S Maghsoodi
email: soheib.maghsoodi@univ-lorraine.fr

O Cuisinier

F Masrouri

Thermal effects on the mechanical behaviour of the soil-structure interface

Keywords: Shear strength, Constant normal stiffness (CNS), Soil-structure interface, Temperature, Thermo-active structures. Résumé Resistance au cisaillement, rigidité normale constante, interface sol-structure, température, geostructures thermo-actives

The mechanical behaviour of the soil-structure interface plays a major role in the shear characteristics and bearing capacity of foundations. In thermo-active structures, due to non-isothermal conditions, the interface behaviour becomes more complex. The objective of this study is to investigate the effects of temperature variations on the mechanical behaviour of soils and soil-structure interface. Constant normal load (CNL) and constant normal stiffness (CNS) tests were performed on soil and soil-structure interface in a direct shear device at temperatures of 5, 22 and 60 o C. Fontainebleau sand and kaolin clay were used as proxies for sandy and clayey soils. The sandy soil was prepared in a dense state, and the clayey soil was prepared in a normally consolidated state. The results showed that the applied thermal variations have a negligible effect on the shear strength of the sand and sand-structure interface under CNL and CNS conditions, and the soil and soilstructure interface behaviour could be considered thermally independent. In clay samples the temperature increase, increased the cohesion and consequently the shear strength, due to thermal contraction during heating. The temperature rise had less impact on the shear strength in the case of the clay-structure interface than in the clay samples. The adhesion of the clay-structure interface, is less than the cohesion of the clay samples.

Introduction

The bearing capacity of foundations is highly dependent on the mechanical behaviour of the soil-structure interface. Therefore, the soil-structure interactions at the interface are of primary importance in foundation designs. Due to the recent developments of clean energy, traditional geostructures such as piles and diaphragm walls have been converted to energy geostructures by installing heat exchanger tubes inside the concrete element [START_REF] Brandl | Energy foundations and other thermo-active ground structures[END_REF]. In energy geostructures the mechanical loads applied to the structure on one hand, and the effect of heat exchange between structure and surrounding soil on the other hand, modify the behaviour of the soil-structure interface. These thermal variations and mechanical loads affect the bearing capacity and frictional resistance of these thermo-active structures. Therefore, the effects of temperature on the soil-structure interface mechanical parameters should be investigated. In this section, a state of the art about the behaviour of the soil-structure interface and its influencing parameters under isothermal conditions are presented. Then, existing studies on the effects of temperature on the mechanical behaviour of soils and soil-structure interfaces are discussed.

Grain size, grain crushability, grain roundness, soil density, initial stress state, structure roughness and shearing rate based on interface tests were addressed as the parameters influencing the soil-structure interface mechanical behaviour [START_REF] Potyondy | Skin friction between various soils and construction materials[END_REF][START_REF] Desai | Cyclic testing and modeling of interfaces[END_REF][START_REF] Boulon | Physical and numerical simulation of lateral shaft friction along offshore piles in sand[END_REF][START_REF] Uesugi | Frictional resistance at yield between dry sand and mild steel[END_REF][START_REF] Poulos | Influence of soil density on pile skin friction in calcareous sediments[END_REF][START_REF] Jardine | Friction coefficients for piles in sands and silts, Offshore site investigation and foundation behaviour[END_REF][START_REF] Lehane | Mechanisms of shaft friction in sand from instrumented pile tests[END_REF][START_REF] Fakharian | Cyclic simple-shear behavior of sand-steel interfaces under constant normal stiffness condition[END_REF][START_REF] Mortara | An elastoplastic model for sand-structure interface behaviour under monotonic and cyclic loading[END_REF][START_REF] Pra-Ai | Behaviour of soil-structure interfaces subjected to a large number of cycles[END_REF].

An important concept to aid in understanding the interface behaviour is the constant normal stiffness (CNS) conditions, which explains the real shear behaviour of embedded foundations; as discussed in the following. The physical concept of constant normal stiffness (CNS) conditions at the soil-structure interface, was introduced by [START_REF] Wernick | Skin friction of cylindrical anchors in non-cohesive soils[END_REF] (Fig. 1). Depending on the volumetric response of the soil at the interface during shearing (dilative or contractive), the surrounding soil stiffness constrains the volumetric response of the interface and acts as a virtual spring with a given stiffness (Eq. 1). ∆σ = -K.∆U

(1)

Where ∆σ(kPa) is the normal stress difference, K(kPa/mm) is the stiffness of the adjacent soil (stiffness of the spring) and ∆U (mm) is the normal displacement difference of the interface.

The tendency of the interface to dilate is counteracted by the elastic reaction of the adjacent soil [START_REF] Hoteit | Etude exprimentale du comportement physique et mcanique des interfaces sols-structures[END_REF][START_REF] Tabucanon | Frictional behaviour between normally consolidated clay and steel by two direct shear type apparatuses[END_REF][START_REF] Fioravante | A constant normal stiffness direct shear box for soil-solid interface tests[END_REF]. [START_REF] Porcino | Interface behavior of sands from constant normal stiffness direct shear tests[END_REF] performed constant normal load (CNL) and constant normal stiffness (CNS) tests on sand-steel interface and showed that the effect of the normal stiffness (K) on the mobilized shear resistance of the interfaces in CNS tests depends on the volumetric response exhibited by the interfaces in the CNL tests. They showed that in dilative regimes, the increase in the current normal stress (σ n ) when sheared in the CNS tests causes an increase in the current shear stress (τ ). On the other hand, in the contractive regimes (smooth interface or loose soil), a decrease in the normal and shear stresses is observed. They also concluded that the increase or decrease in the mobilized shear resistance during the CNS tests are a consequence of the current normal stress evolution, and these changes in mobilized shear resistance are not an effect of the mobilized sand-structure friction angle modification, which remains unchanged.

Another important factor that influences the soil-structure interactions is the structure surface roughness [START_REF] Kishida | Tests of the interface between sand and steel in the simple shear apparatus[END_REF][START_REF] Porcino | Interface behavior of sands from constant normal stiffness direct shear tests[END_REF][START_REF] Hu | Application of damage model for soil-structure interface[END_REF].

Normalized roughness (R n ), as reported by [START_REF] Uesugi | Frictional resistance at yield between dry sand and mild steel[END_REF], was defined by measuring R max (vertical distance between the highest peak and lowest valley) along a profile length L equal to the mean grain size D 50 and then normalized by D 50 :

R n = R max (L = D 50 ) D 50 (2)
Previous investigations [START_REF] Uesugi | Frictional resistance at yield between dry sand and mild steel[END_REF][START_REF] Uesugi | Friction between sand and steel under repeated loading[END_REF][START_REF] Hu | Application of damage model for soil-structure interface[END_REF] indicate a range for the smooth and rough surfaces. The critical roughness (R crit = 0.1 -0.13) was chosen as a range that (R n > R crit ) is a rough surface and (R n < R crit ) is considered as a smooth one.

