Composite Structures 123 (2015) 88–97

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Mixed-mode fracture in lightweight aggregate concrete by usinga moving mesh approach within a multiscale framework

http://dx.doi.org/10.1016/j.compstruct.2014.12.0370263-8223/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Department of Civil Engineering, University ofCalabria, 87036 Rende, Cosenza, Italy. Tel./fax: +39 0984 496916.

E-mail address: f.greco@unical.it (F. Greco).

Luciano Feo a, Fabrizio Greco b,⇑, Lorenzo Leonetti b, Raimondo Luciano c

a Department of Civil Engineering, University of Salerno, Italyb Department of Civil Engineering, University of Calabria, Italyc Department of Civil and Mechanical Engineering, University of Cassino and Southern Lazio, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Available online 24 December 2014

Keywords:Multiscale methodsLinear elastic fracture mechanicsLightweight aggregate concreteMixed modeCrack propagation analysisThree-point bending test

Lightweight aggregate concrete (LWAC) has gained popularity as an alternative to ordinary concrete forstructural purposes, due to its higher strength-to-weight ratio. The present work aims to present novelnumerical results of complete failure simulations performed on pre-cracked beams made of LWAC, sub-jected to a mixed-mode fracture test. To this end, an innovative simulation algorithm for crack propaga-tion within a multiscale framework has been adopted, specifically conceived for predicting micro-cracking in quasi-brittle heterogeneous materials under general loading conditions; such a strategyallows to take into account both the continuous crack propagation along a non-prescribed path andthe crack penetration through a material interface. Path tracking for continuous crack propagation hasbeen performed by using an advanced geometry optimization method coupling a moving mesh approachand a gradient-free optimization solver, whereas crack penetration has been simulated by means of asimplified re-initiation criterion at the interface, involving a material characteristic length. Severalnumerical experiments have been carried out, in order to investigate the influence of the Young’s mod-ulus of lightweight aggregates on the peak and post-peak behavior. These results have been validated bycomparing them with those obtained from fully homogenized analyses based on the LEFM approach.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays, the demand of lightweight aggregate concrete(LWAC) as a structural material in place of the ordinary one isincreasing, due to its peculiar features such as high strength-to-weight ratio, durability and fire resistance. Due to the possibilityof reducing dead loads, lightweight aggregate concrete is clearlysuitable for applications calling for small structural masses, as inearthquake engineering.

High-performance lightweight concretes for structural pur-poses are typically produced using expanded clay, shale or slateas aggregates. These aggregates exhibit a lower density withrespect to natural ones, which is closely related to their porousmicrostructure. Nevertheless, lightweight concretes whose oven-dry density is smaller than 1500 kg/m3 could not be employedfor structural purposes [1,2]; in fact, a correct balance betweenaggregates’ density and crushing strength should be reached, inorder to assure an adequate compressive strength for the finished

concrete (which is typically more than 20 MPa). Also fly ash light-weight aggregates can be adopted, whose properties have beenrecently investigated for producing self-compacting concrete [3].

Unlike for normal weight concretes, the weakest component ofthe system is neither the cement paste nor the interfacial transi-tion zone (ITZ), but rather the aggregate [4–6]; this fact has astrong connection with the final aspect of the crack trajectory,being characterized by a small tortuosity, since, once trapped byan aggregate due to the elastic mismatch, a propagating crack doesnot propagate along the interface, but rather penetrates into theaggregate itself.

As a matter of fact, the mechanical properties of the lightweightaggregate are more similar to those of the cement paste matrixthan to the normal weight aggregate, and variations in aggregatequality and content will be more directly reflected in the propertiesof the finished concrete. The influence of different design variableson the peak load and the softening behavior has been investigatedby several works, mainly the aggregate’s quality [7–10],size [11,12], and volume fraction [7,9,13–15]. Most of these worksare based on experiments, but there exist a few studies devotedto analytical and/or numerical homogenization approaches[7,10,11,13,15].

L. Feo et al. / Composite Structures 123 (2015) 88–97 89

The purpose of this work is to propose a more sophisticatednumerical approach for predicting the highly nonlinear mechanicalbehavior of lightweight aggregate concretes up to failure, takinginto account the effect of the underlying microstructure. To thisend, a simplified model for LWAC has been considered, in whichconcrete is represented as a biphasic material; according to thismodel, the coarse lightweight aggregates (LWAs) are embeddedinto the mortar matrix, which can be thought as the assembly ofthe cement paste and the fine normal weight aggregates (usuallysand). Both phases are susceptible to be damaged, but there existsa perfect mechanical bond between them; such a simplifyinghypothesis has been supported by experience.

