Amna Rekik
email: amna.rekik@univ-orleans.fr

Thi Thu

Nga Nguyen

Alain Gasser

Multi-level modeling of viscoelastic microcracked masonry

published or not. The documents may come

• The Modified Maxwell model is relevant for the mortar's creep at short and long terms

• Both short and long terms, constant and time-dependent crack density were investigated order to provide rigorous analytical global estimates, only the mortar is assumed to be viscoelastic and microcracked. Bricks are assumed to be undamaged, elastic or quasi-rigid. The distribution of microcracks is assumed to be isotropic. The effective behaviour of the viscoelastic microcracked masonry is provided by two steps. The first one relies on the coupling between the Griffith's brittle fracture theory and linear mean-field homogenization scheme in order to account for the effect of microcracks on the macroscopic deformation of the mortar and establishes a linear relation between apparent macroscopic stress and strain. This step allows to easily and fast determine the effective creep function of the microcracked mortar without recourse to 'complex' or heavy numerical inversion of the Laplace-Carson transform. The second step is based on the coupling between asymptotic analysis and homogenization theory applied for a periodic masonry. The proposed models provide analytical solutions -explicit functions of the crack density parameter -for the effective orthotropic behaviour of a microcracked viscoelastic periodic masonry cell. This study proves that the Cecchi & Tralli's and Cecchi & Taliercio's extension estimates are close and that the later are softer. Such overall properties are used to perform finite element computations on a compressed masonry panel as a first application. These models allow then the prediction of mostly stressed and deformed areas in microcracked masonry structures. This study demonstrates that modeling a mortar (at its undamaged or microcracked state) with this Burgers formulation is only suitable for a masonry with too high values of the Maxwell's relaxation time otherwise it yields to vanishing effective properties with the increase of time and crack density leading thus to a premature collapse of the masonry. On the other hand, the Modified Maxwell model permits the masonry to preserve a certain resistance for every range value of Maxwell's relaxation time. These conclusions are valid for masonry either Creep phenomenon and its effects are not only restricted to new structures but they involve also historical monuments as postulated in [START_REF] Shrive | Letter to the Editor[END_REF] and confirmed later in [START_REF] Binda | The collapse of civic tower of Pavia[END_REF][START_REF] Papa | Modeling of the long-term behaviour of historical masonry towers, Chapter 7 in Book: Learning from Failure: Long-term Behaviour of Heavy Masonry Structures[END_REF]. Indeed heavy persistent compressive stresses added to the presence of aggressive environment (temperature/humidity conditions) and the self-weight of historical monuments induce stress redistribution and creep strains which can lead to damage accumulation and partial or total collapse of these monuments as it is the case of several famous examples: the Civic Tower of Pavia (Italy), the St. Magdalena bell-tower in Goch (Germany), the Noto and Monza Cathedrals (Italy) [START_REF] Papa | Modeling of the long-term behaviour of historical masonry towers, Chapter 7 in Book: Learning from Failure: Long-term Behaviour of Heavy Masonry Structures[END_REF]. On the other hand, in the iron and steel industry, the refractory lining of furnaces made up of masonry with bricks and either mortared or dry (i.e. without mortar [START_REF] Andreev | Compressive behaviour of dry joints in refractory ceramic masonry[END_REF]) joints is often exposed to compressive (monotonic or cyclic) loads.

These compressive loads result from constrained thermal expansion since the temperature inside these structures can reach 1650 degrees. This could induce creep behaviour and diffused damage due to initiation and propagation of micro-cracks mainly in the joints [START_REF] Andreev | Compressive fatigue behaviour of refractories with carbonaceous binders[END_REF]. Generally, viscous behaviour of vitreous materials in gaz turbine engines has been modeled by temperature independently rheological models like Maxwell and the generalized Maxwell models [START_REF] Dufferene | Viscoelastic constants of a soda-lime silica glass[END_REF]. Models with a temperature parameter have also been defined since the steady-state creep deformation of ceramics under sustained loading conditions normally exhibits a power-law stress-dependent behaviour [START_REF] Harmuth | Compressive creep testing of refractories at elevated loads -Device, material law and evaluation techniques[END_REF][START_REF] Andreev | Compressive behaviour of ACS torpedo bricks, 11 th Biennial Worldwide Conference on Unified International Technical Conference Refractories[END_REF][START_REF] Ferber | Compressive creep and thermo-physical performance of refractory materials[END_REF][START_REF] Koushik | Creep and visco-elastic behaviour of LPS-SiC sintered with Lu 2 O 3 -AlN additive[END_REF]. Traditionally, both tensile and compressive creep in ceramic materials have been characterized by an empirical creep equation that takes on the form of the Norton-Bailey-Arrhenius equation. Note that a thick mortar tends to decrease the stiffness of structure and increases the likelihood of the possible penetration of process materials into the joints, resulting in the deterioration of the lining. So, the use of thin mortar joint is appropriate and necessary in designing the refractory brick lining system.

Concerning the creep behaviour of traditional mortar, various rheological models namely the USBR, Feng, Ross, typical and modified versions of the Burgers and Modified-Maxwell models may be investigated [START_REF] Choi | Rheological modelling of masonry creep[END_REF][START_REF] Ignoul | Creep behaviour of masonry structures -failure prediction based on a rheological model and laboratory tests[END_REF]. On the other hand, there exist several approaches accounting for damage in viscoelastic materials [START_REF] Ignoul | Creep behaviour of masonry structures -failure prediction based on a rheological model and laboratory tests[END_REF][START_REF] Nedjar | An approach to the modeling of viscoelastic damage: application to the long-term creep of gypsum rock materials[END_REF]. Indeed, the approach presented in [START_REF] Nedjar | An approach to the modeling of viscoelastic damage: application to the long-term creep of gypsum rock materials[END_REF] is based on a coupling between continuum damage mechanics and viscoelasticity through the generalized Kelvin-Voigt model.

Accordingly, a three-dimensional phenomenological model was developed to describe the long-term creep of gypsum rock. The main disadvantage of this model is that it requires experimental investigation [START_REF] Ignoul | Creep behaviour of masonry structures -failure prediction based on a rheological model and laboratory tests[END_REF] or computational efforts to resolve nonlinear equation [START_REF] Nedjar | An approach to the modeling of viscoelastic damage: application to the long-term creep of gypsum rock materials[END_REF] function of internal damage variables. In the works of Nguyen et al. [START_REF] Nguyen | Propagation de fissures et endommagement par microfissures dans un milieu viscoélastique linéaire non veillissant[END_REF][START_REF] Nguyen | Fracturing of viscoelastic geomaterials and application to sedimentary layered rocks[END_REF], the effective behaviour of microcracked linear viscoelastic concrete was derived from a combination of the Griffith's theory [START_REF] Huy Duong | Mécanique de la rupture fragile[END_REF] and the Eshelbybased homogenization scheme [START_REF] Bornert | Homogénéisation en mécanique des matériaux, Tome 1 : Matériaux aléatoires élastiques et milieux périodiques[END_REF]. The undamaged concrete was assumed to obey to the typical and Taliercio [START_REF] Brooks | Composite modeling of masonry deformation[END_REF][START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF][START_REF] Cecchi | A comparison between numerical and analytical homogenized models for visco-elastic brickwork[END_REF]45] proposed models to predict creep coefficients according to the properties of each masonry constituent. These models are based on analytical or numerical homogenization using the finite elements method (FEM) in order to deduce the macroscopic creep of undamaged (without cracks) masonry. In the present study, the coupling between the Griffith's theory and the dilute scheme [START_REF] Bornert | Homogénéisation en mécanique des matériaux, Tome 1 : Matériaux aléatoires élastiques et milieux périodiques[END_REF] will be applied to provide the effective behaviour of a micro-cracked mortar [START_REF] Nguyen | Propagation de fissures et endommagement par microfissures dans un milieu viscoélastique linéaire non veillissant[END_REF].

In a second step, the expressions proposed in [START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF][START_REF] Cecchi | A comparison between numerical and analytical homogenized models for visco-elastic brickwork[END_REF] are extended to determine the effective behaviour of a periodic microcracked viscoelastic masonry cell. It is worth noting that even the dilute scheme is useful for dilute concentrations of cracks, it has been demonstrated in [START_REF] Dormieux | MicroporoMechanics[END_REF][START_REF] Dormieux | Stress-based estimates and bounds of effective elastic properties: the case of cracked media with unilateral effects[END_REF] Taliercio extension models to a 2D masonry with microcracked viscoelastic hybrid mortar. This section studies the effects of crack density parameter, time and ratio between the bricks and mortar

Young's moduli on the evolutions of mortar's creep function and masonry periodic cell's stiffness.

At last, section 6 presents local and global mechanical fields predictions for a compressed masonry panel studied in [START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF]. In addition to the crack density parameter and time, this section studies the effects of the mortar's rheology model and panel's boundary conditions.

A C C E P T E D M

A N U S C R I P T

Main objective and hypothesis

The objective of this study is to evaluate the effective and local behaviour of masonries exhibiting nonlinear behaviours mainly viscoelastic at short and/or long times especially when they are sub- it is an extension of these models to microcracked viscoelastic masonries which considers apart the spherical and deviatoric parts of the creep behaviour unlike applications in [START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF]. Since this model provides stress distribution throughout studied masonry wall or structure with low computational efforts, it allows to predict cracked areas or failure zones mainly if compressive, tensile and shear masonry's strengths are beforehand known. For first applications and for the sake of simplicity, it can be assumed that only the mortar is a micro-cracked viscoelastic material [START_REF] Sacco | A nonlinear homogenization procedure for periodic masonry[END_REF][START_REF] Luciano | Homogenization technique and damage model for old masonry material[END_REF]. Its behaviour (at the undamaged state) obeys to the Modified Maxwell or typical Burgers models. The blocks or bricks are assumed to be undamaged and to have either a rigid or elastic isotropic behaviour. In the mortar, the cracks are assumed to be penny-shaped and to have an isotropic distribution. The proposed approach is based on two main steps. Firstly, the homogenization technique is applied in order to assess the effective behaviour of the micro-cracked mortar. The results of brittle fracture mechanics -the Griffith's theory -could be useful if we move from the real temporal space to the symbolic one due the Laplace-Carson (LC) transform. In the later space, the apparent behaviour of the mortar is linear elastic. This procedure allows the use of expressions available in the literature for the displacement's jump induced by the crack [START_REF] Nguyen | Propagation de fissures et endommagement par microfissures dans un milieu viscoélastique linéaire non veillissant[END_REF]. Assuming again that the displacement jump field depends linearly on the macroscopic stress, it is possible to define an effective linear behaviour for the micro-cracked mortar in the symbolic space. To determine the global behaviour in the real space time, it is possible to apply the inverse of the LC transform in some simple cases. It is then interesting to approach in the symbolic space, at least in short and long terms, the symbolic effective stiffness (or compliance) by an existing rheological model. For example, if the undamaged mortar behaves as the Modified Maxwell model, we try to approach the symbolic effective behaviour of the corresponding microcracked mortar by the same model. After validation of this approximation at short and long terms, the inversion of the apparent effective stiffness will be straightforward.

