Morgane Geyer
email: morgane.geyer@mines-albi.fr

Vanessa Vidal

Thomas Pottier

Christine Boher

Farhad Rézaï-Aria

Investigations on the material flow and the role of the resulting hooks on the mechanical behaviour of dissimilar friction stir welded Al2024-T3 to Ti-6Al-4V overlap joints

Keywords:

During friction stir lap joint welding, hooks are systematically formed at the interface of dissimilar Al2024/ Ti-6Al-4V plates. Various hooks dimensions and shapes are observed according to the welding parameters (welding speed, rotational speed and double-pass). Based on digital image correlation (DIC) during tensile shear tests, it is found that hooks features govern the fracture behaviours and, hence, the mechanical properties of the overall welds. In particular, three fracture behaviours are obtained: at the interface, in the Al2024 and across the Ti-6Al-4V plate, which was never reported in dissimilar Al/Ti studies. The aims of this work are (i) to study the influence of FSW parameters on the mechanisms of hooks formation and (ii) to understand the role of both AS and RS hooks on the mechanical behaviour of the welds, by the means of two new approaches. First, hooks are investigated by X-ray tomography which provided some innovative insight into the material flow (3D viewing). Based on these qualitative and quantitative results, hooks are used as tracers and two new scenarios of AS and RS generation are proposed. Then, the contribution of both hooks on the stress distribution is clearly revealed by some calculations of the stresses reached around AS and RS hooks. Contrary to RS hook, AS hook is sharp and, hence, the stress reached at the top of AS hook is characterized by a stress concentration. The results showed that the fracture behaviours are greatly influenced by hooks size and the formation of microstructural defects inside the Ti-6Al-4V, which depend on the welding parameters. In conclusions, this work revealed the relationship between the achievable parameters, the resulting hooks dimensions and the fracture behaviours. In addition, based on the understanding of the material flow, some ways of optimization of the mechanical properties are proposed.

Introduction

One of the major challenges of the aeronautical industry is the weight reduction in order to decrease fuel consumption, greenhouse gas emission and increase the flying range. It represents both environmental and economical issues. To fulfil these requirements, the aircraft manufacturers tend to replace riveting by advanced joining technologies [START_REF] Mendez | Welding processes for aeronautics[END_REF]. In particular, the demand for a suitable joining process for dissimilar materials has increased along with newer designs of multi-materials structures [START_REF] Kulkarni | Friction Stir Welding of Dissimilar Alloys and Materials[END_REF]. By placing the right material at the right place, the advantage of each material could be exploited. Welding aluminium alloys to titanium alloys could be recommended to combine low density and cost with high specific strength and corrosion resistance requirements [START_REF] Polmear | Light Alloys: Metallurgy of the Light Metals[END_REF]. Nevertheless, due to their large difference in physical properties, such as melting point, thermal conductivity and thermal expansion coefficient, the welding of aluminium alloys to titanium alloys constitutes a significant challenge, especially through fusion welding processes [START_REF] Tomashchuk | Direct keyhole laser welding of aluminum alloy aa5754 to titanium alloy ti6al4v[END_REF].

Friction stir welding (FSW) is an emerging solid state joining process. Materials are joined through mechanical deformation by the use of a rotating tool which locally heats the assembly by friction. Large plastic deformations also contribute to the heating and lead materials to a soft state allowing plastic flow around the tool and then create a metallurgical bond [START_REF] Mishra | Friction stir welding and processing[END_REF]. FSW was firstly developed to join precipitation hardening aluminium alloys, considered "unweldable" with conventional fusion welding processes. Currently, FSW tends to be a promising candidate for the production of hybrids structures, as shown by numerous recent studies devoted to the welding of dissimilar alloys [START_REF] Simar | State of the art about dissimilar metal friction stir welding[END_REF] such as aluminium to titanium [START_REF] Kar | Multi-length scale characterization of microstructure evolution and its consequence on mechanical properties in dissimilar friction stir welding of titanium to aluminum[END_REF], aluminium to copper [START_REF] Kush | A review on dissimilar friction stir welding of copper to aluminum: process, properties, and variants[END_REF], aluminium to magnesium [START_REF] Shah | Review of research progress on aluminium to magnesium dissimilar friction stir welding[END_REF], aluminium to steel [START_REF] Wan | Friction stir welding of dissimilar aluminum alloys and steels: a review[END_REF] and magnesium to steel [START_REF] Singh | Parametric effect on dissimilar friction stir welded steel-magnesium alloys joints: a review[END_REF].

FSW of aluminium to titanium is studied in the two main welding configurations: butt and lap joints. In the butt configuration, welds with high joint efficiency can be obtained with a ductile mechanical behaviour. For instance, [START_REF] Dressler | Friction stir welding of titanium alloy tial6v4 to aluminium alloy aa2024-t3[END_REF] successfully welded Al2024-T3 to Ti-6Al-4V. They obtained a joint with 73% ultimate tensile strength of the base Al2024-T3 which failed at the interface. [START_REF] Wu | Interface and properties of the friction stir welded joints of titanium alloy ti6al4v with aluminum alloy 6061[END_REF] also achieved an optimized Al6061-T6/Ti-6Al-4V butt weld which reached 68% of Al6061-T6 base metal and failed in the thermomechanical affected zone (TMAZ). They found that mechanical properties and fracture location depend on the characteristics of the intermetallics compounds (IMC) layer formed at the interface. In the lap configuration, mechanical properties of welds are generally lower than butt joints and mechanical behaviour is systematically found to be brittle [START_REF] Zhou | Dissimilar friction stir welding of aa6061 and ti6al4v alloys: a study on microstructure and mechanical properties[END_REF][START_REF] Picot | Influence of friction stir welding parameters on titanium-aluminum heterogeneous lap joining configuration[END_REF]. The authors also studied the influence of FSW parameters on interfacial microstructure and the mechanical properties. [START_REF] Yu | Influence of welding parameters on interface evolution and mechanical properties of FSW Al/Ti lap joints[END_REF] showed that two types of Al/Ti interface can be produced: a diffusive or a mixed interface, for low or high heat input respectively. However, regardless of the interfacial microstructure, all welds fail along the interface. Therefore, interfacial microstructure does not seem to correlate with failure and other influential parameter shall be investigated. Contrary to butt joints, lap joints are very sensitive to material flow which leads to the formation of two hooks on both sides of the stir zone. In particular, [START_REF] Yu | Influence of welding parameters on interface evolution and mechanical properties of FSW Al/Ti lap joints[END_REF] observed hooks formation under several welding conditions and do not assess their contribution on the mechanical behaviour of welds despite the fact it is commonly acknowledged that the mechanical strength of FSW lap joints strongly depends on the characteristics of the hooks [START_REF] Liu | The effect of interface defect on mechanical properties and its formation mechanism in friction stir lap welded joints of aluminum alloys[END_REF]. Few studies about Al/Ti friction stir lap welding (FSLW) investigated the impact of hooks on mechanical properties.

Large hooks are generally detrimental to the mechanical properties of lap joints due to the reduction of the effective top plate thickness which induces the decrease of load-carrying capacity of the welds [START_REF] Yang | Effect of tool geometry and process condition on static strength of a magnesium friction stir lap linear weld[END_REF]. In addition, [START_REF] Wang | Effect of hook characteristics on the fracture behaviour of dissimilar friction stir welded aluminium alloy and mild steel sheets[END_REF] reported three different fracture locations in Al6022/mild steel lap joints that were governed by the relative size of hooks on both advancing side (AS) and retreating side (RS). When the height of the RS hook exceeds that of the AS hook, welds failure is observed in the base mild steel and lead to an optimal joint strength. Hence, optimization of mechanical properties of lap joints requires to control the dimensions of the hooks which are determined by the material flow. Although studies showed that size of hooks increases with the reduction of the welding speed and the increase of the rotational speed [START_REF] Dubourg | Process optimisation and mechanical properties of friction stir lap welds of 7075-t6 stringers on 2024-t3 skin[END_REF], the mechanisms of AS and RS hooks generation need to be further investigated. Besides, most studies characterize hooks from a single cross section. However, FSW is a continuous deformation process and hooks dimensions and shape vary along the weld joint. 3D-techniques would be more relevant since they provide a valuable insight to the material flow. Such 3D representations have not been reported to characterize hooks. Finally, even though it is well known that lap joints are very sensitive to the characteristics of hooks, the contribution of AS and RS hooks on stress distribution remains unclear.

The aim of this work is to study the influence of the welding parameters on the material flow and the role of the resulting hooks on the mechanical behaviour of the welds. Dissimilar FSW lap joints between Al2024-T3 and Ti-6Al-4V alloys are produced at different welding and rotational speeds. In addition, one weld is performed in double-pass. Lap joints are investigated by X-ray computed tomography in order to assess the influence of the process parameters on each hook shape and size. Microstructure is investigated by scanning electron microscopy (SEM). The contribution of hooks and microstructure evolutions is assessed through digital image correlation (DIC) during tensile shear tests. The influence of hooks features and damage mechanisms of Ti-6Al-4V are found critical to determine the mechanical properties.

Experimental

Materials and processings

Dissimilar Alclad 2024-T3 aluminium alloy (1.8 mm thickness) and Ti-6Al-4V titanium alloy (0.8 mm thickness) were friction stir welded in a lap configuration. Chemical compositions of these alloys are presented in Table 1.