Micro-crack propagation within both the mortar matrix and thelightweight aggregates has been taken into account by using anenergy-based fracture criterion relying on linear elastic fracturemechanics (LEFM); crack advancing is simulated within a finiteelement framework by taking advantage of a remeshing strategyfor incrementally updating the crack geometry, in conjunctionwith a novel moving mesh approach for selecting the actual crackpath according to the maximum energy release rate criterion; therelated crack propagation algorithm has been developed by theauthors in previous works (see, for instance, [16–19]), andhere extended, accounting for the crack penetration through amaterial interface; in the presence of soft inclusions (weak singu-larity case), crack penetration toward the matrix phase is modeledas an abrupt crack re-initiation based on the introduction of a char-acteristic length depending on the elastic mismatch of the bimate-rial system.

Obviously, a complete failure analysis performed on a fullymeshed model, able to consider the effect of all the microstructuraldetails on the final damage configuration, would require a hugecomputational effort, being unbearable for practical engineeringpurposes. Therefore, a multiscale strategy has been pursed, alreadyused in [19], in conjunction with the previously introduced frac-ture framework; a concurrent multiscale model for lightweightaggregate concrete has been developed, containing a fully micro-scopic description within a target zone, referred to as ‘‘zone ofinterest’’, coinciding with the subdomain directly influenced bythe presence of micro-cracks, whereas outside this region, in whichthe material is sound, the microstructure is replaced by an equiv-alent homogenized material, whose elastic properties are obtainedvia a first-order computational homogenization scheme (a recentsimilar approach can be found in [20], with reference to compositelaminates). For the sake of completeness, other multiscaleapproaches exist in the literature in addition to concurrent meth-ods, i.e. hierarchical and semi-concurrent multiscale methods, likethe heterogeneous multiscale method [21] and the multiscaleasymptotic expansion method [22].

In this work the influence of the aggregate’s quality on theeffective strength and toughness of lightweight concrete is investi-gated, by performing numerical experiments on a mixed modebending test, introduced in [23] and already analyzed in manyworks (both experimental and numerical) for the case of ordinarynormal weight concrete (see, for instance, [24–29]). The relatednumerical results, in terms of overall structural response and crackpath prediction, obtained by using the above-mentioned multi-scale approach, have been compared with those coming from apurely homogenized analysis, in order to assess the validity andthe accuracy of the proposed numerical method.

The paper is structured as follows: in Section 2, the adoptedmultiscale framework is described; Section 3 presents the maintheoretical concepts and the numerical techniques here developedfor simulating crack propagation in LWAC, taking into account theinteraction with its microstructure; then, in Section 4 the mainnumerical results obtained by using the proposed approach are

illustrated, in terms of peak and post-peak behavior for the consid-ered test; finally, Section 5 is devoted to the concluding remarks,together with some future perspectives.

2. Multiscale framework for the failure analysis in LWAC

Lightweight aggregate concrete (LWAC) as well as ordinary nor-mal weight concrete is characterized by a highly heterogeneousstructure, consisting of three distinguishable phases: a continuousphase, i.e. the cement paste matrix, a discontinuous phase, made ofcoarse and fine aggregates, distributed into the continuousphase according to a given granulometry, and an interfacial phaseplace in between, usually referred to as interfacial transition zone(ITZ).

As a matter of fact, our attention is devoted to a lightweightaggregate concrete with natural sand (also referred to as semi-lightweight aggregate concrete or SLWAC), where only the coarseaggregate is made of lightweight material; thus, a simplified modelfor such a multiphasic material can be obtained by explicitly con-sidering only coarse aggregates; cement paste and sand particles,together with their ITZ, can be replaced by an equivalent homoge-nized materials, whose mechanical properties are those of anordinary cement mortar. Due to the small thickness of the ITZbetween cement paste and lightweight aggregates, the microstruc-ture under consideration can be modeled as a bimaterial systemmade of randomly distributed particles perfectly bonded to theembedding matrix. Furthermore, unlike natural aggregates, LWAstend to possess a well-rounded shape, so that they can be regardedas randomly sized spherical particles (see Fig. 1(a)); this fact has adirect implication for the numerical simulations, since there is noneed to account for the influence of the aggregate shape on the glo-bal structural response.

Several mesoscale models exist in the literature for ordinaryconcretes, most of which are based on a random generation of itsunderlying microstructure (see, for instance, [30–33]). Such mod-els are characterized by an extremely complex geometrical config-uration and require the use of very accurate meshes, if used withina finite element framework. Moreover, the spatial distribution ofaggregates must be as macroscopically homogeneous and isotropicas possible; such a requirement can be fulfilled using, for instance,the Random Sequential Adsorption (RSA) model, able to generatethe spatial arrangement of randomly sized particles according toa given granulometry [31]. Other mesomodeling approaches takealso into account the effects of voids on the overall structuralresponse (see, for instance, [34]).

