Valentin Partula

Mélanie Deschasaux-Tanguy

Stanislas Mondot

Agnès Victor-Bala

Nadia Bouchemal

Lucie Lecuyer

Christine Bobin-Dubigeon

Marion J Torres

Emmanuelle Kesse-Guyot

Bruno Charbit

Etienne Patin

Karen E Assmann

Paule Latino-Martel

Chantal Julia

Pilar Galan

Serge Hercberg

Lluis Quintana-Murci

Matthew L Albert

Darragh Duffy

Olivier Lantz

Philippe Savarin

Mohamed Nawfal Triba

Mathilde Touvier

The Milieu

Associations between untargeted plasma metabolomic signatures and gut microbiota composition in the Milieu Intérieur population of healthy adults

Keywords: Gut microbiota, host-gut microbiota relationships, NMR metabolomics, cross-sectional population-based study, epidemiology discovery rate, FDR, Mass spectrometry, MS, Milieu Intérieur, MI, Multivariate analyses with linear models, MaAsLin, Nuclear magnetic resonance, NMR, 1H 1D NMR pulse sequence Nuclear Overhauser Effect Spectroscopy, NOESY1D, Operational taxonomic units, OTU, Permutational analysis of variance, PERMANOVA, Polymerase chain reaction, PCR, correlated spectroscopy, TOCSY, Trimethylamine, TMA, Trimethyl-

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The gut metagenome has been estimated to be 150 to 400 times larger than the human genome [START_REF] Consortium | A human gut microbial gene catalogue established by metagenomic sequencing[END_REF][START_REF] Consortium | An integrated catalog of reference genes in the human gut microbiome[END_REF] , yielding a colossal metabolic potential [START_REF] Yadav | A review of metabolic potential of human gut microbiome in human nutrition[END_REF] . Products of the microbial metabolism comprise a vast range of molecules involved in a variety of biological functions [START_REF] Nicholson | Host-Gut Microbiota Metabolic Interactions[END_REF][START_REF] Schroeder | Signals from the gut microbiota to distant organs in physiology and disease[END_REF] and integrated in proposed "host-microbial metabolic axes". These molecules complement the host metabolism and subsequently influence the host health through potential beneficial or harmful effects [START_REF] Schroeder | Signals from the gut microbiota to distant organs in physiology and disease[END_REF][START_REF] Brown | The Gut Microbial Endocrine Organ: Bacterially Derived Signals Driving Cardiometabolic Diseases[END_REF][START_REF] Rajani | Disruptions in gut microbial-host co-metabolism and the development of metabolic disorders[END_REF][START_REF] Postler | Understanding the Holobiont: How Microbial Metabolites Affect Human Health and Shape the Immune System[END_REF][START_REF] Li | Symbiotic gut microbes modulate human metabolic phenotypes[END_REF] . Consequently, the gut microbiota is now fully considered as an endocrine pseudo-organ within the mammalian holobiont superorganism [START_REF] Brown | The Gut Microbial Endocrine Organ: Bacterially Derived Signals Driving Cardiometabolic Diseases[END_REF][START_REF] Kundu | Our Gut Microbiome: The Evolving Inner Self[END_REF] . Specific studies investigating targeted moleculessuch as short-chain fatty acids, bile acids, choline metabolites, etc.have considerably increased the mechanistic knowledge of microbialhuman co-metabolism [START_REF] Nicholson | Host-Gut Microbiota Metabolic Interactions[END_REF][START_REF] Louis | The gut microbiota, bacterial metabolites and colorectal cancer[END_REF][START_REF] Ryan | Bile acids at the cross-roads of gut microbiome-host cardiometabolic interactions[END_REF][START_REF] Fu | The Gut Microbiome Contributes to a Substantial Proportion of the Variation in Blood Lipids[END_REF] . However, studies addressing more comprehensively the highly complex microbial-host metabolic interactions remain scarce [START_REF] Li | Symbiotic gut microbes modulate human metabolic phenotypes[END_REF] . Metabolomicsthe systematic identification and quantification of the low-molecular weight metabolic products of a biological system at a specific point in timeenables the simultaneous detection and relative quantification of hundreds of molecules, and thus holds promise to elucidate microbial-host interactions [START_REF] Smirnov | Challenges of metabolomics in human gut microbiota research[END_REF] . Studies integrating a metabolomic approach have started to expand. Notably, gut microbiota has been reported to associate with the serum metabolome. Pedersen et al. indeed showed that the serum metabolome of insulin-resistant individuals presents increased levels of branched-chain amino acids (BCAAs), which were further correlated with gut microbial genetic characteristics (namely the potential for biosynthesis of BCAAs and lack of genes encoding bacterial inward transporters BCAAs) [START_REF] Pedersen | Human gut microbes impact host serum metabolome and insulin sensitivity[END_REF] . Additionally, comparisons of the host metabolomes (in plasma or urine) between diseased individuals [START_REF] Kelly | Plasma metabolite profiles in children with current asthma[END_REF][START_REF] Naviaux | Metabolic features of chronic fatigue syndrome[END_REF][START_REF] Urpi-Sarda | Non-targeted metabolomic biomarkers and metabotypes of type 2 diabetes: A cross-sectional study of PREDIMED trial participants[END_REF] or subjects included in nutritional interventions [START_REF] Derkach | Effects of dietary sodium on metabolites: the Dietary Approaches to Stop Hypertension (DASH)-Sodium Feeding Study[END_REF][START_REF] Mora-Cubillos | Plasma metabolomic biomarkers of mixed nuts exposure inversely correlate with severity of metabolic syndrome[END_REF][START_REF] Pantophlet | Metabolic Profiling Reveals Differences in Plasma Concentrations of Arabinose and Xylose after Consumption of Fiber-Rich Pasta and Wheat Bread with Differential Rates of Systemic Appearance of Exogenous Glucose in Healthy Men[END_REF][START_REF] Tovar | Reduction in cardiometabolic risk factors by a multifunctional diet is mediated via several branches of metabolism as evidenced by nontargeted metabolite profiling approach[END_REF] and controls have detected molecules stemming exclusively from microbial activity (e.g. hippurate or p-cresol) as discriminant metabolites. However, in these studies, neither the gut microbiota composition nor its activity were analyzed, and direct associations between the gut microbiota and the host metabolome were not investigated. In fact, research investigating such associations is scarce and often have been Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0007114520004870 conducted on limited sample sizes (N≤100) [START_REF] Li | Symbiotic gut microbes modulate human metabolic phenotypes[END_REF][START_REF] Cui | Metagenomic and metabolomic analyses unveil dysbiosis of gut microbiota in chronic heart failure patients[END_REF][START_REF] Dewulf | Insight into the prebiotic concept: lessons from an exploratory, double blind intervention study with inulin-type fructans in obese women[END_REF][START_REF] Karl | Changes in intestinal microbiota composition and metabolism coincide with increased intestinal permeability in young adults under prolonged physiological stress[END_REF][START_REF] Lin | Integrative metabolic and microbial profiling on patients with Spleenyang-deficiency syndrome[END_REF][START_REF] Liu | Gut microbiome and serum metabolome alterations in obesity and after weight-loss intervention[END_REF] , or focused on subjects with specific characteristics or conditions like chronic heart failure [START_REF] Cui | Metagenomic and metabolomic analyses unveil dysbiosis of gut microbiota in chronic heart failure patients[END_REF] , spleen-yang-deficiency syndrome [START_REF] Lin | Integrative metabolic and microbial profiling on patients with Spleenyang-deficiency syndrome[END_REF] , obesity [START_REF] Dewulf | Insight into the prebiotic concept: lessons from an exploratory, double blind intervention study with inulin-type fructans in obese women[END_REF][START_REF] Liu | Gut microbiome and serum metabolome alterations in obesity and after weight-loss intervention[END_REF] , physiological and metabolic stress [START_REF] Karl | Changes in intestinal microbiota composition and metabolism coincide with increased intestinal permeability in young adults under prolonged physiological stress[END_REF] , etc. More recently, studies were performed on larger population-based samples, including several hundreds of participants and reporting associations between the composition and diversity of the gut microbiota and circulating metabolites, using metabolomics [START_REF] Org | Relationships between gut microbiota, plasma metabolites, and metabolic syndrome traits in the METSIM cohort[END_REF][START_REF] Vojinovic | Relationship between gut microbiota and circulating metabolites in population-based cohorts[END_REF][START_REF] Menni | Serum metabolites reflecting gut microbiome alpha diversity predict type 2 diabetes[END_REF][START_REF] Pallister | Hippurate as a metabolomic marker of gut microbiome diversity: Modulation by diet and relationship to metabolic syndrome[END_REF][START_REF] Wilmanski | Blood metabolome predicts gut microbiome αdiversity in humans[END_REF] . Following these approaches, our objective was to investigate the associations between untargeted plasma metabolomic signatures and gut microbiota composition in a large sample of healthy French subjects.

