Xiang Wang

Prof Jia Li

Fabien Cazes

Guy Dirras

Azziz Hocini

Numerical modeling on strengthening mechanisms of the harmonic structured design on CP-Ti and Ti-6Al-4V

Keywords: Microstructure, Crystal plasticity, Harmonic structure, Ti-6Al-4V, Finite elements

The present work investigates and compares the deformation mechanism of harmonic structured (HS) CP-Ti and Ti-6Al-4V alloy to that of homogeneous coarse-grained (CG) counterparts. A set of simple shear tests on homogeneous CG and HS specimens for CP-Ti and Ti-6Al-4V was conducted under monotonic and cyclic simple shear loading conditions. Efforts have been made to develop a numerical model which is aimed to investigate the microstructurerelated mechanical behavior of the materials. Numerical and experimental results were confronted to confirm the validity of the proposed model. To better understand the strengthening mechanisms of the HS design, comparisons between HS CP-Ti and HS Ti-6Al-4V were conducted and discussed in terms of macroscopic stress-strain response and distribution of stress/strain fields. Furthermore, forecasting mechanical properties of HS CP-Ti and HS Ti-6Al-4V including various volume fractions of fine-grained (FG) regions were simulated by using the developed model. It was demonstrated that strengthening mechanisms of harmonic designed CP-Ti and Ti-6Al-4V structures are different. This result explains why the HS design induces superior strengthening effect on CP-Ti than on Ti-6Al-4V.

Introduction

Grain refinement via microstructure manipulation is an efficient method to strengthen metallic materials. However, the fine-grained (FG) and ultrafine-grained (UFG) materials generally suffer from significant loss of ductility compared to their coarse-grained (CG) counterparts in spite of their high strength. As a result, materials with homogeneous FG or UFG microstructure lack the improved strength-ductility combination, which is crucial in evaluating the performance of metals of interest.

To overcome the disadvantage of homogeneous FG/UFG materials, many researchers have attempted to propose approaches to construct heterogeneous microstructures, which exhibit high strength and proper ductility [START_REF] Kang | Outstanding mechanical properties of higipressure torsion processed multiscale TWIP-cored three layer steel sheet[END_REF][START_REF] Lu | Making strong nanomaterials ductile with gradients[END_REF][START_REF] Fang | Revealing extraordinary intrinsic tensile plasticity in gradient nanograined copper[END_REF][START_REF] Ma | Towards strength-ductility synergy through the design of heterogeneous nanostructures in metals[END_REF][START_REF] Tingaud | Nickel-Tungsten Composite-Like Microstructures Processed by Spark Plasma Sintering for Structural Applications[END_REF][START_REF] Langlois | Spark Plasma Sintering as a route for producing in-demand microstructures: Application to the tensile-ductility enhancement of polycrystalline nickel[END_REF]. Among those works, Ameyama and coworkers proposed a new microstructure called 'harmonic structure' (HS) [START_REF] Fujiwara | Effects of Nano / Meso Harmonic Microstructure on Mechanical Properties in Austenitic Stainless Steel Produced by MM / HRS Process[END_REF][START_REF] Orlov | Obtaining copper with harmonic structure for the optimal balance of structure-performance relationship[END_REF]). It's actually a bimodal microstructure, where an interwoven structure of finegrained (FG) regions and separately distributed coarse-grained (CG) regions are deliberately arranged to create structural heterogeneity. Different from the conventional way of preparing bimodal microstructure materials, the HS manufacturing process is able to control the grain size, volume fraction and even spatial distribution of FG and CG regions, thus achieving reliable mechanical properties and performance.

However, the general deformation mechanisms of different harmonic structured materials are still under exploration up to now. Over the past decade, considerable effort has been put into the processing and characterization of HS materials. High strength with slight ductility loss has been reported in the experimental investigation of commercial-purity titanium within monotonic tensile tests (Vajpai et al., 2016a). The experimental observations in (Vajpai et al., 2016b) illustrated that the peculiar spatial arrangement of FG and CG regions in the HS Co-Cr-Mo promotes uniformity of strain distribution and suppresses the localized plastic deformation, thus leading to improved mechanical properties. Dirras and co-workers (2017) fabricated HS Ti-25Nb-25Zr β-titanium alloy and evaluated its mechanical properties through cyclic shear experiments in comparison with the homogeneous counterpart. For the HS Ti-25Nb-25Zr, it was reported that plastic incompatibilities because of grain size difference were accommodated through accumulating GNS appearing in the FG region/CG region vicinity. Kikuchi and coworkers (2018) studied the fatigue properties of harmonic structured Ti-6Al-4V and observed an improved fatigue resistance.

Although great progress has been achieved in applying the concept of harmonic structure design to different metals, it is clear that this progress is still insufficient for practical engineering needs. For the moment, as different materials generally have different complex microstructure features, such as spatial arrangement of grains and phases, slip system geometries or distribution of grain sizes, etc., the fundamental mechanisms of deformation and strengthening of HS materials have not been well understood. This issue is of crucial importance in optimizing the microstructure design and quantitative prediction on mechanical properties. Theoretical and numerical modeling can play an important role in this material design optimization.

It was noticed that grain sizes and volume fractions of FG/CG regions have a direct influence on mechanical properties of HS materials. Zhang and co-workers performed research on HS SUS304L and SUS316L stainless steels [START_REF] Zhang | Improvement of mechanical properties in SUS304L steel through the control of bimodal microstructure characteristics[END_REF][START_REF] Zheng | Enhanced ductility in harmonic structure designed SUS316L produced by high energy ball milling and isostatic sintering[END_REF], which depicted the dependence of material's strength and ductility on average size of all grains and the volume fraction of FG regions. The experimental study in (Vajpai et al., 2016a) on HS CP-Ti also revealed the combined effect of refined grain size and volume fraction of FG regions in material strengthening. Park and colleagues (2018) conducted research on deformation mechanism of HS 304L stainless steel by using the micro-digital image correlation (micro-DIC) technique to precisely measure the strain distribution. It was reported that the strain concentrations were detected near the FG/CG region boundaries and higher strain level was found in CG regions than in FG regions. In addition to these experimental studies, some numerical works have been carried out to investigate the deformation mechanism of a few HS materials. Liu and colleagues (2018) used a polycrystalline multi-scale approach to build a plasticity-damage finite element model for HS CP-Ti. The Hall-Petch relation was used to simulate the grain size effect of tensile specimens under monotonic loading. In a recent work [START_REF] Wang | A 3D crystal plasticity model of monotonic and cyclic simple shear deformation for commercial-purity polycrystalline Ti with a harmonic structure[END_REF] the hardening behavior of harmonic structured CP-Ti in cyclic shear tests was assessed by developing a crystal plasticity based numerical model. The strain concentration in the CG regions was well reproduced in the simulation. The numerical results indicate that the strength improvement of harmonic structured CP-Ti was chiefly caused by the FG regions. Ti-6Al-4V is the most widely used titanium alloys in lots of industrial applications by virtue of its excellent combination of high strength, corrosion property, ductility, and low weight.

Inspired by the work of Kikuchi and colleagues (2018), according to which the HS Ti-6Al-4V showed a higher fatigue resistance, we carried out a similar investigation on HS Ti-6Al-4V, attempting a better strength-ductility combination under monotonous and cyclic loading.

However, the mechanical tests showed that the improvement is not as significant as expected, contrariwise to the remarkable performance of HS CP-Ti. Therefore, besides microstructure specificities, the strengthening effect of the HS design should be different for these two metals.

Therefore, the main motivation of the present work is to better understand these mechanisms by means of a detailed numerical investigation. This work is focused on understanding the general deformation mechanism of HS materials by performing numerical simulations of strengthening behavior under monotonous and cyclic simple shear loading condition for two typical materials, namely CP-Ti and Ti-6Al-4V. To this end, a new approach combining a crystal plasticity model for lamellar α+β colonies [START_REF] Mayeur | Three-dimensional modeling of titanium-aluminum alloys with application to attachment fatigue[END_REF] with the numerical model developed in authors' recent work [START_REF] Wang | A 3D crystal plasticity model of monotonic and cyclic simple shear deformation for commercial-purity polycrystalline Ti with a harmonic structure[END_REF] has been explored and implemented.

In Section 2 of this paper, two sets of monotonous and cyclic simple shear tests on homogeneous coarse-grained and harmonic structured CP-Ti and Ti-6Al-4V are presented. The characteristics of microstructure for all specimens and the measurement of key microstructure lengths are also described in this section. Section 3 describes the crystal plasticity model for lamellar α+β colonies and for equiaxed α phases as well as the scale transition rule to develop a numerical model which can be used to simulate the deformation behavior of HS CP-Ti and HS Ti-6Al-4V. Section 4 demonstrates details of numerical simulations including the used slip systems, the identification procedure of model parameters and a set of finite element model validations for specimens. Section 5 presents comparisons of numerical and experimental results and comparisons of simulated deformation behaviors between CP-Ti and Ti-6Al-4V. The emphasis is on the discussion of the deformation mechanism of HS CP-Ti and HS Ti-6Al-4V as well as the general strengthening mechanism of HS materials. Finally, some conclusions are made in Section 6.

Materials and experiments

The study of general deformation mechanism of HS materials requires the combination of experimental and numerical research on different materials, thus CP-Ti and Ti-6Al-4V are chosen for the experiments due to the typical microstructures of their HS and homogeneous CG specimens.

Plasma Rotating Electrode Process (PREP) has been applied to fabricate the CP-Ti and Ti-6Al-4V powders whose average particle size was about 140μm and 180μm respectively. The amount of titanium in CP-Ti is 99.81% (wt%). The chemical composition (wt%) of Ti-6Al-4V powders is as follows: Al(6.51) V(4.26) O(0.18) Fe(0.17) C(0.01) N(0.003) H(0.002) Ti(balance).

To produce bulk HS CP-Ti and Ti-6Al-4V specimens, the PREP powders were further processed by mechanical milling (MM) in a planetary ball mill at room temperature under argon gas atmosphere for 100 hours and 25 hours respectively. To make the bulk CP-Ti material, the as-PREP powders and MM powders were consolidated respectively by spark plasma sintering (SPS) at 1073K (800 ) under vacuum atmosphere for 10 min. While for Ti-6Al-4V, the SPS condition was adjusted to 1123K (850 ) and 30 min. More information about the processing techniques can be found in (Ota et al., 2014a(Ota et al., , 2014b)). Three methods, namely SEM, EBSD and Energydispersive X-ray spectroscopy (EDS) are used to provide microstructural information for numerical study. [START_REF] Hocini | Étude comparative du comportement mécanique et des mécanismes de déformation sous cisaillement simple et cyclique des alliages de titane élaborés par métallurgie des poudres: structures harmoniques versus alliages conventionnels[END_REF].

The microstructures of HS and homogeneous samples are analyzed by scanning electron microscopy (SEM) coupled with EBSD analysis. For CG CP-Ti, Figure 1b shows that the microstructure is quite 'homogeneous' with an average grain size of approximately 25μm. In showing the grain sizes and their distribution for homogeneous coarse-grained Ti-6Al-4V and harmonic structured Ti-6Al-4V, respectively [START_REF] Hocini | Étude comparative du comportement mécanique et des mécanismes de déformation sous cisaillement simple et cyclique des alliages de titane élaborés par métallurgie des poudres: structures harmoniques versus alliages conventionnels[END_REF]. Notice the grain size gradient in the fine-grained regions (blue ≈0.45-2 µm;

green ≈2-5 µm).

