A Madame le Professeur Maud Robert Qui me fait le très grand plaisir de venir jusqu'à Grenoble pour juger ce travail, Merci pour ce semestre lyonnais où j'ai tant appris et où j'ai passé de si bons moments avec toute l'équipe du Pavillon D. Merci pour ton énergie, ta pédagogie et tes encouragements.

A Monsieur le Professeur Mircea Chirica Merci pour ton soutien et ton expérience que tu nous transmets tous les jours. Un semestre à peine que nous nous connaissons mais ta présence compte déjà beaucoup pour moi.

Au Pr Fabian Reche Merci pour ta confiance et ton enseignement au quotidien. Ton souci du détail permet de mesurer toute la complexité de la chirurgie et de progresser pas à pas. A ma femme Emmanuelle, que j'aime de tout mon coeur. Merci pour ton efficacité extraordinaire qui te permet de gérer ton petit mari chirurgien, son emploi du temps et tout le reste ! J'espère être digne de toute cette confiance que tu m'apportes chaque jour. Tu es mon rayon de soleil, ma raison d'être, mon bonheur. Je t'aime the main reasons for sigmoid resection in complicated diverticular disease. [START_REF] Chapman | Complicated diverticulitis: is it time to rethink the rules?[END_REF][START_REF] Chapman | Diverticulitis: a progressive disease? Do multiple recurrences predict less favorable outcomes?[END_REF] Significant improvement in quality of life and social functioning accrues from elective sigmoid colectomy in the majority of patients also represents another reason to offer operation. [START_REF] Forgione | Prospective evaluation of functional outcome after laparoscopic sigmoid colectomy[END_REF] It is essential to ascertain exactly the potential long-term iatrogenic sequelae of surgery for the often benign course of the disease, especially in younger patients. [START_REF] Forgione | Prospective evaluation of functional outcome after laparoscopic sigmoid colectomy[END_REF][START_REF] Rafferty | Practice parameters for sigmoid diverticulitis. Standards committee of American Society of Colon and Rectal Surgeons[END_REF] As much as relieving symptoms arising from affected colonic segment, postoperative defecation patterns may be unsatisfactory because of the operation with the result that the intervention has merely replaced one set of symptoms with another. [START_REF] Masoni | Preservation of the inferior mesenteric artery via laparoscopic sigmoid colectomy performed for diverticular disease: real benefit or technical challenge: a randomized controlled clinical trial[END_REF] Furthermore, left colectomy may also detrimentally affect both bladder function and sexual performance through the inadvertent traumatization of the superior hypogastric plexus. [START_REF] Stevenson | Laparoscopically-assisted anterior resection for diverticular disease: follow-up of 100 consecutive patients[END_REF] Dissection of the inferior mesenteric artery (IMA) at its origin is needed to obtain complete lymph node removal in the treatment of left colonic or rectal cancer, but in the surgical management of diverticulitis, the IMA may be spared. [START_REF] Tocchi | Preservation of the inferior mesenteric artery in colorectal resection for complicated diverticular disease[END_REF] Up to now there is still no consensus concerning the vascular approach. [START_REF] Ambrosetti | Management of sigmoid diverticulitis: an update[END_REF] Preservation of the IMA during sigmoid colectomy for diverticular disease theoretically preserves the natural blood supply of the colonic and rectal stump, and the ensuing performance of a healthy and well-nourished anastomosis may prevent anastomotic leakage. Furthermore, sectioning the sigmoid arteries close to the colonic wall without sectioning of the IMA at its origin may preserve the superior hypogastric plexus and could reduce defecatory disorder, as well as urinary and sexual dysfunction. [START_REF] Lesurtel | Does laparoscopic colorectal resection for diverticular disease impair male urinary and sexual function?[END_REF] The elective surgical procedure for sigmoid diverticulitis has long been standardized in our institution and includes high ligation of the IMA proximal to the left colic artery. [START_REF] Trilling | Energy vessel sealing systems versus mechanical ligature of the inferior mesenteric artery in laparoscopic sigmoidectomie[END_REF] Like Senagore et al., we consider that if laparoscopic sigmoid colectomy is to be a common and reproducible procedure regardless of the indications, the surgical technique must be standardized. [START_REF] Senagore | Results of a standardized technique and postoperative care plan for laparoscopic sigmoid colectomy. A 30-month experience[END_REF] The potential benefits of a standardized technique include 1) standardization of surgical instrument sets, 2) definition of benchmarks for the decision to convert before inappropriate investments in time and equipment, and 3) provision of an easily transferred skill set for residents in training, and nurses.

The aim of this study was to prospectively determine the functional impact of elective laparoscopic sigmoid colectomy performed with ligation of the IMA at its origin for complicated diverticular disease in consecutive male patients. To achieve this goal, we therefore assessed intestinal, urinary and sexual function by standardized questionnaires administered at baseline and 6 months postoperatively. Furthermore, these functional results

were correlated to relevant operative parameters, morbidity, and quality of life.

Methods

Study design and setting

We conducted a prospective study at a single 1836-bed university-affiliated hospital serving a predominantly urban population of 675,000 inhabitants in France. The study was conducted in accordance with the principles of the Declaration of Helsinki. According to the recommendations relative to the steps and means to be used for evaluation and diffusion of innovation in surgery published by an international panel of methodologists and surgeons, this study of ambulatory colectomy for consecutive patients can be classified as a phase 1 study.

Informed consent was obtained from all individual participants included in the study.

Patients

From May 2010 to July 2017, all consecutive male patients scheduled for elective laparoscopic sigmoid colectomy for at least one documented episode of diverticulitis at our institution that is both a tertiary referral center and university hospital were included.

Diverticulitis attack was suspected on abdominal symptoms such as left inferior quadrant pain and constipation or diarrhea with fever, and biologic inflammatory syndrome. The diagnosis of diverticulitis was confirmed on at least one CT scan of the abdomen and pelvis at the time of an attack in every instance and included thickening of the sigmoid colon, presence of diverticulas, and corresponding inflammatory mesocolon. Thus, all patients had their disease confirmed by the current gold standard selection modality. [START_REF] Ambrosetti | Management of sigmoid diverticulitis: an update[END_REF][START_REF] Lawrimore | Computed tomography evaluation of diverticulitis[END_REF] Exclusion criteria were patients with colorectal cancer, ulcerative colitis, volvulus, or other sigmoid conditions, along with patients with diverticular bleeding, recurrent diverticulitis after colectomy, or sigmoid undetermined stenosis. Patients with a previous intervention that could have modified the pelvic nerve pattern (i.e., rectal surgery, left nephrectomy, aortic surgery…) and those with peritonitis necessitating an emergency surgery (Hinchey 3-4) [START_REF] Hinchey | Treatment of perforated diverticular disease of the colon[END_REF] were also excluded from the study.

Operative technique

All the interventions were performed by experienced laparoscopic surgeons. The surgical technique has been previously reported. [START_REF] Trilling | Energy vessel sealing systems versus mechanical ligature of the inferior mesenteric artery in laparoscopic sigmoidectomie[END_REF] No mechanical bowel preparation was prescribed in accordance with the current guidelines. [START_REF] Slim | Updated systematic review and metaanalysis of randomized clinical trials on the role of mechanical bowel preparation before colorectal surgery[END_REF] All eligible patients were asked to perform "modern fasting" which is a normal meal the previous evening, followed by oral intake of a 600 mL carbohydrate-loaded drink until 2 hours before surgery. [START_REF] Vlug | Laparoscopy in combination with fast track multimodal management is the best perioperative strategy in patients undergoing colonic surgery -A randomized clinical trial (LAFA-study)[END_REF] A standardized antibiotic prophylaxis with cefoxitin 2 grams was administered at the induction of anesthesia. The patient was placed on the operating table with open legs and both thighs at the same level of the abdomen to enhance maneuverability of the laparoscopic instruments. A urinary catheter was inserted and removed at the end of the procedure.

The first 10-mm balloon port was placed in the umbilicus under direct vision for the 0° laparoscope and a pneumoperitoneum was created and maintained at a pressure ranging between 9 and 12 mm Hg, depending on the physiological state of the patient. The second 12mm port was placed in the right iliaca fossa, the third 5-mm port in the left iliaca fossa and the fourth 5-mm port in the right upper quadrant of the abdomen under direct vision. The abdominal cavity was inspected to confirm the diagnosis and rule out unexpected difficulties.

With the table titled toward the right, the procedure started by transecting the peritoneum on the right of the inferior mesenteric vein to find the avascular plane behind the mesocolon. A medial to lateral dissection was used in every patient. The IMA was identified proximal to the left colic artery, and transected once squeletized with an electrothermal bipolar vessel sealer, after systematic identification of the left ureter and superior hypogastric plexus, lying behind the fascia. Complete mobilization of the left colon including the splenic flexure and the rectosigmoid junction was performed, with division of the inferior mesenteric vein at the inferior border of the pancreas. With the table titled in Trendelenburg position, the distal resection was performed in every case with a laparoscopic articulating linear stapler always aiming to include the rectosigmoid junction. The colorectal anastomosis was performed at upper part of the rectum, side to end, ensuring compliance of the proximal side and was always created intracorporeally using a circular stapler passed transanally. The specimen was removed in every case through a transverse, suprapubic incision without sectioning the rectus abdominis muscles. No drain was left behind. Operating time was considered from first incision to end of wound closure. On return to the surgical unit, patients received no intravenous perfusion; only oral analgesia was administered. Within the following 2 hours, they were encouraged to walk and food was allowed.

