Compressive study on recycled concrete: experiment and numerical homogenization modelling

This paper describes a study of recycled concrete under compressive loads. The study was conducted in two main parts. In the first part, experimental tests were carried out on concrete samples with varying levels of substitution (25%, 50%, and 75%) with recycled aggregate in order to measure the mechanical properties of the recycled concrete. In the second part of the study, a nonlinear homogenization model was developed on the basis of a classical secant approach to predict the behavior of recycled concrete. In this model, we assume that the behavior of the mortar phase, the concrete, and the recycled aggregates follow Mazars damage law. Comparison with the experimental data shows that the proposed homogenization model is accurate and efficient in predicting the correct nonlinear behavior of the recycled concrete. By better understanding the properties and behavior of recycled concrete, it will be possible to develop more effective methods for incorporating recycled materials into concrete structures.


INTRODUCTION
ecycled concrete (RC) made with recycled aggregates from crushing demolition waste offers economic, social, and environmental benefits compared to conventional concrete made with natural aggregates [1,2].Generally, RC exhibits weaker mechanical properties than conventional concrete due to the properties of the recycled aggregates [3,4].The compressive strength of recycled concrete (RC) is related to the properties and rate of recycled aggregates (RA).According to [3,5,6] , recycled aggregates cause a decrease in the compressive strength of recycled concrete.The mechanical properties and performance of such recycled aggregates are at least equal to the properties of natural concrete, which are in any case weaker than those of natural aggregates (NA).The decrease in mechanical properties of RC is also attributed to changes in the interfacial transition zone (ITZ) between aggregate and mortar resulting from the substitution of natural aggregates with recycled aggregates [7].Experimental investigations have shown that the thickness of ITZ in RC is around 55 µm, leading to a decrease in the average modulus of ITZs by approximately 10-30 % compared to the matrix mortar surrounding the aggregates [8].It is essential to examine the mechanical properties of recycled aggregates, old mortar, the interfacial phase between aggregates and old mortar, and the interfacial phase between old mortar and new mortar.These factors significantly influence the mechanical properties of RC.To understand the compressive behavior of RC as a fundamental characterization, various theoretical and numerical models have been developed, including empirical models, fracture mechanics approaches, and continuum damage mechanics [7,9] .These macroscopic models cannot clarify the link between the overall multiscale behavior of RC and its heterogeneities related to its constituent phases such as RA, NA, and pores.Therefore, advanced models based on homogenization theories, such as homogenization using the finite element method, have been proposed [10,11].These models consider the behavior of individual phases and their interactions and can yield valuable insights into the mechanical behavior of recycled concrete.However, it is worth noting that these finite element calculations can be computationally intensive and time consuming.Recent research has introduced a combined finite element method and analytical multiscale model that evaluate the effective elastic properties of recycled concrete using a MoriTanaka homogenization technique [11].This model captures the behavior of the material at different scales, limited to only the elastic domain.The developed multiscale modeling demonstrates good agreement with experimental results in elasticity.Another micromechanic analytical model predicts the strength of recycled concrete by considering the microstructure at different scales and incorporating a Drucker-Prager criterion for hydrate failure mechanisms at the microscopic scale [12].In this work, the study concerns the sensitivity of several parameters, such as old mortar content, water/cement ratio, and aggregate replacement ratio, but the behavior and evolution of the mechanical properties under loading have not been discussed.Recently, Gupta et al. [11]developed an analytical multiscale model based on a coated inclusion to predict the concrete's behavior.The considered microstructure of the whole concrete is constituted by multicoated aggregate embedded in the cement paste [13].These coated layers, or interfacial transition zones (ITZ), considered at the scale of cement paste as a composite formed with fine inclusions (clinker phase, hydration products, pores) embedded in C-S-H gel forms are numerically specified using the cement hydrate platform [14] to deduce the thickness of the ITZ and homogenized using the Mori-Tanaka model.The damage in each ITZ layer and cement paste is modeled based on a threshold strain value of cement paste, which results in a reduction of the elastic modulus of concrete and leads to nonlinear stress-strain behavior.The approach of Gupta et al. predicts the behavior of concrete up to peak stress but does not study the post-peak phase of the stress-strain curve.In addition, obtaining accurate mechanical properties at the lowest scale may be challenging and lead to additional difficulties.Based on experimental compressive study, in this work we aim to develop a numerical homogenization modelling using multiphase mean field model in a general consideration as the form of aggregate to catch the complex behavior of the recycled concrete beyond elastic state.We propose on the basis of continuum micromechanics a simple analytical method able to consider the ITZ in the concrete to predict stress strain compression behavior of several recycled concretes.Initially, the homogenization development concerns the elastic stage of recycled concrete [15,16].The model has demonstrated its potential in predicting the elastic properties of several recycled concretes, drawing inspiration from the work of Cammacho [17] and Pierard [18].Using the secant linearization, the prediction of the nonlinear behavior of composite can be established from the behavior of their individual constituents and the secant properties of each phase can be evaluated at each loading step according their local behavior law.At each step, the linear homogenization model can be applied.The main advantage of the secant homogenization methods is (i ) the good qualitative agreement between their predictions and the results obtained by finite element method, and (ii) the suitable description of the composite response under monotone loading [19,20] .The approach presented in this work, combined with planned experimental investigations, can be highly beneficial in predicting the mechanical behavior of recycled concrete structures.This integration of theoretical analysis with practical R experimentation is crucial for achieving a comprehensive understanding of the performance of such structures, ultimately contributing to their safe and sustainable design.The forthcoming experimental work will involve conducting a series of tests and measurements on recycled concrete specimens.These experiments will encompass various mechanical properties, including Young moduli, compressive strength and workability.By subjecting the specimens to the compressive loading and carefully monitoring their response, valuable data will be gathered to assess their structural behavior and performance.The results obtained from these experiments will provide critical insights into the mechanical characteristics of recycled concrete, aiding in the prediction of its performance in real-world applications.Several aspects of RC have been investigated in previous studies within the literature.These include examining the stress-strain relationship, determining compressive strength, analyzing the effects of uniaxial compression, studying different strengths of parent concrete [5,6], measuring water absorption capacity [19], exploring various types of coarse aggregates [16] , investigating different mix designs and mixing methods, and assessing the levels of recycled aggregate replacement.A multiscale numerical model is also developed and applied to analyse the creep of the recycled concrete [16].This paper is organized as follows: the first part is devoted to the experimental campaign at the macroscopic scale (method, material, tests, and results).Whereas the second part presents the modeling of the linear elastic behavior of RC and the proposed extension to simulate uniaxial loading.The numerical algorithm is also exposed in a third section.In Section 4, several applications are proposed.Firstly, the effective Young modulus is compared to experimental measures.Secondly, the simulations of the uniaxial compressive loading are confronted with the test results of RC with 25, 50, and 75 % of RA.The proposed model is finally used to simulate other RCs from the literature.In the end, some concluding remarks and perspectives are stated.

