Eric Paroissien
email: eric.paroissien@isae-supaero.fr

Frédéric Lachaud

Lucas F M Da Silva

Salah Seddiki

A comparison between macro-element and finite element solutions for the stress analysis of functionally graded single-lap joints

Keywords: functionally graded adhesive, single-lap bonded joint, stress analysis, finite element, macro-element, design

The interest in functionally graded adhesive (FGA) joints has been increasing in recent years. For example, FGAs offer the opportunity to optimize the strength of multimaterial bonded joints by locally tailoring the adhesive properties and without modifying the design of the adherends to be joined. The development of dedicated stress analyses to predict the stress distribution is then of the highest interest to control the strength of such joints. The Finite Element (FE) method is able to address the stress analysis of FGA joints but is computationally costly. Simplified stress analyses have then been developed. The objective of this paper is to assess the prediction of simplified stress analyses, solved through the macroelement (ME) technique, with respect to those of FE models. It is shown that the predictions of ME models are in a sufficient agreement with the FE models to be employed at a pre-sizing stage. The influence of the overlap length is then investigated by the means of the simplified stress analyses. A noticeable result is the existence of an overlap length for which the adhesive peak shear stress is minimal, in the 1D-bar kinematics framework.

NOMENCLATURE AND UNITS

The adhesive bonding technology is attractive since it offers high mechanical performance in terms of stiffness, static strength and fatigue strength [START_REF] Hart-Smith | Design methodology for bonded-bolted composite joints[END_REF][START_REF] Kelly | Quasi-static strength and fatigue life of hybrid (bonded/bolted) composite single-lap joints[END_REF][START_REF] Da Silva | Handbook of Adhesion Technology (2 volumes)[END_REF]. In addition, this joining technology is particularly suitable to the joining of adherends made of dissimilar materials.

The presence of the adhesive layer all along the overlap allows a continuous load transfer between the adherends, contrarily to mechanical fastenings where the load transfer takes place at the fastener location. However, adhesive stress gradients are present at both overlap ends due to the deformability of the adhesive and adherends, so that the load transfer is mainly restricted to small lengths at both overlap ends. Design solutions exist to homogenize the adhesive stress distribution along the overlap. For example, the specimen design for the thick adherend shear test [START_REF] Astm D | Standard Test Method for Thick-Adherend Metal Lap-Shear Joints for the Stress-strain Behavior of adhesives by Tension Loading[END_REF] leads to a homogenization of adhesive shear stress distribution, due to the increase of the ratio between the adherend membrane stiffness relatively to the adhesive shear stiffness. In the case of joining adherends made of dissimilar materials, the adhesive stress peak is more pronounced at one overlap end than at the other one. In such a case, an idea to reduce the adhesive peak stresses is to vary the adhesive properties all along the overlap, referring to the functionally graded adhesive (FGA) joints [START_REF] Carbas | Adhesively bonded functionally graded joints by induction heating[END_REF][START_REF] Carbas | Modelling of functionally graded adhesive joints[END_REF][START_REF] Carbas | Functionally graded adhesive joints by graded mixing of nanoparticles[END_REF][START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF].

The interest in FGAs has increased in recent years [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]. FGA joints involve a continuous variation of the adhesive properties along the overlap allowing for the homogenization of the stress distribution and load transfer. 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 adherends to be joined.

The development of dedicated stress analyses to predict the stress distribution is then of the highest interest to tailor the strength of such joints. The Finite Element (FE) method is able to address the stress analysis of FGA joints [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] Breto | Finite Element Analysis of Functionally Graded Bond-Lines for Metal/Composite Joints[END_REF]. However, FE analyses are computationally costly, so that simplified stress analyses would be useful to allow for extensive parametrical studies, optimization processes and fast decision-making at the pre-sizing stage. In 2014, Carbas et al. published a first analytical approach for 1D-bar stress analysis of FGA joints [START_REF] Carbas | Modelling of functionally graded adhesive joints[END_REF],

based on the shear-lag model by Volkersen [START_REF] Volkersen | Die Nietkraftverteilung in Zugbeanspruchten Nietverbindungen mit konstanten Laschenquerschnitten[END_REF] for homogeneous adhesive (HA) joints. In the case of HA joints, the simplifying hypotheses involved lead to linear ordinary differential equations (ODEs). In the case of FGA joints, the ODEs are not linear anymore since the adhesive properties depend on the location along the overlap. the stress analysis of FGA joints under various geometrical configurations, including nonlinear material behavior [START_REF] Stapleton | Design of functionally graded joints using a polyurethane-based adhesive with varying amounts of acrylate[END_REF]. A JE is a 4-nodes brick element allowing for the modelling of two bonded adherends [START_REF] Stapleton | Design of functionally graded joints using a polyurethane-based adhesive with varying amounts of acrylate[END_REF][START_REF] Gustafson | Dimensionless parameters in symmetric double lap joints: An orthotropic solution for thermomechanical loading[END_REF][START_REF] Stapleton | The analysis of adhesively bonded advanced composite joints using joint finite elements[END_REF]. Over a similar period of time, the first and third authors of the present papers and co-workers have been working on the development of the macro-element (ME) technique for the simplified stress analysis of bonded, bolted and hybrid (bonded/bolted) joints made of dissimilar and possibly laminated beams [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 one-dimensional models of a single-lap joint[END_REF][START_REF] Paroissien | Analytical two-dimensional model of a hybrid (bolted/bonded) single-lap joint[END_REF][START_REF] Paroissien | Elasto-Plastic Analysis of Bonded Joints with Macro-Elements[END_REF][START_REF] Lélias | An extended semi-analytical formulation for fast and reliable mode I/II stress analysis of adhesively bonded joints[END_REF][START_REF] Paroissien | Simplified Stress Analysis of Hybrid (Bolted/Bonded) Joints[END_REF][START_REF] Lélias | Experimental Characterization of Cohesive Zone Models for Thin Adhesive Layers Loaded in Mode I, Mode II, and Mixed-Mode I/II by the use of a Direct Method[END_REF]. The JE and ME are based on the same modelling approach. In 2018, Paroissien et al. published a 1D-bar and 1D-beam analysis of FGA joints subjected to combined thermal and mechanical loads, the resolution schemes of which make use of TEPS and of the ME technique [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF].

The objective of this paper is to present a comparison between the predictions of adhesive stress distributions from the ME technique and the FE method in single-lap FGA joints.

Firstly, simplified stress analyses based on ME technique in 1D-bar and 1D-beam framework are explained. The detailed mathematical description can be found in [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF]. In a second part, the 1D and 3D FE models are presented and validated through dedicated convergence studies. A parabolic graduation of adhesive properties is considered such as:

𝐸 𝑎 (𝑥) = 𝐸 𝑎,𝑚𝑎𝑥 -(𝐸 𝑎,𝑚𝑎𝑥 -𝐸 𝑎,𝑚𝑖𝑛 ) ( 𝑥 𝐿 2 -1) 2 (1)
𝐺 𝑎 (𝑥) = 𝐺 𝑎,𝑚𝑎𝑥 -(𝐺 𝑎,𝑚𝑎𝑥 -𝐺 𝑎,𝑚𝑖𝑛 ) ( 𝑥 𝐿 2 -1) 2 ( 2
)
where L is the length of the overlap, E a,max (E a,min ) is the maximal (minimal) adhesive peel modulus in the graduation and G a,max (G a,min ) is the maximal (minimal) adhesive shear modulus in the graduation. The ratio between the maximal (minimal) adhesive peel modulus and the maximal (minimal) adhesive shear modulus through is chosen, for simplicity purpose, as constant for this paper and equal to 2(1+ a ), where  a is the adhesive Poisson's ratio. It is indicated that this choice is not a restriction imposed by the modelling, since the models are fully parametrical. In this work, the adhesive peel modulus is then represented by the adhesive Young's modulus. These geometrical and material configurations are the same as those used in [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF] inspired by [START_REF] Carbas | Modelling of functionally graded adhesive joints[END_REF][START_REF] Stein | Analytical models for functionally graded adhesive joints: A comparative study[END_REF]. The material and geometrical parameters are provided in Table 1 to Table 3.

Overview of the macro-element technique and application

The ME technique is inspired by the FE method and differs in the sense that the interpolation functions are not assumed. Indeed, they take the shape of solutions of the governing ODEs system, coming from the constitutive equations of the adhesive and adherends and from the local equilibrium equations, related to the simplifying hypotheses. A straightforward consequence is that only one ME is sufficient to model the full length of the bonded overlap (Figure 2). The main work is thus the formulation of the elementary stiffness matrix of the ME. The ME technique can be regarded as a mathematical procedure allowing for the resolution of the system of ODEs, under a less restricted application field of simplifying hypotheses, in terms of geometry, material behaviours, kinematics, boundary conditions and loads. Dedicated 4-nodes bonded-bars (BBa) and bonded-beams (BBe) have been formulated [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 one-dimensional models of a single-lap joint[END_REF][START_REF] Paroissien | Analytical two-dimensional model of a hybrid (bolted/bonded) single-lap joint[END_REF][START_REF] Paroissien | Elasto-Plastic Analysis of Bonded Joints with Macro-Elements[END_REF][START_REF] Lélias | An extended semi-analytical formulation for fast and reliable mode I/II stress analysis of adhesively bonded joints[END_REF][START_REF] Paroissien | Simplified Stress Analysis of Hybrid (Bolted/Bonded) Joints[END_REF][START_REF] Lélias | Experimental Characterization of Cohesive Zone Models for Thin Adhesive Layers Loaded in Mode I, Mode II, and Mixed-Mode I/II by the use of a Direct Method[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, in terms of adhesive stress distributions along the overlap, internal forces and displacements in the adherends at any abscissa along the joint. The boundary conditions and loading are applied through the Augmented Lagrangian method [START_REF] Curnier | A generalized Newton method for contact problems with friction[END_REF][START_REF] Simo | An augmented lagrangian treatment of contact problems involving friction[END_REF]. Besides, an algorithm based on the Newton-Raphson iterative procedure has been developed to support nonlinear material behaviors of the adhesive and adherends [START_REF] Paroissien | Elasto-Plastic Analysis of Bonded Joints with Macro-Elements[END_REF][START_REF] Lélias | An extended semi-analytical formulation for fast and reliable mode I/II stress analysis of adhesively bonded joints[END_REF][START_REF] Paroissien | Simplified Stress Analysis of Hybrid (Bolted/Bonded) Joints[END_REF][START_REF] Lélias | Experimental Characterization of Cohesive Zone Models for Thin Adhesive Layers Loaded in Mode I, Mode II, and Mixed-Mode I/II by the use of a Direct Method[END_REF]. In the case of nonlinear computation, the overlap length needs to be meshed with n_ME MEs along the overlap length (only) to allow for the update of the material properties within the iterative procedure (Figure 2). The meshing of the overlap length is used for the FGA joints.

In this paper, a uniform mesh was chosen, so that the length of each ME is L/n_ME. The actual graduation of the adhesive properties is then approximated by a stepped function. The adhesive peel and shear modulus assigned to the i th ME is taken equal to the value of the modulus graduation function at the abscissa located at the middle of the ME:

𝐸 𝑎 (𝑥 𝑖 = (𝑖 -

In this paper, the ME technique is applied to a particular configuration, under a particular set of boundary conditions and loading. However, this is not a restriction once again. Hereafter, the subscript j=1 (j=2) is related to the upper (lower) adherend. The computations were performed thanks to house-made computer programs developed on MATLAB. They are available on [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF] related to a previous paper [START_REF] Paroissien | Matlab codes attached to[END_REF].

Hypotheses and governing equations 2.2.1 Bonded-bars element: 1D-bar kinematics

The following hypotheses were taken (i) the adherends are linear elastic materials simulated as bars, (ii) the adhesive layer is simulated by an infinite number of linear elastic shear springs linking both adherends, and (iii) the shape of graduation of the adhesive layer shear modulus is considered. As a result, it is supposed that all the adhesive stress components vanish except the in-plane shear. The local equilibrium of both adherends (Figure 3) provides the following equations:

𝑑𝑁 𝑗 𝑏𝑑𝑥 = (-1) 𝑗 𝑇(𝑥), 𝑗 = 1,2 (5)
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 total strain is equal to the mechanical strain plus the thermal strain such as:

𝑑𝑢 𝑗 𝑑𝑥 = 𝑁 𝑗 𝐴 𝑗 , 𝑗 = 1,2 (6)
where A j is the membrane stiffness of the adherend j, given by:

N 1 (x+dx) N 1 (x) T.bdx N 2 (x+dx) N 2 (x)
𝐴 𝑗 = 𝐸 𝑗 𝑏𝑒 𝑗 [START_REF] Carbas | Functionally graded adhesive joints by graded mixing of nanoparticles[END_REF] where e j is the thickness of the adherend j and E j the Young's modulus of the adherend. The displacement u j (x) is the normal displacement of points located at the abscissa x on the neutral line of adherend j.

