H B Boubaker

C Mareau

Y Ayed

G Germain

A Tidu

A crystal plasticity-based constitutive model for near-𝛽 titanium alloys under extreme loading conditions: Application to the Ti17 alloy

Keywords:

A crystal plasticity-based constitutive model is proposed to describe the thermo-mechanical behavior of the Ti17 titanium alloy subjected to extreme loading conditions. The model explicitly incorporates the effect of the crystallographic orientation of the hcp 𝛼 and bcc 𝛽 phases. The constitutive equations are built in the context of continuum thermodynamics with internal variables. The general framework of continuum damage mechanics is used to consider the impact of ductile damage on the mechanical behavior. The proposed model is implemented in a finite element method solver. The material parameters are identified from an extensive experimental dataset with an inverse method. According to the results, the impact of the strain rate and the temperature on the mechanical behavior is correctly depicted. The model is then used to evaluate the impact of temperature on strain localization. The role of the local texture on the development of ductile damage is also discussed for different specimen geometries. Finally, the impact of heat exchanges on the mechanical behavior at low and high temperatures is investigated.

Introduction

Due to their high strength-to-weight ratio, 𝛽-titanium alloys are largely used as structural materials for aerospace applications [START_REF] Boyer | The use of 𝛽 titanium alloys in the aerospace industry[END_REF][START_REF] Arrazola | Machinability of titanium alloys (Ti6Al4V and Ti5553)[END_REF]. This group of titanium alloys is interesting for aerospace applications owing to their high ultimate strength with a high fracture strain Weiss andSemiatin (1998, 1999), [START_REF] Kolli | A review of metastable beta titanium alloys[END_REF]. During fabrication operations, titanium alloys are subjected to severe loading conditions, namely high temperatures (higher than 1000 • C (de [START_REF] De Buruaga | Determining tool/chip temperatures from thermography measurements in metal cutting[END_REF]), high strain rates (around 10 5 s -1 [START_REF] Thimm | Determination of chip speed and shear strain rate in primary shear zone using digital image correlation (DIC) in linear-orthogonal cutting experiments[END_REF]) and large strains [START_REF] Abukhshim | Heat generation and temperature prediction in metal cutting: A review and implications for high speed machining[END_REF][START_REF] Wu | Serrated chip formation and their adiabatic analysis by using the constitutive model of titanium alloy in high speed cutting[END_REF][START_REF] Childs | Ti6Al4V Metal cutting chip formation experiments and modelling over a wide range of cutting speeds[END_REF]. The thermomechanical behavior of titanium alloys under such conditions is quite complex, mostly because many metallurgical phenomena may impact microstructure development. For instance, some experimental studies have discussed the role of crystallographic slip on texture development [START_REF] Dawson | Hot rolling and superplastic forming response of net shape processed Ti-6AI-4V produced by centrifugal spray deposition[END_REF], the impact of recrystallization on the softening behavior [START_REF] Momeni | Effect of hot working on flow behavior of Ti-6Al-4V alloy in single phase and two phase regions[END_REF][START_REF] Guo | Dynamic transformation during the high temperature deformation of two-phase titanium alloys[END_REF] or the consequences of strain localization [START_REF] Semiatin | Plastic flow, microstructure evolution, and defect formation during primary hot working of titanium and titanium aluminide alloys with lamellar colony microstructures[END_REF].

For engineering purposes, numerical simulation is often used to determine the most appropriate processing conditions for 𝛽-titanium alloys [START_REF] Baker | Finite element simulation of high-speed cutting forces[END_REF][START_REF] Zhang | Microstructure-based crystal plasticity modeling of cyclic deformation of Ti-6Al-4V[END_REF][START_REF] Ye | Modeling periodic adiabatic shear band evolution during high speed machining Ti-6Al-4V alloy[END_REF][START_REF] Barbe | Intergranular and intragranular behavior of polycrystalline aggregates[END_REF]. The robustness of such simulations is largely dependent on the accuracy of the associated set of constitutive relations. For severe loading conditions, the description of the mechanical behavior usually relies on empirical thermo-viscoplastic models [START_REF] Johnson | A constitutive model and data for materials subjected to large strains, high strain rates, and high temperatures[END_REF][START_REF] Zerilli | Dislocation-mechanics-based constitutive relations for material dynamics calculations[END_REF]. For instance, the flow rule proposed by [START_REF] Johnson | A constitutive model and data for materials subjected to large strains, high strain rates, and high temperatures[END_REF] has been applied for the Ti6Al4V [START_REF] Yaich | Numerical analysis of constitutive coefficients effects on FE simulation of the 2D orthogonal cutting process: application to the Ti6Al4V[END_REF][START_REF] Harzallah | Numerical and experimental investigations of ti-6al-4V chip generation and thermomechanical couplings in orthogonal cutting[END_REF], Ti17 [START_REF] Ayed | Development of a numerical model for the understanding of the chip formation in high-pressure waterjet assisted machining[END_REF][START_REF] Trabelsi | Tool wear and cutting forces under cryogenic machining of titanium alloy (Ti17)[END_REF] and Ti5553 [START_REF] Wagner | Experimental and numerical investigations of the heating influence on the Ti5553 titanium alloy machinability[END_REF] alloys. Several modified Johnson-Cook flow rules have also been proposed to account for the softening behavior [START_REF] Lee | Plastic deformation and fracture behaviour of Ti-6Al-4V alloy loaded with high strain rate under various temperatures[END_REF][START_REF] Cai | A modified Johnson-Cook constitutive equation to predict hot deformation behavior of Ti-6Al-4V alloy[END_REF][START_REF] Zhou | Inverse identification of modified Johnson-Cook model for cutting titanium alloy Ti6Al4V using firefly algorithm[END_REF] and the microstructural changes [START_REF] Calamaz | A new material model for 2D numerical simulation of serrated chip formation when machining titanium alloy Ti-6Al-4V[END_REF][START_REF] Sima | Modified material constitutive models for serrated chip formation simulations and experimental validation in machining of titanium alloy Ti-6Al-4V[END_REF] of 𝛽-titanium alloys. The model of [START_REF] Zerilli | Dislocation-mechanics-based constitutive relations for material dynamics calculations[END_REF] has been used by [START_REF] Cai | A comparative study on Johnson Cook, Modified Zerilli-Armstrong and Arrhenius-type constitutive models to predict hightemperature flow behavior of Ti-6Al-4V alloy in 𝛼+ 𝛽 phase[END_REF] and [START_REF] Che | An integrated Johnson-Cook and Zerilli-Armstrong model for material flow behavior of Ti-6Al-4V at high strain rate and elevated temperature[END_REF] to describe the flow behavior of the Ti6Al4V alloy. A comparative study on the ability of the Johnson-Cook and the Zerilli-Armstrong models to predict the thermo-mechanical behavior of the Ti6554 𝛽titanium alloy has been performed by [START_REF] Zhan | Constitutive modelling of the flow behaviour of a 𝛽 titanium alloy at high strain rates and elevated temperatures using the Johnson-Cook and modified Zerilli-Armstrong models[END_REF]. According to the authors, while more experimental data is required to calibrate the Zerilli-Armstrong model, it provides a better description of strain softening when compared to the Johnson-Cook model.

While the aforementioned models account for the effect of temperature and strain rate on the mechanical behavior, they ignore the impact of microstructural heterogeneities. These models can therefore be referred to as macroscopic constitutive models in the sense that they https://doi.org/10.1016/j.mechmat.2021.104198 represent the average response of the material of interest. However, when the characteristic length scale of the structure is similar to that of the microstructure, macroscopic constitutive models are inappropriate. This type of situation is commonly encountered for coarsegrained materials and/or processes with strong spatial localization (e.g. machining).

For metallic alloys, the general framework of crystal plasticity [START_REF] Roters | Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications[END_REF] provides a convenient method for incorporating the role of microstructural heterogeneities. While early theories [START_REF] Mandel | Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques[END_REF][START_REF] Asaro | Micromechanics of crystals and polycrystals[END_REF] only considered the contribution of crystallographic slip to plastic deformation, some efforts have been made to include the effect of deformation twinning [START_REF] Kalidindi | Incorporation of deformation twinning in crystal plasticity models[END_REF][START_REF] Salem | Strain hardening due to deformation twinning in 𝛼-titanium: Constitutive relations and crystal-plasticity modeling[END_REF][START_REF] Abdolvand | On the nucleation of deformation twins at the early stages of plasticity[END_REF] and phase transformations [START_REF] Turteltaub | A multiscale thermomechanical model for cubic to tetragonal martensitic phase transformations[END_REF][START_REF] Suiker | Computational modelling of plasticity induced by martensitic phase transformations[END_REF]. Such models have been used to investigate strain localization in a near-𝛼 titanium alloy [START_REF] Bache | Experimental and crystal plasticity studies of deformation and crack nucleation in a titanium alloy[END_REF] and in a 𝛼 + 𝛽 alloy [START_REF] Kapoor | Modeling strain localization in microtextured regions in a titanium alloy: Ti-6Al-4V. Integrating[END_REF]. A crystal plasticitybased model was proposed by [START_REF] Kapoor | Modeling Ti-6Al-4V using crystal plasticity, calibrated with multi-scale experiments, to understand the effect of the orientation and morphology of the 𝛼 and 𝛽 phases on time dependent cyclic loading[END_REF] to investigate the effect of the orientation and the morphology of the 𝛼 and 𝛽 as well as their Burgers orientation relationship (BOR) on the deformation behavior of the Ti6Al4V alloy. A crystal plasticity-based model has been used by [START_REF] Goh | Crystallographic plasticity in fretting of Ti-6AL-4V[END_REF] to study the effect of crystallographic orientation on the deformation response of Ti6Al4V alloy under fretting conditions. Furthermore, several studies have been performed using the framework of crystal plasticity to understand the effect of slip activity on low cycle fatigue behavior [START_REF] Bridier | Slip and fatigue crack formation processes in an 𝛼/𝛽 titanium alloy in relation to crystallographic texture on different scales[END_REF][START_REF] Bridier | Crystal plasticity modeling of slip activity in Ti-6Al-4V under high cycle fatigue loading[END_REF][START_REF] Farooq | Crystal plasticity modeling of the cyclic behavior of polycrystalline aggregates under non-symmetric uniaxial loading: Global and local analyses[END_REF].

