Nadège Bouchonneau
email: bouchonneau@gmail.com

Valérie Sauvant-Moynot

Dominique Choqueuse

François Grosjean

Emmanuel Poncet

Dominique Perreux

Experimental testing and modelling of an industrial insulated pipeline for deep sea application

Keywords: layered structure, subsea, thermal properties, modelling, syntactic foam

Ultra-deep water (up to 3000 m) is one of the next frontiers for oil offshore exploitation. It requires the use of conduits having to resist in the long run (durability about 25 years) the mechanical and environmental requests. One of the key points is the thermal insulation of the structure to avoid the formation of hydrates and paraffin plugs inside of the steel pipe. Over the past 10 years, many studies were performed to better understand the behaviour of the syntactic foams used as thermal insulation of pipes for deepwater production, but few tests were run on industrial prototypes to reach the actual thermal properties of the systems. This paper presents the numerical and experimental characterizations of an industrial multilayered insulated pipeline tested in service conditions. Two thermomechanical finite element modellings of the coated pipeline have been developed to predict its behaviour during service condition tests. The first model considers pure conduction through the inner air inside of the structure and the second model considers convection phenomenon between the inner air and the metallic surfaces inside of the structure. In parallel, industrial pipe tests on an immersed instrumented pipeline, internally heated to temperatures up to 95 °C and subjected externally to hydrostatic pressure up to 300 bar are presented. Experimental data obtained during industrial pipe tests and related model predictions are compared and discussed. Thermal properties of the syntactic foam are determined with steady and transient states analysis. In complement, a study of the model results sensitivity to the input Poisson coefficient is presented.

Introduction

Since new hydrocarbon reserves discoveries are more and more rare in the conventional offshore, the industry is looking into new hydrocarbon reserves perspectives located in deep sea (between 500 and 1 500 m) and ultra-deep sea (between 1 500 and 3 000 m) as underlined by Robertson et al. (2005). Thus, one of the currently most challenging projects in the petroleum industry consists of exploiting oil resources at great depths, where production infrastructures are submitted to high hydrostatic pressures (up to 300bar) and to low external temperatures (about 4 °C at 3000 m). To limit heat losses and so avoid the formation of hydrate and wax plugs inside subsea production flowlines or risers under such pressure and temperature conditions, even during production shutdowns, the pipelines need to be thermally insulated. One of the most efficient type of thermal insulation systems is the multilayered structure made of several materials of different thicknesses directly applied to the external surface of the steel pipe. Currently used materials in thermal insulated multilayered systems for deep sea applications include massive polymers and syntactic foams, composed of hollow glass microspheres embedded in a polymer matrix. These composites must combine thermal insulation function, low buoyancy while providing good compressive strength.

To date, most of the studies conducted in this domain were performed on small samples in order to evaluate material performance under representative conditions of pressure, temperature and ageing media, as studied by [START_REF] Choqueuse | Insulation materials for ultra deep sea flow assurance: Evaluation of the material properties[END_REF]Choqueuse et al. ( ), (2003)), (2005) and [START_REF] Gimenez | Wet ageing of syntactic foams under high pressure / high temperature in deionized and artificial sea water[END_REF]. Up to now, just few experimental recent studies deal with thermal performance evaluation on large-scale multilayered insulated pipelines submitted to severe conditions. Such upscale structure tests were performed by [START_REF] Haldane | A Direct measurement system to obtain the thermal conductivity of pipeline insulation coating systems under simulated service conditions[END_REF][START_REF] Haldane | Development of an alternative approach to the testing of thermal insulation materials for subsea applications[END_REF] in the Heriot-Watt and TNO Institute of Applied Physics work, who developed a direct measurement system to determine in-situ the thermal characteristics of insulated pipes submitted to hydrostatic pressures up to 145 bar, internally heated with circulating oil (up to 140 °C) and with external cooling at 8 °C.

With the development of ultra-deep offshore fields high hydrostatic loads are reached which could have drastic consequences on the insulation coating performance. In this paper, experimental tests on an industrial pipe under higher hydrostatic pressures (up to 300bar) have been developped and performed in order to study the multilayered structure thermal performances in ultra-deep service conditions. The results obtained with two tested prototypes have previously been compared to validate the experimental protocol by [START_REF] Bouchonneau | Thermal Insulation Material for Subsea Pipelines: Benefits of Instrumented Full Scale Testing to Predict the Long Term Thermo-mechanical Behaviour[END_REF].

A thermomechanical modelling of multilayered structures in simulated severe service conditions, based on experimental data collected on samples, is presented. In fact, two hypotheses (internal conduction and internal convection) are presented and compared. These numerical models were developed in order to predict the thermomechanical behaviour of a complex industrial structure composed of different materials.

In parallel, tests on industrial structure up to 300bar hydrostatic pressure (simulating ultradeep water) have been conducted to study the influence of high pressure loads on thermal performances of the syntactic foam insulation coating.

The comparison between model simulations and experimental results is presented and discussed, that contributes to develop and complete a multiscale approach about multilayered coatings thermal performances characterisation. In complement, the sensitivities of the estimated thermal conductivity and the heat capacity values of the syntactic foam to several input parameters in the modelling have been studied.

One of the main objectives is to develop a useful model to predict the thermomechanical behaviour of complex multilayered structures submitted to ultra-deep service conditions, and thus determine if the coating system is well adapted to a particular field.

Experiments

Insulated pipe structure

The industrial structure (1.2m initial length) consists of a steel pipe (internal diameter of about 180mm, thickness about 18mm) and a 5-layers insulating coating (total thickness 61mm) which were industrially applied by side extrusion process, detailed by [START_REF] Berti | Syntactic polypropylene coating solution provides thermal insulation for Bonga risers[END_REF]. The coating is composed of several material types: solid, adhesive polymers and syntactic foam, which is composed of hollow glass microspheres embedded in a polypropylene matrix. One advantage to combine varied materials is the possibility to obtain mechanical and thermal properties as well as good long term behaviour in water required for deep sea application, all these properties being difficult to obtain with a single material. The multilayered system and the materials thicknesses are presented in Figure 1.

Material properties

Thermal and mechanical properties of each constitutive material of the test section are reported, respectively, in Table 1 andTable 2.

Material Thermal conductivity a) (W.m -1 .K -1 )
Heat capacity a) (J.kg -1 .K The values given for the PP syntactic foam were collected from experimental measurements performed at 1bar on small samples by [START_REF] Lefèbvre | Durabilité des[END_REF]. For the steel pipe and the polymers, values from the literature were used as input data in the simulation.

In service tests

Experimental tests on the industrial prototype contribute to develop and complete a multiscale approach by collecting experimental data on an entire structure tested under representative service conditions of pressure and temperature. The tests also contribute to discuss the validity of the numerical modelling.

Testing facilities

For the prototype testing, a high pressure vessel (1m diameter, 2m height, up to 1000bar) located in IFREMER Brest was used (Figure 2).

The pressure inside the vessel is regulated and monitored using a pressure transducer mounted at the top of the pressure vessel. The water temperature inside the pressure tank (about 15°C in this study) is also regulated and monitored during the tests. A schematic view of the coated test pipe in vertical position during testing in the pressure vessel is given in the Figure 3. The number of sensors for the prototype instrumentation was limited by the number of connections available on the pressure vessel flange.

Equipment

The insulated pipe section was machined at both ends to adapt two metallic steel caps in Stainless steel (APX4) covered by 100mm thick Polytetrafluoroethylene (PTFE) insulating caps in order to limit axial heat flow losses. The metallic caps were also equipped with connectors resistant to high external pressure. Three 10-channels connectors were necessary to allow the electrical supply of the inner heating system (one connector) and to collect inner sensor data (two connectors).

In this study, an original heating system was developed instead of the classical circulating oil to limit the convection effects inside of the pipe. The heating system consisting of heating elements (Nickel-Chrome wires) embedded in a thin silicone layer was placed on the internal diameter of the steel pipe and kept in place in contact with internal surface of the pipe by an inflatable silicon system. It is worth noting that such an inner 'dry' configuration with no pressure and no liquid is also very beneficial for the instrumentation used inside the pipe and will simplify the modelling of the inner heat flux boundary condition. In addition, this solution allows an easier control of the heating flow through the coating. The different elements that equipped the prototype are shown on Figure 4.

Instrumentation

The schematic representation of the fully instrumented pipe section and the location of sensors are given in Figure 5.

The insulated pipe section was instrumented with six commercial temperature sensors (Pt100) specified up to 200°C minimum (accuracy of about 0.3 % at 100 °C), located in both inner and outer parts along the pipe length and on the caps (Figure 5):

-T i (°C): inner temperature of the steel surface in the center of the pipe (one Besides, the outer temperature of the water in the vessel, T water (°C), was also measured using a platinium sensor.

The insulated pipe section was also instrumented with four commercial heat flux sensors located in both inner and outer parts along the pipe length and the caps:

-φ i (W.m -2 ): inner thermal flux density on the steel surface in the center of the pipe (one measurement with soft circular fluxmeter of 5µV.m 2 .W -1 sensitivity specified up to 200°C);

-φ e (W.m -2 ): outer thermal flux density on the coating surface in the center of the pipe (three measurements with a semi-rigid fluxmeter of 50µV.m 2 .W -1 sensitivity specified up to 100°C and 100bar (one measurement with a rigid fluxmeter of 30µV.m 2 .W -1 sensitivity specified up to 250°C and 150bar);

-φ b (W.m -2 ): inner thermal flux density on the steel surface in the center of the steel cap (one measurement with rigid rectangular fluxmeter of 36µV.m 2 .W -1 sensitivity specified up to 200°C).

Heat flux sensors with soft flat form were selected for the internal and external pipe surfaces to reduce errors related to the difficulties in mounting rigid flat sensors.

The readings obtained from all the sensors were recorded by a data acquisition unit which could be programmed to take recordings at appropriate time intervals throughout the duration of the test.

Testing procedures

For the experimental testing, a prototype structure has been instrumented and tested successively without additional pressure (external pressure of 1bar -test A) and under 300bar hydrostatic pressure (test B), simulating in-service conditions at about 3000m depth, at two different conditions of inner temperature, in order to study the influence of pressure and temperature on the thermal performances of the structure. Both test sequences programs are

shown in Figure 6.

The details of each step are given in Table 3. The tests sequences have been chosen in order to be representative of service conditions (pressure then temperature). In both cases, it should be emphasized that a test lasting approximately 10 days cannot be used to predict the long term evolution of the insulation coating systems for which water uptake and creep cannot be neglected.

Theory of heat transfer

In parallel to experimental testing, a finite element modelling of the insulated pipeline has been developed in order to satisfactorily predict the thermomechanical behaviour of the structure when submitted to hydrostatic pressure and internal heating. Then an optimisation program based on analytical considerations has been developed to assess both thermal conductivity and heat capacity of the syntactic foam for several test conditions of temperature and pressure. These two complementary modelling approaches notably permit to evaluate the thermal parameters of the material studied (syntactic foam) even in the absence of any external heat flux sensor, in particular under 300bar hydrostatic pressure since commercial sensors are limited to a lower pressure range.

Transient state

The following analytical model describes the transient heat transfer by conduction in the prototype structure. In the case of a one-dimensional radial transfer in a one-layer structure limited by radii r=r int and r=r ext , the heat equations for temperature and heat flux are:

t T a 1 r T r r r 1              for r int < r < r ext (1) and r T S       (2)
with T=T 0 for t=0, Applying a Laplace transform to the variable t, these equations lead to:

     a p dr d r 1 dr d 2 2 (3) and r S        (4)
The quadrupoles notation, given by Maillet et al. (2000), is well suited to relate the Laplace transforms of the temperatures and fluxes at inner and external boundaries obtained by solving the above equations: (2000).

                   ext ext D C B A     int int (5) θ int et θ ext correspond
By applying this development to a multilayered pipeline structure submitted to outer convective losses, the equation (5) becomes:

                             convective w ater ext ext i i i i i S h D C B A     1 0 1 1 6 1 int int (6)
with θ water and  convective the Laplace transforms of the water temperature and the convective heat flux respectively.

