Nicolas Meyer
email: nicolas.meyer@inra.fr

Jacques-Eric Bergez

Julie Constantin

Paul Belleville

Eric Justes

Cover crops reduce drainage but not always soil water content due to interactions between rainfall distribution and management

Intensive agriculture has reached a critical point, particularly in Europe and temperate climate zones.

Conventional agriculture is increasingly shown to be responsible for multiple problems, such as stagnating yields, a decrease in soil fertility, pollution of groundwater and rivers, and soil erosion.

Agriculture must adapt in response to climate change, which is causing more droughts, less water availability, and more extreme climatic events (IPCC, 2013). Therefore, scientists, farmers and policy makers urgently need to find more sustainable and resilient cropping and farming systems. The use of cover crops could help cropping systems become more agroecological and diversified. Cover crops are sown after one cash crop is harvested and terminated before the next one is sown. Their residues are retained as a mulch or incorporated into the soil by plowing or shallow tillage, such as disking. Cover crops provide a wide range of ecosystem services, including reducing nitrate leaching (Tonitto et al.,

significant differences between these two treatments at sowing of the next crop. Several studies reported more water in the topsoil (0-20 cm depth) [START_REF] Blanco-Canqui | Addition of Cover Crops Enhances No-Till Potential for Improving Soil Physical Properties[END_REF][START_REF] Wells | Cultural Strategies for Managing Weeds and Soil Moisture in Cover Crop Based No-Till Soybean Production[END_REF].

Likewise, no consensus exists about the impact of management of cover crop residues after cover crops are terminated. Several studies reported an increase in soil water content (SWC) up to 20 cm deep [START_REF] Alliaume | Reduced tillage and cover crops improve water capture and reduce erosion of fine textured soils in raised bed tomato systems[END_REF][START_REF] Moschler | Winter Cover Crops for Sod-Planted Corn: Their Selection and Management1[END_REF][START_REF] Stipešević | Effects of winter wheat cover crop desiccation times on soil moisture, temperature and early maize growth[END_REF]. [START_REF] Williams | Crop cover root channels may alleviate soil compaction effects on soybean crop[END_REF] observed no differences in soil moisture 20-50 cm deep between cover crop mulch and bare soil. Pedrosa De [START_REF] De Azevedo | The effect of cover crop and crop rotation on soil water storage and on sorghum yield[END_REF] observed no significant difference in soil moisture between keeping cover crops alive during the fallow period and crushing them and leaving their residues as mulch. [START_REF] Kornecki | Effects of recurrent rolling/crimping operations on cover crop termination, soil moisture, and soil strength for conservation organic systems[END_REF] observed a small difference between these two treatments, with more soil moisture under the mulched cover crop.

It is also difficult to generalize the impact of cover crops on water balance because studies are performed in different contexts of fallow periods (e.g. August-November vs. October-April), climate conditions, and cover crop management. In some studies, cover crops are terminated in late winter or early spring, while in others they are terminated the day before sowing the next cash crop, especially when herbicide termination is used. These differences are crucial to understanding cover crop impacts, since multiple processes, of variable intensity, interact in these dynamics. Thus, spring rainfall can recharge the soil and mask differences in SWC and in the topsoil (seedbed of the next cash crop), even though water fluxes (e.g. drainage, evapotranspiration) flow simultaneously. Dynamic soil-crop models can be useful tools to estimate water fluxes that are difficult to measure in field experiments, such as drainage, evaporation, and transpiration, and can do so accurately over long periods. These models also can simulate a wide range of cover crop management practices under several soil and climate conditions and extrapolate results of field experiments that are expensive, time consuming, and rare [START_REF] Bergez | Designing crop management systems by simulation[END_REF].

The goal of our study was to understand and quantify impacts of cover crops and their management on water flux dynamics (e.g. soil evaporation, plant transpiration, drainage) and balance at the crop rotation scale. Two components of the water balance were a particular focus of analysis: (1) drainage, which governs groundwater recharge, and (2) water availability for the next cash crop, which determines the risk of water deficit for crop emergence and early growth. We combined a two-year field experiment with simulation modeling to address this issue. The field experiment was performed to analyze impacts of cover crop management on dynamics of SWC and cover crop biomass. This field experiment was supplemented with a modeling approach using the STICS soil-crop model to understand and quantify the water flux dynamics (drainage, evaporation, and transpiration) not measured during the experiment.

Materials and methods

Methodological approach

To determine the impact of various cover crop management practices on soil water balance during the fallow period and soil water availability for the next spring/summer cash crop, the following methodological approach was used:

1. A two-year field experiment was performed to quantify the impact of three cover crop management practices on SWC in multiple soil layers during the fallow period compared to the SWC under bare soil, used as a control.

2. The dynamics of SWC and cover crop biomass measured in the experiment were used to calibrate the STICS soil-crop model for the mixed cover crop sown in the experiment and evaluate the model's ability to predict the experiment. The objective was to simulate soil and plant dynamics sufficiently well to obtain accurate simulations of water fluxes.

3. We assumed that if dynamics of SWC and crop biomass were simulated accurately, then those of water fluxes would also be simulated accurately. After verifying satisfactory agreement between model predictions and experimental observations, STICS was used to predict dynamics of soil evaporation, plant transpiration, and water drainage (processes not measured in the field) and the complete water balance for the two experimental years. 4. To place the weather during the two experimental years in the study site's interannual variability in climate, we used the calibrated STICS to simulate a 20-year climate series to better understand and generalize the impact of cover crop management on water fluxes and the complete water balance.

Case study site

Climate characteristics

A two-year experiment was performed from 2017-2018 and 2018-2019 (hereafter, "2017" and "2018", respectively) in southwestern France near Toulouse (43°31' N, 1°30' E). The site has a temperate climate corresponding to Cfb in the Köppen climate classification. Over the past two decades , the site's mean (± 1 standard deviation (SD)) annual temperature was 13.8 ± 0.5°C, annual potential evapotranspiration (PET) (Penman equation) was 962 ± 51 mm, and annual rainfall was 655 ± 105 mm. The weather conditions from cover crop sowing in July-August to termination in March-April differed during the two years of the experiment (Fig. 1). For 2017 and 2018, cumulative rainfall was 504 mm and 343 mm, and cumulative PET was 437 mm and 334 mm, respectively. In 2017, autumn was dry and winter and spring were rainy, with a high PET in the three months following the sowing of cover crops. Conversely, in 2018, autumn and early winter were rainy, with a low PET, and late winter and spring were dry. Cumulative temperature (calculated with base 0) from the day of cover crop sowing (31 July in 2017 and 28 August in 2018) until 9 April was 2920°C in 2017 and 2490°C in 2018.

(Figure 1)

Soil characteristics

The two-year experiment was performed at an Institut National de la Recherche Agronomique (INRA) experimental station on two field plots with similar soil characteristics: clay loam in 2017 and loam in 2018 (Table 1). Soil moisture at field capacity (FC) corresponded to the maximum observed during the experiment. Soil moisture at wilting point (WP) and bulk density (BD) were estimated based on soil texture classification and measurements at the experimental station [START_REF] Jamagne | Quelques donnees sur la variabilite dans le milieu naturel de la reserve en eau des sols[END_REF][START_REF] Tribouillois | Cover crop mixtures including legume produce ecosystem services of nitrate capture and green manuring: assessment combining experimentation and modelling[END_REF]. Total available water capacity for plants (TAWC, SWC between FC and WP) was calculated down to 120 cm deep (the maximum root depth observed at this site ;[START_REF] Tribouillois | Cover crop mixtures including legume produce ecosystem services of nitrate capture and green manuring: assessment combining experimentation and modelling[END_REF]). TAWC was nearly the same for the two plots: 152 and 164 mm in 2017 and 2018, respectively. The SWC at FC was 347 and 328 mm in 2017 and 2018, respectively.

(Table 1) 2.3 Field experiment

Experimental design

The experimental design was the same for both years: four treatments and four replicates per treatment. The treatments were (1) bare soil (BS) as a control without plant transpiration; (2) cover crops mechanically terminated by crushing in autumn and left as mulch on the soil surface (CC M ) until spring; (3) cover crops mechanically terminated by crushing in autumn and buried by plowing (CC P ) during winter when soil conditions were suitable (to avoid compaction); and (4) cover crops mechanically terminated by crushing in April (CC L ) immediately before sowing the next spring/summer cash crop. The surface area of an elementary plot was 44 and 70 m² in 2017 and 2018, respectively. The preceding cash crop for both years was durum wheat harvested in late June/early July.

Cover crop species were selected for their ability to grow rapidly during the summer and their tolerance to freezing temperatures. They were also selected to favor complementary abiotic resource acquisition, based on previous studies performed under the same climatic conditions (Couëdel et al., 2018b;[START_REF] Tribouillois | Cover crop mixtures including legume produce ecosystem services of nitrate capture and green manuring: assessment combining experimentation and modelling[END_REF] and to cover the soil during the entire fallow period due to complementary development. A bispecific mixture was chosen: one crucifer, Ethiopan mustard (Brassica carinata), and one legume, crimson clover (Trifolium incarnatum). The crucifer grew rapidly after sowing and during autumn. The legume was able to grow during winter under the mustard and could grow more rapidly in early spring, maintaining soil cover and plant transpiration until April.

Seeds were sown respectively at 3 kg.ha -1 and 7.5 kg.ha -1 of seeds. Shallow tillage was performed in both years before sowing. In both years, 70 mm of irrigation was applied twice after sowing to ensure homogenous emergence and establishment of cover crops to guarantee spatial homogeneity of the soilcrop system. Weeds were controlled on BS using a herbicidal spray in October of both years.

Cover crop biomass and soil water content measurements

Aerial biomass (Mg.ha -1 ) was sampled from a 0.5 m² area in each replicate plot of the cover crop treatments on each termination date (autumn and spring). Mustard and clover biomass was separated and measured. When present, weed biomass was also measured in the cover crop treatments, but it was always low. Dry matter was weighed after 48 h in an 80°C oven. SWC (mm) was measured once per month for 8-9 months from cover crop sowing to the following April. The soil profile was sampled from 0-120 cm deep in layers 20 cm thick. Five samples were collected per replicate plot and then pooled by layer for each plot. Measurements in each of the four replicates were taken independently. Soil samples were weighed before and after 48 h in a 105°C oven to measure gravimetric soil moisture (cg water g -1 soil).

Simulation approach

Model overview

We used the soil-crop model STICS [START_REF] Brisson | An overview of the crop model STICS[END_REF], which simulates daily crop growth, light, water, C and nitrogen (N) balances based on soil, climate, crop species, and agricultural management.