In the present work, a simplified approach is adopted, based onthe hypothesis of neglected polydispersity (which is admissibledue to the sintered nature of coarse lightweight aggregates); in thiscase, the LWAC’s microstructure is characterized by a regular pack-ing of identically sized cylindrical particles. Such a model isbelieved to be valid for our purposes, since only 2D fracture prob-lems have been considered, thus neglecting the through-thicknessdimension. In detail, a hexagonal packing has been chosen (seeFig. 1(b)), been the only one able to preserve the required in-planeisotropy at the macroscopic scale [35].

Let us consider the elasticity problem of a 2D structure made ofLWAC, whose idealized microstructure contains a hexagonalarrangement of identical circular aggregates, subjected to quasi-static loading under small displacements, for a fixed traction-freecrack set Cc ¼

SiCc;i (i ¼ 1; . . . ;n), as shown in Fig. 2(a). Its spatial

domain is the open set X � R2, enclosed by a Lipschitz continuousboundary @X ¼ @tX [ @uX, such that @tX \ @uX ¼£ and the mea-sure of @uX is greater than zero to avoid any rigid-body motions.Such a problem can be mathematically formulated by an ellipticPDE systems with associated boundary conditions:

(a) (b)

Fig. 1. LWAC’s idealized microstructures: (a) 2D slice of a microstructure with randomly distributed spherical aggregates; (b) idealized 2D model based on a hexagonalpacking of cylindrical aggregates.

(a) (b) (c)

Fig. 2. Boundary value problem (BVP) for a 2D model of cracked LWAC structure: (a) original single-domain formulation; (b) identification of the zone of interest and relatedmulti-domain formulation; (c) first-order homogenization step outside the zone of interest.

90 L. Feo et al. / Composite Structures 123 (2015) 88–97

�DivðCðXÞeðuÞÞ ¼ f in X n Cc;

u ¼ �u on @uX;

½CðXÞeðuÞ�n ¼ �t on @tX

8><>: ð1Þ

being f and �t the assigned body forces and tractions, respectively,and �u the prescribed displacement; C is the fourth-order elasticitytensor, supposed to be rapidly varying over the macroscopic vari-able X. Therefore, fully microscopic models are not appropriate forpractical applications, requiring a large computational cost, espe-cially in the case of nonlinear constitutive responses due to evolu-tion of the crack configuration, for which an incrementalformulation is needed; on the other hand, classical homogenizationapproaches cannot be applied, because the assumption of periodic-ity ceases to hold in the presence of evolving microstructure. As aconsequence, other methods are currently preferred to predict fail-ure in heterogeneous materials, most of which belong to the wideclass of multiscale approaches (see [36] for a brief review).

Here, a concurrent multiscale method is adopted, based on amultilevel domain decomposition approach, according to whichthe original problem is split into smaller and more manageablesub-problems to be solved in a coupled manner, as described inthe author’s previous work [19]; in this context, the originaldomain X is partitioned into two sets of non-overlapping subdo-mains, i.e. elastic and damaged domains, denoted as Xe and Xd,respectively (see Fig. 2(b)). The tensor C is assumed to be periodiconly in Xe, i.e. CðXÞ ¼ CeðXÞ for all X 2 Xe, where the superscript edenotes the dependence on a small period; therefore, the givenundamaged microstructure can be replaced by an equivalenthomogenized medium, whose elastic properties are deduced froman asymptotic homogenization method, after identifying a suitablerepeating unit cell (RUC). According to a first-order computation

homogenization scheme, the tensor �C ¼ ð�CijhkÞ of homogenized(effective) moduli (i; j; h; k ¼ 1;2;3) is given by:

�Cijhk ¼1jVRUCj

ZVRUC

ð~CijhkðxÞ þ ~CijlmðxÞ@vhk

l

@xmÞdx; ð2Þ

where V is the volume of the RUC, ~Cijhk are the elastic moduli corre-sponding to the different material phases, and @vhk

l are the so-calledcharacteristic functions of the RUC. After homogenizing the desiredmechanical properties, a coarse-scale resolution is sufficient withinthe elastic subdomain Xe, whereas a fine-scale model is kept withinthe damaging subdomain Xd, here referred to as ‘‘zone of interest’’(see Fig. 2(c)). Within the finite element setting, the continuity con-ditions between the fine- and coarse-scale subdomains (character-ized by an inherent separation in the spatial resolutions) areimposed by using a conforming formulation, which is assured bya gradual transition between the two considered mesh portions.