The constitutive equation for the adhesive layer is provided by:

𝑇 = 𝐺 𝑎 𝑢 2 -𝑢 1 𝑒 𝑎 = 𝑘 𝐼𝐼 Δ𝑢 (8)
with:

Δ𝑢 = 𝑢 2 -𝑢 1 (9)
where e a is the adhesive thickness, G a the adhesive shear modulus and k II =G a /e a the adhesive shear relative stiffness. u is the differential displacement of the adherend interface.

Bonded-beams element: 1D-beam kinematics

The model is based on the following hypotheses: (i) the thickness of the adhesive layer is constant along the overlap, (ii) the adherends are simulated by linear elastic Euler-Bernoulli laminated beams and (iii) the adhesive layer is simulated by an infinite number of elastic shear and transverse springs linking both adherends. The eccentricity of the load path inherent to the single-lap joint configuration leads to a nonlinear secondary bending at both overlap ends [START_REF] Goland | The stresses in cemented joints[END_REF]. To take into account this nonlinear effect without employing a dedicated iterative algorithm, the approach by Luo and Tong was selected [START_REF] Luo | Fully-coupled nonlinear analysis of single lap adhesive joints[END_REF]. The local equilibrium selected for the formulation element is then related to the one used by Luo and Tong [START_REF] Luo | Fully-coupled nonlinear analysis of single lap adhesive joints[END_REF]. It allows for a coupling between the normal force and the bending moment, which leads to a dependency of the ME elementary stiffness matric K BBe on the applied tensile force termed f ME for the ME formulation. The global equilibrium of the single-lap joint leads then to:

𝑁 1 + 𝑁 2 = 𝑓 𝑀𝐸 ( 10
)
This equation is required to apply the Luo and Tong approach of the geometrically nonlinearities. Besides, it is easily possible to take into account the adhesive thickness in the local equilibrium bending equation as suggested by Hart-Smith [START_REF] Hart-Smith | Adhesive-bonded single-lap joints[END_REF]. The ME elementary stiffness matrix K BBe is then dependent on f ME and the adhesive thickness, termed e aME for the ME formulation. As a result, the ME formulation presented provides four different models, since f ME =f or f ME =0 and e aME =e a or e aME =0.

The constitutive equations for the adhesive layer are provided by:

𝑆 = 𝐸 𝑎 𝑒 𝑎 [𝑣 1 -𝑣 2 ] = 𝑘 𝐼 Δ𝑣 (11)
𝑇 = 𝐺 𝑎 𝑒 𝑎 [𝑢 2 -ℎ 2 𝜃 2 -(𝑢 1 + ℎ 1 𝜃 1 )] = 𝑘 𝐼𝐼 Δ𝑢 (12)
with:

Δ𝑢 = 𝑢 2 -𝑢 1 -ℎ 2 𝜃 2 -ℎ 1 𝜃 1 (13)
Δ𝑣 = 𝑣 1 -𝑣 2 (14)
where E a is the adhesive peel modulus and k I =E a /e a the adhesive peel relative stiffness. v is representative of the opening displacement of the adherend interface.

The local equilibrium of both adherends (Figure 4) provides the six following equations:

𝑑𝑁 𝑗 𝑑𝑥 = (-1) 𝑗 cos 𝜃 𝑗 𝑏𝑇, 𝑗 = 1,2 (15)
𝑑𝑉 𝑗 𝑑𝑥 = (-1) 𝑗+1 𝑏𝑆 + (-1) 𝑗 sin 𝜃 𝑗 𝑏𝑇, 𝑗 = 1,2 (16)
𝑑𝑀 𝑗 𝑑𝑥 + 𝑉 𝑗 + cos 𝜃 𝑗 𝑏 (ℎ 𝑗 + 𝑒 𝑎𝑀𝐸 2 ) 𝑇 -sin 𝜃 𝑗 𝑁 𝑗 = 0, 𝑗 = 1,2 (17)
with:

ℎ 𝑗 = 𝑒 𝑗 2 , 𝑗 = 1,2 (18)
where V j is the shear force in the adherend j, M j the bending moment in the adherend j,  j the bending angle in the adherend j and S is the adhesive peel stress. were employed in this paper. First, it is supposed that the adherend bending angles are small allowing for the 1 st order Taylor expansion of sine and cosine functions. Moreover, the product of the adhesive shear stress with the adherend bending angle is neglected. Under these both first assumptions, the Eqs. [START_REF] Stapleton | Design of functionally graded joints using a polyurethane-based adhesive with varying amounts of acrylate[END_REF][START_REF] Gustafson | Dimensionless parameters in symmetric double lap joints: An orthotropic solution for thermomechanical loading[END_REF][START_REF] Stapleton | The analysis of adhesively bonded advanced composite joints using joint finite elements[END_REF] become:

𝑑𝑁 𝑗 𝑑𝑥 = (-1) 𝑗 𝑏𝑇, 𝑗 = 1,2 (19)
𝑑𝑉 𝑗 𝑑𝑥 = (-1) 𝑗+1 𝑏𝑆, 𝑗 = 1,2 (20)
𝑑𝑀 𝑗 𝑑𝑥 + 𝑉 𝑗 + 𝑏 (ℎ 𝑗 + 𝑒 𝑎𝑀𝐸 2 ) 𝑇 -𝜃 𝑗 𝑁 𝑗 = 0, 𝑗 = 1,2 (21)
The following algebraic transformation is applied to each parameters related to the subscript j (i.e.: N j , V j , M j , u j , v j , q j , h j ) quoted X:

( 𝑋 + 𝑋 - ) = Λ ( 𝑋 1 𝑋 2 ) = ( 1 1 1 -1 ) ( 𝑋 1 𝑋 2 ) (22) x y + N 1 +dN 1 M 1 +dM 1 V 1 +dV 1 V 2 +dV 2 N 2 +dN 2 M 2 +dM 2 M 1 V 1 V 2 M 2 N 1 N 2 S.bdx T.bdx T.bdx  1  1  2  1 e a /2
It is then assumed that:

𝑁 - 2 (𝜃 1 ∓ 𝜃 2 ) ≪ 1, 𝑗 = 1,2 (23)
Considering the Eq. ( 10) and Eq. ( 23), the Eqs. [START_REF] Paroissien | Hybrid (bolted/bonded) joints applied to aeronautic parts: Analytical one-dimensional models of a single-lap joint[END_REF][START_REF] Paroissien | Analytical two-dimensional model of a hybrid (bolted/bonded) single-lap joint[END_REF][START_REF] Paroissien | Elasto-Plastic Analysis of Bonded Joints with Macro-Elements[END_REF] become finally:

𝑑𝑁 + 𝑑𝑋 = 0 (24)
𝑑𝑉 + 𝑑𝑋 = 0 (25)

𝑑𝑀 + 𝑑𝑋 = -𝑉 + + 𝑘 𝐼𝐼 𝑏ℎ + 𝑢 -+ ( 1 2 𝑘 𝐼𝐼 𝑏(ℎ + + 𝑒 𝑎𝑀𝐸 ) 2 + 𝑓 𝑀𝐸 2 ) 𝜃 + + ( 1 2 𝑘 𝐼𝐼 𝑏(ℎ + + 𝑒 𝑎𝑀𝐸 )ℎ -) 𝜃 - (26)
𝑑𝑁 - 𝑑𝑋 = 2𝑘 𝐼𝐼 𝑏𝑢 -+ 𝑘 𝐼𝐼 𝑏(ℎ + + 𝑒 𝑎𝑀𝐸 )𝜃 + + 𝑘 𝐼𝐼 𝑏ℎ -𝜃 - (27)
𝑑𝑉 - 𝑑𝑋 = 2𝑘 𝐼 𝑏𝑤 - ( 28
) 𝑑𝑀 - 𝑑𝑋 = -𝑉 -+ 𝑘 𝐼𝐼 𝑏ℎ -𝑢 -+ ( 1 2 𝑘 𝐼𝐼 𝑏(ℎ + + 𝑒 𝑎𝑀𝐸 )ℎ -) 𝜃 + + ( 1 2 𝑘 𝐼𝐼 𝑏ℎ - 2 + 𝑓 𝑀𝐸 2 ) 𝜃 - (29)
The constitutive equations can then be written as:

𝑁 𝑗 = 𝐴 𝑗 𝑑𝑢 𝑗 𝑑𝑥 -𝐵 𝑗 𝑑𝜃 𝑗 𝑑𝑥 , 𝑗 = 1,2 (30)
𝑀 𝑗 = -𝐵 𝑗 𝑑𝑢 𝑗 𝑑𝑥 + 𝐷 𝑗 𝑑𝜃 𝑗 𝑑𝑥 , 𝑗 = 1,2 (31)
𝜃 𝑗 = 𝑑𝑣 𝑗 𝑑𝑥 (32)
where A j is the membrane stiffness of adherend j, B j the coupling membrane-bending stiffness of adherend j, D j the bending stiffness of adherend j (see Appendix A).

The algebraic transformation defined in Eq. ( 22) is applied so that the Eqs. [START_REF] Hart-Smith | Adhesive-bonded single-lap joints[END_REF][START_REF] Dechwayukul | Analysis of the Effects of thin sealant layers in aircraft structural joints[END_REF][START_REF] Hart-Smith | Adhesive-bonded double-lap joints[END_REF] become:

𝑑𝑢 + 𝑑𝑋 = 1 2 ( 𝐷 1 Δ 1 + 𝐷 2 Δ 2 ) 𝑁 + + 1 2 ( 𝐷 1 Δ 1 - 𝐷 2 Δ 2 ) 𝑁 -+ 1 2 ( 𝐵 1 Δ 1 + 𝐵 2 Δ 2 ) 𝑀 + + 1 2 ( 𝐵 1 Δ 1 - 𝐵 2 Δ 2 ) 𝑀 - ( 33
)
𝑑𝑣 + 𝑑𝑋 = 𝜃 + ( 34
) 𝑑𝜃 + 𝑑𝑋 = 1 2 ( 𝐵 1 Δ 1 + 𝐵 2 Δ 2 ) 𝑁 + + 1 2 ( 𝐵 1 Δ 1 - 𝐵 2 Δ 2 ) 𝑁 -+ 1 2 ( 𝐴 1 Δ 1 + 𝐴 2 Δ 2 ) 𝑀 + + 1 2 ( 𝐴 1 Δ 1 - 𝐴 2 Δ 2 ) 𝑀 - ( 35
) 𝑑𝑢 - 𝑑𝑋 = 1 2 ( 𝐷 1 Δ 1 - 𝐷 2 Δ 2 ) 𝑁 + + 1 2 ( 𝐷 1 Δ 1 + 𝐷 2 Δ 2 ) 𝑁 -+ 1 2 ( 𝐵 1 Δ 1 - 𝐵 2 Δ 2 ) 𝑀 + + 1 2 ( 𝐵 1 Δ 1 + 𝐵 2 Δ 2 ) 𝑀 - ( 36
) 𝑑𝑣 - 𝑑𝑋 = 𝜃 - (37) 𝑑𝜃 - 𝑑𝑋 = 1 2 ( 𝐵 1 Δ 1 - 𝐵 2 Δ 2 ) 𝑁 + + 1 2 ( 𝐵 1 Δ 1 + 𝐵 2 Δ 2 ) 𝑁 -+ 1 2 ( 𝐴 1 Δ 1 - 𝐴 2 Δ 2 ) 𝑀 + + 1 2 ( 𝐴 1 Δ 1 + 𝐴 2 Δ 2 ) 𝑀 - (38)
where  j =A j D j -B j B j ≠0.

Formulation of elementary stiffness matrices

The elementary stiffness matrices of BBa and BBe elements, termed K BBa and K BBe respectively, correspond to the linear relationships between the vector of nodal forces F e and the vector nodal displacements U e . Terming  the length of the ME, the definition of elementary stiffness matrices K BBa and K BBe are provided in Eq. (39) and Eq. ( 40)

respectively.

