A Janin

A Constantinescu
email: andrei.constantinescu@polytechnique.edu

D Weisz-Patrault

R Neviere

M Stackler

W Albouy

A new multiaxial specimen for determining dynamic properties of adhesive joints

Keywords: Adhesive joints, DODECA specimen, Dynamic loading, Digital image correlation

Adhesive joints are increasingly introduced in industrial structures for bonding critical parts. Their mechanical characterization is a key element for design and is, therefore, necessary. A significant work has been done for the characterization under quasi-static conditions, but techniques are rather limited for dynamic conditions. Indeed, existing dynamic studies characterize adhesive assemblies and not the adhesive joint alone, and do not investigate multiaxial loadings. Thus, this paper proposes an innovative experimental technique for the characterization of adhesive joints under dynamic multiaxial loadings. The experimental method relies on three main components: i) a conventional Split Hopkinson Pressure Bar (SHPB) apparatus, ii) a novel specimen named as DODECA enabling to test three distinct multiaxial loadings with the same methodology and iii) local strain and stress measurements performed by Digital Image Correlation (DIC). The paper exhibits all steps of the experimental procedure as well as underlying preparation and measuring methods. Stress and strain in the adhesive joint are estimated from experimental data directly both during loading and at the failure point. Finally, the dynamic material behavior of the adhesive joint is identified from the data.

Introduction

Within the framework of energy efficiency and structure lighting, especially for aeronautical applications, the assembly of composite elements using adhesive joints emerged as an interesting alternative to conventional methods such as riveting or screwing. However, adhesive joints are not sufficiently understood and characterized to obtain certifications for critical structural assemblies. Therefore, it is necessary to have a detailed knowledge of the adhesive joint behavior in order to model assemblies of industrial parts. Moreover, these bonded joints are subjected to wide ranges of loading conditions: quasi-static and dynamic multiaxial loadings, thermomechanical fatigue or aging. The present work focuses on the experimental characterization of adhesive joints under dynamic multiaxial loadings. Significant efforts have already been made for quasi-static loadings [START_REF] Cognard | Analysis of the nonlinear behavior of adhesives in bonded assemblies-comparison of tast and arcan tests[END_REF][START_REF] Carrere | 3d models of specimens with a scarf joint to test the adhesive and cohesive multi-axial behavior of adhesives[END_REF]. However, the existing studies dealing with dynamic loadings rely on the single-lap joint specimen [START_REF] Adamvalli | Dynamic strength of adhesive single lap joints at high temperature[END_REF] or the double-lap joint specimen [START_REF] Challita | Finite-element analysis of SHPB tests on double-lap adhesive joints[END_REF][START_REF] Challita | Experimental investigation of the shear dynamic behavior of double-lap adhesively bonded joints on a wide range of strain rates[END_REF] which only investigate the shear behavior of adhesive joints. Thus, the applicability for more complex loading conditions is still limited. Another argument in favor of testing under multiaxial loading is the stress dependence of the mechanical behavior of polymers, like the yield strength, as described for example in [START_REF] Halary | Mécanique des matériaux polymères[END_REF]. In addition, Fig. 1 Experimental set-up scheme of the SHPB apparatus and the local optical measurement system dynamic adhesive joints tests are usually performed with Hopkinson bar systems that provide only global measurements: displacements and resultant forces at both ends of the specimen. This characterizes the adhesive assembly, but not of the adhesive joint [START_REF] Adamvalli | Dynamic strength of adhesive single lap joints at high temperature[END_REF][START_REF] Challita | Finite-element analysis of SHPB tests on double-lap adhesive joints[END_REF][START_REF] Challita | Experimental investigation of the shear dynamic behavior of double-lap adhesively bonded joints on a wide range of strain rates[END_REF]. Therefore, an additional local measurement system during the dynamic testing is of great interest.

The goal of this paper is to characterize the dynamic behavior of adhesive joints using SHPB system and to overcome the two difficulties:

1) by proposing an experimental setup adapted to different multiaxial loading conditions. 2) by adding a local measurement system with high spatial and temporal resolutions to provide relevant information about the adhesive joint.

The first difficulty is to create multiaxial loading conditions. They can be achieved by torsional [START_REF] Raykhere | Dynamic shear strength of adhesive joints made of metallic and composite adherents[END_REF] or tensile [START_REF] Roth | Compact SHPB system for intermediate and high strain rate plasticity and fracture testing of sheet metal[END_REF] Hopkinson bar systems. However, each experimental apparatus provides only one stress state, and explore several stress states seems difficult. Another option is to perform classic compressive dynamic tests using conventional SHPB systems and to design specific specimen geometries in order to obtain multiaxial stress states depending only on a structural effect. Different specimen geometries have been proposed for homogeneous materials (i.e., without adhesive joint) and optimized for a well-defined stress state. For instance, the Compact Compression Specimen (CCS) [START_REF] Rittel | An investigation of dynamic crack initiation in PMMA[END_REF] is designed to study dynamic fracture with large traction component and low shear stress at the crack tip. The main advantage of this specimen geometry is to necessitate only a conventional compression Hopkinson bar system although significant traction is produced. The Shear Compression Specimen (SCS) [START_REF] Zhao | A new shear-compression-specimen for determining quasistatic and dynamic polymer properties[END_REF] is dedicated to the characterization of 2 mm thick polymer films under a biaxial stress state of compression and shear. The Shear Tension Specimen (STS) [START_REF] Dorogoy | A shear-tension specimen for large strain testing[END_REF] is designed to characterize metals under a biaxial stress state of tension and shear. The Double Edge Notched Compression Specimen (DENCS) [START_REF] Kuhn | Fracture toughness and crack resistance curves for fiber compressive failure mode in polymer composites under high rate loading[END_REF] has been used for measuring compressive fracture toughness in dynamic tests. In addition, the well-known Brazilian Disk (BD) specimen has been widely used for the identification of material toughness within the framework of dynamic tests [START_REF] Zhou | Mode-I fracture toughness of PMMA at high loading rates[END_REF][START_REF] Wang | Measurement of mode I and mode II rock dynamic fracture toughness with cracked straight through flattened Brazilian disc impacted by split Hopkinson pressure bar[END_REF]. Some authors proposed an interesting attempt to adapt the original BD specimen to bonded assemblies in quasi-static regime [START_REF] Banks-Sills | Fracture testing of Brazilian disk sandwich specimens[END_REF][START_REF] Martin | A Brazilian disk test for the evaluation of the shear strength of epoxy-joined ceramics[END_REF] with the Sandwich Brazilian Disk (SBD) specimen design. This specimen offers an infinite number of stress states by varying the impact angle in the Hopkinson bar system. In this contribution, a new specimen inspired from the SBD is proposed for dynamic multiaxial loading conditions. It changes the SBD disk into a dodecagon to assure flat surfaces and a plane-to-plane contact between bars and the specimen to ease experimental and numerical procedures. The obtained specimen is then a sandwich dodecahedron called the DODECA specimen. This geometry has already been presented [START_REF] Janin | An experimental technique for the characterization of adhesive joints under dynamic multiaxial loadings[END_REF]. In the present contribution, a new version is proposed in order to avoid edge effects and regularize stresses at both ends of the joint. A specific shape has been designed for this purpose. The specimen geometry enables three different impact angles leading to three different stress states in the adhesive joint.

