Mahdi Masoumi Khalilabad
email: mahdi.masoumi-khalilabad.1@etsmtl.net

Yasser Zedan

Damien Texier

Mohammad Jahazi

Philippe Bocher

Effect of tool geometry and welding speed on mechanical properties of dissimilar AA2198-AA2024 FSWed joint

Keywords:

Different tool geometries were used to investigate the joining of aluminum alloys (AA2198 to AA2024) by friction stir welding (FSW). Three shoulder profiles (flat, raised spiral, and raised fan) and five different pin profiles (cone, half threaded cylindrical, straight cylindrical, tapered cylindrical and square) were selected. Preliminary investigations were conducted by moving the tool into a seamless sheet made of the AA2024-T3 in order to select the tools that produce defect-free joints. Preliminary investigations showed the raised fan shoulder profile helps the material flow from the edge of shoulder to the pin creating a smooth surface finish with no flash in comparison with flat and raised spiral shoulder profiles. Pins with a minimum diameter equal to half the plate thickness produced lack of penetration (LOP) defects, while increasing minimum pin diameter to the plate thickness eliminates the LOP defects. Half threaded cylindrical pin produced tunneling defect, whereas defect free joint made by straight cylindrical, tapered cylindrical and cubic pin profiles. So they were selected for joining AA2024 to AA2198. Fracture locations of different joint variants were observed the vicinity of the thermomechanical affected zone (TMAZ) of AA2198-T3 alloy, and in the nugget on the AA2198-T3 side which have the minimum hardness and highest strain localization as confirmed by hardness maps and digital image coronation (DIC). Higher measured temperature than dissolution temperature of AA2198 main strengthening precipitates could be the reason of low hardness and fracture in TMAZ and center of nugget. Furthermore a raised fan shoulder with a tapered cylindrical pin produced highest elongation and yield strength and it was selected as the best candidate for optimization of the welding parameters. It was found that higher rotational and traverse speeds enhance the formation of tunneling and kissing bond defects, suggesting that longer pins have to be used for higher traverse speeds. Welding speed 750 rpm with 450 mm min -1 could create joint with highest yield strength.

Introduction

Friction Stir Welding (FSW) is an emerging solid state joining technology that has been successfully used in the aerospace industry as an alternative to riveting for the assembly of airplane fuselage panels made of conventional AA2xxx and AA7xxx alloys [START_REF] Hasan | Influence of machine variables and tool profile on the tensile strength of dissimilar AA7075-AA6061 friction stir welds[END_REF][START_REF] Ks Ak | Optimization of FSW parameters to improve the mechanical properties of AA2024-T351 similar joints using Taguchi method[END_REF][START_REF] Durso | The influence of process parameters on mechanical properties and corrosion behaviour of friction stir welded aluminum joints[END_REF]. The specific properties of structural materials, i.e. the ratio of the selected mechanical property to the material density, constitute one of the main drivers in material selection in the aerospace industry. The addition of one weight percent of lithium to aluminum alloys has been shown in the literature to decrease the density by about 3% and increase the Young's modulus by 6%, leading to a substantial increase of the specific Young modulus (E/ρ) [START_REF] Prasad | Aluminum-lithium alloys: processing, properties, and applications[END_REF]. These benefits after extensive researches led to the successful development and commercialization of the 3rd generation Al-Li alloys (e.g., AA2198), which have excellent combinations of specific properties, as compared to their conventional counterparts (e.g., AA2024) [START_REF] Alexopoulos | Fatigue behavior of the aeronautical Al-Li (2198) aluminum alloy under constant amplitude loading[END_REF].

Aerospace industry is interested to replace conventional aluminum alloys with new generation of Al-Li alloys for mass gain purpose. Nevertheless, Al-Li alloys still remain relatively expensive. An economical weight reduction solution for structural components involves using hybrid designs with AA2198 alloys for critical regions, and keeping the remaining regions of the structure in AA2024. Thus, joining of aluminum alloys to Al-Li alloys appears as a necessity. However, there are relatively few papers on this specific joint, and particularly on optimum conditions for obtaining sound dissimilar FSW joints.

Tunneling defect, flash and kissing bond are common defects of FSWed joint. It is reported that kissing bond decreases the fatigue life time [START_REF] Jolu | Fatigue lifetime and tearing resistance of AA2198 Al-Cu-Li alloy friction stir welds: effect of defects[END_REF]. While, it does not affect significantly the tensile strength [START_REF] Kadlec | Mechanical behaviour of AA 7475 friction stir welds with the kissing bond defect[END_REF][START_REF] Jolu | Microstructural characterization of internal welding defects and their effect on the tensile behavior of FSW joints of AA2198 Al-Cu-Li alloy[END_REF]. The key to producing reliable joints by the FSW process is to concurrently select the optimum welding tool and process parameters [START_REF] Mishra | Friction stir welding and processing[END_REF]. Shoulder profile affects the material flow [START_REF] Kumar | The role of friction stir welding tool on material flow and weld formation[END_REF], and its optimum design is critical for producing defect-free joints. Some studies have reported either shoulder profile [START_REF] Scialpi | Influence of shoulder geometry on microstructure and mechanical properties of friction stir welded 6082 aluminium alloy[END_REF][START_REF] Mohanty | Effect of tool shoulder and pin probe profiles on friction stirred aluminum weldsa comparative study[END_REF][START_REF] Heredia | Effect of tool shoulder features on defects and tensile properties of friction stir welded aluminum 6061-T6[END_REF] or pin design [START_REF] Kumar | Effects of tool pin profile and tool shoulder diameter on the tensile behaviour of friction stir welded joints of aluminium alloys[END_REF][START_REF] Idris | Effect of friction stir welding pin shape on mechanical properties of AA6061 alloy weldment[END_REF][START_REF] Reynolds | Effect of tool pin thread forms on friction stir weldability of different aluminum alloys[END_REF][START_REF] Amini | Pin axis effects on forces in friction stir welding process[END_REF][START_REF] Amirafshar | Effect of tool pin design on the microstructural evolutions and tribological characteristics of friction stir processed structural steel[END_REF][START_REF] Jafarlou | Investigation into the role of pin geometry on the corrosion behavior of multi-pass FSWed joints of Al5086 besides applying Al 2 O 3 nanoparticles[END_REF][START_REF] Kadian | Effect of tool pin profile on the material flow characteristics of AA6061[END_REF] effect on the mechanical properties of FSWed joints. Flat shoulder profile with conical pin is the simplest design and has been used successfully for joining aluminum alloys [START_REF] Mustafa | Tool geometries optimization for friction stir welding of AA6061-T6 aluminum alloy t-joint using Taguchi method to improve the mechanical behavior[END_REF][START_REF] Chen | Effects of pin thread on the inprocess material flow behavior during friction stir welding: a computational fluid dynamics study[END_REF]. Also, adding a step to a conical pin [START_REF] Salari | Influence of tool geometry and rotational speed on mechanical properties and defect formation in friction stir lap welded 5456 aluminum alloy sheets[END_REF] and half screwed pin (in comparison to full screwed) [START_REF] Ji | Design of friction stir welding tool for avoiding root flaws[END_REF] was shown to increase the joint mechanical properties and to eliminate the cavity defect. The weld prepared by square pin profile results in smaller grain size and higher mechanical properties [START_REF] Amirafshar | Effect of tool pin design on the microstructural evolutions and tribological characteristics of friction stir processed structural steel[END_REF][START_REF] Kamble | Effect of tool design and process variables on mechanical properties and microstructure of AA6101-T6 alloy welded by friction stir welding[END_REF]. They were reported to produce defect free joints, irrespective of welding speeds [START_REF] Elangovan | Influences of tool pin profile and welding speed on the formation of friction stir processing zone in AA2219 aluminium alloy[END_REF]. As mentioned, appropriate process parameters have an important role to produce a sound weld. A low rotational speed or a higher traverse speed cause tunneling defect and kissing bond due to insufficient material mixing as the heat input is too low and the material is not soft enough [START_REF] Kim | Three defect types in friction stir welding of aluminum die casting alloy[END_REF]. On the other hand, a combination of high speeds and high traverse speeds causes cavities due to abnormal stirring [START_REF] Mastanaiah | Dissimilar friction stir welds in AA2219-AA5083 aluminium alloys: effect of process parameters on material intermixing, defect formation, and mechanical properties[END_REF]. In the range of optimized welding parameters, increasing the traverse speed in the presence of a constant rotational speed improves joint mechanical properties [START_REF] Ma | Effect of welding parameters on mechanical and fatigue properties of friction stir welded 2198 T8 aluminum-lithium alloy joints[END_REF], while increasing the rotational speed results in higher heat input and lower mechanical properties [START_REF] Jannet | Optimization of process parameters of friction stir welded AA 5083-O aluminum alloy using response surface methodology[END_REF]. It should be noted that close to the solidus temperature of the material, the rotational speed has negligible effect on mechanical properties due to self-stabilizing phenomenon [START_REF] Hattel | Thermomechanical modelling of friction stir welding[END_REF][START_REF] Dubourg | Relationship between FSW parameters, hardness and tensile properties of 7075-T6 and 2098-T851 similar but welds[END_REF].

