Andreea Carpiuc-Prisacari

Clément Jailin

Martin Poncelet

Kyrylo Kazymyrenko

Hugo Leclerc

François Hild

Experimental database of mixed-mode crack propagation tests performed on mortar specimens with a hexapod and full-field measurements. Part II: interactive loading

Keywords: Crack propagation, Digital Image Correlation, Mixed-mode fracture, Mortar, Multiaxial tests

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Introduction

In a companion paper proportional and non-proportional crack propagation tests were performed on mortar specimens [START_REF] Carpiuc-Prisacari | Experimental database of mixed-mode crack propagation tests performed on concrete specimens with a hexapod and full-field measurements[END_REF]. The experimental protocol is a modern version of the so-called Nooru-Mohamed tests [START_REF] Nooru-Mohamed | Mixed-mode fracture of concrete: an experimental approach[END_REF][START_REF] Nooru-Mohamed | Experimental and numerical study on the behavior of concrete subjected to biaxial tension and shear[END_REF]. While keeping the same specimen geometry and loading types (Fig. 1(a-b)), the testing machine and the measurement techniques were improved. Contrary to other tests found in the literature [START_REF] Iosipescu | New accurate procedure for single shear testing of metals[END_REF][START_REF] Arrea | Mixed mode crack propagation in mortar and concrete[END_REF][START_REF] Bažant | Shear fracture tests of concrete[END_REF][START_REF] Biolzi | Mixed mode fracture in concrete beams[END_REF][START_REF] Swartz | Mixed mode crack propagation and fracture in concrete[END_REF][START_REF] Ballatore | Mixed mode fracture energy of concrete[END_REF][START_REF] Bocca | Mixed mode fracture of concrete[END_REF][START_REF] Schlangen | Mixed-mode fracture propagation: a combined numerical and experimental study[END_REF][START_REF] Brokenshire | Torsional fracture tests[END_REF][START_REF] Winkler | Traglastuntersuchungen von unbewehrtenund bewehrten Betonstrukturen[END_REF][START_REF] Galvez | Mixed mode fracture of concrete under proportional and nonproportional loading[END_REF][START_REF] Nooru-Mohamed | Mixed-mode fracture of concrete: an experimental approach[END_REF], these experiments were conducted with an experimental protocol combining two main features, namely, the loading history was applied with a hexapod [START_REF] Nierenberger | Multiaxial testing of materials using a stewart platform: Case study of the nooru-mohamed test[END_REF][START_REF] Le Flohic | A 3D displacement control by digital image correlation for the multiaxial testing of materials with a Stewart platform[END_REF], and the displacement fields on both faces of the samples were measured via Digital Image Correlation (DIC) [START_REF] Tomičević | Mechanics-aided digital image correlation[END_REF]. The main conclusions of this first series of tests were:

Y Z P r 0 Y R X U Z U Y (a) Y Z P r 0 Y R X U Z U Y (b) Y Z P r 0 Y R X U Z U Y (c)
• It was possible to accurately conduct tests on such a testing machine.

• Tests were repeatable at the global level.

• Mean force / mean displacement curves were consistent for identical test conditions, and in good agreement with the crack propagation scenario of each test.

• In the case of proportional loading histories, DIC results were in good agreement with the final crack pattern.

• These tests, close to the Nooru-Mohamed configurations, shed new light on their interpretations.

However, there were some shortcomings:

• The period of stability of crack propagation was rather limited (i.e., few millimeters to centimeters). Consequently, the 'period of interest' was short for each test.

• Final cracks were straight, i.e., no reorientation occurred. It would be interesting to perform discriminating and sensitive tests to such reorientations. The Stress Intensity Factor (SIF) ratio K II K I thus was very limited during the stable propagation period.

• As previously observed [START_REF] Van Mier | Geometrical and structural aspects of concrete fracture[END_REF] on similar specimens and loading paths, the crack propagation differed on the surface and in the bulk of the sample.

It was confirmed [START_REF] Carpiuc-Prisacari | Experimental database of mixed-mode crack propagation tests performed on concrete specimens with a hexapod and full-field measurements[END_REF], since the DIC results were not representative of the cracks in the material bulk for non-proportional loadings. Therefore, a detailed comparison of the propagation regime between experimental and numerical results is expected to be more difficult to perform.

To circumvent these limitations, two important changes are proposed in this paper. First, in addition to the global tension (U Z ) and shear (U Y ) displacements that characterize the previous tests, an in-plane rotation (R X ) was added (Fig. 1(c)) with controlled positions of the instantaneous center of rotation P r . This rotational degree of freedom will induce a tensile (and/or compressive) stress gradient. Consequently, when the crack propagates the mode I SIF (K I ) will decrease and the propagation will possibly stop. It is even possible to have a part of the specimen under compression, thereby preventing any propagation or initiation in that region. Moreover, having a sufficiently heterogeneous stress field prevents surface cracks from initiating away from the notches. This technique is nothing more than the adaptation of the beam bending principle (where a stress gradient exists along the thickness of the beam, due to the moment induced by the forces applied at large distance) to a compact specimen case (i.e., the rotation applied at short distance directly induces a moment).

Second, the prescribed loading history will be changed during the test, depending on the crack tip location. The latter can be assessed on the fly by DIC analyses as shown with the previously reported proportional loadings [START_REF] Carpiuc-Prisacari | Experimental database of mixed-mode crack propagation tests performed on concrete specimens with a hexapod and full-field measurements[END_REF]. The SIF ratio that controls the crack orientation could be tuned by a combination of tension and shear. In the present case, the relationship between SIF and load applied on the boundaries is rather straightforward, namely, modifying the tensile level mainly changes the K I value, while varying the shear level mainly changes K II . This relationship is not exactly proportional, and depends on the crack tip location [START_REF] Jailin | Virtual hybrid test control of sinuous crack[END_REF]. Yet this rule-of-thumb analysis enables the loading path to be chosen in advance to obtain a given crack path. One will then rely on the location of the crack tip during the test to modify the loading history in order to accurately obtain the sought result. Changing the loading conditions (i.e., the SIF ratio) for a crack after non-negligible propagation is much more effective for crack reorientation investigations than changing the SIF ratio at the beginning of the test (as for non-proportional tests [START_REF] Carpiuc-Prisacari | Experimental database of mixed-mode crack propagation tests performed on concrete specimens with a hexapod and full-field measurements[END_REF]). The very beginning of propagation is not representative of a fully formed crack since it evolves in a material affected by crack initiation and its interaction with the underlying material microstructure.

The paper will briefly describe the experimental protocol in a first part.

Then the 4 so-called interactive tests will be described, and the test results will be analyzed.

Experimental setup

Most of the experimental setup is the same as in Ref. [START_REF] Carpiuc-Prisacari | Experimental database of mixed-mode crack propagation tests performed on concrete specimens with a hexapod and full-field measurements[END_REF] with some minor modifications/additions. For a detailed description of the experimental protocol, which focuses on a given test of this campaign, see Ref. [START_REF] Carpiuc-Prisacari | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF].

Testing machine

The testing device is a parallel machine composed of 6-axis electromechanical actuators, which are controlled in displacement mode. Its translation accuracy in the used conditions is about 1 µm [START_REF] Le Flohic | A 3D displacement control by digital image correlation for the multiaxial testing of materials with a Stewart platform[END_REF]. The combination of a very slow displacement rate, the monitoring of the crack tip location (described thereafter) and the ability to stop and change the loading path during the test, leads to the possibility to control crack propagation so that a specific the crack path (e.g., chevron, lightning bolt) is obtained.

