Cécile Arnould

Aline Bertin

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Lucas Colabella
email: lucas.colabella@u-pec.fr

Salah Naili
email: naili@u-pec.fr

Sophie Le Cann
email: sophie.le-cann@cnrs.fr

Guillaume Haiat
email: haiat@u-pec.fr

Effect of collagen fibril orientation on the anisotropic properties of peri-implant bone

Keywords: bone, implant, multiscale, homogenization, collagen orientation 100-500 nm Osteonal canal Hydroxyapatite Hydroxyapatite Foam Tissue

come

Introduction

Longer lifespans, stress, and poor nutritional practices are contributing factors to disorders that worsen bone quality and cause teeth loss [START_REF] Fontaine | Obesity and health-related quality of life[END_REF]. Moreover, active lifestyles and traffic and workplace accidents frequently lead to injuries [START_REF] Kurtz | The determination of stress and strain concentrations at an ellipsoidal inclusion in an anisotropic material[END_REF]. In orthopedic and dental surgery, the implantation of biomaterials inside the bone to restore the integrity of the treated organ has become a standard procedure. Implants and joint prostheses have significantly improved maxillofacial surgery (to restore missing teeth or support craniofacial reconstructions) and the treatment of joint degeneration (hip, knee, etc). A fast, robust, and long-lasting bond between the implant and the bone, i.e. adequate osseointegration, is the goal of current orthopedic and dental implant therapies [START_REF] Gao | Biomechanical behaviours of the bone-implant interface: a review[END_REF].

The growth of bone around and onto the metallic foreign object leads to the formation of a bone-implant interface, which evolves as bone remodels [START_REF] Albrektsson | Osseointegrated Titanium Implants: Requirements for Ensuring a Long-Lasting, Direct Boneto-Implant Anchorage in Man[END_REF][START_REF] Shah | Osseointegration and current interpretations of the bone-implant interface[END_REF]). An inability to obtain a bounded boneimplant contact is a common reason for early implant failure (during the first six months of implantation), which is one of the most common causes of failure in dental surgery [START_REF] Chrcanovic | Factors Influencing Early Dental Implant Failures[END_REF][START_REF] Grisar | Retrospective Analysis of Dental Implants Placed Between 2012 and 2014: Indications, Risk Factors, and Early Survival[END_REF]. It is difficult to determine the exact causes of osseointegration failure, which may result from a variety of implant-, surgery-, and patient-related variables [START_REF] Amengual-Peñafiel | Coupling between Osseointegration and Mechanotransduction to Maintain Foreign Body Equilibrium in the Long-Term: A Comprehensive Overview[END_REF][START_REF] Albrektsson | Initial and long-term crestal bone responses to modern dental implants[END_REF]. A deeper understanding of the biomechanical characteristics of the bone-implant interface and the surrounding bone is needed to increase implant stability and decrease surgical failure risks [START_REF] Haïat | Effects of Biomechanical Properties of the Bone-Implant Interface on Dental Implant Stability: From In Silico Approaches to the Patient's Mouth[END_REF][START_REF] Wirth | Implant stability is affected by local bone microstructural quality[END_REF].

Bone is a hierarchical anisotropic composite material that is strong and lightweight at the same time [START_REF] Gao | Connection between elastic and electrical properties of cortical bone[END_REF]. Mineralized bone (at the scale of 10-100 nm) is made of fundamental elements including water, collagen molecules, and hydroxyapatite crystals. The ultrastructure (at the scale of 1-10 µm), is constituted of mineralized collagen fibers embedded in a mineral foam. At the microscale, bone is organized either in a compact pattern (cortical bone) or a spongy pattern (cancellous bone). Cortical bone stiffness is given mainly by its porosity [START_REF] Cai | Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity[END_REF]. However, bone ultrastructure orientation and arrangement are predictors of mechanical properties such as bone strength or elastic modulus [START_REF] Georgiadis | Techniques to assess bone ultrastructure organization: orientation and arrangement of mineralized collagen fibrils[END_REF]. Moreover, earlier studies have shown that peri-implant bone nanostructure is affected by the presence of ceramic and metallic implants (Hoerth et al, 2014a;[START_REF] Bünger | Strontium and Bone Nanostructure in Normal and Ovariectomized Rats Investigated by Scanning Small-Angle X-Ray Scattering[END_REF][START_REF] Cann | Spatio-temporal evolution of hydroxyapatite crystal thickness at the bone-implant interface[END_REF]. However, little is known about how those lower-scale changes impact the macroscale effective mechanical behavior of the bone-implant system.

A common approach to retrieve bone mechanical or acoustical properties consists in coupling 3-D reconstructions obtained from high-resolution imaging techniques with numerical simulation tools [START_REF] Chevalier | Validation of a voxel-based fe method for prediction of the uniaxial apparent modulus of human trabecular bone using macroscopic mechanical tests and nanoindentation[END_REF][START_REF] Haïat | Variation of ultrasonic parameters with microstructure and material properties of trabecular bone: A 3d model simulation[END_REF][START_REF] Haïat | Fast wave ultrasonic propagation in trabecular bone: Numerical study of the influence of porosity and structural anisotropy[END_REF]. High-resolution Finite Element (FE) models are constructed from bone microstructures which are digitized using micro Computed Tomography (micro-CT) or in vivo high-resolution peripheral quantitative CT (HR-pQCT) scanners. FE analyses are used to compute bone effective elastic properties [START_REF] Rietbergen | Relationships between bone morphology and bone elastic properties can be accurately quantified using high-resolution computer reconstructions[END_REF][START_REF] Rietbergen | Direct mechanics assessment of elastic symmetries and properties of trabecular bone architecture[END_REF][START_REF] Ulrich | The ability of three-dimensional structural indices to reflect mechanical aspects of trabecular bone[END_REF][START_REF] Pistoia | High-Resolution Three-Dimensional-pQCT Images Can Be an Adequate Basis for In-Vivo µFE Analysis of Bone[END_REF], to predict bone strength [START_REF] Bv | A survey of microfinite element analysis for clinical assessment of bone strength: The first decade[END_REF] and to conduct multiscale analyses (Podshivalov et al, 2011a,b). These models assume a linear elastic behavior of the material at the ultrastructure scale, whose mechanical properties are obtained via micro and nanoindentation tests [START_REF] Rodriguez-Florez | Age-related changes in mouse bone permeability[END_REF].