To deal with the degradation of mechanical properties, different options have been explored to couple damage mechanics with crystal plasticity, mostly in the context of fatigue damage (e.g. [START_REF] Aslan | Numerical modeling of fatigue crack growth in single crystals based on microdamage theory[END_REF], [START_REF] Sabnis | Microdamage modelling of crack initiation and propagation in FCC single crystals under complex loading conditions[END_REF], [START_REF] Lindroos | Micromechanical modeling of short crack nucleation and growth in high cycle fatigue of martensitic microstructures[END_REF], [START_REF] Mareau | A non-local damage model for the fatigue behaviour of metallic polycrystals[END_REF]) or ductile damage (e.g. [START_REF] Boudifa | A micromechanical model for inelastic ductile damage prediction in polycrystalline metals for metal forming[END_REF], [START_REF] Kim | Necking behavior of AA 6022-T4 based on the crystal plasticity and damage models[END_REF], [START_REF] Ling | An elastoviscoplastic model for porous single crystals at finite strains and its assessment based on unit cell simulations[END_REF]). While the aforementioned approaches allow considering the impact of loading conditions, the role of temperature on damage development is often ignored.

In this work, a constitutive model is developed to describe the mechanical behavior of a coarse-grained Ti17 titanium alloy over a wide range of both temperatures and strain rates. This model considers not only the role of crystallographic slip but also that of damage on the mechanical behavior. For this purpose, the construction of constitutive relations relies on the crystal plasticity framework and continuum damage mechanics [START_REF] Kachanov | Time of the rupture process under creep conditions[END_REF][START_REF] Lemaitre | Mécanique des matériaux Solides-1 𝑒𝑟𝑒 édition[END_REF][START_REF] Chaboche | A review of some plasticity and viscoplasticity constitutive theories[END_REF][START_REF] Bammann | On kinematic, thermodynamic, and kinetic coupling of a damage theory for polycrystalline material[END_REF]. On the one hand, the crystal plasticity framework provides a convenient method for incorporating the contribution of crystallographic slip to plastic deformation. On the other hand, the introduction of a damage variable allows describing the impact of loading conditions (e.g. temperature, strain rate, loading direction) on ductile fracture. Also, because the Ti17 alloy displays a two-phase microstructure, a specific strategy, which relies on homogenization theory, is used to evaluate the contributions of individual variants to the mechanical behavior.

The present paper is organized as follows. The first section is dedicated to the description of the numerical model developed for the coarse-grained Ti17 titanium alloy. The strategy used to determine the corresponding material parameters is detailed in the second section. The experimental dataset used for parameter identification is also presented. Finally, the impact of loading conditions on microstructure development (crystallographic texture, strain localization) and ductile fracture is discussed in the final section.

Numerical model

Multiscale strategy

The Ti17 alloy (83%Ti,5%Al,2%Sn,2%Zr,4%Mo and 4%Cr) investigated in this study displays a microstructure consisting of 67% of lamellar 𝛼 phase and 33% of equiaxed 𝛽 phase. The corresponding microstructure, which is shown in Fig. 1, spans over multiple length scales. Specifically, while the average size of equiaxed 𝛽 grains is about 1000 μm, the average thickness of 𝛼 lamellae is around 0.8 μm.

Because of the multiple microstructural length scales, a multiscale strategy is proposed to model the mechanical behavior of the Ti17 alloy. On the one hand, 𝛽 grains are directly represented through an ad hoc discretization procedure (see Fig. 2). On the other hand, due to their low thickness, it is not numerically conceivable to explicitly consider the role of 𝛼 lamellae. To circumvent this difficulty, a homogenization strategy is used for the development of constitutive relations. Specifically, both the 𝛼 and 𝛽 phases are assumed to contribute to the mechanical behavior of a material point. As a result, according to the classical averaging relations of homogenization theory in the context of finite strains [START_REF] Hill | The essential structure of constitutive laws for metal composites and polycrystals[END_REF][START_REF] Hill | On constitutive macro-variables for heterogeneous solids at finite strain[END_REF], the deformation gradient tensor 𝑭 and the first Piola-Kirchoff stress tensor 𝑷 are given for each material point by1 :

𝑭 = 𝜁 𝛽 𝑭 𝛽 + ∑ 𝑖 𝜁 𝛼 𝑖 𝑭 𝛼 𝑖 = ∑ 𝜙 𝜁 𝜙 𝑭 𝜙 (1) 𝑷 = 𝜁 𝛽 𝑷 𝛽 + ∑ 𝑖 𝜁 𝛼 𝑖 𝑷 𝛼 𝑖 = ∑ 𝜙 𝜁 𝜙 𝑷 𝜙 (2)
According to the above relations, while multiple 𝛼 phase variants may coexist (with volume fractions 𝜁 𝛼 𝑖 ), a single 𝛽 phase variant (with volume fraction 𝜁 𝛽 ) is considered for a given material point. The averaging relations do not allow uniquely specifying the deformation gradient tensor and the stress tensor associated with each variant. These relations must be supplemented with some constitutive equations and a localization rule. In the present work, a non-linear equivalent to the Voigt's assumption [START_REF] Voigt | Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper[END_REF] is adopted. This assumption leads to the simple localization rule:

𝑭 𝜙 = 𝑭 (3)
It is worth mentioning that more realistic approximations, such as the laminate model [START_REF] Nemat-Nasser | Micromechanics: overall properties of heterogeneous materials[END_REF], can be used. However, because of the small size of 𝛼 particles, this type of approach is not applicable here since a material point consists of multiple, rather than one single, 𝛼 variants. Also, our attention is restricted to severe loading conditions, which involve high strain rates. For such conditions, phase transformations do not contribute to the deformation behavior. The volume fractions 𝜁 𝜙 are therefore assumed to be constant during a deformation process. As a consequence, the specific power developed by internal forces 𝑝 is given by:

𝑝 = 1 𝜌 𝑷 ∶ Ḟ (4) = ∑ 𝜙 𝜁 𝜙 𝜌 𝑷 𝜙 ∶ Ḟ 𝜙 (5) = ∑ 𝜙 𝜁 𝜙 𝑝 𝜙 (6)
where 𝜌 is the initial mass density.

In the context of thermomechanics, quantities such as the specific internal energy 𝑢 and specific entropy 𝑠 should be defined. Neglecting interfacial contributions, these thermodynamic quantities are obtained from:

𝑢 = ∑ 𝜙 𝜁 𝜙 𝑢 𝜙 (7) 𝑠 = ∑ 𝜙 𝜁 𝜙 𝑠 𝜙 (8)
In the following, adiabatic conditions are assumed. For this particular case, the specific dissipation source 𝑑 is given by: The latter relation assumes that the temperature is the same for all the variants associated with a material point.

𝑑 = 𝑇 ṡ (9)

Constitutive equations

Kinematics. For each variant 𝜙, the construction of constitutive relations relies on the multiplicative decomposition of the corresponding gradient tensor 𝑭 𝜙 . Indeed, following the suggestion of [START_REF] Lee | Elastic-plastic deformation at finite strains[END_REF], the deformation gradient tensor of a given variant is decomposed into thermoelastic (subscript 𝑡𝑒) and plastic (subscript 𝑝) contributions:

𝑭 𝜙 = 𝑭 𝜙 𝑡𝑒 ⋅ 𝑭 𝜙 𝑝 (12)
In the following, plastic flow is assumed to be incompressible so that det(𝑭 𝜙 𝑝 ) = 1. Also, the above multiplicative decomposition allows identifying the thermoelastic and plastic contributions to the power developed by internal forces with:

𝑝 𝜙 = 1 𝜌 𝑺 𝜙 ∶ Ė𝜙 𝑡𝑒 + 1 𝜌 𝜮 𝜙 ∶ 𝑳 𝜙 𝑝 ( 13
)
The thermoelastic strain tensor 𝑬 𝜙 𝑡𝑒 is evaluated from the thermoelastic contribution to the deformation gradient tensors as follows:

𝑬 𝜙 𝑡𝑒 = 1 2 ( ( 𝑭 𝜙 𝑡𝑒 ) 𝑇 ⋅ 𝑭 𝜙 𝑡𝑒 -𝟏 ) (14)
The plastic contribution to the velocity gradient, which is denoted by 𝑳 𝜙 𝑝 , is obtained for each variant from:

𝑳 𝜙 𝑝 = Ḟ 𝜙 𝑝 ⋅ ( 𝑭 𝜙 𝑝 ) -1 (15)
Also, for each variant, the second Piola-Kirchoff stress tensor 𝑺 𝜙 is connected to the corresponding first Piola-Kirchoff stress tensor 𝑷 𝜙 according to: In a similar fashion, the Mandel stress tensor 𝜮 𝜙 , which is the power conjugate to the plastic part of the velocity gradient, is given by:

𝑺 𝜙 = ( 𝑭 𝜙 𝑡𝑒 ) -1 ⋅ 𝑷 𝜙 ⋅ ( 𝑭 𝜙 𝑝 ) 𝑇 (16)
𝜮 𝜙 = ( 𝑭 𝜙 𝑡𝑒 ) 𝑇 ⋅ 𝑷 𝜙 ⋅ ( 𝑭 𝜙 𝑝 ) 𝑇 (17)
In the present work, plastic flow is the sole result of crystallographic slip. The framework of crystal plasticity [START_REF] Roters | Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications[END_REF] provides a convenient way of considering the role of crystallographic slip on the deformation behavior. For this purpose, a set of slip systems is associated with each variant. Each slip system 𝑠 consists of a slip plane (with unit normal 𝒏 𝜙 𝑠 ) and a slip direction (denoted by 𝒎 𝜙 𝑠 ). Within this framework, the plastic contribution to the velocity gradient is obtained from:

𝑳 𝜙 𝑝 = ∑ 𝑠 𝒎 𝜙 𝑠 ⊗ 𝒏 𝜙 𝑠 ̇𝛾𝜙 𝑠 ( 18
)
where ̇𝛾𝜙 𝑠 is the shear strain rate for the 𝑠th slip system.