Specimens

The specimens were 200 × 200 × 50 mm rectangular parallelepipeds with different notches sawed at mid height of the sample (Fig. 2), namely, two notches The samples were made of the same mortar (Table 1), casting and batch as in Ref. [START_REF] Carpiuc-Prisacari | Experimental database of mixed-mode crack propagation tests performed on concrete specimens with a hexapod and full-field measurements[END_REF], and were stored in the same water basin with the other ones. To perform the test reported hereafter, the specimens were glued to U-shaped plates at the ends of the machine.

Displacement measurements

Full field kinematic measurements were performed during the test via digital image correlation (DIC). Redundant macroscopic displacement measurements obtained with a contact method (LVDT setup) were used to check the measurement reliability.

2D-DIC setup

2D-DIC was performed on each 200 × 200 mm face of the specimens with 2 2048 × 2048 pixel cameras, triggered every 5 s by a common TTL signal.

Regularized DIC [START_REF] Tomičević | Mechanics-aided digital image correlation[END_REF] was used with 15 pixel-elements, and 250 pixel regularization length, which led to standard displacement uncertainties equal to 1.3 (resp.

2.4) µm along Z (resp. Y ) direction.

StereoDIC setup

A pair of cameras on each face was also used for (global) stereocorrelation [START_REF] Beaubier | CAD-based calibration and shape measurement with stereoDIC -principle and application on test and industrial parts[END_REF][START_REF] Dufour | CAD-based displacement measurements with stereo-DIC. Principle and first validation[END_REF], thereby leading to a complete optical setup made of 6 cameras (Fig. 3). These additional 4 devices were 60D The standard uncertainty of the rigid body motions between the upper and lower U-shaped plates obtained by stereocorrelation were assessed with 101 acquisitions when the hexapod was still. The values are 0.07 µm (resp. 0.68, 1.77) µm for a translation U X (resp. U Y , U Z ), and 3.6 × 10 -4 rad (resp. 7.7 × [START_REF] Carpiuc-Prisacari | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF].

10 -3 , 1.1 × 10 -3 ) for a rotation R X (resp. R Y , R Z )

Measurement redundancy

Six Linear Variable Differential Transducers (LVDT) were used, thereby providing a highly time-resolved measurement but with a limited spatial descrip-tion, namely, only the 3 translations and 3 rotations applied to the upper part of the specimen are obtained. They however provided redundant information for the measurement of the rigid body motions of the boundaries. This configuration enabled the consistency of the measurement results to be checked since these two setups were completely independent (i.e., 2 different measurement 130 techniques and 2 separate frames).

The sensors were located close to the specimen, on the octogonal reference plate (Fig. 4). They were held by a rigid steel frame fixed on the lower plate, so that the relative displacement between the upper and the lower part of the machine were measured. The acquisition frequency was 10 Hz and their own 135 standard uncertainty was 0.1 µm. As assessed in Ref. [START_REF] Carpiuc-Prisacari | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF], the deviation of the LVDT setup is about 1 µm over a 2-mm range because of small biases (e.g., misorientation of the octogonal plate, flatness defect).

Interactive tests

Based on the principle of adding an in-plane rotation R X about a center of

2D boundary conditions

Various types of in-plane loading histories were selected:

• Single reorientation test (IT 1). This test had both technical and scientific goals. From the technical point of view, it had to prove that the stabilization of crack propagation was possible, in addition to performing a clear reorientation after a first step of propagation (Fig. 5(a)). The loading path, schematically shown in Fig. 5(b), is simple. The first step is a proportional in-plane rotation (with the center of rotation P r located at mid-width of the specimen, P r = 0) with global shear in order to initiate and propagate a crack [START_REF] Carpiuc-Prisacari | Experimental database of mixed-mode crack propagation tests performed on concrete specimens with a hexapod and full-field measurements[END_REF]. The shear component is then slowly decreased to change its sign, so that the crack reorients [START_REF] Nooru-Mohamed | Mixed-mode fracture of concrete: an experimental approach[END_REF]. Last, the center of rotation P r is progressively moved to the left, up to a location out of the specimen (i.e., P r = 110 mm, while the left boundary of the specimen is at 100 mm and the left notch at 75 mm) to resume propagation (3). • Crack link-up test (IT 2). The technical interest of this test was to show that it was possible to independently initiate and propagate two cracks (Fig. 6(a)) by adjusting the in-plane rotation. A loading path consisting of two rotations of opposite sign, with their center of rotation P r kept at 170 0, coupled with shear and then a final proportional tension-shear loading was applied (Fig. 6(b)).

P r (mm) U Y (µm) R x ( • ) (3) (2) (1) (0) (b)
U Z (µm) U Y (µm) R x ( • ) (3) (0) (1)
(2) The scientific goal was to investigate crack closure since the first crack will be closed during the initiation of the second crack. The second point of interest was the link-up phenomenon appearing at the end of the test, when the two cracks initiating from each notch interacted and joined. The test was thus ended by a global tensile loading to force crack coalescence.

(b)
• Multiple reorientation test (IT 3). By following the same principle as in test IT 1 several times, several reorientations were expected (Fig. 7(a)).

To ensure that the crack did not propagate too much at each step, differ-180 ent centers of rotation were considered (by increasing values of P r , each time located farther away from the notch, see Fig. 7 (b)). Because several reorientations were expected, potential branchings might also appear.

This test was thus interesting for investigating the branching/bifurcation conditions. It is thoroughly described in Ref. [START_REF] Carpiuc-Prisacari | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF] and, as part of the whole

P r (mm) U Y (µm) R x ( • ) (0) (1) (2) (3) (7) (6) (5) (4) (b)

3D boundary conditions

A first part of the IT 4 test aimed at initiating and propagating a crack in mixed I+II mode. Shear (step I -1 in Fig. 8(a)) followed by in-plane rotation with shear (step I -2) was applied. The center of rotation was located at the center of the specimen (P r = 0) to ensure that the crack does not propagate beyond this point. The specimen was then completely unloaded (step I -3).

During the second part of the test, a rotation along the Z axis was applied to induce mode III propagation (step II -1 in Fig. 8 (b)). Last, a large translation along Z was applied (step II -2) to fully open the crack. The technical interest of this test was to show that the setup was also adapted for out-of-plane loadings inducing mode III propagation. The scientific interest is to perform a complex test with mode III propagation with crack closure/friction.

-1 0 1 -44 0 0 0.012 P r (mm) U Y (µm) R x ( • ) (0) (1) (3) (2) (a) 0 350 0 0.35 0 U z (µm) R Z ( • ) R x ( • ) (0) (1) (2) (b)

Test results

In this section the histories of forces and torques, and displacement fields will be presented and commented for each interactive test. Contrary to the previously reported results [START_REF] Carpiuc-Prisacari | Experimental database of mixed-mode crack propagation tests performed on concrete specimens with a hexapod and full-field measurements[END_REF], forces and torques are no longer displayed as functions of the corresponding mean displacement (e.g., F Z (U Z )) since the rotation contribution is not straightforward. For the sake of clarity, the forces and torques are plotted with respect to time.

