On the potential static strength of hybrid (bolted/bonded) lap joints with functionally graded adhesives

Eric Paroissien, 1 and Frédéric Lachaud 2 Institut Clément Ader (ICA), Université de Toulouse, ISAE-SUPAERO, INSA, IMT MINES ALBI, UTIII, CNRS, [START_REF] Hart-Smith | Adhesive bonding of aircraft primary structures[END_REF] Rue Caroline Aigle, 31400 Toulouse, France

In

I. Introduction

HE joints allow for the transfer of loads between the structural components. Consequently, the structural integrity of the joined structure depends on the ability of joints to sustain the load transferred. However, the joints often lead to a reduction of static and fatigue strength [START_REF] Hart-Smith | Design methodology for bonded-bolted composite joints[END_REF] leading to a structural mass increase. The proper choice of joining technology is then decisive for the strength-to-mass ratio of the manufactured structure. Mechanical fastening or bolting, such as riveting or screwing, is a tailored joining technology, so that it is extensively applied for primary structural components. Since the end of the Second World War, adhesive bonding has been employed by civilian aircraft manufacturers for primary structures under the condition to be associated with a redundant load transfer path to be compliant with damage tolerance certification requirements. Adhesive bonding is not the dominant design solution [START_REF] Higgins | Adhesive bonding of aircraft structures[END_REF], even though, compared to bolting, adhesive bonding may offer higher static and fatigue strength at lower embedded mass [START_REF] Hart-Smith | Adhesive bonding of aircraft primary structures[END_REF]. Adhesive bonding can be used in combination with bolting leading hybrid (bolted/bonded) joiningtermed HBB. The HBB joint associates a discrete load transfer mode through the fasteners with a continuous one through the adhesive layer, each having its own stiffness. By sharing the load to be transferred between the fasteners and the adhesive layer, it is then expected to improve the strength of corresponding pure bolted joints or pure bonded joints. Besides, a thin adhesive sealant layer is commonly applied between the aircraft structural components to be bolted, in order to ensure sealing and prevent corrosion [START_REF] Dechwayukul | Analysis of the effects of thin sealant layers in aircraft structural joints[END_REF].

The HBB joining technology was presented as a relevant concept of fail-safe structures by Hart-Smith in 1982 [START_REF] Hart-Smith | Bonded-bolted composite joints[END_REF]. According to this study, HBB joints with aerospace configurations and material systems do not offer any significant increase in strength compared to bonded joints, which could be explained by the low fraction of load transferred by the fasteners. However, Hartman experimentally tested under fatigue loading single-lap bolted/sealed and HBB joints with three lines of fasteners; it was shown that the fatigue strength of HBB joints was significantly higher than the one of bolted/sealed joints [START_REF] Hartman | Fatigue tests on single-lap joints in clad 2024-T3 aluminium alloy manufactured by a combination of riveting and adhesive bonding[END_REF]. Paroissien experimentally showed that the fatigue strength of doublelap bolted/sealed joints with three lines of fasteners can be increased by replacing the sealant by an adhesive, while removing one third of fastener pitches [START_REF] Paroissien | Contribution aux Assemblages Hybrides (Boulonnés/Collés) -Application aux Jonctions Aéronautiques[END_REF]. The adhesive employed was a flexible polyurethane adhesive. It was one T of adhesives used by Kelly to experimentally show that the judicious choice of the adhesive leads to a higher static and fatigue strength of HBB joints [START_REF] Kelly | Quasi-static strength and fatigue life of hybrid (bonded/bolted) composite single-lap joints[END_REF]. Similar conclusions can be found in [START_REF] Paroissien | Contribution aux Assemblages Hybrides (Boulonnés/Collés) -Application aux Jonctions Aéronautiques[END_REF][START_REF] Kweon | Failure of carbon composite-to-aluminium joints with combined fastening and adhesive bonding[END_REF][START_REF] Choi | Experimental study on failure mechanism of hybrid composite joints with different adhesives[END_REF], while numerous published papers showed higher mechanical performances can be obtained with HBB joints such as in [START_REF] Imanaka | Fatigue Strength of Adhesive/Rivet Combined Lap Joints[END_REF][START_REF] Stewart | An experimental investigation of composite bonded and/or bolted repairs using single lap joint designs[END_REF][START_REF] Fu | Fatigue of hybrid (adhesive/bolted) joints in SRIM composites[END_REF][START_REF] Hoang-Ngoc | Simulation of Single-Lap Bonded and Hybrid (Bolted/Bonded) Joints with Flexible Adhesive[END_REF][START_REF] Sadowski | Damage and failure processes of hybrid joints: adhesive bonded aluminium plates reinforced by rivets[END_REF][START_REF] Moroni | Experimental analysis and comparison of the strength of simple and hybrid structural joints[END_REF][START_REF] Esmaeili | Investigation of bolt clamping force on the fatigue life of double lap simple bolted and hybrid (bolted/bonded) joints via experimental and numerical analysis[END_REF][START_REF] Bodjona | Load sharing in single-lap bonded/bolted composite joints. Part I: Model development and validation[END_REF][START_REF] Bodjona | Hybrid Bonded-fastened Joints and their Application in Composite Structures: A general review[END_REF][START_REF] Lopez-Cruz | Investigation of bolted/bonded composite joint behaviour using design of experiments[END_REF][START_REF] Lim | Evolution of mechanical properties of flexible epoxy adhesives under cyclic loading and its effects on composite hybrid bolted/bonded joint design[END_REF][START_REF] Paroissien | Simplified stress analysis of hybrid (bolted/bonded) joints[END_REF] for example.

The interest in functionally graded adhesive (FGA) joints has been increasing in recent years [START_REF] Lim | Evolution of mechanical properties of flexible epoxy adhesives under cyclic loading and its effects on composite hybrid bolted/bonded joint design[END_REF][START_REF] Paroissien | Simplified stress analysis of hybrid (bolted/bonded) joints[END_REF][START_REF] Kawasaki | Functionally Graded Adhesive Joints Bonded by Honeymoon Adhesion Using Two Types of Second Generation Acrylic Adhesives of Two Components[END_REF][START_REF] Durodola | Functionally graded adhesive joints -A review and prospects[END_REF]. FGAs offer the opportunity to optimize the strength of multi-material bonded joints by locally tailoring the adhesive properties and without modifying the design of the adherends to be joined. FGA joints involve a continuous variation of the adhesive properties along the overlap allowing for the homogenization of the stress distribution and load transfer. Manufacturing methods of FGA joints can be found in [START_REF] Durodola | Functionally graded adhesive joints -A review and prospects[END_REF][START_REF] Carbas | Adhesively bonded functionally graded joints by induction heating[END_REF][START_REF] Carbas | Functionally graded adhesive joints by graded mixing of nanoparticles[END_REF]. The objective of the present paper is then to assess the effect of adhesive graduation on the mechanical behaviour and static strength of HBB joints. The assessment is performed through a simplified 1D-bar stress model, able to quickly provide mechanical trends. It is exemplified on a balanced single-lap HBB joint made of thin aluminium plates and in-plane tensile loaded (Fig. 1), with two lines of identical fasteners and various types of adhesive graduation are employed. The present study could be regarded as an approach allowing for a simple and fast assessment of potential benefit in terms static strength only. The 1D-bar model for HBB joint with FGA is based on the macro-element technique [START_REF] Paroissien | Contribution aux Assemblages Hybrides (Boulonnés/Collés) -Application aux Jonctions Aéronautiques[END_REF][START_REF] Paroissien | Simplified stress analysis of hybrid (bolted/bonded) joints[END_REF][START_REF] Paroissien | Hybrid (bolted/bonded) joints applied to aeronautic parts: Analytical onedimensional models of a single-lap joint[END_REF][START_REF] Paroissien | Analytical two-dimensional model of a hybrid (bolted/bonded) singlelap joint[END_REF][START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF]. The macro-element (ME) technique has been initially developed by the authors and co-workers for the simplified stress analysis of HBB joints with homogeneous adhesives (HA) [START_REF] Paroissien | Contribution aux Assemblages Hybrides (Boulonnés/Collés) -Application aux Jonctions Aéronautiques[END_REF][START_REF] Paroissien | Simplified stress analysis of hybrid (bolted/bonded) joints[END_REF][START_REF] Paroissien | Hybrid (bolted/bonded) joints applied to aeronautic parts: Analytical onedimensional models of a single-lap joint[END_REF][START_REF] Paroissien | Analytical two-dimensional model of a hybrid (bolted/bonded) singlelap joint[END_REF] and successfully applied to the simplified stress analysis of FGA joints [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF].

II. Description of the 1D-bar model A. Hypotheses

The following hypotheses are taken (i) the adherends are linear elastic materials simulated as bars, (ii) the adhesive layer is modelled by an infinite number of linear elastic shear springs linking both adherends, (iii) the adhesive layer thickness remains constant along the overlap, and (iv) the fasteners are modelled by linear elastic shear springs linking both adherends. It results then that (i) only longitudinal displacements and normal forces are considered in the adherends, (ii) all the adhesive stress components vanish except the in-plane shear which is constant in through the thickness and (iii) the fasteners transfer the loading by in-plane shearing deformation only.

The graduation of the adhesive properties is represented by a variation of the shear adhesive modulus, termed G a , as function of its abscissa x along the overlap. It is then indicated that the induced rotation of the overlap as well as of fasteners, due the eccentricity of the load paths are not taken into account in this paper. 1D-beam simplified stress analyses of HBB joints with HA or of bonded joints with FGA can be found in [START_REF] Paroissien | Simplified stress analysis of hybrid (bolted/bonded) joints[END_REF] and [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF] respectively.

B. Modelling with the Macro-Element Technique

The ME technique is inspired by the Finite Element method and differs in the sense that the interpolation functions of MEs are not assumed. Indeed, they take the shape of solutions of the governing ordinary differential equations (ODEs) system, coming from the constitutive equations of materials and from the local equilibrium equations, related to the simplifying hypotheses. A direct consequence is only one ME is needed to simulate a full bonded overlap with a HA. Dedicated 4-nodes bonded-bars (BBa) and bonded-beams have been formulated. The main work consists in the formulation of the ME elementary stiffness matrix. The adherend located outside the overlap are modelled by bar elements, while the fasteners are modelled by shear spring elements. To take into account for the adhesive graduation, the overlap is regularly meshed with n_ME BBa elements, the length of which is then L/n_ME. A constant value of the adhesive shear modulus is assigned to each ME. It is taken equal to the actual value of the modulus graduation at the abscissa located at the middle of the ME. The actual graduation of the adhesive properties is then approximated by a stepped function. In this paper, a mesh density of 10 MEs per mm is chosen, which is sufficient to obtain accurate results as shown in [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF]. Once the stiffness matrix of the complete structure is assembled from the elementary stiffness matrices and the boundary conditions are applied, the minimization of the potential energy provides the solution (Fig. 2). The distribution along the overlap of the adherend displacements and internal forces as well as the adhesive shear stress and the bolt load transfer rate are the output of the analysis. A computer program is needed to obtain the solution. The scientific computing software SCILAB is used [START_REF]beta-2[END_REF]. The SCILAB code is attached to this paper as supplement materials.

