Insaf Meriem Boutemine

Zine-Charaf Amir

Mustapha Pacha

Fahima Ameur

Karim Layaida

Hans Yssel

Bene cial role of Pistacia lentiscus aqueous extract in experimental colitis: Anti-in ammatory and potential therapeutic effects

Keywords: Pistacia lentiscus, aqueous extract, DSS, acute colitis, mucus, cytokines, NO

Background: Pistacia lentiscus L. (PL) is a owering plant traditionally used in the treatment of gastrointestinal disorders. The extracts of this plant are endowed with strong pharmacological activities. The aim of our current study was to investigate the anti-in ammatory and potential therapeutic effects of PL leaves aqueous extract (PLAE) against Dextran Sulfate Sodium (DSS)-induced acute colitis.

Materials and methods: The therapeutic effect of PLAE was evaluated after orally administration of 3% DSS alone or concomitantly with PLAE (50, 100 or 200 mg/Kg). Mucosal lesions were assessed by macroscopic and histopathological examination. In this context, hemorrhage, diarrhea, weight loss, and disease activity index (DAI) were determined daily throughout the experiment. In the same way, hematoxylin-eosin and Alcian blue staining of colonic mucosal were used to evaluate respectively mucosal damages and mucus production. Furthermore, the levels of nitric oxide (NO), and proin ammatory cytokines [tumor necrosis factor-α (TNF-α) and interleukin-6 (IL-6)] were measured in plasma, as well as in colonic explants and peritoneal macrophages cultures supernatants.

Results: Administration of DSS+PLAE indicated a signi cant reduction in clinical score of acute colitis DAI compared to DSS alone administration. Interestingly, histological analysis of the mucosa showed that DSS+PLAE-treated groups exhibited almost normal histology evidenced by an intact epithelium structure and less in ammatory cell in ltration in the mucosa. Alcian bleu staining revealed that DSS+PLAE-treated groups displayed almost normal mucus production. Importantly, a signi cant decrease in pro-in ammatory mediators (NO, IL-6 and TNF-α) levels in dose dependent manner was reported in plasma, and culture supernatants of colonic explants and peritoneal macrophages from DSS+ PLAE-treated mice compared to the DSS group. Conclusion: Our results showed that the systemic and local anti-in ammatory activities of aqueous leaves extract of PL improve the clinical signs of acute colitis. Our data suggest that PLAE has bene cial effects and could constitute a promising approach against acute ulcerative colitis by targeting the deregulated immune response.

Highlights

Administration of PLAE (Pistacia lentiscus aqueous extract) during acute DSS induced-colitis:

-Induces reduction in clinical score of acute colitis.

-Prevents colon histological lesions induced by DSS.

-Inhibits mucus production by Goblet cells in acute colitis.

-Induces a signi cant decrease in pro-in ammatory mediators (NO, IL-6 and TNF-α) levels in plasma, and culture supernatants of colonic explants and peritoneal macrophages.

Introduction

In ammatory bowel disease (IBD) constitute a group of chronic and relapsing idiopathic in ammatory disorder of the intestine that include Crohn's disease (CD) and ulcerative colitis (UC) [START_REF] Caruso | Host-microbiota interactions in in ammatory bowel disease[END_REF][START_REF] Geboes | In ammatory Bowel Diseases[END_REF]. The exact factors that trigger the chronic and relapsing intestinal in ammation remain unknown. Multiple lines of evidence suggest that the disease is caused by a con uence of genetic and environmental factors that alter gut homeostasis. It is also thought that IBD results from an aberrant and continuing immune response against endogenous ora and luminal antigens (Zhang and Li, 2014[START_REF] Sou I | Cytokines and Nitric Oxide in Immunopathogenesis of IBD and Potential Therapeutic Approaches[END_REF][START_REF] Toumi | Role of Probiotics and Their Metabolites in In ammatory Bowel Diseases (IBDs)[END_REF]. Although the pathogenesis of both Crohn's disease and ulcerative colitis involves intestinal in ammation, these two disorders differ in several features, including the association with speci c susceptibility loci and the type of immune response and pathology associated with disease [START_REF] Mcgovern | Genetics of in ammatory bowel diseases[END_REF]. CD has long been considered to be driven by Th1 and Th17 responses, whereas UC has been associated with a non-conventional Th2 response [START_REF] Duché | ulcères gastroduodénaux chez l'enfant-Editions techniques-Encyclo[END_REF][START_REF] Dray | Maladie de Crohn et rectocolite hémorragique[END_REF]. Immunological studies have recently focused on the mucosal innate immune responses, such as epithelial barrier integrity, innate microbial sensing, autophagy and the unfolded protein response. During the innate response, the intestine of patients with IBD is massively in ltrated by in ammatory cells, mostly macrophages and neutrophils that release large amounts of pro-in ammatory cytokines, such as TNF-α, IL-6, IL-17, IL-12, and interferon-γ [START_REF] Abraham | Interactions Between the Host Innate Immune System and Microbes in In ammatory Bowel Disease[END_REF][START_REF] Sou I | Cytokines and Nitric Oxide in Immunopathogenesis of IBD and Potential Therapeutic Approaches[END_REF][START_REF] Toumi | Role of Probiotics and Their Metabolites in In ammatory Bowel Diseases (IBDs)[END_REF]). The pro-in ammatory cytokines are involved in the up-regulation of inducible nitric oxide synthase (iNOS) expression as well as nitric oxide (NO) production. In fact, during IBD, the balance between pro-in ammatory cytokines, anti-in ammatory cytokines , and immunoregulatory cytokines (IL-10 and transforming growth factors-β) is disturbed [START_REF] Rafa | IL-23/IL-17A axis correlates with the nitric oxide pathway in in ammatory bowel disease: immunomodulatory effect of retinoic acid[END_REF].

The most commonly used treatments for IBD are 5-aminosalicylates (5-ASA), corticosteroids, immunesuppressors and immunomodulatory therapies that involve the use of humanized monoclonal antibodies mainly against TNF-α. However, treatments targeting the immune system have only an immunosuppressive effect [START_REF] Taylor | Optimization of conventional therapy in patients with IBD[END_REF], and most of them have limitations in side effectsand safety, such as drug dependence, adverse reaction, decline of immune function, and risk of cancer [START_REF] Danese | New therapies for in ammatory bowel disease: from the bench to the bedside[END_REF].

In this sense, it appears important to research and develop an affordable, cost-effective and side-effectfree therapeutic strategy. Therefore, highlighting a new therapeutic approach that targets the deregulated immune response is of great interest. There is growing evidence and focus that herbal remedies can help cure gastrointestinal illness and prevent relapse [START_REF] Ukwe | Evaluation of the antiulcer activity of Olax subscorpioidea Roots in rats[END_REF][START_REF] Bozorgi | Five Pistacia species ( vera, P. atlantica, P. terebinthus, P. khinjuk, and P. lentiscus): A Review of Their Traditional Uses[END_REF][START_REF] Zahouani | Aqueous Leaf Extract of Pistacia lentiscus Improves Acute Acetic Acid-Induced Colitis in Rats by Reducing In ammation and Oxidative Stress[END_REF]). Among those medicines gures Pistacia lentiscus L. (PL), a owering plant growing in the Mediterranean region. In fact, most members of the Pistacia have chemical and therapeutic similarities. Results from several studies have revealed potent pharmacological activities of various parts of this genus plants [START_REF] Paterniti | The Anti-In ammatory and Antioxidant Potential of Pistachios (Pistacia vera L.) In Vitro and In Vivo[END_REF]. The fruits, nuts, resin and leaves of Pistacia lentiscus are used for the treatment of eczema, asthma, kidney stones, throat infections, stomach and diarrhea ache, with astringent, antipyretic, antibacterial, anti-in ammatory and antiviral activities [START_REF] Bozorgi | Five Pistacia species ( vera, P. atlantica, P. terebinthus, P. khinjuk, and P. lentiscus): A Review of Their Traditional Uses[END_REF][START_REF] Paterniti | The Anti-In ammatory and Antioxidant Potential of Pistachios (Pistacia vera L.) In Vitro and In Vivo[END_REF].

The aim of our present study was to evaluate the anti-in ammatory and potential therapeutic effects of aqueous extract of Pistacia lentiscus leaves (PLAE) in the experimental model of Dextran Sulfate Sodium (DSS)-induced acute colitis.

Materials And Methods

Pistacia lentiscus

Pistacia lentiscus L. leaves were collected in East Algeria in December 2017. A specimen of Pisactia lentiscus L. has been identi ed as previously described [START_REF] Boutemine | Gastro-protective, therapeutic and anti-in ammatory activities of Pistacia lentiscus fatty oil against ethanol-induced gastric ulcers in rats[END_REF]).

Preparation of aqueous extract of PL (PLAE)

PL leaves were rst washed with water, dried in the shade, and then powdered with a blender. The powder obtained was then dissolved in bi-distilled water. The mixture was subjected to continuous stirring for a period of 24 hours and then ltered and centrifuged. The supernatant obtained was lyophilised and the extract has been stored.

Acute cell toxicity of PLAEon hepa1c1c7 cell line a-hepa1c1c7cells line culture

Mouse hepatoma hepa1c1c7 (ATCC ® CRL-2026) cell line was cultured in DMEM (Dulbecco's Modi ed Eagle's Medium, Hyclone Laboratories, Logan, Utah, USA) supplemented with 10% of fetal bovine serum (FBS), 100 unit/mL of penicillin and 100 g/mL of streptomycin (Gibco®, Life Technologies, Carlsbad, CA, USA). Brie y, cells at exponential phase were seeded into 96-well plates at 10 5 cells/mL per well. Then, different concentrations of PLAE were added (0.304,0.914,2.74,8.13,24,74,220 and 666µg/mL). Each treatment was tested in triplicate and the control group was given only culture medium without PLAE. The above plate were placed in a 5% CO 2 humidi ed-atmosphere incubator at 37 °C for 24h.

b-MTT assay

The cytotoxic effect of PLAE on hepa1c1c7cell line was determined by 3-( 4, 5-dimethylthiazol-2-yl)-2, 5difenyltetrazolium (MTT) assay. In this sense, at the end of culture time, 20 μL MTT (5 g/L) was added to each well and the plate was incubated at 37 °C for 4 h. The plate was then shaken for 15 min. Finally, the absorbance of each well was measured by microplate reader at 492 nm. The proliferation rate was calculated as A (absorbance) 492 of treated/A492 of untreated (control) cells × 100% (Tankiewicz-Kwedlo et al. 2010).

