roters | Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finiteelement modeling: Theory, experiments, applications[END_REF][START_REF] Dodla | Finite element simulation of lamellar copper-silver composites[END_REF][START_REF] Li | Development of a thermo-mechanically coupled crystal plasticity modeling framework: Application to polycrystalline homogenization[END_REF][START_REF] Flipon | A full-field crystal-plasticity analysis of bimodal polycrystals[END_REF][START_REF] Kasemer | A finite element methodology to incorporate kinematic activation of discrete deformation twins in a crystal plasticity framework[END_REF]. However, they are still time consuming for the industrial applications. Analytical homogenization-based models remain relevant because they are less time consuming but need to be adapted to more complex microstructures than single phase polycrystalline structures using multiple scale transition methods.

Regarding micromechanical mean-field homogenization models for duplex steels, simple approximations using the simple [START_REF] Taylor | Plastic strain in metals[END_REF]'s model were reported by [START_REF] Verhaeghe | Internal stresses in an austenoferritic duplex stainless steel[END_REF] and [START_REF] M'cirdi | Comportement et eendommagement sous sollicitation mécanique d'un acier austénoferritique moulé vieilli[END_REF]. The first self-consistent model for two-phase austenite/ferrite steels was due to [START_REF] Weng | The overall elastoplastic stress-strain relations of dual-phase metals[END_REF], who used the [START_REF] Berveiller | An extension of the self-consistent scheme to plastically-flowing polycrystals[END_REF] self-consistent model. In this model, ferritic and austenitic grains were supposed to be spherical, which appears to be too simplistic for the present microstructure and isotropic secant moduli were used for proportional loadings only such as tensile loadings. A second numerical model was developed by [START_REF] Sigmund | On the thermomechanical deformation behavior of duplex-type materials[END_REF], who also considered a simple modeling assuming an aggregate of both phases with similar shapes. The Cailletaud-Pilvin's "Beta model" [START_REF] Cailletaud | A micromechanical approach to inelastic behavior of metals[END_REF]Cailletaud and Pilvin, 1994;[START_REF] Coudon | A multiscale model for nickel-based directionally solidified materials[END_REF] was used to describe the elastic-plastic responses of duplex stainless steels under nonproportional and cyclic loadings [START_REF] Evrard | Implementation and validation of a polycrystalline model for a bi-phased steel under non-proportional loading paths[END_REF][START_REF] Evrard | Polycrystalline modeling of the cyclic hardening/softening behavior of an austenitic-ferritic stainless steel[END_REF]. The Berveiller-Zaoui's self-consistent model was also used in combination with a physical intra-granular backstress formulation by [START_REF] Fournier | Micromechanical model of the high temperature cyclic behavior of 9-12%Cr martensitic steels[END_REF] for predicting the high temperature cyclic behavior of martensitic steels. [START_REF] Baczmanski | Elastoplastic deformation and damage process in duplex stainless steels studied using synchrotron and neutron diffractions in comparison with a self-consistent model[END_REF] used synchrotron and neutron diffraction measurements to calibrate materials parameters in an elasto-plastic self-consistent scheme (EPSC) considering damage in duplex steels.

However, in all these above models, only two scales were considered in the scale transition of the micromechanical model between the phases and the aggregates. Later, [START_REF] Bugat | Micromechanical modeling of the behavior of duplex stainless steels[END_REF] and [START_REF] Bugat | Comportement et endommagement des aciers austéno-ferritiques vieillis : une approche micromécanique[END_REF] developed for the first time a three-scale model based on two mean field "Beta model" stress interaction laws for these duplex steels with an intermediate scale, which they called "bi-crystal". However, this model was only developed for isotropic elasticity by [START_REF] Bugat | Micromechanical modeling of the behavior of duplex stainless steels[END_REF] and [START_REF] Bugat | Comportement et endommagement des aciers austéno-ferritiques vieillis : une approche micromécanique[END_REF], and, needed to be calibrated with full field FE unit cell calculations. The model was based on the EBSD data acquired in the 2000's, and three different physical scales were essentially identified [START_REF] Bugat | Micromechanical modeling of the behavior of duplex stainless steels[END_REF][START_REF] Bugat | Comportement et endommagement des aciers austéno-ferritiques vieillis : une approche micromécanique[END_REF]. Another interesting approach corresponding to multiple scale modeling of the local correlation between phases for purely anisotropic viscoplastic complex microstructures can be found in [START_REF] Lebensohn | A self-consistent approach for modelling texture development of two-phase polycrystals: application to titanium alloys[END_REF], [START_REF] Lebensohn | Modelling the role of local correlations in polycrystal plasticity using viscoplastic self-consistent schemes[END_REF], where the viscoplastic self-consistent scheme (VPSC) was extended to consider two-site interactions and lamellar structures in titanium aluminides (TiAl). For the same material, but considering a different modeling approach, [START_REF] Roos | Multiscale modelling of titanium aluminides[END_REF] developed a modeling approach based on a three-scale model using the Transformation Field Analysis (TFA) for plasticity together with the elastic self-consistent solution and lamellar phases. [START_REF] Kowalczyk-Gajewska | Micromechanical model of polycrystalline materials with lamellar substructure[END_REF] developed a micromechanical model of polycrystalline materials with lamellar substructure in the framework of the Hill's EPSC model [START_REF] Hill | Continuum micro-mechanics of elastoplastic polycrystals[END_REF] to study the elastic anisotropies and the initial yield surfaces for polycrystalline microstructures of Ti-Al. [START_REF] Franciosi | Multiple continuity of phases in composite materials: overall property and estimates from a laminate system scheme[END_REF]Berbenni (2007,2008) also developed a multi-scale sequential multi-laminate model based on a non-uniform TFA approach with affine approximation to describe the evolution of heterogeneous plastic domains in metallic polycrystals. Stress/strain estimates were improved at the macroscopic and at the grain levels, as compared to classic EPSC scheme, which assumes piece-wise uniform plastic strains in grains.

The lamellar-based approach of [START_REF] Lebensohn | A self-consistent approach for modelling texture development of two-phase polycrystals: application to titanium alloys[END_REF][START_REF] Lebensohn | Modelling the role of local correlations in polycrystal plasticity using viscoplastic self-consistent schemes[END_REF] was recently applied to a Ti-6Al-4V alloy to accurately model the local crystallographic correlations in such complex multiscale microstructures [START_REF] Riyad | Modeling the role of local crystallographic correlations in microstructures of Ti-6Al-4V using a correlated structure visco-plastic self-consistent polycrystal plasticity formulation[END_REF]. A finite strain lamellar-based model elasto-viscoplastic model was developed by [START_REF] Uchida | Micro-, meso-to macroscopic modeling of deformation behavior of semi-crystalline polymer[END_REF] and was applied to semi-crystalline polymers. Another lamellar-based approach was used in a multi-scale model [START_REF] Maresca | Subgrain lath martensite mechanics: A numerical-experimental analysis[END_REF] to describe the micromechanical behavior of martensitic steels. In their approach, martensitic blocks were seen as a two-phase layered composite material with martensitic laths and retained austenite interlayer.

