Noémie Martin

Anis Hor

Etienne Copin

Philippe Lours

Léon Ratsifandrihana

Fatigue properties of as-built and heat-treated Inconel 625 obtained by the hybridization of two laser-powder based additive processes

Laser Powder Bed Fusion and Direct Energy Deposition processes have complementary characteristics. Their hybridization can increase the variety of applications for additive manufacturing. This paper investigates the static and fatigue behaviors of hybrid Inconel 625 parts using monotonic and high cycle fatigue tests, local strain measurements and self-heating method. Fatigue properties of as-built and recrystallized LPBF, DED and hybrid Inconel 625 were evaluated and compared. The effect of heat treatment on hybrid microstructure homogeneity was investigated. Their consequences on mechanical compatibility and fatigue strength were proved. The fatigue strength was attributed to the competition between defects and the strain compatibilities.

Introduction

The growing interest in metallic additive manufacturing is usually related to the design freedom and the gain in time between the design and the production of the parts [START_REF] Leino | The role of laser additive manufacturing methods of metals in repair, refurbishment and remanufacturing -Enabling circular economy[END_REF][START_REF] Wilson | Remanufacturing of turbine blades by laser direct deposition with its energy and environmental impact analysis[END_REF][START_REF] Jones | Remanufacture of turbine blades by laser cladding, machining and in-process scanning in a single machine[END_REF][START_REF] Zhang | Metallic components repair strategies using the hybrid manufacturing process[END_REF]. Initially used for prototyping [START_REF] Wohlers | [END_REF][START_REF] Leandro | Sculpteo dévoile son étude annuelle sur les tendances du marché de l'impression 3D[END_REF], metallic AM is now common for structural repair and high value parts manufacturing, especially in aeronautics and medical domains [START_REF] Buchanan | Metal 3D printing in construction: a review of methods, research, applications, opportunities and challenges[END_REF]. Nickel-based superalloys are commonly used in aeronautical engines, for example for blade applications, because of their outstanding me-chanical strength at high temperatures. Engine blades are subject to constant innovation in terms of geometry and function integration (such as cooling ducts) and hence stand as a promising application for AM [START_REF] Graf | Additive process chain using selective laser melting and laser metal deposition[END_REF]. The feasibility of such specific geometries, and the behavior of such alloys made by AM is therefore of great interest.

Several metallic AM technologies have been developed, differentiated by the type of raw material used (powder, wire), the physical principle (fusion, sentering) or the energy source (laser, electric arc, electron beam). Each technological choice leads to specific parts characteristics [START_REF] Gao | The status, challenges, and future of additive manufacturing in engineering[END_REF]. The comparison of AM technologies has a clear industrial interest. Among the various existing technologies, Laser Powder Bed Fusion (LPBF) and Direct Energy Deposition (DED) are based on the fusion of metallic powder using laser energy. LPBF beneficiates from a good process maturity, leading to an outstanding geometrical resolution and good material soundness [START_REF] Zhang | Horizontal dimensional accuracy prediction of selective laser melting[END_REF][START_REF] Gibson | Additive Manufacturing Technologies[END_REF]. However, its production rate is limited, and its running costs relatively high [START_REF] Buchanan | Metal 3D printing in construction: a review of methods, research, applications, opportunities and challenges[END_REF]. DED has a better production rate, and a great flexibility. The deposition can be performed on any substrate shape. Literature already accounts for attempts to exploit the complementarity between the two processes [START_REF] Graf | Additive process chain using selective laser melting and laser metal deposition[END_REF][START_REF] Godec | Hybrid additive manufacturing of Inconel 718 for future space applications[END_REF][START_REF] Beam | Case Study: Hybrid Piston -BeAM Machines -Directed Energy Deposition[END_REF]. These industrial proofs of concept highlight the potential of AM hy-bridization, but also identified challenges in process control and mate-rial reliability. The satisfactory mechanical resistance of hybrid parts still requires to be demonstrated. Indeed, Jones et al. observed pores, cracks, and a Heat Affected Zone (HAZ) at the interface between the DED deposit and the substrate [START_REF] Jones | Remanufacture of turbine blades by laser cladding, machining and in-process scanning in a single machine[END_REF]. The impact of such interface on the tensile behavior was also investi-gated for various hybrid configurations: interface at 90 • [START_REF] Qin | Microstructure homogenizations of Ti-6Al-4V alloy manufactured by hybrid selective laser melting and laser deposition manufacturing[END_REF], 45 • [12] or 0 • [START_REF] Oh | Repairing additive-manufactured 316L stainless steel using direct energy deposition[END_REF] relative to the solicitation direction. The mechanical strength of this interface was also analyzed in the case of recharging process [START_REF] Pinkerton | Component repair using laser direct metal deposition[END_REF]. Under monotonic tensile loading, the failure occurs mostly in the DED microstructures [START_REF] Liu | Microstructure and mechanical properties of LMD-SLM hybrid forming Ti6Al4V alloy[END_REF][START_REF] Balit | Self-heating behavior during cyclic loadings of 316L stainless steel specimens manufactured or repaired by Directed Energy Deposition[END_REF]. One exception in literature would be DED-casting hybridization [START_REF] Soffel | Interface strength and mechanical properties of Inconel 718 processed sequentially by casting, milling, and direct metal deposition[END_REF], for which the tensile strength of the coarse microstructure induced by casting is weaker than that of the DED microstructure.

In the case of LPBF-DED hybridization, the microstructures induced in both sections are highly dependent on process parameters [START_REF] Marchese | Characterization and Comparison of Inconel 625 Processed by Selective Laser Melting and Laser Metal Deposition[END_REF][START_REF] Nguejio | Comparison of microstructure features and mechanical properties for additive manufactured and wrought nickel alloys 625[END_REF][START_REF] Dinda | Laser aided direct metal deposition of Inconel 625 superalloy: Microstructural evolution and thermal stability[END_REF], but literature generally agrees that the LPBF process induces a finer microstructure and a higher mechanical strength than the DED process does. Marchese et al. compared Inconel 625 obtained by LPBF and DED, and confirmed that LPBF Inconel 625 shows a finer microstructure leading to higher hardness [START_REF] Marchese | Characterization and Comparison of Inconel 625 Processed by Selective Laser Melting and Laser Metal Deposition[END_REF]. The rare publications concerning LPBF-DED hybridization focus on the technological feasibility [START_REF] Gong | Microstructure and mechanical properties of AlSi10Mg alloy built by laser powder bed fusion/direct energy deposition hybrid laser additive manufacturing[END_REF][START_REF] Uhlmann | Effects on the distortion of Inconel 718 components along a hybrid laser-based additive manufacturing process chain using laser powder bed fusion and laser metal deposition[END_REF], but some propose a characterization of the impact of the interface as well as the microstructural heterogeneity on the mechanical strength. Li et al. produced hybrid LPBF-DED Ti-6Al-4V samples and showed that, despite a HAZ reaching several millimeters of depth and a heterogeneous microstructure, the metallurgical bonding at the interface was satisfac-tory [START_REF] Li | Microstructural evolution of an aged Ni-based superalloy under two-stage hot compression with different strain rates[END_REF]. During the tensile loading, the failure occurred systematically in the DED deposit, between two layers. These conclusions were sup-ported by the in-situ SEM tensile observations on the Inconel 718 wrought substrate and DED deposit presented by Guévenoux et al. [START_REF] Guévenoux | Plastic strain localization induced by microstructural gradient in laser cladding repaired structures[END_REF], attributing the strain localization at the DED inter-layer to the Hall-Petch effect and to the suspected presence of small defects and discon-tinuities [START_REF] Guévenoux | Plastic strain localization induced by microstructural gradient in laser cladding repaired structures[END_REF]. Qin et al. confirmed that for a LPBF-DED Ti-6Al-4V hybrid parts, the DED microstructure was the limiting factor for tensile strength even after a heat treatment [START_REF] Qin | Microstructure homogenizations of Ti-6Al-4V alloy manufactured by hybrid selective laser melting and laser deposition manufacturing[END_REF]. Oh et al. showed that the depth and geometry of the notch in the case of the repair of a 316L steel part has an impact on the interface soundness [START_REF] Oh | Repairing additive-manufactured 316L stainless steel using direct energy deposition[END_REF]. They observed a HAZ, as well as cracks or pores between the substrate and the deposit which were considered responsible for the tensile failure.

More recently, Godec et al. analyzed the tensile behavior of hybrid LPBF-DED Inconel 718 samples [START_REF] Godec | Hybrid additive manufacturing of Inconel 718 for future space applications[END_REF]. The authors observed a fine microstructure with a high dislocation density in the LPBF section, while the DED microstructure displayed large columnar grains with occasional fine equiaxial zones. The resulting difference in mechanical properties between the LPBF and DED Inconel 718 was identified as a limitation for hybridization. A homogenizing heat treatment was applied on the hybrid parts, but the DED section was not entirely recrystallized and undesirable secondary phases were observed. As for the previous studies on Ti-6Al-4V, the failure systematically occurred in the DED section for as-built and heat-treated hybrid samples. Optical measurements showed that the strain is concentrated in the DED section. This strain localization has been shown to be the cause of the decrease in tensile elongation as well as responsible for the difference of monotonous tensile behavior between a mono-process part and a hybrid part [START_REF] Martin | Correlation between microstructure heterogeneity and multi-scale mechanical behavior of hybrid LPBF-DED Inconel 625[END_REF].

Thereby, the monotonous tensile behavior of hybrid parts is relatively well documented in the literature, but the fatigue behavior of such parts is still very scarcely studied. Balit et al. investigated the axial fa-tigue properties of 316L stainless steel parts made by hybridization of a wrought substrate and a DED deposit [START_REF] Balit | Self-heating behavior during cyclic loadings of 316L stainless steel specimens manufactured or repaired by Directed Energy Deposition[END_REF]. Using the self-heating technique, they established that the fatigue properties of hybrid parts are not as good as for the mono-process parts. However, they did not clearly show the impact of the hybridization, particularly that of the interface on the fatigue strength, and the deterioration of properties has not yet been explained. The objective of this paper is to investigate the mechanical strength of hybrid LPBF-DED Inconel 625 samples both under static and fatigue loading. The role of the interface and the impact of the microstructural heterogeneity is studied experimentally. The local strain fields are quantified during static and cyclic solicitations and linked with the fa-tigue strength. As-built and heat-treated microstructures are compared, which allows further analysis of the role of microstructure, and sets original perspectives for industrial applications.