Induction of acute colitis by sodium dextran sulfate (DSS)

Female BALB/c mice (18-22g) were allowed free access to food and water and were kept under normal conditions with a 12 h dark/light cycle. All our experiments were approved by Ethics Committee of the Thematic Research Agency in Health Sciences (N°58/DFPR/ATRSS). Induction of colitis was performed according to the Okayasu protocol [START_REF] Okayasu | Novel method in the induction of reliable experimental acute and chronic ulcerative colitis in mice[END_REF], Chassaing et al. 2015). Mice received ad libitum 3% (w/v) of Dextran Sulfate Sodium (DSS) (Salt, Reagent Grade, MP Biomedicals, M.W 36-50 kDa, Ref: 160110) dissolved in drinking water for 7 days.

The mice were randomly divided into 6 groups as follows (Fig. 1):

1) DSS group: mice received DSS dissolved in drinking water , plus 200 μl of distilled water by gavage each day (n = 8).

2) DSS+PLAE 1 group: mice received DSS dissolved in drinking water, plus 200 μl of aqueous extract dissolved in distilled water at 50 mg/Kg by gavage each day (n = 8).

3) DSS+PLAE 2 group: mice received DSS dissolved in drinking water, plus 200 μl of aqueous extract dissolved in distilled water at 100 mg/Kg by gavage each day (n = 8).

Effect of PLAE on clinical manifestations of acute colitis

In order to assess the severity of colitis, body weight, stool consistency, and blood in the stool were determined daily throughout the experiment, using the following classi cation system (Azuma et al. 2010b):

Weight loss was noted as follows: score 0: none; score 1: 1 to 5%; score 2: 5 to 10%; score 3: 10-20%; score 4: > 20% weight loss.

Diarrhea was scored as follows: score 0: normal; score 2: loose stools; score 4: watery diarrhea. Blood in the stool was scored as follows: score 0: normal, score 2: slight bleeding; score 4: rectal hemorrhage.

The sum of the three scores allowed us to calculate the disease activity index (DAI), to assess the severity of the in ammation throughout the experiment (Azuma et al., 2010b). The total DAI score ranged from 0 to 12. On day 7, the mice were euthanized, the blood was collected by cardiac puncture and the plasma was separated and stored at -20°C for subsequent analysis. Next, an intraperitoneal washing was performed for the recovery of peritoneal macrophages. After dissection, the colon and cecum of each mouse were isolated by separating them from the small intestine at the ileocecal junction and from the anus at the distal end of rectum. Next, colon was washed using cold PBS to remove faeces and measured. Raw pictures from all groups of mice were taken. Then, each colon was dichotomized as follows:

A fragment was immersed in 4% formaldehyde for histological analysis.

Small fragments were placed in DMEM supplemented with 10% FCS, 100 unit/mL of penicillin and 100 g/mL of streptomycin for explant culture.

2.4.1. Histological evaluation of colonic lesions a-Histological Staining of colonic mucosal with hematoxylin-eosin (H&E) Colon mucosa of mice were xed in 4% buffered formaldehyde and embedded in para n. Tissue sections 2 μm thick were depara nized using xylene and rehydrated through a series of increasingly concentrated ethanol baths. Sections were stained with hematoxylin and eosin. Brie y, the tissue sections were rst immersed in hematoxylin for 3-5 min.After rinsing, they were stained with eosin for 2 min. The histological criteria were based on the degree of tissue structure changes, as well as in ammatory cells in ltration.

Colonic in ammation scores were determined in the basis of previously described system for DSSinduced colitis (Azuma et al. 2010b[START_REF] Sou I | Crude extract of hydatid laminated layer from Echinococcusgranulosus cyst attenuates mucosal intestinal damage and in ammatory responses in Dextran Sulfate Sodium induced colitis in mice[END_REF]. The histological criteria were noted as follows:

For the epithelium: score 0: normal morphology; score 1: Goblet cell loss; score 2: loss of crypt and presence of epithelium; score 3: loss of crypt and epithelium; score 4: loss of crypt and epithelium (large area).

For in ltration: score 0: occasional in ammatory cells in the lamina propria; score 1: increased in ltration in the lamina propria mainly at the base of the crypts; score 2: con uence of the in ammatory in ltrate extending to the mucosa; score 3: con uence of the in ammatory in ltrate extending to the submucosa; score 4: transmural extension of the in ltrate (extensive).

b-Histological staining of colonic mucosal with Alcian Blue

Each piece of colon was stained with Alcian Blue to assess mucus production. This staining will color the contents of the cells pink and the mucus blue. For this, after depara nization and rehydration, colon sections of 2 μm thick were placed in the Alcian Blue dye for 15 min and then in nuclear red for 2 to 3 min. The intensity of the blue color in the crypts re ects a signi cant production of mucus.

Interpretation of all results (H&E and Alcian blue staining) was performed double-blind in the presence of a pathologist. The images were captured with a digital camera (Casio) on an optical microscope (Motic).

Culture of mice peritoneal macrophages

Peritoneal macrophages were obtained by washing the peritoneal cavity of mice using 5 mL of cold PBS (pH = 7.4). The cells suspensions were washed thrice with PBS by centrifugation at 2800 rpm for 5 min. Pellets recovered were re-suspended in DMEM, supplemented with 10% FCS. Cells were allowed to adhere at a density of 10 5 macrophages/mL in a 48-well microplate. After 2 hours, the wells were washed with PBS to remove non-adherent cells. Then, all macrophages were incubated in complete DMEM at 37°C and 5% CO 2 . After 24 hours, the culture supernatants were recovered and then stored at -20°C for the subsequent determination of NO and cytokines levels.

Ex vivo Culture of colonic explants

Ex vivo culture of two pieces of the colon will provide key information regarding the severity of in ammation. In this context, the colonic sections (approximately 0.5 cm), cutlongitudinally, were washed thrice serially in mixture of antibiotics (100 U/mL of penicillin and 0.1 mg/mL of streptomycin). The washed colons explants was then placed in a well (24 well plate) containing a drop of complete DMEM with 1% antibiotics (penicillin & streptomycin), supplemented with 10% FCS. After 2 hours, 500 μL of DMEM were added and the explants were incubated at 37°C and 5% CO 2 for 20h. Then, supernatants were collected and centrifuged for 10 min at 4°C and store at -20°C until determination of NO and cytokines levels.

Nitrite concentrations measurement

The Griess reaction was used to determine nitrite levels as an indicator of NO production, in plasma and culture supernatants of Ctrl, DSS, DSS+PLAE 2 and DSS+PLAE 3 groups, as described by [START_REF] Touil-Boukoffa | Relationship among circulating interferon, tumor necrosis factor-α, and interleukin-6 and serologic reaction against parasitic antigen in human hydatidosis[END_REF]. Brie y, samples were mixed with Griess reagent and distilled water. After incubation, the absorbance was determined at 543 nm by a spectrophotometer. The concentration of nitrites was determined using a standard curve with sodium nitrite.

Measurement of Il-6 and TNF-a levels

The levels of IL-6 and TNF-α were measured in the plasma and the culture supernatants of colonic explants, and peritoneal macrophages of Ctrl, DSS, DSS+PLAE 2 and DSS+PLAE 3 groups, using speci c enzyme immunoassay kits (ELISA), according to the manufacturer's speci cations (Invitrogen ThermoFisher scienti c). Brie y, capture antibodies were xed on 96-well plates. After 24h, the samples were added followed by detection antibodies conjugated with biotin. Then Avidin-HRP followed by tetramethylbenzidine (TMB) was added to reveal the reaction. Finally, the absorbance was measured at 450 nm with an ELISA reader. A standard curve was performed for each assay using recombinant cytokines (Analytical sensitivity: 3.7 pg/mL for TNF-α and < 3 pg/mL for IL-6).

Statistical analysis

Statistical analysis was performed using GraphPad Prism. All data were expressed as mean ± SEM. The statistical signi cance was assessed using Student's t-test, with p < 0.05 being considered signi cant.

Results

Effect of PLAE on cell proliferation and viability

The Acute cell toxicity of PLAE on hepa1c1c7cell line (Fig. 2) was determined by 3-( 4, 5-dimethylthiazol-2yl)-2,5-difenyltetrazolium (MTT) assay. We observed that exposure of cells to up 220µg/mL PLAE (equivalent to raw medicinal material) for 24 h did not reduce the cellular proliferation and viability (Fig. 2) compared to the control group (0µg/mL of PLAE). In addition, our results indicated that exposure of cells to 666µg/mL PLAE (three times more concentrated than raw medicinal material) for 24 h did not alter the cellular proliferation and viability. Our results clearly showed that PLAE has no cytotoxic activity on hepa1c1c7cell line at concentrations used in our experiments.

PLAE decreases the clinical manifestations of DSS-induced acute colitis

We determined daily throughout the experiment hemorrhage, diarrhea and weight loss. These parameters were used to determine the clinical score of acute disease activity index (DAI) (Fig. 3). Our results suggest that concomitant administration of PLAE at 200 mg/Kg with DSS signi cantly reduced the clinical manifestations and clinical score of acute colitis DAI in comparison to DSS administration alone (Fig. 3).

These symptoms and signs of morbidity were more noticeable from day 5 to the end of the experimental period. Administration of PLAE at 100 mg/Kg highly reduced the clinical manifestations and clinical scores of DAI but did not improve mice weight loss induced by DSS (Fig. 3). However, we observed that the concentration of 50 mg/Kg did not improve the DSS-induced clinical manifestations (Fig. 3). Interestingly, the administration of the aqueous extract alone in absence of DSS had no effect on animals and the results were comparable to those obtained for the control group (Fig. 3).

PLAE prevents shortening of the colon during acute DSS-induced colitis

The colon shortening is a marker of in ammation in DSS-induced acute colitis model [START_REF] Okayasu | Novel method in the induction of reliable experimental acute and chronic ulcerative colitis in mice[END_REF]Chassaing et al. 2015). Indeed, we noticed that in DSS group, the colon length (5.38 ± 0.18 cm) was signi cantly shortened as compared to control group (10.63 ± 0.36 cm, p <0.0001). Administration of DSS+ PLAE signi cantly prevented the colon shortening in dose-dependent manner compared to DSS group (DSS+ PLAE 2: 7.66 ± 0.44cm; DSS+ PLAE 3: 8.25 ± 0.61 cm vs DSS group: 5.38 ± 0.18 cm, p <0.001) (Fig. 4). However, the administration of PLAE at 50mg/Kg (DSS+ PLAE 1 group) has no effect on DSS induced-colon shortening (DSS: 5.38 ± 0.18 cm vs DSS+ PLAE 1:5.52 ± 0.26 cm, p>0.5). In addition, the administration of aqueous extract alone (PLAE group) had no effect on colon size and the results were comparable to those obtained for control group (Fig. 4).

