Frédéric Lebon
email: frederic.lebon@univ-amu.fr

F Fouchal

C Pelissou

A Rekik

R Rizzoni

Modeling the interfaces in masonry structures

This chapter deals with some models for interfaces in the case of masonry structures. Some experimental studies are recalled in the first part. In the second part, four interface models are presented.

Introduction

The aim of the chapter is to present some ideas for modeling interfaces in masonry structures. Devising means of modeling interfaces between solids in structural assemblies is obviously now of great importance in the fields of mechanical technology and civil engineering. These interfaces contribute crucially to the strength of many structures, such as optics lenses, airplanes, asphalt pavements and masonry, for example. It is therefore necessary to develop rather fine models. One of the main problems which often arises in this context is that of developing a unified theory: from the tribological point of view, the contact is often unique, in the sense that it depends on the materials, roughness, wear, etc., and especially on the mechanical system involved. The problem of cracking in pavements is obviously quite different from that of the cornering of an airplane tire. Another problem is due to the smallness of the interface in comparison with the size of the structure, as well as the possibly weak mechanical characteristics (in the case of old mortar, for example). A large number of studies have been devoted to the behavior of interfaces. Two main modeling approaches used for this purpose are phenomenological modeling and deductive modeling. In the first approach, the thickness of the interface is taken to be zero and the mechanical properties are obtained from physical considerations and experiments (see for example [START_REF] Frémond | Adhesion of solids[END_REF]; [START_REF] Point | A delamination model for laminated composite[END_REF]; [START_REF] Freddi | Damage in domains and interfaces: A coupled predictive theory[END_REF]; [START_REF] Raous | Interface models coupling adhesion and friction[END_REF]; [START_REF] Bonetti | Analytical results on a model for damaging in domains and interfaces[END_REF] and references therein). The second approach consists in focusing on the thin layers of material at the micro-mechanical level, which are usually called the interphase. The mechanical parameters of the interface model, characterized by zero thickness, are identified on the basis of the parameters of the material constituting this interphase. These boundary conditions have been extensively used to model imperfect interface properties. One of the method commonly used to model interface conditions is based on the use of asymptotic techniques (see for example [START_REF] Klarbring | Derivation of the adhesively bonded joints by the asymptotic expansion method[END_REF]; [START_REF] Licht | A modeling of elastic adhesive bonded joints[END_REF]; [START_REF] Lebon | Numerical study of soft adhesively bonded joints in finite elasticity[END_REF]Lebon et al. ( , 2004)); [START_REF] Lebon | First order numerical analysis of linear thin layers[END_REF]; [START_REF] Lebon | Asymptotic study of soft thin layer: the non convex case[END_REF] and references therein), in order to include microscopic considerations in the interface model. Obviously, one of the problem which arises when modeling interfaces in masonry is due to the the irregularity of the structures (see figure 1). In this chapter, we will deal only with fairly regular structures (see figure 2). Another problem arises with the constitutive equation for the blocks because stones are quite rigid blocks (see [START_REF] Lucchesi | Masonry constructions: mechanical models and numerical applications[END_REF]; [START_REF] Como | Statica delle costruzioni storiche in muratura[END_REF]). In what follows, we will deal only with deformable elastic bricks. The last main problem depends on whether or not mortar is present in the structure and if so, how to write the constitutive equation for the mortar. If there is no mortar, the contact between blocks can be assimilated to dry friction. In the presence of mortar, it is necessary to model this component. The thickness of the mortar is also an important modeling parameter. If the mortar is thin, an interface law can be used directly for this purpose (see figure 3). If the thickness of the mortar is not negligible, it is necessary to introduce some additional considerations in order to account for the interactions between the bricks and the mortar (see figure 4).

In this chapter, it is proposed to present some interface laws. In the first part, the classical phenomenological law of unilateral contact with dry friction is recalled. In the second part, a phenomenological law of adhesion is presented and modeled based on the adhesion variable introduced by Fremond [START_REF] Frémond | Adhesion of solids[END_REF]). Two deductive interface laws are presented: the first one is linear and the second one is non linear and takes the damage which occurs at interface of this kind into account. The last modeling approach presented here includes mechanical processes responsible for the failure, such as the unilateral contact, the friction at the microscopic scale and the normal and tangential damage. The four models discussed in this chapter were developed in the context of multibody mechanics. This paper is composed of three parts. In the first part, experimental results are recalled. The second part is devoted to the modeling of friction between blocks with no mortar. In the third part of the paper, the phenomenological model of adhesion is presented in the form of two imperfect interface models based on the use of asymptotic techniques. Some numerical examples are provided.

Some comments on experimental results

Brick-brick interactions: dry friction

Michel Jean [START_REF] Jean | Monuments under seismic action[END_REF]) conducted experiments on a small wall (figure 5) consisting of 105 rigid blocks. each of which was 49 mm high, and 124 mm wide (62 mm in the case of half bricks). The wall was set on Figure 5 shows the displacement of the blocks. Note that the interactions between the blocks are limited to contact (there is no penetration) and dry friction. The displacement of the blocks is strongly heterogeneous, especially along the diagonal.

Brick-mortar interactions

This section deals with the behavior of deformable blocks. The local behavior of the interfaces between full and hollow bricks and mortar joints, which are typical quasi-brittle interfaces (see [START_REF] Gabor | Contribution à la caractérisation et à la modélisation des maçonneries non renforcées et renforcées par materiaux composites[END_REF]; [START_REF] Gabor | Modelling approaches of the in-plane shear behaviour of unreinforced and frp strengthened masonry panels[END_REF]), has been studied by various authors. We will attempt here to summarize the results obtained. The experimental device developed by [START_REF]Contribution à la modélisation numérique des interfaces dans les structures maçonnées[END_REF]; [START_REF] Fouchal | Contribution to the modeling of interfaces in masonry construction[END_REF] (figures. 6 and 7) was designed to study on the local scale the shear behavior of a simple assembly consisting of two and three full or hollow bricks (210 × 50 mm) connected by a mortar joint 10 mm thick. The samples were subjected to a monotonous increasing load up to failure. The following findings were obtained (see also figure 8):

• rigid elastic behavior up to the failure, followed by friction sliding behavior; • the behavior of full bricks was fragile beyond the limit strength;

• the behavior of hollow bricks was quasi-fragile beyond the limit strength;

• hollow brick samples showed great dispersion, mainly due to the non uniform distribution of the mortar spikes and local defects in the com- • samples consisting of two and three bricks showed similar behavior.

The choice of basic cell therefore has no effect on the local scale.

Masonry structures without mortar: dry friction modeling (model 1)

This section focuses on structures devoid of mortar, or including mortar with a very low stiffness. In a first approach, the contact is assumed to involve dry friction with no penetration between the blocks. For the sake of simplicity, we have adopted the framework of contact between two deformable solids (see figure 9). The contact can be defined by a punctual correspondence between two surfaces in contact Γ 1 c and Γ 2 c belonging to the domains Ω 1 and Ω 2 of d (d = 2, 3), respectively. We assume that initially

Γ c = Γ 1 c = Γ 2 c .
The relative displacement between two points located on the two surfaces in contact is denoted by [u] with [u] = u 1u 2 . Let F be the density of the contact forces. We take n 1 and n 2 to denote the external unit normal vectors to the boundaries of the two domains. The decomposition into normal and tangential parts is written:

[u] = [u N ]n 1 + [u T ] with [u N ] = [u].n 1
(1)

F = F N n 1 + F T with F N = F.n 1 (2)

Unilateral contact

The contact conditions (Signorini conditions) are written:

F N ≥ 0; [u N ] ≥ 0; F N [u N ] = 0 (3)

Dry Friction

The friction conditions (Coulomb's law) are written:

||F T || ≤ μ|F N | ||F T || < μ|F N | ⇒ [ uT ] = 0 ||F T || = μ|F N | ⇒ ∃λ ≥ 0, [ uT ] = λF T ( 4
)
where μ is the friction coefficient.