Fig. 2 Modelling of the single-lap HBB joint with FGA by using the macro-element technique C. Elementary Stiffness Matrix Formulation

Even if the ME elementary stiffness matrix has already been described in details in [START_REF] Paroissien | Contribution aux Assemblages Hybrides (Boulonnés/Collés) -Application aux Jonctions Aéronautiques[END_REF][START_REF] Paroissien | Simplified stress analysis of hybrid (bolted/bonded) joints[END_REF][START_REF] Paroissien | Hybrid (bolted/bonded) joints applied to aeronautic parts: Analytical onedimensional models of a single-lap joint[END_REF][START_REF] Paroissien | Analytical two-dimensional model of a hybrid (bolted/bonded) singlelap joint[END_REF], the main steps are provided for the comfort of the reader. The length of the ME is quoted . The local equilibrium of both adherends (Fig. 3a) provides the following equations:

𝑑𝑁 𝑗 𝑏𝑑𝑥 = (-1) 𝑗 𝑇(𝑥), 𝑗 = 1,2 (1)
where b is the overlap width, N j the normal force in the adherend j and T the adhesive shear stress. It refers to the local equilibrium employed by Volkersen [START_REF] Volkersen | Die Nietkraftverteilung in Zugbeanspruchten Nietverbindungen mit konstanten Laschenquerschnitten[END_REF]. The constitutive equations for the adherends are such as:

𝑑𝑢 𝑗 𝑑𝑥 = 𝑁 𝑗 𝐴 𝑗 , 𝑗 = 1,2 (2)
where u j is the longitudinal displacement of the adherend j and A j is the membrane stiffness of adherend j equal to e j bE j with e j the thickness of adherend j and E j the Young's modulus of the adherend j. The constitutive equation for the adhesive layer is given by:

𝑇 = 𝐺 𝑎 𝑒 𝑎 (𝑢 2 -𝑢 1 ) (3)
The elementary stiffness matrix of the BBa element, termed K BBa , represents for the linear relationships between the nodal forces and the displacements (Fig. 3b):

( -𝑁 1 (0) -𝑁 2 (0) 𝑁 1 (Δ) 𝑁 2 (Δ) ) = 𝐾 𝐵𝐵𝑎 ( 𝑢 1 (0) 𝑢 2 (0) 𝑢 1 (Δ) 𝑢 2 (Δ)) (4)
From Eqs (1-3) a system of coupled of ODEs in u 1 and u 2 can be written and solved. The four integration constants are identified using the four nodal displacements. The normal forces N 1 and N 2 are then deduced from u 1 and u 2 with the constitutive equation in Eq. ( 2) as function of the nodal displacements. The nodal forces are finally then expressed as functions of the nodal displacements, leading to the closed-form expressions for the K BBa components, such as:

𝐾 𝐵𝐵𝑎 = 1 1+𝜒 𝐴 𝐴 2 Δ ( 𝜂Δ tanh 𝜂Δ + 1 𝜒 𝐴 1 - 𝜂Δ tanh 𝜂Δ - 𝜂Δ sinh 𝜂Δ - 1 𝜒 𝐴 𝜂Δ sinh 𝜂Δ -1 1 - 𝜂Δ tanh 𝜂Δ 𝜂Δ tanh 𝜂Δ + 𝜒 𝐴 𝜂Δ sinh 𝜂Δ -1 - 𝜂Δ sinh 𝜂Δ -𝜒 𝐴 - 𝜂Δ sinh 𝜂Δ - 1 𝜒 𝐴 𝜂Δ sinh 𝜂Δ -1 𝜂Δ tanh 𝜂Δ + 1 𝜒 𝐴 1 - 𝜂Δ tanh 𝜂Δ 𝜂Δ sinh 𝜂Δ -1 - 𝜂Δ sinh 𝜂Δ -𝜒 𝐴 1 - 𝜂Δ tanh 𝜂Δ 𝜂Δ tanh 𝜂Δ + 𝜒 𝐴 ) (5)
where:

𝜒 𝐴 = 𝐴 2 𝐴 1 (6)
𝜂 = √ 𝐺 𝑎 𝑒 𝑎 𝑏 𝐴 2 (𝜒 𝐴 + 1) (7)
The stiffness matrices of the bar elements and of the shear spring element are given by:

𝐾 𝑏𝑎𝑟,𝑗 = 𝐴 𝑗 𝑙 𝑗 ( 1 -1 -1 1 ) (8)
𝐾 𝑠𝑝𝑟𝑖𝑛𝑔 = 𝐶 ( 1 -1 -1 1 ) (9)
where l j is the adherend length outside the overlap and C the fastener stiffness. In the case of balanced HBB joints with two lines of fastener with HA, it is possible to derive closed-form expressions for the bolt load transfer rate as well as the maximal adhesive shear stress. The presence of two lines of fasteners implies that there are three bonded bays (Fig. 1). On each bay i, an ODE in N 2,i can be written from the governing equations Eqs. (1-3) and from the global equilibrium N 1,i +N 2,i =f in any abscissa x:

b) a)

u 1 () u 2 () u 1 (0) u 2 (0) -N 1 (0) -N 2 (0) N 1 () N 2 () N 1 (x+dx) N 1 (x) T.bdx N 2 (x+dx) N 2 (x)
𝑑 2 𝑁 2,𝑖 𝑑𝑥 2 -𝜂 2 𝑁 2,𝑖 = -𝜂 2 𝜒 𝐴 1+𝜒 𝐴 𝑓, 𝑖 = 1. .3 (10)
The solution of this previous ODE takes the following shape:

𝑁 2,𝑖 = 𝑐 𝑖 𝑒 -𝜂𝑥 + 𝑘 𝑖 𝑒 𝜂𝑥 + 𝜒 𝐴 1+𝜒 𝐴 𝑓, , 𝑖 = 1. .3 (11)
where c i and k i are integration constants. From the local equilibrium equation Eq. ( 1), the expression for the shear stress on each bay i is deduced:

𝑇 𝑖 = 𝜂 𝑏 (𝑘 𝑖 𝑒 𝜂𝑥 -𝑐 𝑖 𝑒 -𝜂𝑥 ), 𝑖 = 1. .3 (12)
Six boundary constants have to be written to determine the six integration constants. -d)+ 2 f. The bolt load transfer rates  i are written as functions of the relative displacements of adherends and then of the adhesive shear stress at fastener location:

𝜏 1 𝑓 = 𝐶(𝑢 2 (𝑑) -𝑢 1 (𝑑)) = 𝐶 𝑒 𝑎 𝐺 𝑎 𝑇 1 (𝑑) (13)
𝜏 2 𝑓 = 𝐶(𝑢 2 (𝐿 -𝑑) -𝑢 1 (𝐿 -𝑑)) = 𝐶 𝑒 𝑎 𝐺 𝑎 𝑇 2 (𝐿 -𝑑) (14)
The following linear system is then found:

(

1 1 0 0 0 0 𝑒 -𝜂𝑑 -𝑒 𝜂𝑑 -𝑒 -𝜂𝑑 𝑒 𝜂𝑑 0 0 𝑞𝑒 -𝜂𝑑 𝑟𝑒 𝜂𝑑
-𝑒 -𝜂𝑑 -𝑒 𝜂𝑑 0 0 0 0 𝑒 -𝜂(𝐿-𝑑) -𝑒 𝜂(𝐿-𝑑) -𝑒 -𝜂(𝐿-𝑑) 𝑒 𝜂(𝐿-𝑑) 0 0 𝑞𝑒 -𝜂(𝐿-𝑑) 𝑟𝑒 𝜂(𝐿-𝑑) -𝑒 -𝜂(𝐿-𝑑) -𝑒 𝜂(𝐿-𝑑) 0 0 0 0 𝑒 -𝜂𝐿 𝑒 𝜂𝐿 ) (

𝑐 1 𝑘 1 𝑐 2 𝑘 2 𝑐 3 𝑘 3 ) = ( - 𝜒 𝐴 1+𝜒 𝐴 𝑓 0 0 0 0 1 1+𝜒 𝐴 𝑓 ) (15)
where:

𝑞 = 1 -𝐶 𝑒 𝑎 𝐺 𝑎 𝜂 𝑏 (16
)

𝑟 = 1 + 𝐶 𝑒 𝑎 𝐺 𝑎 𝜂 𝑏 (17)
Since the joint is assumed to be balanced,  A =1 and both fasteners carry the same load. From Eqs. [START_REF] Fu | Fatigue of hybrid (adhesive/bolted) joints in SRIM composites[END_REF][START_REF] Hoang-Ngoc | Simulation of Single-Lap Bonded and Hybrid (Bolted/Bonded) Joints with Flexible Adhesive[END_REF] and from the continuity of adhesive shear stress, it comes then

T 1 (d)=T 2 (d)=T 2 (L-d)=T 3 (L-d)
, so that the linear system is reduced to the following one:

(

1 1 0 0 𝑒 -𝜂𝑑 -𝑒 𝜂𝑑 -𝑒 -𝜂𝑑 𝑒 𝜂𝑑 𝑞𝑒 -𝜂𝑑 𝑟𝑒 𝜂𝑑 -𝑒 -𝜂𝑑 𝑒 𝜂𝑑 𝑒 -𝜂𝑑 -𝑒 𝜂𝑑 𝑒 -𝜂(𝐿-𝑑) 𝑒 𝜂(𝐿-𝑑) ) ( 𝑐 1 𝑘 1 𝑐 2 𝑘 2 ) = ( - 𝑓 2 0 0 0 ) (18)
After linear combinations, the system can be reduced to:

( 1 1 𝑞 ̅𝑒 -𝜂𝑑 𝑟̅ 𝑒 𝜂𝑑 ) ( 𝑐 1 𝑘 1 ) = ( - 𝑓 2 0 ) ⇒ ( 𝑐 1 𝑘 1 ) = 1 𝑟̅ 𝑒 𝜂𝑑 -𝑞 ̅𝑒 -𝜂𝑑 ( 𝑟̅ 𝑒 𝜂𝑑 -1 -𝑞 ̅𝑒 -𝜂𝑑 1 ) ( - 𝑓 2 0 ) (19)
where:

𝑞 ̅ = 1 -cosh 𝜂(𝐿 -2𝑑) + 𝑞 sinh 𝜂(𝐿 -2𝑑) (20)
𝑟̅ = -1 + cosh 𝜂(𝐿 -2𝑑) -𝑟 sinh 𝜂(𝐿 -2𝑑) (21)
The closed-form expressions for the bolt transfer rates are then deduced from Eq. ( 13) and Eqs. (20-21) such as:

𝜏 1 = 𝐶 𝑒 𝑎 𝐺 𝑎 𝜂 𝑏 (𝑘 1 𝑒 𝜂𝑑 -𝑐 1 𝑒 -𝜂𝑑 ) 1 𝑓 = 1 2 𝐶 𝑒 𝑎 𝐺 𝑎 𝜂 𝑏 𝑟̅ +𝑞 ̅ 𝑟̅ 𝑒 𝜂𝑑 -𝑞 ̅𝑒 -𝜂𝑑 = 𝜏 2 (22)
As the maximal shear stress is located at both overlap ends in the case of HA [START_REF] Paroissien | Contribution aux Assemblages Hybrides (Boulonnés/Collés) -Application aux Jonctions Aéronautiques[END_REF][START_REF] Paroissien | Simplified stress analysis of hybrid (bolted/bonded) joints[END_REF][START_REF] Paroissien | Hybrid (bolted/bonded) joints applied to aeronautic parts: Analytical onedimensional models of a single-lap joint[END_REF][START_REF] Paroissien | Analytical two-dimensional model of a hybrid (bolted/bonded) singlelap joint[END_REF][START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF], the closed-form solution for the maximal stress T max is:

𝑇 𝑚𝑎𝑥 = 𝑇 1 (𝑥 = 0) = 𝜂 𝑏 (𝑘 1 -𝑐 1 ) = 1 2 𝑟̅ 𝑒 𝜂𝑑 +𝑞 ̅𝑒 -𝜂𝑑 𝑟̅ 𝑒 𝜂𝑑 -𝑞 ̅𝑒 -𝜂𝑑 𝜂 𝑓 𝑏 (23)

III. Numerical Testing

A. Definition of the Test Case

The example of a balanced single-lap HBB joint made of thin aluminium and two identical fasteners is chosen. It is indicated that, in terms of modelling, the fasteners could be different from each other. The geometry is then built from the diameter of the fastener shank equal to 4.8 mm. The geometrical and mechanical parameters are provided in Table 1. The fastener stiffness C chosen has the same order of magnitude obtained using the Tate and Rosenfeld [START_REF] Tate | Preliminary investigation of the loads carried by individual bolts in bolted joints[END_REF] formula for a titanium fastener. In this paper, the adhesive shear modulus is assumed to follow three different graduation laws: (i) a bilinear law (Eq. ( 26)), (ii) a power law at p=1 (Eq. ( 25)) and (iii) a power law at p=2 (Eq. ( 25)):

𝐺 𝑎 (𝑥) = 𝐺 𝑎,𝑚𝑎𝑥 (𝑥) -(𝐺 𝑎,𝑚𝑎𝑥 (𝑥) -𝐺 𝑎,𝑚𝑖𝑛 (𝑥)) | 𝑥 𝑐 -1| (24)
𝐺 𝑎 (𝑥) = 𝐺 𝑎,𝑚𝑎𝑥 (𝑥) -(𝐺 𝑎,𝑚𝑎𝑥 (𝑥) -𝐺 𝑎,𝑚𝑖𝑛 (𝑥)) ( 𝑥 𝑐 -1) 2𝑝

These graduation laws are symmetrical with respect to the middle of the overlap, where the maximal adhesive shear modulus G a,max is reached. The minimal adhesive shear modulus G a,min is obtained at both overlap ends. Both power laws show a horizontal slope at the middle of the overlap. These graduation laws are then fully defined from G a,max and G a,min , and from a parameter p modifying the shape. In this paper, the minimal adhesive shear modulus is fixed at G a_min =1 MPa. An illustration of these graduation laws along the overlap can be found in Fig. 4 for the particular case where G a,min =1 MPa and G a,max =10 MPa.

Fig. 4 Adhesive shear modulus graduation for the particular case of G a,min =1 MPa and G a,max =10 MPa

B. Load Sharing

Firstly, the load sharing between the fasteners and the adhesive layer is investigated. The bolt load transfer rate at each fastener is then computed as a function of the adhesive shear modulus G a,max for HA and FGAs (Fig. 5). For HA, it means that G a =G a,max . The selected range of variation G a,max is 1 MPa to 300 MPa. For HA, the results come from the formulae in Eq. ( 22) whereas for FGAs the ME model is used. It is shown that a similar evolution of the bolt load transfer as function of G a,max is obtained for HA and FGAs. The more the adhesive shear modulus is elevated, the less the fasteners transfer load. As for the HBB joint with HA, the HBB joint with the considered FGAs tends to behave like a pure bonded joint for higher G a,max and like a pure bolted joint for lower G a,max . Compared to the HBB joint with HA at a given G a,max , the HBB joint with FGAs at that G a,max are characterized by a higher transfer load by the fasteners. The range of G a,max for which the load is shared between the adhesive layer and the fasteners is restricted. In other words, the adhesive layer can be regarded as a third load path in addition to both load paths at fastener location is confined to low values of G a,max and the FGAs tend to enlarge the range of G a,max . For the remainder of the paper G a,max is fixed at 10 MPa, or ten times G a,min , which is an assumed level of maximal reinforcement. The concept of effective modulus presented by Adams and Mallick [START_REF] Adams | The effect of temperature on the strength of adhesively-bonded composite-aluminium joints[END_REF] allows for the use of linear structural sealants [START_REF] Hoang-Ngoc | Simulation of Single-Lap Bonded and Hybrid (Bolted/Bonded) Joints with Flexible Adhesive[END_REF][START_REF] Ramière | Jonctions hybrides boulonnées-collées. Application aux cas des structures d'avions[END_REF].

Fig. 5 Bolt load transfer rate as function of G a,max

C. Adhesive Behavior Under Loading

The adhesive shear stress distribution along the overlap for HA and FGAs are provided in Fig. 6. For HA, the results comes from the formulae in Eq. ( 23) is used whereas for FGAs the ME model is used. If both overlap ends are the location for the maximal adhesive shear stress for the HBB joint with HA, they are the location of the minimal adhesive shear stress for the HBB joint with FGA. The maximal adhesive shear stress is located at the middle of the overlap for the bilinear law and power law at p=1. It is located around the fastener location for the power law at p=2. The use of FGAs allow for this atypical shear stress distribution along the overlap [START_REF] Stein | Analytical models for functionally graded adhesive joints: A comparative study[END_REF][START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF].

Moreover, the fastener location (x=12 mm and x=36 mm) are points of discontinuity of the first derivative of the adhesive shear stress distribution for the HBB joint with HA and FGAs, due to the discrete load transfer mode at fasteners [START_REF] Paroissien | Contribution aux Assemblages Hybrides (Boulonnés/Collés) -Application aux Jonctions Aéronautiques[END_REF][START_REF] Carbas | Adhesively bonded functionally graded joints by induction heating[END_REF][START_REF] Carbas | Functionally graded adhesive joints by graded mixing of nanoparticles[END_REF]. For the geometrical and mechanical parameters selected in this paper, the HBB joint with FGAs exhibit maximal adhesive shear stress in a region of the overlap where the horizontal relative difference of longitudinal adherendthen related to the adhesive shear strainis minimal as shown in Fig. 7. It is noticed that the adhesive stresses and strains at both overlap ends in the FGA configurations appears as more or less independent on the considered graduation. The effects of the adhesive shear modulus graduation appears on the slope of the adhesive shear stress and strain distributions along the overlap as well as on the value and the position on the overlap of the maximal adhesive stresses. This behavior has already been observed by Stein et al. [START_REF] Stein | Analytical models for functionally graded adhesive joints: A comparative study[END_REF] for the case of pure bonded joints.

Besides, for ductile adhesives, it is well established that the adhesive stress is not the suitable strength criteria [START_REF] Adams | The effect of temperature on the strength of adhesively-bonded composite-aluminium joints[END_REF][START_REF] Stein | Analytical models for functionally graded adhesive joints: A comparative study[END_REF][START_REF] Hart-Smith | Adhesive-Bonded Double-Lap Joints[END_REF][START_REF] Niu | Airframe stress analysis and sizing[END_REF]. The adhesive shear strain and strain energy density distributions along the overlap for HA is then provided in Fig. 8. The adhesive shear strain energy density is defined from the adhesive shear strain and stress as it follows: In order to assess the potential benefit in static strength of the HBB joint with HA and FGAs compared to the pure bonded joint with HAtermed B s (HBB/A)it is assumed that the fastener behaviour is not affected during the loading. The potential benefit B s (HBB/A) is computed from the square root of the ratio between the maximal adhesive shear strain energy density of the pure bonded joint with HA to the maximal adhesive shear strain energy density of the pure bonded joint with HA or FGAs at fixed G a,max . The potential benefit B s (HBB/A) as function of G a,max , varying between 1 MPa and 10 MPa, is provided in Fig. 9. It is shown that, for the HBB joint with HA and FGA, the potential benefit increases with decreasing G a,maxrelated to an increase of the load transferred by the bolts. The use of FGAs instead of HA leads to a significant increase in the benefit of around 20% depending on G a,max and the graduation law. The potential benefit B s (HBB/A) does not seem to significantly vary with the graduation law, even if power law with p=1 and p=2 leads to higher benefit at higher G a,max than the bilinear graduation law. It is related to the fact that at fixed G a,max , the load carried by the fastener is higher with HA than with FGAs. Nevertheless, it has been shown that for the HBB joint with FGAs the adhesive shear strain energy density could vary the overlap abscissa and then with the actual value of G a , so that a potential benefit based on the maximal strain energy density could not be suitable. The adhesive shear strain energy density of the HBB joint with FGAs at G a,max =10 MPa is then drawn as function of the actual value of the adhesive shear modulus along the overlap and compared to the maximal adhesive shear strain energy density of the HBB joint with HA in Fig. 10, which is taken as reference. It is shown that the three graduation laws lead to adhesive shear strain energy density significantly lower than the one of the HBB joint with the HA at any G a . As a result, compared to the HBB joint with HA, the HBB joint with FGAs have a potential benefit higher than one, the value of which depends on G a,max and the graduation law. In order to assess the potential benefit in static strength of the HBB joint with HA and FGAs compared to the pure bolted jointtermed B s (HBB/F)it is assumed that the adhesive layer behaviour is not affected during the loading. The three main failure static modes of metallic bolted joints are (i) the net section failure, (ii) the bearing failure and the fastener shearing failure [START_REF] Niu | Airframe stress analysis and sizing[END_REF]. The net section failure depends on the passing load in the plate at the fastener location and the bearing and fastener shearing failure depends on the load carried by the fastener. As a result, the bolt transfer rate is a suitable parameter for the basis of a potential benefit assessment. The potential benefit B s (HBB/F) is then computed as the ratio between 0.5 and the actual bolt load transfer and is given as function of G a,max in Fig. 11. As expected from Fig. 4, the HBB joint with HA provides the maximal potential benefit B s (HBB/F). As for the HBB joint with HA, the potential benefit B s (HBB/F) for the HBB joint with FGAs tends to linearly increase with increasing G a,max . Among the three graduation law the power law at p=2 leads to higher B s (HBB/F). Besides, the potential benefit B s (HBB/F+) in static strength of the HBB joint with two lines of fasteners and with HA and FGAs compared to a pure bolted joints with three lines of fasteners is assessed. The objective is to 12. It is shown that for G a,max lightly higher than 4 MPa, a HBB joint with HA and two lines of fasteners has potentially has the same static strength as the pure bolted joint with three lines of fasteners. For HBB with FGAs, G a,max has to be taken equal from 5 MPa to 7 MPa, depending on the graduation law. Besides, following the Niu uni-axial analysis [START_REF] Niu | Airframe stress analysis and sizing[END_REF], the fatigue strength of bolted joints is related to the stress concentration factor, computed as the ratio of the peak stress at fastener hole with the remote applied stress. The peak stress is computed from the by-passed load and the transferred load so that the stress concentration factor appears a linearly increasing function of the bolt load transfer rate. As a result, assuming that the adhesive behaviour is not degraded under fatigue loaded, it could be possible, with the previous simple analysis, to simply assess the potential benefit in terms of fatigue strength of HBB joints with HA or FGA. It is indicated that Lim et al.

𝑈(𝑥) = 1 2 𝑒 𝑎 [ 𝑢 2 (𝑥) -𝑢 1 (𝑥) 𝑒 𝑎 ] 𝐺 𝑎 (𝑥) [ 𝑢 2 (𝑥) -𝑢 1 (𝑥) 𝑒 𝑎 ] = 1 2 𝐺 𝑎 (𝑥) 𝑒 𝑎 [𝑢 2 (𝑥) -𝑢 1 (𝑥)] 2 (26)
[21] has recently shown that, in the case of HBB joint with a flexible epoxy adhesive, the load carried by the fastening system increases under cyclic loading up to a constant value, allowing for a load with the adhesive layer.