METHODS

Study population

This study was conducted in the framework of the Milieu Intérieur project, which enrolled 1,000 healthy French adults with primary aim to assess the determinants of immunologic variance within the general healthy population. The objectives of the present study are secondary objectives to the Milieu Intérieur project, for which the rationale, design, and methods have been extensively described elsewhere [START_REF] Thomas | The Milieu Intérieur study -An integrative approach for study of human immunological variance[END_REF] . Briefly, 500 females and 500 males equally distributed across 5 classes of age (from 20 to 69 years old), and of mainland French descent for at least 3 generations, were recruited in the suburban area of Rennes (Ille-et-Vilaine, Bretagne, France), between September 2012 and August 2013. Participants were considered "healthy" upon recruitment, as defined per stringent exclusion criteria comprising any chronic disease or condition involving the immune system, abnormal physical examination, or any abnormal clinical-biological analysis [START_REF] Thomas | The Milieu Intérieur study -An integrative approach for study of human immunological variance[END_REF] .

Data collection

Covariates

During the medical visit, investigators measured body-mass index (BMI) using standardized methods, and collected information related to age, sex, smoking status, educational level, employment status, occupational category, income level, and physical activity.

Gut microbiota profiling from stool samples

Gut microbiota composition was determined through the sequencing of the 16S ribosomal RNA gene in stool samples produced at home maximum 24 hours before the scheduled medical visit, collected in a double-lined sealable bag maintaining strict anaerobic conditions, and aliquoted and stored at -80°C upon reception at the clinical site. Detailed information regarding DNA preparation, barcoding Polymerase Chain Reaction (PCR) protocol, sequencing, and computation of microbial diversity indexes are provided in Supplementary Material and Methods 1.

Untargeted NMR metabolomics analysis in plasma samples

Fasting whole blood was collected in heparin tubes and centrifuged (300g, 10min, room temperature) maximum 6h after blood draw, and supernatants were stored at -80°C maximum 10min after centrifugation. Plasma samples underwent 2 freeze-thaw cycles before proton nuclear magnetic resonance (NMR) analysis, the additional freeze-thaw cycle being done for all samples in order to prepare internal quality control samples necessary for the study [START_REF] Gika | Protocol for quality control in metabolic profiling of biological fluids by U(H)PLC-MS[END_REF][START_REF] Dunn | Procedures for large-scale metabolic profiling of serum and plasma using gas chromatography and liquid chromatography coupled to mass spectrometry[END_REF] and so that that all samples would go through the same process. The NMR analysis was performed blindly on samples (randomized for age and sex) according to a previously published protocol [START_REF] Lécuyer | NMR metabolomic signatures reveal predictive plasma metabolites associated with long-term risk of developing breast cancer[END_REF] . Briefly, two complementary one-dimensional NMR sequences were acquired per plasma sample, on a 500MHz-Bruker Avance III spectrometer (Bruker, Billerica, MA, USA) at 300Knamely the [START_REF] Consortium | A human gut microbial gene catalogue established by metagenomic sequencing[END_REF] pulse sequence Nuclear Overhauser Effect Spectroscopy (NOESY1D) with z-gradient (Bruker pulse program noesygppr1d) and the Carr-Purcell-Meiboom-Gill (CPMG) sequences. Consequently, to acquisition, all spectra were pre-processed, calibrated, and ultimately "sliced" into "buckets" with pre-defined limits so as to maximize recovery of peak entities. The buckets were then scaled to the total summed integrals for each spectrum, and the integrals of peak entities were calculated to obtain continuous NMR variables that were used in further statistical analyses. Finally, NMR signals were assigned using ad hoc literature [START_REF] Nicholson | 750 MHz 1H and 1H-13C NMR Spectroscopy of Human Blood Plasma[END_REF] . CHENOMX software (CHENOMX, Edmonton, Alberta, Canada) and Human Metabolome Database [START_REF] Wishart | HMDB 3.0-The Human Metabolome Database in 2013[END_REF] were used to confirm the assignments using the medium spectrum (Supplementary Figure 3) [START_REF] Nicholson | 750 MHz 1H and 1H-13C NMR Spectroscopy of Human Blood Plasma[END_REF] . After the statistical analyses, the assignments of the buckets that associated with the gut microbiota were newly carefully looked and confirmed to be unambiguously assigned by gathering information from S-TOCSY, additional two-dimension NMR experiments (recorded on random samples) such as 1H-1H Total Correlated Spectroscopy (TOCSY) and J-resolved experiments. The detailed NMR metabolomic analysis protocol (sample preparation, post-acquisition spectra pre-processing, and intelligent bucketing of spectra and obtaining of NMR variables) is provided in Supplementary Material and Methods 2.

Statistical analyses

Associations between each NMR variable and each α-diversity index were tested using nonparametric partial Spearman correlations. Associations between each NMR variable and interindividual dissimilarities in gut microbiota composition (Bray-Curtis index of β-diversity) were tested using permutational analysis of variance (PERMANOVA) with 999 permutations.

Associations between each NMR variable and the relative abundances of genera and species were tested using multivariate associations with linear models (MaAsLin). This statistical framework performs boosted, additive general linear models and is suited to test the associations between the relative abundance of microbial community members and metadata (here the NMR variables and covariates). Relative abundances are first transformed through a variance-stabilizing arcsin-square Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0007114520004870 root transformation. Genera and species were included only if their relative abundance and prevalence were higher than 0.01% and 1%, respectively. Partial Spearman correlations, PERMANOVAs and MaAsLins were all multi-adjusted for age (20 to <30, 30 to <40, 40 to <50, 50 to <60, and 60 to <70 years old), sex, BMI (continuous variable, in kg.m -2 ), smoking status (non-smoker, ex-smoker, current smoker), and physical activity (continuous variable, in h.wk -1 ). These covariates were indeed reported in the literature to have a potential impact on gut microbiota composition or metabolomic signature, or significantly associated with microbial or metabolomic features in our analyses. These associations between covariates and gut microbial and metabolomic features were tested using PERMANOVAs and Principal Component Partial R-square (PC-PR2) analysis, respectively, enabling to measure the percentage of variation in microbial composition or metabolomic data explained by the factors tested. Additionally, α-and βdiversity models (partial Spearman correlations and PERMANOVAs) were adjusted for sequencing depth. P-values were adjusted for multiple testing using a Benjamini-Hochberg correction with a 10%-False Discovery Rate (FDR).

To better visualize the interindividual variation in metabolomics signatures in our study sample, Principal Component Analysis (PCA) of the metabolomic datasets were plotted and color-coded based on age, sex, and BMI. Spearman correlations were performed using SAS 9.4; PERMANOVAs, MaAsLins, PC-PR2, and PCA representations of metabolomic datasets were performed using R 3.3.2 (packages vegan, Maaslin, FactoMineR, and FactoExtra).

RESULTS

Characteristics of the study sample

Among 1,000 participants to the Milieu Intérieur study, 138 were excluded from the study sample because of missing gut microbiota composition datadue to failure during PCR, insufficient number of detected reads, or insufficient quantity of stool aliquot or DNA extracted. Among the 862 participants with available gut microbiota data, 16 were further excluded because their Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0007114520004870 metabolomic signatures could not be determineddue to insufficient quantity of plasma aliquot, or failure during NMR spectra acquisition. Participant flow-chart is presented in Supplementary Figure 1. Characteristics of the 846 subjects are presented in Table 1. Mean ± standard deviation (SD) BMI was 24.3 ± 3.3 kg.m -2 and mean ± SD physical activity (including both professional and leisure physical activity) was 5.4 ± 6.3 h.wk -1 . The proportion of smokers was 19.9%.

Characteristics of the gut microbiota in the Milieu Intérieur study have been previously described [START_REF] Scepanovic | A comprehensive assessment of demographic, environmental, and host genetic associations with gut microbiome diversity in healthy individuals[END_REF][START_REF] Partula | Associations between usual diet and gut microbiota composition: results from the Milieu Intérieur cross-sectional study[END_REF] . Microbiota analysis yielded sequencing depth ranging from 5,064 to 240,472 reads per sample (mean sequencing depth ± SD: 21,427 ± 19,180 reads). Detected reads clustered into 8,422

OTUs classified into 11 phyla, 328 genera, and 698 species. Firmicutes (mean relative abundance ± SD: 0.68 ± 0.13), Bacteroidetes (0.29 ± 0.13), Proteobacteria (0.02 ± 0.03), Actinobacteria (0.007 ± 0.01), and Verrucomicrobia (0.004 ± 0.008) were the 5 most abundant phyla. Supplementary Table 1 presents the 22 genera constituting the "core microbiota"defined as the genera shared by at least 95% of samples [START_REF] Falony | Population-level analysis of gut microbiome variation[END_REF] in our study population. A complete description of genera and species composing the gut microbiota of our sample is available in Supplementary Table 2 and NOESY1D metabolomic datasets are shown in Figure 1 and Supplementary Figure 2, respectively. Although no clear clustering appears, grouping patterns along gradient of sex, age, and BMI can be observed. Results of the PC-PR2 analysis measuring the percentage of variation in the metabolomic dataset that is explained by the covariates (age, sex, BMI, smoking status, and physical activity) are shown in Supplementary Table 6.

Associations between plasma metabolomic signals and α-and β-diversity indexes

The associations between metabolomic variables and α-diversity indexes are presented in Figure 2.

