BOND BEHAVIOUR OF A MULTI-FILAMENT YARN EMBEDDED IN A CEMENTITIOUS MATRIX

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1 BOND BEHAVIOUR OF A MULTI-FILAMENT YARN EMBEDDED IN A CEMENTITIOUS MATRIX Von er Fakultät für Bauingenieurwesen er Rheinisch-Westfälischen Technischen Hochschule Aachen zur Erlangung es akaemischen Graes eines Doktors er Ingenieurwissenschaften genehmigte Dissertation vorgelegt von Diplom-Ingenieur Björn Banholzer aus Essen Berichter: Universitätsprofessor Dr.-Ing. Wolfgang Brameshuber Universitätsprofessor Dr.-Ing. Manfre Curbach Universitätsprofessor Dr.-Ing. Hans-Wolf Reinhart Tag er münlichen Prüfung: 19. August 24 Diese Dissertation ist auf en Internetseiten er Hochschulbibliothek online verfügbar.

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3 o ABSTRACT Although the bon of a stran in a cementitious matrix is certainly preominate by the bon properties between filaments an matrix, more etaile information is neee to evaluate the failure mechanisms of such a complex system uner a pull-out loa an hence to allow an analytical an numerical simulation of this composite. Thus, in this stuy ifferent innovative test methos are evelope an use to ientify the failure process of a stran as a result of the pull-out process an ascertain the contact faces between the iniviual filaments an the matrix. Aitionally, base on these finings, numerical proceures are propose to allow, for the first time, to establish a irect relationship between the loa history of a pull-out test, to the failure process of an AR-glass stran, by means of a mathematical function; the so-calle active filament versus isplacement relation N F (Ω). Together with the loa versus isplacement relationship P(Ω) also erive uring the pull-out test, an analytical characterization an simulation of the bon between an AR-glass stran an a cement base matrix will be possible. ZUSAMMENFASSUNG Obwohl as Verbunverhalten eines Multi-Filament-Garn/Feinbeton-Systems sicherlich maßgeblich urch en Verbun zwischen Filament un Matrix beeinflusst wir, müssen wesentlich etailliertere Erkenntnisse vorliegen, um en Versagensprozess ieses komplexen Systems infolge einer Pull-Out-Belastung zu beschreiben un somit eine analytische un numerische Moellierung es Verbunwerkstoffes zu erlauben. Aus iesem Grun wuren innerhalb ieser Arbeit innovative Untersuchungsmethoen entwickelt un angewenet, ie es nun ermöglichen, iesen Versagensprozess eines Multi-Filament-Garn/Feinbeton-Systems infolge einer einwirkenen Pull-Out-Belastung zu ientifizieren un ie Grenzflächen zwischen en iniviuellen Filamenten un er umgebenen Matrix zu bestimmen. Zusätzlich wuren numerische Auswerteroutinen entwickelt, ie es zum ersten Mal ermöglichen, en Versagensprozess eines Multi-Filament-Garn/Feinbeton-Systems mit Hilfe er so genannten Aktiven Filamente/Ausziehweg-Funktion N F (Ω) mathematisch abzubilen un somit eine Verbinung zwischen er Belastungsgeschichte es Verbunwerkstoffes un em Versagen er iniviuellen Filamente aufzustellen. Zusammen mit er ebenfalls im Pull-Out-Versuch ermittelten Kraft/Ausziehweg-Beziehung P(Ω) wir amit erstmals eine analytische Beschreibung un Simulation es Verbunes zwischen Multi-Filament-Garn un zementgebunener Matrix ermöglicht.

4 i ACKNOWLEDGEMENT The research presente in this thesis was carrie out uring my appointment as a research assistant an PhD stuent at the Institute of Builing Materials Research (ibac) of RWTH Aachen University. I am in particular grateful to Professor W. Brameshuber for his support an avice that I have receive on so many occasions. I thank, also, the members of my committee, Professor H.-W. Reinhart (University of Stuttgart) an Professor M. Curbach (Dresen University of Technology) for their interest an valuable suggestions. Finally I woul like to thank my colleagues an friens for the generous support receive uring all the years at this institute. Most importantly, I wish to thank my parents who were a constant source of love an support. This work was carrie out within the Collaborative Research Center 532 Textile reinforce concrete Basics for the evelopment of a new technology an sponsore by the Deutsche Forschungsgemeinschaft (DFG). The support is gratefully acknowlege. Aachen, Summer 24

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6 iii TABLE OF CONTENTS 1 Introuction Objective Proceeing Overview Analytical moeling of a single fiber pull-out process Introuction Analytical bon moels The perfect interface moel (stress approach) The fracture mechanical moel (energy approach) The cohesive interface moel (stress approach) Comparison of the moels Which moel to use? The irect bounary value problem - τ ( s) P( ω) The inverse bounary value problem - P( ω ) τ(s) Summary Application an valiation of the cohesive interface moel Introuction Materials composition an specimen preparation Experimental sequences Experimental methos Test results Application an valiation of the moel Stochastic size effects Summary Bon between glass filaments an a cement base matrix Introuction Materials composition an specimen preparation AR-glass filaments Fine-graine concrete Specimen preparation tensile test... 5

7 iv Table of Contents Specimen preparation pull-out test Experiment sequences Experimental methos Tensile test Pull-out test Optical microscopy Test results Tensile tests on filaments Pull-out tests on filaments Bon stress versus slip relations τ(s) Discussion Summary Bon between glass strans an a cement base matrix Introuction Materials composition an specimen preparation AR-glass strans Fine-graine concrete Specimen preparation pull-out test Experimental sequences Testing proceure Experimental methos The pull-out test The FILT test Laser scanning microscopy (LSM) Scanning electron microscopy (SEM) Tests results The pull-out test The FILT test Laser-scanning microscopy (LSM) Scanning electron microscopy (SEM) Discussion Summary Three imensional arrangement of a stran in a cement base matrix Introuction Materials composition an specimen preparation AR-glass strans / fine-graine concrete Specimen preparation... 94

8 Table of Contents v Experiment sequences Experimental methos X-ray microtomography (XRT) Environmental scanning electron microscopy (ESEM) Results X-ray microtomography (XRT) Environmental scanning electron microscopy (ESEM) Computer aie image analyzing system Summary Moelling of a stran pull-out Introuction Analytical moel Results of Simulation Discussion Summary Application an verification of the stran pull-out moel Materials composition an specimen preparation Experimental methos Test results an simulation outputs Varying material composition Varying embee length Discussion Summary Summary an Conclusion References Appenix A: Application of the cohesive interface moel

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10 1 CHAPTER 1 INTRODUCTION Composite materials currently have a wie range of technical applications, for instance in the automotive inustry, aerospace inustry, or in human meicine. One of the most successful examples of composite usage in civil engineering is in steel reinforce concrete. A more recent evelopment, textile reinforce concrete [Heg1, Heg2a], is an exciting new approach towars the use of composites in builings an other structures an may become a possible future supplementation of materials use in the builing inustry. Placing multiimensional fabrics mae of alkali resistant glass (AR-glass) an processe with the ai of moern textile technology [Gri2, Roy3] instea of the usual steel bars in the main loa irections of a complementary fine graine concrete [Bro1], might enable the engineer in the future to esign an buil light an slener structures with a high loa carrying capacity. However, to unerstan an use the avantages of this new material it is necessary to gain funamental information on the mechanical properties of the composite. These mechanical properties are ientifie within the collaborative research center Textile reinforce concrete technical basis for the evelopment of a new technology (SFB 532) unertaken at RWTH Aachen University [Chu4b, Heg4, Off4, Sch4]. A substantial aspect in these investigations is the characterization of the shear force transmission between the concrete matrix an the AR-glass strans which a fabric consists of; e.g. [Heg3a, Kon3a]. In general the so-calle pull-out test is a wiely accepte experimental technique to etermine the basic shear bon characteristics between a single monolithic reinforcing element, for example a re-bar, an its surrouning matrix. In the present case, however, the AR-glass strans themselves alreay consist of several hunre iniviual filaments approximately 1 to 3 µm in iameter. Thus the way such a reinforcement functions is quite ifferent from other materials use in cement base composites; especially steel fibers or re-bars. Fig. 1.1 shows a cross-section of such an AR-glass stran embee in an OPC matrix. Obviously the non-uniform nature of the stran an the ranom penetration of the concrete make the microstructure of this composite inherently variable an cause the substantial scatter in the force-isplacement relationships etermine in pull-out tests on stran / matrix systems [Bar82, Law86]. So far only a very limite amount of ata is available on these microstructural properties, which strongly affect the transfer of shear forces between stran an matrix an hence the composite s mechanical behavior. Because of this lack of

11 2 1 Introuction information, the interpretation of experimental results even for the apparently simple pull-out test yiels consierable ifficulties since an analytical or numerical moeling of a stran / matrix system is restricte. 1 mm Fig SEM micrograph of an AR-glass stran in a cement base matrix. 1.1 Objective Base on the limite available ata escribe above, the main objective of this stuy is therefore to elaborate on experimental tests an analytical tools to etermine an evaluate the micromechanical an microstructural properties, which efine the principal shear boning characteristics of an AR-glass stran / cement base matrix system. The aim is to subsequently erive an analytical moel which is capable of simulating the pull-out response of this composite. 1.2 Proceeing In orer to achieve the objective, the following sub-problems are taken into consieration: (I) (II) (III) (IV) The principal shear boning characteristics an mechanisms between AR-glass an a cement base matrix are ientifie an evaluate. Detaile information on how an AR-glass stran in a cement base matrix reacts uner a pull-out loa is gaine an the ominating failure mechanisms are to be etermine. The loa carrying cross-section of the stran an the corresponing longituinal contact area with the surrouning matrix are both qualitatively an quantitatively etermine at each loa step uring the pull-out test. Information is gaine on the three imensional arrangement of an AR-glass stran in a cement base matrix with regar to the arrangement of the filaments, the locations an amounts of matrix which has penetrate the stran, as well as vois, pores, an imperfections influencing the interfacial characteristics.