Test IT 1

Following the planned loading history depicted in Fig. 5, the final crack paths are shown in Figs. 10 and11. As expected, they exhibit a chevron-like shape.

The load history was correctly followed by the machine (Fig. 9). The test started with an elastic loading-unloading ([0 -1, 400] s period), then the initiation-propagation stage was composed of three loading steps as planned ( [1, 400 -7, 200] s, [7, 200 -14, 400] s and [14, 400 -27, 000] s periods). An overall good agreement is observed between the mean relative displacements measured by LVDTs, DIC, and the control signal. Since very small values of 215 U X were measured and the out-of-plane loads F X , M Y and M Z were close to zero, it is concluded that the test had no out-of-plane components.

Test IT 2

By applying the planned loading history shown in Fig. 6, the crack paths shown in Figs. 13 and 14 were obtained. As expected, two parallel cracks initiated each one from one notch and propagated one after the other. At the end of the test, link-up occurred but it was not symmetric. The first crack had been reoriented toward the beginning of the second one, instead of both cracks propagating up to joining themselves.

In terms of loading, it is concluded from Fig. 12 that the control signal globally followed the elastic loading-unloading from the beginning to the end of the test, even though some offsets and drifts appeared. First, for a control displacement equal to zero, F Z was equal to about 2.5 kN at the beginning of The similarity of steps 2 and 3 is noticeable. First, the symmetry of the two 265 cracks at the end of step 2 is striking. The fact that their propagation length was very close was due to the fact that each step was stopped for the same crack length, which was assessed by DIC during the test. The initial stiffness (Fig. 12(a)) was very close in both cases, and the subsequent decrease of the F Y and F Z forces was alike. This similarity was due to the successful closure 270 of the first initiated crack (see Figs. 13 and14) at the end of step 2. The displacement field where the first crack was located was very smooth, nearly as if no crack had initiated. The fact that the F Z force tended toward zero when the displacements and M X torque magnitudes were maximum for both steps (respectively at ca; 9,400 s and 17,200 s) did not mean that the crack 275 completely propagated through the specimen (as confirmed by the displacement fields shown in Figs. 13 and14), but that the 'compressive' stress σ ZZ on one side of the ligament exactly compensated the 'tensile' stresses of the other side.

Both steps ended with a quick unload (ca. 600 to 800 s). The propagation was stable during steps 1 and 2, but not during step 3. For this last step, the displacement was increased slowly, damaging the specimen (the force and torque slopes decreased slightly prior to the sudden failure). The maximum opening of both cracks prior to the propagation is labelled (3) in Figs. 13 and14. Right after this time step, sudden propagation of the first crack occurred, leading to the final crack depicted with label (3e) in Figs. 13 and14. Based on the this experimental result, two scenarios may be considered:

(1) the lower crack has branched near the notch and propagated in an unstable manner up to the tip of the upper crack, or (2) unstable propagation had started from the upper crack tip and ended in the lower crack, but not at its tip.

Test IT 3

This complex test is only briefly described hereafter since it has been the scope of a dedicated paper, where an in-depth analysis can be found [START_REF] Carpiuc-Prisacari | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF]. The overall results were positively satisfactory. The loading was chosen to offer several specific possibilities of crack path and the final crack path corresponded to one of them. A first reorientation instead of branching (see row labelled (3) in Figs. 15 and16), then a branching instead of a reorientation (row labelled ( 5)).

The displacements and forces (Fig. 17) showed that the case can be considered as an in-plane test, which was confirmed by the very similar displacement fields obtained on each face. Concerning the propagation history, several plateaus are visible in Fig. 17 It may be concluded that even if the (normal) crack opening was important, up to 50 (resp. 75) µm during step 2 (resp. 4), there still was (lateral) contact due to the crack roughness, except during step 6.

Test IT 4 320

This last more 'exotic' test tried to propagate a crack in mixed-mode I-II, and then in mixed-mode I-III. Because of its complexity, this test is not specifically intended for experimental-numerical comparison, but for proving that it can be conducted with such setup. The mixed-mode I-II loading is shown Fig. 19. The second part was simple in terms of prescribed displacement (Fig. 22). A first step II1 was a R Z rotation while the second step II2 was a U Z translation. The result turned out to be more complex than expected. During step II1 335 the existing crack on face 1 (Fig. 20) propagated because its orientation matched the natural tendency to grow. Conversely, on face 2 (Fig. 21) the original crack did not grow because the mode III loading induced its closure. Aside from the propagation of the original crack, it was noticed that additional cracks initiated and propagated on both faces to adapt the large prescribed rotation. Figure 20 340

shows that one crack initiated from each of the notches on face 1, while on face 2 (Fig. 21), 7 appear with 2 initiating from the notches. Other cracks started from the U-shape plates or from no particular location of face 1. Despite the numerous visible cracks, an important load level was still measured at the end of the first step II1 (Fig. 22) probably because of a 'jamming' phenomenon, and 345 the fact that the cracks were not all transverse through the bulk. By applying the last U Z translation during step II2, existing cracks opened even more while new cracks appeared, thereby creating a complex network. The load decreased during this last step, down to complete failure.

Conclusion

This paper presented a series of 4 crack propagation tests on identical compact mortar specimens. The setup was identical to that used in so-called proportional and non-proportional tests [START_REF] Carpiuc-Prisacari | Experimental database of mixed-mode crack propagation tests performed on concrete specimens with a hexapod and full-field measurements[END_REF]. However the loading history was now enriched with rotations, which led to important improvements in two different domains.

From the experimental techniques point of view, several points are noticeable. First, it was proven that in-plane rotations helped stabilize crack propagation, or control the propagation of 2 different cracks. Consequently, the tests were richer because the information can be acquired during the whole propagation steps contrary to an early unstable propagation [START_REF] Carpiuc-Prisacari | Experimental database of mixed-mode crack propagation tests performed on concrete specimens with a hexapod and full-field measurements[END_REF]. Second, the loadings with additional rotation induced crack propagations through the material bulk, contrary to the non-proportional loadings only relying on two translations [START_REF] Carpiuc-Prisacari | Experimental database of mixed-mode crack propagation tests performed on concrete specimens with a hexapod and full-field measurements[END_REF].

The displacements measured by DIC on both faces was thus meaningful from the mechanical standpoint, thereby enabling for experimental-numerical comparisons. Third, the crack tip position was evaluated during the tests with full-field measurements in order to reorient it by changing the loading history.

Studies of crack reorientation or branching are thus possible. Last, since the testing machine had 6-axes, mode III was also presented as a proof of the versatility of the whole setup. The gathering of all these different enhancements (i.e., stability, versatility, interactivity) pave the way for rich hybrid test generations [START_REF] Jailin | Virtual hybrid test control of sinuous crack[END_REF].

For model identification and validation purposes, the results of this series of tests constitute a rich database (data of tests IT 1 IT 2 and IT 3 are available online (https://doi.org/10.5281/zenodo.2625006). The propagation of cracks was extensively monitored for all the reported tests, as well as the overall force and torque history, which enable for thorough experimental/numerical comparisons. With the chosen loading paths and observed crack paths, these data are an interesting challenge for damage and fracture models, in particular the prediction of reorientation, branching, instability and crack arrest.