Another strategy to characterize the bone mechanical behavior consists in using its hierarchical organization together with empirical relations on its composition to build a multiscale homogenization model able to determine the elastic tensor at the scale of the organ [START_REF] Hellmich | Micromechanics-based conversion of ct data into anisotropic elasticity tensors, applied to fe simulations of a mandible[END_REF][START_REF] Sansalone | Determination of the heterogeneous anisotropic elastic properties of human femoral bone: From nanoscopic to organ scale[END_REF][START_REF] Sansalone | Anatomical distribution of the degree of mineralization of bone tissue in human femoral neck: Impact on biomechanical properties[END_REF][START_REF] Gagliardi | Estimation of the effective boneelasticity tensor based on µCT imaging by a stochastic model. A multi-method validation[END_REF][START_REF] Hellmich | Hierarchical biomechanics: Concepts, bone as prominent example, and perspectives beyond[END_REF]Hamed and Jasiuk, 2012). In this way, bone stiffness can be obtained using only the information on the volume fraction of each constituent assuming a simplified structure at each scale considered. Several authors used this model to compute the elastic properties of trabecular (Hellmich et al, 2004b;Hamed et al, 2012) and cortical [START_REF] Vaughan | A three-scale finite element investigation into the effects of tissue mineralisation and lamellar organisation in human cortical and trabecular bone[END_REF][START_REF] Hamed | Multiscale modeling of elastic properties of cortical bone[END_REF][START_REF] Martínez-Reina | Effect of porosity and mineral content on the elastic constants of cortical bone: a multiscale approach[END_REF][START_REF] Barkaoui | Multiscale approach including microfibril scale to assess elastic constants of cortical bone based on neural network computation and homogenization method[END_REF] bone and to predict bone remodeling [START_REF] Colloca | A multiscale analytical approach for bone remodeling simulations: Linking scales from collagen to trabeculae[END_REF]. However, to the best of the authors' knowledge, it was never used to analyze the peri-implant bone. Furthermore, even if classical computed tomography provides information on the microstructure, its spatial resolution is insufficient to get information on the geometrical arrangement of the ultrastructure. X-ray-based scattering techniques, such as small-angle X-ray scattering (SAXS), are sensitive to the crystal structure and periodic arrangement of bone minerals and can be used to retrieve information about mineral platelet orientation and thickness [START_REF] Wagermaier | Materials Design Inspired by Nature : Function Through Inner Architecture[END_REF][START_REF] Turunen | Bone mineral crystal size and organization vary across mature rat bone cortex[END_REF][START_REF] Mathavan | The compositional and nano-structural basis of fracture healing in healthy and osteoporotic bone[END_REF][START_REF] Liu | Size and habit of mineral particles in bone and mineralized callus during bone healing in sheep[END_REF]Hoerth et al, 2014b) which can be linked to the collagen fibrils' orientation.

This paper aims to determine the spatial variation of the elastic tensor of the peri-implant bone during healing using a three-scale homogenization method that uses as input experimental data spatial variations of porosity and platelet orientation obtained from SAXS measurements of animal samples. Such an approach can be used to relate anatomical variations of porosity and platelet orientation to changes in peri-implant bone biomechanical properties.

Materials and methods

Multiscale homogenization model

A multiscale homogenization model was developed to compute the effective elastic properties of bone at the organ scale as a function of the distance from the implant. This model is based on the previous works of [START_REF] Haïat | Effects of Biomechanical Properties of the Bone-Implant Interface on Dental Implant Stability: From In Silico Approaches to the Patient's Mouth[END_REF] and [START_REF] Sansalone | Determination of the heterogeneous anisotropic elastic properties of human femoral bone: From nanoscopic to organ scale[END_REF]. Three scales are considered: hydroxyapatite foam (HF), ultrastructure (US), and tissue (T) (see Fig. 1) which are respectively associated with three length scales (several hundred nanometers, several micrometers, and several hundred micrometers to several millimeters).

The elastic properties and the volume fraction of the elementary constituents of the bone matrix (hydroxyapatite (HA), collagen (col), and water (W)), the orientation of the collagen fibrils relative to the implant surface (θ), and the bone porosity (ϕ P ) constitute the input data of the model. At each scale, the continuum micromechanics theory based on the Eshelby's representation of the uniform elastic field inside the ellipsoidal inclusion is applied [START_REF] Suquet | Continuum Micromechanics[END_REF]. Inclusions of different shapes (cylindrical or spherical) with different material properties (Laws, 1977a[START_REF] Laws | A note on penny-shaped cracks in transversely isotropic materials[END_REF] are considered at each scale. In what follows, the position vector x is specified through the Cartesian coordinates (x 1 , x 2 , x 3 ) with respect to a Cartesian reference frame R(O; e 1 , e 2 , e 3 ), where O is the origin of the space and (e 1 , e 2 , e 3 ) is an orthonormal basis of this space.

The homogenization methods based on continuum micromechanics can estimate the homogenized elasticity tensor of a representative volume element (RVE) comprised of several phases with different distributions and shapes, which can be organized in a matrix/inclusion-like microstructure or entirely disordered. The homogenized elasticity tensor turns out to be a weighted sum over the phases:

C Hom = r ϕ r C r : A r , (1)
where ϕ r , C r , and A r are the volume fraction, the elastic tensor, and the localization tensor of the phase r, respectively. The symbol ":" in Eq. 1 denotes a double contraction of adjacent indexes of any couple of tensors. Note that the following condition is required: r ϕ r = 1. The fourthorder localization tensor links the microscopic infinitesimal strain field ϵ(x) with the macroscopic infinitesimal strain E, i.e. ϵ(x) = A r (x) : E. It can be shown that the expression of A r reads (see for instance Zaoui (2002)):

A r = I + P r : (C r -C 0 ) -1 : s ϕ s I + P s : (C s -C 0 ) -1 -1 , (2)
where P r is the fourth-order Hill tensor of the phase r embedded in the matrix phase, C 0 is the elastic tensor of the reference material, I is the fourth-order identity tensor and s refers to the phase. The expression of the Hill tensor depends on the shape of the phase and on the elastic properties of the homogeneous host phase which is also called matrix phase.

The choice of the value of C 0 leads to different estimates of the homogenized elasticity tensor C Hom . Two estimates are relevant in the context of this work [START_REF] Nemat-Nasser | Micromechanics: overall properties of heterogeneous materials[END_REF] at the bottom shows for each scale the phases that constitute it, the homogenized elastic tensor C Hom , the elastic tensor of the reference material C 0 , and the estimate used.

when the material microstructure is made up of uniformly dispersed inclusions in a homogeneous matrix. In this case, the effective matrix is an actual, well-identified phase and therefore C 0 = C matrix . Then, the MT estimate of C Hom can be readily computed from Eq. ( 1). The SC estimate is well suited when no actual matrix can be identified but the microstructure is rather made of interpenetrating (continuous or discontinuous) phases. In this case, the effective matrix is assumed to be the homogenized material itself and therefore C 0 = C Hom . It follows that the SC estimate of C Hom leads to an implicit equation (see Eq. ( 1)) which is solved iteratively using a fixed-point method.

In what follows, the equations to compute the effective elastic properties at each scale are described.

Hydroxyapatite foam

The hydroxyapatite foam can be seen as constituted by spherical inclusions of hydroxyapatite, water, and non-collageneous matter filling the entire space. This disordered structure motivates the use of an SC estimate to compute the homogenized elastic properties of the hydroxyapatite foam. Therefore, the elastic tensor is defined from Eq. (1) and Eq. ( 2) with r, s ∈ [HA, W ] as:

C HF = [ϕ HA C HA : I + P HA : (C HA -C HF ) -1 + ϕ W C W : I + P W : (C W -C HF ) -1 ] : [ϕ HA I + P HA : (C HA -C HF ) ϕ W I + P W : (C W -C HF ) -1 ] -1 , (3)
The non-zero components of the Hill tensor P r for spherical inclusions embedded in the isotropic host phase C M F can be expressed in Voigt notation as [START_REF] Parnell | The Eshelby, Hill, Moment and Concentration Tensors for Ellipsoidal Inhomogeneities in the Newtonian Potential Problem and Linear Elastostatics[END_REF] .