Primary outcomes: digestive, urinary and sexual functions

Patients were asked to answer a standardized questionnaire to evaluate preoperative and 6month postoperative bowel symptomatology, urinary and sexual function. Concerning bowel symptomatology, we evaluated the presence and consistency of incontinence and constipation using the Wexner Incontinence Score [START_REF] Jorge | Etiology and management of fecal incontinence[END_REF] and the KESS Score. [START_REF] Knowles | Level of classification performance of KESS (symptom scoring system for constipation) validated in a prospective series of 105 patients[END_REF] The Wexner Incontinence Score consists of five questions scored from 0 to 4 points (0 for no symptom and 20 for total incontinence). The KESS (Knowles Eccersley Scott Symptom) Score consists of thirteen questions scored from 0 to 3 or 4 points (0 for no symptom and 45 for maximum constipation).

The evaluation of urinary function was completed with the International Prostatic Symptom Score (IPSS). [START_REF] Barry | The American Urological Association symptom index for benign prostatic hyperplasia. The Measurement Committee of the American Urological Association[END_REF] The IPSS consists of seven questions scored from 0 to 5 points (0 for no symptom and 35 for the maximum symptom score).

The evaluation of sexual function was completed with the International Index of Erectile Function (IIEF-5). [START_REF] Rosen | The international index of erectile function (IIEF): a multidimensional scale for assessment or erectile dysfunction[END_REF] The IIEF-5 consists of five questions on erection scored from 0 to 5 points (0 for the maximum symptom score and 25 for no symptom).

Secondary outcomes: Surgical data and quality of life

The demographics of all patients (age, body mass index, comorbidity), specific past history of diverticular disease (severity, frequency), and surgical data (operating time, morbidity, mortality, and length of hospital stay), were detailed on a dedicated prospective database.

Finally, Quality of life and the impact of the functional outcome on normal life habits were assessed by the standard Medical Outcomes Survey Short Form 36 (SF-36) questionnaire. [START_REF] Ware | Overview of the SF-36 Health Survey and the International Quality of Life Assessment (IQOLA) Project[END_REF] Divided into three categories: general health, medical status over the last 4 weeks and limitation of activities. General health was evaluated from 6 to 30 points (6 points being the best general health feeling). Medical status over the last 4 weeks was evaluated from 20 to 89 points (20 points being the best medical status). Limitation of activities was completed from 10 to 30 points (10 for the maximum limitation of activities and 30 for no limitation). As for functional scores, patients were evaluated preoperatively and 6 months postoperatively.

Statistical analysis

Baseline patient characteristics were reported as medians and ranges for continuous variables and number and percentages for categorical variables. Fluctuations in the Jorge-Wexner Incontinence, KESS, IPSS, IIEF-5 and SF-36 scores after the six-month follow-up were analyzed using Wilcoxon signed-rank test. Two-sided p-values lower than 0.05 were considered statistically significant. All analyses were performed using Stata version 14.0 (Stata Corporation, College Station, TX, USA).

Results

Patients

Between May 2010 and July 2017, 19 consecutive male patients scheduled for elective laparoscopic sigmoid colectomy for diverticular disease were prospectively enrolled in the study. Median age was 51 years (range, 32-80 years) and a median BMI was 25 kg/m² (range, 16-33 kg/m²). Ten patients (47%) had at least one past digestive surgical history.

Concerning diverticular disease, median number of diverticulitis episodes before surgery was 2 per patient (range, 1-12) with a majority of diverticulitis grade 1 or 2 (95%) (Table 1).

Primary outcomes: digestive, urinary and sexual functions

Concerning the scores about bowel symptomatology, urinary function and sexual function, there were no significant differences between the preoperative and 6 months postoperative total scores.

The digestive scores, with the median of the KESS score and the Wexner score were respectively: 7 and 0 points before surgery, versus 6 and 0 points after (p = 0.39 for KESS score and p = 0.73 for Wexner score) (Tables 2 and3). For KESS score, the only item significantly different was abdominal pain, with 2 points (range, 0-4) before surgery versus 1 point (range, 0-2) 6 months after (p < 0.01).

The urinary function with the median IPSS was 2 points before surgery versus 3 points 6 months later (p = 0.39) (Table 4).

About the sexual function, the median IIEF-5 score before the surgery was 17.5 points versus 18 points 6 months after (p = 0.55) (Table 5). No patient reported sexual dysfunction following such a proximal ligature of the IMA in our series.

No patient was lost to follow-up, but three patients did not answer completely the IIEF-5 score.

Discussion

Historically, Parks' work in 1969 recommended sigmoid colectomy after the second episode of diverticulitis, to prevent the occurrence of a subsequent serious complication. [START_REF] Parks | Natural history of diverticular disease of the colon. A review of 521 cases[END_REF] Works in the early 2000s completely challenged these practice. [START_REF] Chapman | Complicated diverticulitis: is it time to rethink the rules?[END_REF] Indeed, in all patients operated after the second episode, many are operated erroneously, with the consequent risk of morbidity, mortality and impairment of quality of life. Current recommendations tend to propose a conservative medical treatment as far as possible, and delay the surgical treatment for specific, well-defined cases of complications or frail patients in the natural history of diverticular disease, except emergency or doubtful stenosis. [START_REF] Chapman | Complicated diverticulitis: is it time to rethink the rules?[END_REF][START_REF] Salem | The timing of elective colectomy in diverticulitis: a decision analysis[END_REF] This explains in part why this series has small number of patients.

While recent guidelines exist for indication of sigmoid colectomy in the case of complicated diverticular disease, 8 the surgical technique is still debated. If it is admitted that laparoscopy must be performed preferentially, and sigmoid colectomy must resected the rectosigmoid junction, the level of the section of the IMA is not clearly established. [START_REF] Ambrosetti | Laparoscopic elective sigmoidectomy for diverticular disease: a plea for standardization of the procedure[END_REF] In our Department of Colorectal Surgery of Grenoble Alps University Hospital, the IMA is sectioned at its origin, since 1995, for every left colon resection (cancer, inflammatory bowel disease, diverticular disease, etc...). Our experience did not seem to show any increase in functional disorders. [START_REF] Trilling | Energy vessel sealing systems versus mechanical ligature of the inferior mesenteric artery in laparoscopic sigmoidectomie[END_REF] In our prospective study, we included consecutive male patients scheduled for elective laparoscopic sigmoid colectomy with ligature of the IMA at its origin for diverticular disease. All patients answered to the same validated questionnaire before and 6 months after surgery in order to detect the occurrence of a functional disorder. Thus each patient was his own control. There were no significant differences for the digestive, urinary and sexual functions scores. Actually, during our surgery, the IMA was sectioned at 2 cm from its origin, leaving absolutely free the sympathetic innervation lying on the aorta.

Because we were quite far from the plexus, it could not be injured.

In the literature, functional troubles following laparoscopic sigmoid colectomy varies from 8% to 20%, whatever the type of vascular ligature. [START_REF] Lesurtel | Does laparoscopic colorectal resection for diverticular disease impair male urinary and sexual function?[END_REF] It includes anorectal disorders like in the study of Levack et al, urinary disorders, [START_REF] Levack | Sigmoidectomy syndrome? Patients' perspectives on the functional outcomes following surgery for diverticulitis[END_REF] and sexual disorders in 3-9% of cases, mostly with ejaculation disorders.

The frequency of these functional troubles could be increased with the ligature of the IMA at its origin, because of a lesion of the pre-aortic plexus around the origin of the IMA during the dissection. These para-aortic plexus leading sympathetic innervation impulses into the pelvis for anorectal, urinary and sexual functions. Few studies have investigated functional outcomes after sigmoid colectomy with ligature of the IMA at its origin for diverticular disease, with discordant results. In a prospective randomized study of 107 patients, Masoni et al. concluded to a better quality of life and less digestive disorders when the ligature of the IMA is not at its origin. [START_REF] Masoni | Preservation of the inferior mesenteric artery via laparoscopic sigmoid colectomy performed for diverticular disease: real benefit or technical challenge: a randomized controlled clinical trial[END_REF] Similarly, Dobrowski et al., in a non randomized controlled study on a smaller population found fewer anorectal disorders when the IMA is preserved. [START_REF] Dobrowolski | Should we preserve the inferior mesenteric artery during sigmoid colectomy?[END_REF] At the opposite, Lesurtel et al., like this series, did not find any significant difference according to the ligature height of the IMA, for urinary and sexual functions. [START_REF] Lesurtel | Does laparoscopic colorectal resection for diverticular disease impair male urinary and sexual function?[END_REF] Although we systematically perform complete mobilization of the left colon including the splenic flexure, the median operative time in our series was comparable with the one of others. [START_REF] Gaertner | The evolving role of laparoscopy in colonic diverticular disease: a systematic review[END_REF][START_REF] Klarenbeek | Laparoscopic sigmoid resection for diverticulitis decreases major morbidity rates: a randomized control trial. Short term results of the Sigma Trial[END_REF][START_REF] Rotholtz | Patients with less than three episodes of diverticulitis may benefit from elective laparoscopic sigmoidectomy[END_REF] In addition to this, our population was constituted by only male patients in whom pelvic dissection is presumably more difficult due to a narrower pelvis than in female patients.

Of course the teaching mission of our institution supposes to spend time training the medical and paramedical staff.