To this aim, the measured data of IT 2 and IT 3 tests were used in the socalled CARPIUC Benchmark [START_REF] Carpiuc | CARPIUC Benchmark. crack advance, reorientation, propagation and initiation under complex loadings[END_REF] of CFRAC 2017 and CFRAC 2019 conferences (the benchmark data can be found online 1 ) in which several teams participated [START_REF] Oliver-Leblond | Beam-particle approach to model the quasi-brittle behaviour of concrete. carpiuc benchmark[END_REF][START_REF] Delaume | Local adaptive refinement method applied to cohesive zone models for heterogeneous materials[END_REF][START_REF] Nguyen | A phase field method for modelling of crack propagation and initiation under complex loadings[END_REF][START_REF] Ǒzbolt | Modelling mixed mode fracture of cementitious materials using microplane model with relaxed kinematic constraint -carpiuc benchmark[END_REF][START_REF] Salzman | Simulation of the carpiuc test with the thick level set model[END_REF][START_REF] Wu | Phase-field simulation of interactive mixed-mode fracture tests on cement mortar with fullfield displacement boundary conditions[END_REF][START_REF] Carpiuc | Non-local damage model to simulate the quasi-brittle behaviour of concrete carpiuc bench-490 mark[END_REF].

Last, these complex tests (i.e., based on a combination of multiaxial loading conditions and full-field measurements) relying on an interactive procedure (i.e., the decision to change or not, and to which extent, the orientation of the crack at each step of the test) pave the way for future hybrid tests [START_REF] Jailin | Virtual hybrid test control of sinuous crack[END_REF], which are driven 'on the fly' by computed quantities.

Figure 1 :

 1 Figure 1: Different loading histories. Global in-plane tension U Z (a), shear U Y (b) and rotation R X about a point of horizontal coordinate Pr (c). The orientation of the axis for the coordinate of Pr is inverted (-Y instead of Y ) to have increasing values of Pr during the tests, since Pr is moved away from the crack tip to control its propagation.

( 5 ×

 5 25 mm, or 5 × 40 mm) or only one (5 × 25 mm).

Figure 2 :

 2 Figure 2: The 3 studied geometries with shallow double notches (a), deep double notches (b), and single shallow notch.

Figure 3 :

 3 Figure 3: Location of the cameras with respect to the specimen: S l 1 & S r 1 , S l 2 & S r 2 for stereoDIC and T 1 and T 2 for 2D-DIC analyses.

Figure 4 :

 4 Figure 4: Location of the LVDT sensors with respect to the specimen.

Figure 5 :

 5 Figure 5: Single reorientation test IT 1. Expected crack path (a) and corresponding loading path (b). Pr stands for the position along Y of the center of rotation.

10

 10

Figure 6 :

 6 Figure 6: Single reorientation test IT 2. Expected crack path (a) and corresponding loading path (b).

Figure 7 :

 7 Figure 7: Multiple reorientation test IT 3. Expected crack path (a) and corresponding loading path (b).

Figure 8 :

 8 Figure 8: First (a) and second (b) loading steps of IT 4 test.

Figure 9 :

 9 Figure 9: Translation and force vs. time (a). Rotation and torque vs. time (b) for IT 1 test (steps are marked by white and light gray backgrounds).

Figure 10 :

 10 Figure 10: Displacement fields (expressed in µm) on face 1 just before unstable propagation during step 1 (1u) and next stable position (1s), and at the end of step 3 for test IT 1.

Figure 11 :

 11 Figure 11: Displacement fields (expressed in µm) on face 2 just before unstable propagation during step 1 (1u) and next stable position (1s), and at the end of step 3 for test IT 1. Face 2 images are flipped to make the comparison easier.

steps 1 ,Figure 12 :

 112 Figure 12: Translation and force curves vs. time (a). Rotation and torque vs. time (b) for IT 2 test. The first crack initiated after 5,500 s, and the second after 13,500 s (steps are marked by white and light gray backgrounds).

Figure 13 :

 13 Figure 13: Displacement fields (expressed in µm) on face 1 at the end of step 1 (1), step 2 (2), during step 3 (3) for test IT 2. The first crack initiated after 5,500 s and the second after 13,500 s. Final macroscopic crack at the end of step 3 (3e).

Figure 14 :

 14 Figure 14: Displacement fields (expressed in µm) on face 2 at the end of step 1 (1), step 2 (2), and during step 3 for test IT 2. Final macroscopic crack at the end of step 3 (3e). Face 2 images are flipped to make the comparison easier.

Figure 15 :Figure 16 :

 1516 Figure 15: Displacement fields (expressed in µm) on face 1 at the end of steps 1 to 5 for test

.6 10 Force

 10

.6 10 ForceFigure 17 :

 1017 Figure 17: Translation and force vs. time (a). Rotation and torque vs. time (b) for IT 3 test (steps are marked by white and light gray backgrounds).

Figure 18 :

 18 Figure 18: F Z force versus U Z translation (a). F Y force versus U Y translation during IT 3 test.

Figure 19 :Figure 20 :

 1920 Figure 19: Translation and force vs. time (a). Rotation and torque vs. time (b) for the first part of IT 4 test (steps are marked by white and light gray backgrounds).

Figure 21 :

 21 Figure 21: Displacement fields (expressed in µm) on face 2 at the end of steps I -2, II -1 and II -2 for test IT 4. Face 2 images are flipped to make the comparison easier. Final main crack is highlighted in red because the final opening is barely visible on the image.

Figure 22 :

 22 Figure 22: Translation and force vs. time (a). Rotation and torque vs. time (b) for the second part of IT 4 test. The force axis is inverted for a better visibility (steps are marked by white and light gray backgrounds).

Table 1 :

 1 Mortar mix details.

	Effective water Portland cement	Sand 0/4	Plasticizer
	319 kg/m 3	611 kg/m 3	1235 kg/m 3 5.25 kg/m 3
	The main mechanical properties of the mortar (Tab. 2) were obtained follow-
	ing the NF EN 196-1 standard [21]. To check if the measured Young's modulus
	was rate-dependent, 2 other specimens were tested with a load rate 100 times
	lower, but no noticeable variation of Young's modulus or tensile strength oc-

curred. The standard compressive test NF EN 196-1 provided a strength equal to 80 MPa. Last, the fracture energy G f was obtained with 6 70 × 70 × 280 mm specimens following the 50-FMC Draft Recommendation [22], and is equal to 115 ± 19 N/m.

Table 2 :

 2 Average mortar properties obtained for two loading rate.

	Load rate Young modulus Tensile strength
	50 N/s	17.3 ± 0.7 GPa	3.8 ± 0.4 MPa
	0.5 N/s	20.0 ± 0.1 GPa	4.1 ± 0.4 MPa

  Concerning crack closure and friction, progressive softening was visible on F Z and M X curves during step 3, while the F Y magnitude also decreased, albeit to a much lesser extent. This observation tends to prove that crack friction occurred, as well as the fact that the displacement field u Y were continuous on some parts of the crack mouth at the end of the test (see Figs.10 and 11). It is noteworthy that the sharp variation of slope at t ≈ 10, 000 s was only due to chosen changes in the control signal speed, not to crack closure. A last noticeable point was the appearance of a late second crack. It initiated from the second notch and

slowly propagated. The first one did not seem to be influenced by the second crack, and propagated in a straight way. It thus seems reasonable to consider that this second crack is shallow.

test campaign, it will only be summarized herein.