The interface behaviour under non-isothermal conditions is a coupling between the above-mentioned parameters and temperature variations. In the following section, the thermal effects on the mechanical parameters of soils and the soil-structure interface are discussed. Different studies have been performed on the effects of temperature on the mechanical parameters of soils [START_REF] Campanella | Influence of temperature variations on soil behavior[END_REF]Mitchell 1968, Hueckel and[START_REF] Hueckel | Thermoplasticity of saturated clays: experimental constitutive study[END_REF][START_REF] Kuntiwattanakul | Temperature effects on undrained shear characteristics of clay[END_REF][START_REF] Burghignoli | A laboratory study on the thermomechanical behaviour of clayey soils[END_REF][START_REF] Cui | A thermomechanical model for saturated clays[END_REF][START_REF] Delage | On the thermal consolidation of boom clay[END_REF][START_REF] Cekerevac | Experimental study of thermal effects on the mechanical behaviour of a clay[END_REF][START_REF] Abuel-Naga | Experimental evaluation of engineering behavior of soft bangkok clay under elevated temperature[END_REF][START_REF] Boukelia | Effect of temperature and initial state on variation of thermal parameters of fine compacted soils[END_REF][START_REF] Eslami | Pressuremeter test parameters of a compacted illitic soil under thermal cycling[END_REF][START_REF] Jarad | Effect of temperature and strain rate on the consolidation behaviour of compacted clayey soils[END_REF], and these studies indicated that the thermo-mechanical be-haviour of soils is highly dependent on the stress and thermal history of the material.

However, only a few studies have been performed on the soil-structure interactions under non-isothermal conditions [START_REF] Di Donna | Experimental investigations of the soil-concrete interface: physical mechanisms, cyclic mobilization, and behaviour at different temperatures[END_REF][START_REF] Yavari | Effect of temperature on the shear strength of soils and the soil-structure interface[END_REF] According to the literature, the effects of temperature on the friction angle and adhesion of the soil-structure interface, are poorly understood under both CNL and CNS conditions. In this study, a temperature-controlled direct shear device was used to perform interface tests on Fontainebleau sand and kaolin clay on a rough surface under CNL and CNS conditions, to better understand the following:

• The effects of temperature on the shear strength (friction angle, cohesion and adhesion) of soil and soil-structure interface under CNL and CNS conditions.

• The effect of surrounding soil stiffness on the soil-structure interface mechanical behaviour at different temperatures.

• The soil and soil-structure interface volumetric changes during heating (from 22 to 60 o C) and cooling (from 22 to 5 o C) under constant isotropic stress.

Material properties, device and experimental programme

In this section first, the materials used in this study are presented. Then, the details of the temperature-controlled direct shear device, CNL and CNS tests with the device, and calibration are discussed. Finally, the experimental programme is presented.

Material properties

The grain size distributions and physical properties of Fontainebleau sand (siliceous) and kaolin clay used, in this study are presented in Fig. 2, Table 1 andTable 2.

To perform soil-structure interface direct shear tests, a stainless steel plate (80 x 60 x 10 mm) with the desired roughness was designed and used as the structure. This steel plate is used to, avoid abrasion of the surface due to test repetition. The roughness of the steel plate was measured with a laser profilometer (Fig. 3a). Four profiles with lengths of 32 mm (Fig. 3b) parallel to the shear direction were measured. The heights of these four profiles were obtained with the laser are presented in Temperature-controlled direct shear device Fig. 5 shows the temperature-controlled direct shear device. The shear box (60 x 60

x 35 mm) was placed inside a container filled with water to reach saturated conditions (Fig. 5). The heating system consisted of a heater that controlled the fluid temperature circulating in the lower part of the container. Therefore, the water temperature in the container reached the same temperature as the circulating fluid. Three thermocouples, one in the lower half of the shear box, another on the upper half of the shear box and the last in the container, controlled the applied temperature. In this direct shear device, normal stress σ n (kPa), shear displacement W (mm), circulating fluid temperature T ( o C)

and stiffness value K (kPa/mm) were applied, and normal displacement U (mm), shear stress τ (kPa), and sample temperature T ( o C) were measured (Fig. 5).

Constant normal load application

To perform CNL tests, the normal load was applied with a loading frame and kept constant during the tests. To start the shear, a shear displacement rate (mm/min) was applied to the lower half of the shear box and the shear stress was measured. The different parts of the device were connected to a data logger and a commanding system, which enabled the operator to apply different thermo-mechanical paths. Calibrations were performed to account for any temperature effects on different parts of the device.

Constant normal stiffness application

Under CNS conditions, two general behaviours are observed in soils: dilative (dense or overconsolidated soils) and contractive (loose or normally consolidated soils). In the first case, with starting the shear the soil at the interface starts to contract slightly (∆U > 0)

at the beginning of the test, and the amount of normal stress decreases (due to the stiffness of the surrounding soil (virtual springs)) (Eq. 1). After this slight compression, the soil starts to dilate (∆U < 0), and this dilation acts on the surrounding soil. Due to the compression of the surrounding soil, the amount of the normal stress increases (∆σ > 0). This normal stress rise, consequently increases the shear strength of the soil at the interface. Conversely, in the second case (loose or normally consolidated soils), the soil at the interface contracts (∆U > 0), and the normal stress decreases (∆σ < 0) until the shear ceases.

To apply CNS condition to the temperature-controlled direct shear device, the follow-ing procedure was implemented in the command software:

1. The total desired shear displacement, W (8 mm) was divided into 100 segments (W/100 = 0.08 mm).

2. In order to reach the desired W (8 mm) value: (W/100) , 2, 3, ..., 100] where i is the number of segments.

× i i = [1
3. At the end of each segment, the device measures the vertical displacement difference between the beginning of the segment and the end of the segment (∆U = ∆U i 2 -∆U i 1 ).

4. Then, according to Eq. 1, this difference (∆U (mm)) is multiplied by the value of stiffness (K (kPa/mm)), and the consequent normal stress (∆σ n ) that should be applied is calculated.

5. This process is repeated for all segments i(100) until the total shear displacement is reached.

Normal stiffness verification

To calibrate the device for the stiffness application, the variations of normal stress (∆σ)

with normal displacement (∆U ) are presented in Fig. 6. The slope of these curves represents the stiffness value. For tested values of stiffness, a satisfactory correlation is obtained (1-2% precision).

Experimental programme

The experimental programme consisted of soil and soil-structure direct shear tests at different temperatures (Table 3). Soil tests were performed as reference cases for comparison with soil-structure tests to better clarify the role of interface.

The sand programme consisted of a series of constant normal load (CNL) tests at different temperatures to investigate the effects of temperature on the mechanical characteristics. In sand-structure tests, CNL and CNS tests were performed at 22 and 60 o C.

For CNS tests, different stiffness values (K = 500, 1000, and 5000 kPa/mm) were chosen, that were used in previous studies [START_REF] Boulon | Physical and numerical simulation of lateral shaft friction along offshore piles in sand[END_REF][START_REF] Mortara | An elastoplastic model for sand-structure interface behaviour under monotonic and cyclic loading[END_REF][START_REF] Pra-Ai | Behaviour of soil-structure interfaces subjected to a large number of cycles[END_REF].