Overall, and after correcting for multiple testing with a 10%-FDR, observed richness was associated with 24 CMPG and 8 NOESY1D signals. Negative associations were observed with lipid-related signals (namely lipoproteins or unsaturated lipids) as well as with ester-and ketone-related metabolites (partial Spearman's ρ range: -0.133 to -0.097), while positive associations were detected with signals pertaining to amino acids (polar glutamine and histidine and aromatic tyrosine, as well as related metabolites) and proteins, creatinine, as well as with choline, and glycolysis-related metabolites (glucose and citrate) (partial Spearman's ρ range: 0.087 to 0.126).

Similar results were obtained with Chao1 richness estimate but no association was detected with Simpson's α-diversity index. Likewise, no association was observed between NMR metabolomic variables and inter-individual dissimilarities in gut microbiota composition (Bray-Curtis β-diversity index).

Associations between plasma metabolomic signals and bacterial taxa (relative abundances)

The associations (significant at 10%-FDR) between NMR metabolomic variable and the relative abundance of gut microbiota genera and species are presented in Table 2. At the genus and species levels, 25 and 19 associations were detected, respectively. Associated taxa comprised genera and species with prevalence ranging from 20% to 100% of the study population. Creatinine was the metabolite associated with most taxa (16 genera and 11 species). Notably, consistent positive associations were found between the CPMG and the NOESY1D analyses for genera Catabacter, Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0007114520004870 Saccharofermentans, and Oscillibacter and a negative association was observed for Faecalibacterium prausnitzii.

DISCUSSION

In the present work, we investigated the associations between characteristics of the gut microbiotanamely α-diversity, inter-individual dissimilarities, and the relative abundances of genera and speciesand plasma NMR metabolomic signatures of the host in a large population-based sample of 846 healthy French adults. Overall, our results suggest that gut bacterial features are associated with the systemic metabolism of the host. In particular, we found that bacterial richness was positively associated with signals pertaining to amino acids (glutamine, histidine, tyrosine), as well as with creatinine, choline and glycolysis-related metabolites, while negative associations were observed with lipid-, as well as ester-, and ketone-related signals. Finally, specific associations between metabolomic signals and bacterial members of the gut microbiota were detected.

The metabolic relevance and importance of the gut microbiota for the host are now widely admitted, and research scrutinizing host-gut microbial metabolic relationships is an active area of investigation [START_REF] Daliri | The human microbiome and metabolomics: Current concepts and applications[END_REF] . However, studies investigating direct associations between gut microbiota composition and host blood metabolome in large population-based samples are scarce while more available studies have been conducted on a limited number of participants or focused on specific populations [START_REF] Pedersen | Human gut microbes impact host serum metabolome and insulin sensitivity[END_REF][START_REF] Cui | Metagenomic and metabolomic analyses unveil dysbiosis of gut microbiota in chronic heart failure patients[END_REF][START_REF] Lin | Integrative metabolic and microbial profiling on patients with Spleenyang-deficiency syndrome[END_REF][START_REF] Liu | Gut microbiome and serum metabolome alterations in obesity and after weight-loss intervention[END_REF] . In particular, a study by Org et al. assessed the relationships between gut microbiota composition and circulating metabolites in 531 healthy Finnish males from the METSIM cohort [START_REF] Org | Relationships between gut microbiota, plasma metabolites, and metabolic syndrome traits in the METSIM cohort[END_REF] . Consistently with their study, we reported positive correlations between bacterial richness and glutamine, histidine, and creatininewith similar Spearman correlation coefficients. Furthermore, a study on 399 subscribers to a US Scientific Wellness program was able to explain 45% of the gut microbiota alpha-diversity with 40 plasma metabolites, among which a high frequency of phenylalanine and tyrosine metabolites (including p-cresol sulfate, a potentially uremic toxin) and hippurate [START_REF] Wilmanski | Blood metabolome predicts gut microbiome αdiversity in humans[END_REF] while a study in 1529 females from the TwinsUK cohort found five metabolites that consistently correlated with gut microbiota alpha-diversity (Shannon index) Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0007114520004870 including hippurate, p-cresol sulfate, phenylacetylglutamine, 3-phenylpropionate, and hyodeoxycholate [START_REF] Pallister | Hippurate as a metabolomic marker of gut microbiome diversity: Modulation by diet and relationship to metabolic syndrome[END_REF] .

In a study conducted on 893 participants from the Dutch LifeLines-DEEP cohort, Fu et al. [START_REF] Fu | The Gut Microbiome Contributes to a Substantial Proportion of the Variation in Blood Lipids[END_REF] measured specific lipid levels through targeted colorimetric methods, and found that triglycerides and high-density lipoproteins were respectively negatively and positively associated with both bacterial richness and diversity. Consistently, a more recent study using NMR metabolomics in 2309 individuals from the Rotterdam Study and the LifeLines-DEEP cohort observed inverse associations between the diversity of the gut microbiota and serum levels of VLDL particles, triglycerides, total fatty acids, monounsaturated fatty acids and saturated fatty acids but also glycoprotein acetyl, alanine, isoleucine, and lactate and positive associations with HDL particles. In addition, Le Chatelier et al. associated a decreased gut microbial richness with metabolic impairments such as increased fat mass and body weight, inflammation of the adipose tissue and dyslipidemia [START_REF] Metahit Consortium | Richness of human gut microbiome correlates with metabolic markers[END_REF] . In the present work, we consistently found that lipid-related metabolomic signals were negatively associated with bacterial richness. However, we were unable to further characterize these lipid-related signals and discriminate between the different lipid classes potentially hampering the detection of specific associations previously reported [START_REF] Fu | The Gut Microbiome Contributes to a Substantial Proportion of the Variation in Blood Lipids[END_REF][START_REF] Org | Relationships between gut microbiota, plasma metabolites, and metabolic syndrome traits in the METSIM cohort[END_REF] . Indeed, long-chain metabolites such as fatty acids result in an overlap of signals of protons in NMR, limiting a more specific assignment (44) . Still, it should be noted that, in our sample, NMR signals assigned to lipids demonstrated excellent correlation with blood triglycerides (Spearman's ρ > 0.65, p-value < 0.0001 for spectral regions 1.207-1.311ppm, 2.151-2.166ppm, and 2.166-2.167ppm), adding support to the consistency of our results with those previous findings. Furthermore, high-fat Western diet has consistently been associated with gut microbiota impairments [START_REF] Davis | Gut microbiome diversity influenced more by the Westernized dietary regime than the body mass index as assessed using effect size statistic[END_REF][START_REF] De Filippo | Impact of diet in shaping gut microbiota revealed by a comparative study in children from Europe and rural Africa[END_REF] , and lipid-related signals in our study correlated positively with dietary intakes of processed meat (Spearman's ρ=0.09, p-value=0.006 for spectral regions 2.166-2.167ppm and ρ=0.08, p-value=0.02 for 1.207-1.311ppm and 2.151-2.166ppm) and fried products (ρ=0.07, p-value=0.04 for 2.166-2.167ppm), and negatively correlated with fruit (ρ= -0.09, p-value=0.01 for 1.207-1.311ppm and 2.166-2.167ppm and ρ= -0.08, p-value=0.02 for 2.151-Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0007114520004870 2.166ppm) and vegetables (ρ= -0.09, p-value=0.007 for 2.166-2.167ppm and ρ= -0.07, p-value=0.04 for 2.151-2.166ppm). Nonetheless, these assumptions should be considered with caution as additional exploration and targeted identification of these lipid signals using mass spectrometry (MS) or lipidomics is necessary to get more detailed insights into the associations we observed.

Our finding of a positive association between glutamine and gut bacterial richness echoes a result from De Souza et al. who suggested that glutamine might have an anti-inflammatory effect through a modulation of specific gut bacterial members following the observation that an oral supplementation with L-glutamine decreased the Firmicutes/Bacteroidetes ratio in obese and overweight adults [START_REF] De Souza | Oral supplementation with l-glutamine alters gut microbiota of obese and overweight adults: A pilot study[END_REF] .

In our study, creatinine was the metabolomic signal most associated with the relative abundances of bacterial taxa. This by-product of muscle metabolism is exclusively excreted through the kidneys and a high creatinine level in blood is notably used as an indicator of kidney dysfunction [START_REF] Chhibber-Goel | The Metabolite Trimethylamine-N-Oxide is an Emergent Biomarker of Human Health[END_REF] . Our finding of positive associations between plasma creatinine and the relative abundance of various representatives of the Clostridiales order and the Ruminococcaceae family matches results from Org et al. [START_REF] Org | Relationships between gut microbiota, plasma metabolites, and metabolic syndrome traits in the METSIM cohort[END_REF] . We also reported a negative association between creatinine and Faecalibacterium prausnitzii, a species drawing a particular attention from the research community as it has been shown to exert anti-inflammatory properties through its ability to produce anti-inflammatory metabolites (e.g. butyrate), or to inhibit of the production of pro-inflammatory cytokines [START_REF] Ferreira-Halder | Action and function of Faecalibacterium prausnitzii in health and disease[END_REF] .

