Sylvie Giger-Reverdin
email: sylvie.giger-reverdin@agroparistech.fr

Lina Jaber

Joseph-Alexandre Tessier

Christine Duvaux-Ponter

Effect of a heat stress episode on feed and water intake in dairy goats bred under temperate climate

Keywords:

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Effet d'un stress de chaleur sur l'ingestion d'aliment et d'eau chez des chèvres laitières élevées en conditions tempérées

Résumé. L'effet d'un stress de chaleur a été étudié chez huit chèvres laitières élevées en climat tempéré et disposant d'eau à volonté. L'augmentation de la température de 19 à 28°C (mesurée à 17 h) n'a modifié ni la quantité de ration ingérée, ni l'allure globale de la cinétique d'ingestion des aliments et d'eau. Par contre, la quantité d'eau bue a été accrue de 40 % et le temps de latence entre le début du repas et la première buvée, réduit de 35 à 26 min. La teneur en CO 2 du sang a diminué, car les animaux ont hyperventilé pour réduire leur température corporelle, alors que leur température rectale a augmenté de 0,6°C. La production laitière n'a pas été modifiée, mais le taux butyreux du lait a diminué. Un effet chèvre significatif a été observé sur la majorité des paramètres étudiés. En moyenne, les chèvres de race Alpine ont bu plus d'eau par rapport à la quantité de matière sèche ingérée que les chèvres de race Saanen. Ceci pourrait être dû à leur niveau d'ingestion plus élevé qui accroîtrait le dégagement de chaleur post-prandial, ou à une différence de chaleur latente dégagée par sudation liée à la couleur de leur pelage.

Mots

I -Introduction

In recent years, temperate European regions suffered from unusually hot periods, as for example during the summer of 2003. The aim of this work was to study the effect of a hot period on feed and water intake of high-producing dairy goats bred in the Paris area, because most previous studies concerned ruminants under hot climates in areas where water is scarce. Moreover, drinking behavior is seldom studied in ruminants [START_REF] Cardot | Les facteurs de variation du comportement d'abreuvement des vaches laitières[END_REF].

II -Material and methods

This study was conducted at the Experimental Farm of the research Unit INRA-AgroParisTech MoSAR (Thiverval-Grignon, France (48°51' N; 1°55' E); 70 m above sea level). Four Alpine and four Saanen goats (160 DIM at the start of the experiment) were housed in individual pens with free access to feed and water. Goats were fed a total mixed ration (TMR) twice daily after milking, in the proportion of two thirds at 4 p.m. and one third at 8 a.m., according to the milking intervals. The ingredients of the diet, on a dry matter basis, were dehydrated lucerne (30%), meadow hay (20%), pressed sugar beet pulp (30%) and compound concentrate feed (20%).

Animals were weighed weekly. Individual amounts of feed offered and refusals were weighed daily. Intakes of feed and water were recorded separately using, for each animal, two weighingscales (Balea, Saint-Mathieu de Tréviers, France): one was fitted under the feeding trough and the other one under a water container linked to the drinking trough. This system continuously recorded every 2 min the weight of the feed in the feeding trough and of the water in the container with a precision of 5 g. Cumulative dry matter intake (DMI) or water drunk (WD) per kilogram of body weight (BW) was calculated for the 22 hours following the afternoon feeding, using the body weight and the dry matter percentage of the diet (ISO, 1983) measured weekly. Animals had no access to food during the two remaining hours of the 24 h period (around afternoon milking). The evolution of the percentages of DMI and water drunk was studied during the nycthemere by 20-min intervals. The percentage of DMI or water drunk during the first three hours after the afternoon feed allowance was also calculated and called P180DMI and P180WD respectively, because it was the period during which the maximum of variability between goats was observed for food and water intakes [START_REF] Giger-Reverdin | Variabilité interindividuelle des ingestions de ration et d'eau mesurées simultanément chez des chèvres en lactation[END_REF]. The latency from the beginning of the afternoon meal to the first water intake was also calculated.

Two periods of five consecutive days were compared: the first one was between the 17 th and the 22 th of June 2010, and the second one between the 26 th of June and the 1 st of July 2010. Milk production was measured at each milking and averaged for 5 days. Milk samples were analysed for fat and protein contents at the beginning and end of each period. Blood samples were also analysed twice at each period for gas composition with a blood gas analyzer ABL77 (Radiometer®). Temperature (T in °C) and humidity (H %) were measured in the experimental facility at three times each day: 9 am, 1.30 pm and 5 pm (GMT + 2 h). The temperaturehumidity index (THI) was calculated according to the formula of [START_REF] West | Interactions of energy and bovine somatotropin with heat stress[END_REF]: THI = 0.81 T + (H/100) * (T -14.3) + 46.3. Data were analysed using the MIXED procedure of SAS ® for repeated measurements with the following model: Y ij = µ + P i + G j + P i *G j + i ij where represents the overall mean, P i the period effect, G j the goat effect. P i *G j the interaction between the period and the goat effects.

The model included the repetition of the day within the period with the use of a mixed procedure First autoregressive AR(1). For the temperature and humidity data, the only effect tested was that of the period.

III -Results

Temperature and humidity in the goat unit

During both periods, the temperature increased during the day while the humidity decreased. However, the modifications were very small in the first period and more marked in the second one. This last period can be considered as a heat episode for the country with a mean temperature of 27.5°C at 5 pm (26.7 to 28.4°C) and a mean humidity of 41% (35 to 50%). The Feeding and management strategies to improve livestock productivity, welfare and product quality under climate change 273 period effect was significant except for the humidity in the morning (Table 1).