EXPERIMENTAL STUDY
he experimental program takes into account the volume ratio at which recycled aggregates are replacing natural aggregates.Four types of concrete samples are prepared.The first one is a natural concrete NC prepared only with natural aggregates; the other three types of samples are recycled concrete RC, in which the natural aggregates NA are substituted by recycled aggregates RA with different substitution rates (25%, 50%, and 75%), in order to study and analyse the influence of recycled aggregates on the properties of the concrete.

Materials
The cement used in this study is produced by a local company of the CPJ-CEM II/B 42.5 N type, with a characteristic hardened strength of 42.5 MPa and also complying with the Algerian (AN) and European (EN 197-1) standards.Its fineness modulus, according to the Blaine method, is 3700-5200 g/cm2.The natural aggregates NA are provided by a quarry and are produced after crushing rocks from quarries.They are divided into three granular classes: sand (0/3mm), gravel (3/8 mm), and gravel (8/15mm).The recycled aggregates RA are obtained after crushing and grinding demolition waste.The production of the recycled aggregates is done at the level of a local quarry.Recycled aggregates are divided into two classes: gravel (3/8mm) and gravel (8/15mm).The recycled aggregate is made up of two phases: an old mortar with a fraction of 0.6 and an original aggregate with a fraction of 0.4.T Sieve analysis was first performed in order to determine the distribution by weight of the particles of a material according to their dimensions (NF P 18-560).a)-Sieve analysis of natural sand 0/3 mm.b)-Sieve analysis of the NA and RA. Figure 2: sieve analysis of used materials.Fig. 2.a shows the sieve analysis of the sand (0/3mm) while Fig. 3.b shows the sieve analysis of the NA and the RA of fractions (3/8mm) and (8/15mm) respectively.We notice a slight difference in percentage of sieve between the NA and the RA [17].Some physical properties of the used NA and RA are shown in Tab.According to Tab. 1, the bulk and absolute density of the RA are lower than those of the NA, the absorption of NA (3/8mm) and NA (8/15mm) is 1.42% and 0.89%, and the absorption of RA (3/8mm) and RA (8/15mm) is 5.5% and 4.95% [21].It can be seen that the absorption of RA is five times that of NA.The decrease in weight and the increase in absorption of RA are directly attributed to the old mortar attached around the original aggregate.The old mortar contains voids and pores that, on the one hand, reduce the weight and, on the other hand, increase the absorption capacity.The Los Angeles coefficient (Lac) of RA (8/15 mm) is higher than that of NA (8/15mm) by 4.4%, which is attributed to the low wear resistance of the old mortar attached to the recycled aggregates [20].On the other hand, the Lacs of NA (3/8mm) and RA (3/8mm) are slightly different.Therefore, their fragmentation resistance is almost the same.The water used is the water available at the laboratory level, which is potable tap water.