Various studies have reported that F. prausnitzii is reduced in conditions such as ulcerative colitis, inflammatory bowel disease, colorectal cancer, diabetes, psoriasis, etc. [START_REF] Ferreira-Halder | Action and function of Faecalibacterium prausnitzii in health and disease[END_REF][START_REF] Sokol | Faecalibacterium prausnitzii is an anti-inflammatory commensal bacterium identified by gut microbiota analysis of Crohn disease patients[END_REF][START_REF] Machiels | A decrease of the butyrate-producing species Roseburia hominis and Faecalibacterium prausnitzii defines dysbiosis in patients with ulcerative colitis[END_REF] . Consequently, this species has been proposed as a potential biomarker of gut health [START_REF] Lopez-Siles | Faecalibacterium prausnitzii: from microbiology to diagnostics and prognostics[END_REF] and is even considered as an interesting probiotic candidate [START_REF] Martín | Functional Characterization of Novel Faecalibacterium prausnitzii Strains Isolated from Healthy Volunteers: A Step Forward in the Use of F. prausnitzii as a Next-Generation Probiotic[END_REF] . In particular, a decrease in F. praunitzii was previously reported in chronic kidney disease patients compared to healthy controls [START_REF] Jiang | A reduction in the butyrate producing species Roseburia spp. and Faecalibacterium prausnitzii is associated with chronic kidney disease progression[END_REF][START_REF] Jiang | Alteration of the gut microbiota in Chinese population with chronic kidney disease[END_REF] . This echoes our result of a negative association between F. praunitzii and creatinine obtained in a healthy sample without renal dysfunction. Overall, the numerous associations we detected between bacterial taxa and creatinine are in support of a gut-kidney conversation [START_REF] Chen | Microbiome-metabolome reveals the contribution of gutkidney axis on kidney disease[END_REF][START_REF] Evenepoel | The gut-kidney axis[END_REF] . In this previously described bi-directional organ Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0007114520004870 axis, gut contributes to inflammation and renal injury through microbiota dysbiosis and dysregulations in microbial metabolite production. Conversely, kidney perturbations lead to dysbiosis and disruptions in tight junction function therefore promoting inflammatory conditions in the gut. Notably, Jiang et al. reported that indicators of the renal function such as serum creatinine, estimated glomerular filtration rate, and cystatin C were the most important environmental parameters to influence the overall microbial communities in end-stage renal disease patients [START_REF] Jiang | Alteration of the gut microbiota in Chinese population with chronic kidney disease[END_REF] . In our study a link between the gut and the kidneys seemed to be already observed even in a healthy setting where individuals were free of kidney or gut impairments. However, further exploration of renal function indicators (e.g. estimated glomerular filtration rate, cystatin C, and urine creatinine) would be of utmost importance to confirm such a link. If we hypothesize a gut-kidney axis to interpret our results, alternate interpretations could also be proposed. Indeed, negative associations have also been reported between trymethylamine N-oxide (TMAO) and F. prausnitzii [START_REF] Org | Relationships between gut microbiota, plasma metabolites, and metabolic syndrome traits in the METSIM cohort[END_REF][START_REF] Sanchez-Alcoholado | Role of Gut Microbiota on Cardio-Metabolic Parameters and Immunity in Coronary Artery Disease Patients with and without Type-2 Diabetes Mellitus[END_REF] and creatinine is a precursor for the gut bacterial synthesis of trimethylamine (TMA), which is further oxidized into TMAO in the liver [START_REF] Chhibber-Goel | The Metabolite Trimethylamine-N-Oxide is an Emergent Biomarker of Human Health[END_REF] . Therefore, we cannot rule out that the association we observed between creatinine and F. prausnitzii may actually reflect the overall TMAO metabolic pathway instead of being a true association. However, because TMAO could not be detected by NMR in our study, it was not possible to verify this assumption.

An important strength of this work pertains to its designnamely the acquisition of both detailed metabolomics data and gut microbiome data in a large population-based sample of healthy adults.

While individual or specific approaches consider biomolecules separately, metabolomics allows for the simultaneous detection and relative quantification of hundreds of molecules hence the monitoring of subtle system-wide metabolic interactions [START_REF] Patti | Metabolomics: the apogee of the omics trilogy[END_REF] . In this study we used NMRmetabolomics with standardized and optimized protocols yielding high-quality data [START_REF] Beckonert | Metabolic profiling, metabolomic and metabonomic procedures for NMR spectroscopy of urine, plasma, serum and tissue extracts[END_REF] . This robust and stable technology has demonstrated its suitability for long-term epidemiological studies [START_REF] Dumas | Assessment of Analytical Reproducibility of 1 H NMR Spectroscopy Based Metabonomics for Large-Scale Epidemiological Research: the INTERMAP Study[END_REF] , notably by allowing to analyze large samples quickly with a high reproducibility (62) . From a statistical standpoint, the analyses implemented here allowed to explore detailed associations Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0007114520004870 between gut bacterial and metabolomic datasets, beyond the sole assessment of global connections obtained through multivariate correlation-based approaches (e.g. co-inertia or Procrustes analyses).

In addition, we were able to take into account a range of factors reported to influence the gut microbiota composition or the systemic metabolism of the host in our models, thus mitigating potential confounding. Nonetheless, residual confounding cannot be ruled out. For instance, intestinal transit time was shown to be a major confounding factor of gut microbiota composition [START_REF] Roager | Colonic transit time is related to bacterial metabolism and mucosal turnover in the gut[END_REF] , and we were not able to collect this information. Another important strength and originality of this work is that it was based on a large sample from the general population including only healthy men and women selected using strict criteria. Hence, our results were not confounded by underlying disease and reflected normal range. Yet, this may also have induced less variations in plasma metabolites levels or in gut microbiota composition and therefore resulted in associations of smaller magnitude. Some limitations should also be acknowledged for this study. First, the cross-sectional design of our study, in which blood and stool samples were taken at the same time, prevents from drawing any conclusion regarding the sequential causality of events between host metabolism (e.g., creatinine levels) and gut microbiota composition (e.g., F. prausnitzii). Further, plasma metabolomic signatures only give a metabolic "snapshot" that results from both the endogenous and the gut microbial metabolisms, which makes it is not possible to discriminate signals from human or microbial origin. In future studies, subsequent time points with both metabolomic and microbiota data collected would be of utmost interest to investigate the stability over time and the chronology of the associations observed in the present cross-sectional work. In addition, the fact that most detected metabolites result from multiple metabolic pathways that may be related to gut microbiota or not may also explain the modest correlation coefficients observed between plasma metabolites and gut microbial richness. It should also be acknowledged that, although highly informative, the use of 16S rRNA gene sequencing and NMR metabolomics presents technical limitations. Shotgun metagenomics for instance would give a more precise characterization of microbiota features and also bring additional information regarding the functional metabolic capacity of the gut microbiota.

Accepted manuscript

This would allow to go beyond the compositional approach, limited by the redundancy of gene pool and metabolic functionalities between distinct gut microbial members [START_REF] Abubucker | Metabolic Reconstruction for Metagenomic Data and Its Application to the Human Microbiome[END_REF][START_REF] Holmes | Gut Microbiota Composition and Activity in Relation to Host Metabolic Phenotype and Disease Risk[END_REF] , but still relevant according to recent studies arguing that some metabolic pathways pertain to a handful of species [START_REF] Morrison | Formation of short chain fatty acids by the gut microbiota and their impact on human metabolism[END_REF] . Likewise, while NMR detects only the most abundant metabolites (µM to mM range), mass spectrometry (MS) has lower detection limits and higher sensitivity for signal assignment, and would therefore allow the detection of metabolites undetectable with NMR [START_REF] Riekeberg | New frontiers in metabolomics: from measurement to insight[END_REF] . This technical limitation of NMR may explain why several low-concentration metabolites deriving specifically from gut microbiota were not detected in our study. As already mentioned, the precision of NMR bucket assignment was also limited for some metabolites, especially for lipids and proteins. For instance, the unsaturation corresponds to a chemical function in the aliphatic chain of various lipids, and it is impossible to differentiate among the different lipid types within the plasma sample.

Hence, this limits the comparability of our results to prior studies. In addition, plasma samples underwent two freeze-thaw cycles prior to NMR analyses (necessary to prepare internal quality controls) so that the possibility of increased metabolites cleavage cannot be ruled out. Still, this would similarly affect all samples as they all went through the same process.

Although the Milieu Intérieur population is somewhat representative of the French Ille-et-Vilaine region [START_REF] Thomas | The Milieu Intérieur study -An integrative approach for study of human immunological variance[END_REF] , our study sample is not representative of the whole French adult populationcaution is therefore needed in the generalization of our results. However, it is important to acknowledge that from a gut microbiota composition standpoint, our study sample was consistent with other studies performed in Western settings [START_REF] Falony | Population-level analysis of gut microbiome variation[END_REF][START_REF] Rothschild | Environment dominates over host genetics in shaping human gut microbiota[END_REF] , with the same 5 most abundant phyla and shared "core genera" (Supplementary Table 1). Finally, the results highlighted here were mostly exploratory as they were obtained from a single population. These results will need to be replicated and confirmed in the future, using independent samples and/or analytical techniques allowing a better refinement of the detected metabolites (e.g., MS, lipidomics). In addition, the hypotheses we proposed to interpret our results were based on existing literature and would need to be developed and confirmed in experimental settings in vitro and/or in vivo. In particular, detailed mechanistic studies Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0007114520004870 comprehensively elucidating the metabolic capability of gut microorganisms are needed to enlighten the associations we detected.