The FSW parameters were optimized for several dissimilar joints between AA2024 to 5xxx series [START_REF] Venkateshkannan | Influences of tool geometry on metallurgical and mechanical properties of friction stir welded dissimilar AA 2024 and AA 5052[END_REF], 6xxx series [START_REF] Santhosh | Practices in fsw welding and investigate effect of tool turning speed on tensile properties and microhardness of dissimilar aluminium alloys 2024 and 6061[END_REF][START_REF] Moradi | Microstructure and texture evolution of friction stir welded dissimilar aluminum alloys: AA2024 and AA6061[END_REF] and 7xxx series [START_REF] Kumar | A bottom-up optimization approach for friction stir welding parameters of dissimilar Aa2024-T351 and Aa7075-T651 alloys[END_REF]. Dissimilar friction stir welding of AA2024-T3 and AA2198-T3 in butt-joint configuration has been already investigated in terms of microstructural and specific mechanical characterizations for a particular configuration [START_REF] Texier | Near-surface mechanical heterogeneities in a dissimilar aluminum alloys friction stir welded joint[END_REF][START_REF] Robe | Microstructural and mechanical characterization of a dissimilar friction stir welded butt joint made of AA2024-T3 and AA2198-T3[END_REF], but no research on optimization of friction stir welding tool geometry and parameters for joining dissimilar AA2024 and AA2198 joints has been reported in the open literature. Optimized welding parameters for producing similar butt joints for each alloy, AA2024-T3 and AA2198-T3, have been proposed in the literature [START_REF] Ks Ak | Optimization of FSW parameters to improve the mechanical properties of AA2024-T351 similar joints using Taguchi method[END_REF][START_REF] Radisavljevic | Influence of FSW parameters on formation quality and mechanical properties of Al 2024-T351 butt welded joints[END_REF][START_REF] Bitondo | Friction-stir welding of AA 2198 butt joints: mechanical characterization of the process and of the welds through DOE analysis[END_REF][START_REF] Kasman | Effects of FSW parameters and pin geometry on the weldability of EN AW 2024 alloy[END_REF][START_REF] Bitondo | Friction stir welding of AA2198-T3 butt joints for aeronautical applications[END_REF]. The optimized rotational speeds are actually close for both alloys in the thickness less than 10 mm, while the optimized traverse speed for AA2024 is yet lower (73 mm min -1 vs. 300 mm min -1 ). Therefore, a trade-off in terms of welding parameters must be found for the welding of AA2024 to AA2198.

In this study, the effect of the shoulder geometry and pin profile on the microstructure and mechanical properties of FSWed AA2024-T3/ AA2198-T3 joints is documented. To that end, three different shoulder and five pin geometries were used to produce a FSW pass in an AA2024 plate for a given set of welding parameters. Macrograph observations were done to evaluate the occurrence of defects such as tunneling, kissing bond, and flash. Tensile properties (yield strength, ultimate tensile strength, and ductility) of the different welds were investigated to understand the effect of tool geometry and select an optimal tool design. To assess the influence of FSW welding speed on the mechanical properties of the joints, different traverse speeds in the range of 50-450 mm min -1 with two different rotational speeds, 750 and 1000 rpm, were investigated with the optimal tool. Fractographic analyses in combination with microhardness maps and Digital Image Correlation (DIC) technique were used to investigate microstructural features responsible for the fracture.

Experimental procedures, materials and methods

Base materials

Base materials consisting of AA2024-T3 and AA2198-T3 rolled sheets with a thickness of 3.2 mm were used. The nominal chemical compositions of these base materials are reported in Table 1.

Tools and fixture designs

A specific fixture was designed to increase the repeatability rate of the friction stir welding process, as shown in Fig. 1. The welding direction, the cross-weld direction and the normal direction are denoted WD, CWD and ND, respectively. Stoppers, supporters, and clamps were used to fix the plates in these directions, respectively. Clamps and supporters are equipped with screws, and so they apply opposite forces to the ND and CWD directions, while there is no need to apply any force in the welding direction due to the application of the welding force. The supporters and clamps were fixed with a torque wrench to ensure that reproducible forces were applied during the different welding tests.

Seven tool designs, denoted from A to G in Fig. 2, were designed and manufactured based on literature recommendation. The formulas for calculating the specific tool dimensions are shown in Table 2. AISI 4340 steel hardened up to 49 HRC by quenching and tempering before machining was selected for the tool material, as recommended by Ref. [START_REF] Zhang | Review of tools for friction stir welding and processing[END_REF].

Welding process

The first step in the present study was to find the appropriate tool geometry for friction stir welding of AA2024-T3 and AA2198-T3. Two series of welds were produced and investigated. The first of these series were made by moving the tool into a seamless one piece of the AA2024-T3 sheet (i.e., bead-on-plate weld) in order to select the tools that produce defect-free joints on the basis of metallography and visual examinations. The second series was produced using dissimilar welding of AA2024-T3 and AA2198-T3 plates. Then, the tool that could produce a joint between AA2024 and AA2198 with the highest yield strength was selected as the best candidate for further optimization. To optimize the welding for the selected tool, combinations of traverse (ν) and rotational speeds (ω) used in the present study and their related sample codes are presented in Table 3 with heat input index [START_REF] Dixit | Influence of process parameters on microstructural evolution and mechanical properties in friction stirred Al-2024 (T3) alloy[END_REF][START_REF] Arbegast | Hot deformation of aluminum alloys III[END_REF] which is a candidate for the representation of the average thermal profile during welding. The rotation speed, the welding traverse speed, the plunge depth and the tilt angle were 750 rpm, 50 mm min -1 , 0.2 mm, and 0°, respectively. Sodium hydroxide solution (20 g NaOH + 100 mL H 2 O) was used in order to remove the material stuck to the tools between experiments. For the dissimilar configuration, AA2024 plates were placed on the retreating side (RS), with the weld direction perpendicular to the transverse direction (TD), and on the advancing side (AS), AA2198 plates were placed with their rolling direction (RD) perpendicular to the welding direction. This configuration was chosen because the tensile properties of AA2024-T3 in transverse direction are comparable with those of AA2198-T3 in the rolling direction [START_REF] Robe | Microstructural and mechanical characterization of a dissimilar friction stir welded butt joint made of AA2024-T3 and AA2198-T3[END_REF]. Consequently, this configuration could maximize the joint efficiency in the tension condition.

Sample preparation and mechanical tests

The FSWed samples were prepared for metallographic observations using standard polishing procedures down to 1 μm diamond paste followed by BUEHLER Vibromet polishing for 48 h, with a 0.05 μm colloidal silica solution. A Keller etchant was employed for 15 s to reveal the microstructure. Optical micrographs were obtained with an OLYMPUS Lext OLS4100 laser scanning confocal microscope. Tensile test samples for base metals were done on TD and RD directions for AA2024 and AA2198, respectively. Tensile specimens were machined from the welded plates such that the loading direction was parallel to the cross-welding direction of the joined plates. Specimens were extracted from both base materials to provide a reference, and from the joints, as specified in Fig. 3a (the joint being centered in the specimen gage). Tensile tests were performed on dog-bone tensile specimens with the geometry displayed in Fig. 3b. As the tool plunging into the material surface produces sharp edges which are stress concentration sites, the tensile samples were polished on both sides to remove any possible surface defects including kissing bond. Tensile tests were conducted with a 5 kN Kammrath and Weiss micro-tensile device at a constant crosshead displacement rate of 7 μm s -1 . The specimen elongation along the loading direction was continuously recorded using a Keyence LS-7030M optical extensometer, measuring the displacement in the region between the two TMAZ regions (L 0 = 16 mm). Tensile tests were repeated three times for each welding condition. In parallel, in-situ optical microscope tensile tests were conducted for different loading conditions to calculate the local strain fields from measured displacement fields via an optical high resolution-digital image correlation technique (OHR-DIC), as explained in Ref. [START_REF] Texier | Near-surface mechanical heterogeneities in a dissimilar aluminum alloys friction stir welded joint[END_REF]. Optical micrographs were taken in dark field mode with an OLYMPUS DSX-500 optical microscope in an unloaded state to investigate the irreversible deformation of the joints. In order to track the local kinematic field, a surface code was generated at the surface of the sample with the Keller etchant, the so-called speckle. In-plane local displacement fields were assessed for each individual micrograph region with OpenDIC, an open source software application [START_REF] Bridier | Microscopic strain and crystal rotation measurement within metallurgical grains[END_REF]. Strain fields were then calculated with a JAVA companion working with Fiji software based on a twodimensional and iso-parametric finite element formalism, with nodes being the center of the subsets, and with four Gauss bilinear interpolation points per element.