A tipping-bucket approach is used to model the soil, which is divided into five layers with specific characteristics, such as BD, as well as SWC at FC and WP. The water balance was simulated daily by adding soil water supply and subtracting plant transpiration and soil evaporation [START_REF] Brisson | Conceptual Basis, Formalisations and Parameterization of the Stics Crop Model[END_REF]. STICS was evaluated as accurate for a wide range of agro-environmental contexts in France for plant, water, and N outputs for bare soil and many types of cash crops [START_REF] Brisson | An overview of the crop model STICS[END_REF]2009). STICS was also used to simulate cover crops and analyze water, C and N balances and the associated ecosystem services (Tribouillois et al., 2018a). STICS was also successfully evaluated for water drainage [START_REF] Beaudoin | Evaluation of the soil crop model STICS over 8 years against the "on farm" database of Bruyères catchment[END_REF][START_REF] Constantin | Long-term nitrogen dynamics in various catch crop scenarios: Test and simulations with STICS model in a temperate climate[END_REF].

Model initialization and calibration on-site

Soil parameterization and initialization

There was no calibration done on bare soil. Two soils were parameterized separately to simulate each year of the field experiment. For model evaluation, the SWC measured at the beginning of the experiment for each treatment was used, and the soil moisture at FC of each layer was adjusted for each treatment to correspond as closely as possible to field observations. To initialize each soil layer in the model for simulations, the mean SWC and soil mineral N measured in the four treatments at the beginning of the experiment was used for each treatment (Table 1).

(Table 2) 2.4.2.2 Crop calibration STICS has been calibrated for a wide range of species used as cover crops, such as mustard, rapeseed, radish, ryegrass, oat, pea, vetch, and clover [START_REF] Constantin | Long-term nitrogen dynamics in various catch crop scenarios: Test and simulations with STICS model in a temperate climate[END_REF]Tribouillois et al., 2018b); however, no parameterization for a mixture of species was available. To simulate the field experiment accurately, we parameterized STICS for the mustard-clover cover crop mixture. The objective was to simulate, as accurately as possible, the main processes governing water balance dynamics, such as cover crop development, biomass growth, changes in soil moisture over time in each soil layer, and total SWC. Starting with parameters already calibrated for rapeseed (the crop most similar to Ethiopian mustard) and crimson clover, we optimized three parameters for cover crop development, N 2 fixation (for clover), and cover crop water requirements to generate the most accurate joint predictions of the dynamics of biomass and SWC in each layer and in total for both experimental years. Mathematical optimization was based on the method developed by [START_REF] Wallach | A package of parameter estimation methods and implementation for the STICS crop-soil model[END_REF] using the Simplex algorithm, which is available with the STICS model software (Java interface; https://www6.paca.inra.fr/stics_eng).

Water flux simulations of the field experiment

After calibration, model predictions of water drainage, soil evaporation, and plant transpiration were available for each of the four treatments for each year of the field experiment (Table 2); they were used to calculate total water balance and quantify the fluxes. For the 2017 experimental year, simulations began 28 July and ended 28 March. For the 2018 experimental year, simulations began on 1 August and ended on 5 April. Since crimson clover was observed to regrow in both years after being crushed in the CC M treatment, the model was calibrated to simulate this regrowth as well, using only crimson clover parameters as the crop in a second simulation step.

Simulation of water fluxes over 20 years

To evaluate the impact of cover crops on soil water variables over a long time series, a 20-year simulation was performed using a climate series from 1999-2018, from an INRA weather station installed at the study site. The soil was parameterized and initialized with the field data measured in 2017. Then we run simulations twice, once with dates of technical operations of 2017-crop management for all 20 years and once with dates of the technical operations of 2018-crop management for all 20 years, since each year had different field-operation dates (Table 2). The simulations started on 1 August and finished on 1 April for each year of the 20-year simulation. Predicted SWC and cumulative water drainage on 1 April were also recorded. As in the two-year experiment, potential regrowth of crimson clover after being crushed in the CC M treatment was simulated.

Statistical analysis

All statistical analyses were performed using R software (R Core Team, 2018).

STICS evaluation and prediction

Model evaluation criteria

Simulations were performed using the model calibrated for the six cover crop treatments (3 treatments × 2 experimental years). Three statistical criteria were used to evaluate the quality of agreement between observed and simulated variables after calibrating STICS for the cover crop mixturemean deviation (MD), relative root mean square error (rRMSE), model efficiency (EF)calculated as follows:

∑ ̅ with √ ∑ ∑ ̅ ∑ ∑ ̅
where n is the number of observations; and are the simulated and observed values, respectively, and ̅ is the mean value of the observed data.

The three criteria provide a good overview of model performance. MD and rRMSE indicate systematic bias and the dispersal of model simulations compared to observations, respectively. The lower their value, the better is the model prediction. EF, which measures agreement between simulations and observations, varies from 1 (perfect prediction) to -∞. A negative value indicates that the mean of observations is a better predictor than the model.

STICS prediction quality

To assess the ability of STICS to simulate biomass, each soil layer's moisture, and total SWC, the relative root mean square error of prediction (rRMSEP) was calculated using cross-validation and a leave-one-out method [START_REF] Wallach | Working with Dynamic Crop Models 2nd Edition[END_REF]. The six cover crop treatments were separated into two subsamples: one containing five treatments, to estimate crop parameters, and the other containing the sixth treatment to evaluate the prediction quality of STICS compared to observed data. This procedure was applied six times to assess the prediction quality of STICS for the six treatments by calculating individual RMSEp for each treatment.

√ ∑ ̅ ̅̅̅̅̅̅̅̅̅̅
where are the predicted values.

2.5.2 Field experiment and 20-year simulations 2.5.2.1 Field experiment Statistical analyses of SWC and soil moisture for each layer in the experiment were performed. Based on the dataset, and after testing the homogeneity of the variables, a non-parametric Kruskall-Wallis test was used to test the impact of the cover crop management practices on the experimental data. A non-parametric Wilcoxon test was used to compare each pair of cover crop treatments. For all analyses, differences among treatments were considered significant at P < 0.05.

Twenty-year simulations

Statistical analyses of SWC (total and in the 0-10 cm layer) and the difference in drainage predicted in the 20-year simulations were performed. A non-parametric Kruskall-Wallis test was used to test the impact of the cover crop management practices on the simulated data. A non-parametric Wilcoxon test was used to compare each pair of cover crop treatments. For all analyses, differences among treatments were considered significant at P < 0.05.

Results

Field experiment

Aerial biomass of cover crop mixtures

On the date of autumn termination for CC M and CC P , cover crops reached a mean aerial biomass of 4.2

Mg.ha -1 (76% mustard, 24% clover) in 2017 and 3.5 Mg.ha -1 (83% mustard, 17% clover) in 2018 (Fig. 2). Afterwards, the clover regrew in CC M , reaching a biomass of 3.6 and 3.1 Mg.ha -1 at the end of the 2017 and 2018 experimental years (early spring), respectively. In contrast, for CC L at the end of the experimental years, aerial biomass was 7.5 Mg.ha -1 (35% mustard, 65% clover) in 2017 and 8.9

Mg.ha -1 (88% mustard, 12% clover) in 2018. These results illustrate the temporal complementarity of the two species' growth, which covered the soil throughout the fallow period. Ethiopian mustard develops very quickly in late summer and autumn, while clover growth is particularly important in winter and spring.

( (Figure 3a&b)

The change in SWC depended on the yearly rainfall distribution (Fig. 4). In 2017, October and November were dry, each receiving less than 50 mm of rainfall. During these months, measured SWC differed significantly between BS and the cover crop treatments. In December, the difference was ca.

50 mm, due to cover crops having higher actual evapotranspiration (AET) than BS. December and January each received ca. 200 mm of rainfall, which caused soils of all treatments to reach FC. With lower rainfall in March, a significant difference in SWC was observed between BS and CC P , which had no vegetation, and CC L and CC M , which did (Fig. 4a). In 2018, rainfall distribution was more regular than that in 2017. From August-February, 50-100 mm of rain fell each month, which explains the lack of a significant difference among the four treatments during this period. February and March were dry, but resulted in non-significant differences in SWC in the upper layers among the four treatments (Fig. 4b).

(Figure 4a&b)

3.2 STICS calibration and evaluation 3.2.1 Calibration of the cover crop mixture and quality of calibration of STICS for the two years of the experiment STICS simulated aerial cover crop biomass well: EF was high (0.82) with a non-significant low MD of 0.2 Mg.ha -1 of biomass, and rRMSE was only 18.2%. STICS also predicted SWC well, with a high EF (0.87), no significant MD (1 mm of water), and a low rRMSE (4.7%) (Table 3 & Fig. 5) (Supplementary Materials, Figs. S1-S8). STICS predicted water distribution in the soil well, with MD less than 1 cg.g -1 and EF always greater than 0.4 for the five soil layers. Dynamics of the upper layers subject to soil evaporation and root uptake of water were simulated accurately, as indicated by an rRMSE of 13.8% and 9.1% for the 0-20 and 20-40 cm layers, respectively. The rRMSE for the three deepest layers was low (< 9%). The dynamics of SWC and the soil water profile were satisfactory (Supplementary Materials, Figs. S1-S8). SWC differed between BS and cover crops somewhat more in the simulations than in the experiment. However, simulation results lay within 1 SD of the observations, indicating that model error was in the same range as the variability in the field measurements.

(Figure 5)

Evaluation of the predictive quality

Using cross-validation to assess the predictive quality of STICS for simulating biomass, soil moisture in each layer, and SWC provided satisfactory results, indicating an acceptable accuracy of the simulations. The rRMSEP of soil moisture were ca. 15% for soil layers 0-40 cm deep and less than 10% for the deeper layers. The rRMSEP of cover crop biomass were less than 10%, and those of SWC were ca. 5% (Table 3).

(Table 3)

Simulated water fluxes

Like the measurements, predicted SWC was lower for the three cover crop treatments than for BS from late August until the date when FC was reached after winter rainfall (Fig. 6). In 2018, the SWC of cover crops treatments was no more than 15 mm less than that of BS. SWC was 60 mm lower from 50 days after sowing (DAS) to 150 DAS in 2017, while it was 10 mm lower from 20-50 DAS in 2018.

Differences in SWC among the three cover crop treatments depended on the field experiment. No difference in SWC was observed between the three cover crop treatments in 2017, while several differences were measured and simulated in 2018 from 150 DAS. Difference in SWC between CC M and CC L first appeared at 150 DAS, resulting in SWC 18 mm higher for CC M than CC L in late March.

