domain: spi sdv

text:
Karim Kandil

Sid Ali Kaoua

Amar Mesbah

Yuri Voznyak

Fahmi Zaïri

Fahed Zaïri

A novel bio-inspired hydrogel-based lattice structure to mechanically mimic human disc annulus: A finite element study

Keywords: Artificial annulus fibrosus, Lattice structure, Hydrogel, Transversal behavior, Regional dependence

Intervertebral disc hernia and dysfunctions are the most common health problems that affect humans. For this reason, the need of disc prosthetics, and especially annulus fibrosus replacements, is becoming increasingly mandatory and urgent. In the present paper, we present a novel bio-inspired hydrogel-based replacement system with rational mechanically mimetic designs of human annulus fibrosus. The new system consists of hydrogel-based lattice structure with octagonal cells separated by polylactide fibers reinforced hydrogel-based sheets. The mechanics of the new designs may be rationally controlled by tailoring microstructure, in terms of polylactide fibers orientation and hydrogel intrinsic viscosity, and mesostructure in terms of cell walls/sheets thickness. The new designs are successfully compared to the natural annulus fibrosus mechanics in terms of nonlinear stiffness and transversal behavior while considering regional dependency. The impressive biomimetic capabilities of the proposed hydrogel-based lattice structure allow to foresee the possibility of designing personalized disc prosthesis by advanced 3D printing technologies.

Introduction

Intervertebral disc hernia and dysfunctions are the most common health problems that affect both young and old human bodies. Although the intervertebral disc replacements design and prosthetics production have known a remarkable evolution over recent years [START_REF] Galbusera | Biomechanical studies on cervical total disc arthroplasty: a literature review[END_REF][START_REF] Amoretti | Imaging of intervertebral disc prostheses[END_REF][START_REF] Jiang | Crystallinity dependency of the time-dependent mechanical response of polyethylene: application in total disc replacement[END_REF][START_REF] Jiang | Biomechanical response of a novel intervertebral disc prosthesis using functionally graded polymers: a finite element study[END_REF][START_REF] Jacobs | Viscoelastic cervical total disc replacement devices: Design concepts[END_REF], discectomy and spinal fusional surgeries remain the most commonly preferred and used by surgeons [START_REF] Vishteh | Anterior lumbar microdiscectomy and interbody fusion for the treatment of recurrent disc herniation[END_REF][START_REF] Van Ooij | Complications of artificial disc replacement: a report of 27 patients with the SB Charite disc[END_REF][START_REF] Zechmeister | Artificial total disc replacement versus fusion for the cervical spine: a systematic review[END_REF][START_REF] Lopez | Recent trends in medicare utilization and reimbursement for lumbar spine fusion and discectomy procedures[END_REF]. Many generations of replacements have been developed. Firstly, metal or metal and polymers articulating ball and socket prosthetics appeared trying to ensure relative movements between the vertebrae regardless the real local stress-strain and local strain response of the prosthetic. They are often constrained with about three degrees of freedom and gave similar clinical outcomes of fusion surgeries without any significant progression.

After, many prosthetics appeared trying to add time-dependent energy storing properties by using viscoelastic core separating two parallel highly stiff endplates that connect the adjacent vertebrae [START_REF] Jacobs | Viscoelastic cervical total disc replacement devices: Design concepts[END_REF]. The last could guarantee six degrees of freedom to the prosthetic, mimic the damping features of the natural disc and enhance a global strain of the prosthetic under the different physiological movements. The new replacement generations use different concepts regarding the fixation method with the adjacent vertebrae as well as the core design and material. Although fixation method between prosthetic/vertebrae and the core/endplates inside the prosthetic is a main point that differentiate the existing disc implants, the core design and material remain the most important points characterizing their response and allowing them to reproduce the same behavior of the natural disc. Many concepts such in case of the CP ESP prosthetic tried to reproduce the correct disc response by designing an optimized core able to replicate the kinematics of the real disc [START_REF] Lazennec | The innovative viscoelastic CP ESP cervical disk prosthesis with six degrees of freedom: biomechanical concepts, development program and preliminary clinical experience[END_REF]. Other prosthetics construct a core inspired from the natural disc structure such as M6-C [START_REF] Patwardhan | Primary and coupled motions after cervical total disc replacement using a compressible six-degree-of-freedom prosthesis[END_REF][START_REF] Thomas | The M6-C cervical disk prosthesis: first clinical experience in 33 patients[END_REF]. Similar to the real disc, M6-C core is composed of two distinct main parts in the heart of the prosthetic core we can find a compressible polycarbonate urethane polymer replacing the nucleus that is surrounded by ultra-high molecular-weight-polyethylene woven fibers replacing the disc annulus. This type of natural disc structure-based prosthetics demonstrates very promising clinical results [START_REF] Thomas | The M6-C cervical disk prosthesis: first clinical experience in 33 patients[END_REF][START_REF] Patwardhan | Prosthesis design influences segmental contribution to total cervical motion after cervical disc arthroplasty[END_REF]. However, the disc annulus fibrosus remains more complex, with more delicate features that are not considered in replacement modeling and manufacturing until now and that could affect the spinal mobility, recovery and long-term durability after surgery.

The intervertebral disc annulus is known to be highly heterogeneous with regional dependent microstructure/properties relationships [START_REF] Eyre | Types I and II collagens in intervertebral disc. Interchanging radial distributions in annulus fibrosus[END_REF][START_REF] Eyre | Quantitative analysis of types I and II collagens in the human intervertebral discs at various ages[END_REF][START_REF] Skaggs | Regional variation in tensile properties and biochemical composition of the human lumbar anulus fibrosus[END_REF][START_REF] Ebara | Tensile properties of nondegenerate human lumbar anulus fibrosus[END_REF][START_REF] Holzapfel | Single lamellar mechanics of the human lumbar[END_REF]. The annulus mechanics changes between the different radial and circumferential regions of the same disc due to the different local quantities of the constituent elements (collagen fibers, proteoglycans, water) and the collagen organization along with interactions with the surrounding physiological medium [START_REF] Eyre | Types I and II collagens in intervertebral disc. Interchanging radial distributions in annulus fibrosus[END_REF][START_REF] Eyre | Quantitative analysis of types I and II collagens in the human intervertebral discs at various ages[END_REF][START_REF] Holzapfel | Single lamellar mechanics of the human lumbar[END_REF].

Especially, the transversal strains have been proved recently to show unusual behavior with negative Poisson's ratios in the disc radial direction and Poisson's ratios superior to 0.5 in the disc axial direction under a circumferential loading of the annulus fibrosus [START_REF] Baldit | Experimental analysis of the transverse mechanical behaviour of annulus fibrosus tissue[END_REF][START_REF] Derrouiche | A chemo-mechanical model for osmo-inelastic effects in the annulus fibrosus[END_REF][START_REF] Derrouiche | The two Poisson's ratios in annulus fibrosus: relation with the osmo-inelastic features[END_REF][START_REF] Dusfour | Heterogeneous mechanical hyperelastic behavior in the porcine annulus fibrosus explained by fiber orientation: an experimental and numerical approach[END_REF]. Those unusual transversal strains increase the damping characteristic of the disc under sudden and rough loads and their consideration in a replacement device could highly enhance its performance and efficiency. Many replacement systems have been proposed in the literature trying to replicate the annulus fibrosus response materials and geometries inspired from the real annulus micro and macro structures [START_REF] Nerurkar | Nanofibrous biologic laminates replicate the form and function of the annulus fibrosus[END_REF][START_REF] Du | Engineering a biomimetic integrated scaffold for intervertebral disc replacement[END_REF][START_REF] Frost | Replication of annulus fibrosus through fabrication and characterization of polyurethane and cellulose nanocrystal composite scaffolds[END_REF][START_REF] Bhunia | 3D printing of annulus fibrosus anatomical equivalents recapitulating angle-ply architecture for intervertebral disc replacement[END_REF]. In the most recent research contributions highlighted from the literature we can see that Frost and Foster [START_REF] Frost | Replication of annulus fibrosus through fabrication and characterization of polyurethane and cellulose nanocrystal composite scaffolds[END_REF] were inspired from the lamellar structure and the composite composition of the annulus fibrosus to construct an annulus fibrosus scaffold from polyurethane reinforced with cellulose nanocrystals fabricated into ribbons and radially layered structure. In Nerurkar et al. [START_REF] Nerurkar | Nanofibrous biologic laminates replicate the form and function of the annulus fibrosus[END_REF] nanofibrous biologic laminates were suggested to replicate the form and function of the annulus fibrosus. More recently, in Ref [START_REF] Bhunia | 3D printing of annulus fibrosus anatomical equivalents recapitulating angle-ply architecture for intervertebral disc replacement[END_REF] a cross-aligned 3D printable structure was designed to mimic the oriented fibers and the angle-ply architecture of the annulus fibrosus. However, all these contributions tried to replicate mechanically the rigidity and the stress-strain response of the annulus fibrous but neglected the effect of the transversal strains and the unusual Poisson's ratios of the soft tissue which will highly affect the reproduced mechanical behavior. In this study, a novel bio-inspired hydrogel-based substitute is designed in order to mimic the accurate local mechanical response of the disc. The new substitute should be able not only to allow reproducing the complete disc stiffness but permit also controlling and varying locally this response in the different regions of the annulus and replicating accurately the unusual transversal local response of each disc region enhancing its damping and energy storage properties. The substitute design is based and inspired from the accurate microstructure and the most recent findings about the disc structure [20-22; 27-32], using most advanced biocompatible 3D printable materials such as hydrogel and polylactide [START_REF] Bakarich | Threedimensional printing fiber reinforced hydrogel composites[END_REF][START_REF] Bakarich | 3D/4D printing hydrogel composites: a pathway to functional devices[END_REF][START_REF] Bakarich | 3D printing of tough hydrogel composites with spatially varying materials properties[END_REF][START_REF] Wang | 3D printing of polymer matrix composites: a review and prospective[END_REF][START_REF] Herman | Elastically anisotropic phantoms constructed from 3D-printed PLA fibers[END_REF]. The current contribution represents a first step of a very promising research point of interest about biocompatible composites that could mimic the unusual transversal and volumetric strains of the soft tissues. This would guarantee not only the same mechanical resistance of the real soft tissues as usually done but also the same natural feeling, damping movement and flexibility.