Acknowledgements

This work has been supported and carried out within the EnerCampus framework.

0, thereby proving that the load was (mainly) transmitted through the ligament.

with the same orientation, and the forces had significantly decreased, meaning that the crack also propagated through the bulk.


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Mathias Legrand

INVESTIGATION OF ROTOR-CASING INTERACTIONS IN THE CENTRIFUGAL COMPRESSOR OF A HELICOPTER ENGINE

The phenomenology of rotor-casing setups experiencing contact interactions is still poorly understood, particularly when complex geometries such as centrifugal compressors are involved. Although interaction phenomena have been witnessed and recorded during industrial experiments, the physical understanding of what occurs during these interactions is limited. The usual design approach is to consider possible modal interaction points in a linear framework and move these outside of normal operating conditions by means of minor geometric changes. Based on this linear approach, no information on the severity of these interactions is available to the designer. Besides, a possible interaction point appearing in the linear framework may not produce any harmful interactions, thereby increasing design restrictions.

Based on an in-house numerical strategy previously presented, contact interactions for a flexible centrifugal compressor from a helicopter engine and rigid casing setup are investigated. By imposing a small deformation on the casing geometry, blade/casing contact is initiated and subsequent interactions feature complex phenomena that are analyzed. In comparison to previous interaction simulations involving axial compressors, a higher degree of complexity of the numerical simulations stems from a strong curvature of the blade and very significant bladedisk coupling. This coupling presents itself towards the trailing edge where compressor mode shapes indicate a significant component normal to the casing surface. Accordingly, these modes may lead to large amplitude contact forces.

Time simulation results are confronted with experimental observations, and the consistency of the behavior of the numerical model with respect to industrial observations is underlined. A frequency domain post-processing of the results reveals specific engine order interactions and frequency spectra are plotted in order to interpret the phenomenon of interest. Such methodology will enable designers to more efficiently discriminate potential critical interaction speeds as compared to the classical linear frequency approach.

INTRODUCTION

Rotor/stator interactions [START_REF] Morfey | Rotating pressure patterns in ducts: their generation and transmission[END_REF] involve a large variety of complex phenomena. In modern turbo-machinery clearances be-tween the rotor and casing are usually designed at normal operating conditions, allowing for optimal performance without significant interactions between the two parts. Nevertheless, in certain cases an engine may experience operating conditions for which clearances can be overcome and interactions can become non-negligible. As a result, contacts, such as blade/casing contacts, may occur. Subsequent interaction phenomena -which are a specific type of rotor/stator interactions -that were also observed for axial compressors both experimentally [START_REF] Millecamps | Influence of thermal effects during bladecasing contact experiments[END_REF][START_REF] Padova | Experimental results from controlled blade tip/shroud rubs at engine speed[END_REF] and numerically [START_REF] Batailly | Numerical-experimental comparison in the simulation of rotor/stator interaction through blade-tip/abradable coating contact[END_REF] are still under investigation.

Several references may be found regarding the study of modal dynamics of centrifugal compressor geometries [START_REF] Michimura | Vibration of impellers. III-an analysis of coupled vibrations between disk and blades using a reduced impedance method[END_REF][START_REF] Michimura | Vibration of impellers. V-measurement of resonant vibratory stresses of an impeller and pressure distribution due to aerodynamic excitation[END_REF][START_REF] Michimura | Vibration of impellers. VI-vibration analysis at moving coordinates. vii-vibration analysis at moving coordinates[END_REF][START_REF] Michimura | Vibration of impellers. VII-vibration analysis at moving coordinates fixed in rotating impellers [2nd])[END_REF]. More recently, a detailed numerical model of the geometry of interest in this paper was presented in [START_REF] Gibert | Modal tests and analysis of a radial impeller at rest: Influence of surrounding air on damping[END_REF]. Along with other interaction phenomena such as impeller-vaned diffuser interaction [START_REF] Gaetani | Impeller-vaned diffuser interaction in a centrifugal compressor at off design conditions[END_REF], the understanding of interaction phenomena arising from structural contacts is a growing field of investigation. Consequences of these phenomena may result in strong energy vibration between rotor and stator which may cause damage to the structure, as seen in [START_REF]Aircraft Accident Report[END_REF].

The proposed study focuses on blade/casing contacts within the centrifugal compressor of a helicopter engine. One specificity of centrifugal compressors lies in a complicated blade-tip geometry. Due to the high curvature of almost 90 • between inlet and outlet as well as very high stiffness blade areas such as trailing edge, as compared to axial compressors, the computational challenge of contact simulations is significantly increased. Based on an explicit time integration procedure combined with Lagrange multiplier contact treatment [START_REF] Legrand | Full three-dimensional investigation of structural contact interactions in turbomachines[END_REF], a specific contact configuration between the centrifugal compressor and its surrounding casing is investigated throughout a wide rotational speed range. It is assumed that the casing is rigid and subject to a time-invariant deformation along a two-nodal diameter shape -which means it is ovalized, see Fig. 1. The interest lies in the vibratory level of the impeller depending on its rotational speed.

The modeling of the system is detailed in the first section of the paper: both the computation of the reduced order model of the impeller and the contact algorithm are outlined. The second section presents the modal analysis of the impeller and the typical linear criterion for the definition of critical rotational speeds is recalled. In the third section, convergence in time and space is addressed. Results are post-processed in the frequency domain with a 2D-Fourier transform. The two dimensions involved -time and space -are referred to with a specific vocabulary: nodal diameters refer to harmonics in space while harmonics refer to harmonics in time.

MODELING

Note 1: symbols and notations used in the following are all referenced in the nomenclature at the end of the paper.

Note 2: for the sake of confidentiality, all values in the space, time and frequency domains presented have been normalized.

In this section, the methodology used to treat unilateral contact between the flexible rotor and rigid casing are outlined. Contact between the two structures is initiated by a static ovalization of the casing, restricting the tip clearance equally along the entire blade tip for certain angular positions. The reference clearance is about 3% at the LE to 2.2% at the TE normalized to the TE blade height. The ovalization of the casing is implemented by using a Gaussian bell distortion on top of the reference geometry. The restriction amplitude is computed uniformly along the chord at 4.25% the TE blade height. The penetration depth will hence vary from 1.25% at the LE to 2.1% at the TE. The abradable coating that is frequently deposited on the casing is not accounted for in our study. The focus is on direct structural contacts in order to get a better understanding of possible interaction phenomena.

Bladed Disk Model

The high-pressure radial compressor stage of a modern highperformance helicopter engine is investigated. It consists of ten sectors containing each a main and a splitter blades. Contact is only managed on the main blade of each sector. Contact is managed on ten nodes evenly spaced along the chord of each main blade as pictured in Fig. 2. Consequently, a total of a hundred contact nodes are accounted for on the impeller.

The challenge in facing the impeller geometry in comparison with axial compressors lies in the varying blade tip contact normal along the chord. That, in combimation with impeller design specificities such as a low height to chord ratio and significant influence of the disk for very low frequency free-vibration modes, leads to more complex numerical simulations.