(

-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) ) ⟺ 𝐹 𝑒 = 𝐾 𝐵𝐵𝑎 𝑈 𝑒 (39) ( -𝑁 1 (0) -𝑁 2 (0) 𝑁 1 (Δ) 𝑁 2 (Δ) -𝑉 1 (0) -𝑉 2 (0) 𝑉 1 (Δ) 𝑉 2 (Δ) -𝑀 1 (0) -𝑀 2 (0) 𝑀 1 (Δ) 𝑀 2 (Δ) ) = 𝐾 𝐵𝐵𝑒 ( 𝑢 1 (0) 𝑢 2 (0) 𝑢 1 (Δ) 𝑢 2 (Δ) 𝑣 1 (0) 𝑣 2 (0) 𝑣 1 (Δ) 𝑣 2 (Δ) 𝜃 1 (0) 𝜃 2 (0) 𝜃 1 (Δ) 𝜃 2 (Δ)) ⟺ 𝐹 𝑒 = 𝐾 𝐵𝐵𝑒 𝑈 𝑒 (40)
In the case of the 1D-bar kinematics, the closed-form expressions for the components of K BBa can be determined analytically [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 one-dimensional models of a single-lap joint[END_REF][START_REF] Paroissien | Analytical two-dimensional model of a hybrid (bolted/bonded) single-lap joint[END_REF], such as:

𝐾 𝐵𝐵𝑎 =
where:

𝜂 2 = 𝐺 𝑒 ( 1 𝑒 1 𝐸 1 + 1 𝑒 2 𝐸 2 ) ( 42
)
𝜒 𝐴 = 𝐴 2 𝐴 1 (43)
In the case of the 1D-beam kinematics, the components of K BBe cannot be determined analytically and several methodologies have been suggested depending on the simplifying hypotheses [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF][START_REF] Paroissien | Contribution aux Assemblages Hybrides (Boulonnés/Collés) -Application aux Jonctions Aéronautiques[END_REF][START_REF] Paroissien | Analytical two-dimensional model of a hybrid (bolted/bonded) single-lap joint[END_REF][START_REF] Paroissien | Elasto-Plastic Analysis of Bonded Joints with Macro-Elements[END_REF]. In the present paper, a methodology based on the exponential matrix was used [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF][START_REF] Lélias | Experimental Characterization of Cohesive Zone Models for Thin Adhesive Layers Loaded in Mode I, Mode II, and Mixed-Mode I/II by the use of a Direct Method[END_REF] = 𝐴𝑉 where A is 12x12 matrix with real constant components and the unknown

vector V is such that t V=(u 1 u 2 v 1 v 2  1  2 N 1 N 2 V 1 V 2 M 1 M 2 )
. With this formulation methodology, the predictions in terms of adhesive stresses, adherend displacements and internal forces are given at nodes. A mesh is then required to obtain distributions at any abscissa along the overlap. As a result, this formulation methodology is suitable when FGA or nonlinear adhesives are used but less efficient in the case of HA or linear adhesives in terms of distribution predictions. However, this methodology allows for an easy formulation of BBe under various hypotheses in terms of local equilibrium or constitutive behavior. Finally, the elementary stiffness matrix of the bar element, modelling the adherend j outside the overlap is:

𝐾 𝑏𝑎𝑟,𝑗 = 𝐴 𝑗 𝑙 𝑗 ( 1 -1 -1 1 ) , 𝑗 = 1,2 (44)
In the case of the 1D-beam kinematics, the corresponding beam element is formulated following the same methodology as for the BB e (Appendix C in [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF]).

Uniform variation of temperature

As classically done in the FE method, a nodal force vector equivalent to the thermal loading was added at the left-hand side of Eqs. (39-40). The nodal force vector equivalent to the thermal loading is computed from the elementary nodal force vector equivalent to thermal load, given in Eq. ( 45) and Eq. ( 46) for the 1D-bar and 1D-beam kinematics [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF]:

𝐹 𝑒,𝑡ℎ𝑒𝑟𝑚 = ( -𝐴 1 𝛼 1 -𝐴 2 𝛼 2 𝐴 1 𝛼 1 𝐴 2 𝛼 2 ) Δ 𝑇 (45) 𝐹 𝑒,𝑡ℎ𝑒𝑟𝑚 = ( -𝑁 1 Δ 𝑇 -𝑁 2 Δ 𝑇 𝑁 1 Δ 𝑇 𝑁 2 Δ 𝑇 0 0 0 0 𝑀 1 Δ 𝑇 𝑀 2 Δ 𝑇 -𝑀 1 Δ 𝑇 -𝑀 2 Δ 𝑇 ) (46)
where 𝑁 𝑗 Δ 𝑇 the thermal normal force in the adherend j and 𝑀 𝑗 Δ 𝑇 is the thermal bending moment in the adherend j (see Appendix A). In the case of a lay-up characterized by a mirror symmetry, 𝑀 𝑗 Δ 𝑇 = 0 (as well as B j =0).

Convergence study

A convergence study was performed on the adhesive peak stresses as function on the number of MEs n_ME, assuming FGA properties. For the 1D-bar kinematics, the balanced joint under a pure mechanical loading was considered. For the 1D-beam kinematics, the unbalanced joint under a pure mechanical loading was considered, using the ME formulation hypotheses f ME =f and e aME =0. The conclusions for all the other configurations are the same. The maximal adhesive stresses were then recorded as a function of the number of ME. They were then normalized by their corresponding values at n_ME=1000. The normalized adhesive peak stresses are provided as functions of n_ME in Figure 5 and Figure 6, for the 1D-bar and 1Dbeam kinematics respectively. For both kinematics, it is shown that the adhesive peak stresses tend to a finite value. Moreover, for the 1D-beam kinematics, the curves related to peel stresses and shear stresses are very close. In the present paper, the number of MEs is then selected equal to 500 for L=25 mm leading to a mesh density of 20 MEs per mm. 1 and Table 1 to Table 3.

The FE models were developed using the SAMCEF v171 FE code (SIEMENS).

Modelling

To be as close as possible the modelling hypotheses of ME models, the 1D FE models were built from bar or beam elements for the adherends and spring elements for the adhesive layer.

The nodes associated with bar or beam elements were located at the actual neutral line of adherends. The nodes associated with the spring elements were located at the actual interfaces of adherends. For each adherend along the overlap, rigid body elements were used to link the nodes of the neutral lines and to the nodes of the adherend interface. A scheme of the 1D FE models is provided in Figure 7, including the prescribed displacements and loading.

It is then possible to take into account the geometrical effect of the adhesive thickness by setting e aEF =e a instead of e aEF =0 (Figure 7). Linear or nonlinear computation can be used.

The bar and beam elements are based on degree 3 interpolating functions under the Euler-Bernoulli kinematics. The overlap length of adherends was then regularly meshed with n_BE.

The stiffnesses of springs k u and k v are directly related to the mesh density along the overlap [START_REF] Dechwayukul | Analysis of the Effects of thin sealant layers in aircraft structural joints[END_REF]. For a spring element located at an abscissa x along the overlap, the stiffnesses are computed from the actual value of adhesive peel and shear modulus at the abscissa x, the adhesive thickness e a , the width b and the mesh density L/n_BE such as:

𝑘

Convergence study

The adhesive stresses predicted by the 1D FE models are expected to be dependent on the

3D FE models

Modelling

To assess the prediction accuracy of ME models, refined 3D FE models were developed for linear or nonlinear geometrically cases. The adherends were with 3D linear brick 8nodes elements (24 degrees of freedom per element). The selected elements have one internal mode to improve the behaviour under bending. The normal integration scheme was chosen.

The mesh of the 3D-model consists in an extrusion in the width direction of the FE model mesh. However, as the (xy) plane at half width is a symmetry plane (Figure 1), only one-half of the single-lap joint was modelled and symmetry conditions were applied. The adhesive layer was modelled according to different ways. For the 3D FE VM, the same brick elements as the adherends were used for the adhesive layer. For the 3D FE CZM, the adhesive layer was modelled by interface elements without any geometrical thickness. In order to additionally reduce the number of elements, a distributed mesh strategy was employed, leading to the creation of various useful mesh domains. Indeed, according to [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF], the adhesive stress distribution along the overlap predicted by the ME models in the single-lap joint configuration is characterized by high stress gradients at both overlap ends for both HA and FGA joints. It is indicated that for FGA joints, and contrary to the HA joints, it is not always the case as highlighted by Stein et al. [START_REF] Stein | Analytical models for functionally graded adhesive joints: A comparative study[END_REF]. As a result, the mesh was more refined at both overlap ends than at the middle of the overlap (Figure 9). Outside the overlap, the mesh was distributed along the length direction, so that the mesh was more refined near the overlap region. The mesh was distributed along the adherend thickness between the interface with the adhesive layer and its neutral line. Above the neutral line towards the adherend free surface, the mesh is regular. The mesh of the single-lap joint was then designed according to the following rules: (i) coincident meshes between the mesh domains, (ii) transition ratio equal to one between the mesh domains, (iii) element aspect ratio lower than 10, (iv) regular mesh in the width direction, (v) the aspect ratio of the smallest element in the adherendlocated at the interface with the adhesive at the overlap end (Figure 9)equal to one, (vi) regular mesh in the adhesive thickness along the adhesive thickness for the 3D FE VM. Under these rules, the mesh of 3D FE CZM is driven by the thickness of the smallest adherend element at the interface layer s and by the element longitudinal slenderness r (ratio between the maximal and the minimal length in the mesh distribution), while the mesh of 3D FE VM is driven by the number of element in the adhesive thickness n, s and r (Figure 9).

Convergence study

In order to justify the 3D FE models as well as the associated choice of mesh parameters n, s and r, a convergence study of adhesive peak stresses is presented. The single-lap unbalanced FGA joint under a combined mechanical and thermal loading was considered. Nonlinear geometrical computations were carried out for various combinations of n, s and r and the adhesive peak stresses were recorded along the line of adhesive layer mid-plane located in the n elements in the adhesive thickness smallest adherend element r * s s Mesh distribution direction: from the smallest to the largest symmetry plane. Firstly, for the 3D FE CZM, the maximal adhesive shear and peel stresses as functions of s and r are provided in Figure 10 and Figure 11, respectively. It is then shown that a convergence is obtained from a combination of s=0.05 mm and r=5. These parameters were fixed for this paper. Secondly, for the 3D FE VM, the maximal adhesive shear and peel stresses as a function of n, s and r are provided in Figure 12 and Figure 13, respectively. The adhesive stresses were taken along the neutral line of the adhesive layer. The choice n=20, s=0.05 mm and r=5 is sufficient to obtain converged values. These parameters were fixed for this paper. is assessed at the centre of each ME and not at the extremity. Moreover, in terms of distribution along the overlap, the predictions of 1D FE and 1D ME models appear as superimposed, as illustrated in Figure 14, for the unbalanced case under combined mechanical and thermal loadings, for example. It can be concluded that a very good agreement is obtained.

Figure 14. Adhesive shear stress distribution along the overlap from the 1D-bar ME and FE models for the FGA unbalanced joint under combined mechanical and thermal loadings.