(a) (b)

(c) (d)
The identification of key microstructure length scales is directly dependent on the microstructure characterizations. For Ti-6Al-4V specimens, the average size of the lamellar colony is assumed to equal to the average length of α laths. EDS is applied to measure the relative abundance of presented chemical elements in α lath and β lath respectively. This method provides clearer microstructure images, permitting a convincing and precise measurement of the average widths of α and β laths, as shown in Figure 3. For the homogeneous CG Ti-6Al-4V, the measured average lamellar α+β colony size, widths of α lath and β lath are 49.4 , 2.5 and 0.2 . For the HS Ti-6Al-4V, the volume fraction of FG regions is 13.34% and the average grain size of equiaxed α phase is 2.05 . Based on experimental identification, the lamellar α+β colonies in CG regions can be assumed to have the same microstructure lengths as measured for homogeneous CG Ti-6Al-4V. 1. The application of the harmonic structure design on both CP-Ti and Ti-6Al-4V

improves the material strength in comparison with their homogeneous CG counterparts, in both the monotonous and cyclic shear tests.

2. However, the increase of Ti-6Al-4V's strength resulting from the effect of harmonic structure design is less significant than that of CP-Ti. When monotonic loading imposed, the strength of the HS CP-Ti increase more than 25% with respect to the homogeneous CG counterpart, while only about 7% is gained for HS Ti-6Al-4V specimens. Under cyclic loading, this comparison is about 25% vs. 15%. In terms of absolute values, the strength increase provided from the HS design is twice bigger for CP-Ti than for Ti-6Al-4V under monotonic loading.

Under cyclic the absolute increases are similar for the both metals.

In the following sections, the above-obtained experimental results will be analyzed by developing a 3D numerical model with which detailed numerical simulations were carried out.

Simulation methods

General consideration

In numerical simulations, Voronoi's tessellations are regarded as an efficient method to generate random polycrystals [START_REF] Barbe | Intergranular and intragranular behavior of polycrystalline aggregates[END_REF]. However, direct meshing of individual grains or α+β colonies for HS Ti-6Al-4V results in a representative volume element including an extremely huge number of elements due to the huge contrast between the grain sizes in FG and CG regions, leading to thus an impractical and inefficient finite element (FE) model. To overcome this difficulty, a scale transition rule is necessary to develop a realistic 3D numerical model. Among the multi-scale transition methods, the concept of self-consistent framework has proven to be a good candidate to predict the intergranular heterogeneity, where an individual grain is regarded as an inclusion in a homogeneous medium which has the average properties of the polycrystals. In this work, an explicit self-consistent method, called 'β-rule' developed by [START_REF] Cailletaud | Une Approche Micromécanique Phénoménologique du Comportement Inélastique des Métaux[END_REF] and reshaped by [START_REF] Pilvin | Approaches Multiéchelles pour la Prévision du Comportement Inélastique des Métaux[END_REF], was employed in our numerical model.

Therefore, the direct meshing of a large number of grains for HS Ti-6Al-4V is averted and the gigantesque computation task is significantly reduced.

To implement the proposed numerical model in simulating the behavior of HS materials, three operating levels can be defined according to the scale on which the deformation is described by the constitutive relations. The scale of RVE is regarded as the first level where a sufficient number of coarse grained regions and interconnected network of fine grained regions are (a) (b)

considered. And at this scale, the macroscopic responses in terms of stress and strain are assumed to follow the linear elastic relation. The second level called mesoscopic level is then defined at each continuum point. The number of grains contained at every Gauss point is determined through either the convergence test or the microstructure characterization result. Here the 'β-rule' is used to calculate local stresses of the underlying grains from the macroscopic stresses of Gauss points. With respect to the plastic strain, the macroscopic plastic strains are estimated from the granular plastic strains through a homogenization procedure. Finally, the plasticity development is estimated at the third level corresponding to the scale of crystallographic slip system (CSS).

Crystal plasticity is applied at this level to relate slip activities to resolved shear stresses on all slip systems. The implementation of the scale transition rule 'β-rule' is then fulfilled and the three operating levels are depicted in Figure 6. Another difficulty is the complex microstructure of Ti-6Al-4V with lamellar colonies. In addition to employing the high fidelity finite element modelling in which the α phase and the β phase in lamellar α+β colonies are explicitly represented by using fin finite element mesh [START_REF] Obstalecki | Quantitative analysis of crystal scale deformation heterogeneity during cyclic plasticity using high-energy X-ray diffraction and finite-element simulation[END_REF][START_REF] Kasemer | The influence of mechanical constraints introduced by β annealed microstructures on the yield strength and ductility of Ti-6Al-4V[END_REF], an efficient alternative is to construct equivalent slip geometry of multi-phase materials with the help of simplified representations of microstructures [START_REF] Morrissey | Strain Accumulation and Shakedown in Fatigue of Ti-6Al-4V[END_REF][START_REF] Goh | Polycrystal Plasticity Simulations of Fretting Fatigue[END_REF][START_REF] Goh | Crystallographic plasticity in fretting of Ti-6Al-4V[END_REF][START_REF] Schoenfeld | Texture Effects on Shear Response in Ti-6Al-4V Plates[END_REF][START_REF] Mayeur | A three-dimensional crystal plasticity model for duplex Ti-6Al-4V[END_REF]. [START_REF] Goh | Polycrystal Plasticity Simulations of Fretting Fatigue[END_REF] made use of a 2D planar triple slip idealization where the 3D slip geometry was simplified to three active slip systems representing deformation only by prismatic slip. [START_REF] Schoenfeld | Texture Effects on Shear Response in Ti-6Al-4V Plates[END_REF] also employed a planar triple slip idealization with one system representing prismatic slip and two systems representing idealized pyramidal slip. However, such models are incapable of modeling the full spectrum of realistic textures that are commonly attained in Ti-6Al-4V polycrystals. The construction of a 3D constitutive model is of significant importance to study the effects of a wide range of orientation distributions, 3D slip systems and slip system anisotropy on deformation behavior. Direct meshing of such a complex microstructure must contain a great many of grains, and therefore is extremely costly. In the present study, a 3D crystal plasticity model [START_REF] Mayeur | A three-dimensional crystal plasticity model for duplex Ti-6Al-4V[END_REF], which can be referred as equivalent constitutive model (ECM), is adopted. The ECM integrates the slip systems of α and β phases of a laminar + colony into an equivalent super grain by following certain geometrical relationships, named as the Burgers orientation relation (BOR). The consequence of this lamellar structure and the BOR is that certain deformation modes are considered as "hard"

when the slip directions intersect the α/β interface due to the relatively small effective slip distance and other deformation modes are considered as "soft" when the slip systems slide parallel to the α/β interface or bear parallel slip planes in both phases.

By using the ECM, there is no need to explicitly mesh the individual lamellae in a FE model.

The microstructural aspect of the ECM is briefly presented in Appendix 1. More details of this model can be found in the original work of [START_REF] Mayeur | Three-dimensional modeling of titanium-aluminum alloys with application to attachment fatigue[END_REF].

The global scheme of the numerical calculations is as follows:

For CG CP-Ti and Ti6Al4V, polycrystalline RVEs are built by means of Voronoi's tessellation with a large number of grains for CG CP-Ti, or α + β colonies for CG Ti-6Al-4V.

Crystal plasticity theory is used to establish the constitutive laws. Besides, polycrystalline models are also used to verify the validity of the β-rule by applying this rule to a homogeneous RVE.

For HS CP-Ti and HS Ti-6Al-4V, the Voronoi tessellation technique is used for the RVE's formation. CG regions and FG regions are separately meshed. At each Gauss point, a large number of grains are considered for HS CP-Ti while only a finite number of α + β colonies are examined for HS Ti6Al4V in order to establish the corresponding constitutive law. The β-rule is used as a homogenization-localization procedure for macro-micro stress/strain transition.

Constitutive model

In the framework of the large deformation plasticity, the total Green-Lagrange strain tensor can be split as the sum of an elastic part and a plastic part . At the macroscopic level, the elastic portion of deformation is assumed to obey a linear elastic constitutive relationship, namely:

( -" ) (1)
where is the elastic stiffness tensor, and is the second Piola-Kirchhoff stress tensor. The stress measure is Cauchy stress $ which can be deduced from the second Piola-Kirchhoff stress as:

$ 1 det ( ( ( (2)
where ( is the deformation gradient tensor. The yield function used in this work for each slip system is chosen as [START_REF] Chaboche | Constitutive equations for cyclic plasticity and cyclic viscoplasticity[END_REF]:

) * +, * --* + -. * -/ * (3)
where i represents the slip system, -* indicates the back stress, , * denotes the resolved shear stress, / * is the threshold stress representing the initial yielding, . * being its evolution. A rate dependent formulation is employed to describe the relationship between the plastic slip rate of slip systems and the resolved shear stress. The flow rule calculating the plastic slip rate 01 * of slip system i is given by:

01 * 01 2 〈 ) * 4 * 〉 6 789, * --* : (4)
where 01 2 denotes the reference shearing rate, 4 * is the drag stress representing the slip resistance, m describes the strain-rate sensitivity, 789, * --* : represents the direction of plastic flow. Slip activity can only appear when the stresses are above the threshold for all slip systems.

For each slip system i, whose slip direction vector and slip plane normal vector are * and 8 * respectively, the corresponding resolved shear stress , * can be calculated as:

, * $ ; : = * (5) with = * 1 2 9 * ⨂8 * + 8 * ⨂ * : (6)
The back stress is assumed to evolve as a nonlinear Armstrong-Frederick direct hardening / dynamic recovery type relationship [START_REF] Mayeur | A three-dimensional crystal plasticity model for duplex Ti-6Al-4V[END_REF]:

-1 * ?01 * -@-* +01 * + (7)

According to [START_REF] Chaboche | Constitutive equations for cyclic plasticity and cyclic viscoplasticity[END_REF], isotropic hardening is the governing factor to influence the cyclic hardening. Then a phenomenological isotropic hardening rule is applied:

. 1 * A9B -. * :+01 * + (8)

Where . * is isotropic hardening variable, Q stands for its saturation, a denotes the isotropic hardening modulus.

The threshold stress / * is closely related to the free slip length of each slip. Thus a Hall-Petch type relationship is used to assess the scale effects:

/ * / 2 (1 + / C 9D * : 2.E ) (9)
where / 2 is a referential critical resolved shear stress, / C is fixed coefficient. D * is the characteristic microstructure length. For CP-Ti specimens, D * denotes the grain sizes of equiaxed α phases. While when simulating Ti-6Al-4V specimens, the D * values with respect to α+β colonies correspond to either the colony size, the α lath width or the β lath width.

The main source of intergranular heterogeneity is related to a phenomenological variable F ; which is able to correctly capture the plastic accommodation [START_REF] Cailletaud | Utilisation de modéles polycristallins pour le calcul par éléments finis[END_REF].

The granular stress tensor G H is determined by: $ ; $ + @ H (I -F ; ) (10)

I J ) H F ; K HL ( 11
)
where B is the mean of variable F ; for all grains, g represents the equiaxed α grains or lamellar α+β colonies, G indicates all grains (or colonies) considered at a continuum point, @ H denotes a material parameter, ) H denotes the volume fraction for every grain (or every colony). The evolution of F ; is calculated through a differential equation [START_REF] Cailletaud | Utilisation de modéles polycristallins pour le calcul par éléments finis[END_REF]:

F 1 H 1 H -@ N F H J+01 * + OPO *L (12)
where @ N denotes a material parameter, 1 H represents the granular plastic strain rate, tot denotes the corresponding total number of slip systems in the slip geometry, the summation is over the slip systems i of the grain g (or colony g).

