Micromechanics-Based Simulations of Compressive and Tensile Testing on Lime-Based Mortars
MMicromechanics-Based Simulations of Compressive and TensileTesting on Lime-Based Mortars
V. Neˇzerka a, ∗ , J. Nˇemeˇcek a , J. Zeman a a Faculty of Civil Engineering, Czech Technical University in Prague, Th´akurova 7, 166 29 Praha 6, CzechRepublic
Abstract
The purpose of this paper is to propose a continuum micromechanics model for the simula-tion of uniaxial compressive and tensile tests on lime-based mortars, in order to predict theirstiffness, compressive and tensile strengths, and tensile fracture energy. In tension, we adoptan incremental strain-controlled form of the Mori-Tanaka scheme with a damageable matrixphase, while a simple J yield criterion is employed in compression. To reproduce the be-havior of lime-based mortars correctly, the scheme must take into account shrinkage crackingamong aggregates. This phenomenon is introduced into the model via penny-shaped cracks,whose density is estimated on the basis of a particle size distribution combined with the re-sults of finite element analyses of a single crack formation between two spherical inclusions.Our predictions show a good agreement with experimental data and explain the advantages ofcompliant crushed brick fragments, often encountered in ancient mortars, over stiff sand par-ticles. The validated model provides a reliable tool for optimizing the composition of modernlime-based mortars with applications in conservation and restoration of architectural heritage. Keywords: micromechanics, stiffness, strength, fracture energy, Mori-Tanaka method,mortar, shrinkage cracking
1. Introduction
The lime-based mortars were widely used as masonry binder in ancient times [1, 2]. Nowa-days, they are often required by the authorities for cultural heritage for repairs of old masonrybecause of their compatibility with the original materials [3, 4]. The substitution of lime-based mortars with binders based on Portland cement turned out to be inappropriate, becauseof the damage to the original masonry due to high stiffness contrast and presence of solutablesalts [5]. On the other hand, the calcitic matrix of pure lime mortars is relatively weak, morecompliant [6, 7], and susceptible to shrinkage up to 13 % [8, 9].To improve the durability and strength of ancient mortars, masons often used additives richin silica (SiO ) and alumina (Al O ) [6, 10, 11] in the form of volcanic ash or crushed ceramicbricks, tiles or pottery [12, 13]. In recent years, metakaolin has become a very popular alterna-tive to these ancient additives, because of its high reactivity [6, 8, 10]. On the other hand, the ∗ Corresponding author
Email addresses: [email protected] (V. Neˇzerka), [email protected] (J. Nˇemeˇcek), [email protected] (J. Zeman)
Preprint submitted to arXiv December 8, 2015 a r X i v : . [ c ond - m a t . m t r l - s c i ] D ec rushed brick fragments are considered rather as an inert aggregate since the hydraulic reac-tion, if any, can take place only at the interface between the fragments and surrounding matrix.Moreover, the formation of hydraulic products requires the presence of moisture [14], signifi-cant amount of time [15], and ceramic clay fired at appropriate temperatures [16]. Beside thematrix-enhancing additives, the mechanical properties of lime mortars can be also improved byoptimizing the amount and composition of aggregates [6, 7, 17], mostly via a time-demandingtrial-and-error procedure.The goal of this paper is to render the design process more efficient by proposing a simplemodel for the prediction of basic properties of lime-based mortars in tension and compression.Our developments have been inspired by earlier studies dealing with micromechanics of cementpastes in compression [18] and in tension [19]. The first study by Pichler and Hellmich [18]exploits a two-level homogenization approach combining the self-consistent [20, 21] and Mori-Tanaka [22, 23] schemes and a J -based criterion to estimate the compressive strength. In thelatter work, Vorel et al. [19] estimated the tensile strength and fracture energy by combining theincremental form of the Mori-Tanaka method at a single-level with the crack band model [24]to account for the distributed matrix cracking. However, the crucial feature missing in bothmodels is the effect of shrinkage-induced cracks that are intrinsic to the mechanical propertiesof lime-based mortars.The shrinkage-induced cracking in cement-like materials has been addressed by both mod-elling and experimental studies. Detailed analytical investigation into shrinkage cracking arounda single cylindrical aggregate was performed by Dela and Stang [25] to estimate crack growthin high-shrinkage cement paste. Behavior of multi-aggregate systems was addressed numeri-cally by Grassl and Wong [26] using a discrete lattice model; their findings were in agreementwith the cracking patterns observed by Bisschop and Mier [27]. Backscattered electron mi-croscopy (BSE) confirmed that the lime-based mortars rich in sand suffer from an extensivematrix cracking [17]. Based on these studies and also on our independent experimental inves-tigations [8, 28], we decided to represent the shrinkage cracks as penny shaped polydispersevoids in our homgenization scheme.Based on these considerations, the model proposed in Section 2 operates at two scales,see Figure 1. At Level I, we account for the individual components of mortar, such as limematrix, sand or brick particles, and distributed voids. At Level II, the shrinkage cracks areintroduced into the homogenized material from Level I. Details of this procedure are providedin Section 2.1, with the goal to estimate initial elastic properties by the Mori-Tanaka procedureat Level I, Section 2.1.1, and the dilute approximation at Level II, Section 2.1.2. The den-sity and size distribution of the penny-shaped shrinkage cracks are determined from a crackformation criteria, proposed in Section 2.1.3 on the basis of three-dimensional finite elementanalyses of shrinkage-induced cracking between two isolated inclusions. Two extensions of theelastic model are presented next. The strength under stress-controlled uniaxial compression isestimated in Section 2.2 on the basis of the J stress invariant in the matrix phase. Under strain-controlled uniaxial tension, Section 2.3, we employ the incremental form of the Mori-Tanakascheme coupled with an isotropic damage constitutive model to estimate the tensile strengthand fracture energy.Having introduced our model, in Section 3 we specify the input data for individual compo-nents, along with the experimental procedures used to acquire them, and the composition of thetested mortar samples. Section 4 is dedicated mostly to the model validation, concluded by thedetermination of the optimal mix composition. Finally, we summarize our results in Section 5and outline the strategy how to translate them to the structural scale.2n the following text, the condensed Mandel representation of symmetric tensorial quanti-ties is employed, e.g. [29]. In particular, the scalar quantities are written in the italic font, e.g. a or A , and the boldface font, e.g. a or A , is used for vectors or matrices representing second-or fourth-order tensors. A T and ( A ) − denote the matrix transpose and the inverse matrix,respectively. Other symbols are introduced later, when needed.