Several alternative RUCs can be chosen for the consideredmicrostructure. If only translational symmetry is employed, anyperiodical elements shown in Fig. 3 can be selected as repeatingunit cell for homogenization purposes (see [37] for additionaldetails); among them, elements C1 and C6 lead to complicated par-titions of the regions occupied by each material phase as the par-ticle volume fraction increases, whereas elements C4 and C5 areassociated to a complicated way to express periodic boundary con-ditions, if a classical rectangular coordinate system is used. As aconsequence, in this work the rectangular-shaped unit cell C2 isadopted (the choice of the element C3 would lead to the sameimplementation effort and the same computational cost at fixedrequired numerical accuracy). In this case, the equivalent homoge-nized medium possesses transverse isotropy; therefore, in a two-dimensional setting (i.e. under plane strain or stress state acting

Fig. 3. Alternative repeating unit cells (RUCs) for the LWAC’s idealized hexagonalmicrostructure.

L. Feo et al. / Composite Structures 123 (2015) 88–97 91

on the plane of isotropy, here denoted as x1x2 plane), the relatedmacroscopic moduli tensor �C can be written in Voigt notation asfollows:

�C ��C1111

�C1122 0�C1122

�C1111 00 0 ð�C1111 � �C1122Þ=2

264

375; ð3Þ

being characterized by only two independent elastic constants, i.e.�C1111 and �C1122 .

3. Numerical tools for simulating crack propagation in LWAC

Crack propagation analysis in heterogeneous materials isalways a complex task, due to the presence of multiple interactingcracks, subjected to propagation, bifurcation and coalescence phe-nomena. In this section, the adopted numerical tools for simulatingcrack propagation in LWAC, taking into account the competitionbetween different mechanisms at the microscopic scale, will bedescribed.

Although a 3D fracture model is generally required for handlingrandomly oriented microstructures, our attention is restricted tothe case of planar micro-cracks, i.e. through-width cracks propa-gating parallel to the mid-section of the reference 2D specimen.With reference to a single crack of finite length l, Griffith’s energycriterion (in the LEFM setting) can be written in a quasistatic rate-independent formulation by using the following KKT conditions:

_l P 0;GðlÞ � Gc 6 0;

ðGðlÞ � GcÞ_l ¼ 0;

8><>: ð4Þ

where _l is the rate of crack length, G is the strain energy release rateassociated with _l, and Gc is the fracture energy of the constituent inwhich the crack tip is embedded. In order to avoid computationaldifficulties arising from the nonsmooth character of the KKT condi-tions, a crack length control scheme is adopted, according to whichthe nonlinear crack propagation problem is formulated as thesequence of several linear elasticity problems solved in cascade(see [38] for additional details).

3.1. An ALE formulation for the crack path selection

In this section, the adopted numerical tool for simulating thecrack advance along a non-prescribed path is presented. The pointof departure is the expression of the energy release rate G as afunction of the crack length l and the kinking angle h:

Gðl; hÞ ¼ �P�l ðl; hÞ � �P�ðlþ Dl; hÞ �P�ðlÞ

Dl; ð5Þ

where Dl denotes the chosen crack length increment, and P⁄ is thetotal potential energy at equilibrium:

P�ðlÞ ¼ PðuðlÞ; lÞ ¼ infu2UaðlÞ

Pðu; lÞ; ð6Þ

being Ua the set of admissible displacement fields. Thus, by combin-ing Eqs. (5) and (6), the maximum energy release rate (MERR) crite-rion can be enforced in a variational setting as a doubleminimization of P with respect to both the displacement fieldand the crack direction:

suph

Gðl; hÞ ) infh

P�ðlþ Dl; hÞ ¼ infh

infu2UaðlþDl;hÞ

Pðu; lþ Dl; hÞ� �

; ð7Þ

which can be performed by testing different trial directions.The crack update procedure during the searching for the actual

crack direction is performed by taking advantage of a novel movingmesh approach, introduced in [19], based on an Arbitrary Lagrang-ian–Eulerian (ALE) formulation, so that classical remeshing opera-tions are needed only for the crack advance (see [19] for a completedescription of the adopted crack simulation algorithm, and [39] foran ALE-based formulation of delamination problems in reinforcedconcrete beams).