Therefore, we propose here to extend the EVPSC (Elasto-Visco-Plastic Self-Consistent) scheme based on the "Translated Field" (TF) method with affine linearization of the viscoplastic flow rule as developed by [START_REF] Mareau | An affine formulation for the self-consistent modeling of elastoviscoplastic heterogeneous materials based on the translated field method[END_REF] for such duplex steels complex microstructure. This advanced EVPSC scheme was successfully applied to near-β titanium alloys in Lahdi et al. (2018Lahdi et al. ( , 2020)). While the original approach based on the TF method of [START_REF] Paquin | Integral formulation and self-consistent modelling of elastoviscoplastic behavior of heterogeneous materials[END_REF][START_REF] Paquin | Micromechanical modeling of the elastic-viscoplastic behavior of polycrystalline steels[END_REF] was derived from a classical secant approximation, this work adopts the first order affine linearization (see e.g. [START_REF] Masson | Affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals[END_REF][START_REF] Doghri | Mean-field homogenization of elastoviscoplastic composites based on a general incrementally affine linearization method[END_REF] of the viscoplastic flow rule, i.e. a first order Taylor expansion of the viscoplastic strain rate at the local scale. It has been observed in [START_REF] Mareau | An affine formulation for the self-consistent modeling of elastoviscoplastic heterogeneous materials based on the translated field method[END_REF] that the first order affine formulation yields softer responses than the secant formulation for different elastoviscoplastic polycrystals during monotonic and cyclic loadings. Other EVPSC approaches using the "additive interaction law" [START_REF] Molinari | On the self-consistent modeling of elastic-plastic behavior of polycrystals[END_REF][START_REF] Mercier | Homogenization of elastic-viscoplastic heterogeneous materials: self-consistent and Mori-Tanaka schemes[END_REF] were adapted for polycrystals with "affine"-type approximation at large strains [START_REF] Wang | A finite strain elastic-viscoplastic selfconsistent model for polycrystalline materials[END_REF][START_REF] Zecevic | New robust self-consistent homogenization schemes of elasto-viscoplastic polycrystals[END_REF]. Another numerically efficient affine-type EVPSC scheme was also recently derived by [START_REF] Jeong | An efficient elasto-visco-plastic self-consistent formulation: Application to steel subjected to loading path changes[END_REF] and applied to steels subjected to loading path changes. According to [START_REF] Mercier | Comparison of different homogenization approaches for elastic-viscoplastic materials[END_REF], [START_REF] Berbenni | A Mori-Tanaka homogenization scheme for non-linear elastoviscoplastic heterogeneous materials based on Translated Fields: an "affine" extension[END_REF] and [START_REF] Mareau | An affine formulation for the self-consistent modeling of elastoviscoplastic heterogeneous materials based on the translated field method[END_REF], the use of the first order affine formulation improves the estimate of the effective behavior, in comparisons with full-field calculations (FEM, FFT), even when the viscoplastic flow rule is highly nonlinear. Homogenization methods for elastoviscoplastic composite materials considering field statistics, i.e. second moments of stresses, were derived (see e.g. Lahellec andSuquet, 2007, 2013;Brassart et al., 2012;[START_REF] Wu | An incremental-secant mean field homogenization method with second statistical moments for elasto-viscoplastic composite materials[END_REF][START_REF] Masson | A modified secant formulation to predict the overall behavior of elasto-viscoplastic particulate composites[END_REF][START_REF] Berbenni | A time-incremental homogenization method for elasto-viscoplastic particulate composites based on a modified secant formulation[END_REF].

A recent modeling approach for elastoviscoplastic polycrystals based on a differential variational procedure (Das and Ponte Castañeda, 2020) further improves the numerical results compared to FFT full field results. However, it needs a more complex numerical treatment of cyclic and nonproportional loadings, which has not been solved yet.

Following the recent findings and EBSD analyses of Mollens et al. (2022b), a multiscale modeling strategy accounting for the relevant scales must be considered when dealing with the complex microstructure of austenito-ferritic (duplex) steels. In the present paper, a new microstructurebased three-scale homogenization model including two-scale transitions will be developed and discussed. The paper is organized as follows. In section 2, the duplex stainless steel is presented.

The chemical composition, the microstructure and the tensile mechanical properties are reported.

In addition, an EBSD analysis is proposed to identify the different variants and the heterogeneities in the microstructure. In section 3, the micromechanical model based on a three-scale analytical homogenization-based method for elastoviscoplastic heterogeneous materials is developed and specifically applied to duplex stainless steels. In section 4, the results of the model are presented at different scales denominated micro-, meso-, and macro-scales, and the strain/stress fields in both phases are predicted for the different KS-OR variants present in the primary ferritic grains.

The effects of ferrite grain orientations and aging are numerically studied. The different internal stresses present in the microstructure are analyzed in the light of this new approach through cyclic loading responses and a backstress analysis.

Material

Microstructure and mechanical properties

The material of interest is a CF8M alloy (AFNOR Z3CND1910), which is representative of cast duplex stainless steels used in the primary loop components of Pressurized Water Reactors (PWR) in France. Its chemical composition is given in Table 1. Its microstructure has been characterized by EBSD in the previous study of Mollens (2022a), see Figure 1. It is made of primary polygonal ferritic grains of 200 to 1000 micrometers partially transformed with respect of an orientation relationship into Widmanstätten austenite laths of 10 to 50 micrometers width. The retained ferrite phase accounts for about 25% of the total volume fraction. It is located between the austenite laths. Here, only the cast microstructure will be thoroughly modeled, but we will discuss the prospective effect of aging by considering a higher flow stress for ferrite, which may be due to precipitation during aging. The mechanical behavior of the material at the as-received state and after thermal aging at 350°C during 1000h has been characterized by Mollens (2022a) with tensile tests using normalized TC6 specimens. The resulting tensile curves are shown in Figure 2. Thermal aging induces a small increase of the yield stress, an increase of the ultimate tensile stress and a decrease of the elongation at fracture, consistent with previous studies [START_REF] Bugat | Comportement et endommagement des aciers austéno-ferritiques vieillis : une approche micromécanique[END_REF][START_REF] Michaud | Tensile Property Characterization of Thermally Aged Cast Stainless Steels[END_REF]1994).

Figure 2. Tensile behavior measured at ambient temperature on the studied material at the asreceived state and aged 1000h at 350°C (Mollens, 2022a).

Crystallographic and morphological features of the microstructure

EBSD data shown in Figure 1 were first analyzed to characterize the local orientation correlations between both ferrite/austenite phases. Figure 3 displays the local deviation from KS-OR at the ferrite/austenite interfaces. The duplex microstructure was found to display a significantly higher content of close KS-OR interfaces, which are represented with white and yellow interfaces in Figure 3. However, no unique and strict orientation relationship exist between both phases, as already mentioned in previous work (Mollens, 2022b). Moreover, close to some primary ferrite boundaries, the austenite phase exhibit no longer a lath-like morphology and no OR is observed.

This happens mainly for austenite phase that have grown at each side of a primary ferrite grain with close orientation. The EBSD data were further investigated to analyze the trace of ferrite/austenite interfaces respecting KS-OR. The result shows that most of the traces are consistent with a {110} α plane of the primary ferrite grain, which acts as Habit Plane (denoted HP in the following) in the reverse KS -OR (see an example in Figure 4). Assuming close KS-OR for the ferrite to austenite transformation, Table 2 gives the 24 potential variants obtained from a primary ferrite grain (defined by the orientation (0,0,0) for ferrite). For each variant, the corresponding HP is indicated as well as the Bain group it belongs to. Packets of 4 variants sharing the same HP are identified by a similar color. Inside such a packet, the 4 variants belong 2 by 2 to two Bain domains. Finally, from the analysis of the experimental EBSD data (Mollens et al. 2022b), four physical scales can be identified to describe the complex microstructure of the as received cast duplex steel. Figure 5 reports the microstructure with four identified physical scales from EBSD analysis, and the schematic principle of the three-scale homogenization model based on two scale transitions assuming that macro-homogeneity conditions are satisfied. The first physical scale is defined by the single crystalline austenite lath or ferrite interlayer, where classic crystal plasticity will be used in the model to describe the small strain viscoplastic behavior of both phases at the microscale. The austenite lath is characterized by a smooth orientation spread. This lath results from a displacive transformation according to a close {110} α HP and respecting closely the KS-OR with the surrounding ferrite grain. The second physical scale is dictated by a cluster of adjacent austenite laths, characterized by constrained morphological and crystallographic parameters. In a cluster, the austenite laths often share the same orientation with an increased orientation spread.