(4)

The obtained elastic tensor for this mixture is isotropic.

Ultrastructure

At the scale of several micrometers, the matrix phase is the hydroxyapatite foam described in the first step and the collagen fibrils are modeled as cylindrical inclusions, which motivates the use of the MT estimate. To account for the observed variation of the orientation of the collagen fibrils (see Section 2.3.3), a rotation around the common e 1axis is introduced, resulting in the e ′ 3 -axis forming an angle θ with the e 3 -axis (see Fig. 1). The elastic tensor of the ultrastructure C U S ′ referred to a local orthonormal basis defined by (e ′ 1 , e ′ 2 , e ′ 3 ) is computed from Eq. (1) and Eq. ( 2) with r, s ∈ [HF, col] as [START_REF] Parnell | The Eshelby, Hill, Moment and Concentration Tensors for Ellipsoidal Inhomogeneities in the Newtonian Potential Problem and Linear Elastostatics[END_REF]:

C U S ′ = ϕ HF C HF + ϕ col C col : I + P col : (C col -C HF )
-1 :

ϕ HF I + ϕ col I + P col : (C col -C HF ) -1 -1
.

(5)

The non-zero components of the Hill tensor for cylindrical inclusions in an isotropic host phase can be expressed in Voigt notation as: The obtained elastic tensor for the ultrastructure is transversely isotropic with its plane of isotropy the (e ′ 1 , e ′ 2 )-plane (see Fig. 1). Thus, a change of basis is needed to express the elastic tensor C U S in the reference basis (e 1 , e 2 , e 3 ). This tensor is defined by:

P col 11 = P col 22 = P col 1111 = 1 8 C HF 11 + 3C
C U S ijkl = U im U jn U kp U lq C U S ′ mnpq , (7)
where U ij (for i, j = 1, ..., 3) are the components of the transformation matrix U for a rotation around the e 1 -axis which are defined by:

U =   1 0 0 0 cos θ -sin θ 0 sin θ cos θ   . (8)
In the relation ( 7), Einstein's summation convention is used, that is, repeated subscripts are implicitly summed over. Moreover, it should be noticed that while C U S ′ is transversely isotropic, C U S is anisotropic.

Tissue

At the organ scale, bone tissue is considered as composed of cylindrical pores embedded in the homogeneous continuous ultrastructure described in the previous scale. We assumed that the orientation of the vascular porosity is parallel to the implant surface, which is due to the design of the animal model described in Section 2.3.3. Under these conditions, the effective elastic tensor C T of the bone is, as in the previous scales, computed from Eq. (1) and Eq. ( 2) with r, s ∈ [U S, W, P ] as:

C T = ϕ U S C U S + ϕ W P C W : I + P P : (C W -C U S )
-1 :

ϕ U S I + ϕ W I + P P : (C W -C U S ) -1 -1 , (9)
where ϕ W P is the volume fraction of the mesoscopic pores (or porosity), which are considered filled with water.

As described in the previous section, the elastic properties C U S of the homogeneous host phase are anisotropic. Therefore, the Hill tensor has no closed-form expression for any inclusion shape. So, for a cylindrical inclusion, the components of the Hill's tensor can be expressed in the integral form (Laws, 1977b):

P P ijkl = 1 8 [M ijkl + M jikl + M ijlk + M jilk ] , (10)
where the components of the fourth-order tensor M are given by:

M ijkl = 1 π 2π 0 ξ i K -1 jk ξ l ξ 2 1 + ξ 2 2 dψ, (11)
where ξ = cos ψ e 1 + sin ψ e 2 is a unit vector of plane centered at the origin, and K = ξ • C U S • ξ is the acoustic tensor which can be written as follows:

K = cos 2 ψ Q + cos ψ sin ψ (R + R T ) + sin 2 ψ T, (12)
where the superscript T designates the transpose operator. The components of the 3 × 3matrices Q, R and T are defined by the following relations:

Q ik = C U S i1k1 , R ik = C U S i1k2 , for i, k = 1, ..., 3. T ik = C U S i2k2 , (13)
The obtained elastic tensor for the bone C T is anisotropic. It should be noticed that when the collagen fibers are oriented with their principal axis coincident with the principal axis of the pores, i.e. θ = 0, C T has transversely isotropic symmetry.

Elastic tensor decomposition

To study the anisotropic nature of bone at the organ scale, it is useful to decompose its elastic tensor into a sum of orthogonal tensors belonging to the different symmetry classes. Hence, elastic symmetries of the effective elastic tensor C T of the bone were computed using the Matlab Seismic Anisotropy Toolkit (MSAT) by [START_REF] Walker | MSAT-A new toolkit for the analysis of elastic and seismic anisotropy[END_REF]. This is an implementation of the method by [START_REF] Browaeys | Decomposition of the elastic tensor and geophysical applications[END_REF], who considered the elasticity tensor C as an elastic vector X with 21 orthogonal components:

X = (C 11 , C 22 , C 33 , √ 2C 23 , √ 2C 13 , √ 2C 12 , 2C 44 , 2C 55 , 2C 66 , 2C 14 , 2C 25 , 2C 36 , 2C 34 , 2C 15 , 2C 26 , 2C 24 , 2C 35 , 2C 16 , 2 √ 2C 56 , 2 √ 2C 46 , 2 √ 2C 45 ). (14)
Then, X is decomposed by a cascade of projections into a sum of vectors belonging to the symmetry classes triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and isotropic, X = X tri + X mon + X ort + X tet + X hex + X iso .

(15) Finally, the symmetry class fractions are computed as

c c = ∥X c ∥/∥C∥, (16)
where c ∈ [iso, hex, tet, ort, mon, tri].

Experimental input

The model developed in Section 2.1 requires as input data the individual constituents' elastic properties, the description of the phase of each constituent in each RVE (composition of the tissue), the orientation of the collagen fibrils at the ultrastructure scale and the average porosity of the bone at the tissue scale.

Elastic properties of the experimental constituents

The elastic properties of the elementary constituents (collagen, hydroxyapatite, and water) were taken from Hellmich et al (2004a) and are shown in Tab. 1. Both water and mineral are isotropic materials, while the collagen fibrils are transversely isotropic, the isotropy plane being orthogonal to the e 3 -axis of the fibril.