Overall, the present study provides interesting insights regarding the associations between gut microbiota composition and plasma metabolic signatures of the host. Our results contribute to a better understanding of the host-gut microbiota relationships, and notably highlight a possible gutkidney axis in healthy subjects. Because the associations we found in healthy individuals were already reported in chronic kidney disease patients, this builds up to the concept that onset of chronic diseases is a lengthy process that could be detected at early stages. This appears all the more interesting that microbial-host co-metabolites were shown to be excellent prodromal markers of future divergence in metabolic and behavioral outcomes in mice [START_REF] Dumas | Microbial-Host Co-metabolites Are Prodromal Markers Predicting Phenotypic Heterogeneity in Behavior, Obesity, and Impaired Glucose Tolerance[END_REF] . However, our results remain to be confirmed in independent populations, underlying mechanisms to be elucidated, and causal implications for the host health to be established.

  Questionnaires and biological samples were administered and collected by trained medical investigators supported by a full clinical team. The study is sponsored by Institut Pasteur (Pasteur ID-RCB Number: 2012-A00238-35). It was conducted as a single center study and without any investigational product, and was approved by Comité de Protection des Personnes -Ouest 6 on 06/13/2012 (CPP Ouest 6-728/MS2) and by Agence Nationale de Sécurité du Médicament on 06/22/2012 (ID-RCB Number: 2012-A00238-35, ref. ANSM: B120239-70). The protocol, which is Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0007114520004870 registered under ClinicalTrials.gov (NCT01699893), was designed and conducted in accordance with the Declaration of Helsinki and good clinical practice as outlined in the International Conference on Harmonization of Technical Requirements for Registration of Pharmaceuticals for Human Use Guidelines for Good Clinical Practice.

Figure 1 .

 1 Figure 1. Interindividual variation in metabolomic signatures represented by principal components

Figure 2 .

 2 Figure 2. Associations between NMR variables and α-diversity indexes (observed richness and Chao1 estimate of richness) from Spearman partial correlations adjusted for age, sex, BMI, smoking status, physical activity, and sequencing depth, Milieu Intérieur study, France, 2012 (N=846). Qvalues were obtained applying a multiple-testing correction (Benjamini-Hochberg false discovery rate method). Only associations with Q-value ≤ 0.1 for observed richness, as well as subsequent associations with Chao1 richness are presented. ' ** ' P-value ≤ 0.003 and Q-value ≤ 0.05; ' * ' Pvalue ≤ 0.01 and Q-value ≤ ' -' P-value ≤ 0.05 and Q-value >0.1. Corresponding values are shown in Supplementary Table7.

7

 7 Figure 2. Associations between NMR variables and α-diversity indexes (observed richness and Chao1 estimate of richness) from Spearman partial correlations adjusted for age, sex, BMI, smoking status, physical activity, and sequencing depth, Milieu Intérieur study, France, 2012 (N=846). Qvalues were obtained applying a multiple-testing correction (Benjamini-Hochberg false discovery rate method). Only associations with Q-value ≤ 0.1 for observed richness, as well as subsequent associations with Chao1 richness are presented. ' ** ' P-value ≤ 0.003 and Q-value ≤ 0.05; ' * ' Pvalue ≤ 0.01 and Q-value ≤ ' -' P-value ≤ 0.05 and Q-value >0.1. Corresponding values are shown in Supplementary Table7.

  Figure 2. Associations between NMR variables and α-diversity indexes (observed richness and Chao1 estimate of richness) from Spearman partial correlations adjusted for age, sex, BMI, smoking status, physical activity, and sequencing depth, Milieu Intérieur study, France, 2012 (N=846). Qvalues were obtained applying a multiple-testing correction (Benjamini-Hochberg false discovery rate method). Only associations with Q-value ≤ 0.1 for observed richness, as well as subsequent associations with Chao1 richness are presented. ' ** ' P-value ≤ 0.003 and Q-value ≤ 0.05; ' * ' Pvalue ≤ 0.01 and Q-value ≤ ' -' P-value ≤ 0.05 and Q-value >0.1. Corresponding values are shown in Supplementary Table7.

  H 1D NMR Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms.

	https://doi.org/10.1017/S0007114520004870

Table 3 .

 3 NMR analysis on plasma samples generated 260 CPMG and 269 NOESY1D variables. Spectral Results of the PCA performed on the CPMG Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms.

	. Regarding α-

signals and 215 NOESY1D: spectral regions with their metabolite assignments, as well as mean and SD) is presented in Supplementary Tables

4 and 5

.

  Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms.

	https://doi.org/10.1017/S0007114520004870

Table 2 .

 2 Associations between NMR variables and relative abundances of taxa with a Q-value ≤ 0.1 after multiple-testing correction, using MaAsLins, Milieu Intérieur study, France, 2012 (N=846).

		Shift (ppm)	Signal assignation	Phylum| Class| Order| Family|	Genus	Species	MaAsLin coefficient	Coverage 1	p-value 2	Q-value 3
		1.021-1.051 Unassigned signal	Firmicutes| Negativicutes| Selenomonadales| Veillonellaceae|	Sporomusa*		-68.5	35.22	<0.0001	0.07
		1.527-1.631 CH 2 CH 2 COOC	Firmicutes| Clostridia| Clostridiales| Lachnospiraceae| Clostridium XlVa|	Clostridium	C. bolteae	74.2	99.17	<0.0001	0.06
		2.831-2.848 Citrate/=CH-CH 2 -CH= Firmicutes| Bacilli| Bacillales| Bacillaceae 1|	Caldibacillus		-198.2	35.34	<0.0001	0.07
		2.967-3.071 Creatine/albumin lysyl Firmicutes| Clostridia| Clostridiales| Ruminococcaceae|	Hydrogenoanaerobacterium H. saccharovorans	-31.8	90.07	<0.0001	0.07
				Firmicutes| Clostridia| Clostridiales| Ruminococcaceae|	Faecalibacterium		-777.4	99.41	<0.0001	0.02
				Firmicutes| Clostridia| Clostridiales| Ruminococcaceae|	Faecalibacterium	F. prausnitzii	-797.3	99.41	<0.0001	0.03
				Firmicutes| Negativicutes| Selenomonadales| Veillonellaceae|	Sporomusa*		31.8	35.22	<0.0001	0.06
				Firmicutes| Negativicutes| Selenomonadales| Veillonellaceae|	Sporomusa	S. termitida	31.5	34.75	<0.0001	0.06
				Tenericutes| Mollicutes| Acholeplasmatales| Acholeplasmataceae|	Acholeplasma		42.0	32.98	0.0001	0.09
				Firmicutes| Clostridia| Clostridiales| Clostridiaceae 1|	Caloramator		63.9	58.63	<0.0001	0.06
				Firmicutes| Clostridia| Clostridiales| Catabacteriaceae|	Catabacter §		67.0	68.56	<0.0001 0.001
				Firmicutes| Clostridia| Clostridiales| Catabacteriaceae|	Catabacter	C. hongkongensis	64.0	68.56	<0.0001 0.002
				Firmicutes| Clostridia| Clostridiales| Defluviitaleaceae|	Defluviitalea		70.2	81.32	<0.0001	0.07
				Firmicutes| Clostridia| Clostridiales| Defluviitaleaceae|	Defluviitalea	D. saccharophila	71.1	81.32	<0.0001	0.07
				Firmicutes| Clostridia| Clostridiales| Peptococcaceae 2|	Pelotomaculum		74.5	52.36	<0.0001	0.07
	CPMG	3.071-3.111 Creatinine	Firmicutes| Clostridia| Clostridiales| Peptococcaceae 2| Firmicutes| Clostridia| Clostridiales| Christensenellaceae| Firmicutes| Clostridia| Clostridiales| Christensenellaceae|	Pelotomaculum Christensenella Christensenella	P. propionicicum C. minuta	82.6 138.5 170.2	51.30 93.14 93.14	<0.0001 <0.0001 <0.0001	0.01 0.06 0.01
				Firmicutes| Clostridia| Clostridiales| Ruminococcaceae|	Saccharofermentans* §		187.6	65.13	<0.0001 0.005
				Firmicutes| Clostridia| Clostridiales| Ruminococcaceae|	Saccharofermentans	S. acetigenes §	189.3	65.13	<0.0001	0.01
				Firmicutes| Clostridia| Clostridiales| Ruminococcaceae|	Sporobacter		305.1	93.50	<0.0001	0.02
				Firmicutes| Clostridia| Clostridiales| Ruminococcaceae|	Sporobacter	S. termitidis	273.1	93.50	<0.0001	0.07
				Firmicutes| Clostridia| Clostridiales| Ruminococcaceae|	Intestinimonas		320.8	98.70	<0.0001 0.003
				Firmicutes| Clostridia| Clostridiales| Ruminococcaceae|	Intestinimonas	I. butyriciproducens	352.4	98.70	<0.0001 0.002
				Firmicutes| Clostridia| Clostridiales| Ruminococcaceae|	Oscillibacter §		396.2	99.76	<0.0001	0.03
				Firmicutes| Clostridia| Clostridiales| Ruminococcaceae|	Oscillibacter	O. valericigenes	460.2	95.98	<0.0001	0.03
				Firmicutes| Clostridia| Clostridiales| Eubacteriaceae|	Eubacterium		447.6	99.88	<0.0001	0.06
				Firmicutes| Clostridia| Clostridiales| Eubacteriaceae|	Eubacterium	E. coprostanoligenes	580.5	94.44	<0.0001	0.01
		3.111-3.129 3.367-3.437	Unassigned signal	Firmicutes| Clostridia| Clostridiales| Lachnospiraceae| Clostridium XlVa| Firmicutes| Clostridia| Clostridiales| Eubacteriaceae|	Clostridium Eubacterium	C. viride E. rectale*	-133.2 266.4	90.31 98.94	<0.0001 <0.0001	0.03 0.1
		3.437-3.496		Firmicutes| Clostridia| Clostridiales| Eubacteriaceae|	Eubacterium	E. rectale*	-470.9	98.94	<0.0001	0.03
		3.496-3.587 Glucose	Firmicutes| Clostridia| Clostridiales| Eubacteriaceae|	Alkalibaculum		-20.1	74.35	<0.0001	0.07
		5.687-5.727 Urea	Firmicutes| Clostridia| Clostridiales| Ruminococcaceae| Firmicutes| Clostridia| Clostridiales| Ruminococcaceae|	Saccharofermentans* § Saccharofermentans	S. acetigenes §	-219.2 -207.7	65.13 65.13	<0.0001 <0.0001	0.07 0.09