They are separated by layers of primary ferrite. In this contribution, this cluster is modeled at the mesoscale with a two-phase laminate structure (LS) homogenization model. The LS model will be used as scale transition rule from microscale to mesoscale, see Figure 5. The normal vector to the interface between both phases in the cluster is of [110] α type (see Table 2). The third physical scale is defined by the primary ferritic grain that contains all the austenite lath clusters. We can notice that adjacent lath clusters often include two austenite variants belonging to the same Bain group. We will consider all the possible 24 KS-OR variants in the modeling. Finally, the fourth physical scale is defined as an ensemble of primary ferritic grains. Here, the EVPSC scheme based on the Translated Field (TF) method will be used as scale transition rule from mesoscale to macroscale with different representative volume elements (RVE) defined as a single primary ferritic grain (third physical scale) or as an ensemble of primary ferritic grains (fourth physical scale), see Figure 5. The crystal plasticity parameters of the three-scale model will be calibrated from experimental tensile curves considering an ensemble of 15 primary ferritic grains (see Table A.2 in Appendix A).

Three-scale model

The modeling of the elastoviscoplastic behavior of duplex stainless steels is derived following the different scales and modeling assumptions reported in Figure 5. The TF-based EVPSC model is reconsidered with two homogenization steps in a three-scale model. The first step is a homogenization procedure for the two-phase (ferrite/austenite) LS considering an elastoviscoplastic framework with "affine"-type formulation for both phases. Stress and strain concentration equations allow to give an estimate of mean fields in both phases accounting for the stress/strain incompatibilities due to the presence of the HP. The homogenized properties of the LS obtained at this first step (mesoscale model) will be incorporated in the EVPSC scheme for the second homogenization procedure. This one considers an ensemble of two-phase LS constrained by spherical domains corresponding to the different KS-OR variants. The 24 variants (see Table 2) will be considered with equal volume fraction for each primary ferritic grain. Then, the macroscale model is built up with the homogenized elastoviscoplastic behavior for one or 15 primary ferritic grains as reported in Table A.2 in Appendix A. In the following, the model is constituted of four pillars. The elastoviscoplastic framework with "affine"-type formulation is reported in section 3.1.

Then, the meso-to macro-scale model is first derived using the TF-EVPSC scheme in section 3.2. All the equations of the micro-to meso-scale model based on the two-phase LS is detailed in section 3.3. Finally, the constitutive laws for austenite and ferrite at the single crystalline level, i.e., the microscale, are given in section 3.4.

The computational flowchart of the three-scale homogenization model highlighting both scale transitions (TF-EVPC scheme and LS-model) is reported in Figure 6, using the same notations as in the text. Boundary conditions at the different scales are derived from those at the macroscopic scale using macro-homogeneity condition. We assume that the average fields in two-phase LS grains, obtained from stress and strain concentration equations derived from the TF-EVPSC serve as boundary conditions at the mesoscale assuming homogeneity condition at this scale.

Throughout the development of the model, bold characters are used to define rank 2 and 4 tensors in vector notations, ":" denotes the contracted product between 2 tensors. The symbols and denote the ensemble volume averages at the macroscale and the mesoscale, respectively. Indicial notations are used using Cartesian coordinates with the Einstein's summation convention.

Figure 6. Flowchart of the developed three-scale elasto-viscoplastic homogenization model (see notations as in the text).

Elasto-viscoplastic framework and "affine" linearization

Within the infinitesimal strain framework, the total local strain rate tensor relative to an elastoviscoplastic behavior of Maxwell type is decomposed into:

(1) with the elastic strain rate tensor obtained from the generalized Hooke's law, where is the local elastic compliance tensor, is the Cauchy stress tensor, and is the nonlinear stress dependent constitutive law (plastic flow rule) to derive the viscoplastic strain rate tensor. It will be determined from the constitutive laws of single crystalline austenite laths and ferrite interlayer (see subsection 3.4). As performed in [START_REF] Mareau | An affine formulation for the self-consistent modeling of elastoviscoplastic heterogeneous materials based on the translated field method[END_REF], its first order "affine" linearization can be written as follows at a given reference stress:

(2) where is the tangent viscoplastic compliance tensor (respectively the tangent viscoplastic modulus tensor) defined by:

(3) and corresponds to the back-extrapolated viscoplastic strain rate tensor. The Cauchy stress tensor can also be expressed as a nonlinear function of and then could be written in its linearized form at a given reference viscoplastic strain rate as:

(4) where is the back-extrapolated stress tensor related to the back-extrapolated viscoplastic strain rate by . Therefore, the final linearized form of the elastoviscoplastic constitutive law is:

(5)

Meso-to macroscale transition: TF-based EVPSC

Schematic principle and macroscopic averages

In this paper, the scale transition from a single two-phase LS at the mesoscale to the macroscale (one primary ferritic grain or an ensemble of ferritic primary grains) is assessed through the Translated Field (TF) based EVPSC method with affine-type extension. This TF-EVPSC scheme was first developed in [START_REF] Mareau | An affine formulation for the self-consistent modeling of elastoviscoplastic heterogeneous materials based on the translated field method[END_REF] and more recently applied to polycrystalline -Ti alloys by Lahdi et al. (2018Lahdi et al. ( , 2020)). For each laminate denoted "L", the average elastic strain rate tensor is given by the linear elastic Hooke's law: , and, the viscoplastic strain rate tensor is formulated using an affine-type approximation: , or, : + , see equations (2-5).

Strain and stress rate concentration equations

In the TF-EVPSC, the strain rate concentration law is obtained at the level of each LS (here "L" is used to denote the averaged fields on one LS) following [START_REF] Mareau | An affine formulation for the self-consistent modeling of elastoviscoplastic heterogeneous materials based on the translated field method[END_REF] and Lhadi et al. ( 2018):

where the Eshelby tensors [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF] associated to and are given by the expressions respectively with the local parts of the modified Green operators [START_REF] Kröner | Modified green functions in the theory of heterogeneous and/or anisotropic linearly elastic media[END_REF]. In this equation, is the applied strain rate tensor. The 4 th rank elastic and viscoplastic strain concentrations tensors and are defined as and respectively, with the fourth order unit tensor, and and

The effective elastic and viscoplastic moduli, and , are given by one site Self-Consistent (SC) scheme iteration loops with a fixed-point algorithm:

(7a) (7b)
The effective back-extrapolated strain rate in equation ( 6) is obtained from:

(8)

Then, from equation ( 8):

(9)

Substituting the strain rate equation (6) in the local constitutive law gives the stress rate in each laminate "L":

(10)

Macroscopic behavior

The effective properties obtained from the averaging equations of homogenization theory ( , and ). The macroscopic effective behavior is given by: , where the effective viscoplastic strain rate which is chosen to impose the macrohomogeneity condition is obtained as follows:

(11)

with .