Estimation of the volume fraction of each component of bone tissue

Bone tissue is considered to be composed of hydroxyapatite, collagen, and water leading to:

ϕ HA + ϕ col + ϕ W = 1. ( 17
)
The estimation of ϕ HA is done using:

ϕ HA = TMD ρ HA ( 18
)
where ρ HA is the mass density of hydroxyapatite and T M D is the tissue mineral density. In this work, the mineralization is assumed to be constant and T M D = 1 [START_REF] Gagliardi | Estimation of the effective boneelasticity tensor based on µCT imaging by a stochastic model. A multi-method validation[END_REF]. The reader is referred to [START_REF] Sansalone | Determination of the heterogeneous anisotropic elastic properties of human femoral bone: From nanoscopic to organ scale[END_REF] for a detailed description of the influence of mineralization on bone mechanical properties. Volume fractions of hydroxyapatite, collagen and water are computed from an empirical relation obtained through measurements of volume fractions in demineralized bone by solving the nonlinear system [START_REF] Broz | Material and compositional properties of selectively demineralized cortical bone[END_REF]:

ϕ col ϕ W = α + β × e γ×ϕ HA (19)
where α = 0.36, β = 0.084 and γ = 6.7.

Mineralized collagen fibrils orientation

The collagen orientation in the bone tissue close to an implant surface varies during osseointegration. To model this variation, experimental data were extracted from a study formerly conducted by our group [START_REF] Cann | Spatio-temporal evolution of hydroxyapatite crystal thickness at the bone-implant interface[END_REF]. The techniques and relevant results are briefly described hereafter, the reader is referred to the publication for more detailed information.

A standardized Ti6Al4V coin-shaped implant model with a dedicated bone chamber (diameter 5 mm, height 3 mm) had been osseointegrated in rabbit tibiae for 7 weeks (n = 4) and 13 weeks (n = 4) (see Fig. 2A). The newly-formed bone close to the implant surface was exposed (2D slices) and scanned using Scanning small-angle Xray scattering (SAXS), with a micro-focused setup (around 4000 × 250 µm 2 with a pixel size of 5 µm) (see Fig. 2B). From the 2D scattering patterns, the average predominant mineral plate orientation (direction of the crystallographic c-axis) was determined, which is commonly accepted to reflect the mineralized collagen fibril principal orientation, as minerals align and wrap around the fibrils [START_REF] Reznikov | Fractallike hierarchical organization of bone begins at the nanoscale[END_REF]. To extract the spatial variation of the collagen fibril orientation in relation to the implant distance, data were averaged within 20 regions of interest (ROIs) corresponding to 10 µm thick horizontal bands parallel to the implant surface (see Fig. 2B).

The experimental data depicts a spatial variation of the collagen fibril orientation (θ) with the distance to the implant surface (see Fig. 2C,D, dashed lines). Both at 7 weeks and 13 weeks of healing time, the angle between the fibrils and the implant surface is close to zero in the proximity of the implant (10 • < θ < 25 • in the first 80 µm), gradually increasing the further we move away. The variation in the collagen fibrils' orientation is higher at earlier stages of healing.

Bone tissue porosity

From the same experimental data set and similar ROIs (see Section 2.3.3), bone tissue porosity ϕ W P was estimated from the Bone Area/Total Area ratio, i.e. the number of pixels corresponding to the bone tissue over the total number of pixels in the ROI (see Fig. 2C,D, solid lines). ϕ W P is relatively stable at 7 weeks (averaged porosity within the chamber of 84%) and globally decreases with healing time (average of 65% at 13 weeks), depicting a strong drop in the very first 40 µm close to the implant surface. As reference data, a bone porosity value of 8% was identified as a healthy target value (human cortical bone, [START_REF] Gagliardi | Estimation of the effective boneelasticity tensor based on µCT imaging by a stochastic model. A multi-method validation[END_REF] (Hellmich et al, 2004a). Water and mineral are isotropic materials, while the collagen fibril is a transversely isotropic material, the isotropy plane being orthogonal to the e 3 -axis of the fibril.

Results

Response of the model to variations in collagen fibril orientation and porosity

The evolution of the elastic tensor of the tissue C T was computed as a function of the collagen fibrils orientation θ (0 • < θ < 90 • ) and for three different values of ϕ W P : 8% (mature cortical bone), 65%, and 84% (bone during healing time) (see section 2.3.4) (Fig. 3). Additionally, Fig. 4 shows the evolution of C T as a function of bone porosity ϕ W P (0 ≤ ϕ W P ≤ 100%) for different values of collagen fibrils orientation θ = 0 • , 45 • and 90 • . When ϕ W P increases, the values of each component of the elastic tensor of the bone C T decrease (see Fig. 4), and, in particular, the value of C T 44 decreases faster than for the tensile elastic tensor components (i.e. C T 11 , C T 22 , and C T 33 ), which is explained by the fact that the pores are filled with water, which has a lower stiffness compared to the ultrastructure and for which the shear components are equal to zero.

For a given porosity, C T 33 is maximum when θ = 0 • i.e. when the principal axis of the collagen fibrils is aligned with the e 3 -axis. When the principal axis of the collagen fibrils is aligned with the e 2 -axis (i.e. for θ = 90 • ), the behavior depends on the bone porosity. For low porosity, i.e. ϕ W P = 0% to 15%, C T 22 > C T 33 , while for higher porosities (ϕ W P > 15%), C T 22 < C T 33 , which is in agreement with the behavior of the phases at the lower scale, i.e., the ultrastructure. For a low value of ϕ W P , i.e. a high content of ultrastructure, the anisotropic behavior is dominated by the collagen fibril orientation. Moreover, when ϕ W P = 0%, i.e., the only phase is the ultrastructure, C T 22 is higher than C T 33 for θ > 45 • , which indicates that the stiffness is conditioned by the collagen fibril orientation only. In contrast, when ϕ W P is high, the effect of the orientation of the collagen fibrils diminishes, which results in a low variation of the bone elastic tensor as a function of θ.

Regarding C T 44 (see Fig. 3), it is maximum when θ ∼ 45 • since at this point, the collagen fibrils are aligned with the principal stresses obtained with a pure shear load [START_REF] Kaw | Mechanics of Composite Materials[END_REF].

Tensile components

Analysis of the bone-implant interface

The variation of the elastic tensor of the tissue C T was computed as a function of the distance from the implant based on experimental data (see Fig. 2C,D). Different scenarios were tested: the effect of a variation of i) bone porosity ϕ W P alone, ii) collagen fibril orientation θ alone, and iii) ϕ W P and θ combined (see Fig. 5). When analyzing the effect of a variation of ϕ W P alone, the collagen fibrils were considered aligned with the pores (θ = 0).

When analyzing the effect of a variation of collagen fibril orientation alone, ϕ W P was set as the average value in the bone chamber (ϕ W P = 84% for 7 weeks and ϕ W P = 65% for 13 weeks). Only components in the plane of the cross-section of the analyzed implants and for whom the collagen fibril orientation and ϕ W P have a significant effect were considered (i.e. C T 22 , C T 33 and C T 44 ). The variation of the elastic tensor components as a function of the distance from the implant is similar when considering variations of ϕ W P alone or when considering combined variations of ϕ W P and θ (see Fig. 5). In both those cases, the spatial mechanical responses globally follow the spatial variation of the bone porosity (see Fig. 2). At 7 weeks of healing time, the maxima of all components are located at 20 µm from the implant, while the minima are in the ROI furthest from the implant (200 µm), towards mature bone. At 13 weeks, all minima have moved to the ROI closest to the implant, and all maxima are at 40 µm from the implant. There is no variation of the elastic tensor components when only θ varies, for a given ϕ W P .