3.2. Crack penetration through the LWA/mortar interface: a simplifiedapproach

A main crack propagating in LWAC structures may interact withembedded LWAs in different ways; under the assumption of per-fectly bonded material interfaces, the only allowed mechanismsare penetration from mortar to aggregates, and conversely, pene-tration from aggregates to mortar. In the first case, referred to asstrong singularity case, a crack propagating inside the matrix tendsto be attracted by the aggregates due to the elastic mismatch,experiencing at the same time an unstable advancing behavior;this is due to the fact that the energy release rate at fixed loadsgoes to infinity as the crack ligament goes to zero. Conversely,for a micro-crack propagating inside an aggregate (weak singular-ity case), the energy release rate vanishes as the crack ligamentgoes to zero, and a crack arrest event may take place (see [40]for additional details).

In the latter case, if the external loads are increased, the crackmay re-initiate ahead the interface; crack re-nucleation criteriausually require the introduction of a characteristic length, whichis strictly related to the size of the process zone, in the spirit ofthe theory of critical distances [41]. In this setting, an existingcrack is subjected to an abrupt advance of a finite increment.

In order to predict the critical load level corresponding to thecrack re-initiation at interface, a simplified approach is pursued;such a load threshold is derived by applying classical Griffith’s cri-terion to a new crack configuration obtained extending the crackterminating at interface by a newly defined critical distance (seeFig. 4), accounting for the elastic mismatch in bimaterial systems:

lc ¼1p

GcDEr2

cð8Þ

where Gc and rc are the fracture energy and the tensile criticalstress of the mortar, and DE is the difference Em � Ea; being Em

(a) (b)

Fig. 4. Schematic representation of crack penetration through the LWA/mortar interface: (a) configuration before crack re-initiation; (b) configuration after crack re-initiation.

Fig. 5. Geometry and boundary conditions for mixed-mode fracture tests on asingle-notched LWAC beam.

Table 1Material properties of the LWAC’s constituents.

Component Material E (GPa) m rc [MPa] Gc [N/m]

Matrix Cement mortar 24 0.20 15 40Inclusion Expanded clay 4.8 0.08 – 10

92 L. Feo et al. / Composite Structures 123 (2015) 88–97

and Ea the values for Young’s modulus of the mortar and theaggregates, respectively.

4. Numerical experiments: asymmetric three-point bendingtests on a single-notched LWAC beam

Various numerical experiments have been carried out with ref-erence to the complete failure analysis of a single-notched LWACbeam subjected to an asymmetric three point bending test, firstlyintroduced in [23]. Such a test has been simulated for ordinary con-crete in many works neglecting the effects of random microscopicheterogeneities on the overall structural response (see, forinstance, [24–29]). In order to perform the present numerical sim-ulations, the multiscale model described in Section 2 has beenadopted, which combines the accuracy of fully microscopic modelsand the efficiency of homogenized ones.

Section 4.1 describes the considered geometric configurationand material properties; Section 4.2 presents the principal numer-ical results in terms of peak and post-peak behaviors; in Section4.3, these numerical results are validated by comparing them withthose obtained by a purely homogenized model; finally, in Section4.4 the influence of the Young’s modulus of lightweight aggregateson the macroscopic response of the LWAC specimen is investi-gated, with reference to both the considered numerical models.

4.1. Description of geometric configuration and material properties

The considered test involves a proportional quasi-static loadingon a single-notched beam made of LWAC; the specimen is asym-metrically loaded in order to force the initial crack to propagateunder mixed mode conditions. Geometry and boundary conditionsof the test are shown in Fig. 5; the geometrical configuration hasbeen originally designed to be parameterized with respect to thebeam’s height D, in order to investigate possible structural sizeeffects. As such effects are not considered in this work, D is keptfixed and set equal to 75 mm, so that the resulting dimensionsare 337.5 � 75 � 50 mm. The applied point force is balanced bythe two vertical reactions at supports, modeled as point con-straints, as in the most of numerical works about this fracture test.The numerical simulations are carried out under the assumption ofplane stress conditions.

A preexisting crack has been inserted into the model, startingfrom the bottom of the beam in its midsection and running verti-cally for 37.5 mm. The crack trajectory is determined by meansof the iterative-incremental procedure illustrated in Section 3.1,

based on a moving mesh approach; thus, crack propagation is sim-ulated in a discrete manner, by extending a current crack by a con-stant increment Dl, set as 1 mm.

Expanded clay lightweight aggregates with diameter of 8 mmare considered, whose total volume fraction is equal to 40%. Thematerial properties of both mortar matrix and lightweight aggre-gates are listed in Table 1, where E and m are the elastic constants,Gc is the fracture toughness, and rc is the tensile critical stress,needed to compute the characteristic length defined by Eq. (8).