Materials and methods

Inconel 625 powders

The powders were chosen according to the machine manufacturer's recommendations. The LPBF powder was supplied by SLM Solution, and the DED powder by Höganäs. Table 1 details their granulometry and chemical composition as given by the suppliers.

Additional LPBF samples were printed using the Höganäs powder to check for a possible impact of the powder. LPBF samples obtained using H öganäs and SLM Solution powders both showed identical metallurgical Maximal stress for a cyclic loading and mechanical properties. In the following, the powder's impact will be neglected. The differences between LPBF and DED parts will therefore be attributed to the manufacturing process.

LPBF, DED and hybrid processes

All samples were printed vertically by LPBF, DED or both processes for hybridization. The LPBF samples were built as near-net-shape using a SLM Solution 125HL machine. The DED samples were built as cylinders using a BeAM Modulo 500 machine, and then machined to the final geometry. The recommended optimized process parameters were used (given on Table 2).

The hybrid samples were built in two steps using the same LPBF and DED equipment and parameters. First, vertical cylinders with 12 mm diameters and 50 mm height were built by LPBF. Then, they were completed by DED directly on the top LPBF layer, up to a height of 100 mm (Fig. 1).

Reference materials: as-built and heat treated Inconel 625

A good metallurgical bonding is observed between the LPBF substrate and the DED deposit (Fig. 3a), with only occasional entrapped gas pores, observed in mono-process samples as well. At the interface, rare lacks of fusion (Fig. 2b) are observed. The microstructure of the LPBF substrate remains identical to a LPBF-only sample, with dendritic structures and chemical micro-segregations of the heavy alloying ele-ments Nb and Mo. The first layers of the DED deposit display a slightly finer microstructure than that of mono-DED samples, with fine columnar grains. After several millimeters, the large columnar grains and high crystallographic texture typical of the DED microstructure obtained by the chosen parameters are observed (Fig. 3a). In comparison, the LPBF microstructure is fine with a low level of texture and is homogeneous throughout the sample.

A heat-treatment of 4 h at 1150 • C followed by a water-quench successfully recrystallized the microstructure and erased the chemical heterogeneity (Fig. 3b). Both the DED deposit and the LPBF substrate display equiaxed grains, with a high density of twin boundaries, commonly observed in conventional nickel-based superalloys [START_REF] Malej | The influence of segregation bands and hot rolling on the precipitation of secondary phases during aging at 750 • c for nickel alloy 625[END_REF][START_REF] Gao | Grain-size dependent elasticplastic deformation behaviour of inconel 625 alloy studied by in-situ neutron diffraction[END_REF].

Both the heat-treated LPBF and DED Inconel 625 microstructures have similar fully recrystallized aspect, but a significant scale difference remains (Fig. 4b). The mean grain size of the heat-treated DED section is 121 ± 90 µm while the heat-treated LPBF has an average grain size of 32 ± 38 µm.

The difference in microstructures of LPBF and DED Inconel 625 results in different mechanical properties. The induced specific monotonous behavior is detailed in a previous study [START_REF] Martin | Correlation between microstructure heterogeneity and multi-scale mechanical behavior of hybrid LPBF-DED Inconel 625[END_REF], and the main properties of the as-built and heat-treated samples are recalled in the Table 3. The heat treatment reduces the gap in properties between LPBF and DED Inconel 625, which is beneficial to the monotonous behavior of a hybrid part.

Test specimen geometries and elaboration

For the following study, three sample geometries were used depending on the mechanical test performed. Cylindrical samples of diameter 8 mm were used for the fatigue tests of the mono-process LPBF and DED, as-built and treated (Fig. 4.a1). Similar cylindrical samples, but with a thinner diameter of 5 mm were used for the fatigue tests of hybrid specimens, as-built and treated (Fig. 4.a2). Finally, flat specimens (Fig. 4.b) were used for the monotonic tests of hybrid samples as-built and treated, because this geometry allows for image correlation measurements.

The LPBF fatigue specimens were printed as near-net-shapes geometries. Some of them were then heat-treated. The DED fatigue specimens and Hybrid fatigue specimens were obtained by machining of as-built or treated raw cylinders (Fig. 1). All the cylindrical samples, near-net-shape and machined, were polished on a semiautomated set-up. The polishing set up is made of a rotative felt drill gently put in contact with the turning sample. The abrasive medium used are diamond suspen-sions, and the finest size used is 10 µm. The obtained surface roughness is under 5 µm, and has no influence on the fatigue behavior of the samples.

The flat samples were extracted from the raw hybrid cylinders by EDM to the geometry given in the Fig. 4.b. They were manually polished to remove the surface oxidation, and a speckle paint was applied on the surface.

Tensile tests conditions

Hybrid flat tensile samples (Fig. 4.b) were instrumented with an optical extensometer on the full gauge length, and speckle paint for digital image correlation (DIC) analysis [START_REF] Martin | Correlation between microstructure heterogeneity and multi-scale mechanical behavior of hybrid LPBF-DED Inconel 625[END_REF]. The strain ratecontrolled tests were conducted using the optical extensometer.

According to the NF EN 2002-001 standard, two strain rates were used: 8x10 -5 s -1 until 0.5% strain, and then 1.5x10 -3 s -1 until failure. Images were recorded throughout the test with two cameras at an acquisition frequency of 2 Hz. The strain fields were computed using VIC3D software.

Fatigue tests conditions

All fatigue investigations were carried out on a MTS 810 machine, using cylindrical samples (Fig. facilitate the comparison between the loading and the elastic domain [START_REF] Martin | Impact of annealing treatment on the fatigue behavior of Inconel 625 produced by laser-based powder bed fusion[END_REF][START_REF] Stinville | On the origins of fatigue strength in crystalline metallic materials[END_REF]. One can easily convert σ max into the conventional alternating stress σ a by applying a multiplying factor of 0.45.

• Local strains monitoring on hybrid samples

The hybrid samples were equipped with adhesive local strain gauges in the LPBF and DED sections. The samples were solicited for 3000 cycles at load levels σ max between 100 and 700 MPa. The local strains of hybrid samples were recorded and compared to the global stress-strain response of the samples.

• High cycle fatigue tests up-to-failure

Other samples were solicited at a constant load level up-to failure. The S-N diagram was plotted for as-built and heat-treated samples batches (LPBF, DED and hybrid). If the sample reached the conventional fatigue life of 2.2 10 6 cycles without failure, the test was stopped. Subsequently, SEM observations were performed on the fracture surface to investigate the crack initiation and propagation mechanisms.

• Self-heating method

The high cycle fatigue limit was investigated using the self-heating method, which is based on the microplasticity mechanisms, detected through thermal energy dissipation [START_REF] Florin | Fast Fatigue Properties Identification by "Self-heating" Method: Application to Automotive Welded Joints[END_REF][START_REF] Ezanno | A new approach to characterizing and modeling the high cycle fatigue properties of cast materials based on self-heating measurements under cyclic loadings[END_REF][START_REF] Doudard | A probabilistic two-scale model for high-cycle fatigue life predictions[END_REF][START_REF] Ezanno | Fast characterization of highcycle fatigue properties of a cast copperaluminum alloy by self-heating measurements under cyclic loadings[END_REF]. Fig. 5 illustrates one sample mounted and ready to be tested. The samples used for the selfheating tests were instrumented with two thermocouples each. On the mono-process samples, one thermocouple was positioned in the middle of the sample, and one other at one third of the gauge length. The change in temperature measured in both thermocouples were similar. For hybrid samples, one thermocouple was in the middle of the LPBF sec-tion, and a second one in the middle of the DED section.

Two additional thermocouples were also positioned on the grips to monitor the evolution of their temperature. The samples were loaded at different stress levels for 3000 to 5000 cycles until their temperature stabilized (Fig. 5). The sample was left to cool down between two solicitations.

The temperature ΔT is computed for each stress level, using the following equation [START_REF] Amini | Self-heating of metastable 304L austenitic stainless steel under cyclic loading: Influence of initial martensite volume fraction, testing temperature and pre-strain[END_REF][START_REF] Peyrac | Self-heating Method for Fatigue Limit Determination on Thermoplastic Composites[END_REF]:

ΔT = ( T f -T 0 ) - ( T uf + T lf ) -(T u0 + T l0 ) 2 (1)
where T f is the stabilized temperature at the surface of the specimen, T 0 is the initial temperature at the surface of the specimen, T l0 and T u0 the initial temperatures of the upper and lower grips, and finally T lf and T uf are the final lower and upper grip temperatures. For all tests performed, the temperature of the upper and lower grips did not change. Equation ( 1) is therefore reduced to:

ΔT = T f -T 0 (2)
The temperatures ΔT are plotted as a function of the corresponding applied maximum stresses to determine the primary and secondary regimes (Fig. 6) [START_REF] Doudard | Determination of an HCF criterion by thermal measurements under biaxial cyclic loading[END_REF]. The intersection of the two regimes is used to identify graphically the fatigue limit. Several identification techniques are reported in the literature: (i) the intersection of the two asymptotes on Cartesian or logarithmic [START_REF] Florin | Fast Fatigue Properties Identification by "Self-heating" Method: Application to Automotive Welded Joints[END_REF][START_REF] Peyrac | Self-heating Method for Fatigue Limit Determination on Thermoplastic Composites[END_REF] plots were traditionally used, (ii) the first significant temperature raise and slope of temperature change [START_REF] Gornet | A study of various indicators to determine the fatigue limit for woven carbon/epoxy composites under self heating methodology[END_REF], (iii) the development of numerical models to further understand and characterize this technique [START_REF] Doudard | A probabilistic two-scale model for high-cycle fatigue life predictions[END_REF][START_REF] Ezanno | Fast characterization of highcycle fatigue properties of a cast copperaluminum alloy by self-heating measurements under cyclic loadings[END_REF]. In this study, the first method is used. It assumes that fatigue failure results from plastic strain accumu-lation mechanisms and is theoretically demonstrated [START_REF] Roué | Rapid determination of the high cycle fatigue properties of high temperature aeronautical alloys by self-heating measurements[END_REF]. In this method, the first raises in temperature is considered as transition be-tween the two asymptotic regimes. the fatigue strength is identified at the abscissa of the intersection of the two asymptotes.