Comparison of 1D-beam ME models with 1D-beam FE models

In the frame of the

Comparison of 1D ME and FE models with 3D FE models

To compare with the 3D FE CZM and with the 3D FE VM, the 1D-beam FE and ME models with e aME =e aFE =0 and with e aME =e aFE =e a were considered, respectively. The linear and nonlinear geometrical analysis hypotheses was associated for the 1D-beam model with f ME =0

and f ME =f, respectively. The maximal adhesive shear stress T max and peel stress S max predicted by the 1D-beam ME and FE models and by the 3D FE models as well as the differences in T max and S max predicted by the 1D-beam ME and FE models relatively to those predicted by the related 3D FE model are provided in Table B-3 (Appendix B). The FGA balanced and unbalanced joints subjected to pure mechanical or combined mechanical and thermal loadings were considered. Concerning the adhesive peak shear stress, the absolute relative differences are small and lower than 8% in the worst case. It appears that the 1D ME and FE models provide the same order of magnitude of absolute relative differences. Concerning the adhesive peak peel stress, the absolute relative differences are larger than for the shear stress and reach 17.5% in the worst case. The 1D ME and FE models provide almost the same order of magnitude of absolute relative differences. But the largest absolute relative differences are obtained with the 1D ME models. For both shear and peel stresses, the absolute relative differences are lower when the geometrical effect of the adhesive thickness (i.e.: e aME =e aFE =0

and CZM) is neglected. The nonlinear geometrical analysis hypothesis increases the absolute relative differences in terms of adhesive peak peel stress, all the more for the 1D ME models and independently on the adhesive thickness geometrical effect. The adhesive peak shear stress appears as less sensitive to geometrical nonlinearities. The comparison of adhesive shear and peel stress distributions along the overlap for the FGA unbalanced joint subjected to mechanical and thermal loadings and pure mechanical loadings are then provided in Figure 17 to Figure 18 and Figure 19 to Figure 20, respectively. For the 3D FE VM, the adhesive shear tends to zero in x=0 and x=L, in order to comply with the free stress conditions at overlap ends (Figure 19). The adhesive peel stress is decreasing too. The adhesive peak stresses are obtained for an abscissa located within the overlap near the overlap ends. The hypotheses underlying the 1D-beam ME and FE models as well as the 3D FE CZM are not able to produce this behaviour. Nevertheless, a good agreement is globally show in terms of distribution shape. The differences shown are representative for the effect of simplification hypotheses on kinematics and constitutive behaviours. A last remark concerns the choice the adhesive Young's modulus as the adhesive peel modulus in this paper. Other choices could have been made to represent for the confinement of the adhesive layer between both adherends and subjected to out-of-plan tensile. The contribution of adherend out-of-plane stiffnesses in the adhesive peel modulus [START_REF] Hart-Smith | Adhesive-bonded double-lap joints[END_REF] or the plane strain adhesive Young's modulus instead of the adhesive Young's modulus as the adhesive peel modulus [START_REF] Alfredsson | A closed-form solution to statically indeterminate adhesive joint problemsexemplified on ELS-specimens[END_REF][START_REF] Stein | Analytical models for functionally graded adhesive joints: A comparative study[END_REF] are possible choice examples. Nevertheless, this discussion is out of the scope of this paper, even if it could be addressed with the provided models and results presented in this paper. (e aME =e a and f ME =f) and FE (e aFE =e a and nonlinear geometrical) models and 3D FE VM for the unbalanced case under a pure mechanical loading.

Application: influence of overlap length

As an application of the simplified stress analyses based on ME models, the influence of the overlap length on the maximal adhesive stresses of balanced FGA joints was studied. In the case of HA joints, the increase of overlap length tends to decrease the maximal adhesive stresses [START_REF] Hart-Smith | Adhesive-bonded single-lap joints[END_REF][START_REF] Hart-Smith | Adhesive-bonded double-lap joints[END_REF].

1D-bar kinematics

Within the 1D-bar framework the peak shear stress tends to a finite value [START_REF] Hart-Smith | Adhesive-bonded double-lap joints[END_REF] for HA joints.

The analytical expression for the critical applied tensile load can then be deduced. Moreover, it is not useful to increase the overlap length above a certain value to increase the strength.

When using FGA instead of HA, the distribution of adhesive shear stress could predict an adhesive peak stress at the centre of the overlap [START_REF] Stein | Analytical models for functionally graded adhesive joints: A comparative study[END_REF] instead of at both overlap ends. The . For small overlap lengths, the adhesive peak shear stress is located at the centre of the overlap. For long overlap lengths, the adhesive peak shear stress is located at both overlap ends. For intermediate overlap lengths, the adhesive peak stress is reached between the overlap centre and the overlap end. The adhesive peak shear stress as function of the overlap length is provided in Figure 23 for the FGA and HA joints. As for the HA joints, the increase of the overlap length tends to reduce the adhesive peak stresses. Nevertheless, contrary to the HA joints, there is an overlap length for which the adhesive peak shear stress is minimal, and is equal here to around 30 mm.

1D-beam kinematics

The 1D-beam ME model, under the ME formulation hypotheses e aME =0 and f ME =f is used to provide the adhesive peak shear and peel stress as a function of overlap length for FGA and HA joints, in Figure 24 and Figure 25 respectively. The evolutions appear as similar between the FGA and HA joints. However, the adhesive peak shear and peel stresses are lower than those with the minimal modulus HA joint, over the range of overlap length selected. The reduction in adhesive peak shear and peel stress tends is more significant for shorter overlap lengths. The reduction in adhesive peel stress, which mainly drives the failure of singe-lap bonded joints [START_REF] Hart-Smith | Adhesive-bonded single-lap joints[END_REF], is less significant than in adhesive shear stress.

Conclusions

In this paper, a comparison between simplified and refined stress analyses of simply supported FGA single-lap balanced and unbalanced joints subjected to pure mechanical tensile in-plane loading or combined with thermal loading is presented. The simplified stress analyses are considered in the 1D-bar and 1D-beam framework. The ME technique is the resolution scheme used for the simplified stress analysis. The 1D ME models are then described in this paper including a convergence study, while the detailed mathematical derivation can be found in an already published paper [START_REF] Paroissien | Simplified stress analysis of functionally graded single-lap joints subjected to combined thermal and mechanical loads[END_REF]. It is indicated that the 1D-beam ME model allows for the simple consideration of various and classical local equilibriums [START_REF] Goland | The stresses in cemented joints[END_REF][START_REF] Luo | Fully-coupled nonlinear analysis of single lap adhesive joints[END_REF][START_REF] Hart-Smith | Adhesive-bonded single-lap joints[END_REF] offering the consideration or not of the adhesive thickness on the local equilibrium bending Secondly, the 1D-beam ME models provide predictions close to 1D-beam FE models: lower than 4% and 8% on the adhesive peak shear and peel stresses respectively. The geometrical effect of the adhesive thickness and the geometrical nonlinearities cause an increase of absolute relative differences. Thirdly, compared to the predictions of 3D FE models, the 1Dbeam ME models provide predictions lower than 8% on the adhesive peak shear stress and up to 17.5% for the adhesive peak peel stress. Once again, the geometrical effect of the adhesive thickness and the geometrical nonlinearities cause a significant increase of absolute relative differences on the adhesive peak peel stress. The use of 1D-beam FE models instead of 1Dbeam ME models restricts the level of increase of absolute relative differences only; both 1D ME and FE models result in similar order of magnitude of absolute relative difference.

However, the measured relative differences are representative for the some units of MPas. As a result, the simplified stress analyses could be used at the pre-sizing stage to perform early design, possibly based on comparative studies. Fourthly, it is shown that the increase of the overlap length reduce the adhesive peak stresses as for the HA joints. Moreover, there is a range of overlap lengths for which the adhesive peak stresses of FGA joints are smaller than those minimal moduli HA joints. Finally, in the 1D-bar kinematics framework, there is an overlap length for which the adhesive peak shear stress is minimal. (A-7)

The stiffness parameters are thus defined such as for j=1,2:

𝐴 𝑗 = 𝑏 ∑ 𝑄 𝑗 𝑝 𝑗 (ℎ 𝑝 𝑗 -ℎ 𝑝 𝑗 -1 ) 𝑛 𝑗 𝑝 𝑗 =1 (A-8) 𝐵 𝑗 = 𝑏 2 ∑ 𝑄 𝑗 𝑝 𝑗 (ℎ 𝑝 𝑗 2 -ℎ 𝑝 𝑗 - 1 2 )
𝑛 𝑗 𝑝 𝑗 =1

(A-9)

𝐷 𝑗 = 𝑏 3 ∑ 𝑄 𝑗 𝑝 𝑗 (ℎ 𝑝 𝑗 3 -ℎ 𝑝 𝑗 - 1 3 )

Appendix B

This appendix provide the tables relating to the predicted adhesive peak stresses and their relative differences between the 1D-bar ME model and the 1D-bar FE model (Table B-1), between the 1D-beam ME models and 1D-beam FE models (Table B-2) and between the 1Dbeam ME and FE models and the 3D FE models (Table B-3).

  A j extensional stiffness (N) of adherend j B j extensional and bending coupling stiffness (N.mm) of adherend j D j bending stiffness (N.mm 2 ) of adherend j E a adhesive peel modulus (MPa) E a,min adhesive shear modulus (MPa) E a,max adhesive shear modulus (MPa) E j adherend Young's modulus (MPa) of adherend j F e element nodal force vector F e,therm elementary nodal force vector equivalent to thermal load G a adhesive shear modulus (MPa) G a,max maximal adhesive shear modulus (MPa) G a,min minimal adhesive shear modulus (MPa) K BBa elementary stiffness matrix of a bonded-bars element K BBe elementary stiffness matrix of a bonded-beams element K bar,j elementary stiffness matrix of a bar for the adherend j L length (mm) of bonded overlap M j bending moment (N.mm) in adherend j around the z direction 𝑀 𝑗 Δ 𝑇 thermal bending moment (N.mm) in adherend j around the z direction N j normal force (N) in adherend j in the x direction 𝑁 𝑗 Δ 𝑇 thermal normal force (N) in adherend j in the x direction S adhesive peel stress (MPa) S max maximal adhesive peel stress (MPa) T adhesive shear stress (MPa) T max maximal adhesive shear stress (MPa) U e element nodal displacement vector V j shear force (N) in adherend j in the y direction b width (mm) of the adherends e a thickness (mm) of the adhesive layer e aME thickness (mm) of the adhesive layer used in the ME formulation e aFE thickness (mm) of the adhesive layer used in the FE model f magnitude of applied tensile force (N) f ME magnitude of applied tensile force (N) used in the ME formulation h j half thickness (mm) of adherend j k I adhesive elastic stiffness (MPa/mm) in peel k II adhesive elastic stiffness (MPa/mm) in sheark u spring element stiffness (N/mm) along the x-axis k v spring element stiffness (N/mm) along the y-axis n number of elements in the adhesive thickness n_BE number of bar or beam elements n_ME number of macro-elements r element longitudinal slenderness s length of the smallest element u j displacement (mm) of adherend j in the x direction v j displacement (mm) of adherend j in the y direction  overlap length (mm) of a macro-element   T variation of temperature (K) u slipping displacement (mm) v opening displacement (mm)  j characteristic parameter (N 2 .mm 2 ) of adherend j  j coefficient of thermal expansion (K -1 ) of adherend j  characteristic argument (mm -1 ) in 1D-bar analysis   j bending angle (rad) of the adherend j around the z direction 𝜒 𝐴 adherend stiffness unbalance parameter (-) (x,y,z) global reference system of

  The 1D FE models are developed following the 1D-bar kinematics and the 1D-beam kinematics, and termed 1D-bar FE models and 1D-beam FE models. Bar or beam elements are then used to model the adherend, while spring elements model the adhesive layer. 3D models are also studied where 3D-brick elements are used to model the adherends. The adhesive layer is modelled either with brick elements in the 3D FE volume model (VM) or with interface elements in the 3D FE cohesive zone model (CZM). The adhesive stress distribution along the overlap predicted by the 1D ME, 1D FE and 3D FE models are then compared. As an application of the ME models, the influence of the overlap length on the adhesive peak stresses of FGA joints is presented and compared to those of homogeneous adhesive (HA) joints. In this paper, a simply-supported single-lap joint subjected to in-plane tensile loading with a force f=5 kN and a uniform variation of temperature  T =+50°C is considered (Figure1), involving a balanced joint and an unbalanced joint cases. The balanced case corresponds to the joining of two adherends made in steel. The unbalanced case differs from the balanced one by the material of the lower adherend only: aluminium instead of steel.

Figure 1 .

 1 Figure 1. Simply supported single-lap involving the geometrical parameters, boundary

2 .

 2 Description of simplified stress analyses of FGA single-lap joints

  𝑎,𝑚𝑎𝑥 -(𝐸 𝑎,𝑚𝑎𝑥 -𝐸 𝑎,𝑚𝑖𝑛 ) ( 𝑎,𝑚𝑎𝑥 -(𝐺 𝑎,𝑚𝑎𝑥 -𝐺 𝑎,𝑚𝑖𝑛 ) (

Figure 2 .

 2 Figure 2. ME model for a single-lap joint using HA, FGA or nonlinear behavior, including the

Figure 3 .

 3 Figure 3. Free body diagram of infinitesimal pieces included between x and x+dx of both

Figure 4 .

 4 Figure 4. Free body diagram of infinitesimal pieces included between x and x+dx of both

Figure 5 .Figure 6 .