Therefore, the effective behaviour of the micro-cracked viscoelastic mortar could be expressed in the real space time. In a second step, the global behaviour proposed for undamaged linear viscoelastic periodic masonries [START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF][START_REF] Cecchi | Homogenisation procedure to evaluate the effectiveness of masonry strengthening by CFRP repointing technique[END_REF][START_REF] Cecchi | A comparison between numerical and analytical homogenized models for visco-elastic brickwork[END_REF] is extended in this study for similar masonry cell with microcracks.

A C C E P T E D M

A N U S C R I P T Basic steps followed by the proposed models -extension of the Cecchi & Tralli [START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF] and Cecchi &

Taliercio [START_REF] Cecchi | A comparison between numerical and analytical homogenized models for visco-elastic brickwork[END_REF] models to microcracked viscoelastic masonries -are summarized in Figure [START_REF] Andreev | Compressive behaviour of dry joints in refractory ceramic masonry[END_REF]. Hereafter, it is proposed to explicit the main hypothesis adopted to carry out and perform the principle steps ((s 1 ) and (s 2 )) of the proposed model. periodic homogenization (if dimensions of joints are finite [START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF][START_REF] Cecchi | Homogenisation procedure to evaluate the effectiveness of masonry strengthening by CFRP repointing technique[END_REF]) and asymptotic analysis (if these dimensions tend to 0 [START_REF] Cecchi | Out of plane model for heterogeneous periodic materials: the case of masonry[END_REF][START_REF] Cecchi | A comparison between numerical and analytical homogenized models for visco-elastic brickwork[END_REF]) in order to provide the effective stiffness of the homogeneous material MHE-2 (e) equivalent to the masonry's cell (d). The rheology of the mortar with penny-shaped micro-cracks follows either the Modified Maxwell (j) or typical Burgers (f) model.

Homogenization of a microcracked viscoelastic mortar: coupling between Griffith's theory and mean-field homogenization in the Laplace-Carson space

Let us firstly recall that the effective stiffness of an elastic porous medium with a homogeneous solid phase tensor C s is:

C = C s : (I -φA p ) (1)
where A p is the average of the strain localisation tensor A(z) over the pore space Ω p and φ is the porosity volume fraction. Classical estimates of A p are based on the solution of the Eshelby single

A C C E P T E D M A N U S C R I P T
ellipsoidal inhomogeneity problem. Indeed this inhomogeneity Ω p is assumed to be embedded in an infinite homogeneous medium made up of an elastic material subjected to linear displacement boundary conditions of the form:

u(z) = E.z where z → ∞ (2)
In that case, the strain ε p in the inhomogeneity proves to be uniform. In the particular case of a cavity embedded in an infinite solid medium, ε p is given by:

ε p = (I -S E ) -1 : E (3)
where S E is the Eshelby tensor of the cavity function of the cavity's geometry and the bulk elasticity tensor C s . According to equation (3), the dilute scheme's estimate of the localization tensor A p reads:

A p DL = (I -S E ) -1 (4)
For a flat spheroid -usual 3D crack model -of aspect ratio w << 1, the Eshelby tensor S E is a function of w. Accordingly the components A p iikl and A p ikik of the tensor (4) are of the order of 1/w and therefore is the ratio of the normal strain ε nn to the macroscopic strain E. Possible nonenegligible variations of 1/w is in contradiction with the assumption of linearity of the localization relationship (3). To overcome this difficulty, Deudé et al. [START_REF] Deudé | Propriétés élastiques non linéaires d'un milieu mésofissuré[END_REF] have proposed to consider the rate-type formulation of the problem i.e. the strain localisation tensor (3) should be replaced by a strain rate localization as in the following:

ε(z) = A(z) : Ė
Similarly, the rate-type formulation of the Eshelby problem (3) for a spheroidal cavity reads:

εp = (I -S E (w)) -1 : Ė (5)
Recall that w here refers to the aspect ratio in the current configuration of the spheroidal cavity.

Such hypothesis implies that the use of A p in the homogenized stiffness (1) leads to an estimate of the tangent effective stiffness. Moreover, since the crack porosity φ is proportional to w [21], the tangent effective stiffness is mathematically independent of w. This renders the effective behaviour linear elastic. Note that the rate-type hypothesis is indispensable [START_REF] Deudé | Propriétés élastiques non linéaires d'un milieu mésofissuré[END_REF] to also avoid troubles related to possible large strain in the direction normal to the crack.

On the other hand, the extension of the linear homogenization schemes to non-aging viscoelasticity is based on the Laplace-Carson (LC) transform [START_REF] Schapery | Approximate methods of transform inversion for viscoelastic stress analysis[END_REF][START_REF] Rekik | Optimization of the collocation inversion method for the linear viscoelastic homogenization[END_REF]. The effective stiffness C =< C :

A > becomes C * =< C * : A > (6) A C C E P T E D M A N U S C R I P T
in the LC space where C * is the apparent elastic stiffness. The presence of microcracks implies the existence of nonlinearity at the local scale in the relationship between the crack strain and the macroscopic strain. Accordingly the homogenization of a viscoelastic cracked medium is not as straightforward as [START_REF] Binda | The collapse of civic tower of Pavia[END_REF]. The basic idea consists in anticipating that both the microscopic strain field and the displacement discontinuity vectors [u] i linearly depend on the macroscopic stress. Such an hypothesis, confirmed by [START_REF] Nguyen | Propagation de fissures et endommagement par microfissures dans un milieu viscoélastique linéaire non veillissant[END_REF], justifies the use of the LC transform which can be applied to the macroscopic strain related to the microscopic strain and the displacement discontinuity vectors [u] i between two lips of crack.

Homogenization of regular masonry with viscoelastic mortar joints: coupling between asymptotic analysis and periodic homogenization in the real temporal space

Since the procedure explained in the previous paragraph (see section (2.1)) provides the apparent creep function J * of the microcracked viscoelastic mortar, it is possible to derive the real creep function using an appropriate (analytical due to the Bromwich integral defined in the complex plane by [START_REF] Beurthey | Structural morphology and relaxation spectra of viscoelastic heterogeneous materials[END_REF] or numerical as the collocation [START_REF] Rekik | Optimization of the collocation inversion method for the linear viscoelastic homogenization[END_REF] or multi-data [START_REF] Cost | A multidata method of approximate laplace transform inversion[END_REF] methods) inversion procedure of the LC transform.

f (t) = 1 2π f * (p) p e pt dp

Linear elastic case

Classically, to determine the global behaviour of regular masonry with linear elastic constituents, it is useful to apply periodic homogenization technique. For that purpose, Cecchi and Rizzi [START_REF] Cecchi | Heterogeneous material: a mixed homogenization rigidification technique[END_REF], De Felice [START_REF] De Felice | Metodi di omogeneizzazione per sistemi regolari di corpi rigidi[END_REF][START_REF] De Felice | Détermination des coefficients d'élasticité de la maçonnerie par une méthode d'homogénéisation[END_REF] and Cecchi and Sab [START_REF] Cecchi | Out of plane model for heterogeneous periodic materials: the case of masonry[END_REF] have proposed to apply asymptotic homogenization technique with a perturbative analysis as a function of several parameters: ε 0 that defines the relationship between the overall dimensions of the element and those of the characteristic pattern, α which defines the relationship between the dimension of the mortar joint and that of the characteristic pattern and β defining the ratio of the mortar stiffness to the brick stiffness in order to provide constitutive functions depending on the said parameters. Besides these authors have investigated the obtained solutions as these parameters tend towards zero which is frequently the case for the bed joints. These studies are motivated by the fact that a mortar in historical masonry is much more deformable than the block and that its thickness is often negligible compared to the dimensions of the blocks. The homogenized stiffness tensor A i, j is computed on Y * by solving the following auxiliary problem on the periodic linear elastic masonry pattern Y * (see Figure (2)-a).

A C C E P T E D M

A N U S C R I P T Y * (a) followed by Cecchi and Sab in [START_REF] Cecchi | Out of plane model for heterogeneous periodic materials: the case of masonry[END_REF].

                 div(σ * ) = 0 ε * (y * ) = E + sym(grad u * per ) σ * = a i, j : ε(y * ) σ * .n anti-periodic on ∂Y * u * per periodic on ∂Y *
where a i, j (y * ) = a b for y * ∈ block a m for y * ∈ mortar [START_REF] Borderie | Crack closure effect in continuum damage mechanics: numerical implementation[END_REF] Here a b and a m are respectively the elastic constitutive functions pertaining to the blocks and mortar.

σ * and ε * are respectively the local stress and strain in the cell Y * . n is a unit vector normal to the boundary ∂Y * of this cell and u * is the local displacement field of a point M pertaining to this cell.