Experimental results

Heterogeneous response of hybrid samples to monotonic and cyclic loading

The difference of microstructure between Inconel 625 made by LPBF

Table 3

Mechanical and microstructural properties of as-built and heat-treated Inconel 625 obtained by LPBF and DED [START_REF] Martin | Correlation between microstructure heterogeneity and multi-scale mechanical behavior of hybrid LPBF-DED Inconel 625[END_REF].

corresponding global strain of the sample measured by the optical extensometer.

The color scale bar is not linear because of the change of strain-rate during the tensile test. Two strain rates were applied according to the NF EN 2002-001 standard: 8x10 -5 s -1 until 0.5% strain, and then 1.5x10 -3 s -1 until failure. Images were recorded throughout the test with two cameras at 2 Hz acquisition frequency. Hence at the beginning of the test, the slow strain-rate allows for pictures in a lower deformation range than at the end of the test when the strain rate is increased.

For all samples, the localization of the strain in the DED section is obvious for global strains exceeding 0.2 %. This heterogeneity of behavior is even more significant in the case of as-built samples, where the deformation of the DED section exceeds 15 %, whereas the global deformation barely reaches 5 %. Simultaneously, less than 2 % strain is reached in the LPBF section.

After heat-treatment, the heterogeneity is still observed, but to a lesser extent with a maximal strain of 10 % in the DED and more than 4 % in the LPBF section for a global strain of 5 %.

Furthermore, the focus on low global strains given in Fig. 7a.ii and b. ii reveal that the strain localization in the DED section only starts when the global strain exceeds 0.2 %. For global strain levels below 0.2 %, the behavior of LPBF-DED hybrid samples is homogeneous.

Fig. 8 shows the behavior of hybrid samples under cyclic stresscontrolled loading. All cyclic tests are performed with a minimum stress to maximum stress ratio of 0.1. The strain-stress evolution of the LPBF section (in red) and the DED section (in blue) are plotted for 3000 F ig. 6. Example of a temperature raise versus load level plot obtained and determination of fatigue limit. and by DED causes heterogeneous mechanical behavior of hybrid samples under tensile tests [START_REF] Martin | Correlation between microstructure heterogeneity and multi-scale mechanical behavior of hybrid LPBF-DED Inconel 625[END_REF]. The recrystallisation achieved by the heat treatment reduces the gap in properties but does not erase the difference. In both cases, the failure occurs in the DED section, which has a lower yield resistance than the LPBF.

Further investigation of the strain fields during the monotonic test is presented in Fig. 7 for as-built and heat-treated samples respectively. The strain along the median line of the sample is plotted for each frame during the tensile test. The color of the plotted line indicates the stabilizes at a smaller strain level, but it remains greater than the LPBF's.

Finally, the Young modulus of LPBF and DED Inconel 625 were different. The Young modulus observed on the LPBF sections of the samples is approximately 210 GPa, versus approximately 150 GPa for the DED in both states, as-built and treated. It must be emphasized here that these tests do not permit a reliable measurement of the Young modulus. The samples are hybrid samples, with highly heterogeneous properties, and can be considered as structures. Nevertheless, inferior modulus of the as-built DED could be explained by the significant crystallographic texture [START_REF] Liang | Correlation between microstructure and cyclic behavior of 316L stainless steel obtained by Laser Powder Bed Fusion[END_REF]. However, the difference remains after the recrystallization by heat treatment, which rules out texture as the only cause for this difference.

High cycle fatigue behavior

The S-N plots obtained by stress-controlled cyclic tests with a stress ratio R = 0.1 are given in Fig. 9a, for the as-built samples, and Fig. 9b, for heat-treated samples. For the higher stress levels, all samples display similar fatigue strength. However, the LPBF as-built samples show lower fatigue strength at 2.2 10 6 cycles compared to DED and hybrid samples. After the heat treatment, all six batches (as-built and heat-treated, LPBF, DED and hybrid) reveal a lower scattering dispersion for all load levels and similar fatigue resistance at 2.2 10 6 cycles. The significant improvement of LPBF fatigue strength due to heat-treatment is already documented in another paper and attributed to the significant improvement in microstructure ductility [START_REF] Martin | Impact of annealing treatment on the fatigue behavior of Inconel 625 produced by laser-based powder bed fusion[END_REF].

In the case of DED and hybrid samples, the fatigue strength is not cycles at various load cases (σ max = 300, 400 and 500 MPa). The loads investigated are both lower and higher than the yield stress of DED (AB 433 MPa; HT 360 MPa) and LPBF (AB 667 MPa; HT 363 MPa) Inconel 625.

The case of loading between σ min = 30 MPa and σ max = 300 MPa (Fig. 8a1,b1) is under the yield tensile stress of LPBF as-built (45% of YTS), DED as-built (69%) and LPBF heat-treated (83%), DED heat treated (83%). And indeed, as-built and heat treated hybrid samples display a quasi-elastic cyclic response in both LPBF and DED sections. A slight plastic strain accumulation of about + 0.02% is observed on the as-built sample, but not after the heat treatment. The as-built micro-structure of LPBF and DED Inconel 625 is heterogeneous with a high dislocation density. This specific microstructure causes local variations in mechanical properties, and therefore plastic strains localization [START_REF] Guévenoux | Plastic strain localization induced by microstructural gradient in laser cladding repaired structures[END_REF]. After the heat treatments, these heterogeneities are erased.

For higher loading cases (Fig. 8a2, b2 and a3, b3), both LPBF and DED sections are subject to elastic shakedown phenomena. However, the residual deformation of the LPBF section is lower than that of the DED section. For the loading at σ max = 400 MPa, the mean strain of the DED section increases by + 0.5 %, versus less than + 0.15 % for the LPBF section. After heat treatment, the gap is reduced with + 0.45 % for the DED and + 0.2 % for the LPBF. For the loading at σ max = 500 MPa, the strain gauges suffered adhesive failure, preventing from quantifying the gap between the responses of the LPBF and DED sections, but it is clearly worsening. The as-built LPBF stabilizes under +1 % mean strain, while the DED exceeds +3 %, with up to +0.2 % strain accumulation per cycle at the beginning of the fatigue test. After heat treatment, the DED notably impacted by the heat treatment. The tendency and order of magnitude of fatigue strength are similar between Fig. 9a and Fig. 9b, despite the evolution in monotonic behavior and properties detailed previously. The decreases in yield tensile stress of LPBF, DED and hybrid heat-treated samples do not alter the fatigue strengths, nor improve it significantly. Fig. 9 also shows that the hybrid samples, both as-built and heattreated, systematically fail in the DED section, or at the interface. As illustrated by the low dispersion scattering in Fig. 9, the fatigue strength is independent on the fracture localization in both case: hybrid as-built and hybrid heat treated. Moreover, no trend is identified between the loading level and these failure localizations. samples, the temperature increases are close to the asymptotes for all loads levels. In the case of LPBF samples, the temperature raises in the vicinity of the transition is higher than the asymptotes. Furthermore, in the case of as-built LPBF (Fig. 12a), the sample failed following the 2000 first cycles at σ max 700 MPa. Hence it is suspected that the second regime is not yet fully reached, and the resulting intersection is slightly underestimated, and that the transition is even farther from the asymptotes.

Nevertheless, Fig. 12a shows that the self-heating fatigue limit of the as-built LPBF (600 MPa) samples was significantly higher than the DED as-built samples (450 MPa). After the recrystallization by heat treat-ment, the gap between the two microstructures was drastically reduced, with 450 MPa for LPBF samples and 400 MPa for DED samples. The limits found by the asymptotes are significantly superior to those found by the S-N curves. This difference could be caused by the presence of defects, not taken in account in the asymptote analysis.

Contrary to the measurements performed on mono-process samples where self-heating curves are computed separately on different specimens, the two self-heating fatigue curves of hybrid samples are obtained from a single specimen by positioning two thermocouples: the first in the LPBF section and the second in the DED section. The measurements performed on these hybrid samples are illustrated in Fig. 12a for the as-built state and in Fig. 12b for the heat-treated state. The empty symbols represent the previous measurements on the LPBF and DED mono-process samples. Filled symbols represent each section of the hybrid samples (red for LPBF and blue for DED). Likewise mono-process sam-ples, the LPBF section displayed a fatigue limit slightly higher than the DED section. However, the difference between the LPBF and DED sec-tions was significantly lower than that between the mono-process LPBF and DED samples. The heat transfers between the LPBF section and the DED section in hybrid samples may explain this decrease. In the case of heat-treated samples, a given load case caused a lower global increase in temperature of hybrid samples compared to that of mono-process sam-ples. This can be explained by the difference in sample geometry. The hybrid samples have a smaller diameter and hence exhibit less energy dissipation. However, because the asymptotes intersection is used and not the absolute temperature increase, the results remain comparable.

As-built hybrid samples (Fig. 13a) displayed a fatigue limit slightly superior to 450 MPa for the DED section, which was similar to the limit identified on DED mono-process as-built samples. The LPBF section shows a slightly better fatigue limit, above 500 MPa, but this limit was significantly lower than the 600 MPa limit identified on LPBF samples.

Failure analysis

The fatigue failures were investigated by SEM. The analysis revealed the presence of defects at all initiation sites for LPBF and hybrid samples. These defects are systematically located on or near the surface of the samples. In the DED samples, the crack initiated on the microstructure. The initiation mechanisms were identical for asbuilt and heat-treated samples which may explain the small difference between the two states (as-built and heat-treated). Only the as-built samples are illus-trated in Fig. 10.

In hybrid samples failed in the DED section, defects were systematically observed at the crack initiation sites (Fig. 10a). The irregular shape of these defects allows to link them to fluid movements of the melt pool, commonly described as "Marangoni defects" [START_REF] Gao | Effect of processing parameters on solidification defects behavior of laser deposited AlSi10Mg alloy[END_REF]. This type of defects was unexpected as none had been observed in the mono-process DED samples obtained with the same process parameters.

All cracks that initiated at the interface between the DED deposit and the LPBF substrate on hybrid samples revealed a circular lack of fusion (Fig. 10b). Its dimension and consistency recall the circular lasing strategy used for the DED deposit (Table 2). The first layers are depos-ited on substrate at a room-temperature, and the bonding in between DED melting pools is not regular in comparison to higher layers because the steady state is not yet established.

