Sylvain Tourancheau
email: sylvain.tourancheau@univ-nantes.fr

Patrick Le Callet

Dominique Barba

PREDICTION OF PERCEIVED QUALITY DIFFERENCES BETWEEN CRT AND LCD DISPLAYS BASED ON MOTION BLUR

In this paper, a mathematical model of LCD motion blur is used to measure the magnitude of perceived blur in a sequence displayed on LCD. Subjective quality assessment tests on CRT and LCD displays are described. A difference of visual quality between the two types of displays is observed for sequences with movements. This loss of quality can be predicted from the importance of motion blur measured in the sequences. An estimation of LCD perceived quality can thus be made from CRT perceived quality. Technical solutions to LCD motion blur problem can be evaluated by this mean.

INTRODUCTION

High definition television (HDTV) is soon to be introduced in Europe. With the new resolutions of pictures, 1920×1080 in interlaced mode (1080i) and 1280×720 in progressive mode (720p), observers can reduce viewing distance down to 3H (where H is the screen's height) to create a cinemalike experience with immediacy, presence and impact [START_REF] Fujio | Future broadcasting and high definition television[END_REF]. Of course, this increase in resolution leads to an increase in display size which was not possible with standard television. As a result CRT displays, which become heavy and bulky with increased screen size, are doomed to disappear. New display technologies (LCD, PDP) are improving and will soon replace old CRT technology.

It seems that LCD is more likely to succeed because of problems attached to plasma large resolution displays. However, it has been shown that subjective quality of a sequence displayed on LCD is globally lower than subjective quality of the same sequence displayed on CRT [START_REF] Itu | Report on results of comparative subjective picture quality assessment test between CRT and LCD[END_REF]. Among all the defects mentioned by observers, motion blur seems to be the most annoying one. This appears in sequences with rapid movements. Other shortcomings have been enumerated such as colour differences, degradations in dark areas and de-interlacing artefacts for interlaced sequences.

The objective of this study is to quantify the impact of LCD motion blur on the perceived quality on LCD with respect to the perceived quality on CRT (which is considered to be the reference here). Subjective quality assessment tests on the two types of displays are described. Results show a loss of quality between CRT and LCD. Then, a mathematical model of motion blur is used to measure the magnitude of perceived blur on each sequence. A relationship between importance of blur and quality loss can be highlighted, which enables the prediction of this quality loss.

SUBJECTIVE QUALITY ASSESSMENT

Material

In order to develop and evaluate new HDTV quality metrics, psychovisual tests have been designed [START_REF] Péchard | Towards video quality metrics for HDTV[END_REF]. Twelve sequences in 1080i format have been used. Each of them contain 250 frames which correspond to 10 second duration. Each reference (uncompressed) sequence has been distorted with H.264 compression standard at seven different bitrates, to cover the entire quality range. Tests have been conducted both on a CRT and on a LCD display.

In this paper, quality difference introduced by LCD motion blur is studied. For that, only the eight sequences with significant movements, and for which motion blur is the main perceived defect when displayed on LCD, have been selected. Furthermore, only quality scores of reference uncompressed sequences are considered.

Tests have been performed in a specific showroom. Lighting conditions and display parameters have been precisely measured and adjusted according to BT.500-11 and BT.710-4 ITU recommendations. The HDTV displays used were a JVC DT-V 1910CG and a Philips T370 HW01 which both can display 1080i sequences. Viewing distance was set to 3H, where H is the height of the screen.

Observers

Observers were mostly male students in their mid twenties. All are familiar with standard television and cinema but not with HDTV. Every candidate is first checked for color blindness with Ishihara test and for acuity with Monoyer's plates. People with at least one error in Ishihara's test or less than 9/10 in Monoyer's test are rejected. 21 people took part in these tests in the CRT session and 19 in the LCD session.

Protocol

The assessment method required here should allow observers to precisely construct their judgment. As very little quality differences must be detected, the method must force the quality discrimination. A well known stable method for this purpose is the SAMVIQ method [START_REF] Ebu | SAMVIQ -Subjective assessment methodology for video quality[END_REF], developed by France Telecom R&D and standardized by the European Broadcasting Union (EBU). Observers compare sequences (seven distorted sequences and one hidden reference) both between them and with the explicit reference. Notation scale is continuous, each score can take a value between 0 and 100.

SAMVIQ is a multi stimuli continuous quality scale protocol. It provides a precise and reliable [START_REF] Blin | New quality evaluation method suited to multimedia context: Samviq[END_REF] measure of the subjective video quality which can be compared directly to the reference. It is important to note that this reference may or may not be the original video signal. As the observers can directly compare the impaired sequences among themselves and against the reference, they can grade them accordingly. This feature permits a high degree of resolution in the grades given to the system. Moreover, observers have a random access to the sequences, which permits to choose exactly the sequence they want to assess. This allows them to precisely build their assessment opinion. This is particularly interesting in this context where very little quality differences have to be identified.

The consistency of the individual scores is evaluated after the tests have been completed by all the subjects. It is done by applying a suitable "rejection" technique. This is a process in which all scores from a particular subject are omitted from the analysis of data. Following the application of the rejection process, 15 valid subjects should be retained at minimum.

Results

Mean opinion scores (MOS) of observers for the eight reference sequences are shown in Table 1. ∆MOS is the difference of MOS from CRT and LCD :

∆MOS = MOS CRT -MOS LCD (1)
Thus, subjective quality measured on LCD is lower than the one measured on CRT. It can be observed that ∆MOS is varying strongly with the sequences, as a result CRT MOS and LCD MOS are not well correlated. Correlation coefficient and root mean square error between the two MOS sets are given below :

CC(MOS CRT, MOS LCD) = 0.751, (2) RM SE(MOS CRT, MOS LCD) = 8.58. (3)
To predict perceived quality on LCD from perceived quality on CRT isn't an easy task. Here the hypothesis is made that the quality difference ∆MOS depends on LCD motion blur. In the next part, LCD motion blur is described and measured on each sequences using a mathematical model.

Séquence

LCD MOTION BLUR

Causes

Despite recent improvements to LCD technology such as response time compensation [START_REF] Mccartney | A liquid crystal display response time compensation feature integrated into an LCD panel timing controller[END_REF], LCD motion blur remains very annoying for sequences with rapid movements. In fact, even if the response time of a liquid crystal matrix was reduced to zero, motion blur would still appear. This is due to sample-and-hold behaviour of the display : the light intensity is sustained on the screen for the duration of the frame, whereas on CRT light intensity is a pulse which fades over the frame duration (cf. Figure 1). LCD displays are so called hold-type displays. The main difference happens when the eye of the observer is tracking a moving object on the screen: for a given frame, the picture is still on the screen while the eye is still moving slightly anticipating the movement of the object. Edges of this object are displaced on the retina resulting in a blur [START_REF] Kurita | Moving picture quality improvement for hold-type AM-LCDs[END_REF].

Measurements

Psychophysic experiments have been conducted in order to measure blur width as a function of motion speed [START_REF] Péchard | Towards video quality metrics for HDTV[END_REF]. Since the perception of motion blur is directly related to the tracking of a moving object, measurements of blur must be done during tracking task. As a result, designed test must permit the measure of the blur while perceiving it.

Experiments consist in displaying a periodical structure of bars moving on a black background at a constant speed. The scrolling is continuous. Due to motion blur, edges of the bars don't appear sharp like shown in Figure 2-a but spread in the gap between two bars like in Figure 2-b. During the test, the observer has to modify the space between the bars until the two blurred areas begin to blend together. The space between two bars for which two blurs are just merging gives the measure of the motion blur width.

Results and model

In the explored range of speed, the width of blur is proportional to motion velocity. There is no significant difference between horizontal and vertical movements. A linear fitting (with a 0.99957 correlation coefficient) of the results leads to the following relationship between the velocity V of the motion (in unit of length by frames) and the blur width W (in unit of length):

W = 1.039V. (4)
These results agree with the theoretical model developed by Pan et al. [START_REF] Hao Pan | LCD motion blur modeling and analysis[END_REF] which is as following:

W = aV. (5)
In this model, the parameter a depends on the type of temporal reconstruction function of the display. Thus, the blur width due to the displacement of an edge can be measured for different shapes of response. For example, when using a sinusoidal response, the model gives a = 1.044.

Solutions

The blur width depends on the temporal reconstruction function of the display. A material solution to reduce it is to modify the temporal aperture of the display in order to reduce the parameter a. Different methods have been proposed, such as backlight flashing [START_REF] Kurita | Moving picture quality improvement for hold-type AM-LCDs[END_REF][START_REF] Feng | LCD motion-blur analysis, perception, and reduction using synchronized backlight flashing[END_REF], frame rate doubling [START_REF] Kurita | Moving picture quality improvement for hold-type AM-LCDs[END_REF][START_REF] Hong | Motion image enhancement of LCDs[END_REF], black data insertion [START_REF] Hong | Motion image enhancement of LCDs[END_REF] and motion-compensated inverse filtering [START_REF] Michiel | Motion blur reduction for liquid crystal displays: motion-compensated inverse filtering[END_REF].

All these methods lead to different reconstruction functions. Pan's model permits to determine parameter a, and then the blur width W , for all of these functions. Our objective here is to determine the influence of the perceived motion blur on the difference of quality between CRT and LCD. If ∆MOS could be predicted from the importance of perceived motion blur, technical solutions to LCD motion blur problem could be evaluated. In the next part, a relation between LCD motion blur and ∆MOS is highlighted.

QUALITY DIFFERENCE PREDICTION

Figure 3 illustrates the work done in this paper. Since subjective assessment gives a value of ∆MOS for each sequence, an objective model is developed in order to predict this quality loss ∆MOS p from the magnitude of blur in the sequence. Prediction is made in four steps. First, a motion estimation is performed on the sequence. This leads to the construction of tubes which are the sets of blocks positions along the direction of motion. Each tube is classified according to his spatial content. Second, tubes categorized as textures and contours are selected and an average motion vector is computed from all the vectors of these selected tubes. Third, an average magnitude of motion blur is deducted from Pan's model. Fourth, ∆MOS p is computed from a prediction model.