At the level of LS, we have with and are homogenized elastic compliances (resp. moduli), viscoplastic tangent compliances (resp. moduli) and viscoplastic back-extrapolated term for each LS. These tensors are now determined from the homogenization procedure at the mesoscale.

Micro-to mesoscale transition: two-phase LS model

LS model and averaging procedure

At the mesoscale, we consider an infinite two-phase LS model (see Figure 5). Both phases are ferrite (denoted " ") and austenite (denoted " ") with volume fractions and . For the present application, , therefore . The volume averages of strain and stress tensors over each LS are given by the homogeneity conditions at the mesoscale:

(12a) (12b)
where with . The tensors , , and represent the elastic compliances, the strains, the Cauchy stresses and the viscoplastic strains of both phases at a given stress state during loading. Under the assumption of mean viscoplastic strain field in each phase, following strain compatibility and balance of linear momentum in the absence of body force, the strains and stresses are therefore piecewise uniform in the two-phase LS model (see e.g. El [START_REF] El Omri | Elastic-plastic homogenization for layered composites[END_REF][START_REF] Stupkiewicz | Modelling of laminated microstructures in stress-induced martensitic transformations[END_REF][START_REF] Franciosi | Multiple continuity of phases in composite materials: overall property and estimates from a laminate system scheme[END_REF]Berbenni, 2007, 2008;[START_REF] Berbenni | A micromechanics-based model for shear-coupled grain boundary migration in bicrystals[END_REF][START_REF] Richeton | Effects of heterogeneous elasticity coupled to plasticity on stresses and lattice rotations in bicrystals: A Field Dislocation Mechanics viewpoint[END_REF][START_REF] Tiba | Incompatibility stresses at grain boundaries in Ni bicrystalline micropillars analyzed by an anisotropic model and slip activity[END_REF][START_REF] Franciosi | Multiple continuity of phases in composite materials: overall property and estimates from a laminate system scheme[END_REF][START_REF] Richeton | Stress partitioning and effective behavior of N-phase laminates in anisotropic elasticity from a fast explicit method[END_REF].

Strain/stress concentration equations for the LS model

The relationships between the strains and the stresses in each phase and and the strains and stresses at the level of the LS: are first calculated in the frame associated with the planar interface, i.e., the HP. Its unit outward normal n is supposed to be parallel to the e 2 direction of a Cartesian frame (e 1 , e 2 , e 3 ) and is directed from to (see the mesoscale model in Figure 5).

For the 24 KS-OR, the crystallographic planes and directions of the HP are defined in the ferrite frame. Then, the stress and strain tensors calculated in the HP frame are successively rotated in the ferrite frame and in the global frame. The rotation matrix used for the transformation from the HP frame to the ferrite frame is given by the transformations of the HP plane normal vector [h k l] and in-plane direction vector [u v w] (see Table 2) into the unit vectors e 2 and e 3 , respectively (see Figure 5). The rotation matrix from the ferrite frame to global frame is defined by the three Euler-Bunge angles of the primary ferritic grain (see Table A.2).

In the HP frame, we adopt the following tensor to matrix convention [START_REF] Nye | Physical properties of crystals[END_REF]) also known as the Voigt's notation, in which pairs of subscripts (ij) and (kl) are converted to single subscripts: (11)=1, (22)=2, (33)=3, ( 23) and (32)=4, ( 13) and (31)=5, ( 12) and ( 21)=6. In this notation, we will use indices ranging from 1 to 6. The conventions for strains are the following:

and the components of the compliance matrix contain a multiplying factor of 2 if either or is equal to 4,5 or 6 and the other index is equal to 1, 2, 3, and, a multiplying factor of 4 if both and are equal to 4,5 or 6. Thus, each of the tensor components is associated with a matrix or a vector component.

To determine the strain and stress concentration equations for the mesoscale model, the continuity conditions for stress and strain at the interface are invoked due to traction vector continuity and tangential strain continuity for perfectly bonded interface:

(13a) (13b)
Where the in-plane components of strain and stress tensors are denoted P = 1, 3, 5, and the antiplane ones are denoted A = 2,4,6, with the following constitutive laws using index notations:

(14a) (14b)
Starting from equations ( 14) and using equation (12a) together with equations ( 13), the strain concentration laws for both phases are obtained:

(15a) (15b)
where denotes the jump of fields or property, and, where the non-zero components of the symmetric matrix associated with are given in Appendix B, see equation

(B.1).
Conversely, starting from equations ( 14) and using equation (12b) together with equations (13), the stress concentration laws for both phases are given by (see [START_REF] Richeton | Effects of heterogeneous elasticity coupled to plasticity on stresses and lattice rotations in bicrystals: A Field Dislocation Mechanics viewpoint[END_REF][START_REF] Tiba | Incompatibility stresses at grain boundaries in Ni bicrystalline micropillars analyzed by an anisotropic model and slip activity[END_REF]:

(16a) (16b)
where the non-zero components of the symmetric matrix associated with are given in Appendix B, see equation (B.2).

Purely elastic homogenized behavior

Now, let us determine the effective elastic compliance (resp. modulus ) of each LS such that or . Using equation (12a) together with the linear elastic Hooke's law, we obtain:

(17)

Then using both equations ( 16) with , together with equation ( 17), the homogenized elastic compliance for "L" reads (see [START_REF] Richeton | Effects of heterogeneous elasticity coupled to plasticity on stresses and lattice rotations in bicrystals: A Field Dislocation Mechanics viewpoint[END_REF][START_REF] Franciosi | Multiple continuity of phases in composite materials: overall property and estimates from a laminate system scheme[END_REF][START_REF] Richeton | Stress partitioning and effective behavior of N-phase laminates in anisotropic elasticity from a fast explicit method[END_REF]):

(18)

with .

Using equation (12b) together with the linear elastic Hooke's law, we obtain:

(19)
Then, using both equations ( 15) with together with equation ( 19), the homogenized elastic stiffness for "L" reads:

(20) with .

Purely affine viscoplastic homogenized behavior

For purely affine viscoplastic two-phase LS, we consider the following thermo-elastic like linear comparison material for both phases and :

(21a) (21b)
with the following continuity conditions for the purely viscoplastic problem:

(22a) (22b)
and the volume averages on the laminate:

(23a) (23b)
For this purely viscoplastic problem, similar strain concentration equations are found by substituting in equation ( 15 It is noteworthy that equations ( 18) and ( 20) and equations ( 26) and ( 30) are consistent from the homogenization theory since (see e.g. [START_REF] Qu | Fundamentals of micromechanics of solids[END_REF].

Furthermore, it can be seen from these expressions that the method is not restricted to iso-stress (Reuss) or iso-strain (Voigt) assumptions. These ones are used to model duplex steels in the literature. For instance, [START_REF] Verhaeghe | Internal stresses in an austenoferritic duplex stainless steel[END_REF] and M'Cirdi (2000) used a Taylor (1938)'s model, which is equivalent to iso-strain assumption. In section 4, we will show that some HP configurations do not lead to iso-strain states.

Microscale: single crystal's behavior

The local elastoviscoplastic behavior is described within a crystal plasticity framework. The constitutive equations developed by Méric and Cailletaud (1991a,b) are used to model the behavior of crystalline laths in both the f.c.c. structure for austenitic laths and in the b.c.c. structure for ferritic laths. This model depends on a limited number of materials parameters. The viscoplastic strain rate is given by crystal plasticity in both phases:

(32) Equation ( 32) describes the sum of the contributions of all active slip systems (i.e. in each single crystalline phase. The driving force for slip is given by the resolved shear stress , defined as:

(33) with the symmetric Schmid orientation tensor:

where and are unit vectors normal to slip plane and parallel to slip directions.