To conclude, the initiation sites observed on the hybrid samples were different than that of the mono-processes samples. On the one hand, the DED samples, as-built or treated, revealed systematic fracture initiation on microstructure without identifiable defects (Fig. 10c). On the other hand, one or several lacks of fusion defects were identified at the initi-ation site in each LPBF sample (Fig. 10d).

The Fig. 11 shows the two main propagation mechanisms observed in all samples: crystallographic failure around the initiation site, for small crack size, followed by ductile failure, easily noticeable by the fatigue striations [START_REF] Martin | Impact of annealing treatment on the fatigue behavior of Inconel 625 produced by laser-based powder bed fusion[END_REF].

High cycle fatigue strength evaluated by self-heating method

The self-heating results are presented in Fig. 12a for the monoprocess as-built samples and in Fig. 12b for the mono-process heattreated samples. The temperature raises are plotted versus the applied stress levels of LPBF (red) and DED (blue) samples.

The LPBF and DED self-heating curves present a difference in the transition between the two regimes' asymptotes. In the case of DED After heat treatment, all limits identified were in the range 400-450 MPa (Fig. 13.b), whether it was measured on LPBF, DED, hybrid or mono-process.

Finally, Table 4 summarizes experimental yield tensile stresses (YTS) and fatigue strengths estimated from S-N curves and from selfheating curves. The fatigue strengths estimated from S-N plots are lower than the theoretical fatigue limits identified from self-heating method. This difference is particularly significant for LPBF samples, as well as as-built DED samples.

Analysis and discussion

Impact of defects and damage initiation on fatigue strength

The simultaneous analysis of S-N plots and fractographic study of the tested samples can highlight the impacts of defects on the fatigue behavior of samples [START_REF] Rackwitz | Effects of cryogenic temperature and grain size on fatigue-crack propagation in the mediumentropy CrCoNi alloy[END_REF][START_REF] Zhu | Influence of grain size on the small fatigue crack initiation and propagation behaviors of a nickel-based superalloy at 650 • C[END_REF]. For the low stress levels, the crack initia-tion phase, highly impacted by defects, is preponderant in fatigue life as compared to the propagation [START_REF] Yadollahi | Additive manufacturing of fatigue resistant materials: Challenges and opportunities[END_REF]. The fatigue strength at 2.2 10 6 cycles of as-built LPBF is found to be 250 MPa versus 350 MPa after heat treatment. This improvement is even more significant considering the difference in the yield tensile stress between the as-built and the heat treated state. In the S-N plots illustrated in Fig. 13, the maximum stress was normalized by the yield tensile stress. As-built LPBF samples displayed fatigue strength, expressed in terms of maximum stress, lower than 50 % of its YTS, whereas after the heat treatment it can withstand a load case of more than 90 % of its YTS. The improvement was mainly attributed to the higher defect tolerance in the recrystallized and homogenized microstructure compared to the as-built microstructure [START_REF] Martin | Impact of annealing treatment on the fatigue behavior of Inconel 625 produced by laser-based powder bed fusion[END_REF].

Fig. 14b shows that the DED samples displayed fatigue strength at 2.2 10 6 cycles close to 100 % of their YTS, whether they were as-built or heat treated. Unlike the LPBF samples which were printed nearnet-shape and polished, the DED samples were machined and then pol-ished. The fractographic analysis of the samples confirmed that defects were systematically observed on the initiation sites for the LPBF attributed the difference to the interlayer defects. In this study, the self-heating fatigue limit and the fatigue strength, expressed in terms of maximum stress at 2.2 10 6 cycles for as-built and heat-treated DED and hybrid samples are comparable, and close to the respective yield tensile stress of the material (Table 4). All those samples were machined and polished. In the case of as-built LPBF samples, this fatigue strength is below 250 MPa, while the self-heating tests identified a theoretical fa-tigue limit around 600 MPa, close to 100% YTS. After the heat treat-ment, the gap is reduced but remains. All LPBF samples were tested as polished near-net shape samples. In addition, the self-heating results from this study's LPBF samples show a different trend compared to DED samples (Fig. 12). The increases of the sample's temperature are significantly higher than the asymptotes in the transition between the first and second regime. These premature temperature raises can be attributed to the localization of plasticity at the tip of the defects, causing heat dissipation prone to increase the global temperature of the sample before the generalization of microplasticity. The presence of harmful defects is hence detectable by the self-heating method. Yet, the theoretical analysis of the two regimes to determine the fatigue limit is not suitable to the LPBF Inconel 625 microstructure with defects. Using the first temperature increase, asymptote of the transi-tion, or monitoring local temperature fields would be interesting per-spectives to capture the impact of defects by self-heating. Furthermore, the transition between the two regimes could be representative of the defects and their impact on the fatigue strength of metallic materials.

Initiation and propagation mechanisms in hybrid specimen

The crack leading to the fatigue failure of the hybrid specimen always initiated on a defect. When the failure happened at the LPBF-DED interface, a circular lack of fusion was observed. When the initiation took place in the DED section, Marangoni defects [START_REF] Gao | Effect of processing parameters on solidification defects behavior of laser deposited AlSi10Mg alloy[END_REF] were observed (Fig. 10). On the S-N plot Fig. 9, the samples that failed at the interface or in the DED section present similar fatigue lives and similar dispersion. The type of initiation and its localization does not affect the fatigue life of the samples. Only the as-built samples that were solicited at high load levels (greater than450 MPa) display a shorter fatigue life than the mono-process samples.

For higher load levels, the contribution to fatigue life is mainly coming from the crack propagation compared to the crack initiation [START_REF] Rackwitz | Effects of cryogenic temperature and grain size on fatigue-crack propagation in the mediumentropy CrCoNi alloy[END_REF][START_REF] Zhu | Influence of grain size on the small fatigue crack initiation and propagation behaviors of a nickel-based superalloy at 650 • C[END_REF]. The samples that had a shorter fatigue life at high load levels both failed at the LPBF-DED interface, and both displayed a large circular lack-of-fusion mentioned above (Fig. 10). The shape and size of this defect induced stress concentration throughout the crack propagation and caused the fatigue life to drop. The propagation mechanisms observed in the hybrid samples Fig. 11 were identical to that observed in mono-process samples [START_REF] Martin | Impact of annealing treatment on the fatigue behavior of Inconel 625 produced by laser-based powder bed fusion[END_REF]. samples, while the DED samples mostly showed initiation sites within the bulk microstructure (Fig. 10b). The lack of surface defects is sufficient to improve the fatigue resistance of as-built DED Inconel 625 microstructure despite its heterogeneity and high dislocation density. This dislocation density is even higher in the case of LPBF samples and is assumed to be responsible for the low damage tolerance of this micro-structure [START_REF] Martin | Impact of annealing treatment on the fatigue behavior of Inconel 625 produced by laser-based powder bed fusion[END_REF].

According to these results on as-built mono-process samples, the hybrid samples should fail in the LPBF section, or at the interface. Yet, all the hybrid samples tested up-to-failure broke in the DED section or at the LPBF-DED interface (Fig. 9). The hybrid samples were machined and polished, unlike LPBF samples that were printed as near-net-shapes and just polished. LPBF is known to induce most defects around the surface of the samples [START_REF] Yadollahi | Additive manufacturing of fatigue resistant materials: Challenges and opportunities[END_REF]. Machining has removed this specific zone and the microstructure of the LPBF section in hybrid samples is believed to be almost free of defects. Hence, considering the results from self-heating, the fatigue strength of defect-free LPBF is superior to DED Inconel 625, which is consistent with the failures of hybrid samples taking place in the DED section.

This major impact of surface defects is also supported by a short review of the fatigue properties of Inconel 625 reported in the literature [START_REF] Theriault | Fatigue behavior of laser consolidated IN-625 at room and elevated temperatures[END_REF][START_REF] Poulin | Fatigue strength prediction of laser powder bed fusion processed Inconel 625 specimens with intentionallyseeded porosity: Feasibility study[END_REF][START_REF] Koutiri | Influence of SLM process parameters on the surface finish, porosity rate and fatigue behavior of as-built Inconel 625 parts[END_REF][START_REF] Pereira | Fracture behavior and fatigue performance of inconel 625[END_REF][START_REF] Trester | Fatigue Strength if Inconel 625 plate and weldments used in the DIII-D configuration vacuum vessel[END_REF][51]. S-N plots issued from various fabrication processes are given in Fig. 15 (red for as-built LPBF, blue for as-built DED, black for wrought and grey for cast microstructures). The empty symbols correspond to results obtained with symmetric alternating loading (R = -1), and full markers are for conditions identical to this study's (R = 0.1). The fatigue strengths are represented in terms of maximal stress σ max in Fig. 15a, and σ max normalized by the respective YTS of each study in Fig. 15b.

Fig. 15a shows a large variability in fatigue properties. However, when comparing to the respective YTS Fig. 15b, most of the results show fatigue strength between 10 6 and 10 7 cycles to be close to 100% of the YTS, except this study's as-built LPBF and LPBF samples from Koutiri et al. [START_REF] Koutiri | Influence of SLM process parameters on the surface finish, porosity rate and fatigue behavior of as-built Inconel 625 parts[END_REF]. Indeed, those are the only as-built near-net-shape samples (not machined and just polished). It therefore appears that a polishing step alone is not sufficient to remove the defect-dense area near the surface of the sample, which is detrimental to fatigue life. However, heat treatment improves the defects tolerance of the LPBF microstructure and raises his fatigue strength to 100% of the YTS.

The impact of defects on the self-heating response of metallic materials is subject to discussions in the literature, but no firm conclusion has been reached. Defects are often considered as preferential sites for microplasticity in the models used to explore self-heating experimental results and predict fatigue life [START_REF] Ezanno | Fast characterization of highcycle fatigue properties of a cast copperaluminum alloy by self-heating measurements under cyclic loadings[END_REF], but their impact on the selfheating curve shape were not discussed. Bercelli et al. [START_REF] Bercelli | A probabilistic approach for high cycle fatigue of Wire and Arc Additive Manufactured parts taking into account process-induced pores[END_REF] showed that the rare defects detected in a WAAM bronze-Al alloy had no impact on the experimental self-heating curves. Balit et al. [START_REF] Balit | Self-heating behavior during cyclic loadings of 316L stainless steel specimens manufactured or repaired by Directed Energy Deposition[END_REF] showed different self-heating curves for vertical and horizontal 316L steel DED samples and For lower solicitation levels, the initiation step is preponderant, and no clear difference between mono-process and hybrid sample fatigue resistance were observed. Further tests of crack propagation monitoring could confirm this impact of the large defects in the samples section on the fatigue life. In addition, the fatigue resistance of metallic alloys has been mostly studied on samples with homogeneous characteristics. The hybrid samples studied here have two sections with a significant gap in properties and behavior, for example strainstress relations. The description of initiation, propagation and final failure might not be adequate to describe fully the fatigue resistance of hybrid samples.

