Nicolaine Agofack

Siavash Ghabezloo
email: siavash.ghabezloo@enpc.fr

Jean Sulem

André Garnier

Christophe Urbanczyk

Experimental investigation of the early-age mechanical behaviour of oil-well cement paste

Keywords: early-age, cement hydration, shrinkage, stress-strain response, cement sheath, residual strain

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

Introduction

In oil and gas industry, when drilling exploration or development wells, a mud is used to support the walls of the geological formation, to maintain the wellbore stability, to cool and clean the drill bit, to carry out the cuttings and to provide hydrostatic pressure which helps in preventing formation fluids to enter into the wellbore. Steel tubular (called "casing") is then run into the well and a cement slurry is pumped to fill the annulus space between two casings or between the casing and the geological formation. The cement slurry hardens progressively and forms a cement sheath, which plays an important role for well integrity. Along the well, it provides zonal isolation of different fluids (i.e. separation of different fluids, water, gas and oil); it protects the tubular against corrosion and provides mechanical support. The loss of cement-sheath integrity can result in the pressurization of the annulus, in gas migration up to a shallower formation or to the surface, and, in catastrophic cases, in a blowout and significant damage of the infrastructure.

The cement paste is an evolving porous material, its state changes progressively from fluid to solid during the hydration process. Along a well, different compositions of cement slurries can be used and their hydration takes place in different conditions of temperature and pressure depending on depth. Therefore, at a given time, cement paste behaviour and properties depend on the slurry composition of the cement paste including water-to-cement mass ratio (w/c), presence of additive agents, hydration conditions in terms of temperature and pressure. In addition, during the life of a well, from drilling to completion, production and P&A (plug and abandonment), the cement sheath is submitted to various mechanical and thermal loadings that can potentially damage its principal properties and alter its performance. According to some authors [START_REF] Behrmann | Borehole Dynamics During Underbalanced Perforating[END_REF][START_REF] Thiercelin | Cement Design Besed on Cement Mechanical Response[END_REF] the increase of pressure during drilling can excess 40 MPa in the oil wells, and the applied pressure during a casing test can vary from a few tens of MPa in normally pressurized reservoirs to more than 80 MPa in high pressure reservoirs. These loadings may result in non-reversible deformation of cement sheath and consequently the creation of a microannulus in the cement-casing or cement-rock interface. This is particularly important when these loadings are applied at relatively early age, when the mechanical properties and strength of the cement paste are not yet completely developed. As it was reported in the literature [START_REF] Parcevaux | Cement Shrinkage and Elasticity: A New Approach for a Good[END_REF][START_REF] Justnes | Chemical shrinkage of oil well cement slurries[END_REF][START_REF] Lyomov | Shrinkage of Oil Well Cement Slurries[END_REF], at the early-age, the shrinkage of cement paste and the decrease of its pore pressure are considered to be the main causes of leakage in oil and gas wells. However, very few experimental data exist on the mechanical properties (macroscopic shrinkage, elastic modulus, the stress-strain behavior) of the hydrating oilcement in field conditions and at very early age. Some properties, particularly the macroscopic shrinkage, have been extensively studied for civil engineering applications in which the cement paste is hydrated under ambient pressure and temperature conditions. The situation is quite different for oilwell applications in which the cement paste is hydrated under various conditions of pressure and temperature, depending mainly on the depth and reservoir conditions. This work is dedicated to an experimental study of the mechanical behavior of early-age cement paste, during the first 6 days of hydration under various pressures and temperatures. A specially designed experimental device is used to investigate the macroscopic shrinkage of cement paste and its stressstrain response under oedometric conditions, from the first hour of hydration. To investigate the effect of early-age loading history on long-term properties, classical triaxial experiments are performed on hardened samples submitted to various loadings at their early age. The tested material is a class G oilwell cement, with water-to-cement ratio equal to 0.44. The same material has been continuously studied by the co-authors providing a rich database of mechanical, hydraulic, thermal and microstructure properties [6, 7, 8, 9, 10, 11, 12, 13, 14] [15]. The hydration conditions correspond to a temperature range from 7 to 30°C and a pressure range from 0.3 to 45 MPa. These relatively low temperatures are used to slow down the cement hydration kinetics. These experiments are associated with UCA (Ultrasonic Cement Analyser) and isothermal calorimetry experiments for a deeper insight into the behaviour of early age cement paste.

Experimental devices and tests procedures

To study the mechanical behavior of the cement paste at very early-age, one should be able to apply mechanical loadings on the hydrating paste without modifying its hydration conditions of temperature and pressure. This is a difficult task in a classical testing procedure in which the sample's temperature and pressure are generally brought back to ambient conditions before installing it in the loading cell.

The hydration progress during this procedure and the dependence of the cement paste microstructure on temperature during hydration make it difficult to control the conditions at which the experiment is performed. To overcome these difficulties, a special uniaxial strain (oedometric) cell was designed by TOTAL in which the cement paste is poured immediately after mixing, and hydrated under controlled conditions of temperature and pressure. The cement paste can be submitted to mechanical loading cycles, at any time even before setting, without modifying its hydration conditions, what is impossible in a standard triaxial cell.

Figure 1. Uniaxial-strain (oedometric) equipment

The oedometric cell, shown in Figure 1, is composed of a hollow cylinder (50.3 mm of internal diameter, 90 mm of external diameter and 125 mm height) made of steel. Within its elastic domain, the hollow cylinder can support a maximum pressure of 123 MPa applied on its inner surface. The hollow cylinder is surrounded by a cooling/heating system, connected to a cryostat, whereby cold/hot fluid is circulated through a small-diameter tubing. The axial displacement of the sample is measured by a Linear Variable Differential Transformer (LVDT). An electromechanical ram, able to apply 650 kN, is used to apply the axial stress. The oedometric cell is cooled or heated to the test temperature before the preparation of the slurry.

The tested specimen has a diameter of 50.3 mm and a height of 85mm. The axial displacement of the cement paste is measured continuously during hydration. At a given age, a mechanical loading cycle is applied to the hydrating paste to measure its oedometric (P-wave) modulus and to study its stressstrain response. In order to compare this "static" oedometric modulus with the dynamic (P-wave modulus), experiments are performed in similar hydration conditions using an Ultrasonic Cement Analyzer (UCA) apparatus which permits to continuously measure the P-wave velocity (at ultrasonic frequency) on the hydrating cement paste. In oedometric cell, as well as in UCA apparatus, the properties of the hydrating cement paste are measured as function of time. In order to analyze the experimental results as a function of the degree of hydration, a high pressure high temperature (HPHT) calorimeter is used to evaluate the hydration kinetics from the measurement of hydration heat. A standard triaxial cell is also used to analyse the effect of loading history at very early-age on the longterm properties.

To analyze the pore size distribution, the mercury intrusion porosimetry (MIP) is performed on samples with different histories of hydration. The sample dimensions are 25 mm in diameter and 20 mm in height. After drying the samples by freeze-drying, the mercury is injected into the sample under All tests were performed on a slurry made of CEMOIL class G cement (API classification), the chemical composition of which is presented in Table 1. The cement slurry is composed of class G cement powder, distilled and deaerated water with w/c = 0.44, and three additive agents (anti-foaming, dispersing and anti-settling), as presented in Table 2. Distilled and deaerated water is poured into the mixing container, which is then placed on the mixer base. The dry anti-settling agent is added to the water and the motor is turned on and maintained to 4000 ± 200 rotations per minutes for 5 minutes of agitation. Then the liquid anti-foaming and dispersing agents (poured before cement) and the cement powder are put successively into the container in 20 ± 5 seconds, (according to API [START_REF]API-10B-2, Recommended Practice for Testing Well Cementing[END_REF], cement powder shall be added in 15s). The speed is immediately increased to about 12 000 ± 500 rotations per minute and mixing ended automatically after 35 ± 1 seconds.