Concrete specimens
The Dreux-Gorisse method is used for the formulation of concrete [22].The objective of this method is to determine, according to workability and resistance, the nature and quantity of materials necessary for the construction of one cubic meter of concrete (water, cement, sand, and gravel in kg/m 3 ).The prepared concrete is plastic concrete, which has a resistance of 25 MPa at the age of 28 days.In order to have clean materials devoid of all impurities, the aggregates are washed and then dried in a drying room for 24 hours at a temperature of 105 °C.In this work, four types of samples of concrete were realized, noted as NC, RC25, RC50, and RC75, respectively, in which 0%, 25%, 50%, and 75% of NC were replaced by RA.The formulation of the different types of samples is shown in Tab. 2.  For each type of concrete, three cylindrical samples (diameter 16cm and height 32cm) are prepared to determine the properties of the concrete in its hardened state.All these samples are cast in steel molds and prepared under the same conditions.We start with the preparation of materials (water, cement, sand, and gravel), then moisten the walls of the mixer.
The materials are put into the mixer successively, from the largest diameters to the smallest.After the materials are kneaded in a dry state for 30 seconds, add 10% mixing water and mix for 30 seconds.Finally, the rest of the mixing water is added and mixed for one minute and 30 seconds.The molds filled with concrete are vibrated using a vibrating table.The concrete specimens are kept for one day in molds in the laboratory at 25°C, removed 24 hours after casting, and kept in water until the day of testing, i.e. at the age of 28 days.

Experimental tests
Slump test.The slump test is used to determine the consistency of concrete by measuring its slump value under its own weight.This test is carried out on the different concrete formulations (NC, RC25, RC50 and RC75).The slump test results allow us to evaluate the effect of RA on fresh concrete parameters as well as its rheological properties.Ultrasonic pulse velocity (UPV) test.This test is used to determine the quality of the material, by measuring the speed of an ultrasonic wave passing through the tested element.This velocity is high when the material is compact and resistant, and vice versa.A piezoceramic source is electrically pulsed to generate ultrasonic waves that travel in the structural element and are then  Compression test.This test aims to determine the maximum compressive strength and the initial Young's modulus of the concrete samples at the age of 28 days.The compression test is carried out using an hydraulic press of load capacity of 2000 kN, the concrete specimen is subjected to a progressive loading, with a step of loading of 0.5 kN/second, the load applied is displayed on the machine screen and the shortening of the concrete is given by comparators, for each step of loading, one records the force and the corresponding deformation.The experimental device does not make it possible to follow the evolution of the curve of behaviour after the peak of deformation, because the reading of the deformation on the comparators in the phase after peak becomes difficult.

Results and discussion
According to the results presented in Tab. 3, the NC prepared only with NA has a slump of 8.5cm and the RC25 has a slump of 7.5cm, then both are plastic concrete, the RC50 has a slump of 5mm and the RC75 has a slump of 2.5mm, then both are firm concrete.From these results, it can be seen that the NC and the RC25 have the same consistency, so a substitution of 25% of NA by RA does not affect the condition of fresh concrete [23] .However, the further increase in RA leads to a significant decrease in slump and a change in consistency [24].These remarks are attributed to the addition of recycled aggregates (RA), which have a high absorption due to the voids present in the old mortar adhered to the original aggregate, and therefore, a higher rate of RA will increase the porosity, which absorbs more water from the mixing, leading to a decrease in slump and changing the consistency of the concrete [25].The results of the ultrasonic pulse velocity tests are also shown in Tab. 3. The NC shows better quality than the other concretes.As indicated in some previous investigations [26], the speed of the ultrasonic waves decreases with the increase of the RA ratio.The increase in the RA ratio leads to increased porosity in recycled concrete, which affects the transmission of ultrasound waves in concrete and causes a velocity slowdown [27].From Tab. 3, we can observe that the compressive strength of the samples decreases as the RA ratio increases from 2.6% for RC25 to 8.44% for RC75 compared with the NC.This decrease in strength is directly attributed to the mechanical and physical properties of recycled aggregate RA [28], as these are of lower quality than those of natural aggregate NA [29].
Fig. 5 shows the stress-strain curve of all samples recorded in the uniaxial compression test.According to this figure, the behavior curves can be divided into three parts.The first part is presented by the linear zone; the second is the nonlinear zone of the ascending branch; and the third part is represented by the descending branch [30] .We can notice that the concrete behavior curve changes with the incorporation of RA [31], from which we see an influence of RA on the concrete behavior.It is clear that the increase in the RA ratio induces a reduction of the initial elastic modulus as well as a reduction of the peak stress [3].The increase in the RA ratio increases the peak strain as well as the curvature flattening in the postpeak phase.In fact, when the NA is replaced by the RA, the presence of interphase ITZ's such as new mortar-natural aggregate, old mortar-original aggregate, and old mortar-new mortar induces more easily the development of microcracks and therefore reduces the concrete resistance as well as the elastic modulus [31].

