Benoit Lattuca

MD Nesrine Bouziri

MD Mathieu Kerneis

MD; Jean-Jacques J Portal

PhD Jioannong Zhou

BSc Marie Hauguel-Moreau

MD Amel Mameri

MD; Michel Zeitouni

MD Paul Guedeney

MD Nadjib Hammoudi

PhD Richard Isnard

PhD Françoise Pousset

MD; Jean-Philippe Collet

MD, PhD Eric Vicaut

PhD Gilles Montalescot
email: gilles.montalescot@aphp.fr

MD, PhD Johanne Silvain

Jude Medical Bouziri

Antithrombotic Therapy for Patients with Left Ventricular Mural Thrombus

Keywords: Left ventricular thrombus, myocardial infarction, antithrombotic, thrombus regression, mortality, direct oral anticoagulant ARVD: arrhythmogenic right ventricular dysplasia 4 BARC: bleeding academic research consortium left ventricular thrombus MACE: major adverse cardiovascular events MI: myocardial infarction MRI: magnetic resonance imaging STEMI: ST-segment elevation myocardial infarction TIA: Transient Ischemic Attack VKA: vitamin-K antagonist

Ventricular Thrombus and Cardiovascular Events Tweet: Left ventricular thrombus is associated with a very high risk of major cardiovascular events. Total thrombus regression can be obtained with different anticoagulants including DOACs and is a major determinant of improved clinical outcomes. @ActionCoeur

Introduction

The advent of reperfusion therapy and the widespread use of primary percutaneous coronary intervention (PCI) have markedly reduced the incidence of post myocardial infarction (MI) left ventricular thrombus (LVT) over the last decades (1-3). Nevertheless, contemporary epidemiologic studies suggest that the incidence of LVT may remain as high as 15% to 25% in patients with ST-segment elevation MI (STEMI) [START_REF] Mccarthy | Left Ventricular Thrombus After Acute Myocardial Infarction: Screening, Prevention, and Treatment[END_REF][START_REF] Delewi | Left ventricular thrombus formation after acute myocardial infarction[END_REF][START_REF] Robinson | Left ventricular thrombi after STEMI in the primary PCI era: A systematic review and meta-analysis[END_REF] and up to 36% in the setting of dilated cardiomyopathy [START_REF] Gottdiener | Frequency and embolic potential of left ventricular thrombus in dilated cardiomyopathy: assessment by 2-dimensional echocardiography[END_REF][START_REF] Ezekowitz | Comparison of Indium-111 Platelet Scintigraphy and Two-Dimensional Echocardiography in the Diagnosis of Left Ventricular Thrombi[END_REF] when detected with optimal imaging modalities.

Despite adequate anticoagulation therapy, LVT remains a severe complication associated with a high risk of cerebral and peripheral arterial embolism and subsequent mortality [START_REF] Maniwa | Anticoagulation combined with antiplatelet therapy in patients with left ventricular thrombus after first acute myocardial infarction[END_REF][START_REF] Johannessen | Risk factors for embolisation in patients with left ventricular thrombi and acute myocardial infarction[END_REF][START_REF] Mccarthy | Left Ventricular Thrombus: Contemporary Etiologies, Treatment Strategies, and Outcomes[END_REF].

American and European Guidelines [START_REF] Gara | ACCF/AHA guideline for the management of ST-elevation myocardial infarction: a report of the American College of Cardiology Foundation/American Heart Association Task Force on Practice Guidelines[END_REF][START_REF] Ibanez | ESC Guidelines for the management of acute myocardial infarction in patients presenting with ST-segment elevation: The Task Force for the management of acute myocardial infarction in patients presenting with ST-segment elevation of the European Society of Cardiology (ESC)[END_REF] recommend vitamin K antagonist (VKA) for at least 3 to 6 months. The duration must be individualized according to bleeding risk. The need for concomitant antiplatelet therapy may be considered. There is limited supportive evidence [START_REF] Vaitkus | Embolic potential, prevention and management of mural thrombus complicating anterior myocardial infarction: a meta-analysis[END_REF][START_REF] Weinreich | Left ventricular mural thrombi complicating acute myocardial infarction. Long-term follow-up with serial echocardiography[END_REF]. The effect of anticoagulation on LVT regression as well as the optimal anticoagulation therapy duration and their potential impact on clinical outcomes are lacking. Moreover, no study has, to date, evaluated the efficacy and the safety of direct oral anticoagulants (DOACs) in this clinical setting.

The purposes of this study were, first, to describe the effect of anticoagulation therapy on LVT evolution and secondly, to identify the independent predictors of LVT regression and clinical outcomes.

METHODS

Study design and population

From January 2011 to December 2017, all reports of consecutive echocardiogram performed at the Institute of Cardiology, Pitié-Salpêtrière hospital (Paris, France), were analyzed using a computerized case sensitive search. All patients with a reported LVT, regardless of the underlying disease, were screened and only patients with a confirmed LVT by two independent experts, were included. In case of discordance, an additional imaging modality was performed [i.e., contrast trans-thoracic echocardiography, cardiac computed tomography scan (CT scan) or cardiac magnetic resonance imaging (MRI)] and a third expert was requested to confirm the diagnosis. Patients with right ventricular thrombus and atrial thrombus were excluded from the study.

Thrombus evaluation

LVT was defined as an echodense mass adjacent to a hypokinetic or akinetic myocardial segment. To be distinguishable from the underlying myocardium, a clear thrombus-blood interface was required and LVT had to be visible at least on two views during all the cardiac cycle. LVT diameters (mm), area (mm²) and volume (mm 3 ) were assessed for each trans-thoracic echocardiogram considering the mean of 3 measures to study LVT evolution over time (Online Figure 1). Mobility and location of LVT were also defined. LVT was considered mobile if thrombus had a pedunculated non-mural component. A calcified LVT was defined as a persistent left ventricular mural thrombus encapsulated by thickened and calcified endocardium. Left ventricular characteristics including left ventricular ejection fraction (LVEF), left ventricular volume, wall motion, cardiac output and potential mechanical complications were also collected.

All echocardiograms were reviewed using the dedicated TOMTEC imaging system tools (TOMTEC, Unterschleissheim Germany).

Baseline and data collection

All patient baseline characteristics including cardiovascular risk factors, past medical history of major bleeding or ischemic event, underlying disease and long-term use of anticoagulation therapy before the diagnosis of LVT were collected through medical reports.

Special attention was given to the type (VKA, DOACs, low-molecular-weight heparin, unfractionated heparin) of anticoagulant used and duration of anticoagulation therapy and the association with antiplatelet therapy. Treatment duration was obtained through medical reviews of each hospital visit and by attending physician or patient interviews.

Study objectives

The objectives of this study were to evaluate LVT regression under anticoagulation therapy, the time needed for total LVT regression using sequential noninvasive imaging and the independent correlates associated with total LVT regression. The impact of LVT regression on thromboembolism, bleeding and mortality and factors associated with major cardiovascular adverse events were determined.

Endpoint definitions

Total LVT regression was defined by a complete disappearance of LVT on all echocardiography views at the last available follow-up. LVT persistence was classified as an increased thrombus dimension, a stable thrombus or a partial thrombus regression at the last available follow-up.

We collected the following clinical outcomes during the period of observation: major adverse cardiovascular events (MACE) defined as the composite of all cause death, ischemic stroke or transient ischemic attack (TIA), myocardial infarction (MI) or acute peripheral artery emboli., any embolic complications defined as the composite of ischemic stroke or TIA, acute coronary emboli or acute peripheral artery emboli (limb, renal or digestive arteries), all-cause death and bleeding events defined as clinically relevant bleeding events (BARC≥2) and major bleeding events (BARC≥3) according to the international Bleeding Academy Research Consortium (BARC) [START_REF] Mehran | Standardized bleeding definitions for cardiovascular clinical trials: a consensus report from the Bleeding Academic Research Consortium[END_REF].