2. Model
We consider an RVE occupying a domain Ω , composed of m phases indexed by r at Level I,and penny-shaped shrinkage-induced cracks reflected at Level II. The matrix is representedby r = 0 and indexes r = 1 , ..., m refer to heterogeneities of spherical shape or sphericalshell in the case of interfacial transition zone (ITZ) around sand grains, see Figure 1. Thevolume fraction of r -th phase, having the volume | Ω ( r ) | , is provided by c ( r ) = | Ω ( r ) | / | Ω | . Notethe representation of sand / crushed brick particles by spheres is less realistic then e.g. byellipsoids [30], but the effect of the introduced errors is minor relative to the accuracy of theinput data, cf. Section 3. sand grain (2)ITZ (3)void (1) matrix (0)brick fragment (2)void (1) Level I: crack
Level II:
Figure 1: Scheme of the micromechanical model of mortars with various aggregate-types; thenumbers in parentheses refer to the indexes of individual phases.
The elastic response of individual phases is described by the material stiffness matrix L ( r ) .Since all phases are considered as geometrically and materially isotropic, the matrix L ( r ) canbe decomposed using the orthogonal volumetric and deviatoric projections I V and I D , e.g. [29,p. 23], L ( r ) = 3 K ( r ) I V + 2 G ( r ) I D , (1)where K ( r ) and G ( r ) denote the bulk and shear moduli of the r -th phase.3nder the dilute approximation, the mean strain in individual phases, ε ( r ) , is related to themacroscopic strain, ε , via the dilute concentration factors, ε ( r ) = A ( r )dil ε . (2)In the Mori-Tanaka scheme, the strain in individual phases can be found as ε ( r ) = A ( r )dil ε (0) ,where ε (0) is the strain within the matrix found as ε (0) = A MT ε , (3)where the strain concentration factor A MT is provided by A MT = (cid:18) c (0) I + m (cid:88) r =1 c ( r ) A ( r )dil (cid:19) − . (4)Because of isotropy, the effective stiffness at Level I is fully specified by the effective bulk, K Ieff , and shear, G Ieff , moduli K Ieff = c (0) K (0) + m (cid:88) r =1 c ( r ) K ( r ) A ( r )dil , V c (0) + m (cid:88) r =1 c ( r ) A ( r )dil , V , G Ieff = c (0) G (0) + m (cid:88) r =1 c ( r ) G ( r ) A ( r )dil , D c (0) + m (cid:88) r =1 c ( r ) A ( r )dil , D (5)that depend on the volumetric and deviatoric components of the dilute concentration factors A ( r )dil = A ( r )dil , V I V + A ( r )dil , D I D , r = 1 , ..., m. (6) Uncoated Inclusions.
The dilute concentration factors for spherical particles follow from theseminal Eshelby work [31]: A ( r )dil , V = K (0) K (0) + α (0) ( K ( r ) − K (0) ) , A ( r )dil , D = G (0) G (0) + β (0) ( G ( r ) − G (0) ) , (7)where α (0) and β (0) depend on the matrix Poisson’s ratio, ν (0) , as: α (0) = 1 + ν (0) ν (0) ) , β (0) = 2(4 − ν (0) )15(1 − ν (0) ) . (8) Coated Inclusions.