FSW experiments were conducted using a FANUC M900iB/700 robot at Institut Maupertuis (Bruz, France). This equipment is able to weld up to 10 kN of axial forging force. Each investigated welds were done using force control to ensure a good repeatability of the process (±50 N). To avoid overheating of Al2024 and in order to prevent an excessive wear of the tool, the aluminium alloy plate was placed atop the titanium alloy plate. The advancing side was placed on the side of the aluminium alloy plate free edge of the joint. A schematic illustration of the set up is depicted in Fig. 1. The welds were performed using a tool with a 2.0 mm height conical pin. The pin entirely penetrates through the Al2024 plate all the way to the upper surface of the Ti-6Al-4V plate. The purpose being to enable the mixture of the two alloys.

In terms of welding parameters, values of rotational speeds ranged from 500 to 1500 rpm, welding speeds ranged from 60 to 240 mm/ min, while the tilt angle was kept constant at 2°. Two series of welds were produced. The first of these was made with a compressive load of 8250 N. Some welds could not be performed due to improper welding conditions leading the tool to plunge through the plates and perforate the Ti-6Al-4V plate. The second serie was made with a lower force of 7000 N in order reduce the heat input and enables the production of the previously unachieved welds. In addition to these two series, one weld was performed in double-pass. Table 2 lists the welding parameters and specifies the welding conditions leading to an excessive heat input.

Welding characterization

Tomographic analysis

The material flow at the interface between the two alloys is investigated through X-ray computed tomography (CT-scanning). EasyTom 130 tomograph is used for this purpose. The analysed specimen consist in 20 mm wide tensile test coupons cut from the above mentioned samples. The X-ray voltage is set to 113 kV and the power source to 233 μA. Scan resolution, in terms of voxel size, is 20 μm. A serie of 1000 slices in the welding direction is obtained for each sample. Two image processing programs are developed in order to assess both qualitatively and quantitatively the influence of the welding parameters on the material flow. Thanks to the chemical contrast between aluminium and titanium alloys, their X-ray absorption differ. As a result, greyscale images are obtained where aluminium and titanium alloys can be distinguished (Fig. 2a). Segmentation is firstly performed to obtain binary images (Fig. 2b), the titanium alloy can therefore be visualized, alone, from thresholding. Grey threshold values, used for this image processing, were determined for each slice by automatically considering the differences in X-ray exposures of successive slices. This is a normal procedure commonly employed in X-ray tomography experiments. This automatic procedure was developed using an in-house Matlab code to obtain 3D reconstruction images. The quantitative analysis was carried out from a second image processing program developed on Aphelion software. From the segmented images (Fig. 2b), typical parts of the material flow were isolated (Fig. 2d-f) and measured (hooks dimensions and area, and area of the Ti fragments and the cavity). The average values for the whole weld are based on image analysis of 200 slices.

Microstructural characterization

Cross sections of welded joints were prepared for metallographic observations. A standard polishing procedure down to 3 μm diamond paste followed by a finishing with 0.05 μm colloidal silica solution was applied. Macrographs for general observations of hooks were obtained with a tabletop scanning electron microscope (SEM) Hitachi TM3030-Plus. Microstructural observations of the titanium alloy near the interface were achieved on samples etched with a weck reagent to reveal the microstructure. Another scanning electron microscope (SEM) No-vaNanoSEM 450 was used to examine the plastic deformation of the Ti-6Al-4V alloy.

Mechanical characterization

The mechanical strength of the welds was measured from tensile shear tests because it is the most widely mechanical test used in the friction stir lap welding (FSLW) literature [START_REF] Cederqvist | Factors affecting the properties of friction stir welded aluminum lap joints[END_REF]. Since no standard exists for testing FSLW, specimens dimensions and test conditions were chosen according to the most commonly used in the literature [START_REF] Chen | Friction stir lap joining aluminum and magnesium alloys[END_REF]. Rectangular lap shear specimens (20 mm width) were water jet cut perpendicular to the welding direction. Spacers were used for weld alignment of specimens in Intron 5800 R tensile machine (Fig. 3). All tests were conducted at room temperature with a constant crosshead speed of 1 mm/min. Considering the specific tensile specimen geometry, no extensometer was used and only the force and the crosshead displacement were recorded. Three specimens for each welding conditions were tested and failure loads were averaged. During these tensile shear tests, the local displacement fields were determined on the cross section of the specimens, using digital image correlations (DIC -VIC-2D). Before testing, the surface of these specimens was painted with black and white sprays in order to exhibit a contrasted random pattern. Strain fields are calculated from spatial derivation of the successive displacement fields measured at a 10Hz frequency.

Results

Weld macrostructure

During friction stir lap welding, the pin (2.0 mm height) penetrates entirely within the Al2024-T3 top plate (1.8 mm thickness) and reaches the upper surface of the Ti-6Al-4V bottom plate. Assuming that plastic strain is volume-conservative, the Ti-6Al-4V is geometrically forced to flow into the aluminium alloy plate on both side of the pin. The combination of this penetration and the rotational material flow around the pin leads to the formation of two Ti-6Al-4V hooks which delimit the stir zone on the advancing side (AS) and the retreating side (RS). Their different aspects are linked with the asymmetrical material flow on the AS and the RS. On RS, hooks are usually larger and have more complex shapes since the material motion and transfer are more significant in RS as demonstrated by [START_REF] Nandan | Three-dimensional heat and material flow during friction stir welding of mild steel[END_REF]. Six typical interface macrographs are presented in Fig. 4. For all welds, AS hooks extend upward and are bent outward the stir zone. These hooks result from the plastic shear strain of Ti-6Al-4V along the conical side of the pin. The accumulation of deformation leads to an outgrowth of the titanium alloy which tears complex and coupled modes of ductile rupture within the aluminium alloy plate. RS hooks also extend upward before bending inward to penetrate the stir zone. This specific shape could reflect excessive plastic deformations. In addition of hooks, Ti-6Al-4V fragments can be observed. They penetrate inside the stir zone near the interface (Fig. 4c) or can even migrate to the middle of the stir zone (Fig. 4b andd). FSW parameters act on hook height and hook shape. In particular, RS hook could be continuous (Fig. 4b-e) or discontinuous (Fig. 4a andf) and present a smooth (Fig. 4a-c) or segmented morphology (Fig. 4d-f).

It is visible from Fig. 4, that the process parameters of FSW impact both the AS and the RS hooks height. However, the 2D observations of a single cross section may be insufficient to assess every aspects and characteristics of the hooks. Hence, successive polishing (Fig. 5) on the same samples may emphasizing the evolution of hooks in terms of height and shape along the joint direction. However, another technique providing 3D visualization, namely X-ray computed tomography, seems more suited to describe the material flow scenario.

Hooks characterization

Fig. 6 shows the post-weld 3D-reconstructed images according to welding and rotational speeds. Images corresponds to the Ti-6Al-4V distribution inside Al2024 and reveal AS hook, RS hook and Ti-6Al-4V fragments. It is worth noticing that because of the scan resolution (voxel size = 20 μm), Ti-6Al-4V fragments with volume lower than 20 3 μm 3 cannot be detected. As shown in Fig. 6, the Ti-6Al-4V is not steadily distributed along the joint. Both AS and RS hooks height and aspect vary along welds. The same statement can be reported for Ti-6Al-4V fragmentation. These observations confirm the interest of using X-ray tomography to characterize more precisely hooks and Ti-6Al-4V fragments which, then, can be used as tracers to investigate the influence of the welding parameters on the material flow.

In addition to the qualitative results, X-ray computed tomography allows a quantitative evaluation of the Ti-6Al-4V distribution. Based on data from the image processing program (refer to Section 2.2.1), several significant relations are noticed between process parameters and characteristics of the material flow (Fig. 7). Firstly, it is found that penetration depth is proportional to the increase of ω 2 /ν (Fig. 7a). This expression corresponds to the definition of the pseudo heat index, widely used in FSW works to describe the heat input provided by the welding parameters [START_REF] Wang | Effect of pin length on hook size and joint properties in friction stir lap welding of 7b04 aluminum alloy[END_REF]. [START_REF] Arbegast | Characteristics of friction stir welded aluminum alloys[END_REF] defined this unitized parameter to describe the FSW parameters dependence on the heat input. They have shown that for several alu minium alloys, a general relationship between maximum temperature and FSW parameters can be established:

(1) where T m is the melting point of the metals, the constant K is ranging from 0.65 and 0.75, α is between 0.04 and 0.06 and ω and ν are respectively the rotational and the welding speeds.

In dissimilar FSLWed Al6061/Ti-6Al-4V, Yu et al. ( 2019) confirmed that the variation of peak temperatures, measured by thermocouples embedded closed to the interface, is also related to ω 2 /ν. Thus, the first trend (Fig. 7a) is a consequence of the force control mode which affects the amount of Ti-6Al-4V extruded. It has a direct impact on AS hook height which also increases with ω 2 /ν, proportionally to penetration depth of the pin on Ti-6Al-4V (Fig. 7b). On RS, a more complex evolution of RS hook height is noted. It increases first until ω 2 /ν ≃ 5000 rad 2 mm -1 s -1 and then declines for higher heat input (Fig. 7c). On the 3D-reconstructed images it can be observed that this trend is related to an intensification of the bending of RS hook as the heat input increases (Fig. 6). It could be attributed to a severe plastic deformation promoted by the loss of the mechanical properties of the Ti-6Al-4V with the increase of the heat input. For the welding condition leading to the highest heat input, the severe plastic deformation of the Ti-6Al-4V could explain the failure of the RS hook all along the weld direction (Fig. 6a).