Motion estimation

As sequences are interlaced, motion estimation is made on each field. A block 16×8 of an odd (resp. even) field is simultaneously compared to blocks of the two previous and the two next odd (resp. even) fields. The position which minimizes the mean square error is chosen. Thus, a vector is obtained for each 16×8 block of each group of five consecutive odd (resp.even) fields.

For each group of five frames, the motion vectors of even and odd fields are then merged in order to obtain a vector for each 16×16 block. These blocks which are followed along five frames are so called spatio-temporal tubes. Each tubes are classified into categories: contours, textures or uniform areas.

Average motion blur computation

Since motion blur is only visible with a sufficient contrast [START_REF] Laird | Spatio-velocity CSF as a function of retinal velocity using unstabilized stimuli[END_REF], tubes classified contours and textures are selected. For each group of five frames, a spatial vector is computed averaging the vectors of selected tubes. These spatial vectors are then temporally averaged along the sequence. A global motion vector is obtained for each sequence.

The norm V of this global vector is finally used to compute the width of perceived motion blur according to Pan's model (cf. Equation 4). This value W is an indicator of the magnitude of perceived blur along the sequence.

Prediction model

A indicator of the quantity of perceived motion blur has been computed for each sequence. The main objective of this work is to determine the relation between the LCD motion blur and the loss of quality observed between CRT and LCD displays. A non linear function ∆MOS p = f (W ) has been constructed in order to predict the quality difference ∆MOS from the average blur quantity W . It has been assumed that this function has the following shape. In the first part, the magnitude of motion blur is too small to influence perceived quality. In the second one, the quality loss increases with magnitude of motion blur. Finally, in the third part, the quality difference saturates despite the increase of perceived blur. This saturation may be due to contextual effects such as limited assessment scale and presence of quite distorted sequences during quality assessment.

Figure 4 presents ∆MOS as a function of the quantity of motion blur for each of the eight sequences. The prediction model ∆MOS p = f (W ) is represented by the dashed line. This model can be used to predict the quality loss ∆MOS between CRT and LCD from the average magnitude of blur measured on a sequence.

RESULTS

The whole objective model presented in Figure 3 enables the prediction of the difference of perceived quality between CRT and LCD for sequences with significant movements. This difference depends on the average motion blur measured on a sequence. It has been designed using sequences for which motion blur is the main perceived defect when displayed on LCD display. An estimation of the subjective quality scores on LCD from the subjective quality scores on CRT can be made using the following relation:

MOS LCD est = MOS CRT -∆MOS p . (6)
The quality of the model can be measured by the linear correlation and the root mean square error between estimated LCD scores and actual LCD scores: CC(MOS LCD, MOS LCD est ) = 0.953, (7) RM SE(MOS LCD, MOS LCD est ) = 1.30.

(

) 8
These values can be compared with those from Equations 2 and 3. Estimated LCD quality scores are well correlated with actual scores and mean square error is quite small.

CONCLUSION

Subjective assessment quality tests have highlighted an important loss of quality between perceived quality on CRT and perceived quality on LCD for sequences with significant movements. An average magnitude of motion blur has been measured for each sequence. A model has been constructed to predict the quality difference between the two types of displays as a function of motion blur. The perceived quality on LCD has been estimated from the perceived quality of CRT.

Since quantity of perceived blur depends on LCD temporal aperture, the prediction model permits to evaluate the solutions to LCD motion blur defect. However, other aspects must be considered in order to carry out a fine prediction of perceived quality difference between CRT and LCD. Colour differences, gamma and luminosity range for example lead to differences of perception. In future, they should be incorporated into our model to finely characterize quality differences between the two types of displays.

Fig. 1 .

 1 Fig. 1. Temporal evolution of a pixel's intensity on a CRT display (a), on an LCD display (b). (from [8])

Fig. 2 .

 2 Fig. 2. Displayed (a) and perceived (b) images for a horizontal movement from left to right.

Fig. 3 .

 3 Fig. 3. Objective prediction of quality loss ∆MOS.

Fig. 4 .

 4 Fig. 4. ∆MOS as a function of motion blur and prediction model.

Table 1 .

 1 Mean opinion scores by sequences and displays.

		MOS CRT MOS LCD ∆MOS
	PARKRUN	86.28	81.32	4.96
	SHIELDS	84.68	77.95	6.73
	STOCKHOLM	83.56	81.74	1.82
	CONCERT	80.33	72.05	8.29
	FOOT	83.56	73.05	10.51
	VOILE	83.83	73.09	10.74
	SHOW	81.15	69.28	11.87
	CREDITS	82.7	73.76	8.94

ACKNOWLEDGMENT

This work is supported by HD4U European project. The aim of HD4U is to study best conditions to deploy HDTV in Europe. Several manufacturers from consumer devices industries (Philips, Thomson, etc.) and broadcasters (TF1, Euro1080) are involved. Authors would like to thank SVT for the open HDTV sequences, and Thomson and Philips for the HDTV screens. They would also like to thank Mathieu Carnec for the development of the motion estimation algorithm and Arnaud Tirel for his assistance in performing the experiments described in the paper.


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

Rigorous Estimates for Effective Creep-coefficients of Micro-cracked Masonry Accounting for Cracks Interactions

Keywords: masonry, creep, microcracks, finite elements, c ontinuum micromechanics, Ponte-Castañeda & Willis model

Rigorous Estimates for Effective Creep-coefficients of Microcracked Masonry Accounting for Cracks Interactions