Feed and water intake

Dry matter intake (DMI) was not modified by temperature, but water intake increased by 40% when the temperature increased (Table 2). The percentage of dry matter intake during the first three hours after the afternoon feeding did not vary, but there was a decrease in the percentage of water drunk. All the goats started to eat before drinking. Moreover, the latency to the first water intake after feeding allowance decreased with the increase in temperature. Feeding behavior expressed as % DMI per 20min intervals was not modified by heat stress (Fig. 1).

Blood parameters

Blood PCO 2 decreased with the increase in temperature, whereas rectal temperature increased:

Milk production

Raw milk yield was not modified by heat stress, but the fat content and the fat yield decreased, (Table 4). Protein content and protein yield were not affected by the increase in temperature.

Breed and goat effects

There was a breed effect on live-weight (Alpine: 58.3 kg vs Saanen: 64.2 kg; SEM = 0.52; P < 0.001) and intake: Alpine goats ate more dry matter per kg of body weight than Saanen (49.3 vs 45.9 g DMI/kg BW; SEM = 1.15; P = 0.02), and drank more water when expressed on a dry matter intake basis: 2.82 vs 2.46 L/kg DMI; SEM = 0.073; P = 0.01). The increase in this latter ratio with the temperature was more pronounced for the Alpine goats (Period 1: 2.38; Period 2: 3.26) than for the Saanen goats (Period 1: 2.15; Period 2: 2.76).

IV -Discussion

In this experiment, heat stress can be considered as a very moderate one, because the THI index was slightly above 72 during the afternoon in Period 2 which is the threshold level above which there is heat stress [START_REF] West | Interactions of energy and bovine somatotropin with heat stress[END_REF]. It did not modify eating behavior on the contrary to other situations where goats were bred in harsh conditions [START_REF] Silanikove | The physiological basis of adaptation in goats to harsh environments[END_REF]. Nevertheless, total water intake was increased and the latency between the afternoon feeding and the first water intake was decreased, meaning that the animals felt thirsty earlier after TMR allowance

Feeding and management strategies to improve livestock productivity, welfare and product quality under climate change 275 [START_REF] Olsson | Regulation of fluid balance in goats and sheep from dry areas[END_REF]. The respiratory alkalosis (decrease in PCO 2 ) observed during Period 2 could be explained by the hyperventilation of the animals during the heat stress as already observed in cows [START_REF] Kadzere | Heat stress in lactating dairy cows: a review[END_REF] or in goats [START_REF] Augustinsson | Influence of heat exposure on acid-base and fluid balance in hyperhydrated goats[END_REF]. When the animals are panting, which is a way to decrease their inner temperature, they release CO 2 .

Although moderately heat-stressed, the experimental goats showed increased rectal temperature as previously observed in other animals exposed to hot periods [START_REF] Augustinsson | Influence of heat exposure on acid-base and fluid balance in hyperhydrated goats[END_REF][START_REF] Kadzere | Heat stress in lactating dairy cows: a review[END_REF]. The decrease in milk fat content has already been observed in dairy cows under a mild heat stress [START_REF] Bandaranayaka | Changes in the composition of milk and rumen contents in cows exposed to a high ambient temperature with controlled feeding[END_REF]. The difference between breeds in terms of the water/DMI ratio might be explained by the higher level of intake of Alpine compared to Saanen goats and probably by a need to dissipate more heat from rumen fermentation [START_REF] West | Interactions of energy and bovine somatotropin with heat stress[END_REF][START_REF] Kadzere | Heat stress in lactating dairy cows: a review[END_REF]. It could also be due to the difference in coat colours: in Holstein cows, the temperature of the skin is higher in black areas than in white ones, leading to an increase in sweating rate in the black areas compared to the white ones (Da [START_REF] Silva | Evaporative cooling and cutaneous surface temperature of Holstein cows in tropical conditions[END_REF]. This could explain why Alpine goats with their dark-brown coat drank more water/kg DMI than the white coated Saanen goats.

V -Conclusion

High-producing European dairy goats seemed to cope well with short periods of moderate heat stress when water was available ad libitum. This study, which included feeding and drinking patterns, needs to be completed with a larger scale experiment and with measurements of the water fluxes and water losses via urine, faeces and milk. Long term adaptation and effects on production warrant future research in view of rampant global warming.

  -clés. Stress de chaleur -Comportement d'ingestion -Chèvres en lactation -Climat tempéré -Ingestion d'eau.

  Fig. 1. Evolution of the quantities of water drunk and feed intake during the nycthemere.

Table 1 . Temperature and humidity in the goat unit during Periods 1 (control) and 2 (heat stress)

 1

		Time	Period 1	Period 2	SEM	Period effect
	Temperature (°C)	9 am	17.0	21.3	0.52	0.001
		1.30 pm 18.0	25.4	0.56	<0.0001
		5 pm	19.0	27.5	0.41	<0.0001
	Humidity (%)	9 am	66.4	64.8	1.32	NS
		1.30 pm 60.0	45.4	2.95	0.01
		5 pm	57.4	40.6	2.83	0.01
	NS: not significant.