Formulations

The bodies are assumed to be elastic and the deformations are assumed to be small. In this case, formulation of three kinds can be used: primal (the unknowns are the displacements), dual (the unknowns are the stresses) and mixed formulations (the unknowns are the displacements and the stresses).

Figure 9. Dry contact between two blocks

In this chapter, two kinds of formulations are presented. Note that there exists a large class of methods for solving problems of this kind (see [START_REF] Raous | Numerical methods for solving unilateral contact problem with friction[END_REF]; [START_REF] Lebon | Friction modelling of a bolted junction under internal pressure loading[END_REF]; [START_REF] Chabrand | Various numerical methods for solving unilateral contact problems with friction[END_REF]; [START_REF] Lebon | Contact problems with friction: Models and simulations[END_REF]; [START_REF] Fortin | An improved discrete element method based on a variational formulation of the frictional contact law[END_REF]; [START_REF] Alart | A mixed formulation for frictional contact problems prone to Newton like solution methods[END_REF]; [START_REF] Alart | Solution of frictional contact problems using ILU and coarse/fine preconditioners[END_REF]; [START_REF] Wriggers | Computational Contact Mechanics[END_REF] and references therein). Two corresponding algorithms are presented below.

Primal formulation (see [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF])

Let us take H 1 (Ω) to denote the order 1 Sobolev space, and H and H 0 to denote the subspaces of (H 1 (Ω)) 2 defined by

H = {v ∈ (H 1 (Ω)) 2 , v = U on Γ d }, H 0 = {v ∈ (H 1 (Ω)) 2 , v = 0 on Γ d }, ( 5
)
We define K as the convex of contact of kinematically admissible fields

K = {v ∈ H, [v N ] ≥ 0 on Γ c }. u 0 is taken to denote the initial displacements fields. ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ u 0 ∈ H be given, find u ∈ V = H 1 (0, T ; K) such that u(0) = u 0 in Ω a(u, v -u) + j(u, v) -j(u, u) ≤ l(v -u), ∀v ∈ H 0 Γc σ N (u)([z N ] -[u N ])dl ≤ 0, ∀z ∈ K (6) with j(u, v) = - Γc μσ N (u)||[ vT ]||dl, l(v) = Ω F vdx + Γ f f vdl a(u, v) = Ω e(u)Ae(v)dx. ( 7
)
Using a finite difference, u(t k+1 )

u k+1 -u k Δt = Δu k Δt , with u k = u(t k ), at time t k+1 , we have Find u k+1 = u k + Δu k ∈ K such that a(u k+1 , v -Δu k ) + j(u k+1 , v) -j(u k+1 , Δu k ) ≤ l(v -Δu k ), ∀v ∈ K.
(8) Note that the results obtained on the existence, unicity and regularity of the solution for this problem have been discussed in [START_REF] Cocu | Existence of solutions of Signorini problems with friction[END_REF]. An example of non uniqueness is given in [START_REF] Hild | An example of nonuniqueness for the continuous static unilateral contact model with coulomb friction[END_REF]. [START_REF] Raous | Numerical methods for solving unilateral contact problem with friction[END_REF]; [START_REF] Lebon | Friction modelling of a bolted junction under internal pressure loading[END_REF] In the following sections, the indices k (time) will be omitted (this is a static formulation). The above problem is expressed as a fixed point problem on the sliding limit [START_REF] Duvaut | Inequalities in mechanics and physics[END_REF]; [START_REF] Cocu | Existence of solutions of Signorini problems with friction[END_REF])

Fixed point Method FPM (see

λ -→ -μσ N (u(λ)), (9)
u is the solution of a variational inequation similar to the above one, where j(., .) is replaced by j(.) defined by

j(v) = Γc λ|v T |dl. ( 10
)
The latter problem is equivalent to the minimization of

L(v) = 1 2 a(v, v) - l(v) + j(v) in K.
It is possible to regularize the non-differentiable term in the minimization problem, by replacing the absolute value by a smooth function. For example, a suitable value of √ x 2 + 2 or Ln(ch(x/ )) can be chosen [START_REF] Lebon | Two-grid method for regularized frictional elastostatics problems[END_REF]). This regularization procedure does not affect numerical structures. The problem is usually discretized by a linear interpolation (Finite Element Method). Let h be the discretization step. The displacement gives to a vector of dimension N h . If I h is the set of degrees of freedom involved in the friction, the fixed point iteration consists in finding the vector of dimension I h , λ h , which is the fixed point of the previously discretized problem. The algorithm, which is convergent, is written:

0 Initialization u 0 be given 1 Fixed Point (Iteration k) λ k -→ -μσ N (u(λ k-1 )) 2 Minimization (Relaxation) A Initialization u k,0 = u k B Resolution (iteration n, (a ij ) is the stiffness matrix) Case 1 With a normal component of a contact node u k,n+ 1 2 i = 1 aii (f i - j=i-1 j=1 a ij u k,n j - j=N h j=i+1 a ij u k,n+1 j ) If u k,n+ 1 2 i ≥ 0 then u k,n+1 i = 0. ( 11
)
Case 2 With a tangential component of a contact node

u k,n+1 i = 1 a ii (f i - j=i-1 j=1 a ij u k,n j - j=N h j=i+1 a ij u k,n+1 j +λ k i (u k,n+1 i )) If x ≥ 0, (x) = 1, if x ≤ 0, (x) = -1, if x = 0, (x) = 0.
(12) Case 3 With a component of a free node (not involved in the contact)

u k,n+1 i = 1 a ii (f i - j=i-1 j=1 a ij u k,n j - j=N h j=i+1 a ij u k,n+1 j
). ( 13 At each iteration k, the energy present in the discretized convex K h has to be minimized. This step is performed using the relaxation procedure presented above. Since the functional is strictly convex, the minimization procedure has a unique solution.

Mixed formulation (see [START_REF] Alart | A mixed formulation for frictional contact problems prone to Newton like solution methods[END_REF]; [START_REF] Alart | Solution of frictional contact problems using ILU and coarse/fine preconditioners[END_REF])

The idea underlying this method is to write the problem as an equilibrium equation, ⎧ ⎨ ⎩ F ind u and λ such that

F int (u) + F ext + F([u], F ) = 0, a(F -F([u], F )) = 0, ( 14
)
where a is a given coefficient, F ext are the given external forces, F int are internal forces which depend on the constitutive equation and the kinematics, and F([u], λ) is the friction map. In the case of a node in contact with a rigid obstacle (i.e.