Introduction

The rich heritage of historic masonry buildings in different parts in the world bears the idea that masonry is one of the oldest building materials used by man. Moreover, now days, it is still the most widely used building material. Several models have been developed and presented in the literature for studying and predicting the behavior of masonry structures. Depending on the level of accuracy and simplicity required, either macro-(classical notension models [START_REF] Marfia | Numerical procedure for elastoplastic no-tension model[END_REF][START_REF] Lourenço | Continuum model for masonry: parameter estimation and validation[END_REF][START_REF] Pasquale | New trends in the analysis of masonry structures[END_REF][START_REF] Como | Statics of Historic Masonry Constructions[END_REF][START_REF] Angelillo | Constitutive relations for no-tension materials[END_REF][START_REF] Briccola | Analysis of 3D linear elastic masonry-like structures through the API of a finite element software[END_REF][START_REF] Briccola | Analysis of 3D notension masonry-like walls[END_REF][START_REF] Bruggi | Analysis of no-tension structures under monotonic loading through an energy-based method[END_REF] or micromodeling [START_REF] Sulem | A continuum model for periodic two-dimensional block structures[END_REF][START_REF] Masiani | Cosserat and Cauchy materials as continuum models of brick masonry[END_REF][START_REF] Anthoine | Derivation of in-plane elastic characteristics of masonry through homogenization theory[END_REF][START_REF] Ushaksarei | Failure criterion for structural masonry based on critical plane approach[END_REF][START_REF] Luciano | Homogenisation technique and damage model for old masonry material[END_REF] strategies can be used for this purpose. On the other hand, some authors have combined homogenization techniques with a continuum damage mechanics approach [START_REF] Zucchini | A coupled homogenizationdamage model for masonry cracking[END_REF][START_REF] Pegon | Numerical strategies for solving continuum damage problems with softening: application to the homogenization of masonry[END_REF][START_REF] Chengqing | Derivation of 3D masonry properties using numerical homogenization technique[END_REF]. Other authors [START_REF] Alpa | Microstructural model for dry block masonry walls with in-plane loading[END_REF][START_REF] De Buhan | A homogenisation approach to the ultimate strength of brick masonry[END_REF], have defined suitably macroscopic yield failure surfaces. Macroapproaches obviously require a preliminary mechanical characterization of the model, based on experimental laboratory or in situ tests [START_REF] Gabor | Analysis of the inplane shear behaviour of FRP reinforced hollow brick masonry walls[END_REF][START_REF] Gabor | Modelling approaches of the in-plane shear behaviour of unreinforced and FRP strengthened masonry panels[END_REF]. In studies based on micro-analysis, two main approaches have been used: the simplified approach, which is the more refined, and the detailed micromodeling approach. Simplified methods consist in modeling the bricks, mortar and interface separately by adopting suitable constitutive laws for each component. This approach gives highly accurate results, especially at local level. Several authors [START_REF] Lotfi | Interface model applied to fracture of masonry structures[END_REF][START_REF] Lourenço | A multi-surface interface model for the analysis of masonry structures[END_REF][START_REF] Pelissou | Asymptotic modeling of quasibrittle interfaces[END_REF][START_REF] Pegon | Numerical modeling of stone-block monumental structures[END_REF][START_REF] Giambanco | A cohesive interface model for the structural mechanics of block masonry[END_REF][START_REF] Rekik | Homogenization methods for interface modeling in damaged masonry[END_REF] have established that the interface elements reflect the main interactions occurring between bricks and mortar. A simplified micromodel is an intermediate approach, where the properties of the mortar and the interface are lumped into a common element, while expanded elements are used to model the brick units. However this model reduces the computational cost of the analysis, some accuracy is obviously lost. Although creep effects in masonry are far from being negligible, in the literature, little attention [START_REF] Shrive | Letter to the Editor[END_REF][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[END_REF] is devoted to the prediction of the macroscopic creep behavior of masonry under sustained loads. In this context, [START_REF] Brooks | Composite modeling of masonry deformation[END_REF][START_REF] Cecchi | A comparison between numerical and analytical homogenized models for viscoelastic brickwork[END_REF][START_REF] Cecchi | A homogenized viscoelastic model for masonry structures[END_REF][START_REF] Taliercio | Closed-form expressions for the macroscopic in-plane elastic and creep coefficients of brick masonry[END_REF] proposed models to predict creep coefficients according to the properties of each masonry constituent. These models are based on analytical or numerical homogenization using finite elements method (FEM) in order to deduce the macroscopic creep of undamaged (without cracks) masonry. Moreover, most of these studies, except some works such as Cecchi & Taliercio's paper [START_REF] Cecchi | A comparison between numerical and analytical homogenized models for viscoelastic brickwork[END_REF], neglects the brickwork creep which can be argued by the fact that most of the creep effects take place in the bed joints, which are responsible for 60 to 80 % of masonry creep even if their volume fraction represent only 20 % of the brickwork [START_REF] Sayed-Ahmed | Creep deformation of clay masonry structures: a parametric study[END_REF]. Nevertheless, the creep deformation of the brick layers may be far from being negligible. One of the objects of this work is to account for creep of each masonry constituents (brick units and mortar joints). Concerning the creep behavior 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]. Besides, it is well known that cracking is frequently concentrated in mortar joints in case of historical masonry and, in particular, in case of in-plane shear actions with low compression levels. This work takes into account presence of microcracks in the viscoelastic mortar joints. It is interesting to note that in the literature, 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] based for example on a coupling between continuum damage mechanics and viscoelasticity through the generalized Kelvin Voigt model [START_REF] Nedjar | An approach to the modeling of viscoelastic damage: application to the long-term creep of gypsum rock materials[END_REF] in order 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, Dormieux et al. [START_REF] Nguyen | Propagation de fissures et endommagement par microfissures dans un milieu viscoélastique linéaire non vieillissant[END_REF][START_REF] Nguyen | Crack propagation in viscoelastic structures: theoretical and numerical analyses[END_REF], the effective behavior 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 Eshelby-based 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 Burgers model. In Choi et al. [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 [START_REF] Choi | Rheological modelling of masonry creep[END_REF], only the Modified Maxwell model (a parallel combination of the Maxwell model and a spring) is adopted in this paper to describe the creep of the mortar joints. Similarly to the mortar and for the sake of simplicity, bricks are assumed to follow the Generalized Maxwell (GM) model. Now, for microcracked mortar, by contrary to recent works Rekik et al. [START_REF] Rekik | Multi-level modeling of microcracked viscoelastic masonry[END_REF][START_REF] Rekik | Numerical homogenization model for effective creep properties of microcracked masonry[END_REF][START_REF] Nguyen | Approche multi-échelles pour des maçonneries viscoélastiques[END_REF] and Nguyen et al. [START_REF] Nguyen | Propagation de fissures et endommagement par microfissures dans un milieu viscoélastique linéaire non vieillissant[END_REF][START_REF] Nguyen | A Burger Model for the Effective Behavior of a Microcracked Viscoelastic Solid[END_REF] adopting respectively the dilute and dilute/Mori-Tanaka schemes in order to estimate the effective creep function of the mortar, it is proposed in this paper to adopt a rigorous homogenization model -the Ponte-Castañeda & Willis scheme [START_REF] Castañeda | The effect of spatial distribution on the effective behavior of composite materials and cracked media[END_REF] -accounting for cracks interactions. Indeed, it is worth noting that even the dilute scheme is useful for dilute concentrations of cracks, it has been demonstrated in Dormieux and Kondo [START_REF] Dormieux | Stress -based estimates and bounds of effective elastic properties: the case of cracked media with unilateral effects[END_REF][START_REF] Dormieux | MicroporoMechanics[END_REF] that its estimates coincide with those derived from the Mori-Tanaka (MT) scheme [START_REF] Bornert | Homogénéisation en mécanique des matériaux, Tome 1 : Matériaux aléatoires élastiques et milieux périodiques[END_REF]. Moreover, they are close to the Ponte-CastaedaWillis (PCW) estimates for crack density parameter dc ≤ 0.15 [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF]. However the MT predictions violate rigorous bounds of the Hashin-Shtrikmann (HS) [START_REF] Bornert | Homogénéisation en mécanique des matériaux, Tome 1 : Matériaux aléatoires élastiques et milieux périodiques[END_REF] type for composites with isotropic distributions of randomly oriented cracks [START_REF] Castañeda | The effect of spatial distribution on the effective behavior of composite materials and cracked media[END_REF]. It is worth noting that the adopted parameter dc allows the measurement of the effect of cracks on the bulk and shear moduli of the considered microcraked material. According to [START_REF] Dormieux | MicroporoMechanics[END_REF], dc is the relevant parameter with regard to the elastic energy as the internal variable governing the effect of damage evolution instead of the crack surface or its length. In a second step, it is necessary to determine the global behavior of the masonry with viscoelastic bricks and microcracked viscoelastic mortar. This step relies on homogenization technique based on the assumption that a statistically homogeneous medium represented by a 'representative volume element' RVE, or a material with periodic structure represented by a 'repeated unit cell' RUC, can be defined. The RVE, firstly used by Hill [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF], corresponds to a microstructural subregion which is representative of the entire sub scale. Generally, the choice or the modeling of the RVE or RUC affects the homogenization results. Indeed the RVE depends on the investigated morphological or physical property, the contrast in the properties of the constituents, and their volume fractions. For given wanted precision and number of realizations, it is possible to provide a minimal volume size for the computation of effective properties. For composites, the RVE is the smallest material volume element which must contain a sufficient number of inclusions, which makes the effective moduli independent of assumed homogeneous forces or displacements on the RVE boundary. For regular masonry, as it is the case in the present work, it is useful to consider the pattern which is repeated periodically inside the masonry structure and to apply numerical periodic homogenization approach using FEM. At last, the obtained orthotropic effective properties are used to compute the behavior of a masonry wall subjected to compressive loadings [START_REF] De Buhan | A homogenisation approach to the ultimate strength of brick masonry[END_REF][START_REF] Rekik | Multi-level modeling of microcracked viscoelastic masonry[END_REF][START_REF] Rekik | Numerical homogenization model for effective creep properties of microcracked masonry[END_REF]. The layout of the paper is as follows. The first step (see Fig. 1) of the proposed model is detailed in Section 2.

A recall of the PCW estimates for effective moduli of a microcracked media with isotropic distributions of randomly oriented (closed or opened) cracks is provided. After that the identification procedure of the MM's six parameters for the microcracked mortar at short and long terms allowing the expression of its creep function is presented. The last part in Section 2 shows an illustrative example for the followed methodology in step 1. Section 3 deals with the second step (Fig. 1) of the proposed model based on numerical periodic homogenization of regular masonry accounting for creep of both brick units and mortar joints (bricks are assumed to be safe unlike mortar). Section 4 validates the proposed numerical model by comparison with analytical expression for local displacement at the top of a compressed masonry wall with rigid bricks and 'standard' viscoelastic mortar joints. At last, the relevance of the proposed model is illustrated by investigating the case of a masonry wall subjected to compressive loadings.

Instantaneous global properties of viscoelastic microcracked mortar (step -1)

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 symbolic space, the apparent behavior 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] Rekik | Numerical homogenization model for effective creep properties of microcracked masonry[END_REF]. Assuming again that the displacement jump field depends linearly on the macroscopic stress, it is possible to define an effective linear behavior for the microcracked mortar in the symbolic space using one of the available homogenization schemes (dilute, Mori-Tanaka, self-consistent, PCW or others [START_REF] Bornert | Homogénéisation en mécanique des matériaux, Tome 1 : Matériaux aléatoires élastiques et milieux périodiques[END_REF]). To determine the global behavior 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 behavior 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 behavior of the microcracked viscoelastic mortar could be expressed in the real space time. In this work, in the symbolic space, the Ponte-Castañeda and Willis homogenization scheme (PCW) [START_REF] Castañeda | The effect of spatial distribution on the effective behavior of composite materials and cracked media[END_REF] is adopted for its ability of properly accounting for interactions and influence of spatial distribution of microcracks by contrary to the dilute scheme.

Ponte-Castañeda & Willis model

We distinguish the case of closed from that of opened cracks as shown in the following

Closed cracks

We assume that the mortar contains a set of closed cracks with unit normal n and that the contact between the two crack lips is frictionless. In contrast to open cracks, it is necessary to take into account the fact that a compressive normal stress can be transmitted through the contact of the crack lips, while the shear stress in the crack plane remains zero. Adopting a continuum micromechanics approach, the closed crack is represented by a flat ellipsoidal inhomogeneity filled with an isotropic fictitious material having a zero shear modulus, µ c = 0 and a non-zero bulk modulus, k c ≠ 0. Accordingly C c = 3k c J where J is the spherical fourth-order projector. For randomly oriented closed cracks requiring an averaging process on crack orientations, the effective symbolic bulk and shear moduli read

Where is the symbolic Poisson's ratio, and are respectively the matrix's symbolic bulk and shear moduli.

Open cracks

𝐾

(2) Where K is the deviatoric fourth-order projector defined by K = I -J, effective bulk and shear moduli are given in the symbolic space by

Opening-closure transition criteria

A crack is opened means that its lips are free of stress which can be expressed as follows σxy = σyy = 0 :

[uy] ≥0 (4) where [uy] is the displacement jump across the upper and lower crack surfaces. The last condition implies that the asymptotic stress at infinity is a tensile stress 0 otherwise crack closure occurs. Accordingly, a frictionless closed crack, which is a model of unilateral contact, can be defined as follows σxy = 0, σyy ≤ 0 :

[uy] = 0 (5) The transition between the two regimes is defined by the relations σxy = σyy = 0 :

[uy] = 0 (6)

Identification procedure of existent rheological model at short and long times

Whatever the state of the crack (open or closed), the following described identification procedure remains available. Only mortar properties at the macroscopic scale change owing to the state of the crack and the adopted homogenization scheme accounting for the presence of cracks.