Effect of PLAE on colon histological lesions induced by DSS

The observed blooding and changes in colon size in DSS group suggest signi cant tissue alterations. In order to con rm these observations, we performed histological analysis. Colon tissue sections of control mice revealed a well-differentiated, normal-polarity epithelium and well-preserved crypts, as well as a complete absence of in ammatory in ltrate (score 0) (Fig. 5a). In contrast, DSS-treated mice colon displayed signi cant tissue damage as compared to control group (Fig. 5b). These damages are characterized by crypt deletion, complete goblet cell depletion, ulceration, mucosal erosion and replacement of epithelial cells by granulous tissue (score 4). Importantly, an intense in ltration of in ammatory cells into the mucosa and submucosa with a strongly increased size of mucosa-associated lymphoid tissue was observed, as well as the occasional presence of pseudo-polyps (score 4) (Fig. 5b). Interestingly, we observed that treatment with DSS+PLAE 3 signi cantly reduced these tissue alterations (Fig. 5e). In fact, normal crypts were observed, with almost an intact epithelium structure, evidenced by a focal decrease in the height of the epithelium and less cellular in ltration in the mucosa. We noted in some cases the presence of some regenerative ulceration corresponding to remissions (score 1) (Fig. 5e). In DSS+ PLAE 2 group, a locally altered epithelium was observed accompanied by a focal deletion of crypts, and less in ammatory in ltrate compared to DSS group (score 2) (Fig. 5d). However, treatment with PLAE at 50 mg/Kg concomitantly with DSS has no effect on DSS-induced histopathological lesions (score 4) (Fig. 5c). In addition, PLAE group (Fig. 5f) presented histological structure comparable to that of control group (score 0).

PLAE inhibits mucus production by Goblet cells in acute colitis

In order to test the effect of PLAE on mucus production, we stained the colonic tissue sections with the Alcian blue stain. Mucosa of control mice had a normal structure and crypts composed of several welldifferentiated Goblet cells stained blue, re ecting normal mucus production (Fig. 6a). On the other hand, the colonic mucosa of the DSS group was characterized by a hypocrine effect due to the disappearance of Goblet cells and their secretory vesicles. A considerable reduction in blue coloration was observed re ecting a decrease of mucus production (Fig. 6b). Interestingly, colonic mucosa in DSS+PLAE 2 and 3treated mice (Fig 6 d,e) exhibited almost normal mucus production with some degree of focal loss of crypts in the DSS+ PLAE 2 group mucosa (Fig 6d). In addition, the PLAE group showed similar mucus production to the control group (Fig. 6f). Nevertheless, the DSS + PLAE 1 group was characterized by a hypocrine effect and disappearance of Goblet cells comparable to that observed in the DSS group (Fig. 6c).

Our macroscopic and histopathological evaluation revealed clearly that the dose 50 mg/Kg is too low and have no effect on DSS-induced colitis. Hence, the doses 100 mg/Kg and 200 mg/Kg were chosen for additional investigations.

Effect of PLAE on systemic, local and peritoneal macrophages productions of NO in acute colitis

It is well known that DSS-induced colitis is associated with prolonged activation of iNOS and increased NO production at systemic and local levels [START_REF] Sou I | Crude extract of hydatid laminated layer from Echinococcusgranulosus cyst attenuates mucosal intestinal damage and in ammatory responses in Dextran Sulfate Sodium induced colitis in mice[END_REF][START_REF] Sou I | Cytokines and Nitric Oxide in Immunopathogenesis of IBD and Potential Therapeutic Approaches[END_REF]. In order to evaluate the effect of PLAE on systemic and local production of NO, we measured nitrites in plasma and culture supernatants of colon explants. Our results showed that the plasma levels of nitrites in DSS group were signi cantly higher compared to control group (22.04 ± 3.5 vs 4.70 ± 1.95 μM, P <0.05) (Fig. 7a). In contrast, mice treated with DSS+PLAE showed a dose-dependent decrease in systemic levels of nitrites (DSS+PLAE 2: 11.65 ± 6.76 µM, DSS+PLAE 3: 5.38 ± 1.99 μM). In addition, DSS+PLAE 3 group indicated a signi cantly reduced NO level compared to DSS group (5.38 ± 1.99 μM vs 22.04 ± 3.5, p <0.05).

In addition, our data corroborated those measured in culture supernatants of colon explants. Indeed, local NO levels in DSS group were signi cantly higher compared to control group (0.6467 ± 0.1270 vs 0.1698 ± 0.0763 μM /mg of explant) (Fig 7b). On the other hand, treatment with DSS+PLAE signi cantly reduced the level of nitrite in culture supernatants of colonic explants in dose-dependent manner (DSS+PLAE 2: 0.5884 ± 0.1441μM/mg of explant; DSS+PLAE 3: 0.1263 ± 0.0465 μM /mg of explant). Furthermore, when compared to DSS group, local nitrite levels in DSS+PLAE 3 group decrease signi cantly compared to DSS group (0.1263 ± 0.046 vs 0.6467 ± 0.1270 μM /mg of explant, p <0.01) (Fig. 7b).

We also measured the nitrite levels produced by peritoneal macrophages cultured in vitro. Our results showed that nitrite levels in the culture supernatants of peritoneal macrophages obtained from DSS group was signi cantly increased as compared to those from control group (13.64 ± 2.78 vs 6.2 ± 0.32 μM/10 5 cells, p <0.05). Interestingly, NO production by peritoneal macrophages from DSS+PLAE-treated mice was signi cantly reduced compared to DSS group in a dose-dependent manner (DSS+PLAE 2: 6.53 ± 1.69 μM/10 5 cells; DSS+PLAE 3: 4.45 ± 1.98 μM/10 5 cells) (Fig. 7c). In addition, we observed a signi cant decrease of nitrite production in DSS+PLAE 3 treated-group compared to DSS group (4.45 ± 1.98 vs 13.64 ± 2.78 μM/10 5 cells, p <0.05) (Fig. 7c).

Effect of PLAE on systemic and local production of pro-in ammatory cytokines in acute colitis

DSS causes systemic and local in ammation characterized by increased production of pro-in ammatory cytokines [START_REF] Sanchez-Fidalgo | Dietary supplementation of resveratrol attenuates chronic colonic in ammation in mice[END_REF], Shi et al. 2011). Therefore, in order to evaluate the effect of PLAE on systemic production of pro-in ammatory cytokines, we measured plasma levels of IL-6 and TNF-α.

Our results show that the plasma levels of both cytokines in DSS group were signi cantly higher compared to those of control group (IL-6: 31.55 ± 9.95 vs 8.18 ± 1.2 pg/mL; p<0.01, TNF-α: 85.20 ± 9.27 vs 42.5 ± 9.27 pg/mL ; p<0.05) (Figure 8 a and b). In contrast, we observed a dose-dependent decrease in systemic levels of IL-6 and TNF-α in DSS+PLAE-treated mice [DSS+PLAE 2: IL-6: 39.26 ±13.55, TNF-α: 18.85 ± 15.0 pg/mL; DSS+PLAE 3: IL-6: 11.20 ± 1.68 pg/mL, TNF-α: 36.71 ± 0.39 pg/mL)] (Figure 8 a, andb). Furthermore, we observed a signi cant decrease of both pro-in ammatory cytokines in DSS+PLAE 3 treated-group compared to DSS group (IL-6: 11.20 ± 1.68 vs 31.55 ± 9.95 pg/mL; p<0.01; TNF-α: 36.71 ± 0.39 vs 85.20 ± 9.27 pg/mL; p<0.05).

Local production of IL-6 and TNF-α in colonic explant culture were strongly inhibited by PLAE. Levels of both cytokines secreted by colon explants of DSS group (IL-6: 13.54 ± 2.19, TNF-α: 2.28 ± 0.68 pg/mL/mg of explant) were signi cantly higher in comparison to those of control group (IL-6: 3.09 ± 1.2, TNF-α: 0.39 ± 0.083 pg/mL/mg of explant) (Figure 8 c andd). In contrast, treatment with PLAE signi cantly reduced IL-6 and TNF-αlevels in culture of colonic explants in a dose-dependent manner (DSS+PLA 2: IL-6: 10.31 ± 4.34, TNF-α: 1.17 ±1.1 pg/mL/mg of explant; DSS+PLAE 3: IL-6: 6.30 ± 1.46, TNF-α: 0.89 ± 0.21 pg/mL/mg of explant) (Fig. 8 c andd). Interestingly and in comparison to DSS group, DSS+PLAE 3 group shows a signi cant decrease in local production of pg/mL/mg of explant, p<0.001, TNF-α: 2.28 ± 0.68 vs 0.89 ± 0.21 pg/mL/mg of explants; p<0.05).

Effect of PLAE on pro-in ammatory cytokines production by peritoneal macrophages in acute colitis

We then measured the rate of pro-in ammatory cytokines IL-6 and TNF-α secreted by peritoneal macrophages cultured without any form of stimulation. The results showed that IL-6 and TNF-αlevels in culture of peritoneal macrophages obtained from DSS group were higher than those from control group pg/mL/10 5 macrophages, p<0.05 ; TNF-α: 58.11 ± 38.2 vs 32.7 ± 2.8 pg/mL/10 5 macrophages). In contrast, the production of IL-6 by peritoneal macrophages from DSS+PLAE 2 and 3-treated mice was signi cantly reduced as compared to DSS group (Fig. 9 a). In addition, we observed that TNF-α levels in DSS+PLAE 3 treated group was highly reduced compared to DSS group (58.11 ± 38.2 vs 39.28 ± 12.42 pg/mL/10 5 macrophages) (Fig. 9 b).

Discussion

Due to the side effects of the drugs currently available, the use of medicinal plants for treatment of acute and chronic intestinal in ammatory diseases is attracting increasing interest. In our current study, we demonstrated for the rst time the anti-in ammatory and therapeutic activities of the aqueous extract of Pistacia lentiscus leaves (PLAE) on DSS-induced acute colitis.

Crohn's disease and ulcerative colitis are the major forms of intestinal bowel diseases (IBD) characterized by bowel in ammation and abdominal pain. In recent decades, many animal models have been developed to characterize the complexity of IBD pathogenesis, de ning the underlying molecular mechanisms and evaluating potential therapeutics for humans (Chassaing et al. 2015). In the present study, we chose to use experimental colitis model based on administration of dextran sodium sulphate (DSS) in mice. This model has been recognized for its speed, simplicity and reproducibility.