12 1 Introuction 3 The finings an etermine solutions for each sub-problem liste above as well as the relationships an principles iscovere between these ifferent fiels are finally use to assemble a solution to the overall problem, which escribes the shear force transmission an moels the pull-out response of an AR-glass stran / cement base matrix system. 1.3 Overview There are currently no straightforwar experimental methos to etermine the shear bon properties between two materials of a system, i.e. in the present case between AR-glass an cement base matrix (sub-problem I). Thus in chapter 2 analytical expressions an moels publishe in previous stuies use to ientify these shear bon properties are escribe, analyze, an compare. Base on this literature review a consierably enhance moel is propose which relates to an axisymmetric iealization of a single fiber pull-out problem. This moel allows, for the first time, the straightforwar calculation of an N-piecewise linear bon law τ(s) on the basis of an experimentally etermine loa versus isplacement relation P(ω). No optimization routines or fitting proceures are require. In chapter 3 the propose analytical moel is verifie on the basis of experimental tests. However, because a filament / matrix system is very fragile ue to its small imensions an oes not allow an easy variation of embee length an filament iameter, which is inispensable for a verification of the moel, another single fiber / matrix arrangement is selecte. A steel fiber / cement base matrix system is chosen in this thesis to carry out the verification, because this composite allows an easy hanling an variation of the above mentione geometric parameters, i.e. the embee length an filament iameter. The final solution to sub-problem I, i.e. the bon stress versus slip relation τ(s) for an ARglass / cement base matrix system, is then erive in chapter 4 using the propose analytical moel to evaluate the loa versus isplacement responses P(ω) of single filament pull-out tests. Further material properties neee as input parameters for this analysis, for example the tensile strength f t as well as the Young s moulus E F of the filament, an the filament iameter, are also etermine. In chapter 5 a pull-out test on a stran is introuce to erive the loa versus isplacement response P(Ω). Aitionally a testing proceure, the so-calle FILT-test (Failure Investigation using Light Transmission properties) is evelope, to work out how a glass stran in a cement base matrix reacts uner a pull-out loa an to etermine the ifferent failure mechanisms which occur. Investigations using the laser-scanning microscopy an scanning electron microscopy give aitional information on the pull-out processes. All results are assemble to fin a solution to sub-problem II. The results of the FILT test are further evaluate by means of numerical image analyzing proceures, an the loa step epenent length of the longituinal contact areas U C (Ω) as well as the actual loa carrying cross-sections of the stran N F (Ω) are etermine (sub-problem III). Thus a relation is

13 4 1 Introuction establishe between the loa versus isplacement response of the stran / matrix system an the number of filaments failing uring the pull-out process. In chapter 6 the last sub-problem IV is consiere an some information is gaine on the three imensional arrangement of a stran in a cement base matrix with regar to the locations an amounts of matrix which has penetrate the stran as well as the istribution of vois, pores, an imperfections within the stran by means of computer tomography (CT). For a plane cross-section of the system a mathematical relation A M (v) is erive which escribes the ecreasing amount of penetrate matrix towars the core of the stran. Finally in chapter 7 all the solutions to the sub-problems which have been etermine are assemble together to give an answer to the overall problem, of how the response of a stran in a cementitious matrix uner a pull-out loa can be characterize an moelle. Similar to chapter 2, analytical expressions an moels of previous stuies are referre to an examine. Base on these existing moels an the results of the previous chapters, an enhance moel is erive which, for the first time, allows a etaile simulation of the response of a stran / matrix system uner a pull-out loa. The propose moel is applie an finally verifie in chapter 8.

14 5 CHAPTER 2 ANALYTICAL MODELING OF A SINGLE FIBER PULL-OUT PROCESS In this chapter an analytical moel is erive to etermine the shear bon properties between two materials in the form of a bon stress versus slip relation τ(s) base on results of single fiber pull-out tests. After a valiation in chapter 3, the propose moel allows the evaluation of the principal boning characteristics between AR-glass an a cementitious matrix in chapter 4 (sub-problem I). 2.1 Introuction As alreay mentione, there are currently no straightforwar experimental methos, even in moern materials research, to etermine the shear bon properties between two materials. Nevertheless, there is growing recognition among researchers that these properties are of primary importance for the unerstaning of a composite s overall behavior an structural performance. Therefore in recent years many ifferent alternative test set-ups an experimental techniques have been evelope to gain more insight into the basic mechanisms ominating this shear boning behavior. D r x Matrix Fig D Fiber, = L L Axisymmetric moel Iealization Pull-out specimen Original pull-out specimen an axisymmetric 3-D an 2-D iealization respectively. In civil engineering one of the most common tests to investigate the shear force transmission between e.g. a re-bar or a fiber an a cement base matrix is the so-calle pull-out test. In this test a force P pulls a single, monolithic reinforcing element with iameter out of a cuboi of matrix with imensions D D L (Fig. 2.1) while the corresponing isplacement ω is recore. The matrix is either fixe at the top (pull-push test) or fixe at the rear (pull-pull test); see Fig Because an AR-glass stran is mae up of many single, cylinrical,

15 6 2 Analytical moeling of a single fiber pull-out process monolithic structures in the form of filaments (Fig. 1.1), the pull-out test on a single filament may be feasible to investigate the shear bon characteristics between AR-glass an cementitious matrix (sub-problem I). A1 P,ω P,ω A2 P,ω P,ω x x L L x F F F/2 M F/2 M x F F F/2 M F/2 M Fig Iealize test configurations A1 (pull-push test) an A2 (pull-pull test) respectively, an corresponing global force equilibrium. The result of this test is a loa versus isplacement relation P(ω) which is, however, only characteristic for the teste geometric arrangement an oes not represent a general material property of the teste composite. In orer to yiel shear bon parameters which generally efine the shear force transmission between the reinforcing element material an the matrix material an which are inepenent of the geometric arrangement, numerical evaluation proceures to moel the stress transfer have to be applie. These proceures use systems of equations to analytically escribe the processes occurring uring a pull-out test. A 3-D mathematical representation of the pull-out test, however, is ifficult to solve an requires a large amount of computational resources or the application of the Finite Element Metho. Assumptions can be mae about the system to reuce this complexity. One way to simplify an analytical moel is to use symmetry attributes within the investigate system. If the original matrix cuboi is assume as a matrix cyliner of iameter D, the symmetry about the longituinal axis even allows in the present case a 2-D iealization of the 3-D system (Fig. 2.1). Such an axisymmetric iealization of the pull-out problem has been analytically escribe by many researchers over the past years. In general, three ifferent kins of approach to moel this problem can be foun in literature: (I) The perfect interface moel an (II) the cohesive interface moel which are both base on force equilibrium consierations, an (III) the fracture mechanical moel which is base on energy balance principles. All three moels have one aspect in common: that is they are base on a so-calle irect bounary value problem, i.e. they use given shear bon properties to simulate a loa versus isplacement relation P(ω) of a pull-out test. Only in combination with an optimization algorithm which fits the simulate pull-out curve to the experimental ata by aapting the material parameters, is it possible to actually erive unknown shear bon properties for a material combination [Chu3]. But even with moern optimization routines this is quite a ifficult task where many parameters exist simultaneously. Furthermore, the input parameters within the literature moels, which efine the shear force transmission between reinforcing element an matrix, are initially limite in number an allow only a restricte functional

16 2 Analytical moeling of a single fiber pull-out process 7 escription of the material properties, because the more arbitrary material parameters that are implemente in the moel, the more complex the resulting system of equations becomes. This again has a substantial effect on the accuracy of the simulate pull-out response, if the shear bon properties preominating the real structure cannot be escribe sufficiently by the limite moel parameters. To fin the most appropriate analytical moel to evaluate the principal shear bon characteristics between AR-glass an a cement base matrix on basis of the results of the single filament pull-out tests presente in chapter 4, the following proceure is chosen: Firstly all three moels are briefly introuce an subsequently compare in etail. Base on this literature review a consierably enhance moel is propose in which the functional restrictions of the parameters escribing the shear bon properties of the composite are eliminate. Finally a mathematical formulation is introuce which allows the inverse bounary value problem to be solve, i.e. the shear bon properties can be euce irectly from a loa isplacement curve P(ω) recore from a pull-out test, an hence optimization routines are no longer require. 2.2 Analytical bon moels The perfect interface moel (stress approach) As early as 1952 the so-calle perfect interface moel was evelope by Cox, who assume that the shear bon between the reinforcing element an matrix is perfect an hence that isplacements an tractions are continuous at the interface [Cox52]. The assumption of the iealization an the resulting axisymmetric moel mentione in chapter 2.1 an epicte in Fig. 2.1 makes it possible to use equations of elasticity for an axisymmetric stress state as one so by [Tim84], [McC89], an [Nai97]. The original three imensional problem is then reuce to two imensions an can be represente by a set of two imensional fiel equations [Nay77]. Nevertheless, an explicit solution of these equations is actually extremely ifficult to obtain an if erive, is in many cases not feasible for a straightforwar analytical reflection of a pull-out problem because it is still very complex. One of the main reasons for this complexity is that the x an r irections exist in the same unit shear strain equation of a plane element (Eq. 2.1). Only if these two variables are separate from each other oes this permit the two imensional analysis to be simplifie to a one imensional analysis an thus allow a practical application. A stanar assumption for simplification which is use in literature is therefore γ xr u u r x = +, x r u r x u x << r which implies that u r, the eformation in r irection, is a function of r only. γ xr in Eq. (2.1) refers to the in-plane shear strain an u x is the isplacement in x irection (see Fig. 2.1). Another assumption use in virtually all reviewe stuies pertaining to an analytical moeling (2.1)