At the same time in March, the SWC for CC P was the same as that for BS and 70 mm higher than that for CC M (Table 3). From 170 DAS to the end of the experiment, the difference in simulated SWC increased between BS and CC P and between CC L and CC M , which was similar to the measurements.

Simulated SWC decreased more for CC L and CC M than for BS and CC P . The maximum difference in simulated SWC at the end of experiment was 89 mm, which was similar to the difference measured between CC L and BS (Fig. 6). Simulated cumulative water fluxes for the entire experiment are summarized in Table 4.

For both years of the experiment, higher AET was simulated for all cover crop treatments than for BS, due to an increase in leaf transpiration and a concomitant decrease in soil evaporation due to plant cover. In 2017, the three cover crop treatments had similar simulated evapotranspiration, which was 70 mm higher than BS evaporation. The relative amounts of soil evaporation and plant transpiration differed among the cover crop treatments. CC P had 77 mm less simulated transpiration than CC M and 87 mm less than CC L , which followed the trends for dynamics and amount of aerial biomass. The difference in soil evaporation was inversely proportional to transpiration. In 2018, BS had simulated evapotranspiration 36, 93 and 111 mm lower than those for CC P , CC M and CC L , respectively. The relative amounts of simulated evaporation and transpiration were similar in both years of the experiment for a given treatment (Table 3).

Consequently, cover crops reduced predicted water drainage by the same degree, regardless of their management in a given year. Compared to BS, reductions of ca. 60 and 15 mm were predicted in 2017 and 2018, respectively. Water drained 14 and 10 days later under cover crop treatments than under BS in 2017 and 2018, respectively, indicating a delay in filling in the soil water reserve due to higher AET.

(Figure 6)

Water balance in the 20-year simulations

For the two 20-year simulations, cover crop biomass simulated for CC L ranged from 1.9-8.0 ha -1

(mean = 5.0 Mg.ha -1 ), indicating that the biomass measured during the experiment corresponded to the largest range of biomass that could be expected at this site. Simulated cover crop biomass for CC M and CC P at autumn termination ranged from 2.2-6.6 Mg.ha -1 (mean = 4.1 Mg.ha -1 ), indicating that the biomass measured during the experiment lay around the mean biomass that could be expected at this site. The clover regrowth simulated for CC M ranged from 1.0-3.5 Mg.ha -1 (mean = 2.0 Mg.ha -1 ),

indicating that its regrowth measured during the experiment corresponded to the largest range of regrowth that could be expected at this site.

Over 20 years, the SWC simulated on 1 April for CC L and CC M differed significantly from that for BS (Fig. 7), confirming the potential impact of cover crops on the water supply of the next cash crop in certain years. Simulated mean SWC was 50 mm between CC L and BS (significantly different) and 60 mm between CC M and BS, corresponding to the range simulated for the experiment. Simulated mean SWC of CC P did not differ significantly from that of BS over the 20 years, regardless the management year used (2017 or 2018), and was often near FC. For the 2017 management, mean SWC of CC L differed significantly from those of CC P and BS. The impact of CC M depended greatly on the climate year, with high variability and a mean reduction compared to BS of ca. 60 mm. For the 2018 management, the difference in SWC between BS and CC M was ca. 20 mm. Differences in simulated SWC were compared for 1 April. BS and CC P reached FC (or nearly so), with little variability among climate years. Simulated SWC varied greatly for CC L and CC M : for 25% of climate years, it was less than 55% of TAWC. Only three of the 20 simulated years showed no difference in SWC between treatments. These years were particularly rainy in winter and spring, which frequently increased SWC to FC. The influence of sowing and termination dates was large, with the difference between SWC for CC L and CCM M inverted depending on the year of cover crop management simulated: median SWC for CC M was ca. 25 mm higher and ca. 15 mm lower than that for CC L for 2017 and 2018 management, respectively. However, these differences were not statistically significant over the 20 simulated years. Differences in simulated SWC were not due to differences in soil moisture of the 0-20 cm layer, which did not differ significantly on 1 April among the BS, CC P , CC M , and CC L treatments (14.8, 14.8, 15.4, and 15.0 cg.g -1 , respectively). SWC simulated over the 20 years differed from that measured in 2017, during which all treatments reached FC, and in 2018, during which cover crops reduced SWC in spring to a degree found at the higher end of the range simulated over the 20 years.

Water drainage for BS ranged from 0-235 mm for the 20 simulated years. Three years had no drainage for both the 2017 and 2018 management years. Mean and median simulated drainage was 67 mm and 48 mm, respectively, which correspond to low drainage at that site. This indicates that the drainage observed during the experiment was among the highest 25% in 20 years, due to the high winter rainfall. The mean difference in drainage of the cover crop treatments compared to BS was 45 mm and 20 mm for the 2017 and 2018 management years, respectively (Fig. 7). These differences represented a reduction in total drainage of ca. 50% and 25% for 2017 and 2018, respectively. For 25% of the 20 years, no drainage was simulated for the three cover crop treatments, unlike under BS, regardless of the management year used. Over 20 years with the 2018 management, CC M had three years with the same or slightly higher drainage than BS (mean difference = +4 mm). Drainage under cover crops treatments had greater variability than that under BS: in some years, simulated drainage started on the same date under the cover crops and BS, due to a rainy autumn, while in others, drainage started up to 3 months later under cover crops than under BS, especially after a dry autumn. However, the median difference for the beginning of drainage between BS and the cover crop treatments was 14 days, which corresponds roughly to that observed in the experiment.

Like in the experiment, BS had significantly lower simulated AET over the 20 years than the cover crop treatments due to the lack of transpiration. The difference in AET between the three cover crop treatments was small but significant for 2017 management, but not significant for 2018 management.

Compared to the other fluxes, AET varied the least over the 20 years.

(Figure 7) Some studies indicated that cover crops did not influence SWC in spring, as in 2017 in our field experiment [START_REF] Daigh | Soil water during the drought of 2012 as affected by rye cover crops in fields in Iowa and Indiana[END_REF], while others reported a decrease in SWC due to low rainfall in spring, as in 2018 in our field experiment [START_REF] Corak | Accuracy, robustness and behavior of the STICS soil-crop model for plant, water and nitrogen outputs: Evaluation over a wide range of agro-environmental conditions in France[END_REF][START_REF] Restovich | Introduction of cover crops in a maize-soybean rotation of the Humid Pampas: Effect on nitrogen and water dynamics[END_REF]. Blanco-Canqui et al.

(2011) suggested that soil moisture increases from 0-20 cm deep, but we found no difference among the treatments over the experiments or the 20-year simulations. Little rainfall before the last sampling date could have removed differences in soil moisture from 0-20 cm, which could explain why a difference was observed only from 20-60 cm.

The impact of cover crops on the yield of the following cash crop differs in the literature: some studies reported a decrease in yield (Nielsen et al., 2015b), while others reported no negative impacts [START_REF] Eshel | Benefits of growing potatoes under cover crops in a Mediterranean climate[END_REF]. In some cases, yield can increase after a cover crop, especially when legumes are used [START_REF] Tonitto | Replacing bare fallows with cover crops in fertilizer-intensive cropping systems: A meta-analysis of crop yield and N dynamics[END_REF]. However, these studies did not always identify why yield differed. Yield may decrease due to water stress at the beginning of the cash crop, but it can also be caused by allelopathic effects [START_REF] Kessavalou | Winter rye cover crop following soybean under conservation tillage[END_REF] or a decrease in soil N content compared to that of bare soil [START_REF] Tonitto | Replacing bare fallows with cover crops in fertilizer-intensive cropping systems: A meta-analysis of crop yield and N dynamics[END_REF]. In both years of the experiment, CC L and CC M had lower SWC in soil layers 20-60 cm deep than BS and CC P at the end of the experiment, which suggests that late termination of a cover crop decreases SWC due to spring transpiration. If spring rainfall is too low to make up for this difference, growth of the next cash crop could decrease. [START_REF] Krueger | Winter rye cover crop management influences on soil water, soil nitrate, and corn development[END_REF] highlighted the importance of the termination date on reducing the negative impact of cover crops on yields.

The 20-year simulations showed that cover crops reduced both mean and median SWC simulated on 1

April and that spring rainfall was not sufficient to make up for the cover crops' lower SWC compared to that of BS. Indeed, the effective rainfall: which can indicate refilling of soil water, was positive for only five of the 20 years, and of these five, the soil was refilled up to FC in only two of them. SWC of CC P did not differ significantly from that of BS, which indicates that sufficient rain fell from termination to early spring to compensate for cover crop transpiration. This indicates that early cover crop termination avoids large differences in SWC in spring between cover crops and bare soil in temperate and Mediterranean climates, as suggested by [START_REF] Alonso-Ayuso | The kill date as a management tool for cover cropping success[END_REF]. Early termination could be a good way to avoid negative impact, such as pre-emptive competition for water, on the next crop. Termination dates between November and April could be a good compromise to avoid a negative impact on SWC and also could reduce nitrate leaching and soil erosion better and maximize soil C storage. Terminating cover crops before winter could be the best solution in dry regions where water is scarce. Since we tested only two termination dates per year, however, we could not determine an optimal termination date. This should be investigated in the future, such as in a site-specific simulation study.

Sowing date of cover crops is key to minimizing reduction in drainage

In a recent meta-analysis, [START_REF] Meyer | Cover crops reduce water drainage in temperate climates : A meta-analysis[END_REF] reported that cover crops reduce water drainage by a mean of 27 mm compared to that under bare soil, but the reduction varies widely among studies.

Depending on the year and regardless of the cover crop management, our field experiment indicated that cover crops reduce drainage by 20-60 mm, which lies in the range found in the meta-analysis.

Early cover crop termination in autumn or early winter did not influence the reduction greatly, since the same drainage reduction was observed regardless of the termination date or residue management in the two years of the experiment. The 20-year simulations confirmed results of the experiment, showing no difference in drainage reduction among the cover crop treatments for a given sowing date and year. The range of annual reduction in predicted drainage over the 20 years (0-80 mm) was consistent with the variability in results reported in the literature and can explain this variability, since the impact on drainage depends on the interaction between weather conditions and cover crop management. These differences can represent a reduction of nearly 50% in years with high drainage and a complete lack of drainage in drier years. In our experiment, the decrease in drainage due to cover crops was one-third the amount with later sowing (late August) than early sowing (late July).

This indicates that sowing date could be a key point to consider when analyzing the impact of cover crops on drainage, as demonstrated by [START_REF] Justes | Cover crops for sustainable farming[END_REF].