Our results showed that after DSS administration, all animals developed the symptoms of acute colitis characterized by progressive weight loss, diarrhea and rectal bleeding. In fact, we observed rst clinical signs of colitis on day 3 after starting DSS administration, whereas on day 5 most animals started to present heavy bleeding. At macroscopic scale, we observed hemorrhagic erosion zones and ulcerations scattered throughout the colon, the cecum, and the rectum accompanied by colon size shortening. From histological point of view, we observed that DSS causes erosions and thinning of the epithelium surface, edematous zones, crypts dilation, as well as the in ltration of immune cells, mainly polynuclear neutrophils (PNN) and macrophages. The deterioration of the lower intestinal tract was accompanied by a depletion of Goblet cells and a decrease in mucus secretion. These results are consistent with those reported byseveral authorswho speculated that DSS is a chemical colitogen with anti-coagulant properties capable of causing severe murine colitis which closely resembles human ulcerative colitis [START_REF] Okayasu | Novel method in the induction of reliable experimental acute and chronic ulcerative colitis in mice[END_REF][START_REF] Abdelouhab | Mucosal intestinal alteration in experimental colitis correlates with nitric oxide production by peritoneal macrophages: Effect of probiotics and prebiotics[END_REF], Chassaing et al. 2015). The mechanism by which DSS induces intestinal in ammation has not yet been fully described. However, some authors suggest that DSS carries a highly negative charge provided by sulfate groups, toxic to the colon epithelia monolayer. It thus induces erosions that compromise the barrier integrity leading to increased colonic epithelial permeability and allow to the dissemination of the pro-in ammatory intestinal contents in the underlying tissues. [START_REF] Poritz | Loss of the tight junction protein ZO-1 in dextran sulfate sodium induced colitis[END_REF] observed that the administration of DSS induces changes in tight junction protein expression as early as one day after treatment and an in ammatory response characterized by leukocyte recruitment and increased production of pro-in ammatory cytokines [START_REF] Toumi | Bene cial role of the probiotic mixture Ultrabiotique on maintaining the integrity of intestinal mucosal barrier in DSS-induced experimental colitis[END_REF]. These initial effects are followed by increasingly severe symptoms with severe diarrhea, bleeding and acute anemia. In addition, DSS anticoagulant property aggravates intestinal bleeding [START_REF] Chassaing | Dextran sulfate sodium (DSS)induced colitis in mice[END_REF].

All of these reactions cause narrowing of the colon size, progressive weight loss and death of the animals.

Importantly, our study showed that the oral administration of PLAE concomitantly with DSS signi cantly reduced the clinical, macroscopic and histopathological colitis scores in comparison to the administration of DSS alone. Our results are in line with those of several studies. [START_REF] Zahouani | Aqueous Leaf Extract of Pistacia lentiscus Improves Acute Acetic Acid-Induced Colitis in Rats by Reducing In ammation and Oxidative Stress[END_REF] revealed that pre-treatment with PLAE signi cantly reduced the lesion areas in acetic acid-induced acute colitis. In addition, Dellai et al. (2013) suggested that pre-treatment with PLAE at 200mg/Kg decreased the intensity of gastric mucosal damages at 74.9% compared with ethanol-HCl group in an experimental model of gastric ulcer. Furthermore, several other studies showed the bene t of Pistacia lentiscus (PL) extracts on ulcerative colitis. Recently, [START_REF] Ostovan | The Anti-In ammatory Effect of Pistacia lentiscus in a Rat Model of Colitis[END_REF] demonstrated that Mastic Oil of PL signi cantly reduced the total colitis index, including in ammation severity, in ammation extent, and crypt damage scores. Razi et al. (2000) suggested that supplementation with oleogum resin from PL delayed the onset and progression of the disease and helped prevent weight loss in the DSS model of colitis. In addition, Kim et al. (2009) showed that PL mastic decreased the macroscopic and histological damage in TNBS-induced colitis in rats. Our results are also in concordance with our previous study establishing that PL fatty oil pre-treatment or treatment signi cantly reduced ulcerated and hemorrhagic areas in ethanol-induced gastric ulcer [START_REF] Boutemine | Gastro-protective, therapeutic and anti-in ammatory activities of Pistacia lentiscus fatty oil against ethanol-induced gastric ulcers in rats[END_REF].

Otherwise, our results revealed that DSS-administration induces a signi cant increase in systemic, local, and peritoneal macrophages production of NO compared to control group. This result suggests that NO is implicated in the intestinal in ammation. Our results are in agreement with previous studies conducted in our laboratory demonstrating that the administration of 3% DSS induced a signi cant decreases in NO production in plasma, colon and macrophages cultures compared to control group [START_REF] Abdelouhab | Mucosal intestinal alteration in experimental colitis correlates with nitric oxide production by peritoneal macrophages: Effect of probiotics and prebiotics[END_REF], Toumi et al. 2013[START_REF] Toumi | Probiotic bacteria lactobacillus and bi dobacterium attenuate in ammation in dextran sulfate sodium-induced experimental colitis in mice[END_REF]. Other authors demonstrated that the administration of 3% or 5% DSS increases iNOS expression and NO level in rat colon [START_REF] Arafa | Prophylactic role of curcumin in dextran sulfate sodium (DSS)-induced ulcerative colitis murine model[END_REF], Larrosaet al. 2009). In addition, it was shown that inhibition of iNOS expression blunted DSS colitis in mice (Rahimi et al. 2010) In the second part of our experiments, we aimed to determine the systemic, local, and macrophage production of the pro-in ammatory cytokines IL-6, and TNF-α. In agreement with previous studies, our results suggest that DSS administration signi cantly increase pro-in ammatory cytokines production (Zicong et al. 2020;[START_REF] Toumi | Probiotic bacteria lactobacillus and bi dobacterium attenuate in ammation in dextran sulfate sodium-induced experimental colitis in mice[END_REF]. The in ltrated in ammatory cells observed in the mucosa and submucosa after treatment by DSS may secret pro-in ammatory cytokines . These cytokines lead with endotoxin (LPS) and many bacterial components present in the colon to activation of the transcription factor NF-κB, and then the expression of iNOS [START_REF] Toumi | Bene cial role of the probiotic mixture Ultrabiotique on maintaining the integrity of intestinal mucosal barrier in DSS-induced experimental colitis[END_REF][START_REF] Sou I | Cytokines and Nitric Oxide in Immunopathogenesis of IBD and Potential Therapeutic Approaches[END_REF]. NO produced in large quantities by this enzyme, joins the superoxide anion (O -2 ) to generate peroxynitrite ion (OONO -) known for its deleterious effects on tissues via the nitration of proteins and lipid peroxidation [START_REF] Korhonen | Nitric oxide production and signaling in in ammation[END_REF].The latter mechanisms could be involved in the DSS-induced mucosal injuries. In addition, TNF-α can also activate NF-κB and stimulate the production of proin ammatory cytokines (such as IFN-γ) that ampli es its own production [START_REF] Zhou | Effects of ginKgo biloba extract on in ammatory mediators (SOD, MDA, TNF-a, NF-jBp65, IL-6) in TNBS-induced colitis[END_REF]. IL-6 is another important pro-in ammatory cytokine that plays a central role in acute in ammation [START_REF] Kishimoto | Interleukin-6: from basic science to medicine 40 years in immunology[END_REF]). Thus, IL-6 can activate neutrophils, monocytes/macrophages, and lymphocytes at the site of in ammation. More importantly, IFN-γ, TNF-α, or IL-6 can disrupt the epithelial barrier function by apoptosis-independent mechanisms [START_REF] Suzuki | Interleukin-6 (Il-6) regulates claudin-2 expression and tight junction permeability in intestinal epithelium[END_REF]. Furthermore, these cytokines production upregulates the Cyclooxygenase-2 (COX 2) expression. This enzyme is responsible for producing prostaglandins associated with the mediation of in ammation [START_REF] Vane | Cyclooxygenases 1 and 2[END_REF]. [START_REF] Kim | Bene cial effects of the traditional medicine Igongsan and its constituent ergosterol on dextran sulfate sodium-induced colitis in mice[END_REF] showed that the COX-2 and prostaglandins-2 levels were increased in the colon tissue concomitantly with TNF-α and IL-6 levels increases in DSS-treated mice as compared to those of control mice. Interestingly, we observed that systemic, local, and peritoneal macrophages production of NO, IL-6 and TNF-α were signi cantly down-regulated when the DSS is administered concomitantly with PLAE. Our results are in agreement with previous studies. [START_REF] Zahouani | Aqueous Leaf Extract of Pistacia lentiscus Improves Acute Acetic Acid-Induced Colitis in Rats by Reducing In ammation and Oxidative Stress[END_REF] underlined the anti-in ammatory effect of PLAE oral pre-treatment in acetic acid-induced ulcerative colitis, by reducing IL-6 production. [START_REF] Ostovan | The Anti-In ammatory Effect of Pistacia lentiscus in a Rat Model of Colitis[END_REF] showed that PL mastic oil administered intra-rectally signi cantly decreased TNF-α level after 7 days of treatment in acetic acid-induced colitis. Our present nding are also in line with our previous results onethanol-induced gastric ulcer model showing that pre-treatment or treatment with PL fatty oil signi cantly reduced iNOS expression in the gastric mucosa. Systemic and local production of NO was also reduced. Furthermore, a signi cant decrease in IL-6 and TNF-α level was observed in gastric explants culture [START_REF] Boutemine | Gastro-protective, therapeutic and anti-in ammatory activities of Pistacia lentiscus fatty oil against ethanol-induced gastric ulcers in rats[END_REF].

The therapeutic and anti-in ammatory activities of PLAE observed during our study could be due to the presence of many bioactive phyto-chemical compounds. Interestingly, recent phytochemical analyses of PL leavesaqueous extract allowed the identi cation of numerous phenolic compounds such as avonoids (isoquercetin and luterolin), phenolic acids (ellagic and dicaffeoylquinic), avonols (catechin, rutin, and kaempferol) and tannins [START_REF] Zahouani | Aqueous Leaf Extract of Pistacia lentiscus Improves Acute Acetic Acid-Induced Colitis in Rats by Reducing In ammation and Oxidative Stress[END_REF]. Another study conducted on leaves powder, by [START_REF] Arab | Etude phytochimique et l'évolution de l'activité antimicrobienne et antioxydant de l'huile essentielle et des composés phénolique du pistachier lentisque (Pistacia lentiscus )[END_REF] revealed the presence of very high alkaloid and polyphenol content, particularly avonoids (isoquercetin and luterolin), leucoanthocyanins and total tannins (mainly gallic tannins). In addition, [START_REF] Bozorgi | Five Pistacia species ( vera, P. atlantica, P. terebinthus, P. khinjuk, and P. lentiscus): A Review of Their Traditional Uses[END_REF] showed that PL leaves are rich in monoterpenes (α-pinene, limonene, sabinene and mycene), and in sesquiterpenes (β-caryophyllene). In addition, PL leaves also contain sterols and triterpenes, saponosides as well as reducing compounds (oses, holosides and mucilage).