Statistical analysis

All data were shown as means with standard deviations or medians with interquartile ranges for continuous variables and as numbers and percentages of patients for categorical variables. Non-Gaussian variables were compared with the use of the Mann-Whitney test. Chisquare testing was used for frequency comparisons. The 95% confidence interval (CI) for the hazard ratio (HR) is presented. The population was evaluated according to total LVT regression or incomplete regression during follow-up. Time to first clinical endpoints was estimated using the Kaplan-Meier method and differences between total LVT regression and persistent LVT were assessed with the log-rank test. Univariate and multivariate Cox regression analyses were used to identify independent correlates for total LVT regression. The following variables related to clinical outcomes in univariate analysis (p<0.1) were included in the first multivariate model : age, underlying disease (ischemic vs non-ischemic cardiomyopathy), LVEF, anticoagulation by DOACs or heparin (considering VKA treatment as reference), combined antithrombotic treatment by anticoagulant and antiplatelet therapy (in comparison to anticoagulation therapy alone), baseline thrombus area (defined as a continuous variable), history of atrial fibrillation, atrial fibrillation complication (occurring after thrombus diagnosis) and major bleeding (BARC≥3). A second multivariate Cox model, including the aforementioned variables with antithrombotic treatment duration and total LVT regression as additional variables, was performed to identify independent correlates of MACE, all-cause death and all embolic complications in this specific population of LVT patients. All tests had a two-sided significance level of 5% and were performed with the use of SAS software, version 9.2 (SAS Institute).

RESULTS

Characteristics of the patients and thrombus description

Of the 90,065 echocardiograms performed during the study period, a total of 174 patients had a reported LVT and 159 patients were definitively included in this analysis after imaging review by two independent experts (Figure 1). The baseline characteristics of the patients with confirmed LVT are described in Table 1. The population was relatively young (mean age of 58±13 years) with a high prevalence of cardiovascular risk factors leading to ischemic cardiomyopathy (78.6%, n=125) including STEMI in one out of three patients (35.2%, n=56).

Severe left ventricular dysfunction was common (mean LVEF of 31.9±12.5%) with more than half of the population which had a mean cardiac output index of less than 2.6 L/min. The most frequent left ventricular wall motion abnormality was akinesis of the apical segments where LVT were predominantly located (98.1%). Left ventricular aneurysms were found in 15.1% (n=24) of patients. LVT were classified as mobile in 34.6% (n=55) of patients and calcified LVT were observed in 1.3% (n=2) of patients. The largest diameter, surface area and thrombus volume as well as the detailed baseline echocardiographic measurements are displayed in Table 2.

Antithrombotic therapy

Only two patients (1.2%) were treated with antiplatelet therapy alone due to high bleeding risk. The vast majority of the study population (98.8%) was exposed to anticoagulation therapy including VKA (48.4%, n=77) mostly fluindione (46.5%, n=74), DOACs (22.6%, n=36), low molecular weight heparin (23.3%, n=37) or unfractionated heparin (4.4%, n=7).

Concomitant antiplatelet therapy was prescribed in 67.9% (n=108) of patients (Table 3). Median duration of anticoagulation therapy was 508 [115-986] days without significant difference regarding the anticoagulant drug administrated (Online Table 1).

Thrombus regression

A reduction of the thrombus area from the baseline echocardiography to the final echocardiography (median delay of 338 [38-907] days) was observed in 76.1% (n=121) of patients. The time-dependent cumulative total regression of LVT is represented in Figure 2 showing a progressive decrease of thrombus area. Total LVT regression was achieved in 62.3% (n=99) of patients within a median time of 103 [32-392] days.

After multivariate adjustment, the presence of a non-ischemic cardiomyopathy (HR 2.74; 95%CI [1.43-5.26], p=0.002) and a smaller baseline thrombus area (HR 0.66; 95%CI [0.45-0.96], p=0.031) were independently associated with total LVT regression. A thrombus recurrence or an increase of thrombus area was observed in 14.5% (n=23) of patients. A review of medical records reported that recurrent or increased size of LVT was associated with poor treatment adherence, pro-thrombotic conditions such as active cancer, inflammatory, hematologic diseases or chronic renal failure (Online Table 2).

Cardiovascular events

MACE and embolic complications occurred in 37.1% (n=59) and 22.2% (n=35) of patients, respectively, within a median follow-up period of 632 [187-1126] days. All-cause mortality was 18.9% (n=30). Clinically relevant (BARC≥2) and major (BARC≥3) bleeding events occurred in 17% (n=27) and 13.2% (n=21) of patients, respectively. Reduced mortality was observed in patients with total LVT regression (15.2% vs. 25.0%; HR 0.48; 95%CI [0.23-0.98]; p=0.039) whereas an increased major bleeding risk was observed among patients with persistent LVT (9.1% vs. 12%; HR 0.34; 95%CI [0.14-0.82], p=0.011) (Table 4, Figure 3). MACE occurred more frequently in patients treated with a dual antithrombotic strategy (54%, n=30 for the association of single antiplatelet therapy and anticoagulant) than those treated with a triple antithrombotic strategy (29%, n=16 for the association of dual antiplatelet therapy and anticoagulant) with similar rate of major bleedings being 14%, n=8 and 12.5%, n=7 respectively.

After multivariate adjustment, a LVEF ≥35% (HR 0.46; 95%CI [0.23-0.93], p=0.029) and a prolonged anticoagulation therapy over 3 months (HR 0.42; 95%CI [0.20-0.88], p=0.021) were independently associated with a reduced risk of MACE (Figure 4). Moreover, a numerically lower rate of embolic complications was observed in patients treated by a prolonged anticoagulation therapy over 3 months (HR=0.46 [0.18-1.14]; p-value=0.093) (data not shown).

It has to be noted that the type of anticoagulant therapy was not independently associated with cardiovascular events in our study (data not shown).

DISCUSSION

Although the incidence of LVT has decreased in modern times, it continues to complicate both ischemic and non-ischemic cardiomyopathies and is associated with poor outcomes. This analysis provides new insights from a large and well-characterized contemporary cohort of patients with LVT (Central Illustration). For patients with LVT, total regression was only achieved in two thirds of our population within 3 months for half of them using full anticoagulation. The use of antiplatelet agents was common. The independent correlates of total LVT regression were an underlying non-ischemic cardiomyopathy and a smaller baseline thrombus area. LVT patients had a very high risk of MACE, embolic or major bleeding complications and mortality. Most importantly, total LVT regression was associated with a reduced mortality and which was obtained with a full anticoagulation therapy using either heparin, VKA or DOACs.

Thrombus regression and suboptimal anticoagulation therapy

This study highlights that the current antithrombotic regimen needs to be improved since one third of patients did not achieve total LVT regression and remained exposed to a high risk of clinical complications even when combining with antiplatelet agents. Consistent with our findings, a recent study screened more than 140,000 echocardiograms and found that LVT was a rare complication as 128 LVT patients were identified. Total LVT regression was reached in 71% of patients treated by anticoagulation therapy but, unfortunately, echocardiographic followup and data concerning LVT regression were not available for all of the patients [START_REF] Mccarthy | Left Ventricular Thrombus: Contemporary Etiologies, Treatment Strategies, and Outcomes[END_REF]. In a single cohort of 92 post-MI patients treated with VKA, the authors found that LVT regression was dependent on time in therapeutic range. Unlike our study, there was no experience using DOACs [START_REF] Maniwa | Anticoagulation combined with antiplatelet therapy in patients with left ventricular thrombus after first acute myocardial infarction[END_REF].