When dealing with equiaxed α phases, the plastic deformation rate at the grain level 1 H is a linear summation of all activated slip systems described by the symmetrical part of the Schmid tensor in the slip geometry:

1 H J 01 * OPO *L = * (13)
When dealing with the lamellar α+β colony, the granular plastic deformation rate 1 H should be calculated through an equation considering the average volume fractions of and phases for lamellar α+β colonies:

1 H Q R J 01 * S *L = * + Q N J 01 T O TL = T (14)
where Q R and Q N denote the respective volume fractions of and phases and are in direct proportion to their widths, s is the total number of the slip systems corresponding to hcp slip geometry, t is the total number of the slip systems corresponding to bcc slip geometry, and s + t = tot. The macroscopic plastic strain rate can be estimated as the weighted average of the granular plastic strain rate as follows:

1 J ) H K HL 1 H (15)
4.0 Numerical simulations

Slip systems

There are 24 HCP slip systems considered in the model for equiaxed α phase. And for lamellar α+β colonies, 36 slip systems are considered to form the equivalent slip geometry including 24 HCP slip systems and 12 BCC slip systems. These slip systems are shown in Table A1. It is noticed that both the BCC slip systems and 12 〈U + A〉 first-order pyramidal V101 W 1X〈112 W 3〉 are able to give needed degrees of freedom to accommodate plastic deformation along (0001) R direction in the equivalent slip geometry of lamellar colonies. Besides, deformation twinning is very rare and consequently is not considered in the present model.

According to the inverse pole figure (IPF), no preferential grain orientation is observed. For the sake of brevity, only the analysis on IPF data of harmonic structured Ti-6Al-4V is presented as an example. It can be seen in Figure 7 that the maximum pole density is quite low (about 1.6). The same results same have been observed for other samples since the high temperature fabrication process used is the same. Therefore, the Euler angles which represent the grain orientations for both CP-Ti and Ti-6Al-4V can be considered as randomly distributed.

Parameter determination

Model parameters describing the crystal plasticity properties and the scale transition can be gathered from the literature or identified by the experimental tests. Due to the inherent difference between CP-Ti and Ti-6Al-4V, some parameters have to be identified on the basis of their respective mechanical tests. Experimental data of monotonous shear tests for homogeneous coarse-grained specimens and harmonic structured specimens and the cyclic shear tests with ΔΓ 1.155% for homogeneous CG specimens were emplyed to determine the model parameters.

Whereas the data from the cyclic shear tests with ΔΓ 1.155% for HS specimens were then used to examine the performance of the numerical models with the calibrated parameters.

For Ti-6Al-4V, the value of Young's modulus measured according to our experiments is akin to that in the work [START_REF] Simmons | Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook[END_REF]. For CP-Ti, the Young's modulus was set equal to that of Ti-6Al-4V and the value was also close to the measurement reported in the work of [START_REF] Seagle | Titanium and titanium alloys. Kirk-Othmer Encyle[END_REF]. The approximate value of / C was first determined by referring to that reported in (Liu et al., 2019) in which they identified the parameter as 3.8 based on experimental studies presented in ( [START_REF] Naka | Cross-slip on the first order pyramidal plane (1011 W ) of atype dislocations [121 W 0] in the plastic deformation of α-titanium single crystals[END_REF][START_REF] Naka | The low-temperature plastic deformation of α-titanium and the core structure of a-type screw dislocations[END_REF]. By fitting the yield strength of monotonic shear tests for CG CP-Ti and HS CP-Ti samples, a new value for / C which was readjusted from the approximate value and the value of / 2 were estimated. Afterwards, these two parameters were tested again and slightly changed by fitting the cyclic shear test. The same procedure was then repeated for CG Ti-6Al-4V and HS Ti-6Al-4V samples to identify the coefficients / 2 and / C .

The values of 01 2 and m in the flow rule were gathered from different literatures, i.e., from the work of Kasemer and colleagues (2017) for Ti-6Al-4V and from the study [START_REF] Hama | Crystal-plasticity finite-element analysis of anisotropic deformation in a commercially pure titanium Grade 1 sheet[END_REF] for CP-Ti, respectively. Parameters related to isotropic hardening and kinematic hardening, namely C, B, Q and a, were calibrated by fitting the monotonic hardening and cyclic hardening of homogeneous CG specimens for CP-Ti and Ti-6Al-4V respectively. The values of the slip strengths 4 * for different slip systems were determined by referring to the work of Song and colleagues (2008) and akin to those listed in the articles [START_REF] Yang | Characterization and modeling of heterogeneous deformation in commercial purity titanium[END_REF]. The slip strength of bcc slip systems was set equal to that of the basal systems of the hcp slip systems.

For CP-Ti, the values of parameters @ H and @ N are determined by referring to Barkia's work (2014) which is based on his experiments about single crystal titanium. However, due to the lack of direct reference in the literature, the parameters @ H and @ N for Ti-6Al-4V were identified by fitting the numerical overall response of monotonous shear test obtained by polycrystalline simulation for homogeneous CG Ti-6Al-4V.

The calibrated values of the used parameters are listed in Table 2. To better simulate the mechanical responses of the investigated materials, namely, homogeneous coarse-grained CP-Ti and harmonic structured CP-Ti, an array of FE model tests are conducted to determine the proper RVEs in a way similar to that described in [START_REF] Wang | A 3D crystal plasticity model of monotonic and cyclic simple shear deformation for commercial-purity polycrystalline Ti with a harmonic structure[END_REF]. A balance must be found during the construction of a representative microstructure between two opposite requirements (i) having a sufficiently large number of grains or CG regions to get a reasonable description of the local heterogeneity and to preserve the macroscopic isotropy of the RVE, (ii) reducing the time and memory cost to make the simulation computationally feasible. This allows us to efficiently carry out the computation with sufficient reliability. To this end, we carried out numerical tests by doubling successively the value of such a parameter, the number of grains, for example. When the discrepancies between the macroscopic stress-strain curves do not exceed 2.5%, we considered that the convergence is achieved.

The details of numerical tests with respect to CP-Ti material can be found in [START_REF] Wang | A 3D crystal plasticity model of monotonic and cyclic simple shear deformation for commercial-purity polycrystalline Ti with a harmonic structure[END_REF] and are not repeated for the sake of brevity. The microstructure parameters thus optimized for homogeneous coarse-grained CP-Ti and harmonic structured CP-Ti are presented in Figure 9 together with the generated RVEs. Here only the procedures of numerical tests on Ti-6Al-4V specimens are presented in the following.

Due to the lack of existing approach to generate finite element mesh for harmonic structured materials, authors developed an algorithm based on the Voronoi tessellations. The developed algorithm is able to construct many random coarse-grained regions surrounded by an interconnected 3D network of fine-grained regions whose thickness can be easily changed to control the volume fraction. However, the drawback of the developed algorithm is the faces of the generated RVE are slightly uneven. These slightly uneven surfaces have no influence on the accuracy of the numerical simulations as homogeneous strain boundary conditions are imposed on the contour by imposing a displacement vector b in the form of c * d *e f e , f e ∈ iΩ, where is the average strain tensor and iΩ represents the RVE's boundary. unnecessary to directly mesh the fine details of aggregate of lamellar α+β colonies, thus making it feasible to construct the FE meshes for CG regions in HS Ti-6Al-4V.

The values of material parameters describing the crystal plasticity properties will remain the same as that in the previous polycrystalline simulation. And only parameters @ H and @ N need to be identified by fitting the overall response obtained in the previous polycrystalline simulation using the same set of random grain orientation. In Figure 12b, it can be seen that the obtained overall responses matches well with each other when the parameters @ H and @ N are calibrated as 15.0 GPa and 0.0 MPa respectively. We can remark that the ECM coupled with 'β-rule' reproduces faithfully the overall response obtained by polycrystalline simulation. After the validation of the proposed numerical model as mentioned above, FE model tests are carried out to determine a reasonable RVE to simulate the mechanical properties of HS Ti-6Al-4V. Based on the Voronoi tessellations, a method is developed to generate a microstructure which can represent the geometric features of FG and CG regions for HS Ti-6Al-4V. The volume fraction of FG regions can be accurately controlled in the generated FE model by its 'thickness' between two neighboring CG regions. For FG regions, 50 equiaxed α grains are considered at each Gauss point and are supposed to be sufficient for the accuracy of the numerical model. The average number of lamellar α+β colonies in every CG regions can be determined by using the EBSD maps. In this simulation, 7 lamellar α+β colonies are considered for each CG region.

Another important parameter which influences the accuracy of the FE model is the number of CG regions included in the RVE. Figure 13 shows three RVEs with edge length of 1mm, which

Numerical results and discussions

The above developed model is used to simulate the microstructure-related mechanical behavior of homogeneous CG and HS specimens for CP-Ti and Ti-6Al-4V. Priority is given to

(a) (b) (c)
the representation of the numerical results on HS Ti-6Al-4V as these results cannot be found in previous literature.

Comparison with experiments for CP-Ti

The simulations of monotonous and cyclic simple shear deformation for CP-Ti with homogeneous coarse-grained and harmonic structured microstructures had been conducted in authors' previous work [START_REF] Wang | A 3D crystal plasticity model of monotonic and cyclic simple shear deformation for commercial-purity polycrystalline Ti with a harmonic structure[END_REF]. While in this study, for the sake of using the same set of constitutive relations in simulating CP-Ti and Ti-6Al-4V properties, a flow rule which could be found in numerical works about these two materials is used. Consequently, all the simulations are newly carried out and the simulation results are qualitatively similar to those presented in [START_REF] Wang | A 3D crystal plasticity model of monotonic and cyclic simple shear deformation for commercial-purity polycrystalline Ti with a harmonic structure[END_REF]. For the purpose of further analysis of difference in strengthening effect, the comparisons between numerical results and experimental data for monotonic shear test are presented here in brief. More details can be found by referring to [START_REF] Wang | A 3D crystal plasticity model of monotonic and cyclic simple shear deformation for commercial-purity polycrystalline Ti with a harmonic structure[END_REF]

Comparison with experiments for Ti-6Al-4V

The monotonous and cyclic overall stress-strain responses obtained by simulations for homogeneous CG Ti-6Al-4V and HS Ti-6Al-4V using the proposed numerical model are shown in Figure 17 harmonic structured Ti-6Al-4V in the case of cyclic simple shear condition when ΔΓ 1.155% every cycle.

Comparison between Ti-6Al-4V and CP-Ti

Strengthening effects

Referring to experimental data (Figure 15), HS CP-Ti shows a much higher ultimate shear stress (684 MPa) compared with that of homogeneous CG CP-Ti (581 MPa). The material strength is increased by 17.7%. However, Figure 17 shows that the ultimate shear stress of Ti-6Al-4V is only augmented by 6.6%, from 1021 MPa (homogeneous CG Ti-6Al-4V) to 1088 MPa (HS Ti-6Al-4V). It's apparent that the improvement of Ti-6Al-4V's strength by applying (a) (b) harmonic structure design is not as significant as that observed in CP-Ti. This phenomenon consists of the key point of discussions.