Increasing the stiffness value restrains the volumetric response of the interface until a certain case of constant normal stiffness which is called the constant volume condition (CV). These values were chosen to cover the entire range of constant normal stiffness conditions, from very small ranges close to CNL and up to very high values close to CV.

To perform the CNS tests, two scenarios were considered.

Sand programme

To prepare the sand samples for the shear tests, the Fontainebleau sand with a target dry density of 1.67 M g/m 3 was poured into the shear box and compacted using a tamper.

This dry density corresponded to 90% of the relative density (D r ), and the sample was considered to be a dense sand (Table 1). Then, the normal stress was applied to the sand sample (path 0-1 in Fig. 7). After applying the normal stress, to shear the samples in CNL condition at 22 o C, a shear rate of 0.1 mm/min was applied (path 1-2). For the CNL tests at 60 o C, the heating phase (path 1-5, Fig. 7) was applied with a rate of 10 o C/hr, and the shearing phase (path 5-2 ) started.

For the sand-structure tests, the same procedure was performed, except the interface was placed at the lower half of the shear box. For the sand-structure CNS tests, due to the dense state of the soil, path 1-3 at 22 o C and path 5-3 at 60 o C were observed (Fig.

7).

Clay programme

To perform the clay and clay-structure shear tests, kaolin clay was prepared with a water content of 63%, which was slightly higher than its liquid limit (LL = 57%) and the sample was left for 24 hours for homogenization. Subsequently, the clay was poured into the shear box and special attention was paid to avoid any air trap. To perform the CNL tests at 22 o C,, the normal stress was applied slowly and incrementally during the consolidation phase, and each load increment lasted 2 hours, to ensure full consolidation at each step [START_REF] Mortezaie | Effect of frequency and vertical stress on cyclic degradation and pore water pressure in clay in the ngi simple shear device[END_REF]. Two values of initial effective normal stresses (σ n0 = 100, 300 kPa) were chosen for the clay programme. Based on the consolidation tests performed on this kaolin clay, the target void ratios after consolidation for σ n = 100 and 300 kPa were e = 1 and 0.85, respectively. After the consolidation phase for the CNL tests at 22 o C, a displacement rate of 0.006 mm/min that was calculated from the settlement curve and t 50 (time required for the specimen to achieve 50 percent consolidation under the maximum normal stress) of the kaolin, was applied [START_REF] Astm | Standard test method for direct shear test of soils under consolidated drained conditions[END_REF]. This slow rate ensured drained conditions inside the shear box during shearing. The initial heating or cooling phase started at ambient temperature (22 o C). After the consolidation phase, heating or cooling was applied to the samples at a rate of 5 o C/hr. This slow rate avoids a pore pressure increase during the heating phase and was verified by [START_REF] Cekerevac | Experimental study of thermal effects on the mechanical behaviour of a clay[END_REF][START_REF] Cekerevac | Experimental study of thermal effects on the mechanical behaviour of a clay[END_REF][START_REF] Di Donna | Experimental investigations of the soil-concrete interface: physical mechanisms, cyclic mobilization, and behaviour at different temperatures[END_REF]. During the heating or cooling phase in the shear box, thermal vertical deformation of the soil and the soil-structure interface was measured. After these heating or cooling phases, the samples were sheared. For the CNL tests of the clay and clay-structure interface, paths 1-2 and 5-2 were applied, as seen in Fig. 7, but for the CNS clay-structure interface tests, paths 1-4 and 5-4 were observed due to the normally consolidated state of the kaolin samples.

Experimental results for sand

In the following sections, first the CNL sand shear test results, and then, the CNL and CNS sand-structure interface tests are discussed.

Sand

Constant normal stiffness

Fig. 10 shows the sand-structure interface CNS results for the first case (σ n0 = cte and K = 500, 1000, 5000 kPa/mm). With increasing stiffness, the maximum shear strength for CNS tests was increased due to the increase in normal stress (Fig. 10a). The CNS peak shear stress obtained for larger shear displacements and the post-peak softening behaviour was less evident than in the CNL sand-structure tests. In Fig. 10b, at the beginning of shearing, the interface contracted slightly and then started to dilate. For K = 500 kPa/mm, the soil in the interface dilated approximately -0.2 mm, and by increasing the stiffness to 5000 kPa/mm, the dilation was reduced to -0.08 mm (Fig. 10b). Increasing the stiffness restrained the volumetric response of the soil at the interface. These restrained dilations, increased the normal stress (Fig. 10c). The normal stress increased from the initial value (100 kPa) to 180 kPa for K =500 and 1000 kPa/mm. For K = 5000 kPa/mm the normal stress increased to 510 kPa. Therefore, the normal stress variation during shear depended on the volumetric response, and these normal stress increases, consequently increase the shear stress acting on the interface. Fig. 10d shows the Mohr-Coulomb plane of the sand-structure interface. All tests showed, with shear increase, the effective normal stress decreased slightly at the beginning, then followed by an increase until the end of shear. The comparison of curves at 22 and 60 o C shows that the temperature has a negligible effect on the shearing behaviour of Fontainebleau sand-structure interface under CNS conditions.

The second scenario (σ n0 = 100, 200, 300 kPa and K = cte) results are reported in Fig. 11 and are compared with the results of the CNL case (K = 0 kPa/mm). The peak shear stress for CNS tests were obtained for larger shear displacements (1.3 mm), compare to CNL tests (0.8 mm). The increase in peak shear stress for σ n0 = 100, 200, and 300 kPa in the CNS tests were 20, 60 and 80 kPa respectively (Fig. 11a), compared to the values on the CNL tests. The volumetric responses in the CNS tests are more restrained than in CNL tests (Fig. 11b). In Fig. 11c, the normal stress variation during the CNS tests is presented. The normal stress showed a slight decrease until 0.5 mm of W (mm) and then increased. This increase continued until 1.4 mm of W (mm), which corresponds to the peak shear stress; after this peak was reached, the normal stress remained unchanged. The peak and residual friction angles of the interface in Fig. 11d are δ p = 37 o and δ res = 28 o which are smaller than the friction angles obtained in the sand tests (Fig. 8d). This point confirms that the shear occurred exactly in the interface zone, and not in the soil mass.

Sand vs. sand-structure interface

In this section, the CNL test results for the sand and sand-structure interface are compared to better understand the interface behaviour. The shear stress and volumetric response in the CNL sand and sand-structure tests for σ n0 =100 and 300 kPa at T=22 o C are compared in Fig. 12. In the sand-structure tests, the post-peak softening behaviour is more pronounced than in the sand tests (Fig. 8). The peak shear stress is higher (7-15%) in the sand samples than in the sand-structure tests, which confirms the shear failure occurs in the interface zone. The peak shear stresses of the sand-structure tests are obtained at smaller shear displacements (0.7-1 mm) than in the sand (1.6-1.8 mm) case, which could be due to the rearrangement of grains in the interface zone [START_REF] Hoteit | Etude exprimentale du comportement physique et mcanique des interfaces sols-structures[END_REF][START_REF] Tabucanon | Frictional behaviour between normally consolidated clay and steel by two direct shear type apparatuses[END_REF][START_REF] Porcino | Interface behavior of sands from constant normal stiffness direct shear tests[END_REF]. ). The dilation phase began at smaller shear displacements in the sand-structure interface tests (0.7 mm) than in the sand samples

(1.2 mm), and the amount of dilation (0.2 mm) was almost half that of the sand samples (0.4 mm).