With the boundary condition at the prototype-water interface:

)) , ( (

6 t r r T T h r T ext water ext        (7)
The internal temperature variations T int (t) are calculated in the time domain by numerical inversion of the Laplace transform θ int , given by Stehfest (1970).

Steady state

For a one-dimensional conduction problem in a cylindrical structure, the analytical expression for the radial heat flux in steady state conditions is linked to the inner and outer surface temperatures and to the global heat transfer coefficient U of the structure (evaluated on the inner surface of the structure taken as reference) by the following relation:

Q = -U.S. (T ext -T int ) (8)
For a multilayer structure, and by assuming that the thermal contact resistance between each layer can be neglected, the global heat transfer coefficient U of the structure can be expressed in terms of constitutive material thermal conductivities with the following relation:

                           n i i i i L D D 1 1 2 ln   S 1 U (9)
This coefficient is a representative thermal characteristic of the entire system: steel pipe and insulation coating, which represents the thermal performance of the multilayered structure.

Numerical modelling

Model assumptions

The thermomechanical modelling of the prototype structure was performed with the software Comsol Multiphysics ® , based on the finite element method, which is a tool particularly well adapted for coupling problems between several phenomena. The modelled structure is based on the real geometry of the multilayered insulated pipe. The thermal contact resistance between each layer is assumed to be negligible. The structure is assumed to be without structural defects, so the modelling was performed in 2D-axisymetric geometry. The electric connectors placed in the end caps are here not considered to ensure symmetry conditions (same end caps). This allows to model only the half of the prototype structure and thus to increase the number of elements. The finite element model is based on three node elements.

The experimental prototype tests were performed during a relatively short duration, therefore, the water diffusion and also creep aspects are here neglected. These hypotheses also simplify considerately the modelling, decrease the time of calculation, and allow refining the mesh and also increasing the element number for the solution. At this stage of the study, the syntactic foam has been considered as a linear elastic material, with thermal and mechanical properties depending on the temperature. More complete models are under development.

The temperature distribution in the steel pipe and on the insulation coating system was determined during all test sequences. Moreover, thermal heat flux on the external surface of the structure or deformation of each material layer could also be determined.

In order to avaliate the influence of the internal thermal boundaries on the simulation of the thermomechanical behaviour of the structure, two models of the entire structure were developped. The first takes into account the phenomenon of thermal conduction in the air localised inside of the pipe, and the second considers the phenomenon of thermal convection between the metallic surfaces and the internal air.

For both Comsol models (internal conduction or convection), the initial conditions of temperature and pressure depend on each test sequence performed on the structure. The finite element model is based on three node elements and the mesh was locally refined near interfaces and sensors to enhance the resolution. -for the external convection condition, the convection coefficients were calculated from experimental temperatures T e and T water , as described by [START_REF] Eyglunent | [END_REF];

-continuity (of temperature and flow) at the interface between the internal air and the internal surfaces (heating map and metallic surfaces).

-For tests under hydrostatic pressure, an additional stress condition is applied on the external surfaces of the structure. -for the external convection condition, the convection coefficients were calculated from experimental temperatures T e and T water , as described by [START_REF] Eyglunent | [END_REF].

-natural convection along the internal surfaces at both inner end metallic caps in contact with air and natural convection along the external surfaces in contact with water. For the internal convection condition, the convection coefficients were calculated from experimental temperatures -temperature in the centre of the metallic cap (T b ) and initial temperature of the structure (T inicial ) , as described by [START_REF] Evans | An experimental study of transient heat transfer to liquids in cylindrical enclosures[END_REF] or [START_REF] Holman | Heat Transfer[END_REF] -Initial temperature of the entire structure supposed to be stabilized at the water temperature;

-The inner and outer radius of each layer are taken from the numerical modelling results obtained in stationary state with Comsol Multiphysics ® (in order to take into account thermal expansion and pressure effect on each material thickness).

Results and discussion

5.1 Validation of the numerical modelling

Instrumentation limits

During the prototypes testing, some experimental measurement problems of the fluxmeters could be highlighted. The fluxmeters seem to lose sensitivity when the temperature increases.

Figure 9a shows that at 120 W heating power (corresponding to a relative low inner temperature: about 50 °C), the inner fluxmeter measurements are comparative to the heating power registered. But when the heating power increases (inner temperature of about 95 °C), the difference between the heating power value and the inner flow measurement increases as well (Figure 9b). The internal heat flow values are at least approximately 16% smaller than the heat flow that exits from the pipe, which underlines the difficulty to obtain experimentally a precise internal heat flow. The fluxmeter localised at the center of the metallic cap also registered relative low values. This fluxmeter was placed at the interface between the metallic cap and the internal air that could explain the difficulty to obtain pure conductive heat flow. It

shows the limitation of using a single fluxmeter in this position in order to evaluate the heat losses at the end caps. Therefore simulations results will be very used in this work to evaluate external heat flows, and also axial heat losses.

Modelling results

Internal heat flux densities were calculated from the aforementioned heat fluxes (respectively The same remarks could be highlighted for the tests realized under 300bar, illustrated in the Figure 13, where the differences between experimental and simulated temperatures are even higher. The difference observed between experimental and simulated values could be in part explained by the difficulty to determine the thermal convection coefficients (external and internal surfaces). The global heat transfer coefficient U estimated with both numerical models at 1 bar (internal conduction and internal convection) is comparative to the value given by the insulated pipe manufacturer (about 4.2W.m -2 .K -1 at 20°C and 1 bar, according to the thermal characteristics given by the manufacturer), which validates the use of the thermomechanical numerical simulation to determine the global heat transfer coefficient U and the apparent thermal conductivity of insulation materials. The results also highlight that the apparent thermal conductivities of syntactic PP are slightly lower than values measured on small specimens (given in Table 1).

For the test sequences realized at 1bar pressure, simulated results show that U values and apparent thermal conductivities do not seem to be significantly influenced by the high inner steel temperature. For the test sequences realized at 300bar pressure, simulated results show that U values and apparent thermal conductivities increase in relation to the tests realized at 1 bar, and the values also seem to be slightly influenced by the high inner steel temperature.

The relative increase of the U value (about 10% increase between tests at 1 bar and tests at 300bar) and related increase of the apparent thermal conductivity could be explained by damage occurring in the foam microstructure, in particular near the steel pipe where material is subjected to coupled effect of high temperature and complex stress distribution. The Figure 14 represents the stress repartition along the syntactic foam thickness in the middle plan of the structure, obtained with Comsol Mutiphysics ® , when the prototype is submitted to high hydrostatic pressure (300bar) and high inner heating power (240W).

The results show that the von Mises stresses in the syntactic foam are not negligible, in particular near the metallic pipe, and that the values of the three principal stresses are very different, which underlines the anisotropy of the mechanical loads applied to the prototype structure. Therefore it exists deviatoric stresses in the thickness of the multilayer coating that could induce damage, particularly in the structure of the syntactic foam. This damage could conduct to an evolution of the mechanical properties of the foam (Young's modulus E, coefficient de Poisson ν) during the tests and thus explain the relative increase of the thermal conductivity of the foam.

To complete this point, the respective displacements in the radial direction (r) and in the vertical direction (z) obtained with the Comsol Multiphysics ® simulation at the interface between the solid PP layer and the internal surface of the syntactic foam (near the metallic pipe) in the steady state (300bar/240W) are represented along the half structure in Figure 15.

It illustrates the notable differences in the displacements values along the structure, resulting from the coupling of high internal heating and high external hydrostatic pressure. However, it is important to keep in mind that the thermal parameters (global heat transfer coefficient and apparent thermal coonductivity of the syntactic foam) were based on simulation results

(simulated external heat flow at the stationnary state) and on experimental temperatures.

Therefore it is very delicate to evaluate the level of uncertainity of the values obtained.

Transient case

For the simulations based on the Matlab routine, the inner heat flux density feeding the model A parametric study about the heat flux repartition was also realized in order to illustrate the influence of pipe length and thermal conductivity of the material on the radial heat flow percentage (Figure 18). For different pipe length (1.5m, 3m and 6m), and different thermal conductivity of the insulation coating (0.1, 0.15 and 0.2 W.m -2 .K -1 ), the percentage of radial heat flow was determined from numerical simulations. The results show that the nearest from the center of the structure, the higher are the radial heat flow percentage. In the case of a relative short pipe (1.5m length), the radial heat flow percentage in the center of the structure is between 70 and 80%, depending from the thermal conductivity considered. 5. The very similar apparent thermal conductivity values obtained compared to the steady state approaches validate the transient state analysis and its one-dimensional hypothesis.

One can notice that the apparent heat capacity of syntactic PP is lower than the value determined experimentally (heat capacity expression given in Table 1), and this later value should be considered with caution since temperature sensitivity to heat capacity variations is low.

From apparent thermal conductivity and heat capacity values obtained under 120W 1bar and 120W 300bar, there is no significant difference as stated previously from steady state

analysis. Thus, one can conclude that the coupling effect between high hydrostatic pressure (300bar) and inner temperature of the metallic pipe at about 50 °C (corresponding to the inner surface temperature measured in the steady state with 120W heating power input) induces negligible short term consequences on syntactic PP. But in the case of higher temperatures and above coupled with ultra deep service pressure, short term phenomena could occur in the syntactic PP leading to a lowering of insulation performance.

Sensitivity study

One parameter that appears interesting to considerate in the sensitivity study is the Poisson coefficient value of the syntactic foam, which is susceptible to change during the tests. In fact, the influence of temperature increase inside the syntactic foam during the different test sequences could probably induce an increase of the global Poisson coefficient value of the syntactic foam. This coefficient is directly linked to the deformation of the syntactic foam, and thus to the evolution of coating thickness. Therefore, the deformation of each material has been first calculated with the finite elements modelling (developed with Comsol Multiphysics ® ), then reported in the optimization routine to take into account the syntactic foam Poisson coefficient evolution. The optimization results are reported in Table 6, the corresponding coating thickness evolutions during each sequence are related in Figure 19. The notation "νx%" signifies that the input thermal parameter ν has been increased or decreased from x% in comparison to the initial input data in the optimisation routine. The optimised values obtained for the thermal properties of the syntactic foam reveal very low evolutions (up to 0.15% increase), that underlines the fact that with the hypothesis considered in the modelling, the Poisson coefficient variation do not affect significantly optimization results.

These graphics show that the syntactic foam Poisson coefficient evolution did not have a significant influence for the test realized at 1bar (influence of thermal expansion only), whereas at 300bar the coating thickness evolution is more marked due to the influence of thermal expansion and hydrostatic pressure (up to 0.5% increase).

In further studies, the evolution of the Poisson coefficient of the foam with temperature could also be included in the input parameters of the modelling directly as a function of the temperature, as it has been already done for the mechanical modulus.

Conclusions

Two thermomechanical models of a multilayered coated pipeline, alimented with experimental data obtained on small syntactic foam samples, have been developed to predict the behaviour of the structure submitted to internal heating and external pressure.

  (°C): outer temperature of the coating surface in the center of the pipe (one measurement); -T b (°C): inner temperature of the steel surface in the center of one cap (one measurement); -T 100 (°C): inner temperature of the steel surface along the pipe 100mm distant from cap (one measurement); -T 50 (°C): inner temperature of the steel surface along the pipe 50mm distant from cap (one measurement);

  to the transforms of the inner and outer surface temperatures of the cylindrical structure respectively, and A, B, C and D are analytical relations involving Bessel functions and the geometrical characteristics of the structure described by Maillet et al.