4.2. Numerical results obtained by the multiscale model

According to the proposed multiscale method, a zone of interestis properly identified, susceptible to be damaged; within such aregion, the underlying microstructure is explicitly taken intoaccount, and therefore a fine-scale mesh is adopted; conversely,outside the zone of interest, a coarse discretization is sufficient.The whole mesh, including a transition zone for assuring a gradedmesh refinement, is depicted in Fig. 6. An additional local meshrefinement is performed along the crack path and the J-integralcontour, so that the final mesh is composed of about 13,840 qua-dratic triangular elements, resulting in about 55,790 degrees offreedom (DOFs).

Fig. 6. Undeformed configuration of the finite element mesh used for the multiscale model.

0

0.5

1

1.5

2

2.5

3

3.5

4

0.00 0.02 0.04 0.06 0.08 0.10 0.12

mortar propaga�on

LWA propaga�on

re-ini�a�on criterion

F [kN]

δ [mm]

Fig. 7. Load–displacement diagram for mixed-mode fracture test obtained bymeans of a multiscale numerical simulation (MNS).

0

0.5

1

1.5

2

2.5

3

3.5

4

0.00 0.02 0.04 0.06 0.08 0.10 0.12

MNS

HNS

F [kN]

δ [mm]

Fig. 9. Comparisons between the multiscale (MNS) and the homogenized (HNS)numerical simulations for the mixed-mode fracture test in terms of load–displace-

L. Feo et al. / Composite Structures 123 (2015) 88–97 93

As the actual LWAC’s microstructure has been replaced with anidealized periodic microstructure, the statistical effects due to therandomness of inclusion size and distribution are neglected. There-fore, the global structural response obtained by means of thenumerical simulations possesses a deterministic character; never-theless, the force–displacement curve with reference to the loadedpoint, shown in Fig. 7, has a complex behavior.

Each point of this diagram represent the equilibrium state at agiven structural configuration, starting from that containing theinitial crack. Micro-cracking within mortar and aggregates arehighlighted as distinct branches, whereas marked points are asso-ciated to the crack re-initiation process at material interfaces,according the simplified approach illustrated in Section 3.2.

The first equilibrium point determines the peak-load of thedesired structural response, which is equal to 3.8 MPa; after this,the considered specimen exhibits severe snap-back and snap-through behaviors in a cyclic manner, i.e. repeated for each crackpenetration through a lightweight aggregate. It is worth noting

Fig. 8. Undeformed configuration of the finite elem

that this is valid only in the case of a crack passing sufficientlyclose to a material interface, which assures the crack to be trappedby the inclusion due to the elastic mismatch. Each crack penetra-tion from a material to another corresponds to a stability changein the overall structural response; as a consequence, two seriesof turning points can be identified in Fig. 7.

It should be highlighted that the present numerical method isnot able to precisely detect the turning points from unstable to sta-ble branches, corresponding to the crack penetration towards theaggregates. As a matter of fact, no ad-hoc criteria have been pro-posed for the strong singularity case (see Section 3.2), thereforethe exact position of such points is controlled by the chosen valuefor the crack length increment.

The complete load–displacement curve shows a globally unsta-ble behavior, corresponding to a catastrophic collapse if the analy-sis is carried out under displacement control. Fig. 7 also shows thecentral portion of the deformed configuration at the end of simula-tion, together with the final numerically predicted crack trajectory.

ent mesh used for the homogenized model.

ment diagram.

Fig. 10. Comparison between the multiscale (in red) and the homogenized (ingreen) analyses for the mixed-mode fracture test in terms of crack trajectories. (Forthe interpretation of the color reference, the reader is referred to the web version ofthis article.)

94 L. Feo et al. / Composite Structures 123 (2015) 88–97

4.3. Validation of the multiscale model

The proposed multiscale approach for the failure analysis ofLWAC structures is validated by means of suitable comparisons

(a)0

1

2

3

4

5

6

0.00 0.02 0.04 0.06 0.08 0.10

MNS

HNS

F [kN]

δ [mm]

(c) 0

1

2

3

4

5

6

0.00 0.02 0.04 0.06 0.08 0.10

MNS

HNS

F [kN]

δ [mm]

Fig. 11. Load–displacement diagram obtained from both the multiscale (MNS) and the horatio Ea/Em: (a) 0.05; (b) 0.1; (c) 0.5; (d) 1.0.

with a purely homogenized model, referring to the same test.The material is assumed linearly elastic and isotropic, whose prop-erties are the same as those of the equivalent homogenized mate-rial outside the zone of interest in the multiscale model.