C. Remarks and Limits

From the previous analysis, it is possible to represent for the potential benefit B s (HBB/F) as a function of B s (HBB/A) for HBBA with HA and FGAs such as G a,min =1 MPa and G a,max varying between 1 MPa and 10 MPa (Fig. 13). It allows for a simple representation of the potential design choices as functions of prescribed strength objectives expressed in terms of pure bonded or pure bolted joint solutions. For the HBB joint under consideration, the power graduation law at p=2 seems to be the most efficient among the graduation laws tested.

The single-lap configuration leads to the existence of adhesive peel stress due to the eccentricity of the load path.

That peel stress is mainly driven the failure of adhesive layer at both overlap ends, before reaching its maximal adhesive shear strain energy [START_REF] Hart-Smith | Adhesive bonded single lap joints[END_REF]. Several design solutions have been published to reduce the adhesive peel stress [START_REF] Da Silva | Handbook of Adhesion Technology (2 volumes)[END_REF]. The tapering of adherends at overlap ends, which allows for a progressive increase of the neutral line lag, is one solution for example [START_REF] Hart-Smith | Adhesive-bonded scarf and stepped-lap joints[END_REF][START_REF] Oterkus | Nonlinear analysis of bonded composite joints[END_REF]. When applicable to the design requirements, these solutions could promote the failure of the adhesive layer used in the HBB joints by shearing instead of peeling. Besides, the presented analysis of potential does not take into account for the progressive degradation of adhesive properties. The potential increase of the maximal applied tensile load sustainable, compared to the pure bonded configuration, cannot be captured [START_REF] Kelly | Quasi-static strength and fatigue life of hybrid (bonded/bolted) composite single-lap joints[END_REF][START_REF] Paroissien | Simplified stress analysis of hybrid (bolted/bonded) joints[END_REF].

This paper presents only a potential strength benefit which remains to be confirmed in terms of strength benefit, as function of the actual adhesive material behaviour laws and associated relevant strength criteria.

V. Conclusion

In this paper, the potential static strength benefit of HBB joints with functionally graded adhesives (FGAs) is assessed, through a shear lag type simplified stress analysis applied to a particular single-lap joint, made of two thin identical aluminum plates and with two lines of fasteners. Three particular symmetrical graduation laws, defined by the minimum adhesive shear modulus G a,min chosen equal to 1 MPa and by the maximal one G a,max , are tested. The following conclusions could be made. Firstly, compared to a homogeneous adhesive (HA) at a maximal shear modulus G a,max , the FGAs allows for a better load sharing between the adhesive layer and the fasteners on a large range of adhesive modulus. Secondly, the HBB joint with HA offers then a higher potential strength benefit for the fastening part and lower potential strength benefit for the adhesive layer than FGAs. Thirdly, for the mechanical and geometrical configuration tested, the most attractive graduation law is the power law at p=2. In a similar way, this analysis could be performed in the 1D-beam framework instead of 1D-bar analysis, possibly involving adhesive and/or fastener nonlinear material law under various loading, using the ME technique. The models presented can be potentially used to optimize the adhesive graduation law as function of the geometrical and mechanical parameters of the joints to reach fixed potential strength benefits. This work focuses only on the potential mechanical strength and does not take into account for the certification and manufacturing requirements. Finally, an interesting application HBB joining of metallic sheets with HA and FGAs could be the reduction of the load transferred by the fasteners to improve the fatigue life compared to the pure bolted joint, while ensuring the static strength under extreme loads, if the adhesive is able to sustain the applied load over the time.

  AbbreviationsBBa = bonded-bars FE = finite element FGA = functionally graded adhesive

Fig. 1

 1 Fig. 1 Geometrical parametrization of the single-lap joint, boundary conditions and applied loads

Fig. 3 a

 3 Fig. 3 a) Free body diagram of infinitesimal pieces included between x and x+dx of both adherends in the overlap region. Subscript 1 (2) refers to the upper (lower) adherend, b) Nodal forces and nodal displacements on a BBa element

  elastic analyses to take into account nonlinear behaviour. It is based on an equivalency of adhesive shear strain energy. The range 1 to 10 MPa corresponds then to highly ductile adhesives in the previous framework or to

FigFig. 8

 8 Fig. 7 Adhesive shear strain distribution along the overlap

Fig. 9 Fig. 10

 910 Fig. 9 Potential benefit B s (HBB/A) in static strength of the HBB joint with HA and FGAs compared to the pure bonded joint with HA.

Fig. 11 Fig. 12

 1112 Fig. 11 Potential benefit B s (HBB/F) in static strength of the HBB joint with HA and FGAs compared to the pure bolted joint

Fig. 13

 13 Fig. 13 Potential benefit B s (HBB/F) as function of the potential benefit B s (HBB/A) for HBB with HA andFGAs such as G a,min =1 MPa and G a,max varying between 1 MPa and 10 MPa

Table 1 Numerical and geometical parameters

 1

	b (mm)	C (N.mm -1 )	d (mm)	E 1 =E 2 (MPa)	e a (mm)	e j (mm)	f (kN)	L (mm)	l 1 =l 2 (mm)
	24	30000	12	70000	0.2	1.6	5	48	48

Ga in MPa adhesive shear strain energy density in N.mm -1 bilinear (Ga,min;Ga,max)=(1;10)MPa² power_p=1 (Ga,min;Ga,max)=(1;10)MPa² power_p=2 (Ga,min;Ga,max)=(1;10)MPa² homogeneous [maximal] maximal values reached with HBB with HA at corresponding G a evaluate

  the level of G a,max allowing for the removal of one third of fastener pitches. The maximal bolt load transfer rate is located at external fastener lines and is equal to 34.70%, by using the ME model with G a =1E-8 MPa. To compute B s (HBB/F+), this last bolt load transfer rate is taken as the reference bolt load transfer rate instead of 0.5 in B s (HBB/F). It is given as a function of G a,max in Fig.

	10
	9
	8
	7
	6
	5
	4
	3
	2
	1
	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1

Funding Sources

This work has not received any specific grant.

Appendix

This appendix aims at providing elements of validation of the simplified stress analysis presented in this paper.

Firstly, the 1D-bar simplified stress analysis is based on two well-known models: (i) the electrical analogy model for the bolted part by [START_REF] Ross | An electrical computer for the solution of shear-lag and bolted joints problems[END_REF] [START_REF] Ross | An electrical computer for the solution of shear-lag and bolted joints problems[END_REF] and (ii) the shear-lag model for the bonded part with HA by [START_REF] Volkersen | Die Nietkraftverteilung in Zugbeanspruchten Nietverbindungen mit konstanten Laschenquerschnitten[END_REF] [START_REF] Volkersen | Die Nietkraftverteilung in Zugbeanspruchten Nietverbindungen mit konstanten Laschenquerschnitten[END_REF]. In other words, the 1D-bar model for HBB joints with HA is founded on the same hypotheses as those used by Ross and by Volkersen, so that it consists in adding the punctual shear springs by Ross simulating the fasteners within the shear lag model by Volkersen. The ME technique allows for the resolution of the system of ODE's involved. The resolution by the ME technique hase already been compared with a more classical resolution scheme based on the direct integration of ODEs [START_REF] Paroissien | Contribution aux Assemblages Hybrides (Boulonnés/Collés) -Application aux Jonctions Aéronautiques[END_REF][START_REF] Paroissien | Hybrid (bolted/bonded) joints applied to aeronautic parts: Analytical onedimensional models of a single-lap joint[END_REF]. It was shown that both resolution schemes provide the same results for the same set of modelling hypotheses. In order to fit the experimental results as well as the FE predictions, the fastener stiffnesses are tuned as shown in [START_REF] Paroissien | Contribution aux Assemblages Hybrides (Boulonnés/Collés) -Application aux Jonctions Aéronautiques[END_REF]. Moreover, the 1D-bar model for HBB joints was enriched to take in to account for elasto-plastic adhesive material behavior [START_REF] Paroissien | Contribution aux Assemblages Hybrides (Boulonnés/Collés) -Application aux Jonctions Aéronautiques[END_REF][START_REF] Paroissien | Hybrid (bolted/bonded) joints applied to aeronautic parts: Analytical onedimensional models of a single-lap joint[END_REF]. This enriched model was retaken by Bois et al. [START_REF] Bois | An analytical model for the strength prediction of hybrid (bolted/bonded) composite joints[END_REF] and assessed with success against predictions from finite element (FE) models. The same conclusions on the configuration of pure bonded joints with HA can be found in a paper published by the authors of the present paper and co-workers [START_REF] Paroissien | Elasto-Plastic Analysis of Bonded Joints with Macro-Elements[END_REF]. When dealing with the FGA, the 1D-bar HBB model remains founded on both previous well-known models for pure bonded joints and pure bolted joints. In order to take into account for the graduation of adhesive properties, the ME technique is applied involving a mesh strategy. As a result, the 1D-bar model for HBB joints with FGA consists again in adding the punctual shear springs by Ross simulating the fasteners within the shear lag model by Volkersen. In [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF], it is shown that, for the 1D-model for bonded joints with FGA, the resolution scheme using the ME technique provides the same results as the resolution scheme based on Taylor expansion power series by Stein et al. [START_REF] Stein | Analytical models for functionally graded adhesive joints: A comparative study[END_REF], who made available their code as a supplementary materials. A comparison of predictions in terms of the adhesive shear stress distributions along the overlap by the Stein et al. 1D-bar code and the present 1D-bar code is provided in Fig. 14. The material and geometrical parameters described in Table 1 are chosen as well as a power law graduation with p=1 and G a,min =1 MPa and G a,max =10 MPa. Moreover the 1D-ME HBB code is forced to work without any fasteners by setting the fastener stiffnesses C equal to 0 N.mm -1 . The predictions are in very close agreement.

Fig. 14 Comparison of predictions in terms of adhesive shear stress distributions by Stein et al. code and the present 1D-bar ME HBB code with a fastener stiffness C=0 N.mm -1

The authors of the present paper and co-workers have published a recent and detailed paper dedicated to the validation of 1D-bar and 1D-beam simplified stress analyses involving the ME technique as the resolution scheme for the single-lap bonded joints with FGA under combined thermal and mechanical loadings [START_REF] Paroissien | A comparison between macro-element and finite element solutions for the stress analysis of functionally graded single-lap joints[END_REF]. The validation was performed against 1D-bar or 1D-beam FE models and 3D FE models for which the adhesive layer is modelled


M Vellas

C Fualdes

J E Morley

C Dray

L Rodriguez-Manas

A Meyer

Laurent Michel

Y Rolland

Yves Gourinat

Aeroaging -A new collaboration between life sciences experts and aerospace engineers

Keywords: Aging, composites, aircraft, life science, health

published or not. The documents may come

Introduction

A workshop was organized in November, 7th, 2016 in Toulouse behalf of the IAGG-GARN and ISAE on AeroAging between experts from both domains (medicine and aeronautics) to identify possible physical or physiological analogies, common processes and developed methodologies to detect and prevent ageing between human and airframe materials. The underlying objective was to investigate in an open discussion between experts how both area of research overlap and could be relevant each other. After reviewing the similarity between aging in life sciences and in aircraft structure, we will review how we can study aging processes in these two areas and finally we will make some propositions on what can be learned and done in practice. Probably P4 Systems medicine can give some opportunities to both maintain functioning human and aircraft performances.