Model Reduction

The finite element model of an elementary sector of the impeller depicted in Fig. 2 contains about 120,000 degrees of freedom (DOF) and the full impeller contains more than a million DOF. Consequently, time integration on such large models would lead to cumbersome computation times. Accordingly, component mode synthesis (CMS) is used in order to reduce the dimension of the models. It is critical to avoid costly mappings between the finite element and the reduced spaces for contact management. For that reason, the CMS method used in this studywhich is based on the Craig-Bampton method [START_REF] Craig | Coupling of substructures for dynamic analysis[END_REF] -features a mixed reduced space with both physical and modal coordinates.

The combination of a modified Craig-Bampton procedure [START_REF] Sternchüss | On the reduction of quasi-cyclic disks with variable rotation speeds[END_REF] with cyclic symmetry [START_REF] Bladh | Efficient predictions of the vibratory response of mistuned bladed disks by reduced order modeling[END_REF] allows for a fast computation of a reduced order model that accounts for centrifugal stiffening. The key steps of this computation are detailed in the following. The procedure requires three stiffness matrices computed at distinct rotational frequencies: K(0), K( Ω 1 2 ) and K(Ω 1 ) as well as the mass matrix M of an elementary sector of the finite element model.

1. matrices reorganisation: lines and columns of matrices Y = K(Ω) or M are sorted depending on which part of the elementary sector they belong to:

Y =   Y ii Y ib Y ic Y bi Y bb Y bc Y ci Y cb Y cc   and u =   u i u b u c   (1)
2. computation of nodal diameter matrices: each nodal diameter stiffness and mass matrix is computed from the finite element matrices, see [START_REF] Bladh | Efficient predictions of the vibratory response of mistuned bladed disks by reduced order modeling[END_REF].

Ŷ(i) = Y 0 + (Y 1 + Y T 1 ) cos(iα) (Y 1 -Y T 1 ) sin(iα) (Y T 1 -Y 1 ) sin(iα) Y 0 + (Y 1 + Y T 1 ) cos(iα) with Y 0 = Y ii Y ib Y bi Y bb , Y 1 = Y ic 0 n i ,n i -n c +n b Y bc 0 n b ,n i -n c +n b (2)
3. centrifugal stiffening: the three stiffness matrices account-ing for centrifugal stiffening are:

K(i) 0 = K(i) (0) K(i) 1 = 1 3Ω 2 1 16 K(i) Ω 1 2 -K(i) (Ω 1 ) -15 K(i) (0) K(i) 2 = 4 3Ω 4 1 K(i) (Ω 1 ) -4 K(i) Ω 1 2 + 3 K(i) (0) (3)
4. modal reduction: for each of the three rotational speeds (Ω = 0, Ω = Ω 1 /2 and Ω = Ω 1 ), the modal reduction basis Φ(i) (Ω) is:

Φ(i) (Ω) = I 0 Ψ(i) s (Ω) Ψ(i) co (Ω) (4)
For each harmonic, the global reduction basis Υ(i) may then be computed as:

Υ(i) = I 0 Ψ(i) s (0) Ψ (i), * (5)
where Ψ (i), * contains an orthonormal basis generated from the static and constraint modes given in Eq. ( 4). It yields the computation of the reduced harmonic mass and stiffness matrices:

K (i) 0,r = Υ(i),T K(i) 0 Υ(i) K (i) 1,r = Υ(i),T K(i) 1 Υ(i) K (i) 2,r = Υ(i),T K(i) 2 Υ(i) M (i) r = Υ(i),T M(i) Υ(i) (6)
5. recomposition of the reduced model at a given Ω: the final reduced mass and stiffness matrices are:

K r (Ω) = N ∑ i K (i) 0,r + Ω 2 K (i) 1,r + Ω 4 K (i) 2,r M r = N ∑ i M (i) r (7)
For a given impeller, matrices

K (i) 0,r , K (i) 1,r , K (i) 2,r and
M (i)
r are computed only once. Those matrices are relatively small -around a hundred times smaller than the finite element matrices of a sector of the impeller (K and M) depending on the level of modal reduction -thus making the computation of the reduced order model given in Eq. ( 7) particularly fast and easy. The stiffness matrix K r (Ω) of the reduced model is simply denoted K r in the following.

Contact Treatment

Time simulations are carried out based on the strategy presented in [START_REF] Legrand | Full three-dimensional investigation of structural contact interactions in turbomachines[END_REF] which is briefly recalled here for the sake of clarity. Though compatible with the following contact treatment procedure, friction is not accounted for in this study. The considered time-marching procedure involves the explicit central differences scheme combined with a Lagrange multiplier based contact algorithm [START_REF] Carpenter | Lagrange constraints for transcient finite element surface contact[END_REF], assumed on 10 contacting nodes, spread equidistantly along the chord line. At each time step q, the procedure is divided into four steps:

1. prediction at time step q + 1 of the displacements u r :

u q+1,p r = M r h 2 + D r 2h -1 2M r h 2 -K r u q r + D r 2h - M r h 2 u q-1 r (8)
2. determination of the gap function and detection of the contacting nodes of the blade. 3. correction of the predicted displacements leading to a vanishing of the gap function:

g q+1 = C N T u q+1,c r + g p = 0 (9)
Lagrange multipliers (or contact forces) and updated displacements are obtained as follows:

           λ = C N T M r h 2 + D r 2h -1 C N -1 g p u q+1 r = u q+1,p r + M r h 2 + D r 2h -1 C N λ (10)
4. time increment.

MODAL ANALYSIS

The modal analysis of the impeller is carried out considering that the shaft is perfectly rigid and hence the DOF of the impeller that lie on the impeller-shaft interface are clamped, see Fig. 3. The Campbell diagram of the impeller -depicted in Fig. 4 in the relative frame -underlines the evolution of the eigenfrequencies with respect to the rotational speed.

Ω FIGURE 3: CLAMPED INNER SURFACES

Only these first two families of free vibration modes are represented in this paper. Since the first mode families generally have the highest vibration energy participation it is assumed that the first two modal families are dominant in the impeller response even when contact occurs.

In order to identify potential critical velocities, including harmonics, it is convenient to plot the Campbell diagram in the absolute frame, as pictured in Fig. 5, where each eigenfrequency evolution with respect to the rotational speed is given by ( 11),

: i = 0 ( ), i = 1 ( ), i = 2 ( ), i = 3 ( ), i = 4 ( ) AND i = 5 (
) FOR FIRST (SOLID) AND SECOND (DASHED) MODAL FAMILIES where f (i) a and f (i) are the absolute and relative frame rotor modal frequencies of the i-th nodal diameter at a rotational speed of Ω respectively:

f (i) a (Ω) = -f (i) (Ω) + i • n • Ω (11)
Since the casing is perfectly rigid in this study, interactions are considered at the intersection between f (i) a (Ω) and the zero frequency line in the Campbell diagram. Linear considerations predict five interaction points over the operating rotational speed range of interest. Those interaction points are highlightedmarked as Ω i, j , where i is the nodal diameter, and j the modal family respectively -in Fig. 5. For modal coincidence to oc-

0 1 2 3 4 5 0 10 20 Ω 3,1 Ω 4,1 Ω 5,1 Ω 4,2 Ω 5,2 Rotational Speed Ω Frequency f a FIGURE 5: ABSOLUTE FRAME CAMPBELL DIAGRAM: n = 1, i = 0 ( ), i = 1 ( ), i = 2 ( ), i = 3 ( ), i = 4 (
) AND i = 5 ( ) FOR FIRST (SOLID) AND SECOND (DASHED) MODE FAMILIES cur, the rotational speed -in the absolute frame -of the modal shape of the impeller must be null since the casing is perfectly rigid. Under these conditions, in a linear framework, a modal coincidence can be achieved for a rigid and ovalized casing.