The more involved case of particles coated by spherical shells was solvedby Herv´e and Zaoui [32]. Their introduction into the scheme makes the model sensitive to thegrain-size distribution because the strain concentration factors depend on the radius of sandgrains relative to their coating. From that reason the sand aggregates (2) and ITZ (3), Figure 1,must be subdivided into sub-phases m δ corresponding to individual grain-size intervals. Theseare denoted by indices ( , δ ) and ( , δ ), where δ = 1 , ..., m δ .Spatially, the dilute concentration factors for sand grains and surrounding ITZ, both repre-sented by their outer radii, R (2 ,δ ) and R (3 ,δ ) , and Poisson’s ratios, ν (2) and ν (3) , in the form A (2 ,δ )dil , V = 1 Q , A (3 ,δ )dil , V = Q Q (9)4nd A (2 ,δ )dil , D = A − R (2 ,δ )2 − ν (2) B , A (3 ,δ )dil , D = A − R (3 ,δ )5 − R (2 ,δ )5 (1 − ν (3) )( R (3 ,δ )3 − R (2 ,δ )3 ) B , (10)where the auxiliary factors Q , Q , A , A , B and B are provided in [33, Appendix A].The volume fractions of the coatings, { c (3 ,δ ) } m δ δ =1 , are determined from the given volumefractions of the sand sub-phases, { c (3 ,δ ) } m δ δ =1 , as c (3 ,δ ) = (cid:18)(cid:18) R (3 ,δ ) R (2 ,δ ) (cid:19) − (cid:19) c (2 ,δ ) . (11)Because the coatings are assume to replace the matrix surrounding the inclusions, their totalvolume fraction m δ (cid:88) δ =1 c (3 ,δ ) is subtracted from the matrix volume fraction c (0) . The effective stiffness at Level II accounts for the effects of randomly distributed cracksunder the dilute approximation. This reduction is accomplished by introducing additional com-pliances H E and H G [34]: H E = fE Ieff − ν Ieff 2 )(10 − ν Ieff )45(2 − ν Ieff ) , H G = fE Ieff − ν Ieff 2 )(5 − ν Ieff )45(2 − ν Ieff ) , (12)where f is the crack density parameter, determined later in Section 2.1.3, E Ieff and ν Ieff are theeffective Young’s modulus and the Poisson ratio at Level I, obtained from (5) through, e.g. [29,p. 23], E Ieff = 9 K Ieff G Ieff K Ieff + G Ieff , ν
Ieff = 3 K Ieff − G Ieff K Ieff + G Ieff ) , (13)The effective Young’s and shear moduli of the cracked composite then follow from E II = (cid:18) E Ieff + H E (cid:19) − , G IIeff = (cid:18) ν Ieff ) E Ieff + H G (cid:19) − . (14) The criteria for the formation of shrinkage cracks were established based on results of 3DFE model containing 200k to 700k tetrahedral linear elements, depending on the RVE size.The average average element size was equal to 0.1 mm. The RVE consisted of two sphericalinclusions embedded in a matrix. The model was supported in three corners in such a wayto allow contraction of the RVE in all directions without external constraints. The shrinkagewas introduced into the model via incrementally increased matrix eigenstrain and the resultingsystem was solved by the Newton-Raphson algorithm. Because the study was focused onestablishing the critical gap between the inclusions, the simulations were stopped after the crackbetween the inclusions appeared or if the eigenstrain reached , since such value correspondsto the maximum shrinkage of lime-based pastes [8].The simulations were performed in OOFEM software [35], utilizing the anisotropic damagematerial model by Jir´asek [36] for the matrix, and the isotropic elastic model for the aggregates.5he anisotropic damage material model utilizes the concepts from the microplane theory — thedamage variable characterizes the relative compliance of the material for each microplane di-rection and therefore the stiffness is not reduced parallel to the crack. The material parametersof individual phases used for the analysis are summarized in Table 1. ITZ around aggregateswas not explicitly modeled; however, the zone of damage around aggregates developed spon-taneously due to tensile stresses perpendicular to the grains at the interface, Figure 4). Themesh-independence was ensured by the crack band approach [24].Figure 2: FE simulation of the crack formation between two aggregates of a size equal to 0.3and 1.0 mm, respectively; visualized by ParaView software: axonometric view (left), front view(right). The red color indicates the formed cracks (the damage variable is equal to 1.0), whilethe blue regions represent the matrix volume with no damage.The results of the simulations appear in Figure 3, in which we plot the critical shrinkagestrain as a function of the dimensionless gap between the particles, g = 2 l d + d , (15)where l stands for the face-to-face particle distance, and d and d are the particle diameters.Two particle types (compliant brick and stiff sand) and three ratios of the particle diameterswere considered. For the shrinkage strain exceeding ≈ . (which is much below the typicalvalue of ≈ measured in [8]), we observe that a shrinkage crack between the particlesdevelops if the gap is smaller than a critical value g crit ; otherwise no crack forms. The criticalvalue is only weakly dependent on the diameter ratio, but depends strongly on the particlestiffness – in what follows we consider g crit = 0 . for the sand aggregates and g crit = 0 . forthe brick aggregates, respectively. Crack Density Parameter.