The tissue elastic tensor is shown to be mostly orthotropic with 0.1% < c tri < 1.1% and c mon = 0 (see decomposition in Fig. 6). At 7 weeks of healing time, the isotropic symmetry is almost constant, with a mean value of 91 % and a standard deviation of 2%. At 13 weeks, the isotropic symmetry shows a sharp decrease in the ROIs closest to the implant, going from 97% to 84% at 30 µm from the implant, and a nearly constant value after that region, with a mean value of 86% and a standard deviation of 3%.

Discussion

A deeper understanding of how the bone ultrastructure biomechanical properties change in the presence of an implant is essential to understand why and how an implant could later loosen and fail. This study constitutes one of the first attempts to analyze the effect of a variation in the collagen fibrils' orientation on the mechanical properties of the peri-implant bone using data obtained from a micro-focused SAXS setup and a bone chamber model. A multiscale homogenization model was applied, leading to the estimation of the homogenized anisotropic mechanical properties of the bone at the organ level. An originality of this approach is that it takes into account not only the shape of the collagen fibrils and the mesoscopic pores but also the relative orientation between them.

Collagen fibril orientation influence on bone mechanical behavior varies with the porosity

Considering the realistic bone tissue conditions studied within this work (i.e. bone porosity between 41% and 96% and collagen fibril orientation between 10 • and 50 • , see Fig. 2), the variation of all elastic tensor components is negligible when only the collagen fibril orientation changes (see Fig. 5), while outside these newly-formed bone experimental values, e.g. when considering mature cortical bone tissue with average porosity of ∼5-10% [START_REF] Gagliardi | Estimation of the effective boneelasticity tensor based on µCT imaging by a stochastic model. A multi-method validation[END_REF], the elastic tensor components are affected by the collagen fibrils orientation (see Fig. 3). As the porosity decreases, the tissue content increases and the anisotropic behavior starts to be dominated by the collagen fibril orientation. Outside these conditions, the pores, filled with water, dominate the mechanical response.

The bone mechanical behavior evolves as a function of the distance from the implant

The elastic tensor undergoes strong spatial variation within the bone chamber, especially at the late healing time point when the bone is denser. Within the first 40 µm, the values of the elastic components are increased by 3 times, which can be explained by the substantial drop in local bone porosity observed experimentally within that region (see Fig. 2D, solid lines). [START_REF] Cai | Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity[END_REF] similarly concluded that porosity explains most of the variations of the non-zero stiffness constants in mature cortical bone (2 to 20% porosity). Even when after 13 weeks bone is not mature, it can be seen that the high porosity ϕ W P variation has a strong impact on its mechanical response. At early time points, despite a larger spatial variation in collagen orientation (see Fig. 2C, dash-dotted lines), the high and stable porosity value (oscillating around 84%) explains the low variation of the tissue stiffness within the bone chamber. The estimation of bone porosity includes not only the porosity in the already formed bone (which has cylindrical shapes), but also the spaces within Fig. 5: Variation of the components of the elastic tensor of the tissue as a function of the distance from the implant at 7 weeks (left) and 14 weeks of healing time (right). ϕ W P (dashed lines) corresponds to the response of the model to the variation of bone porosity alone, with θ = 0. θ (dash-dotted lines) is the response to the variation of the collagen fibril orientation alone, for a fixed ϕ W P (ϕ W P = 84% for 7 weeks and ϕ W P = 65% for 13 weeks). ϕ W P and θ (solid lines) correspond to the combined variation of bone porosity and collagen fibril orientation. the bone chamber that the bone has not yet filled (bone is both formed and remodeled after introducing the implant). Thus, the high porosity observed at the close interface (first 40 µm) is likely to be caused not only by the bone porosity itself (cylindrical pores) but also by a low bone-implant contact ratio. This results in the mechanical response of the bone tissue at early healing times being mainly that of the pores (water-filled) and not that of the ultrastructure (composed of a mineral phase and collagen fibers). This limitation of the model is discussed in more detail below.

The nano-and micro-structural changes during healing have an impact on the mechanical behavior at higher scales As healing proceeds, the reduction in ϕ W P reflects both the formation of new bone and a decrease of porosity in the previously formed bone, as the tissue grows and densifies [START_REF] Cann | Spatio-temporal evolution of hydroxyapatite crystal thickness at the bone-implant interface[END_REF]. At the nanoscale, the collagen fibrils tend to align better with the implant surface. Mechanically, the alignment of the collagen fibrils leads to a global increase in the elastic tensor component values in the reference frame R(O; e 1 , e 2 , e 3 ). Moreover, the hexagonal symmetry of the elastic tensor increases with healing (see Fig. 6). An elastic medium with hexagonal symmetry is equivalent to a transversely isotropic medium [START_REF] Browaeys | Decomposition of the elastic tensor and geophysical applications[END_REF]. This equivalence suggests that at early healing stages, a combination of a higher porosity and a stronger misalignment between the collagen fibrils and the vascular porosity (see Fig. 2) results in the bone tissue being isotropic rather than orthotropic, which is the expected mechanical behavior in mature bone [START_REF] Cowin | Bone Mechanics Handbook[END_REF].

Limitations

Limits on experimental data. This study is based on 2D experimental data from 8 boneimplant specimens (n = 4 per time point) which represent a limited dataset. However, this data still represents the largest and most refined SAXS study to investigate the spatial evolution of the bone ultrastructure at the bone-implant interface. Moreover, as only mineral crystals diffracting in the beam direction contributed to the average orientation measured here, the analysis is currently limited to the analyzed radial direction and needs to be extended to other cutting directions in the future, particularly along the longitudinal direction, where bone presents a higher degree of orientation [START_REF] Törnquist | Mineralization of cortical bone during maturation and growth in rabbits[END_REF]. Effect of bone mineralization Bone mineralization is not included in the analysis and would be needed to clarify the mechanical response close to the implant. It was not possible to measure the degree of mineralization of the samples used in this work, and a constant value was assumed for all samples and healing times. Xray micro-tomography is frequently used to obtain the degree of tissue mineralization as the grey levels obtained in the reconstructed images are linked to the local content of minerals. However, such a technique is difficult to conduct close to metallic implants due to frequent imaging artifacts [START_REF] Barrett | Artifacts in ct: Recognition and avoidance[END_REF][START_REF] Törnquist | Dual modality neutron and x-ray tomography for enhanced image analysis of the bonemetal interface[END_REF]. Previous numerical work has shown an increase in bone tissue stiffness with the increase of the mineralization on cortical bone [START_REF] Sansalone | Determination of the heterogeneous anisotropic elastic properties of human femoral bone: From nanoscopic to organ scale[END_REF]. Thus, since bone formation and remodeling close to implants leads to significant changes in the degree of mineralization (in particular reduction) [START_REF] Allen | Chapter 4bone modeling and remodeling[END_REF] the assumption made in this work can lead to an overestimation of the tissue stiffness.

