Evaluation of mean performance of cracks bridged by multi-filament yarns

M. Konrad, J. Jerabek, M. Vorechovsky & R. ChudobaChair of Structural Statics and Dynamics, RWTH-Aachen University, Germany

This paper introduces an approach to modeling crack bridge behavior by evaluation of the mean performanceof a multifilament yarn. The influence of imperfections in the material structure is discussed using severalparametric studies. The disorder in the filament bundle and the heterogeneity in the interface layer are describedin form of statistical distributions. The correlation between these distributions is analyzed. Based on thisanalysis an approach to derive an objective statistical representation of the material structure is proposed. Finallythe application to the experiment is described.

1 INTRODUCTIONTextile reinforced concrete (TRC) has emerged in thelast decade as a new composite material combiningthe textile reinforcement with cementitious matrix.Its appealing feature is the possibility to produce fil-igree high-performance structural elements that arenot prone to corrosion as it is the case for steel re-inforced concrete. In comparison with other compos-ite materials, textile reinforced concrete exhibits ahigh degree of heterogeneity and imperfections thatrequires special treatment in the development of nu-merical models.

In particular, the material structure at the microscale exhibits high amount of irregularities and imper-fections in the geometrical layout of the basic compo-nents with varying local properties and in the qualityof local bindings between them. As an example, theseimperfections may be represented by non-parallel ori-entation of filaments within the bundle or by varyingbond quality between filaments and matrix across thebundle. As a result, the damage localization processof textile reinforced concrete exhibits interactions be-tween elementary failure mechanisms in the matrix,in the reinforcement and in the bond.

The presence of imperfections in the yarn and in thebond structure makes it necessary to account for theeffect of scatter of the material properties on the per-formance of the crack bridges. The crack bridges arethe hot-spots of deformation and interaction of dam-age with localized damage at the material interfaces ofmatrix and reinforcement. In a way the crack bridge isequivalent to a tensile test on yarn with an extremelyshort length equipped with a shear-lag-like clampingof filaments. Due to the varying penetration profile

along the yarn, the quality of the shear lag clampingexhibits high scatter. What is sought is a true statis-tical representation of the material structure able toreproduce the crack bridge behavior in different con-figurations and loading histories.

The paper describes the construction of crackbridge models and shows several parametric stud-ies classifying the influence of variations in filamentand bond properties on the performance of the crackbridge.

2 APPROACHES TO MODELING CRACKBRIDGE BEHAVIOR

The filament bundle models provide the steppingstone for robust modeling of the failure process in thebond layer between the multifilament yarn and the ce-mentitious matrix. In the following the existing mod-els for cracks bridged by fibers or fiber bundles will beclassified based on the considered variability of mate-rial parameters.

The effect of the irregular bond structure on theperformance of the crack bridge has been representedby a distribution of the effective filament lengths in(Schorn 2003). (Banholzer 2004) additionally consid-ers a finite bond and the abrasion of the filaments dur-ing the debonding. In (Lin et al. 1999) a microme-chanical model has been developed to characterize theinterfacial properties at single fiber pullout. The effectof fiber alignment on the pullout load was investigatedin (Li 1992). The influence of fiber alignment on in-situ strength has been characterized in (Kanda and Li1998). In (Lin et al. 1999) Eqn. (1) has been used toderive the crack bridging stress versus crack openingrelation by averaging over the contributions of the fil-

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aments.With the strain based formulation of statistical fiber

bundle model (Phoenix and Taylor 1973) it is possi-ble to evaluate the mean response of the bundle undertensile loading analytically:

µθ (e) =

∫

θ

qe (e; θ) dGθ (θ) . (1)

Where qe(e, θ) represents the filament constitutivelaw with global strain e and the vector of material pa-rameters θ and Gθ (θ) is the cumulative probabilitydistribution function of the parameter(s) θ.

We note that this class of models is limited to thelocal load sharing between filaments upon filamentbreak. In other words, the load of the lost filamentgets uniformly distributed across the bundle. Based onthe experimental evidence of low friction between fil-aments presented in (Chudoba, Konrad, Vorechovsky,and Roye 2006) we neglect the interaction effectbetween the filaments with effective lengths the inrange of several millimeters as they occur in the crackbridge.

A thorough study of sources of random-ness/disorder in the multi-filament yarn that arerelevant for its performance in textile reinforcedconcrete has been presented in (Chudoba et al. 2006).For the analysis of bundles with infinite number offibers a continuous analytical model with refinedkinematic relation has been developed. The consid-ered distributions of material properties include boththe variations of parameters from filament to filamentand the variations of local strength and materialstiffness across the bundle. In the companion paper(Vorechovsky and Chudoba 2006) the variations ofmaterial properties over the length are considered.

The scatter of filament stiffness parameters leadsto a reduced strength of a bundle with a very shortlength. In a crack bridge the strength reduction is inparticular caused by the scatter of filament lengthsand their delayed activation (slack). Besides that, thescatter of filament strength across the bundle leads toa further reduction of the bundle peak force (Smithand Phoenix 1981).

In order to demonstrate the effect on an exam-ple we consider a crack bridge with a scatter of fil-ament lengths. In particular, we introduce the rela-tive difference of the filament length with respect tothe minimum length as λ = (`max − `min)/`min witha uniform distribution, i.e. Gλ (λ) = λ/λmax where0 ≤ λ ≤ λmax. The other parameters of the filament,i.e. Young’s modulus E, area A and breaking strain ξare considered constant. For the chosen distribution, itis possible to derive analytical formulas for the bundlemean strength and its variance (Chudoba et al. 2006)

at a given control strain e as

µλ (e;λ) =∫

λqe (e;λ) dGλ (λ)

=

EAe ln (1 + λmax)/λmax

for 0 ≤ e ≤ ξ

EAeln (1 + λmax)− ln(e/ξ)

λmaxfor e ≥ ξ

(2)

The calculated mean load strain diagrams for a bun-dle with 1743 filaments and constant diameter D =25.5µm, Young’s modulus E = 70GPa and breakingstrain ξ = 1.79% are exemplified in Fig. 1 for severallevels of scatter represented by λmax .

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Figure 1: Mean load-displacement diagrams of one crackbridge with uniform distribution of additional fiber lengthλ ∈ (0, λmax). λmax = a) 0.0, b) 0.25, c) 0.5, d) 0.75, e)1.0, f) 1.25 and g) 1.5 plotted with a scatterband (mean ±standard deviation).

The introducing example demonstrates the model-ing concept used in the first part of this paper. In addi-tion to the scatter of filament lengths, we shall includethe scatter of the bond quality across the bundle. In or-der to justify this decision we first discuss the issue ofappropriate choice of material parameters in general.

3 OBJECTIVITY OF THE MATERIAL PARAM-ETERS

The overall goal in the construction of the model of acrack bridge is to predict its performance in a broadapplication context, i.e. in an arbitrary configurationof cracks and textile fabrics. These configurations arerepresented by the crack spacing on the one hand andby the angle at which the roving crosses the crack onthe other hand. In this paper we consider only ten-sile response and limit the discussion to the aspects ofbond parameters that should be independent of crackspacing (Fig. 2).

The minimum number of structural parameters isgiven by their capability to reflect the changes in thematerial structure during the evolution of local dam-age. As shown in the example in Fig. 1, it is possible

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Figure 2: Heterogeneity in the material structure with the sought distributions of micromechanical parameters.

to find such a distribution of filament lengths that canreasonably reproduce the experimental response. In-deed, the performance of a single crack can be repro-duced by a proper choice of length profile resultingin the corresponding number of breaks at each levelof control displacement. However, this distribution ofsmeared effect of length and bond variations cannotbe considered representative in the context of multi-ple cracks. There are especially two reasons:

• The crack bridge with structural variations as-cribed solely to the effective length would ex-hibit a shorter stress transfer length than underthe assumption of finite bond. Therefore the fit-ted crack bridge stiffness would not be consis-tent with the bond stiffness and, thus, inconsis-tent with the crack spacing.

• Even if the crack spacing would be defined apriori, e.g. using the experimentally measuredcrack distribution, the effective length profilesobtained from a single crack bridge experimentwould lead to an underestimated stiffness in thecase of unloading of existing cracks at the for-mation of a new crack. This is due to the factthe the stiffness reduction during the opening ofa crack bridge can only be appointed to the fail-ure of filaments. Thefore the crack will unloadwith the resulting secant stiffness. A considera-tion of plastic deformations in the interface leadsto significantly higher stiffness.

Separate consideration of the scatter of lengths andof the bond quality across the bundle in the vicinityof a crack bridge allows us to capture interactions be-tween cracks. It would certainly be possible to resolvethe material structure in even more detail. On theother hand, a pragmatic material resolution requiresonly to capture the components and interfaces thatdominate in the disintegration of the material struc-ture.

4 EFFECT OF FINITE BOND STRENGTHThe procedure for evaluating the total strength de-scribed in Sec. 2 can be used with more complex ide-alizations of a crack bridge taking into account fur-ther failure and damage mechanisms. In addition tofilament rupture considered in the previous model wenow include the influence of debonding between fil-ament and matrix. The performance of a multifila-ment yarn bridging a crack is evaluated by replacingthe constitutive law qe(e, θ) in Eq. (1) with a filamentpull-out response pu (u, θ) that could be derived ana-lytically using a shear lag model (Stang et al. 1990)with a cohesive interface between the filament andmatrix.

With Gλ (λ) defined as in the last example, wenow include the debonding of a filament from thematrix as an additional effect characterizing the in-terface between a perfectly embedded filament andthe cementitious matrix. In (Banholzer 2004) fila-ment matrix bond laws for perfectly embedded fil-aments have been derived as multi linear functionsτ1 (s) (with s representing the slip) using the fila-ment pull-out test and the cohesive interface model.In the particular case of the Vetrotex AR-glass rov-ing with 2400 tex the derived function shows that noadhesional bond is developing and thus the interfa-cial properties are predominated by friction. There-fore τ1 (s) can be represented in the following bilinearform:

τ1 (s) = τfr ·{

s/scrit 0 ≤ s ≤ scrit1 s ≥ scrit

(3)

Again we keep all other parameters including theinterface characteristics, constant. Fig. 3.1 showsthe load displacement diagrams for the varied rangeof lengths with λmax ∈ [0,0.25,0.5,0.75,1,1.25,1.5]and a constant bond stress τ1 = 3.38 MPa. It exam-plifies that the crack opening at peak load increasesdue to the debonding of the filaments. To demon-strate this effect we now keep λmax = 1.5 constant andvary the bond stress in the range τ1 = c · 3.38MPa,

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3.1: τfr = 3.38MPa and λmax = a) 0.0, b) 0.25, c) 0.5, d)0.75, e) 1.0, f) 1.25 and g) 1.5 plotted with a scatterband(mean ± standard deviation)

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3.2: λmax = 1.5 and τfr = c · 3.38MPa. c = a) 10.0, b) 1.0,c) 0.5, d) 0.25, e) 0.125 plotted with a scatterband (mean ±standard deviation)

Figure 3: Mean load-displacement diagrams of one crack bridge with debonding τfr and with uniform distribution ofadditional fiber length λ ∈ (0, λmax).

c ∈ [10,1,0.5,0.25,0.125]. The resulting load dis-placement diagram Fig. 3.2 shows that the mean crackbridge strength increases significantly with decreas-ing bond stress τ1.

The reason for this increased strength is the homog-enizing effect of the debonding causing a more uni-form stress distribution across the bundle (more fila-ments can act simultaneously before they break). Wealso note that there is a significant reduction in thescatter of strength compared to Fig. 3.1: 184 N forperfect bond and 50 N with included debonding andthe lowest bond stress.

5 EFFECT OF SCATTER IN BOND STRENGTHDue to the irregular penetration of the matrix into theyarn the quality of the interface between matrix andfillment varies. The reason for this scatter of bondquality are the varying contact area between filamentand matrix and variations in the quality of the matrixitself due to the heterogeneous infiltration of aggre-gate particles. To evaluate the influence of the scatterin the interface we introduce the bond quality ϕ of anindividual filament as a dimensionless scaling factorfor the bond law. The bond stress slip relation for thisfilament in the bundle is defined as follows:

τ (s,ϕ) = ϕτ1 (s) (4)

The influence of a bond quality uniformly distributedover the bundle cross section with Gϕ (ϕ) and λmax =0 (all filaments have the same bond free length) is ex-emplified in Fig. 4. The bond quality ϕ has signifi-cant influence on the stiffness. With decreasing meanbond quality µϕ the stiffness decreases due to thelarge amount of debonding (Fig. 4.1. The peak loadis reached at larger displacements. An increased scat-

ter (σϕ) leads to reduced peak load and more ductilepost peak behavior as a result of more inhomgeniousstress transfer in the interface (Fig. 4.2).

6 EFFECT OF CORRELATIONThe micrographs in Fig. 5 suggest that there is apositional dependence for the two studied parame-ters ` and ϕ. Regarding the outside filaments of thecross section we see that they are very well embedded(ϕ ≈ 1) while their free length is very short (λ ≈ 0).For the inner filaments we can observe the opposite.This raises the question of correlation between thesetwo random parameters. In order to evaluate the in-fluence of this correlation we shall consider ϕ and λnormally distributed using the bivariate normal distri-bution:

G(ϕ,λ) =e−

z

2− 2ρ2ϕ,λ

πσϕσλ1√

1− ρ2ϕ,λ

(5)

z =(ϕ− µϕ)2

σϕ2

− 2ρϕ,λ (ϕ− µϕ) (λ− µλ)

σϕ σλ

+(λ− µλ)

2

σλ2

with ρϕ,λ standing for the correlation coefficient.Figs. 6 and 7 demonstrate that the qualitative effects

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4.2: µphi = 0.5, σphi= a) 0.125, b) 0.0625, c) 0.03175

Figure 4: Mean load-displacement diagrams of one crack bridge with constant bond free length (λmax = 0) and uniformdistribution of bond quality ϕ[−].

5.1: cross section 5.2: longitudinal section

Figure 5: SEM micrographs of a mutlifilament yarn embedded in the cementitious matrix.

described in the sections above can be reproducedusing normal distributions. The load strain diagramsshow smoothed response at the peak load and at thetails due to the concentration of the normal distribu-tion at the mean values.

The effect of the correlation between the bond qual-ity ϕ and the bond free length λ · `min is demonstratedin Fig. 8 for a correlation coefficient ρϕ,λ ∈ (−1,1).The correlation coefficient ρϕ,λ = 0 indicates that ϕand λ are completely independent. For ρϕ,λ = 1 theminimum values of ϕ coincide with the minimum val-ues of λ for ρϕ,λ =−1 the minimum values of ϕ coin-cide with the maximum values of λ. A strong negativecorrelation (Fig. 8 a): ρϕ,λ ≈ −1.0) leads to a slightlyreduced crack bridge performance. The negative cor-relation leads to more inhomogeneous stress trans-

fer reducing the peak load an associate crack open-ing to higher stress in the softening branch. This cor-responds to the combination of short filaments withhigh bond and long filaments with low bond. We re-mark if it would be technically possible to achievethe other extreme and combine low bond quality withlarge bond free lengths and vice versa, it would bepossible to obtain an increased bond performance.

7 APPLICATION TO THE EXPERIMENTIn oder to relate the discussed issues to experimen-tal data we shall identify the parameters characteriz-ing the material structure using a pull-out experiment.We shall simplify the modell in such a way that thevariability is considered only deterministically. Thisis justified by relatively low effect of correlation dis-

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Figure 7: Mean load-displacement diagrams of one crack bridge with constant bond free length (λmax = 0) and normaldistribution of bond quality ϕ[−].

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Figure 6: Mean load-displacement diagrams of one crackbridge with debonding (τfr [MPa] = 3.38) and normal dis-tribution of additional fiber length λ ∈ (0, λmax). λmax =a) 0.0, b) 0.25, c) 0.5, d) 0.75, e) 1.0, f) 1.25 and g) 1.5plotted with a scatterband (mean ± standard deviation).

cused above. If we assume ρϕ,λ → −1.0 we can in-terpret the inverse cumulative density functions ofour distributions as profiles of heterogeneity varyingacross the bundle with the random nature replaced bythe parametric coordinate ξ (Fig. 9) representing theposition of the filament in the bundle.

Based on numerical and experimental studies andfor the sake of completness we introduce the third pa-rameter considered in the model: The delayed activa-tion of filaments (slack) within the bond free length.We further assume negative correlation between bondquality and delayed activation. The interaction of thedelayed activation with the bond free length and thebond quality has been described by (Konrad and Chu-doba 2004).

For the purpose of numerical calibration usingthe experimental data we approximate the introduced

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Figure 8: Effect of correlation between bond free lengthand bond quality(ρϕ,λ = a) -.99, b) 0.0, c) 0.99.

profiles by the following polynomial functions:

ϕ (ξ) = (kϕ,0 + kϕ,1 − 2) ξ3

− (3− 2kϕ,0 − kϕ,1) ξ2

+ kϕ,0 ξ

λ (ξ) = λmax + (λmin − λmax) ξ

θ (ξ) = (2 θmax − kθ,1) ξ3

+ (2kθ,1 − 3 θmax) ξ2

+ kθ,1 ξ + θmax

(6)

The parametric coordinate ξ = 0 represents the ideal-ized center of the bundle and ξ = 1 coincides with theouter layer of the bundle.

With the assumption of uniform distribution of im-perfections in the bond and yarn structure along thespecimen the effect of imperfection on the perfor-mance of each crack bridge can be considered thesame. Thus, the quantification of this effect can beperformed with an isolated analysis of a pull-out ex-periment in connection with the bond layer model

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Figure 9: Profiles describing the heterogeneity in the material structure.

shown in Fig. 2. As can be seen in the figure theprofiles of heterogeneity are directly used for discreterepresentation of the crack bridge. We emphasize thatthe fine resolution of the material structure furtheraway from the crack bridge is not needed due to theincreasingly homogeneous stress profile.

In this way we construct the discrete bond layermodel (Fig. 2) with the interface layer between theyarn and the matrix represented by a set of laminas.Each lamina interacts with the matrix trough a givenbond law. It represents a subset of filaments with thesame characteristics and is coupled with the matrixusing zero thickness interface elements (Kaliakin andLi 1995).

The bond law distributed over the bond layer is thenconstructed as:

τ (s, ξ) = τ1 (s (ξ))ϕ (ξ) (7)

Equipped with the boundary conditions of a pull-out test the calibration (Chudoba et al. 2004) of themodel is performed both using the load-displacementcurve and the curve representing the instantaneousfraction of the unbroken filaments during the loadingprocess. The latter is obtained experimentally usingthe FILT test (Failure Investigation using Light Trans-mission properties) by optical recording of the lighttransmission through the unbroken filaments (Ban-holzer 2004) (Fig. 10).

The parameters describing the profiles are obtainedusing the calibration framework of automated cali-bration procedures based on evolutional strategies toavoid convergence to local minima.

8 SUMMARYA specific feature of the textile reinforced concrete isthe high degree of heterogeneity of the reinforcementand of the matrix on similar scales of the materialstructure. The complex damage and failure process

Figure 10: FILT-Test.

calls for an application of stochastic and microme-chanical models that can reflect the heterogeneity.

In this paper a statistical fiber bundle model forthe evaluation of the mean performance of a crackbridged by a multifilament yarn has been introduced.The effect of the disorder in the filament bundle hasbeen described in form of distributions of relative ex-tra filament length λ. We have shown that the scat-ter of filament lengths leads to a reduced performanceof the crack bridge. The introduction of a finite bondstrength has a homogenizing effect leads to an in-creasing strength at larger displacements. The het-erogeneity of the interface layer was represented inform of a statistical distribution of bond quality ϕ. In-creased scatter of the bond quality leads to reducedcrack bridge strength performance. The negative cor-relation of varriations in the bond quality ϕ and in fil-ament lengths λ amplifies both effects. We have pro-posed an approach to deriving an objective statisti-cal representation of the material structure using crosssection profiles of the described parameters and haveintroduced a model utilizing these profiles. Finally,the evaluation of the material parameters calibratingthe introduced model with experimental data has beendescribed.

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ACKNOWLEDGMENTSThe authors thank the German research foundation(DFG) in context of the Collaborative Research Cen-ter 532 for their financial support.

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