Impact of heterogeneous mechanical properties on fatigue strength

The self-heating tests suggest that the Inconel 625 microstructure induced by LPBF, as-built or heat-treated, should have significantly higher fatigue resistance than the DED microstructure. Yet, the fatigue properties of as-built, near-net-shape LPBF samples are significantly limited by the presence of lacks of fusions around the surface of samples, and their fatigue strength are lower than that of the DED's. The heat-treated microstructure exhibits a better accommodation capacity, but the fatigue strength, expressed in terms of maximum stress at 2.2 10 6 cycles, remains slightly lower than that of DED's (Fig. 9b). Despite this superiority of the DED microstructure compared to the LPBF micro-structure when measured on mono-process samples, the failure of hybrid samples systematically occurs in the DED section, or at the interface. In both cases, defects were observed to cause the crack initiation: lacks of fusion at the interface or Marangoni pores in the DED section. Both are related to the process control of DED deposition.

But the difference of microstructure between LPBF and DED is also responsible. The cyclic stress-controlled tests of hybrid samples monitored with strain gauges illustrated in Fig. 8 show immediate strain localization in the DED section, which concentrates most of the accumulated plastic strain. This is due to the lower yield tensile stress of the DED. In addition, the DED section is subject to higher hardening and dissipates more energy than the LPBF section. This is consistent with the self-heating results, where the increases in temperature are higher for the DED section despite the heat transfers between the two sections. In the ideal case of defect-free hybrid samples, it is assumed that the failure would still occur systematically in the DED section due to the strain localization and unbalanced energy dissipation.

In addition, the gauges measured only the axial normal strain. Shear strains could be induced by the gradient of properties between the LPBF substrate and DED deposition and could also explain the preferential failure of hybrid samples at the interface. Further investigations on thinner and flat test samples monitored with strain fields and tempera-ture field measurement could be of great interest to better understand the challenge of samples with heterogeneous properties.

The heat treatment was shown to increase the ductility of the samples, especially for the LPBF Inconel 625 microstructure [START_REF] Martin | Impact of annealing treatment on the fatigue behavior of Inconel 625 produced by laser-based powder bed fusion[END_REF]. It also reduces the gap between the static tensile properties of LPBF and DED, which increases the mechanical resistance of hybrid samples. For cyclic loading, the heat treatment reduced the difference in plastic strain accumulation between the LPBF and the DED sections. However, a sig-nificant difference remains and the DED dissipated more energy thus displaying the lowest fatigue strength. As for the static tensile proper-ties, the hybrid samples' fatigue resistance was limited by the DED section but beneficiated from similar properties. The fatigue strength of hybrid samples is equivalent to that of the DED mono-process parts.

Conclusion

The simultaneous investigation of the fatigue behavior of monoprocess LPBF or DED Inconel 625 with samples produced by LPBF-DED hybridization allowed to identify the impact of each process on fatigue strength of hybrid samples. The effects of the specific LPBF and DED microstructures were explained through the investigation of asbuilt and heat-treated states. In addition, the study involved near-netshaped and machined samples, which permitted to analyze the impact of near-surface defects on LPBF fatigue behavior. Addressing these multiple effects allowed to draw three main conclusions on fatigue behavior of LPBF and DED Inconel 625, as well as a benchmark for hybrid LPBF-DED process: LPBF-DED Inconel 625, this paper points out the following perspectives:

• Combining the self-heating techniques with a local strain field measurement could provide additional information on the fatigue resistance and weaknesses of the samples. The proposed link between the self-heating regimes' transition and the presence of large defects must be validated and numerical modelling should be proposed to analyse fatigue self-heating curves. • The surface of the LPBF samples were polished, while the hybrid specimens were machined. A comparison of LPBF mono-process machined samples could help in determining the exact impact of the surface defects and roughness. • The LPBF-DED interface and local complex strains generated (in particular shear strain) could be quantified. • The DED process could be optimized to limit the difference in microstructures between the LPBF and DED sections of hybrid samples and homogenize even further the mechanical and fatigue properties.

• The fatigue strength of hybrid LPBF-DED parts was identical to that of the weakest section, which in the particular case of machined samples is the DED section. The transition between LPBF and DED microstructures was not the limiting factor and did not weaken the properties of as-built, nor of heat treated samples. • The near-surface defects are the main limiting factors in the fatigue life of LPBF or DED samples, mono-process or hybrid. The near-netshape LPBF samples showed systematic initiation on defects and machined hybrid samples tested showed initiation on near-surface lacks of fusion or Marangoni defects. • The recrystallization provided by heat treatment successfully increased the defects tolerance of the LPBF and DED microstructures.

The fatigue strength, expressed in terms of maximum stress at 2.2 10 6 cycles, was close to 100 % of the YTS for heat treated LPBF, DED and hybrid samples, which was consistent with results reported in the literature [START_REF] Theriault | Fatigue behavior of laser consolidated IN-625 at room and elevated temperatures[END_REF][START_REF] Poulin | Fatigue strength prediction of laser powder bed fusion processed Inconel 625 specimens with intentionallyseeded porosity: Feasibility study[END_REF][START_REF] Koutiri | Influence of SLM process parameters on the surface finish, porosity rate and fatigue behavior of as-built Inconel 625 parts[END_REF][START_REF] Pereira | Fracture behavior and fatigue performance of inconel 625[END_REF][START_REF] Trester | Fatigue Strength if Inconel 625 plate and weldments used in the DIII-D configuration vacuum vessel[END_REF][51]. In addition, the heat treatment reduced drastically the gap between LPBF and DED microstructures.

In addition to this investigation of the fatigue behavior of hybrid

Fig. 1 .Fig. 2 .

 12 Fig. 1. The two fabrication steps for LPBF-DED hybrid samples: a) LPBF b) DED.

Fig. 3 .

 3 Fig. 3. Z-axis IPF EBSD maps of a) as-built b) heat treated hybrid LPBF-DED Inconel 625 in the vicinity of the interface.

Fig. 4 .

 4 Fig. 4. Geometries of the a) cylindrical and b) flat specimens, dimensions in [mm].

Fig. 5 .

 5 Fig. 5. Illustration of the self-heating set-experimental up and example of temperature measurement.

Fig. 7 .

 7 Fig. 7. Evolution of the local strain profile along the a) as-built and b) heat-treated hybrid samples during tensile test. The color scales give the global elongation measured by the optical extensometer i. until speckle loss, ii. for small deformations.

Fig. 8 .

 8 Fig. 8. Stress-strain response of the LPBF (red) and DED (blue) sections of a) as-built and b) heat-treated hybrid samples for different alternative stress levels. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 9 .Fig. 10 .

 910 Fig. 9. S-N plots for a) as-built and b) heat-treated LPBF (red), DED (blue) and hybrid (black) samples tested up-to-failure. Empty markers represent none-broken samples stopped at 2.2 10 6 cycles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 11 .

 11 Fig. 11. Fractographic SEM images of a hybrid fatigue sample failed in the DED section. i. Initiation site, ii. Crystallographic failure, iii. Ductile failure and fatigue striations.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Self-heating fatigue curves for a) as-built, and, b) heat treated LPBF (red) and DED (blue) mono-process samples: self-heating temperatures as a function of their corresponding maximum stress levels. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 14 .

 14 Fig. 14. S-N plots of a) LPBF b) DED as-built and heat-treated samples: the maximum stresses were standardized by their respective yield tensile stresses.

Fig. 15 .

 15 Fig. 15. Comparative S-N plots of fatigue properties of Inconel 625 made by various process: Results from this study and literature represented by a) maximal stress, b) maximal stress standardized by the respective YTS [46-51].

Table 1

 1 LPBF and DED powders' granulometry and chemical composition.

	Powder

size (deciles) Chemical composition (wt %) Ni Cr Mo Nb þ Ta

							Fe	Co
	LPBF	D 10 = 21 µm	Bal.	21.41	8.99	3.69	4.13	0.16
	powder	D 50 = 34 µm
	(SLM	D 90 = 54 µm
	solution)
	DED powder	D 10 = 47 µm	Bal.	20.5	8.5	3.93	4.08	0.17
	(Höganäs)	D 50 = 67 µm
		D 90 = 94 µm

Table 2

 2 LPBF and DED process parameters used for Inconel 625 samples.

Table 4

 4 Summary of Yield Tensile stresses and fatigue properties for LPBF, DED and hybrid Inconel 625 (as-built and heat treated).

			YTS	Fatigue strength determined
				from S-N curves	from self-heating curves
	LPBF	As-built	667 MPa	250 MPa	600 MPa
		Heat-treated	363 MPa	350 MPa	450 MPa
	DED	As-built	433 MPa	400 MPa	450 MPa
		Heat-treated	360 MPa	400 MPa	400 MPa

Acknowledgements

This work was financed and supported by Segula Technologies.

Data availability

Data will be made available on request.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Manuel Leuenberger
email: manuel.leuenberger@inf.unibe.ch

Can I Remove This Method? How Live Feedback from the Ecosystem Supports Co-Evolution

Keywords: CCS Concepts, Software and its engineering → Software maintenance tools, Patterns, co-evolution, API migration, refactoring, IDE, API usage ACM Reference Format:

Albeit open-source projects have been co-evolving since years, upgrading a library can still be a tedious task for developers. APIs change over time, and breaking changes require precious developer time to adapt a dependent project's code. We present our vision on how embracing co-evolution in general, and library upgrades in specific, as a first-class citizen in our IDE, can support both API authors and API users. In a tiny self-experiment we show how API authors can profit from live feedback on the compatibility of their changes. API users can profit by being provided migrations to be applied on the dependent's code.