4. 2

 2 Boundary and initial conditions 4.2.1. Comsol models a) Internal conductionThe geometry of the computational domain, the numerical mesh and the boundary conditions of the two models are shown in Figure7. In this model, the air is considered as a stagnant and purely conductive media. The thermal properties of the air considered in the model are from[START_REF] Holman | Heat Transfer[END_REF].The boundary conditions are:-no displacement in z-direction for the lower section (A-B) and no displacement in rdirection for the upper section (C-D); flow density condition on the inner surface of the steel pipe (heating map);

  a) Internal convectionThe geometry of the computational domain, the numerical mesh and the boundary conditions of the two models are shown in Figure8. Tis second model is based on the hypothesis of thermal convection between the air and the interne metallic surfaces.The boundary conditions are:-no displacement in z-direction for the lower section (A-B) and no displacement in rdirection for the upper section (C-D); -thermal heat flow density condition on the inner surface of the steel pipe (heating map);

  , who described the calcul of the Nusselt coefficient in the case of natural convection phenomenon in closed vertical or horizontal cylinders. The Nusselt coefficient is calculated with the following relation: thermal conductivity of the internal air L: the length of the cylinder -For tests under hydrostatic pressure, an additional stress condition is applied on the external surfaces of the structure. c) Matlab routine Both thermal conductivity and heat capacity can be determined by an optimisation program developed with software Matlab ® with an indirect method relying on an inverse methodology. This routine allowed the evolution of the inner temperature of the steel pipe T i to be adjusted during the testing time simulated by an analytical relation to the experimental data by optimisation (square root method) of the thermal parameters of the material studied, here the syntactic foam. Temperatures and heat fluxes evolutions are obtained in the time space by means of the numerical inversion of each Laplace transform. Transient model hypotheses are: -One-dimensional axisymmetric; -Constant convective heat transfer coefficient and water temperature; -Constant internal heat flux equal to the value of the steady state heat flux in the structure;

  120W or 240W) by dividing flux values by 0.57m 2 (internal heating mat surface area). Both simulations of thermomechanical behaviour of the insulated pipe section were performed with those calculated internal flux densities. The temperature and the displacement distributions simulated within insulated pipe section with the model based on internal conduction is shown in Figure 10a (temperature) and Figure 10b (total displacement), for the case 300bar, 120W. The temperature and the displacement distributions simulated within insulated pipe section with the model based on internal convection is shown in Figure 11a (temperature) and Figure 11b (total displacement), for the case 300bar, 120W.One can notice that for both tests (heating power of 120W and 240W), the simulated interne and externe temperatures Ti and Te fit quite well the experimental values, in particular for the convection model. The other simulated temperatures (Tb, T50 and T100) are significantly overestimated (more than 10 °C difference for T b in step 3 -Figure10b). The results show that for each condition, the convection model fits better the experiments than the conduction model.

  of any external heat flux sensor, the external heat flux could be approached by simulation. For each test sequences, the heat transfer coefficient U has been determined using equation (8) directly from measured temperatures and simulated external radial heat fluxes, for both Comsol models (internal conduction or convection). The external heat flux is determined by integrating the heat flux densities on the external surface of the pipe, considering the length of the pipe that is directly submitted to the heating (1m length as for the heating mat dimension). Then the apparent thermal conductivity of the syntactic PP (corresponding to the thermal conductivity of the syntactic PP averaged on the insulation cross section under thermal gradient) is derived from equation (9). Calculated U values and apparent thermal conductivities of syntactic PP are reported in the

  was determined with Comsol. The model with internal convection relatively better fit experimental temperatures than the conductin model. Therefore, we determined the heat flux density based o the results of internal convection. The hypothesis of 1D-axisymetric used for the Matlab routine could be justificated for a first approach by the repartition of the heat flux through the thickness of the coated pipe. The radial and axial component of the conductive heat flux in the center of the stucture is represented in Figure16, at several times of the modelling (hypothesis of internal convection). We can oberve that in the center of the structure, the axial component of the heat flux through the metallic pipe and the coating is negligeable in regards to the radial component. The axial component of the conductive heat flux in the structure obtained with Comsol is presented in Figure 17. The simulations results show that the axial component of the conductive heat flux increases along the pipe, and is maximum at the interface between the pipe and the metallic cap (circled area).

  Figure19. Optimisation input data and results are reported in the Table5.

Figure 8 .Figure 9 .

 89 Figure 1. Industrial 5-layers insulation system Figure 2. Setting up of the instrumented prototype in the hyperbaric tank Figure 3. Schematic representation of the insulated pipe section during hyperbaric testsFigure 4. Prototype equipment Figure 5. Instrumentation of the industrial prototype Figure 6. Testing programs realized on the prototype -(a) under 1bar pressure (test A), (b) under 300bar hydrostatic pressure (test B) Figure 7. Boundary conditions and mesh of insulated pipe section for the internal conduction model Figure 8. Boundary conditions and mesh of insulated pipe section for the internal convection model Figure 9. Representation of the thermal heat flows during test at 1bar: test A -(a) 120W (step 2), (b) 240W (step 3) Figure 10. Temperature and displacement distributions within insulated pipe section in the steady state 1bar/120W -Model based on internal conduction Figure 11. Temperature and displacement distributions within insulated pipe section in the steady state 1bar/120W -Model based on internal convection Figure 12. Comparison between experimental and simulated temperatures during test at 1bar: test A -(a) 120W (step 2), (b) 240W (step 3)

Table 2 -

 2 Mechanical and thermomechanical properties

Table 3 -

 3 Details of the test sequences

Table 4 .

 4

	Test sequence	Pressure (bar)	Heating mat power (W)	Internal heat flux model (W.m -2 )	External radial heat flow (W)	Total external axial heat flow (W)	Total axial heat losses (%)	Ti (°C)	Te (°C)	U (W.m -2 .K -1 )	Apparent thermal conductivity of syntactic PP (W.m -1 .K -1 )
	TestA
	step2				97.3	22.9	19			4.24	0,167
	conduction TestA	1	120	210				56.4 16.4
	step2				94.1	21.6	18			4.10	0,161
	convection
	TestA
	step3				194.7	45.5	19			4.34	0,171
	conduction TestA	1	240	420				95.8 17.6
	step3				189.7	43.5	18			4.23	0,166
	convection
	TestB
	step3				97.3	22.8	19			4.68	0,186
	conduction TestB step3	300	120	210	93.9	21.5	18	55,6 18.3	4.52	0,179
	convection
	TestB
	step4				194.8	45,3	19			4.96	0,198
	conduction	300	240	420				88.5 20
	TestB step convection				184.22	41.4	17			4.69	0,186

Table 4 -

 4

Heat flows and temperatures in the steady state and related thermal properties (simulation results are in italics)

Table 5 -

 5 Thermal properties of syntactic PP determined by the transient state analysis (simulation results are in italics)

	Ti experiment
	300bar 120W
	Ti Matlab
	optimization
	300bar 120W

Table 6 -

 6 Influence of input Poisson coefficient of the syntactic PP on optimised thermal parameters of the syntactic foam

(a) (b)

Acknowledgments

The authors wish to thank Socotherm for providing the insulated coated pipes, in particular G. P. Guidetti for his interest to this work and N. Lacotte and A. Deuff for the performing of hyperbaric tests.

(b)

The reproductibility of such experimental testing has been verified at 1bar, by testing another similar prototype under same conditions of pressure and temperatures, as described by [START_REF] Bouchonneau | Thermal Insulation Material for Subsea Pipelines: Benefits of Instrumented Full Scale Testing to Predict the Long Term Thermo-mechanical Behaviour[END_REF].

The comparison of experimental and simulated temperature values obtained at 1bar is presented in the Figure 12.


Semi-Analytical Method Based on Magnetic Equivalent Circuit for Synchronous Permanent-Magnet Machines in EV and HEV Applications

S.A. Randi 1 , R. Benlamine 1 , F. Dubas 2 , and C. Espanet 2 1: Renault SAS, 78288 Guyancourt, France 2: FEMTO-ST Institute, UFC, ENERGIE Department, UMR 6174 CNRS, 90010 Belfort, France Abstract: Concentrated Winding Permanent-Magnet (PM) Synchronous Machines (CWPMSM) are more and more used to drive electromechanical systems working in variable speed mode, such as Electric Vehicle (EV) and Hybrid Electric Vehicle (HEV) applications. In order to design and optimize these machines, the authors propose a semi-analytical method based on magnetic equivalent circuit (MEC). The studied structure, 12-slots/8-poles CWPMSM, is an internal rotor topology with surface mounted PMs having radial magnetization. We compare the results obtained with finite-element analysis (FEA) with those given by the MEC. In this paper both precision and time computation are evaluated in order to analyse the possible implementation of the model in an optimal design tool.

Introduction

The design of electric machines for variable speed applications such as EV and HEV is a difficult problem. Indeed when a designer has to define a new electric machine for a given application, he has to solve the problem of sizing parameters (i.e., the nominal and maximum torque, the rated and maximal speed, the nominal and maximal current...). In classical industrial application when the electric machine has only a few operating points in the torque speed plane, the sizing torque and base speed can be found easily. In EV and HEV, there is a large dispersion of operating points in the torque speed plane [START_REF] Randi | Conception systémique de chaînes de traction synchrones pour véhicules électriques[END_REF]. Therefore to find the best solution designers must evaluate a lot of machines characterized by different sizing torques and base speeds. Furthermore when the sizing torque and the base speed are known, designer must answer to a lot of questions: which machine technology, what is the topology (e.g., the number of slots, the number of poles), the winding (e.g., distributed, concentrated...), the major geometric dimensions, etc. Due to the complexity of the sizing, it is clear that that optimization can help the designer to find a new electric machine for EV and HEV. Beside, the size of the optimization problem is related to the number of input parameters (i.e., the sizing parameters, the technology parameters, the topology parameters, the geometric parameters…); finally we must often evaluate hundred thousands of solutions to find the best one. To evaluate the criteria, optimization process must be linked to a model of the electric machine. There is a lot of modelling approach in electric machine, depending on the level of precision and the computation time. Regarding the concordance with measurement, the FEA seems today the best one, if we want to consider complex geometry and magnetic saturation. Unfortunately, this approach cannot be coupled easily with an optimization procedure, because the evaluation of one solution with FEA is quite time consuming (i.e., it can take up to few minutes or even hours).

In this paper, we present a semi-analytical modelling approach for the electric machines based on the MEC [6]- [START_REF] Rasmussen | Magnetic Equivalent Circuit Method For Designing Permanent Magnet Motors[END_REF]. The electric Machine used to present the modelling methodology is a CWPMSM [START_REF] Rasmussen | Magnetic Equivalent Circuit Method For Designing Permanent Magnet Motors[END_REF]. Those machines are more and more used in automotive applications because they enable a great reduction of cooper weight (thus a great reduction of axial length, copper losses and cost) and they offer the advantage of a fully automatized winding process. In section 2.1, we give a brief description of the machine geometry. Then, we explain how the saturation effect is taken into account in section 2.2. In section 2.3, we present the MEC dealing with radial and tangential reluctances The discretization and resolution method are presented in section 2.4. In section 3, we discuss the precision and computation time of the MEC and the section 4 deals with the comparison between the MEC and FEA: we show that the computation time can be divided by 10 with using MEC, without a strong information loss.

Semi-Analytical Modelling via MEC

Electrical Machine Description and Assumptions

The machine considered in this work is a 12slots/8-poles CWPMSM. This studied structure is an internal rotor topology with surface mounted PMs having radial magnetization. To reduce the computation time, the study of the machine can be carried out over one electric period under only 3slots/2-poles, as shown in Figure 1. The electrical steel M330-35A of Arcelor Mittal is used for this machine. The main parameters of the 12-slots/8poles CWPMSM are reported in Table 1. The concentrated winding configuration is presented in Figure 6.