State equations. The list of state variables used for the construction of constitutive equations is presented in Table 1. It consists of the thermoelastic strain tensor 𝑬 𝜙 𝑡𝑒 , the absolute temperature 𝑇 , the damage variable 𝐷 and a set of hardening variables. It is worth mentioning that, in a similar fashion to the temperature, the damage variable is the same for all variants. The underlying assumption is that the development of damage cannot be restricted to a single variant. The damage variable is therefore representative of the degradation of the mechanical properties of a material point, rather than that of a specific variant.

In the following, the hardening variable attached to the 𝑠th slip system is denoted by 𝜆 𝜙 𝑠 . For a given slip system, the hardening variable 𝜆 𝜙 𝑠 is connected to the corresponding dislocation density 𝜚 𝜙 𝑠 with:

𝜆 𝜙 𝑠 = 𝑏 𝜙 𝑠 √ 𝜚 𝜙 𝑠 (19)
where 𝑏 𝜙 𝑠 is the norm of the Burgers vector. For each variant, the specific free energy 𝑎 𝜙 = 𝑢 𝜙 -𝑠 𝜙 𝑇 is decomposed into thermoelastic, thermal and hardening contributions:

𝑎 𝜙 [𝑬 𝜙 𝑡𝑒 , 𝑇 , 𝐷, 𝜆 𝜙 𝑠 ] = 𝑎 𝜙 𝑡𝑒 [𝑬 𝜙 𝑡𝑒 , 𝑇 , 𝐷] + 𝑎 𝜙 𝑡ℎ [𝑇 ] + 𝑎 𝜙 ℎ𝑑 [𝑇 , 𝐷, 𝜆 𝜙 𝑠 ] (20)
According to the above decomposition, the present model assumes that, for each variant, the development of damage affects both the thermoelastic behavior and the hardening behavior. The thermoelastic contribution is given by:

𝑎 𝜙 𝑡𝑒 = 1 2𝜌 𝑬 𝜙 𝑡𝑒 ∶ C 𝜙 ∶ 𝑬 𝜙 𝑡𝑒 - 1 𝜌 𝑬 𝜙 𝑡𝑒 ∶ C 𝜙 ∶ 𝜶 𝜙 ( 𝑇 -𝑇 0 ) + 1 2𝜌 𝜶 𝜙 ∶ ( C 𝜙 -C𝜙 ) ∶ 𝜶 𝜙 ( 𝑇 -𝑇 0 ) 2 (21)
where C 𝜙 (respectively C𝜙 ) is the current (respectively initial) stiffness tensor, 𝜶 𝜙 is the thermal expansion tensor and 𝑇 0 is a reference temperature. While the current stiffness tensor C 𝜙 is impacted by both temperature and damage, the initial stiffness tensor C𝜙 solely depends on temperature. Following the idea of [START_REF] Marigo | Formulation d'une loi d'endommagement d'un matériau élastique[END_REF], the consideration of closure effects relies on the decomposition of the stiffness tensor into spherical and deviatoric contributions. Specifically, in the present work, the stiffness tensor C 𝜙 is calculated from the elastic strain tensor 𝑬 𝜙 𝑒 = 𝑬 𝜙 𝑡𝑒 -𝜶 𝜙 ( 𝑇 -𝑇 0 ) with:

C 𝜙 = { (1 -𝐷) C𝜙 , tr[𝑬 𝜙 𝑒 ] > 0 (1 -𝐷) C𝜙 + 𝐷P 𝑠 ∶ C𝜙 ∶ P 𝑠 , tr[𝑬 𝜙 𝑒 ] ≤ 0 (22)
The spherical projection tensor P 𝑠 is given by:

P 𝑠 = 1 3 𝟏 ⊗ 𝟏 (23)
According to Eq. ( 22), when the spherical elastic strain is positive, stiffness properties are fully degraded for a damaged variant. However, due to closure effects, the spherical contribution to the stiffness tensor is recovered when the spherical elastic strain is negative. For each variant, the thermal contribution 𝑎 𝑡ℎ to the specific free energy is given by:

𝑎 𝜙 𝑡ℎ = c𝑝 ( 𝑇 -𝑇 0 -𝑇 ln [ 𝑇 𝑇 0 ]) ( 24
)
where c𝑝 is the initial specific heat capacity for the phases.

Finally, the contribution 𝑎 𝜙 ℎ𝑑 allows considering the storage of energy associated with the formation of crystallographic defects (e.g. dislocations). This contribution depends on temperature, damage and hardening variables according to:

𝑎 𝜙 ℎ𝑑 = 1 2𝜌 𝐻 𝜙 [𝑇 ] (1 -𝐷) ∑ 𝑠 𝜆 𝜙 𝑠 ∑ 𝑡 ℎ 𝜙 𝑠𝑡 𝜆 𝑡 (25)
In the above equation, 𝐻 𝜙 is the temperature-dependent isotropic hardening modulus and ℎ 𝜙 is a matrix that allows considering the possible interactions between different slip systems. It should be noticed that the development of damage is associated with a softening phenomenon since the contribution of hardening variables to free energy vanishes for a fully damaged material point.

The differentiation of the specific free energy with respect to the thermoelastic strain tensor 𝑬 𝜙 𝑡𝑒 leads to the state equation for the second Piola-Kirchhoff 𝑺 𝜙 :

𝑺 𝜙 = 𝜌 𝜕𝑎 𝜙 𝜕𝑬 𝜙 𝑡𝑒 = C 𝜙 ∶ ( 𝑬 𝜙 𝑡𝑒 -𝜶 𝜙 ( 𝑇 -𝑇 0 ) ) (26)
The specific entropy 𝑠 𝜙 for each variant is obtained from the differentiation of the specific free energy with respect to the absolute temperature:

𝑠 𝜙 = - 𝜕𝑎 𝜙 𝜕𝑇 = - 1 2𝜌 𝑬 𝜙 𝑡𝑒 ∶ 𝜕C 𝜙 𝜕𝑇 ∶ 𝑬 𝜙 𝑡𝑒 + 1 𝜌 𝑬 𝜙 𝑡𝑒 ∶ 𝜕C 𝜙 𝜕𝑇 ∶ 𝜶 𝜙 ( 𝑇 -𝑇 0 ) + 1 𝜌 𝑬 𝜙 𝑡𝑒 ∶ C 𝜙 ∶ 𝜶 𝜙 - 1 2𝜌 𝜶 𝜙 ∶ ( 𝜕C 𝜙 𝜕𝑇 - 𝜕 C𝜙 𝜕𝑇 ) ∶ 𝜶 𝜙 ( 𝑇 -𝑇 0 ) 2 - 1 𝜌 𝜶 𝜙 ∶ ( C 𝜙 -C𝜙 ) ∶ 𝜶 𝜙 ( 𝑇 -𝑇 0 ) + c𝜙 ln [ 𝑇 𝑇 0 ] - 1 2𝜌 𝜕𝐻 𝜙 𝜕𝑇 (1 -𝐷) ∑ 𝑠 𝜆 𝜙 𝑠 ∑ 𝑡 ℎ 𝜙 𝑠𝑡 𝜆 𝑡 (27)
The thermodynamic force associated with the hardening variable 𝜆 𝜙 𝑠 is denoted by 𝑟 𝜙 𝑠 . The corresponding state equation is:

𝑟 𝜙 𝑠 = 𝜌 𝜕𝑎 𝜙 𝜕𝜆 𝜙 𝑠 = 𝐻 𝜙 (1 -𝐷) ∑ 𝑡 ℎ 𝜙 𝑠𝑡 𝜆 𝜙 𝑡 (28)
In the following, the variable 𝑟 𝜙 𝑠 is referred to as the Critical Resolved Shear Stress (CRSS) of the 𝑠th slip system. It measures the resistance to plastic deformation of a given slip system. The energy restitution rate 𝑌 𝜙 , which is the driving force for the development of damage, is given by:

𝑌 𝜙 = -𝜌 𝜕𝑎 𝜙 𝜕𝐷 = 1 2 𝐻 𝜙 ∑ 𝑠 𝜆 𝜙 𝑠 ∑ 𝑡 ℎ 𝜙 𝑠𝑡 𝜆 𝜙 𝑡 - 1 2𝜌 𝑬 𝜙 𝑒 ∶ 𝜕C 𝜙 𝜕𝐷 ∶ 𝑬 𝜙 𝑒 (29)
It is worth mentioning that, though the damage variable is the same for all variants, the energy restitution rate 𝑌 𝜙 is different. Also, the derivative of the stiffness tensor C 𝜙 with respect to the damage variable is given by:

𝜕C 𝜙 𝜕𝐷 = { -C𝜙 , tr[𝑬 𝜙 𝑒 ] > 0 -C𝜙 + P 𝑠 ∶ C𝜙 ∶ P 𝑠 , tr[𝑬 𝜙 𝑒 ] ≤ 0 (30)
The damage variable is therefore representative of the degradation of the mechanical properties of a material point, rather than that of a specific variant. In contrast with physically-based models that explicitly consider the impact nucleation, growth and coalescence of cavities [START_REF] Ling | An elastoviscoplastic model for porous single crystals at finite strains and its assessment based on unit cell simulations[END_REF], [START_REF] Scherer | A strain gradient plasticity model of porous single crystal ductile fracture[END_REF]), the present approach is purely phenomenological and does not consider damage-induced anisotropy.