Polyphenols limit the immune response during intestinal in ammation by limiting tissue colonization by PNN. In vitro and in vivo studies have indicated that polyphenols inhibited phospholipase A2, lipoxygenase and COX 2 involved in the production of in ammatory mediators such as prostaglandins [START_REF] Calder | n-3 polyunsaturated fatty acids, in ammation, and in ammatory diseases[END_REF]. These enzymes act also on NO production by modulating the activity of iNOS [START_REF] Sanchez-Fidalgo | Dietary supplementation of resveratrol attenuates chronic colonic in ammation in mice[END_REF]). In addition, during intestinal diseases, polyphenols have a therapeutic effect similar to that of prebiotics. In fact, most polyphenols pass through the small intestine without being absorbed [START_REF] Scalbert | Dietary intake and bioavailability of polyphenols[END_REF]. Thus, polyphenols are biotransformed to their metabolites by the gut microbiota, resulting in their bioavailability increases. Moreover, polyphenols modulate the composition of the gut microbial community mainly by inhibiting pathogenic bacteria and stimulating bene cial bacteria [START_REF] Lee | Effect of tea phenolics and their aromatic fecal bacterial metabolites on intestinal microbiota[END_REF][START_REF] Ozdal | The Reciprocal Interactions between Polyphenols and Gut Microbiota and Effects on Bioaccessibility[END_REF].

Otherwise, avonoids (such asquercetin) treatment has shown bene cial effects by the amelioration of colonic damage and the reduction of the DAI in the DSS model [START_REF] Oz | Green tea polyphenols and sulfasalazine have parallel antiin ammatory properties in colitis models[END_REF]). The avonoid antidiarrheal effects have been related to their capacity to inhibit muscle contractility, enhance intestinal motility and reduce uid intraluminal accumulation in the gut lumen [START_REF] Di Carlo | Inhibition of intestinal motility and secretion by avonoids in mice and rats: Structure-activity relationships[END_REF][START_REF] Vezza | Flavonoids in In ammatory Bowel Disease[END_REF]. Besides, avonoids signi cantly reduced neutrophil in ltration into the damaged colonic tissue in experimental colitis models [START_REF] Seibel | Oral treatment with genistein reduces the expression of molecular and biochemical markers of in ammation in a rat model of chronic tnbs-induced colitis[END_REF][START_REF] Vezza | Flavonoids in In ammatory Bowel Disease[END_REF]. Flavonoids may inhibit proin ammatory cytokines production by blocking the binding of NF-κB to DNA, as well as inhibiting mitogen-activated protein kinase (MAPK) and STAT activation [START_REF] Salaritabar | Therapeutic potential of avonoids in in ammatory bowel disease: a comprehensive review[END_REF]). In fact, [START_REF] Camuesco | The intestinal anti-in ammatory effect of quercitrin is associated with an inhibition in iNOS expression[END_REF] and [START_REF] Vezza | Flavonoids in In ammatory Bowel Disease[END_REF] proposed that the anti-in ammatory effects of quercitrin might be related to a decrease in iNOS expression through down-regulation of NF-κB in colonic tissue. In addition, the administration of quercetin remarkably decreased the levels of different cytokines in the in amed colon induced by DSS [START_REF] Miller | Nitric oxide. III. A molecular prelude to intestinal in ammation[END_REF][START_REF] Westphal | Green tea polyphenol epigallocatechin-3-gallate shows therapeutic antioxidative effects in a murine model of colitis[END_REF][START_REF] Oz | Green tea polyphenols and sulfasalazine have parallel antiin ammatory properties in colitis models[END_REF]. In another study, [START_REF] Marques | In vitro anti-in ammatory activity of terpenes via suppression of superoxide and nitric oxide generation and the NF-κB signalling pathway[END_REF] suggest that terpenes (monoterpenes and sesquiterpenes) reduce the pro-in ammatory cytokines TNF-α and IL-1α levels and increase the production of IL-10. In addition, terpenes signi cantly inhibit the production of NO by macrophages in vitro and alter the activity of NF-κB [START_REF] Marques | In vitro anti-in ammatory activity of terpenes via suppression of superoxide and nitric oxide generation and the NF-κB signalling pathway[END_REF].

Finally, the study of Andersen and Markham reveal that the total absence of cyanogenic heterosides greatly reduces the toxicological risks associated with the use of Pistacia lentiscus L. These ndings are in agreement with our results suggesting that PLAE has no cytotoxic activity on hepa1c1c7cell line.

In summary, our study and the data on the effects of bioactives components present in PL leaves provides several hypothesis to explain the mechanism of PLAE action.PLAE can act as a therapeutic tool either by i) forming a protective membrane along the colon layers, ii) stimulating secretion of mucus by Goblet cells, iii) interacting directly with the sulfate groups of DSS, thus inhibiting their interaction with the colonic wall, and iv) eliminating pathogenic bacteria (such as Helicobacter pylori) which is involved in the aggravation of DSS-induced colitis. In addition, PLAE may act as an anti-in ammatory agent either at i) cellular level by inhibiting the activation and recruitment of immune cells, ii) membrane level by blocking pro-in ammatory cytokines receptors, or iii) inta-cellular level by reducing the expression of NF-κB, and iNOS, and the production of pro-in ammatory cytokines.

4 )

 4 DSS+PLAE 3 group: mice received DSS dissolved in drinking water, plus 200 μl of aqueous extract dissolved in distilled water at 200 mg/Kg by gavage each day (n = 8). 5) PLAE group: mice received drinking water, plus 200 μl of aqueous extract dissolved in distilled water at 200 mg/Kg by gavage each day (n = 4). 6) Control (Ctrl) Group: mice received drinking water, plus 200 μl of distilled water by gavage each day (n = 8).

Figures Figure 1

 1 Figures

Figure 3

 3 Figure 3

Figure 5 PLAE

 5 Figure 5

Figure 6 PLAE

 6 Figure 6

Figure 7 PLAE

 7 Figure 7

Figure 8 PLAE

 8 Figure 8

Acknowledgements

Many thanks to ATRSS (the thematic research agency in health sciences) how supported this work by a grant. Many thanks to the botanists from "National superior School of Agronomy of El Harrach" (Algiers) how identi ed our plant.

Compliance with ethical standards

Con ict interest: The authors declare that they have no con ict of interest

According to our nding we conclude that PLAE is not e cient at 50mg/kKg, this dose was considered too low for therapeutic effects. However, 100mg/Kg PLAE could be prescript to mild colitis patient. Moreover, 200mg/Kg PLAE could treat the sever forms of colitis.

Conclusion

Collectively, our results suggest that the aqueous extract of the Pistacia lentiscus leaves have a systemic and local anti-in ammatory properties resulting in a therapeutic effect against DSS-induced acute colitis in mice. Thus, the Pistacia lentiscus aqueous extract offers new therapeutic possibilities for the treatment of human IBD.

Supplementary Files

This is a list of supplementary les associated with this preprint. Click to download.


Loïc Chaix
email: lchaix@lma.cnrs-mrs.fr

Mihail Garajeu
email: mihai.garajeu@univ-amu.fr

Pierre-Guy Vincent
email: pierre-guy.vincent@irsn.fr

Ghiath Monnet
email: ghiath.monnet@edf.fr

Martín I Idiart
email: martin.idiart@ing.unlp.edu.ar

Homogenized descriptions for the elastoplastic response of polycrystalline solids with complex hardening laws: application to neutron-irradiated bainitic steels

Keywords: polycrystals, elastoplasticity, neutron radiation, homogenization

Homogenized descriptions are provided for polycrystalline solids deforming in accordance with certain crystal plasticity laws recently proposed for neutron-irradiated bainitic steels. These laws express intragranular plastic slip rates in terms of resolved shear stresses and key microstructural features, such as densities of forest dislocations and of solute clusters, for a wide range of deformation rates, temperatures, and radiation doses. The elastic domain is delimited by thresholds on the resolved stresses that depend on dislocation densities in an intricate manner, and the plastic hardening is described by evolution laws of the Mecking-Kocks type for the dislocation densities with plastic slip. However, thresholds also depend nonlinearly on the resolved stresses themselves. Full-field homogenized descriptions are generated with a Fast Fourier Transform algorithm implemented in the computer code CraFT, while mean-field homogenized descriptions are generated by means of a linear-comparison scheme based on a generalized-secant linearization of the crystal plasticity laws. Multiple ways of accounting for plastic hardening in the mean-field descriptions are explored. Sample results are reported in the form of uniaxial traction curves and concomitant dislocation density evolutions under different scenarios. Overall, the generalized-secant linearization is found to provide an appropriate compromise between precision and mathematical complexity to generate homogenized descriptions for the elastoplastic response of polycrystalline media governed by complex crystal plasticity laws.

Introduction

The elastoplastic behavior of polycrystalline alloys is often described by physically based crystal plasticity laws with complex hardening rules (e.g., [START_REF] Segurado | Computational homogenization of polycrystals[END_REF]. On the other hand, rigorous mean-field homogenization schemes for up-scaling crystal plasticity laws often neglect elastoplastic interactions and plastic hardening to enable the use of variational approximations (e.g., [START_REF] Dendievel | Bounds for the creep behaviour polycrystalline materials[END_REF][START_REF] Debotton | Variational estimates for the creep behaviour polycrystals[END_REF][START_REF] Bornert | Second-order estimates of the Self-Consistent type for viscoplastic polycrystals[END_REF][START_REF] Liu | Second-order theory for the effective behavior and field fluctuations in viscoplastic polycrystals[END_REF][START_REF] Ponte Castañeda | Fully optimized second-order variational estimates for the macroscopic response and field statistics in viscoplastic crystalline composites[END_REF]. A few strategies to incorporate those features into variational mean-field schemes have been pursued notwithstanding, at the expense of introducing ad-hoc approximations adapted to the particular hardening rule being considered (e.g., [START_REF] Masson | An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals[END_REF][START_REF] Lebensohn | Self-consistent modelling of the mechanical behaviour of viscoplastic polycrystals incorporating intragranular field fluctuations[END_REF][START_REF] Suquet | Elasto-plasticity of heterogeneous materials at different scales[END_REF].