[u] = u), in elasto-statics, we have (see [START_REF] Alart | A mixed formulation for frictional contact problems prone to Newton like solution methods[END_REF]; [START_REF] Alart | Solution of frictional contact problems using ILU and coarse/fine preconditioners[END_REF])

F(u, F ) = proj R-(τ N )n + proj C(proj R -(τ N )) (τ T ), ( 15
)
proj is the projection operator, C(F N ) is the Coulomb's cone and r is a parameter (usually equal to -r -1 ). τ N et τ T are given by

τ N = F N + ru N , τ T = F T + ru T . ( 16
)
The non-linear problem is then written

D(x) + U (x) = 0 ( 17
)
where

D(x) = Au + f ext aF and U(x) = F(u, F ) -aF(u, F ) . ( 18
)
This system is solved using a Generalized Newton method

x i+1 = x i -( D(x i ) + ∂U(x i )) -1 (D(x i ) + U (x i )) (19) ∂U (x i
) is a point in the Jacobian set of matrices. The linearized system ( 19) is solved by a solver dedicated to non symmetric systems. The algorithm can be summed up as follows:

0 Initialization x 0 = (u 0 , λ 0 ) be given 1 Resolution (Non-symmetric linear system)

y i+1 = ( D(x i ) + ∂U(x i )) -1 (D(x i ) + U(x i )) 2 Updating x i+1 = x i -y i+1 3 Convergence Test Yes = End; No = Go to Step 1

Concluding comment

The numerical problem can be solved using the open computer code LMGC90 (http://www.lmgc.univ-montp2.fr/˜dubois-/LMGC90/). This code is a numerical platform dedicated to the modeling and simulation of dynamic multibody problems. Problems are approached in the general framework of dynamics (see [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF]; [START_REF] Jean | The non-smooth contact dynamics method[END_REF]). The discretized equations involved in the problem are written: M q = F (q, q) + P (t) + r, + interface conditions + initial conditions and boundary conditions (20

)
where q is a parametrization of the system (degrees of freedom), M is the mass matrix, F (q, q) + P (t) are the internal and external loading vectors, and r is the vector of contact forces. The problem of the 105 blocks presented in the previous section can be solved using this technique. The results are presented in figure 10. These results were obtained by J. J. Moreau [START_REF] Jean | Monuments under seismic action[END_REF]). The correspondence observed seems to be perfect. In this section, the RCCM model is presented [START_REF] Raous | A consistent model coupling adhesion, friction and unilateral contact[END_REF]; [START_REF] Monerie | A model coupling adhesion to friction for the intercation between a crack and a fiber/matrix interface[END_REF]). This adhesion model was used to model mortar/brick interfaces in [START_REF] Fouchal | Contribution to the modeling of interfaces in masonry construction[END_REF]. In this model, the unilateral contact conditions (non penetration between the mortar and the brick, as in the previous section) are combined with dry friction and adhesion between bricks and mortar. The local constitutive equations required for this model are deduced from thermodynamic considerations and based on a material surface hypothesis on the contact zone. The basic idea underlying this model is to introduce a new state variable describing the contact state. This adhesion intensity variable, denoted b, was initially introduced by [START_REF] Frémond | Adhesion of solids[END_REF]. This variable gives the relative proportion of the active links between two bodies in contact. This variable is chosen so that:

• b = 1, total adhesion • 0 < b < 1, partial adhesion • b = 0, no adhesion
As in the previous section, we are working in the framework of the contact between two deformable solids, and the same notations are adopted here. w is taken to denote the Dupré energy, and C N (resp. C T ) to denote the initial normal (resp. tangential) stiffness of the interface. In what follows, p, q and r are three given numbers. The constitutive equations of the interface are given by the following equations, based on state laws and complementarity laws:

Unilateral contact with adhesion R N -C N [u N ]b p ≥ 0; [u N ] ≥ 0; (R N -C N [u N ]b p )[u N ] = 0 (21)
Friction with adhesion

||R T -C T [u T ]b q || ≤ μ|R N -C N [u N ]b p | ||R T -C T [u T ]b q || < μ|R N -C N [u N ]b p | ⇒ [ uT ] = 0 ||R T -C T [u T ]b q || = μ|R N -C N [u N ]b p | ⇒ ∃λ ≥ 0, [ uT ] = λ(R T -C T [u T ]b q ) (22)
Evolution of the intensity of adhesion

0 = -(w -( 1 2 C N [u T ] 2 + 1 2 C T |[u T ]| 2 )b r ) -if b ∈ [0, 1[ (23)
Note that if there is no adhesion (b = 0), this model involves the classical Signorini-Coulomb problem.

A graphic interpretation of the normal part of this RCCM model given in figure 11 (for (p, q, r) = (2, 2, 1)) shows the changes with time in the normal forces depending on the normal displacement jump. The changes in b lead to irreversible effects. If b decreases, the adhesive forces will decrease and eventually disappear. In the case of pure traction ([u N ] > 0), the adhesion resistance (R N = C N [u N ] p ) is activated (elasticity without damage). b decreases when the displacement becomes sufficiently large for the elastic energy to become larger than the adhesion limit w. Problems are usually approached using θ-methods and the Non-Smooth Contact Dynamics (NSCD) method [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF]; [START_REF] Jean | The non-smooth contact dynamics method[END_REF]). Due to the contact conditions, a fairly small time-step is chosen and the problem is condensed in the local frame associated with the contact nodes. The local problem is solved using a non-linear Gauss-Seidel method. The interface is governed by the RCCM law presented above. Contact between bodies is defined by contact nodes. Contact nodes are located between two nodes in the mesh of an element in contact at distances of 0.2 and 0.8, respectively, along each of the segments in contact. The bodies (bricks and mortar) are modeled using Q4 quadrangular finite elements. The numerical tests are performed with a constant time step equal to Δt = 0.5 * 10 -3 s. The computations require 2000 increments in order to reach values resembling the experimental data, and θ (in the time integration method) is fixed and taken to be equal to 0.55. The example presented shows the evolution of a triplet of full bricks and comparisons are made in figures 12 and 13 between two experiments and the numerical results.

Deductive models: linear (and non linear) multi-scale models

Generalities on asymptotic methods There exist a large class of asymptotic methods, such as matched asymptotic expansions [START_REF] Eckhaus | Asymptotic analysis of singular perturbations[END_REF]; Sanchez-Hubert and Sanchez-Palencia (1992)). The general idea is to dilate (blow up) the interphase from the thickness η to 1 (see Figure 14). We have two expansions of the displacement u η , the strain e(u η ) and the stress σ η in the powers of η, that is, an external one in the adherents and an internal in the joint. We have to connect these two expansions along the interface.

In what follows, we study a problem in 2 dimensions in order to simplify the computations. The relations obtained in the internal expansions will be expressed using values that intervene in the external expansions.

a) External expansions The external expansion is a classical expansion

η (x 1 , x 2 ) = u 0 (x 1 , x 2 ) + ηu 1 (x 1 , x 2 ) + ..., e ij (u η )(x 1 , x 2 ) = e 0 ij + ηe 1 ij + ..., e l ij = 1 2 ( ∂u l i ∂xj + ∂u l j ∂xi ), σ η ij (x 1 , x 2 ) = σ 0 ij (x 1 , x 2 ) + eσ 1 ij (x 1 , x 2 ) + ... (24)
b) Internal expansions In the internal expansion, we perform a blow-up of the second variable. Let y 2 = x2 e . The internal expansion gives

u η (x 1 , x 2 ) = v 0 (x 1 , y 2 ) + ηv 1 (x 1 , y 2 ) + ..., ε ij (u η )(x 1 , y 2 ) = η -1 e -1 ij + e 0 ij + eε 1 ij + ..., e l 11 = ∂v l 1 ∂x1 , e l 22 = ∂v l+1 2 ∂y2 , e l 12 = 1 2 ( ∂v l 2 ∂x1 + ∂v l+1 1 ∂y2 ), σ η ij (x 1 , y 2 ) = η -1 τ -1 ij (x 1 , y 2 ) + τ 0 ij (x 1 , y 2 ) + ητ 1 ij (x 1 , y 2 ) + ..., σ η ij,j = ∞ l=-2 η l ( ∂τ l i1 ∂x1 + ∂τ l+1 i2 ∂y2 ).
(25) We use the convention v l = 0, l < 0, τ l = 0, l < -1.