NUMERICAL MODELLING OF RECYCLED CONCRETE MATERIALS
n the subsequent sections, we present a simplest and efficient numerical model based on the multiphase homogenization method.This model enables the prediction of the behavior of recycled concrete RC, encompassing both linear and nonlinear domains.In a previous work [33], the representative volume element RVE of recycled concrete is presented by three phases, new mortar NM, aggregates (recycled and natural aggregates) and voids.In the concrete microstructure, it is important to take into account the thin layers between aggregates and mortar that can include modified elastic properties from mortar called in several work as interfacial transition zone (ITZ).The mechanical identification properties of these interfacial zone required high performance experimental equipment like nanoindentation, Atomic Force Microscopy (AFM), Scanning Electron Microscopy (SEM), and specific procedures for grinding and polishing specimens [8,34].From Xiao et al experimental investigations [8] , the thicknesses of ITZ in recycled aggregate (between old mortar and NA) and in recycled concrete (between new mortar and aggregates) are found to be around 55µm which is very small volume fraction compared to aggregates in concrete.Moreover, the average modulus of ITZs appears to be approximately 70-90% of that of matrix mortar modulus surrounding the aggregates.

Nonlinear homogenization processing
The basic methodology for dealing with the non-linear behavior of heterogeneous materials on the basis of its representative volume element (RVE), in particular the RVE of recycled concrete, requires a set of simplifying assumptions to solve this complex problem.To do so, it is required to substitute the non-linear problem with a series of successive linear problems.It is important (i) to examine and clarify the behavior of each phase of recycled concrete when loads are applied and (ii) to I understand the link between the local strain and stress fields and the macroscopic loading imposed to the representative volume element of the recycled concrete.In the nonlinear framework, the mechanical properties of each constitutive phase of the recycled concrete depend on the strain-stress loading history.To process nonlinear homogenization, a linearization method has to be applied at each loading level.This linearization enables the use of a suitable linear homogenization method designed specifically for recycled concrete [33].Recently, in Barboura's work [35], the secant linearization is used to assess the elastoplastic of co-continuous composite.This secant linearization preserves the overall symmetry of the phases and the composite material throughout the monotonic loading.It establishes a linear relationship between the mean stress and strain of each phase as on can see the Fig. 6 [36].It provides a straightforward and intuitive way to characterize the mechanical response of each phase which the constitutive behavior can be described as follows: (y) = ( (y)) : (y) Figure 6: Secant linearization.
The secant stiffness tensor depends on the mean local strain field of each phase (I=NM, NA and RA) in the two-phase domain i G can be rewritten as follows: (y) = ( (y) ) : (y) Then the effective grains Gi can be estimated by: A , the localization tensor itself depends on the average strain in the phase.
The secant mean phase localization tensor depends on the linearized phase properties at the current loading step and the linear homogenization method of the heterogeneous material described below.

Elastic homogenization method for recycled concrete
The coupling of secant linearization with the three-step homogenization method [33] makes it possible to describe the behavior of multi-phase composites.First, the behavior of each phase of the composite is linearized by a secant approach, and then the behavior of the multi-phase composite is determined by the three-step homogenization method.In this part, we propose to extend the elastic homogenization model [33] to predict the nonlinear behavior of RC.This model is based on the secant linearization procedure of the nonlinear behavior of the constitutive phases.The imposed macroscopic strain field E on the RVE is assumed to be the same for each decomposed grain, which is equal to the average of the local strains in the recycled concrete RVE (see Fig. 7): The symbols < .> indicate the average over the volume of the RVE.
In a nonlinear case, the properties of each phase of grains i G become dependent on the macroscopic loading E applied to their contour.The secant linearization method consists of replacing any nonlinear behavior by a sequence of linearized behavior phases.This secant approximation method links the stress field of each phase of grains i G to the local deformation field by the secant stiffness tensor sct C .In general, this constitutive behavior can be described using deformation theory as follows: As previously described, the representative volume element (RVE) of RC can be viewed as a multiphase composite constituted by a matrix (new mortar), and inclusions, namely RA, NA, and voids.The form and properties of the inclusions can be different and are randomly distributed.To obtain the effective properties of RC, we adapt a multiphase homogenization procedure as described in [18].This elastic homogenization is based on a multistage homogenization method depending on the complexity of the microstructure.In this study, we propose a homogenization strategy composed of three homogenization steps.