A circular penny-shaped crack is assumed to form between twoneighboring particles once their dimensionless gap is smaller than g crit . Given the number ofthe particles, k , distributed within a represenative volume element (RVE), Ω , the crack densityparameter is defined as [34] f = 1 | Ω | (cid:88) { i,j =1 ,...,k : g ij (cid:53) g crit } ( l ij ) , (16)6 gap between particles / mean diameter [-] s h r i n k ag e n ee d e d f o r t h ec r a c k f o r m a t i o n [ % ] sand 1:1sand 1:3sand 1:9crushed bricks 1:1crushed bricks 1:3crushed bricks 1:9 Figure 3: Critical relative gap, g crit , indicated by dashed line for the compliant brick and stiffsand inclusions of three ratios of their diameters, when embedded in high-shrinkage (signifi-cantly exceeding . ) matrix.recall Eq. (15).The input data to the crack density analysis were generated by an in-house packing algo-rithm for polydisperse spheres implemented in MATLAB. For the target volume fraction of theparticles, c (2) , and the given curve, we generate the particle distribution by a random sequentialaddition algorithm, proceeding from the largest particles to the smallest ones. As for the RVEsize, we found that the size of 5-times maximum particle diameter yields the coefficient ofvariation in the density parameter f less than ≈ , Figure 4b, which is sufficient for practicalpurposes. Inspired by phenomenological, e.g. [37], and micromechanical, e.g. [18], models for ce-mentitious materials, we adopt the von-Mises failure criterion for the matrix phase, (cid:113) J (0)2 − f (0)c √ , (17)to estimate the mortar strength in compression. In Eq. (17), f (0)c denotes the matrix compressivestrength, and the second deviatoric invariant in the matrix phase, J (0)2 , is determined from theaverage matrix stress, σ (0) , through J (0)2 = 12 σ (0)T I D σ (0) . (18)The average stress in the matrix phase σ (0) follows from the basic assumption of the Mori-Tanaka method, Eq. (3) as σ (0) = L (0) A MT (cid:0) L IIeff (cid:1) − σ , (19)where σ stands for the applied macroscopic stress. The mortar compressive strength f c is foundby subjecting the sample to the macroscopic uniaxial stress state σ = [ − f c , , , , , T , suchthat the condition (17) is satisfied. 7 a) example of polydisperse system RVEsimulation ( c (2) = 0 . ; the red lines indi-cate diameters of cracks between aggre-gates (if formed) sample size / maximum particle diameter [-] c o e ffi c i e n t o f v a r i a t i o n [ % ] monodispersepolydisperse (b) relationship between RVE size relative to themaximum particle diameter and variation of thecrack density parameters (based on 200 simula-tions) Figure 4: MATLAB simulations of shrinkage-induced cracking within RVE
Primarily the matrix phase is subjected to damage, but also the ITZ stiffness must be re-duced to avoid locking effects. The materials’ softening is driven by the damage evolution inthe matrix phase at Level I σ ( d ) = (1 − ω (0) ) L ( d ) ε ( d ) = L ( d )sec ε ( d ) , (20)where d = 0 , refers to the damageable phases. Because we assume that the strain in thematrix is increasing monotonically, the magnitude of damage in the matrix phase follows from ω (0) = 1 − ε ε (0)eq exp (cid:18) − ε (0)eq − ε ε (0)f − ε (cid:19) , (21)where ε (0)eq denotes the Rankine effective strain, determined from the tensile parts of the princi-pal strains (cid:104) ε (0) I (cid:105) + via ε (0)eq = (cid:107) ε (0) (cid:107) = max I =1 (cid:104) ε (0) I (cid:105) + . (22)The critical equivalent strains at the damage onset, ε (0)0 , and the fracturing strain, ε (0)f , followfrom ε (0)0 = f (0)t E (0) , ε (0)f = G (0)f f (0)t h, (23)where f (0)t is the matrix tensile strength, G (0)f represents the matrix fracture energy, and h isthe crack band width of the strain-softening zone [24], set to . times the maximum aggre-gate [38]. 8n order to estimate the effective tensile properties, we subject the composite to incrementalstrain path ( n = 1 , , . . . ) ε n = [ ε ,n − + ∆ ε, , , , , T , (24)where ∆ ε is a fixed strain increment, equal to − , and ε = . The damage in the matrixphase at the n -th load step, ω (0) n , is determined from the consistency condition between thedamage value in (20), related to the reduction of the matrix and ITZ stiffness, and the valuedetermined from (21) with the local strain ε (0) estimated by the elastic Mori-Tanaka methodfrom Eq. (3) with adjusted phase stiffnesses L ( d )sec and the overall strain set to ε n . The resultingsingle non-linear equation for ω (0) n is solved by the conventional secant method on the interval [ ω (0) n − ; 1] with the accuracy set to − . The simulation is stopped when the damage variable ω (0) reaches the value of . ; that corresponds to the N -th time step.The effective tensile parameters are obtained post-processing of the history of macroscopicstresses σ n = L IIeff ,n ε n for n = 1 , , . . . , N. (25)In particular, the effective tensile strength f t is estimated as f t = N max n =1 (cid:107) σ n (cid:107) , (26)while the effective fracture energy G f follows from G f = h ∆ (cid:15) N (cid:88) n =1 ( σ n − , + σ n, ) . (27)
3. Experimental Analysis
In this section, we gather the results of experimental studies needed to acquire the inputdata to the model, summarized in Table 1, as well as the validation data. After introducingthe mortar components, Section 3.1, in Section 3.2 we discuss the acquisition of the elasticproperties of individual phases. Section 3.3 is dedicated to inelastic properties of the matrixphase and the validation data are introduced in Section 3.4. Note that all experiments werealways carried out on at least six specimens representing the same material or batch, in orderto obtain representative data. In addition, when possible, we also compared our results withindependent results from the literature, in order to ensure their credibility.