(26) c) Continuity conditions The third step in the method consists in the connecting of the two expansions. If the interface between the mortar and the brick is perfect, we have continuity of the displacement and of the stress tensor along this interface. This gives:

(i) v 0 (x 1 , ±1/2) = u 0 (x 1 , ±1/2), (ii) τ -1 (x 1 , ± ± 1/2) = 0, (ii) τ 0 (x 1 , ±1/2).e 2 = σ 0 (x 1 , ±1/2).e 2 . ( 27
)
A linear multi-scale model (model 3) The equilibrium equation becomes:

(η -1 τ -1 ij + τ 0 ij + ητ 1 ij + ...) ,j = 0 (28)
We obtain

τ -1 ij = 0 τ 0 i2,2 = 0 (29) that is τ 0 i2 .e 2 = 0 (30)
The interphase is assumed to be linearly elastic, and the above theory is applied to this kind of material. We have

η -1 τ -1 ij + τ 0 ij + ητ 1 ij + ... = C ijkl η -1 e -1 kl + e 0 kl + ηe 1 kl + (31)
where C denotes the stiffness tensor. We obtain:

τ 0 12 = C 1212 v 0 1,2 τ 0 22 = C 2222 v 0 2,2 (32)
that is

τ 0 12 = C 1212 v 0 1 τ 0 22 = C 2222 v 0 2 (33)
Using the continuity conditions, we obtain σ 0 ij = 0 in the brick and the mortar σ 0 .e 2 = C u 0 along the interface (34

)
where C is a matrix consisting of C 1212 and C 2222 . Note that this technique was used in [START_REF] Rekik | Identification of the representative crack length evolution in a multi-level interface model for quasi-brittle masonry[END_REF] and [START_REF] Rekik | Homogenization methods for interface modeling in damaged masonry[END_REF].

An example of a non linear multi-scale model including microcracks (model 4) (see [START_REF] Pelissou | Asymptotic modeling of quasi-brittle interfaces[END_REF])

General considerations and notations

The model described in this section is an extension of the bulk model introduced in [START_REF] Gambarotta | Damage models for the seismic response of brick masonry walls. part i : The continuum model and its application[END_REF], which takes the damage to the mortar joint into account. The interface modeling procedure (figure 15) consists of three steps:

• Let us take a macroscopic bulk model for quasi-fragile materials;

• The structure is assumed to consist of three phases: material 1 (brick, for example), material 2 (mortar, for example) and a thin interphase between the two materials, consisting of the material described in the first step; • Since the interphase is thin, an interface model is developed by performing an asymptotic analysis, as described above [START_REF] Geymonat | Analyse asymptotique du comportement en flexion de deux plaques collées[END_REF]; [START_REF] Lebon | Asymptotical analysis of Mohr-Coulomb and Drucker-Prager soft thin layers[END_REF]) (the thickness of the interface tends to zero). The model obtained memorizes some of the geometrical and mechanical characteristics of the interphase, such as the thickness, elastic coefficients, normal and tangential stress, damage variable, etc.

For the sake of simplicity, the structure is taken to occupy an open bounded set Ω of 2 with a smooth boundary ∂Ω. The two dimensional space is referred to the orthonormal frame (O, x 1 , x 2 ).

First step: The bulk model In this section, the bulk model introduced by Gambarotta and Lagomarsino (1997) is briefly described. In this model dedicated to masonry structures, the masonry is regarded as a "material" showing nonlinear damage behavior. The macroscopic behavior is accounted for by applying averaging process based on microscopic considerations. In the line with the classical procedure, the strain tensor is decomposed in its linear and nonlinear (anelastic) parts:

e = e el + e an (35)

where σ = Ce el and e an = S σ σ (36)

At the local level (in an elementary volume, see figure 16), the stress vector is decomposed into its normal σ N and tangential σ T parts, and the normal and tangential components of e an are expressed by ( 37) : 17), H is the Heaviside function of the unilateral response of the joint and α is the damage variable. h and k are positive coefficients standing for the opening and sliding compliances of the mortar joint and μ is the internal friction coefficient. where χ I is the characteristic function of the tangential damage undergone by the set I, such that:

e an N = hαH(σ N )σ N e an T = kα(σ T -f ) (37) where f = σ T if σ T ∈ I = ] -μσ N , +μσ N [and f = ±μσ N if σ T / ∈ I (figure
-If σ T ∈ I, χ I (τ ) = 1, there is tangential damage.

-If σ T / ∈ I, χ I (σ T ) = 0, there is no tangential damage. Therefore S σ is given locally in matrix form by:

S σ = hα H(σ N ) 0 ± kα χ I (σ T ) μ kα χ I (σ T ) ( 39
)
The damage is governed by a yield condition Φ(α) ≤ 0, where

Φ(α) = Y -R(α) Y = 1 2 hH(σ N )σ 2 N + 1 2 kχ I (σ T )σ 2 T ( 40
)
where R is the toughness of the material defined by (see figure 18)

R(α) = R 0 α, if 0 ≤ α ≤ 1 R(α) = R 0 /α, if α ≥ 1 (41)
Second step: The interphase In this section, the structure is taken to consist of two materials separated by a thin interphase which is a "mixture" of the other two materials. The interphase is assumed to be parallel to the x 1 -axis and the thickness, which is constant, is denoted by η. In what follows, the constitutive equations are those given in the previous section, i.e., the interface consists of a quasi-brittle material. The elastic part is assumed to be isotropic, and in view of the external normal vector x 2 , the constitutive equation is

σ = Id + CS σ -1 Ce = C σ ε (42)
Upon introducing the Lamé's coefficients, λ and G, we can write:

C σ = ⎛ ⎜ ⎝ λ + 2G - λ 2 hαH 1+(λ+2G)hαH λ -λ(λ+2G)hαH 1+(λ+2G)hαH 0 λ 1+(λ+2G)hαH λ+2G 1+(λ+2G)hαH 0 - ±μλGkαχ I (1+(λ+2G)hαH)(1+Gkαχ I ) - ±μ(λ+2G)Gkαχ I (1+(λ+2G)hαH)(1+Gkαχ I ) G 1+Gkαχ I ⎞ ⎟ ⎠ (43)
Interface behavior Using this asymptotical approach (see the previous section or [START_REF] Lebon | Asymptotical analysis of Mohr-Coulomb and Drucker-Prager soft thin layers[END_REF] in another context), and substituting the asymptotic expansions into the constitutive equations and the equilibrium equations, we obtain expressions linking the stress vector to the jump in the displacement denoted [u]. This gives:

τ 0 22 = lim e→0 λ+2G 1+(λ+2G)hαH ∂v 0 2 ∂y2 τ 0 12 = lim e→0 ±μ(λ+2G)GkαχI (1+(λ+2G)hαH)(1+Gkαχ I ) ∂v 0 2 ∂y2 + lim e→0 G 1+Gkαχ I ∂v 0 1 ∂y2 (44)
By integration, the A-GL (Asymptotic Gambarotta-Lagomarsino) model is therefore given by the following system (in the terms of the normal and tangential components): Due to the geometry and the loading conditions, the behavior of the contact zone is very complex. Figure 20 shows the evolution of the damage along the contact zone, which is strongly non linear. In particular, the damage increases strongly in an intermediate zone, which is subjected to both shear and traction forces. Note that the tangential displacement shows a linear pattern of evolution in the first phase and a non linear pattern corresponding to the failure, in the second phase, as was to be expected.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ σ N = C N 1 + C N hα H(σ N ) [u N ] σ T = ± C N C T μ kα χ I ( 1 + C N hα H(σ N ) ) ( 1 + C T kαχ I ) [u N ] + C T 1 + C T kα χ I [u T ] (

Conclusion

In this chapter, four of the many existing interface models are presented. It is quite difficult to solve large problems using this approach. In Rekik andLebon (2010, 2012), we established that it can be possible to solve larger problems (small walls), however. It will obviously be necessary in future studies to review all the existing models (a simple look at Science Direct T M with the keywords "masonry" and "interface" gives more than 2500 papers), in order to choose the most suitable one and use efficient solvers (DDM, MG, etc.) before it will be possible to solve real physical problems.