The numerical simulation of crack propagation within the LEFMsetting requires the only determination of the overall fractureenergy of LWAC; this material property can be obtained fromexperiments or numerical tests carried out on a fully microscopicmodel. In the present work, the effective toughness of LWAC isassumed to be coinciding with those of the mortar phase. Thischoice allows us to highlight the influence of the LWAs on thepost-peak behavior and the final crack path; this effect is strictlyrelated to the toughening effect provided by the elastic mismatch(see [42] for a complete treatment about toughening mechanismsin heterogeneous materials).

For the failure analysis on a purely homogenized model, acoarse-scale mesh is enough, as shown in Fig. 8. As in the multi-scale model, additional refinements along the crack path and theJ-integral contour are performed, so that the final mesh is com-posed of about 4580 quadratic triangular elements, resulting inabout 18,650 DOFs.

Fig. 9 shows the comparison between the multiscale and thehomogenized analyses in terms of force–displacement curve refer-ring to the loaded point. The numerical results highlight that theLEFM-based homogenized analysis is very accurate with respectto the multiscale one in predicting the peak behavior (with an

(b)0

1

2

3

4

5

6

0.00 0.02 0.04 0.06 0.08 0.10

MNS

HNS

F [kN]

δ [mm]

(d)0

1

2

3

4

5

6

0.00 0.02 0.04 0.06 0.08 0.10

MNS

HNS

F [kN]

δ [mm]

mogenized (HNS) numerical simulations for different values of the Young’s modulus

0

1

2

3

4

5

6

0.0 0.2 0.4 0.6 0.8 1.0

MNS

HNS

F [kN]

Ea /Em

Fig. 12. Behavior of the peak load as a function of Young’s modulus of the aggregatefor both the multiscale (MNS) and the homogenized (HNS) numerical simulations.

Table 2Absolute value of relative percentage errors between the multiscale and thehomogenized numerical simulations in terms of peak load, for different values ofthe Young’s modulus ratio Ea/Em.

Ea/Em 0.05 0.1 0.2 0.5 1.0

ErrorF (%) 4.14 2.62 1.14 0.41 0.22

(a)

(c)

Fig. 13. Crack trajectories obtained from both the multiscale (MNS) and the homogenizeEm: (a) 0.05; (b) 0.1; (c) 0.5; (d) 1.0.

L. Feo et al. / Composite Structures 123 (2015) 88–97 95

absolute relative percentage error in terms of peak load of about1%). Moreover, the displacement value corresponding to the peakload is slightly underestimated by the homogenized analysis; thisis due to the fact that the initial stiffness is overestimated of about5% in the homogenized model. Obviously, the post-peak responsein the homogenized analysis is characterized by a smooth behav-ior, being the microstructural effects completely neglected.

Finally, the crack trajectories obtained by both numerical simu-lations have been superposed, as depicted in Fig. 10. Numericalresults show that the crack path predicted by the homogenizedanalysis deviates from that obtained from the multiscale analysis,resulting in a smaller average kinking angle; nevertheless, such adeviation does not seem to affect the global post-peak behavior.

4.4. Influence of the Young’s modulus of lightweight aggregates

The influence of the quality of LWAs on the effective structuralbehavior during the considered mixed-mode fracture test is inves-tigated by varying their Young’s modulus for both the multiscaleand the homogenized models. The following values of the ratioEa/Em have been considered: 0.05, 0.1, 0.5 and 1, being the caseEa/Em = 0.2 already covered in Section 4.3.

The related numerical results, in terms of load–displacementcurves, are shown in Fig. 11, for both analyses; by comparing thedifferent curves, it should be noted that the initial apparent stiff-ness is characterized by an increasing behavior for increasing val-ues of the ratio Ea/Em; such a result agrees with experiments aswell as with classical micromechanical models. Moreover, the peakload also increases with the increase of the Young’s modulus of

(b)

(d)

d (HNS) numerical simulations for different values of the Young’s modulus ratio Ea/

96 L. Feo et al. / Composite Structures 123 (2015) 88–97

LWAs; this correlation can be better appreciated by consideringFig. 12, which illustrates the behavior of the peak load as a functionof the ratio Ea/Em for both numerical simulations; such a behavioris characterized by a sub-linear correlation between the peak loadand the Young’s modulus of aggregates, for both the multiscale andthe homogenized analyses. Obviously, the peak load for greaterelastic mismatch is expected to assume smaller values, and thelimit value for vanishing Ea/Em should be finite, corresponding tothe ideal case of porous material. Furthermore, the peak loadobtained from the homogenized analysis is always slightly overes-timated, for each considered value of the ratio Ea/Em; the measureof such overestimation is reported in Table 2, where the relativepercentage errors on the peak load between the homogenizedand the multiscale analyses, for each considered case of LWA’selastic modulus, are shown. The maximum relative percentageerror, corresponding to the case Ea/Em = 0.05, is of about 4.1%,whereas the resulting average value of relative percentage errorfor the considered cases is of about 1.7%.