Material and methods

Replacement structure

The proposed design is inspired from the hierarchical structure of the natural annulus from the material scale to the lamellar/interlamellar arrangement scale. A special combination of lattice structure with octagonal cells patterns, separated by oriented fibers reinforced thin parallel sheets, is proposed as shown in Figure 1. The 3D octagonal cells shape is especially designed in order to replicate the 3D response of the natural annulus fibrosus. The special unit cell structure illustrated in Figure 2 allows an auxetic behavior (negative Poisson's ratio) in the Z direction and a standard non-auxetic behavior in the Y direction under stretching in the X direction (equivalent to the circumferential direction of the complete disc). Note that X, Y and Z directions correspond to the circumferential, radial, axial directions of the disc. The compression of the intervertebral disc (which is the most common loading during daily activities due to body and lifted weights) leads to a tension of the annulus in the circumferential direction X of the disc. The parallel sheets give to the structure the required rigidity and govern the transversal strain in the disc axial direction. This is inspired from the real disc structural response under circumferential tension where the rigidity and the transversal behavior are principally governed by the annulus lamellae and the radial strain is mainly controlled by the interlamellar ground substance zone [20-21, 29-32, 38-39].

Two common 3D printable biocompatible materials are selected in the current replacement system as illustrated in Figure 1 for their similarities to natural tissues. Cross-linked hydrogel is selected to act as a replacement of the isotropic proteoglycan extracellular matrix. This material has excellent performance in absorption and energy storage features. The anisotropic type I collagen fibers are replaced by polylactide. This bio-polymer is used as a potential reinforcing material in recent biomechanical applications [START_REF] Critchley | 3D printing of fibre-reinforced cartilaginous tem-plates for the regeneration of osteochondral defects[END_REF]. It will be used in the form of fibers for which the performance can be tailored by deformation-induced phase transformation in the vicinity of the glass transition [START_REF] Mahjoubi | A micro-macro constitutive model for strain-induced molecular ordering in biopolymers: application to polylactide over a wide range of temperatures[END_REF][START_REF] Mahjoubi | Strain-induced phase transformation in poly (lactic acid) across the glass transition: Constitutive model and identification[END_REF]. The 3D printing technologies of hydrogel and polylactide fibers [START_REF] Bakarich | Threedimensional printing fiber reinforced hydrogel composites[END_REF][START_REF] Bakarich | 3D/4D printing hydrogel composites: a pathway to functional devices[END_REF][START_REF] Bakarich | 3D printing of tough hydrogel composites with spatially varying materials properties[END_REF][START_REF] Wang | 3D printing of polymer matrix composites: a review and prospective[END_REF][START_REF] Herman | Elastically anisotropic phantoms constructed from 3D-printed PLA fibers[END_REF] have highly progressed and have become very common through the recent years. Currently, they are considered as a main pathway to functional biomedical devices, and they started to be widely used in many biomedical applications such as cartilage replacements [START_REF] Bakarich | Threedimensional printing fiber reinforced hydrogel composites[END_REF][START_REF] Bakarich | 3D/4D printing hydrogel composites: a pathway to functional devices[END_REF].

Finite element model and boundary conditions

A sample of the octagonal cells replacement system of 25 mm × 10 mm × 10 mm was designed following the same specimen size defined for natural annulus specimens by Derrouiche et al. [START_REF] Derrouiche | A chemo-mechanical model for osmo-inelastic effects in the annulus fibrosus[END_REF]. The sample size was chosen based on the physiological dimensions of the disc and previous simulations established on annulus fibrosus circumferential specimens 1 .

The geometry was implemented in MSC.Marc software and then meshed using isoparametric and arbitrarily hexahedrics 8-node meshing elements. The number of mesh elements was between 25000 and 330000 depending on the thickness of the sample. The average size of elements is 0.4 mm with refinements near contact areas between parallel the parallel sheets and the octogonal cells. An adequate element size was chosen to ensure mesh-independency.

In order to reproduce the same experimental boundary conditions of the literature [START_REF] Ebara | Tensile properties of nondegenerate human lumbar anulus fibrosus[END_REF][START_REF] Baldit | Experimental analysis of the transverse mechanical behaviour of annulus fibrosus tissue[END_REF][START_REF] Derrouiche | The two Poisson's ratios in annulus fibrosus: relation with the osmo-inelastic features[END_REF], the specimen is totally fixed from one side and a 10% circumferential strain is applied from the other side at a stretch rate of 4.2 min -1 . An illustrative example is provided in Figure 3.

Different mesostructure configurations of the replacement system are considered as shown in Figure 3. Their characteristics are listed in Table 1. In a first configuration set, the cell walls thickness is varied while maintaining constant the sheets thickness. In a second configuration set, the parallel sheets thickness is varied while maintaining constant the cell walls thickness.

Microstructure-based material model

Kinematics

Before to present the hydrogel constitutive equations, the kinematics is briefly addressed 2 . Let us consider the hydrogel medium as a homogeneous continuum body. The mapping of the position vector X of a given material point in the initial configuration to the position vector ( )

, t = x φ X
in the current configuration is given by the deformation gradient tensor

X = ∇ F φ
. The time derivative of the deformation gradient tensor F is given by: = & F LF in which L is the gradient tensor of the spatial velocity

t = ∂ ∂ v ϕ : x = ∇ L v .
Introducing the notion of the intermediate configuration, the deformation gradient tensor iso vol = F F F can be multiplicatively split into an isochoric part iso F and a volumetric part vol F :

1/3 iso J - = F F and 1/3 vol J = F I (1)
in which I is the unit tensor and J is the Jacobian, i.e. the determinant of the deformation gradient tensor F .

The corresponding kinematic rate can be additively decomposed as:

1 1 1
iso vol iso iso iso vol vol iso

- - - = + L L L F F F F F F & & 123 14 4244 3 (2)
in which iso L and vol L are the isochoric and volumetric parts of the spatial velocity gradient tensor L .

Using again the basic concept of the deformation gradient separation introducing the notion of the intermediate configuration, the isochoric deformation gradient tensor iso F is further multiplicatively decomposed into an elastic part e F and a viscous part v F :

iso e v = F F F (3)
The elastic and viscous parts e L and v L of the corresponding kinematic rates additively split the velocity gradient

1 iso iso iso - = L F F & : { 1 1 1 e v iso e e e v v e - - - = + L L L F F F F F F & & 1424 3 and ( ) ( ) 1 1 2 2 v v T T v v v v v = + + - D W L L L L L 14243 14243 (4)

Constitutive equations

The hydrogel free energy function Hydrogel ψ is additively split into a volumetric part vol ψ and an isochoric part iso ψ as:

Hydrogel iso vol ψ ψ ψ = + (5)
The non-linear time-dependent response of hydrogel is captured by considering the resistance to deformation as the sum of an equilibrium contribution and a time-dependent contribution deviating from the equilibrium state. The isochoric free energy function iso ψ is thus given by: iso eq neq

ψ ψ ψ = + (6)
where eq ψ is the equilibrium free energy function and neq ψ is the non-equilibrium (viscous) free energy function.