Reliability and repeatability of the oedometric tests

In order to test the reliability of the oedometric test, distilled water is put into the cell. The mechanical loading path of Figure 2a consists of an increase of the axial stress from 5 to 25 MPa, followed by a decrease from 25 to 5 MPa. After maintaining the axial stress at 5 MPa for about 5 minutes, the same loading is again applied. This experiment is equivalent to an isotropic compression on water. The slope of axial stress -axial strain curve gives the bulk modulus of water. The result, presented in Figure 2b, gives the bulk modulus of water equal to 2.09 GPa which is very close to the value found in the literature. Thus, we conclude that the oedometric cell gives reliable results.

Experimental results and discussion

During the cement hydration, the sum of the volumes of consumed cement clinker and water is higher than the volume of the produced hydrates. The difference represents the chemical shrinkage [START_REF] Baumgarte | Case Studies of Expanding Cement To Prevent Microannular Formation[END_REF][START_REF] Reddy | Cement-Shrinkage Measurement in Oilwell Cementing--A Comparative Study of Laboratory Methods and Procedures[END_REF].

The chemical shrinkage results in pore pressure decrease within the cement paste during hydration. As a consequence, the effective stress increases and leads to a decrease of the external macroscopic volume of the hydrating cement paste. When the cement paste is hydrated at constant ambient temperature and under atmospheric pressure, this external volume decrease is called autogenous shrinkage [START_REF] Bentz | Mitigation strategies for autogeneous shrinkage cracking[END_REF][START_REF] Reddy | Cement-Shrinkage Measurement in Oilwell Cementing--A Comparative Study of Laboratory Methods and Procedures[END_REF]. A different term, macroscopic shrinkage, is generally used when the hydration and the resulting volume change take place under pressure higher than the atmospheric pressure and/or temperature different to ambient temperature [START_REF] Reddy | Cement-Shrinkage Measurement in Oilwell Cementing--A Comparative Study of Laboratory Methods and Procedures[END_REF][START_REF] Api | Shrinkage and Expansion in Oilwell Cements[END_REF][START_REF] Baumgarte | Case Studies of Expanding Cement To Prevent Microannular Formation[END_REF]. The macroscopic shrinkage of cement paste was measured in several studies [START_REF] Reddy | Cement-Shrinkage Measurement in Oilwell Cementing--A Comparative Study of Laboratory Methods and Procedures[END_REF][START_REF] Api | Shrinkage and Expansion in Oilwell Cements[END_REF][START_REF] Reddy | Cement Mechanical-Property Measurements Under Wellbore Conditions[END_REF] and more recently by Abuhaikal et al. [START_REF] Abuhaikal | An apparatus for Dissecting Volumetric Changes in Hydrating Cement Paste[END_REF]. In this section, we present a method to measure the macroscopic shrinkage of the cement paste and its stressstrain behaviour without modifying the condition of hydration. The effect of temperature and pressure of hydration on macroscopic shrinkage are analysed. The effect of age and macroscopic shrinkage on stress-strain behaviour of cement paste are also analysed.

Hydration kinetics

The experiments presented in this paper are performed on cement pastes hydrated in a temperature range within 7°C and 22°C, under stresses between 0.3 and 45 MPa. It is known that the hydration kinetics is generally accelerated by increasing the temperature and pressure during hydration [START_REF] Ludwig | Conduction calorimeter for measuring heat of hydration of Portland cement at elevated temperature and pressure[END_REF][START_REF] Taylor | Cement Chemistry[END_REF][START_REF] Thomas | A new approach to modeling the nucleation and growth kinetics of tricalciumsilicate hydration[END_REF][START_REF] Bresson | Hydration of tricalcium silicate (C3S) at high temperature and high pressure[END_REF]. The knowledge of the hydration kinetics of the cement paste in the performed oedometric experiments is important for a better understanding and interpretation of the results. A high pressure high temperature calorimeter is used to evaluate the hydration kinetics from the measurement of hydration heat. Calorimeter data provide the cumulative heat of hydration Q(t) released until a given time t. The degree of hydration can be evaluated through the following equation:

T_CP ( ) ( ) Q t t Q ξ = (1)
where Q T_CP is the total cumulative heat released at complete hydration, which can be estimated by knowing the composition of the cement paste, using the following relation.

_ _ X T CP X CP T X Q m Q = ∑ (2)
were _ represents the mass fraction of the compound X in the cement paste. The Table 3 gives the details of calculation of Q T_CP , assuming the main compounds of cement paste to be: water, alite, belite, aluminate and ferrite. Detailed calculation of the volume fractions of the constituents for the tested cement can be found in [START_REF] Vu | Effet des contraintes et de la température sur l'intégrité des ciments des puits pétroliers[END_REF][START_REF] Agofack | Comportement des ciments pétroliers au jeune âge et integrite des puits[END_REF]. The results are presented in Figure 4 as function of time for different conditions of hydration. For a given temperature during hydration (Figure 4a), the degree of hydration increases with the increase of pressure. The same effect is observed for different temperatures when the pressure during hydration is kept constant (Figure 4b). The presented results are compatible with the well-known pressure and temperature dependency of cement hydration kinetics [START_REF] Taylor | Cement Chemistry[END_REF][START_REF] Schindler | Effect of Temperature on Hydration of Cementitious Materials[END_REF][START_REF] Gallucci | Influence de la température sur le développement microstructural des bétons[END_REF][START_REF] Thomas | A new approach to modeling the nucleation and growth kinetics of tricalciumsilicate hydration[END_REF][START_REF] Gallucci | Effect of temperature on the microstructure of calcium silicate hydrate (C-S-H)[END_REF][START_REF] Vu | Effet des contraintes et de la température sur l'intégrité des ciments des puits pétroliers[END_REF].

Typical results of an oedometric test

The oedometric cell (presented above) is used to investigate the macroscopic shrinkage of cement paste, its dependence on temperature and pressure, and also the stress strain behaviour of the cement paste. For a typical oedometric test, after mixing, the cement slurry is directly put into the oedometric cell which is kept at the test temperature. The axial stress is then increased up to the stress during hydration and keeps constant during the first 144 hours of hydration. At 144 hours, a mechanical loading/unloading cycle is applied with a loading rate of 3 MPa/min. For the samples hydrated under an axial stress higher than 5 MPa (Figure 5a), the loading path consists in a decrease of the axial stress to 5 MPa followed by two cycles with an increase up to 65 MPa and a decrease to 5 MPa and finally of an increase back to the initial axial stress during hydration. For the sample hydrated under axial stress lower than 5 MPa (Figure 5b), the loading path consists in two cycles with an increase up to 65 MPa and a decrease to the axial stress during hydration. The results of two typical tests in terms of axial strain as a function of time are shown in Figure 6 for two cement samples hydrated at 7°C, under 0.3 and 25 MPa, respectively. The experimental response is composed of two main parts. The first part up to 144 h for which the axial stress is kept constant (Figure 6a) shows that the macroscopic shrinkage is higher for the sample hydrated under a higher axial stress. The second part (Figure 6b) presents the response of the samples to the mechanical loading/unloading cycles and shows the lower compressibility of the sample hydrated under a higher stress. These results will be analysed in detail in the following. increase at 144 hours. These results show that at a given time, the macroscopic shrinkage is significantly higher for hydration under higher stress. However, one should note that this observation is influenced to some extent by the pressure dependency of cement hydration kinetics, as presented in The measured axial strain, i.e. the macroscopic shrinkage, at 144 hours of hydration increases with the axial stress during hydration (Figure 9a). [START_REF] Reddy | Cement-Shrinkage Measurement in Oilwell Cementing--A Comparative Study of Laboratory Methods and Procedures[END_REF] also observed this increase of macroscopic shrinkage with the increase of stress applied on the cement paste during its hydration.