The morbidity rate, according to Clavien classification, was similar or even better than the mean values reported in the literature. According to the meta-analysis of Cirocchi et al. [START_REF] Cirocchi | Elective sigmoid colectomy for diverticular disease. Laparoscopic vs open surgery: a systematic review[END_REF] and Gaertner et al. [START_REF] Gaertner | The evolving role of laparoscopy in colonic diverticular disease: a systematic review[END_REF] , postoperative sigmoid colectomy morbidity varies from 8 to 29% with an average of almost 17%. The sigmoid colectomy with the ligation of the IMA at its origin is safe.

Our patients were hospitalized the day before the operation. They systematically followed an enhanced recovery protocol except in case of medical or social reasons. [START_REF] Alfonsi | Working Group of Société Française d'Anesthésie Réanimation (SFAR); Société Française de Chirurgie Digestive (SFCD). French guidelines for enhanced recovery after elective colorectal surgery[END_REF] They were allowed to leave hospital when they ate normally, had bowel motion, and no clinical or biological sign of infection. The total postoperative hospital stay was most of the time around the fourth postoperative day, like in the literature. [START_REF] Vlug | Laparoscopy in combination with fast track multimodal management is the best perioperative strategy in patients undergoing colonic surgery -A randomized clinical trial (LAFA-study)[END_REF][START_REF] Gaertner | The evolving role of laparoscopy in colonic diverticular disease: a systematic review[END_REF] Our results about quality of life, evaluated by the SF-36 questionnaire, demonstrated statistically greater general health and greater medical status after operation. The impact of surgical treatment for diverticular disease on quality of life varies from one study to another.

If Forgione et al. concluded toward an improvement in the quality of life after surgery, [START_REF] Forgione | Prospective evaluation of functional outcome after laparoscopic sigmoid colectomy[END_REF] Scarpa et al. were more moderate, demonstrating a significant benefit from surgery only for severe forms of diverticulitis. [START_REF] Scarpa | Health-related quality of life after colonic resection for diverticular disease: long-term results[END_REF] Well-established indications to elective surgery probably play a major role at reaching the best long-term postoperative functional results and quality of life whatever the surgical technique was carried out. [START_REF] Ambrosetti | Functional results following elective laparoscopic sigmoidectomy after CT-proven diagnosis of acute diverticulitis evaluation of 43 patients and review of the literature[END_REF] Our studies, although prospective on consecutive patients with strict inclusion and exclusion criteria, presents some limitations. First, the number of patients included is small. This can be explained by the decrease tendency in surgical indications for diverticular disease in recent years in our institution, following the recent guidelines. Also, by the monocentric aspect of the study, by the choice to include only male patients, and by the exclusion of patients needing an emergency operation for complicated diverticulitis. A significant difference might have been not detected by a lack of power. Concerning our choice to include only male patients, we found it easier to study sexual function disorders in men. Erection and ejaculation are rather objective and easily measurable phenomenon, which are directly impacted by a lesion of the plexus. Since the consequence of a nervous plexus injury in women is not very well known, we decided not to include female patients in this study. Secondly, this series is a phase 1 study, each patient being his own control. It would have been more powerful to construct a prospective, randomized, controlled study to compare the results of sigmoid colectomy with ligature of the IMA at its origin versus a sigmoid colectomy with preservation of the IMA. But in our group, we considered that all the advantages of a standard operation applicable to all indications of left colectomy overcome the risk of having functional disorders, if any. Moreover, preservation of the superior rectal artery might be responsible for some cases of hemorrhage described after colorectal anastomosis that we will not encounter when sectioning the IMA at its origin. Finally, another limitation of the study is the short follow-up of 6 months, while other studies conducted a follow-up of one year or more. Maybe it was too short, but it seemed to us that 6 months postoperative was sufficient not to confuse immediate outcomes with well-established disorders. In the same line of thought, other questionnaires could have been used like the GIQLI for evaluation of quality of life. Those we have chosen are nevertheless validated and widely used in the literature.

Conclusions

Laparoscopic sigmoid colectomy with division of the IMA at its origin for diverticular disease does not increase functional disorders incidence 6 months after surgery. The benefit of the operation on the quality of life is even significantly greater for general health and medical status. These results probably need to be confirmed by a prospective randomized controlled trial.

Prospective evaluation of functional outcome after laparoscopic sigmoid colectomy with division of the inferior mesenteric artery at its origin for diverticular disease in consecutive male patients ABSTRACT :

This is a monocentric prospective observational study that included consecutively 19 male patients between May 2010 and July 2017, who had a laparoscopic sigmoid colectomy with division of the inferior mesenteric artery at its origin for diverticulitis. Patients completed a questionnaire before and 6 months after surgery to determine the functional impact of this surgical technique. Whether for sexual, urinary or anorectal functions, we did not find any significant difference between preoperative and postoperative results. On the other hand, there is a significant improvement in the quality of life after surgery. The mean operating time was 175 minutes. Mortality was zero. Morbidity was 11% (2 minor events). The length of hospital stay was 6 days. Ligature at the origin of the inferior mesenteric artery does not increase the incidence of functional disorders. This result have to be confirmed by a randomized controlled trial.

Etude prospective des résultats fonctionnels après sigmoïdectomie laparoscopique par ligature à l'origine de l'artère mésentérique inférieure pour diverticulite sigmoïdienne chez des patients masculins consécutifs.

RESUME :

Il s'agit d'une étude observationnelle prospective monocentrique ayant inclus consécutivement 19 patients masculins entre mai 2010 et juillet 2017, opérés d'une sigmoidectomie par voie coelioscopique avec ligature à l'origine de l'artère mésentérique inférieure pour diverticulose colique. Les patients remplissaient un questionnaire avant puis 6 mois après la chirurgie afin de déterminer un impact fonctionnel de cette technique chirurgicale. Que ce soit pour les fonctions sexuelle, urinaire ou bien anorectale nous n'avons pas retrouvé de différence significative entre le pré-opératoire et le post-opératoire. Par contre il existe une amélioration signifcative de la qualité de vie après la chirurgie. Le temps opératoire moyen était de 175 minutes. La mortalité était nulle. La morbidité était de 11% (2 évènements mineurs). La durée moyenne d'hospitalisation était de 6 jours. La sigmoidectomie par ligature à l'origine de l'artère mésentérique inférieure n'augmente pas l'incidence des troubles fonctionnels. Ce résultat est à confirmer par un essai controlé randomisé.

A

  FUNCTIONAL OUTCOME AFTER LAPAROSCOPIC SIGMOID COLECTOMY WITH DIVISION OF THE INFERIOR MESENTERIC ARTERY AT ITS ORIGIN FOR DIVERTICULAR DISEASE IN CONSECUTIVE MALE PATIENTS THESE PRESENTEE POUR L'OBTENTION DU DOCTORAT EN MEDECINE DIPLÔME D'ETAT MATHIEU JOLIVET THESE SOUTENUE PUBLIQUEMENT A LA FACULTE DE MEDECINE DE GRENOBLE* Le : 11 octobre 2017 DEVANT LE JURY COMPOSE DE Président du Jury : Mr le Pr Jean-luc FAUCHERON Membres : Mme le Pr Maud ROBERT Mr le Pr Mircea CHIRICA Mr le Pr Fabian RECHE Mr le Dr Olivier RISSE *La Faculté de Médecine de Grenoble n'entend donner aucune approbation ni improbation aux opinions émises dans les thèses ; ces opinions sont considérées comme propre à leurs auteurs. Monsieur le Professeur Jean-luc Faucheron Veuillez trouver ici l'expression de ma plus profonde gratitude pour m'avoir encadré tout au long de ce travail. Merci également pour ces cinq années d'internat. Votre confiance, votre soutien, votre enseignement et votre bonne humeur comptent énormément pour moi.

10 11

 10 Prospective evaluation of functional outcome after laparoscopic sigmoid colectomy with division of the inferior mesenteric artery at its origin for diverticular disease in consecutive male patients Introduction Avoidance of subsequent acute episodes of diverticulitis and treatment of symptoms represent

  Au Pr Christian Letoublon Merci pour le temps et l'attention que vous avez accordé à ma formation. Veuillez trouver ici l'expression de mes sincères remerciements et de mon profond respect.Aux internes de digestif, A Nico (ses semestres à raler, à partir 1 mois et demi en vacances, et à poser les PAC plus vite que son ombre), Vincent Venkiki (un de mes plus grand regrets de n'avoir pas été ton cointerne… mais on a au moins fait le DU de Traumato ensemble, « non de non » !!), Matthieu, Bertrand (le chat noir de la clinique mutualiste), Adrian, Aline, et les plus jeunes avec qui je n'ai pas encore eu le plaisir de travailler. A mes cointernes actuels : Arnaud, Adrien, Benjamin, Janet, Kash, Mona, Quentin Aux autres internes et anciens internes que j'ai croisé, Camille (la seule personne capable de construire sa maison lors de son passage au 12 e ), Jessica (Morelito), Hannah, Clotilde, Amandine, Delphine, Damien, Julie, Christian, Pierre, Loïc, Gabriel, Nico G., Etienne, Fouad, Nico P. (et les footing à la bastille) Aux équipes d'Annecy, Aux Docteurs D. Duprez, P. Morati, JP. Mestrallet, O. Oulie : merci de m'avoir accueilli pour mon premier semestre et de m'avoir fait découvrir les joies de la chirurgie digestive. mes frangins, Bertrand, Yann et Vincent. Tellement de souvenirs que je garde précieusement ! A quatre garçons ça ne pouvait pas manquer ! Toujours l'un ou l'autre en action pour le meilleur ou pour le pire ! Je crois que je vous aime aussi ! Et bien sûr à Stéphanie, qui fait partie de la tribu à présent !! A Laura, ma presque frangine !! A mes grands-parents. Mémé et Papy qui j'espère nous voient de là où ils sont. Et Mamie la reine de la pizza. A ma marraine Ghyslaine (avec Hervé) et mon parrain Philippe (avec Elisabeth) qui m'accompagnent depuis toujours. Je ne vous remercierai jamais assez pour tout ce que vous avez fait pour moi. A ma belle-famille Mijo et Gérard, Jean-Sébastien et Virginie avec Adrien, qui m'ont accueillis si chaleureusement. Chez vous je me sens comme chez moi ! A tous mes cousins, oncles et tantes : merci pour cette belle famille et ces nombreux moments de bonheur partagés ensemble.