 56 Figure 5. Maximal adhesive shear stresses (T max ) normalized by its value for n_ME=1000 as a

3 .

 3 Description of Finite Element models of FGA single-lap joints 3.1. Overview Two different types of FE models are presented. The first type of FE models are designed to be representative for the ME models in terms of hypotheses. Both 1D-bar and 1D-beam kinematics are then considered leading to one 1D-bar FE model and one 1D-beam model. The second type of FE models are refined 3D models considered as a relevant approximation of the physical reality. The inclusion or not of the adhesive thickness in the local equilibrium bending equations was considered. As a result, the adhesive layer was modelled either with 3D bricks elements or with 3D interface elements; the corresponding 3D FE models are then termed 3D FE volume model (VM) and 3D FE cohesive zone model (CZM) respectively. The geometry, boundary conditions and loading are provided in Figure

  𝑣 (𝑥) = 𝑚(𝑥) 0<x<L)=1 and m(x=0)=m(x=L)=1/2.

Figure 7 .

 7 Figure 7. Principle scheme for the 1D FE models.

  number of spring elements. A convergence study was then undertaken. The results of the balanced case under a pure mechanical loading are presented under the 1D-beam kinematics, such as e aEF =e a and geometrically linear computation. The same conclusions for the other possible configurations hold. For each mesh configuration associated to n_BE, the adhesive peak stresses are then recorded and normalized by their corresponding values at n_BE=500.The normalized adhesive peak stresses are given as a function of n_BE in Figure8. It is shown that the maximal adhesive stresses tend to finite values. In the present paper, the bar or or beam elements is then chosen equal to 500 for L=25 mm leading to a mesh density of 20 elements per mm.

Figure 8 .

 8 Figure 8. Maximal adhesive shear (T max ) and peel (S max ) stresses stress normalized by its value

Figure 9 .

 9 Figure 9. Illustration of the mesh distribution strategy.

Figure 10 .Figure 11 .

 1011 Figure 10. Maximal adhesive shear stress as a function of longitudinal slenderness r for

Figure 12 .

 12 Figure 12. Maximal adhesive shear as functions of the number of elements in the adhesive

  linear geometrical and (iv) e afE =e a and nonlinear geometrical, respectively. The maximal adhesive shear stress T max and peel stress as well as the differences on T max and S max predicted by the 1D-beam ME model relatively to those predicted by the related 1D-beam FE model are reported in

  shear and peel stress distributions along the overlap for the FGA unbalanced joint and for the ME formulation hypothesis e aME =e a and f ME =f (largest relative differences) under combined mechanical and thermal loadings are provided in Figure15ad Figure16. A very good agreement is obtained.

Figure 15 .

 15 Figure 15. Adhesive shear stress distribution along the overlap from the 1D-beam ME

Figure 16 .

 16 Figure 16. Adhesive peel stress distribution along the overlap from the 1D-beam ME (e aME =e a

Figure 17 .

 17 Figure 17. Adhesive shear stress distribution along the overlap from the 1D-beam ME

Figure 18 .

 18 Figure 18. Adhesive peel stress distribution along the overlap from the 1D-beam ME (e aME =0

Figure 19 .

 19 Figure 19. Adhesive shear stress distribution along the overlap from the 1D-beam ME

Figure 20 .

 20 Figure 20. Adhesive shear stress distribution along the overlap from the 1D-beam ME

  Figure 22. For small overlap lengths, the adhesive peak shear stress is located at the centre of

Figure 21 .

 21 Figure 21. Adhesive shear stress distribution along the overlap (normalized abscissa) for

Figure 22 .

 22 Figure 22. Normalized abscissa for which the adhesive shear stress is reached as function of

Figure 23 .

 23 Figure 23. Adhesive peak shear stress as function of the overlap length for FGA and HA

Figure 24 .

 24 Figure 24. Adhesive peak shear stress as a function of the overlap length for FGA and HA

Figure 25 .

 25 Figure 25. Adhesive peak peel stress as a function of the overlap length for FGA and HA

  scheme. Two types of 3D models are employed. The first type models the adhesive layer as a

  Carbas et al. then presented a resolution scheme based on Taylor expansion in power series (TEPS). This stress analysis is restricted to half of the overlap length of balanced joints with a linear graduation of the

adhesive shear modulus. Stein et al. presented a 1D-bar analysis as well as a sandwich-type 1D-beam analysis using TEPS resolution able to address unbalanced bonded joints under any adhesive properties graduations

[START_REF] Stein | Stress solution for functionally graded adhesive joints[END_REF][START_REF] Stein | Analytical models for functionally graded adhesive joints: A comparative study[END_REF]

. In 2017, Stapleton et al. used a joint element (JE) for

Table 1 .

 1 Geometrical parameters of joint configurations

	b (mm)	e a (mm)	e 1 =e 2 (mm)	L (mm)	l 1 =l 2 (mm)
	25	0.2	2	25	75

Table 2 .

 2 Material parameters of adherends.

		Coefficient of thermal expansion (K -1 ) Young's modulus (GPa)
	Steel	12E-6	210
	Aluminium	24E-6	70

Table 3 .

 3 Adhesive material properties.

	E a,max (MPa)	E a,min (MPa)	 a
	6500	2500	0.36

  1D-beam kinematics, the ME model presented cover four different sets of hypotheses involved in the formulation of the ME elementary stiffness matrix: (i) e aME =0 and f ME =0, (ii) e aME =0 and f ME =f, (iii) e aME =e a and f ME =0 and (iv) e aME =e a and f ME =f. To perform

the comparisons, the four different sets of hypotheses were applied to the 1D-beam FE model: (i) e aFE =0 and linear geometrical, (ii) e aFE =0 and nonlinear geometrical, (iii) e aFE =e a and

  Table B-2 (Appendix B). The ME formulation hypothesis e aME =e a and f ME =f leads to the largest relative differences, while the ME formulation hypothesis e aME =0 and f ME =0 leads to the lowest. Within a set of ME formulation hypotheses, the relative differences on the adhesive peak stress are of the same order of magnitude, whatever the adhesive properties, the joint configuration or the applied loadings chosen. Moreover, it appears that the consideration of the adhesive thickness in the local equilibrium bending moment equation (e aME =0 or e aME =e a ) is more influent on the shear stresses than on the peel stresses, independently on the geometrical nonlinearities: around 1% if e aME =0 while around 3% if e aME =e

a . The geometrical nonlinearities have more impact on the peel stresses than on the shear stresses independently of the hypothesis of e aME : around 1% without geometrical nonlinearities while around 7% with geometrical nonlinearities. It is reminded that the ME models provide only an approximation regarding the geometrical nonlinearities effect. The comparison of adhesive

Table B -

 B

	1D-BAR MODELS		T max
	CASE	ADHESIVE PROPERTIES	ME (MPa)	FE (MPa)	ME / FE
	balanced	G a,min HA	15.54 15.54 0,00%
	pure mechanical	G a,max HA	23.97 23.97 0,00%
	loading	FGA	12.59 12.56 0,24%
	unbalanced	G a,min HA	31.49 31.49 0,00%
	pure mechanical	G a,max HA	50.61 50.61 0,00%
	loading	FGA	26.57 26.49 0,32%
	unbalanced	G a,min HA	44.534 44,534 0,00%
	combined mechanical	G a,max HA	71.85 71.85 0,00%
	and thermal loadings	FGA	37.71 37.59 0,32%

1. Adhesive peak shear stresses and difference of 1D-bar ME model prediction relatively to 1D-bar FE model predictions.

Table B -

 B 2. Adhesive peak shear and peel stress and difference of 1D-beam ME model prediction relatively to 1D-beam FE model predictions.

	unbalanced		G a,min HA	45.64	45.66	-0.04%	31.65	34.26	-7.61%
	combined mechanical and thermal loadings	G a,max HA FGA	73.92 41.12	73.94 40.89	-0.03% 0.57%	48.07 30.87	51.42 33.05	-6.52% -6.60%
	1D-BEAM
	MODELS
	1D-BEAM MODELS	ME e aME =0, f ME =0 FE e aFE =0, linear geometrical		T max			S max
	CASE		ADHESIVE PROPERTIES	ME (MPa)	FE (MPa)	ME / FE	ME (MPa)	FE (MPa)	ME / FE
	balanced		G a,min HA	27.31	27.31	0.00%	37.27	37.26	0.03%
	pure mechanical	G a,max HA	43.45	43.45	0.01%	59.74	59.71	0.04%
	loading		FGA	23.79	23.64	0.64%	35.44	35.21	0.66%
	unbalanced		G a,min HA	50.16	50.16	0.00%	61.40	61.39	0.01%
	pure mechanical	G a,max HA	82.36	82.36	0.00%	93.89	93.85	0.03%
	loading		FGA	45.76	45.49	0.58%	59.07	58.74	0.56%
	unbalanced		G a,min HA	57.28	57.28	0.00%	58.67	58.66	0.02%
	combined mechanical and thermal loadings	G a,max HA FGA	93.74 52.15	93.74 51.84	0.00% 0.59%	88.82 56.52	88.80 56.21	0.03% 0.56%
	1D-BEAM MODELS	ME e aME =0, f ME =f FE e aFE =0, nonlinear geometrical		T max			S max
	CASE		ADHESIVE PROPERTIES	ME (MPa)	FE (MPa)	ME / FE	ME (MPa)	FE (MPa)	ME / FE
	balanced		G a,min HA	24.08	24.09	-0.03%	28.89	30.30	-4.68%
	pure mechanical loading	G a,max HA FGA	37.710 20.73	37.72 20.60	-0.01% 0.61%	46.59 27.48	48.35 28.58	-3.65% -3.82%
	unbalanced		G a,min HA	37.22	37.27	-0.12%	31.94	34.31	-6.90%
	pure mechanical	G a,max HA	60.17	60.23	-0.09%	48.84	51.84	-5.80%
	loading		FGA	33.52	33.35	0.49%	31.00	32.96	-5.93%

ME e aME =e a , f ME =0 T max S max FE e aFE = e a , linear geometrical

	CASE	ADHESIVE PROPERTIES	ME (MPa)	FE (MPa)	ME / FE	ME (MPa)	FE (MPa)	ME / FE
	balanced	G a,min HA	28.22	29.18	-3.31%	41.00	40.99	0.03%
	pure mechanical	G a,max HA	44.92	46.47	-3.34%	65.71	65.69	0.04%
	loading	FGA	24.67	25.45	-3.06%	38.98	38.73	0.66%
	unbalanced	G a,min HA	51.86	53.65	-3.34%	67.28	66.91	0.55%
	pure mechanical	G a,max HA	85.21	88.19	-3.38%	102.9	102.3	0.60%
	loading	FGA	47.45	48.97	-3.11%	64.68	63.98	1.09%
	unbalanced	G a,min HA	58.73	60.28	-2.57%	64.41	64.19	0.36%
	combined mechanical and thermal loadings	G a,max HA FGA	96.19 53.63	98.78 54.91	-2.61% -2.33%	97.58 62.00	97.24 61.44	0.35% 0.91%
	1D-BEAM
	MODELS

ME e aME =e a , f ME =f T max S max FE e aFE = e a , nonlinear geometrical

	CASE	ADHESIVE PROPERTIES	ME (MPa)	FE (MPa)	ME / FE	ME (MPa)	FE (MPa)	ME / FE
	balanced	G a,min HA	24.76	25.48	-2.85%	31.78	33.35	-4.70%
	pure mechanical	G a,max HA	38.79	39.94	-2,89%	51.26	53.21	-3.67%
	loading	FGA	21.39	21.97	-2,63%	30.24	31.45	-3.85%
	unbalanced	G a,min HA	37.87	38.64	-1,99%	34.95	37.34	-6.40%
	pure mechanical	G a,max HA	61.26	62.52	-2,02%	53.44	56.42	-5.28%
	loading	FGA	34.22	34.81	-1,72%	33.90	35.81	-5.35%
	unbalanced	G a,min HA	46.06	46.57	-1,10%	34.61	37.22	-7.00%
	combined mechanical and thermal loadings	G a,max HA FGA	74.63 41.63	75.47 41.97	-1,11% -0,81%	52.56 33.71	55.85 35.82	-5.90% -5.88%

Table B -

 B 3. Adhesive peak shear and peel stress and difference of 1D-beam ME and FE model predictions relatively to 3D FE predictions.