Figure 7b presents the evolution of the axial strain of the cement paste as a function of time, for hydration under constant axial stress (25 MPa) but at different temperatures (7, 15 and 22°C). As observed in this figure, the effect of temperature is more pronounced during the first 60 hours of hydration and seems to vanish with the advancement of the reaction. The long-term effects of temperature and pressure on macroscopic shrinkage at 144h are presented in Figure 9 for 7°C temperature (variable stress) and 25 MPa axial stress (variable temperature). It can be seen that the effect of temperature is less significant compared to that of stress (Figure 9b). This is of course partly due to the relatively narrow range of the studied temperatures. The results show that higher pressure during hydration leads to higher compaction of the samples. The density of the hydrated material is thus expected to be higher for samples hydrated under higher pressure. The observed macroscopic shrinkage during hydration results from strongly coupled phenomena: at the very early age and before the percolation the slurry is under an isotropic stress state with the pore pressure equal to the total pressure, i.e. zero Terzaghi effective stress. The water consumption during hydration results in a progressive pore pressure decrease in the samples and thus to an increase of the effective stress.

Moreover, the progressive hardening of the material results in development of shear stress in the sample which is in oedometric condition. These stress changes result in mechanical deformation of the material which can be a combination of deformations with elastic, elastoplastic and viscous nature. For the samples under lowest pressures, 0. The Ultrasonic Cement Analyser (UCA) is used to measure the ultrasonic P-wave velocity in the cement paste during hydration. The dynamic oedometric modulus dyn oed E can then be calculated using the following relation as a function of the P-wave velocity V p , by knowing the density of the sample ρ Error! Reference source not found.. (resp. 3200 m/s) after 300 hours of hydration. The dynamic modulus, calculated using equation ( 3) is presented in Figure 13 and compared with the static modulus measured under similar condition of hydration. The results exhibit a dynamic modulus slightly higher than the static modulus. This is in agreement with existing experimental results showing that the dynamic modulus is generally higher than the static modulus for different materials [START_REF] Kim | Effect of temperature and age on the relationship between dynamic and static elastic modulus of concrete[END_REF][START_REF] Oku | Relation between static and dynamic Young's modulus of nuclear graphites and carbon[END_REF][START_REF] Sabbagh | Dynamic and static moduli of elasticity of resin-based materials[END_REF].

. Effect of age

Seven samples of cement paste, with the same composition (Table 1 & Table 2), hydrated at 7°C and under 25 MPa, are submitted to mechanical unloading/reloading cycles as shown in Figure 5a. Each sample has been loaded at a different age, 15, 18, 20, 40, 60, 103 and 144 h. The results are presented in Figure 14. The loading cycles applied at earlier age show a quite linear stress-strain response with non-recoverable strains in unloading. The results show a progressively non-linear stress-strain response for later loadings particularly after 40h. During the first loading (5 65 MPa), two main slopes can be distinguished on the samples loaded after 20 hours of hydration. These slopes may characterize the elastic and elastoplastic domain of the response with a transition at the elasticity limit.

As presented in Figure 15, this stress limit increases with the degree of hydration (or age) at which the mechanical loadings are applied. Moreover, the slope of the first unloading/reloading cycle (25 5 25 MPa) increases with the age of the cement paste. This is in agreement with the increase of the elastic parameters of the cement paste with hydration as is observed also in the measured oedometric modulus during hydration (Figure 11 & Figure 13).

Effect of loading history

As explained in the introduction, various mechanical loadings can be applied to the cement sheath in an oil-well and it is important to investigate the effect of these early-age loadings on the long-term behaviour of the cement paste. In order to analyse the effect of the previous mechanical loading cycles on the stress-strain behaviour of cement paste at later age of hydration, the samples presented in the Figure 14 have been for a second time submitted to the same mechanical loading at 144 hours. The effect of this first mechanical loading on the cement behaviour is analysed by comparing the result of their second loading with the stress-strain response of the sample submitted for the first time at 144 hours (Figure 14g). On the following figures, the samples are identified by the hydration conditions and the time of their first mechanical loading. For example, T7P25-40 (solid line)

represents the sample hydrated at 7°C and under 25 MPa on which the first mechanical loading is applied at 40 h. The mechanical loading previously applied at 15, 18 and 20 h (Figure 16a, b andc) seem to have little effect on the stress-strain behaviour at 144 hours. The stress-strain curves are quite similar, even though we can observe that the previously loaded samples are a little less deformable. This is much more visible and significant for the samples loaded previously after 40 hours of hydration. The stress-strain response of these samples shows a lower deformability, higher moduli and lower irrecoverable strains in unloading. After the loading at 144 hours, the samples T7P25, T7P25-15, T7P25-18 and T7P25-40 were immediately removed from the oedometric cell and kept into distilled and deaerated water at ambient conditions for at least 3 months. The mercury porosity measurements are then performed and the results are shown in Figure 17. It is worth nothing that the only difference between these samples is the age at which the first mechanical loading was applied. It can be observed that the first loading applied at 15 or 18 h has no significance effect on the mercury porosity, as compared to sample T7P25 on which no further loading cycle has been applied (Figure 17). The loading applied at 40 h reduced significantly the mercury porosity. In Table 4, it is shown that the mercury porosity as well as the total The stress-strain responses of the samples hydrated under different conditions of temperature and axial stress are presented in Figure 18 (the first phases of these experiments are presented in Figure 7). The experiments performed at 7°C under different initial axial stresses are shown in Figure 18a.

Comparing the curves shows that the samples hydrated under higher stress are less deformable during the first loading. For samples hydrated under relatively high stress (10 and 25 MPa), for the first loading until 65 MPa the sample shows a quasi bilinear response corresponding to an elastic response under lower stresses and then an elasto-plastic response for higher stresses with a smooth transition at about 25 to 30 MPa axial stress. The following unloading-reloading cycles result in an almost linear response with a little hysteresis which may be attributed to the viscous behaviour of the cement paste [START_REF] Vu | Time-dependent behaviour of hardened cement paste under isotropic loading[END_REF]. The samples hydrated under relatively low stress (0.3 and 3 MPa) show a more complex response under the first loading up to 65 MPa. A first change in the slope of the stress-strain curve is observed at 20 to 25 MPa axial stress, similar to the one observed for samples hydrated under higher stress, but a second slope change and a significant hardening of the material is observed under higher axial stress of about 45 to 50 MPa. The physical mechanism behind this hardening of the material can be explored considering an elasto-plastic constitutive law for hydrating cement paste based on these experimental results [START_REF] Agofack | Comportement des ciments pétroliers au jeune âge et integrite des puits[END_REF]. Figure 18b presents the results of the experiments performed on cements hydrated under 25 MPa axial stress at different temperatures between 7 and 22°C. The results show quite similar response, which can be related to the narrow range of studied temperatures. One can see however that the samples hydrated under higher temperature are less deformable. This can be related to the increase of the cement hydration rate with temperature. The analysis of the results presented in Figure 7 and Figure 18 shows a strong coupling between the macroscopic shrinkage and stress-strain response of the cement paste. An appropriate constitutive model for the hydrating cement paste should capture this coupling and the physical mechanisms behind the observed response of this material.