The series expansion of the PCW estimates for the mortar's bulk's and shear symbolic modulus at the vicinity of p = 0 gives

the PCW's symbolic bulk's and shear's moduli (Eq. ( 3)) of a non-aging linear viscoelastic (n.a.l.v.) microcracked mortar following the MM's model can be approached, at least at short and long terms, by expressions [START_REF] Zucchini | A coupled homogenizationdamage model for masonry cracking[END_REF] provided in [START_REF] Rekik | Numerical homogenization model for effective creep properties of microcracked masonry[END_REF] available for a

{ 𝑘 ̃𝑃𝐶𝑊 ̃0 0 + 𝑘 ̃0 1 𝑝 + 𝑂(𝑝 2 ) 𝜇 ̃𝑃𝐶𝑊 * =𝜇 ̃0 0 + 𝜇 ̃0 1 𝑝 + 𝑂(𝑝 2 ) 𝑘 ̃0 0 , 𝑘 ̃0 1 , 𝑘 ̃∞ 0 , 𝑘 ̃∞ 1 , 𝜇 ̃0 0 , 𝜇 ̃0 1 , 𝜇 ̃∞ 0 , 𝜇 ̃∞ 1
mortar with a matrix (uncracked mortar) following the MM's rheological model [START_REF] Masiani | Cosserat and Cauchy materials as continuum models of brick masonry[END_REF] The series expansion of Eq. ( 10) at the vicinity of p = 0 gives [START_REF] Anthoine | Derivation of in-plane elastic characteristics of masonry through homogenization theory[END_REF] At the vicinity of p = ∞, MM's symbolic moduli can be approximated by [START_REF] Luciano | Homogenisation technique and damage model for old masonry material[END_REF] respectively for spherical and deviatoric parts. The characteristic times of the spherical and deviatoric parts of the Modified Maxwell model followed by the microcracked viscoelasticortar are respectively [START_REF] Zucchini | A coupled homogenizationdamage model for masonry cracking[END_REF] At last, the approximate creep function of a microcracked mortar which matrix follows the MM's model reads (Eq. ( 15))

The Eqs. ( 7) = ( 11), (8) = ( 12) allow the identification of the following expressions for the six effective MM's parameters for the HEMm (the homogeneous equivalent material to the microcracked mortar)

Illustrative example

In order to illustrate the above described identification procedure whether for mortar with open or closed cracks, it is proposed to consider hereafter the case of a microcracked mortar which properties at the safe state are given in Table 1. 1. For closed cracks, we recall that the bulk's modulus of the microcracked mortar coincides with that of the safe matrix which is independent from the crack density. Accordingly, the PCW and dilute estimates for the spherical part of the mortar's behavior coincide and are also independent from crack density dc.

As shown on Table 2, only expressions of the effective properties related to the deviatoric part of the mortar's behavior differ with the adopted predictive model. Accordingly, also the PCW estimate for the mortar's 'effective Poisson's ratio' is a function of the crack density (but it is time free). This expression is obtained by an exact inversion of the Laplace-Carson transform as follows accordingly (16) which is not heavy to carry out by contrary to the LC inversion [START_REF] Schapery | Approximate methods of transform inversion for viscoelastic stress analysis[END_REF] of PCW estimate for the mortar's effective creep function. Fig. 2-(a) demonstrates that the mortar's 'effective Poisson's ratio' increases with the increase of the crack density in the case of closed cracks. It remains lower than 0.5 showing that the mortar is far from to be incompressible. In case of closed cracks, the damage is assumed to do not evolve [START_REF] Dormieux | Stress -based estimates and bounds of effective elastic properties: the case of cracked media with unilateral effects[END_REF], accordingly, hereafter the crack density dc is assumed to be constant. Fig. 3 reports time-evolution of the mortar's Young's modulus for different crack densities values (dc = 0, 0.2, 0.55, 1). Qualitatively, it shows that the PCW estimate decreases rapidly with time t during a period of about 5.10 6 ≈ 58 days. In a second stage, it stabilizes or reaches a non-null asymptotic limit. This trend is also available for dc = 1. Quantitatively, this asymptotic limit decreases with the increase of the crack density. On Fig. 4 are depicted the PCW and dilute estimates for the mortar's Young's moduli evolutions with crack densities at times t = 20 and 1000 days. It can be observed that PCW and dilute models provide close estimates for the effective modulus 𝐸 ̃PCW (t,dc) for a crack density 0 ≤ dc ≤ 0.25. Beyond this interval, the percent error [START_REF] Alpa | Microstructural model for dry block masonry walls with in-plane loading[END_REF] between these estimates increases to exceed 10 % for dc ≥ 0.55 as shown on Fig. 5. It can also be remarked that the PCW effective estimates are softer than the dilute ones which is an expected result due to the fact that the PCW model accounts for cracks interactions by contrary to the dilute scheme. For mortar with closed cracks, it can be concluded that dilute estimate remains acceptable for every time t and crack density dc < 0.55 (see Figs. 5 and6). Otherwise this model highly overestimates the effective Young's modulus of microcracked mortar. Besides, cumulative errors for both viscoelastic constituents (mortar and bricks) if bricks are also assumed to be microcracked could lead to (highly) stiffer macroscopic estimates for the masonry.

𝑘 𝑀𝑀 * =𝑘 𝑅 𝑠𝑒 + 𝜂 𝑚 𝑠 3 𝑝 + 𝑂(𝑝 2 ) 𝜇 𝑀𝑀 * = 𝜇 𝑘 𝑒 + 𝜂 𝑚 𝑑 2 𝑝 + 𝑂(𝑝 2 ) 𝑘 𝑀𝑀 * =(𝑘 𝑅 𝑠𝑒 + 𝑘 𝑚 𝑠𝑒 ) -3 (𝑘 𝑚 𝑒 ) 2 𝜂 𝑀 𝑠 𝑝 +𝑂(1/𝑝 2 ) 𝜇 𝑀𝑀 * =(𝜇 𝑘 𝑒 + 𝜇 𝑀 𝑒 ) -2 (𝜇 𝑀 𝑒 ) 2 𝜂 𝑀 𝑑 𝑝 +𝑂(1/𝑝 2 ) ( 12
)
𝑘 𝑅 𝑠𝑒 = 𝑘 ̃0 0 , 𝑘 𝑚 𝑠𝑒 =𝑘 ̃∞ 0 -𝑘 𝑅 𝑠𝑒 , 𝜂 𝑀 𝑠 = 3𝑘 ̃0 1 𝜇 𝑘 𝑒 =𝜇 ̃0 0 ,𝜇 𝑀 𝑒 = 𝜇 ̃∞ 0 -𝜇 𝑘 𝑒 , 𝜂 𝑀 𝑑 =2𝜇 ̃0 1 , 𝜏 𝑀𝑀 𝑠 (𝑑 𝑐 )= 𝜂 𝑀 𝑠 (𝑑 𝑐 )(𝑘 𝑅 (𝑑 𝑐 )+𝑘 𝑀 (𝑑 𝑐 )) 3𝑘 𝑅 (𝑑 𝑐 )𝑘 𝑀 (𝑑 𝑐 ) 𝜏 𝑀𝑀 𝑑 (𝑑 𝑐 )= 𝜂 𝑀 𝑑 (𝑑 𝑐 )(𝜇 𝑅 (𝑑 𝑐 )+𝜇 𝑀 (𝑑 𝑐 )) 2𝜇 𝑅 (𝑑 𝑐 )𝜇 𝑀 (𝑑 𝑐 ) J 𝑀𝑀 𝑎𝑝𝑝 (𝑡, 𝑑 𝑐 )= 1 9𝑘 𝑅 (𝑑 𝑐 ) (1 - 𝑘 𝑀 (𝑑 𝑐 ) (𝑘 𝑅 (𝑑 𝑐 )+𝑘 𝑀 (𝑑 𝑐 )) 𝑒 -𝑡/𝜏 𝑀𝑀 𝑠 (𝑑 𝑐 ) ) + 1 3𝜇 𝑅 (𝑑 𝑐 ) (1 - 𝜇 𝑀 (𝑑 𝑐 ) (𝜇 𝑅 (𝑑 𝑐 )+𝜇 𝑀 (𝑑 𝑐 )) 𝑒 -𝑡/𝜏 𝑀𝑀 𝑑 (𝑑 𝑐 ) ) (15)
𝜈 𝑃𝐶𝑊 * (𝑝, 𝑑 𝑐 )= 3𝑘 ̃𝑃𝐶𝑊 * -2𝜇 ̃𝑃𝐶𝑊 * 6𝑘 ̃𝑃𝐶𝑊 * +2𝜇 ̃𝑃𝐶𝑊 * 𝜈 𝑃𝐶𝑊 (𝑑 𝑐 )=1-

Open cracks and time-dependent crack density

By contrary to the case of closed microcracks, both spherical and deviatoric parts of the mortar's behavior are affected by the presence of open cracks. Accordingly, the six effective mortar's (elastic and viscous) parameters identified after averaging process on randomly oriented cracks based on the PCW model are different from those provided by the dilute scheme. Table 3 provides these parameters for both PCW and dilute models when the microcracked mortar's properties at the safe (uncracked) state are given in Table 1. Based on these identified parameters, it is possible to calculate explicitly the effective creep function and hence the mortar's Young's modulus and its effective "Poisson's ratio" in the case of open cracks [START_REF] De Buhan | A homogenisation approach to the ultimate strength of brick masonry[END_REF] By contrary to the case of closed cracks, Fig. 2-(b) shows that the mortar's effective "Poisson's ratio" decreases with the increase of density dc. The later should not exceed 0.41 otherwise the mortar's effective "Poisson's ratio" tends towards negative values which is aberrant. In the following, for the case of microcracked mortar with open cracks, we assume that the condition dc ≤ 0.41 is fulfilled. Moreover, for the sake of showing the trends, we assume firstly that crack density is constant. Secondly, since damage evolves in mortar (or more generally a media) with open cracks, than crack density is assumed to evolve following a power-law time function as prescribed by [54] for the sake of simplicity and as a first assumption.