No direct relationship observed between cover crop biomass and impact on water fluxes above a certain level

Compared to bare soil, cover crops increase evapotranspiration by increasing transpiration even though their cover decreases soil evaporation. In the experiment, cover crops increased evapotranspiration by a mean of 30%. This result agrees with the literature (Nielsen et al., 2015a;[START_REF] Qi | Soil Water dynamics under winter rye cover crop in central Iowa[END_REF], in which several studies mention the relationship between biomass and transpiration [START_REF] Suyker | Evapotranspiration of irrigated and rainfed maize-soybean cropping systems[END_REF][START_REF] Tolk | Transpiration and yield relationships of grain sorghum grown in a field environment[END_REF]. However, no significant relationship was observed between cover crop biomass and evapotranspiration or drainage in either year of the experiment. Tribouillois et al. (2018) reported a correlation between cover crop biomass and an evapotranspiration. They assessed cover crops with lower biomass (0.5-2.5 Mg.ha -1 yr -1 ) than we did (> 8 Mg.ha -1 ). Crops transpiration is linked with the biomass through the leaf area index (LAI) and it is assumed that above a certain level of LAI, transpiration does not increase in proportion, and remains constant from a certain level [START_REF] Kang | Crop coefficient and ratio of transpiration to evapotranspiration of winter wheat and maize in a semi-humid region[END_REF][START_REF] Kristensen | Actual Evapotranspiration in Relation To Leaf Area[END_REF]. This could explain why we did observed no correlation between cover crop biomass and evapotranspiration or drainage. Tribouillois et al. (2018b) also reported a strong negative correlation between the mean decrease in annual drainage and the mean increase in annual evapotranspiration due to cover crops. This latter result was found in the 2017 experimental year. However, in 2018, no significant correlation was observed between the difference in drainage and evapotranspiration. The increase in evapotranspiration influenced not only drainage, which decreased only slightly, but also the soil water content in spring, which was much lower for CC L and CC M than for BS. This could have been due to differences in rainfall distribution during the fallow period.

Study boundaries

Field experiment and simulations

Irrigation was applied after sowing the cover crop to ensure homogenous emergence and establishment. Irrigation in summer, which is a dry period in southwestern France, could explain the large amount of biomass observed, since strong growth of juvenile stages favored crop establishment.

Since we included irrigation in the 20-year simulations, STICS also predicted large amounts of biomass. However, farmers in the region currently do not irrigate cover crops. Consequently, water stress could occur with an early sowing date, such as July or early August in our conditions, which could result in lower biomass. The impact of cover crops on the water balance could change, since the climate in southwestern France is dry in summer, and August rain is not always sufficient to ensure adequate emergence and development of cover crops to produce a sufficient amount of biomass. To evaluate the impact of irrigation, we also performed the 20-year simulations without irrigation. The results (not shown) indicated that drainage was slightly lower, but not significantly so. Predictions for the impact on SWC and the difference in drainage compared to BS were similar with or without irrigation. Without irrigation, the cover crop did not develop in some dry summers, or developed later.

In such cases, cover crops would not influence water fluxes.

The field experiment was performed on two similar soils at the same INRA experimental station. The soils had high SWC due to their great depth. More investigation is required to determine the importance of cover crops on shallow soils. On shallow soils, cover crops could have less impact on drainage due to less water lost via transpiration because their roots would not extend as deep.

In 2018, soil texture in the deeper layers of the field plot varied greatly, even over short distances, which could have masked significant differences in SWC due to cover crop treatments, as predicted in the 20-year simulation. This highlights the ability of the strong complementarity between field experiments and modeling to help understand and quantify dynamic interactions between treatments and weather conditions.

STICS parameterization and initialization

An initial step in our study was to evaluate the ability of STICS to predict aerial biomass of cover crops and SWC over time for the entire fallow period to obtain satisfactory water flux simulations.

Biomass and SWC were simulated sufficiently well for both years and were similar to or even an improvement on results of previous studies [START_REF] Brisson | STICS: a generic model for simulating crops and their water and nitrogen balances. II. Model validation for wheat and maize[END_REF][START_REF] Corak | Accuracy, robustness and behavior of the STICS soil-crop model for plant, water and nitrogen outputs: Evaluation over a wide range of agro-environmental conditions in France[END_REF]. STICS, already used successfully to simulate drainage under cover crops [START_REF] Constantin | Long-term nitrogen dynamics in various catch crop scenarios: Test and simulations with STICS model in a temperate climate[END_REF][START_REF] Tribouillois | Cover crop mixtures including legume produce ecosystem services of nitrate capture and green manuring: assessment combining experimentation and modelling[END_REF], remained accurate in our study. We can therefore assume that STICS simulated the fluxes under different management practices sufficiently well. Nonetheless, we could have improved calibration of the cover crop mixture had we measured dynamics of the leaf area index, since leaf area governs transpiration.

SWC predictions would have been more accurate if soil water content had been initialized for each treatment based on measurements. However, we initialized the soil with the mean SWC measured in the four treatments to ensure that differences in simulated water fluxes would be due only to the management practices, not to different initial states caused by natural field variability. Consequently, STICS slightly overestimated the difference in SWC between CC L and BS in 2018 and thus could have slightly overestimated the difference in drainage and evapotranspiration between cover crops and BS.

Nevertheless, the simulated SWC always lay within 1 SD of observed values, which indicates that overestimates were consistent with experimental results and that predicted differences in water balance among treatments were acceptable.

4.5 Cover crop residues left as mulch can reduce soil evaporation and thus actual evapotranspiration One objective of our study was to analyze the impact of cover crop residues left as mulch via mechanical crushing after the first autumn termination (CC M ) on the water balance, since no reference was available in the literature, despite this practice's great benefit for farmers. Although not planned, the crimson clover regrew after crushing and produced a large amount of biomass. Consequently, CC M caused changes in the water balance similar to those of CC L . Cover crop mulch can increase SWC in the surface layer by reducing evaporation [START_REF] Alliaume | Reduced tillage and cover crops improve water capture and reduce erosion of fine textured soils in raised bed tomato systems[END_REF][START_REF] Moschler | Winter Cover Crops for Sod-Planted Corn: Their Selection and Management1[END_REF][START_REF] Stipešević | Effects of winter wheat cover crop desiccation times on soil moisture, temperature and early maize growth[END_REF]), but we did not observe this effect for clover regrowth in the field experiment. It is well known that residue mulch decreases evaporation. Evapotranspiration under a cover crop terminated early and left as mulch could result in the same cumulative AET as that under bare soil, since compared a bare soil, AET would be higher during cover crop growth, and soil evaporation would be lower after cover crop termination. Thus, a cover crop terminated early and left as mulch could reduce drainage less than a cover crop terminated late. It could also increase soil moisture on the surface (i.e. 0-10 cm) and avoid an overly dry seedbed when the next cash crop is sown. However, it could also make the soil too wet for sowing and cause bearing-capacity problems. This demonstrates that cover crop management, especially the termination date, is important for cover crops to provide a sufficient level of services, such as capturing nitrate or improving soil physical properties [START_REF] Alonso-Ayuso | Assessing cover crop management under actual and climate change conditions[END_REF]. In dry areas where water is scarce, cover crops can reduce drainage and shallow groundwater recharge, and we hypothesis that mulching could be a good practice to maintain the services expected from cover crops, resulting in a good comprise between services and disservices.

Thus, this practice could encourage green manure or catch crop services and avoid disservices by minimizing negative impacts on water-balance fluxes due to pre-emptive competition for water, which reduces drainage and water availability for the next cash crop.

Conclusion

The choice of cover crop termination date and management of cover crop residues could be a way to benefit from all the services they provide, such as the green manure effect or increasing soil physical properties in agroecological systems, while reducing their potential negative effects on the water balance. Our study highlighted the impact of different cover crop management practices and the variability of their water balance compared to that of bare soil. Cover crops clearly increase AET and reduce drainage, but do not always reduce SWC at sowing of the next cash crop, which depends greatly on the rainfall after cover crop termination. A decrease in groundwater recharge must be considered when generalizing cover crops at the regional scale, especially that of shallow groundwater, which is determined by drainage from agricultural soils. We demonstrated that later termination of cover crops could have a negative impact on the next cash crop, even when depending on spring rainfall. However, an optimal solution for cover crop management could include mechanically crushing the cover crop in autumn and leaving the residues as mulch but new experiments must be carried out to verify this hypothesis. Studies that combine field experiments and simulation modeling are required to assess this management practice for other soil and climate conditions, since the issue is also site-specific due to interactions between soil type and depth, and the amount and distribution of rainfall. The potential of cover crop mulch to reduce soil evaporation before sowing the next cash crop could be a good compromise between improving cover crop services and decreasing negative impact on the water balance and its resulting consequences.

  Figure 2) 3.1.2 Soil water measurements during the fallow period Soil water profiles during the fallow period varied by treatment in both years of the experiment, showing no similar trends or final results (Figs. 3a & b). From July-August to November-December, cover crops had significantly less water under them in the soil profile in 2017 than BS did, but they showed no difference compared to BS in 2018. For both years of the experiment, cover crops and BS showed no difference in SWC during winter. From February to March-April, SWC differed significantly between CC L or CC M and BS or CC P in the layers 20-60 cm deep in 2017 and 2018 (Figs. 3a & b).

1

 1 Influence of cover crops on soil water content for the next cash crop depends on their termination date Although the two years of the field experiment had similar cover crop biomass, impact of biomass on the soil water profile differed over time. From sowing to autumn termination in 2017, water depletion was observed along the profile, while in 2018, no difference in the water profile was observed among the treatments. Differences in rainfall dynamics explain why the four treatments showed no differences in the water profile in 2017's spring, while in 2018, there was less water in the soil layers 20-60 cm deep under CC L and CC M .

Figure 1 .

 1 Figure 1. Cumulative (A) rainfall, (B) potential evapotranspiration (PET), and (C) temperature of the study site after sowing cover crops during the 2017 and 2018 experimental years.

Figure 2 .Figure 7 .

 27 Figure 2. Cover crop biomass at (left) autumn or (right) spring destruction in the (top) 2017 and (bottom) 2018 experimental years for the four treatments: bare soil (BS), cover crop crushed in autumn and left as mulch on the soil surface (CCM), cover crop crushed in autumn and buried by plowing (CCP), and cover crop destroyed in April (CCL). Error bars represent ±1standard deviation.

Table 1 .