The three free energy functions are expressed as: )

1 ( 3) 2 eq I µ ψ = -, 1 ( 3) 2
2 1/3 9 1 2 vol K J ψ = - (7)
where µ , v µ and K are material constants to be identified with µ the shear modulus, v µ the viscous constant and K the bulk modulus, 1 tr iso I = C is the first strain invariant for the equilibrium part and 1 tr The viscous stretching rate v D is defined by the following general flow rule:

2 γ ′ = & v v v v D σ σ , ( )
1 tr 2 ′ ′ = v v v σ σ σ (8)
where v σ is the effective value of the viscous Cauchy stress v σ , ′ v σ is the deviator part of v σ and v γ& is the accumulated viscous strain rate [46]:

1 3 1 d m v v v r I γ - = - & σ (9)
in which r , d and m are material constants to be identified and 1 tr

v v I = B is the viscous first strain invariant with = T v v v
B F F the viscous left Cauchy-Green strain tensor.

Hydrogel sheets reinforced with polylactide fibers are described by:

Hydrogel Fibers ψ ψ ψ = + (10)
where Hydrogel ψ is previously introduced in Eq. ( 5) and Fibers ψ is expressed as follows [START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and a comparative study of material models[END_REF][START_REF] Holzapfel | A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects and applications[END_REF]: "unphysical" auxetic behavior [START_REF] Skacel | Poisson's ratio of arterial wall -Inconsistency of constitutive models with experimental data[END_REF]. Nonetheless, it is used to constitutively describe the thin parallel reinforced sheets for which there is no obvious auxeticity. The latter behavior is ascribed to the special configuration and geometry of the unreinforced octagonal cells described using only the expression [START_REF] Van Ooij | Complications of artificial disc replacement: a report of 27 patients with the SB Charite disc[END_REF].

( ) ( ) ( ) ( ) 2 2 1 1 2 4 2 6 2 2 exp 1 1 exp 1 1 2 2 Fibers A A A I A I A A ψ     = - -+ - -     (11)
The Cauchy stress tensor σ is deduced from the differentiation of the free energy functions [START_REF] Van Ooij | Complications of artificial disc replacement: a report of 27 patients with the SB Charite disc[END_REF] and [START_REF] Patwardhan | Primary and coupled motions after cervical total disc replacement using a compressible six-degree-of-freedom prosthesis[END_REF] with respect to the corresponding deformations: The constitutive model has been implemented into MSC.Marc by means of a series of subroutines. Some details are provided in Appendix A.

Model parameters

The hydrogel model parameters were fitted to the experimental data of a P(AAm-co-AAc) nanocomposite hydrogel cross-linked by Fe 3+ carboxyl coordination bonds extracted from [START_REF] Zheng | Metalcoordination complexes mediated physical hydrogels with high toughness, stick-slip tearing behavior, and good processability[END_REF][START_REF] Lin | Constitutive behaviors of tough physical hydrogels with dynamic metal-coordinated bonds[END_REF]. The material system is stretched up to a level of 300% at different stretch rates in the equilibrated and fully swollen state. The rate-dependence and the hysteresis upon a loadingretraction cycle are perfectly reproduced by the constitutive model as illustrated in Figure 3.

The polylactide material parameters were identified using the experimental data of a polylactide fiber extracted from the work of Cicero and Dorgan [START_REF] Cicero | Physical properties and fiber morphology of poly (lactic acid) obtained from continuous two-step melt spinning[END_REF]. The material parameters of the constituent elements are listed in Table 2.

Results

The mechanics of the new designs is examined as a function both of its mesostructure in terms of cell walls/sheets thickness and of its microstructure in terms of fibers orientation.

Mesostructure

The mechanics of the replacement system are reported in Figure 5 considering different mesostructure configurations for fibers orientation θ =±30°. The in-silico results are compared with available experimental data obtained on annulus specimens of the human lumbar intervertebral disc. The experimental stress-strain results are those of Ebara et al. [START_REF] Ebara | Tensile properties of nondegenerate human lumbar anulus fibrosus[END_REF] and the experimental transversal strain results are those of Baldit et al. [START_REF] Baldit | Experimental analysis of the transverse mechanical behaviour of annulus fibrosus tissue[END_REF]. It is very satisfactory to observe that the substitute systems can reproduce the nonlinear stiffness and transversal strains of the natural annulus with auxetic response in the radial direction and unusual response with Poisson's ratios higher than 0.5 in the axial direction. It can be observed that the mechanics is not monotonically dependent on the thickness. The increase in thickness decreases the stiffness but the effect on the transversal strains is very slight. Only the transversal response of the W0.2-S0.2 design slightly diverges from the other configurations when the applied circumferential strain exceeds 0.6.

Microstructure

As observed in Figure 6, the replacement system is able to reproduce perfectly the mechanics of the local annulus specimen. The auxeticity increases when the fibers angle decreases, and the axial strain increases when the fibers angle increases. By varying the orientation of the polylactide fibers, both nonlinear stiffness and transversal behavior, can be modified in the aim to mimic the exact response of the different radial and circumferential regions of the natural annulus reported experimentally in Ebara et al. [START_REF] Ebara | Tensile properties of nondegenerate human lumbar anulus fibrosus[END_REF]. This important result makes it possible to foresee the possibility of designing a complete prosthetic of the disc with gradual variation in local properties and even the creation of personalized tissue-engineered discs.

Discussion

The design of an annulus fibrosus artificial substitute that could perfectly mimic the natural annulus mechanics is a challenging task due to the complexity of the disc tissue hierarchic structure and its unusual transverse response along with regional dependency. However, the challenge was accepted by many researchers in the recent years [START_REF] Nerurkar | Nanofibrous biologic laminates replicate the form and function of the annulus fibrosus[END_REF][START_REF] Pereira | Nanocellulose reinforced gellan-gum hydrogels as potential biological substitutes for annulus fibrosus tissue regeneration[END_REF] and it was hence the main objective of the current contribution. A new annulus replacement system, inspired from the real structure of the natural disc, is designed and implemented in-silico in order to be numerically tested. The system is composed of reinforced parallel sheets separated by a lattice structure with octagonal cells patterns. This configuration permits for the first time reproducing accurately not only the nonlinear stiffness of the annulus [START_REF] Ambard | Mechanical behavior of annulus fibrosus: a microstructural model of fibers reorientation[END_REF] but also its volumetric response [START_REF] Baldit | Experimental analysis of the transverse mechanical behaviour of annulus fibrosus tissue[END_REF][START_REF] Derrouiche | A chemo-mechanical model for osmo-inelastic effects in the annulus fibrosus[END_REF][START_REF] Derrouiche | The two Poisson's ratios in annulus fibrosus: relation with the osmo-inelastic features[END_REF][START_REF] Dusfour | Heterogeneous mechanical hyperelastic behavior in the porcine annulus fibrosus explained by fiber orientation: an experimental and numerical approach[END_REF]. This specific structure optimizes also the consumed material and highly reduces the elaboration cost which is of prime interest since the hydrogel, used as the ground matrix of the designed system, is very expensive. As seen in the simulation results, the system response could be highly controlled by the reinforcing fibers and slightly controlled by the mesostructure thickness. This allows changing the local response of the different annulus regions when constructing a complete prosthetic and permits reproducing the correct regional dependency of the natural disc. It could be thus improved to create advanced personalized replacements. The material choice is very large but as seen in the current contribution, the use of biocompatible materials, such as hydrogel and polylactide fibers, is always essential for biomedical applications [START_REF] Noguchi | Poly(vinyl alcohol) hydrogel as an artificial articular cartilage: evaluation of biocompatibility[END_REF][START_REF] Ramot | Biocompatibility and safety of PLA and its copolymers[END_REF][START_REF] Naahidi | Biocompatibility of hydrogel-based scaffolds for tissue engineering applications[END_REF]. Hydrogel is known to present many mechanical similarities with the human tissues and polylactide is identified as a biodegradable material that starts to be widely used in biomedicine [START_REF] Asghari | Biodegradable and biocompatible polymers for tissue engineering application: a review. Artificial cells[END_REF][START_REF] Da Silva | Biocompatibility, biodegradation and excretion of polylactic acid (PLA) in medical implants and theranostic systems[END_REF]. The possibility of printing those two materials via the most recent 3D printing technologies would simplify the manufacturing of such innovative structures [START_REF] Wei | 3D printing of an extremely tough hydrogel[END_REF][START_REF] Hong | 3D printing of highly stretchable and tough hydrogels into complex, cellularized structures[END_REF][START_REF] Chen | 3D printing of multifunctional hydrogels[END_REF].