Experimental results for clay

This experimental section is divided into the clay and clay-structure results. For the clay tests, the CNL test results and for clay-structure interface tests, the CNL and CNS test results are discussed.

Clay

In this part first, thermal vertical strain of clay samples is presented. Second, the shear curves and volumetric responses for σ n0 =100 and 300 kPa at 5, 22 and 60 o C are presented.

To verify the repeatability of the results, the test with σ n0 =300 kPa at 60 o C was repeated.

After consolidation and reaching the desired void ratio, a heating or cooling phase at a rate of 5 o C/hr, was applied to the clay samples. This heating and cooling caused a thermal vertical deformation under constant effective normal stresses of 100 and 300 kPa (Fig. 13). The heating phase started from 22 to 60 o C and cooling phase was from 22 to 5 o C. The thermal vertical strain was higher for heating cases (0.6 -0.64%) than cooling cases (0.18 -0.2%). Therefore, the slope of heating curves was less than the cooling curves, and heating caused more contraction than cooling.

After heating or cooling, normally consolidated kaolin clay samples were sheared at two different effective normal stresses (σ n0 =100, 300 kPa) at 5, 22 and 60 o C (Fig. 14). In Fig. 14a andc, the shear stress-shear displacement curves for 100 kPa and 300 kPa at 5, 22 and 60 o C are presented. As observed for both effective normal stresses, the shear stress increased with increasing temperature, until the peak values, then it decreased towards the critical state. The residual shear stresses at 5, 22 and 60 o C for σ n0 =100 kPa became convergent after a shear displacement of 5 mm. For σ n0 =300 kPa at 5 and 22 o C the shear stress increased in the same manner, but after a shear displacement of W=3.5 mm, the stress values diverged.

In the volumetric response, the samples that were exposed to higher temperatures showed less contraction during shear (Fig. 14b and Fig. 14d. For example in 100 kPa of effective normal stress, the test at 5 o C showed a contraction approximately 0.89 mm, but for the tests at 22 and 60 o C, this amount decreased to approximately 0.68 mm and 0.38 mm respectively. For 300 kPa, the same trend was observed for the volumetric response (Fig. 14d).

The Mohr-Coulomb plane for the clay tests at different temperatures is presented in

Clay-structure

In the following sections, the CNL and CNS results for the clay-structure are presented.

Constant normal load (CNL)

Fig. 16 shows the thermal vertical strain for σ n0 =300 kPa during the heating phase and after consolidation. The thermal vertical strain caused by the temperature increase from 22 to 60 o C was approximately 0.85%, and for a temperature decrease from 22 to 5 o C in the cooling case, the thermal vertical strain was approximately 0.2% for the clay-structure interface tests.

The thermal volume deformation depends on the volume of the sample [START_REF] Campanella | Influence of temperature variations on soil behavior[END_REF][START_REF] Campanella | Influence of temperature variations on soil behavior[END_REF][START_REF] Baldi | Thermal volume changes of the mineral-water system in low-porosity clay soils[END_REF], and the thermal vertical strain is higher in the clay-structure interface than in the clay samples due to the volume of the clay specimen in the clay-structure tests.

In Fig. 17 the results of the clay-structure interface CNL tests are presented. Fig. 17a shows the shear stress versus shear displacement for σ n0 =100 kPa. The peak shear strength curve was slightly higher at 60 o C than at 5 and 22 o C (∆τ = 8 kPa), but at the critical state, all curves at different temperatures were superimposed. For σ n0 =300 kPa (Fig. 17c), the ∆τ = 10 kPa of difference at the peak was evident for 60 o C compared to 5 and 22 o C. In the critical state, the same behaviour as σ n0 =100 kPa was observed.

In Fig. 17b andd the volumetric behaviours of the clay-structure interface are presented for σ n0 =100 and 300 kPa. For σ n0 =100 kPa (Fig. 17b), the amount of contraction was 0.4 mm for the test at 5 o C, while it was 0.28 mm and 0.12 mm for the tests at 22 and 60 o C, respectively. For σ n0 =300 kPa (Fig. 17d), the same behaviour was observed, and the contraction at 5 o C was higher than those obtained at 22 and 60 o C.

Constant normal stiffness (CNS)

To investigate the shear characteristics of the clay-structure interface, constant normal stiffness (CNS) conditions were applied. The results for K = 1000 kPa/mm that is intermediate value between K = 500 (CNL) and 5000 kPa/mm (CV) are presented.

The shear stress versus shear displacement for two initial normal stresses (σ n0 =100 and 300 kPa) at 22 and 60 o C are presented in Fig. 19a. At σ n0 =100 kPa, the shear stress increased with increasing shear displacement until reaching a value of 1 mm (τ = 33 kPa) then, with a slight decrease, the shear stress continued towards the critical state (τ = 28 kPa). The curves for both 22 and 60 o C followed the same trend. Tests at σ n0 =300 kPa showed a very clear peak and then decreased towards a constant value. As mentioned for 100 kPa, under σ n0 =300 kPa, the shear stresses at 22 and 60 o C are similar.

For both initial normal stresses, kaolin contracted until the end of the shear (Fig. 19b).

For σ n0 =100 kPa at 22 o C, the amount of normal displacement in the critical state was around 0.035 mm. This value was approximately 0.02 mm for tests at 60 o C, and the heated samples showed less contraction. For σ n0 =300 kPa at 22 and 60 o C, the amount of normal displacement in the critical state was 0.9 and 0.6 mm, respectively.

In Fig. 19c the variation of normal stress during CNS tests of clay-structure interface is presented. For both σ n0 =100 and 300 kPa the normal stresses decreased during shearing process. For samples exposed to higher temperatures the amount of reduction was less than samples at 22 o C. For tests at σ n0 =100 kPa at 22 and 60 o C the normal stress decrease was about 42 and 30 kPa respectively.

In Fig. 19d, the normal stress vs. shear stress planes for the clay-structure interface CNS and CNL tests are presented. For σ n0 =100 and 300 kPa in the CNS tests, the shear stress increased with decreasing normal stress, and the shear reached a peak value and then decreased. The heated samples showed less decrease in the normal stress. For example, for σ n0 =300 kPa, the peak shear strength for heated samples was slightly higher than 22 o C, and the normal stress decrease in the heated sample was also less than that at 22 o C. The peak friction angle and adhesion of the CNS tests were 14 o and 13 kPa respectively.

Clay vs. clay-structure interface

In Fig. 20 the CNL tests of clay and clay-structure for σ n0 =100 kPa are compared. The peak shear stress in the clay-structure tests is obtained with smaller shear displacements (1.2-1.6 mm), while for the clay, the peak shear stress is around (2.8-3.2 mm), and the softening behaviour after the peak is more pronounced for the clay-structure interface.

The peak shear stress difference of 6 kPa for clay vs clay-structure at 22 o C and 7.5 kPa at 60 o C for σ n0 =100 kPa confirms that the shear occurs in the interface zone and not in the soil mass.