https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0007114520004870 Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at

Downloaded from https://www.cambridge.org/core. INRAE -Saint Pee sur Nivelle, on 29 Dec 2020 at 14:49:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0007114520004870

Acknowledgements:

The authors wish to thank École doctorale 474 Frontières du Vivant -Programme Bettencourt for the financial support provided to VP's Ph.D. The Milieu Intérieur


I F Kozhevnikov

D Duhamel

H P Yin
email: yin@lami.enpc.fr

J Cesbron

F Anfosso-Lédée

A new algorithm for computing the indentation of a rigid body of arbitrary shape on a viscoelastic half-space

Keywords: Contact mechanics, Viscoelastic contact, Viscoelasticity, Indenter

In this paper the contact problem between a rigid indenter of arbitrary shape and a viscoelastic half-space is considered. Under the action of a normal force the penetration of the indenter and the distribution of contact pressure change. We wish to find the relations which link the pressure distribution, the resultant force on the indenter and the penetration on the assumption that the surfaces are frictionless.

For indenters of arbitrary shape the problem may be solved numerically by using the Matrix Inversion Method (MIM), extended to viscoelastic case. In this method the boundary conditions are satisfied exactly at specified "matching points" (the mid-points of the boundary elements). It can be validated by comparing the numerical results to the analytic solutions in cases of a spherical asperity (loading and unloading) and a conical asperity (loading only).

Introduction

Many contact problems are influenced by the viscoelastic behaviour of the materials. This influence is difficult to be analysed for practical problems. For example, the contact between a tyre and a road is mainly modelled in the frame of the elastic theory. The viscoelastic effects have not been taken into account for computing the pressure distribution. In fact the dynamic modulus of rubber is frequency dependent and more important than the elastic modulus. The aim of our work is to compare the viscoelastic model to the elastic model for a single asperity of arbitrary shape for a increasing and then decreasing vertical loading.

The normal contact between a perfectly rigid indenter and a plane elastic halfspace was first investigated by Boussinesq [START_REF] Boussinesq | Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques[END_REF] using the potential theory. Hertz [START_REF] Hertz | Über die Berührung fester elastischer Körper (On the contact of elastic solids)[END_REF][START_REF] Hertz | Über die Berührung fester elastischer Körper and über die Harte (On the contact of rigid elastic solids and on hardness)[END_REF] also gave an analytical solution of the contact problem in the case of two elastic bodies with smooth and quadratic contacting surfaces. Numerous analytical or semi-analytical solutions were then derived from Boussinesq's theory for a rigid indenter of arbitrary shape on an elastic half-space, especially in axisymmetric contact cases by Sneddon [START_REF] Sneddon | The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile[END_REF]. The paper of Gauthier et al. [START_REF] Gauthier | The Hertz contact problem, coupled Volterra integral equations and a linear complementarity problem[END_REF] is concerned with the indentation of an elastic half-space by an axisymmetric indenter under a monotonically applied normal force and under the assumption of Coulomb friction in the region of contact.

The problem of a rigid indenter pressed into contact with a viscoelastic solid was also investigated by many authors. The simplest approach to this problem follows a suggestion by Radok [START_REF] Radok | Viscoelastic stress analysis[END_REF] for finding the stresses and deformations in cases where the corresponding solution for an elastic material is known. It consists in replacing the elastic constant in the elastic solution by the corresponding integral operator from the viscoelastic stress-strain relations. This approach can be applied to the contact problem provided that the loading program is such that the contact area is increasing. Radok's technique breaks when the contact area decreases. This complication has been studied by Ting [START_REF] Ting | The contact stresses between a rigid indenter and a viscoelastic halfspace[END_REF] for a rigid axisymmetric indenter. Recently Vandamme and Ulm [START_REF] Vandamme | Viscoelastic solutions for conical indentation[END_REF] show that for conical indenter suggestion of Radok remains valid at the very beginning of the unloading phase as well.

Analytical, numerical and experimental studies have been made in case of spherical, conical, pyramid indenters. Ball (Brinell) and flat (Boussinesq) punch indentation was analysed theoretically in the work of Larsson and Carlsson [START_REF] Larsson | On microindentation of viscoelastic polymers[END_REF].

The elliptic indenter has been investigated by Yang [START_REF] Yang | The contact problem for viscoelastic bodies[END_REF]. The sharp indentation tests (the standard shapes of the Vickers, Berkovich and Knoop pyramids) are frequently used to examine hard materials like ceramics. The advantage of these tests is the simplicity of the experimental procedure. The work of Giannakopoulos [START_REF] Giannakopoulos | Elastic and viscoelastic indentation of flat surfaces by pyramid indentors[END_REF] presents the results of frictionless and adhesionless contact of flat surfaces by pyramid indenters. One can also note the numerical works Murakami et al. [START_REF] Murakami | Elastic analysis of triangular pyramid indentation by the finite element method and its application to nanoindentation measurement of glasses[END_REF], Larsson et al. [START_REF] Larsson | Analysis of Berkovich indentation[END_REF] for the analysis of Berkovich indentation; the works of Rabinovich and Savin [START_REF] Rabinovich | Three dimensional modelling of indentation fracture in brittle materials[END_REF] -for the analysis of Knoop indentation.

Cheng et al. [START_REF] Cheng | Spherical-tip indentation of viscoelastic material[END_REF] analyse the indentation of viscoelastic solids by a sphericaltip indenter. Their solutions can apply to the response of compressible as well as incompressible coated layers to a spherical-tip indentation.

In this paper we propose a new algorithm for computing the indentation of a rigid body of arbitrary shape on a viscoelastic half-space. The Matrix Inversion Method (Kalker [START_REF] Kalker | Two algorithms for the contact problem in elastostatics[END_REF]) which is described in the book of Johnson [START_REF] Johnson | Contact Mechanics[END_REF] is used in [START_REF] Yin | A two-scale iterative method for solving the problem of multipunch on an elastic halfspace, Part 1: Numerical comparisons[END_REF][START_REF] Cesbron | A two-scale iterative method for solving the problem of multipunch on an elastic halfspace, Part 2: Experimental comparisons[END_REF] for the analysis of the elastic tyre-road contact. We are extending this method to the viscoelastic problem to compute the pressure distribution history for any load or penetration history.

The paper is structured as follows. The indentation by a rigid indenter of arbitrary shape will be considered first. Then the discretization of the contact problem will be made. Next it will be possible to solve the general contact problem by the Matrix Inversion Method (MIM). The general methodology will be applied to the spherical, conical and cylindrical indenters. The MIM will be compared first to the analytical results for a single spherical indenter (loading and unloading) and secondly to the analytical results for a single conical indenter (loading only). Then the indentation by a rigid cylindrical indenter will be considered. The results will be discussed before concluding remarks.

Algorithm for viscoelastic contacts

The stress-strain relations for an incompressible elastic solid may be written as either:

s ij = 2µ e e ij (1)
and

e ij = 1 2µ e s ij (2)
where µ e is the elastic shear modulus, s ij is the deviatoric components of stress and e ij is the deviatoric components of strain. The corresponding relationships for a linear viscoelastic material can be expressed by the creep and relaxation functions (Volterra equation):

s ij (t) = t 0 G(t -τ ) de ij (τ ) dτ dτ (3)
where G(t) is the relaxation function, which specifies the stress response to a unit step change of strain,

e ij (t) = t 0 J(t -τ ) ds ij (τ ) dτ dτ (4)
where J(t) is the creep function, which specifies the strain response to a unit step change of stress. This approach can be applied if the contact area is increasing.