Table 2 . Feed and water intakes during Periods 1 (control) and 2 (heat stress)

 2

						Period 1	Period 2	SEM	Effect	Interaction
								Period	Goat	Period*Goat
	DMI	1				47.1	47.2	0.36	NS	<0.0001 0.02
	Water drunk	2	107	143	2.3	<0.0001 <0.0001 0.004
	P180DMI	3		0.38	0.37	0.009 NS	NS	NS
	P180WD	3		0.41	0.35	0.012 0.01	0.001	NS
	Latency before the first water
	intake

Table 3 . Blood parameters and rectal temperature during Periods 1 (control) and 2 (heat stress)

 3

		Period 1 Period 2	SEM	Effect		Interaction
					Period	Goat	Period*Goat
	pH	7.39	7.40	0.004	0.04	0.001	NS
	PCO 2 (mm Hg)	44.4	40.6	0.69	0.01	NS	NS
	PO 2 (mm Hg)	35.6	36.6	1.12	NS	0.05	NS
	Bicarbonates (mmol/L)	25.6	24.7	0.37	NS	NS	NS
	Rectal temperature °C	38.5	39.1	0.07	<0.0001	NS	NS
	NS: not significant

Table 4 . Milk yield and composition during Periods 1 (control) and 2 (heat stress)

 4

							Period 1	Period 2	SEM	Effect	Interaction
									Period	Goat	Period*Goat
	Raw milk yield	1	3.55		3.61	0.031	NS	<0.0001	0.03
	Fat content	2				33.3		30.2	0.31	<0.0001	<0.0001	0.01
	Protein content	2	31.9		31.3	0.09	0.01	<0.0001	NS
	Fat yield	3					116		108	1.7	0.01	0.001	NS
	Protein yield	3			113		113	1.0	NS	<0.0001	0.02
	1 in kg/day (mean of 5 days);	2 in g/kg of milk;

3 in g/day; NS: not significant.


Lina S Jaber

Christine Duvaux-Ponter

Shady K Hamadeh

Shadi K Hamadeh

Sylvie Giger-Reverdin

SHORT WATER RESTRICTION EPISODE IN LACTATING ALPINE AND SAANEN GOATS

Keywords: Alpine, Saanen, lactation, water restriction Introducton

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adaptations to these conditions. On the one hand, breeds that are native to arid and semi-arid regions have been well studied and show high resilience to heat and drought through special adaptive mechanisms that include high renal water retention, rapid water replenishment on rehydration with slow release into the bloodstream to prevent haemolysis, and minimization of water loss for evaporative cooling [START_REF] Abdelatif | Effect of state of hydration on body weight, blood constituents and urine excretion in Nubian goats (Capra hircus)[END_REF][START_REF] Silanikove | The physiological basis of adaptation in goats to harsh environments[END_REF][START_REF] Silanikove | The struggle to maintain hydration and osmoregulation in animals experiencing severe dehydration and rapid rehydration: the story of ruminants[END_REF]. On the other hand, European breeds have rarely experienced these conditions in the past and are now being challenged in both their native countries and the countries to which they are exported because of their high productivity. This project aimed to assess the effect of two consecutive periods of water restriction on the physiology and production of Saanen and Alpine goats in mid-lactation raised in their native European climate.

Materials and methods

The experiment was conducted on the experimental farm of the research unit of IN-RA-AgroParisTech MoSAR (Thiverval-Grignon, France; 48° 51' N; 1° 55' E; 70 m above sea level). Eight Alpine and eight Saanen goats (160 days in milk [DIM] at the start of the experiment) were included. The animals were housed in metabolic cages and were well adapted to the experimental setting (including housing, feeding and human manipulation), which was applied from 20 days prior to the beginning of the experiment. The experiment lasted for five days during which water was offered ad libitum during the first day, withheld from immediately after the afternoon milking until the following morning on days 2 and 3 (15 hours of restriction), and offered ad libitum on days 4 and 5. The animals had free access to feed which was composed of dehydrated lucerne (30 percent on a dry matter basis), meadow hay (20 percent), pressed sugar beet pulp (30 percent) and compound concentrate feed (20 percent). The feed was offered ad libitum, making sure that 10 percent was left over; feed quantities were readjusted accordingly, on a weekly basis.

Individual water and feed consumption were measured daily, along with body weight, rectal temperature and milk production. For half of the goats, urine and faecal output was measured on the first three days of the experiment. Daily milk samples were analysed for fat, protein, lactose and urea content. Jugular blood samples were taken each morning, before offering fresh water and feed. A blood gas and mineral analyser (ABL 77, Radiometer, Copenhagen, Denmark) immediately determined sodium (Na+). Plasma was harvested and stored at -20 °C until assays for glucose, non-esterified fatty acids (NEFA), albumin and urea were carried out using a Cobas Mira-Analyzer (Roche, Mannheim, Germany) with commercial kits for glucose (Gluco-quant, Glucose/HK, Roche Diagnostic, Mannheim, Germany), NEFA (NEFA-HR[2], Wako, Chemical GmBH, Neuss, Germany), albumin (80002, Biolabo, 02160, Maizy, France) and urea (Urea/BUN, Roche Diagnostic, Mannheim, Germany). Osmolality was determined with a MARK3, radiometer analytical SA osmometer (manufactured by Fiske® Associates, Norwood, Massachusetts, Untied States of America) applying a method based on the depression in the critical freezing point. Environmental temperature and humidity were recorded three times a day.

The data were analysed by paired-t test using the IBM-SPSS Statistics 21 software. Day 1 was used as the control for each animal and was compared separately with each of the subsequent days of the experiment.

Results and discussion

The water restriction regime caused a significant decrease in water intake (Table 1) on the first day of restriction (day 2 of the experiment). However, on days 4 and 5, water consumption exceeded the control level. This could indicate a reaction of overcom-pensation whereby the animals tend to drink large volumes of water in anticipation of future water shortage episodes. In this experiment, the water drunk in the morning straight after rehydration was more than twice the amount taken on the day of normal watering: 2.88 litres (normal) on day 1 versus 6.4 and 6.8 litres on days 2 and 3, respectively. Such behaviour was previously observed in arid-adapted breeds and denotes the capacity to tolerate rapid rehydration without risking haemolysis [START_REF] Jaber | Responses to repeated cycles of water restriction in lactating Shami goats[END_REF][START_REF] Silanikove | The struggle to maintain hydration and osmoregulation in animals experiencing severe dehydration and rapid rehydration: the story of ruminants[END_REF]. Another factor that could explain the increase in water consumption is the observed increase in ambient temperature during the experimental period, from 15.6 to 19.1 oC in the morning, and from 19.0 to 23.3 oC in the afternoon. This increase in ambient temperature was also reflected by higher rectal temperatures (Table 1), although the temperature range remained within the thermo-neutral zone of the animals [START_REF] Giger-Reverdin | Water metabolism and intake in goats[END_REF]. The increase in water consumption may have been needed for cooling in response to the added heat load.