	Aux Pr M. Adam, O. Boillot, C. Gouillat : merci pour le temps et l'attention que vous avez
	accordé à ma formation.
	Aux anciens CCA, Michel El Bechwaty (« Michmuch bisou ») et Elise Pelascini (la grande
	soeur !) : merci pour tous ces bons moments.
	A mes cointernes lyonnais Jean-Baptiste (le dessinateur aux gouts musicaux douteux),
	Antoinette (tatoueuses à ses heures perdues), Lucas (l'italien « hyper limite ») et Ninon
	(toujours en self-contrôle sauf avec les téléphones)
	A l'équipe de Lyon du centre Léon Bérard,
	Au Pr M. Rivoire : merci de m'avoir accepté tardivement dans votre service, moyennant quoi
	ce semestre a été l'un des plus formateurs et enrichissant de mon internat.
	Au Dr P. Meeus : merci pour votre savoir-faire chirurgical (« magic finger »), votre bonne
	humeur, vos discussions rugby, vos jeux de mots et votre franc parler sur l'hémorroidectomie
	du samedi après-midi !
	Au Dr P. Peyrat : merci pour le temps et l'attention que tu m'as accordé. Tu as dû te faire
	quelques cheveux blancs ?!
	Au Dr M. Tabutin : merci pour cette piqûre de rappel en chirurgie thoracique.
	Au Dr G. Vaz : merci pour ces quelques blocs d'orthopédie au bon souvenir du pavillon T.
	Aux CCA, Marie Poiblanc (la spécialiste de l'oeuf mollet et du dérapage en trottinette) et
	Ophélie Aumont (la clermontoise perdue à Lyon avec le grenoblois !) : merci pour ce
	semestre. Merci surtout pour votre bonne humeur, votre soutien et les nombreux gouters au
	relais H !
	A mes cointernes Paul-No (le référent des séances biblio du vendredi matin), Aminah (si si la
	variété française c'est génial) et Geoffrey (qui j'espère ira visiter la forêt de Saou !)
	A mes Amis,
	A la Bande de BG, Virginie et Romain (les rois de l'apéro et du jardinage au balcon),
	Bérangère et Benoit (les p'ti poulet de Bresse et leur Donuts), Hélène et Julian (les fans de
	Coldplay et du ski de l'extrême sur piste verte ou toit de maison) : merci pour vos apéros du
	mercredi et tous ces fous rires qui animent nos soirées. Sans oublier Titi, Jeanne et Arnaud qui
	nous accompagnent dès que possible.
	Aux amis de la faculté de Lyon, Quentin et Marion, Sophie et Stevan, Lucile et Thibaut,
	Pierre J et Alicia, Pierre Yves et Jennifer, Pierre B, Cécile et Bruno, Elodie et Guillaume,
	tout nous enseigner,
	même la gynécologie !
	Au Dr JB. Putinier : merci pour toutes ces cures de hernie qui ont démystifiées la région Au Dr Julio Abba inguinale, pour tes visites matinales et ta ponctualité au self ! Merci d'être à nos côtés tous les jours, de prendre du temps pour nous encadrer et nous Au Dr F. Stenard : merci pour tes conseils et ta confiance en nous au bloc. « J'y vais mais j'ai former. Ta rigueur professionnelle n'a d'égale que ta gentillesse et ta disponibilité. peur », c'est le début de l'autonomie.
	Au Dr D. Voirin : merci pour ta patience et tous tes conseils. La proctologie devient une Aux assistants ou anciens assistants du 12 e , A Pierre-alexandre Waroquet (mon premier assistant !), Antoine Guillaud (au rire discret et passion !
	A l'équipe de Lyon de l'Hôpital Edouard Herriot,
	Au Pr G. Poncet : merci de m'avoir accueilli dans votre équipe. Votre rigueur et la passion
	que vous investissez dans votre art resteront un exemple pour moi.

Au Docteur Olivier Risse Merci pour tes conseils, ta rigueur et ta volonté de toujours nous tirer vers le haut. Merci pour ces nuits blanches à greffer, ces chapitres d'EMC que je n'aurai peut-être pas encore lus, et cette arme fatale qu'est le « Tazoscan » ! A mes Maitres, Au Pr Catherine Arvieux Merci pour vos conseils et votre engagement dans ma formation. Votre énergie et votre confiance ont animés ces cinq années d'internat. J'espère m'en montrer digne. raffiné), Marion Tavernier (« il vaut mieux un trou dans la peau que la peau dans le trou », je m'en souviens !), Nicolas Mougin (« ça s'autogère, t'occupes ! »), Synthia Guedj (« Cynthia avec un C, tocard !! »), Pierre-yves Sage (le maître à râler de NT), Sandrine Barbois (qui est presque à l'heure le matin maintenant !), Edouard Girard (le fan de Coldplay à n'importe quel prix !). Merci pour votre amitié et votre compagnonnage. Aux Docteurs S. Jager, JM. Chirpaz, G. Allamel, S. Ionescu, V.Verbat, L. Choulet : merci pour ce semestre si formateur. A l'équipe de chirurgie thoracique du CHU de Grenoble, Au Pr PY. Brichon : merci pour ces six mois en thoracique. Votre passion pour le travail bien fait motive toujours plus à se surpasser. Au Pr P. Chaffanjon : merci pour votre enseignement et votre patience au bloc opératoire. L'anatomie cervicale n'a plus de secret pour moi (ou presque !) Aux (anciens) CCA, Albéric De Lambert et Dan Angelescu, et aux Dr A. Pirvu et K. Hireche : merci pour tous vos conseils. A l'équipe de chirurgie pédiatrique de l'HCE, Au Pr C. Piolat : merci pour votre gentillesse, votre expérience et votre patience. Au Dr S. Antoine : merci pour tout. Tu sais à quel point tu as compté pour moi dans ce semestre. Nous avons toujours une randonnée ou un apéro dans la Drôme en suspens ! Au Dr C. Jacquier et Y. Teklali : merci pour votre enthousiasme et votre pédagogie. Aux (anciens) CCA, Yohan Robert et Emeline Bourgeois : merci pour votre amitié, votre bonne humeur et vos conseils.

A l'équipe de la clinique Mutualiste de Grenoble, Au Dr P. Gabelle : merci pour votre bonne humeur et votre envie de Laure Hélène et Guillaume : merci pour ce groupe si animé où chacun de vos caractères vient compléter les autres ! A quand le prochain week end ? Aux copains du Lycée, Martin, Antoine et Andrea avec mon filleul Henri, Thomas et Caroline avec Alexandra et Elea : merci pour cette amitié si forte et si fidèle. J'espère que cela continuera encore longtemps. A Quentin mon ami d'enfance. Les foots au petit square, les invitations incessantes chez l'un ou l'autre, les jeux d'ordinateurs en réseau et à présent les escapades à Munich à la fête de la bière ! Merci d'être toujours là. Notre amitié je l'espère ne cessera jamais. A David Paris : merci de m'avoir fait découvrir et aimer la chirurgie (à l'aube il faut préciser !). J'ai choisi une autre spécialité que l'orthopédie finalement mais rien de tout ça ne serait arrivé sans toi. Merci pour tout.

A ma Famille, A mes parents, qui ont fait de moi ce que je suis ! Qui ont toujours été derrière moi pour donner le meilleur de moi-même. Du bonheur à chaque instant d'avoir un papa et une maman comme vous. Je vous aime. A Merci à tous ceux et celles qui ont croisés ma route un jour. Merci à vous qui êtes venus me soutenir aujourd'hui Enfin, merci tout particulièrement… A ma fille Marie, ma princesse. Chaque seconde à te voir sourire, parler, bouger, grandir… est un émerveillement. Être ton papa me comble d'une immense joie. Rien ne pourrait être plus important.

Remerciements

Secondary outcomes: Surgical data and quality of life

The median operative time was 175 minutes (range, 101-374 minutes). There were no conversions to laparotomy. There were no perioperative deaths. The morbidity rate was 11%, including 2 minor events according to Clavien classification, with a wound abscess evacuated at the patient's bed, and an atrial fibrillation treated medically in one further patient. The median length of hospital stay was 6 days (range, 5-11 days) (Table 6).

Quality of life, evaluated by the SF-36 questionnaire, demonstrated statistically greater general health (p < 0.01) and greater medical status over the last 4 weeks (p < 0.01), 6 months after surgery, compared to baseline. There were no difference about limitation of activities (p = 0.27) (Table 7).