 1 Mean soil texture and physical properties by soil layer at the study site in 2017 and 2018

	Experiment	Depth (cm)	Clay (%)	Loam (%)	Sand (%)	Bulk density (g cm -3 )	Field capacity (cg g -1 )	Wilting point (cg g -1 )
		0 -20	25.8	30.7	43.5	1.50	17.8	8.9
		20 -40	24.8	30.0	45.2	1.50	17.5	8.8
	2017	40 -60 60 -80	28.2 38.9	30.1 33.5	41.7 27.6	1.50 1.45	17.1 20.0	8.5 13.5
		80 -100	38.6	40.0	21.4	1.40	23.6	13.9
		100 -120	31.3	47.4	21.3	1.40	23.6	11.8
		0 -20	24.7	31.4	43.8	1.45	20.0	10.0
		20 -40	23.3	31.0	45.7	1.45	20.0	10.0
	2018	40 -60 60 -80	22.1 21.2	32.0 29.7	45.9 49.1	1.45 1.50	17.5 18.0	8.7 9.0
		80 -100	20.3	26.6	53.1	1.50	17.8	8.9
		100 -120	20.8	28.5	50.7	1.5	17.8	8.9

Table 2 .

 2 Cover crop management at the study site in 2017 and 2018 for the four treatments: bare soil (BS), cover crop crushed in autumn and left as mulch on the soil surface (CCM), cover crop crushed in autumn and buried by plowing (CCP), and cover crop destroyed in April (CCL).

	Characteristic			2017			2018
	Previous crop		Durum wheat			Durum wheat
	Treatment*	BS	CCM	CCP	CCL	BS	CCM	CCP	CCL
	Sowing date	-		31 July		-	28 Aug
	Destruction date	-	15 Nov 15 Nov 11 Apr	-	07 Jan	07 Jan	09 Apr
	Plowing date	13 Dec	-	13 Dec	-	08 Jan	-	08 Jan	-

Table 3 .

 3 Relative root mean square error of predictions (rRMSEP) of interest of the STICS model

		0-20 cm	20-40 cm	Soil moisture (cg g -1 ) 40-60 60-80 cm cm	80-120 cm	0-120 cm	Cover crop biomass (Mg.ha -1 )
	rRMSEP (%)	15.2	13.0	8.1	5.9	8.5	5.6	9.2

Table 4 .

 4 Simulated soil water cycle variables for 2017 and 2018 (all in mm) in the four treatments: bare soil (BS), cover crop crushed in autumn and left as mulch on the soil surface (CCM), cover crop crushed in autumn and buried by plowing (CCP), and cover crop destroyed in April (CCL). SWC = soil water content.

		Initial					Final
	Experiment Treatment	SWC	Evaporation Transpiration	AET	Drainage	SWC
	BS	254	269	0		158	347
	CCP	254	229	111		110	347
	2017
	CCM	254	151	188		97	343
	CCL	254	136	198		95	347
	BS	275	236	0		150	317
	CCP	275	201	71		130	316
	2018
	CCM	275	149	180		131	246
	CCL	275	134	213		127	228

Acknowledgments

This study was funded by the Adour-Garonne water agency (BAGAGES project) and the Occitanie Region (CLE BAGAGES-Cisol project). Many thanks to Eric Lecloux, Benoit Gleizes, and Pierre Perrin for their help with soil water sampling, installation of capacitance probes, and laboratory analysis. We thank Michelle and Michael Corson for proofreading and correcting the English. We also thank the companies Jouffray Drillaud and RAGT Semences for the seeds. We thank also Daniel Wallach for his help for the crop calibration and model evaluation.


Florent Manzoni

Thierry Vidal
email: vidal@insa-toulouse.fr

Alain Sellier

Xavier Bourbon

Guillaume Camps

On the origins of Transient Thermal Deformation of Concrete

Keywords: concrete, creep, modelling, temperature, transient thermal deformation

The conditions that lead to Transient Thermal Deformation (TTD) of concrete subjected to various temperatures and compressive loads in sealed conditions are analysed and a model based on a two-scale porosity is proposed. In this model, water expands in the smaller pores and diffuses progressively to the capillary pores at the upper scale. It is then shown that the water overpressure in smaller pores and the duration of the diffusion process between the two scales of porosity control the TTD. It is also shown that creep or shrinkage before the temperature transition mitigates TTD. The aim is to provide a model able to predict the conditions for TTD to appear and the amplitude of the TTD under various thermo-mechanical conditions. The TTD model is implemented in an existing basic creep model, fitted and compared with different experimental results.

I. Introduction

In the framework of research surveys steered by the French agency for nuclear waste management (Andra) and the Laboratoire Matériaux et Durabilité des Constructions (LMDC) of Toulouse University, the effect of thermo-hydro-mechanical conditions on concrete deformations have been studied in order to predict the long-term behaviour of High On the origins of Transient Thermal Deformation of Concrete 2 Performance Concrete (HPC). As nuclear wastes are exothermic, they could induce a rise in temperature (up to a maximum of 50 °C) and a dilation of concrete containment structures which, in return, could lead to a soil reaction and then a loading of the tunnel during the temperature rise. Concomitance of this temperature rise and mechanical loading are favourable to the appearance of Transient Thermal Deformation (TTD) of concrete, so the phenomenon has to be quantified. The few experimental studies available in the literature for the 20 °C to 100 °C temperature range show that, if concretes are loaded and heated simultaneously, they develop greater deformation than if they are heated first and loaded later [1] [2]. This additional strain, called Transient Thermal Deformation (TTD), has been observed up to 105 °C, which is the threshold temperature for C-S-H dehydration [START_REF] Sabeur | Dehydration Creep of Concrete at High Temperatures[END_REF] [4] [START_REF] Mindeguia | Parametrical study of transient thermal strain of ordinary and high performance concrete[END_REF] [START_REF] Nechnech | An elasto-plastic damage model for plain concrete subjected to high temperatures[END_REF]. Above this temperature, these authors consider TTD as dehydration creep attributable to chemical transformation due to the temperature increase, which leads to a rearrangement of the microstructure.

The aims of the present work are to clarify the conditions needed for TTD to exist, to understand the physical mechanism that controls it, and then to propose a model able to improve existing creep models in order to consider the influence of the chronology of the thermo-mechanical conditions. After an overview of experimental evidence concerning transient thermal deformation phenomena, the creep model proposed by Sellier et al. [START_REF] Sellier | Concrete creep modelling for structural application: non-linearity, multi-axiality, hydration, temperature and drying effects[END_REF] is recalled and is improved so as to take TTD into account. The efficiency of the proposed modelling is tested by comparing numerical results with the experimental data.

II. Experimental evidence

According to available data bases [START_REF] Fahmi | Effects of sustained and cyclic elevated temperature on creep of concrete[END_REF] [10], transient thermal deformation is irreversible and, if concrete is subjected to heating/cooling cycles under load, TTD only occurs during heating periods. Cagnon et al. [START_REF] Cagnon | Transient thermal deformation of high performance concrete in the range 20°C-40[END_REF] have measured the experimental deformations of a mature concrete (made with CEM I cement) subjected to two thermo-mechanical conditions: first heated and then loaded and, conversely, first loaded and then heated. Figure 1 presents the strain evolutions for the two thermo-mechanical conditions. Contraction is conventionally considered as a negative deformation. The concrete heated first and loaded one day later (blue curves) does not develop TTD, unlike the concrete loaded first then heated one day later (green curves) which does develop TTD. This comparison has shown that the TTD component does not modify the long-term creep kinetics but is just a temporary acceleration of creep. These observations are in accordance with the model of transient thermal creep proposed by Bažant and Kaplan [START_REF] Bažant | Concrete at high temperatures: material properties and mathematical models[END_REF]. However, in their model, transient thermal creep can reappear at each temperature variation while the concrete is loaded. Experiments performed by Cagnon did not confirm this aspect, as shown in Figure 1. During the second heating cycle under loading, the two strain evolutions are similar, revealing that TTD does not reappear after the first heating. This non-repeatability in case of heating is one of the main points of discussion. For several authors [START_REF] Mindeguia | Parametrical study of transient thermal strain of ordinary and high performance concrete[END_REF] [10] [START_REF] Khoury | Transient thermal strain of concrete: literature review, conditions within specimen and behaviour of individual constituents[END_REF], TTD appears only once, with the first heating, and it can reappear only if the temperature peak reached in previous heating is exceeded. Fahmi et al. [START_REF] Fahmi | Effects of sustained and cyclic elevated temperature on creep of concrete[END_REF] showed that saturated concrete developed TTD even after several cycles of heating/cooling. However, for concrete stored at 50% RH, this repeatability did not exist. Thus, he showed that the water content modified the amplitude of the TTD. This strain dependence on the degree of saturation has been confirmed by Cagnon et al. [START_REF] Cagnon | Transient thermal deformation of high performance concrete in the range 20°C-40[END_REF] and Hansen and Eriksson [START_REF] Hansen | Temperature Change Effect on Behavior of Cement Paste, Mortar, and Concrete Under Load[END_REF]. Moreover, the loading rate and heating kinetics can also influence TTD [START_REF] Schneider | Behavior of concrete at high temperatures[END_REF].

To explain this phenomenon, various assumptions have been proposed. According to Khoury and Parrot [START_REF] Khoury | Transient thermal strain of concrete: literature review, conditions within specimen and behaviour of individual constituents[END_REF] [START_REF] Parrot | Lateral strains in hardened cement paste under short and long-term loading[END_REF], the origin of TTD could be a differential thermal dilation between the T°C σ various concrete phases and the associated microcracking. However, TTD is also observable on cement paste, so its origin cannot be attributed only to the differential thermal dilation between paste and aggregates. Torrenti [START_REF] Torrenti | Basic creep of concrete-coupling between high stresses and elevated temperatures[END_REF] proposed a damage model coupled with creep to simulate the transient thermal deformation. In his model, the evolution of damage amplifies the stress and thus the viscoelastic strain; the additional strain obtained is considered to be the TTD. Experimentally, the evolution of Young's modulus measured before and after thermomechanical tests at 40 °C in [START_REF] Cagnon | Transient thermal deformation of high performance concrete in the range 20°C-40[END_REF] showed that TTD did not damage the material. Several authors suggest that the transient thermal deformation phenomenon has its origin in water movement at the nanoporous C-S-H scale [10] [12] [15] [16] [START_REF] Bažant | Temperature Effect on Concrete Creep Modeled by Microprestress-Solidification Theory[END_REF]. Some models proposed are usually based on self-desiccation [START_REF] Bažant | Concrete at high temperatures: material properties and mathematical models[END_REF] or thermal activation [START_REF] Nechnech | An elasto-plastic damage model for plain concrete subjected to high temperatures[END_REF] [18] [START_REF] Anderberg | Stress and deformation characteristics of concrete at high temperatures: experimental investigation and material behavior model[END_REF] and are a function of the thermal variation rate (𝑇 ̇). Thus, according to these models, the transient thermal strains will be repeated in case of heating cycles, whatever the maximum temperature reached, which is not in accordance with Cagnon's experimental observations. Thus, there is a need to improve the understanding and the modelling of this phenomenon. For this purpose, a doublescale porosity model with water transfer from one scale to the other is proposed. The overpressure in the finest porosity is assumed to control the material viscosity. This dependence of the viscosity on the pressure at nanoscale is implemented in an existing basic creep model and successfully confronted with literature results.