In order to analyze the hardness distribution through a weld crosssection, 31 profiles of 260 indents were made with an increment of 80 μm in the CWD and ND, respectively. An automatic microhardness machine (CLEMEX), with a 25 g load applied for 10 s was used to obtain the Vickers microhardness maps.

DSC and temperature measurements

Differential Scanning Calorimetry (DSC) measurements were carried out by PerkinElmer equipment model pyris 1. The experiments were conducted under a high purity nitrogen flux. The heating ramp rate was 15 K min -1 from 323 to 773 K. We chose to keep at an initial temperature of 323 K for 10 min in order to allow enough time for the apparatus to stabilize. The samples are 0.5 mm thick discs with a diameter of 4 mm, in order to fit in the dedicated pure aluminum crucibles.

To find the maximum temperature experienced by the base metal, K-type thermocouples located on the weld line (edge of the shoulder) at both advancing and retreating sides were used. Thermocouples were placed inside small holes with a diameter of 1 mm drilled on the surface of base materials. Thermocouples were fixed with Thermigrease TG 20033, which is a silicone-free hard curing paste material known to optimize heat transfer in high temperature conditions. A digital thermometer was used to connect thermocouples to a data acquisition system installed on a personal computer in order to record the temperature evolution during the FSW process.

Results and discussion

3.1. Tool selection for joint between AA2024 and AA2198

Defects for bead on plate of AA2024

The surface pattern resulting from the FSW process with the three different shoulders (flat, spiral and fan) is shown in Fig. 4. The flat shoulder (tool A) tends to produce flash defects on the welded surface as the flat shoulder is not effective for trapping the flowing material under the bottom shoulder [START_REF] Zhang | Review of tools for friction stir welding and processing[END_REF]. Less flash was observed on the advancing side in comparison to retreating side. On the other hand, the raised spiral (tool B) and raised fan (tool C) shoulders prevent the flash defect formation as their surface features return the material to the center during the rotation of the tool. Thus, both spiral and fan shoulder profiles eliminate the need to tilt the tool for avoiding flash defects. Furthermore, the finishing surface produced by spiral shoulder had some degree of galling while the finishing surface produced by fan shoulder was smooth. The results show raised fan is a better selection for tool shoulder profile.

Metallography and visual examination results corresponding to the occurrence of tunneling defects and a lack of penetration (LOP) for a bead on plate of AA2024 are summarized in Table 4. Examples of LOP and tunneling defects observed for tool A are shown in Fig. 5. Metallographic observations demonstrated that tools with a minimum pin (min-pin) diameter equal to half the plate thickness (such as tools A, B and C) produced LOP defects. The pin length should increase for this specific design. When a bigger min-pin diameter was used (equal to the plate thickness), such as in tools D, E, F, and G, no LOP defects were

Table 4

Defects produced as a result of tool design related to bead on plate of AA2024. The rotation speed and the welding traverse speed were 750 rpm and 50 mm min -1 , respectively. found, indicating that material flow across the weld centerline had been amplified, ensuring the mixing of the material at the bottom of the weld [START_REF] Seidel | Visualization of the material flow in AA2195 friction-stir welds using a marker insert technique[END_REF]. Furthermore, the interfacial heat energy generation at the bottom of the joint is a function of the cone diameter, i.e., a higher diameter results in a higher heat input [START_REF] Shi | Transient model of heat transfer and material flow at different stages of friction stir welding process[END_REF][START_REF] Gadakh | Heat generation model for taper cylindrical pin profile in FSW[END_REF]. The results of the half-threaded cylindrical pin (tool D) indicate the presence of a small tunneling defect, whereas no such defects were found in the straight version (tool E).

Microstructure and microhardness maps of AA2024 and AA2198 joints

On the basis of Table 4, only tools E, F and G are capable of producing defect-free beads on plate welds for a set of low welding speeds. They were therefore selected for joining AA2024 to AA2198. The AA2024 and AA2198 joints made by tools E, F, and G were denoted codes E1, F1, and G1, respectively, and their macrograph cross-section observations in planes CWD-ND can be seen in Fig. 6a. The produced joints were free of defects. The three different microstructure zones typically generated by the FSW process, denoted nugget, thermo-mechanically-affected zones (TMAZ), and heat-affected zones (HAZ), as well as the base material, can be identified. Using contrast, grain size, or morphology, the transition between the base metals and the nugget region are straightforward as small recrystallized grain microstructures are found in the nugget region. This transition between the two regions is drawn with a yellow line. Tracking the boundaries between the base metal and HAZ is a challenge. It should be noted from Fig. 6a that the nugget zone of all the joints is composed of three different regions, shown with different colors after etching on the CWD-ND planes. These three zones are denoted Z1, Z2, and Z3. Furthermore, a sharp interface is present for samples E1, whereas the interface of sample F1 is not irregular due to the presence of a cone. It can be seen locally that the material flow at the bottom of the joint is amplified as the pin diameter is increased, causing a sharp interface [START_REF] Seidel | Visualization of the material flow in AA2195 friction-stir welds using a marker insert technique[END_REF].

In order to find the weak regions of the joints, microhardness maps were obtained, and the results are presented in Fig. 6b. They show similar patterns to the weld macrographs shown in Fig. 6a. By considering the hardness of the as-received materials (AA2024 ≈ 150 HV 0.25 and AA2198 ≈ 125 HV 0.25 ), it can be seen that Zones Z2 and TMAZ/HAZ, and the shoulder-affected region (top of the nugget) on the AA2198 side, have the minimum hardness values (90 HV 0.25 ), regardless of the pin geometry. Fig. 7 shows the hardness evolution along the joint is different at AA2024 side in comparison to AA2198 side. The hardness of AA2198 drops in the middle of joint and TMAZ/HAZ area; while, the AA2024 hardness drops only in TMAZ/ HAZ area. This trend could be seen in all the joint variants. Furthermore, the TMAZ/HAZ of the retreating side (AA2024) has a higher hardness than the advancing side (AA2198).

To gather more details about the joint heterogeneity, the metallography images with higher magnifications from the Z1/Z2 boundary, inside the Z2 and Z3 regions, are shown in Fig. 8. It can be seen that the grains of the AA2198 side (Z2) are coarser than those from the AA2024 side (Z1) in the nugget region (Fig. 8a). The delimitation between both materials in the nugget is very clear. Comparing the microstructure of the Z2 and Z3 regions in Fig. 8b andc shows that there is no significant difference between the grain sizes of the two zones. Furthermore, SEM micrographs and the related EDS maps (Fig. 9) show the presence of Mg-Cu precipitates in the Z1 zone and Cu precipitates in Zones Z2 and Z3. These precipitates could be S(Al 2 CuMg) [START_REF] Bousquet | Relationship between microstructure, microhardness and corrosion sensitivity of an AA 2024-T3 friction stir welded joint[END_REF] and T1(Al 2 CuLi) [START_REF] Cavaliere | Al-Li plates joined by friction stir welding: mechanical and microstructural behavior[END_REF] which are the main hardening phase of AA2024 and 2198, respectively. The grain structure and nature of precipitates shows that the joint is not mixed completely and zone Z2 and Z3 are related to the AA2198 material side.