In order to investigate the effect of the elastic mismatch on thepredicted crack path, the crack trajectories obtained by means ofboth the multiscale and the homogenized approaches have beensuperposed, for all the considered values of the ratio Ea/Em. Numer-ical results have highlighted that the homogenized model leads toa significant deviation in the global crack path; such a deviation isessentially triggered by the presence of softer inclusions than thesurrounding mortar, which tend to attract a propagating matrixcrack (see Fig. 13); it is worth noting that different distributionsand positions of lightweight aggregates with respect to the maincrack would lead to different amounts of this path deviation, whichis believed to be highly dependent on statistical effects, which havebeen not considered in this work.

Nevertheless, the accuracy of the homogenized model in pre-dicting the actual crack trajectory increases for increasing valuesof the ratio Ea/Em; As a matter of fact, in the absence of elastic mis-match, the crack trajectories predicted by the two models are com-pletely superposed with no appreciable deviations (see Fig. 13(d)).

5. Conclusions

This work has presented the numerical results of various failuresimulations on a pre-cracked LWAC beam, subjected to a mixedmode bending test, taking into account the effects of its underlyingmicrostructure on the overall structural response. To this end, anovel multiscale model has been developed, which combines theefficiency of homogenized models and the accuracy of fully micro-scopic ones.

Such a model is based on a non-overlapping domain decompo-sition method implemented in a classical finite element setting,relying on the proper identification of a zone of interest, i.e. thezone susceptible to damage growth; outside this region the mate-rial is supposed to be completely sound and the underlying micro-structure is replaced by an equivalent homogenized material,whose properties are derived from a first-order computationalhomogenization scheme.

Such a model is used in conjunction with a crack propagationframework, conceived for predicting micro-cracking in quasi-brit-tle heterogeneous materials under general loading conditions.The proposed fracture approach relies on three main ingredients:(i) the crack length control scheme used in conjunction with aremeshing strategy, capable to handle with possible unstablebranches of the equilibrium path; (ii) a moving mesh techniquewhich enforces the maximum energy release rate criterion in a var-iational setting; (iii) a simplified re-initiation criterion at interface,based on the introduction of a critical crack length, beyond whichthe LEFM approach is supposed to be valid.

The proposed approach has been applied to obtain the load–dis-placement curve for a mixed-mode fracture test. Moreover, a sen-sitivity analysis has been performed by varying the Young’smodulus of aggregates; the related numerical results have shownan overall structural response exhibiting a well-defined peak load,corresponding to the initial crack configuration, and a globallyunstable post-peak behavior. Nevertheless, for each consideredcase, the equilibrium path is locally characterized by severalrepeated stable and unstable branches, intimately related to themicrostructural effects: crack penetration within the LWAs corre-sponds to a jump in the macroscopic structural response in anunstable manner, whereas an opposite jump behavior is experi-enced during penetration within the mortar phase.

The proposed multiscale model has been validated by perform-ing comparisons with a purely homogenized model, whose elasticproperties have been obtained in the same way as for the multi-scale analysis. The LWAC’s equivalent toughness has been set equalto that of the mortar phase; this choice comes from the need fortaking into account the effects of the elastic mismatch.

The results have highlighted that the homogenized model tendsto slightly overestimate the peak load with respect to the moresophisticated multiscale model. Therefore, the LEFM-basedhomogenized analysis is reputed suitable for simulating the con-sidered mixed-mode fracture test in LWAC, provided that the over-all toughness is suitably chosen, in order to consider the effects ofthe microscopic heterogeneities on the overall behavior.

The fundamentals strengths of the proposed approach are: (i)versatility, mainly related to the possibility of investigating dam-age and other nonlinear phenomena in other quasi-brittle materi-als subjected to crack propagation under general loadingconditions; (ii) ease of implementation, essentially due to adoptionof a standard finite element setting, without requiring advancednumerical methods, like the cohesive zone FE method or the X-FEM approach.

A possible future perspective to this work may consist in devel-oping a more general multiscale framework for dealing multiplediscrete crack initiation and propagation together with diffusedamage and/or plasticity models in a unified manner, in order toinvestigate the structural size effects, which are usually capturedby experiments in both ordinary and lightweight aggregate con-crete structures.

Acknowledgments

The work of F. Greco and L. Leonetti was supported by theP.A.R.C.O. Project under funding of Regione Calabria and the workof R. Luciano and L. Feo was supported by the DPC-ReLUIS 2014project (Reinforced concrete sector) under funding of the ItalianCivil Protection.

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