We consider a rigid indenter of arbitrary shape z(x, y) being pressed into contact with a viscoelastic solid. Under the action of a normal force the penetration of the indenter δ(t) and the contact area will both grow with time and the distribution of contact pressure p(x, y, t) will change. We wish to find the relations which link the pressure distribution, the resultant force F (t) on the indenter and the penetration on the assumption that the surfaces are frictionless. Using Cartesian coordinates, the surface of the viscoelastic half-space before loading corresponds to the (x, y) plane, as illustrated in Fig. 1. The boundary conditions (z = 0) of our problem are given by

               u z (x, y, t) = δ(t) -h(t)|z(x, y)|, (x, y) ∈ Ω(t),
σ yz = σ zz = σ zx = 0, (x, y) ∈ Ω(t).

(

) 5
where u z (x, y, t) is the normal component of the displacement of the point on the surface of the half-space, h(t) is the Heaviside step function, σ ij is the stress tensor, Ω(t) is the current contact area.

The Boussinesq problem [START_REF] Boussinesq | Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques[END_REF], involving the normal displacement of the point (x, y) on the surface of the half-space, due to the normal concentrated force p, for the elastic case may be written as:

u z (x, y) = (1 -ν) π 1 2µ e p ρ , ρ = (x -ξ) 2 + (y -η) 2 (6)
where Poisson's ratio ν has been taken as a constant.

Definition 1: The Laplace transform f (s) of function f (t) (f is a continuous function on t ∈ [0, ∞)
and has exponential order as t → ∞):

f (s) = L{f (t)} = ∞ 0 f (t)e -st dt
Definition 2: The Stieltjes convolution ϕ * dψ of two function ϕ(t) and ψ(t) is defined by:

ϕ * dψ = t -∞ ϕ(t -τ ) dψ(τ ) dτ dτ where ψ(t) → 0, t → -∞ and ϕ(t) is continuous for t ∈ [0, ∞)
Theorem 1 (Convolution theorem):

L{ϕ * dψ} = sL{ϕ}L{ψ} or ϕ * dψ = sϕ ψ
The Laplace transform of viscoelastic stress-strain relation is given by:

u z (x, y, s) = (1 -ν) π sJ(s) p(s) ρ (7) by J(s) = 1 s 2 G .
Using the convolution theorem and generalizing the result to include a distribution of forces, we get:

u z (x, y, t) = (1 -ν) π L -1 sJ(s) p(s) ρ = (1 -ν) π J * d    Ω m p(ξ, η, τ )dξdη ρ    (8)
where Ω m is the maximum contact area (the contact area can "pulsate" when a load is applied and then removed).

Thus, the analogue problem for the viscoelastic case with the boundary condition may be written as:

u z (x, y, t) = (1 -ν) π t 0 J(t -τ ) d dτ    Ω m p(ξ, η, τ )dξdη ρ    dτ = = (1 -ν) π t 0 J(t -τ )    Ω m 1 ρ dp(ξ, η, τ ) dτ dξdη    dτ = δ(t) -h(t)|z(x, y)|, (x, y) ∈ Ω(t) (9)
A surface of the half-space of size L x × L y was meshed using N = n x n y rectangular elements of dimensions dx = Lx nx and dy = Ly ny with centres in (x i , y i ) and with uniform pressures acting on each of them (Fig. 2). The time discretization is τ = n τ dτ . We will represent the equation ( 9) in the form

l k=1 J (l -k)dτ n j=1 T ij p k j -p k-1 j dτ dτ = δ l -z i , i = 1 . . . n (10)
where p k j = p(x j , y j , kdτ ), δ l = δ(ldτ ), z i = z(x i , y i ) and n = n(t) is the number of contact elements for the time t. The influence coefficient T ij , which

expresses the displacement at a general point i due to a unit pressure element centred at point j, was calculated using Love's results [START_REF] Love | Stress produced in a semi-infinite solid by pressure on part of the boundary[END_REF] for a rectangular surface acting on an elastic half-space

T ij = (1 -ν) π y j + dy 2 y j -dy 2 x j + dx 2 x j -dx 2 dξdη (x i -ξ) 2 + (y i -η) 2 (11)
Then equation ( 10) is written in the form

J(0) n j=1 T ij p l j = δ l -z i -v l-1 i , i = 1 . . . n v l-1 i = l-1 k=1 J (l -k)dτ n j=1 T ij p k j -p k-1 j dτ dτ -J(0) n j=1 T ij p l-1 j ( 12
)
Here it is necessary to take note of a correct comparison of p k j with p k-1 j since the number of contact elements is different for k and k -1. For solving this problem one can introduce the matrix T of dimension n m × n m and the vector p of dimension n m (the maximum number of contact elements, corresponding

to Ω m ). Now it is possible to determine p l j for a given history of loadings using the MIM which was programmed following the algorithm described in [START_REF] Johnson | Contact Mechanics[END_REF].

According to this algorithm, after solving equation ( 12), some values of p l j may be negative and for the next iteration these mesh points are excluded from the assumed contact area and the pressure there is taken to be zero.

Here δ l is imposed, but if we add the equation

n j=1 p l j dxdy = F l (13)
it is possible to determine p l j and δ l for a given history of the resultant normal contact force on the indenter F l .

Equations ( 11), ( 12) and ( 13) have been programmed and numerical examples will be presented in the next section and compared with analytical solutions when available.

3 Numerical examples

The indentation of a viscoelastic half-space by a rigid spherical indenter

The MIM was first compared to the analytical results for a single spherical indenter of radius R acting on a viscoelastic half-space. The analytical result of Ting [START_REF] Ting | The contact stresses between a rigid indenter and a viscoelastic halfspace[END_REF] and Graham [START_REF] Graham | The contact problem in the linear theory of viscoelasticity[END_REF] which was used for comparisons is described in the book of Christensen [START_REF] Christensen | Theory of Viscoelasticity. An Introduction[END_REF].

δ(t) = a 2 (t) R - 1 R t tm J(t -θ) d dθ    θ t 1 (θ) G(θ -τ ) da 2 (τ ) dτ dτ    dθ (14) p(x, y, t) = 2 π(1 -ν)R t 1 (t) 0 G(t -τ ) d a 2 (τ ) -x 2 -y 2 dτ dτ (15)
where a(t) is the contact radius. The resultant normal contact force F (t) on the indenter acting on a viscoelastic half-space can be calculated as follows:

F (t) = 2π 0 a(t) 0 rp(r, t)dϕdr = 4 3(1 -ν)R t 1 (t) 0 G(t -τ ) da 3 (τ ) dτ dτ (16)
Here t 1 is a new time variable (Fig. 3)

               t 1 (t) = t, t ≤ t m a(t 1 ) = a(t), t ≥ t m (t 1 < t m ) ( 17
)
t m is the maximum of a(t). So, t 1 is the time prior to t m , whereby the contact radius a(t) (when the contact area is decreasing) is equal to the prior contact radius a(t 1 ) (when the contact area was increasing). Thus, the contact pressure and the contact force depend only upon the variation of contact area prior to

t 1 .
In the following examples the stress history is represented in the form of a Prony's series:

σ(t) = E ∞ + (E 0 -E ∞ )e -t τc ε 0 (18)
where τ c is the characteristic time, E 0 is the instantaneous Young's modulus, E ∞ is the statical Young's modulus, ε 0 is the instantaneous strain. Then the creep and the relaxation functions may be written as follows:

                           G(t) = E ∞ + (E 0 -E ∞ )e -t τc , J(t) = 1 E ∞ - (E 0 -E ∞ ) E 0 E ∞ e -E∞ E 0 t τc (19)
Now for the numerical calculations we take the following parameter values E ∞ = 7.5 MPa, E 0 = 15 MPa, τ c = 1 s and R = 5 mm. The contact radius a(t) = 2.2 sin( πt Lt ) is imposed with the duration of loading L t = 5 s, so that the penetration δ(t) varies between 0 and 1 mm. A comparison between the contact forces computed by the MIM with n τ = 201, n x = n y = 81, L t = 5 s, L x = L y = 20 mm and the analytical formula is made in Fig. 4.

The difference between analytical and numerical solutions is small, showing a very good agreement. One can see the asymmetry between increasing and decreasing loadings of the viscoelastic curve. At the beginning both solutions are close to the elastic solution with Young's modulus E 0 and at the end are close to the elastic solution with Young's modulus E ∞ . Fig. 5 shows an example of a strongly time dependent material behaviour with the parameters E ∞ = 0.9 MPa, E 0 = 9 MPa, τ c = 0. s are chosen and will not be changed. Now the penetration δ(t) = sin( πt L t ) is imposed. Again, at the beginning the numerical solution is close to the elastic solution with Young's modulus E 0 (Fig. 6) and at the end is close to the elastic solution with Young's modulus E ∞ . The reason of intersection of viscoelastic and elastic solutions is the memory of the viscoelastic material and for the same reason, the contact force vanishes already while the penetration does not yet equal zero.