17 8 2 Analytical moeling of a single fiber pull-out process of a pull-out test is to replace the correct axial Hooke s law with a one imensional version that ignores transverse stresses. Lastly the axial stress is taken as an average axial stress an is therefore assume to be inepenent of the raial irection. To allow for a close form mathematical solution, bounary conitions are now impose. As shown in Fig. 2.2, the loa application is iealize as a single pull-out loa P at the tip of the reinforcing element (x = L) with a corresponing isplacement ω applie in x irection. The global equilibrium of forces in an arbitrary chosen cross-section leas to FF + FM = (A1) FF + FM = P (A2) (2.2) where F F correspons to the force in the reinforcing element an F M to the force in the matrix at a location x. Utilizing the above-mentione set of fiel equations an using the state assumptions together with symmetry an continuity equations [Nay77], a solution subject to the appropriate bounary conition can be erive in terms of a secon orer ifferential equation, e.g. for test configuration A1: 2 F β F β F 2 F =, 1 = π E FA F F (L) = P, F + E M 1 A M F () = F ηm 4G F + 1 2G M 2ηM 1 1 ln ηm η F ηf 1 2 (2.3) where β is the so-calle shear-lag parameter [Zha]. E F an G F are the axial an shear mouli of the reinforcing element respectively an E M an G M are, respectively, the axial an shear mouli of the matrix; η F an η M are, respectively, the volume fractions of the reinforcing element an the matrix within the specimen; an finally A F an A M are the crosssectional areas of the reinforcing element an matrix respectively ( is the iameter of the fiber). See also [Nai97], [Kim98], an [Nai1]. Eq. (2.3) represents the so-calle shear-lag metho an is often use for analysis of stress transfer problems in polymer composites. It is base on linear elastic material behavior an further assumes a perfect interface, i.e. no slip between reinforcing element an matrix is allowe an isplacements an tractions are continuous at the interface. In subsequent stuies a uniform post-elastic constant bon strength has been applie, e.g. [Nai97], to consier a frictional stress transfer between reinforcing element an surrouning matrix in the ebone region of the system, i.e. in the region x > L-a (Fig. 2.3). The criterion in this approach for the ebone zone to avance from its original length a by an amount a is that the interfacial shear stress must reach a critical value, terme the bon strength τ cr. Generally it can be state that even more recent stuies, e.g. [Nai97], concentrate only on the etermination of the interfacial stress istribution which yiels the shear strength of the system an not on the evaluation an simulation of a complete pull-out response, i.e. a loa isplacement curve P(ω).

18 2 Analytical moeling of a single fiber pull-out process 9 As a result of the iealization of the composite (sharp eges) escribe above as well as the assume linear elastic material properties in combination with a perfect bon, infinite stresses or so-calle stress singularities appear at the crack tip (also in finite element calculations [Mar94]), which are unlikely to arise in a real structure. Due to these singular stresses which are inflicte by the mathematical moel, several authors, e.g. [Sta85, Sha91a] have criticize the use of the perfect interface moel to preict the eboning process. They propose to use a fracture mechanical analysis to overcome this problem The fracture mechanical moel (energy approach) Fracture is generally efine in terms of a surface change in a boy, that is in this case a evelopment of a crack along the reinforcing element an the matrix. The crack propagates along the interface of the composite an creates two new fracture surfaces. Chemical an mechanical bons must be broken in orer to create these new surfaces an this requires energy. Griffith [Gri2] was the first to realize that the energy neee for a crack propagation coul be equate to the increase in surface energy ue to the aforementione increase in surface area, although it was later iscovere that energy is issipate (e.g. as heat) uring the breaking of mechanical an chemical bons when the crack extens. Hence, the fracture mechanical moel is characterize by the assumption that the propagation of the ebone zone by a requires a certain amount of energy G a an this energy is characteristic for the bon between a certain reinforcing element an matrix (among others [Gao87, Zha]). The concept of a critical energy release rate G cr is use, which is base on Griffith s original an well known hypothesis, an requires the knowlege of the external loas an resulting overall eformations of the system. However, even though the transformation of energy from one form into another is involve uring the pull-out process, the total energy of the system always has to remain constant. The Law of Energy Conservation (first law of thermoynamics) states this fact by W = U + K E + U S (2.4) where W is the work one by external loaing, U is the internal energy which inclues elastic an inelastic eformations (U = U e +U ne ), K E is the kinetic energy, an U S is the surface energy of the boy, ue to cracking over a length a. Energy issipate, e.g. ue to changes in thermal an chemical energy uring the formation of a crack or to overcome friction in the broken part of the interface, is inclue in U S an U ne respectively. In the case of the most viewe Moe I (opening) failure, only the elastic eformation of the system U = U e is consiere an the inelastic fraction is neglecte. But in the present case of a pull-out system (Moe II), the energy W f neee to overcome friction if the reinforcing element is ebone an pulle out of the matrix has to be consiere as well (U = U e +W f ). As long as the system is in quasi-static equilibrium an the crack propagation is not catastrophic the kinetic energy of the system is small an can therefore be neglecte ( K E ). Hence, if the reinforcing

19 1 2 Analytical moeling of a single fiber pull-out process element is assume to alreay be ebone from x = L to x = L-a, the global energy equilibrium for a pull-out system can be written as [Leu92] W = U + W + U e f S (2.5) If the system is in equilibrium, the change in energy of the system per unit area when the crack extens by a is given by W U e = a a W + a f US + a U e = a W + a f + π G (2.6) where U S / a = π G is the energy release rate at the interface per unit area of crack progression, ae to the energy of the system ue to the formation of aitional surface area in the boy. Note that π is the circumference of the reinforcing element. During a pull-out test a isplacement ω is applie at the loae en of the reinforcing element an the corresponing, resulting force P is measure an recore as a function of ω. Hence the external work neee for a crack avance a can be written as W = P ω an substitute in Eq. (2.6). This yiels P ω = U e + W f + π G a (2.7) Assuming that plane faces remain planar, u r (M,F,I) / x << u x (M,F,I) / r, i.e. that the raial isplacement u r (M,F,I) is a function of r only (see above an e.g. [Nay77, Nai97]), a uniaxial an constant stress state exists because σ F, an σ M respectively, are taken as average axial stresses an inepenent of r, transverse stresses may be ignore because Hooke s law is aopte in a one imensional version (µ =, e.g. [Nai97]), the reinforcing element an matrix behave linear elastic, an an interface of finite thickness ζ allows for a shear stress transfer between these two materials but oes not experience any longituinal eformation (Fig. 2.4) we get U e = A 2 F L σ F ε F A x + 2 M L σ M ε M x + ζ π L a s(x) τ s ζ () s x (2.8) The first an secon term of Eq. (2.8) represent the elastic strain energy store in the reinforcing element an the matrix respectively. The thir term of Eq. (2.8) represents the elastic region of the interface an is hence integrate over the interval [, L-a], see Fig. 2.3, because from there on the interface is no longer assume to be elastic but is preominate by friction, i.e. macro-cracking. A possible amount of elastic strain energy in this part is neglecte. Applying loaing configuration A1, i.e. substituting Eq. (2.2) into Eq. (2.8) yiels

20 2 Analytical moeling of a single fiber pull-out process 11 U e = 1 2 L F F L a s(x) ( ε ε ) x + π τ() s s x F M (2.9) a τ τ cr ω, P x L τ fr Fiber Crack Interface Matrix Fig. 2.3 Bon stress versus embee length relation τ(x) uner the assumption of linear elastic pre-cracking an constant frictional post-cracking shear bon properties. The evaluation of the surface integral for the friction surface in the interval [L-a, L], the energy W f neee to overcome friction if the reinforcing element is ebone can be given as the ouble integral W f = π L L a s(x) s fr τ () s s x (2.1) Substituting Eq. (2.9) an Eq. (2.1) in Eq. (2.7) yiels 1 P ω = 2 L + π F F L L a ( ε ε ) x + π τ() s F s(x) sfr τ M L a s(x) () s s x + π G a s x (2.11) Such a linear fracture mechanical moel is foun in the early stuy by [Out69], who likewise separate the interface into an elastic region x < L-a, where no amage has occurre an a ebone zone x > L-a, with an interfacial amage leaing to a separation of the reinforcing element an matrix over a crack length a. However, in this ebone zone only frictional stress transfer is consiere ( ( s) = τ = const. τ for s > s fr, see also [Gao88]). Hence, similar to fr the perfect interface moel a linear elastic composite is assume until the pull-out force reaches a critical value an the reinforcing element starts to ebon, because the resulting ouble integrals in Eq. (2.11) are otherwise very har to evaluate. For similar reasons in all reviewe stuies, the consieration of a frictional stress transfer in the ebone region of the reinforcing element assumes only a constant an uniform post-elastic bon strength τ fr. Many authors use this moel since that time to etermine the energy release rate G for τ fr = const. (solving Eq. (2.11) for G), among others [Bu86], [Gao87], [Kim92], [Liu99], an [Leu2]. A possible way to experimentally etermine the energy release rate is given in [Jen85].