The investigated mortars were reinforced by sand or crushed brick fragments, and the de-sired grain size distributions were obtained via sieving. The binder was composed of com-monly available white air-slaked lime (CL90) ˇCertovy schody, Czech Republic, of a high pu-rity (98.98 % of CaO + MgO), and metakaolin (finely ground burnt claystone, commercialname Mefisto L05) in a mass ratio equal to 7:3. The amount of water was adjusted so that thefresh mortars fulfilled the workability slump test, which is set according to ˇCSN EN 1015-3as 15 ± ρ . The matrix mass was determined based on theresult of our previous study [8], which revealed that 1 kg of slaked lime powder and metakaolin9able 1: Material properties of individual phases; ρ , E , ν , f c , f t , G f , and PSD denote themass density, Young’s modulus, Poisson’s ratio, compressive strength, tensile strength, fractureenergy, and particle-size distribution, respectively.phase ρ E ν f c f t G f PSD[kg/m ] [GPa] [-] [MPa] [MPa] [J/m ]matrix 1,066 8.00 0.25 7.0 2.0 12.0 × voids × × × × no needcrushed bricks 1,761 3.50 0.2 × × × neededsiliceous sand 2,720 90.0 0.17 × × × neededITZ 1,066 2.67 0.25 × × × thickness of 10 µ mmixed in the ratio 7:3 produce 1.46 kg of the paste. The volume of voids was obtained exper-imentally using the pycnometer method. For detailed information on the conversion of massfractions to the volume volume fractions for individual phases, the reader is referred to [33,Section 3]. The model validation was done on two kinds of samples, considering monodisperse andpolydisperse particle-size distribution of the aggregates. The term “monodisperse” refers to asingle to fraction 0.5–1.0 mm, obtained by sieve separation during preparation of the samples.The polydisperse particles were spanning the diameter range from 0.063 mm to 4.0 mm; forillustration see the grading curves plotted in Figure 5. sieve size [mm] p a r t i c l e s p a ss i n g [ % o f c ( ) ] monodisperse: sieve analysismonodisperse: modelpolydisperse: sieve analysispolydisperse: model Figure 5: Grading curves representing the grain-size distributions of monodisperse and poly-disperse aggregates. The sieve analysis provides the distribution into five rather wide intervals,which are further decomposed each into 9 sub-intervals by the linear interpolation for the cal-culation purposes. 10 .2. Elastic Parameters3.2.1. Matrix Young’s Modulus
The matrix elastic stiffness (Young’s modulus) was studied by means of quasi-static nanoin-dentation, the nanohardness tester (CSM Instruments, Switzerland) equipped with Berkovichpyramidal diamond tip. The mortar sample containing sand aggregates was sectioned andpolished before the measurement, and a suitable location (away from aggregates) with min-imum roughness was selected, Figure 6. Load-controlled quasi-static indentation test was em-ployed for all imprints, with the load function containing three segments (constant loading at24 mN/min, 10 seconds holding period, and unloading at 24 mN/min). The indents were eval-uated according to the Oliver and Pharr methodology [39], utilizing the unloading part for theassessment of the material elastic modulus. The holding period was introduced to reduce thecreep effects on the elastic unloading [40].The penetration depth of individual indents varied in order to find the relationship betweenthe penetration depth and the measured Young’s modulus. A rapid decrease in elastic stiffnesswith increasing penetration depth, Figure 6 indicated that the evaluated Young’s modulus wasaffected by the presence of microscale porosity occurring within the indented material volume.According to our measurements, based on mercury intrusion porosimetry, the maximum poresize was established as 900 nm and virtually no pores occurred beyond that limit. As a con-sequence, the penetration depth about 2,200 nm appeared sufficient to include all nano- andmicro-pores, yielding the effective matrix stiffness approximately 8 GPa, see Figure 6. (a) location and size of indents within the lime-metakaolin matrix E (0) = 21.584 GPa E (0) = 12.266 GPa E (0) = 8.974 GPa E (0) = 8.054 GPadepth of indents [nm] Y oung ’ s m odu l u s [ G P a ] (b) dependence of the effective matrix Young’smodulus on the depth of indents; the vertical er-ror bars indicate standard deviations in the mea-sured Young’s moduli, horizontal ones repre-sent the deviations in indentation depth Figure 6: Assessment of the matrix stiffness by means of nanoindentation.Note that a similar study has been performed by Neˇzerka et al. [8] for pure lime-based pastesand their mechanical properties. However, the measured values of Young’s modus 3.3 GPareported therein cannot be used for the modeling of mortars, since the pastes were porous andcontained larger voids due to the lack of aggregates. The inclusions contribute to consolidationof the fresh mortar, which results in a much denser matrix.11 .2.2. Young’s Modulus of Aggregates and ITZ