Interaction points associated with other modal families are not considered here for two reasons:

1. because the other modal families involve higher frequency modes, the interaction points occur outside of the rotational speed range of interest, 2. as mentioned above, even when contacts occur, recent studies suggest that the first free vibration modes or families of free vibration modes are predominant in the dynamics of the structure.

NUMERICAL SIMULATIONS

The rotational speed range of interest covers the typical rotational speed range of the impeller with Ω ∈ [1.969 ; 5.951]. As mentioned above, numerical simulations on the full finite element model are not conceivable because of its very large dimension. As a consequence, a reference finite element solution cannot be obtained. Asymptotic space and time convergence are respectively assessed versus the dimension of the modal reduction basis and the time step.

Convergence

Computation times grow nonlinearly with the dimension of the reduced order model and it is thus critical to find the smallest reduction basis allowing to obtain space convergence of the simulations. Figure 6 pictures the radial displacement on the leading edge of the first sector of the impeller over the first five milliseconds of the simulation for different reduced order models. It appears quite clearly that too small reduction bases (330 and 540 DOF) lead to inaccurate results early in the interaction. Larger models (900, 1140 and 1560 DOF) lead to almost perfectly superimposed displacements. Identical observations are made for the tangential displacement of the contact nodes on the trailing edge in Fig. 7.

Similar results are obtained over a longer time interval. A reduced order model containing 900 DOF is well suited and allows for space converged results throughout the rotational speed range of interest. With 900 DOF, the gap between the ten first eigenfrequencies for each nodal diameter of the reduced order model and those of the full finite element model is less than 1%. Simultaneously, time convergence is assessed by refining the time step size and h = 10 -4 allows for convergence.

Case Study

First, a contact simulation for Ω = 5.4 is considered. Results feature displacements, speeds, contact forces and observed contact areas for which both time and frequency domain analyses may be carried out. In addition, displacement and stress fields within a sector at any time step can easily be retrieved. Figure 8 pictures the radial displacement of the trailing edge contact node during the simulation. Over the simulation, the impeller makes 50 rotations and it is visible that a steady state -a response with stable amplitude -is reached for t > 20. This is also visible in the frequency map Fig. 10, as peaks of the FFT signal lie on EO lines.

Once a steady state is reached, a fast Fourier transform (FFT) of the signal is performed, see Fig. 9. It should first be noticed that only the significant peaks are detected for f < 50. Among these peaks, highest amplitudes are observed for f ≃ 10.9, 16.3, 21.8 and 32.7. However, because of the cyclic symmetry of the structure, the first modal families exhibit a clustering in eigenfrequency, particularly around f ≃ 10.9 where the gap in frequencies for the first mode family is negligible, see Fig. 4. Consequently, the precision of the FFT may not be sufficient for a clear identification of the dominant free vibration mode at a given frequency.

It is thus of interest to analyse the results in terms of nodal diameter content. The displacement vector u(t) may be projected in the cyclic symmetry space using the Fourier matrix F:

û(t) = F T u(t) (12)
The displacement vector in the cyclic symmetry space û(t) contains terms associated with each nodal diameter i, i = 0, 1 . . . 5:

ûT (t) = û(0) (t), û(1) (t), û(2) (t), û(3) (t), û(4) (t), û(5) (t) T (13)
The frequency content of nodal diameter i may then be analysed with a FFT of the component û(i) (t). Results provided in the following feature both FFT of finite element displacements u(t) and nodal diameter displacements û(i) (t). The combination of the analysis of both types of displacements allows for an indepth investigation of the interaction phenomenon.

Interaction Map

Figure 10 shows the radial displacement response of the trailing edge node in the frequency domain. This interaction map is drawn putting side by side the spectrum obtained for each rotational speed once a steady state is reached. Highest peaks of amplitude are depicted in red and engine order lines are pictured as black dashed lines defined by f = kΩ, k = 1, 2 . . . 18.

Consistently with the casing ovalization -that implies an excitation frequency of the impeller double the rotational speed Ω -the peaks highest in magnitude are detected around the even engine order lines k = 2, 4 . . . 18. The impeller response varies significantly with Ω and some critical velocities are identified in Tab. 1. For these rotational speeds, significantly higher displacements are found. Some of the critical rotational speeds identified with a modal analysis of the impeller given in Fig. 5 do match with slightly higher amplitudes of vibration in the frequency map. For instance, around Ω ≃ Ω 5,1 = 2.12, significant peaks are visible around the 12 th and 14 th engine order lines (see highlighted areas A and B in Fig. 10). Nevertheless, none of these predicted critical rotational speeds match the highest peak of amplitudes.

Overall, the discrepancy between the linear prediction of the critical rotational speeds -mentioned in Fig. 5 -and those identified with numerical simulations is patent. For highly nonlinear phenomena such as blade-tip/casing contacts, linear interaction criteria often fail [START_REF] Millecamps | Influence of thermal effects during bladecasing contact experiments[END_REF][START_REF] Batailly | Numerical-experimental comparison in the simulation of rotor/stator interaction through blade-tip/abradable coating contact[END_REF]. In particular, contact simulations on axial compressors have revealed the key role of the structure super harmonics in potential interaction areas: rotational speeds Ω = f 1 /n, n = 1, 2 . . . are privileged critical speeds. Accordingly, it seems of great interest to consider super harmonics of the first free vibration modes of the impeller in order to identify more critical rotational speeds. Super harmonics n = 2 of the two first stages of modes and the associated predicted critical rotational speeds are depicted in the Campbell diagram in Fig. 11. For the radial compressors, the number of critical speeds increases significantly: these speeds are listed in Tab. 2.

In addition, critical speeds associated with super harmonics of high free vibration modes are prone to fall within the rotational speed range of interest. Only the critical rotational speeds associated with super harmonics of the two first families of modes are

0 1 2 3 4 5 0 10 20 Ω 2 2,1 Ω 2 2,2 Ω 2 3,1 Ω 2 3,2 Ω 2 4,1 Ω 2 4,2 Ω 2 5,1 Ω 2 5,2 Rotational Speed Ω Frequency f a FIGURE 11: FIRST SUPER HARMONIC ABSOLUTE FRAME CAMPBELL DIAGRAM: n = 2, i = 0 ( ), i = 1 ( ), i = 2 ( ), i = 3 ( ), i = 4

Frequency Split Study

Looking closely at the results in Fig. 10 around area (E), the odd engine order lines feature a response that was previously unknown to the authors in the speed range ∆Ω = [5.33 ; 5.58]. Firstly, a response on the odd engine order lines does not comply with the linear interaction considerations, as well as secondly, a symmetric frequency split around the 1st, 3rd and 5th engine order is observed for Ω > 5.50 (see area E in Fig. 10). A refinement of the aforementioned speed range -with an increased resolution since 150 time simulations1 are carried out over ∆Ω -is depicted in Fig. 12(a). This section aims at identifying the nature of this frequency split. Figure 12(a) represents the frequency domain representation of the radial displacement of trailing edge contact node of the first sector and Figs. 12(b)-(g) show the frequency content for each harmonic component. It is visible that each harmonic dominates on distinct engine order lines. More precisely, the response of harmonic i = 0 is dominant at the zero frequency line, the response of harmonic i = 2 dominates along the 2nd engine order line, the response of harmonic i = 4 is dominant along the 4th and 6th engine order lines due to the aliasing effect, while odd engine orders show an almost negligible participation.