Figure 1 .

 1 Figure 1. Irregular type of masonry a) b) d) Typical Lozerian structures, France c) Miramas-Le-Vieux, France.

Figure 2 .

 2 Figure 2. Examples of regular masonry a)Lozere, France b) Bologne, Italy c) Albi, France d) Ferrara, Italy.

Figure 3 .

 3 Figure 3. A wall without mortar: a study at local level.

Figure 4 .

 4 Figure 4. A wall with mortar: a study at local level.

Figure 5 .Figure 6 .

 56 Figure 5. Experimental device involving 105 blocks (Jean and Moreau (1994))

Figure 7 .

 7 Figure 7. Experimental device involving three bricks

Figure 8 .

 8 Figure 8. Typical behavior of the mortar/brick interface (with full and hollow bricks)

)C

  Convergence test : Yes : u k+1 = u k,n+1 , Go to 3; No : Go to B 3 Convergence Test Yes = End; No = Go to 1

Figure 10 .

 10 Figure10. Comparison between experimental and numerical findings on the example of 105 blocks[START_REF] Jean | Monuments under seismic action[END_REF]

Figure 11 .

 11 Figure 11. Normalized normal forces versus the normalized normal jump in the displacement

Figure 12 .

 12 Figure 12. Damage evolution in a triplet of full-bricks

Figure 13 .Figure 14 .

 1314 Figure 13. Comparison between experimental and numerical results obtained on full brick triplets

Figure 15 .

 15 Figure 15. Interface modeling in three steps

Figure 16 .

 16 Figure 16. Normal and tangential stress vector components

Figure 17 .Figure 18 .

 1718 Figure 17. Relation between f and σ T (friction threshold)

R

  is not diagonal, contrary to the classical case: a non symmetric coupling term occurs between the normal compliance term C N and the tangential compliance term C T . It can be noted that the configuration of traction when f = 0 is combined in this general expression with that of the compression when f = σ T and when the friction threshold is reached at f = ± μ σ N .The damage is governed by a yield condition Φ(α) = Ȳ -R(α) (σ T ) (σ T -±μσ N ) 2 : shear test on a simple brick Let us perform an academic shear test on a single rectangular piece. The piece is bonded at the bottom and subjected to horizontal forces on the left. Details of this test are presented in figure19. This horizontal force which is equal to 20 kN , is applied progressively. The load is applied on the right and left sides. The (academic) coefficients are C N = C T = 500 kN cm -3 , μ = 0.3, h = k = 0.04kN -1 cm 3 , R0 = 10kN cm -1 .

Figure 19 .

 19 Figure 19. Shear test

Figure 20 .

 20 Figure 20. Damage evolution for four typical elements (shear test).


A method is presented here for modeling and predicting the rolling and yaw behavior of an aircraft tire which is subjected to a strong inflation pressure and a concentrated load on the axle, in contact with a flat, rigid surface. Finite element methods were used to model and simulate the aircraft tire/ground interactions. The incompressibility of the material, the large transformations and the unilateral contact with Coulomb friction law were all taken into account. Imaging methods were used to examine the complex structure of the tire cross-section. Comparisons are made between the data obtained with the model, the experimental data and those provided by the manufacturer. The tire response predictions were found to depend considerably on the material and the geometrical characteristics of the tire.

Introduction

Modeling the geometry of tires and predicting their behavior are complex tasks. The loading, which can be either quasi-static or dynamic, also involves severe contact conditions.

Review of the literature

Most studies on tires have been performed on automobile tires [START_REF] Ghoreish | Finite element analysis of steel-belted radial tyre with tread pattern under contact load[END_REF]. Several authors [START_REF] Lacombe | Tire model for simulations of vehicle motion on high and low friction road surfaces[END_REF][START_REF] Ghoreish | Finite element analysis of steel-belted radial tyre with tread pattern under contact load[END_REF][START_REF] Narasimha Rao | A finite element algorithm for the prediction of steady-state temperatures of rolling tires[END_REF][START_REF] Olatunbosum | Finite element simulation of the tyre burst test[END_REF] have focused on truck tires, especially those used on military trucks, and the motion of vehicles has been simulated on various types of soil (dry, wet, muddy), whereas some authors have dealt with bike tires [START_REF] Versteden | Improving a tire for motorcycle simulation[END_REF].

In the aerospatial and aeronautical field, the first studies on these lines were conducted by NASA [START_REF] Tanner | Tire tread temperatures during antiskid braking and "cornering" on a dry runway[END_REF][START_REF] Mccarthy | Temperature distribution in a aircraft tire at low ground speed[END_REF][START_REF] Clark | Heat generation in aircraft tires under free rolling conditions[END_REF]. These studies were mostly experimental and dealt with the thermal aspects (the influence of the temperature on tires dynamic responses, its distribution within the tire thickness, and the evolution of the friction coefficient in tire/ground contact).

In recent studies [START_REF] Navarro | Contribution la modélisation du pneumatique de l'Avion[END_REF], numerical tools have been used to simulate the behavior of aircraft tires, and some modeling aspects still cause engineers and researchers major problems because there are too many non-linear phenomena involved.

Under operating conditions, tire's behavior is highly non-linear due to their constituents, their geometry and shape, the loading conditions, contact with friction and many other interconnected parameters.

The operating conditions and the structure of aircraft tires are very different from those of automobiles. For example, automobile tires are inflated at a pressure of 2 bars, whereas the nominal inflation pressure of aircraft tires is around 15 bars. The vertical loads are also extremely different: 1 to 6 tonnes in the case of automobiles and trucks, as compared with about 20 tonnes in that of aircraft [START_REF] Navarro | Contribution la modélisation du pneumatique de l'Avion[END_REF].

Developing a fine, efficient mesh is difficult and requires lots of resources and it is necessary to simplify the geometry of numerical models (the grooves, asperities, etc.). The challenge is how to find a compromise between material models, geometrical models and the quality of the expected results. Finite Element Analysis (FEA) is now being used routinely to analyze and predict the various aspects of tire behavior. For these predictions to be successful, accurate 3-D tire models are essential. Another fundamental requirement for tire-modeling is the need for accurate, relevant information about the tire cross-section, especially as regards the layout of the composite structure.

Aims of the study

In this study, a numerical model for a smooth tire (without any grooves) is developed for static and dynamic aircraft tire simulations. A method of accurately modeling aircraft tires is presented. Based on experiments performed on samples taken from each of the layers in the tire, three models are developed: an orthotropic elastic model, an isotropic hyper elastic model and a composite model involving embedded elements called the "Rebar" model [START_REF]ABAQUS/Standard, Theory Manual and Example Problems Manual[END_REF]. The mesh is obtained by rotating the axisymmetric 2-D section about the axle of the rim. The grooves are not included in this geometric model, for reasons shown previously in [START_REF] Kongo Kondé | Etude du comportement en roulement d'un pneu d'avion[END_REF]. The grooves

On the modeling of aircraft tire

A. Kongo Kondé a,b , I. Rosu b , F.Lebon b , O.Brardo a , B.Devésa a are not essential because this study focused on the overall behavior of the tire.