Comparison between HS CP-Ti and HS Ti-6Al-4V

Figure 19 displays the simulated effective shear stress distributions on the surfaces of the HS CP-Ti and HS Ti-6Al-4V RVEs in the case of monotonic shear condition when the overall strain reaches the largest imposed value. From Figure 19a, a sharp contrast of stress levels can be observed between the FG and CG regions in the harmonic structured CP-Ti RVE. The stress level in FG regions is apparently much higher than that in CG regions, which only approximates the overall stress. It can be deduced that for HS CP-Ti the strengthening effect is mainly caused by the FG regions. especially obvious when the imposed overall strain is small, as shown in Figure 20a. This deformation behavior shown in the simulation is in agreement with experimental observations reported in [START_REF] Park | Additional hardening in harmonic structured materials by strain partioning and back stress[END_REF]. However, for HS Ti-6Al-4V, a close inspection of Figure 20c reveals no strong correlation between the spatial arrangement of 'hard' and 'soft' CG regions and the strain distribution.

Comparison between CG CP-Ti and CG Ti-6Al-4V

Looking at the effective shear stress distribution for homogeneous CG CP-Ti and homogeneous CG Ti-6Al-4V in Figure 21, it can be noticed that there exists 'hard' grains (or colonies for Ti-6Al-4V) with high stress concentration neighboring with 'soft' grains (or colonies)

with relatively low stress level. This arrangement of neighboring 'hard' and 'soft' grains (or colonies) inevitably creates mismatches of deformation and generates heterogeneous strain distribution among the neighboring grains (or colonies), as displayed by Figure 22.

Prevision for different volume fractions of FG regions

Discussions on the strengthening mechanism

In the present numerical model, the length scale effect as depicted by Eq. ( 9) is considered as one of the main sources of strengthening effect caused by HS design, which essentially represents the obstacles formed by grain or α/β interface boundaries to dislocation motions.

Another underlying source of strengthening effect is the spatial arrangement of FG and CG regions due to the HS design, which is implicitly represented through the corresponding FE meshes.

As pointed previously, the microstructure effects of the HS design on the material properties are quite different CP-Ti and Ti-6Al-4V. For HS CP-Ti, numerical studies show that the FG regions tend to perform like a thin but relatively hard skeleton enveloping the soft CG regions. Therefore, CP-Ti with HS design is able to bear higher level of external load and endure large deformation, exhibiting a higher strength without great loss of ductility.

However, the enhancement of material strength by applying HS design to Ti-6Al-4V is not as efficient as observed in the case of HS CP-Ti. This is most likely attributed to the following two facts:

(1) From the viewpoint of length scale effect, in CG regions the average size of the lamellar α+β colonies is about 50 . However, the mechanical properties of an individual lamellar α+β colony are not determined only by this length scale due to the special deformation mechanism caused by the lamellar α+β microstructure. When the hard deformation mode is predominant in certain cases, the widths of α and β laths (2.5 and 0.2 respectively) could have a major influence on the mechanical behavior of the lamellar α+β colony [START_REF] Mayeur | Three-dimensional modeling of titanium-aluminum alloys with application to attachment fatigue[END_REF].

Consequently, even though the average size of equiaxed α grains in FG regions is about 2 , the strengthening effect caused by its relatively small grain sizes is no longer as significant as in the case of HS CP-Ti.

(2) Among the 36 slips systems in a lamellar α+β colony, 6 slip systems are related to the soft deformation modes. As shown in Figure A1, they are three basal, 1 prismatic and 2 〈111〉V110X slip systems. It can be noticed that, by obeying the Burgers orientation relationship, the distribution of these slip planes and slip directions are quite concentrated in three-dimensional space. Besides, there is a big difference between the length scales concerning the soft and hard deformation modes. Because of these two aspects, an individual lamellar α+β colony becomes significantly anisotropic. In each CG region, only 7 lamellar α+β colonies are taken into account in the numerical simulation and the grain orientation for every lamellar α+β colony is randomly generated. Therefore a CG region is very likely to be anisotropic too. When a load in a fixed direction is applied to the RVE, some CG regions become 'hard' and some other become 'soft.'

Consequently, the local stress is not concentrated in FG regions. This anisotropy strengthening mechanism, which is not significant in CP-Ti, hides the strengthening effect of the FG regions.

Concluding remarks

In the present work, we proposed a multi-scale numerical model, which incorporated a crystal plasticity model and a micro-macro transition scheme, to simulate the microstructure dependent behavior of CP-Ti and Ti-6Al-4V with or without HS design. This model is successfully implemented into a three-dimensional finite element code. It is shown that experimental results can be well reproduced for homogeneous CG and HS specimens under monotonous and cyclic simple shear loading conditions. The present study leads us to make the following concluding remarks:

1. Based on the investigations of HS CP-Ti and HS Ti-6Al-4V, a general strengthening mechanism of harmonic structure designed materials is proposed. The underlying sources of strengthening effect are mainly the length scale effect and the special arrangement of FG and CG regions. Meanwhile, the strengthening of HS materials also is directly influenced by grain orientations, the number of grains (or colonies) in CG regions and the anisotropy of slip system strength.

2. The enhancement of material strength caused by the HS design is more significant for CP-Ti than Ti-6Al-4V. For HS CP-Ti, numerical simulations clearly indicate that the strengthening is mainly caused by the FG regions, which carry an big part of external load.

However, for HS Ti-6Al-4V, even the average size of equiaxed α grains in FG regions is much small compared with the lamellar α+β colony size in CG regions, the improvement of the mechanical property is not as significant as expected. The main reason βor this counter performance is the hard deformation model in lamellar α+β colonies in Ti-6Al-4V. In fact, the small microstructure lengths in lamellar α+β colonies, i.e., the widths of α and β laths, are similar to the average size of equiaxed α grains in FG regions. Therefore, the strengthening effect of FG regions is largely hidden, resulting in a slight improvement of material strength.

Figure 1 .

 1 Figure 1. Figures of the as-processed specimens: (a) and (c) IPF maps showing grain orientation ([0001] projection) for homogeneous coarse-grained CP-Ti and harmonic structured CP-Ti, respectively. The color code is relative to the standard stereographic projection triangle (inset); (b) and (d) figures showing the grain sizes and their distribution for homogeneous coarse-grained CP-Ti and harmonic structured CP-Ti, respectively[START_REF] Hocini | Étude comparative du comportement mécanique et des mécanismes de déformation sous cisaillement simple et cyclique des alliages de titane élaborés par métallurgie des poudres: structures harmoniques versus alliages conventionnels[END_REF].

  CP-Ti exhibits a specific bimodal microstructure where FG regions (with an average grain size of 5μm) form an interconnected 3D network surrounding CG regions (with an average grain size of 30μm), seen in Figure1d. The volume fraction of FG regions is computed to be about 16.0%. Figures2a and 2bshow that the microstructure of the investigated homogeneous CG Ti-6Al-4V is characterized by the lamellar α+β colonies where the β phase lamellas are embedded in α phase transformation structures. In Figures2c and 2dharmonic microstructure can be clearly observed for HS Ti-6Al-4V. However, FG regions are mainly composed of relatively small equiaxed α phase grains and CG regions mainly consist of lamellar α+β colonies, thus significantly adding complexity to the microstructure.

Figure 2 .

 2 Figure 2. Figures illustrating the microstructural of the as-processed specimens: (a) and (c) IPF maps showing grain orientation ([0001] projection) for homogeneous coarse-grained Ti-6Al-4V and harmonic structured Ti-6Al-4V, respectively. The color code is relative to the standard stereographic projection triangle (inset); (b) and (d) figures

Figure 3 .

 3 Figure 3. EDS map showing the relative abundance of chemical elements in lath and lath. Mechanical properties were studied by performing mechanical tests for the four kinds of specimens under monotonous and cyclic simple shear loading on an MTSM20 testing apparatus with a shearing device. The samples are cylindrical plates of 20 mm in diameter and 1 mm in thickness with 15 2 1 effective sheared volume. The imposed shear strain is controlled by Γ s/ , where is the displacement of the grip, 2.0 represents the width of shear zone. A constant strain rate of 10 was used in the loading procedure. Monotonous simple shear tests were conducted for all specimens. Cyclic simple shear experiments accompanying imposed increasing overall strain amplitudes were also performed. The increment of the overall strain amplitudes in cyclic shear tests was ΔΓ 1.155% per cycle. The obtained engineering stress-strain curves are shown in Figure 4 and Figure 5. Results for CG samples and HS samples are shown in the same figure in order to emphasis the strengthening effect of the HS design. Following remarks can be made from these figures:

Figure 4 .Figure 5 .

 45 Figure 4. Experimental results for (a) harmonic structured CP-Ti and homogeneous coarse-grained CP-Ti, (b) HS Ti-6Al-4V and homogeneous CG Ti-6Al-4V in the case of monotonic simple shear condition.

Figure 6

 6 Figure 6 Images showing the implementation of 'β-rule' in the present numerical model.

Figure 7 .

 7 Figure 7. Grain size (left) and IPF (middle) maps, alongside the texture strength measurement (right) for the asprocessed harmonic structured Ti-6Al-4V. Texture strength is expressed by the scale level at the bottom of (c).

Finite

  element model tests for homogeneous CG and HS specimens are performed to investigate the reliability of the proposed numerical model. These model tests are all based on the monotonous simple shear experiments described in Section 2. The boundary condition is schematically shown in Figure 8, where the planned displacements are imposed on all surfaces of RVEs.

Figure 8 .

 8 Figure 8. Illustration of the boundary conditions of simple shear loading used in this work.

Figure 9 .

 9 Figure 9. Illustration of the used RVEs in the present work for (a) homogeneous coarse-grained CP-Ti, where every color indicates an individual grain and 100 equiaxed α grains are included, and (b) harmonic structured CP-Ti, where 30 CG regions (colorful parts) are generated and 30 equiaxed α grains are considered at each Gauss point, white part represents FG regions.When dealing with simulations of Ti-6Al-4V specimens, a cubic RVE of dimension of 1.0×1.0×1.0mm for homogeneous CG Ti-6Al-4V is constructed by using the Voronoi tessellation technique, as shown in Figure10, which contains 50 arbitrary shaped lamellar α+β colonies with random grain orientations. Here the ECM is implemented in the simulation under the framework of CPFEM. As shown in the figure, each colored region denotes an equivalent grain which essentially represents a lamellar α+β colony. By using ECM, the fine details of alternating laths of α phase and β phase are not necessary to be explicitly represented, which significantly simplifies the construction of representative geometry and makes the simulations in 3D cases more feasible. Ten polycrystalline simulations with ten sets of randomly distributed grain orientations are carried out under simple shear loading conditions. The overall stress-strain curves corresponding to the ten simulations are shown in Figure11, where the grain orientation effect to

Figure 10 .

 10 Figure 10. Illustration of (a) the RVE for homogeneous coarse-grained Ti-6Al-4V, where every colored region is an equivalent grain representing a lamellar α+β colony as shown in (b), and 50 lamellar α+β colonies are created.

Figure 11 .

 11 Figure 11. Simulated overall stress-strain curves under monotonic simple shear loading corresponding to ten sets of random grain orientations. The previous simulated overall response of polycrystalline simulation for homogeneous CG Ti-6Al-4V is taken as a reference to test the performance of 'β-rule' and then calibrate the related parameters, namely @ H and @ N . In this model test, we use a cubic RVE of dimension of 1.0 1.0 1.0 with 125 cubic elements, as shown in Figure 12a. One of the ten sets of random grain orientations and the corresponding volume fractions of the 50 lamellar α+β colonies are extracted from the previous polycrystalline simulation and then input into the UMAT programmed for the proposed numerical model. Consequently, the FE model in Figure 12a can be regarded as a homogenized counterpart of the polycrystalline FE model in Figure 10a. It's clear that the proposed numerical model which has incorporated the ECM and 'β-rule' makes it

Figure 12 .