In order to better understand the nature of the response on the odd engine order lines, two zooms over the frequency spectrum obtained for each harmonic for two different rotational speeds Ω a = 5.33 and Ω b = 5.53 are respectively pictured in Figs. 13 and 14. The spectrum depicted in Fig. 13 is obtained for a rotational speed below the one for which the frequency split occurs: the three peaks of interest on the 1st, 3rd and 5th engine order lines solely involve even harmonics.

: i = 0 ( ), i = 1 ( ), i = 2 ( ), i = 3 ( ), i = 4 ( ) AND i = 5 ( )
Similarly, when looking at the spectrum given in Fig. 14 computed for the rotational speed Ω b , depicted in Fig. 10, for which the frequency split is observed -it is visible that the residual peaks only involve even harmonics.

Because of their position on odd engine order lines, these peaks cannot be linked with fundamental harmonics of free vi-

: i = 0 ( ), i = 1 ( ), i = 2 (
), i = 3 ( ), i = 4 ( ) AND i = 5 ( ) bration modes of the impeller. Accordingly, their very existence is an illustration of the incorrectness of linear considerations for the prediction of critical rotational speeds. Taking into account previous results obtained with axial compressors [START_REF] Legrand | Full three-dimensional investigation of structural contact interactions in turbomachines[END_REF], it seems reasonable to assume that these peaks stem from super harmonics influence. Supporting evidence of this assumption lies in the agreement between a few critical rotational speeds predicted for super harmonics n = 2 and n = 3 and the presented numerical results. The nature of the frequency split remains to be investigated.

CONCLUSION

This study focuses on the numerical simulation of blade/casing structural contacts for the centrifugal compressor of a helicopter engine. The use of cyclic symmetry and component mode synthesis method allows for efficient time simulations over a wide rotational speed range. The contact configuration involves a rigid ovalized casing and unilateral contact is accounted for on ten contact nodes on each main blade of the impeller.

Similarly to what was already witnessed with axial compressors [START_REF] Batailly | Numerical-experimental comparison in the simulation of rotor/stator interaction through blade-tip/abradable coating contact[END_REF] asymptotic space and time convergence are shown. A modal analysis of the impeller leads to the computation of its Campbell diagram and the usual linear criterion -defined by the intersection between frequency lines and engine order lines -is used to predict potential critical rotational speeds.

Results are presented both in time and frequency domains. It is first highlighted that linearly predicted critical rotational speeds do not match with the numerically predicted critical rotational speeds. Based on an in-depth frequency domain analysis as well as previous observations with axial compressors, the assumption is made that super harmonics of the free vibration modes of the impeller play a key role in the occurrence of critical regimes.

Work is in progress for the precise identification of the split frequency phenomenon observed around odd engine order lines for high rotational speeds. Also, further numerical studies will incorporate abradable coating on the casing. Also, accounting for a flexible casing within the presented strategy will allow for the detection of travelling modal coincidence [START_REF] Schmiechen | Travelling wave speed coincidence[END_REF] in the absolute frame.

ACKNOWLEDGEMENT

Thanks go to Turbomeca and Snecma for their technical and financial support. This work takes place in the framework of the MAIA mechanical research and technology program sponsored by CNRS, ONERA and SAFRAN Group.

NOMENCLATURE

FIGURE 1 :

 1 FIGURE 1: TWO-LOBE RADIAL CASING RESTRICTION

Ω

FIGURE 2 :

 2 FIGURE 2: IMPELLER CONTACT MODEL: CONTACT NODES , NODE TRAJECTORIES WITHOUT CONTACT ( )

FIGURE 4 :

 4 FIGURE 4: RELATIVE FRAME CAMPBELL DIAGRAM: i = 0 ( ), i = 1 ( ), i = 2 ( ), i = 3 ( ), i = 4 ( ) AND i = 5 () FOR FIRST (SOLID) AND SECOND (DASHED) MODAL FAMILIES

FIGURE 6 :

 6 FIGURE 6: CONVERGENCE OF THE RADIAL DISPLACE-MENT: 330 DOF ( ), 540 DOF ( ), 900 DOF ( ), 1140 DOF ( ) AND 1560 DOF ( ).

FIGURE 7 :

 7 FIGURE 7: CONVERGENCE OF THE TANGENTIAL DIS-PLACEMENT: 330 DOF ( ), 540 DOF ( ), 900 DOF ( ), 1140 DOF ( ) AND 1560 DOF ( ).

FIGURE 8 :

 8 FIGURE 8: RADIAL DISPLACEMENT OF THE LEADING EDGE OF SECTOR 1 AT Ω = 5.4

FIGURE 9 :

 9 FIGURE 9: FFT OF THE SIGNAL IN FIG. 8 AT Ω = 5.4

FIGURE 10 :

 10 FIGURE 10: SPECTRUM OF THE TRAILING EDGE RADIAL FINITE ELEMENT DISPLACEMENT

  ( ) AND i = 5 ( ) FOR FIRST (SOLID) AND SECOND (DASHED) MODAL FAMILIES considered in the following.

FIGURE 12 :

 12 FIGURE 12: SPECTRUM OVER ∆Ω FOR EACH NODAL DIAMETER COMPONENT

FIGURE 13 :

 13 FIGURE 13: FFT AT Ω a = 5.33: i = 0 ( ), i = 1 ( ), i = 2 ( ), i = 3 ( ), i = 4 ( ) AND i = 5 ( )

3 FIGURE 14 :

 314 FIGURE 14: FFT AT Ω b = 5.53: i = 0 ( ), i = 1 ( ), i = 2 (), i = 3 ( ), i = 4 ( ) AND i = 5 ( )

b

  relative to contact degrees of freedom c relative to degrees of freedom in one cyclic boundary i relative to internal degrees of freedom j relative to the mode family number r refers to matrices written in the reduced space

TABLE 1 :

 1 CRITICAL ROTATIONAL SPEEDS NUMERI-CALLY PREDICTED.

	Ω	2.15	3.25 3.96 5.58-5.92
		(A) and (B)	(C)	(D)	(E)

TABLE 2 :

 2 PREDICTED CRITICAL ROTATIONAL SPEEDS.

	Ω n i, j	(i, j, n) Ω n i, j	(i, j, n)
	0.67 (5,1,3) 2.33 (3,2,3)
	0.83 (4,1,3) 2.67 (4,2,2)
	1.00 (5,1,2) 2.75 (4,1,1)
	1.12 (3,1,3) 2.75 (2,1,2)
	1.29 (4,1,2) 3.37 (2,2,3)
	1.42 (5,2,3) 3.67 (3,2,2)
	1.71 (2,1,3) 4.00 (1,1,3)
	1.75 (3,1,2) 4.04 (3,1,1)
	1.75 (4,2,3) 4.50 (5,2,1)
	2.12 (5,1,1) 5.37 (2,2,2)
	2.12 (5,2,2) 5.83 (4,2,1)

A time simulation is a simulation of the contact configuration over 50 revolutions at a given rotational speed Ω