The heat input also affects Ti-6Al-4V fragmentation (Fig. 6). For the lowest heat input, no fragments is found due to limited pin penetration depth in Ti-6Al-4V (Fig. 6h). The increase of the heat input promotes Ti-6Al-4V displacement inside Al2024. Fragments of Ti-6Al-64V are formed and penetrate inside Al2024 stir zone, preferentially on the AS (Fig. 6e-g). With a further increment of the heat input, and simultaneously with the bending intensification of the RS hook, it can be observed that the fragments remain mainly localized around the interface in between the two hooks (Fig. 6a, b andd).

For the weld performed in double-pass, beam hardening artifacts are obtained and disturb edge detection, making difficult a relevant post-processing (prevent any suitable thresholding between Al2024 and Ti-6Al-4V). However, the partial results from CT-scan suggests that this joint presents larger hooks than the welds produced in single-pass, which was confirmed by the multiple measurements made on cross sections. Hence, the additional deformation provided by the second pass enhances the volume of Ti-6Al-4V being extruded and lead to the largest AS and RS hooks (Fig. 7a andb). x" refer to the welding conditions plotted in Fig. 6.

Ti-6Al-4V deformation

The relative motion of the tool with respect to the metal plates generates complex dynamic solicitations. At the interface of the pin tip and Ti-6Al-4V plate, the sub-surface of Ti-6Al-4V, alike the pin sub-surface, experiences severe shear deformations. Depending on the area in contact with the pin, the 3D-stress state changes. A simplified 2D schema is depicted where three regions can be found (Fig. 8). The area under the pin tip experiences a dominant shear compression loading (region I) whereas near the AS and RS hooks dominant shear tensile loadings are prevailing (region II). These regions are separated by pure shear zones (region III).

Fig. 9 exhibits, on the weld which undergoes the most severe deformations (weld 15), microstructures from region I (Fig. 9c) and region II (Fig. 9d ande). In both regions, the initial equiaxed microstructure of the base Ti-6Al-4V alloy (Fig. 9b) is strongly modified. For region I (Fig. 9c), two levels of microstructural modifications occur. Firstly, near the interface Al2024/Ti-6Al-4V, where the higher temperatures are reached, the β phase becomes hardly distinguishable by SEM. This refinement of the microstructure indicates an intensive plastic deformation. Below, in the second level, grains are deformed in the shear direction. In the same direction, some heterogeneities appear in the form of β grains elongated in shear bands. Region II extends along the AS hook in a conical shape, alike the geometry of the pin (Fig. 9d). Accumulated segments of Ti-6Al-4V (Fig. 9d) are found at the base of the AS hook. Alike in region I, shear bands (Fig. 9e) are observed. However, unlike the region I, porosities and cracks are formed. These damage mechanisms are inherent to the tensile shear stress applied in this region. They take place inside the shear bands in a mechanism of growth of voids which promotes the formation of cracks, as suggested by the zone circled in red in Fig. 9e.

In the zone of shear compression (region I), all the welds present an intensive plastic deformation layer. The thickness of this band depends on the process parameters and more precisely extends with the increase of ω 2 /ν, which corresponds to the pseudo heat index (Fig. 10). It indicates that pin penetration inside the Ti-6Al-4V plate increases with the increment of the heat input. It can be associated to the decrease of the mechanical resistance of the alloys. AS hook height being also controlled by ω 2 /ν, it can be assumed that the amount of Ti-6Al-4V displaced inside the Al2024 is related to the pin penetration inside the lower plate, promoted by the increase of the heat input.

Porosities and cracks are systematically formed in the tensile shear zone (region II). However, depending on the process parameters used, the Ti-6Al-4V distribution is modified. Inspection of the deformation of Ti-6Al-4V was also carry out on weld 13 (Fig. 11a). Instead of Ti-6Al-4V segments formed at the base of the AS hook (Fig. 9d), Ti-6Al-4V fragments are produced and locate inside the Al2024 stir zone (Fig. 11b). The microstructure of these fragments, composed of complex shear bands (Fig. 11c), is similar to the Ti-6Al-4V segments depicted in Fig. 9e. Porosities and cracks, emphasized in Fig. 11c, can also be noted. It suggests that these fragments were originally bonded to the titanium alloy plate, as the Ti-6Al-4V segments in Fig. 9d, before being pulled off and penetrate into the stir zone. Therefore, the creation of Ti-6Al-4V fragments seems to result from damage mechanisms which ini tiate in the tensile shear zone due to its ability to generate voids. Then, Ti-6Al-4V fragments can reach different levels in the thickness of the upper plate according to the welding parameters, as stated from X-ray tomography results (Fig. 6).

U N C O R R E C T E D P R O O F

Fracture behaviour

From tensile shear tests, three different fracture behaviours (FB) are noticed: (i) fracture across the Al2024-T3 plate above the AS hook (FB I); (ii) fracture along the interface and across both hooks (FB II) and (iii) fracture across the Ti-6Al-4V plate below the AS hook (FB III). Fig. 12 illustrates those three FB. During tensile shear tests the top plate bends above the AS hook regardless the fracture behaviour. It indicates that deformation localizes above the AS hook (plastic hinge). Besides, it suggests that AS hook has a significant effect on the mechanical behaviour of the assembly.

The strain field evolution of the three mentioned FB are exhibited in Fig. 13.

For FB I (Fig. 13a), the deformation is first localized at the top of the AS hook while it seems to be of a more diffuse nature on the other side (in and around the large RS hook). Then, the top plate starts to bend while the deformation propagates above AS hook until crack initiation. Finally, fracture occurs across the strained aluminium alloy right above the AS hook.

For FB II (Fig. 13b), the deformation also starts at the top of AS hook within the bending zone of the aluminium alloy plate. Unlike FB I, the deformation localizes across the RS hook section and shear failure of the hook is observed. An interfacial crack then propagates along the weld from the RS hook toward the AS hook. The strain localization on the aluminium alloy plate above the AS hook is visible but do not leads to fracture at this location. Indeed, while the crack reaches to bottom of the AS hook, the change in the interfacial orientation results either in the titanium alloy AS hook to be expelled from the aluminium alloy bulk or a propagation of the crack across the AS hook. Alike in the FB I, the aluminium alloy plate bends above the AS and leads to the align ment of the two base materials plates. Hence, the loading at failure on the AS hook is close to parallel to its indentation direction which leads to a tensile load on the AS hook which prompts its ejection.

U N C O R R E C T E D P R O O F
For FB III (Fig. 13c), deformation also reaches both AS and RS hooks leading to the bending of the top plate above AS hook and the alignment of the two edges of the tensile specimen. The major difference concerns the propagation of the deformation between AS and RS hooks. The crack do not propagates all the way towards the AS hook since the titanium alloy substrate fails before crack completion. It is seen from the DIC measurement that no strain is visible within the titanium alloy prior to failure, hence describing a fracture of brittle nature. Such observation suggests the presence of defects in or around this region. Finally fracture occurs across the titanium alloy below the AS hook.

Regardless the FB involved, it can be noticed that no significant deformation appears in the rest of the specimen, either the aluminium alloy or the titanium alloy plates. This simple statement confirms the previous observation namely that the mechanical properties of the overall weld joint mainly depends on hooks features.

Mechanical behaviour

As previously explained, the fracture behaviours of all the welded samples can be classified in three typical behaviour (FB I, FB II and FB III). FBI and FB III comprise only one sample whereas seven samples failed according to FB II. Load-displacement curves obtained for samples belonging to FB I, FB II and FB III and for the weakest base material Al2024-T3 are presented in Fig. 14. In the case of FB II, only two curves are drawn corresponding to the minimum and maximum failure loads obtained with FB II. Unlike the ductile behaviour of the base aluminium alloy, all welds exhibit a brittle behaviour (Fig. 14a). Both failure loads and displacements at failure are clearly lower for the welded samples, compared to the base material Al2024-T3 (Fig. 14a). This indicates that friction stir welding process generated defects which weakened the joints. The maximum failure load was obtained for the FB II and represents 35% of the ultimate load of the base Al2024-T3 (weld 3). Joints efficiencies (Fig. 14b) were evaluated by the ratio between the failure loads of welds and the ultimate load (UL) of base Al2024-T3. The lowest failure load was obtained for FB III (failure across titanium alloy) and corresponds to the weld 15 which undergoes the most severe plastic deformation (Fig. 9). With an ultimate tensile strength (1100 MPa) more than twice higher than Al2024-T3 (480 MPa), this suggests that under such process parameters, the titanium alloy plate undergoes severe modifications during friction stir welding process leading to the formation of defects. An intermediate failure load was obtained for FBI, with the weld 12 performed in double-pass.

It should be mentioned that the stiffness offset (Fig. 14b) between base Al2024-T3 (measured by extensometer) and welded samples (mea sured by traverse displacement) is related to the difference in displacement acquisition.