	CASE	1D ME (MPa)	1D FE (MPa)	3D FE (MPa)	1D ME / 3D FE	1D FE / 3D FE	1D ME (MPa)	1D FE (MPa)	3D FE (MPa)	1D ME / 3D FE	1D FE / 3D FE
	balanced, pure mechanical loading	23.79	23.64	23.99	-0.82%	-1.45%	35.44	35.21	38.34	-7.57%	-8.17%
	unbalanced, pure mechanical loading	45.76	45.49	46.99	-2.62%	-3.18%	59.07	58.74	66.53	-11.21% -11.70%
	unbalanced combined
	mechanical and thermal	52.15	51.84	55.10	-5.35%	-5.91%	62.00	56.21	61.17	1.36%	-8.11%
	loadings
	1D-BEAM	ME FE		e aME =0 e aFE =0		f ME =f nonlinear geometrical
	3D	FE		CZM		nonlinear geometrical
				T max					S max
	CASE	1D ME (MPa)	1D FE (MPa)	3D FE (MPa)	1D ME / 3D FE	1D FE / 3D FE	1D ME (MPa)	1D FE (MPa)	3D FE (MPa)	1D ME / 3D FE	1D FE / 3D FE
	balanced, pure mechanical loading	20.73	20.60	20.50	1.08%	0.47%	27.48	28.58	30.64	-10.29%	-6.73%
	unbalanced, pure mechanical loading	33.52	33.35	33.24	0.83%	0.33%	31.00	32.96	36.89	-15.95% -10.65%
	unbalanced combined
	mechanical and thermal	41.12	40.89	42.27	-2.72%	-3.28%	30.87	33.05	33.73	-8.49%	-2.03%
	loadings
	1D-BEAM	ME FE		e aME =e a e aFE =e a		f ME =0 linear geometrical
	3D	FE		VM			linear geometrical
				T max					S max
	CASE	1D ME
	1D-BEAM	ME FE		e aME =0 e aFE =0		f ME =0 linear geometrical
	3D	FE		CZM		linear geometrical
				T max					S max

(MPa) 1D FE (MPa) 3D FE (MPa) 1D ME / 3D FE 1D FE / 3D FE 1D ME (MPa) 1D FE (MPa) 3D FE (MPa) 1D ME / 3D FE 1D FE / 3D FE balanced, pure mechanical loading

	unbalanced, pure mechanical loading	34.22	34.81	34.36	-0.41%	1.33%	33.90	35.81	41.08	-17.49% -12.82%
	unbalanced combined
	mechanical and thermal	41.63	41.97	42.79	-2.73%	-1.93%	33.71	35.82	37.42	-9.90%	-4.27%
	loadings
	1D-BEAM	ME FE		e aME =e a e aFE =e a		f ME =f nonlinear geometrical
	3D	FE		VM			nonlinear geometrical
				T max					S max
	CASE	1D ME (MPa)	1D FE (MPa)	3D FE (MPa)	1D ME / 3D FE	1D FE / 3D FE	1D ME (MPa)	1D FE (MPa)	3D FE (MPa)	1D ME / 3D FE	1D FE / 3D FE
	balanced, pure mechanical loading	21.39	21.97	22.60	-5.35%	-2.79%	30.24	31.45	35.80	-15.54% -12.16%
	unbalanced, pure mechanical loading	34.22	34.81	34.36	-0.41%	1.33%	33.90	35.81	41.08	-17.49% -12.82%
	unbalanced combined
	mechanical and thermal	41.63	41.97	42.79	-2.73%	-1.93%	33.71	35.82	37.42	-9.90%	-4.27%
	loadings
		24.67	25.45	26.78	-7.88%	-4.98%	38.98	38.73	44.48	-12.35% -12.93%
	unbalanced, pure mechanical loading	47.45	48.97	50.09	-5.28%	-2.24%	64.68	63.98	73.54	-12.05% -13.01%
	unbalanced combined
	mechanical and thermal	53.63	54.91	57.58	-6.86%	-4.64%	62.00	61.44	67.39	-8.00%	-8.83%
	loadings
	1D-BEAM	ME FE		e aME =e a e aFE =e a		f ME =f nonlinear geometrical
	3D	FE		VM			nonlinear geometrical
				T max					S max
	CASE	1D ME (MPa)	1D FE (MPa)	3D FE (MPa)	1D ME / 3D FE	1D FE / 3D FE	1D ME (MPa)	1D FE (MPa)	3D FE (MPa)	1D ME / 3D FE	1D FE / 3D FE
	balanced, pure mechanical loading	21.39	21.97	22.60	-5.35%	-2.79%	30.24	31.45	35.80	-15.54% -12.16%

Acknowledgement

This work has not received any specific grant.

by the nonlinear geometrically 3D FE VM model of the unbalanced single-lap FGA joint under combined mechanical and thermal loadings.

Comparison of ME and FE models predictions

Comparison of 1D-bar ME model with 1D-bar FE model

The maximal adhesive shear stress T max as well as the differences on T max predicted by the 1Dbar ME model relatively to those predicted by the 1D-bar FE model are provided in Table B-1 (Appendix B). A maximal relative difference of 0.32% was obtained for the FGA joint whereas it is equal to 0.00% for the HA joints. This small difference between the FGA and HA joints could be explained by the fact that, for FGA joints, the adhesive modulus of the ME This appendix provides the derivation of the constitutive equations of laminated beams used in the 1D-beam analysis, in the (X,Y j ,Z) reference local axis of the adherend, the height origin of which is taken on the neutral line. The normal force and the bending moment are such as:

where, in the adherend j n j is the number of layers and h pj is the final height of the p j th layer.

Moreover, the orthotopic behavior provides

where, in the adherend j, 𝑄 𝑗 𝑝 𝑗 is the matrix of reduced stiffness in the p j th layer.

As a result, the normal force and the bending moment are given by:


Thomas Mercier

Marion Bouvet

Emilie Dubois-Deruy

Arthur Dechaumes

Olivia Beseme

Vincent Richard

Paul Mulder

Florence Pinet

Interplay between phosphorylation and O-GlcNAcylation of sarcomeric proteins in ischemic heart failure

Post-translational modifications (PTMs) of sarcomeric proteins could participate to left ventricular (LV) remodeling and contractile dysfunction leading in advanced heart failure (HF) with altered ejection fraction. Using an experimental rat model of HF (ligation of left coronary artery) and phosphoproteomic analysis, we identified an increase of desmin phosphorylation and a decrease of desmin O-N-acetylglucosaminylation (O-GlcNAcylation). We aim to characterize interplay between phosphorylation and O-GlcNAcylation for desmin in primary cultures of cardiomyocyte by specific O-GlcNAcase (OGA) inhibition with thiamet G and silencing O-GlcNAc transferase (OGT) and, in perfused heart perfused with thiamet G in sham-and HF-rats. In each model, we found an efficiency of O-GlcNAcylation modulation characterized by the levels of O-GlcNAcylated proteins and OGT expression (for silencing experiments in cells). In perfused heart, we found an improvement of cardiac function under OGA inhibition. But none of the treatments either in in vitro or ex vivo cardiac models, induced a modulation of desmin, phosphorylated and O-GlcNAcylated desmin expression, despite the presence of O-GlcNAc moities in cardiac desmin. Our data suggests no interplay between phosphorylation and O-GlcNAcylation of desmin in HF post myocardial infarction. The future requires finding the targets in heart involved in cardiac improvement under thiamet G treatment.

1

INTRODUCTION

Heart failure (HF) following myocardial infarction (MI) is characterized by alterations of left ventricle (LV) structure and function, known as LV remodeling [START_REF] Pfeffer | Ventricular remodeling after myocardial infarction. Experimental observations and clinical implications[END_REF]; this pathophysiological process is a strong predictor of both HF and death as we recently showed in two cohorts of patients REVE and REVE2 dedicated to the analysis of LV remodeling [START_REF] Savoye | Left Ventricular Remodeling After Anterior Wall Acute Myocardial Infarction in Modern Clinical Practice (from the REmodelage VEntriculaire [REVE] Study Group)[END_REF][START_REF] Fertin | Usefulness of serial assessment of B-type natriuretic peptide, troponin I, and C-reactive protein to predict left ventricular remodeling after acute myocardial infarction (from the REVE-2 Study)[END_REF]. The long-term (> 10 years) clinical follow-up of patients included in these two cohorts has shown that LV remodeling remains independently associated with HF and cardiovascular death [START_REF] Bauters | Long-term prognostic impact of left ventricular remodeling after a first myocardial infarction in modern clinical practice[END_REF].

Evaluation of post-translational modifications (PTM) of sarcomeric cardiac proteins is a promising new approach to studying the mechanisms of HF. The phosphorylation status of sarcomeric proteins is altered in HF and may thus contribute to the decreased cardiac function [START_REF] Hamdani | Sarcomeric dysfunction in heart failure[END_REF]. Another rapid, dynamic, and reversible PTM is O-N-acetylglucosaminylation (O-GlcNAcylation) [START_REF] Zachara | The roles of O-linked -N-acetylglucosamine in cardiovascular physiology and disease[END_REF]. Both phosphorylation and O-GlcNAcylation regulate numerous cellular functions by reversibly adding either phosphate or O-N-acetylglucosamine (O-GlcNAc) to proteins. The crosstalk between these two PTMs may occur by steric competition for occupancy at the either the same or a proximal amino acid site [START_REF] Wang | Cross-talk between GlcNAcylation and phosphorylation: site-specific phosphorylation dynamics in response to globally elevated O-GlcNAc[END_REF]. Specific sites of O-GlcNAcylation described in some cardiac myofilament proteins have suggested that O-GlcNAc and O-phosphate modifications of these proteins may interact dynamically [START_REF] Ramirez-Correa | O-linked GlcNAc modification of cardiac myofilament proteins: A novel regulator of myocardial contractile function[END_REF][START_REF] Hu | Site-specific interplay between O-GlcNAcylation and phosphorylation in cellular regulation[END_REF].

Recently, we have demonstrated interplay between Ser 208 -phosphorylation and Ser 190 -O-GlcNAcylation of troponin T in ischaemic HF, linked to decreased activity of both PKCε and O-GlcNAcase (OGA) and increased O-GlcNAc transferase (OGT) activity [START_REF] Dubois-Deruy | Interplay between troponin T phosphorylation and O-N-acetylglucosaminylation in ischaemic heart failure[END_REF]. We also showed recently than another sarcomeric protein, desmin has been identified by differential proteomic analysis to have increased levels of phosphorylation in LV of HF-rats compared to the sham-rats [START_REF] Bouvet | Increased level of phosphorylated desmin and its degradation products in heart failure[END_REF]. In addition, we found a 2-fold increased serine-desmin phosphorylation in the LV of HFrats, mainly in the insoluble fraction, suggesting the formation of desmin aggregates, toxic for the cardiomyocyte. Desmin is a 53 kDa protein, particularly localized to the Z-band and is considered as a major integrator of contractile apparatus and a critical factor for maintaining intermediate filament structure. Its increased phosphorylation might lead to the network destabilization and formation of aggregates toxic for the cardiomyocyte [START_REF] Lowery | Intermediate Filaments Play a Pivotal Role in Regulating Cell Architecture and Function[END_REF].

Our objectives were to characterize 1) whether the levels of O-GlcNAcylated desmin are regulated in cardiomyocyte by inhibition or activation of O-GlcNAcylation in in vitro and ex vivo cardiac models of HF and; 2) whether modulation of O-GlcNAcylation impacts the phosphorylation levels of desmin with the aim to decrease the phosphorylation levels of desmin and the formation of desmin aggregates following HF development.