For a macroscopic strain E, it follows that:

Σ = σ * Y * = A α,β : E. After topological transformation of periodic pattern Y * (see pattern Y in Figure (2)-b)
) and assuming that lim α(β)β -1 = w 0 = 0, that means for periodic masonry with cohesive joints, the authors have shown that the auxiliary problem (7) can be rewritten as follows with reference to the pattern

Y b A C C E P T E D M A N U S C R I P T (see Figure (2)-c)        div(σ 0 ) = 0 ε 0 (y) = E + ∇ s u 0 σ 0 = A b : ε 0 (y) with boundary conditions                  σ 0 .n anti-periodic on ∂Y b u 0 periodic on ∂Y b [u 0 ] = 0 discontinuous on Σ I [σ 0 .n] = 0 continuous on Σ I σ 0 .n = K [u 0 ] constitutive relation on Σ I ( 8
)
where σ 0 and ε 0 are local stress and strain fields in the pattern Y b . n is the unit vector in the y 1 and y 2 middle planes of the Y -REV, Σ I is the interface and [u 0 ] is the jump in displacement field at Σ I and the fourth-order tensor K is given by

K h ≈ 1 e h (µ M I + (µ M + λ M )n ⊗ n), K v ≈ 1 e v (µ M I + (µ M + λ M )n ⊗ n) (9)
assuming isotropic behaviour of mortar [START_REF] Klarbring | Derivation of model of adhesively bounded joints by the asymptotic expansion method[END_REF]. The resolution of the auxiliary problem (8) in terms of the variables (σ 0 , u 0 ) allows to determine the homogenized stiffness à in the following form by reference to the patternY b [START_REF] Cecchi | Out of plane model for heterogeneous periodic materials: the case of masonry[END_REF] Ã1111 =

K C b 1111 + B e h a 4K e h a + b a K e v a e h a C + K D , Ã2222 = K K C b 2222 + B e v b e v b e h a B + K C b 2222 e h a + e v b + K 2 Ã1122 = K C b 1122 4K e h a + b a K e v a 4 e h a 2 e v b B + C b 1111 e h a C + K D , Ã1212 = C b 1212 K K e v b + 4 a b K e h b C b 1212 e h a F + K G (10)
where

B = C b 1111 2 -C b 1122 2 , C = 4K e h a + e v a K b a + 4K e v b , D = 4K e h a + e v a K b a , F = K e v b + 4K e h b a b + a b 2 e v b , G = K e v b + 4 e h b K a b , K = 2µ m + λ m (plane strain)
or K = 2µ m + λ * m (plane stress) and K = µ m or equivalently

K h = E m (1 -(ν m h ) 2 ) , K v = E m (1 -(ν m v ) 2 )
(plane stress)

K h = (1 -ν m h )E m (1 + ν m h )(1 -2ν m h ) , K v = (1 -ν m v )E m (1 + ν m v )(1 -2ν m h ) (plane strain) K h = E m 2(1 + ν m h ) , K v = E m 2(1 + ν m v ) (11) since µ = E 2(1 + ν) , λ = ν E (1 + ν)(1 -2ν) and λ * = ν E (1 -ν 2 )
. a and b are respectively the height and length of the brick in Y b . e h and e v are thicknesses of the mortar joints in the horizontal and vertical

A C C E P T E D M A N U S C R I P T
directions, respectively. These results are, in the limit, perfectly consistent with those obtained by

De Felice [START_REF] De Felice | Détermination des coefficients d'élasticité de la maçonnerie par une méthode d'homogénéisation[END_REF] in the case of rigid blocks and joints modeled as an elastic interface. The analytical solution [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF] is accurate when ratios between bricks and joints mortar dimensions tend toward 0 which is generally the case for bed joints. It is less accurate in the vertical direction since e v /h b is generally not negligible compared to e h /l b . Motivated by this reason, Cecchi & Barbieri and Cecchi & Tralli have proposed an alternative analytical solution valid for finite joint thickness. Indeed, following the procedure of Cecchi and Tralli [START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF], it is noted that the homogenized moduli for an elastic brick and joint are obtained in the pattern Y , and not in Y b as done in Cecchi and Sab [START_REF] Cecchi | Out of plane model for heterogeneous periodic materials: the case of masonry[END_REF].

The joint thickness is taken into account in the following mathematical procedure

E.(ã H E) ≤ E.( Ã E) ≤ min E.( ÃH E), E.( ÃR E) ( 12
)
where ãH is the homogenized 2D elasticity tensor obtained with plane stress in the bricks and 2D

restriction of K at the plane strain interface in the mortar. ÃH is the homogenized plane strain elasticity tensor i.e. with plane strain hypothesis in both blocks and mortar. ÃR is the homogenized in-plane tensor defined by:

( ÃR ) -1 = (A b * ) -1 + (A F ) -1 (13)
where A b * is the plane stress elasticity tensor of blocks and A F is the homogenized membrane tensor for rigid blocks connected by elastic interfaces. The homogenized tensor ÃR reads [START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF] ÃR

1111 = K h C b 1111 + B e h a + e h 4K v e h a + e h + b + e v a + e h K h e v a + e h 4 e h a + e h 2 e v b + e v B + C b 1111 e h a + e h C + K h D , ÃR 2222 = K h 4K v e h a + e h + b + e v a + e h K h e v a + e h C b 1111 + B e h a + e h e v a + e v 4 e h a + e h 2 e v b + e v B + C b 1111 e h a + e h C + K h D ÃR 1122 = K h C b 1122 4K v e h a + e h + b + e v a + e h K h e v a + e h 4 e h a + e h 2 e v b + e v B + C b 1111 e h a + e h C + K h D , ÃR 1212 = C b 1212 K h K h e v b + e v + 4 a + e h b + e v K h e h b + e v C b 1212 e h a + e h F + 4K h G (14)
where, here

C = 4K v e h a + e h + e v a + e h K h b + e v a + e h +4K h a a + e h e v b + e v
. The differences between equations ( 14) and those obtained by Cecchi and Sab [START_REF] Cecchi | Out of plane model for heterogeneous periodic materials: the case of masonry[END_REF] (see equations [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF]

ÃR 2222 = K K C b 2222 + B e v b + e v e v b + e v e h a + e h B + K C b 2222 e h a + e h + e v b + e v + K 2 , ÃR 1212 = C b 1212 K K e v b + e v + 4 a + e h b + e v K e h b + e v C b 1212 e h a + e h F + 2K G . (15)

Extension to viscoelatic case

For the viscoelastic periodic problem, the auxiliary problem (7) related to the elastic constituents can be rewritten as follows by reference to the periodic pattern

Y *                  Ė + ε( uper ) = ε( u) div(σ(t)) = 0 σ(t).n anti-periodic on ∂Y u per periodic on ∂Y σ(t) = Σ(t) (16)
Here, σ(t) is the microscopic stress tensor state; ε( u(y)) is the microscopic strain tensor state; u per is the periodic displacement field and Ė is the macroscopic in-plane strain tensor rate.

Since in the Cecchi & Tralli work, the hypothesis of cohesive joints was also adopted, an asymptotic auxiliary problem (similar to equations ( 8)) can be written with reference to the pattern Y b . The mortar appears merely as a boundary condition. In that case, assuming also that the viscoelastic mortar is isotropic, then the K tensor can be rewritten as function of the viscous function of the mortar, i.e.:

K h (t) = 1 e h E m h (1 + φ m h (t)) 2(1 + ν m ) I + E m h (1 + φ m h (t)) 2(1 + ν m ) + ν m E m h (1 + φ m h (t)) (1 + ν m )(1 -2ν m ) (n ⊗ n) K v (t) = 1 e v E m v (1 + φ m v (t)) 2(1 + ν m ) I + E m v (1 + φ m v (t)) 2(1 + ν m ) + ν m E m v (1 + φ m v (t)) (1 + ν m )(1 -2ν m ) (n ⊗ n) ( 17
)
where the K tensor has a diagonal form in this case. K = K h for the horizontal interface and K = K v for the vertical interface.

If only the mortar is assumed to be viscoelastic such that its stiffness tensor reads

A m (t) = A m (1 + A C C E P T E D M A N U S C R I P T φ m (t)
), the periodic constitutive function A i,j is similar to that shown in [START_REF] Cecchi | Out of plane model for heterogeneous periodic materials: the case of masonry[END_REF] but by substituting the Young's modulus E m by the inverse creep function

J -1 m (t) = E m (1 + φ m (t))
. It is noted that here only the Young's modulus is assumed to be a function of viscosity; the only characteristic parameters for mortar are the bulk modulus and shear modulus, rather than the Poisson's ratio.

For a microcracked mortar, it is proposed in step (s 2 ) (see Figure [START_REF] Andreev | Compressive behaviour of dry joints in refractory ceramic masonry[END_REF]) to estimate the effective moduli of the homogeneous equivalent material MHE-2 (see Figure (1)-e). These properties could be deduced from the macroscopic plane stress law Σ = C : ε or equivalently ε = S : Σ which reads

    εnn εtt 2ε nt     =        1 Ẽn - νnt Ẽn 0 - νtn Ẽt 1 Ẽt 0 0 0 1 μnt            Σ nn Σ tt Σ nt     (18)
where S and C are respectively the effective compliance and stiffness of the masonry periodic cell.

Accordingly

1 Ẽt (t, d c ) = Cnn Cnn Ctt -Cttnn Cnntt , νnt (t, d c ) = Cnntt Ctt , νtn (t, d c ) = Cttnn Cnn 1 Ẽn (t, d c ) = Ctt Cnn Ctt -Cttnn Cnntt , μnt (t, d c ) = Cntnt (19)
where, in the case of undamaged elastic bricks, the stiffness components Cijkl are provided by the extension of the Cecchi and Tralli expressions [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF] to the cracked state of the mortar joints. Accordingly viscous Young's modulus 'E m (1 + φ m (t))' or equivalently the inverse creep function J -1 m (t) is simply substituted by J -1 m (t, d c ) which accounts for the crack density parameter. Similarly to the undamaged elastic case, the proposed solution is expected to be an upper bound for the masonry stiffness. Recently, based on the works of Cecchi & Sab, Cecchi & Barbieri [START_REF] Cecchi | Homogenisation procedure to evaluate the effectiveness of masonry strengthening by CFRP repointing technique[END_REF] and

Cecchi & Tralli, Cecchi & Taliercio [START_REF] Cecchi | A comparison between numerical and analytical homogenized models for visco-elastic brickwork[END_REF] have proposed a homogenized compliance for viscoelastic undamaged masonry with mortar joints assimilated to interfaces. In this paper we propose to extend also this model to microcracked masonry with finite dimensions of microcracked mortar joints. The

A C C E P T E D M A N U S C R I P T

effective compliance of the damaged viscoelastic masonry reads then:

SR 1111 (t, dc) = S b 1111 e v a + e h J v (t, dc) + 4S b 1111 e h b + e v J h (t, dc) + 4 e v e h (b + e v ) 2 J v (t, dc)J h (t, dc) 4 e h b + e v J h (t, dc) + e h a + e h J v (t, dc) SR 2222 (t, dc) = S b 2222 + e h a + e h J h (t, dc) SR 1212 (t, dc) = S b 1212 + e h a + e h J h (t, d c ) + e v b + e v J v (t, dc) - e 2 v (b + e v )(a + e h ) J 2 v (t, dc) 4 e h b + e v J h (t, dc) + e v a + e h J v (t, dc) SR 1122 (t, dc) = S b 1122 ( 20
)
The damaged masonry's effective moduli are then the following: Ẽ11 (t,

d c ) = 1 SR 1111 (t, d c ) , Ẽ22 (t, d c ) = 1 SR 2222 (t, d c ) , μ12 (t, d c ) = 1 SR 1212 (t, d c ) , νij (t, d c ) = -Ẽii (t, d c ) SR iijj (t, d c ),

Rheological models for undamaged mortar

The rate-dependent mechanical behaviour of mortar is often approximated by a linear viscoelastic model [START_REF] Choi | Rheological modelling of masonry creep[END_REF][START_REF] Ignoul | Creep behaviour of masonry structures -failure prediction based on a rheological model and laboratory tests[END_REF]. For the sake of simplicity, only non-aging formulation will be considered in this work.

