Yiyi Xu
email: yiyi.xu.16@neoma-bs.com

M'hammed Sahnoun

Fouad Ben Abdelaziz
email: fouad.ben.abdelaziz@neoma-bs.com

David Baudry
email: dbaudry@cesi.fr

Anne Louis
email: alouis@cesi.fr

Multi-objective Flexible Job Shop Scheduling Transportation Problem: Simulation Approach

Keywords: Multi-objective Programming, Flexible Job Shop Scheduling, Simulation

Introduction

Influenced by a capitalized concept, "maximum utility with minimum cost", more and more industries in manufacturing area have focused on flexibility and adopted themselves to flexible job shop environments. In traditional Job Shop Scheduling Problem (JSP), there are n jobs and m machines in manufacturing system, where each job consists of j operations that need to be executed on machines by a given order. [START_REF] Kacem | Pareto-optimality approach for flexible job-shop scheduling problems: hybridization of evolutionary algorithms and fuzzy logic[END_REF]. It extends the assumption that only one machine is able to run a particular operation. Since JSP can be considered as a special case of FJSP and JSP is well-known NP-hard, FJSP is also regarded as an NP-hard problem.

Considering its computational complexity, various metaheuristic have been extensively applied to FJSP. Zendieh et al. [START_REF] Zandieh | Solving the flexible job-shop scheduling problem by a genetic algorithm[END_REF] proposed a Genetic Algorithm for FJSP by using several different rules for generating the initial population and several strategies for producing new population for next generation. Gao et al. [START_REF] Gao | Pareto-based grouping discrete harmony search algorithm for multi-objective flexible job shop scheduling[END_REF] introduced a Pareto-based grouping discrete harmony search algorithm (PGDHS) to solve FJSP. Chamber et al. [START_REF] Chambers | Tabu search for the flexible-routing job shop problem[END_REF] extended their Tabu Search strategy previously described for job shops and applied it to FJSP. Besides metaheuristic algorithm, researchers also employ traditional polynomial algorithms and hybrid algorithms, like GA combined with a variable neighborhood descent [START_REF] Gao | Discrete harmony search algorithm for flexible job shop scheduling problem with multiple objectives[END_REF], in FJSP.

The objective of Flexible Job Shop Scheduling Problem is to determine a feasible schedule S by minimizing a given objective function [START_REF] Thörnblad | A competitive iterative procedure using a time-indexed model for solving flexible job shop scheduling problems[END_REF]. In early work, the wide-used objective is from a single dimension, considering only one objective once. Recently, researchers have addressed their study on multiple objectives. Karthikeyan et al. [START_REF] Karthikeyan | A hybrid discrete firefly algorithm for solving multi-objective flexible job shop scheduling problems[END_REF] introduced a hybrid discrete firefly algorithm (HDFA) in which the objectives include the minimization of makespan, maximal workload and total workload of machines. Gao et al. [START_REF] Gao | Discrete harmony search algorithm for flexible job shop scheduling problem with multiple objectives[END_REF] proposed a Discrete Harmony Search algorithm for FJS with weighted combination of makespan, the mean of earliness and tardiness criteria.

In this paper, we propose a multi-objective dynamic scheduling algorithm based on entropy to optimize the transportation task in a flexible job shop manufacturing environment. The system is composed of machines, input/output buffers, transporters and traceable products, where the machines are not identical and have ability to execute more than one operation. A simulation based on multiagent approach is used in order to evaluate proposed solutions regarding multiple objectives, such as makespan, routing flexibility. For programming, we use Netlogo 6.0.1 to code the simulation model with real production data from a Normandy company in France. Netlogo is a multi-agent programmable modeling environment, which is able to present the layout of system and all results directly to industrial users. In previous study, part of research study have used Matlab (Karthikeyan et al. [START_REF] Karthikeyan | A hybrid discrete firefly algorithm for solving multi-objective flexible job shop scheduling problems[END_REF]) or C ++ (Zadieh et al. [START_REF] Zandieh | Solving the flexible job-shop scheduling problem by a genetic algorithm[END_REF]) to test their algorithms in case environment. However, few of them derives their algorithms from simulation of cases. This study is up to fill this gap and aims to offer a more intuitive way for industrial users who may not have programming background [START_REF] Sahnoun | Modelling and simulation of operation and maintenance strategy for offshore wind farms based on multi-agent system[END_REF].

Simulation Model

Multiple-agent modelling

For simulating different scenarios of a flexible manufacturing systems in Netlogo, we developed a multi-agent model composed of 4 interdependent parts (agents): product, transporter, machine and stock (FIG. 1). In current 4 machines 3 jobs flexible job shop manufacturing system model (FIG. 2), there are three types of products. Each type has its own list of operations in a given order. The agent property offers the opportunity to trace each product during whole production.

The function of agent transporter is more than to transfer products between different stocks. In this model, once each transporter will pick one product from stocks separately under our dynamic scheduling algorithm.

In flexible job shop environments, each machine can run more than one operation. Users are able to control the running time of different machine on the same operation with sliders.

Agent stock is capable of recognizing each product in it and the input-stock has a parameter to provide "waiting time" of correspond machine, as one of criteria, for system to do dynamic scheduling.

FIG. 1 -Multi-agent: agents and interactions

Simulation

The model built so far is a base for our future study. It already stimulates all steps during production, such as selection of products under dynamic scheduling algorithm, transportation and monitoring. The plots in FIG. 2 shows various indexes of different agents. For example, machine balance in the upper-right corner takes total working ticks of each machine into account, which presents not only the workload of each machine, but also the difference among all machines.

Dynamic scheduling algorithm

During production, each transporter needs to take decisions separately concerning start point and end point of transferring a product. The dynamic algorithm we propose in this paper, which is developed from a previous study in the LINEACT laboratory [START_REF] Eloundou | Modélisation multi-contraintes dún système de production flexible[END_REF] [10], is based on the entropy concept to classify the rules of selection. In previous study [START_REF] Sahnoun | Simulation-based optimization of sampling plans to reduce inspections while mastering the risk exposure in semiconductor manufacturing[END_REF] other use of simulation were proposed to optimize manufacturing system. The most important problem in this approach is that the decisions given by the algorithm, however, are always trapped to local solutions because the transporters lack of a global vision of the system. In order to overcome this problem we will propose an algorithm to solve dynamic scheduling problem in order to minimize all objectives at the same time.