The purpose of this work is to adapt a particular mean-field homogenization scheme to crystal plasticity laws of the type proposed by [START_REF] Monnet | Multiscale modeling of crystal plasticity in Reactor Pressure Vessel steels: Prediction of irradiation hardening[END_REF] for certain neutron-irradiated bainitic steels employed in nuclear reactor pressure vessels. These laws express intragranular plastic slip rates in terms of resolved shear stresses and key microstructural features, such as densities of forest dislocations and of solute clusters, for a wide range of deformation rates, temperatures, and radiation doses. The elastic domain is delimited by thresholds on the resolved stresses that depend on dislocation densities in an intricate manner, and the plastic hardening is described by evolution laws for the dislocation densities with plastic slip of the type proposed by [START_REF] Kocks | Physics and phenomenology of strain hardening: the FCC case[END_REF]. However, thresholds also depend nonlinearly on the resolved stresses themselves. Mean-field homogenized descriptions are generated here by means of a linear-comparison scheme based on a generalized secant linearization as proposed by Ponte Castañeda (2015). When applied to crystal plasticity laws that derive from a dissipation potential, this scheme exhibits full stationarity with respect to the properties of the linear-comparison polycrystal. The stress-strain-rate relation estimated directly from the linear-comparison polycrystal is thus expected to provide meaningful predictions even when the underlying crystal plasticity law does not derive from a potential. The scheme also delivers estimates for intragranular statistics of the mechanical fields of first and second order. This additional information is required to estimate the evolution of dislocation densities and ensuing plastic hardening. Following common practice, first-order statistics of plastic slip rates are employed. However, three alternative strategies incorporating second-order statistics are explored. Elementary schemes of the Sachs and Taylor type are also considered. Finally, full-field homogenized descriptions are generated with a Fast Fourier Transform-based algorithm implemented in the computer code CraFT of Moulinec andSuquet (1994, 1998). Elementary schemes and full-field descriptions provide references against which the mean-field descriptions can be assessed. To that end, quantitative comparisons are reported in the form of uniaxial traction curves and concomitant dislocation density evolutions under different scenarios.

Throughout the presentation, the symbol ∇ denotes the nabla operator, the symbol "•" denotes the scalar product between vectors as well as tensors, the symbols ⊗ and ⊗ s denote the tensor product between vectors as well as tensors and its symmetric part, respectively, the symbols I and I denote the second-order and fourth-order symmetric identity tensors, J and K are the standard fourth-order hydrostatic and shear projection tensors, and the overdot ( •) denotes time differentiation.

The polycrystalline solid model

Microgeometry

Polycrystals are regarded here as random aggregates of perfectly bonded single crystals or grains. Individual grains are assumed to be of a similar size, much smaller than the specimen size and the scale of variation of the applied loads. Furthermore, the aggregates are assumed to have statistically uniform and ergodic microstructures.

Let the grain orientations within a representative volume element take on a set of N discrete values, characterized by rotation tensors Q (r) (r = 1, ..., N ) relative to a reference system. All grains with a given orientation occupy a disconnected domain Ω (r) and are collectively referred to as 'phase' r. The domain occupied by the polycrystal is then Ω = ∪ N r=1 Ω (r) . Volume averages over the aggregate Ω and over each phase Ω (r) will be denoted by ⟨•⟩ and ⟨•⟩ (r) , respectively. The domains Ω (r) can be described by a set of characteristic functions χ (r) (x), which take the value 1 if the position vector x is in Ω (r) and 0 otherwise. In view of the microstructural randomness, the functions χ (r) are random variables that must be characterized in terms of ensemble averages [START_REF] Willis | The overall elastic response of composite materials[END_REF]. Due to the assumed statistical uniformity and ergodicity, one-point probabilities can be identified with the volume fractions -or concentrationsc (r) = ⟨χ (r) (x)⟩ of each phase r, two-point probabilities can be identified with the volume averages ⟨χ (r) (x)χ (s) (x ′ )⟩, and so on.

Microscopic elastoplastic response

Grains are assumed to individually deform by multi-glide along K slip systems so that the strain rate at any given point within a crystal with orientation r is given by the additive composition

ε = εel + K k=1 γ(r,k) µ (r,k) , (1)
where the first term refers to an elastic strain rate, and the second term refers to a plastic strain rate that results from dislocation gliding along the K slip systems with Schmid tensors µ (r,k) = m (r,k) ⊗ s n (r,k) defined in terms of the unit normals n (r,k) = Q (r) n (k) and glide directions m (r,k) = Q (r) m (k) . For simplicity, the elastic response is assumed linear and isotropic [START_REF] Monnet | Multiscale modeling of crystal plasticity in Reactor Pressure Vessel steels: Prediction of irradiation hardening[END_REF], so

εel = S σ with S = 1 3κ J + 1 2µ K, ( 2
)
where σ is the Cauchy stress tensor, and κ and µ are the bulk and shear moduli of the crystal, respectively. The elastic response is therefore spatially homogeneous. It is emphasized that elastic anisotropy could be easily accommodated by all the homogenized descriptions considered below but would not alter the conclusions of this study. The plastic response, on the other hand, is strongly nonlinear and anisotropic.

The shear rate along any given system is taken of the peculiar form [START_REF] Monnet | Multiscale modeling of crystal plasticity in Reactor Pressure Vessel steels: Prediction of irradiation hardening[END_REF])

γ(r,k) = γ0 sinh   τ (r,k) + τ 0   sgn(σ • µ (r,k) ), (3)
where γ0 and τ 0 denote a reference strain rate and stress, respectively, and τ (r,k) + is an effective stress that depends non-linearly on the resolved shear stress σ • µ (r,k) through the equation

τ (r,k) + = |σ • µ (r,k) | -τ (r,k) c (τ (r,k) + ) + , (4)
where [•] + = max{•, 0} denote the Macaulay brackets. In this last expression, τ

(r,k) c (τ (r,k) +
) is a critical stress level that depends on τ (r,k) + as

τ (r,k) c (τ (r,k) + ) = τ HP + τ 2 SC + τ (r,k) 2 self + β τ (r,k) obs -τ (r,k) + 2 + , (5)
where β is a non-dimensional parameter introduced for numerical purposes, while

τ HP = τ cr d d 0 -1/2 and τ SC = τ cr c SC c SC0 4/7 (6)
are, respectively, characteristic stresses due to the Hall-Petch effect and the presence of solute clusters produced by the neutron radiation. These stresses are proportional to a reference critical stress τ cr , and depend on ferrite lath package size d and solute cluster concentration c SC ; d 0 and c SC0 are characteristic values for these quantities. In turn,

τ (r,k) self = τ cr a (k,k) ρ (r,k) ρ 0 1/2 and τ (r,k) obs = τ cr     K k ′ =1 k ′ ̸ =k a (k,k ′ ) ρ (r,k ′ ) ρ 0     1/2 (7)
are, respectively, stresses due to the dipolar interactions between dislocations belonging to the same slip system and to different slip systems as characterized by the interaction matrix a (k,k ′ ) and a characteristic

k 1 2 3 4 5 6 7 8 9 10 11 12 1 c 1 c 2 c 2 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 2 c 1 c 2 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 3 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 4 c 1 c 2 c 2 c 1 c 1 c 1 c 1 c 1 c 1 5 c 1 c 2 c 1 c 1 c 1 c 1 c 1 c 1 6 c 1 c 1 c 1 c 1 c 1 c 1 c 1 7 c 1 c 2 c 2 c 1 c 1 c 1 8 c 1 c 2 c 1 c 1 c 1 9 c 1 c 1 c 1 c 1 10 c 1 c 2 c 2 11 c 1 c 2 12 c 1
Table 1: Interaction matrix for the ⟨111⟩{110} slip familly dislocation density ρ 0 . The densities of dislocations ρ (r,k) belonging to each slip system, in turn, evolve with slip according to a classical Kocks-Mecking law of the form [START_REF] Kocks | Physics and phenomenology of strain hardening: the FCC case[END_REF])

ρ(r,k) = | γ(r,k) | ρ 0 A d d d 0 -1 + A SC c SC c SC0 4/7 + A self a (k,k) ρ (r,k) ρ 0 1/2 +     K k ′ =1 k ′ ̸ =k A obs a (k,k ′ ) ρ (r,k ′ ) ρ 0     1/2 -A dis ρ (r,k) ρ 0      (8)
with non-dimensional coefficients A d , A SC , A self , A obs and A dis characterizing different interaction statistics.

Macroscopic elastoplastic response

The macroscopic response of the polycrystalline solid is defined as the relation between the time histories of the average strain ε(t) = ⟨ε(x, t)⟩ and average stress σ(t) = ⟨σ(x, t)⟩ over the representative volume element and a given time interval 0 ≤ t ≤ T (e.g., [START_REF] Michel | Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis[END_REF]. Typically, the macroscopic response is determined by prescribing one of the time histories and computing the other. If ε(t) is prescribed, σ(t) must be computed by averaging the stress field that results from solving the field equations for the displacement field u(x, t) within the representative volume element with affine boundary conditions, integrated over time from a certain initial state. Specifically, the field equations are

∇ • σ = 0, ε = ∇ ⊗ s u, u = εx + u′ , (9)
where σ and ε must satisfy the constitutive relations spelled out in the previous subsection, and u ′ is a displacement field that vanishes on the boundary of the representative volume element. The internal state of the solid is characterized by the collection of N × K dislocation density fields {ρ (r,k) }, which play the role of internal variables evolving with a law of the form (8). The total dislocation density within the representative volume element is then given by r) . (10)

ρ(t) = N r=1 K k=1 c (r) ⟨ρ (r,k) (x, t)⟩ (

Material parameters for untextured bainitic steels at low temperatures

The present study is restricted to untextured equi-axed polycrystals. Thus, volume fractions are set to c (r) = 1/N , multi-point probabilities are assumed isotropic functions of relative positions within the representative volume element, and rotation tensors Q (r) with a Sobol sequence of N entries (e.g., [START_REF] Lebensohn | Dilatational viscoplasticity of polycrystalline solids with intergranular cavities[END_REF]. The number of grain orientations is set to N = 100. The above crystal plasticity law pertains to neutron-irradiated bainitic steels deforming at subfreezing temperatures. Thus, crystals are assumed to glide along the twelve slip systems ⟨111⟩{110} (K = 12) and numerical values for the various material parameters are taken from the works of [START_REF] Monnet | Multiscale modeling of crystal plasticity in Reactor Pressure Vessel steels: Prediction of irradiation hardening[END_REF] and [START_REF] Chaix | Modélisation micromécanique du comportement viscoplastique d'un polycristal[END_REF] for -100 • C and are quoted in Tables 1 &2. The ratio c SC /c SC0 , in turn, is set in Section 5 below for specific radiation doses. Material behavior at such low temperatures is particularly relevant for characterizing ductile-to-brittle transitions (e.g., [START_REF] Chakraborty | A unified cohesive zone approach to model the ductile to brittle transition of fracture toughness in reactor pressure vessel steels[END_REF]. Finally, the original law of [START_REF] Monnet | Multiscale modeling of crystal plasticity in Reactor Pressure Vessel steels: Prediction of irradiation hardening[END_REF] defines the critical stress (5) with β = 1, but the lower value β = 0.7 is adopted here to ease numerical computations. In this connection, it is noted that expression (5) produces a critical stress level that varies with resolved shear stress from an initial value

τ (r,k) c = τ HP + τ 2 SC + τ (r,k) 2 self + βτ (r,k) 2 obs (11)
to a saturation value

τ (r,k) c = τ HP + τ 2 SC + τ (r,k) 2 self . (12)
Thus, the incremental response is independent of the coefficient β deep in the plastic range.

