Alternative estimation of effective Young’s Modulus for Lightweight Aggregate Concrete LWAC

The prediction of effective mechanical properties of composite materials using analytical models is of significant practical interest in situations in which tests are impossible, difficult, or costly. Many experimental and numerical works are attempting to predict the elastic properties of Lightweight Aggregate Concrete (LWAC). In order to choose the optimized prediction composite model, the purpose of this paper is to appraise the effective Young’s modulus of LWAC using two-phase composite models. To this effect, results of previous experimental research have used as a platform, upon which, 07 two-phase composite models were applied. The outcomes of this comparative analysis show that not all two-phase analytical models can be directly used for predicting Young’s modulus of LWAC. The Maxwell, Counto1 and Hashin-Hansen models are in close concordance with the experimental Young’s modulus of all LWAC used for comparison in this study (119 values). They were found more appropriate for reasonable prediction of elasticity modules of the LWAC.


INTRODUCTION
ecently, special attention has been paid to the development of Lightweight Aggregate Concrete (LWAC) 1, 2, 3 which offers many advantages as a building material, including low weight, easier construction and better resistance compared with ordinary concrete. Lightweight Aggregate Concrete (LWAC) primarily improves the thermal and sound insulation properties of buildings next to its basic applications 4. The lightweight concrete are created by substituting the natural aggregates with the lightweight aggregates (LWA), which are classified into two fundamental R categories: natural (like pumice, diatomite, volcanic ash, etc.) and manufactured (such as perlite, extended schist, clay, slate, sintered powdered fuel ash (PFA), etc.) 3, 5. Beside its technical and financial interests, LWAC can be integrated into the demarche of sustainable improvement by utilizing in specific artificial aggregates which are lighter than natural aggregates 6. The Young's modulus (elastic modulus) is a very important material property which is measured directly on concrete. Engineers need to know the value of this parameter to conduct any computer simulation of structure. Various experimental works have concerned the study of behavior of LWAC 1, 7, 8, 9, 10, 11, 12. However from an experimental point of view, this is not always easy. Therefore when the tests are impossible, difficult, costly, or timeconsuming, the research about prediction models for the elastic modulus using properly validated composite models is of great practical interest. The aim of the composites materials approach is to develop a model that will enable expression of average properties of the mixtures through properties and volume fractions of its constituents 11. Diverse explicit models of the literature are utilized. Their application to the prediction of LWAC behaviors shows a wide dissimilarity between the different approaches particularly when the volume fraction of reinforcement is more than 40% and when the contrast between the phases grows 9. For this purpose and to distinguish the most appropriate two-phase composite model for predicting LWAC's effective modulus of elasticity, the estimation of the Young's modulus of LWAC using two-phase composite models was applied. Furthermore, an efficient and accurate model is useful to reduce the cost and duration of the experimental mix design studies. In this present work, a large bibliography data for different LWAC tested experimentally and published in the literature are used: De Larrard 7, Yang and Huang 8, and Ke Y et al. 9. For LWAC test results investigated in this study, the volume fraction Vg of the lightweight aggregate varies from 0% (the matrix) to 47.8% and the contrast of the characteristics of the phases E g /E m (Young's modulus of lightweight aggregate and matrix) varies between 0.20% and 95% except for four types of concretes for which this ratio exceeds 1 because of a very low value of E m (E g  E m ) 7. In order to determine the models likely to yield the lowest number of errors; the results of effective Young's modulus of LWAC obtained by using 07 two-phase composite models were compared with the experimental results obtained by De Larrard 7, Yang and Huang 8 and Ke Y et al. 9 (119 values) and discussed. Therefore, prediction possibilities using composite material models in determination of modulus of elasticity were sought and some suggestions were made accordingly to a statistical study.

Two-phase composite models
Ore attention has been paid to lightweight aggregate concrete. The weakest component of LWAC is not the cement matrix or the interfacial transition zone (ITZ) but the aggregates. Therefore, the research about prediction model for LWAC's Young modulus is valuable for the concrete application 6. Lo and Cui 13 illustrate that the ''Wall effect'' does not exist on the surface of expanded clay aggregates in lightweight concrete by SEM and BSEI imaging, resulting in a better bond and much more slender interfacial zone than the ordinary concrete 14. So, materials which are produced can be considered a two-phase composite material. The purpose of the composites materials approach is to develop a model that will enable expression of average properties of the mixtures through properties and volume fractions of its constituents 1, 11. We look for the models to estimate the Young modulus for Lightweight Aggregate Concrete (LWAC) in terms of the properties and volume fractions of its constituents. These include the mortar matrix and the lightweight aggregate as reinforcing material. Before analyzing Lightweight Aggregate Concrete as a composite material, some assumptions must be considered. First, the heterogeneous composite material (LWAC) is considered to be comprised of only two linear-elastic phases (the mortar and the lightweight aggregate). Second, the unit cell is assumed sufficiently large to account for the heterogeneity of the system, and the deferring geometry of the phases. However, it is extremely small so that the composite is described homogeneous on a macro scale 10, 15, 16. Fig. 1 presents the models for an idealized unit cell of a two-phase composite material 10, 11, 17. The LWAC comprises a dispersed phase of lightweight aggregate with a Young's modulus Eg and volume fraction Vg and a continuous phase of the mortar matrix, with a Young's modulus E m and volume fraction V m . M As explained by Gilormini and Brechert 18, the choice of a model is governed by several parameters including the geometry of the heterogonous medium, the mechanical contrast between the phases (E g /E m ) and the volume fraction of reinforcement (V g ). Therefore, the equivalent homogenous behavior of LWAC depends of the characteristics of the mortar (matrix, phase m) and lightweight aggregate (dispersed phase, phase g).   (9)). This Study try to figure out that these composite material models, mentioned above, got reliable prediction abilities for the modulus of elasticity of LWAC. The modulus of elasticity values were predicted utilizing the composite models and then, the predicted results were compared to the experimental results of De Larrard 7, Yang and Huang 8 and Ke Y et al. 9 respectively.