For selecting one product from all in stocks, the first step of the algorithm is to build a decision matrix (FIG. 3) basing on all ready products. The matrix includes the information of each product, such as product ID, and several indexes, like makespan, showing the prediction results of transferring this product. All these indexes are our objectives, hence we are able to consider all objectives simultaneously at each time of selection. On the next step, we normalize all values and get entropy of each of them, Hi,j. Then, all objective columns are sorted by sum of column values, which means the column with max ∑ 𝐻 𝑖,𝑗 𝐼 𝑖=0 lists on the first left position. Each selection may have different priority sequence of objectives, showing the superiority of the algorithm. After this, the matrix will screen out products by each objective criterion until the best product is picked up. Once a products is picked by one of transporter, even it's still in the stock, the other transporter would not consider it into decision matrix any more.

Findings

In current model, we only consider two criteria in decision matrix, makespan and distance, which are two most widely-used objectives in FJSP. Still, the model shows its competitiveness when we compare our proposed model with FIFO (First In First Out) algorithm model and No Priority Decision Matrix algorithm model.

Part of comparison of results is shown in FIG. 4. From the tests we have done, we can find that: firstly, compared to other two algorithms, the average improvement of makespan is 2% when we applied our proposed model in the same condition and the performance of distance objective is always better. Secondly, if we add more transporters when producing same quantity of product, the objective makespan would be better under the same algorithm. While the distance that all transporters passed would increase a bit since more transporters work in.

FIG. 4 -Results Comparison

The comparisons above prove our proposed competitiveness roughly. We believe that if we consider more criteria like routing flexibility in decision matrix, the results would be even much better. After we put all objectives into decision matrix, we will do benchmarking to compare our model with other well-known algorithms like GA or Tabu search and do significance test to illustrate the effectiveness of our simulation approach.

Conclusions and perspectives

Compared to other algorithms, simulation algorithm is able to deal with arbitrary scenarios and monitor all elements in the system simultaneously. In existing literature, few researchers combined routing flexibility with FJSP, even regarding routing flexibility as an objective. In this paper, we introduce a novel simulation approach to solve flexible job shop scheduling problem. Besides, we also propose an algorithm to compute routing flexibility and achieve its optimization. The proposed method of computation of routing flexibility can be extended to other job shop environments.

Acknowledgement: Acknowledgement is made to the Normandy region and the European Union for the support of this research through the European FEDER program by funding project entitled Xterm.


Carmen Argmann

Radu Dobrin

Sami ¤a

Heikkinen

Aure ¤b

´lie Auburtin

Laurent Pouilly

Terrie-Anne Cock

Hana Koutnikova

Jun Zhu

Eric E Schadt

Johan Auwerx

Pparc2 Is a Key Driver of Longevity in the Mouse

Aging involves a progressive physiological remodeling that is controlled by both genetic and environmental factors. Many of these factors impact also on white adipose tissue (WAT), which has been shown to be a determinant of lifespan. Interrogating a transcriptional network for predicted causal regulatory interactions in a collection of mouse WAT from F2 crosses with a seed set of 60 known longevity genes, we identified a novel transcriptional subnetwork of 742 genes which represent thus-far-unknown longevity genes. Within this subnetwork, one gene was Pparg (Nr1c3), an adipose-enriched nuclear receptor previously not associated with longevity. In silico, both the PPAR signaling pathway and the transcriptional signature of Pparc agonist rosiglitazone overlapped with the longevity subnetwork, while in vivo, lowered expression of Pparg reduced lifespan in both the lipodystrophic Pparg1/2-hypomorphic and the Pparg2-deficient mice. These results establish Pparc2 as one of the determinants of longevity and suggest that lifespan may be rather determined by a purposeful genetic program than a random process.

Introduction

Aging is not a disease, but a natural evolution characterized by declining biological function, whose timeline is sensitive to both environmental and genetic factors. Several longevity candidate genes have been identified, including the insulin/IGF1 signaling pathway [START_REF] Selman | Evidence for lifespan extension and delayed age-related biomarkers in insulin receptor substrate 1 null mice[END_REF][START_REF] Taguchi | Brain IRS2 signaling coordinates life span and nutrient homeostasis[END_REF][START_REF] Kenyon | The plasticity of aging: insights from long-lived mutants[END_REF]. With the use of dietary regimens, such as caloric restriction (CR) and by modulating core body temperature, the control of energy metabolism has been implicated as a critical determinant of the aging phenotype [START_REF] Bishop | Genetic links between diet and lifespan: shared mechanisms from yeast to humans[END_REF][START_REF] Wolf | Calorie restriction increases life span: a molecular mechanism[END_REF][START_REF] Conti | Transgenic mice with a reduced core body temperature have an increased life span[END_REF]. A central physiological component of energy metabolism, involved in energy preservation, is the white adipose tissue (WAT), which has also been directly associated with the determination of lifespan [START_REF] Blu ¨her | Extended longevity in mice lacking the insulin receptor in adipose tissue[END_REF][START_REF] Moitra | Life without white fat: a transgenic mouse[END_REF]. However, it is still uncertain whether WAT modulates aging via its ability to e.g. store fat, sensitize towards insulin, or produce adipocyte hormones. Also unknown is the nature of the involved genetic players and importantly, whether they function in a purposeful program or as random genetic events.

Using a systems approach we identified a novel subnetwork of genes in mouse WAT, which potentially impacts longevity, suggesting that aging is the result of a determined transcriptional network program and not entirely accidental. Furthermore, the most significantly enriched biological pathway revealed within this aging subnetwork was the PPAR signaling pathway. The aging subnetwork also contained the nuclear receptor Pparg (Nr1c3), a transcription factor well associated with adipocyte biology [START_REF] Knouff | Peroxisome proliferator-activated receptor-gamma calls for activation in moderation: lessons from genetics and pharmacology[END_REF][START_REF] Chawla | Peroxisome proliferator-activated receptor (PPAR) gamma: adipose-predominant expression and induction early in adipocyte differentiation[END_REF], but whose contribution to longevity has not been previously assessed. In this study, we support our network theory of aging by demonstrating a significantly altered lifespan in 2 independent genetic mouse models expressing reduced levels of Pparg. Thus, in addition to providing novel candidate 'longevity genes' such as Pparg2, this study also provides further insight into the potential role of WAT biology and genetics as determinants of lifespan.