21 12 2 Analytical moeling of a single fiber pull-out process The cohesive interface moel (stress approach) Somehow parallel to the main evelopment of the fracture mechanical moel, [Wan88] was among the first to realize, that solutions to the pull-out problem base only on a linear elastic pre-eboning behavior an a constant, uniform post-elastic frictional shear bon coul not cope with nonlinear abrasion effects observe in experiments conucte on specimens containing nylon monofibers embee in a cementitious matrix. Base on the work by [Law72] who propose a three material composite consisting of a reinforcing element, a matrix an a so-calle interface, [Gop88] introuce the imperfect interface moel accounting for nonlinear elastic material properties. This imperfect interface has nontrivial properties, which cannot be inferre from material parameters of the reinforcing element an matrix, nor measure irectly, but must be obtaine inirectly from the evaluation of pull-out test results. a b P= P=P i P,ω x F F F/2 M F/2 M τ F+ F FF F F x u(x) F u(x) M ζ Fig (a) Global equilibrium of forces an equilibrium of forces on a reinforcing element of length x (test configuration A1). (b) Absolute isplacements of the reinforcing element u F (x) an matrix u M (x) as result of loaing. It is the isplacement continuity requirement vali in the perfect interface moel which is abanone when the interface is assume to be imperfect. To unerstan this imperfection let us imagine a thin region of thickness ζ an ζ << between the constituents, referre to by many authors as the interphase, which has properties ifferent from the reinforcing element an the matrix. For the example of a steel fiber / cementitious matrix system such an interphase was observe by [Pin78, Ben86, Iga96], consisting of a calcium hyroxie layer (CH), an a porous layer of calcium silicate hyrates an ettringite; see Fig Corresponing microharness tests showe [Ben86], that this transition zone, which may exten from the surface of the fiber up to about 5 µm, has ifferent material properties compare to the bulk material, for example the strength of the interphase is quote to be up to 3% less than that of the common cement matrix. If the stiffness of this interphase is much smaller in comparison to the ajoining constituents, the eformation in this zone may be of equal or greater orer than the eformations of the stiffer reinforcing element u F an matrix u M respectively. This interphase eformation can be expresse by the eformation ifference between the ajoining

22 2 Analytical moeling of a single fiber pull-out process 13 reinforcing element an matrix, the slip s = u F - u M. This iea is clarifie in Fig. 2.4 for the test configuration A1 Bulk matrix Interphase 5 µ m Bulk matrix Porous layer Fig Steel fiber CH layer Duplex film Transition zone between a steel fiber an surrouning OPC matrix after 28 ays. If this real existent interphase now is iealize ue to its small thickness to become a surface, i.e. an interface (note the f instea of the ph an the c instea of the s), then this isplacement ifference becomes a isplacement iscontinuity. The istinctive feature of this iealization lies in the recognition that the shear stress between the reinforcing element an matrix τ at any point x is a function of the slip s at that segment x. This function τ(s) is calle the bon stress versus slip relation (BSR) or bon law. An enhancement of the imperfect interface moel is the cohesive interface moel (e.g. [Abr96, Foc]), assuming that an interface failure occurs an hence eboning takes place when the maximum stress at the interface reaches a critical value, e.g. the shear strength of the interface τ cr. After reaching τ cr at x = L-a-c the shear stress graually ecays over a length c as the material weakens (cohesive or softening zone) an can be assume completely ebone from x = L-a onwars, when only friction preominates the stress transfer between the reinforcing element an matrix. A shear stress istribution over the embee length of the reinforcing element L is schematically shown for this moel in Fig. 2.6 (compare Fig. 2.3). a c τ τ cr τ BSR ω, P x L τ fr s cr s fr Fiber s Fig Crack Cohesive interface Matrix Bon stress versus embee length relation τ(x) assuming non-linear elastic precracking an non-uniform frictional post-cracking shear bon properties. This shear stress istribution an hence the progressive eboning of the reinforcing element from the matrix is governe by the unerlying BSR τ(s). So generally it can be state, that as long as the slip between the reinforcing element an matrix is smaller than s cr the reinforcing

23 14 2 Analytical moeling of a single fiber pull-out process element is bone, a crack is initiate when s(x) reaches the value s cr or equivalently when the shear stress reaches the value τ cr. In the range s cr to s fr the bon softens (formation of micro cracks) an when a value s(x) > s fr is reache a macroscopic crack is forme. A goo review on existing cohesive interface moels is given in [Yua1]. The corresponing mathematical representation of the pull-out problem is generally expresse in the literature by a secon orer ifferential equation erive on basis of two equations of equilibrium, an equation of compatibility an Hooke s law, an can be euce as shown in the following: If an axial loa P is applie on the reinforcing element at a location x = L an the system is restraine as shown in Fig. 2.4, the change in loa F F over a istance x along the reinforcing element is a function of the introuce shear stress in terms of x an the circumference of the reinforcing element in contact with the matrix: F F x = π τ (2.12) Assuming that the interface is sheare ue to an applie pull-out loa as shown in Fig. 2.4, an efining the absolute isplacement of the matrix at a point x in relation to its origin as u M an that of the reinforcing element as u F, the slip s can be efine, which is the ifference of u F an u M, i.e. s = u F u. Differentiating s with respect to x yiels s = ε x F ε M M (2.13) where ε F an ε M are the reinforcing element an matrix strains respectively at a point x. Using Hooke s law this yiels s FF = x E A F F FM E A M M (2.14) For emonstration purposes the relationship between τ an s is given as τ = κs where κ is constant, see [Nam88]. By ifferentiating Eq. (2.12) an τ = κs, combining them an using Eq. (2.14) in combination with the static equilibrium from Eq. (2.2) the secon orer ifferential equation of the cohesive interface moel can be erive as: 2 F λ F λ F 2 F =, 1 = π E FA F F (L) = P, F + E M 1 A M F κ F () = (2.15) See also [Naa91], [Baz94], [Abr96], an [Aka99]. Another possible approach is to moel the experimental situation in respect to the local slip s, see e.g. [Som81] an [Foc]. After substituting Hooke s law for the reinforcing element an matrix respectively into Eq. (2.13), ifferentiating the result with respect to x an using Eq. (2.2) in combination with Eq. (2.12), the following secon orer ifferential equation in terms of s is obtaine.

24 2 Analytical moeling of a single fiber pull-out process s = π + A E A E F F M T(s) = π γ τ(s), 1 1 γ = +, A E A E F s = γ F F F M M M τ(s) = T(s) (2.16) Eq. (2.16) represents the basic relationship between the secon orer erivative of the local slip s an the local bon stress τ which is also itself assume to be a function of the local slip. A normalize bon law T(s) is introuce to simplify the following comparisons, iscussions an erivations. To istinguish between the BSR τ(s) an T(s), T(s) shall hereafter be referre to as the normalize bon flow versus slip relation (NBSR). Knowing that at x = the force in the fiber is zero ( s () = ), assuming the slip s at the loae fiber en is ω (s(l) = ω) an enoting the coorinate of the fiber by x L, the mathematical representation of the pull-out problem can be expresse either as bounary value problem (BVP) - see [Foc] - s = T(s), s () =, s(l) = ω (2.17) or as a corresponing initial value problem (IVP) for a fiber en slip υ at x = s = T(s), s () =, s() = υ (2.18) τ τ τ τ τ cr [Naa91] τ cr [Abr96] τ cr [Foc] τ cr theoretical BSR τ fr τ fr τ fr τ fr s=s s cr fr s cr s s s s fr cr s cr s s fr Fig Propose BSR an theoretical BSR. Various proposals about possible BSR τ(s) progressions have been mae - see Fig Many authors (e.g. [Sta9] an [Naa91]) consier the stress-slip relation to consist of a linear elastic part followe by a suen stress rop an a resiual constant friction (compare to the perfect bon an the fracture mechanical moel). However, it can be assume to be more realistic to consier a graual softening after a eboning of the reinforcing element, similar to a Moe I failure. Stuies of [Gop88] an [Abr96] aress issues relating to such a linear softening process uring eboning an pull-out. [Foc] proposes a nonlinear bon stress versus slip relationship superposing a moifie exponential expression (Bertero-Eligehausen-Popov) an a linear ecreasing function. All of the above mentione BSR have in common that only a limite number of parameters is use to escribe the functional relationship between τ an s, an therefore a general

25 16 2 Analytical moeling of a single fiber pull-out process mathematical escription is restricte. This is ue to the fact that until recently it has not been possible to implement such a general mathematical escription of the BSR in an analytical moel to preict the pull-out response of a reinforcing element, because: (I) it is quite ifficult if not impossible to fin a close form analytical solution for a general τ(s) function, (II) numerical simulation has only remarkably avance in recent years, an (III) the simultaneous calibration of a great number of parameters escribing the BSR still nees a consierable amount of computational time or can even not be possible. However, the BSR has to represent the presence an eventual combine action of the following several bon mechanisms in the analytical moel [Bar81]: (I) physical an chemical ahesion between the reinforcing element an the matrix, (II) mechanical interlock ue to eforme, crimpe an hooke reinforcing elements, an (III) friction, which is greatly influence by abrasion [Age99]. Therefore the BSR must be consiere highly nonlinear [Som81, Krü3] as for example shown on the right of Fig Comparison of the moels Any researcher who wants to simulate the pull-out process or yiel the failure criterion for a reinforcing element / matrix system uner a pull-out loa on basis of experimental results, has to ecie which of the above presente moels to use. Therefore the main istinctions between the three moels are outline in the following Perfect interface versus cohesive interface Apparently there has been some confusion with regar to the principal ifferences between the perfect interface moel an the cohesive interface moel as the basic mathematical expressions of both moels for linear elastic pre- an constant frictional post-cracking bon properties look very much alike, see Eq. (2.3) an Eq. (2.15). This confusion an the fact that the former is mainly use by chemical engineers to moel the interfaces of polymer composites, whereas the latter is use by mechanical or civil engineers to characterize the shear bon properties between cement base matrices an certain reinforcing elements, are the main reasons that a closer comparison between both moels has not been mae. However, the main istinctions can be summarize as follows: Whereas the perfect interface moel assumes a perfect contact surface between the reinforcing element an matrix an thus a continuous isplacement fiel within the composite before the actual eboning process, the cohesive interface moel proposes a very thin interphase of thickness ζ between the constituents an abanons the isplacement continuity requirement within the structure. A comparison of Eq. (2.3) (perfect interface moel) to Eq. (2.15) an Eq. (2.16) (cohesive interface moel) outlines these ifferences in the moels. The bon moulus κ in Eq. (2.15) or more generally the BSR τ(s) in Eq. (2.16) which refer to the shear bon properties between reinforcing element an matrix in the cohesive interface moel, contain implicitly the Young s moulus, the tensile strength, the shear moulus, an the shear strength of the transition zone. It is further assume that the total shear eformation of the system between