In the current paper, a specimen of the new substitute system was designed replicating the same form and dimensions of experimental specimens in the literature and tested under tension loading in the circumferential direction of the intervertebral disc. The system succeeded to reproduce the same physiological nonlinear stiffness of the real annulus. What is very interesting is the ability of the system to reproduce the exact stress-strain response of the different intervertebral disc regions. The hydrogel octagonal cells patterns succeed to give the replacement structure the auxetic feature and mimic the correct nonlinear annulus fibrosus radial strain and the radial swelling. Reinforced by the oriented polylactide fibers the parallel sheets shrink axially giving the substitute the natural particular Poisson's ratio higher than 0.5 of the real annulus. The mesostructure (cell walls or parallel sheets) would affect the scaffold response but the effect is negligible compared to the effect of the different constituent elements, in particular the fibers orientation.

Conclusion

A novel hydrogel-based annulus fibrosus substitute system is designed, constructed in-silico and numerically tested. The special structure, bio-inspired from the natural annulus lamellar/interlamellar arrangement, is tailored by the lattice structure and the fibers orientation. It succeeds to mimic accurately the nonlinear stiffness and the transversal behavior of the different annulus regions. By using 3D printable biocompatible materials such as hydrogel and polylactide fibers to construct our system, a complete heterogeneous annulus could be created. In addition, the system permits manufacturing personalized tissueengineered discs miming the mechanics of each individual and each disc level which will be the objective of the following steps of the project.

The verification of the capabilities of our systems to mimic the multiaxial mechanics of the natural annulus remains an important issue [START_REF] Kandil | Interlamellar matrix governs human annulus fibrosus multiaxial behavior[END_REF][START_REF] Tamoud | A multiscale and multiaxial model for anisotropic damage and failure of human annulus fibrosus[END_REF]. In this regard, the constitutive representation of the hydrogel will be enriched in future works, especially to get out of the limitations of the used free energy functions vis-à-vis their predictive capabilities upon multiaxial loading [START_REF] Amores | On the network orientational affinity assumption in polymers and the micro-macro connection through the chain stretch[END_REF]. More sophisticated expressions will be used to accurately describe the multiaxial characteristics of our systems in connection to loading history and hydrogel microstructure [START_REF] Guo | A micromechanics-based model for deformation-induced damage and failure in elastomeric media[END_REF][START_REF] Saadedine | A micromechanics-based model for visco-super-elastic hydrogel-based nanocomposites[END_REF]. The identification of the elastic strain energy function and its derivatives with respect to the volume preserving parts of the first invariant 1 I % , the second invariant 2 I % , the fourth invariant 4 I % and the sixth invariant 6 I % are required by the hyperelastic subroutine as follows:

( ) ( ) ( ) ( ) ( ) 1 2 4 6
; ; ; ;

dev i i i i i I I I I ψ ψ ψ ψ ψ ∂ ∂ ∂ ∂     ∂ ∂ ∂ ∂   % % % % (A2)
where ( ) ( ) ( )

dev i i u i ψ ψ = +
defined for the parallel sheets as:

( ) ( ) ( ) dev Hydrogel Fibers i i i ψ ψ ψ = + and ( ) ( ) vol u i i ψ = (A3)
and for the octagonal cell as:

( ) ( )

dev Hydrogel i i ψ ψ = and ( ) ( ) vol u i i ψ = (A4)
After the definition of the strain energy function and the derivatives, the software defines the elastic second Piola-Kirchhoff stress and the volumetric stress for the parallel sheets as:

( ) ( ) ( ) ( ) 2 eq Fibers e iso i i i i ψ ψ ∂ + ∂ = ∂ S C and
( ) ( )

1 3 vol u i i J J ∂ = ∂ S (A5)
and for the octogonal cells as:

( ) ( ) ( )

2 eq e iso i i i ψ ∂ = ∂ S C and
( ) ( )

1 3 vol u i i J J ∂ = S (A6)

A.2. The viscous subroutine

To compute the current elastic deformation gradient ( ) e i F , the viscous deformation gradient ( )

1 v i - F
user-stored from the previous increment is called:

( ) ( ) ( ) 1 1 e iso i i i ν - = - F F F (A7)
The calculated elastic deformation gradient ( ) e i F is then used to calculate the viscous velocity gradient ( ) ( )

v v i i = L D with ( ) v i D
the viscous stretching rate at the end of the current increment. The rate of the viscous deformation gradient writes:

( ) ( ) ( ) ( ) 1 e iso i i i i ν ν - = F F D F & (A8)
The current viscous deformation gradient ( )

i ν F is iteratively calculated using: ( ) ( ) ( )
1 i i t i ν ν ν = ∆ + - F F F & (A9)
where t ∆ is the time increment.

At the beginning of the mechanical loading, the singularity of the viscous stretching rate is avoided by adding a perturbation coefficient κ =0.01 to The strain invariants are calculated from the Cauchy-Green deformation components in order to be used next in the stress calculation. The non-equilibrium (viscous) stress is calculated in the subroutine by the following expression:

( ) ( )

2 neq v e i i ψ ∂ = ∂ S C (A10)
Finally, the elastic and volumetric stresses are called from the hyperelastic subroutine, and the total second Piola-Kirchhoff stress is calculated by the summation of the different stress components: 5. Mesostructure effects on nonlinear stiffness and transversal strain history in axial/radial directions: (a) cell walls thickness and (b) sheets thickness. The experimental stress-strain data are extracted from [START_REF] Ebara | Tensile properties of nondegenerate human lumbar anulus fibrosus[END_REF] and correspond to the anterior outer disc region; the experimental transversal strain data correspond to the extremum values reported in [START_REF] Baldit | Experimental analysis of the transverse mechanical behaviour of annulus fibrosus tissue[END_REF].

( ) ( ) ( ) ( ) e v vol i i i i = + + S S S S (

  in which the viscous velocity gradient v L is further additively decomposed into a viscous stretching rate v D and a viscous spin rate v W which drops out, i.e. v = W 0 , if irrotationality of the viscous flow is assumed 3 [45]. The evolution of the viscous deformation gradient v & knowing the viscous stretching rate v D .

I

  = C is the elastic first strain invariant for the viscous part. The tensors are, respectively, the isochoric right Cauchy-Green strain tensor and the elastic part. The evolution of the elastic deformation gradient driven by the viscous deformation gradient v F that represents the obtained state during a spontaneous virtual elastic unloading to a free stress state.

where 1 A and 2 A

 2 are two fibers parameters, 4 iso I = C a a and 6 iso I = C b b are the fourth and sixth invariants, a and b are unitary vectors defining the orientation of two families of oriented fibers of angle θ and -θ. The free energy function (11) contributes to the overall function (10) only when the superimposed fibers are in loaded in tension, i.e. 4 1 I > or 6 1 I > , since they are inactive in compression. Note that the expression (11) may lead to an

  is the elastic right Cauchy-Green strain tensor.

  the numerical code.

A11) 1 Figure 1 .Figure 2 .

 112 Figure 1. Hydrogel-based annulus replacement system consisting in a lattice structure with special octagonal cells and parallel thin sheets reinforced by oriented polylactide fibers.

Figure 3 .Figure 4 .

 34 Figure 3. Mesh of the annulus replacement system and deformed mesh showing an auxetic deformation in the Z direction and an ordinary non-auxetic deformation in the Y direction in the course of deformation along the X direction. Different mesostructure configurations are considered to tailor the transverse deformations in the Y and Z directions.

Figure 6 .

 6 Figure 6. Fibers angle variation to mimic the regional variation in nonlinear stiffness and transversal behavior of the natural annulus. The square symbols correspond to the four disc regions extracted from the experiments of Ebara et al.[START_REF] Ebara | Tensile properties of nondegenerate human lumbar anulus fibrosus[END_REF]: Anterior outer (AO), Anterior inner (AI), Posterior outer (PO) and Posterior inner (PI). The illustrative example is for the configuration W0.2-S0.2.

  constituted of m number of elements, nn number of nodes and p number of tensor directions.

	stored in the form of a 3-D matrix [	, , m nn p	]
			ν F	i	is user-

1 3 v I . The value of ( )

Table 1 .