Results/Discussion

We hypothesized that the age-dependent physiological remodeling that leads to phenotypic aging is caused by concerted changes in a longevity-determining genetic network rather than by random changes at the level of individual genes. This hypothesis was tested using a mouse transcriptional network that consists of a union of 4 individual Bayesian networks of predicted causal regulatory interactions in the WAT generated from individual F2 crosses. We interrogated this network of 13088 genes with a seed set of 60 genes, derived from public resources, which either increase or reduce lifespan when genetically perturbed in the mouse (Table S1). Out of these 60 'known' longevity genes, 33 were also present within the adipose tissue network (Table S1; Figure 1A). The pair-wise shortest path analysis against 10 6 randomly selected sets of 33 genes showed that these 33 genes on average were much more tightly connected than expected by chance (p = 0.00149) (Figure 1B). Furthermore, the distribution of the shortest paths within the set of 33 'known' longevity genes was significantly tighter than that for the randomly selected sets as .99% of all Kolmogorov-Smirnov two-sided test p-values were less than 0.05. This tight, non-random interconnection of known aging-linked genes suggests that the associated biological phenomena are deliberate such that other 'unknown' age-related genes and/or biological processes may be predicted. This network theory is reminiscent of the transcriptional consequences of single genetic perturbations, such as knock-out mouse models or DNA polymorphisms, which result in concentrations of transcriptional changes in the genes functionally relating to the perturbed gene rather than altering genes diffusely distributed across the whole network [START_REF] Chen | Variations in DNA elucidate molecular networks that cause disease[END_REF]. Following the concept of using the 'known' to discover the 'unknown', we thus expanded the subnetwork beyond the 33 longevity genes to other genes most highly connected to them, and obtained a larger subnetwork, containing 742 genes (Table 1, Table S2). By assigning importance to the closeness of connection with known longevity genes, we were thus able to suggest several hundred additional genes that may influence the aging process. One such example, among the top 20 genes for the closeness of connectivity with the 33 'known' longevity genes (Table 1), was the eukaryotic translation initiation factor 4E (eIF4E) binding protein 1 (Eif4ebp1, or 4E-BP1) which, in the unphosphorylated state, represses mRNA translation by binding to eIF4E. Since it regulates adipogenesis and metabolism [START_REF] Tsukiyama-Kohara | Adipose tissue reduction in mice lacking the translational inhibitor 4E-BP1[END_REF], and one of the mediators of its phosphorylation is insulin signaling

Author Summary

The progression of aging is controlled by both genetic and environmental factors. Many of these factors are present also in adipose tissue, which itself has been shown to determine lifespan. Applying advanced bioinformatics methods on a large mouse gene expression data set, we identified Pparg (Nr1c3), an important metabolic controller that regulates the expression of many other genes particularly in adipose tissue, to be associated with longevity. This association was verified in experimental mouse models where the lowered expression of Pparg reduced lifespan. In addition to Pparg, our analysis identified .700 potential novel aging genes in mouse adipose tissue. More generally, these findings suggest that lifespan may not be a random process but controlled by a purposeful genetic program.

Figure 1. A subnetwork of likely longevity genes in mouse adipose tissue. (A) Longevity-related subnetwork of 775 genes, extracted from the mouse adipose transcriptional network of 13,088 genes. The 33 ''known'' longevity genes used as a seed set are depicted as green diamonds, and the 213 genes overlapping from the mouse WAT rosiglitazone signature in blue circles. The 5 gene overlap of ''known'' longevity genes and rosiglitazone signature is shown as blue diamonds. Pparg, shown as a red circle, is part of the rosiglitazone signature. (B) The distribution of mean shortest path lengths (m) for the set of 33 ''known'' longevity genes and 10 6 randomly selected sets of 33 genes within the mouse consensus network. Red arrow marks the mean shortest path (m = 6.7102) for the ''known'' longevity genes. doi:10.1371/journal.pgen.1000752.g001 [START_REF] Lin | PHAS-I as a link between mitogen-activated protein kinase and translation initiation[END_REF], Eif4ebp1 can be linked to the established effects of insulin signaling on longevity. Moreover, in Drosophila 4E-BP plays an important role in lifespan extension upon dietary restriction [START_REF] Zid | 4E-BP extends lifespan upon dietary restriction by enhancing mitochondrial activity in Drosophila[END_REF]. Eif4ebp1 has furthermore been identified as a ''funnel factor'' in cancer, through which several oncogenic pathways converge [START_REF] Armengol | 4Ebinding protein 1: a key molecular ''funnel factor'' in human cancer with clinical implications[END_REF].

Biological pathway enrichment analysis is a powerful tool to uncover functional associations within an a priori selected set of genes. When applied to the aging subnetwork of 742 genes (excluding the 33 'known' longevity genes from the full set of 775 genes to eliminate bias), significant enrichment was revealed in several ontology classes with established links to aging such as complement and coagulation cascade (i.e. inflammation), insulin signaling, and ubiquinone pathway (i.e. oxidative stress) (Table 2). Importantly, however, several pathways lacking previously demonstrated association with longevity also appeared among the significantly enriched ontologies. One of these, the PPAR signaling pathway, was actually ranked the highest for the enrichment of all potential longevity genes. Although direct in vivo evidence linking Ppars to aging are scarce, conceptual evidence does exist [START_REF] Chung | Molecular mechanism of PPAR in the regulation of age-related inflammation[END_REF][START_REF] Erol | The Functions of PPARs in Aging and Longevity[END_REF], including links to age-related changes in inflammatory response, insulin sensitivity, distribution and proportion of body fat, oxidative stress [START_REF] Luo | Adipose tissue-specific PPARgamma deficiency increases resistance to oxidative stress[END_REF], and fatty acid oxidation rate. Of the three actual Ppar family members, the only one that was present within the aging subnetwork was Pparc (Table S2). Given that signaling through Pparc is also of vital importance to proper adipose tissue development and function [START_REF] Knouff | Peroxisome proliferator-activated receptor-gamma calls for activation in moderation: lessons from genetics and pharmacology[END_REF][START_REF] Chawla | Peroxisome proliferator-activated receptor (PPAR) gamma: adipose-predominant expression and induction early in adipocyte differentiation[END_REF][START_REF] Koutnikova | Compensation by the muscle limits the metabolic consequences of lipodystrophy in PPAR gamma hypomorphic mice[END_REF][START_REF] Jones | Deletion of PPARgamma in adipose tissues of mice protects against high fat diet-induced obesity and insulin resistance[END_REF][START_REF] He | Adipose-specific peroxisome proliferator-activated receptor gamma knockout causes insulin resistance in fat and liver but not in muscle[END_REF], and that Pparc is regulated in WAT by one of the best established longevity determinants, mammalian SIR2 orthologue sirtuin 1 (Sirt1) [START_REF] Picard | Sirt1 promotes fat mobilization in white adipocytes by repressing PPAR-gamma[END_REF], we hypothesized that perturbing Pparc signaling might affect longevity.

We first tested this hypothesis in silico by using the WAT gene expression signature generated from mice with chemically modulated Pparc activity through the administration of the Pparc agonist, rosiglitazone [START_REF] Chen | Variations in DNA elucidate molecular networks that cause disease[END_REF]. Notably, 213 out of the 1669 genes whose transcriptional expression was altered by Pparc activation, overlapped with the genes in the aging subnetwork at a very high significance level (p = 5.2028*10 230 ) (Table S2). This finding thus validates the association of Pparg with the aging subnetwork and further implicates it as a potential determinant of the aging phenotype.