Introduction

When a library's API changes over time, its dependents have to adapt to the new version of the API. While many of these changes are refactorings that can be recovered and replayed [START_REF] Dig | How do APIs evolve? A story of refactoring[END_REF], some changes are of a more complex nature and require careful changes in a dependent [START_REF] Brito | Why and how Java developers break APIs[END_REF]. Due to the non-triviality of the co-evolution of a library and its dependents, library developers are reluctant to change their public APIs [START_REF] Brito | Why and how Java developers break APIs[END_REF], and developers of dependents are reluctant to upgrade to new versions of a library [START_REF] Gaikovina Kula | Do developers update their library dependencies?[END_REF]. This situation leaves unexploited potential for improvement on both sides. Once a version of an API is defined and published, library developers may value backwards-compatibility higher than changing and improving an API. Developers of dependent projects stick to older versions of a library because an upgrade would require changes in their code and extensive retesting. Sticking to an old version of a library is not without risk, e.g., outdated dependencies potentially reveal known security issues. 1Although a library and its dependent projects co-evolve, communication of changes across projects is usually only possible by rudimentary means. Libraries may publish a new version along with a change log or a migration guide in textual form. 2 Developers of dependents have to filter out the relevant changes and assess the impact on their projects individually and mostly without tool support. Developers of a library dependent can provide bug reports in case they observe broken behavior in a new version. Library developers on the other side are eager to know how their APIs are used [START_REF] Haenni | A Quantitative Analysis of Developer Information Needs in Software Ecosystems[END_REF]. Yet, to learn about how their library is used, developers have to explore the ecosystem. All of the aforementioned tasks lack proper support by the IDE, therefore developers have to rely on external resources, e.g., issue trackers, wikis, mailing lists, Q&A communities, which can be hard to search and extract the relevant aspects for a specific project. We believe that is in the interest of the actors on both sides of the story to establish a tighter feedback loop that is well integrated into their tooling. Adapted tools for the ecosystem-aware co-evolution context should support library developers in exploring the ecosystem and assessing the impact of changes. Library dependents should be able to collect the changes affecting themselves quickly, as well as they could profit from a migration path reified as an executable, inspectable, and debuggable entity.

In this paper we present our vision on how an ecosystemaware, tool-supported co-evolution could look like. We present a small self-experiment around the post-mortem analysis of a breaking change introduced by the author of this paper. The redesign of a dropdown widget in a widget library lead to the removal of a method, which in turn broke a dependent that relied on this method. As a mean to avoid or mitigate the effects of a breaking change, we propose an integrated approach that enables the observation and control over the effects of a change in a library on its dependents. By running the dependent's examples or tests relevant to a library change automatically, we can provide live feedback on the effects of the change to the library developer. This makes the dependent system both inspectable and debuggable for the library developer, which in turn opens the opportunity to create migration scripts to be used by the developers of the dependent.

The rest of the paper is structured as follows. In section 2 we present a tiny case study of a breaking change. Our implementation of a live feedback prototype for co-evolution is outlined and discussed in section 3. Related work is summarized in section 4, and conclusions and future work can be found in section 5.

A Tiny Case Study

Our tiny case study is based on the experience of the author of this paper.

While working on a UI widget library for Pharo Smalltalk3 , the dropdown widget required a redesign to make it more flexible and easier to use. As the widget included some duplicated code as well as unused state and methods, the goal of the first phase was to clean up the dropdown code. In the course of this clean up, the method BrDropdownLook >> #dropdownTarget: was removed, which sets the element the dropdown is attached to. Running hundreds of examples that exercise the widget library confirmed that the method was indeed unused by these examples. Thus, the change was considered viable and the changes were pushed. A few days later, it was discovered that the removal of the method broke the tests in a dependent project that used the method. As a consequence, the initially removed method was added again to support the needs of the dependent project.

An analysis of this story reveals multiple moments where a lack of information leads to suboptimal decisions. First, it was unclear that the removed method was public. The method was removed because it was believed to be internal, as it was neither documented as public, nor exercised in the library's tests. Better tests and better documentation could help to mitigate this problem. Second, it was unclear that a dependent used the method. This is generally the point where the integration in IDEs breaks. IDEs usually assume a closed ecosystem around the projects in the local workspace. This assumption does not hold in the case of libraries. The ecosystem includes its dependents that live outside of the common IDE workspace.

In total it took several days from the initial question "Can I remove this method?" to the conclusive answer "No, it is used.". We want to answer this question and similar ones quicker, live in the best case.

Live Feedback from the Ecosystem

We want to give library developers quick feedback on the effects of their changes, with only minimal changes to their development setup. To achieve this we need to be able to query the ecosystem in a similar way as it is done in the Ecosystem Monitoring Framework. Our prototype is based on a library image, in which the library developer performs the library changes, and a dependent image, which has a dependent installed together with the library. Figure 1 shows how the two images are connected. The dependent image runs on the same machine as the library image, but it could also run on another machine, e.g., as on-demand images on a continuous integration server. In our prototype the dependent image offers two endpoints. Through the changes endpoint the library image pushes all code changes to the dependent image to synchronize the library code in both images. The example runner endpoint provides an interface to evaluate a code snippet in the dependent image and answers the snippet's result or exception. The communication relies on Seamless4 , which enables interaction with remote objects by proxying message sends through a network tunnel.

Tool Integration

Our main goal is to provide live feedback on the effects of library changes. We extend the system browser with badges on classes and methods, indicating whether the class or method is used by the dependent, as well as a green/red colors representing success or failure of the executed dependent examples, see Figure 2 and Figure 3.

Whenever some code changes in the library image, the changes are automatically pushed and applied on the dependent image. Additionally, a change also triggers the execution of examples to exercise the dependent system and gather feedback on the effects of the change. In our case study, we are only interested in examples exercising the code of the dropdown widget. Hence we instrument the methods in the dropdown widget package and track which examples in the dependent use a specific method. After the first change in the library image the 75 relevant method are instrumented and all of the 189 examples of the dependent are run in 25.2s, yielding 51 examples that exercise the dropdown widget. The instrumentation can be removed for subsequent As the whole process is automated, the library developer receives live feedback on the effects on a dependent on of every change in the library.

Discussion

The presented prototype for live feedback from the ecosystem has several drawbacks. First, library dependents have to be detected. This could be solved by mining dependency relation in package repositories, such as GitHub. Second, running multiple images besides a library development image requires resources that can be allocated. In the setup of the case study the dependent image is running on the same machine as the library development image, which makes inspection and debugging of the effects of changes easy. We could also imagine a setup where dependent images are run on-demand on a continuous integration server, which makes inspection and debugging more complex though. Third, the feedback has to be treated with care as the quality depends on the quality of the dependent tests. Depending on how exhaustive the examples and tests of a dependent exercise the relevant library code can vary. No observed exceptions do not necessarily imply a non-breaking change. Likewise, not breaking one specific dependent cannot guarantee that no dependent is affected by a change. As a counter measure, static analysis techniques could be combined with the dynamic technique currently used to gain a more complete picture of the library usage in the ecosystem.

Embracing co-evolution as a first-class citizen in our IDE also offers new opportunities. First, live feedback tightens the feedback loop from a change in a library to its effects in its dependents. Breaking changes become immediately evident as such, given the information is gathered by running examples representative for the library use in the ecosystem. Second, exceptions can be inspected and debugged. This allows the library developer to design a migration script that can be applied when upgrading the library in the dependent. These scripts can then be distributed along with the new version of the library, potentially easing the upgrade of other dependents as well. Instead of readding a temporarily removed method again, such a script could rewrite the call sites of this method either statically or dynamically. Pushing the migration idea further, it might also be possible to design a library upgrade in a dependent as a pull request submitted automatically on the release of a new library version. The pull request could already apply the necessary code transformations on the dependent code, the maintainers of the dependent could review the changes.

Related Work

Our work is motivated by the observation that, besides the fact that many open-source project have been co-evolving and depending on each other for years, upgrading a library dependency is still costly. Kula et al. performed a study on over 4 600 GitHub Maven projects, concluding that the majority of the projects have outdated dependencies, revealing security vulnerabilities. [START_REF] Gaikovina Kula | Do developers update their library dependencies?[END_REF] Bavota et al. studied the evolution of the Apache ecosystem and find that dependency upgrades are mostly performed when many bugs are fixed or a new major version of the dependency provides new features or services. [START_REF] Bavota | The Evolution of Project Inter-dependencies in a Software Ecosystem: The Case of Apache[END_REF] Both studies report the reluctance of developers to upgrade dependencies as it is presumed to be risky and costly.

Automating the upgrade process has been an ongoing subject in the field of API usage research. These approaches attempt to derive the necessary code tranformations on the dependent of an API. Dig et al. reported that many API changes can be modelled by reconstructing basic refactorings on the API, which in turn can then be replayed on the code of a dependent. [START_REF] Dig | How do APIs evolve? A story of refactoring[END_REF][START_REF] Fowler | Refactoring: Improving the Design of Existing Code[END_REF][START_REF] Roberts | A Refactoring Tool for Smalltalk[END_REF] Hora et al., as well as Robilliard et al., inferred additional API properties that support the migration from one version to another one. [START_REF] Hora | APIEvolutionMiner: Keeping API Evolution under Control[END_REF][START_REF] Robillard | Automated API Property Inference Techniques[END_REF]. All of these approaches have in common that they are applied post-mortem, i.e., after a change in a library has already been published. Our approach tries to make the effects of changes more imminent, with the goal to enable more informed decisions when changing code. Tymchuk showed that timing for feedback matters, and developers appreciate contextualized and live feedback for code quality measures. [START_REF] Tymchuk | Quality-Aware Tooling[END_REF] Our current implementation of live ecosystem feedback draws a lot of similarities to the more generally applicable Ecosystem Monitoring Framework by SpasojeviÄĞ. [START_REF] Spasojević | Developing Ecosystem-aware Tools[END_REF] 5 Conclusions and Future Work

We present how the integration of co-evolution as a firstclass citizen of the IDE can look like, and we discuss its benefits and drawbacks. Exemplified by a tiny case study of a breaking change, we show how live feedback on the effects of a change on ecosystem supports library developers by immediately revealing a change as a breaking change. We propose to further explore the integration of co-evolution with the goal to lower to cost of upgrading a library dependency.