First step
In the first step, we determine the effective properties of the recycled aggregate (RA), by using a homogenization model recently developed in [37] named GEEE model, serves as a powerful estimator for calculating the equivalent stiffness of multiphase ellipsoidal heterogeneities.The estimator is formulated explicitly, which facilitates its implementation in a computational program.The recycled aggregate is composed of two phases, old mortar (OM) with a stiffness tensor of OM C and original aggregate (OA) with a stiffness tensor of OA C .At this stage, we can determine an equivalent stiffness tensor RA eq C of the recycled aggregate (RAeq), which can be explicitly expressed as follows: with:

ITZ
) can be taken into account by the GEEE model using an iterative procedure, as explained in Gazavizadeh et al.'s work [37].This interphase is located between the original aggregate (OA) and the old mortar (OM).In this case, to obtain an equivalent recycled aggregate, we use two iterations of the GEEE model.Similarly, by using the GEEE model, the interface ( NM NA ITZ ) between the natural aggregate NA and the new mortar can be taken into account, so the natural aggregate is considered an inclusion surrounded by a layer of new mortar.Thus, we obtain an equivalent natural aggregate NA eq .

Second step
The RVE of recycled aggregate is decomposed into grains ( i G ).Each grains is constitute by two phases, new mortar NM as a matrix and aggregates as inclusion (NA eq or RA eq ), the volume fraction of NM in the VER is the same for all grains ( i G ) The inclusions of the same form are distributed randomly in the matrix.To homogenize each grain on its local axis linked to the inclusion form, we can use the Mori Tanaka model and obtain: where i G C denotes the stiffness of grain, C I denotes the stiffness of inclusion.In the case of porous phases, the homogenized tensor of grain can be expressed as follows: with I denotes the identity tensor of order four and NM P the Green operator related to Eshelby tensor NM S relative to the form of the aggregate (RAeq or NA) or voids surrounded by new mortar matrix (NM) .For a random distribution of inclusion in grains ( i G ), we need to take into account the effect of the inclusion orientation in the matrix on the macroscopic properties.The random orientation distribution is represented by the Euler angles (θ, φ, ψ).We simply require average over the all direction of orientation of the effective stiffness tensor of the two-phase grain Where ij a are the coefficients of the rotation matrix, allowing the transformation of a fourth-order tensor from one frame into another following the Euler angles.

Third step
The RVE of recycled concrete is made up of homogenized grains i G , which constitutes a multi-phase composite, according to Pierrad [18], the macroscopic behaviour of recycled concrete can be determined by the Voigt model: with Gi f denotes the volume fraction of gain

Assumption for natural and recycled aggregates in the concrete
The real RA or NA can be assumed to be of spheroidal form and randomly distributed in the new mortar.A statistical measure of the minimum and maximum width of natural or recycled aggregate after sieve analysis shows that the aspect ratio is between 0.5 and 2. To justify the choice of spherical aggregates, we compare the effective properties of two concretes: the first contains spherical aggregates, and the second contains spheroidal aggregates.Both aggregates are randomly distributed in the new mortar.Fig. 8 presents the normalized Young modulus with respect to aspect ratio, from oblate to prolate forms of inclusion for the two concretes.We observe that the values of the normalized Young modulus of concrete with randomly oriented spheroidal aggregates converge to those with spherical aggregates.We can note that when the aspect ratio is between [0.5 and 2], the difference is not significant, as shown in Fig. 8.In the simulation, for the sake of simplicity, the form of aggregates is taken as spherical.To resume, the homogenized stiffness tensor G i hom sct

C
relates the macroscopic strain load to the average stress of grain i G .
That is the key for the secant homogenization procedure to correctly estimate the constitutive behavior.The homogeneous secant stiffness tensor of each grain is determined by Mori Tanaka's two-phase homogenization scheme: The aggregate obtained is a multi-phase composite, which is homogenized with the Voigt model as follows:

Behaviour of the constituent phases of recycled concrete
In this section, we describe the essential behaviour laws that govern each phase of concrete, covering aggregates and mortar.These laws are crucial to understanding how concrete behaves under different loading conditions, including compression.