Saturation effect

To introduce the saturation effect in the model, the Marrocco's function is used [1]- [START_REF] Dubas | Conception d'un moteur rapide à aimants permanents pour l'entraînement de compresseurs de piles à combustibles[END_REF]. Four interpolation coefficients (i.e. k 1 ~ k 4 ) are used in the formula. They are calculated according to the vendor data and with using the least squares method. The Figure 2 shows that the used coefficients give a good agreement between the B(H) curve given by the manufacturer and the interpolated function given by the Marrocco's function and described herein by

( ) ( ) 1 1 2 4 1 2 2 0 3 k i i i S S S S k S b k k B H B k b k µ   ⋅ - = ⋅ +   +     . ( 1
)
The relative permeability function is presented by ( ) ( )

0 i i i S r S i i S S B B H B µ µ = ⋅ . (2)

Reluctances computation

Each region i of the machine, having the same magnetic permeability, can be described by two tangential and two radial reluctances, with a midpoint, as shown in Figure 3. Then, the MEC of the machine can be implemented. For the stator yoke, a boundary condition (i.e., Dirichlet) is imposed to the outer surface of the machine at r = R sout . Thus, there is no radial reluctance above the midpoint. The same condition is imposed on the rotor yoke at the inner surface at r = R inner . Thus there is no radial reluctance below the midpoint. The tangential and radial reluctances used in the MEC are calculated by

0 1 1 ln i i out r i i i r iron inner R L R µ µ Θ   ℜ = ⋅ ⋅     ⋅ ⋅   , (3)
0 1 1 ln i i i i r iron out i inner L R R Θ Θ µ µ ℜ = ⋅ ⋅   ⋅ ⋅       . ( 4
)
Figure 1: Cross section of 12-slots/8-poles CWPMSM.

] 0 S S F ⋅ ℜ ⋅ ⋅ - = .
(

) 5
The connection between the loop fluxes vector and the reluctance fluxes vector [φ] is described [START_REF] Ostovic | A Simplified Approach to Magnetic Equivalent Circuit Modeling of Induction Machines[END_REF] The sources vector [F], is composed by both the Magneto-Motive Force (MMF) due to the PMs and the MMF due to the current winding. The MMF provided by a PM is given by

0 rm PM m rm B MMF h µ µ = ⋅ ⋅ , ( 7
)
while the MMF provided by one coil, containing turns and fed by a current , is expressed by

I t s MMF n I = ⋅ . ( 7
)
The The non-linear system taking into account the saturation effect is solved iteratively. The vector [F] is divided into N parts (equal or not). Then, (5) and (6) are solved in order to obtain the reluctance fluxes vector [φ]. Thus the reluctance flux density vector [B], is calculated using the vector [φ] and the reluctance surfaces vector [S R ]. The new magnetic permeability can be calculated using (1) and (2), and the vector [B] is added to that obtained during the previous iteration. The different resolution steps are described in Figure 5.

The magnetic permeability is considered as constant in a range of a MMF equal to [F]/N for each iteration. Higher is the number of discretization N, higher the accuracy is.

Numerical Tool

The numerical tool used in this work to evaluate the efficiency and the performances of the MEC modeling approach is Flux 2D developed by Cedrat [START_REF] Flux2d | General operating instructions -Version 10[END_REF]. One pole pairs and three slots are considered for the simulation. A boundary condition is imposed to the inner and outer surface of the machine. The mesh considered is shown in Figure 6. The mesh is finer at the areas where variations of flux density are high (air gap and PMs) and larger elsewhere (stator yoke, rotor yoke, tooth and slot).

Precision based on the Magnetic Flux Density

In Figure ( 7) ~ (14) are presented the radial and tangential components of the no-load magnetic flux density in the different parts of the machine (i.e., the air-gap, the stator yoke, the PMs and the rotor yoke). The solid lines represent the magnetic flux density computed by FEA without saturation, the dashed lines shows the FEA results with saturation taken into account, and the circles corresponds to the magnetic flux density computed by MEC. One can see that there is a good agreement between the magnetic flux densities computed by FEA (with saturation effect) and those evaluated by the MEC. It proves the ability of the MEC to take into account the slotting effect and the magnetic saturation, even with a relatively small network.

Convergence and Computation Time

It must be noted that the resolution of the system of non-linear equation ( 5) and (6), can be done using Newton Raphson algorithm. The fact that the MEC is mesh based lead to a fast convergence and low number of iterations. In Table 2, we present the computation time for the MEC compared with the time needed to solve the problem using FEA. It is important to notice that we divide the computation time roughly by 10. This should enable easy implementation of the MEC with saturation in an optimization process for electric machines. Figure 15: Phase Back EMF @ 3,000 rpm.

Figure 16: Electromagnetic torque computed @ 240 Amax.

Comparison Based on Integral Quantities -Back-EMF and Electromagnetic Torque

Back-EMF in phase-A

In Figure 15 we represent the Back-EMF in phase-A at 3,000 rpm computed by the three different methods: the solid line represents the Back-EMF computed by FEA assuming a linear characteristic of the iron, the dashed line corresponds to the FEA taken into account saturation, and circles represent the points obtained with the MEC. One can see that also for integral value of the machine, there is a good agreement between the results given by FEA and those calculated with MEC.

Electromagnetic Torque

The electromagnetic torque is an important output of our modeling. Indeed this value must be computed with a very good accuracy, since the torque produced by the drive should match with the technical specifications. In Figure 16 we depict three torque curves. The solid line shows the torque computed by FEA in linear condition, the dashed line shows the torque computed by FEA assuming the real B(H) characteristic and the circles corresponds to the points of torque computed by the MEC where the saturation effect is taken into account. It must be noted that the average torque when the saturation is taken into account is lower than the torque when we assume a linear iron steel. One can see also that the MEC modeling give us a quite satisfying correspondence in term of torque computation.

Conclusion

In this paper, we present a semi-analytical modeling approach suitable for the design of electric machine used in EV and HEV applications. The method presented is based on the MEC. The parametric modeling of a new electric machine may take time for the designer. Therefore once the model is built, the parametric study and machine optimization should be achieved easily. To illustrate the efficiency of the method, a 12-slots/8-poles cconcentrated winding PM synchronous machines is used as an example. The comparison between local quantities computed by FEA and those obtained by MEC shows a good agreement, and obviously the integral quantities (i.e., the back-EMF, the electromagnetic torque) correspond too. The main advantage of the MEC is that we can get a precision comparable to what we can obtain by FEA, and with computation time divide by 10. For this reason it should be easy to implement the MEC models in an optimization tool.

  the magnetic field calculated by Marrocco interpolation B s , the magnetic flux density b s , reduced value of the flux density B s ( ) , the vacuum permeability k j , the Morrocco interpolation coefficients (j=1,…4) , radial reluctances , tangential reluctances , the reluctances matrix [S], the loop matrix [F], the sources vector [φ], the loop fluxes vector [ψ], the reluctance fluxes vector [S R ], the reluctance surface vector MMF PM , the MMF provided by PMs MMF I , the MMF provided by a coil h m , the thickness of the PM , the number of turns per coil , the phase current N L , the loops number in the reluctance network N R , the reluctances number in the MEC

Table 1 :

 1 Parameters of 12-slots/8-poles CWPMSM Parameters Number of phases, m [-] Number of series turns per phase, Nsts [-] Maximum current for a waveform sinusoidal, Ismax Rotation speed of the currents, N0 [rpm] Mechanical angle of a pole-pitch, Θp [deg.] Mechanical angle of a tooth-pitch, Θt [deg.] Mechanical angle of a stator slot-isthmus, Θsi [deg.] Mechanical angle of a stator slot-opening, Θso [deg.] Magnet pole-arc to pole-pitch ratio, αp [%] Stator slot-isthmus to tooth-pitch ratio, ζsi [%] Stator slot-opening to tooth-pitch ratio, ζso [%] Relative magnetic permeability of iron (Linear), µ Relative magnetic permeability of the PMs, µrm [-Remanent flux density of the PMs, Brm [T] Outer radius of the machine, Rsout [mm] Radius of the stator yoke surface, Rsy [mm] Radius of the stator isthmus, Ris [mm] Radius of the stator surface, Rs [mm] Radius of the PMs surface, Rm [mm] Radius of the rotor yoke surface, Rr [mm] Inner radius of the machine, Rinner [mm] Machine axial length, Liron [mm]

Figure 2 :

 2 Figure 2: B(H) curves given by the manufacturer and Marrocco interpolation

Figure 3 :

 3 Figure 3 : Reluctance for each region machine and boundary conditions

Figure 4 :Figure 5 :

 45 Figure 4: Discretization of the machine for one pitch and one pole

Figure 4 :

 4 Figure 4: Discretization of the machine for one toothpitch and one pole-pitch.

Figure 5 :

 5 Figure 5: Different steps of the non-linear system resolution

  reluctances matrix [ ] is a diagonal matrix [N R XN R ], where the diagonal is formed by the different reluctances. The loop fluxes matrix [ψ] is a vector [N L X1] formed by the different loop fluxes. The sources matrix [F] has the same dimensions as the matrix [ψ]. The loop matrix [S] is constituted by N R columns and N L lines ([N L XN R ]).

Figure 6 :

 6 Figure 6: Mesh, winding configuration and magnetic field lines.

Figure 7 :

 7 Figure 7: Radial component of the air-gap flux density.

Figure 8 :

 8 Figure 8: Tangential component of the air-gap flux density.

Figure 9 :

 9 Figure 9: Tangential component of the stator yoke flux density.

Figure 10 :

 10 Figure 10: Radial component of the stator yoke flux density.

Figure 11 :

 11 Figure 11: Tangential component of PMs flux density.

Figure 12 :

 12 Figure 12: Radial component of PMs flux density.

Figure 13 :

 13 Figure 13: Tangential component of rotor yoke flux density.

Figure 14 :

 14 Figure 14: Radial component of rotor yoke flux density.

Table 2 :

 2 Computation Time for the Different Method

	No Load	Load @
		240 Amax


Mohamed Rachid Laydi

Christian Lexcellent

Yield Criteria for Shape Memory Materials: Convexity Conditions and Surface Transport

Keywords: Yield surface, shape memory alloy, convexity, duality, phase transformation

We study the yield conditions of phase transformation initiation for shape memory alloys exhibiting asymmetry between tension and compression. An extension of the choice of the classical invariant parameters such as those of Lode is proposed. A necessary and sufficient condition of convexity of these surfaces representing the elastic domain of austenite in the stress space, is established. Moreover the transport of these surfaces in the space of effective transformations strains of martensite is done. Hence, the duality between these two spaces is built. Some applications involving Cu-Al-Be and NiTi shape memory alloys end the purpose.

INTRODUCTION

As is the case in classical plasticity, the determination of the yield surfaces of phase transformation initiation is a key point for modeling shape memory alloys (SMA) behavior. The SMA exhibit an asymmetric behavior between tension and compression. Namely the yield stress is different between these two different uniaxial loadings [1]. Moreover, the SMA are considered as pressure-insensitive alloys. It means that the phase transformation between austenite and martensite is accompanied with no volume change. As pointed out by Raniecki and Lexcellent [START_REF] Raniecki | Thermodynamics of isotropic pseudoelasticity in shape memory alloys[END_REF], the dependence of the yield criterion on the first invariant of stress tensor 1 (I 1 1 tr 112) is negligible. Only the second and third invariants of the deviator of 1 are involved. As for phase transformation initiation, in the tensor 13 the material is only sensitive to the stress deviator S 4 1 dev 112.

In addition to these two classical Lode variables, i.e. the invariant and the angle, a third variable which is the greatest eigenvalue of reduced stress deviator N 4 is introduced in order to simplify the calculations and to generalize the formulations. In the first part, yield criteria for isotropic materials will be stated with a special attention devoted to the convexity of the yield surface of phase transformation initiation in the stress space 1. The second part will concern the transport of these surfaces in the space of effective transformation strain [START_REF] Lode | Versuche fiber den EinluB der mittlerea Hauptspannung auf das FlieBen. der Metalle Eisen, Kupfer und Nickel[END_REF] 6 m associated to the produced martensitic phase. Some applications on NiTi and Cu-Al-Be isotropic SMA end the paper.