Sensitivity studies will be performed in order to determine the influence of geometrical and material parameters on the response of the tire. For reasons of confidentiality, the figures presented here will show only general numerical results.

Tire simulation using finite element methods

In this study, finite element analyses were performed in several stages. In the first stage, the tire was modeled under inflation pressure using an axisymmetric model 2-D. A 3-D model was then developed for performing static vertical loading simulations (i.e. in the footprint stage). To study the lateral, torsional and longitudinal stiffness of the tire, a quasi-static analysis was performed after the static analysis. The thermal effects due to contact with friction were not taken into account in this study.

Tire structure

Modern pneumatic tires consist of a specific combination of rubber compounds, cord and steel belts (see Fig. 1). The main parts of a modern pneumatic tire are its body, sidewalls, beads, and tread. The body is made of rubberized fabric layers called plies, that give the tire its strength and flexibility. The fabric used is rayon, nylon, or polyester cord. The sidewalls and tread are made of chemically treated rubber. Embedded in the two inner edges of the tire are steel loops called beads, supporting the rim of the tire. The rubber components have different characteristics depending on their functional role. The tread, for example, comes into direct contact with the ground and has to be much harder than the sidewalls.

Tire geometry

Some imaging methods developed for inspection purposes are now being used by researchers to acquire the geometric data when no CAD tire geometry data are available. These methods include those based on charge-coupled devices (CDD), cameras, X-ray tomography devices and laser holography equipment, on both the micro and macro scales.

Scanned images give an accurate description of the perimeter of tire cross-section and the locations of the ply lines and cord ends.

The tire cross-section was cut in this study using a water-jet, as shown in Fig. 2. This method gives a highly accurate cutting plane and a very detailed image of all inner layers of the tire, as shown in Fig. 3. Image processing methods, were used to discretize the crosssectional image is order to obtain the real 2-D tire structure, as shown in Fig. 4.

Constitutive models for tire materials

Since rubber is a highly extensible material, small-strain elasticity theory is not suitable for describing the responses of tires to large strains. A useful means of measuring the response consists in using the mechanical energy W stored in the unit volume by deformation.

In tire applications, rubber compounds are often assumed to be isotropic, and cord/rubber composites are assumed to be orthotropic. These components combined can withstand the structural and thermal working conditions to which tires are exposed. In this section we will briefly describe the hyperelastic models and the linear orthotropic properties used so far to define the mechanical characteristics of the rubber, the belt and the ply layers. In this study, the material behavior will not be taken to depend on the temperature, as established experimentally in [START_REF] Kondé | Modélisation du roulement d'un pneumatique d'avion[END_REF]. 1Orthotropic elastic properties have often been used in numerical modeling studies based on methods such as FEA and closedform methods. Since a tire undergoes large deformations, the overall structural problem is non-linear. However, the stiffness predictions obtained in the framework of linear orthotropic elasticity are useful for design purposes. In addition, when these predictions are combined with non-linear FEA methods, a good approximation of the tire's structural response is obtained. The model developed here for composite lamina and laminates was based on those presented in [START_REF] Jones | Mechanics of Composite Materials[END_REF] and [START_REF] Walter | Approximate expressions for the elastic constants of cord-rubber laminates[END_REF].

Hyperelastic constitutive models for rubbers

In tire modeling studies, the incompressible Mooney-Rivlin model is commonly and widely used [START_REF] Ghoreish | Finite element analysis of steel-belted radial tyre with tread pattern under contact load[END_REF][START_REF] Korunovic | Finite element model for steady state rolling tire analysis[END_REF][START_REF] Navarro | Contribution la modélisation du pneumatique de l'Avion[END_REF][START_REF] Olatunbosum | Finite element simulation of the tyre burst test[END_REF][START_REF] Yanjin | Influence of belt cord angle on radial tire under different rolling states[END_REF].

The stresses for hyperelastic materials can be obtained from the partial derivatives of the strain energy functions [START_REF] Lahellec | Second-order estimate of the macroscopic behaviour of periodic hyperelastic composites: theory and experimental validation[END_REF]. It is well known that in the case of pure stress state like uniaxial tension, biaxial tension and planar tension, the stress can simply be described in terms of stretch ratios.

Under uniaxial tensile stress conditions, we have:

λ 1 = λ, λ 2 = λ 3 = λ -1 2 , λ= 1 + ε ( 1
)
where λ i are the principal stretch ratios, λ is the stretch ratio and ε is the strain.

The two first deviatoric strain invariants are

Ī1 = λ 2 + 2λ -1 , Ī2 = λ -2 + 2λ (2)
The principle of virtual work and the material incompressibility (D = ∞) are used to obtain the nominal stress-strain relationship, δU = T δλ = ∂U ∂λ δλ [START_REF] Clark | Heat generation in aircraft tires under free rolling conditions[END_REF] and it follows that

T = ∂U ∂λ = ∂U ∂ Ī1 • ∂ Ī1 ∂λ + ∂U ∂ Ī2 • ∂ Ī2 ∂λ = 2 1 -λ -3 λ ∂U ∂ Ī1 + ∂U ∂ Ī2 ( 4
)
where U is the strain energy function of incompressible hyperelastic materials and T is the stress. Based on experimental data, the coefficients of the constitutive function can be determined for each rubber by performing curve fitting.

Since the Mooney-Rivlin model is widely used in tire modeling studies, this model will be compared with Neo-Hookean and Yeoh models in terms of their predictive power and their accuracy. The criterion on which the choice of models was based was the deformation rate reached (35 to 50%) in aircraft applications [START_REF]Michelin dans le Benelux[END_REF].

The strain energy used in the three models is:

Neo-Hookean:

U = C 10 ( Ī1 -3) (5)
Mooney-Rivlin:

U = C 10 ( Ī1 -3) + C 01 ( Ī2 -3) (6)
Yeoh:

U = C 10 + C 20 ( Ī1 -3) 1 + C 30 ( Ī1 -3) 2 (7)
As regards the predictive validity of these models, we used only the uniaxial tension test data to identify all the coefficients by performing curve fitting before comparing the three models. The results are presented in the next section.

Mechanical behavior of composite cord/rubber materials

Composite materials consist of at least two different constituents or components that are bonded together, giving a structure that meets specific thermo-mechanical requirements. The composite materials used for structural applications often include either continuous or chopped fibers embedded in a softer matrix.

The geometric complexity of tires makes it difficult to cut samples for performing classical experimental tests. Samples were therefore cut here in the radial and circumferential directions, but not in that of the thickness.

The reinforcement of a tire can be modeled in three ways:

1. Orthotropic elastic properties (O): When reinforced rubber is modeled using the orthotropic elastic approach, the reinforcement (steel cords, nylon cords, etc.) is taken to be a homogeneous component. This means cord/rubber composites such as belts and carcass and bead reinforcements are modeled in the form of orthotropic elastic components for which all the elastic modules are determined by performing tension tests on various samples in the radial and/or circumferential directions.