DSC and temperature measurements

DSC test on the base metal AA2198-T3 with temperature measurements at the TMAZ of the advancing and retreating sides were performed to help clarify the root cause of the low hardness levels observed in Fig. 6b. It could be seen that the dissolution of GP-zones occurs at low temperatures in the first endothermic peak A in Fig. 10a [START_REF] Ringer | Effects of cold work on precipitation in Al-Cu-Mg-(Ag) and Al-Cu-Li-(Mg-Ag) alloys[END_REF]. The peaks B and C indicate the precipitation and thickening of T1(Al 2 CuLi) phase, respectfully [START_REF] Dorin | Quantification and modelling of the microstructure/strength relationship by tailoring the morphological parameters of the T1 phase in an Al-Cu-Li alloy[END_REF]. Both of precipitation of Θ(Al 2 Cu) and dissolution of T1(Al 2 CuLi) phases concurrently start from Peak D (640 K) [START_REF] Dorin | Quantitative description of the T1 formation kinetics in an Al-Cu-Li alloy using differential scanning calorimetry small-angle X-ray scattering and transmission electron microscopy[END_REF]. The maximum measured temperature of the TMAZ (Fig. 12b) on the AA2198 side is higher than the dissolution temperature of T1(Al 2 CuLi) precipitates. Therefore, even if the exposed time is short, the welding process can partially dissolve them in TMAZ region. The temperature is also high enough for precipitate growth in the TMAZ region on the AA2024 side, as already reported by Genevois et al. [START_REF] Genevois | Quantitative investigation of precipitation and mechanical behaviour for AA2024 friction stir welds[END_REF]. Hence, the coarsening of S phase in AA2024 and coarsening/partial dissolution of T1(Al 2 CuLi) precipitates during the FSW process are probably the reasons for the observed lower hardness of the TMAZ areas in Fig. 6b. It shows that the AA2198 material is more sensitive to high temperature exposure. The sound joints produced by tools E, F and, G were evaluated using tensile tests to select the best design. The elongations (A%), the 0.2% offset yield strengths (YS) and the ultimate tensile strengths (UTS) of the joints tested in the as-welded condition are shown in Fig. 11. It can be seen that the tensile properties follow the same trend and are similar for all the joints. It is worth noting that the joint produced by tool F demonstrated both higher yield strength and elongation. The strain localization during the tensile test occurred in TMAZ/HAZ of AA2198 and center of nugget for all joint variants (Fig. 12). Besides, the main strain localization is in the center of nugget for sample F1; while, main strain localization occurred in TMAZ/HAZ of AA2198 for sample E1 and G1. All tensile specimens of samples E1 and G1 failed on the advancing side (AA2198) in the TMAZ. F1 specimens failed simultaneously, at both TMAZ of AA2198 and the middle of the joint (Z2 zone); however, final rupture occurred in the middle of the joint. Both fracture modes are shown in Fig. 13, confirming the results of DIC and microhardness maps that the nugget (zone Z2) and TMAZ of AA2198 are the weak regions of the joints. The tool dynamic volume defined by Ref. [START_REF] Elangovan | Influences of tool pin profile and welding speed on the formation of friction stir processing zone in AA2219 aluminium alloy[END_REF] could explain the different local mechanical response of tool F. Tool E and G have the same dynamic shape and volume which, is bigger than dynamic volume of tool F (Table 5). Less dynamic volume of tool F results in less material participating in the joint area and can be the reason of fracture in the middle of the joint.

Fractography

Fractography results using SEM obtained after monotonic tensile tests are reported in Fig. 14. The fracture surface in the middle of the joint (Fig. 14a) shows a ductile fracture with the small dimples (Fig. 14b). On the other hand, fracture surfaces in the TMAZ of AA-2198 (Fig. 14c) show a ductile and shear fracture with no evidence of dimples, even at higher magnification, probably suggesting the full dissolution of precipitates in this region of the weld (Fig. 14d).

Optimization of welding speed parameters

The joint produced by tool F demonstrated slightly better yield strength and elongation. Furthermore, the machining of tool F is easy and its wear vs. cubic pin is less. Also, it requires less force to penetrate into the material in the beginning of welding vs. straight cylindrical tool. So, tool F was considered as the best tool for further optimization.

Joint macrograph

Macrographs obtained on the joints produced with various welding parameters (Table 3) are shown in Fig. 15. It can be seen that increasing the rotational or traverse speed raises the possibility of tunneling defects (sample F6). On the other hand, for all new conditions, kissing bond defects were found in the welds, showing that the mixing conditions were not high enough in temperature at the root of the weld to allow proper material flow. Thus, it can be concluded that the pin length must be increased as the traverse speed increases in order to avoid the formation of kissing bond defect. The tool geometry can still be optimized in the present investigation for higher traverse speed parameters.

Microhardness map

Two-dimensional hardness maps across the ND-CWD section for various welding traverse speeds (Fig. 16) revealed that increasing the traverse speed significantly improves the hardness of the joints, in particular, that of the TMAZ. Specifically, the hardness increased by about 16 and 35 HV 0.25 for AA2198 and AA2024 sides, respectively, while the hardness of the nugget increased by about 17 HV 0.25 (Table 6).

Mechanical response

Increasing the traverse speed from 50 to 450 mm min -1 significantly increased the joint yield strength from 235 to 286 MPa, respectively (Fig. 17). This increase is consistent with a previous study on AA2198-T8 [START_REF] Ma | Effect of welding parameters on mechanical and fatigue properties of friction stir welded 2198 T8 aluminum-lithium alloy joints[END_REF]. While increasing the rotational speed for a given traverse speed (sample F5 vs. F3) did not show any significant effect on the mechanical properties of the joint, as reported in Ref. [START_REF] Dubourg | Relationship between FSW parameters, hardness and tensile properties of 7075-T6 and 2098-T851 similar but welds[END_REF], it decreased significantly the UTS in sample F6 vs. F4 due to the presence of tunneling defect in the joint. The fracture locations of all samples were in the nugget area inside the AA2198 material. Results showed that the optimum rotational and traverse speeds to achieve a defect-free joint with the highest yield strength were 750 rpm and 450 mm min -1 , respectively, corresponding to the cold weld condition (the lowest HI index in Table 3). It should be noted that further improvement of joint strength could be investigated by applying post weld heat treatment, considering the fact that 3rd generation of Al-Li alloys are very responsive to age hardening. Also, using coolant such as air, water or liquid CO 2 , during welding may result in lower welding temperatures and higher joint mechanical properties. These two strategies could be considered for future works.

Conclusions

In the present study, different tool designs were tested for joining dissimilar aluminum alloys AA2024 and AA2198. The raised fan shoulder with a tapered cylindrical pin geometry that is easy to manufacture and provide good mechanical properties was selected to document the effect of welding speeds on joint mechanical properties. Even, if the selected geometry was found not to be the optimized for high production speed, relevant conclusions could be drawn:

• The weak regions of dissimilar joints AA2024/AA2198 are located on the AA2198 material in the nugget and on the TMAZ, justify that colder welding conditions may improve the joint mechanical properties.

• In fact, higher traverse speeds increase the hardness of the TMAZ/ HAZ and nugget, resulting better mechanical properties, however, a kissing bond defect was observed showing that the pin length should be increased for higher manufacturing conditions.

• The optimum welding speed parameters in the range of the tested parameters for achieving the highest yield strength corresponding the coldest weld are 750 rpm and 450 mm min -1 . Fig. 15. Cross-sectional macrographs of joints produced by tool F with parameters of Table The tunneling defect was found in only joint F6 with the highest rational speed (1000 rpm) and highest traverse speed (450 mm min -1 ). Fig. 16. Two-dimensional hardness map across the ND-CW section through the weld joints of samples F1, F2 and F4. A higher welding speed increases the general hardness of the joint.

Table 6

Microhardness of samples F1, F2 and F4 at different locations of the joint. 3. Parameters 750 rpm and 450 mm min -1 (sample F4) produced the joint with highest yield strength.

Fig. 1 .

 1 Fig. 1. Fixture designed for the repeatability of the FSWed joints. The welding plates are fixed by stoppers, supporters, and clamps in the directions WD, CWD, and ND respectively.

Fig. 2 .

 2 Fig. 2. Different tool designs: (A) flat shoulder with conical pin, (B) raised spiral shoulder with stepped conical pin, (C) raised fan shoulder with stepped conical pin, (D) raised fan shoulder with threaded cylindrical pin, (E) raised fan shoulder with cylindrical pin, (F) raised fan shoulder with tapered cylindrical pin, and (G) raised fan shoulder with cubic pin.

Fig. 3 .

 3 Fig. 3. Experimental dissimilar friction stir welding process setup: (a) plates size and locations of the samples extraction and (b) geometry of the dog-bone tensile specimens; all the dimensions are in millimeters.

Fig. 4 .

 4 Fig. 4. Surface morphology related to bead on plate of AA2024 material produced by tools A, B, and C. The raised spiral (tool B) and raised fan (tool C) shoulders prevent weld flashes.

Fig. 5 .

 5 Fig. 5. Defects produced in the bead-on-plate welds: (a) tunneling defect (tool A, advancing side in the middle), (b) LOP defect (tool A) and (c) without defect (tool E, advancing side).

Fig. 6 .

 6 Fig. 6. Cross-section observations of samples E1, F1, and G1. (a) Nugget region delimited by yellow solid lines on CWD-ND cross-section where the different colors resulting from etching are denoted zones Z1, Z2, and Z3. (b) Corresponding 2D microhardness maps showing the minimal hardness values in Z2 zone, shoulder affected area, and TMAZ/HAZ on AA2198 side.

Fig. 7 .

 7 Fig. 7. Micro-hardness distributions performed at mid-thickness (middle) of different joint variants, (a) sample E1, (b) sample F1 and (c) sample G1.

Fig. 8 .

 8 Fig. 8. Cross-section optical metallography images of samples F1, (a) Z1/Z2 boundary, (b) inside Z2 and (c) inside Z3. Zone Z2 and Z3 have coarser grain sizes than zone Z1, while there is no significant grain size difference between zones Z2 and Z3.

Fig. 9 .

 9 Fig. 9. SEM micrographs of Z1, Z2 and Z3 zones and the related EDS maps for elements Mg and Cu. (a) Zone Z1, (b) Zone Z2 and (c) Zone Z3. Zone Z1 has different precipitates in comparison with Zone Z2 and Z3.

Fig. 10 .

 10 Fig. 10. DSC and temperature measurements, (a) DSC result of base metal AA2198-T3 and (b) temperature measurement of TMAZ on both joint sides.