In contrast, the animals reduced their feed intake under water restriction (Table 1). This response is widely observed in small ruminants, which need water for proper digestive function [START_REF] Hadjigeorgiou | The effect of water availability on feed intake and digestion in sheep[END_REF]; it is thought that this is to prevent feed accumulation and compaction in the digestive tract. However, the animals' body weight seemed to increase throughout the experimental period, which may indicate that water consumption has a greater effect on body weight than feed consumption during the short term of this experiment. Water restriction led to a decrease in urine output during the first day. In contrast, the faecal output was not significantly affected (Table 1), but faecal dry matter content increased. These observations indicate that the animals activated their water conservation mechanisms at the level of the kidneys and through the digestive tract, to minimize water loss and further dehydration. Highly adapted sheep breeds have been reported to produce very small amounts of highly concentrated urine when subjected to prolonged water restriction [START_REF] Laden | Dehydration tolerance in Awassi fat-tailed sheep[END_REF]. The Alpine and Saanen goats of this experiment seem to show similar trends in short-term water restriction episodes.

The water restriction protocol applied in this experiment was effective in causing the physiological signs of dehydration as commonly reported in the literature (Table 2). The first sign was the increase in blood albumin observed on days 3 and 4, following water restriction and just before watering. This increase denotes a reduction in blood volume, whereby blood water may have been circulated to other body compartments to maintain normal function [START_REF] Jaber | Responses to repeated cycles of water restriction in lactating Shami goats[END_REF][START_REF] Mengistu | Mechanisms of water economy in lactating Ethiopian Somali goats during repeated cycles of intermittent watering[END_REF]. Blood Na+ and urea also increased under water restriction. Previous studies have indicated that increase in these parameters is a sign of the activation of water conservation at the level of the kidneys, as Na+ and urea reabsorption by the nephrons leads to water reabsorption [START_REF] Mehta | Why does the plasma urea increase during acute dehydration? Advances in[END_REF]. This also explains the previously observed decrease in water urine output. The increase in blood electrolytes, in turn, causes an increase in osmolality, which is observed in this experiment as in the literature [START_REF] Dahlborn | Effect of temporary food or water deprivation on milk secretion and milk composition in the goat[END_REF][START_REF] Mengistu | Mechanisms of water economy in lactating Ethiopian Somali goats during repeated cycles of intermittent watering[END_REF]. It is worth noting that osmolality, urea and albumin dropped slightly below the control levels on day 5, following 24 hours of rehydration. This could indicate that the animals maintained the over-drinking behaviour for one day after the water restriction episode, probably in anticipation of possible future restrictions. It was previously reported that small ruminants may show a slow return to normal blood composition on rehydration, although they ingest large volumes of water as an adaptation to prevent haemolysis [START_REF] Jaber | Water stress in small ruminants[END_REF]. The slightly lower values observed on day 5 indicate a return to normal blood volume, or perhaps more than normal blood volume because of the water influx from the rumen. It would be interesting to evaluate the blood rehydration status in relation to the urinary output in order to assess whether kidney function was restored simultaneously to rid the body of the excess water. 294 ± 0.6 293 ± 0.7 306 3 ± 0.9 307 3 ± 0.7 2923 ± 0.5 Na+ (mmol/litre) 142 ± 0.3 143 3 ± 0.4 149 3 ± 0.4 148 3 ± 0.5 143 ± 0.3

1 Blood was sampled in the morning before offering feed and water.

2 Water removed immediately the afternoon milking until the next morning.

3 Means are significantly different from day 1 (P < 0.05).

The glucose and NEFA results (Table 2) seem to indicate a status of positive energy balance during water restriction although the animals decreased their overall feed intake. It could be speculated that the significant increase in glucose on days 3 and 4 is an indication of altered eating patterns by the thirsty animals, which caused the surge in glucose at the time of measurement, while under normal watering the animals would delay eating until fresh feed and water were offered. Previous reports on glucose levels under dehydration are inconclusive, but it is generally noted that a decrease in feed intake usually leads to a decrease in glucose levels [START_REF] Jaber | Water stress in small ruminants[END_REF]. Behavioural observations on the timing of feed intake under water restriction are needed to clarify the results obtained in this experiment. The NEFA tended to decline over the experimental period, further denoting that the animals were in positive energy balance [START_REF] Jaber | Fat mobilization in water restricted Awassi ewes supplemented with vitamin C[END_REF], which may also explain why no weight loss was observed.

Milk production was maintained under water restriction, with a slight surge in production on day 5 (Table 3), but lactose and urea tended to increase following water restriction on days 3 and 4. It was previously observed that lactose plays an important role in keeping the milk isotonic with the blood [START_REF] Dahlborn | Effect of temporary food or water deprivation on milk secretion and milk composition in the goat[END_REF]. Therefore, in view of the increased blood urea, albumin and osmolality, the changes observed in milk composition are a reflection of changes in the blood. Similarly, the surge in production and the slight drop in major milk components on day 5 may be another reflection of a transient state of over-hydration as noted above, whereby the animals tended to produce a larger, but more dilute, volume of milk. Milk fat was not significantly affected by water restriction but seemed to decline slightly throughout the experimental period. [START_REF] Alamer | Effect of water restriction on lactation performance of Aardi goats under heat stress conditions[END_REF] reported a decrease in the fat content of milk in 25-percent water-restricted goats, while those that were 50-percent restricted maintained normal fat content. Longer-term experiments may be helpful in determining the effect of water restriction on milk composition of high-producing European goats.