CONSIDERATIONS ON THE YIELD CRITERIA

General Formulations and Haigh-Westergaard Representation

As the analysis is restricted to isotropic behavior, the Haigh-Westergaard representation is used (see Figure 1).

Let us consider the second order symmetric tensor 1 (defined in 1 3 ) and the deviatoric tensor S [START_REF] Podgorski | General failure criterion for isotropic media[END_REF] The yield surface equation of phase transformation initiation reads

1 112 1 1 f 1x 4 2 3 4 c 3 (1)
where f 9 03 is a smooth function (the second derivative is continuous) defined on the interval I x (which will be defined later). In fact, f is some "correction function" which allows to take into account the asymmetry between tension and compression called SD effect (stress differential effect), see Raniecki and Mroz [START_REF] Raniecki | Yield or martensitic phase transformation conditions and dissipation functions for isotropic, pressure-insensitive alloys exhibiting SD effect[END_REF]. In Podgorski [START_REF] Podgorski | General failure criterion for isotropic media[END_REF][START_REF] Podgorski | Limit state condition and the dissipation function for isotropic materials[END_REF], f is referred to as the "shape function". The positive quantity 4 c is the yield phase transformation initiation value and is given by

4 c 1 b1T 4 M 0 S 2 (2)
where M 0 S is the martensite start temperature at stress-free state. The interval I x is defined following the choice of the argument x 4 of f7 In fact, there are at least three possibilities for x 4 :

1. The Lode invariant y 4 3 chosen for instance by Raniecki and Mroz [START_REF] Raniecki | Yield or martensitic phase transformation conditions and dissipation functions for isotropic, pressure-insensitive alloys exhibiting SD effect[END_REF], which is an homogeneous function of degree zero of stress deviator

x 4 1 y 4 1 4 det 1N 4 2 6 I y 4 1 [413 1] (3)
and where N 4 is the reduced deviatoric part of 1 given by

N 4 1 8 2 1 S 4 3 (4)
with

N 4 : N 4 3 8 2 7 (5)
2. The Lode angle 5 4 (see [START_REF] Lode | Versuche fiber den EinluB der mittlerea Hauptspannung auf das FlieBen. der Metalle Eisen, Kupfer und Nickel[END_REF]) defined by This latter variable was chosen as variable by Bigoni and Piccolroaz [START_REF] Bigoni | Yield criteria for quasibrittle and functional materials[END_REF] together with

f 1x 4 2 1 1 f 7 15 4 2 7 (6)
as the shape function. 3. In the present paper, a third expression of x 4 is introduced in order to simplify the calculations and the formulations, notably the one associated with the convexity condition. We take the greatest eigenvalue of N 4 3 i.e.

x 4 1 4 1 cos

6 1 3 arccos y 4 7 6 I 4 1 8 1 2 3 1 9 7 (7)

Yield Condition Resolution in Terms of Admissible Stress

Let us recall the yield equation ( 1):

1 3 4 c f 1x 4 2
the solution of which can be written as

1 3 1 3 tr 112 18 4 c f 1x 4 2 8 42 N 4 7 (8)
One can show that N 4 can be split as (see (55) in Proposition 1)

N 4 3 4 e R 4 1 4
1 1 4 2 4 2 1 2 e R 4 2 (9)
where 4 1 1x 4 2 3 with a bijective function which maps I x on

I 1 8 1 2 3 1 9
3
and which is defined by In ( 9)3 e R 4 1 and e R 4 2 are two orthogonal elementary deviators associated to 1 by

e R 4 1 1 R 4 e 1 R T 4 and e R 4 2 1 R 4 e 2 R T 4 3
where R 4 is the rotation matrix transforming 1 in its diagonal form 1), can be written as

4 1 R T 4 1R 4 3 1 112 0 0 0 2 112 0 0 0 3 112
4 3 1 3 tr 1 4 2 1 8 4 c f 1x2 8 42 e 1 4 1 1 4 2 2 1 2 e 2 6 I3 (14)
x being linked to by the relations (11). Therefore the eigenvalues k of the reduced stress tensor 1 4 c are solutions of the following system:

P R O O F O N L Y 6 M. R. LAYDI and C. LEXCELLENT 2 1 4 2 4 3 3 1 1 1 f 1x2 4 4 3 1 1 4 2 2 1 2 2 2 4 3 4 1 3 2 1 1 f 1x2 4 8 3 1 1 4 2 2 1 2 2 3 4 1 4 2 3 3 1 2 f 1x2 3 (15)
for all 6 I7 Now let us examine the dual space associated to the martensite strain.

TENSOR OF EFFECTIVE TRANSFORMATION STRAIN OF MARTENSITE

The concept of "effective transformation strain of martensite" has been introduced by Sadjadpour and Bhattacharya [START_REF] Sadjadpour | A micromechanics inspired model for shape-memory alloy. Smart Material Systems and Structures[END_REF]. This strain tensor called m is the average transformation strain of the different variants of martensite averaged over a representative volume containing multiple grains, after the material has formed an allowable microstructure.

For proportional loadings the experiments show that the phase transformation strain rate is normal to the yield surface 1 112 3 4 c (see Bouvet et al. [START_REF] Bouvet | Mechanical behavior of a Cu-Al-Be shape memory alloy under multiaxial proportional and non-proportional loadings[END_REF]).

Thus one can assume that m 112 admits a potential 1 112 9 03 which means

6 6 m 1 1 1 112 7
Let us introduce the reduced strain tensor

6 6 4 1 6 6 m 3 1 1 112 7
We give two decompositions for the tensor 6 6 4 as follows.

First Decomposition

The first one is given for x 3 y 4 3 5 4 or 4 (see (59) in Proposition 2):

6 6 4 3 f 1x2 N 4 8 a 1x2 f 1 1x2 N 4 ( 16
)
where

N 4 1 1 1 4 2 4 2 1 2 e R

Second Decomposition

This decomposition is done in the basis e R 4 1 3 e R 4

2

. One obtains easily (see Proposition 3)

6 6 4 3 f 1 1 4 2 e R 4 1 4 f 2 1 4 2 e R 4 2 3 ( 19
)
where

f 1 12 3 1 1 4 2 2 d f d 8 f 1x2 f 2 12 3 1 1 4 2 2 1 2 g 12 (20)
with x 3 x 12 (see (11)) and

g 12 1 4 d f d 8 f 1x2 7 (21)
Now let us examine the convexity conditions of the yield function 1.

CONVEXITY CONDITIONS

In order to establish the convexity conditions, g and d for all 6 I7 This latter condition is very useful for the numerical verifications. Now we are going to explicit the convexity conditions following the choice of the variables. [START_REF] Podgorski | General failure criterion for isotropic media[END_REF] The inequalities in (25) for x 3 4 are the simplest because 3 x. One obtains f 112 4 f 1 112 0

Conditions in Terms of

f 11 12 03 (27)
for all 6

8 1 2 3 1 9 7

Conditions in Terms of y 4

When f is a function of x 3 y 4 with 3 cos 1 1 3 arccos 1y 4 2

2

, the expression of the sufficient and necessary conditions for the convexity of 1 is more complicated. In fact it verifies (see (67) and (68)) andC. LEXCELLENT for all y 6 [413 1] 3 or by using (25)

g 12 3 f 1y2 4 3 1 4 2 4 1 2 f 1 1y2 0 d 2 f d 2 3 24 f 1 1y2 8 9 1 4 2 4 1 2 2 f 11 1y2 03 (28) P R O O F O N L Y 10 M. R. LAYDI
g 112 3 f 112 4 9 f 1 112 0 d 2 f d 2 3 24 f 1 1y2 8 9 1 4 2 4 1 2 2 f 11 1y2 07 (29)
Note that due to the identities 4 3 4 3 3 y and

1 1 4 2 2 1 4 2 4 1 2 2 3 1 4 y 2 3 (23) becomes simply d f 1 d 3 1 9 f 4 y f 1 1y2 8 1 1 4 y 2 2 f 11 1y2 0 y 6 [413 1] 3 (30)
which is the convexity condition given by Raniecki and Mroz [START_REF] Raniecki | Yield or martensitic phase transformation conditions and dissipation functions for isotropic, pressure-insensitive alloys exhibiting SD effect[END_REF]. It is clear that this latter condition is just necessary but not sufficient as will be seen below.

Conditions in Terms of 5 4

A supplementary condition on f is required, namely the continuity of g and of d 2 f d 2 at the neighborhood of 5 3 0 (see ( 67) and ( 68)).

It results from ( 22) for x 3 5 3 arccos 12 (see ( 67) and ( 68)) that We thus recover the condition established by Bigoni and Piccolroaz, see [START_REF] Bigoni | Yield criteria for quasibrittle and functional materials[END_REF].

g

Counter-example

The necessary conditions established by Raniecki and Mroz [START_REF] Raniecki | Yield or martensitic phase transformation conditions and dissipation functions for isotropic, pressure-insensitive alloys exhibiting SD effect[END_REF], Bigoni and Piccolroaz [START_REF] Bigoni | Yield criteria for quasibrittle and functional materials[END_REF] and (30), (32) and (33) which come from (23) are not sufficient. In order to convince the reader let us give a counter-example:

f 3 2 4 cos 2 152 3 2 4 cos 2 6 1 3 arccos y 7 3 2 4 2 9 0 6 8 1 2 3 1 9 7
The condition (23)

d f 1 d 3 f 4 d f d 8 1 1 4 2 2 d 2 f d 2 3 3 2 9 03
is fulfilled but not the positivity of the second derivative of f : One finds

1 1a2 3 1 1b2 3 31 28 428 2 4 4 c and 1 1c2 3 7 2 68 2 4 4 c
and it is clear that

1 1c2 2 1 2 11 1a2 8 1 1b22 7
This case is illustrated in Figure 2. In the next section, we return to the expression of the phase transformation strain tensor 4 .

P R O

O F O N L Y YIELD CRITERIA FOR SHAPE MEMORY MATERIALS 13

REDUCED SHAPE OF THE PHASE TRANSFORMATION STRAIN TENSOR 4

As before one determines the reduced shape of 6 6 4 , with the reduced formulae (4):

n 4 1 8
with f 1x 4 2 1 5 f 1x 4 25 2 8 5a 1x 4 25 2 f 1 1x 4 2 2 1 2 3 1 5 f 1 1 4 25 2 8 5 f 2 1 4 2 1 2 3
x 4 and 4 being linked in an univocal sense by (11). n 4 can be written as

n 4 3 f 1x 4 2 f 1x 4 2 N 4 8 a 1x 4 2 f 1 1x 4 2 f 1x 4 2 N 4 3 f 1 1 4 2 f 1x 4 2 e R 4 1 4
f 2 1 4 2 f 1x 4 2 e R 4 2
From now on, we will always suppose that the convexity conditions (22) are satisfied.

First Decomposition

The function f 2 , as defined in (20), being positive, one can write

n 4 3 e R 4 1 4
1 1 4 2 2 1 2 e R 4 2 3 ( 36
)
where

3 f 1 1 4 2 f 1x 4 2 and f 2 1 4 2 f 1x 4 2 3 1 1 4 2 2 1 2 7
In the above, denotes the greatest eigenvalue of n 4 characterized by (56) or in a more explicit way (see (18))

1 1 4 2 6 2 1 2 dh d 6 h 8 1 1 4 2 4 2 1 2 d f d 4 f 3 07 (48)
Particularly, for x 4 3 y 4 , one obtains

1 1 4 y 2 2 1 2 h 1 1y 2 h 1y 2 8 1 1 4 y 2 4 2 1 2 f 1 1y 4 2 f 1y 4 2 3 07
In a general way, by using (42), one deduces the inverse application: where

5 4 3
n 4 3 6 e R 4 1 4
1 1 4 2 6 2 1 2 e R 4 2 n 4 1 1 1 4 2 6 2 1 2 e R 4 1 8 6 e R 4 2 7
(50)

We show (see Proposition 9) that

2 1 16 6 4 2 3 1 4 c S 4 3 S 1 4 c 4 (51)
for all 1 which are solutions of the yield phase transformation surface in the stress space, see

.