The second difficulty is the coupling of a local measurement system with the global measurement system of the dynamic loading. In dynamic testing, several options are available : Moir interferometry [START_REF] Thevamaran | An experimental technique for the dynamic characterization of soft complex materials[END_REF], crack growth photography [START_REF] Zhang | Quasi-static and dynamic fracture behaviour of rock materials: phenomena and mechanisms[END_REF], Digital Image Correlation (DIC). The DIC method has proved its potential in quasi-static measurements [START_REF] Hild | Digital image correlation: from displacement measurement to identification of elastic properties-a review[END_REF][START_REF] Roux | Stress intensity factor measurements from digital image correlation: postprocessing and integrated approaches[END_REF], and dynamic measurements [START_REF] Roth | Compact SHPB system for intermediate and high strain rate plasticity and fracture testing of sheet metal[END_REF][START_REF] Gilat | Full field strain measurement in compression and tensile split Hopkinson bar experiments[END_REF][START_REF] Koerber | High strain rate characterisation of unidirectional carbonepoxy IM7-8552 in transverse compression and in-plane shear using digital image correlation[END_REF][START_REF] Seidt | Synchronous full-field strain and temperature measurement in tensile tests at low, intermediate and high strain rates[END_REF] in recent years. This transition has been possible thanks to technological advances in high-speed imaging systems. In this work, a high-speed camera has been used in order to measure local displacements by DIC. Due to technological limitations, a compromise has to be done between acquisition frequency and spatial resolution. To get optimal measurements, a particular attention has been given to speckle pattern techniques [START_REF] Reu | All about speckles: aliasing[END_REF]. Micrometric resolution has been obtained in the present work.

The paper is organized as follows. The SHPB system is briefly recalled in section 1. Then, the innovative DODECA specimen is presented in section 2. The local measurement setup in the adhesive joint is detailed in section 3. Finally, results are presented and discussed in section 4.

Dynamic testing

The dynamic testing is done using the SHPB apparatus and traditional analysis methods. For a detailed review of this method, the reader is referred to [START_REF] Kolsky | An investigation of the mechanical properties of materials at very high rates of loading[END_REF][START_REF] Gary | Comportement des métaux à grande vitesse de déformation[END_REF][START_REF] Jiang | Hopkinson bar loaded fracture experimental technique: a critical review of dynamic fracture toughness tests[END_REF][START_REF] Chen | Split Hopkinson (Kolsky) bar: design, testing and applications[END_REF]. The conventional configuration of the Hopkinson bar system is made of two 40 mm-diameter high strength aluminum bars. The 86-120 cm-long striker bars are made of the same material and have the same diameter as both input and output bars. The output bar is 2 m long, and the input bar is 3 m long in order to allow longer loading duration. The DODECA specimen is placed between the two bars as shown in Fig. 1. Displacements and forces at bars/specimen interfaces are determined from strain measurements recorded at input and output gauge stations with 1 MHz acquisition frequency. The post-processing of strain signals was done with DAVID software [30] which includes noise filtering, checking force and energy balances, elastic simulation for an accurate transport of elastic waves to interfaces, wave dispersion and local punching correction [START_REF] Safa | Displacement correction for punching at a dynamically loaded bar end[END_REF]. This Hopkinson bar configuration was used for striker impact velocities between 5 and 15 m/s measured with a laser. Silicon grease is applied between the specimen and the bars in order to avoid shear contact stress. In addition, a 50 µm-thick pulse shaper is fixed on the input bar. This technique produces a more constant strain rate by limiting sharp strain rates at the beginning of the test. Thus, repeatability is improved.

DODECA specimen

Geometry

In this section, the DODECA specimen is presented. It consists in a sandwich-dodecahedron (dodecahedron cut in two parts by the adhesive joint) which can be inscribed in a 40 mm diameter disk. The proposed specimen is made of 10 mm-thick aluminum bonded by a 300 µm-thick adhesive joint. A previous design has been proposed in [START_REF] Janin | An experimental technique for the characterization of adhesive joints under dynamic multiaxial loadings[END_REF]. However, the design leaded to edge effects in the adhesive joint. This is a known phenomenon in the field of adhesive joints characterization [START_REF] Cognard | Analysis of the nonlinear behavior of adhesives in bonded assemblies-comparison of tast and arcan tests[END_REF][START_REF] Carrere | 3d models of specimens with a scarf joint to test the adhesive and cohesive multi-axial behavior of adhesives[END_REF]. The present version has been improved by introducing beaks near both edges of the joint as shown in Fig. 2, in order to regularize the stress field in these zones. Three impact angles are defined and labeled 15°45°and 75°. This geometry is well suited for 45°and 75°conditions. However, the 15°condition needs an adjustment. Beaks reduce the contact surface between the specimen and the Hopkinson bars. Contact surfaces are not aligned with the mass center of the specimen which leads to a torque as shown on the left part of Fig. 3. This torque causes specimen rotation from the beginning of the impact loading. A simple solution consists in reducing contact surfaces in order to restore the alignment of contact pressure with the mass center of the specimen as shown on the right part of Fig. 3.

Furthermore, as shown in section Numerical simulation, the stress state for 75°loading condition is highly compressive and joint failure can be difficult. Therefore, if failure is studied, samples dedicated to this loading condition are not bonded all along the specimen length, but only on 12 mm at the center.

Fabrication

The DODECA specimen is made by gluing the two aluminum parts with the adhesive joint. In order to control the thickness of the joint, a special manufacturing device has been conceived. The bonding device is presented in Fig. 4. The initial bonding device presented in [START_REF] Janin | An experimental technique for the characterization of adhesive joints under dynamic multiaxial loadings[END_REF] has changed to improve accuracy and homogeneity of the adhesive thickness all along the specimen. Five samples can be prepared at the same time. Two lateral screws apply pressure on two faces of the dodecahedron in order to maintain adherents in the jig. Two vertical centering screws are used to apply pressure on specimens in the normal direction of the adhesive joint during the whole curing process. Satisfying alignment of both adherents is ensured. The joint thickness is controlled by introducing 300 µm-thick shims shown in Fig. 4. The bonding procedure is summarized as follows:

1) Release agent is applied with a brush on the shims and inside the jig where there is contact between the jig and the adherent. Then, they are left 10 minutes under a hood for drying. 2) Adherents are mounted in the jig and the bottom parts should be proud of 200 µm above the reference flat surface of the jig. If this value is not reached due to machining tolerances, Teflon adhesive tape is applied underneath in order to raise the lower adherent to the nominal value. 3) Lateral screws are tightened in order to avoid slips during the curing process. 4) Surfaces to be bonded are still proud of 200 µm. Thus, they are ground till the reference level of the jig with a P600 carbide sand paper. 5) Surfaces are roughly washed with water, then washed three times with isopropanol and left to dry for 10 minutes under a hood. 6) Release agent is applied on remaining surfaces that should not be bonded. 7) The adhesive joint is a thin film that is cut into 10 × 39 mm pieces with a sharp knife. The film is then applied on one surface to be bonded. 8) Vertical screws are tightened until shims are blocked. 9) The joint is cured at 150°C during 3 hours, and the excess of glue is cut off at the end of the curing process. 10) Lateral faces are polished in order to facilitate the Digital Image Correlation.