Left ventricular thrombus and clinical outcomes

The main result of the present study is the high rate of cardiovascular events as four out of ten patients had a MACE and one out of five patients died during the follow-up period, a dramatically higher case mortality than reported in STEMI patients without LVT [START_REF] Jernberg | Association between adoption of evidence-based treatment and survival for patients with ST-elevation myocardial infarction[END_REF][START_REF] Rosamond | Twenty-two-year trends in incidence of myocardial infarction, coronary heart disease mortality, and case fatality in 4 US communities, 1987-2008[END_REF]. These findings are here again difficult to compare with the few prior observational studies published so far [START_REF] Delewi | Left ventricular thrombus formation after acute myocardial infarction[END_REF][START_REF] Vaitkus | Embolic potential, prevention and management of mural thrombus complicating anterior myocardial infarction: a meta-analysis[END_REF][START_REF] Weinreich | Left ventricular mural thrombi complicating acute myocardial infarction. Long-term follow-up with serial echocardiography[END_REF][START_REF] Weinsaft | Echocardiographic Algorithm for Post-Myocardial Infarction LV Thrombus: A Gatekeeper for Thrombus Evaluation by Delayed Enhancement CMR[END_REF][START_REF] Visser | Embolic potential of left ventricular thrombus after myocardial infarction: a two-dimensional echocardiographic study of 119 patients[END_REF] but are consistent with two recent studies [START_REF] Maniwa | Anticoagulation combined with antiplatelet therapy in patients with left ventricular thrombus after first acute myocardial infarction[END_REF][START_REF] Mccarthy | Left Ventricular Thrombus: Contemporary Etiologies, Treatment Strategies, and Outcomes[END_REF] and highlight the severity of this population and the need for more efficient therapeutic strategy to reverse such poor prognosis. Several approaches to improve the prognosis of LVT patients may be considered. One of them would be to refine the antithrombotic regimen to improve and accelerate LVT regression, a factor associated with lower mortality in our study. It could be argued that LVT regression could at least partially be the consequence of thrombus embolization. However, although we cannot exclude asymptomatic embolization, we did not find significant difference in embolic complications between patients with or without LVT regression. These embolic complications could occur at any time under antithrombotic therapy, as a consequence of thrombus fragmentation, quick thrombus regression or, on the contrary, persistence of the thrombus. The achievement of total LVT regression seems to be a marker of morbidity and mortality of the patient more than the associated embolic risk. It should not be a predominant factor to shorten the antithrombotic treatment. Moreover, after adjustment for confounding factors, atrial fibrillation was not associated with a higher rate of clinical outcomes and our data support that thrombus regression is mostly due to the duration of antithrombotic therapy and the size of the LVT. Beyond the efficacy of each anticoagulant drug, the optimal treatment duration is a major clinical issue. In our study, we observed a lower occurrence of MACE among patients who were treated by an anticoagulation therapy for a longer duration than 3 months. We can hypothesize that a longer antithrombotic treatment could lead to a better prevention of embolic complications or myocardial infarction and prevent LVT recurrence. Such findings need to be confirmed by adequately sized randomized trials. However, any intensification of the antithrombotic treatment may be compromised by more frequent bleeding complications that are already reported in more than 10% of the patients in our study. Although not designed and powered for a direct comparison between DOACs versus VKA, our study suggests a similar rate of total LVT regression, MACE and bleeding complications with both anticoagulant strategies.

Limitations

We acknowledge several limitations. This was a retrospective study from a single centre albeit the largest volume centre in the Paris area. We also acknowledge that the diagnosis of LVT was mainly based on echocardiography which may have a lower sensitivity and specificity for the detection compared to other imaging modalities such as cardiac MRI [START_REF] Mccarthy | Left Ventricular Thrombus After Acute Myocardial Infarction: Screening, Prevention, and Treatment[END_REF][START_REF] Weinsaft | Echocardiographic Algorithm for Post-Myocardial Infarction LV Thrombus: A Gatekeeper for Thrombus Evaluation by Delayed Enhancement CMR[END_REF][START_REF] Srichai | Clinical, imaging, and pathological characteristics of left ventricular thrombus: a comparison of contrast-enhanced magnetic resonance imaging, transthoracic echocardiography, and transesophageal echocardiography with surgical or pathological validation[END_REF]. We also acknowledge that the comparison between the different antithrombotic drugs was limited by the small sample size and that our results should be considered exploratory rather than definitive. An individualized risk stratification based on the patient characteristics, the underlying disease and the LVT evolution under antithrombotic treatment would be ideal to guide physician decisionmaking and to refine the management of LVT with the main objective of reducing the risk of clinical outcomes.

CONCLUSIONS

We conclude that the clinical prognosis of patients with LVT is poor with a very high risk of major cardiovascular events and mortality. Refinement in the antithrombotic management is needed to improve clinical outcomes of these relatively young patients. Total LVT regression can be obtained with different anticoagulation regimens including DOACs and thereby reduce mortality.

CLINICAL PERSPECTIVES

Competency in Patient Care: Left ventricular thrombus is an uncommon complication of ischemic and non-ischemic cardiomyopathies associated with a high risk of adverse events and mortality. Regression of thrombus is associated with lower mortality, but in about one-third of cases, thrombus does not resolve despite anticoagulation with or without antiplatelet therapy.

Translational Outlook: Randomized studies should assess the safety and efficacy of targetspecific oral anticoagulants, clarify the optimum duration of treatment, and identify predictors of recurrence after thrombus resolution.

Figure 1 .

 1 Figure 1. Study flow chart. The figure represents the patients screened and included in the final

Figure 2 .

 2 Figure 2. Time-dependent cumulative total left ventricular thrombus regression. Each point

Figure 3 .

 3 Figure 3. Time-to-event curves for major adverse cardiovascular events (A), all-cause

Figure 4 .

 4 Figure 4. Independent correlates of major adverse cardiovascular events. LVEF indicates

Table 1 :

 1 Baseline clinical characteristicsVariables are expressed in number (%) or mean ± interquartiles. ARVD indicates Arrhythmogenic Right Ventricular Dysplasia; NSTEMI, Non ST-segment Elevation Myocardial Infarction; LVT, Left Ventricular Thrombus and STEMI, ST-segment Elevation Myocardial Infarction.

	Baseline characteristics	Patients with LVT (n = 159)
	Age (years)	58 ±13
	Female sex	28 (17.6)
	Body Mass Index (kg/m 2 )	25.0 (22.6 -27.4)
	Active smoking	73 (45.9)
	Hypertension	57 (35.8)
	Family history of cardiovascular disease	19 (11.9)
	Diabetes mellitus	38 (23.9)
	Dyslipidemia	65 (40.9)
	Creatinine clearance < 60 ml/min	33 (20.8)
	Prior coronary stenting	93 (58.5)
	Prior coronary artery bypass graft	8 (5)
	Peripheral artery disease	18 (11.3)
	Prior stroke or transient ischemic attack	20 (12.6)
	Prior atrial fibrillation	18 (11.3)
	Prior major bleeding	8 (5)
	Previous anticoagulant therapy	20 (12.6)
	Vitamin K antagonist	14 (8.8)
	Direct oral anticoagulants	6 (3.8)
	Underlying disease
	Coronary artery disease	125 (78.6)
	NSTEMI	18 (11.3)
	STEMI	56 (35.2)
	Successful revascularization	45 (80.3)
	STEMI>24h	12 (21.4)
	Dilated cardiomyopathy	23 (14.5)
	Hypertrophic cardiomyopathy	2 (1.3)
	Myocarditis	6 (3.8)
	Tako-tsubo cardiomyopathy	2 (1.3)
	ARVD with associated LV impairment	1 (0.6)

Table 2 .