Evolution equations

Dissipation source. In the absence of heat conduction, the specific dissipation source 𝑑 𝜙 for variant 𝜙 is given by:

𝑑 𝜙 = 𝑝 𝜙 -ȧ𝜙 -𝑠 𝜙 Ṫ (31)
Combining the expression of the specific power developed by internal forces (13) with the state Eqs. ( 26), ( 27), ( 28) and ( 29) allows re-writing the specific dissipation source as follows:

𝑑 𝜙 = 1 𝜌 𝜮 𝜙 ∶ 𝑳 𝜙 𝑝 - 1 𝜌 ∑ 𝑠 𝑟 𝜙 𝑠 λ𝜙 𝑠 + 1 𝜌 𝑌 𝜙 Ḋ (32)
According to the above equation, heat dissipation is the consequence of the accumulation of plastic deformation and/or the development of damage. Also, this equation indicates that the plastic work is not entirely dissipated into heat because of the contribution of hardening to energy storage. Introducing the resolved shear stress 𝜏 𝜙 𝑠 = 𝒎 𝜙 𝑠 ⋅𝜮 𝜙 ⋅𝒏 𝜙 𝑠 leads to the following expression of the specific dissipation source:

𝑑 𝜙 = 1 𝜌 ∑ 𝑠 ( 𝜏 𝜙 𝑠 ̇𝛾𝑠 -𝑟 𝜙 𝑠 λ𝜙 𝑠 ) + 1 𝜌 𝑌 𝜙 Ḋ (33)
According to Eq. ( 11), the specific dissipation source 𝑑 for a material point is obtained from the addition of the contributions of the different variants, that is:

𝑑 = ∑ 𝜙 ( 1 𝜌 ∑ 𝑠 ( 𝜏 𝜙 𝑠 ̇𝛾𝜙 𝑠 -𝑟 𝜙 𝑠 λ𝜙 𝑠 ) ) + 1 𝜌 𝑌 Ḋ ( 34
)
where the energy restitution rate 𝑌 is defined from:

𝑌 = ∑ 𝜙 𝜁 𝜙 𝑌 𝜙 (35)
The above equation indicates that the flux variables ̇𝛾𝜙 𝑠 , λ𝜙 𝑠 and Ḋ are connected to the dissipative forces 𝜏 𝜙 𝑠 , 𝑟 𝜙 𝑠 and 𝑌 .

Viscoplastic flow rule.

In the context of rate-dependent plasticity, it is possible to construct the flow rule from Orowan's equation. Indeed, according to Orowan's equation, the plastic shear strain rate for each slip system is connected to the average dislocation velocity 𝑣 𝜙 𝑠 and the mobile dislocation density 𝜚 𝜙 𝑚,𝑠 according to:

̇𝛾𝜙 𝑠 = 𝜚 𝜙 𝑚,𝑠 𝑏 𝜙 𝑠 𝑣 𝜙 𝑠 ( 36
)
In the present work, the mobile dislocation density is assumed to be equal to some fraction 𝛽 of the total dislocation density so that:

𝜚 𝜙 𝑚,𝑠 = 𝛽𝜚 𝜙 𝑠 = 𝛽 ( 𝜆 𝜙 𝑠 𝑏 𝜙 𝑠 ) 2 (37)
Also, for the estimation of the average dislocation velocity, a simple power law is used:

𝑣 𝜙 𝑠 = 𝑣 𝜙 0 ⟨ | | | 𝜏 𝜙 𝑠 | | | -𝑟 𝜙 𝑠 𝐾 𝜙 𝑠 ⟩ 𝑛 𝜙 𝑠 ( 38
)
where 𝑛 𝜙 𝑠 and 𝐾 𝜙 𝑠 are some temperature-dependent viscosity parameters and 𝑣 𝜙 0 is a reference dislocation velocity. The viscoplastic rule is obtained by combining Eqs. ( 36), ( 37) and ( 38), which leads to:

̇𝛾𝜙 𝑠 = 𝛽 (𝜆 𝜙 𝑠 ) 2 𝑏 𝜙 𝑠 𝑣 𝜙 0 ⟨ | | | 𝜏 𝜙 𝑠 | | | -𝑟 𝜙 𝑠 𝐾 𝜙 𝑠 ⟩ 𝑛 𝜙 𝑠 (39) = ( 𝜆 𝜙 𝑠 ) 2 ⟨ | | | 𝜏 𝜙 𝑠 | | | -𝑟 𝜙 𝑠 𝑘 𝜙 𝑠 ⟩ 𝑛 𝜙 𝑠 ( 40
)
In the above equation, 𝑘 𝜙 𝑠 is a temperature-dependent material parameter that needs to be defined for each slip system. The description of hardening assumes that the evolution of the dislocation density is controlled by the competition between dislocation multiplication and dislocation annihilation. Adopting the proposition of [START_REF] Mecking | Kinetics of flow and strain-hardening[END_REF], the evolution equation for the dislocation density is:

ρ𝜙 𝑠 = ( 𝜅 𝜙 𝑚 √ 𝜚 𝜙 𝑠 -𝜅 𝜙 𝑎 𝜚 𝜙 𝑠 ) | ̇𝛾𝜙 𝑠 | (41)
where 𝜅 𝜙 𝑚 (respectively 𝜅 𝜙 𝑎 ) is a material parameter that controls dislocation multiplication (respectively dislocation annihilation). Using the definition (19) of the isotropic hardening variable, one obtains the corresponding evolution equation:

λ𝜙 𝑠 = ( 1 -𝐵 𝜙 𝑠 𝜆 𝜙 𝑠 ) | ̇𝛾𝜙 𝑠 | (42)
with:

𝜅 𝜙 𝑚 = 2 𝑏 𝜙 𝑠 and 𝜅 𝜙 𝑎 = 2𝐵 𝜙 𝑠 (43)
It should be mentioned that the material parameter 𝐵 𝜙 𝑠 , which controls the asymptotic value of the hardening variable, is assumed to depend on temperature.

Damage rule. For the evolution of the damage variable, a power-law is used:

Ḋ = ⟨ 𝑌 -𝑌 𝑟 𝑆 ⟩ 𝑚 (1 -𝐷) (44)
In the above equation, 𝑆, 𝑚 and 𝑌 𝑟 are material parameters controlling the rate of damage accumulation. Specifically, 𝑌 𝑟 is a threshold that defines the minimum energy restitution rate for the development of damage. Also, the factor (1 -𝐷) is introduced to prevent the damage variable from exceeding unity.

Heat sources

To model the thermo-mechanical behavior of the Ti17 alloy, the heat diffusion equation must be established. For the construction of the heat diffusion equation, it is convenient to introduce the following quantities. First, for a variant 𝜙, the thermoelastic source 𝜑 𝜙 𝑡𝑒 is obtained from:

𝜑 𝜙 𝑡𝑒 = -𝑇 𝜕𝑠 𝜙 𝜕𝑬 𝜙 𝑡𝑒 ∶ Ė𝜙 𝑡𝑒 (45)
Also, the internal coupling source 𝜑 𝜙 𝑖𝑐 associated with the 𝜙th variant is expressed as follows:

𝜑 𝜙 𝑖𝑐 = -𝑇 ∑ 𝑠 𝜕𝑠 𝜙 𝜕𝜆 𝜙 𝑠 λ𝜙 𝑠 -𝑇 𝜕𝑠 𝜙 𝜕𝐷 Ḋ (46)
Finally, the specific heat capacity 𝑐 𝜙 is given by:

𝑐 𝜙 = 𝑇 𝜕𝑠 𝜙 𝜕𝑇 (47)
It is worth mentioning that, because of the impact of temperature on stiffness and hardening properties, the specific heat capacity 𝑐 𝜙 of a given variant is not constant during a deformation process. From the state Eq. ( 27), one obtains the following expression of specific dissipation source:

ḋ𝜙 = ṡ𝜙 𝑇 (48) = 𝑇 𝜕𝑠 𝜙 𝜕𝑇 Ṫ + 𝑇 𝜕𝑠 𝜙 𝜕𝑬 𝜙 𝑡𝑒 ∶ Ė𝜙 𝑡𝑒 + 𝑇 ∑ 𝑠 𝜕𝑠 𝜙 𝜕𝜆 𝜙 𝑠 λ𝜙 𝑠 + 𝑇 𝜕𝑠 𝜙 𝜕𝐷 Ḋ (49)
Using the definitions ( 45), ( 46) and ( 47), the above equation is rearranged as follows:

𝑐 𝜙 Ṫ = 𝑑 𝜙 + 𝜑 𝜙 𝑖𝑐 + 𝜑 𝜙 𝑡𝑒 (50)
It is worth mentioning that, because of the impact of temperature on stiffness and hardening properties, the specific heat capacity 𝑐 𝜙 of a given variant is not constant during a deformation process. The specific heat capacity 𝑐 𝜙 is equal to the initial specific heat capacity c𝜙 when (i) the current temperature is equal to the reference temperature 𝑇 0 , (ii) the thermoelastic strain tensor is zero and (iii) the hardening variables associated with the different slip systems vanish. The heat diffusion equation for a material point is obtained by summing the contributions of the different constituents:

𝑐 Ṫ = 𝑑 + 𝜑 𝑖𝑐 + 𝜑 𝑡𝑒 (51)
with:

𝑐 = ∑ 𝜙 𝜁 𝜙 𝑐 𝜙 (52) 𝜑 𝑡𝑒 = ∑ 𝜙 𝜁 𝜙 𝜑 𝜙 𝑡𝑒 ( 53
)
𝜑 𝑖𝑐 = ∑ 𝜙 𝜁 𝜙 𝜑 𝜙 𝑖𝑐 (54)

Parameter identification

Summary of experimental results

To investigate both the flow behavior and the damage behavior of the present Ti17 alloy, different experimental tests have been carried out. The experimental procedure used for such tests is briefly described here. A detailed description of the experimental setup can be found in [START_REF] Ben-Boubaker | Impact of the initial microstructure and the loading conditions on the deformation behavior of the Ti17 titanium alloy[END_REF].