Fig. 11 .

 11 Fig. 11. Mechanical properties of FSW joints produced by tools E, F and G at 750 rpm and 50 mm min -1 . The joint produced with tool F demonstrated higher mechanical strength and elongation.

Fig. 12 .

 12 Fig. 12. Local in-plane strain distribution obtained with DIC for joints E1, F1 and G1. The strain localization for all tools are in TMAZ/HAZ of AA2198 and center of nugget.

Fig. 13 .

 13 Fig. 13. Fracture locations after monotonic tensile tests: (a) sample F1, middle of the joint, or (b) sample G1 on the advancing side in the TMAZ. The fracture locations correspond to the locations with the lowest hardness values in Fig. 6b and strain localization in Fig. 12.

Fig. 14 .

 14 Fig. 14. Typical fractography micrographs observed by SEM after monotonic tensile tests: (a) sample F1 with a fracture in the middle of the joint, (b) higher magnification from (a), (c) sample G1 with a fracture on the advancing side in the TMAZ of AA2198, and (d) higher magnification from (c).

Fig. 17 .

 17 Fig. 17. Mechanical properties of FSW joints produced by tool F according to welding speed parameters in Table3. Parameters 750 rpm and 450 mm min -1 (sample F4) produced the joint with highest yield strength.

Table 1

 1 Chemical compositions of AA2024 and AA2198 alloys in wt.%[START_REF] Bussu | The role of residual stress and heat affected zone properties on fatigue crack propagation in friction stir welded 2024-t351 aluminium joints[END_REF][START_REF] Chen | Plastic flow and ductile rupture of a 2198 Al-Cu-Li aluminum alloy[END_REF].

	Alloy	Cu	Li	Mg	Ag	Mn	Fe	Zn	Si	Ti	Al
	AA2024	3.8-4.9	-	1.2-1.8	-	0.3-0.9	⩽0.5	0.2	⩽0.5	0.15	Bal.
	AA2198	2.9-3.5	0.8-1.1	0.25-0.8	0.1-0.5	⩽0.5	⩽0.01	⩽0.35	⩽0.08	-	Bal.

Table 2

 2 Selected dimensions for FSW tool design (*PT = plate thickness = 3.2 mm).

	Parameter	Suggested formula Ref.	Used value	Types
				(mm)
	Shoulder diameter 2.2PT + 7.3	[45] 15.8	All tools
	Pin length	PT-0.3	[46] 2.7	All tools
	Max-pin diameter	PT	[23] 3.2	A, B and C
		0.8PT + 2.2	[45] 4.7	D, E, F and G
	Min-pin diameter	0.5PT	-	1.6	A, B and C
		PT	-	3.2	F

Table 3

 3 Welding speed parameters.

	Sample code	Rotational speed	Traverse speed	Heat input index
	F1	(ω) rpm 750	(ν) mm min -1 50	10 × -4 (HI) 1.1250 ω ν 2
	F2	750	150	0.3750
	F3	750	300	0.1875
	F4	750	450	0.1250
	F5	1000	300	0.3333
	F6	1000	450	0.2222

Table 5

 5 Dynamic volume of tool pins. Tool rotation creates dynamic volume e.g. cubic tool pin transforms to a cylindrical tool pin by rotation.

	Tool code	Dynamic pin volume (mm 3 )
	Tool E	47.7
	Tool F	40.0
	Tool G	47.7

Acknowledgments

We would like to express our appreciation to Hossein Monajati for his help during this research. Without his valuable assistance, this work would not have been completed. A part of the research presented in this paper was financed by the Fonds de recherche du Qubec -Nature et technologies through the Aluminum Research Centre -REGAL.


L Duigou

E M Daya

M Potier-Ferry
email: michel.potier-ferry@univ-metz.fr

Equivalent stiffness and damping of sandwich piezoelectric beams submitted to active control

In this paper, a modal approach is realized to define the damping properties of piezoelectric/elastic/piezoelectric beams. This method is based on the classical laminated beam theory and some simple assumptions about the electric fields. This leads to an electromechanical beam constitutive law. The piezoelectric layers play the role of sensor and actuator and two feedback control laws are considered. The equivalent stiffness, eigenfrequencies and loss factors of the whole system beam/control device are obtained.

Introduction

It is well known that piezoelectric materials can provide an effective way to reduce vibrations and noise. Thanks to this property, they are actually added to already existing structures in order to damp and control vibrations at low frequencies in many areas, such as in the aerospace industry. Thus, they can recover completely (layer) or partially (patch) the host structure. Many theoretical and experimental studies have been devoted to get accurate modelling and analysis of sandwich structures with piezoelectric layers such as beams or plates [START_REF] Benjeddou | Free vibrations of simply-supported piezoelectric adaptative plate: an exact sandwich formulation Thin[END_REF][START_REF] Benjeddou | A unified beam finite element model for extension and shear piezoelectric actuation mechanisms[END_REF][START_REF] Fernandes | An accurate modelling of piezoelectric multi-layer plates[END_REF][START_REF] Gopinathan | A review and critique of theories for piezoelectric laminates Smart[END_REF][START_REF] Trindade | Hybrid active-passive damping treatments using viscoelastic and piezoelectric materials: review and assessment[END_REF][START_REF] Zhang | Formulation of an adaptive sandwich beam Smart[END_REF]. Also, the piezoelectric materials can be coupled with viscoelastic damped structures in order to yield hybrid active-passive damping. Indeed, the piezoelectric material permits damping the vibration modes at low frequency whereas the viscoelastic one yields to damping the modes at medium and high frequencies. So, a lot of hybrid (active-passive) damping configurations were developed in the literature to find the best solution which is able to decrease vibration amplitudes for a large number of modes. A review and assessment of these configurations can be found in [START_REF] Trindade | Hybrid active-passive damping treatments using viscoelastic and piezoelectric materials: review and assessment[END_REF].

From an engineering point of view, the damping properties can be defined by two modal quantities [START_REF] Duigou | Iterative algorithms for non-linear eigenvalue problems. Application to vibrations of viscoelastic shells[END_REF][START_REF] Geradin | Mechanical Vibrations. Theory and Applications to Structural Dynamics[END_REF][START_REF] Soni | Finite element analysis of viscoelastically damped sandwich structures Shock Vib[END_REF]: the loss factor (damping) and frequency. This approach of the 3 Author to whom any correspondence should be addressed. dynamical responses based on complex eigenfrequencies has been extensively applied in the case of viscoelastic damped sandwich structures and several numerical techniques have been proposed to characterize and to compute the complex eigenpairs [START_REF] Duigou | Iterative algorithms for non-linear eigenvalue problems. Application to vibrations of viscoelastic shells[END_REF][START_REF] Soni | Finite element analysis of viscoelastically damped sandwich structures Shock Vib[END_REF]. To our best knowledge, this simple modal approach has not yet been applied in the case of smart structures, involving piezoelectric laminates and active control.

In this paper, it is established that the modal approach can be useful in the case of elementary control laws. A piezoelectric/elastic/piezoelectric beam is studied. The piezoelectric layers play the role of sensor (s) and actuator (a) (see figure 1). Two feedback control laws are considered. A simple analytical study of this structure is carried out by using the classical laminate theory for the mechanical properties and by assuming a linearly varying electric field. Next, the electromechanical problem is reduced to a purely mechanical problem, which defines the elastic or viscoelastic equivalent stiffness properties. In the case of thin piezoelectric layers and of bending modes, some more simplifications can be introduced.

Another contribution of the present paper is the introduction of equivalent mechanical problems that include both the electromechanical behaviour of the thin piezoelectric layers and the control law. There are purely mechanical models in the literature, but only at a finite element level. Generally the electric quantities are condensed from an electromechanical finite element model as in [START_REF] Benjeddou | A unified beam finite element model for extension and shear piezoelectric actuation mechanisms[END_REF][START_REF] Trindade | Hybrid active-passive damping treatments using viscoelastic and piezoelectric materials: review and assessment[END_REF],. . .. In [START_REF] Galucio | A fractional derivative viscoelastic model for hybrid active-passive damping treatments in time domain-application to sandwich beams[END_REF], an analytical expression of the electric field is inserted into a finite element mechanical model. Here, we perform an analytical condensation of the electric field to get equivalent beam equations that are of elastic or viscoelastic type.

An electromechanical model for piezoelectric laminated beams

In this part, a model for a piezoelectric/elastic/piezoelectric beam is presented. In view of analytical studies, this model is designed to be as simple as possible including Euler-Bernoulli kinematics and the assumption of transverse electric fields. Various levels of approximations are discussed, as well as the treatment of the electrostatic problem in the sensor.