The Young’s modulus of crushed brick fragments was assessed using the resonance methodas an average of six measurements on the uncrushed prismatic specimens (40 × ×
160 mm).The Young’s modulus of river sand was provided by Nilsen and Monteiro [41] and the valueis in agreement with Daphalapurkar et al. [42] who used nanoindentation for the sand elasticstiffness assessment.According to Yang [43], the ITZ stiffness reduction in the thickness of 10 µ m around stiffaggregates is approximately 30 %, compared to the surrounding matrix. The formation ofdamaged zone due to shrinkage cracking was also predicted by the numerical model as demon-strated in Figure 2. Therefore, the 30 % matrix stiffness reduction was also adopted in our mi-cromechanical model. The reinforcement of the interface between the crushed brick fragmentsand the surrounding matrix by the formation of hydration products [14, 16] was not consideredin the model, because their impact on mechanical properties was found to be negligible [8]. The values of the Poisson’s ratio were set according to literature survey. Namely, the valueof 0.25 was proposed for the lime-based pastes by Drd´ack´y and Michoniov´a [44], and weemploy the same value also for voids and ITZ. Vorel et al. [19] considered the value of thePoisson’s ratio equal to 0.17 for siliceous sand and the same value was suggested in [33] forclay brick fragments.
The onset of plastic deformation is assumed to take place exclusively in the matrix. Itsstrength was determined from the destructive uniaxial compression tests carried out on cubic40 × ×
40 mm specimens of lime-metakaolin pastes, as described in [8].
Since the damage evolution in our model is restricted to the matrix, the tensile strength andfracture energy of other phases was not investigated. Because of a complicated clamping of thesamples during the uniaxial tension test and huge scatter of the measured data [8], the tensile(formally flexural) strength was determined from the three-point bending tests on unnotchedsimply supported 160 × ×
40 mm beams with the distance between supports equal to120 mm.The same experimental set-up was employed for the determination of the fracture energy,however, the beams were weakened by a 10 mm notch in the midspan in order to capture thedescending part of the load-displacement diagram and avoid snap-back. The fracture energy, G f , was evaluated directly from the recorded load-displacement diagram using the RILEMapproach [45]. Beside the acquisition of the input data, the purpose of the experimental analysis was to val-idate the proposed model. To that goal, six mortar specimens representing each batch were castin prismatic molds 40 × ×
160 mm, compacted using shaking table to get rid of excessiveair bubbles, and removed from the molds after 24 hours. The amount of water was adjustedso that the fresh mortars fulfilled workability defined by ˇCSN EN 1015-3 and the mortar coneexpansion was in the range of 13 ± ± ◦ C and relative humidity ranging between 65 and 75 %. The material properties ofeach mortar mix were assessed after the curing period of 3 months, using the same methods asdescribed in Sections 3.2.1, 3.3.1, and 3.3.2.The binder was composed of the same constituents as described in Section 3.1, i.e. lime andmetakaolin in the mass ratio equal to 7:3. The aggregates were either sand or crushed bricks,both of monodisperse and polydisperse particle-size distribution as indicated in Figure 5. Inorder to test the model predictions, the samples with polydisperse distribution were prepared invariable binder to aggregate ratio, yielding the aggregate volume fractions equal to c (2) = 0.238,0.384, 0.483, 0.555, and 0.609, respectively.