In

Discussion

In the following section, the discussion about the effects of temperature on sand and clay interface shear strength is presented.

Effect of temperature on sand Fig. 21 shows the effect of temperature on cohesion, adhesion and friction angle of different soils that have been studied for interface direct shear tests at different temperature variations in the literature, which is compared to results obtained in this study. As can be seen, the effect of temperature on the peak and residual friction angle of sand and sand-structure interface are negligible (Fig. 21b). The similar shear curves, volumetric behaviour and Mohr-Coulomb plane at different temperatures (Fig. 8) for sand samples also confirm that, the effect of temperature on the shear strength of sand is negligible.

These observations are in line with existing studies [START_REF] Di Donna | Experimental investigations of the soil-concrete interface: physical mechanisms, cyclic mobilization, and behaviour at different temperatures[END_REF][START_REF] Yavari | Effect of temperature on the shear strength of soils and the soil-structure interface[END_REF]. For sand-structure interface tests with different stiffness and temperatures, it was observed that, the interface behaviour in CNS condition does not change at different temperatures. For sandy soil used in this study, in the context of energy foundations, heating a sand-structure interface in this range of temperatures does not change the mechanical properties of the interface, and in the design calculations no further precautions should be considered for thermal effects.

Effect of temperature on clay

In clay tests, heating under drained conditions led to a contraction of the samples, and consequently the shear strength increased due to thermal hardening (Fig. 14). Several authors observed this thermally induced contraction and shear strength increase during drained heating of normally consolidated clays [START_REF] Campanella | Influence of temperature variations on soil behavior[END_REF][START_REF] Hueckel | Thermoplasticity of saturated clays: experimental constitutive study[END_REF][START_REF] Kuntiwattanakul | Temperature effects on undrained shear characteristics of clay[END_REF][START_REF] Chiu | Behaviour of normally consolidated clay at elevated temperature[END_REF][START_REF] Cekerevac | Experimental study of thermal effects on the mechanical behaviour of a clay[END_REF][START_REF] Abuel-Naga | Experimental evaluation of engineering behavior of soft bangkok clay under elevated temperature[END_REF]. [START_REF] Baldi | Thermal volume changes of the mineral-water system in low-porosity clay soils[END_REF] have explained, the reason of this thermoplastic strain may be in the micro-structural changes as the tendency of clay flakes to group together which increases the mineral-to-mineral contact and generates irreversible strain. This may be due to the sliding, or partially sliding, shear mode along the interface.

Indeed, [START_REF] Lemos | Clay-interface shear resistance[END_REF] showed that, in clays with high clay content in which residual soil shear is in the sliding mode, peak interface shear strength normally is close to the soil residual strength and is independent of roughness (Fig. 20). Therefore, in the clay-structure interface tests performed in this study, the sliding or partially sliding shear mode at interface occurred for all tested temperatures.

The peak cohesion of clay samples increases from 17 to 23 kPa while the peak adhesion of clay-structure increases from 12 to 18 kPa with temperature increase from 22 to 60 o C while the residual adhesion remained stable (Fig. 21a). Therefore, it can be concluded that cohesion is more affected by temperature modification. This reflects the interaction between the structure and the clay. For kaolin clay, heating the interface tend to slightly increase the shear strength of the interface. Therefore, in terms of structural safety of energy geostructures, temperature increase in normally consolidated kaolin clay can be considered as a positive aspect.

In isothermal conditions, Potyondy 1961 have performed direct shear tests on different soils and different structural materials like steel, concrete and wood. He found that the friction angle and adhesion of a smooth steel interface is less than a smooth concrete interface. Therefore, it can be concluded that the nature of the structural material plays a major role in the interface behaviour and further works should be carried out to investigate the nature the interface behaviour on different construction materials.

Conclusions

Constant normal load (CNL) and constant normal stiffness (CNS) interface tests were conducted on soil and soil-structure samples at different temperatures (5,22 and 60 o C).

The results showed that the mechanical properties of sand are independent of temperature In kaolin clay, temperature does not affect the friction angle and the main effect is the increase of the cohesion or adhesion. For clay tests, due to thermal contraction of kaolin during heating, the soil becomes denser and shows a higher shear strength. It was found that temperature increases the cohesion of clay samples. In clay-structure contact, due to difference in the nature of materials (clay vs. metal) the adhesion is not as much as clay case, therefore the shear strength increase with temperature increase, is not as much as clay case. In CNS tests on clay-structure interface, the soil exposed to higher temperatures, showed less contraction during shearing, and consequently less normal stress decrease due to the denser state of the heated clay-structure samples prior to shearing.

Therefore, in the interface the soil becomes denser with heating and the shear strength increases slightly.

Further work will be carried out to investigate the effects of thermo-mechanical cycling on the mechanical behaviour of soil-structure interface.

  . Di[START_REF] Di Donna | Experimental investigations of the soil-concrete interface: physical mechanisms, cyclic mobilization, and behaviour at different temperatures[END_REF] performed interface direct shear tests on quartz sand and illite clay at different temperatures (22, 50 and 60 o C). These tests showed that the sand-concrete interface behaviour was not directly affected by temperature changes, but the clay-concrete interface showed higher shear strength at higher temperatures. The residual interface friction angle of the clay-concrete decreased slightly at high temperatures, but the adhesion (cohesion between soil and structure) increased with increasing temperature. The authors suggested that this result is related to the thermal consolidation of the clay, which results in an increase of the contact surface between the clay and concrete.[START_REF] Yavari | Effect of temperature on the shear strength of soils and the soil-structure interface[END_REF] conducted soilstructure interface direct shear tests on Fontainebleau sand and kaolin clay samples at 5, 20 and 40 o C. The shear strength of the clay samples was higher than that of the clay-concrete interface, and the effects of temperature (in the range of 5-40 o C) on the shear strength and friction angle were negligible in the sand, clay and clay-concrete interface. They pre-consolidated all the samples to 100 kPa of vertical stress and heated to 40 o C prior to the application of the initial conditions. Therefore, they found that the effect of temperature on the clayconcrete interface, which was mainly related to thermal consolidation, was negligible.

  Fig. 4a. To determine the roughness of the interface, each profile was divided into the D 50 of Fontainebleau sand (0.23 mm) and at each D 50 , the R max was measured. The values of R max were divided by D 50 to obtain the normalized roughness (R n ). For Fontainebleau sand, the normalized roughness R n is presented in Fig. 4b. Most of the normalized values are between 0.02 and 0.3. The largest value of normalized roughness R n (0.32) was determined; therefore, the stainless steel plate is considered as a rough and very rough surface for Fontainebleau sand and kaolin clay.

  First, shear tests with different stiffness values (K = 500, 1000, and 5000 kPa/mm) and constant effective normal stress (σ n0 = 100 kPa) were performed at 22 and 60 o C. The aim of this part was to determine the effect of different stiffness values at 22 and 60 o C on sand-structure interface. The second scenario, included interface shear tests at three different effective normal stress values (σ n0 =100, 200 and 300 kPa) with a constant stiffness value (K = 1000 kPa/mm). This scenario was performed to determine the friction angle of the interface and also compare the CNS and CNL tests.