U N C O R R E C T E D P R O O F
Typical points of the load-displacement curves for the three FB obtained and the strain fields associated are presented in Fig. 15. Each load-displacement curve presents two typical points marked by circles (Fig. 15a-d). The first point corresponds to a change in slope and coincides with the crack initiation along the AS hook (Fig. 15a1, b1, c1 and d1). The second point corresponds to the final crack propagation leading to fracture (Fig. 15a2, b2, c2 and d2). For FB I and FB II, a decline in slope (Fig. 15a-c) is observed after crack initiation along AS hook. This could be attributed to the reduction of the load carrying capacity of the joints, related to the decrease of the effective thickness of the aluminium alloy plate. For FB III, two sudden load drops are noticed (Fig. 15d). The first one coincides with the simultaneous crack initiation along the AS hook and along the interface (Fig. 15d1). The second sudden load drop corresponds to the crack initiation across the titanium alloy plate where final fracture occurs (Fig. 15d2).

Relative size of AS and RS hooks influences the mechanical behaviour (see tables in Fig. 15). AS hook size impacts the first change in slope while RS hook size affects the second typical point of the curves. In particular, a large AS hook causes a change in slope at low displace ments (Fig. 15a andc). A large RS hook enables a higher displacement between the two typical points (Fig. 15a) whereas a reduced RS hook provokes a sudden load drop (Fig. 15d).

The influence of the process parameters on the failure load and the fracture behaviours is plotted in Fig. 16. A trend can be noted between failure loads of FB II and ω 2 /ν which indicates that the increase of the heat input is detrimental to the mechanical properties. With further increase of the heat input, the mechanical properties decrease significantly and lead to the lowest failure load with FB III, across the titanium alloy plate (weld 15). An intermediate failure load is obtained with FB I. It corresponds to the weld produced in double-pass (weld 12).

The heat input, related to the pseudo heat index ω 2 /ν [START_REF] Arbegast | Characteristics of friction stir welded aluminum alloys[END_REF], significantly affects hooks shape (Fig. 6), hooks height (Fig. 7) and the Ti-6Al-4V plastic deformation (Fig. 10). As a consequence, the mechanical properties are impacted. In particular, DIC results revealed that AS and RS hooks impact the mechanical behaviour. The influence of the welding parameters on the material flow and the role of the resulting hooks on the stress distribution will be discussed.

Discussion

Influence of the welding parameters on the material flow

In the overlap configuration, the generated material flow is complex and leads to the generation of hooks. These features are tightly related to the geometry of the tool and the in-process parameters such as velocities and load or depth. Indeed, the input heat (mainly related to the pseudo heat index ω 2 /ν and F) prompts the loss of the mechanical resistance of the alloys which promotes their deformation and further material motion. As shown in Fig. 6 (particularly d, f and g), titanium alloy fragments can be transported several millimeters away from their original location. Accordingly it is reasonable to assume that (i) due to its high thermal conductivity, the aluminium alloy of the top plate is significantly heated at such distance from the tool and (ii) the material embedded in the so-called thermo-mechanically affected zone behave like a fluid.

The heat generated by friction (ϕ/m) is classically described by the following relation [START_REF] Kagnaya | Wear mechanisms of wc-co cutting tools from high-speed tribological tests[END_REF]:

(2) where τ s is the shear stress, V s is the sliding velocity, α is the heat partition coefficient that represents the portion of heat flux transmitted to the materials and η is the coefficient which specifies fraction of frictional power converted into heat. Note that (i) α is defined as the ratio of the effusivity of both materials [START_REF] Bonnet | Improvement of the numerical modeling in orthogonal dry cutting of an aisi 316l stainless steel by the introduction of a new friction model[END_REF] and (ii) it is conventionally consider that 90% of the frictional power is converted into heat and so 10% is devoted to strain hardening and further microstructural evolutions [START_REF] Bonnet | Improvement of the numerical modeling in orthogonal dry cutting of an aisi 316l stainless steel by the introduction of a new friction model[END_REF].

Under such formalism, the axial load (here 8250 N) and the rotational velocity of the tool ω are responsible for most of the generated heat. More specifically, owing to the surface ratio, the shoulder may be singled out in this matter. Once heated, the material of both alloys can flow around and beneath the tool and then relocate material in order to fill the hole generated behind the tool pathway. The circular flow combined with the forward motion of the tool lead to asymmetrical distribution of velocity (Magnus effect) and therefore an asymmetrical distribution of the pressure field (Bernoulli's principle) [START_REF] Lloyd | An introduction to fluid mechanics and heat transfer[END_REF]. The assessment of the location of the stagnation points for every investigated process parameters show that all the incoming material of the AS side is transported to the RS side (Fig. 17). Simultaneously, the axial force and the planar surfaces beneath the shoulder and the pin lead to an upward material extrusion. At the contact Al2024/shoulder, this flow is responsible for the flashes visible on the top free surface, whereas at the contact Ti-6Al-4V/pin, this flow results in the formation of hooks. In particular, due to the force control of the process the amount of Ti-6Al-4V displaced into Al2024 is controlled by the penetration depth of the pin, which is enhanced with the increase of the heat generated (related to the pseudo heat index ω 2 /ν).

Thus, the formation of both hooks results from a combined contribution of the circular and upward flows. Assuming that the velocity composition is different on both side, two scenarios emerge:

• AS hook generation: the material is lifted and deposited as such. The extrusion results mostly from the centrifugal force and is very comparable to the flashes observed at the top of the aluminium alloy plate. However owing to the feed motion of the tool only the section away from the tool pathway remains post-process: namely the AS hook (Fig. 17).

• RS hook generation: the stirred material is first lifted upward by the pin penetration, then transported around the tool and finally deposited behind it when its velocity decreases (Fig. 17). For this reason the Ti-6Al-4V fragments, which penetrated inside the stir zone, preferentially located on this side (refer to Section 3.2 and Fig. 6).

In addition, the higher circumferential velocity on this side prevents from any outward extrusion. On the contrary the shape of the velocity field behind the tool leads to locate the generation zone around the central axis of the tool pathway [START_REF] Hamilton | A coupled thermal/material flow model of friction stir welding applied to Sc-modified aluminum alloys[END_REF]. The

U N C O R R E C T E D P R O O F
schematic of Fig. 17 also point out the fact that the velocity composition of ω and ν controls the angle at which the material is ejected behind the tool and thus the length of the penetrating titanium extrusion.

Beyond these rough considerations, the wide variety of shapes and size of the generated hooks of the present study leads to consider that the process parameter play a major role in the material extrusion and deposition. It is therefore relevant to use hooks as marker of the material flow even though post-mortem observations lead to a drastic simplification of a way more complex phenomenon.

Fig. 18 presents optical microscope images of the welds cross sections produced with increasing heat input. It is first seen that the force control of the process imposes that when the heat input increases, the penetration depth of the tool increases as well. Consequently, it is seen from Fig. 18 that the amount of extruded material being lifted up also increases. On the AS side, the hooks exhibit a flash like shaped regardless of the heat input. On this side height the hook profile increases with the heat input (Figs. 18 and7a) which is consistent with larger amount material being extruded. On the RS side the thickness and the length of the hooks increases with the heat input, here again owing to the amount of material being transported. However, on this side, the shape of the hooks and its continuous nature seem significantly affected by the plastic deformation capacity of the Ti-6Al-4V.

At low heat input, the material behaves with a high viscosity. Its plastic flow capacity around the tool is affected by the lower temperature. On the RS side, the relative pressure is lower and prompts a tensile-like loading responsible for local micro-tearing, crack initiation and material failure.

At intermediate heat input, clear hooks are formed on RS, the heat is sufficient to allow material extrusion and subsequent motion. At such temperature, the strain at failure is increased and allows large strain within the titanium alloy [START_REF] Lee | Plastic deformation and fracture behaviour of Ti6AJ4V alloy loaded with high strain rate under various temperatures[END_REF]. This is why RS bending intensifies.

At high heat input, the stirred titanium alloy is less viscous and is extruded in large quantities owing to the force control of the process which leads to deeper penetrations of the pin. In addition, the aluminium alloy being also over heated offers no resistance to titanium alloy particle motion. Hence, the extruded material is easily transported around the tool and ejected behind it. Even though the rise of temperature tends to increase the strain at failure, the circumferential velocity leads to particle projection responsible for the generation of a discontinuous hook (Fig. 18c anda).

Post-mortem analysis from CT-scan reconstruction of the post-process titanium alloy plate (Fig. 6) offers a valuable insight at the material flows involve in friction stir welding. A clear dependency of the hooks shape and size to the process parameters is observed. More specifically, the pseudo heat index [START_REF] Arbegast | Characteristics of friction stir welded aluminum alloys[END_REF] seems to act as a key factor in this matter.

Influence of hooks on stress distribution

The most obvious weld characteristic observed having an influence on the mechanical behaviour is the formation of hooks. As pointed out by DIC results, hooks disturb stress distribution during tensile shear tests and generate two strain localization areas (Fig. 13). The first one is located on the Al2024 above the AS hook, whereas the second one is located on the Ti-6Al-4V at the base of the RS hook. Fig. 19 defines these two areas as respectively "critical zone A" and "critical zone B". During tensile shear tests, the critical zone A mainly undergoes tension whereas critical zone B undergoes shearing.

The mechanical properties of the overall welds are govern by the peak stresses reached in critical zones A and B, which are influenced by hooks features. The joint efficiency previously defined (part 3.5) expresses the global load bearing capacity of the joint with respect to the weakest material, i.e. the aluminium alloy. Thus, this "macro" joint efficiency does not reflect the stresses locally reached in critical zones A and B. To better understand the role of hooks on the stress distribution during tensile shear tests, the stress reached in critical zones A and B are estimated and two new "local joint efficiencies A and B" will be defined.