First, some uniaxial compression tests have been conducted on a fully 𝛽 microstructure at different temperatures (from 25 • C to 900 • C) and different strain rates (from 0.1 s -1 to 10 s -1 ). For these tests, cylindrical specimens (with a diameter of 6 mm and a height of 9 mm) have been used. To obtain a fully 𝛽-microstructure, these specimens have been heat treated at 950 • C for 30 min, then rapidly cooled to the test temperature and deformed under uniaxial compression. The corresponding stress-strain curves are plotted in Fig. 7.

Second, some uniaxial compression tests have also been performed on cylindrical specimens with the as-received 𝛼 + 𝛽 microstructure presented in Fig. 1. To avoid phase transformation, these specimens have been heated from room temperature to the test temperature (from 25 • C to 800 • C) with a rate of 100 • C/s. Once the test temperature was reached, the specimen has been deformed with a constant nominal axial strain rate (from 0.1 s -1 to 10 s -1 ). The results of these tests are shown in Fig. 8.

Third, to characterize the impact of loading conditions (i.e. temperature, strain rate and triaxiality) on ductile damage, some tension tests have been carried out on some flat specimens with the as-received 𝛼 +𝛽 microstructure. As illustrated by Fig. 4, some smooth and notched specimens have been used for these tests. Depending on the notch geometry, the triaxiality ratio changes from zero (pure shear) to 0.72 (close to biaxial tension). These specimens have been deformed at various strain rates and various temperatures until fracture was detected. The results of these tests are presented in Fig. 9. For the purpose of illustration, a typical fracture surface showing the ductile aspect of fracture is shown in Fig. 3. This feature of fracture surfaces is observed whatever the loading conditions (temperature and strain rate) are.

Material parameters

Numerical models

To identify the material parameters of the proposed model for the Ti17 alloy, an inverse method has been used. First, the results of the uniaxial compression tests have been used to adjust the parameters of the viscoplastic flow rule and the isotropic hardening rule. For the application of the inverse method, the proposed constitutive model has been implemented within the ABAQUS Explicit finite element solver. The numerical model used to simulate uniaxial compression tests is presented in Fig. 5. The corresponding microstructure, which has been generated with the NEPER software [START_REF] Quey | Large-scale 3D random polycrystals for the finite element method: Generation, meshing and remeshing[END_REF], contains approximately 1200 grains. The average number of elements per grain is about 170, which is sufficient to achieve convergence. To account for friction at the interface between the specimen and the anvils, a Coulomb model with a friction coefficient of 0.3 is used [START_REF] Li | An experiment study of the lubrication behavior of graphite in hot compression tests of Ti-6Al-4V alloy[END_REF].

Second, the tension test results have been used to estimate the parameters of the damage model. The numerical models presented in Fig. 6 are used for the simulation of tension tests. These models, which only consider the gauge volume of tension specimens, have first been generated with the NEPER software [START_REF] Quey | Large-scale 3D random polycrystals for the finite element method: Generation, meshing and remeshing[END_REF]. Then, the notches have been introduced by removing elements and adjusting node positions in the vicinity of the notch. The numerical model contains approximately 500 grains. For the simulation of tension tests, the real grain size of 800 μm is used. Each grain is meshed with C3D4 tetrahedral elements, with an average number of 500 elements per grain. As mentioned earlier, in the context of adiabaticity, heat exchanges are neglected when performing numerical simulations of compression and tension tests.

Viscoplastic and hardening parameters of the 𝛽 phase

To model the viscoplastic behavior of the 𝛽 phase, crystallographic slip is assumed to occur on the ⟨111⟩{110} and ⟨111⟩{112} slip systems [START_REF] Orlans-Joliet | Yield surfaces of b.c.c. crystals for slip on the ⟨111⟩ {110} and ⟨111⟩ {112} systems[END_REF][START_REF] Liang | A critical review of experimental results and constitutive models for BCC and FCC metals over a wide range of strain rates and temperatures[END_REF][START_REF] Lewis | Slip systems and initiation of plasticity in a body-centered-cubic titanium alloy[END_REF][START_REF] Lhadi | Micromechanical modeling of the effect of elastic and plastic anisotropies on the mechanical behavior of 𝛽-Ti alloys[END_REF]. The material parameters for the 𝛽 phase are listed in Table 2. An initial dislocation density of 4 × 10 15 m -2 is assumed for all slip systems. Following the suggestion of [START_REF] Martin | A multiscale model for the elastoviscoplastic behavior of Directionally Solidified alloys: Application to FE structural computations[END_REF], no distinction is made between self-and latent hardening. As a result, the coefficients of the interaction matrix ℎ are all equal to unity. Also, some parameters, including the elastic constants [START_REF] Martin | A multiscale model for the elastoviscoplastic behavior of Directionally Solidified alloys: Application to FE structural computations[END_REF], the thermal expansion coefficient [START_REF] Liang | A critical review of experimental results and constitutive models for BCC and FCC metals over a wide range of strain rates and temperatures[END_REF] and the initial specific heat [START_REF] Teixeira | Modeling of the effect of the 𝛽 phase deformation on the 𝛼 phase precipitation in near-𝛽 titanium alloys[END_REF], have been extracted from literature. The remaining viscoplastic and hardening parameters of the 𝛽 phase have been adjusted as a function of temperature to reproduce the experimental nominal stress versus nominal strain curves obtained for the 𝛽-treated microstructure. For this purpose, each grain of the numerical model is composed of a single 𝛽 phase variant, which has been assigned a random crystallographic orientation. As illustrated by Fig. 7, the flow behavior of the 𝛽-treated microstructure is generally well described by the numerical model. For low temperatures (𝑇 ≤ 450 • C), the maximum error between the experimental and numerical curves does not exceed 6%. For temperatures above 700 • C, a yield drop phenomenon, which is attributed to either dynamic recrystallization [START_REF] Ben-Boubaker | Impact of the initial microstructure and the loading conditions on the deformation behavior of the Ti17 titanium alloy[END_REF] or the multiplication of mobile dislocations [START_REF] Teixeira | Transformation kinetics and microstructures of Ti17 titanium alloy during continuous cooling[END_REF], is observed. This phenomenon is partly captured by the proposed model. Specifically, while the contribution of dynamic recrystallization is ignored, the role of mobile dislocations on the development of plasticity is explicitly accounted in the flow rule as (see Eqs. ( 37)-( 40)).

Viscoplastic and hardening parameters of the 𝛼 phase

The description of the viscoplastic behavior of the 𝛼 phase assumes that crystallographic slip is restricted to the basal ⟨11 20⟩{0001}, prismatic ⟨11 20⟩{10 10}, pyramidal ⟨𝑎⟩ ⟨11 20⟩{1 101} and pyramidal ⟨𝑐 + 𝑎⟩

Table 3

Material parameters for the 𝛼 phase. The Kelvin scale is used for temperature.

Parameters

Values

𝑘 𝛼 𝑠 (MPa) 539-0.4 𝑇 𝑛 𝛼 172 exp (-0.002 𝑇 ) 𝐻 𝛼 (MPa) 1970 exp (-0.0042 𝑇 ) 𝐵 𝛼 𝑠 9 exp (0.002 𝑇 )
⟨11 23⟩{ 1011} slip systems [START_REF] Zhang | Determination of Ti-6242 𝛼 and 𝛽 slip properties using micro-pillar test and computational crystal plasticity[END_REF][START_REF] Kasemer | On slip initiation in equiaxed 𝛼/𝛽 Ti-6Al-4V[END_REF]. The same value as for the 𝛽-phase is used for the initial dislocation density for all slip systems. The material parameters of the 𝛼 phase are presented in Table 3. The elastic constants [START_REF] Martin | A multiscale model for the elastoviscoplastic behavior of Directionally Solidified alloys: Application to FE structural computations[END_REF], the thermal expansion coefficients [START_REF] Martin | A multiscale model for the elastoviscoplastic behavior of Directionally Solidified alloys: Application to FE structural computations[END_REF] and initial specific heat [START_REF] Teixeira | Modeling of the effect of the 𝛽 phase deformation on the 𝛼 phase precipitation in near-𝛽 titanium alloys[END_REF] have been extracted from literature. The material parameters involved in the viscoplastic flow rule and the hardening rule have been selected to reproduce the results of uniaxial compression tests obtained for the as-received 𝛼 + 𝛽 microstructure. To perform the corresponding simulations, each grain is composed of a single 𝛽 phase variant with a random crystallographic orientation and multiple 𝛼 phase variants. Based on the experimental observations conducted by [START_REF] Salib | Influence of transformation temperature on microtexture formation associated with 𝛼 precipitation at 𝛽 grain boundaries in a 𝛽 metastable titanium alloy[END_REF], no 𝛼 phase variant selection phenomenon is considered. As a result, the 𝛼 phase variants contained within an element are randomly selected amongst the twelve possible variants obtained from the Burgers orientation relationship:

{110} 𝛽 ‖ (0001) 𝛼 and ⟨111⟩ 𝛽 ‖ [ 11 20 ] 𝛼 ( 55
)
The numerical nominal stress versus nominal strain curves obtained after parameter identification are plotted in Fig. 8. The agreement with experimental results is correct. Specifically, the decrease of the hardening rate with an increasing temperature is correctly depicted.

Damage parameters for the as-received 𝛼 + 𝛽 Ti17 microstructure

The damage parameters have been adjusted to reproduce the nominal stress-strain curves obtained from tension tests. As for uniaxial compression specimens, each grain consists of a single 𝛽 phase variant with a random crystallographic orientation and multiple 𝛼 phase variants. The resulting damage parameters are listed in . As illustrated by Fig. 9, the beneficial effect of an increasing temperature and/or a decreasing strain rate on ductility is correctly captured by the numerical model.