Constant crack density. Fig. 7 shows that the PCW estimates for the mortar's Young's modulus decreases with the increase of damage but does not vanish for dc ≤ 0.4. Moreover as predicted by the dilute scheme [START_REF] Rekik | Multi-level modeling of microcracked viscoelastic masonry[END_REF], the PCW estimates stabilizes beyond t ≥ 2.10 7 ≈ 231 days and tend towards none null asymptotic limits even if the PCW model provides softer estimates than the dilute ones for t ≤ 1000 days (see Fig. 8). The huge difference between the PCW and dilute estimates mainly beyond dc = 0.2 (see Figs. 9 and 10) demonstrates PCW model

k R (MPa) k M (MPa) η s (MPa.s) µ R (MPa) µ M (MPa) η d (MPa.s) Dilute model k R k M η s µ R µ M η d
Fig. 8 Open cracks: PCW and dilute estimates for the microcracked mortar's Young's modulus versus crack density at instants t = 20 and 1000 days.

Time-dependent crack density evolution.

As noted above, for the sake of simplicity, the crack density is assumed to accumulate in the following form as proposed by Shrive et al. [START_REF] Shrive | Chapter 4, Book: Learning fro m Failure -Long-term Behaviour of Heavy Masonry Structures[END_REF] ( [START_REF] Gabor | Analysis of the inplane shear behaviour of FRP reinforced hollow brick masonry walls[END_REF] where τD is a constant damage time which refers to the time where most damage would occur. The coefficients are taken here as τD = 800, η = 0.3, and n = 10. dc(t) represents the level of damage accumulated from the time at which damage starts, t0, to the time of evaluation t. In the calculations here, damage is assumed to begin at 400 days. The rate of damage accumulation with this model is slow initially, but accelerates over time, as shown on Fig. 11 reporting Fig 4 .4 in [START_REF] Shrive | Chapter 4, Book: Learning fro m Failure -Long-term Behaviour of Heavy Masonry Structures[END_REF]. For the sake of showing the trends and effects, quite considerable damage is assumed to occur in a relatively short time in this example. Here damage factor attains about 0.33 after 1000 days with the damage starting at 400 days. Damage initially accumulates at a very low rate but then increases rapidly [START_REF] Shrive | Chapter 4, Book: Learning fro m Failure -Long-term Behaviour of Heavy Masonry Structures[END_REF]. Fig. 12 shows a rapid decrease of the mortar's Young's modulus for t ≤ 5.10 7 (s) ≈ 578 (days). After that it tends towards an asymptotic limit of about 2000 (MPa). At a third step, it decreases rapidly for t ≥ 8.10 7 (s). It can also be observed that for the considered time power-law crack density, dilute and PCW estimates for time evolution of the mortar's effective Young's modulus are close for t ≤ t1 = 8.10 7 (s) ≈ 925 (days) (Fig. 13). This can be argued by the fact that dc is negligible below t1. After that (that means when t > t1) this damage parameter increases rapidly. Accordingly the mortar's effective Young's modulus decreases rapidly and it can be observed that the PCW estimates are (highly) softer than the dilute ones. Moreover, errors between these estimates exceed 10 % beyond t2 = 8.4 10 7 (s) ≈ 972 (days). This trend is consistent with that observed on Fig. 9 3. Actual global properties of microcracked masonry using periodic homogenization (step-2): viscoelastic bricks and mortar In this part, both of brick and mortar are assumed to follow the Generalized Maxwell (GM) model. For the sake of simplicity, only one term is considered for the GM model [START_REF] Sayed-Ahmed | Creep deformation of clay masonry structures: a parametric study[END_REF]. Accordingly rheological model followed by bricks and mortar coincide with the MM's model which tends towards a 'purely' elastic behavior both at the vicinity of 0 or at infinity as shown below .

(

) 20
The mortar's behavior can then be considered elastic and damaged at short and long terms as it )

(𝐽 𝑀𝑀 (𝑡 → ∞) = 1 9𝑘 𝑅 + 1 3𝜇 𝑅 )
evolves with crack density variation. Its behavior, at each time t, can be described by two effective parameters: the Poisson's ratio (relation ( 16)) if cracks are closed (respectively, relation [START_REF] Alpa | Microstructural model for dry block masonry walls with in-plane loading[END_REF], if cracks are opened) and 'Young's modulus' 𝐸 ̃PCW(t, dc) = 1//𝐽 ̃PCW(t, dc) if damaged or 𝐸 ̃PCW(t) = 1/𝐽 ̃PCW(t) otherwise (dc = 0). The considered bricks are assumed to be isotropic, safe (uncracked) with a viscoelastic behavior following the MM's model with parameters data given in Table 4. Bricks can than be considered to be "elastic" with a time-dependent 'Young's modulus' (see Fig. 14).

(21) Dimensions of the bricks are the height a = 55 mm, width b = 120 mm and thickness s = 250 mm. joints are 10 mm thick. For the viscous rheological model, since the instantaneous Young's moduli E0 for the MM's model is given by E0 = E(t = 0) = ER +EM where the relaxation moduli is set equal to EM = eiE0 then the spring's Young's moduli ER reads ER = (1ei)E0. ei is a dimensionless parameter. Now, according to the fact that the considered masonry shows regular arrangement of bricks and mortar's joints, it is possible to consider only a periodic cell and carry out numerical periodic homogenization using FE method. Moreover, as this periodic cell presents two axes of symmetry, normal and tangential directions along the unit vectors n and t, respectively, only its quarter (see Fig. 15) will be retained for computation.

When applying a constant macroscopic stress and assuming that the per phase localization tensor 𝐴 𝑟 is time independent as previously mentioned in [START_REF] Rekik | Numerical homogenization model for effective creep properties of microcracked masonry[END_REF] following the hypothesis of Deudé et al. [START_REF] Deudé | Propriétés élastiques non linéaires d'un milieu mésofissuré[END_REF] then the average strain 𝜀̅ 𝑟 per phase r and the masonry's overall behavior reduce respectively to 𝜀̅ 𝑟 = 𝐴 𝑟 :𝜀̅ and 𝜎 ̅ = 𝐶 ̃: 𝜀̅ where the overall stiffness is given by 𝐶 ̃=< 𝐶: 𝐴 >. Recall that the average strain localization over the periodic cell reads < A >= I. It is then important to determine components of the localization strain tensor A r ijkl. To assess the effective 'elastic engineering constants', it is proposed to subject the unit cell to three types of loadings: axial compression along n, axial compression along t and shear loading as shown on Fig. 15). In this case, strain localization components A r ijkl are given by the following equations: [START_REF] Lourenço | A multi-surface interface model for the analysis of masonry structures[END_REF] The localization strain tensor A is assumed to be orthotropic. Since the symmetry of the Cauchy strain tensor both in the anisotropic and isotropic spaces is required, it follows that A r ijkl = A r jikl = A r jilk (minor symmetry). The major symmetry of A r is also necessary A r ijkl = A r klji. Accordingly only the components Axxxx, Ayyyy, Axyxy = Axyyx and Ayxxy = Ayxyx are not null. Owing to the classical Voigt notation, the constitutive behavior law of the unit cell reads [START_REF] Pelissou | Asymptotic modeling of quasibrittle interfaces[END_REF] where 𝜎 ̅ =< 𝜎 > 𝑉 is the overall applied stress on the periodic cell. The software Cast3M [54] has been used to provide local mechanical fields and mainly average mechanical fields such as strain 𝜀̅ 𝑟 , stress 𝜎 ̅ 𝑟 over each phase r (r = b for bricks, m for mortar) and macroscopic strain 𝜀̅ = 𝑓 𝑏 𝜀̅ 𝑏 + 𝑓 𝑚 𝜀̅ 𝑚 calculated in order to deduce components of the effective stiffness tensor 𝐶 ̃ (Eq. ( 23)). The five engineering 'constants' are then given by [START_REF] Pegon | Numerical modeling of stone-block monumental structures[END_REF] Recall that for an isotropic material (brick and mortar), components of the stiffness tensor 𝐶 𝑟 (r = b, m) read [START_REF] Giambanco | A cohesive interface model for the structural mechanics of block masonry[END_REF] Where are respectively the bulk and shear moduli. Fig. 16 reports the time evolution of PCW estimates for the masonry's effective Young's 𝐸 ̃𝑥𝑥 , 𝐸 ̃𝑦𝑦 , and shear 𝜇 ̃𝑥𝑦 moduli. Qualitatively, it can be observed that the Young's moduli decrease quickly during the first 100 days to attain almost their half initial values (at t = 0). Beyond 100 days, these moduli stabilize during a period of about 600 days before decreasing again rapidly beyond t = 800 (days), but less quickly compared to the first 100 days period. Recall that similar trends were observed for dilute estimates of the curves 'masonry's effective Young's moduli versus time' where considered bricks were rigid [START_REF] Rekik | Multi-level modeling of microcracked viscoelastic masonry[END_REF]. Effective shear moduli is softer than effective Young's moduli (is 𝐸 ̃𝑥𝑥 /6, and 𝐸 ̃𝑦𝑦 /4). Moreover, by contrary to Young's moduli, it decreases progressively for t ≤ 100 days and t ≥ 800 days (there is no sudden nor intensive decrease). Fig. 17 shows that masonry's macroscopic estimates derived from PCW and dilute schemes are very close for t ≤ 800 days (error lower than 4 %). Beyond this instant the percent error between the two estimates increases rapidly to exceed 10 % and even reach 24 % at t = 1000 days. At each time t, the PCW estimates are a softer than the dilute ones. On Fig. 18 are depicted time-evolutions of percent error [START_REF] Rekik | Homogenization methods for interface modeling in damaged masonry[END_REF] between macroscopic properties of masonry with elastic and viscous bricks. The highest gap between the two cases is observed for the 𝐸 ̃𝑦𝑦 modulus. The brick's viscosity increases stiffness along the normal direction y. This difference stabilizes around 12 % during a period of about 600 days after a rapid increase during the first 200 days. Beyond t = 800 days, 𝐸 𝑟𝑟 (%) increases to exceed 15 %. For 𝐸 ̃𝑥𝑥 modulus, differences between the elastic and viscous cases is almost negligible since it is lower than 4 % for t ≤ 800 days. Similarly to the modulus 𝐸 ̃𝑦𝑦 , the error for 𝐸 ̃𝑥𝑥 increases rapidly for t ≥ 800 days to exceed 15 % at t = 1000 days. By contrary to moduli 𝐸 ̃𝑥𝑥 and 𝐸 ̃𝑦𝑦 , shear modulus of masonry with viscous bricks is softer than masonry with elastic bricks. Beyond t = 900 days, this trend is reversed. However difference between effective modulus 𝜇 ̃𝑥𝑦 for the two cases (elastic and viscous) remains acceptable (lower than 8 % for t ≤ 1000 days).