III. Modelling of experiments

This section first presents the creep model in which the TTD is to be implemented. A lot of creep models are proposed in the literature. Benboudjema et al. [START_REF] Benboudjema | Interaction between drying, shrinkage, creep and cracking phenomena in concrete[END_REF] split the basic creep into a spherical and a deviatoric part. Thus de spherical creep corresponding to a migration of adsorbed water in micro and nanoporosity and the deviatoric creep is the sliding of C-S-H layers. The model is based on the assumption that the basic creep kinetics of concrete is controlled by two antagonistic phenomena: the material viscosity, which allows creep, and a consolidation phenomenon, which corresponds to a transfer of stresses from viscous particles (typically C-S-H bundles) to elastic particles (other hydrates and aggregates). As the stresses in the non-viscous particles increase, the C-S-H are unloaded, which slows down their strain rates. TTD is assumed to take place in the nanoporisity of C-S-H bundles: due to the temperature rise, the embedded water dilates and its pressure increases, transiently modifying the bond between C-S-H particles and so their viscosity. Once this overpressure has been depleted by diffusion of water from the C-S-H to capillary pores, the C-S-H bundles return to their initial configurations and the creep velocity returns to that of basic creep.

III.1. Principles of the creep model

The model chosen to implement the TTD phenomenon is described in [START_REF] Sellier | Concrete creep modelling for structural application: non-linearity, multi-axiality, hydration, temperature and drying effects[END_REF]. As explained above, in this model the slowdown of creep kinetics is due to a consolidation phenomenon that increases the apparent material viscosity. The consolidation function assumes that there is a strain potential that is intrinsic to the concrete, and each deformation (shrinkage and creep) consumes this potential and consolidates the material, thus slowing down creep. The strain is modelled with a Maxwell model in which the viscosity evolves with the strain [START_REF] Sellier | Concrete creep modelling for structural application: non-linearity, multi-axiality, hydration, temperature and drying effects[END_REF]; this strain corresponds to 𝜀 𝑀𝐶 in Eq. (1).

Meanwhile, Cagnon et al. [START_REF] Cagnon | Transient thermal deformation of high performance concrete in the range 20°C-40[END_REF] observed experimentally that TTD did not modify later creep kinetics. In other words, the creep kinetics is the same at the end of the TTD whatever the TTD magnitude. Everything happens as if the concrete fluidifies momentarily, without any effect of TTD on the consolidation process. From a macroscopic point of view, this is similar to a temporary fluidification of concrete that disappears approximately with the end of heating. As basic creep and TTD are both viscous strains but only the basic creep creates a consolidation, the creep strain 𝜀 𝑀 in Eq. ( 1) has be decomposed into two parts: a consolidating strain 𝜀 𝑀𝐶 (corresponding to the initial basic creep model proposed in [START_REF] Sellier | Concrete creep modelling for structural application: non-linearity, multi-axiality, hydration, temperature and drying effects[END_REF]) and an un-consolidating strain 𝜀 𝑀𝐶 ̅ for the TTD. The former corresponds to the basic creep, which depends and acts on the consolidation process, and the latter, corresponding to the TTD, depends on the consolidation but does not act on it.

𝜕𝜀 𝑀 𝜕𝑡 = 𝜕𝜀 𝑀𝐶 𝜕𝑡 + 𝜕𝜀 𝑀𝐶 ̅ 𝜕𝑡 (1)
The two components of this creep model are explained below: first the basic creep part, then the TTD.

III.2. Basic creep equations

The strain 𝜀 𝑀𝐶 corresponding to basic creep comes from Sellier et al. [START_REF] Sellier | Concrete creep modelling for structural application: non-linearity, multi-axiality, hydration, temperature and drying effects[END_REF]. It is briefly recalled in this part.

III.2.1. Permanent creep

In each main direction, I , of elastic strain, the basic creep velocity, presented in Eq. ( 2), is proportional to the elastic strain 𝜀 𝐼 𝐸 via a characteristic time 𝜏 𝐼 𝑀 :

𝜕𝜀 𝐼 𝑀𝐶 𝜕𝑡 = 𝜀 𝐼 𝐸 𝜏 𝐼 𝑀 . 𝐶 𝑤 (2)
In Eq. ( 2), 𝐶 𝑤 stands for the influence of water content on creep velocity. It is given by Eq.

(3). The temperature acts through the moisture effect because water viscosity changes with temperature [START_REF] Ladaoui | Effect of a temperature change from 20 to 50°C on the basic creep of HPC and HPFRC[END_REF]. This effect is modelled with an Arrhenius law in Eq. ( 4). In this equation, 𝐸 𝑊 𝑎 corresponds to the water viscosity activation energy (≈17000 J/mol), 𝑅 is the perfect gas constant (8.31 J/mol/K) and 𝑇 𝑟𝑒𝑓 is the reference temperature (for which 𝜏 𝑟𝑒𝑓 𝑀 (Eq. 6) is fitted). As C-S-H properties also depend directly on the water content, the term 𝐶 𝑠 𝑤 (Eq. 5) is used. It corresponds to the water saturation of the porosity, 𝑆𝑟, and allows creep velocity reduction when the concrete dries.

𝐶 𝑤 = 𝐶 𝑠 𝑤 . 𝐶 𝑇 𝑤 𝐶 𝑇 𝑤 = 𝑒𝑥𝑝 (- 𝐸 𝑊 𝑎 𝑅 ( 1 𝑇 - 1 𝑇 𝑟𝑒𝑓 )) (4)
𝐶 𝑠 𝑤 = 𝑆𝑟 (5)
The characteristic time 𝜏 𝐼 𝑀 of Eq. ( 2) is defined by Eq. ( 6) where 𝜏 𝑟𝑒𝑓 𝑀 is a fitting parameter (reference characteristic time) characterizing the initial creep velocity of a saturated concrete never loaded before.

𝜏 𝐼 𝑀 = 𝜏 𝑟𝑒𝑓 𝑀 . 𝐶 𝐼 𝑐 (6)
𝐶 𝐼 𝑐 is the consolidation function. It allows the creep velocity to be modified according to the creep that has already occurred and the thermo-mechanical conditions that are integrated into the creep coefficient 𝑘 presented in Eq. ( 8)

𝐶 𝐼 𝑐 = 1 𝑘 𝑒𝑥𝑝 ( 1 𝑘 ( 𝜀 𝐼 𝑀𝐶 𝜀 𝐼 𝐸 ) + ) (7)
The function 𝐶 𝐼 𝑐 increases with the permanent strains 𝜀 𝐼 𝑀𝐶 due to shrinkage and creep, expressing a material consolidation. Thus, it induces a function enhancing the characteristic time and leading to a creep velocity decrease.

𝑘 = 𝑘 𝑟𝑒𝑓 . 𝐶 𝑝 𝑇 . 𝐶 𝑀 (8)
In this equation, 𝐶 𝑝 𝑇 (Eq. 10) and 𝐶 𝑀 (Eq. 11) stand for the thermal and the mechanical effects on creep potential, respectively, according to Sellier et al. [START_REF] Sellier | Concrete creep modelling for structural application: non-linearity, multi-axiality, hydration, temperature and drying effects[END_REF]. 𝐶 𝑝 𝑇 considers the effect of thermal damage and 𝐶 𝑀 the effect of mechanical damage. Thus, in case of thermo-mechanical variations, the potential is automatically adapted. 𝑘 𝑟𝑒𝑓 is the creep coefficient calculated in Eq. ( 9). Note that these temperature effects are not the causes of TTD but only the causes of basic creep velocity dependence on the temperature.

𝑘 𝑟𝑒𝑓 = 𝜀 𝑟𝑒𝑓 𝑀 𝜀 𝑟𝑒𝑓 𝐸 (9)
𝜀 𝑟𝑒𝑓 𝑀 is a fitting parameter called the reference creep potential, which represents an inherent permanent delayed strain characteristic of the material; 𝜀 𝑟𝑒𝑓 𝐸 is the reference elastic strain defined as the material deformation under uniaxial compressive stress of 33% of its compressive strength.

According to Ladaoui et al. [START_REF] Ladaoui | Analysis of interactions between damage and basic creep of HPC and HPFRC heated between 20 and 80°C[END_REF], the differential dilation between paste and aggregates could cause thermal damage and increase creep potential. This damage is modelled with an Arrhenius law, as presented in Eq. [START_REF] Cagnon | Transient thermal deformation of high performance concrete in the range 20°C-40[END_REF].

𝐶 𝑝 𝑇 = ( 𝑒𝑥𝑝 (- 𝐸 𝑝 𝑎 𝑅 ( 1 𝑇 - 1 𝑇 𝑡ℎ𝑟 )) 𝑖𝑓 𝑇 > 𝑇 𝑡ℎ𝑟 1 𝑖𝑓 𝑇 ≤ 𝑇 𝑡ℎ𝑟 (10)
𝑇 𝑡ℎ𝑟 is the threshold temperature (≈45 °C for the material studied in [START_REF] Ladaoui | Analysis of interactions between damage and basic creep of HPC and HPFRC heated between 20 and 80°C[END_REF]) above which thermal damage occurs, and 𝐸 𝑝 𝑎 is a fitted activation energy (≈25000 J/mol).

The 𝐶 𝑀 coefficient takes the possible non-linear effect between creep potential and mechanical loading level into account, through an equivalent Drucker-Prager shear stress 𝜏 𝐷𝑃 .

In Eq. ( 11), the critical shear stress 𝜏 𝑐𝑟 𝐷𝑃 is the critical value causing tertiary creep by microstructure damage.

𝐶 𝑀 = 𝜏 𝑐𝑟 𝐷𝑃 𝜏 𝑐𝑟 𝐷𝑃 -𝜏 𝐷𝑃 (11)
To consider the beneficial effects of triaxial confinement in reducing the impact of deviatoric stress, 𝜏 𝐷𝑃 is calculated in Eq. ( 12), which corresponds to a Drucker-Prager criterion:

𝜏 𝐷𝑃 = √ 𝜎 𝑑 :𝜎 𝑑 2 + 𝛿 𝑇𝑟(𝜎) 3 (12)
where 𝜎 𝑑 is the deviatoric part of the stress tensor and 𝛿 is the confinement effect coefficient.