Fig. 8a presents

  Fig.8apresents the results of the sand CNL tests at 22 and 60 o C, which will be used as a reference for the sand-structure tests. The dense sand samples show a peak shear stress at a small shear displacement, and then, with a decrease, reach a critical state at both 22 and 60 o C. In Fig.8b, the volumetric behaviour of the sand is presented. The amount of contraction (∆U > 0) is around 0.06 mm; then, at 1.2 mm of shear displacement, the dilation (∆U < 0) phase starts and continues until U = -0.4 mm, and finally, a constant value corresponding to the critical state of the soil is reached. Fig.8cshows the stress ratio (η = τ /σ n ) variations with shear displacement, and the peak shear strengths are reached at similar shear displacements (W = 1.5 -1.6 mm) for different normal stresses. Fig. 8d shows the Mohr-Coulomb plane for the sand samples. The peak friction angle for the tests is 41.6 o at 22 and 60 o C, while the residual friction angle is 34 o . The same peak and residual friction angles at both temperatures show the negligible effect of thermal variations on the shearing behaviour of the studied sand.

  Pra-Ai 2013 conducted interface direct shear tests on dense Fontainebleau sand samples on a rough steel plate under isothermal conditions, and they found 38 o and 29 o for the peak and residual friction angles of the interface respectively. In the CNS sand-structure interface tests, the peak and residual friction angle of the interface (40.7 o and 32.4 o ) are so close to those in the CNL sand-structure tests (40.4 o and 32.7 o ). The friction angle of the sand-structure interface is not affected by the CNS condition, which was also observed by[START_REF] Porcino | Interface behavior of sands from constant normal stiffness direct shear tests[END_REF]

Fig. 15 .

 15 Fig. 15. The internal friction angle of the clay soil obtained at different temperatures, shows a slight increase with temperature increase (14.4 o to 15.3 o ) which can be considered negligible, but the main difference was the cohesion increase from 11 to 17 kPa and then to 23 kPa for tests at 5, 22 and 60 o C, which could be due to thermal hardening during the heating phase.

Fig. 18

 18 Fig. 18 shows the Mohr-Coulomb plane for the clay-structure interface CNL tests at different temperatures. The peak friction angle for all studied temperatures was 14 o . The main difference between the Mohr-Coulomb envelopes for different temperatures was the adhesion. The increase in temperature, increased the peak adhesion (cohesion between soil and structure) from 12.5 kPa to 18 kPa while the residual adhesion remained constant (16.5 kPa).

  terms of volumetric behaviour, the clay tests are completely different with claystructure tests (Fig. 20b). In clay tests the contraction increases until the end of test but in the clay-structure tests the contraction increases more rapidly compare to the clay tests, and then remains constant. Tsubakihara and Kishida 1993 have performed direct shear tests on Kawasaki clay and mild steel. They found that, the reason for the volumetric behaviour difference between clay and clay-structure tests, can be the sliding shear mode that occurs in the interface.

Fig. 20

 20 Fig. 20 shows that the peak and residual shear strength in the clay-structure tests, were always less than those obtained during clay tests. This showed that the shearing occurred at the interface zone. Moreover, the peak shear stresses of the clay-structure tests are close to the residual shear stress values of clay tests at all tested temperatures.

(

  22 and 60 o C) for both sand and sand-structure tests. Different stiffness values were applied under CNS conditions at different temperatures, and it was observed that such as the CNL tests, the temperature does not change the interface behaviour under the CNS condition. Additionally the same interface friction angle was obtained in both CNL and CNS tests for sand-structure interface tests.

Figure 1 :Figure 1 :Figure 2 :

 112 Figure 1: CNS concept of the soil-structure interface (after Wernick 1978).

Figure 3 :Figure 4 :Figure 5 :Figure 5 :

 3455 Figure 3: (a) Steel mould dimensions and, laser setup (b) direction and dimensions of laser profiles.

Figure 6 :Figure 7 :Figure 8 :Figure 9 :Figure 11 :Figure 12 :Figure 13 :

 6789111213 Figure 6: Imposed stiffness verification. (a) K = 500 kPa/mm; (b) K = 1000 kPa/mm; (c) K = 5000 kPa/mm. σ n0 = 100 kPa.

Figure 14 :

 14 Figure 14: CNL results for clay samples at T = 22 o C, T = 60 o C and T = 5 o C. (a) Shear stress vs. shear displacement (σ n0 = 100 kPa); (b) normal displacement vs. shear displacement (σ n0 = 100 kPa); (c) shear stress vs. shear displacement (σ n0 = 300 kPa); (d) normal displacement vs. shear displacement (σ n0 = 300 kPa)

Figure 15 :Figure 16 :Figure 17 :

 151617 Figure 15: Shear stress vs. effective normal stress of CNL clay tests at T = 5 o C, T = 22 o C and T = 60 o C

Figure 18 :Figure 19 :

 1819 Figure 18: Shear stress vs. effective normal stress of CNL tests of clay-structure interface at T = 5 o C, T = 22 o C and T = 60 o C.

Figure 20 :

 20 Figure 20: Comparison of CNL results for clay and clay-structure interface with σ n0 = 100 kPa at T = 5 o C, T = 22 o C and T = 60 o C .

Table 1 :

 1 Fontainebleau sand physical properties[START_REF] Pra-Ai | Behaviour of soil-structure interfaces subjected to a large number of cycles[END_REF]

	D 50 (mm)	ρ s (g/cm 3 )	γ dmax (kN/m 3 )	γ dmin (kN/m 3 )	e max	e min	= D 60 /D 10 C u
	0.23	2.65	17.2	14.2	0.866	0.545	1.72

Table 2 :

 2 Kaolin clay physical and thermal properties[START_REF] Yavari | Effect of temperature on the shear strength of soils and the soil-structure interface[END_REF]

	LL (%)	P L (%)	I p (%)	(M g/m 3 ) ρ s	(W/mK) λ	(J/m 3 K) C	k (m/s)
	57	33	24	2.60	1.5	3.3	10 -8

Table 3 :

 3 Experimental programme of soil and soil-structure interface tests the soil-structure interface (after Wernick 1978). . . . . . . . . . . . . . . 2 Grain size distribution of Fontainebleau sand and kaolin clay. . . . . . . . . . . . . . . . .