Another general observation, which can be made based on these experimental results, corresponds to the significantly higher deformability of the cement paste hydrated under relatively low stress. This higher deformability shows the higher risk of creation of a micro-annulus at shallower depths of a well.

• After 3 months of hydration

Three samples hydrated at the same temperature (7°C) but under different axial stresses 3, 10 and 15 MPa (T7P3, T7P10, T7P15) have been submitted to triaxial testing after 3 months of hydration.

The samples were submitted to the same mechanical loading path at 144 hours as presented in Figure 5. The samples were then kept at the ambient temperature in saturated condition for 3 months before performing triaxial experiments. These experiments have been performed under a confining pressure of 25 MPa. The differential stress as a function of axial and radial strain is presented in Figure 19. The results clearly show that the stress-strain behaviour of the cement paste is affected by the condition of hydration during the first 144 hours. Samples hydrated under lower stress show higher deformability and lower shear strength. This confirms once again the higher potential of micro-annulus formation at shallower depths. The mechanical static and dynamic properties of the samples are presented in Table 5. The static moduli have been evaluated in the unloading-reloading cycles. The static drained bulk and Young moduli increase with the increase of the axial stress during hydration applied on the samples during the first 144 hours of hydration. As commonly observed, the dynamic bulk modulus and Poisson's ratio are higher compared to the static values. The explanation of this difference can be traced to the fact that the dynamic parameters are measured under quasi-undrained conditions whereas the statics ones are obtained under drained conditions. It can also be due to the amplitude of deformation which is much smaller for dynamic measurements (around 10 -7 -10 -6 in dynamic measurements, and 10 -3 in static measurements [START_REF] Mavko | The Rock Physics Handbook : Tools for seismic Analysis of Porous Media[END_REF]) as also mentioned by [START_REF] Oku | Relation between static and dynamic Young's modulus of nuclear graphites and carbon[END_REF]. The effect of hydration conditions in these samples shows that the cement paste stiffness in long-term increases with the increase of the stress applied during the early-age of hydration.

Conclusion

The early-age mechanical behaviour of a class G oil-well cement paste is experimentally investigated using a specially designed cell, which permits mechanical loading application from the first hour of hydration, even before the setting of the cement. The macroscopic shrinkage of the cement paste as well as its stress-strain response under oedometric loading are studied for various pressures during hydration, from 0.3 to 45 MPa, and for temperatures during hydration from 7 to 30°C. The results showed a significant increase of the macroscopic shrinkage with the pressure during hydration. At a given age, the cement paste hydrated under lower pressures exhibits a significantly higher deformability when submitted to a mechanical loading cycle. This result shows a higher risk of creation of micro-annulus at shallower depth during the mechanical loadings, for example during a casing test. The stress-strain response of the cement paste strongly depends on the cement age. When submitted to a loading cycle at different times, a mostly linear and reversible response observed at the very early age is progressively transformed to a highly nonlinear response and increasing irreversible deformations. At a given time, the mechanical response of the cement paste submitted to a loading cycle depends on the loading history. This is an important point showing that the simulation of the cement sheath response in an oil well needs a complete modelling of its behaviour from the placement to the current state, including various applied loads. This is confirmed also for the long-term behaviour by performing classical triaxial experiments after 3 months hydration on cement samples with different loading history at early age. The samples submitted to lower stress at early age show higher deformability. In the studied ranges of pressure and temperature, the pressure has a much more significant effect on the mechanical response of the cement, compared to the temperature effect.

However, one should note that a relatively narrow range of mostly low temperature are investigated here, mainly to slow down the hydration rate for performing loading cycles application in a relatively constant hydration degree. This study will be extended in the future by exploring a broader range of higher temperatures.

Figure 2 .

 2 Figure 2. Reliability test: (a) loading path, (b) stress-strain behaviour and bulk modulus.

Figure 3 .

 3 Figure 3. Repeatability of the oedometric tests: three cement sample hydrated at 7°C and under 25 MPa of axial stress.

Figure 4 .

 4 Figure 4. Degree of hydration as function of time for different conditions of hydration in term of temperature and pressure.

Figure 5 .

 5 Figure 5. Loading path at 144 hours of hydration for different stress during hydration: (a) more than 5 MPa (b) less than or equal to 5 MPa. Duration of the cycles around 1h30min, loading rate 3 MPa/min.

Figure 6 .

 6 Figure 6. Typical result for oedometric tests: two samples hydrated at 7°C and under 0.3 and 25 MPa

Figure 4 .

 4 Figure 4. The results of three experiments under various pressures, for which calorimeter results were available, are plotted in Figure8as a function the degree of hydration. Two distinguished responses can be observed for macroscopic shrinkage under high and low pressures: the samples hydrated under high pressure, i.e. 25 and 45 MPa, show almost similar response in terms of a quasi linear macroscopic shrinkage evolution with the degree of hydration, while the sample hydrated under a low pressure of 0.3 MPa, shows a more nonlinear response. The initial slope of the axial strain -degree of hydration response at the very beginning is very close to the one of the high-pressure samples. This should correspond to the chemical shrinkage of the cement hydration, as before percolation and formation of a solid skeleton all chemical shrinkage is transformed to the macroscopic shrinkage. A value of 0.044 -0.060 can be found for the chemical shrinkage of this cement which is very close to the values presented in the literature[START_REF] Tazawa | Chemical shrinkage and autogenous shrinkage of hydrating cement paste[END_REF][START_REF] Lyomov | Shrinkage of Oil Well Cement Slurries[END_REF][START_REF] Mounanga | Predicting Ca(OH)2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF][START_REF] Bouasker | Chemical shrinkage of cement pastes and mortars at very early age: Effect of limestone filler and granular inclusions[END_REF][START_REF] Reddy | Cement-Shrinkage Measurement in Oilwell Cementing--A Comparative Study of Laboratory Methods and Procedures[END_REF][START_REF] Yodsudjai | Chemical shrinkage behavior of pastes made with different types of cements[END_REF].

3

 3 MPa and 3 MPa, the water consumption may result in a partial desaturation of the material. A better understanding of the mechanisms of deformation of the samples needs a coupled chemo-poro-mechanical simulation of the presented experiments, taking into account the above-mentioned phenomena.

Figure 7 .

 7 Figure 7. Experimental measures of macroscopic shrinkage of cement paste during the first 144 hours of hydration: (a) constant temperature during hydration (7°C) [T7P10 and T7P3 are from [15]], (b) constant axial stress during hydration (25 MPa).

Figure 10 .

 10 Figure 10. Loading path and stress-strain response for a cement hydrated at 7°C and under 25 MPa (sample M-T7P25)

Figure 11 .

 11 Figure 11. Elastic, static oedometric modulus as a function of time for three cement samples hydrating at 7°C and under 15, 25 and 45 MPa.