In fact, it is proved that the space of yield surface of phase transformation initiation (i.e. the elastic domain in the stress space of the mother phase: the austenite) is dual to the space occupied by the domain associated to the effective transformation strain of the produced phase: the martensite. The functions f 1x 4 2 and h 1x 6 2 inherit the same properties, notably that of the convexity.

The parameter x 6 can be obtained starting from x 4 (42) and

4 3 arctan 6 a 1x 4 2 f 1 1x 4 2 f 1x 4 2 7 7
Reversely, the parameter x 4 can be obtained from x 6 (49) and

6 3 arctan 6 a 1x 2 h 1 1x 2 h 1x 2 7 7
The transport from one space to the another can be made in a very simple way by using the relation

4 8 6 3 07 (52)
The variables 6 4 and S 4 can be considered as dual with the normality rules associated to their yield surfaces respectively.

In the next section, we give a few applications of the surface determination concerning two different shape memory alloys.

APPLICATIONS

We consider two experimental series concerning Cu-Al-Be and NiTi alloys, whose yield values are given (in MPa) in the tables below.

Concerning the Cu-Al-Be alloy, the yield points are obtained by bi-compression tests on cubes and tension (compression)-internal pressure on tubes (see Bouvet et al [START_REF] Bouvet | Mechanical behavior of a Cu-Al-Be shape memory alloy under multiaxial proportional and non-proportional loadings[END_REF]).

Cu-Al-Be:

1 i 3 4 1 0 0 0 4 2 0 0 0 0 3 1 9 i 9 n 3 11,

Shape Function Choice

We will use the shape function f which has been introduced in [START_REF] Bouvet | Mechanical behavior of a Cu-Al-Be shape memory alloy under multiaxial proportional and non-proportional loadings[END_REF], namely They are determined by the intersection of the two curves 4 c 1a2 and the "reverse" formulae a 14 c 2 with the following definitions: In fact, the function defined in (53) behaves as f 12 3 1 4 8 q where and q are given by the formulae (86) and ( 87

f 1y2 3 cos 6
4 c 1a2 1 n " i31 1 i cos 6 1 3 arccos 1 1 4

Application 1: Cu-Al-Be

The normalized solutions of (1) given by (15) are characterized by the couples

14 1 3 4 2 2 6 $ 1 3 j 8 2 i 3 i 8 2 j 5 6 I 1 1 9 i 3 j 9 3

Application 2: NiTi

In this case, the eigenvalues of 1 are The normalized solutions of (1) given by (15) are such that where

1 1 3 3 2 3 1 3
1 2 3 8 1 3 2 1 8 3 2 3
i.e.

4 zz 3 4 2 4 z5 3 1 3 3 1 3 4 2 2 1 2 4 1 2 7
As for the solutions given by (44), they are characterized by the couples

6 zz 3 4 6 2 6 z5 3 1 3 3 1 6 3 4 6 2 2 1 6 2 4 6

CONCLUSION

We have performed the modelling of the yield curves of phase transformation initiation for isotropic SMA. materials, in the stress space, by taking into account the asymmetry between tension and compression. Many choices of the shape function f 1y2 are possible notably a linear expression, one with a "cosinus" formulation or a power law in terms of , and in every case, the fitting of experimental results are good. We give a necessary and sufficient condition for convexity of the yield curve thus completing the necessary one as stated by Raniecki and Mroz [START_REF] Raniecki | Yield or martensitic phase transformation conditions and dissipation functions for isotropic, pressure-insensitive alloys exhibiting SD effect[END_REF] and Bigoni and Piccolroaz [START_REF] Bigoni | Yield criteria for quasibrittle and functional materials[END_REF]. Special attention is paid to the transport of these yield surfaces from the stress space to the one of effective transformation strain of the produced phase (i.e. the martensite). A duality is established between these two surfaces. An extension of the investigation is possible concerning anisotropic shape memory alloys as drawn or rolled SMA bars.

PROOFS OF THE PROPOSITIONS

Proposition 1. For any choice of x (y 4 , 5 4 or 4 2, N 4 can be written as

N 4 3 1x2 e R 4 1 4
1 1 4 5 1x25 2 2 1 2 e R 4 2 ( 55
)
Moreover, is equal to One can verify that y 4 1 4 det 1N 4 2 6 [413 1] and that the Cardan discriminant is equal to In the general case 03 the equation has three real distinct roots given by 18) and ( 17). The derivative of 1 is given by ( 16)

3 8 42 N 4 : e R 4
3 4 1 4 2 1 1 4 y 2
1 1 3 f 1x2 N 4 8 a 1x2 f 1 1x2 N 4 7 (59)
Moreover

1 1 112 : 1 3 1 112 17 (60)
Proof. We are going to use the following equation of the derivative of 1

1 1 3 f 1x2 1 1 8 f 1 1x2 1 x 1 ( 61
)
and the following identity:

1 3 S 4 4 1 3 tr S 4
1. The first term in (61) is classical:

1 1 3 8 1 15S 4 52 3 8 S 4 15S 4 52 3 8 2 S 4 8 5S 4 5 3 8 2 S 4 1 3 N 4 7 (62)
The second term is based on the characterization of see (56):

3 1 41 S 4 : e R 4 1 7
One obtains

1 3 1 41 e R 4 1 4 1 42 1 1 S 4 : e R 4 1 3 1 41 e R 4 1 4 N 4 1 1 3 e R 4 1 4 e R 4 1 4
1 1 4 2 2 1 2 e R 4 2
i.e.

1 d dx x 1 3 1 1 4 2 2 1 2 1 1 4 2 2 1 2 e R 4 1 8 e R 4 2 3 1 1 4 2 2 1 2 N 4 7
This leads to

1 x 1 3 1 1 4 2 2 1 2 dx d 3 a 1x2 N 4 7 (63)
Finally one obtains (60) by observing that

1 1 112 : 1 3 1 1 112 : S 4 3 18 42 1 1 1 112 : N 4 2 3 18 42 f 1x2 1N 4 : N 4 2 3 1 112 7 P R O O F O N L Y YIELD CRITERIA FOR SHAPE MEMORY MATERIALS 29
Proposition 3. The derivative of 1 is equal to

1 1 112 3 f 1 1 4 2 e R 4 1 4 f 2 1 4 2 e R 4 2 3 (64)
with

f 1 12 3 f 1x2 8 1 1 4 2 2 1 2 a 1x2 f 1 1x2 3 f 1x2 8 1 1 4 2 2 d f d f 2 12 3 1 1 4 2 2 1 2 f 1x2 4 a 1x2 f 1 1x2 3 1 1 4 2 2 1 2 6 4 d f d 8 f 1x2 7 7
(65) And the inverse system reads

a 1x2 f 1 1x2 3 1 1 4 2 2 1 2 f 1 12 4 f 2 12 f 1x2 3 f 1 12 8 1 1 4 2 2 1 2 f 2 12 7 (66)
Proof. By substituting N 4 and N 4 in (59) as a function of e R 4 1 and e R 4 2 , see ( 9) and ( 17), one obtains (64) with

f 1 12 3 f 1x2 8 1 1 4 2 2 1 2 a 1x2 f 1 1x2 f 2 12 3 1 1 4 2 2 1 2 f 1x2 4 a 1x2 f 1 1x2 7
The final expression is obtained by substitution of a 1x2

f 1x2 4 3 1 4 2 4 1 2 f 1 1x2 if x 3 y 4 f 1x2 8 1 1 4 2 2 4 1 2 f 1 1x2 if x 3 5 4 f 1x2 4 f 1 1x2 if x 3 4 3 (67)
and

d 2 f d 2 12 3 24 f 1 1x2 8 9 1 4 2 4 1 2 2 f 11 1x2 if x 3 y 4 4 1 1 4 2 2 4 3 2 f 1 1x2 8 1 1 4 2 2 41 f 11 1x2 if x 3 5 4 f 11 1x2 if x 3 4 7 (68) P R O O F O N L Y 30 M. R. LAYDI and C. LEXCELLENT
In particular, g and d 2 f d 2 are two continuous functions on I for x 3 y 4 or x 3 4 3 and just on 1 2 3 1 for x 3 5 4 7

Proof. On one hand we have

g 12 1 f 1x2 4 dx d f 1 1x2
and on the other

d 2 f d 2 3 d 2 x d 2 f 1 1x2 8 6 dx d 7 2 f 11 1x2 7
So by substituting

dx d 3
3 1 4 2 4 1 2 if x 3 y 4 4 1 1 4 2 2 4 1 2 if x 3 5 4 1 i f x 3 4 and d 2 x d 2 3 24 if x 3 y 4 4 1 1 4 2 2 4 3 2 if x 3 5 4 0 i f x 3 4
in the above, we finally recover the two identities (67) and ( 68). The continuity of the functions g and Proof. A necessary and sufficient condition for the convexity of 1 is based on the following inequality:

1 112 1 1b2 8 1 1 1 1b2 : 11 4 b 2 23 13 b7 (70)
With tr11 1 1b22 3 0 and 1 1 1b2 : b 3 1 1b2 (see (60)), the inequality is reduced to

1 112 1 1 1 1b2 : S 4 2 3 13 b3
or by multiplying by 8 2 1 41 ,

f 1x2 8 42 1 1 1 1b2 : N 4 2 7 (71) P R O O F O N L Y YIELD CRITERIA FOR SHAPE MEMORY MATERIALS 31
The calculation of 1 1 1 1b2 : N 4 2 gives

1 1 1b2 : N 4 3 f 1 1 b 2 e 1 4 f 2 1 b 2 e 2 : e 1 4 1 1 4 2 2 1 2 e 2 3 8 2 f 1 1 b 2 8 f 2 1 b 2 1 1 4 2 2 1 2 7
The inequality (71) then becomes

f 1x2 f 1 1 b 2 8 1 1 4 2 2 1 2 f 2 1 b 2 3 b 6 I7 (72)
To show (72) let us introduce the function u 12 as

u 12 3 f 1 1 b 2 8 1 1 4 2 2 1 2 f 2 1 b 2 4 f 1x2 3
with b fixed on I7 We must prove that u 12 9 03 6 I7 Two cases will be investigated:

Case b 3 1. For 3 b , one obtains (66):

u 1 b 2 3 b f 1 1 b 2 8 1 1 4 2 b 2 1 2 f 2 1 b 2 4 f 1x b 2 3 07
As the first derivative of u is

du d 3 f 1 1 b 2 4 1 1 4 2 2 4 1 2 f 2 1 b 2 4 d f d 3
by using (18), du d can be written as

du d 3 1 1 4 2 2 4 1 2 1 1 4 2 2 1 2 f 1 1 b 2 4 f 2 1 b 2 4 a 1x2 f 1 1x2 7
So, for 3 b 3 with b 3 1, thanks to (66), one obtains du d

1 b 2 3 07 A second derivation gives d 2 u d 2 3 4 1 1 4 2 2 4 3 2 f 2 1 b 2 4 d 2 f d 2 7
By the Taylor formula at the neighborhood of b , one has

P R O O F O N L Y 32 M. R. LAYDI and C. LEXCELLENT u 12 3 u 1 b 2 8 du d 1 b 2 1 4 b 2 8 1 2 d 2 u d 2 12 1 4 b 2 2 3 1 2 d 2 u d 2 12 1 4 b 2 2
where is between and b . So the necessary and sufficient convexity condition becomes

d 2 u d 2 9 0 in 8 1 2 3 1 8
3
which can be written as

4 f 2 1 b 2 3 4 1 1 4 2 b 2 1 2 g 1 b 2 9 1 1 4 2 2 3 2 d 2 f d 2 3 6 8 1 2 3 1 8 3 b 6 8 1 2 3 1 8 7 ( 73
)
The function d 2 f d 2 being continuous on I , (73) implies when tends to 1

f 2 1 b 2 03 b 6 8 1 2 3 1 8 3 (74)
and by the boundary continuity

g 1 b 2 0 b 6 8 1 2 3 1 9 7 ( 75
)
The function g being continuous on I , by using (73), one also have when b 1 which is always true thanks to (75). In fact the function

1 1 4 2 2 3 2 d 2 f d 2 03 6 8 1 2 3
) f 12 3 41 f 1x2 is decreasing d ) f d 3 41 d f d 4 42 f 3 4 42 6 f 4 d f d 7 3 4 42 g 12 9 07 So ) f 12 ) f 112 41 f 1x2 f 1x 1122 7
This ends the proof of the proposition. 2 Proposition 6. Let (22) hold with g non-identically equal to zero. Then 3 as defined by (37), is a continuous bijective application from I to I , admitting a first derivative (see ( 43)) which is continuous and increasing on 1 2 3 1 .