 1 Different mesostructure configurations of the annulus replacement system.

	fibers angle ±42°	fibers angle ±30°
	Mimic AI region	Mimic AO region
	Without fibers	fibers angle ±35°
	Mimic PI region	Mimic PO region
	region
	Transversal strains	30 degree 35 degree	Axial Radial
		42 degree
		Without fibers
		Circumferential strain

Table 2 .

 2 Parameters of the constituent elements.

The mean non-degenerate lumbar human intervertebral disc heights were found experimentally to be around 10 mm[START_REF] Roberts | MRI analysis of lumbar intervertebral disc height in young and older populations[END_REF] and about 40 mm of anteroposterior diameter and 50 mm of transverse diameter[START_REF] Wang | A morphological study of lumbar vertebral endplates: radiographic, visual and digital measurements[END_REF]. Hence, the crosssection dimensions of our replacement system circumferential specimen were 10 mm × 10 mm, and the length

was 25 mm, these values are totally physiological and represent the real dimensions of an annulus fibrosus tensile specimen. These realistic dimensions were previously adopted and validated in simulations of the real intervertebral disc annulus fibrosus[START_REF] Derrouiche | A chemo-mechanical model for osmo-inelastic effects in the annulus fibrosus[END_REF][START_REF] Kandil | Interlamellar-induced timedependent response of intervertebral disc annulus: a microstructure-based chemoviscoelastic model[END_REF][START_REF] Kandil | Interlamellar matrix governs human annulus fibrosus multiaxial behavior[END_REF][START_REF] Kandil | A microstructure-based modeling approach to assess aging-sensitive mechanics of human intervertebral disc[END_REF]].[START_REF] Amoretti | Imaging of intervertebral disc prostheses[END_REF] The following notation is used throughout the section. Tensors and vectors are denoted by normal boldfaced letters and italicized boldfaced letters, respectively, while scalars and individual components of vectors and tensors are denoted by normal italicized letters. The superposed dot designates the time derivative.

The multiplicative decomposition of the deformation gradient implies that the unloading process (relating the deformed configuration with the relaxed configuration) is not uniquely defined. Indeed, an arbitrary rigid body rotation of the relaxed configuration still leaves the relaxed configuration stress free. One way to make unique the unloaded configuration, with no loss in generality, is to assume the viscous flow irrotational.

Appendix A. Numerical implementation

The constitutive model affected to the replacement system is implemented into MSC.Marc software by means of a series of subroutines. The reader is also referred to complementary references [START_REF] Kandil | Interlamellar-induced timedependent response of intervertebral disc annulus: a microstructure-based chemoviscoelastic model[END_REF][START_REF] Ovalle-Rodas | A thermo-visco-hyperelastic model for the heat build-up during low-cycle fatigue of filled rubbers: formulation, implementation and experimental verification[END_REF][START_REF] Guo | A thermo-viscoelastic-damage constitutive model for cyclically loaded rubbers. Part I: model formulation and numerical examples[END_REF] for the general implementation procedure of coupled models. An anisotropic hyperelastic subroutine is here coupled to a viscous subroutine. A total Lagrange formulation was adopted for the present calculations. When the total Lagrange is selected, MSC.Marc software deals with the second Piola-Kirchhoff stress S and the Green-Lagrange strain E . An updated Lagrange formulation could be also used and in that case the software would deal with the Cauchy stress σ and the logarithmic strain ( ) ln F .

A.1. The anisotropic hyperelastic subroutine

The deformation gradient ( ) i F is computed at the beginning of each increment by [START_REF] De Souza Neto | Computational Methods for Plasticity: Theory and Applications[END_REF]:

Replacement system

Replication of the natural behavior

Resulted response

New design

Hydrogel reinforced by polylactide fibers Hydrogel 3D cell special configuration After deformation -

FE computations

Regional-dependent non-linear stiffness Regional-dependent transversal strain


P Chateau-Cornu

M R Ardigo-Besnard
email: maria-rosa.ardigo-besnard@u-bourgogne.fr

A Tellier

A Besnard

J.-P Chateau-Cornu

Effect of the microstructure on the tribological properties of HIPed and PTA-welded Fe-based hardfacing alloy

Keywords: hardfacing, coating, microstructure, wear resistance, friction, powder metallurgy, PTAW

Introduction

Fe-based hardfacing alloys with high carbon and chromium contents possess good wear resistance properties in severe environmental conditions, such as high pressure, high temperature and corrosive atmosphere [START_REF] Pawar | Effect of different carbides on the wear resistance of Fe-based hardfacing alloys[END_REF][START_REF] Ocken | The galling wear resistance of new iron-base hardfacing alloys: a comparison with established cobalt-and nickel-base alloys[END_REF]. Among all the Fe-based alloys, Norem02 is typically used as hardfacing for sliding components in nuclear power plants, where temperatures can range from 90 to about 300 °C under a normal contact stress of few hundreds MPa [START_REF] Persson | Effect of temperature on friction and galling of laser processed Norem02 and Stellite 21[END_REF][START_REF] Kim | The temperature dependence of the wear resistance of iron-base NOREM 02 hardfacing alloy[END_REF][START_REF] Ocken | Reducing the Cobalt Inventory in Light Water Reactors[END_REF]. Several authors report that Norem02 elaborated by different techniques, such as laser processing, plasma transferred arc welding (PTAW), gas tungsten arc welding (GTAW) and laser-aided direct metal rapid tooling (DMT) process, presents satisfactory wear resistance properties at room temperature [START_REF] Ocken | The galling wear resistance of new iron-base hardfacing alloys: a comparison with established cobalt-and nickel-base alloys[END_REF][START_REF] Persson | Effect of temperature on friction and galling of laser processed Norem02 and Stellite 21[END_REF][START_REF] Kim | The temperature dependence of the wear resistance of iron-base NOREM 02 hardfacing alloy[END_REF][START_REF] Park | Wear characteristics of Stellite6 and NOREM02 hardfaced SKD61 hot working tool steel at the elevated temperature[END_REF]. This behaviour is explained by a low stacking fault energy (SFE) of the matrix [START_REF] Ohriner | The chemistry and structure of wear resistant, iron-base hardfacing alloys[END_REF], that suppresses the cross-slip of dislocations and then increases the work hardening rate of the alloy [START_REF] Vaidya | The role of twinning in the cavitation erosion of cobalt single crystals[END_REF][START_REF] Bhansal | The role of stacking fault energy on galling and wear behaviour[END_REF]. Generally, the increase of the material hardness by work hardening improves the galling resistance, as observed in the case of the Norem02 [START_REF] Persson | Effect of temperature on friction and galling of laser processed Norem02 and Stellite 21[END_REF][START_REF] Vikström | Galling resistance of hardfacing alloys replacing Stellite[END_REF]. However, a significant degradation of the wear resistance properties of Norem02 is reported for temperatures higher than 150 °C, when adhesive wear becomes the dominant wear mechanism [START_REF] Persson | Effect of temperature on friction and galling of laser processed Norem02 and Stellite 21[END_REF][START_REF] Kim | The temperature dependence of the wear resistance of iron-base NOREM 02 hardfacing alloy[END_REF]. Actually, the development of strain-induced martensite is considered of primary importance for the wear resistance of Norem02. The suppression of strain-induced phase transformation from austenite to martensite above 150 °C, known to be related to the increase of the stacking fault energy (SFE) of the matrix with temperature, is supposed to be one of the main reasons of the change in the dominant wear mechanism at high temperature and of the significant loss of the wear resistance of Norem02.

It is well known that the tribological behaviour of steels is influenced by their microstructure, particularly the size and the distribution of the hard phases, typically carbides [START_REF] Xu | Effect of slippage rate on frictional wear behaviors of high-speed steel with dual-scale tungsten carbides (M6C) under highpressure sliding-rolling condition[END_REF][START_REF] Chaus | Role of microstructural factor in wear resistance and cutting performance of high-speed steel end mills[END_REF][START_REF] Bourithis | Comparison of wear properties of tool steels AISI D2 and O1 with the same hardness[END_REF][START_REF] Korade | Study of effect of population density of carbides on surface roughness and wear rate of H21 tool steel[END_REF][START_REF] Iakovakis | Evaluation of wear mechanisms in additive manufactured carbide-rich tool steels[END_REF]. Karlsson et al. [16,[START_REF] Karlsson | Influence of tool steel microstructure on friction and initial material transfer[END_REF] showed that fine carbides, homogenously distributed into the matrix, typically enhance the wear performance of stainless and tool steels.

