Composites: Part B 55 (2013) 518–527

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Composites: Part B

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Non-linear analytical model of composites based on basalttextile reinforced mortar under uniaxial tension

1359-8368/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compositesb.2013.06.043

⇑ Corresponding author. Tel.: +34 667 178 992/946 430 850; fax: +34 946 460900.

E-mail address: pello.larrinaga@tecnalia.com (P. Larrinaga).

Pello Larrinaga a,⇑, Carlos Chastre b, José T. San-José c, Leire Garmendia a

a TECNALIA, Parque Tecnológico de Bizkaia, c/Geldo Ed. 700, 48160 Derio, Spainb UNIC, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugalc UPV/EHU, Department of Engineering of Materials, c/Alameda Urquijo s/n, 48013 Bilbao, Spain

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

Article history:Received 13 February 2013Received in revised form 18 April 2013Accepted 22 June 2013Available online 5 July 2013

Keywords:A. Fabrics/textilesB. Mechanical propertiesC. Analytical modellingD. Mechanical testing

The recent development of inorganic based composites as low-cost materials in reinforced concretestructural strengthening and precast thin-walled components, requires the creation of models that pre-dict the mechanical behaviour of these materials.

Textile Reinforced Mortar (TRM) shows complex stress–strain behaviour in tension derived from theheterogeneity of its constituent materials. This complexity is mainly caused by the formation of severalcracks in the inorganic matrix. The multiple cracking leads to a decrease in structural stiffness. Due to thesevere conditions of the serviceability limit state in structural elements, the prediction of the stress–strain curve is essential for design and calculation purposes. After checking other models, an empiricalnonlinear approach, which is based on the crack control expression included in the Eurocode 2, is pro-posed in this paper.

Following this scope, this paper presents an experimental campaign focused on 31 TRM specimensreinforced with four different reinforcing ratios. The results are analysed and satisfactorily contrastedwith the presented non-linear approach.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

By means of several research projects, the mechanical possibil-ities and advantages of Textile Reinforced Mortar have been pro-ven, both as strengthening material and as precast material (inthis field this composite is also called Textile Reinforced Concrete)[1–5]. However, its implementation as a regular technique is stillfar. One important step to reach this objective is modelling themechanical behaviour of TRM for future applications in real situa-tions. As TRM usually bears tensile loads, it is very relevant to mod-el the behaviour of this material under pure tensile loads, i.e. todefine its stress–strain relationship.

In the bibliography, there is broad information referred tonumerical or analytical TRM models which show excellent results[6–9]. Nevertheless, most of these analyses require the use of spe-cific softwares [9] or are costly in terms of time. Moreover, in somecases additional information is required, which involves the devel-opment of additional tests [7,8].

For these reasons, it is convenient to produce simple and easy-to-implement models. Stress–strain mathematical expressions are

usually proposed to model the materials, both for linear and non-linear analysis. These expressions are based on experimental dataand can be used as constitutive equations in simple numericalmodels. Concrete and steel are obvious examples of stress model-ling by means of mathematical expressions, in fact, both materialsare modelled in design codes as the Eurocode 2.

A nonlinear approach is presented in this paper to define thestress–strain relationship of Basalt Textile Reinforced Mortar(BTRM) under uniaxial tensile loads. The model is based on theconcrete crack control expression included in the Eurocode 2[10]. The developed expression is calibrated empirically withexperimental data included in the article, the results of 31 TRMspecimens subjected to uniaxial tensile loads.

2. Materials, basalt textile and mortar characterisation

Basalt appears to be a material which can offer interestingopportunities in the future of the construction industry [11]. Re-cent studies have included basalt fabrics as reinforcement in FRPcomposites [11,12] and basalt textile as TRM internal core [13,14].

The basalt textile used in this study consists of rovings woven inthe two principal directions, i.e. a bidirectional mesh which geom-etry is detailed in Table 1. Basalt rovings are covered by a bitumencoat in order to improve the bond between the mortar matrix and

Table 1Basalt textile geometry.