MATERIALS AND METHODS

Experimental rat model of ischemic heart failure

All animal experiments were performed according to the Guide for the Care and Use of Laboratory Animals published by the US National Institutes of Health (NIH publication NO1-OD-4-2-139, revised in 2011). Animals were used and experimental protocols performed under the supervision of a person authorized to perform experiments on live animals (F. Pinet: 59-350126). Approval was granted by the institutional ethics review board (CEEA Nord Pas-de-Calais N°242011, January 2012). Before surgery, rats were anaesthetized (sodium methohexital, 50 mg/kg intraperitoneal (IP)), while analgesia was administered before (xylazine 5 mg/kg IP) and 1 hr after surgery (xylazine 50 mg/kg subcutaneously) as previously described [START_REF] Mulder | Early versus delayed angiotensin-converting enzyme inhibition in experimental chronic heart failure. Effects on survival, hemodynamics, and cardiovascular remodeling[END_REF]. MI was induced in 10-week-old male Wistar rats (Janvier, Le Genest St isle, France) by ligation of the left anterior descending coronary artery according to the method previously described [START_REF] Pfeffer | Ventricular remodeling after myocardial infarction. Experimental observations and clinical implications[END_REF][START_REF] Mulder | Early versus delayed angiotensin-converting enzyme inhibition in experimental chronic heart failure. Effects on survival, hemodynamics, and cardiovascular remodeling[END_REF]. Haemodynamic and echocardiographic measurements were taken at 2 months after surgery, followed by heart excision as previously described [START_REF] Cieniewski-Bernard | Proteomic analysis of left ventricular remodeling in an experimental model of heart failure[END_REF]. Tissues were kept at -80°C until analysis.

Isolated heart perfusion

In vitro LV function was determined in randomly selected control male Wistar rats either untreated (bodyweight: 453 ± 7 g) or treated (bodyweight: 444 ± 14 g) with 100 µL OGA inhibitor, Thiamet G (200 µM diluted in saline buffer, Sigma-Aldrich, Lyon, France) for 2 hours. After anaesthesia (50 mg/kg of sodium pentobarbital injected intraperitoneally), the heart was rapidly excised and plunged in ice-cold oxygenated KH buffer (5.5 mM glucose, 1.25 mMCaCl 2 , 120 mM NaCl, 31 mM NaHCO 3 , 4.7 mM KCl, 1.2 mM MgSO 4 , 1.2 mM KH2PO4, [pH 7.4]) as previously described [START_REF] Dubois-Deruy | Interplay between troponin T phosphorylation and O-N-acetylglucosaminylation in ischaemic heart failure[END_REF]. The heart was transferred within 30 seconds to a Langendorf heart perfusion apparatus and perfused at constant hydrostatic pressure (90 mm Hg). A balloon was inserted into the LV and connected to a pressure transducer to record LV (systolic and diastolic) pressure, LV developed pressure and heart rate for 25 minutes. The balloon was inflated with water, allowing a similar and constant LV distending pressure of 10 mm Hg. At the end of each experiment, the LV was snap-frozen in liquid nitrogen and stored at -80°C until analysis.

Cell Culture 2.3.1 Primary cultures of neonatal rat cardiomyocytes

Primary cultures of rat neonatal contractile cardiac myocytes (NCM) were prepared from heart ventricles of 1-or 2-day-old rats as previously described [START_REF] Dubois-Deruy | Interplay between troponin T phosphorylation and O-N-acetylglucosaminylation in ischaemic heart failure[END_REF]. Briefly, cardiac cells of newborn rats' ventricles were dissociated by enzymatic digestion with 0.04% collagenase II (Worthington, Lakewood, NJ, USA) and 0.05% pancreatin (Sigma-Aldrich). Non-NCM were removed by 30 min centrifugation at 1,600 g in a discontinuous Percoll gradient (bottom 58.5%, top 40.5% [v/v], Sigma-Aldrich). NCM were then seeded at a density of 4×10 5 cells per well in 6-well plates coated with 0.01% of collagen (Sigma-Aldrich) (8×10 5 cells per well when they are seeded on coverslip) and cultured in a medium containing DMEM/Medium199 (4:1), 10% horse serum (Life Technologies), 5% fetal bovine serum (FBS) (ATCC), 1% penicillin and streptomycin (10,000 U/mL, Life Technologies) for 7 days at 37°C under 5% CO 2 atmosphere.

Small interfering RNA transfection

The first two individual pre-designed specific siRNA specifically targeting rat OGT mRNA, rat OGT and non-targeting control were used (ON-TARGETplus siRNA, Dharmacon, GE Healthcare). NCMs were plated (100,000 cells/well) in 6-well plates and were allowed to grow for 24h without antibiotics. The first 2 individual OGT (OGT1 and OGT2) siRNAs (5 nmol/L) were transfected with the DharmaFECT® reagent (4 µL) according to the manufacturer's recommendations. Total cell extracts were collected 72 h after transfection.

Protein extraction

Proteins from human hearts and rat LVs were extracted from frozen tissues (after removing the infarcted area) with Dounce-Potter homogenization into ice-cold RIPA buffer (50 mM Tris [pH7.4], 150 mM NaCl, 1% Igepal CA-630, 50 mM deoxycholate, and 0.1% SDS) containing antiproteases (Complete TM EDTA-free, Roche Diagnostics), serine/threonine and tyrosine protein phosphatase inhibitors (Phosphatase inhibitor Cocktail 2 and 3, Sigma-Aldrich), 1 mM Na 3 VO 4 and PUGNAC (50 µM). After 1 hr incubation at 4°C, the homogenate was centrifuged at 15,300 g for 15 min at 4°C and the supernatant containing soluble proteins was collected. After treatments, cells were rinsed twice with PBS before being mechanically scraped from the plate in 50 µL of ice-cold RIPA buffer. Soluble and insoluble proteins were extracted as describe above. Protein concentrations for all samples were determined with a Bradford-based method protein assay (Biorad, Marnes-la-Coquette, France).

Immunoprecipitation, Western blot, Phos-Tag TM and WGA gels 2.5.1 Immunoprecipitation

Immunoprecipitation was performed with 50 µg of NCM proteins or 100 µg of LV proteins pre-cleared by incubation with protein A/G magnetic beads (88802, Pierce) for 1 hr at 4°C with gentle shaking. The pre-cleared proteins were then mixed with 2 µL of anti-desmin antibody (ab3236, Abcam) diluted in 100 µL of RIPA 1X buffer as previously described [START_REF] Bouvet | Increased level of phosphorylated desmin and its degradation products in heart failure[END_REF]. After overnight incubation at 4°C on a rotating device, immune complexes were precipitated at 4°C for only 2 hrs on a rotating device with 35 µL of protein A/G magnetic beads. Immunoprecipitated (IP) complexes were then washed four times with 750 µL of RIPA 1X buffer before denaturation in Laemmli buffer at RT for western blot analysis.

409496-R1

Western blot

Soluble proteins (10-50 μg) were analysed on 12% SDS-PAGE gels. Proteins were transferred to nitrocellulose membranes and blocked for 1 hr in Tris-buffered saline with 0.1% [v/v] Tween-20 (TBS-T) containing 5% [w/v] skim milk or BSA with constant shaking. Membranes were then incubated with primary antibodies diluted in TBS-T with 5% skim milk or BSA overnight at 4°C with constant shaking. The blots were then washed with TBS-T and incubated at RT with horseradish peroxidase labelled secondary antibodies diluted in 5% skim milk or BSA/TBS-T for 1 hr. Following the secondary incubation, the membranes were washed with TBS-T before blots imaging. Equal protein loading was confirmed using GAPDH and sarcomeric actin immunoblotting.

For western blots of desmin-IP proteins, membranes were first incubated with RL2 antibody diluted in 5% fresh BSA overnight at 4°C before incubation with other antibodies.

Phos-tag TM gels

Soluble proteins were analysed on 10% gels containing 40 µM of Phos-tag™ (Wako, Osaka, Japan) and 100 µM of Mn 2+ at 90V for 2.5 hrs. The excess of metal was removed by washing the gels three times for 10 min in transfer buffer (NuPAGE® Transfer Buffer, Invitrogen) containing 10% methanol and 10 mM EDTA and then three times for 10 min in transfer buffer containing 10% methanol before the proteins were transferred onto 0.2 µm PVDF membrane. To detect desmin and its phosphorylated forms, membranes were blocked 1 hr in 5% skim milk/TBS-T before overnight incubation at 4°C with desmin antibody diluted 1/1000 in blocking solution. The following steps were similar to those described above for western blot. Briefly, the Phos-tag™ molecules incorporated into the SDS-PAGE are able with the cooperation of two Mn 2+ metal cations to slow down the migration of phosphorylated proteins. Therefore, phosphorylated desmin migrates at higher apparent molecular weight than the non-phosphorylated form. However, at equivalent phosphorylation levels, the position of the phosphate group can also influence the apparent molecular weight of a protein in a Phos-tag™ gel.

WGA

Soluble proteins (50 µg) were analysed on 7% gels containing 3.75 mg/mL of Wheat Germ Agglutinin (WGA) (L9640, Sigma-Aldrich) at 4°C at 20 mA for 2 hrs as previously described [START_REF] Kubota | WGA-based lectin affinity gel electrophoresis: A novel method for the detection of O-GlcNAc-modified proteins[END_REF]. After migration, the same protocol for protein transfer and incubation with primary and secondary antibodies as for classical western blot was applied.

List of antibodies

Protein

Blots imaging

The Chemidoc® XRS+ camera (Biorad) and the Image Lab™ software were used for blots imaging and densitometry analysis. Membranes were incubated for 5 min with Clarity™ Western ECL Substrate (Biorad) before imaging. The signal was quantified from the image obtained just before saturation. The band corresponding to the protein of interest was framed within a defined area to express the signal intensity depending of the area. This value was related to the intensity value of the reference protein (GAPDH or sarcomeric actin). The values were expressed in arbitrary units (U.A.).

Statistical Analysis

Data expressed as means ± SEM were analyzed with GraphPad Prism version 6.01 (GraphPad Software, San Diego, CA) and comparisons were made by Student's t test, Mann-Whitney (twotailed), one-or two-way analysis of variance (ANOVA) with Tukey's post hoc test, as appropriate. Results were considered statistically significant if the p<0.05.

RESULTS

Post-translational modification of desmin heart failure rats

Cardiac remodeling and dysfunction in HF-rats was characterized at 2 months post-MI by significant increases in LV end-diastolic pressure, LV end-diastolic and end-systolic diameters, and LV weight as previously shown [START_REF] Dubois-Deruy | Interplay between troponin T phosphorylation and O-N-acetylglucosaminylation in ischaemic heart failure[END_REF]. We previously investigated cardiac phosphoproteomic changes associated with LV remodeling and dysfunction in this HF-rat model. At 2 months after surgery, proteomic analysis revealed different LV phosphoproteomic patterns between the shamand HF-rats [START_REF] Dubois-Deruy | Circulating plasma serine 208 -phosphorylated troponin T levels are indicator of cardiac dysfunction[END_REF]. We previously identified two spots as being desmin [START_REF] Bouvet | Increased level of phosphorylated desmin and its degradation products in heart failure[END_REF]. We highlighted a significant increase of desmin phospho-species in LV of rats 2 months after MI compared to controls (Fig. 1A) without any modulation of total desmin protein levels (Fig. 1B). Then, we looked for interplay between phosphorylation and O-GlcNAcylation of desmin in the same experimental model and found a significant decrease of O-GlcNAcylated desmin in LV of HF rats compared to controls (Fig. 1C). These data showed that desmin may bear O-GlcNAc residues and that the levels of O-GlcNAcylated desmin were inversely related to the levels of phosphorylated desmin in LV of HF rats. To confirm these data, we tested several modulators of O-GlcNAcylation in several in vitro and ex vivo models, such as primary culture of cardiomyocyte (NCM) and isolated perfused heart.

Modulation of O-GlcNAcylation in vitro in primary cultures of neonate cardiomycytes 3.2.1 Inhibition of OGA by thiamet G

First, we studied the impact of OGA inhibition by thiamet G in primary cultures of neonate rat cardiomyocytes (NCM) as designed (Fig. 2A) and observed a significant increase of O-GlcNAcylated proteins (1.16 ± 0.09 vs. 2.55 ± 0.27, P <0.01) (Fig. 2B). Thiamet G treatment has no effect on total desmin levels (Fig. 2C) nor on desmin phosphospecies (Fig. 2D). To quantify the levels of O-GlcNAcylated desmin, we used immunoprecipitation (IP) of desmin species followed by a western blot with RL2 antibody to detect O-GlcNAcylated proteins (Fig. 2E). We verified by desmin western blot the efficiency of IP with desmin detected in input, IP (IP Des) and supernatant of IgG IP (S igG) but not in supernatant of desmin IP (S Des). The specificity of this approach was also verified with no O-GlcNAcylated proteins detected in beads. As observed the levels of O-GlcNAcylated desmin was very low in NCM and we did not observe any modulation in NCM treated with thiamet G (Fig. 2E)

Silencing of OGT

Second, we studied the impact of OGT silencing in primary cultures of neonate rat cardiomyocytes (NCM) as designed (Fig. 3A) and observed a significant decrease of O-GlcNAcylated proteins (1.08 ± 0.03 vs. 0.54 ± 0.08 (OGT1) and 0.52 ± 0.07 (OGT2), P <0.05) that is due to the significant decrease of OGT validating the efficiency of OGT silencing (0.99 ± 0.13 vs. 0.23 ± 0.02 (OGT1) and 0.20 ± 0.05 (OGT2), P <0.01) (Fig. 3B). Both OGT siRNA1 and siRNA 2 did not show any effect on total desmin levels (Fig. 3C) nor in desmin phosphospecies (Fig. 3D). We then quantified the levels of O-GlcNAcylated desmin as described in Figure 2E. Conversely, we observed an unspecific low intensity band for O-GlcNAcylated desmin with beads but we did not observe any modulation of O-GlcNAcylated desmin after OGT silencing in NCM (Fig. 3E).