 2 Baseline echocardiographic evaluation.

	Baseline echocardiographic measurements	Patients with LVT (n =159)
	Left ventricular ejection fraction (%)	31.9 ± 12.5
	Cardiac output (l/min)	4.1 (3.0 -4.9)
	Left ventricular wall motion
	Akinesis in apical segments	130 (81.8)
	Hypokinesis in apical segments	6 (3.8)
	Global hypokinesis	23 (14.5)
	Left Ventricular Thrombus
	Largest diameter (mm)	19 (13 -24)
	Area (cm 2 )	1.34 (0.8 -2.57)
	Volume (mm 3 )	1.06 (0.49 -2.40)
	Mobile thrombus	55 (34.6)
	Apical thrombus	156 (98.1)
	Left ventricular aneurysm	24 (15.1)
	Other mechanical complication	1 (0.6)
	Variables are expressed in number (%) or mean ± interquartiles.
	LVT indicates Left Ventricular Thrombus

Table 3 .

 3 Antithrombotic strategy following left ventricular thrombus diagnosis

	Antithrombotic strategy	Patients with LVT (n=159)
	Antiplatelet therapy only	2 (1.2)
	Anticoagulation only	45 (28.3)
	Anticoagulation + Antiplatelet therapy
	Aspirin with anticoagulant	52 (32.7)
	Clopidogrel with anticoagulant	3 (1.9)
	Ticagrelor with anticoagulant	1 (0.6)
	Anticoagulant + Dual Antiplatelet Therapy	56 (35.2)
	Anticoagulation type
	Vitamin K Antagonist	77 (48.4)
	Coumadin	3 (1.9)
	Fluindione	74 (46.5)
	Direct oral anticoagulant	36 (22.6)
	Apixaban 2.5mg*2	9 (5.7)
	Apixaban 5mg*2	9 (5.7)
	Dabigatran 110mg*2	5 (3.1)
	Dabigatran 150mg*2	0 (0)
	Rivaxoraban 15mg	10 (6.3)
	Rivaxoraban 20mg	3 (1.9)
	Low Molecular Weight Heparin	37 (23.3)
	Unfractionated Heparin	7 (4.4)

Variables are expressed in number (%). LVT indicates Left Ventricular Thrombus.

Table 4 .

 4 Cardiovascular events in the whole population and according to the occurrence of a total thrombus regression

	All population (n=159)	Total thrombus regression (n=99)	Persistent thrombus (n=60)	Log-Rank test p value

This work was supported by Allies in Cardiovascular Trials Initiatives and Organized Networks ACTION Group -www.action-coeur.org Conflict of Interest: Lattuca has received research grants from Biotronik, Boston Scientific, Daiichi-Sankyo, Fédération Française de Cardiologie and Institute of CardioMetabolism and Nutrition; consultant fees from Daiichi-Sankyo and Eli Lilly; and lecture fees from AstraZeneca and Novartis. Kerneis has received research grants from Fédération Française de Cardiologie, European Society of Cardiology, Servier and consultant fees from AstraZeneca, Bayer, Daiichi-Sankyo, Eli-Lilly and Sanofi. Zeitouni has received research grants from Federation Française de Cardiologie and Servier. Collet has received research grants from AstraZeneca, Bayer, BMS, Daiichi-Sankyo, Eli-Lilly, Fédération Française de Cardiologie, Lead-Up, Medtronic, MSD, Sanofi, WebMD. Vicaut has received research grants from Boehringer and Sanofi; consultant fees from Abbott, Eli Lilly, Hexacath, Fresenius, LFB, Medtronic, Novartis, Pfizer and Sanofi. Montalescot has received research grants from Abbott, Amgen, Actelion, American College of Cardiology Foundation, AstraZeneca, Axis-Santé, Bayer, grants from Algorythm and Fondation de France;


Wajdi Zouari
email: wajdi.zouari@univ-reims.fr

Hocine Kebir

Mustapha Assarar

Tewfik Ghomari

Rezak Ayad

Numerical modelling of 2D composite solids accounting for local fibre bending stiffness

Keywords: 2D composite solids, Projected Fibre Approach, Fibre bending stiffness, Finite element analysis

This contribution deals with the development of a four-node membrane composite finite element to analyse two-dimensional (2D) fibre-reinforced solids taking into account the local bending stiffness of fibres. This plane composite element is a combination of a four-node membrane element with rotational degrees of freedom (DOFs), modelling a unit of the matrix domain, and two-node Timoshenko beam elements that represent fibres crossing each matrix element. The so-called 'Projected Fibre Approach' is then considered to express the beam elements variables in terms of their corresponding matrix elements. Accordingly, the obtained system of equations size is equivalent to that of a non-reinforced medium which considerably reduces the computing cost. To assess the accuracy of the proposed plane element, three examples are presented and the obtained results are found to agree well with the reference solutions.

Introduction

Compared to metallic materials, composite materials generally present interesting specific mechanical properties which allow them to meet the requirements of technical specifications in various fields. Besides, thanks to their ability to custom-made, composite materials offer the designers the possibility to manufacture structures whose characteristics could be optimized according to the composite material constituents. To expand the use of composite materials in a reliable way, an accurate prediction of their macroscopic behaviour from those of their constituents is required. To this end, several micromechanical analytical models, based on the definition of a representative volume element (RVE) of the composite materials, have been firstly proposed to estimate their effective elastic properties. Most of these approaches are based on statistical continuum theories and mean-field approaches based on Eshelby's solution [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF] (see the review by Raju et al. [START_REF] Raju | A review of micromechanics based models for effective elastic properties of reinforced polymer matrix composites[END_REF] for more details). In particular, these models include the self-consistent method [START_REF] Hill | A self-consistent mechanics of composite materials[END_REF][START_REF] Budiansky | On the elastic moduli of some heterogeneous materials[END_REF][START_REF] Huang | A generalized self-consistent mechanics method for composite materials with multiphase inclusions[END_REF][START_REF] Benveniste | Revisiting the generalized self-consistent scheme in composites: Clarification of some aspects and a new formulation[END_REF] and the well known Mori-Tanaka model [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] or a combination of these two approaches as reported by Peng et al. [START_REF] Peng | Evaluation of mechanical properties of particulate composites with a combined self-consistent and MoriTanaka approach[END_REF]. Despite their simplicity of implementation, these analytical approaches present some limits such as the difficulty to take account of the microstructural characteristics like the size, the complex shape and the distribution of the reinforcement.

To overcome these limits, other numerical techniques, based essentially on the finite element method and facilitated by the progress of computing techniques and capabilities, have been also proposed to predict the effective mechanical behaviour of heterogeneous composite materials (see [START_REF] Vanalli | A simple way to introduce fibers into FEM models[END_REF][START_REF] Radtke | A partition of unity finite element method for obtaining elastic properties of continua with embedded thin fibres[END_REF][START_REF] Cunha | A finite element model with discrete embedded elements for fibre reinforced composites[END_REF][START_REF] Kebir | A specific finite element procedure for the analysis of elastic behaviour of short fibre reinforced composites. The projected fibre approach[END_REF][START_REF] Tian | Representative volume element for composites reinforced by spatially randomly distributed discontinuous fibers and its applications[END_REF][START_REF] Bernardo | Modeling and simulation techniques for polymer nanoparticle composites A review[END_REF][START_REF] Li | Evaluation of elastic properties of fiber reinforced concrete with homogenization theory and finite element simulation[END_REF][START_REF] Zhang | Discrete modeling of fiber reinforced composites using the scaled boundary finite element method[END_REF] among others). For instance, Vanalli et al. [START_REF] Vanalli | A simple way to introduce fibers into FEM models[END_REF] have developed a finite element formulation to predict the behaviour of arbitrarily placed short or long fibre-reinforced composites without explicitly model the reinforcement. In this case, no additional degree of freedom associated with the reinforcement is introduced in the model and some problems related to the node-by-node geometrical definitions and the complex mesh generation are hence avoided.