Design thickness, tf 0.0349 mmOpening size 25 � 25 mmWeight of the dry sheet 233 g/m2

Density 2.75 g/cm3

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the textile. In addition, the coat improves the textile performancedue to its capacity to transfer load directly to more roving fila-ments [15,16].

According to several previous studies, there is a considerablegap between the main mechanical properties – tensile strength,ultimate strain and Young’s modulus – of a single fibre or filamentand those of the textile mesh [17]. For this reason, the mechanicalcharacteristics of the textile were experimentally determined intensile tests on specimens which length was 600 mm.

ASTM D 5034 [18] standard, used as a reference, suggests astrain rate of 300 mm/min. However, in order to carry out a suit-able data acquisition, a speed of 1 mm/min was selected for thepresent test. In total, seven textile specimens of four rovings weretested. The results are summarised in Table 2.

The difference between the data given by the manufacture andthe experimental results is caused by several factors, mainly theload transfer between filaments and the difficulty to exert an iden-tical initial length and strain for all the strands of the specimens[13,17]. Hence, it was impossible to achieve a simultaneous rup-ture of all the rovings, so the experimental values for ultimate ten-sile strength and ultimate tensile strain cannot be considered asreference data. However, as the tested textile presents linearbehaviour until rupture, the tensile Young’s modulus values ob-tained in this test can be easily calculated [14] and it will be con-sidered for the model presented in this paper. Moreover, despitethe difficulties derived in this kind of tests, the obtained resultspresented low scattering.

A non-commercial cement-based mortar was used as TRM ma-trix. The performance of any externally bonded strengthening sys-tem is clearly influenced by the interaction between the matrixand the inner reinforcement and the capacity of the composite–substrate interface [15]. The maximum grain size of the sand usedin the mortar was 0.6 mm. This factor enhances the workability ofthe fresh mixture and facilitates its interaction with the textilemesh. The amount of redispersable resins was lower than 5%, in or-der to achieve a fire-proof mortar. As can be observed in Table 3,there are not innovative products in the mortar composition be-cause the idea is to present the TRM as a competitive material.However, special attention was paid to its composition so as toachieve appropriate workability, curing time and fire resistance.

Table 2Average results of basalt textile under pure tensile load.

Filament – givenby the manufacturer

Textile –experimentala

Ultimate tensile strength, rfu (MPa) 2100 1160 (0.025)Young’s modulus, Ef (GPa) 89 67 (0.054)Ultimate tensile strain, ef (%) 3.14 1.82 (0.054)

a COV between brackets.

Table 3Mortar dosage by weight (%).

W/C ratio 0.2

Sand (grain size < 0.6 mm) 60–70Grey cement type II 42.5R 30–40Polymeric chopped fibres 3–5Redispersable resins 1–3

After 28-day curing, 40 � 40 � 160 mm prisms were tested todetermine mortar mechanical properties according to UNE-EN1015.11:1999 [19]. Compressive strength is 19.8 MPa while tensileflexural strength is 7.2 MPa.

3. Uniaxial tensile test

The experimental campaign included in this paper is formed by31 uniaxial tensile specimens. Four series, from one to four textilelayers, of seven specimens each one were defined. The effect ofreinforcing ratio was therefore analysed. Specimens were taggedas TBX, where the X represents the number of internal reinforce-ment layers, i.e. in this paper from TB1 to TB4. Besides, three addi-tional specimens were also manufactured without reinforcingmaterial.

3.1. Specimens geometry and manufacturing process

TRM has attracted attention for the last 10 years. There hasbeen a considerable increase in research projects and publicationsrelated to this innovative composite. However, no standard charac-terisation tests are available in the bibliography. Thus, several testproposals have already been made [6,15,20,21] by differentauthors. They differ primarily in: the shape and geometry of thespecimens, the connection with the test machine clamps, the strainrate and the instrumentation.