Limitation of the model. The porosity was modeled by cylindrical canals parallel to the implant surface. However, during bone healing, the pore shape may differ from Haversian canals and may be composed of large cavities filled with marrow. These cavities have irregular form and their mechanical effects might not be suitably described by cylindrical pores. As in the case of the degree of mineralization, accounting for the real shape of the pores requires a much more detailed morphological analysis using X-ray micro-tomography images that should be addressed in further studies.

Conclusions

A multiscale homogenization model was proposed to compute the effective elastic properties of newly-formed bone (macroscale) as a function of the distance from the implant, based on experimentally obtained tissue structure and composition at lower scales. This method can be used to replace the complex structure of the peri-implant bone for a continuum model to analyze the spatiotemporal evolution of bone during the healing time. In this work, the effects of the porosity and the collagen fibrils orientation variations on the bone anisotropic properties in the proximity of the implant were investigated.

The findings revealed a strong variation of the components of the effective elasticity tensor of the bone tissue as a function of the distance from the implant. The effective elasticity is primarily sensitive to the porosity (mesoscale) rather than to the collagen fibrils' orientation (submicroscale). However, the orientation of the fibrils has a significant influence on the symmetry properties of the effective elasticity tensor. A decrease in porosity leads to a decrease in bone isotropy and an increase in the impact of the fibrils' orientation.

These results demonstrate that the collagen fibril orientation affects the effective elastic properties of the bone throughout the remodeling process in the proximity of an implant. Collagen fibril orientation should be taken into account to properly describe the effective elastic anisotropy of bone tissue at the organ scale.

Fig. 2 :

 2 Fig. 2: Sectional scheme of the standardized implant model used in (Le Cann et al, 2020) (A). After osseointegration of 7 weeks and 13 weeks, specimens are explanted and analyzed with SAXS (B) to extract the spatial variation of bone porosity ϕ W P (solid lines) and collagen fibril orientation θ (dashed lines) averaged in 10 µm thick bands parallel to the implant surface (C, D).

Fig. 3 :

 3 Fig.3: Variation of the components of the elastic tensor of the tissue at the macro scale as a function of the collagen fibrils orientation θ for three different values of bone porosity ϕ W P : 8% (mature cortical bone), 65%, and 84% (bone during healing time) .

Fig. 4 :

 4 Fig. 4: Variation of the components of the elastic tensor of the tissue at the macro scale as a function of bone porosity ϕ W P for three different values of the collagen fibril orientation θ: 0 • , 45 • , and 90 •

Fig. 6 :

 6 Fig. 6: Symmetry class decomposition of the elastic tensors as a function of the distance from the implant.

  :

	P r 11 = P r 22 =P r 33 = P r 1111 =
	2C HF 12 + 7C HF 44 15C HF 44 (C HF 12 + 2C HF 44 )	,
	P r 12 = P r 13 =P r 23 = P r 1122 =
	-	C HF 12 + C HF 44 15C HF 44 (C HF 12 + 2C HF 44 )	,
	P r 44 = P r 55 =P r 66 = 4P r 2323 =
		2(3C HF 12 + 8C HF 44 )
	15C HF 44 (C HF 12 + 2C HF 44 )

Table 1 :

 1 ). Elastic properties of the elementary constituents

	Phase		Elastic proprieties given in GPa	ϕ
		C 11	C 33	C 12	C 13	C 44
	Water	2.3	2.3	2.3	2.3	0	0.31
	Hydroxyapatite	142.5	142.5	52.7	52.7	44.9	0.33
	Collagen	11.7	17.9	5.1	7.1	3.3	0.36

Acknowledgements

This work has been supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 101027554. This project has received funding from the project OrthAncil (ANR-21-CE19-0035-03) and from the project OrthoMat (ANR-21-CE17-0004).

Availability of data and materials

The data that support the findings of this study are available from the corresponding author upon request.

Declarations Funding

No funding was received to assist with the preparation of this manuscript.

Competing interests

The authors declare no competing interests.

Conflict of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Ethics approval


J Milanese
email: julien.milanese@ensiacet.fr

E Andrieu

J B Osio

J Alexis

D Bardel

⋅ D Bardel

Study of the oxidation assisted intergranular cracking mechanism on a Ni-Base superalloy

Keywords:

In order to collect accurate information about the widely reported Oxidation Assisted Intergranular Cracking (OAIC) mechanism of the superalloy 718 at 650 °C in air environment, we investigated a new tensile test procedure which enables to have access to a quantitative semi-continuous assessment of the damaging process of the alloy 718 during a standard tensile test. Tensile tests were carried out on a solutionized and aged alloy 718 by varying the deformation mode of the alloy during the ongoing experiment. The semi-continuous quantification of the intergranular damage was performed after a fracture surface analysis of samples tested in both Dynamic Strain Ageing (DSA) domain and DSA-subdomain where Portevin-Le Chatelier (PLC) effect occurs. This innovative testing method allows to characterize the intergranular damage that occurs when the material is stressed with an unserrated DSA deformation mode as the alloy 718 is only sensitive to OAIC in this domain. In these conditions, the onset of the intergranular cracking process was found to be around 10% of total strain. An increase in the cracking kinetic when necking occurs was also noticed. A close connection between the intensity of the intergranular damage and the cumulated strain in DSA deformation mode before the onset of the damaging process was found. The gathered information represents a significant improvement in the understanding of still debated OAIC or Stress Corrosion Cracking (SCC) mechanisms.