Some numerical examples have been considered to assess the formulation's accuracy, including the non-linear elastoplastic analysis of reinforced concrete beam under four-point bending, and the presented results were globally in good agreement with reference solutions. In another work, Cunha et al. [START_REF] Cunha | A finite element model with discrete embedded elements for fibre reinforced composites[END_REF] have proposed a numerical approach to predict the crack behaviour of steel fibre-reinforced concrete. In this model, the steel fibre-reinforced concrete has been treated as a two phase material in which the steel fibres are explicitly modelled with embedded cables within a three-dimensional (3D) mesh representing the matrix. The numerical simulations of tensile and three-point bending tests showed a good agreement with the experimental results. In a third work by Tian et al. [START_REF] Tian | Representative volume element for composites reinforced by spatially randomly distributed discontinuous fibers and its applications[END_REF],

3D representative volume elements of short carbon fibre-reinforced magnesium composites, with relatively high fibre volume fraction, have been generated by the random sequential absorption algorithm and then analysed by the FEA package ABAQUS. To this end, the 4-node tetrahedron element C3D4 has been considered to model the matrix and the short carbon fibres. The presented results showed firstly the increase of Young's modulus and secondly the decrease of Poisson's ratio with the fibre volume fraction. In addition to these works, Kebir and Ayad [START_REF] Kebir | A specific finite element procedure for the analysis of elastic behaviour of short fibre reinforced composites. The projected fibre approach[END_REF] have developed a plane three-node triangle composite element, named PFT3, based on the so-called 'Projected Fibre Approach'. This membrane finite element has been used to predict the elastic properties of short hemp fibre-reinforced polypropylene composite. Short hemp fibres with different volume fractions have been randomly generated within a 2D representative volume element initially discretized by three-node constant strain triangular elements, representing the matrix domain. Then, the intersections between the discrete short hemp fibres and the matrix mesh have been automatically handled by a specific algorithm developed by the authors. After this step, all fibres crossing each matrix element have been represented by two-node truss elements and the variables of these latter have been expressed in terms of their corresponding matrix element based on the assumption of a perfect bonding of the fibres to the matrix. This finite element formulation allowed to correctly estimate the experimental Young's modulus of 30%wt hemp-polypropylene composite and to predict the evolution of Young's modulus and Poisson's ratio with the fibre volume fraction.

Based on the same Projected Fibre Approach, Tiar et al. [START_REF] Tiar | A nonlinear finite element formulation for large deflection analysis of 2D composite structures[END_REF] have developed a geometrically non linear four-node quadrilateral membrane element to study 2D composite solids subjected to large deflection.

In this paper, we propose an extension of the finite element formulation proposed by Kebir and Ayad [START_REF] Kebir | A specific finite element procedure for the analysis of elastic behaviour of short fibre reinforced composites. The projected fibre approach[END_REF] and based on the Projected Fibre Approach. The main objective remains the modelling of 2D composite solids without explicitly model the reinforcement, which allows obtaining a system of equations equivalent to that of a non-reinforced medium. The first novelty of the finite element formulation proposed in this work concerns the modelling of discrete fibres with two-node Timoshenko beam elements to take account of their bending stiffness. In fact, modelling discrete fibres with two-node truss elements as considered in Kebir and Ayad [START_REF] Kebir | A specific finite element procedure for the analysis of elastic behaviour of short fibre reinforced composites. The projected fibre approach[END_REF] could result in erroneous prediction of the 2D composite solids effective behaviour, especially those subjected to bending-dominated loading. The second novelty of this work is the use of a four-node quadrilateral membrane finite element with rotational DOFs to model the matrix domain. This element, named PFR4, is based on a plane adaptation of the so-called Space Fibre Rotation (SFR) concept introduced by Ayad [START_REF] Ayad | Contribution to the numerical modeling of solids and structures and the non-newtonian fluids forming process. Application to packaging materials[END_REF][START_REF] Ayad | Enrichment of linear hexahedral finite elements using rotations of a virtual space fiber[END_REF]. This element has been shown to be much more accurate than the classical four-node quadrilateral membrane element even with coarse and distorted meshes [START_REF] Zouari | Quadrilateral membrane finite elements with rotational DOFs for the analysis of geometrically linear and nonlinear plane problems[END_REF]. The main advantage of modelling the matrix domain with PFR4 elements is the possibility to project the two-node Timoshenko beam elements variables onto those of their corresponding matrix elements.

Finite element approximation of a 2D composite medium

Consider a 2D fibre-reinforced solid B 0 as depicted in Figure 1. In the absence of body forces, the static mechanical equilibrium of B 0 requires locally:

     σ xx,x (x) + σ xy,y (x) = 0 σ xy,x (x) + σ yy,y (x) = 0 ∀ x ∈ B 0 (1)
where σ xx , σ yy and σ xy are the plane components of the stress tensor at a local point x of B 0

(σ xx,x = ∂σ xx ∂x ).
By introducing the virtual displacement vector {δu} = {δu δv} T , the weak form of the equilibrium equation ( 1) is obtained as

δW = B 0 {δε} T {σ} dV δW int - ∂B 0 {δu} T {T }dS δWext = 0 ∀ δu (2)
where {σ} = {σ xx σ yy σ xy } T is the vector of plane stresses, {δε} = {δε xx δε yy 2δε xy } T is the vector of virtual plane strains and {T } = {T x T y } T is the vector of external forces applied to ∂B 0 , the boundary of B 0 .

The finite element approximation of Equation ( 2) is constructed by dividing the 2D composite solid B 0 into elementary domains as shown in Figure 1. Accordingly, we present in this work the formulation of a composite membrane element named PFQ4R (Projected Fibre 4-node Quadrilat- eral element with Rotational variables) representing a unit of the matrix domain and the fibres crossing it (Figure 2). The formulation of PFQ4R is based on the so-called Projected Fibre Approach, introduced by Kebir and Ayad [START_REF] Kebir | A specific finite element procedure for the analysis of elastic behaviour of short fibre reinforced composites. The projected fibre approach[END_REF]. This approach allows taking account of the position, distribution and orientation of discrete fibres within the matrix domain as well as their mechanical properties. Discrete short or long fibres are represented by straight line segments before being introduced in the matrix medium. Then, their intersections with the matrix mesh are handled by a specific algorithm developed by the authors as shown in Figure 1. In a first approximation, Kebir and Ayad [START_REF] Kebir | A specific finite element procedure for the analysis of elastic behaviour of short fibre reinforced composites. The projected fibre approach[END_REF] and Tiar et al. [START_REF] Tiar | A nonlinear finite element formulation for large deflection analysis of 2D composite structures[END_REF] have proposed to model all fibres crossing an element of the matrix mesh by two-node truss elements. In this work, we propose an enrichment of this approximation by representing all fibres crossing a matrix element rather by two-node Timoshenko beam elements. The main advantage of representing fibres with Timoshenko beam elements instead of truss elements is to take account of their local bending stiffness which allows better predicting the behaviour of composite solids under bending loading. In order to reduce the computing cost of 2D fibre-reinforced solids, the beam elements variables, in particular their rotational DOFs, have to be projected onto those of their corresponding matrix elements as initially proposed by Kebir and Ayad [START_REF] Kebir | A specific finite element procedure for the analysis of elastic behaviour of short fibre reinforced composites. The projected fibre approach[END_REF]. To this end, a four-node quadrilateral membrane element with rotational DOFs is chosen to model a unit of the matrix domain (Figure 3). This element is named PFR4 and is based on a plane adaptation of the so-called Space Fibre Rotation (SFR) concept introduced by Ayad [START_REF] Ayad | Contribution to the numerical modeling of solids and structures and the non-newtonian fluids forming process. Application to packaging materials[END_REF] (see also [START_REF] Ayad | Enrichment of linear hexahedral finite elements using rotations of a virtual space fiber[END_REF] for more details about the SFR concept). The formulation of PFR4 is reported in Zouari et al. [START_REF] Zouari | Quadrilateral membrane finite elements with rotational DOFs for the analysis of geometrically linear and nonlinear plane problems[END_REF] and briefly recalled in the following for completeness. To simplify, we consider in the following only one fibre that reinforces a matrix unit as depicted in Figure 2. Besides, a perfect bonding of the fibre to the matrix is assumed and both are supposed to be elastic and isotropic. The proposed plane composite element PFQ4R is a combination of the SFR concept-based four-node membrane element, representing the matrix unit, and the classical two-node Timoshenko plane beam element that models the fibre crossing the matrix element.