Parameters Values

𝑌 𝑟 (MPa) 49 exp(-0.0014 𝑇 (K)) 𝑆 (MPa) 5 𝑚 6.2

Discussion

Impact of temperature on strain localization

The experimental results obtained for compression tests [START_REF] Ben-Boubaker | Impact of the initial microstructure and the loading conditions on the deformation behavior of the Ti17 titanium alloy[END_REF] indicate that the deformation behavior is impacted by strain localization. Specifically, as illustrated by Figs. 10 and 11, two strain localization modes have been observed. The first mode, which is observed at high temperatures, leads to the formation of a Highly Deformed Region (HDR) in the center of the specimen. The second strain localization mode, which is predominant at low temperatures, results from the formation of an Adiabatic Shear Band (ASB).

The numerical results for a compression specimen deformed with a strain rate of 10 s -1 and an initial temperature of 25 • C are shown in Fig. 10. The numerical results are consistent with experimental observations. Indeed, a shear band, with an angle of approximately ±45 • with respect to the compression direction, is clearly detected at the end of the compression test. As shown in Fig. 10, the crack initiates at the two tips of the band and then propagates along the shear direction. The formation of the shear band is mostly explained by the local temperature elevation, which reaches 160 • C at the instant of fracture.

For an initial temperature of 800 • C (see Fig. 11), the temperature elevation is much lower (about 120 • C) and less localized. Since the maximum temperature is obtained within the central region of the specimen, plastic deformation tends to concentrates in this region. The resulting highly deformed region is oriented perpendicular to the compression direction, which is consistent with experimental results.

Impact of local microstructure on ductile fracture

To evaluate the sensitivity of the proposed damage model to the local microstructure, some simulations have been performed using four different sets of crystallographic orientations for smooth and notched specimens. For the different sets of orientations, the crystallographic texture is globally the same (i.e. random) but locally different. The initial temperature is set to 25 • C while the nominal strain rate is fixed to 1 s -1 for all simulations. The nominal stress-strain curves obtained for the different specimen geometries are plotted in Fig. 12. For both smooth and notched specimens, the flow behavior is not much affected by the local microstructure. However, when looking at the fracture strain, some discrepancies between the specimen geometries are observed. For smooth tension specimens, the impact of the local microstructure on the fracture strain is limited. At the opposite, for notched specimens, especially

Impact of crystallographic texture on ductile fracture

To investigate the impact of texture on the development of ductile damage, some uniaxial tension tests, with different crystallographic textures for the 𝛽 phase, have been simulated. Specifically, different sets of orientations, with either the <100>, <110> or <111> directions aligned with the loading direction, have been generated for the 𝛽 phase. For the 𝛼 phase, no specific variant selection procedure has been used.

The nominal stress-strain curves obtained for the different fiber textures for an initial temperature of 25 • C and a nominal strain rate of 1 s -1 are plotted in Fig. 14. The flow behavior is not much affected by the crystallographic texture of the 𝛽 phase. The effect of the crystallographic texture of the 𝛽 phase on the flow behavior is limited mostly for two reasons. First, the development of plasticity is also controlled by the 𝛼 phase, which is predominant for the Ti17 alloy. Second, because many slip systems can be activated, the plastic deformation anisotropy of the 𝛽 phase is not very important.

In contrast with the flow behavior, the development of ductile damage in smooth specimens is dependent on the crystallographic texture. Specifically, the fracture strain is minimal for <111> fiber texture and maximum for <100> fiber texture.

Impact of heat exchanges

To evaluate the impact of heat exchanges, the cases of adiabatic and isothermal uniaxial tension are examined. For this purpose, a nominal strain rate of 1 s -1 is imposed to the smooth tension specimen. Two different initial temperatures, of either 25 • C or 800 • C, are considered. The nominal stress-strain curves obtained for the different conditions are plotted in Fig. 15. For both temperatures, the flow stress obtained under adiabatic conditions is inferior to that obtained for isothermal conditions as a result of thermal softening. However, while the impact of heat exchanges is limited at 800 • C, their role on the deformation behavior cannot be ignored at 25 • C.

The above results suggest that particular care should be taken when investigating the impact of strain rate on the deformation behavior. While isothermal conditions are representative of low strain rates, adiabatic conditions are appropriate for high strain rates. As a result, from an experimental perspective, the strain rate effect cannot be fully decoupled from that of temperature, especially for low temperatures and high strain rates [START_REF] Zaera | On the Taylor-Quinney coefficient in dynamically phase transforming materials. Application to 304 stainless steel[END_REF][START_REF] Jovic | Mechanical behaviour and temperature measurement during dynamic deformation on split Hopkinson bar of 304L stainless steel and 5754 aluminium alloy[END_REF].

Conclusion

In this work, a crystal plasticity-based constitutive model has been proposed to describe the behavior of the 𝛼 + 𝛽 Ti17 alloy under extreme loading conditions. The constitutive relations are developed in the context of finite strains using the general framework of crystal plasticity. To consider ductile fracture, a damage variable is introduced to describe the progressive stiffness reduction when the material is subjected to severe loading conditions. The constitutive relations have been implemented in the ABAQUS/Explicit finite element solver with a user-defined subroutine. The model parameters have been identified for the Ti17 alloy using the experimental results obtained for tension and compression tests. According to the results, the numerical and The proposed model provides a consistent description of the impact of temperature on strain localization. Specifically, the different strain localization modes observed under uniaxial compression depending on the temperature are correctly reproduced. Also, the numerical results obtained for notched specimens indicate that, because of the coarsegrained microstructure, the fracture strain is highly dependent on the local microstructure, which does not guarantee representativity. Finally, the impact of heat exchanges on the deformation behavior cannot be ignored for low temperatures.

Further work will focus on the application of the proposed model to the specific context of machining. Specifically, the model will be used to investigate the impact of microstructure and processing conditions on the chip morphology and cutting forces for the Ti17 alloy. Also, at this stage, no regularization technique is used to limit mesh dependency.

Fig. 1 .

 1 Fig. 1. Metallographic observations of the as-received 𝛼 + 𝛽 Ti17 titanium alloy.

Fig. 2 .

 2 Fig. 2. Representation of the adopted modeling strategy of the 𝛼 + 𝛽 Ti17 microstructure.

Fig. 3 .

 3 Fig. 3. SEM observations of the as-received 𝛼 + 𝛽 Ti17 titanium alloy deformed at a strain rate of 0.1 s -1 , a temperature of 25 • C showing the ductile aspect of fracture.

Fig. 4 .

 4 Fig. 4. Geometries of smooth and notched specimens.

Fig. 5 .

 5 Fig. 5. Geometrical model and boundary conditions for the simulation of compression tests.

Fig. 6 .

 6 Fig. 6. Geometrical model and boundary conditions for the simulation of tension tests.

Fig. 7 .

 7 Fig. 7. Comparison between the numerical and experimental nominal stress-nominal strain curves obtained under uniaxial compression for the 𝛽-treated Ti17 microstructure.

Fig. 8 .

 8 Fig. 8. Comparison between the numerical and experimental nominal stress-nominal strain curves obtained under uniaxial compression for the as-received 𝛼 + 𝛽 Ti17 microstructure.

Fig. 9 .

 9 Fig. 9. Comparison between the numerical and experimental nominal stress-nominal strain curves obtained under tension for the as-received 𝛼 + 𝛽 Ti17 microstructure for smooth and notched specimens.

Fig. 9 .

 9 Fig. 9. (continued).

Fig. 10 .

 10 Fig. 10. Evaluation of instantaneous local variables (equivalent plastic contribution to the deformation gradient field ‖𝑭 𝑝 -𝑰‖ and temperature field 𝑇 ) at the instant of failure for 𝛼 + 𝛽 Ti17 specimen deformed in compression at 10 s -1 and 25 • C.

Fig. 11 .

 11 Fig. 11. Evaluation of instantaneous local variables (equivalent plastic contribution to the deformation gradient field ‖𝑭 𝑝 -𝑰‖ and temperature field 𝑇 ) at an axial strain of 30% for an 𝛼 + 𝛽 Ti17 specimen deformed in compression at 10 s -1 and 800 • C.

Fig. 12 .

 12 Fig. 12. Impact of local microstructure on the numerical nominal stress-nominal strain curves obtained under different tension tests. The temperature and the strain rate are set to 25 • C and 1 s -1 , respectively.

  experimental nominal stress-strain curves of the different tests are in good agreement.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Impact of the 𝛽-grains orientation on the numerical nominal stress-nominal strain curves obtained under tension for 𝛼 + 𝛽 Ti17 smooth specimens deformed at a temperature of 25 • C and a strain rate of 1 s -1 .

Table 1

 1 State variables and associated thermodynamic forces used for the construction of constitutive relations.

		State variable Thermodynamic force
	Thermoelastic strain tensor 𝑬 𝜙 𝑡𝑒	𝑺 𝜙 ∕𝜌
	Absolute temperature	𝑇	-𝑠 𝜙
	Hardening variables	𝜆 𝜙 𝑠	𝑟 𝜙 𝑠 ∕𝜌
	Damage variable	𝐷	-𝑌 𝜙 ∕𝜌

Table 2

 2 Material parameters for the 𝛽 phase. The Kelvin scale is used for temperature.

	Parameters	Values
	𝑘 𝛽 𝑠 (MPa)	157-0.12 𝑇
	𝑛

𝛽 25 exp (-0.001 𝑇 ) 𝐻 𝛽 (MPa) 1320 exp (-0.0032 𝑇 ) 𝐵 𝛽 𝑠 4 exp (0.03 𝑇 )

Throughout this paper, the 𝜙 superscript is used to denote a quantity that is attached to either the 𝛽 phase variant or one of the 𝛼 phase variants (i.e. 𝜙 = {𝛽, 𝛼 1 , 𝛼

, …}).