To put this hypothesis to further rigorous in vivo testing, we investigated the role of Pparg in longevity in two mouse models with genetically altered levels of Pparg expression: the hypomorphic Pparg1/2 knock-out mouse, which lacks Pparg exclusively in WAT (Figure S1A) and is severely lipodystrophic and remains insulin resistant throughout life [START_REF] Koutnikova | Compensation by the muscle limits the metabolic consequences of lipodystrophy in PPAR gamma hypomorphic mice[END_REF]; and the Pparg2 deficient mouse that lacks Pparc2 in all tissues (Figure S1B) and shows some features of moderate lipodystrophy and insulin resistance at a young age [START_REF] Zhang | Selective disruption of PPARgamma 2 impairs the development of adipose tissue and insulin sensitivity[END_REF], but which fully compensates upon aging (see below). The nearly complete knockdown of Pparg1 and Pparg2 in the WAT of male Pparg hyp/hyp mice resulted in a reduction in lifespan by approximately 16 weeks when compared to the wild type mice (93.764.4 vs 109.663.4 weeks, p = 0.03) (Figure 2A). In some respects this observation goes against the prediction that reduced fat mass, as seen during CR [START_REF] Bishop | Genetic links between diet and lifespan: shared mechanisms from yeast to humans[END_REF][START_REF] Wolf | Calorie restriction increases life span: a molecular mechanism[END_REF], would increase longevity; however, if the known insulin sensitizing effects of Pparc were key to mediating the effects of CR, then one would expect reduced longevity in the Pparg hyp/hyp mice, where whole body insulin resistance is prominent. However, one potentially confounding factor in this experiment is the profound lipodystrophy exhibited by the Pparg hyp/hyp mice, which may not represent 'normal' metabolic environment due to the amount of metabolic compensation by the upregulation of other signaling pathways that these mice need for survival [START_REF] Koutnikova | Compensation by the muscle limits the metabolic consequences of lipodystrophy in PPAR gamma hypomorphic mice[END_REF]. Also, although differences in the amount of gross Table 1. Top 20 genes most highly connected to the set of 33 ''known'' longevity genes in male mouse adipose tissue. tumors were not observed upon macroscopic necropsy, we can not exclude the possible contribution of more discrete tumors to the decreased longevity of the Pparg hyp/hyp mice. Interestingly though, the males of an equally lipodystrophic A-ZIP/F-1 mouse model have more than 40% mortality rate before 30 weeks of age [START_REF] Moitra | Life without white fat: a transgenic mouse[END_REF], in comparison to the Pparg hyp/hyp mice which survived 85% of the average ,2 year lifespan of wild type mice. In this sense, Pparg hyp/hyp mouse model is one of the longest living severely lipodystrophic models reported.

In order to assess more directly the effects of Pparg on longevity, without the added complication of reduced adiposity or insulin sensitivity, we made use of Pparg2 2/2 mice that we generated in the laboratory and which lack Pparc2, the WAT enriched Pparc isoform, in all tissues. Although young Pparg2 2/2 mice are lean [START_REF] Zhang | Selective disruption of PPARgamma 2 impairs the development of adipose tissue and insulin sensitivity[END_REF], our ,2 year old Pparg2 2/2 mice had the same total and lean body mass, body fat content (Figure S2A and S2B), and caloric intake (12.3361.53 vs. 14.2461.53 kcal/day/mouse, p = 0.421) as their age-matched littermate controls. Young Pparg2 2/2 mice have also been reported to be insulin resistant [START_REF] Zhang | Selective disruption of PPARgamma 2 impairs the development of adipose tissue and insulin sensitivity[END_REF]. Again in contrast, there were no differences in glucose tolerance, the HOMA index for insulin resistance, nor in circulating insulin or adiponectin levels between our Pparg2 2/2 and Pparg2 +/+ mice at ,2 years of age (Figure S2C, S2D, S2E, S2F). Thus, our aging Pparg2 2/2 mice represent a very metabolically 'clean' model for investigating the role of Pparg2 in longevity.

Consistent with reduced longevity in the Pparg hyp/hyp mouse, we noted a significant decrease in lifespan in Pparg2 2/2 mice. The female Pparg2 2/2 mice lived, on average, 8.8 weeks less than their wild type controls (p = 0.02 when limiting the analysis to those living no more than 120 weeks), although this difference seemed to disappear towards extreme age (Figure 2B). Gross morphological differences that could contribute to mortality were not observed between the genotype groups, although again the contribution of more discrete tumors can not be excluded. Since the Pparg2 2/2 mice had reduced longevity, comparable to that in Pparg hyp/hyp mice, but were not lipodystrophic or insulin resistant, our observations point more towards a specific role for Pparc2 and any of its downstream pathways in the regulation of longevity, rather than mere changes in fat content and/or insulin signaling.