Local behavior of a masonry wall: open cracks

Since in the case of closed cracks, the damage is assumed to not evolve [START_REF] Dormieux | Stress -based estimates and bounds of effective elastic properties: the case of cracked media with unilateral effects[END_REF] and in order to consider interesting problems accounting for crack density evolution, it is proposed in this section to deal only with the case of open cracks.

Rigid bricks: comparison between numerical and analytical solutions

In order to validate the herein proposed numerical model, it is suggested in this part to compare its result in terms of vertical displacement of a compressed wall along the y direction (Fig. 19-(a)) to the analytical solution of a homogeneous plate submitted to the same vertical compressive load and assuming the hypothesis of 'in-plane stress' [START_REF] Cecchi | A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork[END_REF].

Here, bricks are assumed to be rigid while mortar is viscoelastic following the Generalized Maxwell rheological model which elastic and viscous properties (one term) are provided in Table 1. Accordingly, the mortar's Young's modulus evolves at safe (dc = 0) and micro-cracked (dc following time power-law relation ( 18)) states as reported on Fig. 20. It can be noticed that while error between these states (safe and micro-cracked) is lower than 6 % for time t ≤ 750 (days), it increases relatively quickly after this limit to reach 22 % at t = 900 (days).

For a homogeneous vertically compressed plate, we recall that the analytic solution is given by [START_REF] Cecchi | A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork[END_REF] (27) where L and H are respectively the width and height of the panel and s is the brick thickness. Q1 and 𝑞 1 = 𝑄 1 /(𝐿𝑠) denote respectively the total vertical load and pressure applied at the top of the masonry panel. A F ijkl represent approximate expressions for the homogenized relaxation coefficients provided by the kinematically admissible solution given in [START_REF] Cecchi | A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork[END_REF]. Particularly, [START_REF] Binda | The collapse of civic tower of Pavia[END_REF] where a denotes the height of the brick, eh is the thickness of the mortar head joints and ( 29) is the bulk relaxation moduli of horizontal interfaces. In this work, note that viscoelastic behaviors of the bed and head joints (indexed respectively by 'b' and 'h') are assumed to be coincident and are defined by the relaxation function Em(t) (instantaneous "'Young's modulus"' of the mortar given by Em = 1/JMM app (see relation [START_REF] Pegon | Numerical strategies for solving continuum damage problems with softening: application to the homogenization of masonry[END_REF]). On the other hand, to obtain the local numerical solution for this structure, the real heterogeneous problem (Fig. 19-(a)) is substituted by the homogenized one denoted by MHE(t, dc) (Fig. 19-(b)) which orthotropic properties used for finite element resolution of the considered problem are determined due to the periodic homogenization procedure previously described in Section 3. Fig. 21 reports time evolution of the vertical displacement at the top of the compressed masonry wall composed of rigid bricks and (safe or microcracked) viscoelastic mortar. At the safe state of the mortar joints, this figure allows comparison between finite elements solution and analytical one provided by Eq. ( 27) assumed to be the reference solution. It can be seen that numeric and analytic solutions show similar qualitative trends.

𝑢 𝑦 (𝑥, 𝐻, 𝑡) = 𝑄 1 𝐻 𝐴 2222 𝐹 (𝑡)(𝐿𝑠) = 𝑞 1 𝐻 𝐴 2222 𝐹 (𝑡) 𝐾 ℎ (𝑡) = 𝐸 𝑚 (𝑡) (1 + 𝜈 𝑚 )(1 -2𝜈 𝑚 ) 𝐴 2222 𝐹 (𝑡) = (𝑎 + 𝑒 ℎ )𝐾 ℎ (𝑡) 𝑒 ℎ
Quantitatively, numeric solution is very close to the analytical one. Indeed, according to Fig. 22, the error between numerical and analytical (red curve) solutions does not exceed 7 % for all time 0 ≤ t ≤ 1000 (days). This non-null error (at safe state of viscoelastic masonry) can be explained by the fact that analytic model implies approximate expressions for homogenized relaxation coefficient ( ). Moreover, it is available for masonry with rigid bricks connected by interfaces, however the herein proposed numerical model accounts for finite thickness of mortar joints (here set equal to 10 mm). The curve describing the evolution of the displacement uy at the top of the compressed wall (Fig. 21) with microcracked mortar joints shows qualitative trend similar to that observed for masonry wall with safe mortar joints till t = 600 (days). After that the absolute value of the displacement uy does not stabilize as it increases with time. This is consistent with the decrease of the microcracked mortar's Young's modulus observed on Fig. 20. The difference between safe and micro-cracked states is highlighted by the blue curve depicted on Fig. 22 showing that presence of micro-cracks induces a gap between uy results lower than 13 % for time t ≤ 750 (days) but which increases with time to reach (for example) 31,3 % at t = 900 (days). It is worth noting that Fig. 22 reports also curve estimating error between numerical results for uy at the top of the compressed wall with mortar joints at safe and microcracked states. Qualitatively, the trend of this curve is similar to that estimating error between analytic and numeric results at safe state of mortar joints. Quantitatively, this curve shows errors lower than those provided by blue curve. Indeed for t ≤ 760 days, this error is below 7 %. After that, it can reach almost 25 % at t = 900 days which is consistent with the decrease of the microcracked mortar's Young's modulus observed on Fig. 20.

Viscoelastic bricks: numerical solution

This part aims to assess the relevance and efficiency of the proposed model based on the coupling between PCW homogenization model and Griffith's brittle theory. For this purpose, we consider the problem of a viscoelastic masonry wall of dimensions L = 1560 mm (length) and H = 1040 mm (height) submitted to compressive loadings (three distributed loads at the top and two lateral edges and an additional concentrated load F applied on the top) as shown on Fig. 23. Bricks are assumed to be safe and viscous following the MM's rheological model. The mortar's (respectively, brick's) material data used to compute this problem are given in Table 1 (respectively, Table 4). Crack density is assumed to evolve to a time power-law function ( 19)). On the other hand, as the arrangement of the bricks is regular, the effective behavior of the panel is assumed to be well estimated by that of a periodic cell which effective behavior (macroscopic effective stiffness) is determined in Section 3. The wall can then be modeled as a homogeneous material which properties coincide with those of the equivalent material HEM-2 (Fig. 1-(e)). On Fig. 24 are depicted the normal and shear stresses distributions in the compressed wall at t = 900 days. Qualitatively, it can be observed that, under BC-2, distribution of the stress field σyy is symmetric (Fig. 24-( For the normal stress, these estimates are close except around the axis of symmetry of the wall where dilute scheme gives softer estimates than PCW model (a percent error of about 8 % is attained at the peak of stress σyy). This error value is acceptable, however it can be (quite) harmful if we study the risk of failure of the wall. Moreover, this error increases in the vicinity of the application point of the concentrated load (it can exceed 80 % inside the stress localization area). Although Fig.

27-(b)

shows close estimates deriving from dilute and PCW models, the percent error between these estimates reaches 46 % (Fig. 28) at three zones: near right and left wall edges and area surrounding application point of the concentrated load. This difference is clearly harmful and even dangerous when predicting failure of this compressed wall. The PCW model is then greatly recommended rather than the dilute homogenization model in order to anticipate efficiently the collapse of masonry compressed wall. On the other hand neglecting brick's creep can be acceptable when estimating the normal local stress (see Fig. 29) since error between masonries with elastic and viscous bricks is lower than 4 % at the level of the wall's middle height, however it is strongly harmful for the prediction of magnitudes of shear stresses for which evolution of error along the axis x shows a strong jump at the level of the wall's symmetry axis.