26 2 Analytical moeling of a single fiber pull-out process 17 fiber an matrix is passe in this interfacial layer efine by λ or τ(s). λ an τ(s) are inepenent material parameters which efine, for example, the surface epenent ahesion between reinforcing element an matrix an cannot be inferre from material properties of the composite alone. This is in contrast to the interfacial parameter β in Eq. (2.3) which efines the shear bon properties in the perfect interface moel an is base purely upon the composite s material properties. As a result, potential surface treatments or changing transition layers (see Fig. 2.5), which certainly influence the bon characteristics [Wu99], can only be inclue in the perfect interface moel if an effective matrix iameter epening on the fiber surface properties [Zha1] is assume, i.e. by aapting the matrix area A M. Also, potential nonlinear shear bon properties as observe e.g. by [Wan88] can only be consiere by a corresponing aaptation of the geometric parameter A M. This might also be the reason that in contrast to the cohesive interface moel only linear elastic, pre- an constant frictional post cracking shear bon properties have so far been assume in the perfect interface moel, which are the main cause of the inflicte stress singularities at the crack tip. Nevertheless, both moels efine a so-calle stress criterion in the form of a bon shear strength τ cr (see e.g. Fig. 2.7) to efine the onset of a shear crack between reinforcing element an matrix an thus the start of the eboning process Perfect interface versus fracture mechanics In recent years the analytical moeling of the eboning process within a composite either uner the assumption of a perfect interface or uner fracture mechanical consierations has been compare in ifferent stuies, e.g. more lately [Zha98a]. Whereas the fracture mechanical moel (Eq. (2.11)) consiers the eboning to be ue to crack propagation along the interface an the failure criteria as the critical energy release rate G cr, the perfect interface moel (Eq. (2.3)) uses the shear bon strength τ cr as the criteria for an interfacial failure. Accoring to [Zha98a] the principal relation between these two failure criteria can be establishe as τ cr = 1 L 1 E F G cr η + η F M E M (2.19) if the temperature ifference between temperature at a stress-free state an the test specimen temperature is neglecte. In any case, the fracture mechanical moel uses the general energy rate G as the main input characteristic which is again assume to be an inepenent material parameter similar to the τ(s) relation in the cohesive interface moel; compare Eq. (2.11). Although [Zha] showe, that both moels can aequately escribe the experimentally erive loa versus isplacement relationships of pull-out tests carrie out on glass fiber / epoxy resin systems, no general proof coul be given as to whether one moel was more appropriate in escribing the shear force transmission than the other. Some authors even

27 18 2 Analytical moeling of a single fiber pull-out process claim the perfect interface moel to not reflect the reality of a reinforcing element / matrix system uner all conitions, especially not for a brittle failure [Leu9] Cohesive interface versus fracture mechanics Whilst in recent stuies ifferent authors have compare the cohesive interface moel with the fracture mechanical moel, e.g. [Sta9, Leu92, Sch92, an Leu2], an erive, that either uner certain conitions both moels lea to comparable results or that uner other conitions one moel is more favorable than the other- e.g. [Leu9] -, no proof for the general applicability of either moel coul be given. The mathematical representation of these moels, i.e. Eq. (2.11) an Eq. (2.16), are therefore irectly compare below an it is shown in this thesis for the first time that they are in fact equivalent an escribe the interfacial processes in a similar way. In principle the following erivation, which has been carrie out in close cooperation with the Institute for Pure an Applie Mathematics at RWTH Aachen University ([Ban4a], [Ban4b], [Jun4a], [Jun4b] an [Jun4c]), is similar to the comparison of the stress an mechanical approaches publishe in [Sta9]. However, in this following case a general escription of the BSR is chosen instea of a pure linear elastic prean constant frictional post crack relation. As a first step Eq. (2.9) is substitute in Eq. (2.11), the shear stress is normalize, i.e. T(s) = π γ τ(s), an F F is replace by s / γ accoring to Eq. (2.16): 1 P ω = 2 γ 1 + γ L L L a s 2 s(x) sfr 1 x + γ T L a s(x) T () s () s s x + π G a s x (2.2) Note, that the integral of the bon stress versus slip relation is evaluate in the interval [s fr, s(x)] because friction is assume to occur only if s(x) > s fr (see also later chapter 2.2.4). However, in Eq.(2.2) friction is not assume to be constant unlike in every other stuy so far, but it is vali for any pre- an post-failure istribution of τ(s). As a secon step a similar expression for the external work P ω neee for a crack avance a is erive on the basis of the IVP state in Eq. (2.18) representing the stress approach so that both moels can be mathematically compare later on. By expressing the pull-out problem with respect to the local slip s (Eq. (2.16)) instea of the force F F in the reinforcing element (Eq. (2.15)), such a mathematical comparison is simplifie. This is ue to the fact that the BSR is explicitly aopte in the secon orer ifferential equation an relate irectly to all the measurans of the pull-out test by the bounary conitions (Eq. (2.17) an Eq. (2.18) respectively). As the isplacement ω is increase by ω, the slip istribution s(x) is increase by δ = s ω ω (thus δ(l) = ω). Rewriting Eq. (2.18) an multiplying by δ yiels

28 2 Analytical moeling of a single fiber pull-out process 19 δ = δ x s x T(s) Using x x s x s x x s x δ δ + = δ an integrating from to L gives δ δ = δ δ = δ δ = δ L L L L L L L x x x s s x x x s x s x x x s x x s x x T(s) Substituting P (L) s γ =, () s = as well as ω = δ L) ( an rearranging yiels x T(s) x x x s P L L δ + δ ω = γ Recalling ω ω = = δ s s an ω ω = = δ s x s shows γ + γ = ω x s T(s) 1 x s 2 1 P L s(x) L 2 (2.21) The terms in the brackets on the right sie of Eq. (2.21) represent the total strain energy in the system, i.e. for the iniviual components as well as for the interface of the composite. A comparison of Eq.(2.2), which represents the external work P ω neee for a crack avance a in the interface of the fiber / matrix system - erive on the basis of fracture mechanical an energy conservation consierations - an Eq. (2.21) representing the same expression - erive on basis of the stress approach - yiels () () a G x s s T 1 x s s T 1 x s T(s) 1 L a L s(x) s a L s(x) L s(x) fr + π γ + γ = γ (2.22) Solving Eq. (2.22) for G yiels γ π γ π = x s T(s) 1 x s T(s) 1 a G L a L s(x) s L a L s(x) fr (2.23) Hence

29 2 2 Analytical moeling of a single fiber pull-out process s fr G = τ (s) s (2.24) Note, that Eq. (2.24) correspons to the total area uner the BSR curve in the range from s = to s = s fr (see right han sie graph in Fig. 2.8). Hence, the energy release rate for Moe II is base on the same principles, which have been use for a long time, for a Moe I failure. τ [Baz94] τ present stuy τ cr τ cr Fig τ fr G s s cr s fr τ fr G s s cr s fr Relation between the BSR an G accoring to [Baz94] an the present stuy. Note further that the BSR τ(s) is assume to be epenent on the material an not on the crack length a nor the isplacement ω, an it has been foun that τ(s) (cohesive interface moel) can be irectly relate to G (energy approach) an G is constant (Eq. (2.24)). Hence the energy criterion is satisfie. Base on the work by [Bar62] an [Ric68], i.e. assuming a cohesive zone ahea of the actual crack, [Pal73] erive a similar expression, as early as 1973, for the energy release rate G by evaluating a line integral J on a curve Γ surrouning the crack tip only which starts from the lower surface an ens on the upper surface of the cohesive zone of the crack. This iea was picke up by ifferent authors, incluing among others [Baz94, LiV97, an Leu2]. However, in contrast to the present work friction is consiere ifferently in these stuies which assume G to be the area uner the τ(s) curve in the interval [, s fr ] minus a frictional part also ranging from zero to s fr (τ fr assume constant). In the present stuy, G correspons to the total area uner the τ(s) curve in the interval [, s fr ] (see Fig. 2.8 for illustration) an oes not consier a reuction for friction ue to the assumption mae before (see Eq. (2.1)) that friction is only activate when the slip has reache a critical value s fr. The work neee to introuce microcracking, which results in the softening behavior, is explicitly inclue in the fracture energy which is state in Eq. (2.24). In almost all stuies pertaining to a comparison of the stress an the energy approach ([Hsu9], [Sta9], an [LiV94] among others), the failure criterion is chosen to be the shear strength τ cr an hence the corresponing slip s cr of the system. However, as has been shown, this stress criterion only yiels the corresponing fracture energy if a brittle failure occurs, i.e. s cr = s fr, for example the case in [Sta9]. In all other cases, the stress criterion τ cr unerestimates the energy neee to avance a crack of length a by a. Therefore, the criterion in the stress approach for an interfacial failure shoul not be the critical shear stress τ cr, but shoul instea be the critical slip s fr between fiber an matrix, from which on friction preominates the stress transfer in the interface.