Constitutive law

Piezoelectric materials have the capacity to act, to react to an electric field or to a strain. This results in the following constitutive law:

σ = cε -e t E D = eε + E (1)
where σ , ε, D, and E are the stress tensor, strain tensor, electric displacement vector and electric field vector, respectively. c, e, and are the elasticity matrix, whose coefficients are measured at constant electric field, the piezoelectric matrix, and the dielectric permittivity tensor, whose components are measured at constant strain. Orthotropic piezoelectric materials and an extension mechanism are considered. The constitutive relation can be represented in terms of a reduced number of components with respect to the three Cartesian/orthogonal axes, labelled 1, 2 and 3 of piezoelectric material. We suppose that the piezoelectric layers are thin, the stress tensor is uniaxial and the electric fields D and E parallel to the transverse direction. So, the constitutive law is reduced to [START_REF] Benjeddou | A unified beam finite element model for extension and shear piezoelectric actuation mechanisms[END_REF] (2)

Electric description

The electric field-electric potential relation is the following:

E 3 = - ∂φ ∂z . ( 3
)
We consider that the central layer is conductive with an uniform potential, fixed to zero [START_REF] Gopinathan | A review and critique of theories for piezoelectric laminates Smart[END_REF]. Often in the literature, the potential field is assumed to be linearly varying across the thickness, in accordance with the same variation of the stress and strain [START_REF] Kapuria | An efficient coupled zigzag theory for dynamic analysis of piezoelectric composite and sandwich beams with damping[END_REF][START_REF] Liew | Finite element method for the feedback control of FGM shells in the frequency domain via piezoelectric sensors and actuators[END_REF][START_REF] Maurini | On a model of layered piezoelectric beams including transverse stress effect[END_REF]. It may be asked whether a constant electric field gives a sufficient description in the case of a thin layer. An adimensional variable r (0 r 1) is introduced in piezoelectric layers, for instance by z = hc 2 + h s r in the top layer. So, the electric field is written in terms of two unknowns φ 1 and φ 2 :

E 3 = - 1 h s (φ 1 + φ 2 (1 -2r )) ( 4
)
where φ 1 = φ 1 (x) represents the superficial potential and φ 2 = φ 2 (x) the internal potential variation.

Kinematics

To describe the properties of a sandwich piezoelectric/elastic/piezoelectric beam, we use the classical theory of laminates. The kinematics of the structure is represented by the classical hypothesis of Euler-Bernoulli:

u(x, z) = u(x) -zw (x) w(x, z) = w(x) ε 1 = u -zw .
(5)

Electromechanical law of the sandwich beam

Now a global electromechanical behaviour law is established. In this respect, the definitions of the normal force N i and the bending moment M i of each layer are needed:

N i = Si σ i dS and M i = Si (z -z i )σ i dS i = s, c, a.
Because of the constitutive law (2), the uniaxial stress σ 1i is equal to

σ 1c = E c (u -zw )
for the central elastic layer

σ 1i = c * 11 (u -(z -z i )w
)e * 31 E 3 for the piezoelectric layers (i = a, s). [START_REF] Geradin | Mechanical Vibrations. Theory and Applications to Structural Dynamics[END_REF] In accordance with Euler-Bernoulli kinematics, the beam stresses can be represented by a global normal force N and a global bending moment M. Classically, they are expressed as functions of the normal forces and bending moments of each layer in the following form:

N = N s + N c + N a M = M s + M c + M a -z s N s -z a N a ( 7
)
with z s = hc+hs 2 and z a = -hc+ha 2 . By combining ( 6) and [START_REF] Gopinathan | A review and critique of theories for piezoelectric laminates Smart[END_REF], the global electromechanical constitutive law of the laminate is deduced:

N = (E S) mec u + B mec w + e *
with

(E S) mec = E c S c + c * 11 (S s + S a ) B mec = -c * 11 (z s S s + z a S a ) (E I) mec = E c I c + c * 11 (I s + I a ) + c * 11 (S s z 2 s + S a z 2 a ). (9)
So ( 8) is the global constitutive law, that expresses the beam generalized stresses N and M as functions of the membrane strain u , of the curvature w , of the superficial potentials φ 1s and φ 1a and on two additional quantities φ 2s and φ 2a for the internal variations of the electric field. A few remarks can be made about this constitutive law. First, without electric fields, one recognizes the classical laminated beam theory. Then, B mec represents the membrane/bending coupling term. If the beam is symmetrical (z a = -z s ), B mec is equal to zero, that is to say there is no membrane/bending coupling. Finally, the electric quantities φ 1a , φ 2a , φ 1s and φ 2s can be considered as forcing terms of a classical beam model.

Condensation of electric terms in sensor layer

With the help of the Galerkin method, the superficial potential and internal potential variation of the sensor can be written as functions of strain quantities u and w (see the appendix):

φ s = φ 1s r + φ 2s r (1 -r ) φ 1s = e * 31 * 33 h s (u -z s w ) φ 2s = e * 31 2 * 33 h 2 s w . ( 10
)
Next, we replace φ 1s and φ 2s by their expression [START_REF] Maurini | On a model of layered piezoelectric beams including transverse stress effect[END_REF] in [START_REF] Kapuria | An efficient coupled zigzag theory for dynamic analysis of piezoelectric composite and sandwich beams with damping[END_REF]. So, we obtain a new constitutive law, which takes account of electric boundary conditions in the sensor:

N = (E S) eq u + B eq w -e * 31 S a h a φ 1a M = B eq u + (E I) eq w + e * 31 S a z a h a φ 1a + 1 12 φ 2a (11)
with

(E S) eq = (E S) mec + e * 2 31 * 33 S s B eq = B mec - e * 2 31 * 33 z s S s (E I) eq = (E I) mec + e * 2 31 * 33 S s z 2 s - h 2 s 12 . ( 12
)
So the electric behaviour of the sensor modifies the apparent stiffness of the beam, by increasing the membrane stiffness E S and the bending stiffness E I. Considering the expression of B eq , one sees that this electric condensation induces a membrane bending coupling, even with a symmetric beam. This dissymmetry of the laminate behaviour does not arise from a material dissymmetry, but from the difference of electric boundary conditions between the sensor and the actuator.

What approximations are needed in the case of thin piezoelectric layers?

To simplify this discussion, let us consider a symmetrical beam:

h a = h s , S a = S s , z a = -z s . ( 13
)
Then, the term B mec in the constitutive law becomes zero and the others are rewritten in the following form: (14) By comparison with the one-layer beam, we want to evaluate the extra terms due to the direct effect of the piezoelectric layers or those due to the electrostatic condensation in the sensor. To simplify, a rectangular symmetric cross-section is considered. This leads to

(E S) eq = E c S c + 2c * 11 S s + e *
I j → h 3 j 12 S j → h j . ( 15
)
Then, three adimensional parameters are introduced:

r E = c * 11 E c , r p E = e * 2 31 c * 11 * 33 , r h = h s h c ( 16
)
where r E and r p E represent material ratios and r h a thickness ratio.

First, the second term of (E S) eq represents the direct contribution of the piezoelectric layers to the stiffness, and the last term the effect of the electrostatic condensation. Using the hypotheses (15) and ( 16), the following equivalences can be written:

2c

* 11 S s E c S c 2r E r h e * 2 31 * 33 S s (2c * 11 S s ) = r p E /2. ( 17
)
In order to evaluate the adimensional terms, let us consider the sandwich beam studied by Zhang and Sun [START_REF] Zhang | Formulation of an adaptive sandwich beam Smart[END_REF]. It is constituted by PZT5H material in the skins and aluminium in the core. Its geometric characteristics are L = 10 cm, h c = 1.6 cm and h s = h a = 0.1 cm. With these data, the approximated values of the adimensional ratios are

r E 1, r p E 0.5, r h 6 × 10 -2 . ( 18
)
The material ratios (r E , r p E ) are neither small nor big, whereas the thickness ratio is sufficiently little. Therefore, the second and third terms of (E S) eq represent respectively 12% and 3% of the first one. So, they cannot be neglected.

The same study is realized for (E I) eq . In this case, the terms due to the direct effect of piezoelectric layers 2c

E c I c = 2r E (r 3 h + 3r h (1 + r h ) 2 ) e * 2 31 * 33 E c I c S s z 2 s - h 2 s 12 = r p E r E r h (3(1 + r h ) 2 -r 2 h ). ( 19
)
Considering again the sandwich beam studied by Zhang and Sun, these terms are respectively about 36% and 10%. But, they can be simplified by neglecting the terms involving r 3 h and even r 2 h . Two terms are functions of r 3 h : the bending stiffness of the piezoelectric layers 2r E r 3 h and the quadratic term of the potential -r p E r E r 3 h . These last terms represent respectively 0.04% and 0.01% of E c I c . Therefore, two more assumptions can be introduced to get a simpler model. First, the stress in the skins can be reduced to the membrane contribution (i.e., the bending moments M s and M a can be neglected). Second, the quadratic term φ 2s in the electric potential has no influence on the global behaviour and can be dropped. This means that the assumption of constant electric fields is relevant for thin skins (r h lower than 0.2). In what follows, we limit ourselves to this simplified analysis.