Methods

Participants

A prospective observational study with one-year follow-up using the data of the Utrecht Proactive Frailty Intervention Trial (U-PROFIT). The U-PROFIT trial is a single blind threearmed cluster randomized controlled trial that was conducted in the Netherlands between 2010-2012 [START_REF] Odegard | Physical aging of epoxy polylmers and their composites[END_REF]. This trial evaluated the (cost) effectiveness of a proactive personalized primary care program to preserve daily functioning in older people (n=3092). Details and the results have been described elsewhere [START_REF] Odegard | Physical aging of epoxy polylmers and their composites[END_REF][START_REF] Pascault | Influence of hydrothermal ageing and thermal treatments on the viscoelastic behavior of DGEBA-MCDEA epoxy resin[END_REF]. Briefly, the first arm consisted of a screening and monitoring intervention based on routine primary care data to identify patients at risk. General practices in this arm we asked to use this instrument and to provide care based on current guidelines. General practices in the second arm received this instrument combined with a nurse-led care program. Registered practices nurses were extensively trained and delivered this evidence-based care program, conducted a comprehensive geriatric assessment at home, developed a care plan in close collaboration with the GP and other health care professionals [START_REF] Pascault | Influence of hydrothermal ageing and thermal treatments on the viscoelastic behavior of DGEBA-MCDEA epoxy resin[END_REF]. General practices in the third arm provided care as usual.

In the current study, data was used of participants enrolled in the control group (N=805).

Participants were recruited from 35 general practices located in and around Utrecht, the fourth city in the Netherlands, with over 300.000 citizens. The electronic medical record (EMR data) from the general practitioner (GP) was screened for eligible participants, who had to be aged 60 years and over, and at least one of the following criteria: (1) a frailty index score using the accumulated deficit approach18 with a cutoff of >0.20 19; (2) polypharmacy, defined as five or more different medications in chronic use ; (3) a consultation gap, defined as not having consulted the GP in the past three years, except for the yearly influenza vaccination. The frailty index was constructed according to the Rockwood-approach18 and consisted of a list of 50 items of International Classification of Primary Care (ICPC) and anatomic therapeutic chemical (ATC) codes [START_REF] Tsotsis | Thermo-oxidative ageing of composite materials[END_REF]19). Exclusion criteria were living in a nursing home or assisted living facility, and terminal illness. Eligible older adults were invited and participated after written informed consent. Data were collected using questionnaires at baseline, six and twelve months after inclusion.

What is aging?

There are a number of parallels between aging in life sciences and in aircraft structures (Table 1). Comparisons between human aging and airplane aging have been established: Aging, in life science, is defined as a progressive decline in physiological function, leading to an age-dependent decrease in rates of survival and reproduction [START_REF] Rose | Evolutionary Biology of Aging[END_REF]. Aging is a process that is genetically determined and environmentally modulated [START_REF] Kirkwood | Understanding the Odd science of Aging[END_REF].

For human aging, frailty is a clinical state in which there is vulnerability for developing increased dependency and/or mortality when exposed to a stressor [START_REF] Rosado-Artalejo | Global performance of executive function is predictor of risk of frailty and disability in older adults[END_REF]. Frailty can occur as the result of a range of diseases, medical and environmental conditions. Frailty is a pre-disability status still accessible to preventive measures. It needs to be distinguished from disability which is usually an irreversible condition. The disability process can be slowed by physical and mental exercise but can be accelerated by diseases (4).

To extend lifespan without disability of one individual, overcoming some of the consequences of the aging processes, preventing diseases and associated functional limitations is relevant. This approach has been demonstrated to be especially relevant in the frail elderly population. Different tools have been developed to screen frailty in community-dwelling older population: The Fried's phenotype of frailty (5), The Frailty Index [START_REF] Rockwood | Some mathematical models of frailty and their clinical implications[END_REF] or the IANA -FRAIL scale [START_REF] Abellan Van Kan | A Task Force on frailty assessment of older people in clinical practice[END_REF].

In aircraft structures

Aging is defined as a slow alteration of a material chemical or physical structure by the simple effect of time or by special treatment to which it is exposed. These constraints have a detrimental effect on the material properties. It leads to gradual loss of the design function and ultimate failure or unacceptable loss of efficiency [START_REF] David | Aging of polymeric materials: principles[END_REF]. Aging may be broadly categorized by three primary mechanisms: chemical, physical and mechanical. The interaction between these three areas is highly dependent on two variables: material characteristics and aging environment [START_REF] Gates | The physical and chemical ageing of polymeric composites[END_REF]. The long-term properties of composite materials, when exposed to a combination of in-service loads and environments are still not well characterized. The importance to predict and overcome aging is relevant. Similarly to human aging, different methods have been implemented to detect these early problems: NDT (Non destructive testing methods), e.g., visual, magnetic, radiographic and ultrasonic techniques [START_REF] Garnier | The detection of aeronautical defects in situ on composite structures using Non Destructive Testing[END_REF].

How do we predict and study aging?

In life science

Models and prediction -Animal models of aging

In medicine, Aging is the largest risk factor for numerous human diseases, and understanding the aging process may thereby facilitate the development of new treatments for ageassociated diseases. Aging research on humans is complicated by many factors like environmental variability, genetic variability, budget, and ethical factors but moreover, due to their long natural life span. An alternative is to develop and study animal and cellular models of human disease. The cellular models provide relevant information of the mechanisms; however, they do not necessarily replicate the in vivo biology. It is why, as all organisms age, animal models can be useful for studying aging. Exciting research is taking place in variety of species, commonly from yeast to mouse but also in other species closely related to human beings like monkeys.

Several cellular processes are associated to ageing. Among them, cellular senescence imposes permanent proliferative arrest on cells in response to various stressors. Cell senescence has emerged as a potentially important contributor to aging and age-related disease [START_REF] Childs | Cellular senescence in aging and age-related disease: from mechanisms to therapy[END_REF].

Cells can be considered as a clock and each cell cycle promotes damage within the cell such as DNA damage, telomere dysfunction, epigenetic changes, deregulation, metabolic defects, stem cell exhaustion, inflammatory effects, and senescence. These damaging effects bring about cell aging and human aging.

Figure 1

What is cell senescence?

One of the approaches to study human aging is to search for organisms as a model. The chosen organisms can be small, inexpensive and experimentally tractable, allowing the manipulation of the factors affecting aging. As they have a short lifespan, scientists can control the environment and minimize the genetic variation by using inbred animals.

Different species are used in animal models. These species have different lifespans due to potential disparities in several physiological mechanisms including: -Longevity / size relationship -Longevity / temperature relationship -Longevity / metabolism relationship.

The points that have been highlighted are: -The importance of animal model studies for human aging understanding. Simplified models are less complex (phenotype), have shorter lifespan and are affordable (costs). -In animal models, every environmental parameter is challenged in order to better characterize lifespan differences and assess aging. -Animal models of aging are precisely studied, from cellular level to intact animal. -The results can be scaled to human based on mathematical and statistical analysis.

The models that are most used in aging research are yeast, worm (C. elegans), fly (Drosophila melanogaster) and mouse. It has been noted that human and yeast shared pathways of aging [START_REF] Myeong | High-throughput analysis of yeast replicative aging using a microfluidic system[END_REF], that C. elegans lifespan genes have human homologues [START_REF] Kenyon | elegans mutant that lives twice as long as wild type[END_REF]; and the Drosophila insulin-like pathway studies have been highly interesting [START_REF] Tatar | A mutant Drosophila insulin receptor homolog that extends life-span and impairs neuroendocrine function[END_REF].

In aircraft structures

Qualification rules

Engineering structures must be capable of performing their function throughout a specified lifetime while meeting safety and economic objectives. These structures are exposed to a series of events that include loading, environment, and damage threats. These events, either individually or cumulatively, can cause structural degradation, and can affect the ability of the structure to perform its function. Since the beginning of aviation, we have been looking for the material the most efficient, the most sustainable, and the less sensitive to environment. Composite is at present the best material and has been developed and used today on aircraft.

The use of commercial aircraft beyond their original design service objective is a new objective for the airlines. For this reason, it is necessary to understand composite aging. The aging will be mainly caused by damages (oxydation), loading (fatigue) and environment conditions (temperature, humidity, radiation….). These events will modify the structure.

The aging of material can be defined as the transformation or the modification of some properties of the material by the simple effect of time or by the special treatment to which it is subjected. The main objective is to ensure the continued operational safety of composite structures as they age. Structures are certified by authorities (FAA for USA and EASA for European Union) in order to ensure safety. There is a constant evolution of regulations. These rules are done by a close cooperation between the authorities and the aircraft manufacturers. Composite structures long-term aging is more and more taken into consideration for certification.

Figure 2

Recommended general approach to characterization on aging of materials and structures (15

)
Medialization and prediction A composite material is constituted of a matrix (organic) and continuous, long or short fibers (glass, carbon). The matrix holds in position the fibers and ensures the load transfer to the fibers. The fibers provide stiffness, strength and anisotropy to the structure. The failure mechanism starts with the matrix cracks, then a delamination stage and finally fiber failure. The Tg (glass transition temperature) is an important parameter of the material and reflects the resin property. The glass transition is linked to the amorphous phase and has a key role on polymer properties. If the temperature is below Tg, the material is in an amorphous state, the material is stiff and brittle. When the temperature is over Tg, we are in the rubbery state. And the material is visco-elastic (flexible and ductile).

There are different types of composite aging. The first is the physical aging which is linked to this glass transition [START_REF] Odegard | Physical aging of epoxy polylmers and their composites[END_REF] showed that the density of the material increase with ageing time at T>Tg. The material properties thus depend on ageing time. Indeed, the density, stiffness and brittleness increases. The yield stress decrease the toughness decreases severely. This physical ageing is reversible with temperature rejuvenation.

The second aging category is moisture aging which is related to the water absorption in the material modeled by Fick's law. The characteristics of the diffusion in composite is linked to the anisotropy, depends on matrix percentage (~40%). The diffusion is thermal activated and generates a saturation mass (%VH). The characteristic time diffusion is: 8 days -1mm, +2 years / 10mm [START_REF] Odegard | Physical aging of epoxy polylmers and their composites[END_REF]. This moisture degradation is reversible (classically for epoxy resin) and sometimes irreversible (for step-growth polymer and for some thermoplastic). The most relevant effect of moisture aging is the plasticizing effect with a break up of hydrogen bond in the network. The material goes from stiff to flexible, the Tg decreases as well as the matrix young modulus and failure stress.

The next ageing category is Ultraviolet ageing. Composite material is sensitive to photo-oxidation, surface degradation under oxygen or ozone, accelerated with UV exposition. The mechanism is chemical. And the consequences are: a reduction of the polymer's molecular mass, the material is more brittle, leading to a reduction of tensile and impact strength as well as a discoloration and loss of surface smoothness. The effects are limited to surface, and are mainly a concern for painting, for aerodynamics and composite material during production (before painting).

The last ageing category is thermal ageing. For epoxy resin, the degradation mechanisms are post-curing and chain scission. It occurs when the temperature is over 150°C with dioxygen presence. The predominant mechanism is thermo-oxidation. At higher temperatures, there is thermolysis and thermooxidation. The thermo-oxidation is controlled by O2 diffusion and interacts with damage that leads to surface cracks.

Figure 3 Thermo-oxidation

The thermal aging effects are (9): a mass loss, Tg evolution, surface appearance and for the mechanical properties we have the diminution of elastic properties, yield strength and toughness.

Below is presented a summary table of the different aging categories. Aging affects the resin matrix and the fiber/matrix interface and lead to: -A decrease of properties such as stiffness, strength and toughness. -A surface alteration and cracks It is important to note the interaction of environmental effects with mechanical loads, the cyclic environmental conditions. The process of ageing is a time constant process and thus the need of accelerated ageing tests is essential for structures development and certification.