The critical shear stress is linked to a corresponding uniaxial critical compressive stress intensity, 𝜎 𝑐𝑟 , presented in Eq. [START_REF] Torrenti | Basic creep of concrete-coupling between high stresses and elevated temperatures[END_REF].

𝜏 𝑐𝑟 𝐷𝑃 = 𝜎 𝑐𝑟 √3 (1 - 𝛿 √3 ) (13)
𝜎 𝑐𝑟 = 2 3 𝑅 𝑐 ( 𝜒 𝑀 𝜒 𝑀 -1 ) (14)
𝜒 𝑀 is a non-linear creep coefficient corresponding to non-linear creep amplification observed under a compression stress of 66% of the compressive strength and is close to 2 for an Ordinary Portland Cement based concrete [START_REF] Sellier | Concrete creep modelling for structural application: non-linearity, multi-axiality, hydration, temperature and drying effects[END_REF].

III.2.2. Reversible creep

In this model, a reversible creep is modelled using a Kelvin module 𝜀 𝐼 𝐾 (Eq. 15). The maximum amplitude of reversible creep is made proportional to the elastic strain by means of parameter 𝜓 𝐾 .

𝜕𝜀 𝐼 𝐾 𝜕𝑡 = 1 𝜏 𝐾 ( 𝜀 𝐼 𝐸 𝜓 𝐾 -𝜀 𝐼 𝐾 ) (15)
The characteristic time 𝜏 𝐾 is defined by Eq. ( 16), where 𝜏 𝑟𝑒𝑓 𝐾 is a fitting parameter associated with a reference time characterizing the initial material recovery.

𝜏 𝐾 = 𝜏 𝑟𝑒𝑓 𝐾 . 𝐶 𝑇 𝑤 (16)
A thermal effect parameter 𝐶 𝑇 𝑤 affects the reversible creep velocity in the same way as for the permanent creep Eq. (4).

III.3. Modelling of Transient Thermal Deformation

The strain associated with the TTD is 𝜀 𝑀𝐶 ̅ in Eq. ( 1). It is now presented and explained.

III.3.1. Physical phenomenon

According to Vlahinić et al. [START_REF] Vlahinić | Transient creep effects and the lubricating power of water in materials ranging from paper to concrete and Kevlar[END_REF], transient creep under moisture content variation can be modelled by a modification of the material viscosity in relation to the fluid interflow at nanoscale. This flow is activated to balance a pressure difference between two pore regions, macropores and nanopores. In our case, if the material is heated, nanoporous water dilation changes C-S-H organization and consequently modifies their apparent viscosity until the overpressure induced by the water dilation is depleted by the flow from C-S-H intra porosity to the capillary porosity at the upper scale. During this process, the C-S-H viscosity is reduced and the phenomenon can be considered as a transient fluidification of concrete, which causes creep amplification. The pressure difference between capillary and nanoporosity causes water to diffuse from nanopores to capillary pores until the pressures rebalance. Consequently, if the material is heated long enough before mechanical loading, the water transfer is finished before the creep strain develops and TTD does not occur. Thus, the transient thermal deformation can appear only if the loading is applied before the fluidification is over and thus before the system has returned to a steady state.

III.3.2. Water transport equation

To model the nanoporous pressure 𝑃 𝑛 in C-S-H bundles, a poro-mechanics analogy is proposed (Eq. 17).

𝑃 𝑛 = 𝑀 𝑛 (∅ 𝑛 -∅ 𝑟𝑒𝑓 𝑛 ) (17)
For a unit volume of concrete, the water pressure 𝑃 𝑛 is proportional to the difference between the C-S-H nanoporous water volume, ∅ 𝑛 , and to a reference volume, ∅ 𝑟𝑒𝑓 𝑛 . At this scale, the porosity is an adsorbed layer that stays saturated with water even if the hygrometry is low (down to 11% RH) according to Jennings [START_REF] Jennings | Colloid model of C-S-H and implications to the problem of creep and shrinkage[END_REF]. Thus, ∅ 𝑟𝑒𝑓 𝑛 can also be considered equivalent to the initial nanoporous volume of C-S-H. The C-S-H gel represents approximately 70% of the cement paste volume for classical cementitious material (Portland cement), and the average porosity is 28% [START_REF] Powers | The thermodynamics of volume change and creep[END_REF]. According to Jennings [START_REF] Jennings | Refinements to colloid model of C-S-H in cement: CM-II[END_REF], the nanoporosity of C-S-H is of the order of 18% of the global porosity. Consequently, the nanoporous volume is roughly 3.5% of the cement paste volume (0.7x0.28x0.18 = 0.035m 3 /m 3 of cement paste). The proportionality coefficient 𝑀 𝑛 is also defined in the framework of the poro-mechanics analogy, through Eq. [START_REF] Schneider | Concrete at high temperature-A general review[END_REF]. It is a Biot modulus at the C-S-H scale, which considers the interactions between C-S-H and water.

1

𝑀 𝑛 = 𝑏 𝑛 -∅ 𝑟𝑒𝑓 𝑛 𝐾 𝑟 + ∅ 𝑟𝑒𝑓 𝑛 𝐾 𝑤 (18)
where 𝑏 𝑛 is the Biot coefficient for nano sites (≈ 0.65) [START_REF] Ulm | Is concrete a poromechanics material? -A multiscale investigation of poroelastic properties[END_REF]; 𝐾 𝑟 is the C-S-H stiffness; 𝐾 𝑤 is the water compressibility, which can be considered as a constant in the temperature range studied (≈ 2 300MPa). Under the assumption that C-S-H stiffness is about 30 GPa [START_REF] Vandamme | Nanogranular Origin of Concrete Creep[END_REF], the Biot modulus is approximately 28 GPa.

The flow necessary to rebalance the pressure difference between nanoscale and capillary scale is defined through Eq. ( 19) for a unit of material volume. The pressure difference is dissipated by water diffusion until the system is balanced, i.e. P n = P c .

𝜕𝑚 𝑛→𝑐 𝜕𝑡 = - 𝑘 𝑛 .(𝑃 𝑛 -𝑃 𝑐 ) 𝜂(𝑇)
. 𝜌(𝑇)

In this equation, the mass flow 20), is obtained from the Van-Genuchten isotherm, without the gas pressure, according to the assumption proposed by Mensi et al. [START_REF] Mensi | Séchage du béton: analyse et modélisation[END_REF] and presented as a function of capillary saturation state 𝑆𝑟 𝑐 .

𝑃 𝑐 = -𝑀 𝑐 (𝑇). (𝑆𝑟 𝑐 (- 1 𝑚𝑣𝑔 ) -1) (1-𝑚𝑣𝑔) (20)
𝑚𝑣𝑔 is the fitting parameter controlling the shape of the isotherm curve, and 𝑀 𝑐 is a fitting parameter proportional to the capillary tension (Eq. 21), and thus depending on the temperature in order to respect an isothermal model with temperature variation (Figure 2) as proposed by Chhun [START_REF] Chhun | Modélisation du comportement thermo-hydro-chemo-mécanique des enceintes de confinement nucléaire en béton armé-précontraint[END_REF].

𝑀 𝑐 (𝑇) = 𝑀 𝑐, 𝑟𝑒𝑓 [𝑒𝑥𝑝 (-

𝑇-𝑇 𝑟𝑒𝑓 ∆𝑇 𝑟𝑒𝑓 )] (21)
Figure 2: Modelling of desorption isotherms at different temperatures [START_REF] Chhun | Modélisation du comportement thermo-hydro-chemo-mécanique des enceintes de confinement nucléaire en béton armé-précontraint[END_REF] In this equation, 𝑀 𝑐, 𝑟𝑒𝑓 is the capillary tension coefficient at the reference temperature, 𝑇 𝑟𝑒𝑓 , and ∆𝑇 𝑟𝑒𝑓 is a reference temperature variation presented in Table 1. The capillary tension, and consequently the water capillary pressure, decrease when heating is applied.

Constants Value used

∆𝑇 𝑟𝑒𝑓 20 K 𝑇 𝑟𝑒𝑓 293 K 𝑀 𝑐,𝑟𝑒𝑓 10 MPa mvg 0.26
Table 1: Values of constants used to model the capillary tension [START_REF] Chhun | Modélisation du comportement thermo-hydro-chemo-mécanique des enceintes de confinement nucléaire en béton armé-précontraint[END_REF] The effect of temperature, 𝑇, on water density, 𝜌, is modelled using the empirical equation proposed by Thiensen, Eq. ( 22) [START_REF] Tanaka | Recommended Table for the Density of Water between 0 °C and 40 °C Based on Recent Experimental Reports[END_REF].

𝜌(𝑇) = 𝜌 𝑟𝑒𝑓 . [1 - (𝑇-𝑇 1 ) 2 .(𝑇-𝑇 2 ) 𝑇 3 2 .(𝑇-𝑇 4 ) ] (22)
𝜌 𝑟𝑒𝑓 is the reference water density at 273 K and at atmospheric pressure; the other parameters are presented in Table 2. The water density decreases with heating as represented in Figure 3 within the studied temperature range. Ladaoui et al. [START_REF] Ladaoui | Effect of a temperature change from 20 to 50°C on the basic creep of HPC and HPFRC[END_REF] model the water viscosity evolution with temperature, Eq. ( 23), with an Arrhenius law as the 𝐶 𝑇 𝑤 coefficient of Eq. ( 4):

𝜂(𝑇) = 𝜂 𝑟𝑒𝑓 𝑒 - 𝐸 𝑤 𝑎 𝑅 ( 1 𝑇 - 1 𝑇 𝑣,𝑟𝑒𝑓 ) (23)
This law is a function of a reference viscosity 𝜂 𝑟𝑒𝑓 (2.07x10 -14 MPa.d) at a reference temperature 𝑇 𝑣,𝑟𝑒𝑓 (273 K). The evolution of the viscosity is presented in Figure 3. It decreases with temperature rise, which facilitates the C-S-H movement by lubrication.

Figure 3: Modelling of water density and apparent viscosity evolution versus temperature [START_REF] Tanaka | Recommended Table for the Density of Water between 0 °C and 40 °C Based on Recent Experimental Reports[END_REF] and [START_REF] Ladaoui | Effect of a temperature change from 20 to 50°C on the basic creep of HPC and HPFRC[END_REF].

Considering the small quantity of water exchangeable between nanoscale and capillary scale, the water supplied to the capillary porosity is assumed not to affect the capillary pressure, which is mainly controlled by the water exchanges at capillary scale only.