A simplified analysis for bending modes

In this part, a new simplification is introduced to analyse the bending modes of symmetrical sandwich beams. We assume further:

u ≈ 0 and |u | z s |w |. ( 20
)
This means that the membrane/bending coupling is neglected and that the bending constitutive law is reduced to

M = (E I) eq w -e * 31 S s z s h s φ 1a (21)
with (E I) eq defined in (14). Let us recall that φ 2a has been dropped according the argument given in section 2.6. At this level, feedback control laws will be introduced, by connecting the two potentials in the piezoelectric layers. These control laws will be condensed, to get a global bending law, including the three layers of the laminate and the control device.

Direct proportional feedback control

With this control law, the superficial potential applied to the actuator is directly proportional to the superficial potential of the sensor:

φ 1a (x) = G c φ 1s (x) (22)
where G c represents the control gain.

According to the assumptions (20) and φ 2a = 0 and to the law (22), the superficial potential becomes

φ 1a = -G c e * 31 * 33
h s z s w . ( 23
)
By combining ( 21) and ( 23), a sort of equivalent elastic constitutive law for the whole system (beam and control device) is obtained. This yields a classical form for the elastic bending of beams, involving a simplified expression of the system bending stiffness (E I) simp :

M = (E I) simp w (E I) simp = (E I) eq + G c S s e * 2 31 * 33 z 2 s . ( 24
)
Therefore, a symmetrical piezoelectric/elastic/piezoelectric beam can be represented by an equivalent homogeneous beam, whose stiffness matrix increases with the control gain G c . One remarks that the increase of stiffness depends on the following parameter:

g = G c S s e * 2 31 * 33 (25)
that is a function of the control gain and of the dielectric and piezoelectric constants. It could be also convenient to consider a non-dimensional constant r control as follows:

r control = G c S s E c I c e * 2 31 * 33 Z 2 s = gZ 2 s E c I c . ( 26
)
So the simplified system bending stiffness can be rewritten in a more convenient form, where (E I) eq has been defined in [START_REF] Trindade | Hybrid active-passive damping treatments using viscoelastic and piezoelectric materials: review and assessment[END_REF] and ( 14):

(E I) simp = (E I) eq + gZ 2 s = (E c I c ) (E I) eq E c I c + r control .
The balance of momentum leads to the following equation:

M + (ρ S) eq ẅ = 0 ( 27
)
where (ρ S) eq is the equivalent linear density of the sandwich beam:

(ρ S) eq = ρ s S s + ρ c S c + ρ a S a . ( 28
)
The modal solutions of (24) and ( 27) are classical [START_REF] Geradin | Mechanical Vibrations. Theory and Applications to Structural Dynamics[END_REF], and the nth eigenfrequency is given by

ω 2 n = (µ n ) 4 (E I) simp (ρ S) eq L 4 ( 29
)
where the non-dimensional numbers µ n depend on the boundary conditions. For instance, for a bi-articulated beam, µ n = nπ. So the control law (22) leads only to an increase of the global stiffness of the beam (if G c is positive) and therefore of the eigenfrequencies, that are given by ( 24), (28), and (29), for any boundary condition. Such a law does not induce damping, because the eigenfrequencies (29) are real.

This increase of stiffness can be characterized by the parameter g (see (25)) or by the adimensional parameter r control , which is a function of the control gain and of the material and geometrical parameters of the laminate.

Direct and proportional velocity feedback control

In the case of the direct and proportional velocity feedback control, the potential applied to the actuator is proportional to the time derivative of the sensor potential:

φ a (x) = G c φs (x) ( 30
)
with G c positive. As previously, the electric field is assumed to be constant in each piezoelectric layer. Using [START_REF] Maurini | On a model of layered piezoelectric beams including transverse stress effect[END_REF], (20) and considering φ 2s = 0 (see section 2.6), equation (30) leads to

φ 1a = -G c e * 31 * 33 h s z s ẇ . ( 31
)
As before, this φ 1a is replaced in the expression of bending moment (21) and we obtain

M = (E I) eq w + gz 2 s ẇ (32)
where the parameter g represents the term of control and is defined as previously (25). This relation ( 32) is a sort of viscoelastic bending law. The viscoelasticity is of Kelvin type and it is due only to the control law (30).

Considering a problem of free vibrations, the deflection w(x) can be sought in the form w = W (x)e iωt and the constitutive law (32) is rewritten in the following form:

M = [(E I) eq + iωgz 2 s ]W . ( 33
)
Therefore, M is defined like M = E I (ω)w . This corresponds to a complex stiffness matrix linearly dependent on ω. So the behaviour of the whole system is equivalent to that of a viscoelastic beam and a loss factor can be defined for each vibration mode. Using the equivalent constitutive law (32), the dynamic equation ( 27) leads to

W (4) -(ρ S) eq ω 2 E I (ω) W = 0. ( 34
)
From Geradin [START_REF] Geradin | Mechanical Vibrations. Theory and Applications to Structural Dynamics[END_REF], the ratio L 4 (ρ S ) eq ω 2 /E I (ω) is real and depends only on the boundary conditions and on the mode. These values are exactly the adimensional numbers µ n , recalled previously. Because E I (ω) is complex, the solution ω of the free vibration problem is complex and a damping can be defined by the following equation in ω:

ω 2 n = µ 4 n (ρ S ) eq [(E I) eq + iωgz 2 s ]. (35)
To simplify, the eigenfrequency of a beam only constituted by the elastic core is introduced as a reference:

ω 2 nc = µ 4 n E c I c ρ c S c L 4 . ( 36
)
So, equation (35) becomes

λ 2 n -iλ n β n -1 = 0 λ n = ω ω nc β n = g z 2 s E c I c ω nc = r control ω nc (37)
where r control is defined as in (26). Note that, in the present case, r control is not adimensional, because of a different definition of the control gain. The solution of this equation is

λ n = iβ n ± 4 -β 2 n 2 . ( 38
)
The damping η n is defined classically by

ω 2 n = 2 n (1 + iη n ),
where n represents the real frequency. Here, it is equal to

η n = β n 4 -β 2 n 2 -β 2 n ( 39
)
with β n defined in (37).

If β n 1, the damping is about β n ; that is to say,

η n = G c ω nc S s z 2 s I c e * 2 31 E c * 33 = r control ω nc . ( 40
)
In any case, each modal damping η n depends only on a adimensional parameter β n = r control ω nc , a function of a parameter r control similar to that given above and of the reference frequency ω nc . This second type of control law induces a damping that is proportional to G c , but does not modify the eigenfrequency very much.

Are the problems well posed?

With the first control law (22), the resulting equations ( 24), ( 27) are those of an elastic beam. This equation is known to be well posed, at least if the equivalent modulus (E I) simp is positive. So the gain can be positive or negative and not too large. The well posed condition is

r control - (E I) eq E c I c . ( 41
)
With the control law (30), one gets the equations of a Kelvin viscoelastic beam ( 27), (32), that are well posed only when the gain is not negative. These equivalent mechanical models have never been presented in the literature. This result has consequences on the relevant discretization techniques. For instance, the control term in (32) leads to a parabolic equation so that the time derivative must be discretized in an implicit way to get a stable scheme.

A second analysis taking into account the axial terms

In this last part, we study the bending modes of a sandwich beam with active control but without neglecting the axial strain and axial inertia term as in the previous section. This will permit us to define the limits of this simplified analysis. Nevertheless, the assumption of a constant electric field per layer (φ 2s = φ 2a = 0) will be kept, because it has been established in section 2. Within beam theory, the balance of momentum leads to

-N + (ρ S) eq ü = 0 M + (ρ S) eq ẅ = 0 ( 42
)
where M and N are defined by the constitutive laws ( 11), ( 14) and ( 15) and (ρ S) eq by ( 28).

The two previous control laws will be considered again. For simplicity, the analysis is limited to a simply support beam (N = M = w = 0 at x = 0, L).

Direct proportional feedback control

With the direct proportional feedback control law, the equivalent constitutive law of the system is defined from ( 8), ( 22) and ( 10):

N M = (E S) eq -g B eq + gz s B eq -gz s (E I) eq + gz 2 s u w ( 43
)
where g represents the term of control and is defined by (25).

As compared with (24), this constitutive law of the system couples membrane and bending quantities. The constitutive matrix depends on the control parameter g. Note that it is not symmetric, which is due to the non-conservative character of the control law.

The solution of the system (42), ( 43) is sought in the following form:

u = q 1 cos nπ x L e iωt w = q 2 sin nπ x L e iωt .
(44)

Effect of membrane/bending coupling.