The shear modulus and the elastic modulus in the thickness are computed using numerical samples, and the Poisson coefficients are determined by applying the following stability conditions (( 10) and ( 11)

) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 3 γ 1 γ 2 γ 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 E 1 -υ 21 E 2 -υ 31 E 3 0 0 0 -υ 12 E 1 1 E 2 -υ 32 E 3 0 0 0 -υ 13 E 1 -υ 23 E 2 1 E 3 0 0 0 0 0 0 1 G 12 0 0 0 0 0 0 1 G 23 0 0 0 0 0 0 1 G 31 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ σ 1 σ 2 σ 3 τ 1 τ 2 τ 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (8)
The stability of an orthotropic material also requires that [S] must be positive-definite, which results in the criteria:

E 1 , E 2 , E 3 , G 12 , G 23 , G 31 > 0 ( 9 ) |ν ij | < E i E j 1 2 i, j = 1, 2, 3 ( 1 0 ) 1 -ν 12 ν 13 -ν 13 ν 23 -ν 12 ν 23 -2ν 12 ν 13 ν 23 < 0 (11)
2. Isotropic Hyperelastic model (H): Reinforced rubber is modeled using the hyperelastic Yeoh model, and by performing simple tension tests on radially and circumferentially reinforced specimens.

Rebar approach (R):

With the Rebar approach, the reinforcement is treated like a one-dimensional element [START_REF]ABAQUS/Standard, Theory Manual and Example Problems Manual[END_REF] (see Fig. 5).

The Rebar properties depend on the elastic modulus, which is determined performing a simple tension test. The viscosity of the rubber is not taken into account. The rubber is assumed to be purely hyperelastic. The properties of Rebar components differ from those of the underlying component, and their orientation can be defined relative to the local coordinate system as shown in Fig. 5. To define the Rebar reinforcement, one must specify the crosssectional area A of each Rebar component, the spacing S between two consecutive cords, and the orientation angle θ of the Rebar cord in the local frame.

Experimental determination of the properties of rubber

To determine the properties of the tire material studied, uniaxial tension tests were performed on each rubber material, the steel beads and the reinforcement cords. Fig. 6 shows the specimens used in the tire rubber tests.

Experimental results on tire specimens

Uniaxial tension tests were carried out with an INSTRON 10KN

Testing Machine with a large deformation extensometer. Fig. 7 gives the stress-strain curve obtained on the tread rubber in several tests.

Comparisons between the various predictions obtained on tread rubber materials are made in Fig. 8. These comparisons show that:

• The Mooney-Rivlin and Neo-Hookean models were satisfactory up to 50% of the strain rate. Beyond this point, these models considerably underestimated the behavior of the material.

• The Yeoh model predictions matched the experimental test data exactly, although a small error occurred at small deformations.

The Yeoh model was therefore chosen to describe the hyperelastic behavior of tire rubber.

Interactions 2.5.1. Tire/rim contact

The contact of the tire with the rim was simplified by making the assumption that the tire sticks to the rim.

Coulomb friction at tire/ground contact

The tire/ground contact complicates the finite element model, since contact and friction problems are highly non-linear. However, these effects cannot be ignored. The contact problem was described in the FEM model using a "soft" contact approach with exponential regularization [START_REF]ABAQUS/Standard, Theory Manual and Example Problems Manual[END_REF]. The Coulomb friction was modeled using a stiffness method. The stiffness method used for friction in ABAQUS/Standard is a penalty method that permits some relative motion of the surfaces [START_REF]ABAQUS/Standard, Abaqus Analysis User'S Manual[END_REF].

Finite element analysis

All the finite element calculations were carried out using ABA-QUS/Standard Version 9-EF on an HP server with two Xeon quadcore processors cadenced at 3 GHz, using the parallel code version.

Static analysis

The first step in studying the tire's behavior consisted in performing a 2-D inflation analysis. The second step consisted in determining the footprint, which is the static deformed shape of the pressurized tire produced by a vertical dead load (corresponding to the weight of an airplane). A three-dimensional model is required for this analysis. The finite element mesh used with this model is obtained by revolving the axisymmetric cross-section about the axle of the rim.

Inflation analysis

Inflation analysis (see Fig. 9) is mainly performed on the axisymmetric model in order to check the integrity of the model, to determine the deformed shape of the inflated tire, to locate the stresses in the carcass, belts and rubber components and to set the basis for the 3-D FE model. The group element CGAX4(H) and CGAX3(H) from the ABAQUS element library was selected to represent all the rubber parts. These are four-and three-nodded elements with twist and constant pressure accounting for the incompressibility of the rubber material. The reinforcement materials were modeled in three ways:

• with CGAX4(H) and CGAX3(H) elements for orthotropic elastic model;

• with CGAX4(H) and CGAX3(H) elements for isotropic hyperelastic model;

• with SFMGAX1 for Rebar elements (two-nodded linear axisymmetric surface element with twist).

The three models inflated at the nominal pressure are presented in Fig. 9. The deformation of the belt zone given by the orthotropic and hyperelastic models does not correspond to the physical shape of the inflated tire as the Rebar model does.

Analysis of a vertically loaded tire

The inflation analysis of the axisymmetric model was followed by a static 3-D analysis of a vertically loaded tire. This analysis consisted of three steps: • In the second step, the 3-D tire was inflated and the ground surface was gradually moved towards the tire axis until a prescribed ground displacement value was reached.

• In the third step, the prescribed ground displacement value was replaced by a constant force acting as the airplane weight. This step served as the basis for the subsequent quasi-static and/or dynamic analyses. The results of the vertically loaded tire analysis are shown in Fig. 10.

Quasi-static analysis

In the first validation step, the numerical vertical and horizontal deflections and the footprint shape and area were compared with the manufacturer's data.

Next, three types of loading simulations were performed in order to compare the lateral, longitudinal and torsional stiffnesses obtained with the three models tested with the manufacturer's data (see Fig. 11).

Results

Simulations were performed under static vertical loads F z ranging from 0.125 to 0.75 of f 0 . The inflation pressure p 0 was 17.2 bars and the static friction coefficient μ s was 0.65. p 0 and f 0 stand for the references values of the pressure and the vertical load respectively. The orthotropic and hyperelastic models involve about 100 000 degrees of freedom (DOF), while the Rebar model has about 210 000 DOF. Total run time for a complete analysis, including inflation, footprint, lateral, longitudinal and torsional analyses was about 6000 seconds in the case of the most complex Rebar model.

Static results

Load-deflection curves

Fig. 12 shows the load-deflection curves obtained with the three material models, in comparison with the manufacturer's data and the experimental results.

It can be seen from these comparisons that there was a good agreement between the reference data (manufacturer's data and experimental results) and the values obtained with the (R) model. However, the results obtained with the (H) and (O) models differed considerably from the reference data. This means that the (H) and (O) models are much stiffer than the real tire stiffness measured in the vertical loading direction.

Contact area

Fig. 13 gives the contact area vs. loading curves obtained with the three tire models, in comparison with the manufacturer's data and the experimental results. These comparisons show that, with all 3 models (H, O, R), the contact areas predicted were larger than those corresponding to the experimental and manufacturer's data, mainly because we did not take the tire grooves into account. The contact area was about 35% larger without the grooves.

The next two figures (Figs. 14 and15) give the evolution of the contact area depending on the loading force and the shape of the contact area predicted using a grooved model.

As we can see here, the contact area could only be predicted by modeling a grooved tire.

Changes in the tire sidewall deflection during inflation and loading

These experimental tests were conducted with several inflation pressures and vertical loads.

In Table 1 comparisons are made between the experimental data and the three models tested, in terms of the changes in the sidewall geometry at a pressure p 0 and a vertical load F z = 0.585 f 0 (see Fig. 16).

Table 2 presents the lateral deflection of the tire measured at the extreme points of the sidewall (Fig. 17) during vertical loading with a force f 1 = 0.585 f 0 .

These comparisons show that the Rebar model gives the most accurate aircraft tire predictions.