The introduction of the nanoporous water pressure (Eq. 17) in the diffusion equation (Eq. [START_REF] Anderberg | Stress and deformation characteristics of concrete at high temperatures: experimental investigation and material behavior model[END_REF] links the capillarity pressure and the nanoporous water pressure (Eq. [START_REF] Jennings | Colloid model of C-S-H and implications to the problem of creep and shrinkage[END_REF].

𝜕𝑃 𝑛 𝜕𝑡 = (𝑃 𝑐 -𝑃 𝑛 ) 𝜏 𝑛 (24)
with 𝜏 𝑛 the characteristic time for the evacuation of nanoporous pressure defined in Eq. [START_REF] Ulm | Is concrete a poromechanics material? -A multiscale investigation of poroelastic properties[END_REF].

𝜏 𝑛 = 𝜂(𝑇) 𝑀 𝑛 .𝑘 𝑛 . (25)
Therefore, the state variable of this physical phenomenon is simply the C-S-H nanoporous water pressure.

Figure 4 illustrates the pressure evolution of nanoporous water modelled with the parameter values (Table 4) used to fit Cagnon's experimental data [START_REF] Cagnon | Transient thermal deformation of high performance concrete in the range 20°C-40[END_REF] (Figure 1 and Figure 7). When the temperature rises, the water thermal dilation creates an overpressure in nanoporosity until the temperature stabilizes. From that moment, water diffusion allows the nanoporous water pressure to decrease until it returns to equilibrium with the capillary pressure. When cooling, the nanoporosity is in depression and the water diffuses from the capillary to the nanoporous volume. If the material is heated quickly, the water in the nanopores does not have enough time to be evacuated and the pressure is amplified.

III.3.3. Evolution of transient thermal deformation

As for basic creep (Eq. 2), the transient thermal deformation is assumed proportional to the elastic strain 𝜀 𝐼 𝐸 , Eq. ( 26).

𝜕𝜀 𝑀𝐶 ̅ 𝜕𝑡 = 𝜀 𝐼 𝐸 𝜏 𝐼 𝑀 . 𝐶 𝑛 𝑤 (26)
The Maxwell characteristic time τ I M presented in Eq. ( 26) is the same as in Eq. ( 6), but it is affected by the coefficient 𝐶 𝑛 𝑤 used to model the influence of nanoporous water pressure on the C-S-H viscous behaviour. This coefficient has to make a temporary decrease of the material viscosity possible as long as the nanopressure is not depleted, which induces an equation of the form [START_REF] Jennings | Refinements to colloid model of C-S-H in cement: CM-II[END_REF]:

𝐶 𝑛 𝑤 = ( 𝑃 𝑛 𝑃 𝑘 𝑛 𝑖𝑓 𝑃 𝑛 > 0 0 𝑖𝑓 𝑃 𝑛 ≤ 0 (27)
If the C-S-H intra porosity is in depression or at equilibrium with respect to the water capillarity (𝑃 n = 𝑃 c ), 𝐶 𝑛 𝑤 is equal to zero, which means that TTD does not occur. But, in case of nanoporous overpressure, a linear relation between 𝐶 𝑛 𝑤 and 𝑃 n controls the TTD kinetics through the fitting parameter 𝑃 𝑘 n (which has the dimension of a pressure). Thus, if a saturated, mechanically loaded material is subjected to heating, an overpressure appears and leads to TTD. Over time, the nanoporous pressure decreases to balance the capillary pressure, slowing the TTD kinetics.

In Figure 5, the maximum TTD velocity is reached at the maximum temperature and decreases quickly until falling to zero just a few hours later (due to water diffusion). With these equations, if a concrete is heated a long time after its mechanical loading, the TTD is lower than if it was heated just after its mechanical loading. Before the rise in temperature, the shrinkage and creep of concrete will increase the consolidation (and so the characteristic time 𝜏 𝐼 𝑀 ), and consequently decrease the TTD velocity according to Eq. ( 26). As shown by Fahmi et al. [START_REF] Fahmi | Effects of sustained and cyclic elevated temperature on creep of concrete[END_REF], for saturated concrete (𝑃 𝑐 = 0 𝑀𝑃𝑎), the TTD can be repeated if temperature cycles are close together because the consolidation cannot progress enough to significantly affect the characteristic time of these delayed strains.

IV. Application

The equations introduced previously were tested on experimental results from Cagnon et al. [START_REF] Cagnon | Transient thermal deformation of high performance concrete in the range 20°C-40[END_REF]. This part presents the application to sealed samples. The sealing was used to prevent the desiccation of the specimens during the tests. Before the tests, samples were cured in water at

The concrete used was a High Performance Concrete made with CEM I cement. The values of its instantaneous mechanical properties and its coefficient of thermal expansion are presented in The modelling of elastic strain and the basic creep presented in Figure 6 allowed fitting parameters to be determined, which are summed up in Table 4 (𝜏 𝑟𝑒𝑓 𝑀 , 𝜏 𝑟𝑒𝑓 𝐾 , 𝜓 𝐾 and 𝜀 𝑟𝑒𝑓 𝑀 ).

These parameters were maintained for all the following modelling. Figure 7 shows the experimental evolutions of strain corresponding to a second set of thermomechanical conditions from Cagnon's data, which will be used for modelling. In this test, sealed concrete was loaded one day before heating and was then subjected to thermomechanical variations as detailed above the graph.

In this figure, the model is first used without TTD (black curve), in order to show the underestimation of strain when TTD is not considered. It is obvious that, in this case, the basic creep modelling with thermal activation is not sufficient, and the missing part constitutes the TTD. TTD appears early and its amplitude can be estimated at approximately 300 µm/m. However, it is noteworthy that the velocity of strains after 10 days, computed by the model fitted above, is of the same order as found experimentally. This result shows clearly that TTD is a strain shift that occurs during heating without affecting the consolidation phenomenon responsible for the basic creep curve shape.

When the TTD model is activated, the TTD parameters (𝑘 𝑛 and 𝑃 𝑘 𝑛 ) are fitted using a leastsquares method. The parameter values are given in Table 4 and the completed model is presented in Figure 7. To test the model, the first thermo-mechanical evolution presented in Figure 1 was modelled with the same set of parameters (the result of model fitting is shown in Figure 1). Some samples (plotted in blue, mean value) were heated then loaded and others (plotted in green, mean value) were loaded then heated. From the comparison of model and experimental results, the model was found to be able to predict the measured strain evolution with the same set of parameters if TTD occurred. The effect of consolidation on repeatability of TTD in the case of a heating cycle was also taken into account as shown by the green curve. As expected, TTD appears with the first heating. However, during the second heating, the previous consolidation (due to creep and shrinkage), and consequently the characteristic time τ I M , are sufficient to prevent TTD from occurring again. If the thermal cycle had been closer in time, the consolidation would have been less developed and the TTD could have been repeated.

This may be especially relevant for saturated concrete, which thus has not been previously consolidated by capillarity tension.

Model parameter

Conclusion

Based on experimental observations, the TTD known to occur only if heating occurs after or simultaneously with the mechanical loading is similar, at the macroscopic scale, to a sort of a transient fluidification conditioned by two other phenomena: firstly, the concrete should not have undergone creep before loading and, secondly, the water saturation must be sufficient during the heating. If these four conditions (loading, heating, no or slight creep before heating, high saturation ratio) are fulfilled, then TTD can occur. To model the occurrence and the amplitude of the TTD as simply as possible, it is assumed that the origin of the TTD remains in the nanoporous water of C-S-H. When heated, nanoporous water dilates and changes the C-S-H viscosity until the local overpressure at this scale is depleted by diffusion of this water towards the capillary porosity. The proposed model then allows simulations of various experimental strain evolutions to be performed under several thermo-mechanical conditions with a single set of parameters. This model is sufficiently easy to use to be implemented in a basic creep structural model. This possibility will now allow the risk of TTD to be automatically considered in finite element structural analysis. Despite this modelling, which can determine the occurrence and the amplitude of the phenomenon, determining the physical reasons why the viscosity of C-S-H bundles is so dependent on the water pressure at nanoscale remains a challenge, especially for researchers working on homogenization theory and on modelling at atomic scale.

Figure 1 :

 1 Figure 1: 1 st thermo-mechanical conditions: elastic strain, basic creep, thermal dilation and TTD for a sample loaded first (plotted in green) and a sample heated first (plotted in blue), with modelling. t 0 corresponds to the end of water curing and R c is the compressive strength (experimental results from [10])

  to capillary scale depends on the water nanoporous pressure 𝑃 𝑛 , water capillary pressure 𝑃 𝑐 , fluid apparent viscosity 𝜂 and fluid density 𝜌, which depend on the temperature. A fitting parameter 𝑘 n is introduced to control the flow velocity. This parameter can be considered as a dimensionless coefficient of permeability at nanoscale. Its value can vary because the morphologies of C-S-H change according to the concrete formulation (high density or low density C-S-H for instance). At the capillary scale, the water capillary pressure, Eq. (

Figure 4 :

 4 Figure 4: Example of water nanoporous pressure evolution under temperature cycle (20-40-20 °C) with 𝑘 𝑛 =7.16x10 -19 and 𝑃 𝑐 =0 MPa.

Figure 5 :

 5 Figure 5: Example of evolution of moisture coefficient 𝐶 𝑛 𝑤 with time under temperature cycle T (20-40-20°C) with𝑘 𝑛 =7.16x10-19 , 𝑃 𝑘 𝑛 =102kPa and 𝑃 𝑐 =0MPa.

Figure 6 :

 6 Figure 6: Experimental modelled strain evolutions under loading (30%Rc) versus time from t 0 date of loading at 20°C with instantaneous elastic strain and creep strain (experimental results from [10])

Figure 7 :

 7 Figure 7: 2nd thermo-mechanical conditions: elastic strain, basic creep, thermal dilation and TTD, with and without TTD modelling (experimental results from [10])

Table 2 :

 2 Values of constants in Thiensen equation

Table 3 .

 3 The experimental and calculated strains were obtained on uniaxial compressive loading at 24.3 MPa. To model the TTD, the creep parameters were required, thus basic creep was also modelled.

	Value

Table 3 :

 3 Concrete properties

Table 4 :

 4 Fitting parameters for concrete creep and transient thermal deformation

°C for more than 3 months after casting to allow their hydration to stabilize, then they were sealed just before thermo-mechanical conditions were applied.

Acknowledgements

This work was carried out at LMDC (University of Toulouse), in collaboration with and with support from Andra (the French agency for nuclear waste management).