  At ultrasonic frequency, the pore pressure generated within the cement paste during the wave's travelling do not have enough time to diffuse out of the sample. The measured dynamic modulus is thus considered as undrained oedometric modulus. For the samples hydrated under pressure, the average density of the cement paste increases during hydration, due to the compaction effect. For the cement sample hydrated under 25 MPa, the evolution of the average density and of the P-wave velocity during hydration are given in Figure12. The density (resp. the P-wave velocity) of the cement increases from approximately 1910 kg/m 3 (resp. 1400 m/s) for the slurry state to around 2000 kg/m3

Figure 12 .

 12 Figure 12. Average density and ultrasonic P-wave velocity as a function of time for the cement sample hydrating at 7°C and under 25 MPa [28].

Figure 13 .

 13 Figure 13. Comparison between the dynamic and static elastic "undrained" oedometric moduli for samples hydrated at 7°C and under 25 MPa on the UCA and the oedometric cell, respectively.

Figure 14 .Figure 15 .

 1415 Figure 14. Stress-strain behaviour at different age of cement samples hydrated under similar oedometric conditions (7°C and 25 MPa)

Figure 16 .

 16 Figure 16. Effect of previous mechanical loadings applied within 15 h and 103 h on the stress-strain behaviour of the cement paste submitted to the same mechanical loadings at 144 hours.

  porosity are smaller in sample T7P25-40 as compared to the other samples. The mercury and total porosity are in the same order as those found by Ghabezloo et al. Error! Reference source not found.. The decrease of the porosity with the application of the mechanical loading cycles is related to the compaction of the cement sample.

Figure 17 .

 17 Figure 17. Mercury porosity on cement paste as a function of the reached pore diameter

Figure 18 .

 18 Figure 18. Experimental measures of stress-strain behavior of cement paste at 144 hours of hydration: (a) constant temperature during hydration (7°C), (b) constant axial pressure during hydration (25 MPa).

Figure 19 .

 19 Figure 19. Results of the triaxial tests on the cement paste hydrated under oedometric conditions during the first 144 hours and kept at ambient conditions for 3 months. The confining pressure was 25 MPa [15]

Lower header Space for floating system Axial LVDT sensor σ 1 Upper header Cement paste Surrounding pipe for heating or cooling Hollow cylinder 50.3 mm controlled

  pressure. At each pressure level the quantity of injected mercury enables calculating the porosity and the applied pressure is used to estimate the corresponding pore entry size.

Table 1 . Chemical Composition of a class G cement (Blaine = 323 m 2 /kg) Chemical composition Measures API recommendations Spec 10A

 1

	Magnesium oxide (MgO) %	1.8	Max 6.0
	Sulfurtrioxide (SO3) %	2.5	Max 3.0
	Loss on ignition %	0.6	Max 3.0
	Insoluble residue %	0.10	Max 0.75
	C 3 A %	2.1	Max 3.0
	C 4 AF + 2 C 3 A %	17.4	Max 24.0
	C 3 S %	58.9	Min 48.0/ Max 65.0
	(0.658 × K 2 O) + Na 2 O %	0.55	Max 0.75

Table 2 . Composition of 600 ml of cement slurry: w/c = 0.44, density = 1900 ± 10 kg/m3) Components Quantity (grams)

 2

	Cement (class G)	783.53
	water (distilled and deaerated)	339.54
	Anti-foaming (D047) *	6.27
	Dispersing agent (D604AM) *	9.47
	Anti-settling (D153) *	1.18
	* commercial identification of the product

Table 3 . Estimation of hydration heat of a class G cement paste (w/c = 0.44)

 3

	Compounds	Chemical composition	(J/g)	_	_	(J/g)
	Water	H 2 O	0	0.3
	Alite	C 3 S	500	0.427
	Belite	C 2 S	260	0.106		297
	Aluminate	C 3 A	900	0.010
	Ferrite	C 4 AF	420	0.111

Table 4 . Physical properties of four samples conserved in water at ambient conditions for at least 3 months after the application of the mechanical loading cycles at 144 hours.

 4

		T7P25	T7P25-15	T7P25-18	T7P25-40
	Mercury porosity (%)	26.7	27.5	26.2	23.4
	Saturated density (g/cm 3 )	2.022	2.022	2.010	2.005
	Total porosity (%)	43.5	43.1	43.3	42.8
	4.5.3.

Effect of temperature and pressure conditions during hydration • At 144 hours of hydration

Table 5 . Static and dynamic parameters of three samples of cement paste hydrated under oedometric conditions during the first 144 hours and kept at ambient conditions for 3 months.

 5

		Static parameters	Dynamics parameters
	Samples	T7P15	T7P10	T7P3	T7P15	T7P10	T7P3
	K d (GPa)	7.56	7.16	6.95	16.32	16.15	16.25
	G (GPa)	7.08	6.56	5.56	8.21	8.36	8.34
	ν	0.14	0.15	0.18	0.31	0.30	0.30
	E (GPa)	16.18	15.08	13.17
	E oed (GPa)	16.99	15.91	14.36
	∆(σ ax -σ rad ) max (MPa)	81	76	64

Acknowledgments

The authors gratefully acknowledged TOTAL for supporting this research. They also wish to thank Joselin DE-LA-IGLESIA, Edmond POYOL and Sandrine ANNET-WABLE from TOTAL for performing calorimeter experiments and Nicolas CRUZ from ACEI for his assistance in the testing program.

Oedometric moduli

The oedometric moduli are measured in the performed oedometric experiments. The static modulus is measured by performing mechanical unloading and reloading on the hydrating cement paste in undrained condition. The measurements are performed for three samples M-T7P15, M-T7P25 and M-T7P45 hydrating at the same temperature (7°C), but under different axial stresses 15, 25 and 45 MPa, respectively. "M-" before the test conditions is related to "modulus". The loading path is similar for the three conditions. It is presented in Figure 10a for the sample M-T7P25, showing an unloading from the axial stress during hydration (i.e. 25 MPa) to 5 MPa and reloading to the stress during hydration.

The unloading/reloading cycles were applied at different given times with the rate of 3 MPa/min. The duration of one unloading/reloading cycle is around 7, 13 and 27 minutes for the samples M-T7P15, M-T7P25 and M-T7P45, respectively. Considering the relatively slow hydration rate at 7°C, it can be assumed that these loading cycles are performed at constant degree of hydration. The stress-strain


K Kane

Julien Jumel
email: julien.jumel@u-bordeaux.fr

F Lallet

A Mbiakop-Ngassa

J.-M Vacherand

M E R Shanahan

A Novel Inflation Adhesion Test for Elastomeric Matrix / Steel Cord

Keywords: Rubber cord adhesion, hyperelasticity, inflation test

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

Introduction

Tyres are complex multi-layered structures, which maintain the contact between vehicle and soil. Their tribological and mechanical performances are therefore crucial. In addition, for safety and reliability reasons, durability is still an issue since tyres must sustain severe inservice loading conditions as well as aggressive environmental exposure. Their structure mostly consists of rubber layers reinforced with textile or metal cords, which control the overall stiffness as well as the strength of the structure (Lechtenboehmer A.). As it is in all composite structures, mastering the matrix (rubber) to reinforcement (cord) adhesion is very crucial. The most traditional technique to ensure strong chemical adhesion between metal cord and rubber consists of depositing a thin layer of brass on steel wires before embedding them in the raw (un-vulcanized) rubber. During the vulcanization process, Cu x S bonds are produced (van Ooij) which develop strong chemical links between the two materials.

However, this adhesive interface suffers from progressive degradation due to water diffusion that provokes corrosion of the interface (Stevenson A.). Therefore, new coatings having better adhesive and ageing behaviour are under development. In this context, the development of refined experimental characterization procedures to evaluate the mechanical performances of the interface is an issue since the mechanical properties are generally evaluated with standard procedures whose result may suffer from various experimental artefacts.