Ur Rahman et al. [START_REF] Ur Rahman | Wear characterization of multilayer laser cladded high speed steels[END_REF] reported that the wear resistance of high speed steel alloys is increased at 25 °C by a refinement of the microstructure. Malayoglu et al. [START_REF] Malayoglu | Comparing the performance of HIPed and Cast Stellite 6 alloy in liquid-solid slurries[END_REF] compared the performance of hot isostatically pressed (HIPed) and cast Stellite 6, a Co-based alloy. The HIPed alloy, composed of fine grains and small carbides dispersed into the matrix, presents a higher resistance to mechanical degradation (erosion and corrosion-erosion), compared to the cast material. The positive impact of a finer microstructure on the galling properties of some Fe-based alloys was also reported by Ohriner et al. [START_REF] Ohriner | The chemistry and structure of wear resistant, iron-base hardfacing alloys[END_REF]. Wong Kian et al. [START_REF] Wong Kian | Comparison of erosion-corrosion behaviour of hot iso statically pressed and welded Stellite coatings[END_REF] compared the corrosion-erosion behaviour of HIPed and GTAW deposited Stellite coatings and showed that the HIP samples presented higher resistance than the welded ones.

Based on these literature results, it is possible to claim that HIP is an interesting elaboration process for the production of improved hardfacings for nuclear power plants, as the refinement of the microstructure of the Norem02 induced by HIP can represent a solution to improve the tribological properties of the alloy. The aim of this work is to give preliminary results, at laboratory scale, about the tribological properties of HIPed Norem02 and to show the opportunity of considering this powder metallurgy process as an effective alternative to more conventional hardfacing techniques, e.g. PTAW. The Norem02 powder is directly HIPed on the valve seat surface, which represents the surface to be coated and constitutes a part of the container. The wear resistance at room temperature of Norem02 elaborated by HIP and PTAW are investigated and friction mechanisms are compared.

Materials and methods

The Norem02 powder used in this study was supplied by Ducal. Its composition is 1.2 C -24.8 Cr -4.6 Mn -3.5 Ni -2.1 Mo -3.0 Si -0.027 O -bal. Fe (in wt.%). Morphological and chemical characterisations of the powder are reported in a previous work [START_REF] Tellier | Influence of the process parameters on the microstructure of a hardfacing coating elaborated by Hot Isostatic Pressing[END_REF] and are not described here. The powder composition was reproduced in the nominal compositions of the HIP and PTAW hardfacings. The HIPed Norem02 samples were obtained by sintering at 1100 °C and 120 MPa for 3 hours, after a 2 hours heating and pressurizing ramp and natural cooling (process details are available in [START_REF] Tellier | Characterization of the interfacial zone between a HIPed Fe-based alloy and a stainless steel container[END_REF]). The Norem02 elaborated by PTAW was taken from an industrial pressurized water reactor (PWR) shutter for which precise process parameters are not of our knowledge. The thickness of the hardfacings was about two centimetres. The surfaces used for the tribological tests were free from any substrate influence (e.g. dilution). The samples were polished using SiC papers (down to 4000 grit), diamond paste (down to 1 µm), and colloidal silica (30 nm). Microstructures were characterised using a JEOL JSM-7600F scanning electron microscope (SEM) equipped with a field emission gun (FEG) and coupled with an energy dispersive X-ray spectrometer (EDX) operating at an accelerating voltage of 15 kV. Chemical phases identification was done by X-ray diffraction (XRD) using a Bruker D8-A25 diffractometer with Cu Kα (λ = 0.154056 nm), a step of 0.02° and a scan rate of 1.8 seconds per step.

The typical morphologies of samples elaborated by HIP and PTAW are shown in Figure 1.

The HIPed Norem02 (Figure 1 (a)) presents a fine-grained microstructure. XRD coupled with SEM-EDX analyses performed in previous studies [START_REF] Tellier | Influence of the process parameters on the microstructure of a hardfacing coating elaborated by Hot Isostatic Pressing[END_REF][START_REF] Tellier | Characterization of the interfacial zone between a HIPed Fe-based alloy and a stainless steel container[END_REF] revealed that the matrix is primarily austenitic and only contains a small amount of ferrite. Small M23C6-type carbides (M = Cr, Mn, Mo) are homogeneously dispersed in the matrix: globular carbides in zones between the primary powder particles and lath-like interdendritic carbides inside the particles (see supplementary material S1). On the other hand, as shown by XRD and SEM-EDX analyses (see supplementary materials S2 and S3), the Norem02 elaborated by PTAW (Figure 1(b)) is composed of large austenitic grains, a small amount of ferrite, acicular intergranular M7C3-type carbides (see supplementary material S4) and intergranular Si and Mo enriched phases. Table I reports the mechanical properties of the HIPed Norem02 and of the alloy elaborated by PTAW. Tensile tests were performed on flat specimens machined in the HIPed part to determine the Young's modulus, the yield and ultimate tensile strengths and the fracture elongation. Properties of the PTAW alloy were found in ref. [START_REF] Beaurin | Microstructural and mechanical properties evolutions of plasma transferred arc deposited Norem02 hardfacing alloy at high temperature[END_REF]. The hardness of both samples was measured in Vickers configuration under a load of 500 g (4.9 N). It is worth noting that the HIPed Norem02 presents superior mechanical properties than the one elaborated by PTAW.

Wear tests were performed at room temperature using a pin-on-disk tribometer (Anton-Paar Instruments) in a ball-on-plate configuration under different normal loads (2, 5 and 10 N) against a WC-Co ball counterpart (6 mm diameter). These parameters are based on a previous work of Liu et al. [START_REF] Liu | High temperature wear and corrosion resistance of Co-free Ni-based alloy coatings on nuclear valve sealing surfaces[END_REF]. However, in the present study, the linear velocity (0.05 m•s -1 ) was about three times higher than in ref. [START_REF] Liu | High temperature wear and corrosion resistance of Co-free Ni-based alloy coatings on nuclear valve sealing surfaces[END_REF]. The total sliding distance was 100 m. Moreover, WC-Co (instead of the 100C6 steel, for instance, or Si3N4 used in ref. [START_REF] Liu | High temperature wear and corrosion resistance of Co-free Ni-based alloy coatings on nuclear valve sealing surfaces[END_REF]) was selected as counterpart because it is a carbide (as the hard phases present in the Norem02) and has high mechanical properties (nominal hardness of 1570 HV and Young modulus of 610 GPa), higher than that of the Norem02 (Table I). The wear of the ball was consequently avoided.

The WC-Co ball is grade 10, following the ABMA norm, with a roughness of 25 nm. The surface arithmetical roughness Sa of Norem02 elaborated by HIP and PTAW is 15 and 50 nm, respectively.

For each load, a rotation radius of 1.5, 2.5 and 3.5 mm was used, keeping constant the linear speed and the total friction length. It allowed verifying that the rotation radius does not influence neither the shape and size of the wear track, nor the value of the friction coefficient.

Friction force-distance curves were obtained for each applied load. For each curve, zones where the friction force is constant (plateau) were identified. The corresponding mean tangential forces were plotted versus the three normal loads in order to determine the friction coefficient, which is the slope of the line according to the Coulomb's friction law. For each elaboration process (HIP and PTAW), nine measurements were performed (the three loads were applied on three rotation radii) on the different samples to estimate the friction coefficient, in order to assess the repeatability.

After tribological tests, wear tracks were observed by a 3D optical profiler (WYKO NT1100).

Surface maps were obtained and the depths and widths of the worn areas were measured using Gwyddion software [25]. For each track, four zones were chosen in four orthogonal directions and one depth profile per zone was measured. The depth and width of the wear track were finally calculated as the average of the values measured in the four zones. Two samples at a given condition were considered to calculate the mean depth and width of each wear track. In addition, for each measured zone, Gwyddion software calculates the worn area (area of the curve below the initial surface), making it possible to estimate the volume and the weight losses and to calculate the wear rate.

Finally, wear tracks (surfaces and cross-sections) were characterised by SEM-EDX, using the apparatus described above. In the case of the cross sections, only wear tracks obtained under a load of 10 N are presented, as the differences between the two samples are more significant under this load.