The distribution of the pressure p(0, y, t) for t = 2.5 s is represented in Fig. 7 with n τ = 401, n x = n y = 101, L t = 5 s, L x = L y = 10 mm. The differ-ence between the viscoelastic analytical and viscoelastic numerical solutions p(0, y, t) is small showing a good agreement.

In Fig. 8 the dependence of the force versus the penetration δ(t) = V t for different values of loading velocity V = 1, 10, 1000 mm/s and n τ = 401, n x = n y = 81, L t = 1 s, L x = L y = 20 mm is illustrated. The difference between the graphics which correspond to V = 10 mm/s and V = 1000 mm/s is small while the difference between the graphics which correspond to V = 1 mm/s and V = 10 mm/s is much larger. The reason is the characteristic time of material τ c which equals 1 s. Hence for V = 1 mm/s the loading time

t = δ V = 1 s = τ c .
On the contrary for V = 10 mm/s and V = 1000 mm/s the loading time and the characteristic time of the material are very different.

In addition, the iterative method converges, as shown in Fig. 9 3.2 The indentation of a viscoelastic half-space by a rigid conical indenter

The MIM was secondly compared to the analytical result for a single conical indenter of semi-opening angle θ acting on a viscoelastic half-space when a contact radius increases monotonically. The elastic solution of Love [START_REF] Love | Boussinesq's problem for a rigid cone[END_REF] (and later Vilke [START_REF] Vilke | On Non-Hertzian Contact of Wheel and Rail. Research on Problems of Stability and Stabilization of Motion[END_REF]) may be written:

F = 4 π tgθ2µ e δ 2 (20)
Following Radok's suggestion, we rewrite equation [START_REF] Love | Stress produced in a semi-infinite solid by pressure on part of the boundary[END_REF], replacing the elastic constant in the elastic solution by the corresponding integral operator:

F (t) = 4 π tgθ t 0 G(t -τ ) dδ 2 (τ ) dτ dτ (21)
A comparison between the contact forces computed by the MIM with n τ = 401, n x = n y = 201, L t = 5 s, L x = L y = 20 mm, θ = 45 • and the analytical formula ( 21) is shown in Fig. 10. This difference between the contact forces is small, showing a very good agreement. Here the penetration δ(t) = sin( πt Lt ) is imposed and we have the same effect as previously for the spherical case: at the beginning the numerical solution is close to the elastic solution with Young's modulus E 0 and at the end is close to the elastic solution with Young's modulus E ∞ .

The distribution of the pressure p(0, y, t) for t = 2.5 s is represented in Fig. 11 with n τ = 401, n x = n y = 181, L t = 5 s, L x = L y = 7 mm. The maximum pressure is concentrated in the centre of contact area. The absolute value of pressure is much higher than for the spherical case but the contact area is much less. The elastic solutions p(0, y, t) with Young's modulus E 0 and E ∞ are also plotted as a reference.

The indentation of a viscoelastic half-space by a rigid cylindrical indenter

Now a cylindrical indenter of radius R = 5 mm and of width l = 3 mm is considered. The penetration δ(t) = sin( πt Lt ) is imposed. The distribution of the pressure for t = 2.5 s is represented in Fig. 12 with n τ = 401, n x = n y = 101, L t = 5 s, L x = L y = 10 mm. The maximum pressure is concentrated at the boundaries of the contact area (the axis y is the axis of symmetry of the cylinder). The absolute value of pressure is much higher than for the spherical case but less than for the conical case. In Fig. 13 the difference between the viscoelastic solution for t = t 1 = 1.25 s (increasing load) and viscoelastic solution for t = t 2 = 3.05 s (decreasing load) is displayed. The the analytical results for a single spherical indenter acting on a viscoelastic half-space. Then the comparison was made in case of a single conical indenter when the contact radius increases monotonically. The contact force and the pressure distribution computed by the MIM agree well with the analytical solutions. The contact force as a function of time, the pressure distribution, the contact area are found for a single spherical, conical and cylindrical indenter. In all cases the viscoelastic solutions have been compared to the elastic solutions. One can see that viscoelastic curves of contact force versus penetration are different for increasing and decreasing load. Also the influence of the loading velocity on the results has been studied. It is very interesting to notice that when the increasing and decreasing loads almost equal, the corresponding distributions of the pressure are different. The new algorithm allows estimating the influence of the viscoelasticity of materials on the contact. It can for instance be used for tyre-road contact computations. The contact between a rigid indenter of arbitrary shape and a viscoelastic half-space 21

2
The discretization of the contact problem by rectangular elements 22

, ------viscoelastic numerical, • •• •• •• •• •• • elastic with E ∞ = 7.5 MPa, ••••••••••••• elastic with E 0 =
viscoelastic numerical, • •• •• •• •• •• • elastic with E ∞ = 7.5 MPa, ••••••••••••• elastic with E 0 =
•• •• •• •• •• • elastic with E ∞ = 7.5 MPa, ••••••••••••• elastic with E 0 = 15

  3 s and n τ = 601, n x = n y = 81, L t = 5 s, L x = L y = 20 mm. The normal contact force on the indenter equals zero at t = 5 s even if the penetration does not yet equal zero. One can see also the asymmetry of the curve and the rapid change of the slope. For the next examples the parameters E ∞ = 7.5 MPa, E 0 = 15 MPa, τ c = 1

3 A new time variable t 1 23 4

 23 Spherical case. The contact radius a(t) is imposed (a) The difference between the contact forces, --viscoelastic analytical

24 5

 24 15 MPa (b) The penetration δ(t) Spherical case. The contact radius a(t) is imposed. Strongly time dependent material (a) The difference between the contact forces, --viscoelastic analytical, ------viscoelastic numerical (b) The penetration δ(t) 25 6 Spherical case. The penetration δ(t) is imposed (a) The difference between the contact forces, --viscoelastic numerical, ------elastic with E ∞ = 7.5 MPa, • •• •• •• •• •• • elastic with E 0 = 15 MPa (b) The penetration δ(t) 26 7 Spherical case. The pressure distribution for t = 2.5 s and x = 0 mm, --viscoelastic analytical, ------viscoelastic numerical 27 Spherical case. The dependence of force versus the penetration δ(t) for different values of loading velocity, --V = 1 mm/s, ------V = 10 mm/s, • •• •• •• •• •• • V = 1000 mm/s 28 9 Spherical case. Convergence of the algorithm 29 10 Conical case. The penetration δ(t) is imposed. The difference between the contact forces, --viscoelastic analytical, ------

Fig. 1 .Fig. 2 .

 12 Fig. 1. The contact between a rigid indenter of arbitrary shape and a viscoelastic half-space

Fig. 4 .

 4 Fig. 4. Spherical case. The contact radius a(t) is imposed (a) The difference between the contact forces, --viscoelastic analytical, ------viscoelastic numerical, • •• •• •• •• •• •

  MPa (b) The penetration δ(t)

Fig. 5 .

 5 Fig. 5. Spherical case. The contact radius a(t) is imposed. Strongly time dependent material (a) The difference between the contact forces, --viscoelastic analytical, ------viscoelastic numerical (b) The penetration δ(t)

Fig. 6 .Fig. 9 .Fig. 10 .Fig. 11 .

 691011 Fig.6. Spherical case. The penetration δ(t) is imposed (a) The difference between the contact forces, --viscoelastic numerical, ------elastic with E ∞ = 7.5 MPa,• •• •• •• •• •• • elastic with E 0 = 15 MPa (b) The penetration δ(t)

Fig. 12 .

 12 Fig. 12. Cylindrical case. The pressure distribution for t = 2.5 s

Fig. 13 .Fig. 14 .Fig. 15 .

 131415 Fig. 13. Cylindrical case. The pressure distribution for t 1 = 1.25 s and t 2 = 3.05 s (a) y = 0 mm, --viscoelastic for t 1 , ------viscoelastic for t 2 (b) x = 0 mm, --viscoelastic for t 1 , ------viscoelastic for t 2

This work is supported by ADEME in the frame of the French and German Cooperation P2RN project.

pressure distributions p(x, 0, t 1 ) and p(x, 0, t 2 ) are represented in Fig. 13 (a) and the pressure distributions p(0, y, t 1 ) and p(0, y, t 2 ) are represented in Fig.

13 (b). It is very interesting to notice that the resultant normal contact forces on the indenter almost equal F (t 1 ) ≈ F (t 2 ) (Fig. 14), while the distribution of the pressure for t = t 1 and t = t 2 are different i.e. p(x, y, t 1 ) = p(x, y, t 2 ).

In Fig. 15 the dependence of force versus the penetration for two different values of velocity of loading V = 1 mm/s, V = 10 mm/s and n τ = 401,

The penetration is imposed as follows:

In this case the penetration δ(t) increases monotonically to 1 mm and then decreases monotonically to 0 mm. As previously, for the spherical case, the difference between the two curves is large. The reason is the same : the characteristic time of the material τ c equals 1 s. Also here hysteresis is observed in the force-penetration curve as viscoelastic substance loses energy when a load is increasing, then decreasing.

Conclusions

The Matrix Inversion Method (MIM) introduced by Kalker and used for the analysis of elastic contact, has been extended to the equivalent viscoelastic problem so that the problem can be solved for different loading histories and for indenters of arbitrary shape. The numerical results were first compared to