Proof. We examine two cases.

Case 4 3 1. This case is trivial as 1 17 In fact, from (65), we have f 1 112 9 0 and f 2 112 3 03 and so

f 1 3 5 f 1 1125 2 8 5 f 2 1125 2 3 f 1 112
which gives

1 4 2 3 f 1 112 f 3 17
Reciprocally, if 3 13 6

f 2 1 4 2 f 1x 4 2 3 1 1 4 2 2 1 2 3 0 f 2 1 4 2 3 1 1 4 2 4 2
f 1 d f 1 d 8 f 2 d f 2 d 7 3 42 f 2 1 d f 1 d 8 42 f 1 f 2 d f 2 d 3 d f 1 d 8 42 f 2 6 f 1 d f 2 d 4 f 2 d f 1 d 7 so d d 3 43 f 2 6 f 2 d f 1 d 4 f 1 d f 2 d 7 7
Then from (24) and (66) we obtain

d d 3 43 d f 1 d f 2 1 1 4 2 2 4 1 2 1 1 4 2 2 1 2 f 2 8 f 1 3 43 d f 1 d g f3
which is exactly the expression (43). So from (23) and since f et g are strictly positive when 6 1 2 3 1 , one concludes that is strictly increasing on

d 3 3a 1 4 2 4 1 2 3 3 d f d 1 4 f 2 4 1 2 d f d 3 a 1 4 2 4 1 4 f 2 4 0 and d 2 d 2 3 24a 3 3 d 2 f d 2 1 4 f 2 4 1 2 8 8 6 d f d 7 2 f ! 7
Hence following (25), one must verify

g 112 3 f 1y 1122 4 d f d 112 0 a 4 6 d f d 7 2 f 0 7
The first inequality is direct because f 1y 1122 3 f 112 3 1 and d f d 112 3 a. Concerning the second one, it is equivalent to

d f d f 1 2 9 a 1 2 1 2 a 1 2 1 4 2 4 1 2 4 1 2 9 1 4 f 2 4 1 2 f 4 1 2 7
We will show that

u 12 9 u 1 f 2 3 (78) for u 1x2 1 1 4x 2 4 1 2 x 4 1 2 3 x 6 8 1 2 3 1 9 7
In fact, as u is an increasing function

du dx 3 1 2 x 4 3 2 1 1 8
An elementary calculation using (50) shows that cos

1 4 2 n 4 4 sin 1 4 2 n 4 3 Q 1 e R 4 1 4 Q 2 e R 4 2 3
where

Q 1 1 6 cos 1 4 2 4 1 1 4 2 6 2 1 2 sin 1 4 2 Q 2 1 1 1 4 2 6 2 1

APPENDIX A

First we will examine the convexity conditions of a "power law" shape function f 12. Then this function is compared in the Figure 12 with the "cosinus" formulation of f 1y2, see (53).

Proposition 11. Let 1 112 3 1 f 1 4 2 where f is defined by f 12 3 1 4 8 q 3 (81)

where the parameters and q are such that 0 9 q 9 1 if q 1 q 9 0 otherwise7 The couples 13 q2 are chosen such that the two curves coincide at points y 3 413 4 Figure 13 gives the couples 13 q2 as a function of a7 Note that the convexity condition (82) is clearly verified.

APPENDIX B

We are going to give here a necessary and sufficient convexity condition for the shape function introduced by Raniecki and Mroz [START_REF] Raniecki | Yield or martensitic phase transformation conditions and dissipation functions for isotropic, pressure-insensitive alloys exhibiting SD effect[END_REF]. As is a decreasing function on

  R. LAYDI and C. LEXCELLENT

Figure 1 .

 1 Figure 1. Deviatoric stress section: definition of angle 5 4 , 5 6 and yield surface F112 3 4 c 7

where 1 b 1 tr 1 a T b 2 for every second order tensor a and b and 5a5 1 3 tr 1 a T a 2 . 8 5S 4 5 3 with

 1283 is the identity tensor. Two dots denote the scalar product and tr 172 denotes the trace operator, so that a : In a classical way, the Huber-Von Mises equivalent stress 1 is defined by1 1

x 4 1 5 4 1 1 3 arccos y 4 6 I 5

 65

for the first choice 1x 3 y 4 2 cos 1x2 for the second choice 1x 3 5 4 2 x for the third choice 1x 3 4

 24

3 1 112 9 2 112 9 3 4 i: e R 4 j 3 e i : e j 3 8 2 i j e R 4 i: 1 3 e i : 1 3 0 i3 j 3

 334403 13 27 (13) Note that the Lode 5 4 is the angle between N 4 and e R4 1 see (56) and Figure1. Moreover 4 3 the solution of equation (

d 2 f d 2 3

 3 42 07 Let us give a concrete example where the convexity condition is not fulfilled. Take a3 b3 c such that

Figure 2 .

 2 Figure 2. Non convex yield surface(counter example).

  R. LAYDI and C. LEXCELLENT

  As for the Ni-Ti alloy, the tension (compression)-torsion tests on tubes as performed by Laverhne-Taillard et al.[START_REF] Laverhne-Taillard | Multiaxial shape memory effect and superelasticity[END_REF], give the following yield values:

Figure 3 .

 3 Figure 3. Obtention of a and 4 c parameters.

Figure 4 .

 4 Figure 4. Comparison between "cosinus" formulation and straight lines for Cu-Al-Be alloy.

more simply as the classical linear function f 1y2 3 b 1 8 b 2 3 0794 b 2 3 0706 and NiTi : b 1 3 0791 b 2 3 07097 Figures 4

 23070974 Figures4 and 5show the comparison of the "cosinus" formulation with the linear form for the above parameters which are obtained by minimizing the functional

Figure 5 .51 112 4 4 c 5 2 ! 1 2 7b 2 b 1 3 07090791 3

 521233 Figure 5. Comparison between "cosinus" formulation and straight lines for Ni-Ti alloy.

Figure 6 .I 1 1 9 i 3 j 9 3 ' 3 see

 63 Figure 6. Yield surfaces in the stress space for Cu-Al-Be alloy.

4 zz 3 1 8 3 4 z5 3 (Figure 7 .

 37 Figure 7. Yield surfaces in the phase transformation strain dual space for Cu-Al-Be alloy.

Figure 8 .

 8 Figure 8. Univocal correspondence between 4 and 6 for Cu-Al-Be alloy.

Figure 9 .

 9 Figure 9. Yield surfaces in the stress space for Ni-Ti alloy.

  The eigenvalues k of N 4 solve

Figure 10 .

 10 Figure 10. Yield surfaces in the phase transformation strain dual space for Ni-Ti alloy.

4 2 7 If 3 4 3 1 1 3 2 3 4 1 2 and 3 3 3 1 or y 4 3 41 1 3 41 and 3 2 3 3 3 1 2 7

 473417 03 the equation has three real roots: one single and one double. One has y

d 2 f d 2 is trivial. 2 Proposition 5 .

 25 Let g and d 2 f d 2 be continuous on I . Then 1 is a convex function if and only if the condition (22), i.e.

  direct. Indeed, (73) is obtained by the two inequalities (74) and (76). Case b 3 17 Then x b 3 1 and a1x b 2 3 03 and therefore one has f 1 1 b 2 3 f 112 and f 2 1 b 2 3 07 Then proving u 12 9 0 comes to show that u 12 3 f 1x 1122 4 f 1x2 9 03

Then 1 6 I7 2 Figure 12 1y2 3 cos 6 1 3 arccos 11 4 a 11 4 y22 7 7 P

 162127 Figure 12 illustrates, for values of a 3 1 10 3 2 10 3 7 7 7 3 9 10 , the comparison with the function

Proposition 12 .Figure 14 Figure 13 .Figure 14 .

 12141314 Figure14shows the evolution of b as a function of .

  We are now ready to state a result of duality type. Just as for 1, the derivative of 2 with respect to 6 6 4 3 see (16), takes the form21 16 6 4 2 3 h 1x 6 2 n 4 8 a 1x 6 2 h 1 1x 6 2 n

	YIELD CRITERIA FOR SHAPE MEMORY MATERIALS 17
	6.2. Duality
		O N L Y 4
	O O F
	P R
	5 8 6
	4 3 cos	1 arccos 6 8 6	2	(49)
	y 4 3 cos	1 arccos y 6 8 3 6	2	7

  Figure 11. Univocal correspondence between 4 and 6 for Ni-Ti alloy.

	YIELD CRITERIA FOR SHAPE MEMORY MATERIALS 27 28 M. R. LAYDI and C. LEXCELLENT
	Proposition 2. Let a 1x2 and N 4 defined by (
																O N L Y O N L Y
				O O F O O F
	P R k 3 cos 6 arccos 1y 4 2 8 2k 3 P R	7	, k 3 13 23 33		(58)
	i.e.
	k 3 cos	6	1 3	arccos 1y 4 2 7	cos	6	2k 3	7	4 sin	6	1 3	arccos 1y 4 2 7	sin	6	2k 3	7	7
	So, the diagonal matrix composed of the eigenvalues of N 4 can be written as
				4 3 4 e 1 4	1 1 4 2 4	2 1 2 e 2 3
	where 4 3 cos The expression of 4 in (56) is obtained by the orthogonality (13) between e R 4 1 1 3 arccos 1y 4 2 2 3 e 1 3 diag 1 cos 1 2k 3 22 and e 2 3 diag 1 sin 1 2k 22 . 3 1 and e R 4 2 . 2

  Let 1 112 3 1 f 1y 4 2 where f is defined by (53). Then 1 is convex if and only if the inequalities (54) are verified.

	YIELD CRITERIA FOR SHAPE MEMORY MATERIALS 35
	One has to note that
	12 1 1 4 a 11 4 y2 3 4 f 3 4 3 f3
	O N 2 because a 11 4 y2 3 1y2 3 417 2 f 9 17 Indeed f 3 1 where y 3 4 3 4 3 and 1 By deriving , one obtains L Y d
	O O F
	1 2 3 1 bijection. The smoothness of comes from the smoothness of the composition of these and thus we have a functions. 2 P R
	Proposition 7. Proof. (54) is a necessary condition because the function arccos 172 exists only if
	41 9 1 4 a 11 4 y2 9 13 y 6 [413 1] 7
	Let us show that the condition is also sufficient. In fact, the case where a 3 1 (resp. a 3 0) is trivial because it corresponds to f 1y2 3 cos 1 1 3 arccos 1y2 2 3 (resp. f 1y2 3 cos 1 1 3 arccos 112 2 3 1). We must therefore verify for
	0 a 17

  The values of 4 and 6 defined by (39) and (46) are opposite, i.e. 4 8 Let 1 be the solutions of (1). Then, the identity (51) holds.