Introduction

The superalloy 718 is well known for possessing good mechanical properties and corrosion/oxidation resistance over a wide range of temperatures [START_REF]INCONEL alloy[END_REF][START_REF] Ter-Ovanessian | Quantitative assessment of intergranular damage due to PWR primary water exposure in structural Ni-based alloys[END_REF][START_REF] Miglin | Microstructure and stress corrosion resistance of alloys X750, 718, and A286 in light water reactor environments[END_REF]. However, the alloy 718 is widely reported to be subject to an intergranular brittle fracture when loaded in particularly harsh environments [START_REF] Kimmerle | Stress corrosion cracking of alloy 718 in pressurized-water-reactor primary water[END_REF][START_REF] Leonard | Analysis of stress corrosion cracking in alloy 718 following commercial reactor exposure[END_REF][START_REF] Rezende | Oxidation assisted intergranular cracking under loading at dynamic strain aging temperatures in Inconel 718 superalloy[END_REF]. The effect of the environment is essential in the damaging process of the superalloy as no intergranular brittle fracture was observed on samples tested under vacuum or inert gas [START_REF] Garat | Influence of Portevin-Le Chatelier effect on rupture mode of alloy 718 specimens[END_REF][START_REF] Andrieu | Oxidation assisted intergranular cracking in alloy 718: effects of strain rate and temperature[END_REF]. Whether in the water of Pressurized Water Reactors (PWRs) or in air at 650 °C, the resulting intergranular damage of the alloy (assimilated to an InterGranular Stress Corrosion Cracking "IGSCC" mechanism on one side and to an Oxidation Assisted Intergranular Cracking "OAIC" mechanism on the other side) depends on the deformation mode of the alloy. The transition from an unserrated flow typically observed in the Dynamic Strain Aging (DSA) domain to a serrated flow observed when the Portevin-Le Chatelier (PLC) effect occurs in a DSA subdomain lead to a transition in the fracture mode from intergranular fragile to fully transgranular ductile [START_REF] Garat | Influence of Portevin-Le Chatelier effect on rupture mode of alloy 718 specimens[END_REF][START_REF] Andrieu | Oxidation assisted intergranular cracking in alloy 718: effects of strain rate and temperature[END_REF][START_REF] Max | A re-examination of the Portevin-Le Chatelier effect in alloy 718 in connection with oxidation-assisted intergranular cracking[END_REF]. Even if several still-debated mechanisms for the IGSCC or OAIC of the alloy 718 have been reported in literature [START_REF] Andrieu | Intergranular crack tip oxidation mechanism in a nickel-based superalloy[END_REF][START_REF] Gao | Niobium enrichment and environmental enhancement of creep crack growth in Nickel-base superalloys[END_REF][START_REF] Molins | Oxidation assisted crack propagation of alloy 718[END_REF][START_REF] Miller | High temperature oxidation of Nb, NbC and Ni 3 Nb and oxygen enhanced crack growth[END_REF][START_REF] Miller | Mechanism for oxygen enhanced crack growth in Inconel 718[END_REF][START_REF] Kameda | High temperature brittle intergranular cracking in high strength alloys undoped and doped with S, Zr, and/or B-II. Solute segregation analysis[END_REF][START_REF] Bika | A kinetic model for dynamic embrittlement[END_REF][START_REF] Bika | Sulfur-induced dynamic embrittlement in a low-alloy steel[END_REF][START_REF] Heuer | Disorder-induced melting in nickel: implication to intergranular sulfur embrittlement[END_REF], the particular flow induced by the plastic instabilities and the resultant modification of the interaction between dislocations and solutes inhibits the damaging mechanisms of the alloy in which a preferential loading of grain boundaries or segregating elements at grain boundaries are possibly involved [START_REF] Garat | Influence of Portevin-Le Chatelier effect on rupture mode of alloy 718 specimens[END_REF][START_REF] Andrieu | Oxidation assisted intergranular cracking in alloy 718: effects of strain rate and temperature[END_REF][START_REF] Max | A re-examination of the Portevin-Le Chatelier effect in alloy 718 in connection with oxidation-assisted intergranular cracking[END_REF][START_REF] Deleume | Facteurs métallurgiques et mécaniques contrôlant l'amorçage de défauts de corrosion sous contrainte dans l'alliage 718 en milieu primaire des réacteurs à eau sous pression[END_REF].

The analysis of fractured samples after conventional tensile tests at high temperature in air environment is used to characterized and quantify the intergranular damage of the superalloy 718. However, only one information about the total intergranular damage at the breakage of the material can be deducted from this assessment. No information on the damaging process, as its onset or its development during the mechanical test, is available.

The objective of this study was to determine the occurrence and the evolution of the intergranular cracking of a superalloy 718 due to an OAIC mechanism during a tensile test at 650 °C in air environment by means of an innovative method. Tensile tests during which the deformation mode of the superalloy 718 is modified from a DSA mode to a PLC mode at a given strain level and vice versa were carried out. The intergranular damage which exclusively occurs over the section of the tensile curve performed in DSA can be therefore determined by a standard fractography analysis of the tensile samples. Thus, a semi-continuous assessment of the intergranular damaging process of the superalloy 718 at 650 °C in air environment was achieved.

Experimental Procedure

The material used for this research was a 0.260 mm thick nickel base superalloy 718 strip whose chemical composition is given in Table 1.

The alloy was fully solutionized at 1080 °C during 48 s and conventionally aged as follows: 720 °C for 8 h, cooled to 620 °C at 50 °C/h, 620 °C for 8 h then furnace cooled. From the manufacturing route described above, only remains a δ-free γ′/γ″-hardened fully recrystallized solid solution with an average grain size of about 13 µm. As shown in Fig. 1, carbides alignments in the rolling direction and dispersed isolated primary carbides which cannot be redissolved at these temperatures are still observed in the proportions of ∼0.46 surf.%.

Tensile tests were realized under laboratory air testing conditions at a temperature of 650 °C with a strain rate controlled by the travel speed of the testing device. The mechanical testing procedure is automatically defined and allows to change the strain rate at a given deformation level during the ongoing test. Strain rates of 1.10 -3 s -1 and 1.10 -2 s -1 were used as they respectively result in a conventional DSA deformation mode and in a PLC deformation mode for the used material at 650 °C. Immediately prior to the test, samples were hold at 650 °C for 10 min in order to homogenize the sample temperature and allow oxidation phenomena to occur. An exhaustive list of the carried out tests is given in Table 2.

Fractured tensile samples were then observed by secondary electron microscopy using a FEI INSPECT S50 microscope.

Results

Constant Strain Rate Tensile Tests

Stress-strain curves performed at a constant strain rate during the whole test are provided in Fig. 2. As expected, the selected strain rates of 1.10 -3 s -1 and 1.10 -2 s -1 lead respectively to a continuous and serrated flow of the material. A decrease of about 20 MPa in the tensile strength and of 3% in the ultimate strain is observed when the PLC effect occurs.

A mixed transgranular ductile and intergranular fragile rupture mode is observed on the samples strained at 1.10 -3 s -1 (Fig. 3b,c) while the fracture mode of the tensile samples strained at 1.10 -2 s -1 in the PLC subdomain is fully transgranular ductile (Fig. 3a). As reported in literature [START_REF] Garat | Influence of Portevin-Le Chatelier effect on rupture mode of alloy 718 specimens[END_REF][START_REF] Andrieu | Oxidation assisted intergranular cracking in alloy 718: effects of strain rate and temperature[END_REF][START_REF] Max | A re-examination of the Portevin-Le Chatelier effect in alloy 718 in connection with oxidation-assisted intergranular cracking[END_REF], the effects of the combined environment and mechanical load on the damaging process of the superalloy 718 are enabled when the flow is unserrated whereas they are suppressed when the plastic instabilities related to the PLC effect occur.

Two Strain Rate Tensile Tests

Curves of tensile tests realized by switching from DSA to PLC deformation modes and vice versa during the ongoing test are presented in Fig. 4. It is important to note that the second part of the two strain rate test curves (black curves in Fig. 4) fits well with the standard tensile test curves (grey curves in Fig. 4). Despite an important quantity of deformation cumulated under a different deformation mode, the material flow stress in the second part of the test is the same as the one of a constant strain rate tests. As the tensile test mechanical history does not affect the material flow stress, it is thus, possible to estimate that the damaging mechanisms noted above stay identical irrespective of the testing procedure. Therefore, the quantification of the intergranular damage occurring over the only section of the curve realized with a DSA deformation mode is possible.