Annexes


Christophe Dano

Pierre-Yves Hicher

POUR SOLS VIERGES U N Modele De Comportement

Injectes C Dano

P-Y Hicher

A constitutive model for uncemented and grouted sands

Introduction

Constitutive models have been previously proposed in the literature in order to simulate the mechanical behavior of cemented soils [START_REF] Hirai | An elastic-plastic constitutive model for the behavior of improved sandy soils[END_REF][START_REF] Prisco | A mathematical model of grouted sand allowing for strength degradation[END_REF]Reddy et Saxena, 1992;[START_REF] Abdulla | Behavior of cemented sands -Part II: modelling[END_REF][START_REF] Sun | An elastoplastic model for frictional and cohesive materials and its application to cemented sands[END_REF]. However, their degree of complexity usually make these models not easily applicable in engineering design because they are generally characterized by an important number of parameters whose physical meaning could also be questionable. The identification of these parameters could require sophisticated laboratory tests.

Therefore, for a design purpose, a straightforward constitutive model with seven parameters is presented herein. Based on the elastic perfectly plastic model with a Mohr-Coulomb yield surface, it aims at describing the non linear behavior of both uncemented and grouted soils from the onset of shearing to the maximum shear strength by use of only two additional parameters.

Experimental observations

Grout injection in the laboratory

Grout injection allows to reinforce a soil by filling its voids with a cementitious binder. The setting of the cement gives rise to a notable improvement of the stiffness and of the strength of the soil, which subsequently protects the workers and the neighboring building foundations from any structural damage.

In order to study the mechanical behavior of grouted sands, a fine and siliceous Fontainebleau Sand, with a mean grain size D 50 equal to 220 µm, a Hazen's ratio C u equal to 1.4, a minimum and maximum void ratio e min and e max respectively equal to 0.576 and 0.857, was injected in the laboratory by a microfine cement grout with a cement to water ratio equal to 0.172. The injection device was described in [START_REF] Dano | Comportement mécanique des sols injectés[END_REF]. It leads to a good homogeneity of the grouted samples as well as a good repeatability of the injection process.

Mechanical behavior

Drained triaxial tests were performed on samples of uncemented Fontainebleau sand prepared through air pluviation and on samples of grouted sand. Density indexes of the granular skeleton were similar in the two cases (I d ≈ 95 %). Confining pressures σ' c were taken between 0 and 400 kN/m². The comparison of the stress -strain curves (Figure 1a) clearly shows the effect of the grout injection on the mechanical behavior of the tested soils. Similar patterns were observed on alluvial deposits sampled during the tunneling of the 14 th metro line in Paris (France) and injected with the same microfine cement grout. The maximum strength is increased. In the (p', q) representation where p' is the mean effective stress and q the deviatoric stress, the maximum shear strength envelope of the grouted sand is fitted by a straight line whose slope M p is almost parallel to the linear strength envelope obtained for the uncemented sand, for the same density index (Figure 1b). In other words, the friction angle ϕ' is hardly changed by the grout injection. The slope M p is equal to 1.60 for the uncemented sand (ϕ' = 39.1 degrees) and 1.67 for the grouted sand (ϕ' = 40.7 degrees). The slight increase of the friction angle is probably due to the additional compaction of the granular skeleton during injection.

It also induces a cohesion C m whose value mainly depends on the soil (nature, density, grading curve) and on the grout (nature and cement to water ratio C/E) [START_REF] Tailliez | Etude expérimentale du comportement mécanique des sols granulaires injectés[END_REF][START_REF] Dano | Comportement mécanique des sols injectés[END_REF]. The cohesion C m is equal here to 660 kN/m 2 , which corresponds to a cohesion c' in the Mohr diagram equal to 341 kN/m 2 . [START_REF] Tailliez | Etude expérimentale du comportement mécanique des sols granulaires injectés[END_REF] proved experimentally that the mechanical behavior is independent of the strain rate for soils injected with microfine cement grouts.

Grout injection also generates a clear improvement of the stiffness. However, in the case of grouted sands, the effect of the mean stress on the tangent stiffness is less sensitive than in the case of uncemented sands [START_REF] Dano | Comportement mécanique des sols injectés[END_REF].

In the two cases, the volumetric behavior is firstly contracting and then dilating for greater deviatoric stresses. Values of the Poisson's ratio ν are also similar, between 0.2 and 0.3. However, some differences appear due to the cementation between the grains. The dilation angle ψ of grouted sands is at least equal and often greater than the dilation angle of uncemented sands (Figure 1a).

For uncemented sands, the characteristic state concept defined by [START_REF] Luong | Phénomènes cycliques dans les sols pulvérulents[END_REF], which corresponds to the transition between the contracting domain and the dilating one, can be considered. The maximum shear strength coincides with the maximum dilation rate. For grouted sands, it is no longer the case. The transition from the contracting domain to the dilating one is more or less simultaneous to the peak strength. Consequently, the mechanical behavior of grouted sands is rather contracting up to the maximum strength.

Constitutive model

Equations of the model

In order to numerically simulate the previous experimental observations, a strain hardening elastoplastic model was elaborated. This constitutive model was based on the commonly used elastic perfectly plastic model with a Mohr-Coulomb criterion. It aimed at describing the non linear behavior of both uncemented and grouted sands. Compressive stresses and strains are taken negative, except the mean stress p' for convenience.

At the beginning of the loading, these materials are assumed to behave elastically according to the Hooke's law:

( ) ( ) ( ) ( ) ε × ν + × × + Ι ε × ν - × ν + × ν = σ 1 2 E 2 Tr 2 1 1 E o (1)
where E, ν, σ, ε and Ι are respectively the elastic modulus, the Poisson's ratio, the stress tensor, the strain tensor and the identity matrix. However, the elastic domain is limited to a small region around the hydrostatic axis in the principal stress space (Figure 2). By strain hardening, the size of the elastic domain gradually increases as a function of the deviatoric plastic strain ε d p through the yield function f and the hyperbolic hardening function r d :

( ) ( ) ( ) ( ) ( ) 0 r C ' p m M q r , , q , ' p f p d d p p p p d d = ε × + - θ × = ε θ (2) ( ) ( ) p d ve el p d el p d d a r 1 r r ε + - × ε + = ε (3)
where M p is the slope of the maximum shear strength line in the (p', q) representation and C p the hydrostatic tensile strength which is assumed to be fixed on the hydrostatic axis as indicated in Figure 2. The parameter r el defines the initial size of the elastic domain whereas the parameter a ve characterizes the hardening rate. The function r d is equal to 1 when the maximum shear strength line is reached. Previous experimental works on purely frictional materials and on sand -cement mixtures have shown the effect of the third stress invariant on the shear strength [START_REF] Reddy | Constitutive modeling of cemented sand[END_REF]. Assuming that the third stress invariant also affects the behavior of grouted sands, the yield function depends on the Lode's angle θ such as:

( ) ( )       θ × ϕ - θ × ϕ - = θ sin 3 ' sin cos ' sin 3 3 6 m p (4)         × × - = θ 3 2 3 J J 2 3 3 sin Arc 3 1 (5)
where J 2 and J 3 are respectively the second and the third stress invariants of the deviatoric stress tensor s whose components are: The function m(θ) is equal to 1 in axisymetric compression (θ = π/6) and to β = (3-sinϕ')/(3+sinϕ') in axisymetric extension (θ = -π/6). The relation ( 2) is therefore equivalent to the Mohr-Coulomb criterion [START_REF] Bardet | Lode dependences for isotropic pressure-sensitive elastoplastic materials[END_REF]. The constitutive model is also chosen as a non-standard model which requires the formulation of a plastic potential function. Based on the characteristic state concept by [START_REF] Luong | Phénomènes cycliques dans les sols pulvérulents[END_REF], the plastic strain increments are related to the slope M c of the characteristic line in the (p',q) representation and to the cohesion C p by: ( ) ( )

( ) Ι × σ × - σ = Tr 3 1 s ij ij
el p d d p c c p d p v r 1 r 1 C ' p q m M d d - ε - × + - θ = ε ε (7) ( ) ( )       θ × ϕ - θ × ϕ - = θ sin 3 sin cos sin 3 3 6 m c c c (8)
where ϕ c is the friction angle corresponding to the characteristic state. A fictitious degradation of the cohesion which depends on the hardening function r d is introduced in the relation (7). It allows to accurately reproduce the volumetric strains in the dilating domain for grouted sands. The plastic potential function depends on the Lode's angle in order to maintain it inside the yield surface.

Identification of the parameters

The constitutive model is characterized by seven parameters : -the elastic modulus E and the Poisson's ratio ν for the elastic part of the model; -the slope of the shear strength envelope M p and the intercept with the stress deviator axis C m or the pseudo-hydrostatic tensile strength C p (C p = C m / M p ); -the slope of the characteristic state line in the (p',q) representation M c ; -the parameters r el and a ve which respectively define the initial size of the elastic domain and the strain hardening rate. The elastic parameter E is conventionally the tangent modulus at the origin of the stress deviator -axial strain curve. The Poisson's ratio ν is related to the initial slope of the volumetric strain versus axial strain curve in the contracting domain:

        ε ∆ ε ∆ - × = ν 1 v 1 2 1 ∆ε v > 0, ∆ε 1 > 0 (9)
The identification of the three parameters M p , C m and M c requires to perform at least three triaxial tests and to report the stress states corresponding to the maximum shear strength and to the characteristic state in the (p',q) representation, as done in Figure 1.