Conclusions

This experiment demonstrated the capacity of high-producing Alpine and Saanen goats to adapt to a short-term water restriction regime. Although signs of dehydration were observed, these signs were temporary and did not seem to affect the overall status of the animals, as attested by their maintained body weights and milk production. Longerterm studies would be interesting to assess the effect of such watering regimes over an extended period; in addition, behavioural data on water and feed intake would be helpful in explaining all aspects of these breeds' adaptability to water shortage.

TABLE 1 .

 1 WATER AND FEED INTAKES, BODY WEIGHTS, RECTAL TEMPERATURES AND FAECAL AND URINE OUTPUTS OF LACTATING SAANEN AND ALPINE GOATS SUBJECTED TO TWO SUCCESSIVE 15-HOUR PERIODS OF WATER RESTRICTION

	Parameter	Day 1	Day 2 1	Day 3 1	Day 4	Day 5
	Water intake (litres)	7.26 ± 0.301	6.41 2 ± 0.301	6.85 ± 0.282	8.04 2 ± 0.359	8.32 2 ± 0.381
	Feed intake (kg)	5.45 ± 0.129	5.02 2 ± 0.127	5.00 2 ± 0.134	5.29 2 ± 0.136	5.40 ± 0.140
	Body weight (kg)	60.6 ± 1.82	61.4 ± 2.19	62.02 ± 1.86	63.12 ± 1.96	62.52 ± 1.99
	Rectal temperature	38.54 ±	38.70 2 ±	38.80 2 ±	38.84 2 ±	38.83 2 ±
	(oC)	0.056	0.044	0.044	0.050	0.057
	Wet faecal material	3.30 ±	3.45 ±	3.45 ±
	output (kg)	0.171	0.209	0.143
	Faecal material dry	27.3 ±	29.2 2 ±	28.5 2 ±
	matter (%)	0.71	1.02	0.69
	Urine output (litres)	1.11 ± 0.089	0.89 2 ± 0.089	1.25 ± 0.124
	1 Water removed directly after the afternoon milking until the next morning.
	2 Means are significantly different from day 1 (P < 0.05)

TABLE 2 .

 2 BLOOD CHEMISTRY PARAMETERS IN LACTATING SAANEN AND ALPINE GOATS SUBJECTED TO TWO SUCCESSIVE 15-HOUR PERIODS OF WATER RESTRICTION

	Parameter 1	Day 1	Day 2 2	Day 3 2	Day 4	Day 5
	Glucose (g/litre)	0.583 ± 0.0054	0.583 ± 0.0112	0.625 3 ± 0.0167	0.612 3 ± 0.0075	0.571± 0.0071
	NEFA (mmol/litre) 137± 6.6 136 ± 6.4	133.00 ± 4.771	127± 3.3 125 ± 4.2
	Urea (g/litre)	0.256 ± 0.0171	0.235 ± 0.0148	0.331 3 ± 0.0169	0.397 3 ± 0.0280	0.250 ± 0.0136
	Albumin (g/litre)	43.43 ± 0.371	43.98 ± 0.358	46.04 3 ± 0.523	44.61 3 ± 0.382	42.78 ± 0.342
	Osmolality (mOsm/
	kg)

TABLE 3 .

 3 MILK PRODUCTION AND COMPOSITION OF LACTATING SAANEN AND ALPINE GOATS SUBJECTED TO TWO CONSECUTIVE 15-HOUR PERIODS OF WATER RESTRICTION Water removed immediately after the afternoon milking until the next morning. 2 Means are significantly different from D1 (P < 0.05).

	Parameter	Day 1	Day 2 1	Day 3 1	Day 4	Day 5
	Milk production (litres)	3.61 ± 0.119	3.59 ± 0.108	3.57 ± 0.110	3.60 ± 0.110	3.73 ± 0.107
	Milk fat (g/kg)	33.1 ± 1.07	33.0 ± 1.056	32.78 ± 1.137	32.3 ± 1.05 32.0 ± 0.87
	Milk protein (g/kg)	31.9 ± 0.58	31.6 2 ± 0.56	31.8 ± 0.59 31.6 ± 0.46 31.5 2 ± 0.52
	Lactose (g/kg)	43.8 ± 0.36	43.8 ± 0.37	44.6 2 ± 0.41	44.1 ± 0.45 43.5 2 ± 0.37
	Milk urea (g/kg)	0.567 ± 0.0142	0.536 2 ± 0.0141	0.592± 0.0199	0.596 ± 0.0182	0.517 2 ± 0.0167
	1

Acknowledgements

J. Tessier, A. Eymard, J.P. Bonte, J. Chevallier and G. Durandet are gratefully acknowledged for the care of the goats at the experimental farm and for their technical assistance during the experiment. We also thank O. Dhumez and F. Ternois for the analyses of samples.


Jeremy Bleyer
email: jeremy.bleyer@epfl.ch@enpc.fr

Jean-François Molinari
email: jean-francois.molinari@epfl.ch

Microbranching instability in phase-field modelling of dynamic brittle fracture

The microbranching instability occurring for rapidly propagating cracks in brittle materials has been described in various experiments as an intrinsically three-dimensional phenomenon. Using a variational phase-field model, we show that the microbranching process is, indeed, a threedimensional instability which exhibits a strong dependence on the sample width and can be suppressed for very thin samples. We show that the phase-field internal length scale is the decisive variable governing the branching pattern, which can be either localized in the transverse direction as observed in glass for example, or, on the contrary, almost translational invariant with quasiperiodic structures, as observed in PMMA.