Together our studies thus reveal another genetic factor, Pparg2, that affects the basic mechanisms of aging, independent of changes in fat mass or insulin sensitivity [START_REF] Selman | Evidence for lifespan extension and delayed age-related biomarkers in insulin receptor substrate 1 null mice[END_REF][START_REF] Taguchi | Brain IRS2 signaling coordinates life span and nutrient homeostasis[END_REF][START_REF] Blu ¨her | Extended longevity in mice lacking the insulin receptor in adipose tissue[END_REF]. Interestingly, a potential molecular mechanism linking aging and Pparc has recently been suggested to involve a steroid receptor coactivator-1 (SRC-1) as the age-induced loss of PPARc/SRC-1 interactions increased the binding of PPARc to the promoter of a model adipogenic gene for fatty acid binding protein 4 (FABP4, also called aP2) [START_REF] Miard | Aging alters PPARgamma in rodent and human adipose tissue by modulating the balance in steroid receptor coactivator-1[END_REF]. Both our in silico and in vivo results in the mouse tie longevity tightly together with signaling through Pparc, and especially the Pparc2 isoform. We have recently shown increased longevity in knock-in mice carrying the Ala12 allele of the common human genetic variant Pro12Ala variant of PPARG2 [START_REF] Heikkinen | The Pro12Ala PPARgamma2 variant determines metabolism at the geneenvironment interface[END_REF], which associates with leanness and improved insulin sensitivity in both man and mouse [START_REF] Heikkinen | The Pro12Ala PPARgamma2 variant determines metabolism at the geneenvironment interface[END_REF][START_REF] Altshuler | The common PPARgamma Pro12Ala polymorphism is associated with decreased risk of type 2 diabetes[END_REF][START_REF] Deeb | A Pro12Ala substitution in PPARgamma2 associated with decreased receptor activity, lower body mass index and improved insulin sensitivity[END_REF]. The species gap between mice and humans for the role of Pparc2 in longevity is bridged by the observation that lifespan is increased also in human carriers of the Ala12 allele of the Pro12Ala variant of PPARG2 [START_REF] Barbieri | Gender specific association of genetic variation in peroxisome proliferator-activated receptor (PPAR)gamma-2 with longevity[END_REF]. In the clinical setting, therefore, the links we show between longevity and both Pparg and the rosiglitazone signature suggest that thiazolidinediones [START_REF] Lehmann | An antidiabetic thiazolidinedione is a high affinity ligand for peroxisome proliferator-activated receptor gamma (PPAR gamma)[END_REF] (TZDs), like rosiglitazone or pioglitazone which are widely used Pparc agonists and insulin sensitizers in the treatment of type 2 diabetes mellitus (T2D), could be beneficial for longevity. On the face of it, this may in fact seem paradoxical, considering that impaired insulin signaling through insulin receptor or its substrates increases, rather than decreases lifespan in a number of mouse models [START_REF] Selman | Evidence for lifespan extension and delayed age-related biomarkers in insulin receptor substrate 1 null mice[END_REF][START_REF] Taguchi | Brain IRS2 signaling coordinates life span and nutrient homeostasis[END_REF][START_REF] Blu ¨her | Extended longevity in mice lacking the insulin receptor in adipose tissue[END_REF]. However, this can be reconciled by the fact that these models are primarily protected from the detrimental effects of age-induced increase in plasma insulin levels as TZDs lower circulating insulin levels [START_REF] Lebovitz | Rosiglitazone monotherapy is effective in patients with type 2 diabetes[END_REF][START_REF] Smith | Rosiglitazone, but not glyburide, reduces circulating proinsulin and the proinsulin: insulin ratio in type 2 diabetes[END_REF]. Fittingly, low insulin levels and maintained insulin sensitivity characterize human centenarians [START_REF] Barbieri | Glucose regulation and oxidative stress in healthy centenarians[END_REF]. In light of the above, the results from ongoing outcome trials evaluating the long-term health benefits of treatments with PPARc-agonists, i.e. TZDs, are eagerly awaited.

In summary, we have identified a substantial set of potential novel longevity genes in mouse adipose tissue, and demonstrate, as a case study, the significant effects of perturbed Pparc activity on mouse lifespan. Furthermore, our network analysis suggests that, at least in the context of adipose tissue, the determination of longevity may not be a random process, but governed by a concerted effort of a distinct subnetwork of genetic players.

Materials and Methods

Ethics statement

Animal experiments were approved by the local ethics committee and performed according to governmental guidelines.

Compilation of the seed set of 60 ''known'' longevity genes

To obtain a list of genes with known association to longevity, we used the Phenotypes section of the Mouse Genome Informatics (MGI) resource of The Jackson Laboratory (http://www.informatics. jax.org/) [START_REF] Bogue | Mouse Phenome Database (MPD)[END_REF], the GenAge Model Organisms pages for mouse within

The Human Ageing Genomic Resources (HAGR) [START_REF] De Magalhaes | HAGR: the Human Ageing Genomic Resources[END_REF], and a literature search. The list was compiled in October, 2007.

Generation of the transcriptional network for mouse adipose tissue

Detailed description of these methods is given in Text S1. In summary, we obtained male adipose tissue gene expression data from 4 different mouse F2 crosses [START_REF] Schadt | Mapping the genetic architecture of gene expression in human liver[END_REF][START_REF] Wang | Genetic and genomic analysis of a fat mass trait with complex inheritance reveals marked sex specificity[END_REF] using Agilent microarrays, and generated a Bayesian network for each cross by integrating genetic and gene expression data [START_REF] Zhu | Increasing the power to detect causal associations by combining genotypic and expression data in segregating populations[END_REF][START_REF] Zhu | An integrative genomics approach to the reconstruction of gene networks in segregating populations[END_REF][START_REF] Schadt | An integrative genomics approach to infer causal associations between gene expression and disease[END_REF]. The combined network, containing 13088 nodes and 22809 edges, was obtained as the union of all these 4 separate Bayesian networks.

Connectivity of ''known'' longevity genes within the adipose transcriptional network

To assess the degree of connectivity of the 33 'known' longevity genes that were present in the adipose consensus network, mean shortest paths were computed using Dijkstra's algorithm [START_REF] Dijkstra | A note on two problems in connexion with graphs[END_REF] for our set of 33 nodes ( = genes) as well as 10 6 randomly selected sets of 33 nodes. Briefly, the algorithm finds the smallest number of edges we have to ''walk'' in order to ''travel'' from a source node ( = gene) to another node ( = gene) of interest within the map/ network. The probability of finding random sets of 33 nodes with shorter mean paths than with our set was obtained by counting the number of such eventualities within the randomized sets, and amounted to a p-value of 0.00149, demonstrating that indeed our 33 genes are much more connected within the adipose tissue consensus network than expected by change. Kolmogorov-Smirnov (KS) test was used to further assess whether there were any significant differences between the shortest path distribution within our longevity gene-set and those within each of the 10 6 random sets. The resulting p-value distribution demonstrated that indeed the longevity genes shortest path distribution is not a normal occurrence in the network.

Generation of Pparg hyp/hyp and Pparg2 2/2 mice Pparg2 2/2 mice were generated from Pparg hyp/hyp [START_REF] Koutnikova | Compensation by the muscle limits the metabolic consequences of lipodystrophy in PPAR gamma hypomorphic mice[END_REF] mice by successive matings with transgenic C57Bl/6J mice expressing FLP and Cre recombinases to remove the Pparg2 specific exon B. All mice studied were backcrossed a minimum of 9 generations to achieve an essentially pure C57Bl/6J background.

Survival

The original survival cohorts consisted of 38 wild type and 24 Pparg hyp/hyp male, and 25 wild type and 26 Pparg2 2/2 female mice which were maintained on a 12 hour light/dark cycle, fed regular chow, had free access to H 2 O and received standard animal care. The mice were bred locally and were entered into the survival cohort over the course of 23 weeks for male Pparg hyp/hyp mice, amd 19 months for female Pparg2 2/2 mice. For all groups, deaths were recorded weekly. Mice observed as moribund were euthanized and recorded as dead on that week. All Pparg hyp/hyp reached the end-point, but a few Pparg2 2/2 mice survived at the time of analysis.