In the next section, we will present the results obtained using the Rebar model alone. As explained above, the definition of Rebar elements in shells and surface elements is based on three geometric properties: the cross-sectional area ( A) of each individual Rebar element, the spacing (S) between the elements and their orientation (θ ) with respect to the local coordinate system. Since, the properties of Rebar elements depend on the elastic modulus (E), we focused here on the effects of these geometrical and material properties on the lateral, longitudinal and torsional responses of the tires simulated.

Geometrical effects 4.2.1. Carcass orientation effects

Four carcass angles θ (0 • , ±4 • , ±7 • and ±12 • ) were selected as design variables for investigating the effects of the carcass angle on the tire's lateral, longitudinal and torsional stiffness. This sensitivity study was performed to determine the exact orientation of the fibers in the tire because these orientations cannot be measured directly.

Belt orientation effects

Three belt angles, κ = ±70 • , ±80 • and 90 • , were selected. The carcass angle θ was kept equal to ±7 • . Fig. 20 shows k Y and k X at each angle κ. When κ differed from 90 • , k Y was constant and stiffer. k X was slightly less stiff, while k M Z seemed to be insensitive to the orientation of the belt.

This finding means that woven belt cords increase the lateral stiffness of tires, but decrease the longitudinal stiffness. In fact, at κ = 90 • , belt cords have no effect in the lateral direction as the fiber orientation is orthogonal to the loading direction. However, in the case of woven cords, the belt rigidity E belt also contributes to the stiffening in the lateral direction and to the softening in the longitudinal direction.

Rebar spacing effects

Four fiber spacing values (S 1 = S/2, S 2 = S, S 3 = 4S and S 4 = 10S) were selected for investigating the effects of S on the tire's lateral, longitudinal and torsional stiffness. S denotes the distance measured between two fibers in the reinforced layers. θ was assumed to be always equal to ±7 • . Figs. 21 and22 give the k Y , k X and k M Z stiffness depending on the fiber spacing. As was to be expected, increasing the fiber spacing S decreased the stiffness k Y and k X . However, k M Z remained constant at all the spacing values tested.

Effects of the material used

Five materials are mainly used in the field of tire manufacture these days: rayon, nylon, polyester, aramid, and steel [START_REF] Walter | The pneumatic tire[END_REF]. Because of their high modulus, steel and aramid are mostly used in the radial belts and single-ply carcasses of large radial tires. Rayon is used in both the carcasses and belts of passenger cars radial tires, but is less suitable for heavy-duty tires. Modern polyester cord is an excellent carcass textile, which is used along with steel belts in passenger cars and light trucks tires and is becoming the most widely used material for this purpose worldwide. However, it lacks the toughness and heat resistance required for large tires. Nylon is the most suitable textile for large bias truck, earthmover and aircraft tires, but nylon like polyester, is not stiff enough for radial belts.

Some sensitivity studies were performed on fiber moduli. Six belt fiber moduli E belt = (E steel or E rayon or E kvelar or E polyester or E nylon or E nylon /10) and two carcass fiber moduli E c = (E polyester or E nylon ) were selected as the design variables for investigating the effects of the material on the tire response. E belt and E c denote the reference elastic moduli in the belt and carcass layers, respectively.

θ is assumed to be equal to ±7 • . predicted by the Rebar model was therefore very similar to the value specified by the manufacturer. The steel-polyester material showed the greatest lateral stiffness (see Fig. 23).

The tire response seems to be more sensitive to the carcass moduli. Most carcass layers of aircraft tires are presumably made of polyester, as stated in [START_REF] Walter | The pneumatic tire[END_REF].

These sensitivity studies show that the k Y depends on the modulus of the carcass E carcass while k X depends on E belt . k Y and k X seem to be strongly coupled. Although many configurations and simulations were performed, it is difficult, however, to determine exactly how are correlated the lateral and longitudinal stiffness.

Conclusion

In this paper is presented a method for modeling an aircraft tire under severe operating conditions using finite element tools. The Rebar model proved to be the most accurate mean of modeling tire behavior under static and quasi-static loading conditions with a view to dynamic simulations. The Rebar modeling of the reinforced zones also gave the most realistic results. After the many sensitivity studies performed here, the next challenge is to elucidate the strong coupling which seems to exist between the lateral and longitudinal stiffness.

Fig. 1 .

 1 Fig. 1. Aircraft tires components.

Fig. 2 .

 2 Fig. 2. Water-cutting process.

Fig. 3 .

 3 Fig. 3. Layout of the composite structure.

Fig. 4 .

 4 Fig. 4. Discretized tire structure.

Fig. 5 .

 5 Fig. 5. Rebar components defined relative to the local coordinate system.

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig. 6. Samples: CE stands for the belt and FF for the carcass. C and R (circumferential or radial) denote the direction in which the sample was cut.

Fig. 9 .

 9 Fig. 9. 2-D axisymmetric finite element model.

Fig. 10 .

 10 Fig. 10. Loaded and unloaded half-tire geometry given by the Rebar model.

Fig. 11 .

 11 Fig. 11. Lateral, longitudinal and torsional loadings. • In the first step, the 3-D model was created by rotating the axisymmetric model around the tire axis.

Fig. 12 .

 12 Fig. 12. Load/deflection curves.

Fig. 13 .

 13 Fig. 13. Contact area curves.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Evolution of the contact area in the case of a grooved tire.

Fig. 16 .

 16 Fig. 16. Sidewall profiles for 0.15p 0 and p 0 .

Fig. 17 .

 17 Fig. 17. Lateral deflection at f 1 .

Fig. 18 (

 18 a) gives the lateral force as a function of the lateral displacement. In this figure, the slope gives the lateral stiffness k Y of the tire (the force is applied to the center of the rim). It can be seen from this figure that the stiffness increases with θ .

Fig. 18 (Fig. 18 .

 1818 Fig. 18(b) gives the longitudinal force as a function of the longitudinal displacement. In this figure, the slope gives the longitudinal stiffness k X of the tire. Regardless of the orientation, the stiffness predicted by the model was always greater than the manufacturer. Fig. 19 gives the torsional moment as a function of the rotational displacement. In this figure, the curves give the torsional stiffness k M z of the tire. No strong dependence on θ can be observed, and the values predicted were always greater than the manufacturer's. The values θ = ±7 • gave the most accurate lateral behavior k Y .

Fig. 19 .

 19 Fig. 19. θ effects on k Mz tire stiffness.

Figs. 23 Fig. 20 .Fig. 21 .

 232021 Figs.23 and 24show the predicted k Y , k X and k M Z stiffness corresponding to each modulus (E belt and E carcass ). Since E carcass was always greater than E nylon , no significant changes in k Y and k X were observed. The predicted values of k X were always greater than the manufacturer's values. When E carcass was beyond this range, k Y decreased slightly and k X decreased significantly. The k X

Fig. 22 .

 22 Fig. 22. S effects on k Mz tire stiffness.

Fig. 23 .

 23 Fig. 23. Effects of the fiber modulus on k Y and k X tire stiffness.

Fig. 24 .

 24 Fig. 24. Effects of the fiber on the tire stiffness k Mz .

Table 1

 1 Changes in sidewall geometry during inflation.

		Inflation	Left	Tire	Right
		pressure	sidewall	tread	sidewall
	Experimental	p 0	1	1	1
	Orthotropic model	p 0	0.98	9.48	0.98
	Hyperelastic model	p 0	0.97	9.59	0.97
	Rebar model	p 0	0.89	1.08	0.89

Table 2

 2 Lateral deflection.

		Vertical load	Lateral deflection
	Experimental	f 1	1
	Orthotropic model	f 1	0.87
	Hyperelastic model	f 1	0.89
	Rebar model	f 1	0.95

In this experiment, it was established that the temperature variations are restricted to the tread zone.