For comparison purpose between strain fields inside masonry under boundary condition BC-1 and BC-2, Fig. 30 reports εxy and εyy strain evolutions curves versus the x axis. Globally, under these boundary conditions, it is observed that the MM's model predicts small strains. Moreover, qualitatively, these evolutions show similar trends (symmetric and anti-symmetric evolutions by reference to the axis x = L/2 respectively for εyy and εxy) as those observed for the normal and shear stress. Indeed, under BC-1, normal strain is almost constant with variation of the abciss x. However absolute value of the strain εyy presents a peak around the symmetry axis of the wall. The magnitude of this peak is two times higher than values of normal strain attained at the level of the wall's edges. Clearly, normal strains induced by condition BC-2 are higher than those attained under condition BC-1 due to the additional concentrated load F. This remark remains available for the shear strain predicted at the wall line y = H/2. Sign of corresponding shear peaks under BC- for shear strain Err(%) belongs to the interval [10 %, 66.9 %]. Similarly to the shear stress, the maximal error value for shear strain is attained around the symmetry axis of the wall (x = L/2). This result confirms again that neglecting cracks interactions highly underestimates stress and strain local (normal and shear) fields throughout the compressed wall. This could lead to false interpretations of the magnitude of local stress and strain fields throughout the studied wall and accordingly to not properly anticipate the wall's failure. At last, comparison between PCW predictions for the strain fields for masonries with elastic and viscous bricks (Fig. 32) reveals that absolute normal strain values attained inside the masonry with elastic bricks are higher than masonry with viscous bricks. In this context, errors for normal strain varies between 12.4 % and 17.8 %. However evolution of error related to shear strain show a strong jump around the symmetry axis of the wall. Otherwise it varies between about 4.3 % and 18.1 %

Synthesis of the results

In the following, Tables 5 and6 summarize trends of above obtained results at three levels: micro-cracked mortar, masonry periodic cell and compressed wall

Conclusions and perspectives

This work provides a multi-scale improved model to the earlier one proposed in [START_REF] Rekik | Numerical homogenization model for effective creep properties of microcracked masonry[END_REF][START_REF] Rekik | Multi-level modeling of microcracked viscoelastic masonry[END_REF] allowing rigorous assessment at short and long-terms of global and local behavior of microcracked viscoelastic masonry. Advantages of the proposed model are mainly related to the none recourse to numerical inversion of the Laplace-Carson transform [START_REF] Cost | A multidata method of approximate laplace transform inversion[END_REF][START_REF] Rekik | Optimization of the collocation inversion method for the linear viscoelastic homogenization[END_REF] when assessing the overall creep behavior of mortar and periodic masonry cell, to the ability of this model to take into account cracks interactions, creep of bricks units and variation of mortar's "'Poisson's ratio"' with crack density. Assuming that the PCW estimates are the reference solution at local and global levels, this work shows that the dilute scheme overestimates the microcracked mortar's overall properties whether with open or closed cracks. This result is available at the level of the masonry's periodic cell (with open cracks which crack density evolves owing to a time power-law function) in terms of overall orthotropic properties. At the level of a compressed masonry wall with a concentrated load exerted on its top, it has been observed that dilute estimates for local shear stresses highly underestimate the PCW ones. Moreover, even if percent error for dilute and PCW estimates for the local normal stress can be acceptable at the middle height of the wall, it can be significant in the wall area located at the vicinity of application point of the concentrated load. These results demonstrates that using the dilute scheme instead of the PCW model to predict the local behavior of a compressed wall can be harmful when investigating the risk of failure. This work shows also that viscosity of bricks increases the overall stiffness of masonry cell (of about 15 % to 17 % for 𝐸 ̃𝑦𝑦 and 8 % for 𝜇 ̃𝑥𝑦 if dc = 0.33) for none negligible crack density (dc ≥ 0.12) of microcracks inside mortar joints. This effect is more significant along the direction of head joints even if dc is small (0 ≤ dc ≤ 0.12). At the level of the investigated compressed wall, creep of bricks induces higher strains but lower (normal and shear) stresses except around the axis of symmetry of the wall (direction of the concentrated load) where relative error between local normal (shear) stresses in masonry with elastic and viscous bricks attain its peak. This error is (highly) more significant for local shear stresses at the vicinity of wall edges (of about 20 %) and mainly around the axis of the concentrated load (x = L/2). Moreover this demonstrates the fact that neglecting the creep of bricks could false prediction of failure in the wall.

In this work, for the sake of simplicity and trends, the crack density is assumed to follow a time power-law function [START_REF] Shrive | Chapter 4, Book: Learning fro m Failure -Long-term Behaviour of Heavy Masonry Structures[END_REF]. In the future, it will be interesting to investigate damage propagation in a thermodynamics framework [START_REF] Dormieux | MicroporoMechanics[END_REF] or by a stress criterion [START_REF] Massart | Structural damage analysis of masonry walls using computational homogenization[END_REF]. For bricks and mortar, considering creep functions following the Generalized Maxwell (GM) model with two terms could enrich and increase the accuracy of predictions provided by the proposed model. Moreover, it could be interesting to assess the relevance of the proposed model by comparison with a 'full' heterogeneous finite elements model (by modelling all bricks and microcracked viscoelastic mortar joints). At last, accounting for thermal loading could extend this model for refractory linings present for example in blast furnaces subjected to compressive loads resulting from constrained thermal expansion [START_REF] Ferber | Compressive creep and thermo-physical performance of refractory materials[END_REF][START_REF] Andreev | Compressive behaviour of dry joints in refractory ceramic masonry[END_REF].

Fig. 1

 1 Fig.1 Basic steps followed by the propose model: (s1) and (s2).

  When all microcracks in the RVE (Representative Volume Element) are open, the global symbolic deformation is elastic and the effective symbolic stiffness tensor 𝐶 ̃ read 𝐶 ̃ * = 3𝑘 ̃𝑃𝐶𝑊 * 𝐽 + 2 𝜇 ̃𝑃𝐶𝑊 *

( 7 )

 7 and at the vicinity of p = ∞, it reads 𝑘 ̃𝑃𝐶𝑊 * are assumed to be known. For example, for open cracks, using theorems on the initial and final values given

Fig. 2

 2 Fig. 2 Evolution of the PCW estimate for the effective "Poisson's ratio" of the microcracked mortar with crack density dc in the cas e of closed (a) and open (b) cracks.

Fig. 3

 3 Fig. 3 Closed cracks: PCW estimates for the evolution of the mortar's effective Young's modulus versus 'Ln(time)'.

Fig. 4

 4 Fig. 4 Closed cracks: comparison between PCW and dilute estimates for the microcracked mortar's effective Young's modulus (t = 1000 days) versus crack density.

Fig. 6

 6 Fig. 6 Closed cracks: percent error between PCW and dilute estimates for the microcracked mortar's effective Young's modulus versus time.

Fig. 5

 5 Fig. 5 Closed cracks: percent error between PCW and dilute estimates for the microcracked mortar's effective Young's modulus (t = 20 and 1000 days) versus crack density.

Fig. 10

 10 Fig. 10 Open cracks: percent errors between PCW and dilute estimates for the microcracked mortar's Young's modulus with different crack densities (dc = 0, 0.1, 0.2, 0.3 and 0.4) versus time.

Fig. 9

 9 Fig.[START_REF] Sulem | A continuum model for periodic two-dimensional block structures[END_REF] Open cracks: percent errors between PCW and dilute estimates for the micro-cracked mortar's Young's modulus versus crack density either for t = 20 and 1000 days.

Fig. 7

 7 Fig.7 Open cracks: PCW estimates for the mortar's Young's modulus versus 'Ln(Time(s))' for different crack densities

Fig. 12

 12 Fig. 12 Comparison between dilute and PCW estimates : time evolutions of the microcracked mortar's Young's modulus with open cracks which crack density is assumed to be a time power-law function.

Fig. 11

 11 Fig.11Non-linear evolution of damage ratio with time[START_REF] Shrive | Chapter 4, Book: Learning fro m Failure -Long-term Behaviour of Heavy Masonry Structures[END_REF].

Fig. 13

 13 Fig.[START_REF] Luciano | Homogenisation technique and damage model for old masonry material[END_REF] Open cracks which crack density evolves following a time power-law function: percent error between PCW and dilute estimates

Fig. 14

 14 Fig. 14 Time evolutions of the 'effective Young's moduli' of the considered viscous brick and mortar

) 1 𝐸 1 𝐸

 11 ̃𝑡𝑡 (𝑡, 𝑑 𝑐 ) = 𝐶 ̃𝑛𝑛𝑛𝑛 𝐶 ̃𝑛𝑛𝑛𝑛 𝐶 ̃𝑡𝑡𝑡𝑡 -𝐶 ̃𝑡𝑡𝑛𝑛 𝐶 ̃𝑛𝑛𝑡𝑡 , ̃𝑛𝑛 (𝑡, 𝑑 𝑐 ) = 𝐶 ̃𝑡𝑡𝑡𝑡 𝐶 ̃𝑛𝑛𝑛𝑛 𝐶 ̃𝑡𝑡𝑡𝑡 -𝐶 ̃𝑡𝑡𝑛𝑛 𝐶 ̃𝑛𝑛𝑡𝑡 , 𝜇 ̃𝑛𝑡 (𝑡, 𝑑 𝑐 ) = 𝐶 ̃𝑛𝑡𝑛𝑡 , 𝜈 ̃𝑛𝑡 (𝑡, 𝑑 𝑐 ) = 𝐶 ̃𝑛𝑛𝑡𝑡 𝐶 ̃𝑡𝑡𝑡𝑡 , 𝜈 ̃𝑛𝑡 (𝑡, 𝑑 𝑐 ) = 𝐶 ̃𝑡𝑡𝑛𝑛 𝐶 ̃𝑛𝑛𝑛𝑛

Fig. 15 Fig. 16

 1516 Fig. 15 Boundary and symmetry conditions for the considered quarter cell subjected to axial normal (a) or tangential (b) compression or shear (c) loadings

Fig. 17

 17 Fig. 17 Comparison between PCW and dilute estimates for effective properties of a masonry cell constituted by viscous bricks and microcracked mortars joints with open cracks which crack density evolves following a time power-law function.

Fig. 18

 18 Fig. 18 PCW effective estimates for periodic masonry cell with microcracked mortar joints (open cracks): comparison between elastic and viscous bricks.

Fig. 20

 20 Fig. 20 Mortar joints : Time evolutions of the 'instantaneous global Young's moduli Em' of the considered viscoelastic mortar at safe and microcracked states and the error (%) between these moduli.