Modulation of O-GlcNAcylation ex vivo in isolated perfused heart

In isolated perfused hearts, we assessed the functional cardiac modification induced by thiamet G, a specific OGA inhibitor, injected 2 h before the perfusion (Fig. 4A). Thiamet G decreased significantly the coronary flow in only sham-rats. In HF-rats, thiamet G increased significantly LV developed pressure and cardiac output without any effect on heart rate (Table ).

We validated by western blot the inhibitory effect of OGA by thiamet G with significant increase of O-GlcNAcylated LV proteins in sham-and HF-rat heart perfused with the inhibitor (Fig. 4B). We found a significant decrease of total desmin expression in LV from HF-rat perfused compared to the sham-rats independently of thiamet G perfusion (Fig. 4C). We visualized desmin phosphospecies by desmin immunoblot of Phos-tag™ gels and we did not find any modulation with different molecular weight desmin species detected in thiamet G-perfused heart (Fig. 4D).

New technology to detect O-GlcNAcylated proteins

Due to the difficulties to detect specifically O-GlcNAcylated desmin in LV proteins either from cultures of cardiomyocytes or perfused heart, we compare the sensitivity of 2 methods, western blots of O-GlcNAcylated LV proteins separated with classical SDS-PAGE gel (Fig. 5A) and WGA-SDS-PAGE gel (Fig. 5B) as recently described [START_REF] Kubota | WGA-based lectin affinity gel electrophoresis: A novel method for the detection of O-GlcNAc-modified proteins[END_REF]. First, red ponceau staining of the transferred membranes showed a less sensitivity to detect protein profiles with WGA gel. But conversely, we observed for the detection of O-GlcNAcylated LV proteins with RL2 antibody a stronger signal of better quality with WGA gel (Fig. 5B) by comparison to SDS gel (Fig. 5A). We then tested the detection of desmin in WGA gel and only found one band with a stronger signal in thiamet G treated samples either in sham-or HF-rats (Fig. 5B) that we were unable to quantify due to the strong background.

4

DISCUSSION Post-translational modifications of sarcomeric proteins play an important role in HF-induced cardiac dysfunction [START_REF] Wu | Protein Kinase C: a novel regulator of both phosphorylation and dephosphorylation of cardiac sarcomeric proteins[END_REF][START_REF] Dubois | Decreased serine207 phosphorylation of troponin T as a biomarker for left ventricular remodelling after myocardial infarction[END_REF]. We previously found a 2-fold increased phosphorylation of desmin levels by phosphoproteomic analysis [START_REF] Bouvet | Increased level of phosphorylated desmin and its degradation products in heart failure[END_REF] in a well characterized rat ischemic HF experimental model in which the induction of anterior MI leads to LV remodeling and to HF [START_REF] Mulder | Long-term heart rate reduction induced by the selective I(f) current inhibitor ivabradine improves left ventricular function and intrinsic myocardial structure in congestive heart failure[END_REF]. Nine serine resides conserved between species were identified by mass spectrometry to be phosphorylated [START_REF] Bouvet | Increased level of phosphorylated desmin and its degradation products in heart failure[END_REF]. In the present study, we found a decrease of O-GlcNAcylation levels of desmin and we hypothesize that a cross-talk between phosphorylation and O-GlcNAcylation of desmin may occur during HF development as we have recently shown for troponin T in in the same rodent model of HF [START_REF] Dubois-Deruy | Interplay between troponin T phosphorylation and O-N-acetylglucosaminylation in ischaemic heart failure[END_REF]. Here, our aim was to determine whether the levels of O-GlcNAcylated desmin are regulated in cardiomyocytes by inhibition or activation of O-GlcNAcylation and whether modulation of O-GlcNAcylation impacts the phosphorylation levels of desmin, in order to identify the serine residues that could be involved in the interplay.

For that purpose, first, we have modulated the levels of O-GlcNAcylated proteins in primary culture of neonate rat cardiomyocytes by either acting on OGA inhibition by treatment of the cardiomyocytes with a specific OGA inhibitor, thiamet G [START_REF] Vocadlo | O-GlcNAc processing enzymes: Catalytic mechanisms, substrate specificity, and enzyme regulation[END_REF] or by OGT silencing. We verified that both treatments were efficient in primary cultures of cardiomyocytes with significant increase of O-GlcNAcylated proteins after OGA inhibition and significant decrease of O-GlcNAcylated proteins after OGT silencing. With both treatments, we did not observe any changes in total desmin protein levels and in the expression of desmin PTMs: phosphorylation and O-GlcNAcylation.

Second, we used an ex vivo model of perfused rat heart in order to test the efficiency of thiamet G on cardiac contractility. We found a positive impact of OGA inhibition on several cardiac parameters which were different depending on the sham-(coronary flow) or HF-(LV developed pressure and cardiac output) animals, except for the heart rate which was not modified in both groups of animals after a 2 hours perfusion of thiamet G. We verified the efficiency of OGA inhibition by the significant increase of O-GlcNAcylated proteins in perfused heart in both group of animals treated (1.4 fold, P<0.05). And as shown in in vitro experiments, we did not observe any modulation of total desmin and its phosphorylated forms, except a significant decrease of desmin expression between sham-and HF-rats after the 2 hours perfusion, that can be explained by a degradation during the perfusion, such as calpain [START_REF] Bouvet | Increased level of phosphorylated desmin and its degradation products in heart failure[END_REF]. We found these negative results by 2 techniques: IP desmin followed by phospho-serine WB (not shown) and Phos-tag™ gels.

Despite these negative results with in vitro and ex vivo cardiac models of O-GlcNAcylation modulation, we did not want to exclude that the technology used to quantify the levels of O-GlcNAcylated desmin was inappropriate or not enough sensitive. To clarify this hypothesis, we have tested a technique recently published [START_REF] Kubota | WGA-based lectin affinity gel electrophoresis: A novel method for the detection of O-GlcNAc-modified proteins[END_REF]. Despite the less sensitivity of the technique to detect the proteins transferred into the membrane (red ponceau staining), we improved the quality of the profile of O-GlcNAcylated proteins by comparison to a classical western blot with the use of the same antibody against the O-GlcNAc moieties. Unfortunately, the quality of the WGAdesmin blot did not allow quantifying the desmin band detected despite the highest intensity of desmin in thiamet G-treated LV samples. Further improvement will be required to use this promising technique to quantify the levels of O-GlcNAcylation of a specific protein.

Our data have shown that despite the presence of O-GlcNAc moieties on desmin from heart, there is no regulation of O-GlcNAcylated desmin by modulators of O-GlcNAcylation and subsequently there is no interplay between phosphorylation and O-GlcNAcylation of desmin.

FIGURES LEGENDS

Figure 1 :

 1 Figure 1: Post-translational modifications of desmin of LV from HF-rats. Quantification of Ser-phosphorylated desmin (A), total desmin levels (B), and O-GlcNAcylated desmin levels (C) in LV (50 µg) of sham-(n=11) (white box) and HF-(n=11) (black box) rats at 2 months after MI. The positions of molecular weight are indicated as kilodalton (kDa) on the left. An internal standard was loaded in each gel for the standardization and quantification. Graphs show mean ± SEM values expressed in arbitrary units (A.U.). * P <0.05; ** P <0.01.

Figure 2 :

 2 Figure 2: Impact of OGA inhibition by thiamet G in primary cultures of NCM. A: Description of the protocol designed for Thiamet G (TG) treatment of primary culture of neonate cardiomyocytes (NCM). B: Representative western blots (left panel) and quantification (right panel) of O-GlcNAcylated proteins levels in control (Ctrl) and NCM treated with 100 nM of thiamet G (TG) during 24h (n=12). C: Western blots (upper panel) and quantification (lower panel) of total desmin levels in the same samples. D: Phosphorylation profiles of desmin were analyzed in the same samples by Phos-tag™ gel. E: Representative immunoprecipation (IP) with desmin antibody before western blot with RL2 antibody (left panel) and quantification (right panel) of O-GlcNAcylated desmin levels in the same samples. Inp: input, IP: immunoprecipitated proteins, S: IP supernatant, Des: desmin, IgG: Immunoglobulin. The arrow indicates the O-GlcNAcylated desmin. Data are expressed as means of an arbitrary unit (A.U.) ± SEM. The positions of molecular weight are indicated as kilodalton (kDa) on the left. **P<0.01.

Figure 3 :

 3 Figure 3: Effect of OGT silencing in primary cultures of NCM. A: Description of the protocol designed for OGT silencing of primary cultures of NCM. B: Western blots (left panel) and quantification of O-GlcNAcylated proteins levels (upper and right panel) and OGT (lower and right panel) in primary cultures of NCM transfected with non-targeting (NT) siRNA control and OGT1 and OGT2 siRNA (n =12). C: Western blots (upper panel) and quantification (lower panel) of total desmin levels in the same samples. D: Phosphorylation profiles of desmin was analysed in the same samples by Phos-tag™ gel. E: Representative immunoprecipation (IP) with desmin antibody before western blot with RL2 antibody (left panel) and quantification (right panel) of O-GlcNAcylated desmin levels in the same samples. Inp: input, IP: immunoprecipitated proteins, S: IP supernatant, B: beads alone, S*: IP supernatant of beads. The arrow indicates the O-GlcNAcylated desmin. Graphs show mean ± SEM values expressed in arbitrary units (A.U.). The positions of molecular weight are indicated as kilodalton (kDa) on the left. *P<0.005; **P<0.01

Figure 4 :Figure 5 :

 45 Figure 4: Effect of OGA inhibition by thiamet G in isolated perfused heart. A: Description of the protocol designed for thiamet G (TG) perfusion in sham-(n=6) and HF-(n=7) rats 6 weeks post-MI. B: Western blot (left panel) and quantification (right panel) of O-GlcNAcylated proteins levels measured in proteins extracted from LVs of isolated perfused sham-and HF-rat hearts treated or not with 100 µM thiamet G for 2 hr (n =7 in each group). C: Western blots (upper panel) and quantification (lower panel) of total desmin levels in the same samples. D: Phosphorylation profiles of desmin were analyzed in the same samples by Phos-tag™ gel. Graphs show mean ± SEM values expressed in arbitrary units (A.U.). The positions of molecular weight are indicated as kilodalton (kDa) on the left. * P <0.05; ** <0.01 Figure 5: Analysis of O-GlcNAcylated LV proteins by Western blot and WGA-SDS-PAGE gel electrophoresis. A: Red ponceau staining (left panel) and western blot (right panel) of O-GlcNAcylated proteins (50 µg) extracted from sham-and HF-rats treated or not with thiamet G.

ACKNOWLEDGMENTS

We thank Jean-Paul Henry for the surgical induction of MI in rats.

FUNDING

This work was granted by the "Féderation Hospital-Universitaire" FHU REMOD-VHF.

The future is to find the targets of O-GlcNAcylation in heart that can be involved in cardiac improvement and to find the pharmacological agents able to decrease the levels of phosphorylated desmin which is toxic due to the aggregation of phosphorylated desmin in failing heart.

AUTHOR CONTRIBUTION

MB and EDD designed the study, make experiments and wrote the paper. TM, AD and OB. make experiments. VR, PM and FP designed the study and wrote the paper.

DATA AVAILIBILITY STATEMENT

The data, analytic methods, and study materials presented in this study will be been made available to other researchers for purposes of reproducing the results or replicating the procedure.