To evaluate the adhesion between layers, peel tests are generally used such as (Cook J. W.), (D1781-98(2012)), (D1871-04(2014)), (D2229-10(2014)) & (D429-14). Such tests are also used for testing new coating performance but are applicable only in planar configuration.

Brass coating is generally deposited on the steel during the final humid drawing step of the cords. This specific process will confer to the coating a specific structure in terms of roughness, homogeneity, grain size and so on (Rault V.), (Vignal V.), (Zelin M.). As a consequence, using peel test for characterizing rubber adhesion on plated plate could lead to biased results especially if other deposition methods are used such as electroplating, PVD or else." .

Apart from the classical peel test protocol, other standardized procedures are proposed to evaluate rubber adhesion to rigid substrate (ISO 5600 : 2011 -rubber -determination of adhesion to rigid materials using conical shaped parts) ensuring crack propagation along the interface.

Other specialized test protocols and specimens have also been proposed in the literature (Brown R.) for better observation and evaluation of crack propagation conditions. Despite more homogeneous and better controlled loading conditions, plane geometries are used again which are not compatible with cord-rubber adhesion evaluation. In these works (Brown R.), advanced fracture mechanics is proposed to evaluate the mode mixity condition at the crack tip so as to evaluate the influence of the material viscosity.

For a better evaluation of the cord-rubber adhesion, testing for individual cord -rubber mechanical behaviour is fundamental. The pull-out test, as describe in the ASTM standard (D1871-04(2014)), (Jamshidi M.) & (Gent A.N.), is generally used for that purpose. Although this test is simple to implement, quantitative evaluation of adhesive performances is difficult to achieve. Since the failure of the specimen is controlled by many parameters (wire / rubber plasticity, interface failure, mode mixing and so on) as well as artefacts (interface friction, tools friction). In order to achieve more intrinsic evaluation of mechanical performances of cord/rubber adhesive system a new test configuration is proposed here. The protocol is inspired by the more traditional blister test (Dannenberg H.), which is used to evaluate the adhesive properties of coating or painting. The test commences with injecting a pressurized fluid between the substrate and the coating aimed at fracturing the interface; the SERR can therefore be determined by monitoring both the crack extension as well as the pressure evolution during the test. Such protocol has also been used for characterizing rubber/substrate adhesion in planar configuration (Dillard D. A.). In such blister tests however, a coating failure was generally observed due to strong adhesion. Therefore, to prevent the failure of the coating layer, a confinement system is used (Napolitano M.J.), (Chang Y. S.) & (Parsons M.).

To evaluate rubber / cord adhesion, a cylindrical version of the constrained blister test is proposed here. The test specimen consists of a cylindrical rubber envelope partially bonded to a metal wire along its central axis. A crack is forced to propagate along the interface by injecting a liquid under pressure in between the two media from the debonded side, as shown in Figure 1. A concentric confinement envelope is used outside the rubber cylinder to prevent unstable inflation (and therefore unstable failure). A constant pressure value is recorded during the crack propagation associated to self-similar crack propagation process which can be used to evaluate the interface critical SERR.

Inflation of a thick hyperelastic cylinder has been studied for a Neo-Hookean material by (Anani Y.). However, the type of rubber material provided by Michelin was a Mooney-Rivlin solid (Mooney M.). The Mooney-Rivlin constants were also provided along with the specimens. The scope of this work was to develop a successful test setup and formulate a theoretical as well as an analytical model that describes the tests. The detailed analysis of material properties is left for future research prospects.

In this article, the test set-up is described with preliminary experimental results. Subsequently, a thick hyperelastic cylinder inflation model is developed based on Skala's (Skala D. P.) work, to evaluate the effect of fluid pressure on rubber envelope deformation. Therefore . The analytical model is compared with a FE model for verification of the obtained results. Finally, a global energy balance is performed for quantitative evaluation of the interface SERR from various measurable properties (fluid injection pressure, specimen geometry and deformation, material properties).

Rubber cord adhesion inflation test -RCAIT

Prior to the development of the theoretical and numerical modelling of the problem, an

Experimental set-up

The test specimen, presented in Figure 1, consists of a φ=1.3mm steel wire coated with brass.

The steel wire is embedded axially in a φ=9.4mm cylindrical rubber envelope. The initial crack or debonding is created by applying a PTFE tape on a length of 50mm on the steel cord from the fluid injection side, before the vulcanization process. The vulcanization process bonds the rest of the steel cord to the rubber, leaving an initial crack length of 50mm. Prior testing, the specimen is placed in a climatic chamber and exposed to 40°C and 60%RH

atmosphere for 60 hours to simulate accelerated ageing conditions. After the ageing process, the specimen is tightened to the test bench at its fluid injection end imposing radial and axial constrain. While doing so, the rubber envelope is held fixed only while the steel cord is free to move axially during the test, allowing the debonded rubber length to inflate freely. A PMMA tube with 10mm internal diameter and 5mm thickness is used for radial confinement of the specimen. To provoke interface separation, demineralised water is injected at 2ml/min using a KD Scientific 410 Series syringe pump equipped with a 20ml syringe. During the experiment, the pressure is recorded using Swagelok S Model Transducer with a capacity of 250bar.

Experimental Results

Figure 2(a) shows the evolution of pressure versus injected water volume during the test. The test sequence is divided into two phases. Up to approx. 5ml, the pressure is increasing while the water is injected in the rubber envelope. No crack propagation is observed during this phase; but the envelope is visibly inflated and can be seen touching the confinement tube.

After nearly 8ml of fluid injection, a pressure of approx. 66.6 bar is recorded followed by a steady increase in the inflated length of the specimen that corresponds to the crack propagation. Nearly constant pressure is observed during the crack propagation, corresponding to a nearly constant Strain Energy Release Rate, similar to the study by (Chang Y. S.).

Thick rubber tube inflation model

We now consider a thick elastomer envelope or tube under internal pressure as previously studied by (Skala D. P.). The main notations, results and constitutive equations are recalled below before some modification of the boundary conditions are considered to take into account the effect of radial confinement. With these mathematical derivations, we aim to describe the stress state in the rubber envelope of the specimen as well as the strain energy stored considering its incompressible hyperelastic nature. Ultimately, an energy balance analysis of the water injection process is proposed to determine the Strain Energy Release Rate associated with the crack propagation.

Constitutive equations

We consider a thick hollow elastomer tube with inner and outer radii given by v 0 and w 0 respectively. The problem to solve is clearly axisymmetric, therefore only radial and axial displacements are considered here. The radial and circumferential principal stretch ratios are

given by:

= = (1)
= 1 - 1 = -1 (2)
with c being an integration constant to be determined from constitutive and boundary condition equations together with λ z . The general expression of stretch ratios being known, we now assume an incompressible hyperelastic Mooney-Rivlin material behaviour (Mooney M.) for the rubber. This assumption is owing to the material data provided by Michelin, as explained earlier. Such behaviour is suitable for a hyperelastic material when deformation does not exceed 200%. The relation between Cauchy Stresses and stretch ratios is given by:

= 2 -2 + (3)
p being the local hydrostatic pressure component of stresses. Under axisymmetric loading condition, the equilibrium equation gives the relation:

+ - = 0 (4)
Combining equations (2), ( 3) and (4) according to the procedure followed by (Skala D. P.), a general expression of the hydrostatic pressure component, p, is found:

2 = 1 - -1 1 2 - 1 2 + 1 2 -1 - 1 -1 + (5)
with α = C 2 /C 1 . Therefore, the stress distribution in the rubber is fully determined by only three parameters λ z , c and d, which are obtained by solving the boundary condition equations as described hereafter.