Fig. 19

 19 Fig.[START_REF] Gabor | Analysis of the inplane shear behaviour of FRP reinforced hollow brick masonry walls[END_REF] Rigid bricks and viscoelastic mortar joints: Real (a) and equivalent homogenized (b) masonry panels subjected to a vertical distributed load applied at the top[START_REF] Baraldi | Discrete element model for inplane loaded viscoelastic masonry[END_REF].

  a)) by reference to the symmetry axis of the panel (x = L/2) unlike that of the shear stress σxy which is anti-symmetric (Fig.24-(b)). Similar trends are observed on Fig.25related to the case of a wall under compressive loadings without concentrated load F (boundary condition BC-1) showing however different stress localization zones: at the right and left corners of the top of the wall. By contrary, normal and shear stresses concentrate at the vicinity of the application's point of the concentrated load F when considering wall under boundary condition BC-2. Moreover, although under BC-1, compressive loadings induce only compression (σyy ≤ 0) throughout the area of the wall, the additional load F make appears tensile stresses (σyy ≥ 0) in small area surrounding the application point P of the load F as a reaction (response) to the highest values of normal compressive stresses reached around point P. Quantitatively, comparison between Figs.[START_REF] Pegon | Numerical modeling of stone-block monumental structures[END_REF] and 25 shows that the magnitude of the local normal stress throughout the wall under BC-1 is negligible compared to that attained in a wall under BC-2. Under condition BC-1, it can also be observed that the induced local shear stress is negligible compared to the induced local normal stress. For wall under BC-2, the shear stress is not negligible but (almost 4 times) lower than local normal stress. Fig.26illustrating evolutions of stress components at the middle height of the wall (y = H/2) along the x axis shows difference of stresses magnitudes between boundary conditions BC-1 and BC-2 demonstrating that a concentrated load at the middle length of the wall increases the stresses magnitudes and even implies a peak for normal (shear) stress along the axis of the load F.

Fig. 21

 21 Fig. 21 Rigid bricks and viscoelastic mortar joints: Time evolutions of the local vertical displacement uy at the top of the panel.

Fig. 22

 22 Fig. 22 Time evolutions of the error between analytic (safe mortar) and numerical (safe and microcracked mortar) solutions for vertical displacement uy at the top of the compressed wall

Fig. 25 1 Fig. 24

 25124 Fig. 25 Local normal (left) and shear (right) stress snapshots throughout a wall (with viscous bricks) under BC-1

Fig. 23

 23 Fig. 23 Masonry panel submitted to boundary conditions BC-1 (a), BC-2 (b) and equivalent problem (c) for masonry under BC-2.

Fig. 27 allows

 27 Fig.27allows comparison between PCW and dilute estimates for the local normal and shear stresses. For the normal stress, these estimates are close except around the axis of symmetry of the wall where dilute scheme gives softer estimates than PCW model (a percent error of about 8 % is attained at the peak of stress σyy). This error value is acceptable, however it can be (quite) harmful if we study the risk of failure of the wall. Moreover, this error increases in the vicinity of the application point of the concentrated load (it can exceed 80 % inside the stress localization area). Although Fig.27-(b)shows close estimates deriving from dilute and PCW models, the percent error between these estimates reaches 46 % (Fig.28) at three zones: near right and left wall edges and area surrounding application point of the concentrated load. This difference is clearly harmful and even dangerous when predicting failure of this compressed wall. The PCW model is then greatly recommended rather than the dilute homogenization model in order to anticipate efficiently the collapse of masonry compressed wall. On the other hand neglecting brick's creep can be acceptable when estimating the normal local stress (see Fig.29) since error between masonries with elastic and viscous bricks is lower than 4 % at the level of the wall's middle height, however it is strongly harmful for the prediction of magnitudes of shear stresses for which evolution of error along the axis x shows a strong jump at the level of the wall's symmetry axis. For comparison purpose between strain fields inside masonry under boundary condition BC-1 and BC-2, Fig.30reports εxy and εyy strain evolutions curves versus the x axis. Globally, under these boundary conditions, it is observed that the MM's model predicts small strains. Moreover, qualitatively, these evolutions show similar trends (symmetric and anti-symmetric evolutions by reference to the axis x = L/2 respectively for εyy and εxy) as those observed for the normal and shear

Fig. 27

 27 Fig. 27 Boundary condition BC-2: comparison between PCW and dilute estimates for the evolutions of normal and shear stresses versus x axis (wall with viscous bricks) and study of the effect of accounting for creep of bricks.

Fig. 28

 28 Fig. 28 Percent error between dilute and PCW estimates for local normal (left) and shear (right) stresses in the middle height of the wall (case with viscous bricks).

Fig. 29

 29 Fig. 29 Percent error between PCW estimates for local normal (left) and shear (right) stresses in the middle height of the wall: effe ct of bricks' creep.

Fig. 30

 30 Fig. 30 PCW estimates at time t = 900 days for the evolutions of the local strain εyy (left) and εxy (right) versus abciss x in the middle height of a wall with viscous bricks: comparison between boundary conditions BC-1 and BC-2.

Fig. 31

 31 Fig. 31 Boundary condition BC-2: comparison between PCW and dilute estimates for the evolutions of normal and shear stresses versus x axis (wall with viscous bricks) and study of the effect of accounting for creep of bricks

Fig. 32

 32 Fig. 32 Local normal (left) and shear (right) strain snapshots throughout a wall (with viscous bricks) under BC-2

Table 1

 1 Elastic and viscous moduli of the mortar (one term, n = 1) tested by Brooks et al.[START_REF] Brooks | Composite modeling of masonry deformation[END_REF][START_REF] Cecchi | A comparison between numerical and analytical homogenized models for viscoelastic brickwork[END_REF]

		E0 (MPa)	Ν	ei	τM (days)
	Mortar	7700	0.2 0.7602	7.1

.3.1. Closed cracks Dilute

and PCW estimates for effective elastic and viscous properties (explicit functions of the parameter dc) of the considered microcracked viscoelastic mortar with closed cracks are provided in the Table

Table 2

 2 Closed cracks : identified MM's parameters for the microcracked mortar at the macroscopic scale (effective properties) using the PCW and dilute homogenization models.

	𝐸 𝑟𝑟 (%) = 100(	𝐸 ̃𝑃𝐶𝑊 -𝐸 ̃𝐷𝐼𝐿 (𝐸 ̃𝑃𝐶𝑊 +𝐸 ̃𝐷𝐼𝐿 )/2 )
	PCW	kR (MPa)	kM (MPa)	ηs (MPa.s)
	model	1025.81	3251.97	5.9846 10 9
		µR (MPa)	µM (MPa)	ηd (MPa.s)
		-769.358 +	3245 .73 2.10938 + 𝑑 𝑐	-2438.98 +	10289.4 2.10937 + 𝑑 𝑐	-2.99233 10 9 +	1.26239 10 10 2.10938 + 𝑑 𝑐
	Dilute	µR	µM	ηd
	Model	769.358	2438.97	2.99233 10 9
		1 + 0.948148𝑑 𝑐	1 + 0.948148𝑑 𝑐	1 + 0.948148 𝑑 𝑐

  the presence of open cracks implies a high interaction level between the micro-cracks by contrary to the same density of closed cracks. Accordingly, for damaged mortar with open cracks it is important and more rigorous to assess its effective Young's modulus by means of the PCW model instead of the dilute one if dc > 0.2.

	𝜈 ̃𝑃𝐶𝑊 = -0.185394+(0.369141+0.2𝑑 𝑐 )𝑑 𝑐

that

Table 3

 3 Open cracks : identified MM's parameters for the microcracked mortar at the macroscopic scale using the PCW and dilute homogenization models.

Table 4

 4 Elastic and viscous moduli of brick (one term, n = 1)[START_REF] Brooks | Composite modeling of masonry deformation[END_REF][START_REF] Cecchi | A comparison between numerical and analytical homogenized models for viscoelastic brickwork[END_REF]

		E0 (MPa)	Ν	ei	τM (days)
	Brick	17100	0.15 0.5327	33.8

Table 5

 5 Comparison between PCW and dilute estimates at three levels: microcracked mortar, masonry periodic cell and compressed wall. Trends Mortar's effective Young's modulus DIL > PCW, Err ≥ 10 %, for dc ≥ 0.25, if open cracks (0.55 if closed cracks).

	Masonry's effective cell properties (open cracks & dc = time power-law function)
	DIL ≥ PCW (Err → 19 to 24 % for dc → 0.33)
	Local fields in a compressed wall (open cracks & dc = time power-law function)
	and	are close (-8 % ≤ Err ≤4 %])
	slightly stiffer than
	|𝜀 𝑦𝑦 𝐷𝐼𝐿 | ≪ |𝜀 𝑦𝑦 𝑃𝐶𝑊 | (𝐸 𝑟𝑟 ∈ [47 %, 60 %])

Table 6

 6 Comparison between masonry with elastic and viscous bricks: PCW estimates Trends Masonry's effective cell properties (open cracks & dc is a time power-law function) Local fields in a compressed wall (open cracks & dc = time power-law function)

* = 𝜇 ̃𝑚 * (1 -480 𝑑 𝑐 (1 -𝜈 𝑚 * )(5 -𝜈 𝑚 * ) 675(2 -𝜈 𝑚 * ) + 64𝑑 𝑐 (4 -5𝜈 𝑚 * )(5 -𝜈 𝑚 * ) ) 𝜇 𝑚 * (p) = 𝜇 𝑅 𝑒 + 1 1 𝜇 𝑀 𝑒 + 2 𝑝 𝜂 𝑚 𝑑 , 𝑘 𝑚 * (p) = 𝑘 𝑅 𝑠𝑒 + 1 1 𝑘 𝑀 𝑒 + 3 𝑚 * = 3 𝑘 𝑚 * -2𝜇 𝑚 * 6 𝑘 𝑚 * +2𝜇 𝑚 *

𝑘 ̃𝑃𝐶𝑊 *