4. Results and Discussion
To reproduce the experimental outcomes, the computational procedures described in Sec-tion 2 were employed. The Young’s modulus, tensile strength and fracture energy were ob-tained by an inverse analysis of the stress-strain diagrams predicted by the proposed model .The simulations of uniaxial compression and tension tests were carried out considering thesame composition as pursued when preparing the tested mortars presented in Section 3.1. Thestudy was mainly focused on the relationship between the effective mortar properties and thevolume fraction of aggregates, either crushed bricks or quartz sand, both in mono- and poly-disperse configuration. −4 strain [-] s t r e ss [ M P a ] measurementscalculation −4 strain [-] s t r e ss [ M P a ] measurementscalculation Figure 7: Comparison between the experimentally obtained and calculated stress-strain dia-grams in uniaxial tension for the mixes containing monodisperse sand (left) and crushed brick(right) aggregates.The accuracy of our model is demonstrated by means of the uniaxial tensile stress-straincurves in Figure 7, provided by the analysis of the samples with “monodisperse” particle sizedistribution, recall Figure 5. The visual comparison suggests that the model matches wellthe elastic modulus and the peak strength; the post-peak softening curves are reproduced lessaccurately, but still provide reasonably accurate values of the fracture energy. The MATLAB code
Homogenizator MT: Composite with Cracks , can be used for reproduction of resultscontained in this paper and it is freely available at http://mech.fsv.cvut.cz/˜nezerka/software . ξ = (cid:80) ni =1 ( x i − x )( y i − x ) (cid:112)(cid:80) ni =1 ( x i − x ) (cid:112)(cid:80) ni =1 ( y i − y ) (28)to quantify the match between the n measured values x , x , . . . , x n and the correspondingmodel predictions y , y , . . . , y n , with e.g. x = n (cid:80) ni =1 x i . Figure 8(a) demonstrates that the Young’s modulus of mortars containing sand is constantwith the increasing volume fraction of sand particles. This rather surprising behavior is theresult of the development of ITZ around the grains, as suggested by Neubauer et el. [46], andthe presence of cracks among the shrinkage-constraining grains, observed e.g. in the study ofmortar microstructures by Mosquera et al. [47].The mortar stiffness reduction with the increasing volume fraction of crushed bricks is aconsequence their compliance, which in turn results in the reduction shrinkage-induced crack-ing and elimination of ITZ formation. Moreover, the hydration promoting nature of the water-retaining crushed brick fragments and their rough surface contribute to a perfect interfacialbond, which has a positive impact on the inelastic properties discussed next. volume fraction of aggregates, c (2) e ff ec t i v e Y o un g ’ s m o du l u s [ G P a ] sample B, sand (experiments)sample B, sand (calculations)sample B, bricks (experiments)sample B, bricks (calculations) (a) dependence of the effective mortar Young’smodulus on the amount of aggregates effective Young’s modulus (calculated) [GPa] e ff ec t i v e Y o un g ’ s m o du l u s ( e x p e r i m e n t s ) [ G P a ] sandcrushed bricks (b) comparison between the calculated and mea-sured values ( ξ = 0.973) Figure 8: Comparison between the calculated effective Young’s modulus and the experimen-tally obtained data on mortars containing polydisperse aggregates.Given that the values of Pearson correlation coefficient equal 0.973, Figure 8(b), the modelpredictions of the effective Young’s modulus can be considered very accurate. Such accuracycan be attributed to the introduction of penny-shaped cracks between closely packed aggregates14nto the homogenization scheme at Level II. Without this step, the method significantly over-estimate the stiffness of lime-based mortars, especially for stiff sand aggregates, see [48] forfurther details.
As follows from our experimental data, Figure 9(a), and from independent findings byLanas et al. [7], the compressive strength of mortars containing sand grains should be higherthan of those with crushed brick fragments and its value should decrease with increasing ag-gregate volume fractions, because of the stress concentration in the matrix phase. Both trendsare correctly reproduced by our model both quantitatively and qualitatively; as visible fromFigure 9(b), the agreement between the model predictions and experiments are of a similaraccuracy as for the Young’s modulus. volume fraction of aggregates, c (2) e ff ec t i v ec o m p r e ss i v e s t r e n g t h [ M P a ] sample B, sand (experiments)sample B, sand (calculations)sample B, bricks (experiments)sample B, bricks (calculations) (a) dependence of the effective mortar compres-sive strength on the amount of aggregates effective compressive strength (calculated) [MPa] e ff ec t i v ec o m p r e ss i v e s t r e n g t h ( e x p e r i m e n t s ) [ M P a ] sandcrushed bricks (b) comparison the between the calculated andmeasured values ( ξ = 0.846) Figure 9: Comparison between the calculated effective compressive strength and the experi-mentally obtained data on mortars containing polydisperse aggregates.
According to experimental measurements, Figure 10(a), the tensile strength of lime-basedmortars is also reduced with the increasing volume fraction of aggregates, which is also inagreement with the findings of Lanas et al. [7]. The tensile strength reduction with the increas-ing amount of aggregates is more pronounced in the case of mortars containing sand. Thisphenomenon is reflected by higher strain concentrations in the matrix, responsible for the onsetof damage at lower levels of externally applied macroscopic strain. The agreement betweenthe model and experiments for the tensile strength is slightly worse that in the case of elasticstiffness, compare Figure 10(b) to Figure 8(b), but we consider such accuracy to be sufficientfor engineering purposes, especially when taking into account the scatter of experimental data,Figure 10(a). 15 .2 0.3 0.4 0.5 0.60.811.21.41.61.822.2 volume fraction of aggregates, c (2) e ff ec t i v e t e n s il e s t r e n g t h [ M P a ] sample B, sand (experiments)sample B, sand (calculations)sample B, bricks (experiments)sample B, bricks (calculations) (a) dependence of the effective mortar tensilestrength on the amount of aggregates effective tensile strength (calculated) [MPa] e ff ec t i v e t e n s il e s t r e n g t h ( e x p e r i m e n t s ) [ M P a ] sandcrushed bricks (b) comparison the between the calculated andmeasured values ( ξ = 0.916) Figure 10: Comparison between the calculated effective tensile strength and the experimentallyobtained data on mortars containing polydisperse aggregates.