Design solutions

The aircraft structures are exposed to many different stressors (individually or in combination) such as: erosion, aggressive fluids, UV, fire, lightning strike (direct and indirect effect), mechanical loading (including cycling and creep), mechanical impact (in production and/or in-service), temperature and humidity [START_REF] Pascault | Influence of hydrothermal ageing and thermal treatments on the viscoelastic behavior of DGEBA-MCDEA epoxy resin[END_REF][START_REF] Tsotsis | Thermo-oxidative ageing of composite materials[END_REF].

All these parameters must be addressed in the design of the aircraft.

An extensive test survey is undertaken during the development stage for certification of the aircraft program. The tests performed on the components follow a pyramidal approach. Aged specimens are tested as well.

Figure 4

Pyramidal test approach for airframe qualification This approach is similar to the development of animal model to study human aging: influential parameters are identified on airframe aging with the building block approach from elementary coupons to large scale tests to support analogies regarding in-service life.

Airbus has developed a number of prevention solutions or demonstrations summarized below: -Erosion → Anti erosion coating -Aggressive fluids → No effect worse than moisture -UV → Anti-UV coating -Fire → Application of resin as per FST (Fire, Smoke and Toxicity) requirement -Lightning strike → Protection per layer of metallic foil and design principles -Mechanical loading → Fatigue testing and design principles -Mechanical impact → Testing -Temperature → Mechanical testing on aged specimen and design principles -Humidity → Mechanical testing on aged specimen and design principles

All these measures constitute a robust approach demonstrated with no in-service issue on more than 230 million flight hours (~26000 years)

In addition to these prevention solutions, the establishment of criteria and models allowing predicting the long-term behavior of composite structures is an important challenge for aircraft manufacturers and authorities. The optimization of the performance of structure components and thus of the materials, on longer operating lives, need robust prediction models, that consider the total of the applied stresses and external environmental threats.

Perspectives, what can we do in practice

Real Time Bone and Composite Repair

One example where the development of in-flight composite repair, which will be essential for long space flights, is the continuous repair process of human bone. The osteocytes are small cells that act as mechanostressors and can recognize cracks in bone. When this happens they activate osteoclasts to clear the damaged bone and osteoblasts to then repair the area (rebuild bone). Composite repair manufacturers are already experimenting with automatic repair for cracks in the fuselage, e.g., Racks of separated epoxy resin and amidomine or microcapsules that release dicyclopentadiene to interact with a ruthenium catalyst. Nanobots modeled on bone cell remodeling may prove to be an even more effective method for in-flight repair.

Determinants

Many factors combine together to affect the health of individuals and communities. Whether people are healthy or not, health status is determined by circumstances and environment. To a large extent, factors such as where we live, the state of our environment, genetics, our income and education level, and our relationships with friends and family all have considerable impacts on health, whereas the more commonly considered factors such as access and use of health care services often have less impact (19,[START_REF] Hajek | [END_REF].

The determinants of health include: -The social and economic environment, -The physical environment, and -The person's individual characteristics and behaviors.

Extensive studies have been conducted showing that health prospects have been shaped by five domains: genetic and gestational endowments, social circumstances, environmental conditions, behavioral choices, and medical care [START_REF] Mcginnis | United States[END_REF][START_REF] Kucukerdonmez | Comparison of nutritional status in the elderly according to living situations[END_REF]. These determinants of health have been identified and quantified [START_REF] Mcginnis | The case for more active policy attention to health promotion[END_REF][START_REF] Schroeder | We can do better -Improving the health of the American people[END_REF]. It is now quite clear that when the objective is to reduce early deaths, medical care has a relatively minor impact. The greatest opportunity to improve health and reduce premature deaths depends on personal behavior. Behavioral causes account for nearly 40% (obesity, tobacco…) of all deaths in the United States [START_REF] Schroeder | We can do better -Improving the health of the American people[END_REF].

Figure 5

Health determinants in U.S [START_REF] Hood | Systems medicine and pro-active P4 medicine: transforming healthcare with scientific wellness -a personal journey[END_REF] The identification of health determinants is precious information for aging assessment. Whether the parallelism can be made about aircraft aging remain an open question. However, establishing the weight of each determinant on "Health" or "Quality" of airframe and by extension the whole aircraft would be helpful to give direction to research programs.

P4 Systems approach

Medicine is an old discipline but also a fast-evolving sector. Medicine is increasingly moving from a reactive to a proactive approach discipline over the next decade, becoming a predictive, personalized, preventive and participatory, the so-called P4 medicine (Table 3) [START_REF] Hood | Predictive, personalized, preventive, participatory (P4) cancer medicine[END_REF]. Systems medicine abilities are to: -Provide deep insight in the disease mechanisms -Make blood a diagnostic window for viewing health and disease for the individual -Stratify complex diseases into their distinct subtypes for a impedance match against proper drugs -Provide new approaches to drug target discovery -Generate metrics for assessing wellness.

Two of the clinical objectives of P4 medicine are to quantify wellness and to demystify disease [START_REF] Hood | A personal view on systems medicine and the emergence of proactive P4 medicine: predictive, preventive, personalized and participatory[END_REF].The two main challenges utilizing P4 medicine are technical (digital revolution, big data) and societal barriers (sharing data). Societal barriers will be the most challenging. In order to bring different parts (patients, physicians, health-care members) to align with P4 medicine, new strategic partnerships have to be created (large clinical centers and patient-advocate groups).

As for aeronautics, the system approach is a key for a better maintainability. It is important to avoid separated non-crossing tasks like illustrating below that unable actor to see the global issue.

Aeronautics use since long time system approaches.

However, can we develop even more the aircraft system approach with the P4 points? This approach needs a different mindset as earlier check should be done, and aircraft operation can be potentially modified. Success of this approach is based on partnership. P4 medicine is patient participatory-based. This means that support should be given to airlines to improve aircraft life.

Figure 6

 6 Figure 6Aircraft "health" determinants

Table 1

 1 Aging parallelism

	Human Aging	Aircraft Aging
	Genes	Design
	Growth	Manufacturing
	Physician visit	Evaluation (Check)
	Preventive Medicine	Maintenance
	Hip/Heart	Part replacement
	Oxidative damage	Corrosion
	Cancer	Radiation damage
	Frailty	Fatigue

Table 2

 2 Aging type

	Aging type	Where	Effects	T°+Factor
	« Physical »	Bulk	Reversible	Endogene
	Moisture	Bulk	≈ Reversible	H20
	UV Rays	Surface	Irreversible	UV + O2 ,O3
	Thermal with O2 Bulk Surface Irreversible Irreversible	Endogene O2

Table 3

 3 P4 medicine vs Today medicine

	P4 medicine	Today medicine
	Proactive, predictive (very early	Reactive
	stage)
	Individual, precision medicine	Population
	Wellness & diseases	Only diseases
	Personalized data clouds	Average patient population
	Personalized data clouds for clinical	Averaged patient population for
	trials	clinical trials
	P4 medicine is based on systems approaches to disease,
	emerging technologies and analytical tools.

Table 4

 4 P4 maintenance vs Today maintenance

	P4 Maintenance	Today Maintenance
	Proactive, predictive (very early	Reactive
	stage)
	Singular A/C	A/C Fleet
	Performance & damages	Only defects, damages
	Singularized data clouds	Average fleet surveys
	Singularized data clouds for	Averaged fleet population for
	performance surveys	surveys

Conflict of Interest: ???????

Ethical standard: ??????.


S Briot

A Pashkevich

In this paper, a new methodology for the optimal design of parallel kinematic machine tools is proposed. This approach is based on the concept of the maximal inscribed parallelepiped and uses technology-oriented constraints that are motivated by particular applications. This methodology is applied on two translational parallel robots with three degreesof-freedom (DOF): the Y-STAR and the UraneSX. An analysis of the size of their workspace as a function of the design constraints is made. It is shown that, for identical workspaces with similar properties, the size of the legs of the UraneSX are greater than for the Y-STAR, thus leading to larger deformations. However, the footprint surface needed in order to install the Y-STAR is about two times bigger than for the UraneSX. Therefore, it may be interested to use the UraneSX in order to save some place on ground in manufacturing centres.

I. INTRODUCTION

arallel kinematic machines (PKM) are commonly claimed to offer several advantages over their serial counterparts, such as high structural rigidity, better payloadto-weight ratio, high dynamic capacities and high accuracy [START_REF] Merlet | Parallel robots[END_REF][START_REF] Tlusty | Fundamental comparison of the use of serial and parallel kinematics for machine tools[END_REF][START_REF] Chablat | A Comparative Study between Two Three-DOF Parallel Kinematic Machines using Kinetostatic Criteria and Interval Analysis[END_REF]. Therefore, they are prudently considered as promising alternatives for many modern material processing operations, especially in automotive and aerospace industry, in which high accuracy positioning and high-speed motions of a work tool are required. Thus, PKM have gained essential attention of a number of companies and researchers. However, most of the existing PKM still suffer from two major drawbacks, namely, a complex workspace and highly non-linear input/output relations [START_REF] Tsai | Robot analysis: the mechanics of serial and parallel manipulators[END_REF][START_REF] Bonev | The parallel mechanism information center[END_REF].

For most of PKM, the performances vary considerably for different points in the workspace and for different directions at one given point. This is a serious disadvantage for machining applications [START_REF] Kim | Performance analysis of parallel manipulator architectures for CNC machining applications[END_REF][START_REF] Ph | A comparative study of parallel kinematic architectures for machining applications[END_REF], which require regular workspace shape and acceptable kinetostatic performances throughout. In milling applications, for instance, the machining conditions must remain constant along the whole tool path [START_REF] Rehsteiner | Putting parallel kinematics machines (PKM) to productive work[END_REF]. Nevertheless, in many research papers, this criterion is not taken into account in the algorithmic methods used for the optimization of robots [START_REF] Luh | Working capability analysis of Stewart platforms[END_REF][START_REF] Merlet | Determination of 6D workspace of Gough-type parallel manipulator and comparison between different geometries[END_REF].

This work is focused on the optimal design of a robot for given geometric, kinematic and kinetostatic properties derived from technical applications (e.g. size of the workspace, maximum speeds, forces transmission, accuracy). The main contribution of this paper is in the area of application of the operation research methods to the integrated design optimization of complex mechanical structures, such as parallel robots. The proposed approach operates with the 'workspace grid' that is evaluated using a dedicated dynamic-programming-based algorithm allowing, for each particular set of the design parameters, to estimate the largest cuboid-shaped workspace with the desired properties. Further, the workspace parameters are evaluated in the frame of the global optimization. Finally, contrary to many works on optimal design of parallel robots (see for example [START_REF] Liu | Performance atlases and optimum design of planar 5R symmetrical parallel mechanisms[END_REF][START_REF] Pashkevich | Design optimization of parallel manipulators for high-speed precision machining applications[END_REF]), it is proposed in this paper to use technology oriented indices in order to define the optimal design parameters.

The paper will be divided as follows. In the second part, the design problem and methodologies are explained. In the third section, the performance measures used are presented. In part four, the optimization procedure is described and it is applied on an industrial case study in part five. Finally, in the last section, conclusions are drawn.

II. DESIGN PROBLEM AND METHODOLOGY

Manipulator design traditionally begins with the selection of a kinematic framework and achieving certain geometric goals such as workspace size and dexterity. Besides, for particular manufacturing tasks, the manipulator geometry is optimized with respect to the desired velocity, accuracy or force transmission factors. This yields a simplified CAD model of a relevant mechanism that defines basic dimensions of the links and spatial locations of all active and passive joints, as well as joint limits. At the next step, this model should be detailed by providing real shapes of links to produce the solid CAD model.