Future Work

We believe that our approach also has the potential to enable exploration, creation, and validation of necessary migrations on library dependents for library upgrades. In this work we only hint at how the upgrade process might be improved, we intend to further investigate this research direction. Furthermore, the evaluation of the usefulness of our approach needs to be extended by applying it over a longer time within a real world experiment.

Figure 1 .

 1 Figure 1. Architecture to connect and synchronize a library image with a dependent image running dependent code.

Figure 2 .

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Acknowledgments

We gratefully acknowledge the financial support of the Swiss National Science Foundation for the project "Agile Software Assistance" (SNSF project No. 200020-181973, Feb 1, 2019 -Apr 30, 2022).


Philippe Laurenc ¸ot
email: philippe.laurencot@univ-smb.fr

LARGE TIME CONVERGENCE FOR A CHEMOTAXIS MODEL WITH DEGENERATE LOCAL SENSING AND CONSUMPTION

Keywords: Mathematics Subject Classification, 35B40 -35K65 -35K51 -35Q92, Key words and phrases, convergence -Liapunov functional -chemotaxis-consumption model -local sensing

Convergence to a steady state in the long term limit is established for global weak solutions to a chemotaxis model with degenerate local sensing and consumption, when the motility function is C 1 -smooth on [0, ∞), vanishes at zero, and is positive on (0, ∞). A condition excluding that the large time limit is spatially homogeneous is also provided. These results extend previous ones derived for motility functions vanishing algebraically at zero and rely on a completely different approach.

Introduction

The chemotaxis system with local sensing and consumption

∂ t u = ∆(uγ(v)) in (0, ∞) × Ω , (1.1a)
∂ t v = ∆v -uv in (0, ∞) × Ω , (1.1b)
∇(uγ(v)) • n = ∇v • n = 0 on (0, ∞) × ∂Ω , (1.1c)

(u, v)(0) = (u in , v in ) in Ω , (1.1d)
describes the dynamics of a population of cells with density u ≥ 0 living on a nutrient with concentration v ≥ 0 and moving in space under the combined effects of a nutrient-dependent diffusion and a nutrient-induced chemotactic bias [START_REF]Traveling bands of chemotactic bacteria: a theoretical analysis[END_REF]. Here, Ω is a bounded domain of R n , n ≥ 1, and the motility γ is a smooth function which is positive on (0, ∞). Unlike the classical Keller-Segel system with local sensing

∂ t u = ∆(uγ(v)) in (0, ∞) × Ω , (1.2a)
∂ t v = ∆v -v + u in (0, ∞) × Ω , (1.2b)
∇(uγ(v)) • n = ∇v • n = 0 on (0, ∞) × ∂Ω , (1.2c) (u, v)(0) = (u in , v in ) in Ω , (1.2d)
in which the variable v is rather the concentration of a signalling chemical produced by the cells as accounted for in (1.2b) [START_REF] Keller | Initiation of slime mold aggregation viewed as an instability[END_REF], the nutrient is consumed by the cells in the model (1.1) according to the nonlinear absorption term -uv in (1.1b). The dynamics of the two models is thus expected to differ significantly. On the one hand, the long term behaviour of solutions to (1.2) is far from being completely understood. Convergence of (u, v) to the spatially homogeneous steady state (

u in 1 /|Ω|, u in 1 /|Ω|) is established in [3, Theorem 1.5] when γ ′ (s) ≤ 0 ≤ sγ ′ (s) + γ(s)
for s ≥ 0 and a similar result is likely to be true when γ ′ ≥ 0. Such a simple dynamics is however unlikely to be the generic one, as there exist non-constant stationary solutions to (1.2) in some domains, see [START_REF] Lin | Large amplitude stationary solutions to a chemotaxis system[END_REF][START_REF] Wang | Steady states and pattern formation of the density-suppressed motility model[END_REF]. Besides, though solutions to (1.2) are global and even bounded for a large class of motility functions γ [1-5, 7, 9, 17, 19, 25, 26], unbounded solutions do exist [START_REF] Fujie | A note on construction of nonnegative initial data inducing unbounded solutions to some two-dimensional Keller-Segel systems[END_REF][START_REF]Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions[END_REF][START_REF] Jin | Critical mass on the Keller-Segel system with signal-dependent motility[END_REF].

The situation is somewhat simpler for (1.1) as the set of stationary solutions can easily be identified and seen to depend heavily on the value of γ(0) [START_REF]Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities[END_REF][START_REF]Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migrationconsumption interaction[END_REF]. Indeed, if γ(0) > 0, then the only steady states to (1.1) are the spatially homogeneous solutions (M, 0), M ≥ 0, and they attract the dynamics [START_REF] Ph | Long term spatial homogeneity for a chemotaxis model with local sensing and consumption[END_REF][START_REF]Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities[END_REF]. In contrast, the set of stationary solutions to (1.1) is much larger when γ(0) = 0, as (ū, 0) is a stationary solution to (1.1) for any (sufficiently smooth) function ū. Despite this wealth of stationary solutions, Winkler proves in [START_REF]A quantitative strong parabolic maximum principle and application to a taxis-type migration-consumption model involving signal-dependent degenerate diffusion[END_REF][START_REF]Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migrationconsumption interaction[END_REF] that the dynamics of (1.1) selects one and only one steady state in the large time limit: more precisely, given

γ ∈ C 1 ([0, ∞)) , γ(0) = 0, γ > 0 on (0, ∞) , (1.3)
satisfying

γ ∈ C 3 ((0, ∞)) and lim inf s→0 γ(s) s α > 0 (1.4)
for some α ≥ 1 and a suitably constructed global weak solution (u, v) to (1.1) [START_REF] Winkler | Application of the Moser-Trudinger inequality in the construction of global solutions to a strongly degenerate migration model[END_REF][START_REF]Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migrationconsumption interaction[END_REF], there is a non-negative measurable function u ∞ such that (u(t), v(t)) converges to (u ∞ , 0) in a suitable topology if one of the following additional assumptions holds true:

(I): n ∈ {1, 2} and γ ′ (0) > 0, see [START_REF]A quantitative strong parabolic maximum principle and application to a taxis-type migration-consumption model involving signal-dependent degenerate diffusion[END_REF]Theorem 1.4], (II): n ≥ 3 and there is

α ′ ∈ (1, 2] such that lim sup s→0 s 2-α ′ |γ ′′ (s)| < ∞ , (1.5)
see [START_REF]Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migrationconsumption interaction[END_REF]Theorem 1.2]. Observe that, for α ≥ 1, the function γ(s) = s α , s ≥ 0, satisfies (1.3), (1.4), and (1.5) with α ′ = min{α, 2}.

Besides, conditions on γ, u in , and v in are provided to guarantee that u ∞ is not a constant, see [22, Theorem 1.5] and [START_REF]Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migrationconsumption interaction[END_REF]Corollary 1.4].

The aim of this note is to prove that these results are actually valid under the sole assumption (1.3) on γ, without assuming an algebraic behaviour of γ near zero. However, the convergence established below takes place in a weaker topology for the u-component than the one obtained in [START_REF]A quantitative strong parabolic maximum principle and application to a taxis-type migration-consumption model involving signal-dependent degenerate diffusion[END_REF][START_REF]Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migrationconsumption interaction[END_REF]. In addition, the approach used herein is different and thus provides an alternative viewpoint on the stabilization issue for (1.1). We first state the convergence result.

Theorem 1.1. Assume that γ satisfies (1.3) and consider (u in , v in ) ∈ L ∞ + (Ω, R 2 )
, where E + denotes the positive cone of the Banach lattice E. If (u, v) is a global weak solution to (1.1) in the sense of Definition 2.1 below, then

A ∞ (x) := ∞ 0 (uγ(v))(s, x) ds ∈ H 1 + (Ω) , x ∈ Ω , and
u(t) ⇀ u ∞ := ∆A ∞ + u in in H 1 (Ω) ′ as t → ∞ , (1.6) lim t→∞ v(t) p = 0 , p ∈ [1, ∞) . (1.7) Moreover, u ∞ , ϑ (H 1 ) ′ ,H 1 ≥ 0 for all ϑ ∈ H 1 + (Ω) and u ∞ , 1 (H 1 ) ′ ,H 1 = u in , 1 (H 1 ) ′ ,H 1 .
The cornerstone of our approach to the study of the long term behaviour of global weak solutions to (1.1)is the observation that the dynamics of the system (1.1) is somewhat encoded in that of the auxiliary function

A(t, x) := t 0 (uγ(v))(s, x) ds , (t, x) ∈ (0, ∞) × Ω ,
which bears several interesting properties collected in Lemma 2.3. Among others, t → A(t, x) is non-decreasing for a.e. x ∈ Ω and the trajectory {A(t) : t ≥ 0} is bounded in H 1 (Ω), two features which guarantee in particular that the function A ∞ introduced in Theorem 1.1 is well-defined and lies in H 1 (Ω). Moreover, for all t ≥ 0, the function A(t) is a variational solution of the elliptic equation

-∆A(t) = u in -u(t) in Ω , ∇A(t) • n = 0 on ∂Ω ,
so that the large time behaviour of u(t) is driven by that of A(t).

The second contribution of this paper is in the spirit of [START_REF]Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migrationconsumption interaction[END_REF]Theorem 1.3] and provides an estimate on the distance in H 1 (Ω) ′ between the initial condition u in and the final state u ∞ of the u-component of (1.1). Its statement requires additional notation, which we introduce now: for z ∈ H 1 (Ω) ′ , we set z := z, 1 (H 1 ) ′ ,H 1 /|Ω| and note that

z = 1 |Ω| Ω z(x) dx for z ∈ H 1 (Ω) ′ ∩ L 1 (Ω). Now, for z ∈ H 1 (Ω) ′ with z = 0, we define K[z] ∈ H 1 (Ω) as the unique (variational) solution to -∆K[z] = z in Ω , ∇K[z] • n = 0 on ∂Ω , (1.8a) satisfying K[z] = 0 . (1.8b) We then choose the following norm • (H 1 ) ′ on H 1 (Ω) ′ : z (H 1 ) ′ := ∇K[z -z ] 2 + | z | , z ∈ (H 1 )(Ω) ′ . Proposition 1.2. Assume that γ satisfies (1.3) and consider (u in , v in ) ∈ L ∞ + (Ω, R 2 ). If (u, v
) is a global weak solution to (1.1) in the sense of Definition 2.1 below and u ∞ denotes the weak limit in H 1 (Ω) ′ of u(t) as t → ∞ given by Theorem 1.1, then

u ∞ -u in 2 (H 1 ) ′ ≤ u in ∞ v in 1 γ ′ L ∞ (0, v in ∞ ) .
(1.9)

In particular, if u in ∞ v in 1 γ ′ L ∞ (0, v in ∞ ) < u in -u in 2 (H 1 ) ′ , (1.10
)

then u ∞ is not a constant.
An immediate consequence of Proposition 1.2 is that, given u in ∈ L ∞ + (Ω) with u in ≡ u in and a sufficient small v in ∈ L ∞ + (Ω), the first component of the corresponding global weak solution (u, v) to (1.1) has a non-constant limit. A estimate on the required smallness of v in in L ∞ (Ω) is provided by (1.10) and reads

v in ∞ γ ′ L ∞ (0, v in ∞ ) < u in -u in 2 (H 1 ) ′ |Ω| u in ∞ .
Such a result is obviously connected with the fact that the solution (u, v) to (1.1) with initial condition (u in , 0) is the stationary solution (u, v) = (u in , 0), as already mentioned.