		σ n (kP a)	K (kP a/mm)	T o (C)	Type of
					test
	Sand	100, 200, 300	0	o	CNL
	Sand	100, 200, 300	0	o	CNL
	Sand-structure	100, 200, 300	0	o	CNL
	Sand-structure	100, 200, 300	0	o	CNL
	Sand-structure	100	500, 1000, 5000	o	CNS
	Sand-structure	100	500, 1000, 5000	o	CNS
	Sand-structure	100, 200, 300	1000	o	CNS
	Clay	100, 300	0	5 o	CNL
	Clay	100, 300	0	o	CNL
	Clay	100, 300	0	o	CNL
	Clay-structure	100, 300	0	5 o	CNL
	Clay-structure	100, 300	0	o	CNL
	Clay-structure	100, 300	0	o	CNL
	Clay-structure	100, 300	1000	o	CNS
	Clay-structure	100, 300	1000	o	CNS

List of symbols


Véronique Favier
email: veronique.favier@ensam.eu@b

Antoine Blanche
email: antoine.blanche@univ-montp2.fr

Ngoc Lam

Nicolas Ranc
email: nicolas.ranc@ensam.eu

André Chrysochoos
email: eandre.chrysochoos@univ-montp2.fr

Ngoc Lam Phung

Energy Dissipation and Self-Heating due to Microplastic Deformation Mechanisms at Very High Cycle Fatigue for Single-Phase Ductile Metals

Keywords: ultrasonic fatigue, copper, persistent slip bands, heating, fatigue limit

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The very high cycle fatigue (VHCF) regime is associated with stress amplitudes lower than the conventional fatigue limit and as a result, numbers of cycles higher than 10 9 . The present paper deals with a fundamental research project. However, it is motivated by industrial problems. Some mechanical components, such as pistons, rotating axes, in the transportation and more generally engineering industry, have been designed previously using fatigue resistance data at lower numbers of cycles (<10 7 cycles ; the regime of high cycle fatigue, HCF) whereas they must endure cyclic loading for a number of cycles higher than >10 9 cycles [START_REF] Bathias | Gigacycle Fatigue in Mechanical Practice[END_REF]. Many experiments in the literature prove that failure occurs at stress amplitudes lower than the conventional fatigue limit.

In the high cycle fatigue (HCF), cyclic strain localization in Persistent Slip Bands (PSBs) is generally observed and accepted as the critical onset of fatigue damage in ductile single-phase materials and also in many precipitation-hardened alloys [START_REF] Mughrabi | On multi-stage fatigue life diagrams and the relevant life controlling mechanisms in ultrahigh-cycle fatigue[END_REF][START_REF] Mughrabi | Cyclic Slip Irreversibilities and the Evolution of Fatigue Damage[END_REF][START_REF] Risbet | Some comments about fatigue crack initiation in relation to cyclic slip irreversibility[END_REF]. Once formed, PSBs, accompanied by the development of extrusions and notch-like deepenings, commonly referred to intrusions, lead to cracks. For ductile single-phase materials, it is suggested that crack initiation takes place at the surface owing to the accumulation of very small but irreversible plastic deformation over very large number of cycles [START_REF] Mughrabi | Cyclic Slip Irreversibilities and the Evolution of Fatigue Damage[END_REF]. However, there seems to be no possibility for the formation of a pronounced PSB structure at very low stress amplitudes below the PSB threshold in the HCF regime. To better understand the relationship between loading conditions and PSBs, it is necessary to conduct additional investigations after cyclic deformation in the VHCF regime.

In this work, we focused on pure copper which is face-centered cubic (fcc) single-phase metal. This material is a good candidate because its HCF behavior has been largely studied in literature. To perform experiments up to a very high number of cycles in a reasonable time, ultrasonic equipment at a testing frequency of 20 kHz was used. It takes only fourteen hours to perform 10 9 cycles at a testing frequency of 20 kHz, whereas it takes one hundred days at 100 Hz. In contrast to conventional fatigue tests (low and high cycle fatigues), detecting irreversible microstructural changes using cyclic stress-strain curves analysis is no more possible, at least with conventional measurement systems, since the macroscopic behavior is quasi-elastic. To cope with this problem, we proposed to use the energy signature of the deformation mechanisms. The temperature increase of the specimen was measured during the cyclic loading. The dissipated energy was deduced from temperature field measurements by solving the heat equation accounting for exchanges losses and material conductivity. This calorimetric method has been successfully developed for conventional fatigue at testing frequencies ranging from 10-50 Hz [START_REF] Boulanger | Calorimetric analysis of dissipative and thermoelastic effects associated with the fatigue behavior of steels Inter[END_REF][START_REF] Honorat | A calorimetric analysis of polymer behaviour using a pixel calibration of an IRFPA camera[END_REF][START_REF] Mareau | Influence of the free surface and the mean stress on the dissipation in steels under cyclic loading[END_REF][START_REF] Doudard | Identification of heat sources from infrared thermography : Determination of 'self-heating' in a dual-phase steel by using a dog bone sample[END_REF][START_REF] Connesson | Experimental energy balance during the first cycles of cyclically loaded specimens under the conventional yield stress[END_REF]. In this project, it has been adapted to 20 kHz cycling loading. From an industrial point of view, ultrasonic fatigue testing and dissipation estimates during cycling loading may appear as rapid methods to select or to class materials with regard to their fatigue properties.

Material and experimental procedure

Material and specimens. Commercial polycrystalline copper in initial form of a hot rolled plate of 14 mm of thickness was tested. The specimens were extracted from the center of the plate and the microstructure of material is presented in Fig. 1. The mean grain size was about 30 µm with a lot of annealed twins. The material is quasi-isotropic textured. The present study was conducted by using an ultrasonic fatigue machine at 20 kHz with cylindrical and flat hourglass shaped specimens. The specimen dimensions were calculated so that the free resonant frequency of specimen in the first longitudinal mode is 20 kHz. All specimens were treated at 250°C for 60 min to relieve the residual stress (ASM specialty Hand book) without change of microstructure. Then, a procedure of mechanical and electrolytical polishing was executed till remove all hardened layers on surface specimen. As a result, before testing, the specimens were mirror polished and without residual stresses. The dynamic Young modulus in calculations was E d = 130 GPa [START_REF] Stanzl-Tschegg | Life-time and cyclic slip of copper in the VHCF-regime[END_REF]. The stress/strain amplitude along the specimen was estimated by a one-dimensional elastic calculation. It was in a good agreement with the experimental measurement obtained by the strain gages. The geometry and the stress amplitude distribution in specimens are presented in Fig. 2.

Fig.1. Microstructure of material

Experimental procedure. To perform experiments up to the VHCF regime, an ultrasonic equipment at a testing frequency of 20 kHz was used. The S-N curve of the material in the VHCF regime was first established. The fatigue tests were carried-out with cylindrical specimens at various stress amplitudes with a constant stress ratio (σ max /σ min ) R = -1. In order to prevent from a heating effect at high stress amplitudes, the specimens were cooled by a cold air gun during the test till rupture. To study the formation and development of slip markings (SMs) on the specimen Fig. 2. Ultrasonic fatigue specimens (a) cylindrical specimen (b) flat specimen surface at very low stress amplitudes and after very high numbers of cycles, the flat specimens were used. This type of specimen is appropriate to study the microstructure and temperature and intrinsic dissipation by an infrared camera. The experiments were periodically interrupted after several specified numbers of cycles for surface observations.