The adhesive bond quality is estimated by digital microscopy as shown in Fig. 5. The obtained samples present few air bubbles (5% of the lateral surface). The average joint thickness is 300 µm with 6% variations along the joint length. Numerical simulations have been performed in order to design the DODECA specimen. An elastic Finite Element analysis has been performed with the software ABAQUS [32]. Stress distribution along the adhesive joint is evaluated as a function of time for 15°, 45°and 75°loading conditions. Practically, two-dimensional quasi-static simulations have been performed in order to avoid a very long computation time. A very simplified Hopkinson bar impact is modeled by imposing displacements of 100 µm along the normal direction of input specimen surface and by blocking displacements at the output surface. 80 000 4-node reduced integration elements (CPS4R) are used with elastoplastic properties of both adhesive joint and substrate. There are 10 elements in the thickness of the adhesive joint, in order to analyze interface and bulk stresses. The aim is not to reproduce a Hopkinson bar experiment accurately, but to find the stress distribution in the joint with a typical loading condition in terms of amplitude. More representative models [START_REF] Challita | Finite-element analysis of SHPB tests on double-lap adhesive joints[END_REF][START_REF] Zhao | A new shear-compression-specimen for determining quasistatic and dynamic polymer properties[END_REF] could be used to analyze the stress distribution in this experiment, but, they would have a significant calculation time. Results are presented in Fig. 6, 7 and 8. It should be noted that x and y denote the joint length and the joint thickness directions respectively. All stress distributions are heterogeneous but show a critical stress state in the joint centre, with joint edges unloaded. Tab. 1 summarizes the three stress distributions according to the loading angle with the DODECA specimen.

Moreover, as mentioned in section Geometry, the proposed geometry presents beaks at joint edges in order to avoid stress singularities. The effect of this specific geometrical adjustment is presented in Fig. 9 and 10. Adding beaks does not change stress distribution, but eliminate edge effects as expected. Indeed, analytical approaches have proved that substrates with beaks (substrate angle ≤ 50°) reduce stress singularities [START_REF] Cognard | Influence of adhesive thickness on the behaviour of bonded assemblies under shear loadings using a modified TAST fixture[END_REF][START_REF] Leguillon | Computation of singular solutions in elliptic problems and elasticity[END_REF].

Digital Image Correlation

A local displacement measurement method is considered with Digital Image Correlation (DIC). Since the test duration is about a few hundreds of µs, a high-speed imaging system is needed. The temporal resolution should be at least 70,000 frames per second (fps). Thus, in dynamics, there is a compromise to be made between spatial resolution and acquisition frequency. Typical quasi-static studies use very high spatial resolution (for instance, 2048×2048 pixels) pictures to do DIC. In dynamics, this standard cannot be reached because of the very high temporal resolution needed. To get high-quality DIC measurements with a low spatial resolution, images contrast has to be optimal. Thus, a particular attention has been given to lightning techniques and speckle pattern. A very powerful lightning is required on the area of interest because of the low exposure time. In this contribution, a halogen flashlight (triggered with the input strain gauge) is used to overcome this issue. Furthermore, a speckle pattern is applied with an airbrush on the specimen face by using a painting adapted to high strain rates. Best results are obtained by applying first a homogenous white coat and then, a black speckle pattern whose typical size is 150 µm. Speckle size is determined with a digital microscope. Technical information is summarized in Tab. 2. Images are processed with Vic2D using 9 × 9 pixel subset for measuring displacements. This size was chosen to obtain reasonable displacement resolution in substrate areas. Too big subsets propagate high displacements from the adhesive joint to substrates. For measuring strains, bigger subsets (21 × 21 pixel subset) are used because of higher noise present in the strain field. The imaging area is presented in Fig. 11. For 15°and 45°loading conditions, the imaging area is initially shifted from the joint plane of 10 pixels in order to take into account the rigid body motion of the specimen during the SHPB test. For the 75°loading condition, the rigid body motion is more significant and the imaging zone is changed to focus only on the central zone of the specimen. In addition, the imaging area is initially shifted from the joint plane of 30 pixels. A picture of the experimental setup is proposed in Fig. 12 with the camera, the painted specimen, the Hopkinson bar system and a plexiglass protection. A flexible additional lighting system is required to calibrate the imaging area before the test precisely.

Considering the pixel size (52 µm or 55 µm) and the joint thickness (300 µm), this experimental setup does not enable us to measure local displacements in the adhesive joint directly. However, displacement fields in the adherents surrounding the joint are measured accurately. Therefore, a global strain (homogeneous along the joint thickness) can be estimated on this basis.

Results and discussion

This paper focuses on developing a methodology to characterize adhesive joints under dynamic and multiaxial loading conditions. Thus, following experimental results are presented to demonstrate potentialities offered by the proposed methodology but do not aim at contributing to material data bases. Due to confidentiality, most scales are normalized by arbitrary values in the following.

Global measurements

In this section, typical results obtained with the Hopkinson bar system are presented. Several tests have been performed for each loading condition. The (normalized) force obtained at the specimen/output bar interface is presented as a function of time for the three loading conditions in Fig. 13, 14 and 15. Parameters for different tests and loading conditions are summarized in Tab. 3, 4 and 5. Obtained signals are very similar for each loading condition. Differences are mainly due to variability of the striker speed. The higher the striker speed is and the sharper the output force evolves and the shorter the test is. Indeed, the test duration is limited by the joint failure, because otherwise, the striker length determines the test duration completely. Most presented tests have been performed with a 86 cm long striker which corresponds to a 350 µs test. In Fig. 13, 14 and 15, failure is reached before this global value and force drop corresponds to joint failure. In addition, contact conditions, specimen/bar alignment and specimen geometrical (e.g, joint thickness) and material variability explain, to a lesser extent, discrepancies that can be observed in Fig. 13, 14 and 15. Strain rates are evaluated from the Hopkinson tests between 700 and 1500 s -1 depending on the striker speed. The assembly strength (maximal force) is very repeatable with a standard deviation evaluated to 2.6 % of the average normalized strength for the 45°loading condition. The two other angles show similar standard deviations values concerning assembly strength. The 15°loading leads to the lowest strength because of the mixed traction/shear stress state. The 75°loading leads to the highest strength because of the high compression present in the stress state. This global measurement characterizes the whole assembly : adherents and adhesive joint. A local measurement is required to characterize the adhesive joint.

Local measurements

Striker speeds have been chosen so that 15 to 20 images can be taken during the test before joint failure. Images acquisition is triggered on signal detection at the input strain gauge. Horizontal displacements denoted by u are presented in Fig. 16 for the 45°loading condition and V = 5.8 m/s at different times: 1 µs, 14 µs, 41 µs and 180 µs. White squares are used as virtual gauges and indicate the specimen center just above and underneath the joint. Displacement amplitude clearly increases with time. Displacement jump between the two adherents is related to the global shear strain in the adhesive joint. Displacements are qualitatively consistent with numerical simulations presented in Fig. 7, shear stress reaches a maximum at the center and decreases near the specimen edges. On the last displacement field at 180 µs, one can notice zones where the DIC did not work (gray zones along the joint) because deformations were sufficient to make the paint chip off.