In critical zone A, a strain field with a butterfly shape, typically encountered at the notch tip [START_REF] Pilkey | Peterson's Stress Concentration Factors[END_REF], is formed at the top of AS hook due to its higher sharpness compared to RS hook. Therefore, AS hook can be considered as a notch which induces a stress concentration, characterized by a stress concentration factor [START_REF] Pilkey | Peterson's Stress Concentration Factors[END_REF]. The stress concentration factor (K t ) relates the peak stress (stress in the perturbed region) and the nominal stress (stress without notch) as follow:

(3) Considering the welds aspect, K t is estimated on the assumption that a rectangular plate in tension with fillet is the closer geometry of welds (Fig. 20). Geometrical characteristics of the welds are used: the thickness of the base material Al2024 plate D, by taking into account its thinning, the effective thickness above AS hook d, the AS hook height t and the notch tip radius r. The ratio r/t is kept constant at 0.5 for all welds and data from the image processing program (refer to Section 2.2.1) are used for d and t variables. Values ranging between 1.8 and 2.15 are obtained.

A local joint efficiency A is defined as the ratio between the tensile peak stress reached in critical zone A and the ultimate tensile strength of the base Al2024-T3. The results are plotted in Fig. 21 and point out that FBI results from a severe stress concentration in critical zone A. Locally, the peak stress at the top of AS hook reaches the ultimate tensile strength of Al2024 (local joint efficiency A superior to 100%) and leads to crack initiation and fracture across the aluminium alloy. For FBII and FBIII, the peak stresses reached in critical zone A are insufficient (local joint efficiency A lower than 100%) and fracture across the aluminium alloy plate (critical zone A) is not achieved. Size of AS hook mainly govern the stress concentration factor in critical zone A. High AS hook height accentuates the stress concentration resulting in the change in slope observed for low displacement through the plastic hinge phenomenon (Fig. 15). Insufficient AS hook heights do not induce failure across Al2024 and lead to a further increase of the load, which will impact the stress reached in critical zone B.

In critical zone B, the shear stress (τ) reached across the RS hook can be estimated from:

(4) where F is the failure load and S RS the RS hook shear cross section, determined from data from the image processing program.

A local joint efficiency B is defined as the ratio between the shear stress reached in critical zone B and the ultimate shear stress of the base Ti-6Al-4V. The results are plotted in Fig. 22 and show that the stress reached in critical zone B exceeds the ultimate shear stress of Ti-6Al-4V (local joint efficiency B superior to 100%) for FBII. Failure of RS hook leads to crack propagation and fracture across the interface (FBII). The peak stress was not calculated for FB III due to the limited size of RS hook, which results from its failure (Fig. 15d). Crack can easily propagates along the interface due to the lack of any obstacle. This is why a sudden load drop was observed in load-displacement curve (Fig. 15d). It emphasizes the mechanical anchorage action of RS hook. For FB I, the high RS hook size (Fig. 15a) inhibits crack propagation in critical zone B. By its action of mechanical anchoring, RS hook postpones crack initiation and increases the displacement at failure, as presented in Fig. 15a.

Finally, the fracture behaviours mainly result from the influence of both AS and RS hooks on the stress distribution during tensile shear test. It is found that the RS hook plays the role of mechanical anchoring while AS hook acts as a notch due to its high sharpness. Subsequently, the stress concentration at critical zone A, along with the decrease of the effective thickness of the upper plate, lead to impair the load carrying capacity of the weld and thus explains the slope drop observed after crack initiation along AS hook (Fig. 15).

FBI occurs for large AS hook, prompted by the second pass. The peak stress reached in critical zone A is equivalent to the ultimate strength of Al2024. FBII and FBIII occur for lower hooks dimensions. The peak stress reached in critical zone B is equivalent to the ultimate shear strength of Ti-6Al-4V. FB II occurs for complete crack propagation along the interface. FB III results from the presence of defects in Ti-6Al-4V. Microstructural modifications, prompted by the high heat input, impel the crack to deviate though the damaged titanium alloy (weld 15) and ultimately leads to a tensile fracture of the thinned titanium plate.

Relationship between the welding parameters, hooks dimensions and mechanical behaviours

Influence of hooks dimensions on the mechanical behaviour is summarized in Fig. 23. The relationship with the FSW parameters is also highlighted. FB II is the most represented fracture behaviour. Welds performed in single-pass mainly lead to hooks with similar dimensions. The resulting AS hook height is insufficient to initiate crack propagation in critical zone A. Therefore, crack initiates in critical zone B. Larger AS hook, and hence FB I, could not be performed in single-pass. A higher heat input leads to improper welding conditions (tool plunge through the Al2024 top plate and the perforation of the Ti-6Al-4V bottom plate) as obtained with welds 4, 6, 7, 9, 10 and 11 (Table 2). The higher appropriate heat input in single-pass is performed with weld 15. However, a fractured RS hook and a pre-cracked region in the titanium alloy plate (Fig. 9) are observed. These specific modifications, characterized by a high AS hook and a low RS hook, lead to fracture across titanium alloy (FB III) (Fig. 23).

Large AS and RS hooks, promoted by the additional heat input provided by the second pass, lead to crack initiation in critical zone A and, therefore, FB I. Higher hooks dimensions could be obtained by varying the welding parameters in double-pass. However according to the previously exposed considerations, such feature is certainly not desirable in terms of mechanical properties of the joint.

To sum up: a low AS hook and a high RS hook would increase the mechanical properties, which is in good accordance with the results of [START_REF] Wang | Effect of hook characteristics on the fracture behaviour of dissimilar friction stir welded aluminium alloy and mild steel sheets[END_REF]. The suppression of the AS hook could avoid a stress concentration in critical zone A whereas a higher RS hook could promote a mechanical anchoring action. Nevertheless, these optimized hooks dimensions coincide with the non-achievable region (Fig. 23). According to the understanding of the material flow, the feasibility of a reduced AS hook combined with a large RS hook is not expected. Indeed, AS hook height being controlled by the pin penetration depth, the lowest AS hook size achievable would be equivalent to the minimal penetration depth of the pin, i.e. 200 μm (2.0 mm pin height against 1.8 mm top plate thickness). For these conditions, the amount of Ti-6Al-4V extruded is low and would generate a limited RS hook height. It should be mentioned that these conclusions are drawn for a given tool and could drastically differ with another tool geometry and dimensions. In particular, the decrease of the pin length could reduce the amount of extruded Ti-6Al-4V penetrated inside the aluminium alloy plate. Such conditions, would lead to lower hooks size. Therefore the mechanical resistance of the weld is expected to increase. Besides, AS and RS hooks being asymmetrical (different size, shape and sharpness), different results could be obtained in the other loading configuration (modification of the relative position of AS and RS hook), as reported by [START_REF] Yang | Effect of tool geometry and process condition on static strength of a magnesium friction stir lap linear weld[END_REF]. In conclusion, this study pointed out that dissimilar friction stir lap welds are very sensitive to FSW parameters such as the welding speed, the rotational speed, the number of pass and to the force control mode. Modification of the tool and the loading configuration could constitutes some ways to optimize the mechanical resistance of welds.

Conclusion

In this present work, Al2024-T3 to Ti-6Al-4V overlap joints were produced with various welding speeds and rotational speeds. The forge force was modified in order to achieve welds with appropriate heat input. In addition, one weld was performed with double-pass. Three different fracture behaviours were observed: (i) FB I, across the Al2024-T3 plate above the AS hook; (ii) FB II, along the interface and (iii) FB III, across the Ti-6Al-4V plate below the AS hook. The formation of hooks, as well as their contributions on the mechanical behaviour of welds were investigated. Based on these observations and analysis, the following conclusions could be drawn:

1. The formation of hooks results from the contribution of the upward and circular flows. First, due to the force control of the process, the amount of Ti-6Al-4V extruded increases with the heat input, related to the pseudo heat index ω 2 /ν. However, the volume of Ti-6Al-4V transported to form AS and RS hooks is not distributed in the same way due to the asymmetrical distribution of velocity fields. AS hook is formed due to the extrusion action of the tool, alike flashes. RS hook results from material extrusion and deposition behind the tool. 2. FSW parameters impact the material extrusion and deposition, and therefore hooks shape and size. AS hook keeps a flash like shaped and its height increases with the heat input (ω 2 /ν), owing to the force control of the process. Evolution of RS hook height and shape is more complex. The volume of RS hook is also proportional to the heat input (ω 2 /ν). However, the deposition of Ti-6Al-4V is also impacted by its plastic deformation capacity. RS hook height firstly increases with the heat input, which allows a large strain within the Ti-6Al-4V. Subsequent increase leads to an intensification of RS hook bending and, hence, RS hook height decreases. For high heat input, the high circumferential velocity leads to material projection and failure of the RS hook. 3. AS and RS hooks disturb the stress distribution at their vicinity.

They form two critical zones namely critical zone A (shear tensile in Al2024 above AS hook) and critical zone B (shear compression in Ti-6Al-4V at the base of RS hook). Crack initiation in one of these regions directly leads to further crack propagation and structural failure. Hooks dimensions control the stresses reached in critical zones A and B. 4. FB I is obtained for large AS and RS hooks which lead to a peak stress equivalent to the UTS of Al2024-T3 in critical zone A. Crack initiated along AS hook can propagate across the aluminium alloy plate.