The practical interest of this simple formulation is that it allows to transform a time-dependent boundary value problem into a linear elastic one using the well-known correspondence theorem based on the Laplace-Carson transform. Among the simplest formulations used to model the non-aging model is characterized by an isotropic fourth-order tensor related to its elasticity or viscosity

C e K = 3k e K J + 2µ e K K, C v K = η s K J + η d K K, C e M = 3k e M J + 2µ e M K, C v M = η s M J + η d M K (21)
where k α and µ α (α = K or M ) denote the bulk and shear moduli. η s α and η d α represent respectively the bulk and shear viscosity. In the following, only rheological MM's model is presented.

Modified Maxwell model (MM). The constitutive law of the MM's model (see Figure (1)-j) is given by

S v M σ + S e M σ = S v M C e R ε + (I + S e M C e R ) ε (22)
where, for isotropic mortar material, the elastic and viscous compliances of the Maxwell

and allows the definition of the following symbolic Modified Maxwell elastic compliance

S * M M = (S v M C e R + p(I + S e M C e R )) -1 (S v M + pS e M ) (24)
Assuming the isotropy of the mortar behaviour, the symbolic compliance [START_REF] Dufferene | Viscoelastic constants of a soda-lime silica glass[END_REF] reads

S * M M = 1 3k * s J + 1 2µ * s K (25)
The associated apparent creep function is then given by

J * M M = 1 E * M M = 1 9k * s + 1 3µ * s = 1 9 k R + pk M η s M /3 k M + pη s M /3 + 1 3 µ R + pµ M η d M /2 µ M + pη d M /2 (26) since 1 k * s = 1 k R + pk M η s M /3 k M + pη s M /3 , 1 µ * s = 1 µ R + pµ M η d M /2 µ M + pη d M /2 . ( 27
)

A C C E P T E D M A N U S C R I P T

The analytical direct inversion of (26) leads to the MM's real creep function

J M M (t) = 1 9k R + 1 3µ R - k M 9k R (k R + k M ) e -t/τ s M M - µ M 3µ R (µ R + µ M ) e -t/τ d M M (28)
with the characteristic times

τ s M M = η s M (k R + k M ) 3k R k M and τ d M M = η d M (µ R + µ M ) 2µ R µ M
for respectively the spherical and deviatoric parts of the MM's viscous behaviour.

Creep function of microcracked non-aging linear viscoelastic mortar

We consider a RVE Ω made up of non-aging linear viscoelastic material and a network of plane penny-shaped cracks. In contrast to 3D ellipsoidal crack model considered in the Eshelby-type approach, the mathematical penny-shaped crack is a 2D problem in nature which does not refer to an aspect ratio w. During the time period of loading, assuming that the displacement u(z, t)

prescribed on the boundary ∂Ω is given by u(z, t) = ε(t).z where the macroscopic strain ε(t) is the sum of the contribution of the elastic solid and of the fractures as follows

ε = 1 V Ωs εdV + i C i [u] i ⊗ s n i dS (29)
where Ω s is the domain of the undamaged matrix and a ⊗ s b = a ⊗ t b + b ⊗ t a 2 for the vectors a and b. C i and n i denote respectively the boundary of the crack i and the vector normal to its plane. For empty cracks, the overall stress reads Σ =< σ > Ω = 1 V Ωs σdV = C : ε since the local stress in the voids is null.

Symbolic space: coupling between homogenization and Griffith's theory

The Laplace-Carson transform of the macroscopic strain (29) reads:

ε * = 1 V Ωs ε * dV + i C i [u] * i ⊗ s n i dS ε * = C * -1 : Σ * + 1 V i C i [u] * i ⊗ s n i dS (30)
In the framework of the stress-based dilute scheme [START_REF] Dormieux | Stress-based estimates and bounds of effective elastic properties: the case of cracked media with unilateral effects[END_REF], the displacement jump [u] i is linearly related to Σ. Indeed, the normal displacement jump (mode I) at the lips of a crack in an infinite matrix submitted to an isotropic asymptotic macroscopic stress Σ * = Σ * i can be written:

[u n ] * = 4(1 -ν * s ) π Σ * µ * s l 2 -ρ 2 .
(31)

A C C E P T E D M A N U S C R I P T

where l is the crack's radius and ρ is the position of a point M pertaining to the crack's plane.

The tangential displacement jump (mode II) under an asymptotic shear stress Σ * = Σ * n ⊗ s t where t is parallel to the crack's plane reads:

[u t ] * = 4(1 -ν * s ) π(2 -ν * s ) Σ * µ * s l 2 -ρ 2 . ( 32
)
Effective dilute symbolic bulk moduli. Under an isotropic loading (Σ * = Σ : i), the elementary contribution of a crack to the macroscopic strain is purely normal ([u] * = [u n ] * n) as shown in [START_REF] Luciano | Homogenization technique and damage model for old masonry material[END_REF] and reads:

C i [u n ] * i ⊗ s n i dS = 8l 3 3 Σ * (1 -ν * s ) µ * s n ⊗ n (33)
Assuming that all cracks have the same radius l, an integration over all orientations on the unit sphere yields the total crack contribution:

1 V i C i [u n ] * ⊗ s ndS = 8l 3 3 Σ * (1 -ν * s ) µ * s |n|=1 n ⊗ n dS 4π = 8N l 3 9 Σ * (1 -ν * s ) µ * s i ( 34
)
where N denotes the number of cracks per unit volume. Combining ( 30) and ( 34) defines the effective state equation under isotropic loading:

ε * = C * -1 : (Σ * : i) + 8d c 9 Σ * (1 -ν * s ) µ * s i (35)
for the damage parameter d c = N l 3 [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF][START_REF] Dormieux | MicroporoMechanics[END_REF]. At last, since

trace(ε * ) = Σ * k * DL = 1 k * s + 8d c 3 (1 -ν * s ) µ * s Σ * (36) then 1 k * DL = 1 k * s + 8d c 3 (1 -ν * s ) µ * s , or equivalently 1 k * DL = 1 + d c Q * k * s where Q * = 16 9 (1 -ν * 2 s ) (1 -2ν * s ) (37)
in which the symbolic Poisson's ratio of the undamaged matrix reads:

ν * s = 3k * s -2µ * s 6k * s + 2µ * s .
Effective dilute symbolic shear moduli. Under a deviatoric loading in which the macroscopic stress reads Σ * = Σ * (e 1 ⊗ e 1e 3 ⊗ e 3 ), the elementary contribution of a crack to the macroscopic strain is derived from [START_REF] Luciano | Homogenization technique and damage model for old masonry material[END_REF] and [START_REF] Nedjar | An approach to the modeling of viscoelastic damage: application to the long-term creep of gypsum rock materials[END_REF] according to its orientation. In the polar coordinates system (r, θ, Φ), the displacement jump is given by

[u] * = Σ * (1 -ν * ) πµ * X r e r + 2 2 -ν * (X θ e θ + X Φ e Φ ) a 2 -ρ 2 (38)

A C C E P T E D M A N U S C R I P T

if the normal vector n coincides with the radial unit vector e r . Integration over all crack orientations on the unit sphere yields

1 V i C i [u] * i ⊗ s n i dS = d c M * 2µ * Σ * where M * = 32 45 (1 -ν * s )(5 -ν * s ) 2 -ν * s (39)
Accordingly and due to [START_REF] Koushik | Creep and visco-elastic behaviour of LPS-SiC sintered with Lu 2 O 3 -AlN additive[END_REF], the apparent effective shear modulus reads

1 μ * DL = 1 + d c M * 2µ * s (40)

Creep function: identification of existent rheological models at short and long times

Since the expressions ( 37) and ( 40) can not be satisfied rigorously, it was proposed in [START_REF] Nguyen | Propagation de fissures et endommagement par microfissures dans un milieu viscoélastique linéaire non veillissant[END_REF] to identify the best approximation of the effective behaviour in the class of Burgers' (Modified Maxwell's) model if the mortar in its undamaged state follows the Burgers' (Modified Maxwell's) model. The idea is to satisfy the series expansion of the dilute estimates of the bulk's [START_REF] Nguyen | A Burger Model for the Effective Behavior of a Microcracked Viscoelastic Solid[END_REF] and shear [START_REF] Rekik | Optimization of the collocation inversion method for the linear viscoelastic homogenization[END_REF] moduli to the first order at p = 0 and p → ∞ such that

lim t→∞ f (t) = lim p→0 f * (p) and lim t→0 f (t) = lim p→∞ f * (p) (41)
Hereafter, only the identification of the MM's creep parameters function will be detailed. Parameters related to the Burgers creep function at damaged state [START_REF] Nguyen | A Burger Model for the Effective Behavior of a Microcracked Viscoelastic Solid[END_REF][START_REF] Nguyen | Approche multi-échelles pour des maçonneries viscoélastiques[END_REF] are reported in Appendix (A).