Results and discussion

Optical profiler characterisations of wear tracks

Figure 2 shows the 3D surface maps of the tracks after wear tests under the three loads. Figure 3 presents the corresponding 2D profiles. Under a load of 2 N, for both samples (HIP and PTAW), it is possible to observe a partial material redeposition on the edges of the tracks, causing the formation of a relief with a height ranging between 0.5 and 1 µm in the case of the HIPed Norem02 (Figure 2 (a)) and between 1 and 2 µm for the alloy elaborated by PTAW (Figure 2 (b)). The wear track of the sample elaborated by PTAW appears wider and more damaged than the one formed in the case of the HIPed alloy. The depth of the wear track ranges between 0.5 and 1 µm in the case of the HIPed sample and has an average value of 1 µm for the Norem02 obtained by PTAW.

8 Under a load of 5 N, the average depth of the wear track is about 1 µm for the HIPed sample (Figure 2 (c)) and 3 µm for the Norem02 elaborated by PTAW (Figure 2 (d)). For the latter, some areas present an increased depth, ranging between 4 and 5 µm, with some localised zones exceeding 6 µm. A redeposition of material on the edges of the track occurred for both samples during the wear test, as observed under 2 N.

The differences in the depths of the wear tracks when passing from a load of 2 to 5 N, are more significant for the Norem02 elaborated by PTAW than for the HIPed sample.

Finally, under a load of 10 N, the mean depth is about 2.5 µm for the HIPed sample (Figure 2 (e)). In the case of the alloy elaborated by PTAW (Figure 2 (f)), the wear track is formed by parallel grooves and presents a mean depth ranging between 6 and 7 µm with a maximum of 10 µm. As previously observed, the differences in the depth of the worn surfaces when passing from a load of 5 N to 10 N are more significant for the sample elaborated by PTAW.

Concerning the width of the wear tracks, a constant difference of around 115 µm between the HIPed sample and the alloy elaborated by PTAW is observed and seems to be independent on the applied load.

These results indicate that the elaboration by HIP improves the wear resistance of the Norem02 at room temperature. The reason of this behaviour is explained in terms of microstructure. As shown in Figure 1, the average grain size of the HIPed hardfacing is smaller than in the PTAW one. Moreover, it is known that the wear resistance of a material is increased by the presence of hard phases, such as carbides [START_REF] Xu | Effect of slippage rate on frictional wear behaviors of high-speed steel with dual-scale tungsten carbides (M6C) under highpressure sliding-rolling condition[END_REF][START_REF] Chaus | Role of microstructural factor in wear resistance and cutting performance of high-speed steel end mills[END_REF][START_REF] Bourithis | Comparison of wear properties of tool steels AISI D2 and O1 with the same hardness[END_REF][START_REF] Korade | Study of effect of population density of carbides on surface roughness and wear rate of H21 tool steel[END_REF][START_REF] Iakovakis | Evaluation of wear mechanisms in additive manufactured carbide-rich tool steels[END_REF][START_REF] Karlsson | Influence of work material proof stress and tool steel microstructure on galling initiation and critical contact pressure[END_REF][START_REF] Karlsson | Influence of tool steel microstructure on friction and initial material transfer[END_REF]. As reported by Karlsson [START_REF] Karlsson | Galling resistance and wear mechanisms for cold-work tool steels in lubricated sliding against high strength stainless steel sheets[END_REF], the distribution of the carbides plays a major role in wear behaviour. A homogeneous distribution, as in the case of the HIPed sample where carbides are located inside the austenitic grains and between the primary powder particles, will improve the resistance against the surface deterioration. On the contrary, in the case of the alloy elaborated by PTAW, carbides and Si-Mo enriched phases are located in the intergranular zones, leaving large austenitic matrix areas exposed to wear phenomenon.

Estimation of wear rates and friction coefficients.

The wear rate estimated for each sample under the different tested conditions are presented in Figure 4. The wear rates calculated after wear tests under 2 N are similar for the samples elaborate by HIP ((8.1 ± 1.6)•10 -6 mm 3 .m -1 ) and PTAW ((7.8 ± 1.6)•10 -6 mm 3 .m -1 ). The difference becomes significant at higher load: at 5 N, the wear rate of the alloy increased to (24 ± 5)•10 -6 mm 3 .m -1 and (79 ± 16)•10 -6 mm 3 .m -1 for the samples elaborate by HIP and PTAW respectively; at 10 N, the wear rate of the alloy increased again to (86 ± 17)•10 -6 mm 3 .m -1 and (440 ± 90)•10 -6 mm 3 .m -1 for the samples elaborate by HIP and PTAW respectively. At 10 N, the wear rate of the HIPed alloy is close to the values estimated for the sample elaborated by PTAW, but under a load of 5 N. Generally, the wear rate increases with the applied load, but the increase is more significant for the sample elaborated by PTAW. In the case of the HIPed Norem02, the wear rate is multiplied by a factor 10 between 2 N and 10 N, while for the alloy elaborated by PTAW, the difference is of two orders of magnitude.

Even if no work on Norem02 performed in the experimental conditions used in the present study exists in the literature, it is still possible to consider, for comparison, the results reported by Bourithis et al. [START_REF] Bourithis | Comparison of wear properties of tool steels AISI D2 and O1 with the same hardness[END_REF]. The authors tested the wear behaviour of a tool steel containing 1.5 wt. % of carbon. After wear tests of a total length of about 1050 m, under an applied load of 9.8 N and a linear speed ranging from 0.05 m.s -1 to 0.75 m.s -1 , they found wear rates varying between 4•10 -3 and 1.6•10 -2 mm 3 .m -1 . Considering the shorter sliding distance in the present study, the Norem02 wear rates are consistent with these typical values measured for reference steels in the field of wear resistance.

Friction force-distance curves obtained under different loads (Figure 5) allowed calculating the friction coefficients. This parameter corresponds to the slope of the regression line obtained plotting the mean friction force measured as a function of the normal load [START_REF] Stempflé | Influence of intergranular metallic nanoparticles on the fretting wear mechanisms of Fe-Cr-Al2O3 nanocomposites rubbing on Ti-6Al-4V[END_REF] (Figure 5 insets). The advantage of this statistical method is to avoid the problems due to the dispersion of a single measurement. It is interesting to note that, for both HIPed and PTAW samples, friction force-distance curves display the existence of three distinct friction behaviours for all tested loads (Figure 5). The first behaviour corresponds to the running-in period from the start of the test and for few sliding meters. This stage is characterized by a low friction coefficient (0.12 and 0.18 for the HIPed and PTAW hardfacing respectively) and ends with a sharp increase of the friction force. The following behaviour is a quasi-stable regime, where the friction force signal is highly serrated. For both samples, the friction coefficients are equal to 0.39. This first plateau progressively increases to the last behaviour characterized by a higher friction coefficient (0.57 and 0.74 for the HIPed and PTAW hardfacing respectively) and a more stable friction force, except local variations. During the running-in period, the WC-Co ball slides on the micro asperities of the surface, i.e. the hard carbides phases. For the PTAW, the large austenitic grains lead to more adhesive contact and consequently a higher friction coefficient. This period ends with the formation of the third body composed of carbides and austenite debris. In the first plateau, abrasive wear is the dominant mechanism. The progressive transition from the first plateau to the second one can be explained by the increase of the temperature during friction and the formation of an oxide layer protecting the Norem02 from further abrasion. The friction is mainly dominated by adhesive mechanisms characterized by the higher friction coefficient and occasional removal of the transfer layer. The higher friction coefficient (0.74 compared to 0.57) of the PTAW alloy comes from the largest contact surface, as observed by profilometry. Generally, a high friction coefficient is associated to an intensive friction, leading to major damages. Persson et al. [START_REF] Persson | Effect of temperature on friction and galling of laser processed Norem02 and Stellite 21[END_REF] evaluated friction and galling properties of laser processed Norem02 using loadscanning tests. They associate the increase of the friction coefficient to massive galling occurring at high temperature and probably due to the absence of strain-induced martensitic transformation. They reported that at elevated temperatures (> 250 °C) the friction coefficient of the laser processed Norem02 reaches 0.5-0.7 and the predominant wear mechanism becomes adhesive wear. In the present study, the friction coefficient of the HIPed Norem02 calculated considering the second plateau, can be associated to less severe friction, leading to less important surface degradation, than the one obtained for the alloy elaborated by PTAW.

It is worth noting that the transition from the first to the second friction plateau is delayed with the increase of the load. This can be explained by the removal of the oxide layer due to the presence of abrasive particles, which amount increases with the normal load and the wear or the hardfacing. This delay is systematically longer for the PTAW alloy and the second plateau is even not attained under a load of 10 N after 100 m. This is in agreement with the wear rate and the surface maps and depth profiles results, showing that the microstructure obtained by HIP allows the material to better withstand mechanical stress during friction tests.