The additional heat input provided by the second pass results in high hooks sizes and amplifies this very phenomenon. 5. FB II is obtained for limited sizes of AS and RS hooks. The peak stress reached in critical zone A is not sufficient to prompt failure within the upper plate. A further increase of the load applied in critical zone B is observed. Crack initiates in critical zone B due to a peak stress equivalent to the shear stress of Ti-6Al-4V. Then, the crack propagates along the interface due to the lack of any defects. 6. FB III is also obtained for limited sizes of AS and RS hooks. Fracture across Ti-6Al-4V occurs due to the formation of a pre-cracked zone in the Ti-6Al-4V, related to the severe plastic deformation of the Ti-6Al-4V and thinning of the bottom plate.

Fig. 1 .

 1 Fig. 1. Schematic illustration of a FSW Al2024-T3/Ti-6Al-4V lap joint (dimensions in mm).

Fig. 2 .

 2 Fig. 2. (a) CT-scan images used for post-processing procedures; (b) segmentation for binary images and analyse of (c) the Ti-6Al-4V hooks; (d) the Ti fragments penetrated inside the aluminium alloy plate and (e) the cavity formed by the pin penetration in the titanium alloy plate.

Fig. 3 .

 3 Fig. 3. Schematic illustration of lap shear specimen.

Fig. 4 .

 4 Fig. 4. Macrographs (SEM-BSE images) showing different hook features for different weld conditions: (a) weld 3; (b) weld 1; (c) weld 14; (d) weld 8, (e) weld 5 and (f) weld 15.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Macrographs (SEM-BSE images) of the same sample after successive polishing.

  Fig. 7. Influence of the process parameters on: (a) the penetration depth, (b) AS hook height and (c) RS hook height. Annotations "6.x" refer to the welding conditions plotted in Fig. 6.

Fig. 8 .

 8 Fig. 8. Schematic illustration of the stress state exerted by the pin on the Ti-6Al-4V plate.

Fig. 9 .

 9 Fig. 9. SEM images of weld 15 illustrating the severe plastic deformation of Ti-6Al-4V: (a) SEM-BSE macrograph of shear compression (square no. 1) and shear tensile (square no. 2) regions; (b) SEM-ETD image of the base Ti-6Al-4V alloy (equiaxed microstructure); (c) enlarged SEM-ETD image of the square no. 1; (d) enlarged SEM-ETD image of the square no. 2 and (e) enlarged SEM-ETD image of the square no. 3.

Fig. 10 .

 10 Fig. 10. Influence of the process parameters on the thickness of the intensive plastic deformation layer produced near the Al2024/Ti-6Al-4V interface.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. SEM images illustrating the severe plastic deformation of Ti-6Al-4V of the weld 13: (a) SEM-BSE macrograph indicating shear tensile region (square no. 1); (b) enlarged SEM-ETD image of the square no. 1; (c) enlarged SEM-ETD image of the square no. 2.

Fig. 13 .

 13 Fig. 13. Evolution of the strain fields, along cross sections, evaluated by digital image correlation and schematics on the three fracture behaviours: (a) FB I; (b) FB II; (c) FB III. Note that ɛ1 represents the major principle strain.

Fig. 14 .

 14 Fig. 14. (a) Load-displacement curves of the base Al2024-T3 and the three different fracture behaviours and (b) enlarged picture of dash rectangular region marked in (a) with annotation of the joint efficiency, compared to the ultimate load (UL) of Al2024-T3.

Fig. 15 .

 15 Fig. 15. Identification of strain fields associated to two typical points of the load-displacement curves for: (a) FB I (weld 12); (b) FB II -Fmax (weld 3); (c) FB II -Fmin (weld 14) and (d) FB III (weld 15). AS and RS hooks dimensions are exposed. Note that ɛ1 represents the major principle strain.

Fig. 16 .

 16 Fig. 16. Influence of the process parameters on the failure loads and the fracture behaviours.

Fig. 17 .

 17 Fig. 17. Schematics of the distribution of velocity field during FSW process for two distinct conditions: (a) ω ≫ ν and (b) ω ≈ ν.

Fig. 18 .

 18 Fig. 18. Optical microscope images of the welds cross sections: (a) weld 3; (b) weld 1 and (c) weld 11 sorted in ascending order of ω 2 /ν increasing. Note that weld 11 was produced with an improper welding condition (leading the tool plunge through the Al2024 top plate and the perforation of the Ti-6Al-4V bottom plate).

Fig. 19 .

 19 Fig. 19. Definition of critical zones A and B during tensile shear test.

Fig. 20 .

 20 Fig. 20. Definition of the parameters used to estimate the stress concentration factor K t in critical zone A.

Fig. 21 .

 21 Fig. 21. Local joint efficiency estimated in critical zone A for each FB (7 tests failed according to FBII against 1 test according to FBI (weld 12) and 1 test according to FBIII (weld 15)).

Fig. 22 .

 22 Fig. 22. Local joint efficiency estimated in critical zone B for each FB (7 tests failed according to FBII against 1 test according to FBI (weld 12) and 1 test according to FBIII (weld 15)).

Fig. 23 .

 23 Fig. 23. Relationship between the achievable FSW parameters, the resulting AS and RS hooks dimensions and fracture behaviours for a given tool and loading configuration.

Table 1

 1 Chemical composition of the base materials (weight %).

		Al	Cu	Fe	Mg	Mn	Ti	V	Zn
	Al2024-T3 Ti-6Al-4V	Bal. 6.21	4.60 -	0.16 0.13	1.30 -	0.56 -	0.02 Bal.	-3.99	0.22 -

Table 2

 2 List of the investigated samples and their respective welding parameters. Welds produced with improper welding conditions are identified by *.

	Weld number	Force (N)	Rotational speed ω (rpm)	Welding speed ν (mm/ min)
	1 2 3 4* 5 6* 7* 8 9* 10* 11* 12 -double-pass 13 14 15	8250 8250 8250 8250 8250 8250 8250 8250 8250 8250 8250 8250 7000 7000 7000	500 500 500 750 750 1000 1000 1000 1500 1500 1500 500 750 1000 1500	60 120 240 60 120 60 120 240 60 120 240 120 60 120 240

Acknowledgements

This work is part of a National project IMPAACT (Intelligent and Multi-material Process of Assembly: Aluminum alloys Coupled with Titanium) in collaboration with AVANTIS Group and MAUPERTUIS Institute and funded by DGA. The authors thank the different partner as well as the technician team of the ICA institute.

Authors' contribution

Morgane Geyer: conceptualization, formal analysis, investigation, writingoriginal draft, visualization. Vanessa Vidal: conceptualization, writingreview & editing, supervision. Thomas Pottier: conceptualization, writingreview & editing. Christine Boher: conceptualization, writingreview & editing, supervision, funding acquisition, project administration. Farhad Rézaï-Aria: conceptualization, writingreview & editing, supervision.

Declaration of Competing Interest

The authors report no declarations of interest.


Matheus Patrick

Soares Barbosa
email: mpatrick.barbosa@gmail.com

Micky Rakotondrabe

Helon Vicente

Hultmann Ayala

Deep Learning Applied to Data-driven Dynamic Characterization of Hysteretic Piezoelectric Micromanipulators

Keywords: Identification and control methods, Micro and Nano Mechatronic Systems, Perception and sensing, Hysteresis modeling, Nonlinear system identification, Piezoeletric micromanipulators, Artificial neural networks

The presence of nonlinearities such as hysteresis and creep increases the difficulty in the dynamic modeling and control of piezoelectric micromanipulators, in spite of the fact that the application of such devices requires high accuracy. Moreover, sensing in the microscale is expensive, making model feedback the only viable option. On the other hand, data-driven dynamic models are powerful tools within system identification that may be employed to construct models for a given plant. Recently, considerable effort has been devoted in extending the huge success of deep learning models to the identification of dynamic systems. In the present paper, we present the results of the successful application of deep learning based black-box models for characterizing the dynamic behavior of micromanipulators. The excitation signal is a multisine spanning the frequency band of interest and the selected model is validated with semi static individual sinusoidal curves. Various architectures are tested to achieve a reasonable result and we try to summarize the best approach for the fine tuning required for such application. The results indicate the usefulness and predictive power for deep learning based models in the field of system identification and in particular hysteresis modeling and compensation in micromanipulation applications.

INTRODUCTION

System identification is the field within automatic control which encompasses the creation of dynamic models for the purposes of analysis and control of physical systems. Methods from system identification may be applied in a wide range of problems, also other than process control, such as biology in general [START_REF] Bekey | Identification of biological systems: a survey[END_REF], artificial pancreas [START_REF] Hajizadeh | Multivariable recursive subspace identification with application to artificial pancreas systems[END_REF], leg prosthesis [START_REF] Abdelhady | System identification and control optimization of an active prosthetic knee in swing phase[END_REF], and water quality monitoring [START_REF] Beck | In Hvdrological forecasting-Prévisions hydrologiques[END_REF] to cite a few. Black-box modeling stands for the creation of mathematical abstractions of the system under study using solely input and output measurements. That is, no knowledge regarding the physical system is assumed to be available a priori and experiments are performed in order to enable the creation of the models offline. For its general applicability, nonlinear black-box modeling is an important tool for building models for different purposes such as simulation, analysis, and design.