Vertical displacements denoted by v are presented in Fig. 16 for the 45°loading condition and V = 5.8 m/s at 180 µs. This displacement field is more difficult to interpret because there is no significant displacement jump (extrema lies between 353 and 320 µm). However, accuracy is sufficient to observe the compression level evolution along the joint length qualitatively. At the center (white squares), it is characterized by the difference between blue and green zones which result is low compression level. Compression increases toward the edge until it reaches a maximum characterized by the difference between blue/green and red zones and it decreases at the very edge. This qualitative evolution is consistent with numerical simulations presented in Fig. 7.

Consider (x, y, z) Cartesian coordinates corresponding respectively to the joint length, the joint thickness and the joint depth directions. Strains are computed as the symmetric part of the displacement gradient. Strain fields could be obtained directly from the DIC by considering the numerical gradient of displacements. However, measurement noise is amplified by this numerical procedure. In this contribution, a simplified analysis enables us to overcome this difficulty. Since only surface measurements are available, it is assumed that displacements are homogeneous along the z-direction. Thus, displacements along x and y denoted by u * and v * depends only on x and y in the specimen and displacements along z are set to zero. (The superscript * indicates that the quantity is related to the adherent, otherwise the quantity is considered in the adhesive joint). Moreover, as already mentioned, the spatial resolution is not sufficient to measure displacements in the joint directly. Thus, strains in the joint are assumed to be independent on the y-direction and are only related to the following displacement variations (where h denotes the joint thickness):

∆ u(x) = u * (x, h/2) -u * (x, -h/2) ∆ v(x) = v * (x, h/2) -v * (x, -h/2) (1)
And then, displacements u and v in the joint are assumed to be linear as a function of y and reads:

u(x) = ∆ u(x) h y + f u (x) and v(x) = ∆ v(x) h y + f v (x) (2)
Functions f u (x) and f v (x) may be evaluated in Fig. 16. Since strains are evaluated in this study at the joint center (i.e., using displacements in virtual gauges in Fig. 16), f u (x) and f v (x) are neglected. Moreover, variations of h∆ v(x) along the x-directions are assumed to be negligible compared to ∆ u(x), more precisely:

∆ u(x) >> h ∂ ∆ v(x) ∂ x (3)
Thus, by combining (1) and ( 2), the strain tensor that should be identified along the joint length reads:

ε xy (x) = ∆ u(x) 2h and ε yy (x) = ∆ v(x) h (4)
Normalized shear and normal strains are presented as a function of time at the joint center in Fig. 17 and 18 for 15°, 45°and 75°loading conditions and for V = 5.0, V = 5.8, V = 14.2 m/s, respectively. Those time evolutions are consistent with the previous numerical analysis and the three expected stress states described in Tab. 1. Virtual gauges (15-pixel white squares) can solve 1-µm displacements which set the minimal strain detection to 5.10 -3 . Normalized strains at failure are listed for all tests in Tab. 6. For each loading condition, a good repeatability is observed. In addition, both 45°and 75°loading conditions present similar shear strain at failure. The discrepancy may be explained by different compressive strain at failure. Indeed, compressive strain seems to reduce the effectiveness of shear on crack propagation. Much lower shear strains at failure are observed for the 15°loading condition. This happens because tension is also applied and then mixed shear/traction crack propagation occur.

Local stress measurement and joint behavior

Even though the measured force at the specimen/output bar interface gives information only about the whole assembly, one can present the global force against the local shear strain measured with DIC. Thus, an approximate behavior of the joint is obtained and presented in Fig. 19. Good reproducibility is achieved. This analysis supposes a homogeneous shear stress state all along the joint which is unrealistic from Fig. 7.

Proper behavior of the joint can be derived from the proposed measurements. Local stress measurement in the joint can be performed and inferred directly from the DIC measurement. Thus, the adhesive joint behavior is estimated without the development of an inverse numerical method. Indeed, displacement fields can be post-treated by Vic2D to compute strain fields. As already mentioned, this procedure amplifies measurement noise. However, uncertainties related to the numerical derivation can be reduced by increasing the subset size in the correlation. Thus, strain measurements are obtained in the aluminum adherents near the adhesive joint as shown in Fig. 20 for instance. The virtual gauge (white rectangle) is set to 100×15 pixels to measure local stresses in the center. The virtual gauge size has been also increased to reduce measurement noise due to numerical derivation. Several assumptions are needed to infer stresses from measured strains. Since surface measurements are done, plane stress state is assumed. Dynamic characterization of the aluminum adherent alone enabled us to identify material parameters (i.e., the Young modulus E * , the shear modulus G * and the Poisson ratio ν * ). Elastic behavior is assumed in the aluminum. Stresses in the aluminum adherent are given as a function of strains:

σ yy = E * 1 -(ν * ) 2 (ε yy + ν * ε xx ) and σ xy = 2G * ε xy (5)
Thus, one can measure stresses in the aluminum close to the adherent/joint boundary and infer stresses in the joint by continuity of normal stress (i.e., σ yy and σ xy ) through the interface. However, the stress component σ yy is very difficult to measure in the aluminum adherent because ε xx and ε yy are needed and measurement noise of each signal accumulates. In addition, ε yy is very noisy because there is very few pixels along the y-direction. In this contribution, this stress component is not proposed.

Shear strain ε xy in the adherent is presented in Fig. 20 for the 45°loading condition at 175 µs. The red part corresponds to the strain in the adherent and the highly deformed purple part corresponds to the strain in the joint. However, strains in the adhesive part cannot be used because there are too few pixels in the adhesive part. Strains in this part cannot be trusted.

Shear stress in the virtual gauge is then obtained from [START_REF] Challita | Experimental investigation of the shear dynamic behavior of double-lap adhesively bonded joints on a wide range of strain rates[END_REF]. Stresses are assumed to be homogeneous in the joint along the joint thickness (y-direction). By relating this stress measurement and the strain measurement detailed in section Local measurements, an experimental behavior estimation of the joint can be derived. Normalized results are presented in As expected, both 15°and 45°loading conditions exhibit the same adhesive shear modulus, but different yield and failure stresses. This is a well-known phenomenon for polymers : yield and failure stresses increases with hydrostatic pressure. Thus, 45°-yield stress is higher than 15°-yield stress because of the compression in the test at 45°.

Conclusion

This paper deals with the development of a new specimen dedicated to dynamic and multiaxial testing of adhesive joints. The general method consists in using classic split Hopkinson pressure bar combined with local displacement measurements performed by Digital Image Correlation. The innovative DODECA specimen allows three different impact loadings, which means three different stress states to test. Simple assumptions on the local displacement field enabled us to get local strain and stress in the adhesive joint and to obtain an experimental estimation of the joint behavior. In addition, strains at failure are also an interesting result of the proposed experimental procedure.

The introduction of DIC in the dynamic characterization of adhesive joints provide new means for developing new adhesive joints testing experiments. This is particularly interesting because of the nonuniformity of the strain field in specimen designed for adhesive joints testing. It will contribute to a further understanding of the material response of adhesive joints.

Further work is still necessary to interpret the presented measurements. In particular, numerical inverse method would be useful to identify a complete behavior of the adhesive joint by minimization procedures. The inverse calculation would require an accurate numerical dynamic model able to represent both force and local displacements with a reasonable calculation time.