The determination of the initial size of the elastic domain is a complex experimental problem since it involves to follow the evolution of the secant modulus from the very small strain domain to the large strain domain. It is well-know that for many soils, the elastic behavior is limited to strains less than 10 -5 [START_REF] Hicher | Elastic properties of soils[END_REF]. Only specific devices placed on the specimens themselves inside the triaxial cell and fitted out with high performance transducers give access to an accurate measure of the secant modulus. When such an internal device can not be used, the value of the parameter r el can only be assessed. For instance, the value of the parameter r el is set to 0.01 for uncemented sands and to 0.1 for grouted sands.

Finally, assuming that elastic strains are negligible compared to the plastic strains, the value of the parameter a ve is optimized from the stress -strain curves obtained in the laboratory. In the case of uncemented sands, the hyperbolic hardening function is particularly well adapted. Computations reveal that the parameter a ve depends on the nature of the soil and on the confining stress such as a ve = k × σ' c where k is a constant (k = 1.3 × 10 -6 m²/kN for the Fontainebleau sand) (Table I). On the contrary, in the case of grouted sands, an exponential hardening function would be more suitable to describe the non linearity. The implementation of an exponential hardening function constitutes a further development of the model. The hyperbolic function is subsequently considered for grouted sands: optimization computations show a slight dependence of a ve with the confining stress (Table II).

Validation computations

Verification of the predictions of the model for axisymetric loading paths is done in this section. Triaxial tests on saturated uncemented Fontainebleau sand samples (I d = 95 %, B-values greater than 0.96) and on grouted Fontainebleau sand samples saturated with water before injection were carried out. The values of the model parameters are indicated in Table I for the uncemented sand and in Table II for the grouted sand. Figure 3 represents the comparison between experimental data and computational results. The slight scattering can be explained by the fact that the values of the parameters M c , M p and C p are mean values obtained by carrying out at least three triaxial tests. Stress-strain relationships and volumetric strains are correctly assessed.

Conclusions

Based on experimental results, a strain hardening elastoplastic model was elaborated. Seven parameters characterize this model suitable for describing the behavior of both uncemented and grouted sands. The model also constitutes a good compromise between simplicity and performances of the predictions. Further improvements will include the implementation of a continuously differentiable yield surface in order to avoid singularities due to the Mohr-Coulomb formulation and the definition of a new hardening function in order to better fit the non linear behavior of grouted sands.

  Figure 1a. Stress -strain curvesFigure1b. Shear strength envelopes.

Figure 1 .

 1 Figure 1. Results of triaxial tests carried out on uncemented and grouted Fontainebleau sand.

Figure 2 .

 2 Figure 2. Principle of the constitutive model.

Figure 3 .

 3 Figure 3. Numerical computations for uncemented and grouted sands.

Table I .

 I Values of the parameters for the uncemented Fontainebleau sand.

	σ' c (kN/m 2 ) E (MN/m 2 )	ν	M c	M p	C p (kN/m	2 )	r el	a ve
	100	64							0.00049
	200	131	0.25	1.17	1.60	0		0.01	0.00063
	400	222							0.00087

Table II .

 II Values of the parameters for the grouted Fontainebleau sand. σ' c (kN/m 2 ) E (MN/m 2 )

					ν	M c	M p	C p (kN/m	2 )	r el	a ve
	100							0.00012
	200	307	0.20	1.59	1.68	317	0.1	0.00014
	400							0.00018
		1500
	)
	2
	Stress deviator q ( kN/m	500 1000	Symbols : Experiment Continuous lines : Numerical simulation
		0
		0	0.01	0.02 Axial strain ε	0.03 1	0.04	0.05


J Paltrinieri

F Nyssen

M.-O Parent

A Batailly

Towards full 3D numerical simulation of whirl motions stemming from unilateral contact constraints in aircraft engines

Keywords: rotor/stator interaction, rubbing, nonlinear dynamics, unilateral contact dynamics, friction, abradable coating, fan stage interaction rotor/stator, frottement aube/carter, dynamique non-linéaire, dynamique du contact, frottement, revêtement abradable, soufflante

The reduction of nominal clearances between a rotating bladed-disk and its surrounding casing yields a very significant increase of the overall engine efficiency. However, the smaller the clearances, the higher the risk of structural contacts between static and rotating components that may lead to hazardous interaction phenomena. In particular, at the fan stage of an aircraft engine, impacts between the rotating bladed-disk and the casing may generate forward or backward whirl motions induced by the precession of the shaft axis of rotation. In such specific configuration, an accurate modeling of interaction phenomena requires to account for both centrifugal and gyroscopic effects on the rotor. This contribution addresses the development of efficient reduced-order models of industrial finite element models embedding both centrifugal and gyroscopic effects. Proposed developments are validated on an academic model and are then applied on the finite element model of an aircraft engine fan stage. Results obtained with the academic model underline that the impact of gyroscopic effects on the rotor's dynamics is essentially related to the frequency split of 1-nodal diameter free-vibration modes of the first modal family. Results presented on the industrial finite element models are limited to a few case studies as a proof-of-concept.

INTRODUCTION

Recent research efforts focusing on the numerical simulation of rotor/stator interactions stemming from structural contacts between blade-tips and casings in aircraft engines led to the development of an industrial computational tool [START_REF] Millecamps | Snecma's viewpoint on the numerical and experimental simulation of blade-tip/casing unilateral contacts[END_REF] designed for the numerical prediction of such events. These interaction phenomena are intrisically non-linear and indeed require dedicated computational tools to overcome the current limitations of commercial softwares such as: [START_REF] Millecamps | Snecma's viewpoint on the numerical and experimental simulation of blade-tip/casing unilateral contacts[END_REF] the absence of efficient strategies for the modeling of abradable coating removal [START_REF] Legrand | Numerical investigation of abradable coating removal through plastic constitutive law in aircraft engine[END_REF] or, [START_REF] Legrand | Numerical investigation of abradable coating removal through plastic constitutive law in aircraft engine[END_REF] the impossibility to build physically relevant reduced-order models (ROM) of rotating structures subjected to contacts [START_REF] Arzina | Vibration analysis of compressor blade tip-rubbing[END_REF]. The physical relevance of rotor reduced-order models is intimately related to the modeling of both centrifugal and gyroscopic effects. For most stages of an aircraft engine, accounting for centrifugal effects is of prime importance since bladed components are highly sensitive to them, particularly in fan and low-pressure stages featuring longer blades. Gyroscopic effects however typically have low to no influence on the rotor's dynamics as long as its axis of rotation is fixed. Such hypothesis is commonly accepted when investigating low-or high-pressure compressor stages. For that reason, numerical simulations of two types of rotor/stator interactions namely [START_REF] Millecamps | Snecma's viewpoint on the numerical and experimental simulation of blade-tip/casing unilateral contacts[END_REF] rubbing phenomena [START_REF] Roques | Modeling of a rotor speed transient response with radial rubbing[END_REF][START_REF] Ma | Vibration response analysis of blade-disk dovetail structure under blade tip rubbing condition[END_REF][START_REF] Emery | An experimental investigation of temperatures due to rubbing at the blade-seal interface in an aircraft compressor[END_REF] and, (2) modal interaction events [START_REF] Schmiechen | Travelling wave speed coincidence[END_REF][START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for flexible casings with friction and abradable coating[END_REF] have been conducted without accounting for gyroscopic effects. A third type of rotor/stator interactions due to structural contacts is related to the occurence of orbital motions of an aircraft engine fan stage around its axis of rotation. These whirl motions [START_REF] Sinha | Dynamic characteristics of a flexible bladed-rotor with coulomb damping due to tip-rub[END_REF][START_REF] Sinha | Rotordynamic analysis of asymmetric turbofan rotor due to fan blade-loss event with contact-impact rub loads[END_REF] may feature significant displacements of the center of the rotor thus considering a negligible impact of gyroscopic effects is not an acceptable assumption anymore. As a continuation of previous research work on whirl motions based on simplified 2D models [START_REF] Salvat | Two-dimensional modeling of unilateral contact-induced shaft precessional motions in bladed-disk/casing systems[END_REF], this contribution discusses the development of ROMs accounting for both centrifugal and gyroscopic effects over a wide angular speed range. Such models are indeed the missing step required for carrying out the first full 3D numerical simulations of whirl motions with industrial finite element models.

The simulation of whirl motions initiated by blade-tip/casing contacts is intrinsically non-linear and constitute a challenge from a numerical point of view. Such numerical investigations have been the focus of several research work carried out in collaboration with different manufacturers [START_REF] Williams | Simulation of blade casing interaction phenomena in gas turbines resulting from heavy tip rubs using an implicit time marching method[END_REF][START_REF] Sinha | Rotordynamic analysis of asymmetric turbofan rotor due to fan blade-loss event with contact-impact rub loads[END_REF][START_REF] Salvat | Two-dimensional modeling of unilateral contact-induced shaft precessional motions in bladed-disk/casing systems[END_REF]. So far, and to the best of the author's knowledge, these motions have never been predicted with full 3D finite element models and there is currently no way to obtain information specific to the balde design with respect to these interactions. This level of prediction may only be reached by considering industrial finite element models which calls for the use of advanced reduction techniques in order to maintain reasonable computation times.

The first section of the article focuses on the theoretical description of gyroscopic effects modeling and their combination with ROMs of industrial finite element structures. Then, an extensive validation procedure is carried out with an emphasis on the convergence of numerical results with respect to key parameters of the simulation such as: (1) the modal reduction parameter η, (2) the time integration step ∆t and, (3) the spatial discretization of the abradable coating deposited on the casing contact surface. The second section also features a comparison between simulations with and without gyroscopic effects on an academic model. Finally, in the last section of the article, the proposed developments are applied on the industrial finite element model of an aircraft engine fan stage.