Piezoeletric materials are recognized as actuators basis thanks to its high frequency rates, high resolution, portability, and ease of intergration [START_REF] Rakotondrabe | Smart materials-based actuators at the micro/nano-scale[END_REF]. Moreover, in some situation, piezoelectric actuators might be used as their proper sensors allowing feedback control for performances enhancement [START_REF] Rakotondrabe | Combining self-sensing with an unkown-input-observer to estimate the displacement, the force and the state in piezoelectric cantilevered actuator[END_REF][START_REF] Aljanaideh | Observer and robust h-inf control of a 2-dof piezoelectric actuator equipped with selfmeasurement[END_REF][START_REF] Ivan | Quasi-static displacement self-sensing measurement for a 2-dof piezoelectric cantilevered actuator[END_REF]. However, whenever actuating at the micro scale, sensing is not as easy to the uncertainties in the measured signals and to the difficulty to directly access. In such cases, it is mandatory to employ soft sensors to make possible the actuation using artificial feedback loops. In the literature we can find several methods for modeling such systems, e.g Preisach, Prandtl-Ishlinskii or Bouc-Wen models as noted in [START_REF] Gu | Modeling and control of piezo-actuated nanopositioning stages: A survey[END_REF][START_REF] Rakotondrabe | Multivariable classical prandtlishlinskii hysteresis modeling and compensation and sensorless control of a nonlinear 2-dof piezoactuator[END_REF][START_REF] Habineza | Multivariable generalized bouc-wen modeling, identification and feedforward control and its application to a 2-dof piezoelectric multimorph actuator[END_REF]. However, the precise modeling under those models require difficult and ad-hoc tuning which is a complex task in order to represent nonlinearities such as creep and hysteresis. As those characteristics are challenging in control design, the models used to tune the control laws should be equipped to better represent the actual system in simulation. As such, data-driven modeling such as black-box system identification is a general modeling framework which hinders the difficulties mentioned before.

In a previous work by the authors a 2-DOF piezoelectric actuator is modeled in [START_REF] Ayala | Modeling of a 2-dof piezoelectric micromanipulator at high frequency rates through nonlinear black-box system identification[END_REF], using shallow artificial neural networks at high frequency rates in a black-box approach. A piezo-actuated nanopositioning stage is controled in [START_REF] Ling | A robust resonant controller for highspeed scanning of nanopositioners: Design and implementation[END_REF], where the effect of the applied voltage in the displacement is modeled locally as transfer. A multi axle piezo tube is compensated in open-loop in [START_REF] Habineza | Multivariable compensation of hysteresis, creep, badly damped vibration, and cross couplings in multiaxes piezoelectric actuators[END_REF], where the authors use a local linear model for inversion. In [START_REF] Escareno | Robust micro-positionnig control of a 2dof piezocantilever based on an extended-state lkf[END_REF] the authors employ two coupled Hammerstein models to represent a 2-DOF piezocantilever in the task of micropositioning leveraged by an Kalman filter. Finally, [START_REF] Rakotondrabe | Modeling and compensation of multivariable creep in multi-dof piezoelectric actuators[END_REF] use MIMO transfer function to model other nonlinearities in multiaxes piezoelectric actuator and create an approximate inverse to their compensation. Artificial neural networks have been long used for system identification [START_REF] Narendra | Identification and control of dynamical systems using neural networks[END_REF] and have their roots tied with controls community. Radial basis functions [START_REF] Hultmann Ayala | Cascaded evolutionary algorithm for nonlinear system identification based on correlation functions and radial basis functions neural networks[END_REF] and multilayer perceptrons [START_REF] Li | Dynamic system identification via recurrent multilayer perceptrons[END_REF] are valuable tools for the nonlinear black-box system identification practitioner. On the other hand, deep learning has been given considerable attention from the research community and practitioners due to impressive improvements mainly in the field of image processing [START_REF] Lecun | Deep learning[END_REF]. However, deep learning has not been extensively applied in system identification in order to solve complex dynamic systems modeling tasks. In [START_REF] De La Rosa | Randomized algorithms for nonlinear system identification with deep learning modification[END_REF] the authors use a restricted Boltzmann machine trained with random weights and test the methodology with standard benchmarks in system identification such as the gas furnace data [START_REF] Box | Time series analysis, forecasting and control[END_REF], simulated nonlinear system [START_REF] Narendra | Gradient methods for the optimization of dynamical systems containing neural networks[END_REF] and the Wiener-Hammerstein case study [START_REF] Schoukens | Three benchmarks addressing open challenges in nonlinear system identification[END_REF]. In [START_REF] Qiao | A deep belief network with PLSR for nonlinear system modeling[END_REF] the authors employ partial least squares regression in order to estimate deep neural models and evaluated the methodology with a simulated nonlinear system, a chaotic system, and the prediction of total phosphorus quantity in a waste water treatment system with acquired data.

In this work we focus on the application of nonlinear black-box models to characterize the hysteretic behavior of piezoelectric micromanipulators using deep neural models as an alternative to the existing models. To this end, we perform specific data acquisition using a multipurpose signal for the estimation of model parameters that covers the whole frequency band of interest. Then we validate the created model with various individual sinusoidal input signals, in order to assert whether the hysteretic behavior has been adequately captured by the algorithm. The contributions of the paper thus are the following. We adapted and tested state-of-the-art deep neural networks Tensorflow c framework [START_REF] Hope | Learning Ten-sorFlow: A Guide to Building Deep Learning Systems[END_REF] from Google in order to solve identification problems. These changes encompass the creation of input/output pairs for estimation, validation and respective prediction horizon. Moreover, we confirmed that the general purpose multisine signal can be used for the creation of black-box models for micromanipulators, which is very convenient for the designer as the only parameters one has to tune is the frequency range, the sampling time, and the amplitude of interest. It is important to mention that all those parameters may be inferred for the task at hand. We also evaluated a series of deep neural network architectures, which may shed light for other researchers willing to apply the methods herein presented in problems which involve hysteresis which are frequently found in nature or of similar complexity.

The present paper is organized as follows. In Section II the methods employed in the present work related to nonlinear black-box system identification and signal excitation design are given. The case study is presented in Section III while in Section IV we give the results of the application of system identification to the problem of hysteresis characterization in micromanipulators. Lastly, the conclusion and future works are stated in the last Section V.

NONLINEAR BLACK-BOX SYSTEM IDENTIFICATION

Nonlinear black-box system identification consists of building mathematical abstractions of dynamic systems without (or with very little) any assumption about the physical properties of the system, relying purely on measured input and output data [START_REF] Ljung | Perspectives on system identification[END_REF]. It is a powerful and general framework for constructing data-driven dynamic models, with applications that span different fields.

Let u(t) and y(t) denote the input and output of the system at a given discrete-time instant t, and both these quantities are available. The NARX (Nonlinear AutoRegressive with eXogenous inputs) model is defined as

y(t) = F [y(t -1), y(t -2), . . . , y(t -n y ), u(t -1), u(t -2), . . . , u(t -n u )] + ξ(t); (1)
where n y and n u are the maximum lags (or orders of the model) at the output and the input. The residual is defined as the error from the output of the model when compared to the measured output and can be calculated by

ξ(t) = y(t) -ŷ(t) (2)
where ŷ(t) is the predicted value. The function F [•] is a nonlinear function mapping in R n φ → R from the n φ model inputs to the model output. It describes the dynamic relationship with past measured input and output data and thus describes the prediction of the model denoted by ŷ(t). The one-step-ahead (OSA) prediction is denoted by ŷ(t) as it is predicted based on most recent measurement. It differs from free-run simulation (FRS) from the fact that in the latter we use predictions over predictions to calculate the outputs of the model, which in many cases is more indicated to perform model validation as the errors accumulate over time.

Model Validation

We use error based metrics to evaluate the prediction accuracy of the models built. In order to do so, we employ multiple correlation coefficient (R 2 ) defined as [START_REF] Haber | Structure identification of nonlinear dynamic systems-a survey on input/output approaches[END_REF][START_REF] Schaible | Fuzzy logic models for ranking process effects[END_REF]]

R 2 = 1 - N t=1 [ξ(t)] 2 N t=1 [y(t) -ȳ] 2 (3)
where the upper bar denotes the mean value of the the sequence. The R 2 shows in an uniform scale, without regard to the magnitude one is measuring, the prediction quality as its maximum value is one which implies all error sequences equal to zero.

It is important to mention that the R 2 metric is calculated in OSA and FRS, for both the estimation and validation phases, but only the FRS were used to validate the model.

Excitation Signal

For the acquisition of input and output estimation data we may employ open loop experiments as the test bench is available and no issues related to stability are expected. Thus we may design an ad-hoc excitation signal to perform the measurements. It is known that the exciting signal should span over the full band of interest and thus we employ the general purpose multisine excitation signal, for details on these issues see [START_REF] Pintelon | System identification: a frequency domain approach[END_REF]. It may be calculated as

u(t) = n f k=1 A cos[2πf k t + φ k ] ( 4
)
where f k and φ k are the frequency and phase components of each sinusoidal components of the overall multisine signal. The total number of sinusoids should be large enough so that the frequency resolution is large enough. So each of f k may be determined equally spaced between a minimum and maximum value of interest with n f components. The phase, however, may be picked randomly in the range [0, 2π] for each component.

The excitation signals for the validation are pure sinusoidals with the amplitude and frequency of interest.