Acknowledgments

This research was supported by two French organizations from ''Angers Loire Metropole'': ''Angers Loire Development'' and ''Angers Technopole.'' The authors would like to acknowledge their financial support.

While the current approach allows describing the initial stage of ductile failure, the description of the final stage leading to total failure is possibly mesh-dependent. As suggested by [START_REF] Scherer | A strain gradient plasticity model of porous single crystal ductile fracture[END_REF], crystal plasticity-based gradient models can be adopted in the context of ductile damage to circumvent such difficulties.

CRediT authorship contribution

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.


Cesar Adolfo

Escobar Carlos

Cecilia Velino

Magali Brunet
email: magali.brunet@cemes.fr

Luc Robbiola
email: robbiola@univ-tlse2.fr

Elena Bernardi

Carla Martini

Cristina Chiavari
email: cristina.chiavari@unibo.it

Andrea Balbo

Élodie Guilminot

Cesar Escobar Carlos

Elodie Guilminot
email: elodie.guilminot@loire-atlantique.fr

Galvanic corrosion over World War II aircraft wrecks

Keywords: metals in marine environments: corrosion mechanisms and protective treatments WWII cultural heritage, Aluminium alloy, Oxygen consumption

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers,

Introduction

The corrosion behaviour of modern alloys for aircraft applications is extensively studied. However, these analyses are insufficiently performed over real and complex cultural heritage objects, many of which exposed to outdoor conditions and aggressive environments [START_REF] De La Fuente | Studies of long-term weathering of aluminium in the atmosphere[END_REF]. Besides, characterization of the long-term behaviour of materials exposed to complex environmental conditions is of great challenge. It is known that pure aluminium in outdoor conditions corrodes weakly because it formed spontaneously a protective oxide layer [START_REF] Vargel | Corrosion of Aluminium[END_REF]. Unfortunately, aluminium alloys used in WWII planes, especially those from the 2000 series (Al-Cu) known as Duralumin, do not have the same corrosion resistance [START_REF] Blanc | The role of precipitates enriched with copper on the susceptibility to pitting corrosion of the 2024 aluminium alloy[END_REF], and different studies have shown several corrosion forms, such as intergranular, exfoliation, pitting, and galvanic corrosion [START_REF] Tissier | Characterization of Two Propeller Blades from Submerged World War II-Era Wrecks, Aluminum: History, Technology[END_REF]. The galvanic corrosion process is driven by the electrode potential difference between two dissimilar metals and each of the metals acts as either the anode or the cathode [START_REF] Masfeld | Area Relationships in Galvanic Corrosion[END_REF]. Generally, when the galvanic coupling between Al and Fe alloys is established, the dissolution of Al alloys is accelerated [START_REF] Kosaba | Effect of anodizing on galvanic corrosion resistance of Al coupled to Fe or type 430 stainless steel in diluted synthetic seawater[END_REF]. One of the main challenges in the field of cultural heritage conservation is to be able to determine the stability of the objects against corrosion, requiring techniques that are reliable, fast, and reasonably easy to use. For example, as the cathodic reaction of the corrosion process involves the reduction of dioxygen in presence of surface water, placing the object in an airtight recipient and measuring its oxygen concentration over time could be an interesting alternative, assuming that a faster oxygen consumption indicates a higher corrosion rate [START_REF] Matthiesen | In situ measurement of oxygen consumption to estimate corrosion rates[END_REF]. Furthermore, this concept could be extrapolated for analysing the galvanic coupling effect and how the cathode/anode area ratio affects the corrosion rate. In this context, this work aims to analyze the degradation of two aircraft wrecks from WWII exposed to marine environments for more than half a century and the importance of galvanic corrosion on their level of degradation.

Materials and methods

This study presents the corrosion state of two aircraft wrecks from WWII. These objects were discovered occasionally and recovered from French coasts. The first object is a propeller, probably of a DH Mosquito aeroplane (Fig. 1). This object was found by a trawler in the Fécamp channel (French city seaside to the English Channel) in April 2016. The second one is a Supermarine Spitfire wing discovered three miles from the Plérin coast (Near Saint-Brieuc, also a French city seaside to the English Channel) by a fisherman in 1988 (Fig. 2). After that, this wing was stored in a garden, for about 30 years in an uncontrollable and outdoor environment. For the elemental composition, small-size samples were cut from the objects, then they were embedded in epoxy resin, mechanically grounded up to 4000 SiC paper grade and eventually polished on cloths with diamond paste from 3 μm down to 1 μm. Scanning electron microscopy (SEM) JEOL JSM 5800LV, operated at 20 kV and coupled with Energy Dispersive X-Ray Spectroscopy (EDS), was used to analyse the elemental composition. EDS acquisition was performed on six areas of 500x500 μm² with a minimum of 100000 counts on each spectrum. To evaluate the galvanic coupling effect and how the cathode/anode area ratio affects the stability of the objects, six different Fe/Al area ratios were selected, increasing the Fe area (0.01, 0.1, 0.2, 0.5, 1 and 10). Iron nails were hammered and fastened to Al plates in pre-made holes of 1.8 mm diameter (Fig3). The samples were enclosed in a transparent bag of Escal TM film with an optical oxygen sensor (PreSens TM ) steaked at the upper inner part of the bag (seen as a small pink square in Fig. 3). The oxygen content inside the containers was measured through a fiber optic oxygen meter, Fibox 4. Inside the pockets, the atmosphere was controlled using 1 mL of water/glycerol mixture (for pouches up to 1000 cm 3 ), maintaining a relative humidity of 90% [START_REF] Morin | Contrôle de l'humidité relative par un mélange eau/glycerol: deux approches complémentaires[END_REF]. Escal is a durable oxygen barrier film without oxygen consumption tested for at least a year [START_REF] Matthiesen | A novel method to determine oxidation rates of heritage materials in vitro and in situ[END_REF]. Moreover, the flexibility of this bag allows us to place samples inside, with different shapes and sizes, reducing the volume around the samples, which makes it advantageous when low oxygen consumption rates are measured [START_REF] Matthiesen | A novel method to determine oxidation rates of heritage materials in vitro and in situ[END_REF]. Finally, nine conditions were evaluated, the six Fe/Al area ratio of AW2017-T4 aluminium alloy (similar composition as Hiduminium family) plates coupled with low carbon steel nails, the AW2017-T4 aluminium plate and the low carbon steel nail separately and the control condition without metal (Escal bag with 1 mL of glycerol solution and sensor). A portable Bruker tracer III-SD X-ray fluorescence (XRF) instrument was employed to analyse the surface composition of objects. This equipment is incorporated with a Silicon Drift Detectors (SDD), which provide high-speed data acquisition and light element sensitivity. The spectra were acquired through a 4x3 mm window and a vacuum system that improved the detection of light elements. The duration of each analysis was 120 seconds with voltage and current of 40kV and 13µA, respectively.

Results and discussion

In the first step, analysis by SEM-EDS on polished samples provided information on the corrosion features and the bulk composition of the different metallic alloys. The elemental analysis summarised in Table 1 helped us to identify the alloys used to manufacture the different pieces and objects. The propeller blades and the plate of the wing belong to the Duralumin family, containing aluminium as the base, copper (3.5-4.8 wt.%), magnesium (0.4-1.8 wt.%), and manganese (0.4-1.5 wt.%).