The rectangular parallelepiped specimen is easy to manufactureand implement. For the present study it was decided to manufac-ture specimens with a 100 � 10 mm cross-sectional area and600 mm in length. Samples were prepared in plywood formworks.In order to promote the failure of the specimen in its middle thirdportion, both ends of each specimen were extra-reinforced withtwo layers of 200 � 100 mm textile (Fig. 1). The internal reinforc-ing layers (800 � 100 mm) were uniformly positioned within thecross section, e.g. three layers distributed every 2.5 mm on thespecimens reinforced with two or more layers. Textile rovingswere oriented according to the longitudinal axis of the specimens.In the particular case of the unreinforced specimens, only the addi-tional reinforcement was installed at both ends. (see Fig. 2)

Specimens were cured in a saturated atmosphere for sevendays, and, then, they were stored for 21 days at room temperature(18 �C and 60%RH). Tests were carried out between 28 and 34 daysafter the specimens were manufactured.

3.2. Test setup

TRM tensile specimens were tested in a Schenk 100 kN presswhich was programmed to exert a deformation rate of 0.5 mm/min [15]. Tensile force was applied with specially designed metal-lic clamps, in order to avoid any stress concentration point, which

Fig. 1. Uniaxial tensile TRM specimen geometry.

Fig. 2. Two steps of the TRM tensile specimens manufacturing.

Fig. 3. Uniaxial tensile of the test setup.

Fig. 4. Load–strain curves of TRM specimens.

520 P. Larrinaga et al. / Composites: Part B 55 (2013) 518–527

would cause premature brittle failure without reaching the ulti-mate tensile load.

As the formation of cracks is promoted in the specimen centralthird part, two displacement transducers (LVDTs) were placed oneach side of the specimen to measure the elongation of that area.The measured reference length, as can be observed in Fig. 3, was210 mm. All the data was compiled by a data logger at a frequencyof 5 Hz.

3.3. Experimental results

The purpose of testing unreinforced specimens was to charac-terise the behaviour of the mortar under pure tensile loads. As itwas expected, only one crack was formed in the unreinforced spec-imens. The average results of the three specimens are displayed inTable 4.

The load–strain curves of 28 TRM tests with 1–4 layers areshown graphically in Fig. 4. The results show good repeatability

Table 4Average results of unreinforced specimens.

Fmu (N) rmua (MPa) emu (%) Em (GPa)

2480 2.48 0.03 8.25

a rmu = Fmu/(10 � 100).

in contrast with the characteristic scattering of this kind of mate-rials. Four stages are clearly differenced in each curve. This behav-iour is typical in inorganic composites subjected to uniaxial tensileload [22], reinforced above the critical volume fraction.

In the first stage, the load is primarily supported by the mortaruntil its cracking. After this pre-cracking state, a second stage be-gins with a multicracking process where the load is supported byboth materials (mortar and fibres). In the third stage only the fibrescarry the load. This stage could be divided into two subsectionsaccording to its linearity. Firstly, the composite presents linearbehaviour until the progressive rupture of filaments inside the rov-ings [14] and the debonding at the textile–matrix interface causesthe ‘‘softening’’ of the specimens’ stiffness. Thus, the second sub-section presents a non-linear behaviour. The fourth stage beginsafter the maximum load is reached and ends with the rupture ofall rovings.

Table 5 includes the average main values for each series: ulti-mate tensile force and stress, the Young’s modulus of stage III,and the final strain at each stage. The tensile stress of the speci-mens is calculated dividing measured load by the area of internalreinforcement: b � tf � n; where b is the width of the specimen(100 mm), tf is the design thickness (0.0349 mm) and n is the num-ber of textile layers installed as internal reinforcement.

The analysis of the results produces interesting interpretations.According to the bibliography, the stiffness of the third stage Et,III isslightly lower than the Young’s modulus of the textile reinforce-ment Ef (see Table 2). In three of the four series Et,III remained closeto 60 GPa, while the basalt textile Young’s modulus is 67 GPa, i.e.9% higher. This reduction has been quantified (10–30%) by otherauthors [22,23]. Likewise, there was good correlation in series

Table 5Average results for each TRM series.

Series Ftr (N) rtra (MPa) Et,III (GPa) et,I (%) et,II (%) et,III (%)

TB1 3797 1088 43 0.034 0.40 2.15TB2 8772 1256 59 0.041 0.29 1.96TB3 12,515 1195 57 0.028 0.21 2.10TB4 16,679 1194 61 0.028 0.15 2.07

a rtr = Ftr/(0.0349 � 100 � n).

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TB2, TB3 and TB4 in terms of tensile strength and ultimate strain.The main difference between these three series is the length of thestage II (see Table 5). The multiple cracking strain et,II was reducedwith a higher number of reinforcement layers [24].

Nevertheless, series TB1 showed a significant discrepancy whencompared with the other results. The average value for Et,III wasequal to 43GPa. The low amount of internal reinforcement in seriesTB1 can explain this discrepancy; one layer may not be enough toachieve a monolithic material which could be considered as a com-posite. According to Peled and Bentour [16] the critical volumecontent of fibres in cement composite is about 1–3%. At ratiosabove this level the behaviour load is characterised by multiplecracking and the TRM is able to carry the additional load appliedafter the matrix has cracked. However, Series TB2 presents a vol-ume content of 0.7% and the specimens showed multiple crackingand behaved as the rest of the series with higher fibre content.

These statements can be contrasted with the stress–strain re-sults included in Fig. 5. While series TB2, TB3 and TB4 presentedsimilar behaviour after multiple-cracking, series TB1 developed adifferent slope at third stage.

The analysis of the crack pattern also provides interesting infor-mation. Firstly, it is noticeable how the number of cracks riseswhen the number of textile layers increases (see Fig. 6). Moreover,the distance between cracks is reduced when more internalreinforcement is used. At series TB1 only one or three crackswere formed, being an evidence of its incapability for developingmultiple cracking and behave as an inorganic based composite.Finally, as the fibre content increases, the crack pattern tends tostandardise.

In the following section a non-linear analytical model ispresented. The obtained experimental results will be comparedwith those obtained in the model in order to check its validity.

4. TRM modelling

The proposed model is based on the crack control expression in-cluded in the ‘‘Eurocode 2: Design of concrete structures. Part 1–1’’and the Aveston–Cooper–Kelly theory. This well-known theory

Fig. 5. Direct comparison between specimens of different series.

was the first satisfactory explanation of multiple cracking. For thisreason, the authors would like to revise the ACK theory and com-pare its results with the experimental data.

4.1. ACK-theory

This theory defines a theoretical tri-linear stress–strain behav-iour of a composite with a brittle matrix, in which it is assumedthat the fibres are held in the matrix solely by the presence of fric-tion, and that axial sliding along a fibre–matrix interface would oc-cur under a critical, limiting value of longitudinal shear stress[25,26]. The fibre debonding degree and crack spacing are closelylinked to the maximum shear stress at the fibre–matrix interface.Several models for brittle matrix composites have already beenmodelled considering the ACK theory, e.g. [24,27,28]. Its basicassumptions are described in [24].

As it has been experimentally stated, TRM tensile behaviourcould be divided in three different, but complementary, stages.The ACK theory consists on three straight lines which superim-poses the experimental stress–strain curve (see Fig. 7).

According to ACK theory, at the first stage, the composite obeysthe law of mixtures:

Ec1 ¼ Ef Vf þ EmVm ð1Þ

where Ec1 is the composite stiffness, Ef represents the tensileYoung’s modulus of the fibres, Em is the matrix one, and Vf and Vm

are the volumetric fraction of fibres and matrix, respectively. Atthe first stage, the matrix–fibre interface shear behaviour is as-sumed to be elastic.