 12 Figure 12. (a) Illustration of FE mesh used in the test of β-rule, (b) Numerical results obtained by FE model test for β-rule and polycrystalline simulation and the comparison with experimental data for homogeneous CG Ti-6Al-4V under monotonic simple shear loading condition.

  50 and 100 CG regions respectively and have the same volume fraction of FG regions (13.34%). Monotonic simple shear loading is applied in the FE model tests up to Γ 10.0% of the overall shear strain. The numerical results shown in Figure14indicate that the difference between the overall responses is not significant when using the three RVEs. And the accuracy of FE models can be obtained when the number of CG regions is 50, as the overall response of RVEs including 50 CG regions is very close to that including 100 CG regions.

Figure 13 .

 13 Figure 13. Illustration of RVEs including (a) 30, (b) 50 and (c) 100 CG regions for HS Ti-6Al-4V, where every colored part represents a CG region and the white part represents FG regions.

Figure 14 .

 14 Figure 14. Effect of the number of CG regions in the FE model on the overall response of HS Ti-6Al-4V under monotonic simple shear loading condition.

  . It's observed that the simulated overall responses for homogeneous CG and HS CP-Ti under monotonous and cyclic shear loading agree well with the experimental results in terms of the yield strength and strain hardening (Figures 15-16). The numerical model is able to capture the influence of microstructure on macroscopic properties and reproduce well the strengthening effect of harmonic structure design on CP-Ti.

Figure 15 .

 15 Figure 15. Simulated stress-strain curves and experimental results for homogeneous coarse-grained CP-Ti and harmonic structured CP-Ti in the case of monotonic simple shear condition.

Figure 16 .

 16 Figure 16. Simulated curves and experimental results for (a) homogeneous coarse-grained CP-Ti and (b) harmonic structured CP-Ti in the case of cyclic simple shear condition when ΔΓ 1.155% every cycle.

Figure 17 .

 17 Figure 17. Comparison between simulations and experimental results for homogeneous CG Ti-6Al-4V and HS Ti-6Al-4V in the case of monotonic simple shear condition.

Figure 18 .

 18 Figure 18. Simulated curves and experimental results for (a) homogeneous coarse-grained Ti-6Al-4V and (b)

  As a contrasting case, the stress field in HS Ti-6Al-4V is dramatically different, as shown in Figure19b. It can be seen that some neighboring CG regions display completely different stress levels. It appears that some CG regions are much 'harder' than the others. This phenomenon is quite different from the HS CP-Ti sample, in which stress field in the CG regions is relatively homogeneous. Moreover, no special stress concentration can be found in the FG regions. Different from the HS CP-Ti sample, no strong strengthening effect of FG regions is revealed.

Figure 19 .

 19 Figure 19. Effective shear stress distribution for (a) HS CP-Ti when Γ 24.0% and (b) HS Ti-6Al-4V whenΓ 12.7%.

Figure 20

 20 Figure20displays the effective shear strain fields for HS CP-Ti and HS Ti-6Al-4V in the case of monotonic simple shear condition. For HS CP-Ti, the relatively hard FG regions tend to carry lower strains and strain concentration mainly emerges in CG regions. This feature is

Figure 20 .

 20 Figure 20. Effective shear strain distribution for (a) HS CP-Ti when Γ 0.57%, (b) HS CP-Ti when Γ 24.0% and (c) HS Ti-6Al-4V when Γ 12.7%.

Figure 21 .

 21 Figure 21. Effective shear stress distribution for (a) homogeneous CG CP-Ti when Γ 24.0% and (b) homogeneous CG Ti-6Al-4V when Γ 12.7%.

Figure 22 .

 22 Figure 22. Effective shear strain distribution for (a) homogeneous CG CP-Ti when Γ 24.0% and (b) homogeneous CG Ti-6Al-4V when Γ 12.7%.

  It's very challenging to change one microstructure feature without modifying the others experimentally, because the variation of these features is usually highly coupled. However, it can be convenient to vary the microstructure features independently in numerical simulations to obtain a direct insight of their effect on material properties. Thus an array of synthetic specimens is generated in this section by varying the volume fraction of FG regions in HS CP-Ti and HS Ti-6Al-4V. Then simulations are conducted to study the responses to variations in this volume fraction. It should be noted that only the volume fractions of FG and CG regions are varied while and parameter values are held constant in the analysis. Referring to the reported ranges in previous experimental studies, volume fraction of FG regions for each synthetic specimen is fixed at 13.3%, 16.0%, 20.0% and 25.0% respectively. All the simulation results for HS CP-Ti and HS Ti-6Al-4V (Figure23) show a positive correlation between the volume fraction of FG regions and material strength. Material strength follows a rising trend with an increase of volume fraction of FG regions in HS CP-Ti and HS Ti-6Al-4V. As mentioned in the previous comparisons, there is a greater sensitivity to changes of HS features for CP-Ti in comparison with Ti-6Al-4V.

Figure 23 .

 23 Figure 23. Simulated responses to variations in volume fraction of FG regions conducted on synthetic specimens (a) for HS CP-Ti and (b) for HS Ti-6Al-4V in comparison with simulations for experimental specimens.

3.

  The proposed numerical model is capable of simulating deformation behavior of HS CP-Ti and HS Ti-6Al-4V. Numerical results are in good agreement with experimental data. By varying microstructure features independently in simulations, it's possible to gain a quantitative understanding of the key microstructure effects of the HS design on material properties, viz. the influence of some important parameters such as volume fractions of FG and CG regions, average size of grains or colonies, or widths of α and β laths in lamellar α+β colonies.

Table 2

 2 Model parameters used in this work.

	Parameter	CP-Ti	Value Ti-6Al-4V	Parameter	CP-Ti	Value Ti-6Al-4V
	4 Z[S[T (MPa)	353	353	m	50.0	9.0
	4 \*S6 (MPa)	397	397	01 2 ( )	0.001	1.0
	4 C\][^( MPa)	441	441	Q(MPa)	207.0	50.0
	4 C\]_`[^( MPa)	503	503	a	14.5	2.0
	4 Z__ (MPa)	-	353	B(MPa)	815.0	2070.0
	E(GPa)	106.0	106.0	C	10.0	5.0
	a / 2 (MPa)	0.3 40.7	0.3 48.0	@ N (MPa) @ H (GPa)	65.5 45.0	0.0 15.0
	/ C ( 2.E )	4.4	5.15

Acknowledgements

The support by the French National Research Agency in the framework of ANR 14-CE07-0003 "HighS-Ti" program is highly acknowledged. Xiang WANG highly acknowledges the financial support from China Scholarship Council (CSC). G. Dirras is grateful to Dr D. Tingaud and Dr. F. Mompiou for fruitful discussions regarding EBSD and EDS investigations.

Appendix 1: Microstructural aspect of the equivalent constitutive model

In this Appendix, the microstructure aspect of the equivalent constitutive model (ECM) is briefly presented in order to make the paper more self-consistent. More complete description of the ECM concept can be found in the original papers of [START_REF] Mayeur | Three-dimensional modeling of titanium-aluminum alloys with application to attachment fatigue[END_REF][START_REF] Mayeur | A three-dimensional crystal plasticity model for duplex Ti-6Al-4V[END_REF] and [START_REF] Zhang | Microstructure-based crystal plasticity modeling of cyclic deformation of Ti-6Al-4V[END_REF].

A1: Microstructure of a lamellar α+β colony Ti-6Al-4V alloy possess a complex microstructure essentially characterized by the lamellar α+β colonies containing secondary α and β phases arranged in lamellar structure. There are 24 different slip systems in the α phase and 12 slip systems in the β phase. The slip systems in the α+β lamella are listed in Table A1.

Table A1

Slip systems used in this study.

Phase Crystal type

Slip families Indices Number of slip systems

The crystallographic orientation of the α and β lamellae in lamellar α+β colonies are strictly coupled to each other obeying the Burgers orientation relationship [START_REF] Lutjering | Titanium[END_REF][START_REF] Suri | Room temperature deformation and mechanisms of slip transmission in oriented single-colony crystals of an α/β titanium alloy[END_REF]. These relationships are as follows: (0 0 0 1) α // {1 1 0}β and m 1 1 2 W 0 n o // < 1 1 1 >β as shown in Figure A1a.

A3: Equivalent constitutive model (ECM)

Thanks to the special microstructure of the α+β colonies and the Burgers orientation relation, the entire laminar α+β colony can be considered as a single super grain in the crystal plasticity analysis (Figure A1b). Inspired by the notion in the work of [START_REF] Goh | Crystallographic plasticity in fretting of Ti-6Al-4V[END_REF], [START_REF] Mayeur | Three-dimensional modeling of titanium-aluminum alloys with application to attachment fatigue[END_REF] assumed that slip systems having a slip direction intersecting the α/β interfaces are difficult to activate because of the comparatively short effective slip distance. And the threshold stresses of


Adèle De Masson

Jean-David Bouaziz

Luc Sulimovic

Charles Cassius

Marie Jachiet

Marius-Anton Ionescu

Michel Rybojad

Martine Bagot
email: martine.bagot@aphp.fr

Tu-Anh Duong

Dominique Denjean

Marie-Pierre Labarthe

Maud Bézier

Marie Risbourg

Geneviève Payan

Sabrina Alain

Frédéric Mathivon

Anny Cohen-Letessier

Delphine Kerob

Jean-Philippe Hellier

Christelle Comte

Fabielle Keller

Caroline Brue

Paul Lestang

Laurence Allanore

Eliane Pierkarski-Carp

Anne Amoric

Hervé Serpier

Philippe Pruvost

Fabien Guibal

Damien Giacchero

Sandrine Elisa Funck-Brentano

Sierra Fortuny

Isabelle Gallay

Agnès Zavarro

Caroline Bider- Valle

Sylvie Lagrange

Etty Grynberg

Florence Weill

Dominique Penso

Marie Gomel

Jean Schneider

Anne Larabelle

Philippe Bonhomme

Marie-Sophie Gautier

Jean Hatchuel

Imane Mourtada

Charlotte Fite

Catherine Oliveres-Ghouti

Elisabeth Domergue

Sabrina Fourcade-Roch

Sylvie Lecanu

Nathalie Sebban

Bruno Halioua

Anne Bellut

Fabienne Keller

Isabelle Baratte

Françoise Lejoyeux

Laurence Ollivaud

Georges Abirached

Marielle Burnouf

Beatrix Reynayd- Mendel

Jean-Noël Dauendorffer

Joëlle Sebaoun

Hervé Garrat

Marie-Martine Pomper

Anne-Marie Heudes

Isabelle Beaulieu

Hugues Cartier

Amélie Arsouze

Dominique Lons- Danic

Michèle Pelletier

Valérie Gallais

Valérie Piantade

Marlène Risbourg

Georges Reuter

Serge Dahan

Murielle Creusot

Abdallah Kolli

Isabelle Egasse-Broca

Jean-Luc Rigon

Pascale Sabban

Hélène Flacher

Benoît Jaillard

Pierre André

Dominique Debjoux

Elodie Poirier

Bénédicte Solyga

Marc Perrussel

Sabrina Makhloufi

Bertrand Margnier

Clotilde Huzar

Laetitia Vandame

Hortense Thelu

Anne-Claire Chollet

Frédérique Marchal

Michael Naouri

Marion Nadaud

Elodie Boissy

Abdelhamid Lameche

Charles Berdougo

Olivo

Berdougo, Hamelin Audrey Rosado

Marie-Laure Rolland

Gabriel Fléchet

Colonna

Boglio

Rafii

Delphine Lefevre

Isabelle Jouannet

Berdah

Trouche

Françoise Santos

Isabelle Truchot

Lavallée

Prof Martine Bagot

Chilblains is a common cutaneous finding during the COVID-19 pandemic: A retrospective nationwide study from France

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

1

Chilblains are a common cutaneous finding during the COVID-19 pandemic: a retrospective nationwide study from France Adèle de Masson MD PhD 1,2 , Jean-David Bouaziz MD PhD 1,2 , Luc Sulimovic MD [START_REF] Joob | COVID-19 can present with a rash and be mistaken for Dengue[END_REF][START_REF] Alramthan | A case of COVID-19 presenting in clinical picture resembling chilblains disease. First report from the Middle East[END_REF] , Charles Cassius MD [START_REF] Joob | COVID-19 can present with a rash and be mistaken for Dengue[END_REF][START_REF] Recalcati | Cutaneous manifestations in COVID-19: a first perspective[END_REF] , Marie Jachiet MD [START_REF] Joob | COVID-19 can present with a rash and be mistaken for Dengue[END_REF] , Marius-Anton Ionescu MD 1 , Michel Rybojad MD [START_REF] Joob | COVID-19 can present with a rash and be mistaken for Dengue[END_REF] , Martine Bagot MD PhD 1B), acral (n=142, 80%, Figure 1C), morbilliform (n=25, 9%, Figure 1D-E), petechial (n=7, 3%), livedo reticularis (n=4, 1%) and other (n=41, 15%). Some patients presented with cutaneous signs fitting in multiple categories (detailed in Table 1).