To formulate the design problem, let us define the manipulator geometry by the mapping g:   W, where  =  1 × …  n and W = p 1 × … p n denote respectively the configuration space and the workspace.  i are the joint coordinates and p i are the coordinates of the end-effector. n is the number of DOF. Besides, for each workspace point p  W, let us define the matrices K v (p, ), K f (p, ), K a (p, ), that describe various mechanical properties of the manipulator (velocity, force transmission, accuracy, etc.) for any given set of the design parameters . Let us also assume that for each type of the matrices K  ,   {v, f, a, …}, there are defined physically consistent scalar measures   (K  ),   {i, t, …} that may be directly included in the design objectives or constraints. Some examples of such measures (isotropy, transmission factors, etc.) are presented in the following sections.

Sébastien Briot, Anatol Pashkevich and Damien Chablat

Optimal Technology-Oriented Design of Parallel Robots for High-Speed Machining Applications

P

Similarly, for the global evaluation of the manipulator, let us introduce the performance measures   (g, ),   {m, l, w, …}, that depend both on the adopted geometrical structure g and the physical parameters of the links .

Examples of the global measures include the total mass of the manipulator, the length of the principal links, the workspace size, etc. Then, following the general methodology adopted for the considered application area (high-speed machining), the design optimisation problem can be stated as achieving the best value of the performance indices

     , min ) , ( π π g (1)
subject to the constraints

        , , ) , (   S π p K (2)
that must be satisfied for all points of the cuboid workspace W 0 of size a × b × c, which includes the manufacturing task.

It should be noted that the latter assumption (concerning the workspace shape) is essential here and allows considerably speed-up the optimization routines. Since in practice this problem cannot be solved by the direct search methods, in the following subsections, there will be presented the discretisation scheme and relevant optimisation algorithms allowing to obtain desired solutions in reasonable time.

III. PERFORMANCE MEASURES

Let us present here the most essential technology-oriented performance measures that are used in the following of this paper. Traditionally, they are directly included in the design constraints/objectives to be satisfied or optimized throughout the prescribed workspace. However, in this paper, each performance measure is preliminary converted in an alternative form that defines the workspace subset where the relevant criterion is higher/lower of the desired value.

It should be noted that, in the following of this paper, our analysis will be restrained to 3-DOF translational PKM as they are mostly used for machining of materials.

A. Size of the workspace

Using the set of parameters , the workspace W may be generated using the kinematic equations and the joint limits [START_REF] Merlet | Parallel robots[END_REF].

Since, for the considered application, the desired regular workspace is a parallelepiped W 0 of size {a 0 × b 0 × c 0 }, the relevant measure may be defined by the largest similar object W abc = {µa 0 × µb 0 × µc 0 } inscribed in W, i.e.

 

W W T T W T W 0 T 0    ) ( max arg ) , ( ); ( ,      abc (1)
where µ, T are respectively the scalar scaling factor and the coordinate transformation operator in the Cartesian space. This notion is the fundamental issue of this paper and is discussed in details in the following section.

B. Velocity transmission factors

For quantifying the speed capability of a manipulator, several kinematic performance indices are defined using the Jacobian matrix J (see [START_REF] Merlet | Jacobian, manipulability, condition number, and accuracy of parallel robots[END_REF]), such as the condition number, the largest/smallest singular value (also called the maximal/ minimal transmission factor and denoted as  max and  min , respectively -Fig. 1a), the dexterity, the manipulability, etc.

However, as mentioned by Merlet in [START_REF] Merlet | Jacobian, manipulability, condition number, and accuracy of parallel robots[END_REF], the previously cited indices does not take into account the 'technological reality' of the mechanism, as they are based on the use of the Euclidian norm of the input velocity vector Φ  (|| Φ  || being considered equal to 1) while it is clear that each actuator may have a velocity

i    [-max i   , max i   ], where i   and max i  
are the actual and maximal velocities for the actuator i (i = 1 to n). Thus, it is necessary to redefine the transmission factors, of which expressions are presented in the next subsections.

Velocity transmission along all directions of the workspace.

Normalizing the problem and considering that max i   = 1, i.e. replacing the unit sphere by a unit cube (Fig. 1b), one can obtain the image of the unit square by the transformation f: x  J x. The obtained figure is a parallelepiped, where point B j is considered to be the image of point A j by the transformation f.

One can now define the minimal and maximal transmission factors min

v k and max v k . Factor max v k
is the largest distance from the origin of the frame to the faces of the parallelepiped. Its corresponding expression may be written as:

)

) ( ( max max j j v k e q J  , for j = 1 to 4 (2)
where Let us consider that the Jacobian matrix J may be decomposed into three vectors I 1 , I 2 and I 3 such as:

e 1 = [+1, +1, +1] T , e 2 = [+1, -1, +1] T , e 3 = [+1, +1, - 1
  3 2 1 I I I J  . ( 3
) The faces of the parallelepiped are the images by the transformation f of the faces of the unit cube, i.e. there are attained when at least one actuator is at its maximal velocity

max i    ( 1 max  i   , i = 1, 2, 3
). Therefore, the parameterized expressions of the faces of the parallelepiped are, for any

vector  = [ 1 ,  2 ] T , ] 1 , 1 [ , 2 1      :   ) ( ) ( ) ( ) ( ) ( m T m ijk m ijk m ijk m ijk z y x 2 1 J δ J V    , (4)
for m = 1 or 2, i, j, k = 1, 2, 3, i ≠ j, i ≠ k, j ≠ k, where ] [ j i I I J 1  , k m I J 2 ) 1 (  . ( 5
)
The distances || ) (m ijk V || (m = 1, 2) from the origin to any point belonging to the faces are given by

) ( ) ( ) ( m ijk T m ijk m ijk V V V  . ( 6
)
Finding the minimum of this expression remains to computing the minimum of || ) (m ijk V ||². Thus, differentiating expression [START_REF] Kim | Performance analysis of parallel manipulator architectures for CNC machining applications[END_REF] with respect to the parameter , it may be found that the minimum appears when

  k T ij ij T ij m ijk J J J J δ V δ 1 2 ) ( 0       . ( 7
)
Verifying that the components of  belong to the interval [-1, +1] and introducing them into (6), it may be found that:

       2 1 1 1 1 2 2 2 J J J J J J J J V V T T T T m ijk ijkm 1 ) ( min min     (8) Thus, the expression of min v k is ) ( min min , , , min ijkm m k j i v k V  , (98)
for m = 1, 2, i, j, k = 1, 2, 3, i ≠ j, i ≠ k, j ≠ k.
It should be mentioned that, if the components of  do not belong to the interval [-1, +1], thus the obtained value for min ijkm

V

does not correspond to the minimal distance between the face of the parallelepiped and the centre of the frame, but between the plane on which is lying the face under consideration and the centre of the frame. It could be proved that, even if this happens, the value of min v k will be obtained when considering another face for which the components of  will belong to the interval [-1, +1]. Therefore, there is no need to use other expressions than those presented above.

Velocity transmission along some particular directions of the workspace.

In some cases, it is not necessary to guarantee a minimal velocity along all the axes of the workspace, but along some particular directions (e.g., in the xy plane). For demonstration purpose, let us consider that a minimal velocity has to be guaranteed in the xy plane. Vector v xy = [v x , v y ] T corresponds to the velocity of the platform in the xy plane and is of dimension 2, while Φ  is still of dimension 3. These vectors are linked by the relation:

Φ J v xy xy   , (10)
where xy J is a 2 by 3 matrix, of which lines are identical to the two first lines of the matrix J. In such a case, let us consider the transformation g: x  J xy x. The image of the unit sphere by the transformation g is an ellipse, and the image of the unit cube is a hexagon (Fig. 2). Indeed, this hexagon is the projection of the parallelepiped of Fig. 1 onto the (V x , V y ) plane (Fig. 3). Six of the edges of the unit cube are transformed into the six edges of the hexagon. The six others edges of the cube are transformed into six other lines inscribed inside the surface of the hexagon. Moreover, six of the eight vertices of the cube are transformed into the six vertices of the hexagon. The images of the two other vertices of the cube are located inside of the hexagon.

So the problem remains to find the smallest and largest distances from the origin to the edges of the hexagon. Factor max v k is defined as the largest distance from the origin of the frame to the hexagon. Therefore, max v k may be written as:

) ) ( ( max max j j v k e q J xy 
, for j = 1 to 4 [START_REF] Liu | Performance atlases and optimum design of planar 5R symmetrical parallel mechanisms[END_REF] where e j are defined at Eq. ( 2). Factor min v k is the smallest distance between point O and the edges of the hexagon.

Let us decompose the Jacobian matrix J xy into three vectors I 1 , I 2 and I 3 such as:

  3 2 1 xy I I I J  . ( 12
)
The edges of the hexagon are the image by the transformation g of the edges of the unit cube, i.e. there are attained when at least two actuators are at their maximal velocity

max i    ( 1 max  i   , i = 1, 2, 3
). Therefore, the parameterized expressions of the edges of the hexagon may be found among these expressions, for any

] 1 , 1 [     :   ) ( ) ( ) ( ) ( m T m ijk m ijk m ijk y x 2 1 J J V     , ( 13
)
for m = 1 to 4, and i, j, k = 1, 2, 3, i ≠ j, i ≠ k, j ≠ k, where i

I J 1  , ( 14a
) k j I I J 2   ) 1 ( , k j I I J 2   ) 2 ( , ( 14b
) k j I I J 2    ) 3 ( , k j I I J 2    ) 4 ( . ( 14c
)
The minimal distance

min ijkm V =   ) ( min m ijk V
from the origin to any point belonging to the edges of the hexagon may be obtained by a similar way as that presented in the previous section. Its expression is identical to that given at Eq. ( 8), for m = 1 to 4; i, j, k = 1, 2, 3 with i ≠ j, i ≠ k, j ≠ k.

It is necessary to find the value min

Performance measure

Here, let us propose to redefine the global kinematic metric by using the above notion of the "largest inscribed parallelepiped". For instance, if the velocity transmission factors min , where v min and v max represent the minimal and maximal desired velocity transmission factors in the workspace of the manipulator:

); (

0 W T W   abc (15)   W W T T 0 T     ) ( ; and max arg ) , ( max max min min ,     v k v k v v
In the next subsection, we present a performance measure for the accuracy of the robot to optimize.

C. Accuracy transmission factors

The accuracy of a mechanism depends on different factors. However, as pointed out by Merlet [START_REF] Merlet | Computing the worst case accuracy of a PKM over a workspace or a trajectory[END_REF], active-joint errors are the most significant source of errors in a properly designed, manufactured, and calibrated parallel robot.

The classical approach consists in considering the first order approximation that maps the input error to the output error:

Φ J p δ δ  ( 16
) where  represents the vector of the active-joint errors,  p the vector of end-effector errors. This method will give only an approximation of the end-effector maximum error, but at an optimisation stage, this model may be sufficient. However, it is preferable to use a safety coefficient on the desired accuracy of the platform in order not to have too poor accuracy for the finally designed machine.

From Eq. ( 16), it may be noticed that this expression is similar to the relationship for the velocity, which states that:

Φ J v   (17
) Moreover, it seems obvious that, taking account that the accuracy  i of one actuator is comprised between - and + ( being the accuracy of the actuated pair), the method to compute the accuracy transmission factors is totally equivalent to the computation of the velocity transmission factors. Moreover, by normalizing the problem, (i.e.  = 1), the accuracy transmission factors are equivalent to the velocity transmission factors.