Proofs

Let us first make precise the notion of global weak solution to (1.1) to be used in this paper. We emphasize here that, since γ(0) = 0, the equation (1.1a) is degenerate, so that we cannot expect much regularity on u. Definition 2.1. Assume that γ satisfies (1.3) and consider (u in , v in ) ∈ L ∞ + (Ω, R 2 ). A global weak solution to (1.1) is a pair of non-negative functions (u, v) such that

u ∈ C w ([0, ∞), H 1 (Ω) ′ ) ∩ L ∞ ((0, ∞), L 1 + (Ω)) , v ∈ C([0, ∞), L 1 + (Ω)) ∩ L ∞ ((0, ∞) × Ω) ∩ L 2 loc ([0, ∞), H 1 (Ω)) , u γ(v) ∈ L 2 loc ([0, ∞), L 2 (Ω)) , which satisfies u(t), ϑ(t) (H 1 ) ′ ,H 1 - Ω u in ϑ(0) dx = t 0 Ω u(s)γ(v(s))∆ϑ(s) dxds + t 0 u(s), ∂ t ϑ(s) (H 1 ) ′ ,H 1 ds for ϑ ∈ L 2 ((0, t), H 2 N (Ω)) ∩ W 1,2 ((0, t), H 1 (Ω)) and t ≥ 0, where H 2 N (Ω) := {z ∈ H 2 (Ω) : ∇z • n = 0 on ∂Ω} ,
as well as

Ω (v(t)ϑ(t) -v in ϑ(0)) dx + t 0 Ω ∇v(s) • ∇ϑ(s) dxds + t 0 Ω (uv)(s)ϑ(s) dxds = t 0 Ω v(s)∂ t ϑ(s) dxds for ϑ ∈ L 2 ((0, t), H 1 (Ω)) ∩ L ∞ ((0, t) × Ω) and t ≥ 0.
We shall not address the existence issue here and refer to [START_REF] Winkler | Application of the Moser-Trudinger inequality in the construction of global solutions to a strongly degenerate migration model[END_REF][START_REF]A quantitative strong parabolic maximum principle and application to a taxis-type migration-consumption model involving signal-dependent degenerate diffusion[END_REF][START_REF]Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migrationconsumption interaction[END_REF] for results in that direction, the main assumption on γ being that it vanishes in an algebraic way at zero. In the non-degenerate case γ > 0 on [0, ∞), implying in particular that γ(0) > 0, existence results are also available, see [START_REF] Li | Global boundedness and large time behavior of solutions to a chemotaxis-consumption system with signal-dependent motility[END_REF][START_REF] Li | Refined regularity analysis for a Keller-Segel-consumption system involving signaldependent motilities[END_REF][START_REF]Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities[END_REF].

We now fix γ satisfying (1.3) and consider (u in , v in ) ∈ L ∞ + (Ω, R 2 ), along with a global weak solution (u, v) to (1.1) in the sense of Definition 2.1. As a first step towards the identification of the large time limit of (u, v), we collect obvious consequences of (1.1), the non-negativity of u and v, and the comparison principle. Lemma 2.2. For t ≥ 0,

u(t) = M := u in and v(t) ∞ ≤ V := v in ∞ . (2.1) Moreover, ∞ 0 (uv)
(t) 1 dt ≤ v in 1 . (2.2)
Proof. We integrate (1.1a) with respect to space and time and use the no-flux boundary conditions (1.1c) to obtain the first identity in (2.1). Similarly, we infer from (1.1b) and (1.1c) that

d dt v(t) 1 + t 0 (uv)(s) 1 ds = v in 1 , t ≥ 0 , (2.3)
from which (2.2) readily follows. Finally, we use the comparison principle to deduce from (1.1b), (1.1c), and the non-negativity of uv that v(t, x) ≤ V for (t, x) ∈ [0, ∞) × Ω, thereby completing the proof of (2.1).

We now define the auxiliary function

A(t, x) := t 0 (uγ(v))(s, x) ds , (t, x) ∈ (0, ∞) × Ω ,
and devote the next lemma to its properties.

Lemma 2.3. The function A belongs to L ∞ ((0, ∞), H 1 (Ω)) with

A(t) 1 ≤ v in 1 γ ′ L ∞ (0,V ) , t ≥ 0 , (2.4a
)

∇A(t) 2 2 ≤ u in ∞ v in 1 γ ′ L ∞ (0,V ) , t ≥ 0 . (2.4b)
In addition, t → A(t, x) is a non-decreasing function for a.e. x ∈ Ω and

A ∞ (x) := sup t≥0 {A(t, x)} = ∞ 0 (uγ(v))(s, x) ds , x ∈ Ω , (2.5)
is well-defined and belongs to H 1 + (Ω). Also, for any ϑ ∈ H 1 (Ω),

lim t→∞ A(t) -A ∞ 2 = lim t→∞ Ω ∇ϑ • ∇(A(t) -A ∞ ) dx = 0 . (2.6)
Proof. Owing to the non-negativity of u and γ,

A(t 1 , x) ≤ A(t 2 , x) , 0 ≤ t 1 ≤ t 2 , x ∈ Ω ,
while, for t ≥ 0, it follows from (1.3), (2.1), and (2.2) that

A(t) 1 = t 0 Ω (uγ(v))(s, x) dxds ≤ γ ′ L ∞ (0,V ) t 0 Ω (uv)(s, x) dxds ≤ v in 1 γ ′ L ∞ (0,V )
, which proves (2.4a). Furthermore, the monotone convergence theorem implies that the function A ∞ defined by (2.5) belongs to L 1 + (Ω) and lim

t→∞ A(t) -A ∞ 1 = 0 . (2.7)
We next infer from (1.1a), (1.1c), and the definition of A that, for t ≥ 0,

u(t) -∆A(t) = u in in H 1 (Ω) ′ . (2.8)
In particular, A(t) -A(t) = K[u in -u(t)] belongs to H 1 (Ω) and we infer from (2.8) and the non-negativity of u(t) and A(t) that

∇A(t) 2 2 = -∆A(t), A(t) (H 1 ) ′ ,H 1 ≤ u(t) -∆A(t), A(t) (H 1 ) ′ ,H 1 = Ω u in A(t) dx ≤ u in ∞ A(t) 1 . Hence, by (2.4a), ∇A(t) 2 2 ≤ u in ∞ v in 1 γ ′ L ∞ (0,V )
, from which (2.4b) follows. Finally, we deduce the H 1 -regularity of A ∞ from (2.4b) by a weak compactness argument, whereas the convergence (2.6) is an immediate consequence of (2.4a), (2.4b), and (2.7).

We next turn to the convergence of v and begin with a classical energy estimate, which is available here thanks to the non-negativity of the right hand side of (1.1b).

Lemma 2.4. For t ≥ 0, d dt v 2 2 + 2 ∇v 2 2 + 2 v √ u 2 2 = 0 . Lemma 2.5. For each p ∈ [1, ∞), lim t→∞ v(t) p = 0 .
Proof. Introducing P := K[u -M] and P in := K[u in -M], we observe that P = P in -A, so that

∇P 2 ≤ ∇P in 2 + ∇A 2 ≤ c 1 := u in 2 + u in ∞ v in 1 γ ′ L ∞ (0,V ) . (2.9)
We next infer from (1.8a) and (2.3) that

d dt v 1 = - Ω uv dx = - Ω (M -∆P )v dx = -M v 1 + Ω ∇v • ∇P dx .
Hence, using (2.9) and Hölder's inequality,

d dt v 1 + M v 1 ≤ ∇v 2 ∇P 2 ≤ c 1 ∇v 2 .
After integration with respect to time, we obtain

v(t) 1 ≤ v in 1 e -M t

t→∞ v(t) 1

 1 = 0 , thereby proving Lemma 2.5 for p = 1. To complete the proof, we use the above convergence, along with (2.1) and Hölder's inequality. Thanks to the above analysis, we are now in a position to prove Theorem 1.1 and Proposition 1.2.

Acknowledgments

Enlightening (electronic) discussions with Michael Winkler on the topic studied in this paper are gratefully acknowledged. Part of this work was done while enjoying the kind hospitality of the Department of Mathematics, Indian Institute of Technology Roorkee.

Proof of Theorem 1.1. According to Lemma 2.3, the function A ∞ introduced in Theorem 1.1 is welldefined and belongs to H 1 + (Ω). Setting u ∞ = u in + ∆A ∞ ∈ H 1 (Ω) ′ , we infer from (2.8) that, for t ≥ 0 and ϑ ∈ H 1 (Ω),

and the right hand side of the above identity converges to zero as t → ∞ due to (2.6). We have thus proved the convergence (1.6), whereas the convergence (1.7) is established in Lemma 2.5. As for the properties of u ∞ stated at the end of Theorem 1.1, they readily follow from (2.1), the non-negativity, and the convergence (1.6)

Proof of Proposition 1.2. The starting point is the estimate (2.4b) and the convergences (2.6) which imply that

)

Assume now that (u in , v in ) satisfies (1.10). It follows from (1.9) and (1.10) that

which completes the proof after noticing that the property u ∞ = u in established in Theorem 1.1 excludes that u ∞ coincides with any other constant.