Moreover, the index notation will be used in the following to facilitate the presentation of the formulation. The m and f characters are used to distinguish between the matrix and the fibre variables, respectively.

Stiffness matrix of the quadrilateral element presenting a unit of the matrix domain

At the element level, the weak form of the equilibrium (2) can be rewritten as

δW e = δW m int + δW f int -δW e ext (3)
where δW m int and δW f int are, respectively, the unit matrix and the fibre internal virtual works.

For linear elastic problems, the internal virtual work δW m int of the matrix element is given by:

δW m int = V m {δε} T [C m ]{ε} d V m (4)
where [C m ] is the (3 × 3)-sized plane elasticity matrix of the matrix domain.

We recall that the matrix element is based on a plane adaptation of the SFR concept and hence presents displacement and fictive rotational DOFs. We start by choosing one point p of the standard four-node quadrilateral membrane element as depicted in Figure 3. The SFR concept supposes for plane elements a fictive small out-of-plane rotation of the nodal fibre ip around the node i represented by the rotation vector θ i z. This fictive normal rotation leads to an additional vector θ i z∧ip that enhances the displacement vector approximation of the classical four-node quadrilateral element [START_REF] Ayad | Contribution to the numerical modeling of solids and structures and the non-newtonian fluids forming process. Application to packaging materials[END_REF][START_REF] Ayad | Enrichment of linear hexahedral finite elements using rotations of a virtual space fiber[END_REF][START_REF] Zouari | Quadrilateral membrane finite elements with rotational DOFs for the analysis of geometrically linear and nonlinear plane problems[END_REF]:

u(ξ, η) = 4 i=1 N m i (ξ, η) u i + θ i z ∧ iq ⇒      u v      = 4 i=1 N m i (ξ, η)      u i -(y -y i )θ i v i + (x -x i )θ i      (5)
where

N m i (ξ, η) = 1 4
(1+ξ i ξ)(1+η i η) are the classical shape functions associated with the four-node quadrilateral element, x and y are the cartesian coordinates of p, x i and y i (i=1,4) are the nodal coordinates and u i , v i and θ i (i=1,4) are the nodal DOFs of the matrix element based on the SFR concept (two displacements and one fictive rotation per node).

The approximation ( 5) can be rewritten in a matrix form as:

     u v      = [N ] {u m n } ; [N ] =    {N u } T {N v } T    =       {N ui } T • • • • • • i = 1, 4 {N vi } T       (6)
with

{N ui } = {N m i 0 -N m i (y -y i )} T ; {N vi } = {0 N m i N m i (x -x i )} T (7)
and

{u m n } = {• • • | u i v i θ i | • • • i = 1, 4}
T is the nodal DOFs vector of the matrix element.

Using the approximation (6) of the displacement vector, we find the (3 × 6)-sized matrix [B m ] that relates the plane strain vector {ε} to the DOFs vector {u m n }:

{ε} = [B m ]{u m n } with [B m ] =       {N u,x } T {N v,y } T {N u,y } T + {N v,x } T       (8)
The internal virtual work of the matrix element becomes:

δW m int = {δu m n } T V m [B m ] T [C m ][B m ] d V m [K m ] {u m n } (9)
where [K m ] is the linear stiffness matrix of the quadrilateral membrane element presenting a unit of the matrix domain.

Stiffness matrix of the fibre element

In this work, each fibre crossing a matrix element is modelled with a two-node Timoshenko plane beam element to account for its bending stiffness (Figure 4). Local longitudinal and transverse displacements of the beam element are written as:

     u(x, y) = u 0 (x) -y θ(x) v(x, y) = v 0 (x) (10)
where u 0 and v 0 are the local displacements of the reference line and θ is the small normal rotation of the beam element cross-section. u 0 and v 0 can be approximated by the one-dimensional shape The internal virtual work of the beam element δW f int is written as:

functions N f 1 = 1 -ξ 2 and N f 2 = 1 + ξ 2 : u 0 = N f 1 u a + N f 2 u b , v 0 = N f 1 v a + N f 2 v b , x = L 2 ξ with -1 ≤ ξ ≤ 1 (11)
δW f int = L/2 -L/2 δe m N + δγ xy T y + δκ M z d x = L/2 -L/2 {δ Γ} T {Σ} d x (12)
where e m = u 0,x , κ = -θ ,x (the bending strain), γ xy = v 0,x -θ (the shear strain), N = A σ xx dS (the axial force with σ xx is the axial stress), T y = A σ xy dS (the shear force with σ xy is the shear stress) and M z = A y σ xx dS (the bending moment). {Γ} and {Σ} are respectively the generalized strain vector and the stress resultant vector related through a (3 × 3)-sized matrix [C f ] given by:

{Γ} = {e m γ xy κ} T {Σ} = {N T y M z } T , [C f ] =        E f A 0 0 0 kG f A 0 0 0 E f I        , G f = E f 2 ( 13
)
E f is Young's modulus of the fibre crossing the matrix element, I is the area moment of inertia of its cross-section about z and k is a shear correction factor.

Using the approximations [START_REF] Cunha | A finite element model with discrete embedded elements for fibre reinforced composites[END_REF], {Γ} is related to the local DOFs vector of the beam element

{u f n } = {u a v a θ a u b v b θ b } T through a (3 × 6)-sized matrix [B f ] as {Γ} = [B f ]{u f n } with [B f ] =        - 1 L 0 0 1 L 0 0 0 - 1 L -N f 1 0 1 L -N f 2 0 0 1 L 0 0 - 1 L        (14)
Using Equations ( 13) and ( 14), the internal virtual work of the beam element becomes

δW f int = {δu f n } T L/2 -L/2 [B f ] T [C f ][B f ] d x [K f ] {u f n } (15)
where [K f ] is the local stiffness matrix of the beam element. By introducing the following passage matrix [P ], the internal virtual work of the beam element reads:

δW f int = {δu f n } T [P ][K f ][P ] T [K f ] {δu f n } with [P ] =    [P ] 0 0 [P ]    , [P ] =       t x n x 0 t y n y 0 0 0 1       (16)
t x , t y , n x and n y are the global coordinates of the unit vectors t and n, {u

f n } = {u a v a θ a u b v b θ b } T
is the global DOFs vector of the fibre element and [K f ] is its global stiffness matrix.

Stiffness matrix of the composite element PFQ4R

We recall that the internal virtual work of the composite element PFQ4R is given by

δW c int = δW m int + δW f int ( 17
)
where δW m int is approximated by Equation ( 9) and δW f int is approximated by Equation ( 16). It is important to note that the fibre Young's modulus E f in the expression of [K f ] should be replaced by E f * = E f -E m to account for the matrix beam element removed from the matrix domain as explained in Figure 2.