Aggregates
The natural aggregates NA and the equivalent recycled aggregates RAeq, represent the reinforcement or inclusion phases in the recycled concrete.The behaviour of the NA and RAeq is assumed to be linear elastic, so the average stress in those two phases is linearly linked to the average strain.

Matrix
The new mortar is governed by the Mazars [39,40] damaged law, widely used in the literature [41,42], the decrease of the material rigidity under effect of micro cracks is driven by a scalar internal variable D. The stress-strain relationship is given by the following equation: The variable of damage D results from a combination of a traction damage t D and a compression damage c D , it can be written as follow: The coefficient t α which carries out the coupling between the damage in traction and the damage in compression, it is equal to 0 in the total absence of traction, and equal to 1 in the total absence of compression.The damage decompositions under traction or compression are defined by the following equations: ε : The threshold strain of damage.

 , t
A , c A , t B and c B : Material parameters to identify.eq ε : The equivalent strain which controls the damage D in the material and defines the load surface f , such that: As extension is the main reason of the concrete cracking, the equivalent strain is defined as follows: where + < ε > is the positive part of the principal strains defined as follows:

General algorithm
We apply the secant homogenization strategy to multiphase concrete composites by taking the behavior law of each constituent into account.The new mortar matrix (NM) is governed by the Mazars [39,40] damage law.The two types of inclusions, NA and RAeq remain linear and elastic.In the secant formulation, the macroscopic stress and the stiffness tensor sct hom C are calculated at each step of macroscopic loading.In the following, in order to simplify the notation, the domain of a pseudo-grain ( i G ) is denoted by (V), its constituents are denoted by (V NM ) for the mortar matrix, and (V I ) for the inclusion phase (NA) or (RAeq).The numerical algorithm starts with the knowledge of the applied macroscopic strain n+1 n = +Δ E E Eat step (n+1).The properties of each phase of the recycled concrete (NM, NA, and RA eq ) are known in the previous step (n).


The optimization of these parameters involves utilizing a simplex algorithm implemented in Matlab code.The objective is to minimize the residual error between the numerical stress obtained from the homogenization model and the experimental stress depicted in Fig. 5.The residual error is calculated using the following equation: with: er R : Total error.In Fig. 9, the experimental stress strain curve of natural concrete NC and the numerical results from the nonlinear homogenization method using the optimized Mazars parameters relative to the new mortar are herein superposed.In this numerical simulation, we assume that the concrete is composed of a single grain composed of new mortar NM and natural aggregate NA.In this case, the natural aggregates remain elastic, and their elastic properties can be taken from the literature [34].Then, the parameters NM NM NM  Similarly, we apply the identification method to another compressive test of recycled concrete from the work of Xiao [4] in particular the experimental result of natural concrete NC.As mentioned before, new mortar NM obeys to Mazars law [39,40], and the natural aggregate NA remains elastic.Using the identified Mazars's parameters of NM, listed in Tab. 5, the confrontation between the simulation and the experimental results is shown in Fig. 10.As we can observe, the identified Mazars's parameters of NM allow us to correctly reproduce the experimental measurements of NC.

NUMERICAL SIMULATION OF RECYCLED CONCRETE
he multiphase homogenization model proposed in the previous section is used to simulate the compressive tests of different RC specimens and to compare the numerical results with the experimental data.First, we discuss the numerical results in the case of linear elasticity.Secondly, we extend them to nonlinear elasticity.Finally, we confront the proposed model with the experimental results reported in the work of Xiao [4].

Linear properties of recycled concrete
In this first application, all the phases are considered linear, elastic, homogeneous, and isotropic.The shapes of NA and (RAeq) are assumed to be spherical.The mechanical properties of the phases are summarized in Tab.6, and the elastic properties of old mortar are obtained from literature [34,38].In the following simulations, the volume fraction of NA is assumed to be 0.4.We start with the calculation of the effective stiffness tensor of the RA using Eq.10.The effective elastic stiffness tensor of the RC is obtained from the mixture law of the equivalent grains calculated by using the Mori-Tanaka model, (Eq.12).T Fig. 11 presents the effective Young modulus of recycled concrete with respect to recycled aggregate volume rates.The experimental data are also plotted for comparison.One can observe that the maximum difference between the numerical results and the experimental values is less than 5%.Therefore, the proposed homogenization model is highly accurate in predicting the linear elastic behavior of RC.Moreover, the elastic homogenization strategy developed in this study is easy to implement.