Metabolic exploration of Pparg2 2/2 mice

Approximately 2 year old wild type (n = 5) and Pparg2 2/2 (n = 9) mice were subjected to the following analysis according to standardized Eumorphia/EMPReSS (http://empress.har.mrc.ac. uk/) protocols: body composition by quantitative nuclear magnetic resonance on a Minispec analyzer (Bruker Optics, The Woodlands, TX), food intake, intraperitoneal glucose tolerance test (IPGTT), and fasting plasma insulin and adiponectin measure-ments using Ultrasensitive Mouse Insulin ELISA kit (Mercodia, Uppsala, Sweden) and Quantikine Mouse Adiponectin/Acrp30 Immunoassay (R&D systems Inc., Minneapolis, MN), respectively. HOMA index for insulin resistance was calculated from fasting glucose and insulin values [START_REF] Heikkinen | Evaluation of glucose homeostasis[END_REF].

RNA analysis

Pparg1 and Pparg2 gene expression in WAT, BAT, liver and skeletal muscle of Pparg hyp/hyp mice was previously reported [START_REF] Koutnikova | Compensation by the muscle limits the metabolic consequences of lipodystrophy in PPAR gamma hypomorphic mice[END_REF] and is presented for comparative purposes. For Pparg2 2/2 mice, total RNA was extracted from WAT, BAT, liver and skeletal muscle either with RNeasy for Lipid Tissues Mini Kit (Qiagen, Valencia, CA) or Trizol (Invitrogen, Carlsbad, CA), and reverse transcribed to cDNA using SuperScript II System (Invitrogen) and random hexamer primers. Pparc1 and Pparc2 gene expression was quantified by qRT-PCR using isoform-specific primers and SYBR Green chemistry on a LightCycler 480 (Roche, Penzberg, Germany).

Statistical analyses

Statistical methods pertaining to the network and other associated analysis of gene expression and gene set data were as detailed above. Kaplan-Meier survival analysis, which allows for censored cases, was used to analyze the survival data in SPSS (version 14). Metabolic and molecular data for Pparg hyp/hyp and Pparg2 2/2 mice were analyzed using Student's t-test and are presented as means 6 s.e.m. S1 The seed set of 60 ''known'' longevity genes in mouse for the identification of a novel transcriptional longevity subnetwork. The gene list was derived from public resources (see Materials and Methods). The overlaps with the consensus white adipose tissue network (33 genes) and the rosiglitazone signature in the mouse WAT (5 genes) are indicated by a plus sign. Found at: doi:10.1371/journal.pgen.1000752.s003 (0.03 MB XLS)

Supporting Information

Table S2 Listing of 742 potentially novel longevity genes. Genes are ranked for the strength of connectivity with the network of the 33 ''known'' longevity genes within the male mouse adipose tissue transcriptional network of 13,088 genes. Found at: doi:10.1371/journal.pgen.1000752.s004 (0.12 MB XLS)

Text S1 Supporting Methods. Found at: doi:10.1371/journal.pgen.1000752.s005 (0.18 MB DOC)

Figure 2 .

 2 Figure 2. Pparg determines longevity. (A) Lifespan of hypomorphic (hyp) Pparg deficient mice (n = 38 wild type and 24 Pparc hyp/hyp mice). ** p = 0.003. (B) Lifespan of Pparg2 knock-out mice (n = 25 wild type and 26 Pparg2 2/2 mice). * p = 0.020 when mice .120 weeks were excluded from the test. doi:10.1371/journal.pgen.1000752.g002

Figure S1

 S1 Figure S1 Pparg1 and Pparg2 gene expression in WAT, BAT, liver and skeletal muscle in mouse models with altered Pparg locus. Data are presented relative to mean WAT expression in the wild type (Pparg +/+ ) for each Pparg isoform. Note the much lower expression levels in liver and muscle. (A) Hypomorphic Pparg deficient mouse. (B) Pparg2 knock-out mouse. Note that only one mouse per group was analyzed. Found at: doi:10.1371/journal.pgen.1000752.s001 (0.07 MB TIF)

Table 2 .

 2 Pathway analysis of the predicted novel longevity genes in male mouse adipose tissue.

	Pathway	Gene count		p	Overlap genes
		Overlap	Pathway
	PPAR signaling pathway	11	69	0.00066	Apoa5, Apoc3, Aqp7, Cpt2, Fabp3, Gyk, Pck1, Pparg, Rxrg, Slc27a2, Sorbs1
	Ubiquinone biosynthesis	4	9	0.00069	Coq5, MMG00237617, ND5, Ndufa12
	Valine, leucine and isoleucine degradation	8	40	0.00078	Acat1, Aldh6a1, Auh, Dld, Echs1, Hibadh, Mccc1, Mcee
	Pentose phosphate pathway	6	25	0.00133	Aldob, Fbp1, Fbp2, H6pd, Pgm2, Tkt
	Complement and coagulation cascades	10	65	0.00153	C1r, C2, F2, Fgb, Fgg, Hc, Kng1, Plg, Serpine1, Serpinf2
	ECM-receptor interaction	11	80	0.00230	Cd44, Col2a1, Col3a1, Col4a1, Col5a3, Fn1, Fndc1, Itga6, Itga7, Spp1, Thbs2
	Propanoate metabolism	6	29	0.00300	Acacb, Acat1, Aldh6a1, Echs1, Ldhc, Mcee
	Carbon fixation	5	23	0.00536	Aldob, Fbp1, Fbp2, Gpt1, Tkt
	Insulin signaling pathway	14	133	0.00760	Acacb, Eif4ebp1, Fasn, Fbp1, Fbp2, Pck1, Pde3b, Pik3r5, Ppargc1a, Ppp1r3b,
					Pygb, Slc2a4, Socs2, Sorbs1
	Glycolysis/Gluconeogenesis	7	51	0.01434	Aldh1a3, Aldob, Dld, Fbp1, Fbp2, Ldhc, Pgm2
	Pyruvate metabolism	6	40	0.01514	Acacb, Acat1, Dld, Ldhc, Pck1, Pcx
	Alanine and aspartate metabolism	5	32	0.02206	Adssl1, Asns, Dld, Gpt1, Pcx
	Fatty acid biosynthesis	2	6	0.03434	Acacb, Fasn
	Cysteine metabolism	3	15	0.03845	Cars2, Cdo1, Ldhc
	Citrate cycle (TCA cycle)	4	26	0.04157	Dld, Idh1, Pck1, Pcx

The input set of 197 genes was determined by the overlap of the full set of 742 potentially novel longevity genes and the set of 3835 genes for which functional data was available in the KEGG repository at the time of analysis. Note that the 33 ''known'' longevity genes were excluded from the determination of the input set to remove bias. Only those pathways with p,0.05 are shown. doi:10.1371/journal.pgen.1000752.t002

PLoS Genetics | www.plosgenetics.org 1 December 2009 | Volume 5 | Issue 12 | e1000752

PLoS Genetics | www.plosgenetics.org

December 2009 | Volume 5 | Issue 12 | e1000752

Acknowledgments

We thank the members of the Auwerx and Schadt laboratories for helpful discussions.