In order to limit the total number of variables of the problem, the fibre beam element degrees of freedom are projected onto those of its corresponding matrix element. Accordingly, the obtained system of equations size is equivalent to that of a non-reinforced 2D medium. To this end, the geometrical merging of the beam element nodes a and b (Figure 4) into the matrix space is done by finding their isoparametric coordinates (ξ a , η a , ξ b , η b ) in terms of the nodal coordinates of the matrix and the fibre elements using the area coordinates method as discussed in [START_REF] Tiar | A nonlinear finite element formulation for large deflection analysis of 2D composite structures[END_REF]. The beam element isoparametric coordinates are then used to relate its global DOFs vector {u f n } to {u m n } through a (6 × 12)-sized matrix [T ] as:

{u f n } = [T ]{u m n }, [T ] =                        N m i (ξ a , η a ) 0 N m i (ξ a , η a ) × (y a -y i ) 0 N m i (ξ a , η a ) N m i (ξ a , η a ) × (x a -x i ) 0 0 N m i (ξ a , η a ) • • • • • • i = 1, 4
N m i (ξ b , η b ) 0 N m i (ξ b , η b ) × (y b -y i ) 0 N m i (ξ b , η b ) N m i (ξ b , η b ) × (x b -x i ) 0 0 N m i (ξ b , η b )                        (18)
Thanks to this projection, it is possible to write the internal virtual work of the composite membrane element PFQ4R only in terms of the DOFs vector of the matrix element {u m n } as:

δW c int = {δu m n } T [K m ] + [T ] T [K f ][T ] [K c ] {u m n } (19)
where [K c ] is the stiffness matrix of the composite membrane element PFQ4R.

Numerical validation

The four-node quadrilateral composite plane element PFQ4R was implemented into the commercial finite element code ABAQUS via a user element subroutine (UEL). Numerical models (ABAQUS input files) of the following assessment tests were created by a Fortran program developed by the authors that automatically handles the intersections between the matrix mesh and the discrete fibres. To assess the accuracy of PFQ4R, three validation tests are considered in the following. The obtained results of PFQ4R are compared with those of the quadrilateral composite element PFQ4 developed by Tiar et al. [START_REF] Tiar | A nonlinear finite element formulation for large deflection analysis of 2D composite structures[END_REF] (a combination of the classical four-node quadrilateral membrane element and the two-node truss element). In addition to PFQ4, the solution of PFQ4R is compared with PFQ4R ⋆ which is a combination of the four-node quadrilateral membrane element with rotational DOFs (PFR4) and the two-node truss element. The main objective is to clearly evaluate the contribution of modelling discrete fibres by beam elements instead of truss elements.

Three-point bending of a composite beam reinforced with short fibres

In this first example, we study the deflection of a composite beam subjected to a three-point bending test as shown in Figure 5. This beam is constituted of a magnesium alloy AZ91D matrix We propose a plane modelling of this problem in which all short fibres extremities are projected onto the xy plane as explained in [START_REF] Kebir | A specific finite element procedure for the analysis of elastic behaviour of short fibre reinforced composites. The projected fibre approach[END_REF] and depicted in Figure 5. To obtain a reference solution of the composite beam centre deflection, its homogeneous elastic properties are required. We use the effective elastic properties reported by Tian et al. [START_REF] Tian | Numerical evaluation on mechanical properties of short-fiber-reinforced metal matrix composites: Two-step mean-field homogenization procedure[END_REF] based on tensile tests. They found globally an isotropic behaviour of the AZ91D/carbon composite material characterized by E = 50.450 GPa, ν = 0.342 and G = 19.020 GPa. Based on Euler-Bernoulli beam theory, the maximum deflection of the composite beam is given by the following expression:

δ c = F L 3 48EI + F L 4AG (20)
where A corresponds to the beam cross section and I is its moment of inertia. This expression gives the reference central deflection of the composite beam:

δ ref c = 50.868 × 10 -2 mm.
Before studying the bending behaviour of the composite beam, we use the three plane finite elements PFQ4, PFQ4R and PFQ4R ⋆ to determine its homogeneous elastic properties. To this end, a representative volume element (RVE) of dimensions 1 × 1 mm 2 was considered and subjected to three loadings separately as discussed in Tiar et al. [START_REF] Tiar | A nonlinear finite element formulation for large deflection analysis of 2D composite structures[END_REF]: a tension along the x -direction, a tension along the y-direction and finally a shear test in the xy plane. This RVE was modelled with 20 × 20 quadrilateral elements. Table 1 summarizes the obtained effective elastic properties of the AZ91D/carbon composite material compared with the experimental results.

The results of Table 1 indicate a good agreement between the solutions of the three composite elements and the experimental results and confirms that the the AZ91D/carbon composite material is approximately isotropic. We also remark that PFQ4R firstly slightly overestimates the elastic moduli and the shear modulus and secondly slightly underestimates Poisson's ratios when compared with PFQ4 and PFQ4R ⋆ . This could be explained by the fact that the short carbon fibres which are oriented with respect to the x and y directions present a bending stiffness along these directions when they are modelled with beam elements. This brings an additional stiffness along the x and y directions and could explain the relative difference between the results of the three plane elements.

We now study the bending behaviour of the composite beam subjected to three-point bending as depicted in Figure 5. The half of the composite beam was modelled with four regular meshes constituted of 5 × 1, 10 × 2, 20 × 4 and 40 × 8 quadrilateral elements. Table 2 summarizes the obtained results of the three plane elements compared with the analytical central deflection and Figure 6 shows their convergence curves. We remark that the three composite plane elements globally agree well with the reference analytical deflection and convergence is obtained with mesh refinement. Besides, PFQ4R is found to converge more quickly to the reference solution than PFQ4 and PFQ4R ⋆ . Even if carbon fibres are short in this example, we note that modelling them with beam elements leads to slightly more accurate results than truss modelling if we compare PFQ4R and PFQ4R ⋆ solutions.

Total number of elements

Normalized deflection

Transverse bending of three cantilever beams reinforced by long fibres

In this second example, we study the bending behaviour of three cantilever composite beams reinforced respectively by one, two and three long fibres and subjected to a transverse loading of 50 N at their free ends as depicted in Figure 7. The elastic properties of the matrix are E m = 2 GPa and ν m = 0.3 while those of the reinforcement are E f = 50 GPa and ν f = 0.3. For each beam, the diameter of fibres is chosen to obtain approximately the same fibre volume fraction for beams. PFQ4 and PFQ4R ⋆ are found to be inaccurate especially for the first beam as the central fibre placed along the neutral axis of this beam experiences only axial strains when modelled with truss elements. This inaccuracy decreases for the second and third beams as the number of fibres across the thickness increases. This example clearly illustrates the necessity to model flexural stiffness of long fibres to correctly predict the behaviour of composite solids and structures subjected to bending loading.

Cantilever composite beam with an oriented long fibre

In this third example, we study the cantilever composite beam of Figure 8 reinforced by one long fibre and subjected to a tensile loading characterized by a pressure of 200 MPa applied at its free end. The long fibre has a cylindrical shape with length L f = 80 mm and diameter D f = 8 mm (V f ≈ 13.5%). We consider the same material properties as example 2 (E m = 2 GPa, ν m = 0.3, E f = 50 GPa and ν f = 0.3). The long fibre is oriented at an angle α with respect to the x-axis as depicted in Figure 8 and this causes bending of the composite beam as shown in Figure 9. The objective is to assess the accuracy of the three plane elements PFQ4, PFQ4R and PFQ4R ⋆ in the estimation of the longitudinal and transverse displacements of point A belonging to the free end (Figure 8). For comparison, the composite beam was modelled with a refined mesh of 73950 C3D20 elements which was found to assure converged results. For the plane mesh, the composite beam was discretized with 520 membrane elements as depicted in Figure 8. Table 4 summarizes the obtained results of the three plane composite elements for α = 0, 5, 10 and 15 • and Figure 10 shows the evolution of their relative errors with respect to the angle of orientation α. The results of Table 4 and Figure 8 indicate that PFQ4R and PFQ4R ⋆ present approximately the same result in the estimation of the longitudinal displacement of point A. However, PFQ4R is more accurate than PFQ4R ⋆ in the estimation of the transverse displacement of A. In fact, PFQ4R ⋆ overestimates by about 7% the transverse displacement of A when compared with the solution of PFQ4R. This difference is due to the bending stiffness of the long fibre, when this latter is oriented with respect to the x-axis, taken into account in the formulation of PFQ4R.