Simulation of experimental compressive tests
In this second application, several simulations are carried out to predict the compressive behavior of recycled concrete for varied proportion of RA.Using the identified parameters shown in Tab. 4 and 6, we simulate the compressive behavior of the recycled concretes RC25, RC50 and RC75 considered here as a three-phase composites (NM, NA and RAeq).The nonlinear homogenization strategy described in the previous section is used.We assume that the new mortar matrix obeys to the Mazars law [39,40] and the aggregates remain elastic.Fig. 12.a, 12.b and 12.c show the numerical simulation results compared to the experimental ones.A good agreement was obtained.Tab.7 shows the experimental and numerical peak stress and its corresponding strain for the different studied recycled concretes.We can notice that the model slightly underestimate the peak stress around 6% of error for RC25.For RC50 and RC75, the difference between the experimental and numerical stress is around 1%.However, the difference between the strain at the peak estimated by the model and that of experimental is generally between 1% to 9%.From the results shown in the previous figures and Tab. 7, it can be seen that the numerical model describes well the behavior of concrete.The slope of the numerical stress-strain curve is close to that of the experimental curve, the numerical model estimates correctly the maximum stress.However, the numerical strain at the peak is slightly overestimated compared with the experimental results.
Simulation results with literature Xiao et al [4] In this last applications, we apply the nonlinear homogenization modelling to simulate the compressive behavior of recycled concretes studied experimentally in [4].In this work, 5 types of concrete are formulated based on ordinary Portland cement with a compressive strength of 32.5 MPa.The formulation and the physical properties of the natural aggregates and the recycled aggregates are reported in Tabs.1-2 in [4].The RA used is from the waste concrete brought from the runway of an airport in Shanghai, PR China.Tab. 8 presents the experimental and numerical stress and strain peak, from results, the experimental pic stress and strain of NC is 26.07 MPa and 0.0018, respectively, however the values shown by the model are 26.25 MPa and 0.0019, and the model overestimated the pic stress and strain by 0.7% and 5%.For RC100, the experimental pic stress and strain are 23.56MPa and 0.0023, respectively, whereas the numerical model estimates a pic stress and strain of 24.49MPa and 0.0021, respectively, then the pic stress is underestimated by 3% and the pic strain is overestimated by 8%.The maximum stress estimated at the peak is well predicted, as well as the ultimate strain, in the post peak phase, even if the stress decreases the strain increases, and thus the model predicts almost in perfect agreement.[4].In the linear phase, the numerical slope is nearly identical to that obtained experimentally.The proposed nonlinear homogenization model allows to correctly estimate the behavior in compression especially before the peak whatever the fraction of recycled aggregates in the concrete.In addition, the maximum stresses of the different concretes are correctly predicted.It can be seen that even though the nonlinear homogenization model reproduces globally correct softening behavior after the peak, a non-negligible overestimation can be observed.This overestimation decreases when the fraction of recycled aggregates increases.These encouraging results concern the results obtained by an approximation of the local fields by their average that is considering a classical secant linearization which tends to overestimate the nonlinear behavior compared to a modified secant linearization [42] and [43].As already presented previously, this approximation link the stress field linearized by the quadratic mean to the local strain field, for the construction of the nonlinear secant homogenization model.Other ways for improvement can also be envisaged, namely the consideration of a damage law with the consideration of the concrete lateral damage.The difficulty lies in particular in the step of identifying the model parameters, which is an essential step for its prediction efficiency.

CONCLUSION
his paper presents an experimental and numerical study on recycled concrete.The following remarks and conclusions can be drawn from this work: 1.The experimental work involved uniaxial compression tests on natural concrete (NC) and recycled concrete (RC) with different substitution rates of natural aggregates (NA) by recycled aggregates (RA).The stress-strain curves obtained from these tests showed that the addition of recycled aggregates in the concrete affects its mechanical properties in both the fresh and hardened states.Specifically, the incorporation of recycled aggregates in new concrete leads to a decrease of the Young modulus and the compressive strength, and an increase of the maximal strain.2. A linear elastic homogenization model for recycled concrete is established by combining the GEEE-MT-Voigt methods through a three-step homogenization procedure.This model is then extended to the nonlinear regime by using a secant formulation.Mazars damage law is incorporated into the model to describe the softening behavior.The recycled concrete can be simulated as a three-phase composite by using the proposed homogenization procedure.3. The proposed homogenization model was validated by comparing the numerical results with the experimental behavior of NC and different RCs.The numerical predictions show a good agreement with the experimental results for different substitution rates.T Overall, the findings of this study provide valuable insights into the properties and behavior of recycled concrete, and may contribute to the development of more effective methods for incorporating recycled materials into concrete structures.

Figure 1 :
Figure 1: Waste used to obtain recycled aggregates (right): demolition waste from laboratory waste (left).