Expansions of the mortar's effective dilute apparent moduli

At the vicinity of p = 0, the expansions in series of 27)-a) and 27)-b) provide the following approximations

1 k * s = 1 k s R + O(p) (see equation (
1 µ * s = 1 µ s R + O(p) (equation (
1 k * DL = (1 + d c Q 0 0 ) k s R + O(p), 1 μ * DL = (1 + d c M 0 0 ) µ s R + O(p) (42)
where the coefficients Q 0 0 , Q 1 0 , M 0 0 and M 1 0 are given by

Q 0 0 = 4k s R (3k R + 4µ R ) 3µ R (3k R + µ R ) , Q 1 0 = 2 9 (2η s M µ R -3k R η d M )(9k 2 R + 4µ 2 R + 6k R µ R ) µ 2 R (3k R + µ R ) 2 M 0 0 = 16 45 (3k R + 4µ R )(9k R + 4µ R ) (3k R + µ R )(3k R + 2µ R ) , M 1 0 = 8 45 (3k R η d M -2η s M µ R )(63k 2 R + 60k R µ R + 16µ 2 R ) (3k R + µ R ) 2 (3k R + 2µ R ) 2 . ( 43
)

A C C E P T E D M A N U S C R I P T

On the other hand, at the vicinity of p = ∞,

1 k * s = 1 k e K + k e M + 3k 2 M (k e M + k s R ) 2 η s M p + O(p -2 ) and 1 µ * s = 1 µ e K + µ e M + 2µ 2 M (µ e M + µ s R ) 2 η d M p + O(p -2 ), accordingly        1 k * DL = 1 + d c Q ∞ 0 k e M + k s R + 1 p (3k 2 M (1 + d c Q ∞ 0 ) + d c (k e M + k s R )Q ∞ 1 η s M ) η s M (k M + k s R ) 2 + O(1/p 2 ) 1 μ * DL = 1 + d c M ∞ 0 µ e M + µ s R + 1 p 2µ 2 M (1 + d c M ∞ 0 ) + d c (µ e M + µ s R )M ∞ 1 η d M η d M (µ M + µ s R ) 2 + O(1/p 2 ) ( 44
)
where the coefficients

Q 0 ∞ , Q -1 ∞ and M 0 ∞ are given by Q 0 ∞ = 4 3 (k M + k R )( 1 µ M + µ R + 3 3(k M + k R ) + (µ M + µ R )
),

M 0 ∞ = 16 45 8 - 9(k M + k R ) 3(k M + k R ) + (µ M + µ R ) - 6(k M + k R ) 3(k M + k R ) + 2(µ M + µ R ) (45)

Identification of an existent rheological model for the cracked mortar's creep function

The dilute symbolic bulk's (37) and shear's (40) moduli of a non-aging linear viscoelastic (n.a.l.v.) microcracked mortar following the MM's model can be approached by expressions similar respectively to ( 27)-a and ( 27)-b available for a mortar with a matrix following the MM's rheological model

k * M M = k R (d c ) + pk M (d c )η s M (d c )/3 k M (d c ) + pη s M (d c )/3 , µ * M M = µ R (d c ) + pµ M (d c )η d M (d c )/2 µ M (d c ) + pη d M (d c )/2 (46)
The series expansion of the dilute symbolic estimate of the bulk's modulus ) at the vicinity of 0

1 k * M M = 1 k R (d c ) -p η s M (d c ) 3k 2 R (d c ) + O(p 2 ), 1 μ * M M = 1 µ R (d c ) -p η d M (d c ) 2µ 2 R (d c ) + O(p 2 ) (47) and ∞        1 k * M M = 1 k R (d c ) + k M (d c ) + 1 p 3k 2 M (d c ) η s M (d c )(k M (d c ) + k R (d c )) 2 + O( 1 p 2 ) 1 μ * M M = 1 µ R (d c ) + µ M (d c ) + 1 p 2µ 2 M (d c ) η d M (d c )(µ M (d c ) + µ R (d c )) 2 + O( 1 p 2 ) (48) added to the equalities 1 k * DL ≈ 1 k * M M and 1 µ * DL ≈ 1 µ * M M
allow the identification of the following six

MM's parameters k R (d c ) = k R 1 + d c Q 0 0 , µ R (d c ) = µ R 1 + d c M 0 0 k e M (d c ) = k M + k R 1 + d c Q ∞ 0 - k R 1 + d c Q 0 0 , η s M (d c ) = (η s M + d c (η s M Q 0 0 -3k e R Q 1 0 )) (1 + d c Q 0 0 ) 2 µ M (d c ) = µ M + µ R 1 + d c M ∞ 0 - µ R 1 + d c M 0 0 , η d M (d c ) = η d M + d c (η d M M 0 0 -2µ R M 1 0 ) (1 + d c M 0 0 ) 2 (49) A C C E P T E D M A N U S C R I P T
The characteristic times of the spherical and deviatoric parts of the Modified Maxwell model followed by the microcracked viscoelastic mortar are respectively

τ s M M (d c ) = η s M (d c )(k R (d c ) + k M (d c )) 3k R (d c )k M (d c ) , τ d M M = η d M (d c )(µ R (d c ) + µ M (d c )) 2µ R (d c )µ M (d c ) (50)
At last, the approximate creep function of a microcracked mortar which matrix follows the MM's model reads case has been yet proven in [START_REF] Cecchi | A comparison between numerical and analytical homogenized models for visco-elastic brickwork[END_REF] by comparison with more accurate finite element solution. Based on this result, only elastic bricks with Young's moduli E b ≥ 20E m (t 0 ) (here the initial time t 0 is set equal to 0) will be considered in the following. For these evaluations, the adopted mortar's data are given in table-1. The properties of the undamaged bricks are the following: ν b = ν m and

J app M M (t, dc) = 1 9k R (d c ) 1 - k M (d c ) (k R (d c ) + k M (d c )) e -t/τ s M M (dc) + 1 3µ R (d c ) 1 - µ M (d c ) (µ R (d c ) + µ M (d c )) e -t/τ d M M (dc) (51
E b = 100E m (t 0 )
where for this hybrid mortar E m (t 0 ) = J -1 M M (t 0 , 0) = 6150 MPa for the MM's model and J -1 Bu (t 0 , 0) = 4038 MPa for the Burgers' one. Recall here that

J -1 M M (0, 0) = E R + E M and J -1
Bu (0, 0) = E M . Bricks are of dimensions 250x55 mm 2 and mortar's joints thicknesses are e h = e v = 10 mm. Effective masonry's moduli for cases E b = 40E m (t 0 ) and 20E m (t 0 ) allowing to study the effect of the parameter β are reported in Appendix (B). Note that the characteristic times

E M (MPa) τ M (s) E R (MPa) τ R (s) ν m 4038 46490 2112 90866 0.22
Table 1: Elastic and viscous moduli of hybrid mortar [START_REF] Ignoul | Creep behaviour of masonry structures -failure prediction based on a rheological model and laboratory tests[END_REF] for the spherical and deviatoric hybrid mortar's behaviour are assumed to be equal τ s = τ d = τ for the MM's and Burgers' models. The Young's modulus E R and Poisson's ratio ν R coincide respectively with E K and ν K , properties of the Kelvin-Voigt's spring.

A C C E P T E D M A N U S C R I P T

For damaged state, to validate the step (s 1 ), comparisons with results provided by Nguyen et al. [START_REF] Nguyen | Propagation de fissures et endommagement par microfissures dans un milieu viscoélastique linéaire non veillissant[END_REF] for concrete material following the typical Burgers model were carried out. Moreover the evolutions of the MM's model parameters identified in section (4.2.2) versus the crack density d c depicted in Figure [START_REF] Cecchi | Homogenisation procedure to evaluate the effectiveness of masonry strengthening by CFRP repointing technique[END_REF] in Appendix [START_REF] Borderie | Crack closure effect in continuum damage mechanics: numerical implementation[END_REF] validate this identification step since these parameters decrease with the increase of the damage parameter. Similar validation procedure was carried out for the Burgers parameters but is not reported in this paper for the sake of brevity.

In this work, the validity of the step (s 2 ) was ensured by comparisons with Cecchi & Sab [START_REF] Cecchi | Out of plane model for heterogeneous periodic materials: the case of masonry[END_REF] effective stiffnesses available for elastic undamaged masonry either with plane strain or plane stress hypothesis (see table-2 in [START_REF] Cecchi | Out of plane model for heterogeneous periodic materials: the case of masonry[END_REF]). When accounting for the creep effects, this model was validated by making comparisons with Cecchi & Tralli results available for periodic masonry with undamaged (d c = 0) mortar following the USBR rheology model (see Figure [START_REF] Borderie | Crack closure effect in continuum damage mechanics: numerical implementation[END_REF] in [START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF]). Note that in this paper, the implemented effective stiffness components Ã2222 and Ã1212 are similar to those proposed by Cecchi & Sab [START_REF] Cecchi | Out of plane model for heterogeneous periodic materials: the case of masonry[END_REF] (see equation ( 10)) but accounting for creep and finite dimensions of mortar joints. Recall here that this version of the Burgers model is different from the modified or adapted ones proposed in [START_REF] Papa | A visco-damage model for brittle materials under monotonic and sustained stresses[END_REF]46,47]. Parameters of these modified Burgers models -generally up to 15 -need experimental investigations to be identified so as experimental creep curves coincide with modified Burgers predictions. In this part, it is assumed that crack does not propagate following the Nguyen et al. [START_REF] Nguyen | A Burger Model for the Effective Behavior of a Microcracked Viscoelastic Solid[END_REF][START_REF] Nguyen | Generalized Kelvin model for micro-cracked viscoelastic materials[END_REF] researches. Tables [START_REF] Andreev | Compressive behaviour of ACS torpedo bricks, 11 th Biennial Worldwide Conference on Unified International Technical Conference Refractories[END_REF] and (3) provide the effective properties of a periodic masonry cell with hybrid viscoelastic mortar at its undamaged and microcracked states with quasi-rigid (E b = 100E m (t 0 )) bricks. Tables [START_REF] Bornert | Homogénéisation en mécanique des matériaux, Tome 1 : Matériaux aléatoires élastiques et milieux périodiques[END_REF] to [START_REF] Borderie | Crack closure effect in continuum damage mechanics: numerical implementation[END_REF] reported in Appendix [START_REF] Borderie | Crack closure effect in continuum damage mechanics: numerical implementation[END_REF] give the effective properties of a periodic vis- states for the MM's model shows that this error increases with the increase of the ratio β to reach a maximum of about 71% which is an asymptotic limit for E r attained in the case of quasi-rigid bricks (E b = 100E m ). This result demonstrates that the MM's model allows the masonry to preserve a certain resistance for β ≥ 20 unlike this version of the Burgers model for which the masonry seems to fail. Figures [START_REF] Beurthey | Structural morphology and relaxation spectra of viscoelastic heterogeneous materials[END_REF] In this part, for the sake of simplicity, it is assumed that the crack density d c is set equal to zero at the initial time t 0 = 0 and evolves linearly with time t as follows

d c = ḋc t (52)
where, here, the rate of the crack density ḋc is assumed to be a positive constant lower than 0.002/day (i.e. if t = 100 days, then d c reaches a maximum of 0.2). Indeed, beyond this limit, the dilute estimate will not be appropriate. Of course, as well known, the increase of the damage rate reduces the stiffness of the masonry cell.

Conclusions and perspectives

In this paper extensions of the Cecchi & Tralli [START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF] and the Cecchi & Taliercio [START_REF] Cecchi | A comparison between numerical and analytical homogenized models for visco-elastic brickwork[END_REF] models were proposed for microcracked non-aging viscoelastic masonry using the MM's and basic Burgers models.