Full-field descriptions based on a Fast Fourier Transform algorithm

Full-field descriptions for the above polycrystalline solids are generated here by regarding them as periodic media with unit cells of cubic shape and sufficient microgeometrical intricacy. Granular morphologies are generated by Voronoi tessellations with N = 100 seeds distributed in such a way as to optimize uniformity of volume fractions among grains. Each grain is then randomly assigned one of the crystal orientations of the Sobol sequence. Figure 1 shows ten unit cells with different microgeometries.

Following [START_REF] Moulinec | A fast numerical method for computing the linear and nonlinear properties of composites[END_REF], the equilibrium equations of Section 2.3, with the crystal plasticity law of Section 2.2 and periodicity conditions on u ′ , are then discretized in time and space, and solved by a fixed-point method using an implicit Euler scheme and a Fast Fourier Transform algorithm with a cubic grid as implemented in the computer code CraFT. The so-called 'basic' scheme of the code allowing for mixed loading conditions is employed to ensure strain compatibility.

Mean-field descriptions

Mean-field descriptions for the above polycrystalline solids are confected here by resorting to two simplifying approximations of common use. Firstly, the homogenization operation on the elastoplastic response is carried out in a decoupled fashion wherein the elastic and plastic deformations are treated separately. Secondly, the dislocation density fields are assumed uniform within all grains with a given orientation r. To expound the rationale behind these approximations, the elastoplastic strain rate (1) at a given instant of the deformation process within any given crystal with orientation r is written as

ε = σm 3κ I + 1 2µ σd + K k=1 γ(r,k) σ d • µ (r,k) , {ρ (r,k) } µ (r,k) (13)
where σ m = trσ/3 and σ d = σ -σ m I are the mean and deviatoric parts of the stress, respectively, and the functions γ(r,k) give the plastic slip rates in terms of the resolved shear stresses and dislocation densities {ρ (r,k) } by the constitutive relations (3)-( 7). The concomitant evolution laws for the dislocation densities are in turn written as

ρ(r,k) = γ(r,k) σ d • µ (r,k) , {ρ (r,k) } R (k) ρ (r,k) , (14)
where the functions R (k) are defined by the right-hand side of (8). Expression ( 13) is now averaged over the representative volume element to obtain the macroscopic strainrate-stress relation

ε = σm 3κ I + 1 2µ σd + Ė σ d , ρ (r,k) (15)
with

Ė σ d , ρ (r,k) = N r=1 K k=1 c (r) γ(r,k) σ d • µ (r,k) , {ρ (r,k) } µ (r,k) (r) . (16)
Then, the macroscopic strain-rate-stress relation ( 15) is approximated by a similar relation but with the functional (16) replaced by the function

˙ E σ d , ρ (r,k) = N r=1 K k=1 c (r) γ(r,k) σ d • µ (r,k) , {ρ (r,k) } µ (r,k) (r) , (17)
where the pointwise varying fields ρ (r,k) have been replaced by the piecewise uniform fields N r=1 χ (r) (x)ρ (r,k) , the deviatoric stress field σ d (x) is now solution to the equilibrium equations in an elastically rigid polycrystal with ⟨σ d ⟩ = σ d and intraphase isochoric strain rates given by the averaged terms in ( 16), and the dislocation densities evolve with some approximation of ( 14) in terms of piecewise uniform slip-rate measures

N r=1 χ (r) (x) Γ(r,k)
, to be suitably chosen, according to

ρ(r,k) = Γ(r,k) R (k) ρ (r,k) . ( 18
)
Approximate stress-strain-rate relations can now be generated by estimating the function ( 17) with nonlinear homogenization schemes for N -phase media. These schemes can also deliver estimates for the first and second moments of the intraphase field distributions. Estimates for the first and second moments of the intraphase plastic slips can thus be employed in the choice of slip-rate measures for the evolution of the dislocation densities. On the other hand, it is noted that the approximation (17) entails a separate treatment of elastic and plastic deformations in the homogenization process; mean-field descriptions based on this decoupling approximation are therefore unable to provide rigorous intragranular statistics of the strain and stress fields (e.g., [START_REF] Idiart | Estimates for the overall linear properties of pointwise heterogeneous solids with application to elasto-viscoplasticity[END_REF][START_REF] Zecevic | New robust self-consistent homogenization schemes of elasto-viscoplastic polycrystals[END_REF].

Elementary schemes of the Sachs and Taylor type

Elementary schemes provide simple references against which other schemes can be compared. The elementary scheme of the Sachs type amounts to evaluating the function ( 17) and concommitant evolution law (18) at a uniform stress field given by its macroscopic value σ d . Thus, the function ( 17) is approximated by

˙ E S σ d , ρ (r,k) = N r=1 K k=1 c (r) γ(r,k) σ d • µ (r,k) , {ρ (r,k) } µ (r,k) , (19)
and the slip-rate measures in (18) are identified with

Γ(r,k) S = γ(r,k) σ d • µ (r,k) , {ρ (r,k) } . ( 20
)
In turn, the elementary scheme of the Taylor type amounts to evaluating the function ( 17) and concommitant evolution law (18) at a uniform plastic strain field. Thus, the function ( 17) is in this case approximated by a function ˙ E T (σ d , {ρ (r,k) }) defined through the system of algebraic equations

N r=1 c (r) σ (r) d = σ d and K k=1 γ(r,k) σ (r) d • µ (r,k) , {ρ (r,k) } µ (r,k) = ˙ E T r = 1, ..., N, (21)
for the function value ˙ E T and the average deviatoric stresses σ (r)

d , and the slip-rate measures in ( 18) are identified with

Γ(r,k) T = γ(r,k) σ (r) d • µ (r,k) , {ρ (r,k) } . ( 22
)
4.2. A linear-comparison scheme of the Self-Consistent type Elementary schemes of the Sachs and Taylor type depend on intragranular field statistics of first order only. To improve accuracy, refined schemes incorporate statistics of second order. In this work we employ the fully optimized second-order homogenization scheme of Ponte [START_REF] Ponte Castañeda | Fully optimized second-order variational estimates for the macroscopic response and field statistics in viscoplastic crystalline composites[END_REF]. This scheme approximates the function ( 17) by

˙ E F OSO σ d , ρ (r,k) = M σ d , ρ (r,k) σ d + η σ d , ρ (r,k) , ( 23
)
where the tensor functions M and η represent the effective properties of a linear-comparison polycrystal with the same microgeometry as the original polycrystal but with linear local responses of the form εd = M (r) σ d + η (r) characterized by r,k) .

M (r) = K k=1 1 2m (r,k) µ (r,k) ⊗ µ (r,k) and η (r) = K k=1 η (r,k) µ (r,k) (24) with 1 2m (r,k) = γ(r,k) -γ(r,k) τ (r,k) -τ (r,k) and η (r,k) = γ(r,k) - τ (r,k) 2m (
(25)

In these last expressions, the various resolved stresses and slip rates depend on σ d and ρ (r,k) through the first and second moments of the stress field within the grains; they can be written as

τ (r,k) = τ (r,k) + τ (r,k) 2 -τ (r,k) 2 and τ (r,k) = τ (r,k) -τ (r,k) 2 -τ (r,k) 2 , ( 26
) γ(r,k) = γ(r,k) τ (r,k) , {ρ (r,k) } and γ(r,k) = γ(r,k) τ (r,k) , {ρ (r,k ′ ) } , (27)
where

τ (r,k) = µ (r,k) • ⟨σ d ⟩ (r) and τ (r,k) = µ (r,k) • ⟨σ d ⊗ σ d ⟩ (r) µ (r,k) . ( 28
)
The local crystal plasticity law in this linear-comparison polycrystal is of the form γ(r,k) = σ d •µ (r,k) /(2m (r,k) )+ η (r,k) . Thus, relations (25) constitute a secant linearization of the non-linear stress-slip-rate response of the crystals that incorporates intragranular stress statistics of up to second order through the standard deviations of the resolved shear stresses SD (r,k)

(τ ) = (τ (r,k) 2 -τ (r,k) 2 ) 1/2 , see fig.
2. These stress statistics can be computed from the effective properties of the linear-comparison polycrystal through the identities

⟨σ d ⟩ (r) = 1 c (r)
∂ u ∂η (r) and

⟨σ d ⊗ σ d ⟩ (r) = 2 c (r) ∂ u ∂M (r) , ( 29
)
where u is an effective stress potential of the linear-comparison polycrystal of the form

u σ d , {M (r) }, {η (r) } = 1 2 σ d • Mσ d + η • σ d + 1 2 g. ( 30
)
The effective properties M, η and g must be estimated by some suitable linear homogenization scheme for N -phase media. The so-called Self-Consistent scheme of [START_REF] Laws | On the thermostatics of composite materials[END_REF] and [START_REF] Willis | Variational and related methods for the overall properties of composites[END_REF] is known to be particularly suitable for polycrystalline media of the type considered in this work; expressions are recalled in Appendix A for convenience. Finally, the first and second moments of the intragranular slip rates can be computed as

γ (r,k) = τ (r,k) 2m (r,k) + η (r,k) and γ (r,k) = τ (r,k) 2 4m (r,k) 2 + η (r,k) τ (r,k) m (r,k) + η (r,k) 2 . ( 31
)
Therefore, the mathematical structure of this second-order homogenization scheme suggests four natural choices for the slip-rate measures Γ(r,k) F OSO dictating the evolution of dislocation densities: |γ (r,k) |, |γ (r,k) |, |γ (r,k) |, and γ (r,k) . Each choice generates a different set of predictions as discussed in Section 5 below. Note that the nonlinear local behavior of the grains enter the homogenization scheme only through relations ( 27) and evolution equations (18); the scheme is thus easily adapted to other types of local behaviors.