Deep Learning Models

Deep neural models for regression may be described as

r = G[x x x, θ θ θ] ( 5
)
where r is the predicted target value, x x x is the model input vector, θ θ θ is the set of weights which define the network, and G[•] is a nonlinear function mapping from the inputs of the network to the prediction. Following the notation in [START_REF] Haykin | Neural networks and learning machines[END_REF] we may further extend the definition in (5) as

r = φ   k w ok φ   j w kj φ . . . φ i w li x i     (6)
where φ[•] is the neuron activation function, w ok is the synaptic weight from k-th to o-th layer. We can see from both definitions that the neural network is a concatenation of a single entity called neuron, which has different weights and biases throughout the structure.

The activation function φ[•] can be set as sigmoid, hyperbolic tangent or rectified linear unit (ReLU), where z is the input. The latter is described as φ(z) = max(0, z) (7) where z is an input of a ReLU neuron and φ(.) is the activation function.

Once the architecture is defined, one should define the objective function, or loss function, which will be minimized throughout the learning process. We define it as

L [x x x, θ θ θ; p p p] = 1 M M i=1 [r i -ri ] 2 ( 8
)
where M is the total amount of input/output pairs which amounts to the mean squared error (MSE). So by fine tuning the values in θ θ θ we expect to minimize the MSE and thus improve the predictive capability of the deep model.

The neural models so defined are trained through an optimization algorithm via stochastic gradient descent. According to [START_REF] Goodfellow | Deep learning[END_REF] the RMSprop training algorithm has been proposed by [START_REF] Hinton | Neural networks for machine learning[END_REF] and is among the most used options for finding the weights of deep neural networks architectures. A voltage is applied to the active layer of the beam, which deflects due to the piezoelectric effect. (a) MATLAB and dSpace are used to send the input commands and store the data for identification which is performed offline. In (b) we show the real system where the acquisitions were made.

CASE STUDY: PIEZOELECTRIC MICROMANIPULATOR

The system studied in this paper consists of a piezoelectric actuator with cantilever structure with rectangular section and which is classically used in micromanipulation application. The actuator, exhibited in Fig. 1, has dimensions of (length, width, thickness): 15mm x 2mm x 0.3mm. When a voltage is applied to the piezoelectric actuator, it bends. This bending is exploited to push and to manipulate small objects [START_REF] Escareno | Backstepping-based robust-adaptive control of a nonlinear 2-dof piezoactuator[END_REF]. The voltage (u), ranging between [-100V, 100V] is generated from a computer using Matlab-Simulink and a High-Voltage-Amplifier (HVA). The bending (displacement y) of the actuator is measured with an optical displacement sensor having a resolution 10nm and a bandwidth of 5kHz: LK2420 from Keyence.

Between the computer and the remaining elements of the setup (i.e. sensor, HVA amplifer, actuator), an acquisition board is used as converters: dS1104 from dSPACE. The acquisition board is set at 20kHz sampling frequency. This simple device exhibits nonetheless complex dynamic relation from the input voltage and deflection. This makes complicate its use in micropositioning applications, requiring sophisticated estimation and control loops to adhere to the problem requirements.

Measured Data

The data was acquired using the test bench previously described. We recorded two distinct datasets, (i) the multisine signal as in ( 4) and (ii) a pure sinusoidal signal for validation of the hysteresis modeling.

With respect to (i), the multisine was chosen because it is a general purpose excitation signal that can be designed to operate at a predefined frequency band. In Fig. 2 we depict the whole dataset for the estimation of the parameters of the model. It has been constructed with n f = 5000 sinusoidal signals and uniformly distributed random phases, band of interest up to 0.5 kHz frequency and maximum amplitude of 100 V, as limited by the device.

The data was sampled at 10 kHz. We highlight that modeling in such high frequencies is important for rapid and accurate micropositioning. In Fig. 3 we show the power spectrum of the excitation signal and the output, spanning the band of interest up to 0.5 kHz. This band was chosen because it includes both the operation frequency range and the first resonant frequency of the piezo actuator. It is possible to see a peak around 800 Hz in the power spectrum which is outside the excitation signal frequency band. Its presence may be due to non-linearities or unknown disturbances. We are not interested to analyse outside the excitation signal band and the amplitude of this peak is close to -40 dB, while the band of interest is at around 0 dB.

In (ii) we simply applied sinusoidal signals, one at a time, with different frequencies ranging from 0.1 Hz to 0.5 kHz. The pure sinusoidal signal frequently are used to evaluate the adherence of the model to capture the hysteretic behaviour, which is a major issue for the piezo cantilever in micropositioning tasks.

RESULTS

In the present section we depict the results obtained when applying deep neural nets for system identification of the piezoelectric micromanipulator. The input and output data for estimation data have been depicted in Subsection 3.1.

We have normalized all datasets and the results are given in dimensionless scale for sake of simplicity. This has been done to ease the learning process, as the inputs and outputs have different dimensions and this could make it more difficult as the optimization share the learning rate throughout the epochs.

We tested architectures with number of layers spanning from 3 to 5, where each layer is composed of 25, 50, or 100 neurons. We used also 100 epochs, learning rate 10 -4 , a batch of 128 input/output pairs, and ReLU activation function. The order of models n y and n u were 10 and the algorithm used to train the neural network was an RMSprop which divides the gradient by a running average of its recent magnitude. The convergence of the loss and mean absolute error, during training phase, occurred during the first 25 epochs for most of the models, showing small improvements after the succeeding epochs. The choice of best model took in consideration the mean values of R 2 obtained in the validation phase, the number of parameters and the complexity of architecture. In Table 1 it is possible to see all tested architectures and based on that, with 2501 total parameters, we choose the model with 4 layers of 25 neurons as it presents a better compromise between complexity and accuracy. In the following we depict the results of this model with respect to R 2 obtained in the validation phase.

In Table 2 it is possible to see that all the values in the validation phase are close to unity. This shows the excellent prediction capability of the constructed models, as shown when performing FRS on a dataset that has a different nature than that of the estimation phase. Such quality improves our confidence on the model predictive capability and encourages its use in soft sensor applications for example. The values obtained for R 2 mentioned are confirmed in the plots illustrated in Fig. 4. We can see that the lines of prediction and measured data are virtually indistinguishable. and ŷs (t) (red) for the various frequencies tested in validation phase. Note that the hysteretic behavior has been adequately captured for a wide band. Also, it is interesting to note how the hysteresis loop shape changes when the frequency increases.

CONCLUSION

In the present paper we have successfully applied deep neural networks in the task of dynamic modeling of piezoelectric micromanipulators with real-world acquired data.

We have created a dedicated general purpose excitation signal for acquiring data to create the model, giving indication of the importance of multisine signals for hysteresis modeling as also remarked in [START_REF] Noel | A nonlinear state-space approach to hysteresis identification[END_REF] for a hysteretic simulated system. The importance of the steps given in this paper impacts compensator design and simulation, given the high accuracy of the model obtained.

Future research will be devoted to perform a full architecture search and/or extensive hyper parameters tuning, including, but not limited to, use of pyramid structure for the neural network, the automatic creation of the models using for that end neuroevolution techniques [START_REF] Stanley | Designing neural networks through neuroevolution[END_REF], which still lack relevant engineering applications that involve dynamic systems modeling such as monitoring, prediction, compensation, and simulation. It seems that the combination of powerful complex representation and ease of model creation may play an important role in the nonlinear blackbox system identification space. Even if neuroevolution methods in the deep learning space imply greater computational burden, they are applicable in many engineer-ing relevant situations and important for automating the model building activity which is time consuming and thus expensive to be made at scale, if not impossible, when problem dependent decisions are needed.

Fig. 1 .

 1 Fig. 1. Description of the piezoelectric actuator test bench.A voltage is applied to the active layer of the beam, which deflects due to the piezoelectric effect. (a) MATLAB and dSpace are used to send the input commands and store the data for identification which is performed offline. In (b) we show the real system where the acquisitions were made.

Fig. 2 .

 2 Fig. 2. Explorative plots of the measured input and output data. From top to below, we see the input [V] and output [µm] of the whole dataset with eight seconds (8e4 samples at 10 kHz). On the bottom we plotted five thousand samples from zero (left) and eight seconds (right) from the input and output relations.

Fig. 3 .

 3 Fig. 3. Power spectrum for the multisine input and the measured output.

f = 0. 1 Fig. 4 .

 14 Fig. 4. Output predictions for the selected model. Note that the data is normalized. Curves of u(t) versus y(t) (blue)and ŷs (t) (red) for the various frequencies tested in validation phase. Note that the hysteretic behavior has been adequately captured for a wide band. Also, it is interesting to note how the hysteresis loop shape changes when the frequency increases.

Table 1 .

 1 Mean of the values of R 2 for all validation datasets and total number of parameters, according to the different architectures tested.

	Layers Neurons	Mean R 2	Parameters
	3	25	0.984742017	1851
	3	50	0.992217724	6201
	3	100	0.964782331	22401
	4	25	0.989210717	2501
	4	50	0.916190505	8751
	4	100	0.992844673	32501
	5	25	0.992375649	3151
	5	50	0.986898346	11301
	5	100	0.992068821	42601

Table 2 .

 2 Values for R 2 in validation phase, varying frequencies for the excitation signal, using 4 layers of 25 neurons.

	Frequency (Hz)	FRS
	0.1	0.986797839
	1	0.989170955
	10	0.991136621
	50	0.998469432
	100	0.998403016
	150	0.997474415
	200	0.994559081
	250	0.985648806
	300	0.995859260
	400	0.987992324
	450	0.997331124
	500	0.985664909