These extensions are based in a first step on the coupling between the Griffith's theory and the dilute mean-field homogenization scheme in order to derive easily and with low computational effort -without recourse to numerical inversion of the Laplace transform -the effective creep function of a microcracked non-aging viscoelastic mortar. In a second step, the proposed model is based on periodic homogenization accounting for finite thickness of the mortar joints in order to express

•

  We propose a multi-level model to predict the orthotropic behaviour of cracked masonry • This model relies on the coupling between Griffith's theory and homogenization methods • Overall estimates of Cecchi & Taliercio model are softer than the Cecchi & Tralli's ones

  S C R I P T with elastic or quasi-rigid undamaged bricks. Keywords: Mean-field homogenization; Brittle fracture; Periodic homogenization; Viscoelasticity; Laplace-Carson transform; Masonry Nomenclature C effective or homogeneous stiffness tensor φ porosity volume fraction A p average of the strain localization tensor A strain localization tensor Ω p pore space u local displacement field E macroscopic strain ε p average of the strain over the pore S E Eshelby tensor of the cavity function A p DL dilute strain localization tensor A p w aspect ratio of an ellipsoidal cavity C * effective stiffness in the symbolic space ε local strain rate Ė macroscopic strain rate α ratio between mortar's and brick's dimensions β ratio between brick's and mortar's stiffnesses σ local stress field Σ macroscopic stress field σ 0 local stress field in pattern Y b ε 0 macroscopic stress field in pattern Y b [u 0 ] displacement jump at interface Σ I Σ I mortar joint interface in pattern Y b [u] displacement jump between crack's two lips J mortar's creep function Y * periodic cell of a running bond masonry Y topological transferred pattern of the cell Y * Y b periodic masonry Y * with cohesive joints K fourth-order stiffness tensor of cohesive joints [u * ] symbolic crack displacement jump l radius of penny-shaped crack d c crack density parameter k s bulk's modulus of the undamaged matrix µ s shear's modulus of the undamaged matrix ν s undamaged matrix Poisson's ratio k l bulk's modulus of the spring µ l shear's modulus of the spring (l = M or K) η s spherical bulk's viscosity of the dashpot η d deviatoric shear's viscosity of the dashpot τ s spherical relaxation time τ d deviatoric relaxation time < . > r average over phase r of the quantity "." i second-order identity tensor I fourth-order identity tensor J spherical fourth-order projector K fourth-order stiffness tensor of mortar joint µ m shear moduli of mortar a length of the brick b height of the brick e h thickness of the bed joint e v thickness of the head joint λ m Lame's coefficient (plane stress) λ * m Lame's coefficient (plane strain)

  Burgers model. In[START_REF] Choi | Rheological modelling of masonry creep[END_REF], an experimental study was carried in order to investigate the creep of masonry. A number of rheological models (USBR, Feng, Ross, typical Burgers, Modified Maxwell) are examined to assess their ability to predict the creep of masonry. It was proved that the Modified Maxwell model is the most accurate. According to this result, the Burgers model, namely a Maxwell system connected in series with a Kelvin-Voigt one, and the Modified Maxwell scheme (a parallel combination of the Maxwell model and a spring) models are adopted in this paper to describe the mortar joint's creep. Moreover, in the literature, little attention is devoted to the prediction of the macroscopic creep behaviour of masonry. In this context, Brooks, Cecchi & Tralli, Cecchi & Taliercio

  scheme is useful for dilute concentrations of cracks, it has been demonstrated in[START_REF] Dormieux | MicroporoMechanics[END_REF][START_REF] Dormieux | Stress-based estimates and bounds of effective elastic properties: the case of cracked media with unilateral effects[END_REF] that its estimates coincide with the Mori-Tanaka predictions and are close to the Ponte-Castañeda-Willis (PCW) estimates for d c ≤ 0.15). This result is valid for open and closed frictionless cracks. These global properties are used to compute the behaviour of a compressed wall as done in[START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF]. The main hypothesis adopted in this work are explicited in section 2 for both first and second steps based respectively on the approximation of the creep function of a microcracked non-aging linear viscoelastic mortar and the determination of the effective stiffness of a regular masonry with undamaged bricks and microcracked mortar joints with finite dimensions. The modified Maxwell and Burgers models followed by the rheology of undamaged mortar are recalled in section 3. The steps allowing the determination of the effective creep function of a microcracked linear viscoelastic mortar are explained in section 4. Section 5 provides a first application of the proposedCecchi & Tralli and Cecchi &

  jected to severe or long term loading such as historical monuments or refractory masonry linings working under high temperatures. For undamaged state, this work enlarges the applications -USBR and Feng models -treated in the Cecchi & Tralli paper and the Generalized Maxwell model in the Cecchi & Taliercio work to the Modified Maxwell and typical Burgers rheological models. Moreover

Figure 1 :

 1 Figure 1: Main steps of the proposed model: (s 1 ) -approximation of the effective creep function knowing the effective properties of the homogeneous material MHE-1 (c) equivalent to the non-aging linear viscoelastic mortar (b) joints with isotropic distribution of cracks present in the periodic masonry cell (a) -and (s 2 ) based on two techniques:

Figure 2 :

 2 Figure 2: Running bond masonry: finite element discretization (d) and topological transformations (patterns Y (b), Y B with mortar joints assimilated to interfaces or surface discontinuities (c)) of the characteristic elementary pattern

  where (i, j) ∈ {1, 2} . These properties are explicit function of the crack density -damage parameter -, time and ratios e h a + e h , e v b + e v instead of respectively e h a and e v b . It is worth noting, that for the case of undamaged mortar interfaces, Cecchi & Taliercio have proven that this solution is more consistent with a numerically homogenized solution based on the finite elements method for ratios E b /E m ≥ 20 when the mortar follows a Generalized Maxwell model. In the following, we assume that this condition ensuring the accuracy of the Cecchi & Taliercio's analytical model with the additional assumption of finite joints dimensions is also available for damaged mortars. Hereafter mortar will follow typical Burgers and Modified Maxwell models. Moreover, the effective moduli of masonry proposed by Cecchi & Tralli and Cecchi & Taliercio will be extended to the damaged case of masonry with finite dimensions of joints without any recourse to numerical inversion of the Laplace Carson method. The yet proven accuracy of the Cecchi & Taliercio's analytical model for ratios E b /E m ≥ 20 will be useful to assess the accuracy of the Cecchi & Tralli's extension model to damaged mortars.

  S C R I P Tlinear viscoelastic mortar's behaviour, it is possible to quote the USBR, Ross, Feng, Burgers and Modified Maxwell models[START_REF] Choi | Rheological modelling of masonry creep[END_REF][START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF] mainly based on connections in parallel and/or in series of Maxwell and Kelvin-Voigt parts. Each element (spring and dashpot) of the Maxwell (M) or Kelvin-Voigt (KV)

  The elastic stiffness of the spring reads C e R = 3k e R J + 2µ e R K. The Laplace-Carson transform applied to the behaviour law (22) leads to (S v M + pS e M )σ * = (S v M C e R + p(I + S e M C e R ))ε *

) 5 .

 5 Application for a 2D masonry with microcracked viscoelastic hybrid mortar To assess the reliability of the Cecchi & Tralli extension model presented in this paper, it is proposed, in a first step (s 1 ) (see Figure (1)), to implement the creep functions (53) and (51) and to study the effects of the crack density d c , time t and geometrical parameter β = E b /E m on the trends of the creep of the mortar. As mentioned previously (see section (2.2.2)), the estimates of the Cecchi & Taliercio extension model are useful as reference solution since the accuracy of this model at undamaged

5. 1 . 1 DLand J Bu - 1 DLFigure ( 3 )

 1113 Figure (3) and (4) illustrate the evolutions of normalized inverse dilute creep functions of respectively the MM's and Burgers' models with respect to the damage parameter d c and the time t. When the

Figure 3 :

 3 Figure 3: Mortar following the MM's model: Variation of J -1 M M (t, d c )/J -1 M M (0, 0) versus time (b) for undamaged mortar (d c = 0) and microcracked mortars (d c = 0.1, d c = 0.2) and (a) versus the damage parameter d c at different times t = 0, 1 day and t ≥ 5 days.

Figure 4 :

 4 Figure 4: Mortar following the Burgers's model: Variation of J -1 Bu (t, d c )/J -1 Bu (0, 0) versus time (b) for undamaged mortar (d c = 0) and microcracked mortars (d c = 0.1, d c = 0.2) and (a) versus the damage parameter d c at different times t = 0, 1, 5 days and t ≥ 10 days. damage parameter d c increases, Figure(3)-a (respectively (4)-a) shows that the normalized inverse creep function J -1 (t, d c )/J -1 (0, 0) does not evolve beyond t ≥ 5 days for the MM's (respectively, 10 days for the Burgers') model. Moreover, unlike the Burgers' model, the MM's one predicts a slower decrease of the normalized creep function versus the damage parameter d c . Indeed, for the considered mortar with properties gathered by means of short term compressive tests on masonry wallets [28], this function vanishes for times exceeding 10 days according to this version of the Burgers model whereas the MM's estimates tend towards non-zero asymptotic limits for t ≥ 5 days. For a given crack density parameter d c = 0, 0.1 or 0.2, Figure (3)-b demonstrates that the MM's model yields to a constant function J -1 (t, d c )/J -1 (0, 0) with variation of the time beyond t = 10 6 (s) (i.e. almost 11 days). As expected, the increase of d c decreases the level of the asymptotic limits reached by this function. The difference between the MM's curves for different d c is not negligible (around 15%) unlike that observed for the Burgers curves which are very close especially at short and long terms.Figure (4)-b shows again that the Burgers model leads to vanishing inverse creep

Figure ( 4 )

 4 -b shows again that the Burgers model leads to vanishing inverse creep functions for t ≥ 3 10 8 (s) i.e. 1157 days for every crack density value d c ≥ 0.

2 .

 2 Effective behaviour of periodic microcracked viscoelastic masonry cell 5.2.1. Constant crack density d c

  coelastic masonry cell with respectively E b = 20E m and E b = 40E m . For the considered hybrid mortar, these tables demonstrate that the effective properties of a masonry cell with a microcracked mortar following the Burgers model vanish of about 99% by reference to the undamaged state for any range of the stiffnesses ratio β = E b E m ∈ [20, 100]. The evaluation of the absolute error E r between masonry cell's effective properties at undamaged and damaged (t = 1000 days, d c = 0.1)

Figure 5 : 2 A

 52 Figure 5: MM's model: Evolutions of the effective properties of the periodic masonry cell as function of time for different crack density parameters d c = 0, 0.1 and 0.2

Figure 6 :

 6 Figure 6: Burgers' model: Evolutions of the effective properties of the periodic masonry cell as function of the time for different crack density parameters d c = 0, 0.1 and 0.2

Figure 9 :

 9 Figure 9: BC-2 boundary conditions: MM's model predictions for the maps of stresses σ xy (a) and σ yy (b) for d c = 0.15 at t = 1000 days. Here crack density d c = ḋc t where ḋc = 1.5 10 -4 /day.