According to both, model and experiments, the values of fracture energy were higher inthe case of mortars containing crushed brick aggregates, also because of the lower contrastin elastic properties leading to the decrease in stress concentrations. In consequence, thesemortars accumulate more energy by allowing to reach higher values of elastic deformation,Figure 7. Again, the predictions of our model are satisfactory considering the relatively highscatter of the experimentally obtained data, as indicated by ξ equal to 0.879, see Figure 11(b).In summary, our experimental and model-based results confirm that with respect to sand-reinforced lime mortars, the mortars containing crushed brick particles provide lower stiffness,higher tensile and compressive strengths, and fracture energies. These factors combined al-low to achieve significantly larger maximum strain and ductility in tension, which probablycontributes to the increased earthquake resistance. Such conclusions correspond well with thefindings by e.g. Moropoulou et al. or Baronio et al. [49, 50], but our model provides a differentexplanation for the emergent behavior. Specifically, the improved mechanical performance isin [49] primarily attributed to the formation of hydration products at the interface between thelime matrix and fragments of crushed bricks. Even though this phenomenon has been recentlyconfirmed by nano-indentation [28], our model reveals that it is of secondary importance. Thedominant mechanism is found in the reduction of the shrinkage cracks due to more compliantcricks at Level II of the model; recall Figure 3 and see [48] for an explicit demonstration in theelastic regime. Having validated the model, we now proceed with its application to the design of the opti-mal ratio between the amount of sand and crushed brick fragments to achieve maximum elasticdeformation in tension and compressive strength. Figure 12 shows the predicted dependence ofthe target effective mortar properties, from which we suggest the optimum binder / sand / crushedbricks ratio equal to 1:1:1.5. These values are close to the generally established minimum16 .2 0.3 0.4 0.5 0.601234567891011 volume fraction of aggregates, c (2) e ff ec t i v e f r a c t u r ee n e r g y [ J / m ] sample B, sand (experiments)sample B, sand (calculations)sample B, bricks (experiments)sample B, bricks (calculations) (a) dependence of effective mortar fracture en-ergy on the amount of aggregates effective fracture energy (calculated) [J/m ] e ff ec t i v e f r a c t u r ee n e r g y ( e x p e r i m e n t s ) [ J / m ] sandcrushed bricks (b) comparison the between the calculated andmeasured values ( ξ = 0.879) Figure 11: Comparison between the calculated effective fracture energy and the experimentallyobtained data on mortars containing polydisperse aggregates. m a s s p o r t i o n o f s a n d m a ss p o r t i o n o f b r i c k s e ff ec t i v ec o m p r e ss i v e s t r e n g t h [ M P a ] −4 m a s s p o r t i o n o f s a n d m a ss p o r t i o n o f b r i c k s m a x i m u m e l a s t i c t e n s il e s t r a i n [ - ] −4 Figure 12: Dependence of the mortar compressive strength (left) and maximum elastic de-formation in tension (right) on the amount of sand and crushed brick fragments, consideringpolydisperse aggregates. 17inder / sand mass ratio 1:3 [9] in order to avoid excessive cracking, especially when recalcu-lated to volumetric fractions due to low mass density of crushed brick aggregates. Based onour experience, such ratio also provides optimum workability when used as a bed joint masonrymortar.
5. Conclusions
The simple micromechanics-based model of lime-based mortars for the estimation of theirelastic stiffness, compressive and tensile strength, and fracture energy was proposed and vali-dated against experiments. The model consists of two levels, where the lower level describes theinteraction among individual components of a mortar mix, while the upper scale accounts forthe shrinkage-induced cracks that significantly influence the overall mechanical performance.As for the prediction of the effective parameters, the Mori-Tanaka / Dilute Approximation wasused to estimate the overall stiffness, a J -based failure condition involving the average stressin matrix is adopted under compression, and under tension we employed the incremental Mori-Tanaka method coupled with the isotropic damage law and crack bend theory.Based on the presented results, we have found that the model correctly predicts1. the elastic stiffness of mortars containing sand, which does not increase with the increas-ing volume fraction of the aggregate due to the formation of ITZ and shrinkage-inducedcracks between closely packed grains,2. the elastic stiffness of mortars containing crushed brick fragments, which is also reducedby the addition of crushed bricks as a consequence of their low stiffness, and due toformation of shrinkage-induced cracks between the particles, however in lesser extentcompared to the stiff sand grains,3. the mortar compressive strength, which is higher in the case of mortars containing stiffsand grains, and decreases with the increasing amount of the aggregates,4. the mortar tensile strength, significantly reduced by an increase of the amount of aggre-gates; the effect is more pronounced in the case of sand grains, rather than compliantcrushed brick fragments,5. the mortar fracture energy, being higher if the crushed brick fragments replace sandgrains, since the more compliant mortars can reach higher elastic deformation.For all these quantities, we reached both the quantitative and qualitative agreement between theexperimental results and the model predictions.The validated model can be used for obtaining the input data for numerical modeling ofmortars at meso- and macroscale, or to find an optimal composition of the mix with respectto structural performance in masonry structures. As for the latter application, we invite theinterested reader to our follow-up paper [51], where we confirm by full-scale testing that theoptimized mortar delivers superior mechanical performance and durability with respect to con-ventional lime-based mortars with sand reinforcement. Acknowledgments
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