We acknowledge grant support of the NIH (DK067320-01), the European Union (Eugene2; LSHM-CT-2004-512013 and Sirtuins; ERC-2008-AdG-23118), the Ecole Polytechnique Federale de Lausanne, the Swiss National Science Foundation, Universite Louis Pasteur, CNRS, INSERM, the Hopital Universitaire de Strasbourg, France, and the Academy of Finland. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Author Contributions


Torrenti Jean Michel
email: jean-michel.torrenti@univ-eiffel.fr

Aili Abudushalamu

Dependence of basic creep on the relative humidity

Keywords: Concrete, creep, basic creep, relative humidity

come

Dependence of basic creep on the relative humidity

Introduction -Influence of RH on basic creep

The prediction of delayed strains is particularly important for prestressed structures like bridges or nuclear power plants. In modern codes like the fib MC2010 [1], the delayed strains are split into four components: autogenous shrinkage, drying shrinkage, basic creep, and drying creep. Basic creep is the creep of a concrete sample in a test where drying is avoided. It is the difference between the strain of a sealed loaded specimen and the autogenous shrinkage. When drying is allowed, drying creep is the difference between the total strain minus the total shrinkage and the basic creep.

But, in the drying creep test, the relative humidity (RH) is lower than the RH of the test performed to measure the basic creep. And there are several experimental pieces of evidence showing that, when the RH is lower, basic creep is reduced. Abiar [START_REF] Abiar | Cinétique de dessiccation et déformations différées du béton (analyse et modélisation)[END_REF] has compared the basic creep of a sample that was pre-dried with the classic basic creep.

Figure 1 shows that when the sample was pre-dried the basic creep was very small. Wittmann [START_REF] Wittmann | Einfluss des feuchtigkeitsgehaltes auf das kriechen des zementsteines[END_REF] has also performed basic creep tests on cement paste samples that were pre-dried at 105°C then stored in climatic chambers at different relative humidities for at least 3 months and tested only the specimens that did not change their weight. The creep tests were also performed with the corresponding imposed RH for each specimen. It is also clear here that the initial relative humidity has a strong influence on the basic creep, i.e. see Figure 2. Indeed, the water content plays an important role in the delayed deformations of concrete. Concerning basic creep, several theories are existing to explain this phenomenon: Bazant et al. [START_REF] Bažant | Microprestress-solidification theory for concrete creep. I: Aging and drying effects[END_REF] explained it with the famous micro-prestress theory in which the origin of the creep is the shear slip at overstressed creep sites and, recently, Vandamme [START_REF] Vandamme | Two models based on local microscopic relaxations to explain long term basic creep of concrete[END_REF] has proposed basic creep originates from local microscopic relaxations. And the presence of water is an important factor in these phenomena.

The macroscopic basic creep compliance follows a logarithmic evolution of the form:

𝐽 𝑏𝑐 = 1 𝐶 ln (1 + 𝑡-𝑡 0 τ(𝑡 0 ) ) (1)
where 𝐽 𝑏𝑐 is the basic creep compliance, 𝐶 is a parameter that is homogenous to a basic creep modulus, 𝑡 0 is the age at loading, t -𝑡 0 is the duration of loading, and τ(𝑡 0 ) is a characteristic time depending on the age of loading. There is several experimental evidence of this logarithmic evolution in laboratory tests, see for instance [START_REF] Roy | Long term basic creep behavior of high performance concrete: data and modelling[END_REF][START_REF] Torrenti | Analysis of some basic creep tests on concrete and their implications for modeling[END_REF], in real structures like bridges [START_REF] Bažant | Excessive creep deflections: An awakening[END_REF], or nuclear containments [START_REF] Aili | Modeling Long-term Delayed Strains of Prestressed Concrete with Real Temperature and Relative Humidity History[END_REF].

Results of micro-indentation tests

Vandamme [START_REF] Vandamme | Two models based on local microscopic relaxations to explain long term basic creep of concrete[END_REF] has also shown that micro-indentation allows the determination of the basic creep modulus 𝐶 in a short time because the characteristic time τ(𝑡 0 ) is reduced thanks to the high stress generated during such a test. Several authors have studied the creep of concrete using a micro-indentation test with different relative humidities [START_REF] Zhang | Long-term creep properties of cementitious materials: Comparing microindentation testing with macroscopic uniaxial compressive testing[END_REF][START_REF] Frech-Baronet | New evidences on the effect of the internal relative humidity on the creep and relaxation behaviour of a cement paste by microindentation techniques[END_REF][START_REF] Liu | Multi-scale creep analysis of cement paste-Indentation prediction and time correspondence of mechanisms[END_REF][START_REF] Suwanmaneechot | Creep behavior of CSH under different drying relative humidities: Interpretation of microindentation tests and sorption measurements by multi-scale analysis[END_REF]. Of course, the magnitude of the creep modulus depends on the tested materials. But, it could be assumed that the variation of this parameter with RH is the same for all materials (i.e. the possible coupling between the material composition and the effect of RH is neglected). Figure 3 presents the evolution of the ratio C(𝑅𝐻)/C(100%) with RH. Despite a certain variability, there is a bi-linear trend in the evolution: between 80% and 100% RH the variation is limited while it is larger for RH below 80%. Equation (2) gives a possible relation for this evolution:

𝐶(𝑅𝐻) 𝐶(100%) = { 1 + 0.1 (100-𝑅𝐻) 20 , 80% < 𝑅𝐻 < 100% 1.1 + 2.1 (80-𝑅𝐻) 80 , 0 < 𝑅𝐻 < 80% (2)
Fig. 3. Relation between the basic creep modulus and the relative humidity (creep tests from [2, 3 and 14], micro-indentation tests from [START_REF] Zhang | Long-term creep properties of cementitious materials: Comparing microindentation testing with macroscopic uniaxial compressive testing[END_REF][START_REF] Frech-Baronet | New evidences on the effect of the internal relative humidity on the creep and relaxation behaviour of a cement paste by microindentation techniques[END_REF][START_REF] Liu | Multi-scale creep analysis of cement paste-Indentation prediction and time correspondence of mechanisms[END_REF][START_REF] Suwanmaneechot | Creep behavior of CSH under different drying relative humidities: Interpretation of microindentation tests and sorption measurements by multi-scale analysis[END_REF]).