Once again, the manipulator global performance may be evaluated by the size of the parallelepiped workspace that satisfies the design specification , where  min and  max represent the minimal and maximal desired accuracy transmission factors in the workspace of the manipulator:

); (

0 W T W   abc (18)   W W T T 0 T       ) ( ; and max arg ) , ( max max min min ,     v v k k
Let us now deal with the last proposed performance measure.

D. Force transmission factors

For 3-DOF translational parallel manipulators, the input efforts  are related to the forces f applied on the platform by the following relation: 19), it may be noticed that the relationship for the accuracy is similar to the relationship for the velocity (Eq. ( 17)), but J -T is used instead to J.

τ J f T   (19) From expression (
Moreover, it is also clear that the effort  i of one actuator is comprised between - max and + max ,  max being the maximal effort admissible by the actuated pair. So, the force transmission factors may be computed in the same manner as the velocity transmission factors. The maximal and minimal force and moment transmission factors will be denoted as min , where f min and f max represent the minimal and maximal desired accuracy transmission factors in the workspace of the manipulator: ); (

0 W T W   abc (20)   W W T T 0 T     ) ( ; and max arg ) , ( max max min min ,     f k f k f f
In the next subsection, we show that the velocity and force transmission factors are antagonistic and that some zone of dominance for these factors may be defined.

E. Duality between force and velocity transmission factors

From Eqs. ( 17) and ( 19), it may be noticed that if the velocity and force transmission factors were computed as usual as the minimal and maximal eigenvalues of J and J -T , the following equivalence would have existed:

max min / 1 v f k k  and min max / 1 v f k k  ( 21
)
Such reciprocity allows us defining some zones of dominance between the parameters min Theses assertions are quite simple when considering the transmission factors defined as the eigenvalues of J and J -1 . But, as presented above, these factors are defined differently and expression ( 21) is not valuable. But, as it will be observed in part V, some zones of dominance between the parameters may be found and these zones seem to be separated by hyperbolas.

IV. DESIGN PROCEDURE

The first step of the design procedure is to evaluate the value of the scale factor µ defining the size of the workspace W abc for the design parameters  and the constraints   . The problem of the estimation of the largest cuboid-shaped subworkspace, where the relevant criterion is higher or lower of the desired value, may be solved numerically, using the workspace discretisation and applying the dynamic programming.

Let us define the workspace grid {G ijk } that includes the manipulator workspace W 0 = {a 0 × b 0 × c 0 } and possesses uniform but differents steps along the Cartesian axes, namely (a 0 /N 0 ; b 0 /N 0 ; c 0 /N 0 ), where N 0 defines the discretisation precision. Besides, for each node of the grid, let us compute relevant local performance measure and define a 3D binary matrix  ijk  {0, 1}, where  ijk = 1 if the corresponding design constraint/objective is satisfied, and  ijk = 0 otherwise. For computation conveniences, let us also set

 ijk = 0 if G ijk  W 0 .
Thus, the original problem is reduced to searching for the largest cubic submatrix inside of { ijk } containing non-zero values only. The latter can be efficiently solved applying the following algorithm that operates with additional integer matrix { ijk } that defines sizes of the candidate solutions with the vertex (i, j, k):

Step 0. Set  ijk = 0, i, j, k Step 1. Set ijk =  ijk , for {i = 1 & j, k}  {j = 1 & i, k}  {k = 1 & i, j} Step 2. for i = 2:i max do for j = 2:j max do for k = 2:k max do if  ijk = 1 then                             1 , 1 , 1 1 , 1 , 1 , , 1 , 1 , 1 1 , , , 1 , , , 1 , , , , , , min 1
k j i k j i k j i k j i k j i k j i k j i ijk Step 3. Find d = max( ijk ) -1; (i 0 , j 0 , k 0 ) = argmax( ijk )
Step 4. Retrieve from the grid {G ijk } the desired cuboid bounded by the indices (i 0 -d, j 0 -d, k 0 -d) and (i 0 , j 0 , k 0 ).

Validity of this routine and correctness of the relevant recurrent expression can be proved using the standard ideas of the dynamic programming, similar to finding the largest square block in two-dimensional binary matrix.

Hence, for each performance measure and each set of the design parameters, it can be computed a workspace-based metrics composed of the coordinates ranges of the largest cuboid-shaped sub-workspace W abc and of the scale factor µ (= µ()) defining the ratio between the size of W 0 and W abc .

The second step of the design procedure is to find the optimal geometry of the robot according to the desired objectives f i . For instance, for the geometric, kinematic and kinetostatic design, the objective is to minimize the manipulator dimensions (links lengths), while the constraints define the desired workspace size and the range of the velocity, accuracy and effort transmission factors.

Let us denote as  0 a set of normalized design parameters (in most of cases,  0 is not composed of fixed constants only, but also of variables). Moreover, for computational convenience, let us transform the design constraints and present them in the scalar form 0 ) (

k k h h  0 π
. Then, the optimization problem may be rewritten as

0 π 0 π max ) (   (22) subject to 0 ) ( k k h h  0 π , k. (23)
At the end of the design algorithm, it will be obtained only one value of the scale factor µ, which appear for the optimal normalized design parameters opt 0 π . In order to get the real optimal design parameters  opt allowing the real robot to have the desired properties into the entire workspace W 0 , the normalized design parameters opt 0 π have to be scaled using the factor µ. Thus, such a formulation allows considerably simplifying the design optimization problem.

V. APPLICATION EXAMPLES A. Industrial problem

Let us demonstrate the efficiency of our design approach on a concrete problem coming from the industrial sector of the region of Nantes (France). One of the most important activity areas of this region is the manufacturing of bathroom components (shower cabin, washbasin, bathtub, etc. -Fig. 4). Most of parts used during the assembly process of the bathroom component are flat and made of thermosetting materials. The main operations achieved on these parts is trimming, i.e. the suppression of the edges of the parts in order to obtain a good surface roughness.

The machines tools that are used for the trimming of these bathroom components must be designed such as they attain the following characteristics:

workspace W abc of size {2. experience, several types of robots may be envisaged, such as the Y-STAR [START_REF] Hervé | Group mathematics and parallel link mechanisms[END_REF], the UraneSX [START_REF] Company | Modelling and design issues of a 3-axis parallel machine-tool[END_REF], the Orthoglide [START_REF] Chablat | A Comparative Study between Two Three-DOF Parallel Kinematic Machines using Kinetostatic Criteria and Interval Analysis[END_REF], the Hybridglide [START_REF] Company | Machine-Outils Rapides à Structure Parallèle[END_REF], the 3-UPU [START_REF] Tsai | Kinematics and Optimization of a Spatial 3-UPU Parallel Manipulator[END_REF], etc. However, because the workspace is not a cube, but a flat parallelepiped, it appears that only two kinds of architectures, that have non-orthogonal arrangement of legs and that are already used in machining process, seem to be best adapted: the Y-STAR and the UraneSX (Fig. 5). It should be mentioned that, in our study, the prismatic guides of the UraneSX are vertical.

B. Comparison between the alternative solutions

Let us first compare the size of the workspace of the Y-STAR and the UraneSX in general cases, i.e. for many given maximal admissible values of the transmission factors. To apply the proposed design approach, let us normalize the problem by dividing the size of the workspace by 2.5 m such as the normalized workspace W 0 is of dimension {1 × 1 × 0.2} and by taking into account that the length of the legs of the mechanisms are identical and equal to 1.

Taking into account the general design objective, the scale factor µ may be optimized 1 . The results are plotted at Figs. 6 and7.

It follows from these results that the curves may be decomposed into three parts. In the first part (in bright gray), the variation of min UraneSX, which implies that, for identical workspaces, the UraneSX will have legs of size greater than the Y-STAR, thus leading to larger deformations of the robot. But, before giving final conclusions on these two robots, other constraints coming from requirements that take into account the entire manufacturing cell should be considered.

C. Optimization results for bathroom components milling

For the design of these two robots, actuated prismatic tables have to be used. The admissible velocity found in manufacturers' catalogues for long prismatic tables is about v t =150 m/min, and their common minimal payload along their axis is of f t = 3000 N for ball-screw systems. Moreover, the accuracy of these pairs is about a t = 0.05 mm. Therefore, it was found that min Extracting µ from the previous general results and then scaling the robot taking into account that the corresponding normalized workspace is equal to µ W 0 , the lengths of legs are obtained. They are of 2.16 m for the Y-STAR and of 4.62 m for the UraneSX. It is clear here that the Y-STAR will have smaller legs, and as a result, potentially a smaller loss of accuracy due to deformations, than the UraneSX. However, it could be shown that the footprint surface needed in order to install the Y-STAR is about two times bigger than for the UraneSX. Therefore, it looks more attractive to use the UraneSX in order to save some place on ground in manufacturing centres.

VI. CONCLUSIONS

In this paper, it has been proposed a new methodology for the optimal design of parallel kinematic machine tools. It is based on the concept of the maximal inscribed parallelepiped and uses technology-oriented constraints. To compute the parallelepiped, a dedicated combinatorial optimization algorithm is proposed. This approach is applied on two 3-DOF translational parallel robots: the Y-STAR and the UraneSX. An analysis of the size of their workspace as a function of the design constraints is made. It is presented that, for workspaces with similar properties, the legs of the UraneSX are longer than for the Y-STAR, thus leading to greater deformations. However, it is shown that the footprint surface needed in order to install the Y-STAR is about two times bigger than for the UraneSX. Therefore, it may be interesting to use the UraneSX in order to save some place on ground in manufacturing centres.

  ] T and e 4 = [+1, -1, -1] T . To obtain the expression of min v k , more computations are required. Indeed, min v k is the smallest distance between point O and the faces of the parallelepiped.

Fig. 1 .

 1 Fig. 1. Mapping, using the Jacobian matrix, (a) of the unit sphere and (b) of the unit cube.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Mapping of the unit cube using the matrix J xy .

  twelve expressions. The six smallest values for the distances ) (m ijk V should be disregarded, because they do not correspond to distances between the origin of the frame and the edges of the hexagon. The value of min

  the global metrics, the manipulator global performance is evaluated by the size of the parallelepiped workspace that satisfies the design specification

  performance may now be evaluated by the size of the parallelepiped workspace that satisfies the

k

  , resp.) that are separated by a hyperbola. The zone upside the hyperbola corresponds to the zone where max is unnecessary because the limit f min (v min , resp..) is more constraining. On the contrary, the zone downside the hyperbola is the zone where the parameter max v k ( max f k , resp.) is dominant.

  5 m × 2.5 m × 0.5 m}; -||v xy || = 60 m/min, ||f xy || = 300 N and ||p xy || = 0.25 mm (f xy and p xy are the components of vectors f and p in the xy plane). From these requirements, and from our industrial

Fig. 4 .

 4 Fig. 4. Typical examples of bathroom components manufactured in the region of Nantes (France).

Fig. 5 .

 5 Fig. 5. The robots under study: (a) Y-STAR and (b) UraneSX.

  xy || / f t = 0.1, min v k = ||v xy || / v t =0.4 and max v k = ||p xy || / a t = 5.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Values of the scale coefficient µ for the Y-STAR robot as a function of min f k and max v k , for (a) min v k = 0.2, (b) min v k = 0.4 and (c) min v k = 0.6.

In the case of the UraneSX, the scale factor µ depends also on the radius R of circumcircle of the base triangle. But, we consider that the value of the scale factor is defined as the maximum of µ for any given R.

ACKNOWLEDGMENTS

This work has been supported by the French région Pays de la Loire (RoboComposite project).