Fig. 2

 2 Fig. 2 Schematic view of the DODECA specimen

Fig. 3

 3 Fig. 3 Geometrical adjustments to avoid rotation for the 15°condition

Fig. 4 aFig. 5

 45 Fig. 4 a) Specimens before curing and bonding devices. b) Specimen just after curing. c) Final specimen ready to be tested

Fig. 6

 6 Fig. 6 Numerical simulation: normal stress σ yy (black line) and shear stress σ xy (blue line) versus the distance to the joint centre for the 15°loading condition

Fig. 7

 7 Fig. 7 Numerical simulation: normal stress σ yy (black line) and shear stress σ xy (blue line) versus the distance to the joint centre for the 45°loading condition

Fig. 8

 8 Fig. 8 Numerical simulation: normal stress σ yy (black line) and shear stress σ xy (blue line) versus the distance to the joint centre for the 75°loading condition

Fig. 9 Fig. 10

 910 Fig. 9 Numerical simulation : normal stress σ yy as a function of distance to joint centre, with and without beaks

Fig. 11 a

 11 Fig. 11 a) Imaging area for 15°and 45°loading conditions. b) Imaging area for the 75°loading condition

Fig. 14 Fig. 15

 1415 Fig. 14 Normalized force as a function of time for the 75°loading condition

Fig. 16

 16 Fig. 16 Horizontal (top) and vertical (bottom) displacement field measured by DIC for the 45°loading condition at v = 5.8 m/s at different times: 1 µs, 14 µs, 41 µs and 180 µs

Fig. 17 Fig. 18

 1718 Fig. 17 Normalized shear strain ε xy as a function of time for 15°, 45°and 75°loading conditions

Fig. 19

 19 Fig. 19 Normalized force as a function of normalized shear strain ε xy for 45°loading conditions

Fig. 21

 21 Fig. 21 Adhesive joint behavior identified on 15°and 45°loading conditions : normalized shear stress σ xy as a function of normalized shear strain ε xy

Table 1

 1 Summary of available stress distributions with the DODECA specimen

	Loading	Shear	Compression Tension
	angle (°)
	15	++	no	+
	45	++	+	no
	75	+	++	no

Table 2

 2 High-speed imaging parameters

	Camera	Phantom v7.3
	Lighting	Visatec SOLO 1600 B
		(halogen flashlight, 150
		W)
	Lens	Nikon 105 mm f 2.8 1:1
	Exposure time	1 µm
	Time between 2 images	13-14 µs
	Resolution 15°and 45°416×80 pixels (1 px 52
		µm)
	Resolution 75°196×144 pixels (1 px 55
		µm)
	Software DIC	Vic2D
	Subset displacements	9 pixels (step : 1 px)
	Subset strains	21 pixels (step : 1 px)

Table 3

 3 Summary of four tests for the 45°loading condition

	Striker length	Striker speed V	Normalized
	(cm)	(m/s)	strength
	86	7.2	0.94
	86	6.3	0.89
	86	5.8	0.90
	86	5.8	0.895
	Fig. 13 Normalized force as a function of time for the 45°loading condition

Table 4

 4 Summary of three tests for the 75°loading condition

	Striker length	Striker speed V	Normalized
	(cm)	(m/s)	strength
	120	12.4	1.6
	120	11.9	1.37
	86	14.2	1.44
	Table 5 Summary of three tests for the 15°loading condition
	Striker length	Striker speed V	Normalized
	(cm)	(m/s)	strength
	86	5.0	0.65
	86	4.3	0.56
	86	4.8	0.56

Table 6

 6 Summary of normalized failure strain for all tested specimens

	Loading	Shear	Traction/compression
	condition	strain at failure	strain at failure
	15°0.27	0.70
	15°0.13	0.25
	15°0.20	0.35
	45°0.80	-0.15
	45°0.75	-0.25
	45°0.78	-0.20
	45°0.87	-0.15
	75°0.98	-1.30
	75°0.91	-1.10
	75°0.95	-1.20

Table 7

 7 Identification of normalized elasto-plastic parameters from Fig 21

		15°45°G
		5.9	6.2
	σ y	0.35	0.60

Acknowledgements Authors thank Philippe Chevallier (LMS) for his help with experiments and Gerard Gary (LMS) for his useful advice on dynamic experiments.


D Weisz-Patrault
email: daniel.weisz-patrault@polytechnique.edu

Energetic approach coupled with analytic solutions for the evaluation of residual stress

Keywords: Energetic approach, Plasticity, Analytic solution

Cet article contribue plus généralement à une stratégie de calcul mixte analytique/numérique visant à calculer les contraintes résiduelles apparaîssant dans les bandes métaliques après le bobinage. Au cours du refroidissement des bobines on assiste à des transformations multi-phases et à de la plasticité de transformation induite. Ainsi, chaque spire de l'enroulement est soumise à une déformation libre globale qui peut être suffisante pour générer des déformations plastiques macroscopiques. Pour chaque spire on dérive une solution du problème d'un tube cylindrique soumis à une déformation libre arbitraire. Les dévelopements mathématiques reposent sur l'équation linéaire inhomogène de Navier en traitant la plasticité par l'introduction d'une déformation plastique déviatorique inconnue. Une solution analytique est obtenue sous forme de dévelopement en série pour chaque déformation plastique testée. Enfin, un principe énergétique permet de déterminer la déformation plastique solution du problème. En pratique, un optimisation numérique est menée directement sur les coefficients du développement en série de la déformation plastique.

Introduction

The current dynamic of steel manufacturing is to regularly develop new stronger grades enabling users to reduce strips or profile thicknesses and thus reduce produced tonnages, which participates to the energy efficiency by minimizing for instance the total mass of vehicles etc. One of the major issues related to this evolution of steel production is the forming processes that lead to serious residual stress problems which in turn can result in instabilities such as strip buckling during rolling process or coils collapsing on themselves. In addition to heterogeneous plastic deformations, irreversible deformations responsible for these residual stresses are due to different phase transitions under applied loads that occur during most forming processes. In order to establish technological strategies aimed at a better control of residual stress fields, it is essential to understand and to simulate accurately these processes. This paper is situated within the frame work of numerical simulation of the coiling process of steel. Plastic deformations along with multi-phase transitions are responsible for large irreversible strain leading to major residual stress issues. A non-linear mixed analytical/numerical approach has been recently proposed [START_REF] Weisz-Patrault | Residual stress generated by transformation induced plasticity. application to stability of coils[END_REF] in order to compute residual stresses generated by different contributions of inelastic eigenstrain occurring during the coiling process (including both the winding phase and the cooling phase). In particular, transformation induced plasticity has been taken into account following the recent work [START_REF] Weisz-Patrault | Multiphase model for transformation induced plasticity. extended leblond's model[END_REF] based on the classical Leblond's model [START_REF] Leblond | A theoretical and numerical approach to the plastic behaviour of steels during phase transformations i. derivation of general relations[END_REF][START_REF] Leblond | A theoretical and numerical approach to the plastic behaviour of steels during phase transformations ii. study of classical plasticity for idealplastic phases[END_REF][START_REF] Leblond | Mathematical modelling of transformation plasticity in steels i: case of ideal-plastic phases[END_REF][START_REF] Leblond | Mathematical modelling of transformation plasticity in steels ii: coupling with strain hardening phenomena[END_REF]. The mixed analytical/numerical approach [START_REF] Weisz-Patrault | Residual stress generated by transformation induced plasticity. application to stability of coils[END_REF] consists for each time step in applying the overall inelastic eigenstrain (depending on the previous time step) by solving analytically the inhomogeneous Navier equation in each layer of the coil. Contact pressures are updated by numerical optimization. The analytic solution relies on a series expansion of the right side term of the inhomogeneous Navier equation. A specific function basis has been introduced obtain simple identification of the solution. The homogeneous solution is more classically obtained by using harmonic potential theory as exposed in [START_REF] Barber | of Solid Mechanics and Its Applications[END_REF] and bi-harmonic potentials as in [START_REF] Weisz-Patrault | Evaluation of contact stress during rolling process, by three dimensional analytical inverse method[END_REF].