30 2 Analytical moeling of a single fiber pull-out process 21 As mentione before, several authors have criticize the use of the stress criterion to preict fiber eboning, but this is only true if the perfect interface moel is use. In the case of the cohesive interface moel, however, the stress criterion is vali ue to a constant energy release G an because the BSR use in this moel can be irectly relate to G Which moel to use? It has been shown in the previous chapter that the application of either the cohesive interface moel or the fracture mechanical moel to analyze experimental pull-out test results is actually not epenent on the type of failure occurring in the system as propose by [Leu9] or [LiV97], but both moels are in fact equivalent assuming they are base on the same iealizations. However, if the fracture mechanical moel is use to erive the interfacial properties in the form of the energy release rate G as e.g. one by [Jen85], then the full interfacial fracture process will be specifie by this energy release rate only. As it is certain that ifferent BSR relationships can lea to an ientical energy release rate, as G relates to the integral of τ(s) Eq. (2.24), the BSR τ(s) is more explicit than the energy release rate G in escribing the interfacial characteristics. Therefore in contrast to the fracture mechanical moel, the cohesive interface moel seems a better choice to etermine more etaile shear bon properties of a filament / cement base matrix system as escribe later on in chapter 4. A shortcoming of the perfect interface moel is that, as mentione before, this moel consiers possible influences on the bon characteristics between the reinforcing element an the matrix (such as those ue to surface treatments or potential nonlinear shear bon properties) by an aaptation of the geometric parameter A M. However, this aaptation seems inappropriate as A M also preominates the extensional stiffness of the matrix in the moel an thus there is a confusion between two ifferent material parameters of the composite. Furthermore, the perfect interface moel shows stress singularities at the crack tip which in all probability o not occur in the real composite an lastly this moel oes not account for an interphase which has been experimentally observe for steel fiber / cement base systems [Iga96] an is likely also to arise in AR-glass filament / cement base matrix composites. Due to these reasons the cohesive interface moel is favore over the perfect interface moel in this stuy. However, ue to the aforementione reasons all propose cohesive interface moels are so far base on a BSR τ(s) escribe only by a limite number of parameters (see Fig. 2.7) an therefore a general mathematical escription of the bon law is restricte. To allow such a general mathematical escription of the BSR within the cohesive interface moel, an approach base on the work by [Win85] is introuce an consierably enhance in the following (irect bounary value problem, i.e. τ ( s) P( ω) ). An N-piecewise linear relation between the bon stress τ an the slip s as shown in Fig. 2.9 is use to permit a relatively free characterization of the bon stress versus slip relation (N is not limite) an thus a goo approximation of the theoretical BSR state in Fig The following moels

31 22 2 Analytical moeling of a single fiber pull-out process were erive in close cooperation with the Institute for Pure an Applie Mathematics at RWTH Aachen University ([Ban4a], [Ban4b], [Jun4a], [Jun4b] an [Jun4c]). 2.3 The irect bounary value problem - τ( s) P( ω) The mathematical representation of the pull-out problem for the cohesive interface moel can be expresse by a secon orer ifferential equation erive on the basis of two equations of equilibrium, an equation of compatibility an Hooke s law (see Eq. (2.16)) an two initial conitions, resulting in the IVP state in Eq. (2.18). The ifferential equation an the initial conitions state there correspon to the test configuration A1 illustrate in Fig. 2.2, i.e. a pull-push test. The following erivation is therefore also base on this test configuration. Assuming now an N-piecewise linear bon law with no limitation of N (see Fig. 2.9) the function T(s) = π γ τ(s) can be expresse for an interval i ( s i 1 s s i ) as T T T(s) = m i 1 i 1 i s s i i 1 ( s s ) + T, m =, T = s i = i i 1 (2.25) Similar to T(s) the force F F in the fiber is normalize as well. q(x) = γ FF (x) = s (x) (2.26) For simplicity reasons the state of the system x is not explicitly written in the equations any more. Hence, for example q(x) = q. τ cr τ τ i-1 τ i s cr s i-1 s i s Fig N-piecewise linear relation between the bon stress τ an the slip s. From Fig. 2.2 the bounary conitions can be euce; the force in the fiber at x = L correspons to the pull-out loa P, hence s (L) = q(l) = γ P = ϕ. Note that the normalize pull-out force γp is set as ϕ for simplification. Knowing that at x = the force in the fiber is zero, the following is obtaine: s () = q() =. Likewise it can be state that the isplacement at the loae en of the fiber gives s(l) = ω. Assuming the slip s at the free fiber en to be υ (s() = υ), the above mentione bounary an initial conitions mathematically expresse in Eq. (2.17) an Eq. (2.18) respectively can be illustrate as shown in Fig The IVP (Eq. (2.18)) can be solve easily in an iterative process for any given T(s) an υ with a numerical integration proceure, e.g. the RUNGE KUTTA Proceure ([Bro91]), an the help of a popular math - program (e.g. Maple ). However, to allow a better insight into the

32 2 Analytical moeling of a single fiber pull-out process 23 processes uring a pull-out of a fiber, to give a complete solution routine to simulate a pullout test, an primarily to work out the basics for the inverse bounary value problem P( ω ) τ(s) introuce later on, the erivation of a solution proceure is shown as follows. L ω ω = s n s s= n ω si s x x k+1 x i x i+1 x n s (L)=ϕ P, ω P γϕ -1 T T i s k s k-1 T n υ x k s ()= x i x=l n x Fig Slip istribution an bounary conitions. During a pull-out test a isplacement ω is applie at the loae en of the fiber an continuously increase until the fiber is pulle out of the surrouning matrix. The corresponing resulting force P is measure an recore as a function of ω. In the analytical simulation, similar to the actual pull-out test, the isplacement ω is given an the resulting normalize pull-out force γ 1 ϕ has to be etermine in an iterative proceure, such that the bounary conition at the free fiber en s () = is satisfie (compare Fig. 2.1 an Eq. (2.18)). For simplification of the following erivation an iscussion we consier the situation at a certain loa step uring the pull-out test, where a isplacement ω is applie at the loae en of the fiber an a corresponing normalize pullout force of γ 1 ϕ is measure (see Fig. 2.1, graph on left han sie). This loa step is chosen such that the introuce isplacement ω at x = L uring the simulate pull-out test equals the slip of the upper-boun of the interval n of the selecte piecewise linear function T(s), i.e. ω = s n = s(l). The corresponing normalize pull-out force is ϕ = q n = s (L). The graph on the right of Fig. 2.1 shows these two parameters in the slip versus location of embeing curve. The applie isplacement ω correspons to the slip s n at the location x n = L an the normalize pull-out force ϕ to the slope s n at the location x n = L. In a similar way the inex i for i = to n mentione in the following erivation is associate with the slips s i at the upper-boun of the interval i [ si 1,si ] of the piecewise efine NBSR T(s) with the corresponing normalize shear flows s = Ti an fiber forces s = q i as well as slips s = s i at a location x i (see Fig. 2.1 mile graph / right han sie graph). Keeping this in min we can procee as follows: Using a reuction of orer metho s s = 1 2 (s ) = T(s) s 2

33 24 2 Analytical moeling of a single fiber pull-out process similar to [Baz94], knowing that the lower bounary for s is υ (the slip at x =, see Fig. 2.1), an substituting a function A(s) for the integral term of 2 T(s), s = T(s) in Eq. (2.18) yiels s 2 s = 2 T(s) s = A(s) A( υ), A(s) : = 2 υ s T(s) s (2.27) Eq. (2.27) states that the ifference of the integral of T(s) in the interval [, υ] an [, s] respectively correspons to the ifference of the squares of the tensile forces in the fiber at x = an x. Hence the force in the fiber q i-1 at a location x i-1 ( s (x i 1) = q i 1 ) can be etermine for a given s (x ) = q, if Eq. (2.27) is evaluate for s = s i an s = s i-1 an the results are subtracte. This yiels i i s i 2 s 2 i 1 = 2 = 2 s s = m i i υ s T(s) s 2 i i T(s) s 1 2 ( s s ) + 2 T ( s s ) i i 1 s i 1 υ T(s) s i 1 i i 1 (2.28) Substituting q 2 i 1 = q s an i 2 i m with s i = q an s = i i 1 q i 1 an rearranging gives s i 1 i 2 ( s s ) 2 T ( s s ) i i 1 i 1 i i 1 (2.29) Eq. (2.29) states that for a given force q i = s i in the fiber at a location x i a force q i 1 = s i 1 in the fiber at a location x i-1 may be calculate. x i correspons to the upper-boun s i an x i-1 correspons to the lower-boun s i-1 of the interval i of the piecewise linear NBSR T(s). See Fig Fig υ s= n ω s i s k s k-1 s x x s (x)= x k x i s i x i x n s (L)=ϕ x=l n Slip istribution with a zero slope in the interval k+1 at xˆ (conition I). x As mentione before, a solution for the IVP in Eq. (2.18) is foun if a pull-out force ϕ = q n can be etermine such that the slope s an hence the force q at a location xˆ in the fiber is zero (conition I), an that the location where s = is at xˆ =, i.e. the free fiber en (conition II).

34 2 Analytical moeling of a single fiber pull-out process 25 See Fig For i = n the force q n is known to be the normalize pull-out force n s (x = L) = q = ϕ. Because the NBSR is also known, the forces in the fiber can be calculate recursively with the help of Eq. (2.29), starting from i = n. During this recursive process it may occur that a q 2 k 1 < is calculate accoring to Eq. (2.29) but q is still greater zero. This inicates that the slope of the function s(x) has change its 2 k sign between the locations x k-1 an x k ; see Fig Hence the conition s = is satisfie within the range x k 1 < xˆ < x k. Because the locations x k-1 an x k also correspon to the lower-boun s k-1 an upper-boun s k, respectively, of the interval k of the NBSR (Fig. 2.1), it can be further euce that the slip of the free fiber en υ is in the range of s k-1 υ < s k (Fig. 2.11), an thus it can be state that s turns zero somewhere in the interval k. However this recursive etermination of the fiber forces using Eq. (2.29) only guarantees that the slope is zero at a xˆ (conition I) but not that s = at the location xˆ = (conition II) because Eq. (2.29) is inepenent of x. To etermine the location xˆ where s =, the sum of the incremental lengths x i = x i x i 1 has to be etermine (compare Fig. 2.11). Only if this sum yiels the embee length L, is the above mentione conition II that s = at xˆ = satisfie. The summation of the incremental lengths x i can be written as follows (compare Fig. 2.11) n L = x + (2.3) i= k+ 1 x i with k being the interval number corresponing to the interval just before s turns zero. x i an x of Eq. (2.3) can be etermine as follows. Evaluating Eq. (2.27) for s an s = s i-1, subtracting the results, an rearranging yiels (similar to Eq. (2.28)) s = s x = m i 2 2 ( s s ) + 2 T ( s s ) + s i 1 i 1 i 1 i 1 (2.31) After a separation of variables, substituting q i 1 = si 1, an integrating, the following for the lower-boun s i-1 an upper-boun s i of the interval i of the T(s) function is obtaine: s si 1 2 i 1) + 2 Ti 1 (s s i 1 ) + i ŝ x i = (2.32) m (s s q i Accoring to [Bro91] Eq. (2.32) can be evaluate as follows. For m i > : 1 m i q i + Ti x i = ln (2.33) m i m i q i 1 + Ti 1 An for m i < : ( T q T q ) 1 m i i i 1 i 1 i x i = arcsin (2.34) 2 2 m Ti 1 m i i q i 1 2 i 1

35 26 2 Analytical moeling of a single fiber pull-out process The last incremental length x can be etermine knowing that the force in the fiber at xˆ is zero ( ). Substituting q 2 k 1 s k s k 1 Tk T = m k k 1 accoring to Eq. (2.25), an q 2 k 1 in Eq. (2.29) yiels after a few transformations for the interval k T 2 = T 2 k m k q 2 k (2.35) where T k 1 T is the shear stress at xˆ. Using q 2 k 1, T k 1 T, an Eq. (2.35) in either Eq. (2.33) or Eq. (2.34) gives x for m k > an m k < respectively. This yiels for m k > 1 m k q k + Tk x = ln (2.36) 2 2 m k T k m k q k an for m k < 2 2 ( q k Tk m k q k ) 1 m k x = arcsin (2.37) 2 m Tk m k k q k 2 The above liste proceure can be summarize as follows: A ϕ = q n has to be foun in an iterative proceure for a given isplacement s(x n = L) = ω such that the incremental lengths x i (for m i > accoring to Eq. (2.33) an for m i < accoring to Eq. (2.34)) sum up to the embee length L accoring to Eq. (2.3). The lowerboun k in Eq. (2.3) is evaluate by using Eq. (2.29) in a recursive way starting from i = n until a (q k-1 ) 2 < is etermine an hence the interval number k is foun in which the force in the fiber becomes zero. If a ϕ = q n is foun which satisfies the bounary conition s () = the solution is foun to the IVP liste in Eq. (2.18) an hence the resulting force P =γ -1 ϕ is etermine for an applie isplacement ω at the loae en of the fiber. To erive a complete loa isplacement iagram for a given normalize shear flow relation T(s) an hence a given shear stress relation τ(s), initial conitions P = γ -1 ϕ have to be foun for many loa steps of a pull-out test such that Eq. (2.3) is always satisfie. Nevertheless, with the help of a computer routine, where the above escribe numerical iteration process is inclue, the right initial conitions ϕ can be foun quite easily, an hence a complete pull-out test is simulate. An example to clarify the principle solution proceure is given in Appenix A. 2.4 The inverse bounary value problem - P( ω) τ(s) It has been foun an shown in the previous chapter that there is a straightforwar relation between the NBSR T(s) an the corresponing normalize pull-out force versus isplacement relation ϕ(ω) an also that a loa versus isplacement relation can be erive from a given

36 2 Analytical moeling of a single fiber pull-out process 27 BSR. Base on this result it must also be possible to erive a BSR from a given loa versus isplacement relation, i.e. P( ω ) τ(s) for a given P(ω). The same proceure as escribe above is utilize but instea of etermining the pull-out force P in an iterative proceure, the normalize shear stress T i is ientifie, as shown below. ω s s x P γϕ -1 1 ω ω 1 =s 1 T T 1 s s= 1 1 ω s s ()= x s (L)=ϕ 1 x 2 x=l 1 x P γϕ -1 2 ω 2 =s 2 T T 2 T 1 s 2 =ω 2 s 1 s ()= x 1 s (L)=ϕ 2 x=l 2 x Fig Inverse etermination of T(s). To inicate the current loa step, the normalize pull-out force ϕ an the corresponing isplacement ω are labele with the loa step number n, i.e. ω n for the isplacement an ϕ = q n = s n (L) for the normalize pull-out force at a loa step n. See Fig for illustration. Note, that the partition of ω in the loa versus isplacement curve results in the same number of ivisions in the T(s) relation, i.e. ω n always correspons to the slip s n at a location x n = L as well as to the slip s n for the T(s) relation an is therefore the upper-boun of the interval n of the piecewise linear function T(s), with i n. The normalize pull-out force ϕ n again correspons to the slope s n at the location x n = L. The principal solution routine for the inverse bounary value problem is as escribe above, except that now a T n has to be foun in an iterative proceure for a given isplacement n 1 ω = s (x = L) an a given pull-out force P = γ ϕ = γ s (L), such that the incremental lengths x i (for m i > accoring to Eq. (2.33) an for m i < accoring to Eq. (2.34)) sum up to the embee length L accoring to Eq. (2.3) (compare Fig. 2.1). x is again efine as escribe in Eq. (2.36) an Eq. (2.37) respectively. The lower-boun k in Eq. (2.3) an Eq. (2.36) is evaluate in turn by using Eq. (2.29) in a recursive manner from i = n (ϕ n = q n ) until a (q k-1 ) 2 < is calculate an thus the interval is foun in which the force in the fiber turns zero. The point (s n, T n ) an hence the point (s n, τ n ) of the esire BSR is foun if the bounary conition s () = of the IVP, state in Eq. (2.18) an Eq. (2.3), is satisfie. Starting at a very low loa level an therefore with a small initial isplacement ω 1 an assuming a piecewise linear relation between T an s, the corresponing an only unknown variable is T 1 which can easily be etermine with the help of Eq. (2.3), knowing that m 1 > an n = 1 (see Fig. 2.12). Thus, the first point in the BSR is foun. 1

37 28 2 Analytical moeling of a single fiber pull-out process Choosing a following loa step an the corresponing somewhat higher isplacement ω 2 (n = 2), T 2 can be evaluate in a similar iterative proceure using Eq. (2.29) in a recursive manner from i = n = 2 (ϕ 2 = q 2 ) until a (q k-1 ) 2 < is etermine an hence the interval number k in Eq. (2.3) is foun in which the force in the fiber turns zero. If the chosen T 2 further satisfies Eq. (2.3), the next point in the BSR is foun (see Fig. 2.12). Note that for this recursive calculation the next point in the BSR can only be etermine if the previous points are known. Similar to the proceure escribe above the total BSR can be evaluate loa step by loa step. A problem arises from this stepwise etermination of the BSR, because as alreay mentione, for the etermination of a further point n in the T(s) relation, all previously ientifie linear parts of the BSR are neee, which is in contrast to the irect bounary value problem. There, the ientifie pull-out forces at a certain loa step are no longer use for subsequent calculations, an possible errors mae uring the evaluation at this step o not influence the further calculation of the loa isplacement curve to be ientifie. Bon stress τ in N/mm² 5 Oscillation Effects 4 Average τ(s) Slip s in mm Fig Oscillation effects uring the inverse ientification of T(s). However, for the inverse bounary value problem errors mae in the erivation before, o sum up or even potentially increase, which results in oscillation effects an in the worst case in a non converging calculation. See Fig for an example. This phenomenon has previously been observe in many cases for inirect bounary value problems an for ifferent applications. For a general overview on inverse problems in engineering mechanics see [Tan98] an [Tan]. To minimize the influence of this error propagation certain features an regularization methos are implemente in the numerical solution routine of the presente work. Among other things, a regularization metho following [Lam] is use to suppress the aforementione oscillation effects. This proceure introuces certain collocation points S in the chosen intervals [S i, S i+1 ] for i n an seeks a new solution accoring to the above escribe solution routine. An averaging technique is utilize to evaluate a new T i+1 closer to the proper NBSR an thus the error magnification is suppresse to a consierable egree. Another possibility is the implementation of a monotony preconition, i.e. after the evaluation

38 2 Analytical moeling of a single fiber pull-out process 29 of the maximum normalize bon stress T cr in the NBSR, the following etermine T n must be monotonically ecreasing. 2.5 Summary In this chapter a proof is given that the fracture mechanical moel (energy approach) an the cohesive interface moel (stress approach), which are commonly consiere to be alternative or are even exclue for certain reasons, are equivalent - chapter As the so-calle cohesive interface moel allows an easier mathematical representation of the pull-out problem, it is foun to be aequate to simulate the loa versus isplacement response P(ω) of a single fiber pulle out of the surrouning matrix. Further the fracture energy neee to exten the crack between fiber an matrix by an increment a is etermine to represent the integral over the bon stress versus slip relation in the interval [, s fr ] (Eq. (2.24)), where s fr correspons to the slip value between fiber an matrix above which the interface is assume to be fully cracke an only frictional stresses are transmitte. Base on the theory of the cohesive interface moel, an analytical approach to simulate the single fiber pull-out behavior is introuce in which a N-piecewise linear bon stress versus slip relation is aopte with no limitation of the linear intervals assume. Hence, if N is taken large, any possible bon law istribution can be approximate, i.e. a general mathematical escription of this relation is allowe, but until now this has not been possible. As no close form analytical solution of this irect bounary value problem exists for this approach, a simple numerical solution proceure is presente an applie. Finally an analytical simulation proceure is propose, which is base on an inverse bounary problem an allows for the first time the straightforwar calculation of an N-piecewise linear bon law τ(s) with no limitation of N on the basis of an experimentally etermine loa isplacement istribution P(ω), without using any optimization routines or fitting proceures. Although some more work has to be one in the future to allow an automate ientification of the BSR base on experimentally etermine loa isplacement relations P(ω), e.g. the integration of more powerful regularization methos to suppress oscillation effects, the numerical proceure presente in this stuy is a first step towars a more etaile evaluation of the bon properties of composites.