Linear elastic fracture mechanics has been shown to describe accurately the dynamics of a single propagating crack at low velocities i.e. below 0.3 -0.4c R with c R being the Rayleigh wave speed [START_REF] Freund | Dynamic fracture mechanics[END_REF][START_REF] Sharon | [END_REF][3]. However, for higher velocities, various experiments demonstrated that cracks stop being simple [START_REF] Ravi-Chandar | Dynamic fracture[END_REF][START_REF] Scheibert | [END_REF] and exhibit dynamic instabilities characterized by crack velocity oscillations and frustrated branching attempts called microbranches [6]. This microbranching instability has been observed in different materials such as PMMA [6], Homalite [7], glass [8] or neo-Hookean brittle gels [9] and present universal features such as the existence of a critical velocity below which branching cannot occur. However, the crack surface patterns can be quite different depending on the material. In glass or gels for instance, microbranches tend to repeat themselves along lines parallel to the crack propagation direction and are highly localized along the width direction of the sample [9]. Conversely in PMMA, at the onset of the microbranching instability, microbranches also localize along the width direction but rapidly merge, forming an almost invariant pattern and showing a periodic alternance of up and down microbranches with a well-defined length scale. This typical length scale does not seem to exist for glass or gels [10]. It is now clear that the microbranching instability is a three-dimensional phenomenon but many aspects such as the origin of the difference in terms of crack surface patterns or the influence of the sample geometry are not fully understood. Our aim is to show through fully three-dimensional simulations that the crack-tip dissipation length scale is the key parameter controlling the different observed branching regimes.

Although microbranching instability has been reproduced numerically using discrete lattice models [11,12], its occurence seems to be highly dependent on the type of lattice and on the form and intensity of the chosen potential. Similarly, cohesive models are only able to reproduce part of the observed experimental features of dynamic fracture. In particular, cracks do not seem to exhibit a limiting speed below the Rayleigh wave speed despite branching [13]. Besides, the inherent mesh dependency of such an approach makes it ill-suited to an accurate study of the microbranching instability. Other strategies for simulating crack propagation in brittle media consist of non-local continuum models which aim at regularizing the crack topology by a continuous scalar field.

Sharing strong similarities with damage gradient approaches [14], phase-field modelling is a promising method which relies on basic ingredients such as damagedependent elastic moduli, a fracture energy G c and an internal length scale l 0 (see the Supplemental Materials for more details). The modelling of the I+III mixed-mode instability using such an approach has, for instance, shown very promising results [15,16]. Regarding the branching process, a large majority of works concentrated on two-dimensional simulations and, usually, were only able to reproduce macroscopic branching [17][18][19]. In [20], microbranching has been obtained in 3D computations using a Ginzburg-Landau phase-field model after perturbing the initial crack front and relatively good agreement of the fractographic patterns with experimental observations has been obtained. Although the distinction between macro-and microbranching may be somehow arbitrary, we will consider microbranching as situations where branched crack deviations remain of the order of magnitude of the crack tip dissipation length scale l 0 .

A recent study on the crack branching phenomenon in a two-dimensional pre-strained plate configuration [19] has revealed that single crack propagation is characterized by a progressive widening of the damaged band associated to an increase of the energy release rate. Macroscopic branching has been observed to occur when the energy release rate exceeds a critical value close to 2G c , as initially argued by Eshelby [START_REF] Eshelby | Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids[END_REF]. In FIG. 1. Crack surface patterns for different loading levels ∆U for the same plate width (W = 1 mm). The crack propagates as a single straight crack (with a slightly curved crack front) for low loading levels (top), the microbranching instability appears for higher loading with longer branches and shorter distance between successive branches for increasing loading (middle). For even higher loading (bottom), macroscopic branches are formed and are very close to 2D simulation patterns. Inset: sample dimensions and side view of a typical crack pattern in 2D simulations.

particular, no microbranching phase has been identified in such simulations for homogeneous media. We aim here at investigating how three-dimensional simulations are able to reproduce this particular instability in the same pre-strained plate configuration.

We considered a finite-element discretization of a rectangular domain of dimensions L × H × W in the (Oxyz) frame with L = 2H = 32 mm (see Fig. 1) and the width dimension W ranging from 0.2 to 5 mm, a prenotch of 4 mm in the x-direction is located on the left boundary along the middle plane y = H/2. The internal length scale l 0 characterizing the phase-field model has also been varied between 0.02 to 0.1 mm, the mesh size being smaller than l 0 near the main propagation plane to avoid any mesh-dependency. The pre-strain loading consists in fixed vertical displacements of intensity ±∆U on the top and bottom boundaries, both horizontal displacements are free and all other boundaries are stress-free. Unless stated otherwise, the default values are W = 1 mm, l 0 = 0.04 mm and ∆U = 0.06 mm, default material properties are close to PMMA (see [19]). For this specific prestrained plate setting, the initially stored elastic energy per unit length along the x-direction is Ψ = 2(∆U ) 2 E/H. We can then define a critical loading ∆U c below which crack propagation is not possible i.e. when Ψ = G c or equivalently ∆U c = G c H/2E ≈ 0.028 mm.

The first striking result is that microbranching patterns appear as a transition between single straight crack propagation and macroscopic branching patterns (Fig. 1) for three-dimensional simulations with higher stored energy. The patterns obtained for the higher loading ∆U = 0.08 mm = 2.9∆U c are close to those obtained in 2D simulations whereas for ∆U = 0.05 or 0.06 mm (1.8∆U c or 2.1∆U c ), we observe patterns exhibiting an up-and-down alternance of microbranching attempts which are not seen in 2D simulations but are very similar to periodic striation patterns appearing in PMMA (see Fig. 4(c) in [6] and [START_REF] Bouchbinder | [END_REF]). The crack surface for the case ∆U = 0.06 mm exhibits more translational invariance along the z-direction than the lower loading case ∆U = 0.05 mm. The increase of symmetry along the z-direction with the loading is consistent with experimental observations on PMMA [6] and other simulations using lattice models [12]. In such cases, the crack-tip splitting occurs simultaneously along the whole crack front.