Even though non-linear contributions such as microscopic plasticity have been taken into account, macroscopic plasticity has been neglected, that is to say that the macroscopic von Mises equivalent stress does not reach the macroscopic yield stress. However, macroscopic plasticity may occur if the yield stress has already been reached during the winding phase of the process, for rather thick strips for instance. Thus, this paper is an attempt to introduce, under simplifying assumptions, macroscopic plasticity by adding an unknown deviatoric contribution to the imposed eigenstrain. Then, an energetic approach is used to identify this plastic contribution. It consists in minimizing the total stored elastic energy plus the plastic dissipation associated to the plastic strain tensor introduced in the eigenstrain. This plastic contribution can be interpreted as a distance (or a cost) between different states [START_REF] Fedelich | An analysis of stability of equilibrium and of quasi-static transformations on the basis of the dissipation function[END_REF][START_REF] Mielke | Energetic formulation of multiplicative elasto-plasticity using dissipation distances[END_REF]. Thus, the energetic approach consists in seeking the lowest energy state by taking into account the cost by terms of dissipation distance. The proposed solution combines analytical developments for the inhomogeneous Navier equation and numerical optimizations for the identification of the plastic strain tensor.

Decomposition of the problem

In this contribution only one layer is considered, the problem of the determination of contact pressures being addressed in [START_REF] Weisz-Patrault | Residual stress generated by transformation induced plasticity. application to stability of coils[END_REF]. For each time step the body is subjected to an imposed eigenstrain computed on the basis of the extended Leblond's model [START_REF] Weisz-Patrault | Multiphase model for transformation induced plasticity. extended leblond's model[END_REF]. This contribution focuses only on one time step and solves semi-analytically the problem of an elastic-plastic tube subjected to an arbitrary eigenstrain under axi-symmetrical assumption as proposed in [START_REF] Weisz-Patrault | Residual stress generated by transformation induced plasticity. application to stability of coils[END_REF]. Even though a non-linear behavior is considered, the proposed strategy relies on linear solutions. Indeed, the linear inhomogeneous Navier equation is solved for any unknown trial plastic strain ε p (where p stands for plastic). The latter is determined in the end by minimizing the sum of the elastic energy E [ε p ] and the plastic dissipation D [ε p ] associated to ε p . Therefore, the problem can be decomposed into sub-problems as shown in figure 1 and the plastic strain considered as solution denoted by ε p,s (where s stands for solution) is numerically determined by:

ε p,s = argmin ε p , tr ε p =0 E [ε p ] + D [ε p ] (1)
Figure 1: Decomposition

Mathematical preliminaries

This section deals with mathematical definitions needed for the proposed solution. The following functions are proposed for expanding several functions arising in the paper into series:

G (α,β) m (r) = J α x (β) m r r sup Y β x (β) m r inf r sup -J β x (β) m r inf r sup Y α x (β) m r r sup (2)
where J α and Y α are the Bessel functions of the α-th order of the first and second kind respectively and x (β) m are successive positive roots (indexed by m ∈ N) of

x → J β (x) Y β x r inf r sup -J β x r inf r sup Y β (x) (3)
Introducing the following scalar product:

f, g = rsup r inf rf (r)g(r)dr (4)
One obtains the following orthogonality relations:

G (α,α) m , G (α,α) l = G (α,α) m , G (α,α) l if m = l 0 if m = l (5)
In the following, some functions will be projected on the linear span of functions

G (α,α) m denoted by span G (α,α) m (r), 1 ≤ m ≤ M
where M is a fixed integer. However, it should be noted that G (α,α) m vanishes at r = r inf and r sup . Thus, if functions that do not vanish at these points are considered one should add for instance J α (r/r sup ) and Y α (r/r sup ) to the vector space in order to have non-vanishing values at r = r inf and r = r sup . Thus the vector space on which most functions arising in the following will be projected reads:

A (α) = span J α r r sup , Y α r r sup , G (α,α) m (r), 1 ≤ m ≤ M (6) Let f : (r, z) ∈ [r inf , r sup ]×[-L, L] → f (r, z
) be a function sufficiently regular so that scalar products (4) are well defined, one can write for all α ∈ R by first expanding f (r, z) into a Fourier series along the z-direction and then by projecting all r-dependent Fourier coefficients denoted by f k (r) on the vector space A (α) :

f (r, z) ≃ K k=-K A k J α r r sup + B k Y α r r sup + M m=1 C m,k G (α,α) m (r) exp ikπ L z (7)
where the Fourier coefficients according to the z-direction are:

f k (r) = 1 2L L -L f (r, z) exp -ikπ L z dz (8)
where the projection reads:

C m,k = G (α,α) m , f k (r) G (α,α) m , G (α,α) m (9)
and where values at r = r inf and r = r sup gives:

A k B k = J α r inf rsup Y α r inf rsup J α (1) Y α (1) -1 . f k (r inf ) f k (r sup ) (10)

Inhomogeneous problem A

The first problem is a tube subjected to the imposed eigenstrain ε * . Boundary conditions are not specified, that is to say that only a particular solution of the inhomogeneous Navier equation is sought regardless of surface traction T . The obtained stress field is denoted by σ * and the traction vector σ * .n will be corrected by adding the homogenous solution of the problem C. The solution of problem A is already addressed in [START_REF] Weisz-Patrault | Residual stress generated by transformation induced plasticity. application to stability of coils[END_REF]. Main results are stated here for sake of clarity. The inhomogeneous Navier equation reads:

µ∆ u * + (λ + µ)∇div u * = div (λtr (ε * ) I + 2µε * ) (11)
Hence, considering the axi-symmetrical assumption:

       (λ + 2µ) ∂ 2 u * r ∂r 2 + 1 r ∂u * r ∂r - u * r r 2 + µ ∂ 2 u * r ∂z 2 + (λ + µ) ∂ 2 u * z ∂r∂z = f * r (r, z) (λ + 2µ) ∂ 2 u * z ∂z 2 + µ ∂ 2 u * z ∂r 2 + 1 r ∂u * z ∂r + (λ + µ) ∂ 2 u * r ∂r∂z + 1 r ∂u * r ∂z = f * z (r, z) (12)
where:

       f * r (r, z) = (λ + 2µ) ∂ε * rr ∂r + λ ∂ε * θθ ∂r + ∂ε * zz ∂r + 2µ ε * rr r - ε * θθ r + ∂ε * rz ∂z f * z (r, z) = (λ + 2µ) ∂ε * zz ∂z + λ ∂ε * rr ∂z + ∂ε * θθ ∂z + 2µ ∂ε * rz ∂r + ε * rz r (13)
Then by using the procedure described in section 3, following series expansions are considered:

             f * r (r, z) = K k=-K A r k J 1 r r sup + B r k Y 1 r r sup + M m=1 C r m,k G (1,1) m (r) exp ikπ L z f * z (r, z) = K k=-K A z k J 0 r r sup + B z k Y 0 r r sup + M m=1 C z m,k G (0,1) m (r) + D z k exp ikπ L z (14)
where coefficients

A r k , A z k , B r k , B z k , C r m,k , C z m,k
, and D z k are explicitly computed as functions of the imposed eigenstrain ε * following the procedure detailed in section 3. It should be noted that f * r (r, z) and the partial derivative of f * z (r, z) with respect to r have been expanded. Since the link between ε * and f * r (r, z), f * z (r, z) is not essential in this contribution, details are omitted and the reader is simply referred to [START_REF] Weisz-Patrault | Residual stress generated by transformation induced plasticity. application to stability of coils[END_REF]. However, it should be mentioned that numerical derivations of the imposed eigenstrain are necessitated. A particular solution of the inhomogeneous Navier equation is sought as follows:

             u * r (r, z) = K k=-K U r k J 1 r r sup + V r k Y 1 r r sup + M m=1 u r m,k G (1,1) m (r) exp ikπ L z u * z (r, z) = K k=-K U z k J 0 r r sup + V z k Y 0 r r sup + M m=1 u z m,k G (0,1) m (r) + W z k (r) exp ikπ L z (15)
where:

W z k (r) =        D z k r 2 4µ if k = 0 - D z k (kπ/L) 2 (λ + 2µ) if k = 0 (16)
By plugging (15) into ( 12) and identifying:

U r k U z k = S k . A r k A z k and V r k V z k = S k . B r k B z k and u r m,k u z m,k = S (1) m,k . C r m,k C z m,k (17)
where:

S k =   -(λ + 2µ) 1 rsup 2 -µ kπ L 2 -(λ + µ) 1 rsup ikπ L (λ + µ) 1 rsup ikπ L -µ 1 rsup 2 -(λ + 2µ) kπ L 2   -1 (18)
and:

S (β) m,k =    -(λ + 2µ) x (β) m rsup 2 -µ kπ L 2 -(λ + µ) ikπ L x (β) m rsup (λ + µ) x (β) m rsup ikπ L -µ x (β) m rsup 2 -(λ + 2µ) kπ L 2    -1 (19)
A particular displacement field u * r , u * z , solution of the inhomogeneous Navier equation ( 12) has been established. Therefore the associated stress field σ * can be computed as well, using the isotropic behavior. More precisely, a Fourier series expansion of σ * is obtained since the displacement field is known as a Fourier series expansion.

Inhomogeneous problem B

The second problem is similar to the problem B. A tube subjected to the unknown deviatoric plastic strain ε p is considered. For the previous problem A, the relationship between f * r (r, z), f * z (r, z) and the imposed eigenstrain ε * is purely numerical, since ε * is known. One could directly consider the series expansion of f * r (r, z), f * z (r, z) without referring to ε * . However, in this section the plastic strain ε p is unknown and should be determined through [START_REF] Weisz-Patrault | Residual stress generated by transformation induced plasticity. application to stability of coils[END_REF] in the end. The proposed minimization is done directly on ε p and not on the right side term of the inhomogeneous Navier equation denoted by f p r (r, z), f p z (r, z). Therefore one should consider a series expansion of ε p instead of f p r (r, z), f p z (r, z) and solve the inhomogeneous equation.

The unknown plastic strain ε p is deviatoric. Therefore, it remains three independent components, namely ε p rr , ε p θθ and ε p rz . In this contribution, an approximate solution is obtained by introducing an assumption so that the number of independent components is reduced to two. Considering applications to coiling process, shear stresses are much smaller than other components and can be neglected in the plastic strain, thus ε p rz ≃ 0. Therefore considering deviatoric plastic strain (i.e., ε p θθ = -ε p rr -ε p zz ), the inhomogeneous Navier equation associated to the problem B reads:

       (λ + 2µ) ∂ 2 u p r ∂r 2 + 1 r ∂u p r ∂r - u p r r 2 + µ ∂ 2 u p r ∂z 2 + (λ + µ) ∂ 2 u p z ∂r∂z = f p r (r, z) (λ + 2µ) ∂ 2 u p z ∂z 2 + µ ∂ 2 u p z ∂r 2 + 1 r ∂u p z ∂r + (λ + µ) ∂ 2 u p r ∂r∂z + 1 r ∂u p r ∂z = f p z (r, z) (20)
where:

     f p r (r, z) = 2µ ∂ε p rr ∂r + 2 ε p rr r + ε p zz r f p z (r, z) = 2µ ∂ε p zz ∂z (21)
Following ideas developed in section 3 the right side term of the Navier equation is expanded into series:

             f p r (r, z) = K k=-K A p,r k J 1 r r sup + B p,r k Y 1 r r sup + M m=1 C p,r m,k G (1,1) m (r) exp ikπ L z f p z (r, z) = K k=-K A p,z k J 0 r r sup + B p,z k Y 0 r r sup + M m=1 C p,z m,k G (0,1) m (r) + D p,z k exp ikπ L z (22)
where

G (α,β) m
is defined by ( 2) and where f p r (r, z) and the partial derivative of f p z (r, z) with respect to r. As mentioned above, series expansion of the plastic strain should be obtained in order to write the plastic dissipation. Following expressions are solutions of (21) considering that the right side terms are given by ( 22):

ε p zz = K k=-K a p,z k J 0 r r sup + b p,z k Y 0 r r sup + M m=1 c p,z m,k G (0,1) m (r) + d p,z k exp ikπ L z (23)
and:

ε p rr = K k=-K a p,r k J 2 r r sup + b p,r k Y 2 r r sup + M m=1 c p,r m,k G (2,1) m (r) exp ikπ L z - K k=-K r sup r a p,z k J 1 r r sup + b p,z k Y 1 r r sup + M m=1 c p,z m,k x (1) m G (1,1) m (r) + d p,z k 2 exp ikπ L z (24) with:                  A p,r k = 2µa p,r k r sup B p,r k = 2µb p,r k r sup C p,r m,k = 2µc p,r m,k x (1) m r sup and                  A p,z k = 2µa p,z k ikπ L B p,z k = 2µb p,z k ikπ L C p,z m,k = 2µc p,z m,k ikπ L D p,z k = 2µd p,z k ikπ L (25)
Right side terms f p r (r, z) and f p z (r, z) are expressed as Bessel functions of the first and zero orders respectively alike f * r (r, z) and f * z (r, z) in ( 14). It should be noted that coefficients a p,r k , b p,r k , c p,r m,k and a p,z k , b p,z k , c p,z m,k are determined in section 7 through a minimization procedure according to [START_REF] Weisz-Patrault | Residual stress generated by transformation induced plasticity. application to stability of coils[END_REF]. Displacements u p r and u p z are sought in the form:

             u p r (r, z) = K k=-K U p,r k J 1 r r sup + V p,r k Y 1 r r sup + M m=1 u p,r m,k G (1,1) m exp ikπ L z u p z (r, z) = K k=-K U p,z k J 0 r r sup + V p,z k Y 0 r r sup + M m=1 u p,z m,k G (0,1) m (r) + W p,z k exp ikπ L z (26)
Where for k > 0 (clearly D p,z 0 = 0 hence W p,z 0 = 0):