According to the archival studies, several families of aluminium alloys were employed in the United Kingdom for aircraft manufacturing during WWII. The Hiduminium DU Brand characterised by its low magnesium content (below or equal to 1 wt.%) and the Hiduminium 72 with a higher magnesium content (0.8-1.8 wt.%). In this way, Hiduminium DU Brand was probably used to manufacture the propeller blades and Hiduminium 72 for the Al plates of the Supermarine Spitfire wing. [START_REF] Ouissi | Comparison of aluminum alloys from aircraft of four nations involved in the wwii conflict using multiscale analyses and archival study[END_REF]. On the other hand, the Al cover was probably made of cast alloy with a composition quite close to the Hiduminium alloy R.R. 50, described as a general-purpose casting alloy regularly employed for cylinder blocks, crankcases and all parts in automobile and aircraft manufacture [START_REF]Hiduminium: Technical Data[END_REF]. The differences that could make us doubt are the silicon and nickel levels slightly lower and higher than the standards, respectively. However, it should be remembered that these alloys date from a war period when exhaustive quality controls of raw materials were not a priority. The hub of the propeller hub and the bolts of the Spitfire wing were identified as ferrous alloys with low traces of manganese and chromium. As shown in Fig. 1. the propeller blades are in an advanced state of corrosion. covered by white and blue-green corrosion products. They are known to be hydrated variants of aluminium or copper-aluminium oxides and hydroxides [START_REF] Tissier | Characterization of Two Propeller Blades from Submerged World War II-Era Wrecks, Aluminum: History, Technology[END_REF]. In addition, intergranular and exfoliation (lamellar) corrosion features are observed in Fig 4a . These types of corrosion are usually exhibited in archaeological elements made from aluminium alloys containing between 3.5 and 5.5% Cu. which is generally recognised as a harmful constituent of these alloys from a corrosion standpoint [START_REF] Stratton | Aluminum and Its Light Alloys[END_REF]. In Fig. 2. the aluminium part covering the transmission system of the propeller also exhibits an important degree of deterioration. Fig. 4b shows characteristics of intergranular corrosion on this piece favoured by the presence of nobler AlFeMnSi and Cu-rich phases, making this region more reactive than the bulk of the grains, causing this zone to corrode preferentially, and creating micro galvanic corrosion cells between the cathodic phases and the anodic aluminium matrix [START_REF] Zeng | Microstructure, microhardness and corrosion resistance of laser cladding Ni-WC coating on AlSi5Cu1Mg alloy[END_REF]. Contrary to the deterioration state of the aluminium parts from the propeller, where it was almost impossible to find evidence of the original protective paint film, the Spitfire wing shows a better conservation state, even after a long exposure time to the marine environment (Fig. 2). On the aluminium plate were observed regions with the remains of the protective paint, approximately 20 µm thick, measured through SEM (Fig 4c). Moreover, evidence of localised pitting corrosion was observed in the regions where the paint protection was broken. The ferrous pieces from the propeller hub and transmission system showed good homogeneity and uniform corrosion. In contrast, the ferrous bolts from the Spitfire wing showed a more degraded state with thicker corrosion products, as observed in Fig 4e . As shown in Fig. 2, some bolts disappeared, which is attributed to the corrosive process. The exfoliation, intergranular and pitting corrosion forms observed on the different parts of Al alloys can be explained by the aggressive marine environment as the composition and microstructure of the alloy and the aggressive environmental conditions to which they were exposed during the time they were buried. XRF analysis revealed the surface composition of the different objects, showing traces of sulphur and chlorine compounds. This information is crucial because the presence of sulphate ions in a marine environment with chloride ions increases the rate of pitting on Al alloys. This situation can be even more detrimental since some studies have shown that alloys such as 2024-T4 and 2014-T3 underwent more severe exfoliation corrosion processes when they were in contact with steel parts, developing a galvanic corrosion process [START_REF] Deck | Deconstructing the Dymaxion House: Designing Survival Strategies for an Aluminum Tension Structure[END_REF][START_REF] Graedel | Corrosion Mechanisms for Aluminum Exposed to the Atmosphere. The Morphology of Atmospheric Corrosion Layers on Aluminum[END_REF]. Generally, when the galvanic coupling between Fe alloys and Al is established, the dissolution of Al is accelerated, and the rate of this acceleration is controlled in part by the cathode/anode area ratio. The higher this ratio is, the higher the corrosion rate will be, which could be one of the reasons why the Al parts of the propeller suffered more aggressive corrosion due to their galvanic coupling with the big ferrous part (Propeller hub and transmission system). The Fe/Al area ratio was determined geometrically and plotted in Fig. 5, showing the ratio difference between the two objects. In the Spitfire wing, the Al proportion is more significant than the Fe one (Fe/Al areas <1). However, this situation is different in the propeller, and the Fe area is larger than that of Al (Fe/Al areas >1). For this reason, to know how much these area ratios affected the corrosion rate of these objects, the six different Fe/Al area ratios of modern alloys were chosen as indicated in Fig. 5 and their oxygen consumption was analysed. Fig. 5 The six Fe/Al area ratio used for the oxygen consumption analysis and the relative position of the Spitfire wing and the aircraft propeller.

Fig. 6 shows the measurements of oxygen concentration of the different conditions for 28 days. These curves show a higher oxygen consumption in the first days of analysis; however, their slops are accentuated after the third day. From them, the oxygen consumption rate was calculated to compare the different samples' results, normalizing the size effect of the parts and the Escal bags. Thus, the oxygen consumption rate may be calculated as:

Oxygen consumption rate [𝑚𝑔/𝑑𝑎𝑦] = 𝐴𝑖𝑟 𝐼𝑛𝑡 . 𝐾. 𝑑𝐶 𝑑𝑡 ⁄ 100
Where Airint is the internal air volume inside the bags, quantified by subtracting the bags and sample volume. The bag volume was measured through buoyancy in water. In this study, the external and internal volumes were considered equivalent, and the magnitude of the borders thermally sealed was negligible. The sample volume was calculated by geometry. K is the oxygen content in atmospheric air (0.276 mg/cm 3 at 23 °C), dC/dt is the slope of the oxygen curve (%sat/day), and the constant 100 (%) is included to convert the number to mg/day [START_REF] H Matthiesen Henning | A fast and non-destructive method to document and quantify the efficiency of metals conservation[END_REF].

Fig. 6 Oxygen concentration versus time of the six Fe/Al area ratio of Al alloy plates coupled with low carbon steel nails), the Al plate, and Fe nails separately and the control condition (Escal bag with 1 mL of glycerol solution). The slope of the different curves cannot be compared directly as the amount of air inside the bags may vary. Table 2 summarizes the results of the different conditions studied. In it, we can observe that for the 0.5 Fe/Al area ratio, the oxygen consumption is slightly higher than that consumed by the control system. Besides, the Fe/Al ratios of 1 and 10 show the highest oxygen consumption values of all the systems studied. On the other hand, the samples of Al and Fe not galvanically coupled and enclosed separately showed negligible oxygen consumption values under the same atmospheric conditions (90% RH at room temperature). One thing interesting to recall is that the Fe bolts from the Spitfire wing suffered a more advanced degradation than their surrounded Al plate despite the possible cathodic protection provided by their Al galvanic pair. Our first hypothesis was to consider that a paint protective layer on the Al plates acted as an electrical insulator between the pieces, thus avoiding the galvanic interaction. However, this was dismissed as there is still electrical conductivity between the iron bolts and the Al plate 0with a resistance value of around 0.1 ohms (measured after removing the corrosion products on the parts).

A second hypothesis still open is to consider the effect of the protective paint layer on this Al surface, which could have changed its galvanic potential towards more noble values, decreasing the galvanic protection effect over the Fe bolts, or even changing the role of which part is acting as anode or cathode. Furthermore, the good condition of the Al plates has been associated with the protective effect of its paint layer, even after a long exposure time in seawater [START_REF] Kosaba | Effect of anodizing on galvanic corrosion resistance of Al coupled to Fe or type 430 stainless steel in diluted synthetic seawater[END_REF].

The results obtained from the oxygen consumption tests are an interesting indication of the harmful effect of the Fe/Al galvanic coupling. However, this is a preliminary study, and complementary experiments, including surface characterisation, and electrochemical analysis are being carried out to understand better whether the objects are actively corroding, why? and the galvanic corrosion roll on these and other objects from WWII. Besides, this is the first step for using this oxygen consumption test on real artefacts considering the galvanic corrosion effect.

Conclusion

Two aircraft wrecks from WWII were analysed: an aircraft propeller probably from a DH Mosquito and one wing from the Supermarine Spitfire, both found in the seabed. The propeller blades and the cover of the transmission system were made of Hiduminium DU Brand and Hiduminium alloy R.R. 50, respectively. They showed a remarkable degradation with features related to intergranular, exfoliation and pitting corrosion processes mainly developed during the time they were immersed in the aggressive marine environment; these pieces were assembled with the Fe hub and transmission system, which had an area comparable to the Al parts (Fe/Al area ratio 0.4-1). On the other hand, the Spitfire wing with a lower Fe/Al area ratio (0.1) shows a better conservation state, even when it was also exposed to a marine environment for more than half a century. Finally, preliminary results from the oxygen test showed that the oxygen consumption rate increases with a Fe/Al area ratio higher than 0.5. Highlighting, that these oxygen consumption results were obtained in atmospheres without sulfate and chloride aggressive ions and were due to the galvanic coupling interaction in high relative humidity (90% RH). These results gave us new perspectives for future analysis of the galvanic corrosion effect on objects from WWII. Such as using oxygen consumption tests in more aggressive conditions close to the marine environment and utilising complementary electrochemical measures to help us better understand the anode/cathode activity in these objects.

Fig. 1

 1 Fig. 1 Aircraft propeller (blade length 130 cm and maximum chord 26 cm) found in Fécamp (French city seaside to the English Channel), indicating the degradation state of the different components. The Fe/Al area ratio in this object is among 0.4-1, according to the Al piece evaluated.

Fig. 2

 2 Fig. 2 Wreck of a Supermarine Spitfire wing (length 436 cm and maximum chord 67 cm) discovered in 1988 near the Plérin coast (French city seaside to the English Channel). The zooming area shows the bolt's degradation and the Al surrounding area. The Fe/Al area ratio between the Fe-bolt and the al plate is close to 0.1.

Fig. 3

 3 Fig.3Escal bag containing the sample with the Fe/Al area ratio 0.1, a plastic container with 1.0 mL of water/glycerol mixture, 70/30%. The optical oxygen sensor was attached to the upper inner part of the bag using adhesive tape with a central hole of 4mm Ø, allowing oxygen diffusion to the sensor.

Fig. 4

 4 Fig. 4 Cross-section images from the different parts of the artefacts: (a) Propeller blade of Al, (b) Al cover of the transmission system, (c) Al plate from the Spitfire wing, (d) Propeller hub of Fe and (e) Fe bolt from the Spitfire wing.

Table 1 .

 1 Elemental composition (wt%) by SEM-EDS of the blade and the fuselage plate from the Fécamp propeller and the Spit Fire wing, respectively.

	Object	Al	Mg	Si	Mn	Fe	Ni	Cu	Cr
	Propeller blades	91.93 0.47 1.26 0.62 0.29 0.04 4.85
	Cover -transmission system	91.06 0.61 3.78 0.02 0.93 0.87 1.61
	Plate-Spitfire wing	94.34 1.20 0.90 0.55 0.33 0.02 4.04
	Propeller hub	-	0.14 0.22 0.88 99.76	-	-	0.89
	Bolt-Spitfire wing	-	0.09 0.26 0.62 96.74 1.16	-	0.55

Table 2 .

 2 Oxygen consumption measurements for the different conditions studied. The value of oxygen consumed by the control system was subtracted from all samples.

	Fe/Al ratio	Volume of sample (cm 3 )	Internal air volume (cm 3 )	Oxygen consumption rate (mg/day)
	0.01	5.18	22.0	N
	0.1	3.36	33.4	N
	0.2	3.48	51.0	N
	0.5	3.82	51.1	0.020
	1	4.35	65.2	0.184
	10	9.05	34.2	0.194
	Al plate	3.26	13.3	N
	Fe nail	2.68	10.8	N
	Control	2.63	10.5	0.015