Quantification of the Intergranular Damage

The intensity of the damage was evaluated by means of a fractograph analysis. As detailed in Fig. 5, an Oxidation Assisted Intergranular Cracking Index (OAICI) is calculated as the ratio of the sum of the edging lengths of the intergranular fracture zones by the total perimeter of the analyzed specimen. Only is taken into account the intergranular fracture zones whose depth is strictly greater than one grain. The Table 3 shows the results of the OAICI determinations. Whatever the test sequence was, the intensity of the intergranular damage seems to be directly related to the deformation cumulated with an unserrated flow of the material (i.e. DSA deformation mode).

From DSA to PLC tests, an increase from 0 to 16.5% of the total strain cumulated in DSA mode leads to an increase in the intergranular sensitivity of the material from 0 to 9.0%. The same conclusion can be drawn from PLC to DSA tests as an increase from 0 to 12.5% in the cumulated strain in PLC mode (i.e. a decrease in the strain level cumulated in DSA mode for samples with a similar ultimate strain) leads to a decrease from 8.0 to 0% of the OAICI. Furthermore, tests switching from DSA to PLC show that no intergranular damage is generated before 12.5% of total strain.

Discussion

The Fig. 6a shows the evolution of the intergranular damage intensity calculated from the DSA to PLC tensile tests as a function of the total strain. Values are replaced over the standard DSA stress/strain curve of the studied alloy 718 at 650 °C in air environment. The onset of the intergranular damage occurs between 10 and 12.5% of total strain. It also corresponds to the strain when the material strength is maximal. The perceived maximal strength and necking on the stress/ strain curve of the material at 650 °C in air environment seems to be directly related to the appearance of the intergranular damage. It can potentially be considered that the decrease in the load-bearing cross section which occurs when intergranular cracks form is accountable to the perceived flow of the material. As they are widely influenced by the damaging effect of the environment, the noted material strength and elongation from a tensile test carried out at 650 °C in air environment do not correspond to the inherent mechanical properties of the alloy 718 but only reflect the particular behavior of the material in a particular environment. An explosion of the OAICI at the very end of the tensile test/necking is also observed. The OAICI increases from 1% to around 8.5% as the total strain goes from 15% to 18.5%. The intergranular cracking of the alloy 718 is a very rapid mechanism as the majority of the intergranular damage is generated in about 35 s.

The average intergranular cracking propagation speed was calculated at 0.5 µm/s. The Fig. 6b shows the evolution of the OAICI as a function of the cumulated strain in DSA mode calculated from the PLC to DSA tensile tests. As mentioned before, the intensity of the intergranular damage increase with the cumulated strain in DSA mode.

The Fig. 7 shows a standard DSA mode tensile test curve and several tensile tests carried out with a transition from a PLC deformation mode to a DSA deformation mode at various total strains. It was observed before that the intergranular damage occurs and develop after 10% of total strain. However, in the case of PLC to DSA tensile tests, no differences from one test curve to another is noticeable beyond 10% of total strain whereas the intensity of the intergranular damage generated during the same period decreases. This change in the intergranular damaging process can therefore be attributed to the quantity of deformation cumulated in DSA mode before 10% of total strain as longer this part of the curve is, greater the OAICI values are. During this period, deformation induced solute segregation and preferential loading of grain boundaries whose intensities depend on the deformation cumulated in DSA mode can potentially be implemented. Once the cohesion stress of grain boundaries is reached intergranular cracking occurs and only loaded and segregated grain boundary interfaces fail. Thus, the greater the strain cumulated before the onset of the intergranular cracking, the larger the weakened grain boundaries and the OAICI value. The time spent in DSA deformation mode before the onset of intergranular cracks can be assimilated to the "incubation period" of the intergranular cracking mechanism during which solute segregation or preferential loading at grain boundaries may potentially occur and express on the fracture surface of the material.

Conclusions

In order to completely grasp the intergranular cracking process of the alloy 718 during a tensile test at 650 °C in air environment, the following conclusions can be gleaned from this work:

(1) A certain amount of deformation cumulated in the material in DSA deformation mode is needed for the intergranular damage of the superalloy 718 at 650 °C in air environment to occur. In this study, the onset of the intergranular cracking was found to be around 10% of total strain and affects the mechanical properties of the alloy. (2) The intergranular cracking process of the alloy 718 is accelerating at the end of the tensile test and leads to the breakage of the material. (3) The intensity of the intergranular damage is directly correlated to the cumulated strain in DSA mode before the onset of the material cracking during which preferential loading or solute segregation at grain boundaries can express their negative effect. However, additional work on this specific point needs to be carried out in order to confirm these assumptions.

Fig. 1 3 Fig. 2

 132 Fig. 1 Microstructure of the superalloy 718 used in this study. a Optical micrograph of an etched sample (RD: rolling direction), b SEM micrograph showing carbides

Fig. 3 aFig. 4

 34 Fig. 3 a Fully transgranular ductile fracture of the sample strained at 1.10 -2 s -1 in PLC mode at 650 °C in air environment, b mixed transgranular ductile and intergranular fragile fracture mode of the sample strained at 1.10 -3 s -1 at 650 °C in air environment, c magnification of the area enclosed in white in picture (b)

Fig. 5

 5 Fig.[START_REF] Leonard | Analysis of stress corrosion cracking in alloy 718 following commercial reactor exposure[END_REF] Detailed diagram of the OAICI determination method. Where L IG is the edging length of a considered intergranular fracture zone greater than one grain (displayed as the arrows on the diagram) and L Tot the total perimeter of the tensile sample rupture surface

Fig. 6 a

 6 Fig. 6 a Evolution of the IGSSCI values versus strain and standard stress/strain curve of the experimented alloy 718 at 650 °C in air environment (from the DSA to PLC tensile test results). b Evolution of the IGSSCI values versus cumulated strain in DSA mode (from the PLC to DSA tensile test results)

Fig. 7

 7 Fig. 7 Tensile curves of a full DSA mode tensile test and PLC to DSA tests with transition strains of 2.5, 5 and 10% at 650 °C in air environment (stress values are shifted for a clearer view)

Table 1

 1 Chemical composition in wt% of the superalloy 718 heat used in this study

	Ni	Fe	Cr	Nb	Mo	Ti	Al	Mn
	52.1	Bal.	18.3	5.14	2.91	1.02	0.54	0.06
	Ta	Si	Co	Cu	C	S	B	P
	0.01	0.06	0.02	0.01	0.05	0.0002	0.003	0.008

Table 3

 3 Calculated OAICI values for the realized tensile tests

	Deformation mode	Transition strain (%)	Deformation mode	OAICI (%)
	DSA			8.0
	DSA	2.5	PLC	0
	DSA	5	PLC	0
	DSA	10	PLC	0
	DSA	12.5	PLC	0.3
	DSA	15	PLC	1.1
	DSA	16.5	PLC	9.0
	PLC			0
	PLC	2.5	DSA	7.6
	PLC	5	DSA	4.3
	PLC	10	DSA	2.9
	PLC	12.5	DSA	0

Acknowledgements

The financial support of this research project was provided by AREVA NP. Thanks are addressed to MIDIVAL for the provided welcome on its material testing site within this work.