It is important to underline that tribological tests were realised at room temperature. A trend reversal, with a modification of the wear mechanisms, cannot be excluded at more elevated temperatures.

MEB-EDX analyses of wear tracks.

In order to confirm the wear mechanisms of the sample elaborated by HIP, the worn surfaces obtained under different loads at the end of the friction tests were analysed by SEM-EDX.

Figure 6 shows the SEM micrographs of the surface of the wear track after the test under a load of 2 N. The track is composed of grooves originating from the abrasive wear of the first plateau and covered by a non-continuous transfer layer. EDX analyses reveal that this layer is composed of oxides, rich in Cr and Fe (see supplementary material S5), confirming the occurrence of oxidative wear in a second time. Grooves are also visible above the oxide layer.

They are generally associated to abrasive wear [START_REF] Karlsson | Galling resistance and wear mechanisms for cold-work tool steels in lubricated sliding against high strength stainless steel sheets[END_REF] but they also can form from plastic deformation [START_REF] Bourithis | Comparison of wear properties of tool steels AISI D2 and O1 with the same hardness[END_REF]. Figure 6 (b) displays a magnification of an area containing oxides. Close to the zones where the oxide layer was fractured and removed, it is possible to note the presence of tongue-shaped area, all having the same direction and indicating the displacement direction of the second body during the wear test. This kind of morphology is commonly associated to adhesive wear [START_REF] Aoh | On the wear behavior of surface clad layers under high temperature[END_REF]. The worn track presents typical marks of abrasive wear followed by oxidative but also adhesive ones.

The worn track obtained under a load of 5 N is displayed in Figure 7. Again, large grooves covered by an oxide layer (see supplementary material S6) can be observed. The grooves are deeper than those formed under 2 N. Some cavities can also be noticed, which result from the removal of debris off the metallic matrix. Unfortunately, the nature of the debris could not be determined. The magnification (Figure 7 (b)) shows a zone where the oxide layer was fractured and partially removed; some tongue-shaped area, associated to an adhesive wear mechanism, were also revealed. As previously suggested, abrasive followed by both oxidative and adhesive mechanisms are involved in the sample wear. As reported by Bourithis et al. [START_REF] Bourithis | Comparison of wear properties of tool steels AISI D2 and O1 with the same hardness[END_REF], the presence of cavities and deep grooves indicates that oxidation is not the primary wear mechanism.

The surface of the worn track obtained under 10 N is presented in Figure 8. As revealed for lower loads, oxides are present and grooves are clearly visible. Some cavities can be noticed, as observed under 5 N. The oxide layer seems more discontinuous. Tongue-shaped area are more numerous and are larger than those observed under 2 and 5 N (Figure 8 (b)). EDX analyses revealed the presence of Cr and Fe enriched oxides inside some tongue-shaped area (see supplementary material S7). This might be the result of trapping of the oxides during the tongue-shaped area formation or a consequence of the redeposition phenomenon. These results confirm that oxidative and adhesive wear contribute to the global wear mechanism after the abrasive period. Some authors report the formation of strain-induced martensite, generally below 180 °C [START_REF] Kim | The temperature dependence of the wear resistance of iron-base NOREM 02 hardfacing alloy[END_REF][START_REF] Lee | Sliding wear behavior of hardfacing alloys in a pressurized water environment[END_REF]. In the present study, XRD analyses of the worn track after the test under 10 N were performed but the presence of martensite was not revealed. On one hand, the conditions leading to the formation of strain-induced martensite might have been modified by the elaboration by HIP; on the other hand, it is also possible that the stress induced by an applied load of 10 N is not sufficient to trigger the strain-induced martensitic transformations.

The cross-section of the HIPed sample after tests under 10 N was also characterised by SEM-EDX near the middle of the wear track (Figure 9). The sample surface appears plastically deformed; no cracks are visible close to the surface or more deeply in the bulk. In agreement with the previous characterisations, the presence of Cr and Fe enriched oxides was revealed at the surface of the wear track by EDX analyses; no oxide was detected below the surface.

In order to compare the two elaboration processes, a global view of the cross-section of the worn track of the Norem02 elaborated by PTAW after wear test under 10 N is shown in Figure 10 (a). The friction force-distance curve shows that the second friction plateau, mainly dominated by oxidation and adhesive wear, was not reached under this load. A deformation of the sample surface can be observed, as well as grooves. No cracks are visible close to the surface or into the bulk. The magnification (Figure 10 (b)) confirms that the sample is damaged only close to the surface, where a significant plastic deformation occurred. This deformation leaded to a modification of the morphology of the carbides located close to the surface, which became more globular. Locally, a wear debris accumulation was revealed, as well as the presence of some spalled zones. EDX analyses indicated that these debris are mainly a mixture of Cr and Fe enriched oxides; some particles present a high amount of carbon and were identified as carbides. Figure 11 shows that debris are either incorporated in the spalled areas (zone 1) or accumulated at the surface (zone 2). The presence of the debris can be associated to abrasive wear [START_REF] Gahr | Microstructure and wear of materials[END_REF][START_REF] Dalmau | Tribological Behavior of New Martensitic Stainless Steels Using Scratch and Dry Wear Test[END_REF], supporting the results obtained by the friction force-distance curve under this load. Furthermore, these debris certainly participated to the abrasive wear mechanism, acting as a third body formed by phases harder than the metallic matrix (such as oxides or carbides) [START_REF] Denape | Third body concept and wear particle behavior in dry friction sliding conditions[END_REF].

These results prove that, also in the case of the sample elaborated by PTAW, oxidation occurred, as it was observed for the HIPed alloy, but the increased stress during the wear test caused the pull-out of the oxides and of the material itself.

Conclusions

The tribological properties of a Fe-based hardfacing alloy Norem02 elaborated by HIP was investigated and compared to the performance of the same material elaborated by PTAW, a more conventional hardfacing technique. The main findings can be summarized as follow:

• Under loads higher than 2 N, the HIPed Norem02 presents a better wear resistance than the alloy elaborated by PTAW. Wear tracks of HIPed alloy are less deep and wide. The wear rates estimated for the HIPed Norem02 are lower (under a load of 10 N, the difference in wear rates between the two materials is about 5.5 times).

• Three friction coefficients were calculated for the Norem02 elaborated by HIP and by PTAW, considering the running-in period and the two plateaus of the friction forcedistance curves. The friction coefficient of the HIPed Norem02 calculated considering the second plateau is significantly lower compared to the sample elaborated by PTAW.

• MEB-EDX analyses revealed typical marks of oxidative and adhesive wear covering the deep grooves of the initial abrasive wear and confirming that two mechanisms are involved in the wear. Firstly, a dominance of abrasive wear, secondly a dominance of oxidative and adhesive wear.

• These performances confirm that the elaboration by HIP is a promising way to improve the tribological behaviour of Norem02 as hardfacing, thanks to the fine microstructure obtained, shortening the duration and reducing the effect of the abrasive wear period compared to the sample elaborated by PTAW.

Figure captions

Figure 1 :

 1 Figure 1: SEM images of Norem02 elaborated by (a) HIP and (b) PTAW.

Figure 2 :

 2 Figure 2 : Surface maps of Norem02 after wear tests with a load of 2 N (a) HIP and (b)

Figure 3 :

 3 Figure 3: 2D depth profiles under different loads for the Norem 02 elaborated by (a) HIP and

Figure 4 :

 4 Figure 4: Wear rates under different loads for the Norem 02 elaborated by HIP (white bar)

Figure 5 :

 5 Figure 5: Friction force-distance curves under different loads obtained for the (a) HIPed

Figure 6 :

 6 Figure 6: SEM micrographs of the surface of the wear track of the HIPed sample under 2 N,

Figure 7 : 21 Figure 8 :

 7218 Figure 7: SEM micrographs of the surface of the wear track of the HIPed sample under 5 N,

Figure 9 :

 9 Figure 9: SEM micrographs of the cross-section of the HIPed sample after test under 10 N.

Figure 10 :

 10 Figure 10: SEM micrographs of the cross-section of the worn track of Norem02 elaborated by

Figure 11 :

 11 Figure 11: Cross-section of the worn track of Norem02 elaborated by PTAW. Zone 1: debris

Acknowledgements

The authors would like to thank the Bourgogne-Franche-Comté Region for financing part of this study and Framatome, which finances the Chair in which this program is included.

Tables