Stage I finishes when the matrix reaches its tensile failure stressand it shows multiple cracking. Matrix tensile failure stress, rmu,(and its corresponding strain emu) has a direct influence on themultiple cracking stress rmc:

rmc ¼Ec1rmu

Emð2Þ

At this value, the composite presents multiple cracking, i.e. succes-sive cracks are formed as the composite strain increases. When acrack appears in the matrix and reaches a fibre, debonding of thematrix–fibre interface occurs due to weakness of the bond. Then,a constant frictional interface shear stress s is considered. This shearstress provides normal stress transfer from fibres to the inorganicmatrix. The length of the debonded interface d can be written byexpressing the force equilibrium along the loading (longitudinal)axis of the fibres [25]:

d ¼ Vmrrmu

Vf 2sð3Þ

where r is the fibre radius and s represents the frictional shearstress at the matrix–fibre interface. At the multiple cracking stage,distances between cracks are no smaller than d and no larger than2d. The spatial introduction of cracks occurs randomly until nospace remains for new cracks, in a similar way to the geometricalcar parking problem. Widom [29] determined that the average dis-tance between cracks equals X = 1.337d. By means of this value it ispossible to determine the composite strain (emc) when the multiplecracking stops:

emc ¼ ð1þ 0:666aeÞrmu

Emð4Þ

where:

ae ¼EmVm

Ef Vfð5Þ

At third stage only fibres contribute to bear the applied deforma-tion. Thus, the matrix stress remains constant despite the increase

Fig. 6. Crack pattern development. From up-left to down right: TB1, TB2, TB3 and TB4.

Fig. 7. Typical stress–strain curve of TRM in tension (in black) and ACK linearisation(in dotted grey).

Table 6ACK theory results.

Series Vm (–) Vf (–) Ec1 (GPa) rmc (MPa) emc (%)

TB1 0.9965 0.0035 8.46 2.54 0.734TB2 0.9930 0.0070 8.66 2.60 0.381TB3 0.9895 0.0105 8.87 2.67 0.263TB4 0.9860 0.0140 9.07 2.73 0.204

Fig. 8. ACK model – TB1 series.

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of the tensile load. TRM stiffness at this stage is defined by expres-sion (6):

Ec3 ¼ Ef Vf ð6Þ

The data necessary to run the ACK model is taken out of present pa-per’s Section 2:

- Ef: 67 GPa- Em: 8.25 GPa- tf: 0.0369 mm- rmu: 2.48 MPa

Table 6 is completed according expressions (1)–(6) and the pre-vious experimental data. Furthermore, the ACK models obtainedfor each series are directly compared with the experimentalresults. In order to facilitate a clear understanding of the graphs,

the area covered by the experimental curves is highlighted in grey(see Figs. 8–11).

In general terms, mainly at stage III, ACK model proved to beless than fully effective. The main discrepancy point is the strainat the end of the multi-cracking stage, emc. In addition, the stiffnessof the third stage appears to be greater in the model. This discrep-ancy is caused by the loss of stiffness observed at the end of stage

Fig. 9. ACK model – TB2 series.

Fig. 10. ACK model – TB3 series.

Fig. 11. ACK model – TB4 series.

P. Larrinaga et al. / Composites: Part B 55 (2013) 518–527 523

III in all the tested combinations. As has been previously stated, thecombination of progressive rupture of filaments inside the rovings[14] and the debonding at the textile–matrix interface could be themain reason for this ‘‘softening’’ of the specimens’ stiffness. As ACKmodel does not expect this behaviour, it might be possible to sim-ulate that loss of stiffness developing a non-linear model.

Moreover, this model works better with more basalt layers.However, the lack of accuracy of the ACK model in heavily rein-forced organic based composites was studied by [24]. Probably,ACK theory reproduces suitable results in a specific ratio of com-posite internal reinforcement.

4.2. Cracking Model

As previously referred, TRM is a composite formed by a rein-forcement core embedded in an inorganic matrix, similar to rein-forced concrete. If an RC block is subjected to uniaxial tensileload, the following behaviour takes place (see Fig. 12).

This behaviour is very similar to that observed for the TextileReinforced Mortar. Therefore, the assumptions established for rein-forced concrete may be employed in the composite modelling.According to Eurocode 2 [10], the crack width, wk, may be calcu-lated from the following expression:

wk ¼ sr;maxðesm � ecmÞ ð7Þ

where according to Eurocode 2: sr,max is the maximum crack spac-ing, esm is the mean strain in the reinforcement, ecm is the meanstrain in the concrete between cracks.

It can be considered that the total sum of cracks widths is equalto the difference between the elongation of the reinforcement andthe elongation of concrete:X

w ¼ Dl� Dlc ¼ ðesm � ecmÞ � l ð8Þ

esm � ecm may be calculated from the next equation [10]:

esm � ecm ¼rs � kt

fct;eff

qp;effð1þ ae � qp;eff ÞEs

ð9Þ

where rs is the stress in the tension reinforcement assuming acracked section, kt is a factor dependent on the duration of the load,fct,eff is the mean value of the tensile strength of the concrete at thetime when the first crack is formed, qp,eff is the reinforcement ratio,As/Ac, and ae is the ratio Es/Ec.

This concept might be considered to model the TRM tensilebehaviour. The authors have associated the expression (9) withthe stage III. When all the cracks are formed TRM stress–strainrelationship could be defined by means of this formula. Hence,the model is called in this paper Cracking Model. First of all, it isnecessary to adapt the nomenclature of the expression (9) to theTRM’s one. These are the adopted changes:

- The strain in the matrix between cracks remains constant afterthe crack formation. Therefore, ecm is replaced by mortar tensileultimate strain emu.

- The value of the reinforcement strain esm is substituted by theTRM strain at stage III (e). Consequently, the composite tensilestress r replaces the value rs, i.e. general terms to relate bothvalues (stress and strain) at the third stage.

- As the materials are different, their nomenclature is also chan-ged Ec by Em (mortar replaces concrete), and Es by Ef (textilesreplace steel).

- The last terms that are adapted are fct,eff and qp,eff, which aresubstituted by their equivalent rmc (mortar tensile failurestress) and ae (Vf/Vm) respectively.

- Finally, the coefficient kt is defined empirically.

After these changes expression (9) turns into:

e� emc ¼r� kt

rmuVfð1þ ae � Vf ÞEf

ð10Þ

r ¼ ðe� emcÞEf þ ktrmu

Vfð1þ aeVf Þ ð11Þ

After this concept is set, it is possible to define the Cracking Modelas a tri-linear model with the same shape as ACK theory. Stage I isdetermined, as in ACK theory, by the mixture law. This stage will berepresented as a line and the values which define it (Ec1, emu and

Fig. 12. RC block subjected to uniaxial tensile load (a) and its load versus length increase (b).

Fig. 13. Cracking Model in TB1 series.

Fig. 14. Cracking Model in TB2 series.

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rmc) are calculated with expressions (1) and (2) using the sameinitial information.

Expression (11) is, basically, the stress–strain relationship atstage III and can be employed to define the multiple crackingstrain, emc. By means of this expression, and using the multiplecracking stress rmc and the mortar tensile ultimate strain emu, thecomposite strain at the end of stage II, can be calculated as:

emc ¼rmc � kt

rmuVfð1þ ae � Vf Þ

Efþ emu ð12Þ

All the information necessary to run the model is extracted, as inthe ACK model, from the data compiled in the experimental cam-paign. However, there is a factor, kt, that has not been properly de-fined. In Eurocode 2, kt represents the duration of the load and it isequal to 0.6 for short term loading and 0.4 for long term loading[10].

The entire TRM uniaxial tensile tests were conducted with thesame displacement rate. So, it is estimated that this factor kt shouldbe constant for all series. This value has to be empirically adjustedwith the experimental data, so the selected value is 0.2. There is asignificant difference with the values given by the Eurocode for thisfactor kt. Nevertheless, several aspects might influence in this fac-tor besides the test speed; for example the interface behaviour.While the Eurocode 2 is referred to the combination of concretewith continuous and corrugated steel bars, in the tested specimensthe matrix is in contact with fibre rovings and different interfacebehaviour.