MODELING

The gyroscopic effects affect the velocity term in the equation of motion. This term represents the conservation of the angular momentum. The angular speed is assumed to be constant in the following developments.

Equation of motion

Taking into account the centrifugal and gyroscopic effects, the equation of motion which governs the system is given by

Mẍ + (D + G(Ω)) ẋ + K(Ω)x = F (1)
where M, G, K and D denote respectively the mass, gyroscopic, stiffness and damping matrices, Ω is the angular speed, and F contains both the external and the contact forces.

Denoting N the interpolation function matrix, the matrices in Eq. ( 1) are given by

M = V ρN NdV (2) G(Ω) = V 2ρN ΩNdV (3)
K(Ω) = V ε(x) Cε(x)dV (4)
F = V N f dV ( 5
)
where ρ is the density of the material, ε the deformation tensor, V the integration volume and f the external force.

Time integration procedure

An explicit time integration procedure is used in the paper, based on the central finitedifference method. At each time step t, displacements x t+1 are linearly predicted using the following scheme:

x p t+1 = Ax t + Bx t-1 (6)
A = M h 2 + D + G 2h -1 2M h 2 -K (7)
B = M h 2 + D + G 2h -1 D + G 2h - M h 2 (8)
If a contact interaction is detected at the time step t, a correction is added to the predicted displacements such as:

x t+1 = x p t+1 + Cλ (9)
where λ denotes the contact forces and C = M h 2 + D+G 2h -1 . The addition of gyroscopic effects modifies the structural matrices involved in the time integration scheme, but does not affect the computational time of the procedure.

Gyroscopic effects

The gyroscopic matrix is linearly dependent on the angular speed matrix Ω which is skewsymmetric and whose three components characterize the rotation of the structure along three orthogonal axes x, y and z, such as

Ω(t) =   0 -Ω z (t) Ω y (t) Ω z (t) 0 Ω x (t) -Ω y (t) Ω x (t) 0   (10)
In the remainder, it is assumed that the rotor's axis of rotation is z. The angular speed is assumed constant and is denoted: Ω z (t) = Ω. Under this assumption:

K( Ω) = 0 (11)
and the angular speed matrix becomes

Ω =   0 -Ω 0 Ω 0 0 0 0 0   = Ω   0 -1 0 1 0 0 0 0 0   S (12)
Those approximations allow to extract the angular speed from the integral in Eq. (3), which leads to

G(Ω) = 2 Ω (S ⊗ I n ) M (13)
where ⊗ and I n respectively denote the Kronecker product and the identity matrix of dimension n, the number of degrees of freedom of the model.

Centrifugal effects

The centrifugal effects affect the computation of the stiffness matrix. Since the angular speed Ω is constant, as a first approximation, one may write [START_REF] Sternchüss | On the reduction of quasi-cyclic disks with variable rotation speeds[END_REF]

K(Ω) = K a + Ω 2 K b + Ω 4 K c ( 14
)
where the matrices K a , K b and K c are computed by computing Eq. ( 14) for three values of the angular speed Ω, for instance, with Ω = 0, Ω max /2 and Ω max . In this case, the three matrices can be expressed as [START_REF] Sternchüss | On the reduction of quasi-cyclic disks with variable rotation speeds[END_REF][START_REF] Kaptan | The vibrational behavior of coupled bladed disks with variable rotational speed[END_REF]]

K a = K(0) (15)
K b = 1 3Ω 2 max 16K Ω max 2 -K(Ω max ) -15K(0) (16)
K c = 4 3Ω 4 max K Ω max 2 -4K(Ω max ) + 3K(0) (17)
Cyclically symmetric structures In this study, the structures (i.e., the bladed-disk and its surrounding casing) are perfectly tuned and composed of N identical sectors. As a consequence, their modes can be sorted by modal families and nodal diameters. Those properties yield symmetric mass and stiffness matrices. In addition, these matrices are block circulant in a cylindrical coordinate system, such as [START_REF] Bladh | Efficient predictions of the vibratory response of mistuned bladed disks by reduced order modeling[END_REF]

Y =           Y 0 Y 1 0 0 . . . Y 1 Y 1 Y 0 Y 1 0 . . . 0 0 Y 1 Y 0 Y 1 . . . 0 . . . . . . 0 0 . . . Y 1 Y 0 Y 1 Y 1 0 . . . 0 Y 1 Y 0           , Y ∈ {K; M} (18)
where Y 0 corresponds to one sector and Y 1 to the mechanical coupling between two adjacent sectors.

In the same way, considering Eq. ( 13), the gyroscopic matrix is skew-symmetric and block circulant, such as

G(Ω) =           G 0 G 1 0 0 . . . G 2 G 2 G 0 G 1 0 . . . 0 0 G 2 G 0 G 1 . . . 0 . . . . . . 0 0 . . . G 2 G 0 G 1 G 1 0 . . . 0 G 2 G 0           (19) with: G 0 (Ω) = 2 Ω (S ⊗ I n ) M 0 (20) G 1 (Ω) = 2 Ω (S ⊗ I n ) M 1 (21) G 2 (Ω) = 2 Ω (S ⊗ I n ) M 1 ( 22
)
where n represents the number of degrees of freedom of an elementary sector. Additionally, the projection onto the Fourier space (by means of the Fourier matrix F) leads to mass and stiffness matrices M and K, organized by spatial harmonic n d , which are symmetric and block diagonal

Ỹ = F YF = diag( Ŷ(0) , Ŷ(1) , Ŷ(2) , ..., Ŷ( N/2 ) ), Y ∈ {K; M} (23)
in which:

Ŷ(n d ) = Y 0 + (Y 1 + Y 1 ) cos(n d α) (Y 1 -Y 1 ) sin(n d α) (Y 1 -Y 1 ) sin(n d α) Y 0 + (Y 1 + Y 1 ) cos(n d α) (24)
for n d = 0, ..., N 2 . The • symbol means to the integer part. The same projection onto the Fourier space can be done for the gyroscopic matrix, leading to a skew-symmetric and block diagonal matrix

G = F GF = diag( Ĝ(0) , Ĝ(1) , Ĝ(2) , ..., Ĝ( N/2 ) ) (25) with Ĝ(n d ) = 2 Ω (S ⊗ I n ) M (n d ) .

Modal reduction

A modal reduction method is used to reduce the dimension of the model. The Craig-Bampton method [START_REF] Craig | Coupling of substructures for dynamic analyses[END_REF] is considered here because it advantageously retains physical degrees of freedom that are needed for accurate contact management.

Using the change of variable

x = Φq ( 26
)
where Φ is the reduction matrix of the Craig-Bampton method [START_REF] Craig | Coupling of substructures for dynamic analyses[END_REF] in which η modes are retained and q is the reduced coordinates, the equation of motion in the modal space becomes

Φ MΦq + Φ DΦ + Φ GΦ q + Φ K(Ω)Φq = Φ F (27)
Denoting the reduced matrices with the index r (A r = Φ ÃΦ), Eq. ( 27) now reads

M r q + (D r + Ω G r ) D r q + K r (Ω) q = F r (28) with G r = 2 (S ⊗ I n ) M r .
One may note that the reduced gyroscopic matrix G r is independent of the angular speed Ω and thus requires to be computed only once for a given rotating structure. The gyroscopic effects act as a modification of the damping matrix in the equation of motion, with D r the modified damping matrix in Eq. (28). Also, more details with respect to the computation of K r (Ω) and the treatment of centrifugal effects may be found in [START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF].

CONVENTIONS AND NOTATIONS

In the following, all presented data are normalized for the sake of confidentiality. Angular speeds and frequencies are directly comparable: both quantities are expressed in Hertz prior to normalization. In terms of visual representation, the following colour code is employed for all bar graphs and plots below: black is employed for standard results: these are obtained with converged parameters (spatial discretization of the abradable coating, time integration step, and reduction parameter), red refers to the convergence of the results with respect to the modal reduction parameter η, green refers to the convergence of the results with respect to the spatial discretization of the abradable coating n ab , blue indicates the graph provides insight on purely numerical parameters (convergence with respect to the time step ∆t, symmetry of the simulation. . . ).

VALIDATION ON AN ACADEMIC BLADED DISK

For the sake of efficacy, the numerical validation of the integration of gyroscopic effects within a bladed disk ROM is carried out on a simplified academic model, see Fig. 1. Fast computations allow to conduct an in-depth validation featuring convergence analyses with respect to key parameters. This academic model has 12 blades, it is assumed perfectly cyclically symmetric with respect to its axis of rotation. In order to isolate the influence of gyroscopic effects, centrifugal effects are neglected with this academic model. η increases, both the first and the second modal families are very accurately represented throughout the considered angular speed range. Results for higher harmonics (2 to 6) are similar but not detailed here for the sake of brevity.