The behavior of the single crystal (cubic crystals for both phases) is supposed to be elasticviscoplastic for both phases at small strains, where plastic distortion results from crystallographic slips on specific systems defined by a plane normal and a slip direction . Here, the effect of lattice rotation evolution is neglected as in previous studies on the same material (see [START_REF] Bugat | Micromechanical modeling of the behavior of duplex stainless steels[END_REF][START_REF] Bugat | Comportement et endommagement des aciers austéno-ferritiques vieillis : une approche micromécanique[END_REF]. For f.c.c. crystals (austenite), 12 octahedral slip systems of type (111)

<110>. Following previous studies in duplex steels [START_REF] Bugat | Micromechanical modeling of the behavior of duplex stainless steels[END_REF][START_REF] Bugat | Comportement et endommagement des aciers austéno-ferritiques vieillis : une approche micromécanique[END_REF][START_REF] Verhaeghe | Internal stresses in an austenoferritic duplex stainless steel[END_REF][START_REF] Verhaeghe | Damage and rupture mechanisms in an austenoferritic duplex steel[END_REF], we assume that only the dense systems (110) <111> are activated in the ferrite (b.c.c). This assumption is made due to the KS-OR, which favors such family of slip systems in the b.c.c. structure.

The phenomenological model with hardening variables developed by Méric and Cailletaud (1991a,b) is used to describe the viscoplastic slip rate : [START_REF] Molinari | On the self-consistent modeling of elastic-plastic behavior of polycrystals[END_REF] where the Macaulay brackets describe the ramp function. In equation [START_REF] Molinari | On the self-consistent modeling of elastic-plastic behavior of polycrystals[END_REF], and are the kinematic and the isotropic hardening shear stresses, respectively, given by their rates:

(36) and:

(37)

where .

In the above equations, and are two material coefficients characterizing the viscous effect, ,

, and are hardening parameters, is an matrix describing the interactions between the different slip systems, where is the total number of the slip systems in both austenite and ferrite single crystals (here for both the f.c.c. and b.c.c. slip systems, see Appendix C).

Therefore, at the grain level, the viscoplastic strain rate tensor is given by:

(38)

According to the "affine" formulation used in this study (equation ( 2)), the tangent viscoplastic compliance tensor and the back-extrapolated viscoplastic strain rate tensor write in index notation, as:

(39) and:

(40) using equation ( 33). Then, equations ( 39) and ( 40) are used in equations ( 30) and (31).

Simulation results

Model parameters

To determine materials parameters, the tensile response of the CF8M material is predicted with the three-scale homogenization model, where 15 reconstructed primary ferritic grains are considered with volume fractions extracted from EBSD data, and crystallographic orientations given in Table A.2 (Appendix A). These orientations are defined with respect to the global Cartesian frame (X, Y, Z). Each primary ferritic grain is made of 24 laminate structures (LS) (lath clusters) included in spherical domains with equal volume fractions . It is then assumed that the occurrence probability of each variant is equal with no specific variant selection. This assumption follows the EBSD analysis, which does not display the predominance of specific variants. Therefore, the Representative Volume Element (RVE) used for the three-scale model to identify materials parameters contains 360 spherical grains with LS structures (see Figure 5).

First, the elastic properties of the material are supposed to be isotropic with (Young's modulus) and (Poisson ratio). The crystal plasticity hardening parameters were identified on two tensile curves obtained for the non-aged material (Figure 7(a)) and the aged one (Figure 7(b)), and for an applied strain rate of in the X-direction (tensile direction).

Following the parameter identification performed by [START_REF] Bugat | Comportement et endommagement des aciers austéno-ferritiques vieillis : une approche micromécanique[END_REF] for similar duplex steels, we considered the same viscosity parameters: = , K=20, and we also assumed that . Therefore, the identified crystal plasticity parameters are gathered in Table 3 for the nonaged and aged material. Table 3. Viscoplastic flow rule and hardening parameters (Méric-Cailletaud's law) for both phases identified on the non-aged and aged experimental tensile curves. Numerical comparisons with unit cell calculations using the EVPFFT model.

Numerical assessment for one KS-OR variant at mesoscale

Austenite

In a first part, it is important to discuss and validate the micromechanical model chosen at the mesoscale in Figure 5. One KS-OR variant is chosen, i.e. the first variant among the list of the 24 variants denoted V1 in Table 2. For these simulations, the Cartesian coordinate system (X, Y, Z) is fixed with the HP (i.e., the interface between both phases) as shown in Figure 8, with unit normal n parallel to the Y-axis (// 0] ), while the X-axis and the Z-axis (// are parallel to the HP (see Figure 8).

The volume fraction of ferrite (blue colored phase in Figure 8) is set to 25% to be consistent with the experimental data, and the Euler-Bunge angles for ferrite in this configuration is given by:

. The volume fraction of austenite (red colored phase on Figure 10) is set to 75% with Euler-Bunge angles: .

The monotonic tensile responses simulated with the mean field model along the X-, Y-,directions with applied strain rate are first compared to full-field calculations performed with the EVPFFT formulation (i.e. the elastoviscoplastic fast Fourier transform). In Figure 9, the average stresses vs. average strains over the LS (black colored curves), ferrite (blue colored curve) and austenite (red colored curve) are reported. Here, the EVPFFT formulation, first developed by [START_REF] Lebensohn | An elasto-viscoplastic formulation based on fast fourier transforms for the prediction of micromechanical fields in polycrystalline materials[END_REF], was adapted with the numerical implementation of the Méric and Cailletaud (1990a,b)'s crystal plasticity laws reported in section 3.4 (see dotted curves). It is then checked that a very good match is obtained between both methods. However, the analytical model is more relevant for multiscale computations (see flowchart in Figure 6), because it does not need full field calculations and is less computational than EVPFFT.

In Figure 10, it is shown that the largest stress differences between both phases and between the overall behavior of the laminate composite are obtained for the the X-and Z-tensile loading directions, i.e., parallel to the HP. Then, large incompatibility stresses are observed for these tensile directions, while uniform strain is present. In contrast, due to traction continuity at the HP, the same stress is obtained in both phases for the case where the tensile direction is parallel to the Y-direction, but the strains are not uniform anymore. Therefore, the classic iso-strain assumption frequently used in the literature is not valid anymore in this configuration. The responses of the LS model are now compared with the individual single crystalline responses, which are simulated separately for same boundary conditions (see the curves with dotted lines in Figure 10). The objective is to study the influence of the strain-hardening behavior of each phase compared to the role of incompatibility stresses on the stress/strain responses and on slip activity.

Regarding the X-direction, a larger hardening rate is observed in the case of the LS in both phases

Mechanical behavior of one primary ferritic grain containing 24 KS-OR variants: tensile loading along the [100] direction

Let us consider a first case where the calculations are performed in the Cartesian frame related to ferrite. Hence, the crystallographic orientation of ferrite is supposed to be given by the Euler-Bunge angles: . In this case, the Euler-Bunge angles for the 24 KS-OR variants are given in Table 2. The RVE used for the three-scale model contains 24 spherical grains with LS structures (see Figure 5) for one primary ferritic grain. It is subjected to a tensile loading along the X-axis with applied strain rate .

Figure 11 shows the distribution of stresses along the tensile direction in ferrite ( corresponding to the stresses parallel to [100] , as a function of applied tensile strain ( .