Evaluating the function (23) requires, therefore, the solution of a system of non-linear algebraic equations for the 2 × N × K linear-comparison local properties m (r,k) and η (r,k) , coupled with the system of non-linear algebraic equations for the linear-comparison effective properties (A.5). Fixed-point methods are employed for that purpose in the present study. In turn, time integration of the evolution equations ( 18) is carried out by means of an implicit Euler scheme with fixed time stepping.

A simpler linear-comparison scheme of the Self-Consistent type

A simpler version of the above linear-comparison homogenization scheme is available from the earlier work of [START_REF] Debotton | Variational estimates for the creep behaviour polycrystals[END_REF]. This simpler scheme employs a secant linearization about the second moments of the intraphase stresses rather than the generalized-secant linearization (25) based on first and second moments. It approximates the function (17) by

˙ E SEC σ d , ρ (r,k) = M σ d , ρ (r,k) σ d , (32)
where the tensor function M is given by the Self-Consistent estimate (A.5) for the effective property of a linear-comparison polycrystal with the same microgeometry as the original polycrystal but with linear local responses of the form εd = M (r) σ d with M (r) characterized by the above formulae with τ (r,k) = 0 and τ (r,k) = τ (r,k) . The dislocation densities can then be updated by identifying slip-rate measures Γ (r,k)

SEC with |γ (r,k) | as given by (31) 1 .

Sample results and discussion

Sample results are reported for specimens subject to uniaxial tension under four different scenarios: (a) unirradiated specimens loaded at a low rate, (b) irradiated specimens loaded at a low rate, (c) unirradiated specimens loaded at a high rate, (d) irradiated specimens loaded at a high rate. Unirradiated specimens have no solute clusters so that c SC = 0, while irradiated specimens have a solute cluster content c SC = 8.8 × 10 -2 c SC0 associated with a high fluence. In turn, the low and high loading rates are respectively σe = µ ε0 and σe = 100µ ε0 , ε0 = 2 × 10 10 γ0 is a characteristic macroscopic strain rate. Henceforth, σ e = (3/2)σ d • σ d and ε e = (2/3)ε d • ε d denote the von Mises equivalent stress and strain, respectively. Initially, undeformed specimens are free of residual stresses and contain an equal amount of dislocation densities ρ init = 2.0 × 10 -2 ρ 0 on all slip systems.

Full-field simulations for unirradiated specimens subject to uniaxial tension at low loading rate are shown in fig. 3. Traction curves in the form of equivalent stress versus equivalent strain, respectively normalized by τ cr and γ cr = τ cr /µ, together with the concomitant evolution of total dislocation density, normalized by its range, at least for the range of deformations considered. However, this is probably due to a compensation of inaccuracies. Indeed, all second-order estimates overestimate flow stresses in the elastoplastic transition as a consequence of the decoupled homogenization of elastic and plastic deformations. The FOSO(γ) estimates compensate this overestimation with the peculiar evolution of the plastic hardening rate. By contrast, the alternative FOSO(γ) and FOSO(γ) estimates based on intragranular slip rate fluctuations deliver somewhat less accurate predictions for the flow stress but the most accurate predictions for the plastic hardening rate deep in the plastic range. Thus, this accurate hardening rate does not compensate the inaccuracies induced on flow stresses by the elastoplastic decoupling. From a practical standpoint prioritizing accuracy of flow stresses, FOSO(γ) estimates appear to be preferable; from a theoretical standpoint, however, the alternative FOSO(γ) and FOSO(γ) estimates should be regarded as most befitting. If combined with a coupled homogenization of elastic and plastic deformations (e.g., [START_REF] Lahellec | Effective response and field statistics in elasto-plastic and elasto-viscoplastic composites under radial and non-radial loadings[END_REF][START_REF] Das | Differential variational estimates for the macroscopic response and field statistics of elastoviscoplastic polycrystals[END_REF], the alternative choices γ and γ for updating dislocation densities are likely to generate the most accurate predictions. In fact, these alternative estimates are already seen to deliver the most accurate predictions for the total dislocation density within the entire range of loading rates and radiation doses considered. Figure 7 shows plots for the maximum and minimum values among all resolved stresses ( τ (r,k) , τ (r,k) ) and slip rates ( γ(r,k) , γ(r,k) ) associated with the FOSO(γ) and FOSO(γ) estimates in unirradiated specimens loaded at a low rate. An approximately specular symmetry about zero is observed between the two sets of curves. This is a consequence of the opposing signs in the definitions (26) for the resolved stresses τ (r,k) and τ (r,k) , and, together with the fact that dislocation density rates are proportional to the absolute values |γ (r,k) | or |γ (r,k) |, it explains the similitudes between predictions generated by the two types of estimates. Finally, the FOSO(γ) estimates based on the second moments of the intragranular slip rates are seen to be the least accurate among the second-order estimates: they overestimate plastic hardening rate and therefore flow stresses deep in the plastic range, and significantly overestimate the total dislocation density, even in excess of the Taylor estimates. Additional comparisons have also been performed with an alternative set of material parameters corresponding to higher subfreezing temperatures.

The FOSO(γ) and FOSO(γ) estimates are found to outperform the other estimates even more notably, providing accurate predictions not only for the hardening rates but also for the flow stresses [START_REF] Chaix | Modélisation micromécanique du comportement viscoplastique d'un polycristal[END_REF].

It is noted, however, that the most relevant choice of slip-rate measures for the description of hardening may be contingent on the specific crystal plasticity law. For instance, [START_REF] Suquet | Elasto-plasticity of heterogeneous materials at different scales[END_REF] have advocated for the use of the second moments of intragranular plastic slip and have verified its suitability for certain crystal plasticity laws describing transient creep in ice. On the other hand, [START_REF] Castelnau | The effect of strain heterogeneity on the work hardening of polycrystals predicted by mean-field approaches[END_REF] reported mean-field descriptions of the self-consistent type for two-dimensional model polycrystals exhibiting linearly viscous slip rates with linearly hardening flow stresses and observed that the predicted stress-strain response was relatively insensitive to the use of slip-rate measures employing statistical information beyond the first moments. It would not be surprising that higher-order slip-rate measures may result progressively more relevant as the crystal plasticity law is increasingly nonlinear.

In conclusion, second-order homogenization estimates based on a generalized-secant linearization are found to provide an appropriate compromise between precision and mathematical complexity to generate mean-field descriptions for the elastoplastic response of polycrystalline media governed by complex crystal plasticity laws. Most accurate descriptions should result from the combined use of a generalized-secant linearization of the crystal plasticity law, a slip-rate measure incorporating information on intragranular sliprate fluctuations for the evolution of the dislocation densities, and a coupled homogenization of the elastic and plastic deformations. Strategies to incorporate this last feature in the above mean-field descriptions for neutron-irradiated bainitic steels are currently under investigation.

Figure 1 :

 1 Figure 1: Ten microgeometries employed for the full-field descriptions.

Figure 2 :

 2 Figure 2: Schematic of the linearization employed by the second-order homogenization scheme of Ponte Castañeda (2015).

Figure 3 :

 3 Figure 3: Full-field simulations for unirradiated specimens subject to uniaxial tension at low loading rate: (a) traction curves and (b) total dislocation densities for a single realization discretized with 9 3 , 19 3 , 28 3 , 37 3 , 47 3 , and 56 3 voxels per unit cell; (c) traction curves and (d) total dislocation densities for ten realizations discretized with 47 3 ; symbols correspond to the most representative realization.

Figure 4 :Figure 5 :Figure 6 :Figure 7 :

 4567 Figure 4: Traction curves for: (a) unirradiated specimens loaded at a low rate, (b) irradiated specimens loaded at a low rate, (c) unirradiated specimens loaded at a high rate, (d) irradiated specimens loaded at a high rate. Full-field simulations (FFT) are compared with elementary homogenization schemes of the Sachs and Taylor types, and the second-order (FOSO) homogenization scheme of the Self-Consistent type. Also included in part (a) is the secant (SEC) homogenization scheme of the Self-Consistent type.

Table 2 :

 2 are specified in terms of Euler angles set in accordance Material parameters

	symbol parameter	value	unit
	κ	bulk modulus	175	GPa
	µ	shear modulus	81	GPa
	γ0	reference slip rate	6.3 × 10 -16	s -1
	τ 0	reference viscous shear stress	94	kPa
	τcr	reference critical shear stress	300	MPa
	d/d 0	ferrite lath package size parameter	8.3	-
	ρ 0	characteristic dislocations density	2.24 10 14	m -2
	A d	grain size obstacle	0.015	-
	A SC	solute clusters obstacle	94.8	-
	A self	dislocation obstacle (same system)	11.9	-
	A obs	dislocation obstacle (different systems)	107.4	-
	A dis	thermal activation and anhilation parameter 5.93	-
	c 1	interaction matrix coefficient	0.1	-
	c 2	interaction matrix coefficient	0.7	-

Acknowledgments

The work was funded by the Institute of Fusion and Instrumentation Sciences in Nuclear Environments (ISFIN) and by the Institut de Radioprotection et de Sûreté Nucléaire (IRSN). Additional funds from the Universidad Nacional de La Plata (UNLP) were recieved by M.I.I. through grant I-2019-247.

Appendix A. The Self-Consistent scheme Self-consistent estimates for thermoelastic systems are available from the works of [START_REF] Laws | On the thermostatics of composite materials[END_REF] and [START_REF] Willis | Variational and related methods for the overall properties of composites[END_REF]. They can be written as r) , and g

where B (r) and b (r) are concentration tensors given by

In this last expression, M * = Q -1 -M where Q is a microstructural tensor that depends on M and on the 'shape' of the two-point correlation functions for the distribution of the grain orientations. The polycrystals considered in this work are 'equi-axed' and untextured so that the two-point correlation functions exhibit isotropic symmetry. In this case,

where L = M -1 , and the components of H(ξ) are given in terms of the components of the accoustic tensor

Finally, M solves the implicit equation

which follows from the combination of expressions (A.1) 1 and (A.2) 1 . In all the above expressions, the inverse of fourth-order tensors is defined in the subset of incompressible compliance tensors. Expressions for the derivatives required by the identities (29) can be found in [START_REF] Liu | Macroscopic Behavior, Field Fluctuations and Texture Evolution in Viscoplastic Polycrystals[END_REF].