Comparative analysis
omparison between the estimative results of effective elastic modulus of LWAC obtained as a result of calculations of the Eqns. (2-9) and those of experimental data have been presented in Tabs. 5, 6 and 7 respectively. A confrontation of LWAC Young's modulus between experimental results in 7, 8, 9 and the predictions of 07 composite models material models are shown in Fig. 2, Fig. 3 and Fig. 4 respectively. The differences between the various predictive composite models and the experimental results in 7, 8, 9 have been computed according to the proportion of reinforcement Vg in LWAC. When the volume fraction of aggregates Vg grows, the errors between the predictions and the experimental results increase for all composite material models. Since the weakest component of LWAC is not the cement matrix but the lightweight aggregates, the effect of volume fraction of lightweight aggregate on Young's modulus of LWAC is very clear. The increase in the volume fraction of lightweight aggregates Vg substantially reduces the Young's modulus of the LWAC. To compare the experimental and predicted Young's modulus of LWAC, the error percentage E  . is determined using the following expression:       Table 9: Error percentages of composite models and experimental results in 8 (%).
Compared with the experimental data of Yang and Huang 8 (Tab. 6, Tab. 9), Bache and Nepper-Christensen, Counto2, Popovics, Hirsch-Dougill, underestimate the measured Young's modulus. On the other hand, the Maxwell and Counto1 models overestimate the Young's modulus measured by Yang and Huang [8]. As seen in Tab. 9, for the Hashin-Hansen and Counto1 models, 12/12 cases give E smaller than 5%. E ranges from 0.94% to 4.87%. The Maxwell gives 12/12 cases smaller than 10% and 11/12 smaller than 5%. E ranges from 1.66% to 6.14%. In all composite models, the error percentages differ between 0.05% and 12.57%. It can be seen that the most accurate models are those of Hashin-Hansen, Counto1 and Maxwell which give less errors percentages. The predictions of the LWAC Young's modulus using the 07 composite material models are compared with experimental data of Ke Y et al. [9] (Tab. 7 and Tab.10) in Fig. 4. All selected composite models appear applicable to predict the Young's modulus of LWAC tested by Ke Y et al [9]. For the Maxwell model, 50/75 cases give E smaller than 5% and 22/75 cases smaller than 10%. This means that 72/75 cases have E smaller than 10%. This model converges on the experimental values measured by Ke Y et al. [9] with an absolute maximum difference E of 15.72%. For the Counto1 model, 47/75 cases lead to E smaller than 5% and 23/75 smaller than 10%, which gives 70/75 cases with E smaller than 10%, with a maximum difference of 16  The Bache and Nepper-Christensen and Hirsch-Dougill models underestimate the Young's modulus of LWAC measured in [3]. Bache and Nepper-Christensen model, 43/75 cases give E smaller than 10% and E ranges from 31.45% to 1.62%. For the Hirsch-Dougill model, 29/75 cases give E smaller than 10% with 23 cases smaller than 5%. It can be seen by examining Fig. 4 that the most accurate models are those of Maxwell, Counto1 and Hashin-Hansen which give less errors percentages ( Fig. 4 and Tab. 10).

Statistical analysis
In order to confirm what has been announced previously and distinguish the most suitable model for predicting the effective elasticity modulus of the LWAC, a global statistical study was carried out on all the experimental values of the three researchers (119 measures). To this effect, the mean values and standard deviation for all composite models used in this study and experimental data are calculated as seen in Tab. 10. Popovics Hirsch

CONCLUSION
he modulus of elasticity is a very important mechanical parameter, its determination sometimes involves impossible, difficult or costly tests, the alternative use of the biphasic laws in these cases appears very interesting but the choice of a model and not another remains a question which requires a precise examination and strongly depends on the type of materials chosen. In order to choose the optimized prediction composite model for Lightweight Aggregate Concrete, the purpose of this paper was to appraise the effective Young's modulus of LWAC using two-phase composite models. From the obtained numerical predictions, as confronted to existing experimental data and analytical results, the main findings are summarized below: When the Young's modulus of lightweight aggregates E g is much less than the Young's modulus of the mortar matrix in the lightweight aggregate concrete Em, Hirsch-Dougill models remain distant from experimental results and cannot be applied to predict the modulus of elasticity of LWAC. Using Popovics, Counto2 and Bache-Nepper Christensen composite models may not always produce accurate results. For 119 experimental values of Young's modulus for LWAC, the Maxwell, Counto1 and Hashin-Hansen seem the most reasonable for this purpose. The Maxwell model takes into account in the calculation of the effective elastic modulus of the contrast between the two phases (the mortar matrix and the light aggregates) represented by the coefficient  (E g /E m ) which made it possible to simulate the materials well and offered consequently more precise results if compared with other models. Thus, the precision of this prediction model demonstrates its effectiveness and potential application as a model for Lightweight Aggregate Concrete. The Maxwell model remains close from the experimental values with a man value error equal to 0.29 and a standard deviation equal to 5.27. In addition the Counto1 and Hashin-Hansen models provide a good prediction of T