	36 M. R. LAYDI and C. LEXCELLENT YIELD CRITERIA FOR SHAPE MEMORY MATERIALS 37
	i.e. The last term reads (see (42))
	1y2 9 11 4 a 11 4 y22 3 y 6 [413 1] 3 d 6 d 4 3 6 1 1 4 2 4 2 4 1 2 8 d 4 d 4 7 1 1 4 2 2 1 2 3
	P R O O F O N L Y 1y2 3 cos 6 1 3 arccos 1y2 7 7 This is always true, because 1y2 is an increasing function and y 9 1 4 a 11 4 y2 7 This ends the proof of the proposition. 2 Proposition 8. 6 3 07 for Proof. The derivation w.r.t. 4 of the expression cos 4 3 f 1x 4 2 f 1x 4 2 3 8 2 f 1x 4 2 h 1x 2 3 gives 4 d 4 d 4 sin 4 3 8 2 6 d f d 4 h 8 f dh d 6 d 6 d 4 7 7 Let us then multiply by 114 2 6 2 1 2 cos 4 to obtain 4 d 4 d 4 1 1 4 2 6 2 1 2 tan 4 3 1 1 4 2 6 2 1 2 d f d 4 f 8 1 1 4 2 6 2 1 2 dh d 6 h so that we finally have 1 tan 4 8 tan 6 2 d 6 d 4 3 03 L Y i.e. tan 4 8 tan 6 3 07 N 2 O Proposition 9. Proof. Set h 1 5h 1x 6 25 2 8 5a 1x 6 25 2 h 1 1x 6 2 2 1 2 . From (47), F O h 3 h 1x 6 2 1 8 5a 1x 4 25 2 f 1 1x 4 2 f 1x 4 2 2 ! 1 2 3 h 1x 6 2 f 1x 4 2 f 1x 4 2 3 8 42 f 1x 4 2 f 1x 4 2 f 1x 4 2 3 8 42 7 f 1x 4 2 O Hence 2 1 16 6 4 2 3 h 1 cos 1 6 2 n 4 8 sin 1 6 2 n 4 2 3 8 42 f 1x 4 2 1 cos 1 6 2 n 4 8 sin 1 6 2 n 4 2 7 P R Integrating the relation (45) one has also d 6 7 d 4 But from (18) and (39), one can write 2 1 16 6 4 2 3 8 42 f 1x 4 2 1 cos 1 4 2 n 4 4 sin 1 4 2 n
	tan 4 3	a 1x 4 2 f 1 1x 4 2 f 1x 4 2	3	1 1 4 2 4 f 1x 4 2 2 1 2 d f d 4
	tan 6 3	a 1x 6 2 h 1 1x 6 2 h 1x 6 2		12x 2 2 1 9 03 1 4 2 6 3 h 1 1x 6 2 2 1 2 d f d 6
	and thus it is sufficient to verify and 6 1 1 4 2 4 2 4 1 2 8	d 4 d 4	7 1 1 4 2 6	2 1 2 tan 4 8 tan 6	d 6 d 4	3 07	(79)

4 2

  2 2 3 81 (resp. 3 2 41 3 2 4 1 2 3 1), i.e.

		q 3		2 ln 122	ln	f	1 4 f 4 2 2	2 2 4 f 1412 4	(86)
			3		f		4 2 4 q 2 4 1 2 4 1 2
	where
	f 1412 3 cos	6	3 1	arccos 11 4 2a2 7
	f 4	2 2	!	3 cos 1 3 arccos 1 4 a 1 8	2 2	!!!	7	(87)

  Finally, following (25), it is neccesary and sufficient that (see (67)) g 112 3 f 112 4 9 f 1 112 3 1 8 11 4 exp 1422 11 8 922 0

	44 M. R. LAYDI and C. LEXCELLENT
		d d	3 4	8 3	1 4 2 4 1 2 43 1 1 8 12 2 2	03
	so (91) is equivalent to
	i.e.	1	9 112 3 12 1 exp 1422 11 8 92 4 17 8 27 7 O N L Y
	O 3 17 4 182 exp 1422 F 5 3 exp 1422 9 03 O thus for a parameter between 0 and 3 But for 9 8 27 d d
	P R
					1 2 3 1	3 because

7(32)If now, instead of f one uses its inverse f 7 1 f 41 , then in this case, the condition (32) becomes

Then, from (4),

As 1 is a solution of (1), then from (8), one finally obtains


D Choqueuse

P Davies
email: peter.davies@ifremer.fr

D Perreux

L Sohier

J-Y Cognard
email: jean-yves.cognard@ensieta.fr

Mechanical Behavior of Syntactic Foams for Deep Sea Thermally Insulated Pipeline

Keywords: syntactic foam, hydrostatic compression, shear test

Ultra Deep offshore oil exploitation (down to 3000 meters depth) presents new challenges to offshore engineering and operating companies. Flow assurance and particularly the selection of insulation materials to be applied to pipe lines are of primary importance, and are the focus of much industry interest for deepwater applications. Polymeric and composite materials, particularly syntactic foams, are now widely used for this application, so the understanding of their behavior under extreme conditions is essential. These materials, applied as a thick coating (up to 10-15 cm), are subjected in service to:

. A series of tests has been developed to address this point, which includes:

-hydrostatic compression -shear loading using a modified Arcan fixture This paper will describe the different test methods and present results obtained for different types of syntactic foams.

Presentation

Syntactic foams are now widely used as passive thermal insulation material for deep sea pipeline and components (Figure 1). They are made by incorporating hollow glass microspheres (150µm > Ø > 10µm, wall thickness around 1.5 µm) in polymers (PU, PP, Epoxy …). Their composition confers good performance in terms of thermal insulation and hydrostatic compression (nominally 0.18 > k W/(m.K) > 0.12 and max pressure > 30 MPa ). They are applied using different manufacturing process including co extrusion, lateral extrusion, molding….During the lifetime of the structure (installation, in service, …) high mechanical loading can be applied to the material and good knowledge of the mechanical response of such materials is required in order to guarantee long term reliability of the insulated structures Figure 1 : Thermal insulation structure (doc Socotherm) Uniaxial compression properties of syntactic foams have already been addressed by different authors [START_REF] Gupta | Studies on compressive failure features in syntactic foam material[END_REF][START_REF] Koopman | Microstructural failure modes in three-phase glass syntactic foams[END_REF]. In a recent experimental study [START_REF] Adrien | Experimental study of the compression behaviour of syntactic foams by in situ X-ray tomography[END_REF] X ray microtomography has been used to visualize and describe the behavior of these materials during uniaxial loading and confined compression. However all these tests are polluted by edge effects, and a correct description of initial damage appears difficult. In order to model the behavior of the structure during its life time properly it is important to propose mechanical tests giving access to intrinsic properties of the material

Hydrostatic compression

Very few methods are available to determine the hydrostatic compression behavior of materials and "in house" tests are generally used to provide information. The principle is generally to follow the displacement of the piston of a cylinder in which the sample to be tested is placed. The pressure/piston displacement curve can provide a collapse pressure identified by a change of slope of the curve.

In order to improve this characterization Ifremer has developed a specific method based on the direct measurement of the buoyancy of a sample during pressure increase (Figure 2). In order to increase the sensitivity of the measurement the procedure consists of following the evolution of the buoyancy of both the sample and the steel ballast, and knowing the specific gravity of the material at atmospheric pressure it is possible to determine the specific gravity of the material versus the pressure. In addition, knowing the evolution of the density of water with pressure [START_REF]Algorithms for computation of fundamental properties of seawater[END_REF] it is possible to determine accurately, from the evolution of the buoyancy of the material, the evolution of the volume of the sample versus pressure and then to determine intrinsic parameters such as bulk modulus and collapse pressure.

A material sample, volume around 1 dm 3 , is placed below the balance with steel ballast whose weight is calculated in order to give to the ensemble (sample + ballast) a small positive weight in water. All the system is placed in a hyperbaric tank and the pressure is increased by means of an electric pump.

Taking into account the accuracy of the system measurement and of the input data, the accuracy of results obtained for the evolution of buoyancy and bulk modulus can be evaluated : for a 1 dm 3 volume specimen, this is around 1%.

Water uptake of samples during the test can affect the results for bulk modulus. However, the duration of testing, about 1 hour, leads to very little water ingress. Nevertheless systematic weighing of the sample after the test gives an indication of the quantity of water absorbed during the test, to check that it is related to surface absorption only.

Typical curves from the tests are reported below (Figure 3). The curve of variation of volume versus pressure gives access to the intrinsic mechanical behavior of the material. The global behavior of the materials can be characterized by a first linear elastic response, where the bulk modulus of the material is identified (2.9 GPa), and the collapse pressure (95 MPa), called "crush pressure".

A comparison has been made between two materials, Mat A and Mat B, having the same grade of spheres and two different matrix polymers: a rigid matrix (epoxy), Mat A and a soft matrix (PU), Mat B.

It can be noted (Figure 4a) that the initial bulk modulus of the two materials is comparable and that the crush pressure of the material with rigid matrix is significantly higher than for the material with the soft matrix (90 MPa compared to 50 MPa). Up to 80 % of the crush pressure the material with the rigid matrix presents an apparent linear elastic behavior, in strong contrast to the material with the soft matrix which presents a non linear behavior from the beginning of the test. The hydrostatic compression behavior appears to be strongly governed by the nature of matrix.

A study has been made on a material (Mat C: epoxy matrix) at different temperatures (4°C, 50°C and 100 °C) (Figure 4b). This shows a strong influence of the temperature. It can be assumed that the temperature, in terms of micromechanical behavior, mainly affects the behavior of the matrix.

Shear test

To complete the information provided by the hydrostatic compression test it is important to evaluate the shear behavior of the material. For this type of material different solutions have been envisaged: torsion tests on cylinders, Iosipescu test… In a first step, and taking into account the ease of sample preparation (difficulty to machine cylindrical samples) the Iosipescu test has been retained, and tests have been performed on materials A and B (Figure 5). Different material thicknesses (3 and 10 mm) have been tested. These tests provide an apparent shear modulus, but for the rigid matrix the failure appears to be sudden while for soft matrix the strain is too high to be able to reach the failure. In order to address properly non linearity and failure of these materials, which were clearly not obtained with the Iosipescu test, shear tests using the modified Arcan fixture have been evaluated, Figure 6. The Arcan test was initially developed [START_REF] Arcan | A method to produce uniform plane-stress states with application to fiber-reinforced materials[END_REF] to produce uniform plane stress and a modified version has been used recently, associated with image analysis, to characterize bonded joints [START_REF] Davies | Influence of adhesive bond line thickness on joint strength[END_REF]. Different geometries of samples have been investigated and finally 0.5*7,6*66mm 3 specimen bonded on aluminium support have been selected in order to minimize the effect due to the appearance of normal stress. Tests were performed at different loading rates on material B (Figure 8). High sensitivity to loading rate is highlighted, which reveals a viscoelastic/ plastic behavior of the material.

Conclusions

Syntactic foams exhibit complex mechanical behavior (damageable -visco -elasto-plastic behavior) and their characterization needs a careful experimental approach in order to obtain valid data. Two types of tests are proposed in this paper.  Hydrostatic compression test. These are based on the measurement of apparent buoyancy during pressure increase, gives pertinent data allowing characterization of the material.  Shear tests. For these, improvement of the methods is still necessary, but an adapted Arcan fixture appears to be very promising. This experimental study will be completed by multi axial tests using a specific device allowing mechanical loading to be performed under hydrostatic compression loading [START_REF] Cartié | the influence of hydrostatic pressure on the interlaminar fracture toughness of carbon/epoxy composites[END_REF], in order to cross-correlate the results obtained from the different test methods.

Figure 2 :

 2 Figure 2 : Hydrostatic compression test

Figure 3 :

 3 Figure 3: Hydrostatic compression behavior

Figure 4 :

 4 Figure 4: Hydrostatic compression behavior a -effect of the matrix rigidity b-effect of the temperature

Figure 5 :

 5 Figure 5: Iosipescu shear test results

Figure 6 :Figure 7 :

 67 Figure 6: modified Arcan shear test set up