The transition from straight crack propagation, localized events along the z-direction and increasing z-invariance leading to macroscopic branching seems to be a continuous process with respect to the loading amplitude in which the branch lengths progressively increase. This progressive transition is further confirmed by varying the specimen width W for the same loading (Fig. 2). The microbranching instability can be suppressed by decreasing the plate width up to W = 0.2 mm for this loading. The suppression of the microbranching instability for very thin plates has also been reported experimentally for brittle gels [23]. In these experiments, a crack-tip oscillatory instability has been observed for velocities higher than 0.9c s (c s being the shear wave speed), which we do not obtain here and observe macrobranches instead. The question remains to know if this difference can be attributed to nonlinear geometric effects due to the large prestrain level for such gels. Conversely, when increasing the plate width up to 5 mm, the crack surface exhibits much more z-localized events without any well defined quasi-periodic patterns as those for W = 1 mm. By increasing the loading for W = 5 mm, we confirm that the crack surface becomes increasingly z-invariant with the same quasi-periodic microbranching regime. Contrary to [20], we did not observe a strong difference in terms of critical speed associated with branching compared to the 2D simulations of [19]. Interestingly, the crack branching speeds have always been observed to be higher than the theoretical minimal speed of 0.52c R necessary for the propagation of branched cracks found in [24].

By looking closely at the initiation of the instability for the largest width W = 5 mm, the initial propagation phase corresponds to a single flat crack surface with a slightly curved crack front due to 3D effects. The fully damaged zone increases with crack advance, as in 2D simulations, which can be associated to an increasing local damage dissipation rate Γ, identified with an apparent fracture energy (see [19]). We computed this quantity for different two-dimensional slices along the z-direction and at different propagation times. We obtained the local variation of Γ by mapping these values to the current position of the main crack front along the middle plane Y = H/2. The evolution of Γ(x, z) along with the crack front position at different time intervals are represented in Fig. 3-top. The initial flat propagation phase corresponds to a uniform increase of Γ with respect to z until approximately 6 mm from the prenotch. Between 6 and 8 mm, slight variations of Γ along z appear and the first microbranching events take place at locations where Γ(x, z) exceeds 2G c , splitting the front into two different parts. Between the branching events, Γ is smaller and the single crack front continues its propagation. The coexistence of these two different states along the z-direction leads to a strong curvature of the crack front and generates complex dynamics with important variations of crack velocity and energy release rate (Fig. 3-bottom). It is worth noting that Γ shows a non-monotonic behavior with respect to the crack front velocity prior to branching. This can be explained by the fact that the increasing damage dissipation (although the crack front has not branched yet) is compensated by a slowdown of the crack front. This slight slowdown of the crack front prior to branching has also been observed in [20]. As described in [25], the release of the crack front from the branch arrest leads to an increase of crack velocity and apparent fracture energy ahead of the first event and eventually forms a new branching event along FIG. 3. Top: Evolution of local damage dissipation rate Γ(x, z) with crack front advance for W = 5 mm (∆U = 0.06 mm), the thin lines correspond to the crack front position at successive identical time intervals. The first microbranching events occur at localized regions along the z-direction where Γ exceeds 2Gc and result in large local deceleration and a distorted crack front along the main plane. The second branching events occur along the same lines as the first ones associated to an increase of damage dissipation along these preferential directions. Bottom: Evolution of average crack velocity v and damage dissipation rate Γ along the crack front during propagation. For the single crack propagation phase, the standard deviation along the crack front is small whereas large variations occur during the microbranching phase due to non-uniformity of v and Γ along the front.

the same direction.

The robustness of the previous results has been assessed by introducing some randomness on the local Young's modulus or on the local fracture energy G c in the sample (the standard deviation was about 10%). The quasi-periodic regime and, in particular, the microbranches length L branch and their typical spacing along the x-direction ∆x did not change. However, the randomness triggers the first branching event at different time and location. It also modifies the dynamics for the z-localized patterns (for W = 5 mm in particular), resulting in a different fracture surface. These results confirm that the quasi-periodic regime and, in particular, L branch and ∆x, are weakly coupled to 3D dynamics in the zdirection, which differs when adding randomness. In [10], these quantities have been shown to scale with a materialdependent length

l 1 = G c (v)/E or l 2 = G c (v)E/σ 2
c where E is Young's modulus and σ c a critical stress. In our phase-field framework, the material fracture energy is rate-independent G c (v) = G c and l 0 can be related to the following elastic limit stress σ c = 3G c E/(8l 0 ) [14], so that l 2 = 8l 0 /3. We then tested the previous scaling laws by first varying the ratio l 1 = G c /E while keeping l 0 constant and secondly by varying l 0 while keeping l 1 constant. In the first case, we changed either G c (resp. E) while keeping l 0 and E (resp. G c ) fixed. In order to compare similar situations with the same crack driving force, we also changed the loading level ∆U to keep the same ratio between the initial stored energy per unit length Ψ and G c . The results in terms of crack patterns are exactly the same suggesting that l 1 is not the governing length scale of the microbranching instability in the phase-field model. On the contrary, when varying l 0 , we observed a clear dependence of all the crack patterns for various loadings and plate widths: in particular, L branch and ∆x in the quasi-periodic regime increase with l 0 (Fig. 4). By recording these quantities for many patterns exhibiting this quasi-periodic regime we show, first, that L branch and ∆x are positively correlated, suggesting that the nucleation of the next branching event is directly related to the death of the previous one as proposed in [25]. Moreover, the plate width W seems to have no influence on selecting the microbranch length scale contrary to l 0 . More precisely, we find that the mean values of branch length and periodicity scale are given by L branch = (20.5 ± 2.5)l 0 and ∆x = (12.8 ± 1.7)l 0 for the data of Fig. 4.