These calculations are useful to determine the stresses normal to a cleavage plane in ferrite ( which are {100} -type plane [START_REF] Bugat | Micromechanical modeling of the behavior of duplex stainless steels[END_REF][START_REF] Bugat | Microstructure and damage initiation in duplex stainless steels[END_REF][START_REF] Bugat | Comportement et endommagement des aciers austéno-ferritiques vieillis : une approche micromécanique[END_REF][START_REF] M'cirdi | Comportement et eendommagement sous sollicitation mécanique d'un acier austénoferritique moulé vieilli[END_REF][START_REF] M'cirdi | Experimental approach of a crystallographic cleavage criterion in a cast aged duplex stainless steel[END_REF]. Three groups of variants with different mechanical behaviors can be clearly identified: two groups exhibit stresses reaching a range between 500-600 MPa at 10% of tensile strain, and one group reflects higher stresses exceeding 1000 MPa at 10% of tensile strain.

Figure 11. Stresses in ferrite as a function of macroscopic tensile strain (in the X-direction or along

[100] ) for the 24 KS-OR variants.

The analysis of these three groups in terms of local strains and stresses in both phases is reported. The first group corresponds to variants V1, V3, V5, V7, V13, V15, V21, V23 with only Bain group B1 (see Figure 12). The second one corresponds to variants V2, V4, V6, V8, V14, V16, V22, V24 (see Figure 13), and the third one corresponds to variants V9, V10, V11, V12, V17, V18, V19, V20 (see Figure 14). We notice that each of the three groups contains specific HP. According to Table 2, it is seen that the first two groups with lower stresses (Figures 12 and13) contain HP: H1, H2, H3, H4, whereas the third group, which give higher stresses (Figure 14) is only constituted of HP: H5 and H6. Therefore, it is shown that the HP parallel to the loading direction (X-axis) like H5 and H6 give to highest strains and stresses in ferrite associated with these variants. The stresses in ferrite and austenite for these variants are much different from the averaged stress at the mesoscale for each variant (green colored solid lines) and the macroscopic average stress at the scale of the primary ferritic grain (black colored solid lines). The strains in both phases are uniform. This mechanical behavior looks like the single LS responses in Figures 10(a) and 10(c).

In the case of HP: H1 to H4, they are oriented at 45° with respect to the X-tensile loading axis. For each variant with H1 to H4, stress difference is not observed but a large strain discrepancy is present between both phases and the mesoscale and macroscopic average responses (see Figures 12 and13).

Effect of the primary ferritic grain crystallographic orientation on the mechanical responses

In this section, the role of primary ferritic grain orientation is investigated. Then, we consider here the crystallographic orientation of the first ferritic grain in the list:

(see Table A.2 in Appendix A). It corresponds to the largest grain of the specimen (green colored on the left part of Figure 1). Figure 15 shows the stress heterogeneities in ferrite among variants for this orientation (here the stress component along the X-direction). Overall, these stress heterogeneities are more important than for the initial orientation , see Figure 11. The clear separation into three well defined groups of variants as in Figure 11 vanishes. For each previous group, there is now a large scatter of stress responses in ferrite due to primary ferritic grain orientation which breaks the symmetries of the HP in the KS-OR variants with respect to the loading axis. Figure 16 shows the axial stresses for austenite, ferrite and the averaged responses at the mesoscale and at the macroscale for the first group (V1, V3, V5, V7, V13, V15, V21, V23). It is shown that the highest stresses in ferrite are found for variants V5, V7, V21, V23. Figure 17 shows the axial stresses for austenite, ferrite and the averaged responses at the mesoscale and at the macroscale for the second group (V2, V4, V6, V8, V14, V16, V22, V24).

It is shown that the stresses in ferrite are lower than the ones in the first group: the highest stresses are found with V2, V14, V16. In these groups, the variants V6, V8, V13, V15 still exhibit very low strains in ferrite compared to austenite as in the case of the initial orientation (see Figures 12(b) and 13(b)). In the third group of variants (V9, V10, V11, V12, V17, V18, V19, V20), the stresses in ferrite remain high (around 1000 MPa at 10% of strain) and the strains are more uniform, see Figure 18, which corroborates the results found in Figure 14. However, there are now small discrepancies between the different variants of this group due to the crystallographic orientation of the ferritic grain.

. Simulation of aging effects

Since the microstructure of ferrite evolves during aging, it becomes harder and more brittle, these effects can be simulated by considering different hardening parameters for ferrite only ( and q), while assuming the same materials parameters for austenite. The materials parameters were identified using the tensile curve of the aged specimen until 6% of strain (see Table 3 and Figure 19 displays the comparisons between stress/strain responses according to the X-direction obtained for the three groups of variants with materials parameters identified for the as-received and the aged specimen. The first group is represented by variants 1, 3 (H1, B1), see Figures 19(a,b). The second group of variants is represented by variants 2, 4 (H2, B2), see Figures 19(c,d). The third group of variants is represented by variants 9, 11 (H5, B2), see Figures 19(e,f).

It is seen that aging is at the origin of an increase of the stress in ferrite. Therefore, incompatibility stresses and strains between both phases are more important in the case of the aged specimen.

This trend has a consequence in the internal stresses (i.e., backstress) at the different scales (see section 4.7).

Multiscale stress and strain heterogeneities

An interesting application of the present three-scale homogenization model is to study multiscale stress and strain heterogeneities, which is not possible with a simple two-scale homogenization model.

In Figure 20, we report stress/strain states at both micro-and meso-scales recorded at 5% of macroscopic strain, with their component along the tensile direction (X-direction). Two orientations are considered for the single primary ferritic grain:

(tensile loading along the [100] direction), see Figures 20(a,b), and:

in Figures 20(c,d). Figures 20(a,c) correspond to non-aged cases while

Figures 20(a,c) correspond to aged ones. For all cases, it is clearly observed that the highest stress and strain heterogeneities are present at the microscale in ferrite (see blue circles).

Stress and strain heterogeneities in austenite (see red circles) are always lower than for ferrite.

In Figures 20(a,b), three groups of variants are clearly visible to give three different internal mechanical states: one is close to the iso-strain state, while the two others are close to the isostress state. As seen in section 4.3, these groups depend on the HP orientation with respect to the loading direction. Stress and strain heterogeneities at the mesoscale (average stresses and strains in lath clusters, see black circles) are much lower in magnitude but are far from specific internal states like iso-strain or iso-stress. This result proves again that a simple Voigt model (iso-strain assumption) is not applicable to derive the mechanical state at the mesoscale. When the orientation of the single primary ferritic grain is changed, Figures 20(c,d) display more stress/strain heterogeneities at micro-and macro-scales due to a larger crystal orientation spread. Such trend looks like polycrystalline non-uniform deformation and stress distributions (see e.g., [START_REF] Nicaise | Coupled effects of grain size distributions and crystallographic textures on the plastic behaviour of IF steels[END_REF]. However, here, two different scales are considered for strain and stress heterogeneities and those in ferrite (at the microscale) remain the largest. Figure 20 shows that for all studied cases, the multiscale modelling and simulations have been truly

accomplished and the averaging strain and stress relationships between scales are satisfied. (see e.g., [START_REF] Evrard | Polycrystalline modeling of the cyclic hardening/softening behavior of an austenitic-ferritic stainless steel[END_REF][START_REF] Wang | Polycrystal plasticity modeling for load reversals in commercially pure titanium[END_REF]. In this duplex steel, the development of internal stresses at the different scales is analyzed in the light of the present model to study the Bauschinger effect, which is defined as the difference in plastic flow stress after reversed deformation [START_REF] Atkinson | The work-hardening of copper-silica. IV. The Bauschinger effect and plastic relaxation[END_REF][START_REF] Corbin | The Bauschinger effect in a particulate reinforced Al alloy[END_REF]. Such study was investigated in as received and aged duplex stainless steels by [START_REF] Verhaeghe | Internal stresses in an austenoferritic duplex stainless steel[END_REF]. Even though the Méric-Cailletaud's crystal plasticity model can be used with intra-crystalline kinematical hardening (through calibrating parameters c and d in equation ( 36)), we have not considered it here, since no experimental cyclic tests were performed to calibrate these parameters on this specific material.