Figure 10 :

 10 Figure 10: Boundary condition BC-1 or BC-2: trends in σ xy (a) and σ yy (b) at the panel section A-A for the Modified Maxwell's model at time t = 0 with d c = 0 and t = 1000 days with d c = 0.15. Here crack density d c = ḋc t where ḋc = 1.5 10 -4 /day.

Figure 11 :

 11 Figure 11: BC-2 boundary conditions: MM's model predictions for the map of the strain ε xy (a) and ε yy (b) for: d c = 0.15 and 1000 days. Here crack density d c = ḋc t where ḋc = 1.5 10 -4 /day.

Figure 12 :

 12 Figure 12: Boundary condition BC-1 or BC-2: trends in ε xy (a) and ε yy (b) at the panel section A-A for the Modified Maxwell's model at times t = 0 with d c = 0 and t = 1000 days with d c = 0.15. Here crack density d c = ḋc t where ḋc = 1.5 10 -4 /day.

(

  E b = 40E m ) and hybrid viscoelastic mortar accounting for time-dependent d c with a constant rate ḋc = 1.5x10 -4 /day. A C C E P T E D M A N U S C R I P T C. Long-term characteristic times and corresponding effective propertiesE M (MPa) τ M (s) E R (MPa) τ R (s) ν m

  In the Cecchi & Barbieri's paper: D = a a+e h 4K e h a+e h + (b+ev) (a+e h ) K ev a+e h ,

	are substituted respectively by	e v b + e v	and	e h a + e h	, in the components ÃR 2222 , ÃR 1212 and the coefficients
			M A N U S C R I P T
	A C C E P T E D
					) lie in the ratio	e v b	and	e h a	, which

C (mentionned above) and D.

  and (6) depict the evolutions of the effective properties of the periodic

	Mortar: rheological model t (days) d c Ẽtt (MPa) Ẽnn (MPa)	νtn	νnt	μtn (MPa)
	Burgers	0	0	111223.	11792.8	0.037 0.004	8087.07
		25	0	3082.32	142.66	0.002	0	202.6
		25	0.1	2620.62	120.77	0.0016	0	172.17
		100	0	810.81	36.74	0	0	53.18
		100	0.1	688.77	31.17	0	0	45.17
		1000	0	82.37	3.70	0	0	5.4
		1000	0.1	69.95	3.15	0	0	4.58
	Modified Maxwell	0	0	169277	39314.9	0.060 0.0140	14432.7
	Absolute error E r (%)	0 ≥ 100 0.1 ≥ 100	70955.6 61324.4 63.7	14095.1 12011.6 69.4	0.025 0.005 0.022 0.004 63.5 71.4	5150. 4387.1 69.6

Table 2 :

 2 Cecchi & Tralli's extension model: effective properties of a periodic masonry cell with quasi-rigid bricks (E b = 100E m ) and hybrid viscoelastic mortar[START_REF] Ignoul | Creep behaviour of masonry structures -failure prediction based on a rheological model and laboratory tests[END_REF] masonry cell as functions of the crack density parameter d c and time t. As expected, the normal-

	Mortar: rheological model t (days) dc Ẽtt (MPa) Ẽnn (MPa)	νtn	νnt	μtn (MPa)
	Modified Maxwell	0	0	164964.	37535.2	0.060 0.0134	14400.6
	Maxi. absolute error E r (%)	0 ≥ 100 0.1 ≥ 100	68761.2 59395.3 3.24	13428.3 11441.3 4.98	0.0245 0.0050 0.0212 0.004 3.3 5	5138.13 4376.9 0.023
		M A N U S C R I P T
	A C C E P T E D

Table 3 :

 3 Cecchi & Taliercio's extension model: effective properties of a periodic masonry cell with quasi-rigid bricks (E b = 100E m ) and hybrid viscoelastic mortar[START_REF] Ignoul | Creep behaviour of masonry structures -failure prediction based on a rheological model and laboratory tests[END_REF]

Table - 3

 - ). The crack density d c accelerates this softening with an additional reduction of moduli of about 10% (E b = 20E m ) and 13% (E b = 100E m ) for Ẽtt . These results are consistent with those provided by Choi et al.[START_REF] Choi | Rheological modelling of masonry creep[END_REF]. Indeed, unlike the typical Maxwell model, the additional spring in this model's chain provides resistance after full relaxation which makes the long-term creep rate decrease continuously to better represent the realistic long-term behaviour of masonry. Moreover, it has been proven in[START_REF] Choi | Rheological modelling of masonry creep[END_REF] and confirmed by Shrive et al.[START_REF] Shrive | Creep analysis of clay masonry assemblages[END_REF] that the basic Burgers model only gives good results for the first few days (up to 100 days) and then exaggerates the creep strain. Similarly to the USBR model, the Burgers model represents continuously increasing creep deformation unlike the Ross, Feng and Modified Maxwell models which predict finite creep deformation at t = ∞. Indeed, for the MM model, the asymptotic infinite limit isJ M M (t → ∞, d c ) = 1 E R (d c )for every crack densityd c (see equation (51)). Moreover, it has been shown that the Burgers model [17] exhibits a serious convergence problem when identifying the creep parameters of a number of testd experimental specimens. Accordingly, it has been concluded [17] that in most cases the MM's model can more accurately predict masonry creep compared with the Feng, Ross, USBR and even basic Burgers rheological models since when the MM's model, USBR and Ross models were applied to test data reported by Brooks and Bingel [9] and Shrive et al. [43], it provided the lowest prediction errors. The main disadvantage of the modified Burgers models is their practical applicability since they involve significant number of parameters -up to 15 parameters for the global version -which need to be estimated according to experimental data. Moreover, they require a postulation or experimental investigation of the damage variables evolution laws. Although these disadvantages, modified Burgers models are more accurate and able to reproduce creep behaviour of masonry structures compared to the basic version of this model. Note that by analogy to the simple At last, as shown in tables (3), (6) and (8), for undamaged and damaged states, Cecchi & Taliercio's extension effective estimates are close to the Cecchi & Tralli's extension ones when the mortar behaves following the MM's model. The absolute error E r between the two models estimates increases with time and damage to reach a maximum of 5%. Recall that the MM's model leads to finite asymptotic limits for global moduli of damaged masonry (see Figures (5)). It is worth noting that the Cecchi & Taliercio estimates are softer than the Cecchi & Tralli ones for undamaged and damaged masonry's states. Accordingly, the Cecchi & Taliercio extension model seems to be more accurate than the Cecchi & Tralli's extension one. In this context, recall that the Cecchi & Tralli's estimate is expected to be upper bound for the exact overall solution [15].

	M A N U S C R I P T 5.2.2. Time-dependent crack density d c
	modified Burgers version (without the Bingham element) proposed by Verstrynge et al., it will be A C C E P T E D
	interesting to postulate a damage evolution law (similar to equation (5) in [46, 47]) -function of
	stress and time -and a damage rate formulation (as equation (6) [46, 47]) -function of stress -so as
	there is correlation between this version of the Burgers model and experimental creep tests provided
	in [34, 46, 47].

Note that disadvantages presented by creep predictions provided by the basic Burgers model for masonry -but not for concrete materials -have motivated some researchers to insert in this model a 'frictional' (or Bingham) element between the spring and the dashpot of the Maxwell element in order to prevent the activation of secondary creep at low stresses. Moreover they have added (static and viscous) damage parameters respectively in first and second Maxwell's components as proposed by Papa & Taliercio

[START_REF] Papa | A visco-damage model for brittle materials under monotonic and sustained stresses[END_REF]

. Other authors simply added viscous damage parameter as done by Verstrynge et al. For this damage variable (D v ), often evolution laws for brittle materials are used, such as for rock salt

[START_REF] Chan | A constitutive model for inelastic flow and damage evolution in solids under triaxial compression[END_REF]

.

  Table-4 provides time evolution of the effective moduli of a damaged masonry

	Rheological model t (days)	dc	Ẽtt (MPa) Ẽnn (MPa)	νtn	νnt	μtn (MPa)
	Modified Maxwell	0	0	164964.	37535.2	0.060 0.013	14400.6
		1	1.5 10 -4	99349.4	20317.3	0.035 0.007	7780.04
		5	7 10 -4	70366.6	13774.4	0.025 0.0049	5270.76
		40	6 10 -3	68119	13290.3	0.024 0.0047	5085.25
		100	1.5 10 -2	67177.8	13088.5	0.024 0.0047	5007.94
		500	7.4 10 -2	61512.	11885.7	0.022 0.00425	4547.1
		1000	0.15	55645.6	10661.	0.02	0.004	4078.01
		1200	0.2	52163.2	9943.73	0.018 0.003	3803.36

Table 4 :

 4 Exension of the Cecchi & Taliercio's model: effective properties of a periodic masonry cell with elastic bricks (E b = 100E m ) and hybrid viscoelastic mortar accounting for time-dependent d c with a constant rate ḋc = 1.5x10 -4 /day.

cell when considering a linear time-dependent crack density (52) with ḋc = 1.5x10 -4 /day. It can

Table 9 :

 9 Cecchi & Taliercio's extension model: effective properties of a periodic masonry cell with elastic bricks (E b = 20E m ) and hybrid viscoelastic mortar accounting for time-dependent d c with a constant rate ḋc = 1.5x10 -4 /day.