Boundary conditions

Two different sets of boundary conditions should be considered since two phases are observed during the inflation test. Initially, the outer surface of the rubber envelope is free to expand since there is no contact with the confinement tube. This phase is called Unconfined Inflation

Stage (UIS). The following set of boundary conditions is therefore considered.

! = "# = -$ (6) ! = %# = 0 (7) & ! #2' ( ) = $ '" (8)
$ is the fluid pressure imparted by the injected water. The cord immersed in the pressurized fluid is not clamped and can move axially freely. Therefore, the axial force applied to the rubber envelope is given by equation ( 8).

The detailed explanations for the rest of the boundary conditions can be found in Skala's original article (Skala D. P.). Solving the boundary conditions as simultaneous equations, and defining the two dimensionless quantities h = w 0 /v 0 and K=1/(cv 0 2 ), a transcendental relation is found:

-2!* -1# + , -1 + --+ . !1 - # * !1 + . # * + . = 0 (9)
This relation is solved numerically to determine the K versus λ z evolution from which all stretch ratios and stresses can be found w.r.t. the dimensionless pressure P n (= $ /2 ) using equations (2) to (5). P.) for intermediate steps):

0 = 1 + % (11)
Therefore, the constant c is determined directly as a function of the longitudinal elongation λ z , with the relation:

= 0 -% (12)
Similar to the UIS, integration of boundary condition equations ( 6), ( 8) and (10) lead respectively to relations:

$ 1 = - 1 2 2 2 + . 1 + . -!1 + . #3 + 2 3 + 2. 2!1 + . # + 1 2 !1 + . #3 - (13) !1 + . #$ 1 = - + !* -1# + 1 2 . + 1 - - 1 * * + . 1 + .
-1 2 + 1 + -!5* + . 6 5* + . 6 -51 + . 6 51 + . 6# (14)

with P n = P i /2C 1 being the dimensionless pressure. These two simultaneous non-linear equations can be solved numerically for a given value of λ z. Finally, all displacements, stresses and stretch ratios are derived.

Numerical Modelling

Figure 5: Schematic of the Rubber Cylinder Inflation modelled in ABAQUS.

The rubber cylinder inflation test was modelled in ABAQUS using the axisymmetric elements -CAX4RH and considering Mooney-Rivlin material behaviour and contact condition with a rigid confinement envelope (Figure 5). The mesh size gradually increased from the inner radius to the outer radius; 0.02mm at the inner radius (v 0 ) and 0.1mm at the outer radius (w 0 ).

A confinement ratio (R/w 0 ) of 1.06 was chosen as in the experimental conditions. The rubber cylinder was modelled to be of a length of 500mm and the displacements and stresses were calculated at the middle of the length to avoid any edge effects. Figure 6 shows the comparative results of the inflation tests during the UIS and CIS with a confinement ratio of 1.06. The UIS (black curve) and CIS (blue curve) are respectively, solutions to the Equation ( 9) and Equation ( 13)-( 14). After the Confinement Contact point, UIS follows unstable inflation whereas CIS follows stable inflation. Due to the instability condition after the point P max , the numerical simulation (FE model) stopped as maximum pressure is reached. For the confined configuration with contact, the theoretical model is found to be in very good agreement with the FE model. The transition between the two stages is well captured. Once the analytical model is assessed, parametric investigation is performed. In particular,

Figure 7 illustrates the effect of the dimensionless rubber envelope radius h and confinement ratio R/w 0 on the rubber tube inflation. The overall response is the same whatever geometrical parameters are chosen. Obviously, the P n (λ z ) evolution does not depend on parameter R/w0

during the UIS and the specimen is found stiffer when h increases leading to higher P n values.

More interestingly, the dotted lines indicate the transition from UIS to CIS. It can be observed that there is a risk of unstable inflation and necking for large R/w 0 values since large λ z values are needed to achieve contact condition in that case.

Influence of boundary conditions

The Thick Rubber Tube Inflation Theory explained earlier is based on an important assumption that the Cauchy Stresses along the tube radius are not constant. This assumption is similar to the one in the standard Thick Cylinder Inflation Theory for elastic materials. For elastic materials, only Engineering (or Nominal) Stresses are calculated. However, for a hyperelastic material, the Cauchy Stresses are calculated due to the large deformations and displacements values that lead to geometrical nonlinearities. In this section, the stress variation along the rubber tube thickness owing to the various boundary conditions is studied.

Effect of axial stretch ratio (λ z ) on the Cauchy Stresses as well as the Dimensionless Pressure (P n ) is elaborated.

In the previous analysis, a series of dimensionless parameters have been introduced which should be reminded here. α = C 2 /C 1 is a material parameter, defined for a Mooney-Rivlin solid. In this theoretical work, the same value of α = 0.0625 which represents the specimen rubber material is used. The dimensionless pressure Pn is considered here as the driving quantity. All stress/stretch evolutions depend on the initial tube dimension as given by the dimensionless radius, h = w0/v0. Similar plots can also be obtained in CIS (w 0 =w=R). A comparison of the theoretical Cauchy

Stress variation and the one from the FE model is shown in Figure 10. The stress variation at different axial stretch ratios is nearly identical for both methods of calculation. The horizontal dotted lines represent the undeformed tube radii (w 0 and v 0 ). The confinement tube is loaded with a radial pressure during the inflation. This is far from being negligible and possible radial expansion of the confinement tube could be expected in case this envelope would not be thick enough.

Despite the contact pressure, no frictional forces are expected since once the contact condition is reached between rubber and confinement envelope; no relative displacement is expected if the inner pressure remains constant. This might not be the case if large pressure fluctuations are measured during the experiment due, for example, to stick-slip phenomena, or if creep behaviour is present. To minimize, or better, eliminate this effect, lubrication would be needed.

Virtual Crack propagation

In the previous section, rubber envelope inflation was studied considering the two sets of boundary conditions encountered during the experiment. Theoretical equations were developed or calculating both the axisymmetric deformations and stresses in the rubber. In the current section, these equations are employed for calculation of the energy stored in the rubber during crack propagation. An energy balance equation is subsequently derived to evaluate the crack propagation condition in the frame of linear elastic fracture mechanics (LEFM) theory. Then, no large process zone is supposed to develop ahead the crack tip nor plastic deformation in the rubber envelope. Although LEFM is considered here, the process is expected to be highly nonlinear due to the material and geometrical nonlinearities. All possible additional dissipation mechanisms in the rubber due to viscosity or hardening are neglected at this stage.

Energy Release Rate Evaluation

The entire test procedure is driven by injecting a pressurized fluid in between the rubber envelope and the steel cord. The overall energy balance can be described as follows. The energy input is due to the work done by the pressurized liquid. This becomes distributed between the energy stored in the deformed rubber (and the small amount stored in the compressed fluid at high pressures) and that required to cause separation of the rubber / cord interface i.e. fracture energy. Once the initial liquid injection (system loading) is finished, the measured pressure remains reasonably constant during crack propagation (see Figure 2 (a)).