To study the effect of membrane/bending coupling, a membrane static condensation is considered in the first equation of (42). Thus, q 1 can be expressed according to q 2 and it is replaced in the second equation of (42). This permits us to obtain the bending frequency, which is equal to

ω 2 n = n 4 π 4 (ρ S) eq L 4 -(B 2 eq -g 2 z 2 s ) (E S) eq -g + (E I) eq + gz 2 s . (45)
As in the previous study, with this law, the frequency is real and there is no damping. Compared to (29), ω 2 n of (45) includes an extra term because of taking into account the membrane motion. This term is compared to the bending stiffness (E c I c ) of the central layer:

-(B 2 eq -g 2 z 2 s ) [(E S) eq -g]E c I c = - 3r 2 E r p2 E r 2 h (1 + r h ) 2 (1 -G 2 c ) 1 + 2r E r h + r E r p E r h (1 -G c ) = 3 × 10 -3 G 2 c -1 1 -G c /38 . ( 46
)
The last equality holds with the numerical values presented in section 2. Hence the influence of the axial strain is small (<0.3%) for a gain lower than 1. From (46) and figure 2, it appears that this coupling term can be neglected for G c lower than 3.

Effect of the membrane inertia.

Here equations ( 42) and ( 43) are solved without any simplification. This yields, by letting k = nπ/L,

[(E S) eq -g]k 2 -(ρ S) eq ω 2 (B eq + gz s )k 3 (B eq -gz s )k 3 [(E I ) eq + gz 2 s ]k 4 -(ρ S) eq ω 2 × q 1 q 2 = 0 0 . ( 47
)
Thus, the eigenfrequencies are given by an equation of the second degree with (ρ S) eq ω 2 as unknown. Here we only discuss in which conditions the axial inertia term is small. So it is sufficient to compare the first and the last term of the (1, 1) term of the matrix (47). Because only an order of magnitude is sought, we suppose ω = ω nc , (ρS) eq = ρ c S c , (E S) eq = E c S c . (48) Also, the expression (36) of ω nc and hypothesis (15) are considered. Then, the ratio (ρ S) eq ω 2 /(E S) eq k 2 becomes

(ρ S) eq ω 2 (E S) eq k 2 n 2 π 2 I c L 2 S c n 2 π 2 12 h c L 2 . ( 49
)
This ratio depends on the mode (n 2 ) and on the beam slenderness (h c /L). So, the effect of the membrane inertia can be neglected for an aspect ratio L/nh c greater than 10.

Direct and proportional velocity feedback control

In this case, the two potentials φ a and φ s are related by the law (30). Because of [START_REF] Maurini | On a model of layered piezoelectric beams including transverse stress effect[END_REF], the constitutive law of the whole system is then expressed in the following form:

N M = (E S) eq B eq B eq (E I) eq u w + -g gz s -gz s gz 2 s u ẇ (
50) where g, always given by (25), represents the control term. Once more, this global law is viscoelastic of Kelvin type. Note that the damping matrix is not symmetric.

The analysis of the vibration modes of a bi-articulated beam is similar to that in 4.1.

Here, we limit our approach by using a static approximation in the axial equation. Then, the second equation of system (42) is written as

-(ρ S) eq ω 2 + iωgz s n 4 π 4 L 4
B eq (E S) eq + z s

+ n 4 π 4 L 4 - B 2
eq (E S) eq + (E I) eq = 0.

(51)

The solution ω 2 of this last equation is complex. That is to say, a damping can be defined. Compared to (34), equation (51

)
includes two extra terms ( Beq (E S)eq ) and (-

B 2
eq (E S)eq ) due to the membrane strain. To obtain the damping, equation (51) must be solved:

λ 2 n -iλ n β n -γ = 0 λ n = ω ω nc β n = G c ω nc e * 2 31 E c * 33 S s z s I c B eq E c S c + z s γ = 1 - B 2 eq E 2 c S c I c . ( 52
)
The solution of equation ( 52) is

λ n = iβ n ± 4γ -β 2 n 2 . ( 53
)
The damping is defined as previously, and is equal to

η n = β n 4γ -β 2 n 2γ -β 2 n . ( 54
)
If β n 1 and γ 1, the damping is about β n ; that is to say,

η n β n = 3G c ω nc r h r E r p E (1 + r h ) 2 (1 -r h r E r p E ). ( 55
)
With the numerical data (18) for r E and r p E , it becomes

η n 3 2 G c ω nc r h (1 + r h ) 2 1 - r h 2 . ( 56
)
In the case of thin piezoelectric layers (r h 1, then β n 1 and γ 1) the damping (56) is consistent with the values (40) that had been found in the previous section. This establishes than the membrane strains can be neglected for thin skins. Indeed, these comments are in good agreement with the results presented in figure 3. In this, the results from formula (39) and those from (54) are reported for comparison.

Are the problems well posed?

It is not so obvious to decide if the equations (42), (43) lead to a well posed problem, because the stiffness matrix in (43) is not symmetric. Nevertheless the elastic term does not satisfy the ellipticity conditions if the gain g is sufficiently large and especially when g > (E S) eq or g < - In the case of a damping control, the damping matrix in (50) never has a definite sign. So the dynamic problem (42), ( 50) is not well posed, whatever the gain. There is a spillover phenomenon in the membrane problem if G c is positive and in the bending problem if G c is negative. Thus, it would be dangerous to solve numerically the complete dynamical problems established in this part, as well as the electromechanical problems of section 2 associated with the presented control laws. Nevertheless, one can find such calculations in the recent literature. Note that the modal analysis of this paper remains consistent, because it is restricted to few modes, while the instabilities and the ill posed problems are related to high eigenfrequencies.

(E I) eq z 2 s . ( 57
)

Conclusion

In this paper, simple models and analytical modal solutions for piezoelastic/elastic/piezoelastic beams have been presented. Two feedback control laws have been considered and constitutive laws of the whole system beam-control device have been established. These global laws can be of elastic type (conservative or not) or of viscoelastic type.

In the case of thin piezoelectric layers, a pure bending model is sufficient to define bending vibration modes. Within this very simple modelling, the behaviour of a system is summarized by a stiffness bending modulus, that depends on the geometrical and material data of the laminate and on the control gain. The latter modulus can be real or complex, according to the type of control law. This means that the control can modify the stiffness of the system or induce some damping. Such a very simplified analysis can be useful to describe more complicated systems such as hybrid passiveactive damping, laminated plates or more intricate control devices.

Furthermore, a complex modal approach has been considered to characterize active damped systems. This method had been widely applied for viscoelastic damping. Within this method, each complex mode is associated with a complex eigenvalue and this defines an eigenfrequency and a loss factor per mode. In our opinion, the modal approach should be more extensively used in the case of piezoelectric laminates and active control. It permits one to simply characterized the damping properties of an active system, independently of the applied loads.

Appendix: approximate analysis of the electric problem

The electric problem in the sensor is studied according to the assumptions described in section 2. It is reduced to a transversal problem, associating the constitutive law (8) and the following ones:

∂ D z ∂z = 0, φ( r = 0) = 0, D z (r = 1) = 0 (A.1)
where the adimensional variable r (r = 0 refers to the bottom layer and r = 1 refers to the top layer) is used. This assumes that the central layer is conductive and that the electric current due to the central system is neglected. The variational form of (A.1) is It must be noted that φ 2 is proportional to h 2 s . To know if the potential φ 2 has an important part in the bending stiffness (E I) eq , we look at the contribution of the sensor in the bending moment, M s , which is defined by where s denotes the sensor. The last term of (A.8), which comes from the quadratic term φ 2 , is function of h 2 s . That is why it can be neglected for thin skins, as explained in section 2.6.

M s = - 1 0 σ 1 z dz.

Figure 1 .

 1 Figure 1. The piezoelectric/elastic/piezoelectric beam and notations.

Figure 2 .

 2 Figure 2. Modal frequency ω n as function of the control gain constant G c . Simply supported beam. Study of the first vibration mode. Direct proportional feedback control. (×) results from formula (29), (continuous line) results from formula (45).

Figure 3 .

 3 Figure 3. Modal loss factor η n as a function of the control gain constant G c . Simply supported beam. Study of the first vibration mode. Direct and proportional velocity feedback control. (x) results from formula (39), (continuous line) results from formula (54).

1 0ε 1 1 0 ε 1 ( 1 -

 11111 D z (r )δφ (r ) dr = 0 ( A . 2 ) for any δφ(r ) that is zero for r = 0. With account of the constitutive law(r )δφ (r ) dr. (A.3)Next, we assume that the potential φ(r ) varies in a quadratic manner across the layer and a Galerkin approximation is applied:φ(r ) = φ 1 r + φ 2 r (1r ) δφ(r ) = r and δφ = r 2 .(A.4) So equation (A.3) leads to the following approximated system: 2r ) dr .(A.5)As ε = uzw , the system (A.5) has the following solution:

σ 1

 1 defined in (2) is rewritten with account of expressions (A.6):σ 1 = -c *11 zwpart of (E I) eq due to the sensor, denoted (E I) s , is equal to(E I) s = c *