	Mortar	t (days)	dc	Ẽtt (MPa) Ẽnn (MPa)	νtn	νnt	μtn (MPa)
	Modified Maxwell model	0	0	117634.	34387.1	0.105 0.030	13263.9
		1	1.5 10 -4	60058.9	13326.6	0.053 0.012	13326.6
		5	7 10 -4	58413.9	12873.	0.052 0.0116	4935.87
		40	6 10 -3	58413.9	12873.	0.052 0.052	4935.87
		100	1.5 10 -2	57720.4	12683.6	0.0516 0.0113	4863.01
		500	7.4 10 -2	53487.4	11550.8	0.0478 0.010	4427.29
		1000	0.15	48995.8	10390.8	0.043	0	3981.39
		1200	0.2	47403.6	9989.47	0.042	0	3827.2

Table 10 :

 10 Cecchi & Taliercio's extension model: effective properties of a periodic masonry cell with elastic bricks

Table 11 :

 11 Elastic and viscous moduli of tested specimens from Pavia Tower[START_REF] Papa | Modeling of the long-term behaviour of historical masonry towers, Chapter 7 in Book: Learning from Failure: Long-term Behaviour of Heavy Masonry Structures[END_REF] E M (MPa) τ M (s) E R (MPa) τ R (s) ν m

	4000	2x10 8	2112	300000 0.29

Table 12 :

 12 Elastic and viscous moduli of hybrid mortar at long-term[47]

	Mortar: rheological model t (days)	d c	Ẽtt (MPa) Ẽnn (MPa)	νtn	νnt	μtn (MPa)
	Burgers	0	0	66946.1	19569.9	0.105 0.030	7548.55
		1	1.5 10 -4	59434.2	16195.5	0.093 0.025	6236.67
		5	7 10 -4	48204.6	11926.1	0.075 0.0118	4583.06
		40	6 10 -3	18051.84	3580.9	0.028 0.0056	1370.53
		100	1.5 10 -2	8599.9	1606.15	0.0135 0.002	614.14
		500	7.4 10 -2	1754.31	314.32	0.0027	0	120.11
		1000	0.15	797.30	142.	0.0012	0	54.27

Table 13 :

 13 Cecchi & Taliercio's extension model: effective properties of a periodic masonry cell with elastic bricks (E b = 40E m ) and long-term characteristic time viscoelastic mortar (similar to those of tested specimens from the Pavia Tower[START_REF] Papa | Modeling of the long-term behaviour of historical masonry towers, Chapter 7 in Book: Learning from Failure: Long-term Behaviour of Heavy Masonry Structures[END_REF]) accounting for time-dependent d c with a constant rate ḋc = 1.5x10 -4 /day.

	Mortar	t (days)	dc	Ẽtt (MPa) Ẽnn (MPa)	νtn	νnt	μtn (MPa)
	Burgers model	0	0	75875.3	22365.6	0.137 0.040	8177.6
		1	1.5 10 -4	60718.9	15880.3	0.11 0.028	5790.89
		5	7 10 -4	43045.6	9950.05	0.078 0.018	3619.56
		40 100 500 1000 1200	M 6 10 -3 37547.5 1.5 10 -2 36856.9 7.4 10 -2 32678.9 0.15 28339.1 A N U S C R I P T 8376.39 0.068 0.015 3045.14 8186.47 0.066 0.0148 2975.87 7071.83 0.06 0.013 2569.51 5973.14 0.05 0.01 2169.34 0.2 26831. 5604.61 0.038 0.01 2035.19
	A C C E P T E D

Table 14 :

 14 Cecchi & Taliercio's extension model: effective properties of a periodic masonry cell with elastic bricks (E b = 40E m ) and long-term characteristic time viscoelastic hybrid mortar [47] accounting for time-dependent d c with a constant rate ḋc = 1.5x10 -4 /day.

	Mortar	t (days)	d c	Ẽtt (MPa) Ẽnn (MPa)	νtn	νnt	μtn (MPa)
	Modified Maxwell model	0	0	117202.	34174.6	0.096 0.028	13392.3
		1	1.5 10 -4	117169.	34158.7	0.096 0.028	13386.
		5	7 10 -4	117036.	34095.	0.095 0.027	13360.8
		40	6 10 -3	115886.	33545.1	0.094 0.027	13143.3
		100	1.5 10 -2	113942.	32632.	0.093 0.026	12782.2
		500	7.4 10 -2	101894.	27378.5	0.083 0.022	10708.5
		1000	0.15	88940.	22408.9	0.072 0.018	8752.45
		1200	0.2	84363.4	20798.	0.069 0.017	8119.59

Table 15 :

 15 Cecchi & Taliercio's extension model: effective properties of a periodic masonry cell with elastic bricks (E b = 40E m ) and long-term characteristic time viscoelastic hybrid mortar [47] accounting for time-dependent d c with a constant rate ḋc = 1.5x10 -4 /day.

Preprint submitted to International Journal of Solids and StructuresDecember 1, 2015

e v b B + C b

A C C E P T E D M

A N U S C R I P T be remarked that the effective moduli decrease quickly for t ≥ 5 days. Beyond this limit, these moduli decrease slightly with time even if d c increases. As proven previously, this table shows that effective properties of damaged masonry following the MM's model do not vanish for d c ≤ 0.2.

For the considered mortar (see Table -1), similar evaluation of the effective properties of damaged masonry with a linear time-dependent crack density were carried out for the proposed extension to the Burgers model. This result showing vanishing effective moduli within about 20 days is not reported here for brevity. Owing to results depicted on Table-13, similar trends are obtained for mortar with long-term characteristic times (see depicting elastic and viscous properties of tested specimens from the Pavia Tower [START_REF] Papa | Modeling of the long-term behaviour of historical masonry towers, Chapter 7 in Book: Learning from Failure: Long-term Behaviour of Heavy Masonry Structures[END_REF]) but with effective properties tending towards 0 within higher duration (beyond 500 days). These trends change (see when considering mortar with too high (τ M ≥ 10 7 (s)) values of the Maxwell's relaxation time τ M (see as it is the case of hybrid mortar with long-term characteristic times [47]. Indeed, in this case, the masonry's Burgers predictions, even though lower than MM's overall estimates, do not vanish after 1200 days with a crack density d c = 0.2 (see . This demonstrates the relevance of the proposed Burgers model for the prediction of the long-term creep with very high values of the Maxwell's relaxation time τ M ≥ 10 8 (s). Tables-15 and -16 related to mortars with long-term characteristic times with properties of respectively hybrid mortar [47] or similar to tested specimens from the Pavia Tower [START_REF] Papa | Modeling of the long-term behaviour of historical masonry towers, Chapter 7 in Book: Learning from Failure: Long-term Behaviour of Heavy Masonry Structures[END_REF] confirm trends previously mentioned for the MM's model when dealing with mortar's properties gathered by means of short term compressive tests on masonry wallets [START_REF] Ignoul | Creep behaviour of masonry structures -failure prediction based on a rheological model and laboratory tests[END_REF]. The trends of the MM's model are then independent from the values of the Maxwell's relaxation time, short or long-term gathered properties of the mortar. At last, it is worth noting that similar stress and time dependent evolution law and damage rate to those proposed in [START_REF] Papa | A visco-damage model for brittle materials under monotonic and sustained stresses[END_REF]46,47] should be considered similarly in a future work.

Computation of the proposed model to the case of a compressed masonry panel

In this part, only the Cecchi & Taliercio extension estimates will be considered. In order to evaluate the relevance of the proposed extension model, we treat the case of a masonry panel of dimensions L = 1560 mm (length) and H = 1040 mm (height) studied in [START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF] subjected to boundary conditions BC-1 with three distributed loads at the top and two lateral edges (see Figure ( As the arrangement of the bricks is regular, the effective behaviour of the panel is assumed to be

well estimated by that of a periodic cell (see Figure 1-a). The panel can then be modeled as a homogeneous material which properties coincide with those of the equivalent material MHE-2 (see Figure [START_REF] Borderie | Crack closure effect in continuum damage mechanics: numerical implementation[END_REF] -c). The mortar's data used to compute this problem are provided in Table-1.

To validate this computation, it was checked that, under boundary condition BC-1, stress field Quantitatively, similarly to boundary conditions BC-1, stress components are almost coincident under BC-2 for t = 0 and t = 1000 days independently of the damage state: d c = 0 or d c = 0.15. This quantitative aspect is confirmed by Figures [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF] showing the evolutions of stress components along the x axis located at the middle height of the wall (x = H/2). On the other hand, the absolute values of the strain fields components increase with the increase of damage (Figures [START_REF] Cecchi | A comparison between numerical and analytical homogenized models for visco-elastic brickwork[END_REF]). Moreover, under BC-1, strain values are lower than those attained under BC-2. Indeed, for a same damaged state d c = 0.15, ratios between maximum absolute values of ε yy reached under BC-2 and BC-1 at time

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A N U S C R I P T t = 1000 days is about 8. This ratio is around 2 for the strain ε xy . Even lower than ε yy , the strain ε xy is more sensitive to the boundary condition BC-2. Quantitatively, for the considered mortar with sort-term gathered data, while stresses do not change throughout the wall between the two states (t = 0, d c = 0) and (t = 1000 days, d c = 0.15) for the MM's model, this field is amplified according to this version of the Burgers model under the boundary conditions BC-2. At last, under these boundary conditions, it is observed that the MM's model predicts small strains unlike those provided by this version of the Burgers model. For the later, there are great differences between the magnitudes of strain fields at undamaged (t = 0, d c = 0) and damaged (t = 100 days, d c = 0.1) states. Such results motivate to avoid modeling traditional mortars with short-term gathered properties using the Burgers model since it leads to vanishing masonry's stiffnesses and large strains increasing thus the risk of failure. These trends for the Burgers model are not valid if the Maxwell's relaxation time of the mortar is too high (τ M ≥ 10 7 (s)) as it is the case f hybrid mortar with long-term characteristic times (Table -12). In the later case, the Burgers model is expected to provide local fields predictions similar to the MM's estimates. The boundary conditions BC-1 are preferable to BC-2 since the later increases the stress and strain levels throughout the wall. Owing to Figures [START_REF] Brooks | Composite modeling of masonry deformation[END_REF]

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stiffness at short and medium terms and exaggerated local strains. At last, this study should be improved on by accounting for creep of bricks which often occurs under high temperatures. More complex loadings and/or a masonry structure (an assembly of walls) could also be considered and simulated using the proposed model. It could be more rigorous to account for interactions between microcracks using the Mori-Tanaka or Ponte-Castañeda & Willis (PCW) homogenization models, to include crack density evolution law and damage rate similar to those proposed in [START_REF] Papa | A visco-damage model for brittle materials under monotonic and sustained stresses[END_REF]46,[START_REF] Papa | Modeling of the long-term behaviour of historical masonry towers, Chapter 7 in Book: Learning from Failure: Long-term Behaviour of Heavy Masonry Structures[END_REF]47].

These perspectives will be investigated in future works.

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A. Burgers model (Bu): microcracked state

An approximation of the Burgers' type for the creep function of a microcracked non-aging linear viscoelastic mortar with a matrix following the Burgers' rheology is

where

where the identified parameters of the Burgers model at the vicinity of 0 and ∞ are respectively the M µ e M (3η

If spherical and deviatoric relaxtation times are equal (τ s i = τ d i = τ i where i = M or K), then the effective creep function (53) reads

where

Recall that relaxation times at undamaged state are the following: