Influence of the macro-porosity and the meso-structure on the dynamicproperties of concrete

F. Gatuingt & S. PierreLMT-Cachan, ENS Cachan / CNRS / UniverSud Paris, 61 avenue du President Wilson, F-94235 Cachan, France

ABSTRACT : Traditionally, the variability of concrete properties (in statics) is represented - in classical ma-croscopic models - using stochastic tools that can be complex and have to be identify with a large number ofexperimental tests. However, this phenomenon is usually linked to a ”deterministic” characteristic of the ma-terial : the multiple defaults that can be found in the cement paste matrix. This works intends to explore theinfluence of the heterogeneous meso-structure coupled with an explicit representation of the macro-porosity onthe dynamic properties of concrete like material. For this purpose we use FE simulations with a cohesive ap-proach and an explicit representation of the meso-structure, including macro-pores in the cement paste. In thiskind of material, heterogeneities (in our case the biggest aggregates) have sizes of the order of tens of millime-ters with defaults (pores) in the matrix we can imagine that they will influence fracture properties. To obtain arealistic description of the meso-structure we used a distribution of the aggregate according to the Fuller distri-bution (Fuller & Thompson 1907) and the distribution of the macro-porosity is computed by analyzing imagesof real concrete specimens obtained by X-ray microtomography.

1 INTRODUCTION

The aim of this study is to analyze the dynamic be-havior of concrete with FE simulations using a co-hesive approach and an explicit representation of themeso-structure including macro-pores in the cementpaste. The distribution of the macro-porosity is com-puted by analyzing images of real concrete speci-mens obtained by X-ray microtomography. With thisdescription we intend to explore the influence of theheterogeneous meso-structure coupled with an expli-cit representation of the macro-porosity on the dyna-mic properties of concrete like material. For this pur-poses 2D geometrical models of concrete consistingof aggregates, interfacial transition zones and a ma-trix (with holes) are generated. For the simulations,we use a finite element framework with cohesive ele-ments to explicitly represent the crack nucleation andgrowth. The debonding process in the cohesive ele-ments is controlled by a traction separation law basedon the popular linear extrinsic irreversible law propo-sed by (Camacho & Ortiz 1996).We validate our model by simulating direct dynamictension tests of concrete specimens. The role of the

meso-structure and the influence of the loading rateare then analyzed. We especially focus on the evolu-tion of the stress peak and the dissipated energy. Wecan observe that with a traction separation law inde-pendent of the strain rate we are not able to reproducethe macroscopic rate effect experimentally observedwhile the global dissipated energy is correctly predic-ted. We will also show the respective influences ofthe meso-structure and the loading rate on the varia-bility of the peak stress in tension. We finally attemptto give some results on the scale effects in dynamicswith various sizes of our specimens.

2 MESO-STRUCTURE GENERATION2.1 Meso-structure description

The meso-scale allows representing the biggest he-terogeneities and provides a way to better understandthe internal failure mechanisms in the materials. Forthese reasons, the choice of the phases represented inthe meso-structure is assumed to be important to pre-serve the physical meaning of simulations such as theones presented hereafter. The choice made here is torepresent only the millimetric porosity and the aggre-

gates larger than these pores. The inclusions, both ag-gregates and pores, are discretized in classes of identi-cal radius, based on a distribution function (the Fullercurve for the aggregates), as in (Cusatis 2011) :

F (di) =

(di

dmax

)n

(1)

Where di (i=1,...,N) are the diameters of the inclu-sions classes, dmax = dN is the largest inclusions classdiameter and n = 0,5

3. The micro-scale of several micrometers, where the main constituent is the hardened portland cement paste composed ofcement gel and capillary pores containing liquid water and water vapor.

4. The mini-scale of up to several milimeters, where there is mortar, consisting of fine mineral aggregates (sand),with dimensions from 0.1 mm to about 3 mm, embedded in hardened cement paste, of which two kinds must bediscerned: (a) the bulk paste, and (b) the interfacial transition zone (ITZ), with the typical width of 15–40 lm [18],which surrounds the aggregate surfaces and has a higher porosity and lower stiffness and strength than the bulkpaste.

5. The meso-scale, where concrete may be regarded as a two-phase composite material consisting of coarse mineral aggre-gates, with sizes over 50 mm, embedded in a matrix of mortar (the coarse aggregates are also surrounded by ITZ. In thissense, the material meso-scale is better considered as a three-phase composite, although the third phase, consisting of theITZ that is contacting only the coarse aggregates, occupies a small volume fraction).

6. The macro-scale is the usual engineering scale on which concrete is treated as a homogeneous continuum.

In this paper, we adopt and improve a recently developed random lattice model [15], which simulates concrete meso-structure and in which the customary simulation of random configuration of coarse aggregates (particles) is enhancedby introducing particle interaction laws (i.e., meso-scale constitutive laws) that take approximately into account the effectof finer-scale structure, consisting of fine aggregates (sand) in the mortar and the ITZ. These effects include the frictionalshear resistance and fracturing of mortar zones connecting two large aggregate pieces, the lateral confinement of thesezones by the bulk of mortar, and the effect of ITZ on connection stiffness.

The geometry of the model, for a given specimen, is defined by the following data: (1) the position of each coarse aggre-gate particle, (2) the connections between these particles (i.e., the topology of the interactions), and (3) the effective areathrough which the interaction forces between two adjacent coarse aggregate pieces are transmitted.

A typical granulometric distribution is described by the function F(d) = (d/da)n (Fuller curve) where da is the maximumaggregate size and n ! 0.5. This curve can be approximated (Fig. 1) by choosing a discrete number of characteristic aggre-gate sizes di (i = 1, . . . ,N) where dN = da and d1 = 4.75 mm (i.e., the size passing sieve #4) is the threshold that convention-ally separates the fine aggregates (discarded by the present model) and the coarse aggregates. The ratio of the mass ofaggregates with characteristic dimension di to the total mass of aggregates is wi = (Fi+1 " Fi"1)/2 (for i = 2, . . . ,N " 1),w1 = (F2 " F1)/2, and wN = (FN " FN"1)/2, Fi = F(di). The number of aggregates of various sizes, which must be usedto simulate a specimen of volume V, is Ni = wiaV/(qavi) where vi is the volume of one aggregate piece, qa = mass densityof aggregates, and a = total aggregate content, as mass per unit volume of concrete. The water–cement ratio w/c, the massdensity of cement (qc ! 3150 kg/m3) and the content of entrapped or entrained air can be used to get the mass density ofaggregates when it is unknown [15].

To generate a statistically isotropic random microstructure, the centers of particles (aggregate pieces) of specified diam-eters are placed throughout the volume of the specimen one by one (from the largest to the smallest) by using a procedureintroduced by Bazant et al. [6] (and also described in detail in [15]). In this procedure, after generating a new particle posi-tion by a random number generator, a check is made for possible overlaps of this particle with the previously placed par-ticles and with the boundaries of the specimen.

Fig. 1. Typical sieve curve (Fuller curve with n = 0.5) and its approximation with a discrete number of aggregate sizes.

7156 G. Cusatis et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 7154–7171

FIGURE 1: Discretization of the aggregates sizes dis-tribution

As shown in Figure 1, inclusions smaller than d1will not be represented (Cusatis et al. 2006).

This discretization is computed as follow :

Ψi =Fi+1 − Fi−1

2(i = 2, . . . ,N − 1)

Ψ1 =F2 − F1

2

ΨN =FN − FN−1

2

(2)

Where : Ψi is the ratio of the di diameter class in-clusions volume over the total inclusions volume andFi = F (di) is the fraction of the di diameter class.

It is clear that knowing the total granular fractionand the Ψi for each of the represented class is suffi-cient to define the classes.

2.2 Pores distributionThe two types of inclusions are discretized sepa-

rately for the sake of simplicity. This choice impliedthe strong hypothesis that a cumulative distribution

function similar to the Fuller curve for the aggre-gates could be obtained for the macro-porosity. It isclassical to use a Mercury Intrusion Porosimetry testto obtain the pores size, volume, density and otherporosity-related characteristics of a material. But inour case, the assumption of a millimetric porosity im-plies that it is not possible to obtain these pore dis-tribution through this test as the pores are too wide(and not connected anyway). From our knowledge,the data we are interested in are not available for clas-sical concrete. What can be found is related to per-vious concrete in cold weather climates and to thefrost resistance of concrete. In these concretes macro-pores result from entrained air. This parameter can becontrolled in the process of concrete production pro-vided that concrete will be subjected to freeze-thawcycles (Cordon 1966). Unfortunately in these studies,only the total porosity of concrete was determined byconcrete density volume for instance. This fact led usapproximate a macro-pore distribution curve compu-ted through the post-treatment of X-ray tomographyimages of real concrete specimens (see Figure 2) .

FIGURE 2: Original tomography image

FIGURE 3: Modified tomography image obtainedfrom Figure 2

The raw images from the tomography (Figure 2)are filtered using some native Matlab tools to bina-

rize it (Figure 3). To the resulting images is then ap-plied a fire-propagation-like algorithm to identify thepores. The output of this identification is the numberof voxels constituting each pores. We finally use thestrong assumption that all the macro-pores are sphe-rical that allows us to compute an equivalent radiusfor each pore. From this last operation, the expectedcumulative density function can be obtained (see Fi-gure 4) :

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

Pore radius [mm]

Surf

ace

frac

tion

FIGURE 4: Porous cumulative density function

Based on these results only two classes of poresare necessary to represent the 3,42% porosity (seeTable 1).

TABLE 1: Pores distribution obtainedRadius [mm] 0.7 0.5Surface fraction [%] 0.85 0.85

For the aggregates, the represented classes are inTable 2.

TABLE 2: Aggregates distributionRadius [mm] 7 6.25 5 4Surface fraction [%] 2.38 3.98 16.71 14.32

2.3 Meso-structure generationIn order to distribute the inclusions over the surface

of the specimen, the algorithm introduced by (Bazantet al. 1990) was used. Using this algorithm, the pla-cing of the inclusions obeys a simple uniform randomdraw for the spatial coordinates until the fraction ofeach inclusions class is achieved. The position is che-cked for interferences with inclusions already in place

and another position is drawn if necessary until a va-lid one is found. In order to fit the larger inclusionseasily and optimize the accuracy, the classes are com-pleted from the largest to the smallest diameter.Despite its simplicity, this algorithm guarantees theisotropic distribution of each inclusions class and inour case it is accurate enough. Figure 5 shows the re-lative error (volume fraction generated/volume frac-tion expected) for different sizes of the specimen ge-nerated normalized with the diameter of the biggestaggregate. We can see on this Figure that when thespecimen is larger than five time this diameter, theerror of the algorithm is lower than 5%. The discre-pancy plotted corresponds to different mesh genera-tion with the same parameters.In order to study the influence of the macro-porosityon the dynamic properties of concrete, the samemeso-structure is generated with and without poresin order to have the same aggregates arrangement inboth cases.

403020100

Normalized length (L/d_max)

0,001

0,01

0,1

Rel

ativ

e er

ror

FIGURE 5: Relative error on the granular fraction ge-nerated

We can notice that we have to use this algorithmwith precaution for later studies using denser inclu-sions structures. Likewise, for large structures thischoice of algorithm might be very expensive in termof calculation time.

3 NUMERICAL SIMULATION3.1 F.E. mesh generation

For the numerical simulations presented in the nextsection, we have to generate conforming FE meshwith 6-nodes triangular elements in 2D. The finiteelement code used for this study is based on a New-mark explicit time integration scheme. This schemeis conditionally stable so that the time step has to besmaller than a critical value fonction of the element

FIGURE 6: Example of a mesh of a meso-structurewith pores

size (Courant–Friedrichs–Lewy condition). That iswhy it is important to be able to control the size ofthe smallest element in the FE mesh. In order to gene-rate this mesh, we use the GMSH software (Geuzaine& Remacle 2009). The algorithm creating the geo-metry of the meso-structure outputs GMSH-readablefiles containing all the settings for the meshing. Thealgorithm used is the common Delaunay two dimen-sional meshing algorithm and the size of the elementsis chosen to be between 3 to 4 times smaller than thesmallest inclusion diameter. A finer meshing wouldnot increase significantly the accuracy of the resultsas shown by (Gatuingt et al. 2013). It should be notedthat the mesh fits the interfaces so that the elementseach belong to only one material. This is also a re-quirement for the cohesive model used here. Figure 6shows a zoom on a part of a mesh where an aggregateand two pores are shown.

3.2 Cohesive approach

As it has been mentioned before, the numerical fra-mework used in this study is a Finite Element Me-thod using an extrinsic cohesive approach. The co-hesive element method allows us to model dynamiccrack propagation and damage in a brittle materiallike concrete. The fracture process is described by thecohesive approach (introduced by Dugdale (Dugdale1960) and Barenblat (Barenblatt 1962) in the 1960s)as a separation process occurring at the crack tip ina small region of material called cohesive zone. Thiscan be introduced into a standard finite element envi-ronment using interface elements with null thicknessand with a fracture-based constitutive law. We assumethat the bulk material outside the cohesive zone re-

mains elastic. In our case the crack path is not knowna priori and all lines in the mesh are considered as apotential crack path. During the simulation, the stressat the interface between two adjacent continuum ele-ments is computed and compared to the fracture cri-terion at the end of every time step. The interfacialstress, σ, is calculated averaging stresses of the adja-cent Gauss points of the two continuum elements. Ifthe inter-element stress exceeds the critical stress va-lue, the nodes located at the inter-element boundaryare doubled, the two elements are topologically dis-connected and a cohesive element is inserted (see Fi-gure 7). After the nodal disconnection, the interfacialstress starts being controlled by the traction separa-tion law implemented in the cohesive element (Figure8).

n

s

nσ

tt

FIGURE 7: Schematic of the nodal disconnection bet-ween two continuum elements in which the interfacialstress has exceeded the critical stress

The coupling of the above mentioned fracture cri-terion together with an initially rigid cohesive law al-lows capturing the initiation of new cracks withoutan a priori definition of the possible cracks. Never-theless, the cracks are constrained to propagate follo-wing the inter-element boundaries and the fineness ofthe mesh can affect it. The law we use is the linearirreversible softening law proposed by (Camacho &Ortiz 1996). The traction separation law only dependson the effective scalar displacement δ :

δ =√

∆2n + β2∆2

s (3)

where−→∆ = (∆n,∆s) is the relative displacement

vector and the parameter β accounts for the couplingbetween normal and tangential displacements. Thevalue of β has to be estimated (e.g. by correlating ex-perimental results with numerical simulations (Ruiz,Pandolfi, & Ortiz 2001)) but it has been shown by(Snozzi, Caballero, & Molinari 2011) that the peakstrength is only slightly affected by β for a tensile loa-ding. Its value has been kept to one in this study. Thecohesive tractions law is then :

−→t =

t

δ(∆n−→n + β2∆s

−→s ) (4)

where t represents an effective cohesive traction. Thistraction in case of crack opening is given by :

t= fct

(1− δ

δc

)for δ = δmax and δ > 0 (5)

Where fct represents the local material strengthand δc represents the effective relative displacementbeyond which complete decohesion occurs. Whereasfor crack closure or reopening (δ smaller than δmax)the functional form is assumed to have the form :

t =δ

δmax

tmax for δ < δmax (6)

where tmax is the value of the effective tractionwhen δ is equal to δmax, in which is stored the maxi-mal effective opening displacement attained up to themoment. Moreover δmax also accounts for the irrever-sibility of the law allowing successive loading, unloa-ding and reloading. The evolution of the linear de-creasing law is graphically shown in Figure 8. Notethat the definition of fct and δc implicitly establishesthe existence of an effective fracture energyGc, whichcorresponds to the area under the curve of Figure 8 :

Gc =1

2fctδc (7)

Partially damaged cohesive elements have dissipa-ted an energyW <Gc. As the present work is focusedon tensile uniaxial loading, the fracture energy of thedifferent phases of our concrete is identified to the ex-perimental one for Mode I, i.e. Gc = GI

c .

δδcδmax

GGc

fct

tmax

FIGURE 8: Representation of the linear cohesive law.

3.3 Dynamic direct tension testTo analyze the dynamic tensile response of a

concrete specimen (2D plane strain), the specimen isloaded under displacement control with an imposedstrain rate ε. All the nodes of the finite element meshwhich are located on the upper (respectively lower)boundary moved at a constant velocity V0 :

V0 = εh

2(8)

where h is the height of the studied specimen. Toavoid stress wave propagation and an early fracturenear the boundaries, all the nodes of the finite elementmesh are prescribed an initial velocity in accordanceto their vertical position y (Miller et al. 1999) as illus-trated in Figure 9 :

Vy(y) =2V0hy (9)

h vyv0-v0

yv0

v0

FIGURE 9: Velocity initial condition

3.4 Results obtainedIn our simulations we decided to simulate four

strain rates ε = 0.1, 1., 10. and 100. /s. Due to ourexplicite time integration scheme we are not able tocompute a real quasi-static test and we assume that ε= 0.1 is close enough to this test.

Figure 10 shows the response of dynamic tensiontests performed on a 15× 15 cm2 specimen. We cansee that with our approach we are able to obtain a rateeffect in the result of this test without any rate effectin the constitutive laws as already shown in (Gatuingtet al. 2013). This rate effect is only due to the tran-sition of a single crack for law strain rate to diffusecracking for higher one as shown by comparing Fi-gure 11 and Figure 12.

0,0e+00 2,0e-04 4,0e-04 6,0e-04 8,0e-04 1,0e-03

Strain [-]

0

1e+06

2e+06

3e+06

4e+06

5e+06

Stre

ss [P

a]

0.1 [1/s]1 [1/s]10 [1/s]100 [1/s]

FIGURE 10: Stress-Strain responses in tension for dif-ferent strain rates

FIGURE 11: Cracks path for ε = 0.1/s for the samplewithout and with pores

FIGURE 12: Cracks path for ε = 10/s for the samplewithout and with pores

We can also notice in Figure 11 and Figure 12 thatthe macro-pores clearly influence the cracks path both

0.1 1 10 100

Strain rate [1/s]

3e+06

3.5e+06

4e+06

4.5e+06

5e+06

Tens

ile st

reng

th [P

a]

No poreWith pores

FIGURE 13: Rate effect on the tensile strength obtai-ned without and with pores in the sample tested

for the case of a single crack (ε = 0.1/s) and for thecase of diffuse cracks (ε = 10/s). This effect modi-fies the tensile strength obtained in the same sampledepending on the presence of pores as expected. Butthe global influence of the porosity is not yet totallyclear. Figure 13 shows the influence of the porosity onthe rate effect for a 15× 15 cm2 sample. We can seeon this Figure that the tensile strength is lower withpores but the evolution of the curve seems to be thesame without and with pores. But we have to concludewith precaution due to the discrepancy (several meso-structure generations) of the results at ε = 100/s in thesame range of the effect of the porosity.

4 CONCLUSION

In this work we proposed a meso-scopic model forthe analysis of dynamic tensile failure of concrete.This model is based on a 2D finite element descrip-tion with cohesive capability of a mix of aggregates ina mortar paste matrix without and with macro-pores.The influence of the heterogeneous meso-structureand of the porosity of concrete and the loading rateon the tensile response have been studied. With ourspecimen size we observed a small impact of the ag-gregates arrangement on the tensile strength for thesmallest strain rates and a more important one at ε= 100/s. The porosity clearly influenced the tensilestrength but its effects on other aspects of fracture isno so clear up to now. It will be interesting to inves-tigate in more details the influence of the porosity tosee if we will be able to obtain a kind of ”Weibull”response of our specimens.

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