At the mesoscale, the incompatibility stresses are dependent on the HP orientation with respect to the loading direction (see Figure 10), which lead to a backstress originating from inter-phase stress accommodation. Figure 21 reports simulated cyclic responses (2 cycles) performed in tension (forward loading) and compression (reverse loading) for the case of one primary ferritic grain with 24 KS variants (i.e., 24 grains with two-phase LS), and for the case of 15 primary ferritic grains with 24 KS variants in each grain (i.e., 360 grains with two-phase LS). Figures compare the cyclic responses of the case of an ensemble of 15 primary ferritic grains with no aging and with aging, respectively. Following [START_REF] Atkinson | The work-hardening of copper-silica. IV. The Bauschinger effect and plastic relaxation[END_REF], [START_REF] Corbin | The Bauschinger effect in a particulate reinforced Al alloy[END_REF] and [START_REF] Verhaeghe | Internal stresses in an austenoferritic duplex stainless steel[END_REF], the overall backstress at the macroscale ( reflecting the Bauschinger effect is defined as , where is the forward flow stress at 2% of strain and is the reverse stress at the onset of plasticity in compression during the first cycle.

21(a

The results for the simulations are gathered in Table 4. It is shown that the increase in backstress due to aging is important. This increase varies from 51 to 65 MPa, see Figures

Conclusions

In this paper, a three-scale microstructure-based elastoviscoplastic homogenization model is presented for the first time to predict the local and the overall behavior of a CF8M alloy, which is a cast duplex stainless steel used primary loop components of Pressurized Water Reactors (PWR).

Mechanical tensile tests have been performed on as received (cast) and aged (1000h, 350°C) alloy to identify the crystal plasticity parameters based on the Méric-Cailletaud (1991a,b)'s model at the level of single austenite laths and ferrite interlayers. The main conclusions of the paper are the following:

 The model is able to describe the average mechanical fields (stress and strain fields) at the different scales of the duplex steel microstructure: (i) in the austenite laths and in the ferrite interlayer related with a KS-OR, (ii) at the level of a cluster of austenite laths in the twophase LS structure, (iii) at the level of primary ferritic grains, (iv) at the level of an ensemble of primary ferritic grains.

 The morphology due to the cluster of austenite laths is responsible for a strong anisotropic behavior observed from the mechanical responses of LS structure satisfying a pure KS-OR.

This trend was confirmed with comparisons with EVPFFT simulations with same crystal plasticity laws. The slip analysis confirms the important effects of incompatibility stresses dependent on the local configuration defined by the HP and the tensile directions.

Generally, the individual responses of both phases are not sufficient to represent the stress/strain responses of the LS structure, except for the case where the loading direction is parallel the crystallographic direction in ferrite.

 In the case where the tensile loading is performed according to the crystallographic direction in ferrite, it is found that the tensile stresses in ferrite strongly depend on the HP orientation with respect to the loading direction, where we have identified three groups of KS variants with similar mechanical behavior. The effect of primary ferritic grain crystallographic orientation is important and changes the internal mechanical state in ferrite for the different variants belonging to the three above identified groups. It was also observed that stress heterogeneities are more pronounced inside the primary ferritic grain.

 The effect of aging was discussed assuming that the microstructure of ferrite has changed, affecting only the intra-granular isotropic hardening parameters of ferrite. These parameters were identified using the tensile stress/strain curve for the aged specimen. Then, aging was found to enhance the mechanical contrast between both phases and increase the incompatibility stresses.

 Finally, the Bauschinger effect was studied in the light of the model with and without aging for cyclic tension-compression loading for different microstructures. It was shown that the backstress depends on ferritic grain orientation and aging. The largest backstress was found for the aged material constituted of 15 primary ferritic grains constituted of 24 spherical domains with two-phase LS, representing lath clusters in each ferritic grain.

Some interesting research directions remain to do with the present developed three-scale model.

First, we will aim to perform some predictions in comparison with in situ mechanical tests extracting kinematical strain fields like in situ digital image correlation (see e.g. Guéry et al., 2016), high-resolution digital image correlation (HR-DIC) [START_REF] Harte | A statistical study of the relationship between plastic strain and lattice misorientation on the surface of a deformed Nibased superalloy[END_REF][START_REF] Maresca | Measurement and prediction of the transformation strain that controls ductility and toughness in advanced steels[END_REF], or, lattice strain evolution in the elastoplastic transition (in situ X-Ray diffraction, see e.g. [START_REF] Wang | Stress and strain relaxation in magnesium AZ31 rolled plate: In-situ neutron measurement and elastic viscoplastic polycrystal modeling[END_REF][START_REF] Gadalinska | Stress localisation in lamellar cementite and ferrite during elastoplastic deformation of pearlitic steel studied using diffraction and modelling[END_REF]. Bauschinger tests on cast and aged specimen can be performed to identify kinematical hardening parameters at the single crystalline level. The hydrostatic pressure fields, as predicted by the present three-scale model, as well as slip activities can be used to predict damage initiation during aging. The present model can be applied to other two-phase complex microstructures involving laminate structures like pearlitic steels [START_REF] Delannay | Modeling of microscopic strain heterogeneity during wire drawing of pearlite[END_REF] or twophase titanium alloys with Burgers-OR [START_REF] Fan | Modeling the anisotropy of hot plastic deformation of two-phase titanium alloys with a colony microstructure[END_REF]. Lastly, the present three-scale micromechanical model can be incorporated in structural FE-based calculations using the MFront code generator [START_REF] Helfer | Introducing the open-source MFront code generator: Application to mechanical behaviours and material knowledge management within the PLEIADES fuel element modelling platform[END_REF], as first performed in [START_REF] Tsekpuia | Développement et implémentation numérique d'un modèle d'homogénéisation élastoviscoplastique à base microstructurale pour des aciers inoxydables austénito-ferritiques[END_REF]. Appendix C. Crystallographic slip systems for f.c.c. austenite and b.c.c. ferrite

Table C.1. The 12 slip systems in the f.c.c. structure with Schmid and Boas's notation (1968).

Notation Crystallographic plane Crystallographic direction A2

( 1 1) [0 1] A3

( 1 1) [1 0 1] A6

( 1 1) [1 1 0] B2

(1 1 1) [0 1] B4

(1 1 1) [ 0 1] B5

(1 1 1) (1 0 1)

[ 1 0] C1 ( 1) [0 1 1] C3 ( 1) [1 0 1] C5 ( 1) [ 1 0] D1 (1 1) [0 1] D4 (1 1) [ 0 1] D6 (1 1) [1 1 0] Table C.
[ 1 1] 6A ( 1 0) [ 1 1] 2B (0 1) [1 1 1] 4B ( 0 1) [1 1 1] 5B ( 1 0) [1 1 1] 1C ( 1) [ 1] 3C (1 0 1) [ 1] 5C ( 1 0) [ 1] 1D ( 1) [1 1] 4D ( 0 1) [ 1] 6D ( 1 0) [1 1]

Figure 1 .