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Conclusion

In this paper, a four-node quadrilateral membrane composite element, named PFQ4R, was formulated to analyse 2D fiber-reinforced solids. This element constitutes an extension of the composite four-node plane element PFQ4, developed by Tiar et al. [START_REF] Tiar | A nonlinear finite element formulation for large deflection analysis of 2D composite structures[END_REF], as it accounts for the local bending stiffness of short and long fibres. The PFQ4R formulation is based on the so-called Projected Fibre approach that can handle discrete short and long fibres in a continuum matrix without meshing them. In this study, fibres are represented by two-node Timoshenko plane beam elements while the matrix domain is modelled with four-node membrane elements with rotational DOFs. To obtain a numerical model equivalent to that of a non-reinforced medium, the beam elements variables are expressed in terms of their corresponding matrix elements. Three numerical examples were considered to assess the accuracy of PFQ4R and the obtained results were found to be more accurate than those of PFQ4 and to agree globally well with the reference solutions.

Data availability

The code and input files required to reproduce these findings cannot be shared at this time as they also form part of an ongoing study.

+

  The matrix domain meshed with four-node quadrilateral membrane elements A part of the initial discrete fibre represented with a two-node Timoshenko beam elementHandling of the intersections between fibres and the matrix meshThe 2D composite solid meshed with four-node quadrilateral composite elements

Figure 1 :

 1 Figure 1: The different steps followed to model a 2D fibre-reinforced solid based on the Projected Fibre Approach.

Figure 2 :

 2 Figure 2: The composite membrane element PFQ4R resulted from the combination of a four-node quadrilateral membrane element with rotational DOFs, named PFR4, and the classical two-node Timoshenko plane beam element.

Figure 3 :

 3 Figure 3: The four-node quadrilateral element with fictive rotational DOFs, named PFR4, representing a unit of the matrix domain.

Figure 4 :

 4 Figure 4: The Timoshenko beam element representing one fibre crossing a matrix element.

(E m = 45

 45 GPa and ν m = 0.35) reinforced by 10% in volume of randomly distributed T300 short carbon fibres (E f = 230 GPa and ν f = 0.25). The short carbon fibres present a cylindrical shape with length L f = 0.15 mm and diameter d f = 0.01 mm (L f /d f = 15). Thanks to symmetry, only the half of the composite beam is analysed.

Figure 5 :

 5 Figure 5: Short carbon fibre-reinforced magnesium composite beam under three-point bending.

Figure 6 :

 6 Figure 6: Convergence curves of PFQ4, PFQ4R and PFQ4R ⋆ in the estimation of the normalized central deflection of the composite beam subjected to three-point bending.

Figure 8 :

 8 Figure 8: Cantilever composite beam reinforced by one long fibre oriented at an angle α with respect to the x-axis.

Figure 9 :

 9 Figure 9: Deformations of the composite beam for α = 0 (left) and α = 10 • (right).

Figure 10 :

 10 Figure 10: Evolution of the relative errors of PFQ4, PFQ4R and PFQ4R ⋆ in the estimation of u A (left) and v A (right) with respect to the angle of orientation α.

Table 1 :

 1 Effective elastic properties of the AZ91D/carbon composite material predicted by the three composite plane elements and compared with the experimental results of Tian et al.[START_REF] Tian | Numerical evaluation on mechanical properties of short-fiber-reinforced metal matrix composites: Two-step mean-field homogenization procedure[END_REF] (the quantity inside parentheses indicates the relative error in %).

		Experiment PFQ4	PFQ4R	PFQ4R ⋆
		[21]
	E x (GPa)	50.450	51.031	51.577	50.981
			(1.152)	(2.235)	(1.053)
	E y (GPa)	50.450	51.082	51.673	51.016
			(1.253)	(2.426)	(1.123)
	ν xy	0.342	0.347	0.335	0.348
			(1.461)	(-2.046)	(1.754)
	ν yx	0.342	0.348	0.334	0.348
			(1.754)	(-2.339)	(1.754)
	G xy (GPa)	19.020	18.911	19.353	18.884
			(-0.573)	(1.752)	(-0.715)

Table 2 :

 2 Central deflection of the AZ91D/carbon composite beam under three-point bending predicted by the three composite plane elements and compared with the analytical deflection (the quantity inside parentheses indicates the relative error in %).

	Mesh	PFQ4	PFQ4R	PFQ4R ⋆
	5 × 1	0.335	0.364	0.442
		(-34.025)	(-28.344)	(-13.011)
	10 × 2	0.447	0.508	0.562
		(-12.047)	(-0.134)	(10.599)
	20 × 4	0.491	0.5036	0.533
		(-3.476)	(-0.999)	(4.819)
	40 × 8	0.505	0.5042	0.519
		(-0.724)	(-0.881)	(2.027)

Table 3 :

 3 Transverse displacements (in mm) of point A belonging to the free ends of the composite beams reinforced by one, two and three long fibres (the quantity inside parentheses indicates the relative error in % with respect to C3D20 solutions).

		C3D20	PFQ4	PFQ4R	PFQ4R ⋆
	Beam 1	11.591	19.192	11.321	19.863
			(65.576)	(-2.329)	(71.365)
	Beam 2	4.802	5.291	4.831	5.351
			(10.183)	(0.603)	(11.432)
	Beam 3	4.757	5.096	4.809	5.166
			(7.126)	(1.093)	(8.597)

Table 4 :

 4 Longitudinal and transverse displacements (in mm) of point A belonging to the free end of the cantilever composite beam reinforced by one long fibre (the quantity inside parentheses indicates the relative error in % with respect to C3D20 solution).

		C3D20	PFQ4	PFQ4R	PFQ4R ⋆
	u A	4.818	4.399	4.661	4.661
	α = 0		(-8.695)	(-3.258)	(-3.258)
	v A	0	0	0	0
			(0)	(0)	(0)
	u A	5.045	4.909	4.929	4.939
	α = 5 •		(-2.869)	(-2.473)	(-2.275)
	v A	-3.659	-4.054	-3.798	-4.050
			(10.795)	(3.798)	(10.685)
	u A	5.697	5.371	5.537	5.576
	α = 10 •		(-5.722)	(-2.808)	(-2.123)
	v A	-6.346	-7.602	-6.748	-7.208
			(19.791)	(6.334)	(13.583)
	u A	6.487	6.444	6.396	6.455
	α = 15 •		(-0.662)	(-1.402)	(-0.493)
	v A	-7.618	-8.548	-7.973	-8.487
			(12.207)	(4.660)	(11.407)

the three composite beams (V f ≈ 25%). To this end, they were fixed to φ 1 = 4 mm, φ 2 = 2.827 mm and φ 3 = 2.308 mm. To model these composite beams, the matrix domain was discretized with a regular mesh constituted of 33 × 3 four-node quadrilateral elements and the objective is to estimate the transverse displacement of point A belonging to the free ends of the three composite beams using the plane elements PFQ4, PFQ4R and PFQ4R ⋆ . For comparison, the reference transverse displacements of point A were obtained by considering 3D models using the ABAQUS 20-node hexahedral element C3D20. Accordingly, the three composite beams were modelled with three converged meshes of 52800, 59000 and 67400 C3D20 elements, respectively, and both the matrix and the long fibres were modelled with this quadratic element. The reference transverse displacements of point A were found to be equal to 11.591 mm for the first beam, 4.802 mm for the second beam and 4.757 mm for the third beam. Table 3 summarizes the obtained results of the three plane elements compared with the reference solutions. Table 3 shows that PFQ4R results agree well with the reference solutions for the three composite