Acral lesions were unexpectedly common (n=142). Chilblain-like lesions were the most frequent of the acral lesions (n=106/142, 75%, Figure 1C and supplemental Figure 1). Vesicular acral (dyshidrosis-like) lesions were reported in 20 cases (14%). Acrodynia was present in 18 cases (6%), sometimes isolated.

Among the 277 patients, 34 had a SARS-Cov2 PCR test, of which 25 were positive (74%). Seven of these 25 (28%) had acral lesions. Among the patients without a positive PCR test, 115 patients had suggestive extracutaneous symptoms (detailed in Table 1) and/or reported a close contact with a COVID-19 patient. Additionally, 59 patients had isolated chilblains (without any past history of chilblains and in the absence of cold exposure), without associated extracutaneous symptoms, which may suggest cutaneous symptoms of COVID-19, since this finding has been found in some patients with a positive SARS-Cov2 PCR test in our case series and in two recently published case reports (3,4). Histological examination of 3 chilblain-like lesions showed a lichenoid dermatitis with a perivascular and eccrine mononuclear infiltrate, and vascular microthrombi in 2 cases. Due to the retrospective, outpatient setting, and the limited number of available SARS-Cov2 PCR tests in France to date, most patients were not tested. However, the number of observed chilblain-like lesions, in patients without significant past medical history, is extremely unusual during the spring season in France, especially as people are staying inside, and may suggest a link with COVID-19.

Finally, the presence of microthrombi in chilblain patients is consistent with the altered coagulation status observed in severe COVID-19 patients (5). Although the number of tested patients does not allow us to draw firm conclusions regarding a direct link between SARS-Cov2 and these skin lesions, the unexpected outbreak of acral skin lesions in this epidemic context requires further investigation.

Figure 1 .Supplemental Figure 1 .

 11 Figure 1. Different types of skin rashes observed during the COVID-19 pandemic

  1,2* , Tu-Anh Duong MD PhD 4* , on behalf of the SNDV (French Union of Dermatologists-Venereologists) Coronavirus disease 19 (COVID-19), a pneumonia associated with severe acute respiratory syndrome coronavirus 2 (SARS-Cov2), was first identified in Wuhan, China in December 2019, and was characterized as a pandemic by the World Health Organization (WHO) on March 11, 2020.

	Dear Editor,
	Fever, dry cough, dyspnea, fatigue, anorexia, ageusia and anosmia are common symptoms of
	COVID-19. Reported skin manifestations of COVID-19 include erythematous lesions, sometimes
	with dengue-like petechiae (1), and urticaria and chickenpox-like vesicles (2). We performed a
	retrospective observational nationwide study of skin lesions encountered during the COVID-19
	pandemic in France from March 18 th to April 9 th , 2020 in an outpatient setting of French private
	practices. Patients gave informed consent for the publication of their photographs. Two hundred
	seventy-seven patients were enrolled, half of them were men, median age was 27 years (range, 2-
	98). The lesions were classified into six categories: urticarial (n=26, 9%, Figure 1A), vesicular
	(n=41, 15%, Figure
	2

Table 1 . Characteristics of the patients

 1

	Characteristic
	All patients (n)

Acknowledgments: The authors thank all the dermatologists of the French Union of Dermatologists and Venereologists SNDV, general practitioners, pediatricians, who provided data and pictures, and the patients involved in this study.


Huihui Zhang

Pingping Nie

Yongzheng Sun

Yong Shi

Fixed-time flocking problem of a Cucker-Smale type self-propelled particle model

Keywords: flocking, fixed-time, Cucker-Smale system PACS: 89.75.-k, 05.45.Xt, 05.40.Ca

In this paper, we consider the fixed-time flocking problem of the Cucker-Smale system. We present a variation of the standard Cucker-Smale system, and a simple fixed-time controller is designed. Based on the fixed-time stability theory, sufficient conditions for the system to achieve flocking within fixed time are obtained. Both our theoretical and numerical results demonstrate that the convergence time depends on the group size, and the group with high density can transit to ordered collective motion more rapidly.

Introduction

Nowadays, due to emerging applications in sensor networks, robot systems [START_REF] Park | Cucker-smale flocking with inter-particle bonding forces[END_REF] and unmanned aerial vehicles [START_REF] Belz | Spontaneous flocking in human groups[END_REF], research on the collective dynamics has received growing attention from diverse scientific disciplines [START_REF] Deisboeck | Collective behavior in cancer cell populations[END_REF][START_REF] Pinheiro | Linking individual and collective behavior in adaptive social networks[END_REF][START_REF] Tan | Evolutionary dynamics of collective behavior selection and drift: flocking, collapse, and oscillation[END_REF]. Flocking, as a typical example of collective motion, is a phenomenon in which a larger number of agents, using only limited environmental information and simple rules, organize into an ordered motion. Flocking phenomena are ubiquitous in nature and human-made systems, such as, the emergent behavior of birds, schooling of fish, and swarming of bacteria [START_REF] Wang | Collective behaviors through social interactions in bird flocks[END_REF][START_REF] Klopfer | Social interactions in discrimination learning with special reference to feeding behavior in birds[END_REF][START_REF] Weihs | Hydromechanics of fish schooling[END_REF].

Collective motion is often modeled by agent-based models which assume that each individual alters its behavior according to signals of its neighbors. The well-known Vicsek model [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF], originally proposed by Vicsek in 1995, provides a systematical framework to analyze the possible mechanisms by which the flocking may form. In Vicsek's self-propelled particle model, each particle adjusts its velocity in response to the average velocity of its near neighbors. The results obtained from the Vicsek model demonstrate that disordered group movement can transit to ordered motion as the group density increases [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF]. Some theoretical explanations of the Vicsek model can be found in [START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules[END_REF]. Self-propelled particles models are nowadays commonly used to study the collective motion. Several modifications of the Vicsek model have been introduced and analyzed both with or without leaders, with or without noise [START_REF] Vicsek | Collective motion[END_REF].

In 2007, Cucker and Smale proposed a flocking model, which assumes that each particle adjusts its velocity by adding to it a weighted average of the differences of its velocity with those of the other particles [START_REF] Ha | Emergent behavior of a cucker-smale type particle model with nonlinear velocity couplings[END_REF]. By using the graph theory and matrix theory, Cucker and Smale proved that the conditional flocking or unconditional flocking of the Cucker-Smale (C-S) system could occur, which depends on the level of communication rate [START_REF] Ha | Emergent behavior of a cucker-smale type particle model with nonlinear velocity couplings[END_REF][START_REF] Carrillo | Asymptotic flocking dynamics for the kinetic cucker-smale model[END_REF]. Various modifications of the C-S model have been addressed in recent years. For instance, in order to avoid the collisions, a repelling force between particles was added to the generic model [START_REF] Cucker | Avoiding collisions in flocks[END_REF]. The emergent behavior of the C-S system under hierarchical or rooted leadership was investigated [START_REF] Shen | Cucker-smale flocking under hierarchical leadership[END_REF][START_REF] Li | Cucker-smale flocking under rooted leadership with fixed and switching topologies[END_REF][START_REF] Li | Cucker-smale flocking with alternating leaders[END_REF][START_REF] Ha | Flocking behavior of the cucker-smale model under rooted leadership in a large coupling limit[END_REF]. The effect of the processing time delay was analyzed in [START_REF] Liu | Flocking and asymptotic velocity of the cucker-smale model with processing delay[END_REF][START_REF] Erban | A cucker-smale model with noise and delay[END_REF][START_REF] Choi | Emergent behavior of cucker-smale flocking particles with heterogeneous time delays[END_REF]. The flocking dynamics of the C-S system on digraph has been investigated [START_REF] He | Cucker-smale flocking subject to random failure on general digraphs[END_REF][START_REF] Dong | Flocking of the cucker-smale model on general digraphs[END_REF]. Mu et al. investigated the hierarchical flocking of C-S systems under random interactions [START_REF] Mu | Hierarchical cucker-smale flocking under random interactions with timevarying failure probabilities[END_REF]. By adding the multiplicative or additive noise term into the original C-S model, stochastic flocking of C-S systems in noisy environments has been investigated [START_REF] Ha | Emergence of time-asymptotic flocking in a stochastic cucker-smale system[END_REF][START_REF] Ahn | Stochastic flocking dynamics of the cucker-smale model with multiplicative white noises[END_REF][START_REF] Sun | Flocking of multi-agent systems with multiplicative and independent measurement noises[END_REF][START_REF] Sun | A positive role of multiplicative noise on the emergence of flocking in a stochastic cucker-smale system[END_REF].

Most of the previous studies focused on the asymptotical flocking, which means that the convergence time is infinite. However, in the real world, moving animal group can reach the ordered state within finite time. For example, a flock of birds can return to an orderly flight within a finite time after being influenced by occasional interference. The finite-time collective behavior of Vicsek type models has been widely studied [START_REF] Li | Finite-time consensus for nonlinear multi-agent systems with time-varying delay: An auxiliary system approach[END_REF][START_REF]Finite-time and fixed-time bipartite consensus of multi-agent systems under a unified discontinuous control protocol[END_REF][START_REF] Li | Finite-time consensus of second-order multi-agent systems via a structural approach[END_REF][START_REF] Li | Event-based discrete-time multi-agent consensus over signed digraphs with communication delays[END_REF][START_REF] Sun | Convergence time and speed of multi-agent systems in noisy environments[END_REF]. However, there are a few results about the finite-time flocking problem of C-S type systems.

Recently, Han et al. [START_REF] Han | Finite-time flocking problem of a cucker-smale-type self-propelled particle model[END_REF] investigated the finite-time flocking problem of C-S systems. In their work, a simple finite-time controller is designed and sufficient conditions for the finitetime flocking are obtained by using the theory of finite-time stability for ordinary differential equations [START_REF] Sun | A positive role of multiplicative noise on the emergence of flocking in a stochastic cucker-smale system[END_REF][START_REF] Han | Finite-time flocking problem of a cucker-smale-type self-propelled particle model[END_REF]. However, the settling time of the finite-time flocking depends on the initial conditions which are difficult to be estimated or even can not be measured. To overcome this limitation, we will design a new controller which can drive the C-S to the desired state in a finite time and, more importantly, the estimation of the settling time is independent of the initial conditions.