Consequences for the creep of concrete

The fact that the basic creep modulus increases when RH decreases implies that during a creep test where drying is allowed, there is less basic creep than in a test where drying is prevented. The first consequence is that, since basic creep is dependent on drying, there is also a dependence of basic creep on the size of the concrete element (scale effect). The second consequence is that when drying creep is deduced by the subtraction of the basic creep from the total creep, there is an underestimation of the drying creep. It is not possible to know experimentally the error that is induced. In the following part of the paper, we will estimate such an error by a numerical simulation of the creep test of a drying specimen.

After modelling the drying of a concrete sample, the basic creep of a sealed sample is adjusted on experimental results. The results of the concrete of the VeRCoRs experiment are used here [START_REF] Aili | Modeling Long-term Delayed Strains of Prestressed Concrete with Real Temperature and Relative Humidity History[END_REF]15]. Then the basic creep of the same concrete when drying occurs is estimated.

Modelling

As an application of the above findings, we choose to simulate a VerRCorRs test: the drying of a cylindrical sample of diameter D = 16 cm and height H = 1 m, mechanically submitted to a constant compression of magnitude 12 MPa during almost two years. To this end, a coupled hydro-mechanical resolution procedure is performed; a hydric problem for the drying evolution part, followed by a mechanical part that takes into account the RH-dependent creep phenomenon.

Modelling of the drying part of the test

The primary unknown of the problem is chosen to be the RH-field that we denote here for convenience by h(x,t), and that is governed by the following transient equation,

∂𝑆(ℎ) ∂ 𝑡 + div[-𝑑(ℎ) ∇ ℎ] = 0 (3)
where S(h) is the saturation function deduced from the desorption isotherm curve. This latter is herein approximated by the Hailwood-Horrobin function as,

S(ℎ) = ℎ 𝐹 1 + 𝐹 2 ℎ + 𝐹 3 ℎ 2 (4)
with F 3 = 1 -F 1 -F 2 , and where 𝐹 1 and 𝐹 2 are adimensional constants.

Moreover, in Eq. ( 3), 𝑑(ℎ) is the hydric diffusion coefficient that depends on the relative humidity. It is chosen as, 𝑑(ℎ) = 𝐵𝑒 𝐴ℎ [START_REF] Vandamme | Two models based on local microscopic relaxations to explain long term basic creep of concrete[END_REF] where 𝐴 and 𝐵 are parameters to be fitted with the concrete weight loss data obtained experimentally. To evaluate this water mass loss, we need to know the value of the water content at ℎ =100%, and that we denote by W 100 .

For the finite element simulations, the axisymmetry of the problem is exploited. The sample domain we consider is of height 𝐻 = 1𝑚 and radius 𝑅 = 8 cm. The finite element discretization we use consists of 300 quadrilateral linear elements: 30 elements radially, and 10 elements along the axis of the sample.

The initial condition is set to ℎ(𝒙, 𝑡 = 0) = 100%. For the boundary conditions, a relative humidity ℎ ext = 50% is prescribed on the lateral surface during the whole simulation, and the sample is isolated at the top and bottom faces. Hence, the diffusion problem is purely radial. Table 1 summarized the material parameters used for the drying part of the numerical simulation. The predicted curve of water mass loss is plotted in Figure 4. A good agreement is to be noticed when comparing with the experimental data at our disposal. For the sake of illustration, Figure 5 shows the computed RH-field at two different times.

Modelling of the basic creep when drying occurs

For the mechanical part of the problem, the basic creep strain tensor ε bc is here treated as an internal variable, see for instance [START_REF] Nedjar | An approach to the modeling of viscoelastic damage. Application to the long-term creep of gypsum rock materials[END_REF][START_REF] Nedjar | Incremental viscoelasticity at finite strains for the modelling of 3D concrete printing[END_REF] for analogies. The way in which this internal variable evolves is motivated by the basic creep compliance given in Eq. ( 1).

In this case the evolution equation that governs the basic creep strain in its one-dimensional version is given by:

𝜀̇b c + E C(t+τ) 𝜀 bc = E C(t+τ) 𝜀 ( 6
)
where ε is the total strain, and 𝐸 is the elastic Young modulus. Here 𝐶 is the basic creep modulus discussed above, and τ is the initial characteristic time. These two parameters must be identified from the experimental data. Now for the mechanical part of the problem, the above cylindrical sample is submitted to a constant compression of 12 MPa. For the boundary conditions, the bottom face is simply fixed in the vertical direction, and the lateral surface is free. For the elastic parameters, we use 𝐸 = 36 GPa and a Poisson's ratio ν = 0.2. We have identified the following creep parameters for an RH-independent analysis: With the parameters given in Eq. ( 7), the predicted creep evolution is plotted in Figure 6, blue curve. It is in good agreement with the experimental curve (dotted curve). On the other hand, when using the RH-dependent creep law as proposed in Eq. ( 2), the creep evolution is slightly different, see the red curve in Figure 7. This difference appears to grow with drying.

Conclusions

In modern calculation codes such as the fib model code 2020, the creep is the sum of two components: the basic creep and the desiccation creep. The latter is estimated by making the difference between the total creep measured on a drying specimen and the creep measured on a non-drying specimen. Such estimation implies that the basic creep is the same in these two specimens, even though their water contents are different. However, tests on specimens, confirmed by micro-indentation tests, have shown that basic creep depends on the water content of the cementitious material tested. The modelling framework described in this contribution allows considering the difference in creep amplitude with the RH-dependent evolution law. It shows that, in a test where the specimen dries, the basic creep is overestimated, which means that the drying creep is underestimated. As only the latter depends on the size of the parts, this can lead to an error in the prediction of delayed deformations in the very long term.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Basic creep tests of sealed and pre-dried concretes (from Abiar [2])

Table 1 .

 1 Material parameters used for the hydric part of the problem. Parameters Saturation function 𝑆(ℎ), Eq. (4) 𝐹 1 = 1.335 𝐹 2 = 1.391 Hydric diffusion 𝑑(ℎ), Eq. (5) 𝐵 = 11.75 10 -9 𝑚𝑚 2 /𝑑𝑎𝑦 𝐴 = 24.25 Water content at ℎ = 100% W 100 = 163 𝑙/𝑚 3

Fig. 4 .

 4 Fig. 4. Water mass loss of the cylindrical concrete sample. Experimental data and simulated curve issued from the FEM computation.

Fig. 5 .

 5 Fig. 5. Simulated relative humidity fields at two different times: at t = 10 days (left), and at t = 770 days (right).

Fig. 6 .

 6 Fig. 6. Evolution of the longitudinal creep strain: (a) with the RH-independent creep model (blue curve), and (b) with the RH-dependent model as given by Eq. (2) (red curve).