Figure 3 :
Figure 3: Materials used in the manufacture of concrete.
sensed by the receiver on the opposite side of the sample.The source and receiver signals are recorded by an Olson Instruments data collection platform equipped with an UPV system.a) The Abrams cone slump test.b) Ultrasonic pulse velocity (UPV) test.c) Compression test device.

Figure 5 :
Figure 5: Behavior in uniaxial compression of concrete NC, RC25, RC50 and RC75.The Young modulus depends on the phases that constitute the concrete.According to Tab. 3, the Young moduli of recycled concrete (RC25, RC50, and RC75) are lower than those of NC.The decrease in Young's modulus is a function of the replacement rate[32].This decrease is attributed to recycled aggregates (RA), which have lower elastic rigidity compared to natural aggregates (NA).

(
given by the localization tensor of the mean secant strain of order 4 of phase (I) as follows: Eq.4) presents a nonlinear problem, because the average strain in each phase I (y) ε depends on the localization tensor I sct

Figure 7 :
Figure 7: Grains subjected to a macroscopic deformation E.

Figure 8 :
Figure 8: Normalized Young's modulus as a function of aspect ratio.


Loop on each pseudo grain i G  Initialization in the new mortar matrix phase NM < ε > =E . Call the Mazars model with NM < > ε as an argument, this model gives us the secant stiffness of the matrix sct NM C and the average stress NM < > σ . Calculate the secant stiffness tensor sct hom C using the (Eq.12). Calculate the localisation tensor of the strain in the matrix phase Calculate the localisation tensor of the deformation in the inclusion phase.Calculate the deformation in the inclusion phase i i < > = : ε A E Verify the accounting of the deformations in the matrix phase If yes, no more iterations on this pseudo grain. Otherwise, perform a new iteration with Calculate the secant modulus of the RVE and the macroscopic stress.
rom the experimental test on the concrete specimen, we aim to identify the needed parameters of Mazar's model applied to the new mortar.In our analysis, we suppose that all the aggregates introduced in the mortar to prepare the whole concrete remain elastic in all the compressive loading paths.As detailed in the previous section, the parameters of the Mazars model are NM NM NM c c D A , B ,  .As suggested in this law, these parameters fluctuate from reasonable values as stated in Mazars's model identified in our algorithm as a constraint as:

exp σ : Experimental stress at loading direction. hom 11 
: Numerical stress estimated by the homogenization model at loading direction, depending on imposed deformation loading ε(i) and the model parameters NM c A , NM c B and NM D ε .N: Number of loading step.

Figure 9 :
Figure 9: Behavior modelling of natural concrete NC compared to the experimental behavior.

Table 4 :
,ε obtained by the identification method using the homogenization strategy as described previously are shown in Tab.4Mechanical properties and Mazars parameters of different phases (NM, NA).

Figure 11 :
Figure 11: Effective Young modulus of recycled concrete.

Figure 12 :
Figure 12: Behaviour modelling of recycled concrete RC compared to the experimental results.

Fig. 13 .
Fig.13.a, 13.b, 13.c and 13.d show the numerical simulations of the compressive behavior of different RC (RC30, RC50, RC70 and RC100) obtained by the nonlinear secant homogenization model, compared to the experimental data[4].In the linear phase, the numerical slope is nearly identical to that obtained experimentally.The proposed nonlinear homogenization model allows to correctly estimate the behavior in compression especially before the peak whatever the fraction of recycled aggregates in the concrete.In addition, the maximum stresses of the different concretes are correctly predicted.It can be seen that even though the nonlinear homogenization model reproduces globally correct softening behavior after the peak, a non-negligible overestimation can be observed.This overestimation decreases when the fraction of recycled aggregates increases.These encouraging results concern the results obtained by an approximation of the local fields by their average that is considering a classical secant linearization which tends to overestimate the nonlinear behavior compared to a modified secant linearization[42] and[43].As already presented previously, this approximation link the stress field linearized by the quadratic mean to the local strain field, for the construction of the nonlinear secant homogenization model.Other ways for improvement can also be envisaged, namely the consideration of a damage law with the consideration of the concrete lateral damage.The difficulty lies in particular in the step of identifying the model parameters, which is an essential step for its prediction efficiency.

Figure 13 :
Figure 13: Results of numerical modelling for different fraction of recycled aggregates.

Table 1 :
1. Properties of the used aggregates (sand, NA and RA).

Table 5 :
Properties of the constituent phases of recycled concrete.

Table 6 :
Mechanical properties of RC constituents.

Table 7 :
Numerical and experimental pic stress and strain.

Table 8 :
Numerical and experimental pic stress and strain.