The main purpose of this paper is to investigate the fixed-time flocking of a C-S system. A continuous non-Lipschitz C-S type model is proposed. Based on the fixed-time stability theory [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF], sufficient conditions for the fixed-time flocking are established. It is shown that the C-S system can reach the flocking in a fixed time if the communication rate satisfies a lower bound.

The upper bound of the convergence time is also obtained, which shows a power-law relationship between the convergence time and the group size. Both our analytical and numerical results imply that individuals in groups with high density can transit to ordered collective motion more rapidly. Further, the influence of the control parameters on the convergence rate is also investigated.

The remainder of the paper is organized as follows. In section 2, we give the definition of fixed-time flocking of the C-S system and the fundamental lemmas of this article. Then, we perform an analysis for proposing a theorem in section 3; after that, in section 4, we present numerical simulations to verify the theoretical results in section 3. Finally, we summary our the The C-S model can be formulated by the following equations:

         ẋi = v i , 1 ≤ i ≤ N, vi = 1 N N j=1 ψ( x j -x i )(v j -v i ). (1)
Here, x i ∈ R d denotes the position of ith particle. Its velocity is denoted by

v i ∈ R d . The communication rate ψ(•) : [0, ∞) → [0, ∞)
quantifies the influence between ith and jth particles and is a non-increasing function, the communication rate ψ is taken as ψ(y) = K/(σ 2 + y) β . The results obtained in C-S model demonstrate that the phase transition between the unconditional and conditional flocking depends on the parameter β [START_REF] Ha | Emergent behavior of a cucker-smale type particle model with nonlinear velocity couplings[END_REF].

In this paper, we consider an interacting particle system of N identical self-propelled particles. Throughout this paper, we denote by x i the position of ith particle evaluated at t, and by v i the velocity of ith particle evaluated at t. Let ψ ij . = ψ( x j -x i ) denote the communication rate between particles i and j. In this paper, we assume that the communication rate ψ ij is locally Lipschitz continuous and uniformly bounded away from zero. This assumption means that the communications among particles always exist. The C-S type model is governed by the following differential equations:

       ẋi = v i , 1 ≤ i ≤ N vi = u F,p i + u F,q i , (2)
where u F,α i = N j=1 ψ ij sig(v j -v i ) α , sig(v j -v i ) α = (sign(v j1 -v i1 )|v j1 -v i1 | α , . . . , sign(v jd -v id )|v jd -v id | α ) T , 0 < p < 1 < q,
and sign : R → {-1, 0, 1} is the sign function. The motivation and some physical explanations for the design of the controller (2) are presented in the following Remarks 1 and 2.

Remark 1. Cucker and Smale proved that the flocking of the generic C-S model (1) can occur in an infinite time duration [START_REF] Ha | Emergent behavior of a cucker-smale type particle model with nonlinear velocity couplings[END_REF], while in [START_REF] Sun | A positive role of multiplicative noise on the emergence of flocking in a stochastic cucker-smale system[END_REF][START_REF] Han | Finite-time flocking problem of a cucker-smale-type self-propelled particle model[END_REF], the authors investigated the finite-time flocking of the C-S system by applying the theory of finite-time stability. As shown in the Introduction, the settling time of the finite-time flocking depends on the initial conditions of the system, which may be unavailable in advance. In this paper, we extend the results in [START_REF] Sun | A positive role of multiplicative noise on the emergence of flocking in a stochastic cucker-smale system[END_REF][START_REF] Han | Finite-time flocking problem of a cucker-smale-type self-propelled particle model[END_REF] to fixed-time flocking based on the technology of fixed-time control.

Remark 2. It is well known that the linear control presented in the original C-S model [START_REF] Park | Cucker-smale flocking with inter-particle bonding forces[END_REF] can only ensure the asymptotic flocking, i.e., the convergence time is infinite. In order to reach the flocking in a finite-time, the nonlinear control should be used. According to the finite-time control technology, one of the typical nonlinear control is u F,p i which means that the interactions among individuals are attractive (repulsive) when v j > v i ( or v j < v i ) . Using the nonlinear coupling u F,p i , the C-S system can reach the flocking in a finite time, which has been verified by Han et al. in [START_REF] Han | Finite-time flocking problem of a cucker-smale-type self-propelled particle model[END_REF]. However, the convergence time obtained by [START_REF] Han | Finite-time flocking problem of a cucker-smale-type self-propelled particle model[END_REF] depends on the initial states of individuals which is usually unknown or difficult to estimate. It is easy to see that,

for 0 < p < 1 < q, v j -v i p < v j -v i q when v j -v i > 1, while the opposite occurs for v j -v i < 1.
Then, the individuals will receive strong interactions by the control u F,q i (or u F,p i )

when v j -v i > 1 (or v j -v i < 1)
. Therefore, in this paper, we introduce both u F,p i and u F,q i into the protocol (2), which can ensure the C-S systems reach the flocking in a finite time and the settling time is independent of the initial states of individuals. We will show that if the communication rate has a lower bound, the fixed-time flocking of system (2) can be achieved.

The following concepts and lemmas play an important role in the proof of the main results.

Lemma 1. [START_REF] Polya | Inequalities[END_REF]. Let a 1 , a 2 , . . . , a n > 0 and 0 < r < p. Then

( n i=1 a p i ) 1/p ≤ ( n i=1 a r i ) 1/r . Lemma 2. [36] If ξ 1 , ξ 2 , .
. . , ξ n ≥ 0 and 0 < p ≤ 1, then the following inequality holds:

n i=1 ξ i p ≤ n i=1 ξ p i . Lemma 3. [36] If ξ 1 , ξ 2 , .
. . , ξ n ≥ 0 and q > 1, then the following inequality holds:

N 1-q n i=1 ξ i q ≤ n i=1 ξ q i .
Lemma 4. [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] Consider the following equation:

ẋ = f (t, x), x(0) = x 0 ,
where x ∈ R n and f : R + × R n is a nonlinear continuous function. Assume the origin is an equilibrium point of the above equation. If there exists a continuous radially unbounded function

V : R n → R + {0} such that (1) V (x) = 0 ⇐⇒ x = 0;
(2) for some α > 0, β > 0, 0 < p < 1, q > 1, any solution x(t) satisfies the inequality

V (x(t)) ≤ -αV p (x(t)) -βV q (x(t)),
then the origin is globally fixed-time stable and

V (t) ≡ 0 if t ≥ 1 α(1 -p) + 1 β(q -1)
.

Main results

In this section, we present sufficient conditions for the existence of fixed-time flocking of system [START_REF] Belz | Spontaneous flocking in human groups[END_REF]. Firstly, we give the definition of the fixed-time flocking of the C-S system.

Definition 1. We say system (2) reaches a flocking in fixed time if for any initial condition 2) satisfy the following two conditions: (i) The difference of all velocities tends to zero in fixed time, i.e., v i -v j = 0, ∀ t ≥ T max , where

x i (0), v i (0) and 1 ≤ i, j ≤ N , the solutions {x i , v i }(i = 1, . . . , N ) to (
T max = inf{T : v i -v j = 0, ∀ t ≥ T } is called the convergence time. (ii)
The diameter of a group is bounded i.e., sup

0≤t≤∞ x i -x j 2 < ∞.
Theorem 1. Consider the C-S type system [START_REF] Belz | Spontaneous flocking in human groups[END_REF]. Assume that the communication rate function ψ is locally Lipschitz continuous and has a lower bound, i.e., there exists ψ * > 0 such that

ψ * = inf 1≤i,j≤N
ψ ij . Then the system (2) can reach the flocking in fixed time. And the convergence time T 1 is estimated by

T 1 ≤ T max = 2 ψ * (2N ) p+1 2 (1 -p) + 2 ψ * d 1-q 2 2 q+1 2 N q (q -1)
.

Proof. From system (2), we have

vi = N j=1 ψ( x j -x i )[sig(v j -v i ) p + sig(v j -v i ) q ] = N j=1 ψ( x j -x i )sig(v j -v i ) p + N j=1 ψ( x j -x i )sig(v j -v i ) q .
To investigate the flocking condition for system (2), we introduce following average variables

x = 1 N N i=1 x i , v = 1 N N i=1 v i .
From the definition of the function sig(•) p , we have

ψ( x j -x i )sig(v j -v i ) p = -ψ( x i -x j )sig(v i -v j ) p . Thus N i=1 N j=1 ψ( x j -x i )sig(v j -v i ) p = N i,j=1 [ψ( x j -x i )sig(v j -v i ) p +ψ( x i -x j )sig(v i -v j ) p ] = 0.
In the same way

N i=1 N j=1 ψ( x j -x i )sig(v j -v i ) q = 0.
Therefore, we obtain from system (2) that dx = vdt, dv = 0, which yields that x(t) = x(0) + tv(0), v(t) ≡ v(0).

Let xi = x i -x, vi = v i -v be the fluctuations around the center of mass system. It is easy

to see that xc = 1 N N i=1 xi (t) = 0, vc = 1 N N i=1
vi (t) = 0. Then system (2) can be written as

         dx i = vi dt, dv i = N j=1 ψ( xj -xi )[sig(v j -vi ) p + sig(v j -vi ) q ]dt, (3)
which means that the new variables (x i , vi ) satisfy the equations ( 2) and xc ≡ 0, vc ≡ 0. To simplify the notions we drop the hat notion in the microscopic variables and use (x i , v i ) instead of (x i , vi ).

It is easy to see that

N i=1 x i (t) = N i=1 v i (t) = 0. ( 4
) Let x = (x 1 , x 2 , . . . , x N ) ∈ R d×N , v = (v 1 , v 2 , . . . , v N ) ∈ R d×N .
Take the candidate Lyapunov function:

V (t) = N i=1 v i 2 , (5)
and let

X(t) = N i=1 x i 2 . ( 6
)
Let < •, • > represent the inner product between vectors. Then from Eq. ( 4) we have

N i,j=1 v i -v j 2 = N i=1 N j=1 < v i -v j , v i -v j > = N i=1 N j=1 ||v i || 2 -2 N i=1 N j=1 < v i , v j > + N i=1 N j=1 ||v j || 2 = N i=1 N j=1 ||v i || 2 -2 N i=1 < v i , N j=1 v j > + N i=1 N j=1 ||v j || 2 = N i=1 N j=1 ||v i || 2 -2 N i=1 < v i , 0 > + N i=1 N j=1 ||v j || 2 = N i=1 V (t) -0 + N i=1 V (t) = 2N V (t). (7)
In the same way, we have

N i,j=1 x i -x j 2 = 2N X(t). (8)
It is easy to see that the difference of all individuals' velocities will tend to zero in fixed time if the function V (t) tends to 0 in fixed time. And the diameter of a group is bounded if X(t) is bound.