 1 Figure 1. Inverse Pole Figure map//Y of the as-cast duplex steel respectively for retained ferrite (upper map) and austenite (bottom map) using EBSD data from Mollens (2022a).

  Figure 3. (a) EBSD map plotting the KS-OR deviation at the ferrite/austenite interfaces for as received cast material, and (b) the corresponding distribution with the angular deviation color code (in °).

Figure 4 .

 4 Figure 4. (a) Geometric definition of the habit plane with habit plane normal n, (b) {110} pole figure for ferrite, (c) {111} pole figure for austenite.

Figure 5 .

 5 Figure 5. Description of the microstructure with four identified physical scales inferred from the EBSD analysis with the presence of lath clusters inside primary ferritic grains (left); Schematic principle of the three-scale homogenization model based on two scale transitions (right): from microscale to mesoscale with the two-phase LS model, and, from mesoscale to macroscale with the SC scheme based on the TF method.

Figure 7 .

 7 Figure 7. Identification of crystal plasticity parameters for both phases using the tensile curves of the as-received specimen (a); and the aged specimen (b).

Figure 8 .

 8 Figure 8. Two-phase laminate configuration for the first KS-OR in the list of Table 2. The calculations are made with the mean field mesoscale model for one laminate structure (LS).

Figure 9 .

 9 Figure 9. Monotonic tensile responses simulated with the mean field model at mesoscale (solid lines) along the: (a) X-, (b) Y-, with applied stain rate compared to full-field EVPFFT calculations with periodic boundary conditions and same applied strain rate (dotted lines). The reported stresses are the average stresses in ferrite, austenite and in the LS (overall).

(Figure 10 .

 10 Figure 10. Monotonic tensile responses simulated with the mean field model at meso-scale (solid lines) along the: (a) x-, (b) y-, (c) z-directions with applied stain rate compared to single crystalline responses with same boundary conditions (dotted lines). The reported stresses are the average stresses in ferrite, austenite and in the laminate (overall).

Figure 12 .Figure 13 .Figure 14 .

 121314 Figure 12. Stress/strain responses according to the X-direction for the first group of variants: (a) 1, 3 (H1, B1); (b) 5, 7 (H3, B1); (c) 13, 15 (H4, B1); (d) 21, 23 (H2,B1)

Figure 15 .Figure 16 .Figure 17 .Figure 18 .

 15161718 Figure 15. Stresses in ferrite for the 24 different KS-OR variants as a function of macroscopic strain according to the X-direction for the ferrite orientation:.

Figure

Figure 19 .

 19 Figure 19. Comparisons between the stress/strain responses according to the X-direction obtained for the first group with variants 1, 3 (H1, B1): (a) no aging, (b) aging; for the second group with variants 2, 4 (H2, B2): (c) no aging, (d) aging; for the third group with variants 9, 11 (H5, B2): (e) no aging, (f) aging.

Figure 20 .

 20 Figure 20. Macroscopic tensile stress/strain response (solid line), and, corresponding stress/strain states recorded at the mesoscale (black circles), and, at the microscale for ferrite (blue circles) and for austenite (red circles) at 5% of macroscopic strain. Single primary ferritic grain orientation: (a) no aging, (b) aging; Single primary ferritic grain orientation: (c) no aging (d) aging.

  ) and 21(b) compare the cyclic responses with no aging and with aging, respectively, for . Figures 21(c) and 21(d) compare the cyclic responses with no aging and with aging, respectively, for . Figures 21(e) and 21(f)

Figure 21 .

 21 Figure 21. Tension-compression cyclic responses (2 cycles) for one single primary ferritic grain with orientation: (a) no aging, (b) aging; one single primary ferritic grain with orientation given by (c) no aging, (d) aging; ensemble of 15 primary ferritic grains: (e) no aging, (f) aging.

  2. The 12 slip systems in the b.c.c. structure with their notations according toLe et al.

Table 1 .

 1 Chemical composition of the studied CF8M material from Mollens (2022c).

	Element C	Mn	Si	Ni	Cr	Mo	S	P
	Content (%wt)	0.032	0.80	1.04	10.4	20.9	2.68	0.0007 0.025

Table 2

 2

	Variants				HP	HP	HP	Bain
					plane	direction	name	group
					(h k l)	[u v w]
	V1	5.77	48.19	5.77	0)		H1	B1:x
	V2	95.77	48.19	5.77			H2	B2:y
	V3:	185.77	48.19	5.77			H1	B1:x
	V4:	275.77	48.19	5.77	)		H2	B2:y
	V5:	6.41	137.87 14.39	)		H3	B1:x
	V6:	48.05	85.70	99.62			H4	B3:z
	V7:	353.59	42.13 194.39			H3	B1:x
	V8:	311.95	94.30 279.62	)		H4	B3:z
	V9:	96.41	137.87 14.39			H5	B2:y
	V10:	138.05	85.70	99.62	)		H6	B3:z
	V11:	83.59	42.13 194.39	)		H5	B2:y

. 24 KS variant definition for b.c.c. to f.c.c transformation according to the KS-OR. Color code means the HP's type (6 different habit planes). The variants are divided into 3 Bain groups, denoted B1 to B3. The list of the 24 group symmetry matrices to generate these variants is given in Table

A

.1, see Appendix A.

Resulting physical scales and modeling assumptions: towards a three-scale homogenization model

Table 4 .

 4 Values of forward flow stress at 2% of strain, reverse stress at the onset of plasticity in compression, and backstress for the simulations of different microstructures with non-aged or aged condition.

	Microstructures and RVEs
	One primary ferritic grain	397	153	122
	RVE: 24 grains with LS
	Non-aged
	One primary ferritic grain	464	90	187
	RVE: 24 grains with LS
	Aged
	One primary ferritic grain	418	100	159
	RVE: 24 grains with LS
	Non-aged
	One primary ferritic grain	465	45	210
	RVE: 24 grains with LS
	Aged
	Ensemble of 15 primary ferritic grains	439	49	195
	RVE: 360 grains with LS
	Non-aged
	Ensemble of 15 primary ferritic grains	503	9	247
	RVE: 360 grains with LS
	Aged

Table A .

 A 2. Averaged crystallographic orientations (Euler-Bunge angles) of reconstructed ferritic grains from EBSD maps. : largest grain used for simulation of the single primary ferritic grain.

	Primary ferritic grains
	1*	18.11	40.12	7.52
	2	25.86	35.40	46.27
	3	150.47	9.73	0.61
	4	106.38	20.88	75.83
	5	140.37	32.08	66.45
	6	190.89	44.42	51.02
	7	178.52	42.59	28.43
	8	195.83	40.65	43.53
	9	109.88	36.43	19.41
	10	195.21	15.2	40.48
	11	353.19	8.85	356.26
	12	229.57	41.94	40.62
	13	89.43	33.77	60.68
	14	177.31	24.8	8.22
	15	319.43	39.12	58.27

*

Acknowledgments

The authors thank the MAI-SN (Materials Aging Institute -Scientific Network) for financial support.

Fruitful discussions with Maxime Mollens (EDF R&D), Stéphane Roux (ENS Paris Saclay) and Lionel Germain (LEM3 Metz) are acknowledged.

Appendix Appendix A. Crystallographic orientation data