Increasing l 0 for fixed geometry and loading progressively leads to macroscopic branching as the branches get longer and farther apart. On the contrary, reducing l 0 leads to smaller and closer microbranches but also less z-invariance, as when increasing the plate width. Quite interestingly, the total dissipated energy for the three patterns illustrated in Fig. 4 is the same within 2%. Although a precise quantitative investigation is difficult, our simulations indicate that the typical width of a branching event ∆z is also positively correlated to l 0 and therefore explains that translational z-invariance appears when l 0 /W ∝ ∆z/W becomes close to unity, thus separating the two microbranching regimes. revHowever, ∆z is much larger than l 0 by at least one order of magnitude so that the transition occurs at a macroscopic width compared to l 0 . Unfortunately, larger simulations are out-of-reach to further investigate the influcence of W with respect to the extent of localized events in the z-direction.

These different results therefore suggest that the microbranching instability patterns are highly dependent on an intrinsic material length scale characterizing the dissipation processes at the crack tip. In the phasefield framework, this length scale is given by l 0 and can be interpreted as a cohesive zone length over which the LEFM singularity is regularized. Consequently, as a crack front accelerates, the apparent fracture energy increases through a thickening of the fully damaged band which we associate with a velocity-toughening mechanism. When reaching locally a critical value of 2G c , crack tip splitting occurs over a width ∆z. If ∆z/W ≤ 1, splitted and single crack tips can coexist simultaneously along the crack front. This results in complex dynamics in the (x, z)-plane due to crack front curving and release of energy when the microbranch dies. This situation typically occurs in glass for which the critical length scale is very small compared to the typical sample size. Reproducing such localized chains of microbranching is unfortunately too computationally expensive as it would require an extremely fine mesh to resolve the small length scale l 0 but our results in the extreme cases with the smallest l 0 /W confirm this scenario. Conversely, if ∆z/W ≥ 1, crack tip splitting occurs along the whole front and the crack tip dynamics enter a new stable regime of alternating up-and-down microbranches with well defined geometric characteristics related to l 0 . This situation typically occurs in PMMA for which the characteristic length scale is much higher than for glass. When ∆z/W 1, this regime translates into macroscopic branching as branched crack tips are now sufficiently well separated and are less influenced by each other.

Obviously, the previous analysis also depends on the value of the loading or crack velocity although the main ideas still hold. A different crack velocity changes the fracture energy in PMMA for instance and, therefore, influences the length scale. Here, we used a rateindependent phase-field model which does not account for a material dependence of G c on the velocity. Most certainly, the very simple damage model used in the phase-field approach misses some complex mechanisms taking place inside the crack tip process zone, explaining why the obtained crack front velocities are relatively high (around 0.7c R ) compared to critical velocities measured experimentally. However, the results of Fig. 3 show that the onset of branching is well discribed by a local energetic criterion on Γ. Hence, for high loading or crack velocity, there is a higher probability that random perturbations will surpass this critical value, yielding a larger width ∆z of the microbranching event, until eventually reaching a full localization across the sample width. Similarly, at higher loading level, branched crack tips release more energy and may survive longer than for configurations with less energy, resulting in a transition to macroscopic branching.

In conclusion, our simulations reveal a clear 2D-3D-2D transition from a straight crack propagation to z-localized microbranching, further evolving into a z-invariant quasi-periodic regime and eventually into macrobranching, in good agreement with the well-known mirror-mist-hackle transition observed on post-mortem fracture surfaces. We have shown how the material length scale l 0 governs the different branching regimes with respect to the specimen width and the loading level. In particular, the microbranching regime cannot be obtained for very thin samples or 2D simulations due to its intrinsic three-dimensional nature. The onset of this dynamic instability is noise-triggered and occurs when the local energy release rate exceeds a critical value of 2G c , already associated to the onset of branching in 2D simulations. Depending on the size of the typical branching width with respect to the sample width, branching patterns are either translationally invariant and weakly coupled to 3D dynamics as observed in PMMA or localized along the z-direction and highly dependent on complex dynamics, which is close to what is observed in glass and brittle gels. The present work illustrates that phase-field models are a relevant approach for modelling complex crack propagation dynamics, relying on simple well-defined energetic principles. More quantitative results (especially concerning crack velocities) can be expected by modelling rate-dependency in PMMA. Finally, better insight on the branch width selection could be gained from theoretical perturbation analysis of crack-tip splitting processes. Details regarding the phase-field model used for the simulations are given in the supplemental material attached with this paper.

FIG. 2 .

 2 FIG. 2. Crack surface patterns for different plate widths Wfor the same loading level (∆U = 0.06 mm). The microbranching instability is clearly suppressed for the thin plate (top), the macrobranching crack pattern being reminiscent of 2D simulations. The up-and-down quasi-periodic regime of small microbranching is obtained for the intermediate width (middle) whereas more localization in the z-direction is obtained for the larger width (bottom) without exhibiting any well-structured feature. See movies in the Supplemental Materials.

FIG. 4 .

 4 FIG. 4. Correlation between microbranch spacing ∆x and microbranch length L branch in quasi-periodic regime for different W ( : 0.5 mm, • : 1 mm, : 2 mm) and different l0 (blue : 0.02 mm, green : 0.04 mm, red : 0.06 mm). Inset: side view of z-invariant patterns for W = 1 mm and different l0.