Next, we firstly prove that V (t) tends to zero in fixed time. Consider the time derivative

V (t) along the second equation of (2), dV (t) dt = 2 N i=1 < v i , v i > = 2 N i=1 < v i , N j=1 ψ( x j -x i )[sig(v j -v i ) p + sig(v j -v i ) q ] > (9) = 2 N i=1 N j=1 ψ( x j -x i ) < v i , sig(v j -v i ) p > + 2 N i=1 N j=1 ψ( x j -x i ) < v i , sig(v j -v i ) q > Note that N i=1 N j=1 ψ( x j -x i ) < v i , sig(v j -v i ) p > = N i=1 N j=1 ψ( x j -x i ) < v i -v j , sig(v j -v i ) p > + N i=1 N j=1 ψ( x j -x i ) < v j , sig(v j -v i ) p > (10) = - N i=1 N j=1 ψ( x j -x i ) < v j -v i , sig ( v j -v i ) p > - N i=1 N j=1 ψ( x j -x i ) < v j , sig(v i -v j ) p > .
It is clear see that

N i=1 N j=1 ψ ( x j -x i ) < v i , sig (v j -v i ) p >= N i=1 N j=1 ψ ( x j -x i ) < v j , sig (v i -v j ) p > .
Then, one has

N i=1 N j=1 ψ( x j -x i ) < v i , sig(v j -v i ) p > = - 1 2 N i=1 N j=1 ψ( x j -x i ) < v j -v i , sig(v j -v i ) p > (11) = - 1 2 N i=1 N j=1 ψ( x j -x i ) d k=1 |v jk -v ik | p+1 .
In the same way

N i=1 N j=1 ψ( x j -x i ) < v i , sig(v j -v i ) q >= - 1 2 N i=1 N j=1 ψ( x j -x i ) d k=1 |v jk -v ik | q+1 .
Then, we obtain from Eq. ( 9) that

dV (t) dt = - N i=1 N j=1 ψ( x j -x i ) d k=1 |v jk -v ik | p+1 - N i=1 N j=1 ψ( x j -x i ) d k=1 |v jk -v ik | q+1 . ( 12
)
Using Lemma 1, we can then easily show that

d k=1 |v jk -v ik | p+1 1 p+1 ≥ d k=1 |v jk -v ik | 2 1 2 = v j -v i .
10 Thus, we have

d k=1 |v jk -v ik | p+1 ≥ v j -v i p+1 . (13)
Furthermore, we obtain from Eqs. ( 12) and ( 13) that

- N i=1 N j=1 ψ( x j -x i ) d k=1 |v jk -v ik | p+1 ≤ - N i=1 N j=1 ψ( x j -x i ) v j -v i p+1 . ( 14
) If inf s≥0 ψ(s) ≥ ψ * , (15)
we can estimate that

- N i=1 N j=1 ψ( x j -x i ) d k=1 |v jk -v ik | p+1 ≤ - N i=1 N j=1 ψ * v j -v i p+1 = -ψ * N i=1 N j=1 v j -v i 2 p+1 2 .
Based on the Lemma 2, we have

- N i=1 N j=1 ψ( x j -x i ) d k=1 |v jk -v ik | p+1 ≤ -ψ *   N i=1 N j=1 v j -v i 2   p+1 2 = -ψ * 2N V (t) p+1 2 . ( 16
)
Using Lemma 3, we can then easily show that

d k=1 |v jk -v ik | q+1 = d k=1 (|v jk -v ik | 2 ) q+1 2 ≥ d 1-q 2 ( d k=1 |v jk -v ik | 2 ) q+1 2 = d 1-q 2 v j -v i q+1 .
We can estimate that

- N i=1 N j=1 ψ( x j -x i ) d k=1 |v jk -v ik | q+1 ≤ - N i=1 N j=1 ψ( x j -x i )d 1-q 2 v j -v i q+1 ≤ - N i=1 N j=1 ψ * d 1-q 2 v j -v i q+1 (17) = -ψ * d 1-q 2 N i=1 N j=1 v j -v i 2 q+1 2 ≤ -ψ * d 1-q 2 N 1-q 2 N i=1 N j=1 v j -v i 2 q+1 2 = -(dN ) 1-q 2 * 2N V (t) q+1 2 .
Then, we obtain that

dV (t) dt ≤ -ψ * 2N V (t) p+1 2 -(dN ) 1-q 2 ψ * 2N V (t) q+1 2 .
Finally, from Lemma 4, we have

V (t) ≡ 0, t ≥ T 1 ,
and the fixed time is estimated by

T 1 ≤ T max = 2 ψ * (2N ) p+1 2 (1 -p) + 2 ψ * d 1-q 2 2 q+1 2 N q (q -1) . ( 18
)
Accordingly

v i (t) ≡ 0, t ≥ T 1 , i = 1, 2, . . . , N, (19)
which implies that the condition (i) of the definition of fixed-time flocking holds. Now, it is time to show that the function X(t) is bounded. According to the first equation of ( 2), we have

dX(t) dt = 2 n i=1 < x i , v i > ≤ 2 n i=1 x i v i (20) ≤ 2X 1 2 (t)V 1 2 (t).
12 By using the comparison theorem of differential equations, we have X(t) ≤ Γ(t), where Γ(t) satisfies the following equation:

dΓ(t) dt = 2 Γ(t)V 1 2 (t), (21)
with the initial condition Γ(0) = X(0). Simple calculation leads to

Γ 1 2 (t) = Γ 1 2 (0) + t 0 V 1 2 (s)ds.
Then

X 1 2 (t) ≤ X 1 2 (0) + t 0 V 1 2 (s)ds.
Since V (t) ≡ 0 after t > T 1 , thus

X 1 2 (t) ≤ X 1 2 (0) + T 1 0 V 1 2 (s)ds.
Noting the fact that V (t) ≤ V (0), we have

X 1 2 (t) ≤ X 1 2 (0) + V 1 2 (0)T 1 < ∞. or equivalently sup 0≤t≤∞ x i -x j 2 < ∞. (22)
Thus the condition (ii) of the fixed-time flocking is satisfied. This completes the proof.

Remark 3. Clearly, ψ ≥ 1/(1 + D max ) β , where D max denotes the maximum distance between two individuals. And D max has an upper bound if the system (2) can reach the flocking. Thus, the communication rate function has a lower bound, which is a necessary condition for the emergence of flocking dynamics.

Remark 4.

From Theorem 1, we can conclude that the convergence time of system (2) depends on the group size N , the lower bound of the communication rate, and the control parameter p, q.

Especially, the estimation of the convergence time in (18) uncovers a power-law relationship between the convergence time and the group size. Our analytical results show that the group with high density can transit to ordered collective motion more rapidly.

Remark 5. The finite-time flocking of C-S systems has been investigated in [START_REF] Han | Finite-time flocking problem of a cucker-smale-type self-propelled particle model[END_REF], however, our study has some significant differences from Ref. [START_REF] Han | Finite-time flocking problem of a cucker-smale-type self-propelled particle model[END_REF]. On the one hand, the controllers designed are different. The controller in [START_REF] Han | Finite-time flocking problem of a cucker-smale-type self-propelled particle model[END_REF] only has a nonlinear coupling term u F,p i and the interaction from this nonlinear control is weaker than that from the linear control when the velocity errors among individuals are large. While the controller in (2) contains two nonlinear control terms u F,p and u F,q . And the individuals can receive strong interactions from the control u F,q i when the velocity errors among individuals are very large. On the other hand, the main results are different. In the literature [START_REF] Han | Finite-time flocking problem of a cucker-smale-type self-propelled particle model[END_REF], the C-S system can reach the finite-time flocking. However, the settling time of the finite-time flocking depends on the initial conditions which are difficult to be estimated or even can not be measured. While, as a further study, in our manuscript, the setting time can be estimated to a value less than T max , which is independent of the initial conditions.

This indicates that the fixed-time flocking of the C-S system can be realized regardless of the initial conditions.

Simulation results

In this section, some numerical simulations on the C-S model ( 2) are performed to validate our main results in Theorem 1. In the simulations, for simplicity, we present results for systems in the one-dimensional setting, i.e., d = 1. The initial positions and velocities x i (0), v i (0) are uniformly taken from the interval [-3, 3], and the communication rate is taken as ψ(s) = 1/(1 + s 2 ) β . To characterize the transition to ordered flocking behavior we define two indicators:

δ v (t) = (1/N ) N i=1 [v i (t) -v c (t)] 2 , and δ x (t) = (1/N ) N i=1 [x i (t) -x c (t)] 2 .
We say the numerical flocking takes place if δ v (t) < 10 -6 and δ x (t) < ∞.

First, we present results of numerical simulations for system (2) with β = 1/4, p = 0.7, q = 2, N = 30. Next, we show the influence of parameters p, q and β on the convergence speed. We ran the simulations of system (2) with different values of p, q and β. First, we fixed q = 2, β = 1, and took p = 0.2, 0.4, 0.6, 0.8, respectively. From Fig. 2(a) we can observe a decreasing relation between convergence time and control parameter p; we fixed p = 2, β = 1, and took q = 1.5, 2.0, 2.5, 3.0, respectively. Fig. 2(b) displays the decreasing relation between convergence time and control parameter q, which implies that the system (2) can quickly achieve flocking with large value of parameter p, q. In Fig. 3, we fixed p = 0.7, q = 2 , and took β = 0.5, 1, 1.5, 2.

The definition of the communication rate function ψ shows that small values of β means strong interactions between agents. The simulation result suggests that the C-S system with strong coupling can reach flocking with high speed. Understanding the influence of group density on the convergence time is the major goal of this paper. In the following simulations, we study the dependence of the flocking behaviour on the number of individuals N . To gain the clear understanding of the influence of group size, we ran the simulation with N ∈ {2, 4, 6, . . . , 40} and β = 1/4, p = 0.7, q = 2. The convergence time T 1 (N ) derived from the simulations and calculated by using the analytical estimation in [START_REF] Ha | Flocking behavior of the cucker-smale model under rooted leadership in a large coupling limit[END_REF] are shown in Fig. 4. Both the numerical and analytical results show that as the number of individuals increases the convergence time is significantly decreased. Our result implies that the individuals in groups with high density can transit to ordered collective motion more rapidly. This is similar to the simulation results obtained from the Vicsek-type model. It is intuitively imaginable that a group with higher density leads to better connectivity of the network, underpinning the dynamics of the interacting moving agents. Thus, in a group with high density, each individual can receive more explicit information from the group, leading to better information exchanging. Therefore, a group with higher density leads to a faster convergence rate of flocking. This observation is in accord with previous experimental and theoretical studies. For example, Vicsek et al. show that moving groups can transit to ordered motion when the group density increases [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF]. Buhl et al.

found that, as locust density increases, the swarming locusts exhibit highly alignment [START_REF] Somarakis | Convergence analysis of classes of asymmetric networks of cucker-smale type with deterministic perturbations[END_REF].

Conclusion

In this paper, we have studied the flocking problem of the C-S model with a continuous and non-Lipschitz protocol. By using the fixed-time stability theorem of differential equations, sufficient conditions for the fixed-time flocking are given. Our results demonstrate that the convergence time depends on the group size and control parameters. Especially, both our analytical and numerical results uncover a power-law relationship between the convergence time and the group size, which implies that individuals in groups with high density can transit rapidly to ordered collective motion. Here, we focus on the influence of the convergence speed of flocking dynamics. The fixed-time flocking defined in this paper does not require that the collision avoidance of the group. Real bird flocks can not only achieve the fixed-time flocking defined in this paper but also individuals will not collide with each other. Thus, the collision avoidance is another important problem in flocking dynamics, which will be investigated in our future work.

Figs. 1 Figure 1 :Figure 2 :

 112 Figure 1: Flocking of the C-S type model (2) with communication rate ψ(s) = 1/(1 + s 2 ) β . (a)

Figure 3 :

 3 Figure 3: The influence of the parameter β on the convergence speed. The flocking indicator

Figure 4 :

 4 Figure 4: Variation of the convergence time with the number of individuals, N , calculated by

Acknowledgments: This work was supported by the National Natural Science Foundation of China (Grant Nos. 11711530203 and 11771443), the Fundamental Research Funds for the Cen-tral Universities under Grant No. 2017XKZD11, and the China Scholarship Council (Contract No. 201504490047).