W p,z k = - D p,z k (kπ/L) 2 (λ + 2µ) (27)
By inserting (26) into (20) and identifying, one obtains:

U p,r k U p,z k = S k . A p,r k A p,z k and V p,r k V p,z k = S k . B p,r k B p,z k and u p,r m,k u p,z m,k = S (1) m,k . C p,r m,k C p,z m,k (28)
where S k and S

m,k are defined by ( 18) and (19). A particular displacement field u p r , u p z , solution of the inhomogeneous Navier equation (20) has been established. Therefore the associated stress field σ p can be computed as well, using the isotropic behavior. More precisely, a Fourier series expansion of σ p is obtained since the displacement field is known as a Fourier series expansion.

Homogeneous problem C

In this section additional homogenous solutions are derived in order to verify boundary conditions. Indeed only particular solutions of the inhomogeneous Navier equation have been exhibited so far, regardless of the surface traction denoted by T . Thus the surface traction considered in this section is T tot = Tσ * .nσ p .n as shown in figure 1. This part of the solution is classically obtained by using harmonic potential theory and bi-harmonic potentials. The problem being solved in this section is a simple tube subjected to the surface traction T tot (r inf ) and T tot (r sup ) at the inner and outer radii respectively with neither body forces nor imposed eigenstrain. This part of the solution is identical to those derived in [START_REF] Weisz-Patrault | Residual stress generated by transformation induced plasticity. application to stability of coils[END_REF] and similar to those presented in [START_REF] Weisz-Patrault | Evaluation of contact stress during rolling process, by three dimensional analytical inverse method[END_REF]. Classic harmonic and bi-harmonic potentials are used as exposed in [START_REF] Barber | of Solid Mechanics and Its Applications[END_REF]. One can mention that an hypercomplex potential formulation [START_REF] Weisz-Patrault | Three-dimensional elasticity based on quaternion-valued potentials[END_REF] could also have been used. Details are not exposed in this contribution since the originality of the present work relies more on the determination of the unknown plastic strain by minimization procedures. Thus, displacement and stress fields u and σ are assumed to be known for each set of tested coefficients a p,r k , b p,r k , c p,r m,k and a p,z k , b p,z k , c p,z m,k .

Energetic approach

The unknown plastic strain ε p or more precisely C p = a p,r k , b p,r k , c p,r m,k , a p,z k , b p,z k , c p,z m,k are determined in this section using energetic arguments. The total elastic stored energy E [C p ] plus the dissipated energy associated to the plastic strain D [C p ] should be minimized and the problem reduces to [START_REF] Weisz-Patrault | Residual stress generated by transformation induced plasticity. application to stability of coils[END_REF]. It should be noted that the minimization process contains all non-linear aspects of the problem. Indeed, the Navier equation remains linear even with an elastic-plastic behavior, only the determination of the plastic strain is not a linear procedure. In the following we consider that the external work vanishes, then the total stored energy reads:

E [C p ] = 1 2 V σ : εdV (29)
where V denotes the volume of the tube, σ = σ * + σ p + σ is the total stress considering all the three problems A,B and C and ε is the associated strain with the isotropic behavior of Lamé's coefficients λ, µ.

The dependence of the elastic energy on the plastic strain is not explicit in (29), however, σ p obviously depends on ε p as well as σ because of the surface traction Tσ * .nσ p .n.

The equivalent plastic strain rate (or the cumulative plastic strain rate) is defined by:

εeq = 2 3 εp : εp (30)
The dissipated energy is defined as:

D [C p ] = V ε eq σ Y dV (31)
where σ Y denotes the yield stress and ε eq depends explicitly on C p through (24). Thus the minimization problem (1) enabling to determine the solution (denoted with the superscript s) reads:

C p,s = argmin C p E [C p ] + D [C p ] (32)
In practice the gradient free Nelder-Mead algorithm base on simplex updates and included in the free software Scilab [START_REF] Scilab | Scilab: Free and open source software for numerical computation[END_REF] has been used in order to solve (32).

Results

In this section a numerical test is performed in order to validate the proposed approach. An rather arbitrary eigenstrain given by (33) and presented in figure 2 is imposed with free surface traction (i.e., T = 0). Comparisons with a Finite Element computation, performed using Castem [START_REF]Cast3m[END_REF] are proposed. Even though the proposed approach is intended to coil modeling for which layers are very thin compared to their width (usual thicknesses are less than 1 mm for widths greater than 1 m) a very thick hollow cylinder is considered in this paper in order to avoid long computation times for the FEM. Indeed, a very thin layer would have necessitated a refined mesh. Geometrical parameters are listed in table 1.

Practically an axi-symmetrical Finite Element simulation has been performed with linear N r × N z quadrangular elements. A good agreement is observed between the presented solution and the Finite Element computation. The purely elastic solution is added in order to show the effect of plasticity. It should be mentioned that boundary conditions are correctly verified by the proposed approach and not correctly verified by the Finite Element computation, indeed σ rr and σ rz should vanish at r = r inf and r = r sup and σ rz should vanish at Z = 0 considering the symmetry of the problem. This aspect may explain the discrepancy observed for σ rz in figures 3d and 5d. Moreover the equivalent plastic strain (or the cumulative plastic strain) is quite well predicted by the proposed model as shown in figures 4 and 6.

ε * = 0.2 (r -r sup )(r -r inf ) r sup r inf cos π z L (e r ⊗ e r -e θ ⊗ e θ ) (33)

Conclusion

A mixed analytical/numerical method has been developed to solve the problem of an hollow cylinder subjected to arbitrary eigenstrain and surface traction. An unknown deviatoric plastic strain is added to the imposed eigenstrain and the linear inhomogeneous Navier equation is solved analytically by introducing specific series expansions. The unknown plastic strain is then identified eventually by minimizing the sum of the elastic energy and the dissipated energy associated to the unknown plastic strain. This approach enables to use conveniently the linearity of the Navier equation even though plasticity is considered, non-linear aspects being encompassed in the energetic formulation. A comparison with a Finite Element computation has been proposed and good agreement is observed. This paper is part of a more general numerical strategy consisting in modeling residual stresses (generated by phase transitions, transformation induced plasticity etc...) in coils.

Figure 2 :

 2 Figure 2: Imposed eigenstrain

Figure 3 :Figure 4 :

 34 Figure 3: Test 1: Stress comparision with Finite Element simulation

Figure 5 :Figure 6 :

 56 Figure 5: Stress comparision with Finite Element simulation

Table 1 :

 1 Geometrical parameters

	L	40 (mm)
	r inf 15 (mm)
	r sup 20 (mm)
	N r	20
	N z	40