Calorimetric analysis

Heat diffusion model. A numerical model was built to estimate the evolution of intrinsic dissipation fields from the temperature measurement fields along the fatigue test. A making of longitudinal distributions of heat sources within the specimen gage part was proposed. A 1D calorimetric analysis has been justified assuming in a first approximation a uniaxial tensioncompression stress state. From the heat equation:

(1

)
where T is the temperature, ρ the mass density, C the heat capacity, k the thermal conduction coefficient and s(x,y,z,t) symbolizes the volume heat source, following classical hypotheses [START_REF] Boulanger | Calorimetric analysis of dissipative and thermoelastic effects associated with the fatigue behavior of steels Inter[END_REF], the 1D diffusion equation for a non-constant cross-section can be written as [START_REF] Honorat | A calorimetric analysis of polymer behaviour using a pixel calibration of an IRFPA camera[END_REF]:

(2)
where the mean temperature variation over the cross-section is identified with the surface temperature, being the first derivative of the cross-section while and are the longitudinal profiles of dissipation and thermoelastic sources. Moreover the time constant characterizes the lateral heat exchanges by conduction, convection and radiation between the specimen and the surroundings. These exchanges are assumed to be proportional to the temperature variation (linear Fourier boundary conditions). In order to estimate the left-hand side of Eq. ( 2), longitudinal temperature profiles were built by averaging the surface temperature over the specimen width.

Taking the loading frequency (20 kHz) and the maximum frame rate of the IR camera (100 Hz) into account, it must be stressed that the thermoelastic source amplitudes are, here, out of reach. Only the mean dissipation per cycle will be derived from the discrete, noisy thermal data. Dissipation estimate. To compute , the different partial derivative operators were estimated using spatio-temporal fitting. The local approximation functions were chosen as:

(3) where and are second-order polynomials in x. These polynomials are identified over each approximation zone using a local least-squares method. The approximated thermal field is then used to compute the intrinsic dissipation, via an estimate of the time and space partial derivative operators of the heat equation (see Eq.2) [START_REF] Honorat | A calorimetric analysis of polymer behaviour using a pixel calibration of an IRFPA camera[END_REF].

Results

S-N Curve. The S-N curve of the studied pure copper obtained by ultrasonic fatigue machine at 20 kHz is shown in Fig. 3. It displays two regimes. In the first regime, failure occured between 3.6x10 6 and 5x10 7 cycles with stress amplitude ranging from 115 MPa to 97 MPa. The first regime is associated with the high cycle fatigue (HCF) regime. In the second regime, failure took place at numbers of cycles greater than 5x10 7 cycles. The stress amplitude is lower than 97 MPa. The S-N curve slope in VHCF is more than 10 times smaller than in HCF. In other words, the number of cycles very quickly increases with decreasing the stress amplitude. At Δσ/2 = 91.2 MPa, we did not observe any failure up to 5.4x10 9 cycles at which the test was stopped. Our results are in very good agreement with Stanzl-Tschegg et al. [START_REF] Stanzl-Tschegg | Life-time and cyclic slip of copper in the VHCF-regime[END_REF]'s experiment results obtained on similar pure polycrystalline copper. Their results revealed a "fatigue limit at 1x10 10 cycles" also called "fatigue life threshold" equal to at Δσ/2 = 92.2 MPa. Three types of persistent slip markings in fatigue. The appearance and evolution of slip markings (SM) in the VHCF regime for a wide range of stress amplitude from 38 MPa to 115 MPa have been studied. It was found that the lower the stress amplitude, the higher the number of cycles necessary to observe the early SMs. Besides, we distinguished 3 types of SMs: type I, type II and type III which differ from their location within the microstructure and their morphology (Fig. 3). All these SMs were shown to be Persistent Slip Markings (PSM) and are Persistent Slip Bands. The appearance of the different types of slip markings depends essentially on stress amplitude. At high stress amplitudes, i.e. higher than "fatigue limit at 1x10 10 cycles" at Δσ/2 = 92.2 MPa, the majority of the first slip markings is of type I. The amount of the PSM type I decreases with decreasing stress amplitudes and the PSM type II become more and more dominant. At very low stress amplitudes such as 44 MPa, type III is dominant. All kinds of PSMs (I,II and III) have been reported before [e.g. [START_REF] Stanzl-Tschegg | Life-time and cyclic slip of copper in the VHCF-regime[END_REF][START_REF] Stanzl-Tchegg | Mechanisms of strain localization, crack initiation and fracture of polycrystalline copper in the VHCF regime[END_REF]. However, here these three types of PSMs are classified here in terms of the stress amplitude ranges at which they appear predominantly. Whatever the type of PSMs, their amount was found to increase with increasing the number of cycles. Intrinsic dissipation. Figure 5 shows the intrinsic dissipation distribution (divided by in order to have volume energy rate (W/m 3 ) expressed in (°C/s)) along the specimen axis obtained for maximal stress amplitude of 72.1 MPa after 10 7 cycles. The intrinsic dissipation is concentrated in the middle of the specimen and decreases toward the ends. Hence, it is, as expected, related to the distribution of stress (Fig. 2). Optical micrographs revealed that this zone display the highest quantity of slip markings (Fig. 5). This result indicates that dissipation is related to microplasticity. The surface average intrinsic dissipation was estimated from thermal field measurements over fatigue tests. Fig. 6 exhibits the intrinsic dissipation at 10 6 cycles for various stress amplitudes ranging from 20 to 90 MPa. Results stressed that dissipated energy exists whatever the applied load. Higher the stress range, higher the intrinsic dissipation. We can note that even below about 30 MPa, at which no slip marking appear even at number of cycles higher than 10 10 and sometimes considered as an "irreversible threshold" in literature [START_REF] Stanzl-Tschegg | Life-time and cyclic slip of copper in the VHCF-regime[END_REF], a non-zero intrinsic dissipation can be detected (Fig. 6 (b)). It was also found that dissipation always increased with the number of cycles revealing continuous changes within the material. This result is consistent with the gradual increase of the PSM (slip bands) amount on the specimen surface over cycles.

Conclusion

Microstructural change of commercially pure polycrystalline copper after 20 kHz fatigue tests in the very high cycle fatigue regime was characterized by coupling two methods: (1) observing the specimen surface relief during cycling and (2) estimating the intrinsic dissipation from temperature measurements. A good correlation was found between slip activity and dissipation. Dissipation and slip band amount always increased over the number of cycles. At very small stress amplitudes, no slip band appeared up to 10 8 cycles but the material was found to dissipate energy. Results derived from tests performed at high loading frequency on pure cupper specimens showed a drift of dissipative regimes incompatible with concepts of fatigue limit and/or asymptotic cyclic stability. These results reveal that the material never reached a steady state. Therefore it could break at higher number of cycles.

Fig. 3 .

 3 Fig.3. S-N curves to rupture and to form the early slip markings (slip bands) -Open and full circles and squares correspond to the fracture of the specimen and the appearance of the early slip bands, respectively.

Fig. 4 .

 4 Fig.4. Three types of slip markings in VHCF (a) Type I ; (b) Type II ; (c) Type III and its zoom (d)

Fig. 5 :

 5 Fig. 5: Dissipation along the specimen at Δσ/2 = 72.1 MPa, 10 7 cycles and a zoom at the middle of specimen

Fig. 6 -

 6 Fig. 6-(a) Dissipation vs. Stress amplitude Fig. 6-(b) Zoom between 20 and 60 MPa

Acknowledgements

The authors are grateful for financial support from Agence Nationale de la Recherche France ANR-09-BLAN-0025-01 and to company Griset for supplying copper.