Non-linear convergence A computational tool, developed for Safran Aircraft Engines [START_REF] Millecamps | Snecma's viewpoint on the numerical and experimental simulation of blade-tip/casing unilateral contacts[END_REF], dedicated to the numerical simulation of rotor/stator interactions is here employed. As it rotates, the rotor impacts a perfectly rigid casing. Abradable coating deposited on the casing contact surface may be worn off as the blades impact the casing. Contact simulations rely on an explicit time marching procedure for which convergence must be checked with respect to key parameters including:

(1) the time integration step ∆t, (2) the modal reduction parameter η and, (3) the number of elements used for the abradable coating discretization in the circumferential direction n ab . The asymptotical convergence of the predicted results-meaning the observation of the superimposition of the results as one of these parameters is enriched-is detailed below. First, the radial displacement u LE of the first blade's leading edge is depicted for different time steps in Fig. 4. An all-or-nothing type of convergence is evidenced: too large time steps yield numerical divergence of the simulation soon after the first contact events (such simulations are not pictured here) while smaller time steps (∆t ≤ 5 • 10 -7 s) lead to almost perfectly superimposed results. Contrary to the time integration step ∆t, the number of elements used for the abradable coating discretization has low to no effect on the computation times. For that reason, a fairly high number of elements is usually considered. Contact simulations are here performed for n ab = 5 000, 10 000, 15 000 and 20 000. It can be seen in Fig. 5 that for n ab ≥ 10 000 all the results are perfectly superimposed. The same convergence is observed with respect to the predicted wear patterns along the casing contact surface.

Finally, the asymptotical convergence of the predicted results is checked with respect to the modal reduction parameter η. Results depicted in Fig. 6 underline the convergence of the results as η is increased. The participation of free-vibration modes belonging to modal families 2 and above is evidenced as η = 2 yields very inaccurate displacements. For higher values of η, the good superimposition of the obtained results highlight the convergence of the numerical model. In order to confirm the proper numerical behaviour of the contact algorithm, a perfectly symmetric casing deformation is now considered: the casing is ovalized. The even number of blades on the rotor imposes that this configuration shall remain perfectly symmetric over time. For instance, since the academic model possesses 12 blades, the displacements predicted on blades 1 and 7 in their local frame must be identical. Displacements plotted in Fig. 7 are perfectly superimposed and thus confirm the good numerical behaviour of the employed strategy when gyroscopic effects are accounted for.

Contact simulations

Contact simulations between the bladed disk and its surrounding casing are now carried out over a wide angular speed range Ω ∈ [0.5 ; 0.75]. In total, a hundred simulations are performed. For each simulation, the rotor undergoes 100 revolutions and its angular speed is constant. Contact is initiated through a deformation of the rigid casing. Based on existing experimental data [START_REF] Millecamps | Influence of thermal effects during blade-casing contact experiments[END_REF], the assumption is made that the casing is insensitive to contact, it is thus modeled as a perfectly rigid structure. For each simulation, once a steady state has been reached, a Fourier transform of the blade displacements is made. The spectra obtained for each simulation are put side by side in order to draw the interaction maps plotted in Figs. A few critical areas-areas over which peaks of amplitudes of vibration are foundare numerically predicted. Highest amplitudes of vibration are found for Ω 0.62 at f 3.5 which suggest that the rotor vibration response is dominated by free-vibration modes belonging to the second modal family (see Figs. 2 and3(b)). Additional peaks are evidenced at other angular speeds around f 10 and f 27 thus highlighting the key role of higher order modal families in the bladed disk's dynamics. An in-depth analysis of these interactions goes beyond the scope of the current article. Instead, the focus is made on the influence of gyroscopic effects: the subtraction of the spectra obtained with and without gyroscopic effects is plotted in Fig. 9. Non negligible differences are observed in the vicinity of each peak of amplitudes of vibration. This is expected and simply underlines the fact that the rotor natural frequencies are affected by gyroscopic terms. This effect translates into a shift of predicted peaks both in angular speeds and frequencies hence affecting the subtraction made in Fig. 9. For Ω ∈ [0.6 ; 0.65], significant differences are found between the two spectra with respect to the first modal family (f 1), see the red lines in Fig. 9. The first modal family, specifically for harmonic 1, is indeed the modal family for which gyroscopic terms have the strongest effect on natural frequencies. Thus, it seems logical that differences between the spectra are located in this frequency domain. The fact that these differences are restricted to a specific angular speed range indicates that outside of this range, the first modal family has low to no effect on the rotor's steady state.

INDUSTRIAL APPLICATION

The industrial model considered in this article is that of an aircraft engine fan stage, the corresponding finite element model is partially depicted in Fig. 10. Accounting for gyroscopic effects on such stage of an aircraft engine is of particular interest as it may be subjected to orbital motions due to, for instance, residual imbalance. The system is assumed to be perfectly cyclically symmetric, it is clamped in two areas matching bearing locations, see areas in blue in Fig. 10. The finite element model contains 24 sectors and 25 360 nodes per sector.

Linear convergence

A ROM of the full bladed disk is computed based on the methodology presented in [START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF]. Both centrifugal and gyroscopic effects are accounted for based on the methodology presented in this article. The considered angular speed range is Ω ∈ [0 ; 0.5]. The Campbell diagram obtained with the ROM (with a modal reduction parameter η = 30, computational time is a few seconds) is depicted in Fig. 11. The contribution of both centrifugal effects and gyroscopic effects are evidenced in Fig. 11: (1) centrifugal effects yield an overall increase of the fan eigenfrequencies as the angular speed increases and, (2) gyroscopic effects are responsible for the split between the eigenfrequencies of the first harmonic of the first two modal families (hatched areas in Fig. 11). The computed ROM thus allows for an efficient extension of the reduction procedure presented in [START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF] to gyroscopic effects. It also yields very efficient computations for linear applications (forced responses, modal analysis or time integration). ).

The convergence of the eigenfrequencies of the ROM as the reduction parameter η increases is evidenced in Figs. 12(a) and 12(b): as η increases, the computed eigenfrequencies become perfectly superimposed throughout the angular speed range Ω ∈ [0 ; 0.5]. One may note that the considered values of η are much larger than in the case of the academic model, see Fig. 3. This is not related to the model itself but to the fact that centrifugal effects are here accounted for. Indeed, as detailed in [START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF], accounting for centrifugal effects directly within the ROM of a full bladed disk implies a significant enrichment of the modal reduction basis. Hence, η cannot simply be interpreted as a number of contraint modes per harmonic and larger values must be considered to reach convergence all over the angular speed range of interest.

Roadmap for the numerical simulation of whirl motions

The fan stage ROM accounting for both centrifugal and gyroscopic effects may be used for the numerical prediction of sophisticated interaction motions. As an example, contact simulations are carried out over several revolutions of the rotor, the displacement field predicted in the fan stage at a time where contacts occur on two of its blades is depicted in Fig. 13. The chosen color code underlines that the shaft axis of rotation undergoes significant displacements. Providing physically realistic results requires an in-depth convergence analysis that cannot be presented here for the sake of brevity. Work is in progress to simulate whirl motions featuring significant amplitudes of vibration of the rotor around its axis of rotation. Such amplitudes may be a consequence of bearing failure-that may be easily represented numerically by removing the kinematic constraints on clamped nodes close to the bladed-disk, see Fig. 10.

CONCLUSION

This article focuses on the modeling of centrifugal and gyroscopic effects within efficient reduced-order models. The emphasis is on the combination of gyroscopic effects-assuming a constant angular speed-with a modal reduction technique. The resulting reduced-order models advantageously feature a mixed set of reduced degrees of freedom combining both modal and physical quantities that are required for contact treatment. The possibility to embed both centrifugal and gyrocopic effects within these models opens avenues for the efficient numerical simulation of sophisticated interaction phenomena within aircraft engines as well as turbomachine components in general. This modeling strategy is applied on the fan stage of a civil aircraft engine. The accuracy of the modeling is underlined: both centrifugal and gyroscopic effects are accurately captured over a wide angular speed range at low computational cost. The presented work allows for performing the first numerical simulations of whirl motions initiated by structural contacts. The possibility to use full 3D finite element models is of prime importance for manufacturers as this paves the way for the obtention of results specific to the blade design.

ACKNOWLEDGEMENT

This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors are also grateful to the industrial partner for supporting this project, Safran Aircraft Engines.

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Academic model used for validation (only four of the twelve sectors are shown).

Figure 4 .Figure 5 .

 45 Figure 4. Predicted displacement for: ∆t = 5 • 10 -7 s ( ), ∆t = 10 -7 s ( ), ∆t = 5 • 10 -8 s () and ∆t = 10 -8 s ( ). η = 20, n ab = 10 000.

Figure 6 .

 6 Figure 6. Predicted displacements for: η = 2 ( ), η = 12 ( ), η = 20 ( ) and η = 50 ( ). ∆t = 10 -7 , n ab = 10 000.

Figure 7 .

 7 Figure 7. Axially opposed radial displacements of blades 1 ( ) and 7 ( ) leading edges.

  8(a) and 8(b), respectively with and without gyroscopic effects. The same colour code is used for both maps: from white to black where black underlines the peaks in amplitudes of vibration.

Figure 8 .

 8 Figure 8. Interaction maps obtained with the academic model.

Figure 9 .

 9 Figure 9. Subtraction between spectra plotted in Figs. 8(a) and 8(b).

Figure 10 .Figure 11 .

 1011 Figure 10. Partial representation of the finite element model of a civil aircraft engine fan stage. Clamped areas are marked in blue:.

Figure 12 .

 12 Figure 12. Campbell diagrams for different values of the modal reduction parameter: η = 2 ( ), η = 88 ( ), η = 138 ( ) and η = 188 ( ).

Figure 13 .

 13 Figure 13. Displacement field predicted in the fan stage following contacts along blade-tips.

1 -

 1 Department of Mechanical Engineering, École Polytechnique de Montréal, P.O. Box 6079, Succ. Centre-Ville, Montréal, Québec, Canada H3C 3A7 2 -Safran Aircraft Engines, site de Villaroche, Moissy-Cramayel, 77550, France