The failure strain value (et,III) of the basalt TRM specimens is lessthan the basalt fibres one (ef), a value given by the manufacturer.To take this issue into account, a reduction factor, b, is proposedto obtain the failure strain et,III:

et;III ¼ b � ef ð13Þ

This reduction factor is obtained empirically with the average ulti-mate strain of all the tested series. Thus, for these basalt TRM spec-imens b reduction factor is equal to 0.65.

et;III ¼ b � ef ¼ 0:65ef ð14Þ

The obtained results are directly contrasted with the experimentaldata in Figs. 13–16. The area covered by the experimental curvesis highlighted in grey while the numerical curve is represented inblack. This information is also summarised in Table 7.

The proposed Cracking Model presents much better results thanthe ACK model for the multiple cracking strain, emc. There are nosignificant discrepancies between the analytical and experimentaldata on this key point.

However, the behaviour of the model at stage III is still not en-tirely satisfactory. There is agreement on the first phase of this

stage, but, as strain increases, there is a greater difference betweenthe model and the experimental curves. This discrepancy is basedon the linearity of the model. The end of stage III is characterisedby a progressive loss of the composite stiffness; an effect previ-ously commented.

To achieve a good correlation between the model and theexperimental data, it is necessary to change expression (11) intoa nonlinear expression. This effect is achieved by means of theRichard and Abbot approximation, which has been successfullyused in previous numerical models of materials [30].

In this model, a bi-linear stress–strain relationship is assumed,based on an expression of four parameters, E1, E2, f0, n (see Fig. 17),initially proposed by [31]. Expression (15), which defines the

Fig. 15. Cracking Model in TB3 series.

Fig. 16. Cracking Model in TB4 series.

Table 7Cracking Model data and results.

Series Vm (–) Vf (–) Ec1 (GPa) rmc (MPa) emc (%)

TB1 0.9965 0.0035 8.46 2.54 0.896TB2 0.9930 0.0070 8.66 2.60 0.473TB3 0.9895 0.0105 8.87 2.67 0.332TB4 0.9860 0.0140 9.07 2.73 0.261

Fig. 17. Richard and Abbot approximation [30].

Fig. 18. Proposed calibration.

Fig. 19. Cracking Model, modified version, in TB1 series.

P. Larrinaga et al. / Composites: Part B 55 (2013) 518–527 525

approximation, is introduced in the Cracking Model, and its param-eters are adjusted according to the experimental data, as follows:

fc ¼ðE1 � E2Þec

1þ ðE1�E2Þecf0

� �nh i1nþ E2ec ð15Þ

E1 in expression (15) is the main Young’s modulus in the Richardand Abbot Model; therefore, it is substituted by the textile modulus,Ef. After conducting this small adjustment, expressions (12) and(15) are combined leading to a new expression (16).

r ¼ ðEf � E2Þðe� emcÞ

1þ ðEf�E2Þðe�emcÞf0

� �nh i1nþ ðe� emcÞE2 þ kt

rmu

Vfð1þ aeVf Þ ð16Þ

After several considerations, the definitive approximation has beendefined according to Fig. 18. The experimental curves are supposedto be asymptotic to a line parallel to the horizontal coordinate axis,i.e. E2 is equal to 0.

Another key parameter, f0, is defined as the stress at the textileunder the same strain at which TRM reaches the maximum load(et,III). The average of this value in all the series is 2.07%. Thus, bymeans of expression (17), f0 is equal to 1407 MPa. Finally, the valuen = 4 is established empirically, i.e. it has been established accord-ing to the experimental data.

f0 ¼ Ef � et;III ¼ Ef � b � ef ð17Þ

Including these considerations, the stress–strain curve for stage IIIis defined as:

r ¼ Ef ðe� emcÞ

1þ Ef ðe�emcÞf0

� �nh i1nþ kt

rmu

Vfð1þ aeVf Þ ð18Þ

The final results of the new version of the Cracking Model werecompared with the experimental data obtaining satisfactory agree-ment: see Figs. 19–22.

Fig. 20. Cracking Model, modified version, in TB2 series.

Fig. 21. Cracking Model, modified version, in TB3 series.

Fig. 22. Cracking Model, modified version, in TB4 series.

526 P. Larrinaga et al. / Composites: Part B 55 (2013) 518–527

The calibration has clearly optimised the Cracking Model andled to a satisfactory agreement with the obtained experimental re-sults. The resulting curve simulates the real behaviour nicely whenthere is enough reinforcement material to act as a composite. Infact, the Cracking Model does not take into account the problemsderived from the lack of internal reinforcement and generates acurve that shows considerable differences with the experimentalones (series TB1).

The proposed simulation presents optimum results for the mul-tiple cracking strain emc, so there are no significant discrepanciesbetween the analytical and experimental data on this key point.

The behaviour of the model at stage III is satisfactory due to theintroduced non-linearity which improves the model’s performanceat the end of this stage.

As it has been stated this model presents an empirical compo-nent. Its accuracy could be affected by different parameters, suchas specimens’ geometry, test setup, reinforcing material or textilecoating. However, Cracking Model has been successfully provenby the authors with other reinforcing materials such as glass, car-bon and steel wire [14]. This fact enhances the versatility of theCracking Model to materials that could be used as TRM reinforcingcore.

5. Conclusions

Both as new construction material or as structural strengthen-ing solution, inorganic-based composites, mainly TRM and TRC,have increased their relevance within the construction area andare currently focusing the efforts of several research groups.

For this reason, it is important to standardise the use of theTRM. Characterisation test, manufacturing and application guidesor analytical models are among the aspects that should be norma-lised. There are studies that have developed analytical and numer-ical models with satisfactory results. However, most of them arevery sophisticated and their application could be expensive interms of time and budget. Therefore, it is necessary to developmodels easy-to-apply.

TRM, as strengthening material, usually works under tensileloads which are transferred by adherence from the strengthenedelement. It is essential to study the tensile behaviour of TRM to de-fine its possibilities and failure modes. Especially after stating thatthese failure modes have different nature compared to those ob-served in other externally bonded strengthening systems.

Textile Reinforced Mortar is a composite with a very complexmaterial behaviour. In addition, its structural relevance asks foraccurate model with no significant discrepancies with the realbehaviour.

A non-linear model is presented in this paper to describe thestress–strain behaviour of TRM under uniaxial tensile loading.The model is based on the RC crack control expression includedin the Eurocode 2, but takes into account the nature of the compos-ite constituent materials. The goal of this study is to verify the sim-ulation, called Cracking Model, contrasting its results with thoseobtained in an experimental campaign also included in this paper.In addition, a well-known model as ACK theory is also discusses inthis document and its simulations compared with the experimen-tal results.

Due to the good fit between the calculated and the experimen-tal values, the Cracking Model might be employed as TRM consti-tutive equations in nonlinear analysis of reinforced concretestructural elements strengthened by means of this technique, suchas beams working in flexure. Cracking Model has already been em-ployed as a constitutive equation in numerical models developedto simulate the behaviour of reinforced concrete beams strength-ened in flexure using Textile Reinforced Mortar [14]. Moreover,as TRM can be also used as strengthening material in masonrystructures, the Cracking Model could be integrated in the analysismethods developed for that kind of structures.

Nevertheless, due to the nature and the thoroughness of thestructural calculus within construction sector, it is necessary tosupport this empirical model by means of more experimental datain order to contrast its accuracy according to different parametersand, thus, defining the boundaries of Cracking Model. Future re-search lines are focused on testing new reinforcing materials, moreeffective matrices and different reinforcing rates to enlarge theknowledge related to TRM and check the effectiveness of Cracking

P. Larrinaga et al. / Composites: Part B 55 (2013) 518–527 527

Model as a valid tool for new construction design or retrofittingpurposes.

Acknowledgments

This research work was funded through the research projectsDFB 7-12-TK-2009-10 and BIA2010-20789-C04-03/04; the BasqueGovernment’s contract IT781-13; and the scholarship programmeof the Iñaki Goenaga Foundation.

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