The process can therefore reasonably be assumed to be steady state and self-similar. In the following, a virtual crack extension 'δa' of the interface is considered along the cord. Due to the axial deformation, the additional length of free, unattached rubber is stretched toλ z δa. The corresponding work done in pressure injection is then simply determined as the product of the constant pressure P i at which the crack propagates and the injected fluid volume during the crack propagation:

8 = $ '!" -" # δ9 (15)
Taking into account the fluid compressibility constant 'χ', the potential energy stored in a compressible fluid during the crack propagation is given by relation:

: ;< = ' 2 =$ !" -" # >9 (16)
In this analysis, the incompressible Mooney-Rivlin model is used to describe the mechanical behaviour of the rubber. The strain energy density of such a solid is given by the following relation:

>8! , , # = !@ -3# + !@ -3# (17)
where:

@ = + + (18) @ = + + (19)
Therefore, the potential energy stored in the rubber per unit length is obtained by integrating the relation (17) over the tube cross-section as follows:

K = $ !" -" # 2" - = 4" $ !" -" # - " 5{ ! + 2 -3# + ! + -3#I % -" 2 - + { F % " G - 1 2 % -1 " -1 I6 (25)
In this expression, most of the terms are measurable quantities (Pi, λz, v, C 1 , C 2 , …) and/or are determined by solving the 'Rubber Tube Inflation Problem' from the measured inner pressure. The value of G does not depend on the initial crack length or debonding length which is consistent with the constant pressure measured. However, to avoid edge effects and to facilitate homogeneous inflation of the tube, an initial crack length of approx. 50% of the total length of the specimen was chosen in the experimental test.

Depending on the rubber material properties, specimen dimensions, radial clearance for inflation (= R -w 0 ) and the strength of the interface, the crack propagation may occur during UIS or CIS. However, the method followed to calculate the energy balance is unchanged, therefore Equation (25) holds true in both cases.

Figure 11 shows the behaviour of the strain energy release rate -Equation ( 25 Using a VCCT-like approach, SERR was calculated from the FE model for 5 different crack lengths. The difference of energies stored in the system between two crack lengths was used to calculated energy of fracture and subsequently the SERR at given fluid pressure. Figure 12 shows comparison of SERR calculated using Equation ( 25) and the VCCT-like FE approach. rubber envelope continues to store potential energy E pr that gives a steadily decreasing value of G. To avoid such erroneous results, it is imperative to avoid the unstable inflation. A stable inflation prior to the crack propagation can be attained by reducing the ratio R/w 0 , such that the CIS starts before the point P i =P max is reached. The whole test can also be carried out under purely CIS (w 0 =R) which will certainly avoid the unstable inflation. However, the testing equipment should be designed to work under very high pressures.

Conclusion

In the present article, a new test configuration is proposed to evaluate the fracture energy of a rubber/cord interface for tyre application. The test is a cylindrical version of the standard constrained blister test used to evaluate rubber adhesion on rigid substrates. The test procedure commences with injecting a fluid between the metal cord and a cylindrical rubber envelope. The pressure required to inflate the rubber envelope and to propagate the crack along the cord is measured during the experimental test. A mechanical analysis of this test is proposed here to evaluate the fracture energy from the measured pressure. Analytical modelling as well as numerical analyses are performed to take into account the incompressible hyperelastic nature and large deformation of the rubber envelope. The equations developed for the energy balance are strictly applied to an incompressible Mooney-Rivlin solid. However, the method followed can be applied to any nonlinear material if the strain energy density function (Equation ( 17)) and its relations with the Cauchy Stresses are known.

Due to the non-linear deformations, the determination of fracture is not straightforward since rigorous characterization of the rubber mechanical behaviour is needed. However, it is suggested that combining pressure monitoring with additional measurements such as longitudinal elongation of the rubber would enable a more direct evaluation of the critical SERR.

Further developments are still required at this step since the hypotheses made on the rubber mechanical behaviour are simple. In particular, following the analysis proposed on the more traditional rubber peel and blister tests, complementary analysis should be performed to evaluate the additional energy dissipation due to viscoelasticity and/or hardening during the crack propagation. The strain energy density function of the rubber can therefore be altered in

Figure 1 .

 1 Figure 1. (a) Test Specimen (b) Schematic of the specimen cross-section. The initial

Figure 2 :

 2 Figure 2: (a) Fluid Pressure vs Injected Volume. A nearly constant fluid pressure is observed

Figure 3 :

 3 Figure 3: Rubber tube deformation stages. P i is the applied fluid pressure.

Figure 4 :

 4 Figure 4: Unstable inflation of rubber cylinders. P max is the maximum fluid pressure and the

Figure 6 :

 6 Figure 6: Comparison of Unstable and Stable inflation of the rubber envelope. Confinement

Figure 7 :

 7 Figure 7: Effect of dimensionless radius (h = w 0 /v 0 ) and confinement ratio (R/w 0 ) on Rubber

Figure 8 :

 8 Figure 8: Effect of confinement ratio (R/w 0 ) on Rubber Tube Inflation for a fixed

Figure 9 :

 9 Figure 9: Unconfined Inflation Test. Effect of Axial Stretch Ratios on Cauchy Stresses and P n

Figure 10 :

 10 Figure 10: Purely Confined Inflation Test. Effect of Axial Stretch Ratios on Cauchy Stresses

  Figure11shows the behaviour of the strain energy release rate -Equation (25) w.r.t. the fluid

Figure 1 :

 1 Figure 1: Strain Energy Release Rate as a function of injection Fluid Pressure.

Figure 12 :

 12 Figure 12: Evolution of SERR with applied pressure. Comparison between theoretical model

Figure 13 :

 13 Figure 13: Comparison of the energies stored in the fluid and the rubber at crack propagation,

>: ; = & >8. 2' ( ) (20) Using ( 18) and (19), the above equation can be solved to obtain:

It should be noted again that the integration is performed in the deformed state. Such procedure naturally takes into account the large displacement effects. Considering a virtual crack propagation of 'δa' along the cord, the deformation potential energy of the rubber tube is increased to λ z δaδE pr .

Finally, the energy dissipated through the interfacial separation should be considered as given by the relation:

where, K c is the critical SERR of the interface and 2'" >9 is the area of the fractured face.

Applying energy balance to the whole system:

The following expression of the critical strain energy release rate is found: a way that the effects of loading rate on viscous dissipation can be taken into consideration during the system energy balance.

A rigorous study on the effect of various fluids on the crack propagation pressure as well as the fracture energy can be directly applied in the tyre industry. Since a fluid is used to inflate the rubber envelope, specimen-fluid interaction was observed during the tests (cord corrosion, rubber swelling etc.). These difficulties could be avoided by replacing the fluid (water) by other fluids (sqalene, mineral oil etc.). Besides, this also suggests that RCAIT could be of interest for quantitative characterization of rubber / cord durability as an alternative to the strip blister test (Hamadea R.F.).

This article focuses on the global evaluation of the total SERR that is facilitated by the selfsimilar nature of the crack propagation. However, a refined analysis of the transition zone is clearly of interest to evaluate the mode mixity condition as well as possible development of process zone ahead the crack tip. By coupling the pressure injection with an external axial load on the steel cord, a mixed mode crack propagation can be produced. Effect of mode mixing on the fracture energy can thus be studied by varying the proportions of the pressure and the external axial loads. Refined experimental and numerical mechanical analysis is required for precise evaluation of mode mixity condition at the crack tip, as observed in other configurations (Liechti K. M.), (Liechti et Adamjee), (Liechti K.M.), (Mauchien T. K.). This mode mixity will probably vary with the specimen and confinement tube dimensions therefore a parametric analysis is required in the future.