39 3 2 Analytical moeling of a single fiber pull-out process

40 31 CHAPTER 3 APPLICATION AND VALIDATION OF THE COHESIVE INTERFACE MODEL To illustrate the application of the moel erive in the previous chapter an allow an experimental valiation, pull-out tests on steel fiber / cementitious matrix systems with ifferent fiber iameters an bon lengths are carrie out an escribe in this chapter an analyze with regar to the unerlying BSR τ(s). The effects on the experimental results an on the evaluate τ(s) relationships ue to these varying geometrical parameters are investigate, because the propose moel is only vali if it allows an ientification of shear bon parameters which are inepenent of the geometric arrangement. This valiity implies that the material characteristics, which generally efine the shear force transmission between the material of the reinforcing element an the material of the matrix, are moele sufficiently. Furthermore, this verification assures that the shear bon parameters evaluate later on in chapter 4 reflect the effective bon properties of AR-glass filaments embee in a cementitious matrix. 3.1 Introuction In principle the single fiber pull-out test (Fig. 2.2) is base on a relatively simple experimental configuration an has wiely been use not only to investigate the shear bon properties of steel fiber or re-bar / cement base matrix systems (e.g. [Naa91a]) but also of ceramic matrix (e.g. [Bar91]) or polymer matrix composites (e.g. [Zha]). However, this experimental technique also has some limitations associate with the scale of the test. Firstly, there is a maximum embee length of the reinforcing element, L crit, permitte for pull-out without a tensile failure occurring. Seconly, so-calle stochastic size effects can be observe for small iameter fibers embee in a cementitious matrix as this matrix is an inhomogeneous structure [Gut99]. The inhomogeneous ranom arrangement of, for example, pores can inuce a ramatically reuce effective embee length not visible from the outsie, which will in turn result in an unerestimation of the actual shear bon properties, if this effect is not regare in the analytical moel use for the evaluation. Nevertheless, if these aspects are consiere uring the testing proceure an later on uring the analytical evaluation process, the single fiber pull-out test is an applicable tool to investigate the shear bon properties.

41 32 3 Application an valiation of the cohesive interface moel 3.2 Materials composition an specimen preparation The straight an smooth steel fibers with ifferent iameters are cast in a fine-graine concrete matrix (PZ-899-1) which features a maximum grain size of.6 mm, a water / biner ratio of.4, an a biner content of 7 kg/m³. The actual concrete composition is liste in Table 3.1. For more information refer to [Bro1]. Table 3.1. Fine-graine concrete PZ composition. PZ Constituent Content kg/m³ Biner CEM I 52,5 N 49 Fly ash 175 Silica fume 35 Amixtures Superplasticizer 14 Aggregates Siliceous fine san 499 San 714 During casting the fibers are situate vertically in the mol of imensions 5mm 5mm L, with L being the embee length of the fiber (Fig. 3.1). The fibers run over the total height of the specimen an exten on the upper sie about 125 mm an on the lower sie about 5 mm. The hole on the lower sie of the mol allowing the fiber to exten is seale with silicon paste. The embee length L is varie from 12.1 to 8. mm an the fiber iameter from.3 to 2.5 mm (Table 3.2). Steel fiber Formwork Fine graine concrete Silicon paste 125 mm L 5 mm Fig Specimen preparation for steel fiber pull-out tests. All fibers use in the test are previously egrease with aceton. After casting, the specimens are compacte for 6 sec. on a V-B table with a vibration amplitue of.5 mm. The

42 3 Application an valiation of the cohesive interface moel 33 specimens are cure in the mol for 48 hours an then store at 2 C at 65% RH until reay for testing at a concrete age of 7 ays. In tension tests, the Young s moulus of the finegraine concrete was etermine to be 35, N/mm² [Bra2] an that of the steel fiber to be 21, N/mm² [Geu3] Experimental sequences The experimental program carrie out in this investigation to verify the cohesive interface moel an stuy the interfacial bon in cementitious composites comprises a total of 18 series of tests. Each of the 18 series consists of five tests for a certain combination of fiber iameter an embee length, i.e. the test is repeate four times for each combination. In Table 3.2 these combinations are specifie. Table 3.2. Combinations of fiber iameter an embee length. Combination Fiber iameter Embee length /mm L/mm Experimental methos In orer to examine the eboning an pull-out processes in the experiments, the specimen is glue (X6) at its base to a steel block, such that the fiber extens through a hole in the block with its en attache to a linear variable ifferential transucer (LVDT), fixe to the bench of the universal testing machine (Instron 5566). See Fig. 3.2 for a schematic illustration of the

43 34 3 Application an valiation of the cohesive interface moel test set-up. The steel block itself is also fixe by a frame to the bench of the testing machine an the top en of the fiber is clampe by a mechanical grip at a free length of 1 mm to the cross hea of the Instron. The loaing is controlle by cross hea isplacement with a rate of about.1 mm/min. This eformation control is aopte to allow measurements in the resiual stage. If the cross hea isplacement is utilize to gain information on the eformation of the fiber immeiately above the matrix, the elastic strain contribution over the free length of the steel fiber between the clamping point an the point of intersection with the matrix has to be subtracte. To avoi the problems which result from this manual revision of the P(ω) relationship - e.g. ue to a slip in the mechanical clamping - a vieo extensometer is introuce in the experiment. This system consists of two main parts: a vieo camera an a vieo processing part, which is store in a PC containing a frame-grabber interface car an software to analyze the ata. The frame-grabber interface car converts the PAL vieo signal into an 8-bit igital format whilst simultaneously generating a 8 x 6 pixel image. The interface is capable of resolving the gray scale level of each pixel into 256 shaes. An analogue interface is available to connect the loa cell (Instron, measuring range.1 to 5. kn) to the PC an hence both signals can be save simultaneously. Loa Cell 123 Fiber Matrix Steel block LVDT Vieo - extensiometer P ωυ, Fig Set-up for pull-out tests on steel fibers embee in a cementitious matrix. The vieo-extensometer operates irectly as a (non-contact) isplacement measurement evice by etermining the change in istance between two markers, the so-calle targets. The targets prouce rapi contrast changes in gray scale an thus allow the evaluation of the absolute isplacements of a point by tracking these specific gray scale istribution in the x - an y irection respectively in the sequence of the pictures taken (12.5 pictures per secon). The maximal resolution epens on the fiel of view. In the present case the accuracy is etermine for a isplay winow of 1 x 1 mm², which is utilize for the testing of about.6 µm. Two white paint markings, one on the steel fiber itself about 1/1 of a millimeter

44 3 Application an valiation of the cohesive interface moel 35 above the intersection point of fiber an matrix an another on a reference plate fixe to the bench, are use as targets for the vieo extensometer. Hence, instrumentation is provie to measure the isplacement of the fiber at the active (loae) an passive (free) en relative to the en faces of the specimen, i.e. P(ω) an P(υ) relationships are erive at the loae an free fiber ens respectively. The applie force, an the isplacements of the loae as well as free en of the fiber are continuously recore for every 2 N change in force. 3.4 Test results The loa isplacement curves are given separately for the loae as well as for the free fiber en isplacements, i.e. P(ω) on the left han sie of Fig. 3.3 to Fig. 3.2 an P(υ) on the right han sie of Fig. 3.3 to Fig. 3.2 respectively (black lines). Aitionally in each iagram the simulate P(ω) or P(υ) curves are plotte (black squares). As will be explaine later in this stuy these simulate loa versus isplacement curves are base on one an the same BSR τ(s) which is evaluate with the help of the cohesive interface moel erive earlier. Pullout force P in N 8 Simulation P(ω) Experiment P(ω) Displacement ω in mm Pullout force P in N 8 Simulation P(υ) Experiment P(υ) Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of 2.5 mm an an embee length of 35. mm (combination 1 in Table 3.2); Experiment an simulation. Pullout force P in N 6 Simulation P(ω) Experiment P(ω) Displacement ω in mm Pullout force P in N 6 Simulation P(υ) Experiment P(υ) Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of 2. mm an an embee length of 4. mm (combination 2 in Table 3.2); Experiment an simulation.

45 36 3 Application an valiation of the cohesive interface moel Pullout force P in N 6 Simulation P(ω) Experiment P(ω) Displacement ω in mm Pullout force P in N 6 Simulation P(υ) Experiment P(υ) Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of 2. mm an an embee length of 35. mm (combination 3 in Table 3.2); Experiment an simulation. Pullout force P in N Simulation P(ω) Experiment P(ω) Displacement ω in mm Pullout force P in N Simulation P(υ) Experiment P(υ) Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of 2. mm an an embee length of 3. mm (combination 4 in Table 3.2); Experiment an simulation. Pullout force P in N 3 Simulation P(ω) Experiment P(ω) 2 1 Pullout force P in N Simulation P(υ) Experiment P(υ) Displacement ω in mm Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of 2. mm an an embee length of 2. mm (combination 5 in Table 3.2); Experiment an simulation.

46 3 Application an valiation of the cohesive interface moel 37 Pullout force P in N 2 Simulation P(ω) Experiment P(ω) Displacement ω in mm Pullout force P in N 2 Simualtion P(υ) Experiment P(υ) Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of 2. mm an an embee length of 1. mm (combination 6 in Table 3.2); Experiment an simulation. Pullout force P in N 8 Simulation ω Experiment ω Displacement ω in mm Pullout force P in N 8 Simulation υ Experiment υ Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of 1.5 mm an an embee length of 8. mm (combination 7 in Table 3.2); Experiment an simulation. Pullout force P in N 4 Simulation P(ω) Experiment P(ω) Displacement ω in mm Pullout force P in N 4 Simulation P(υ) 3 Experiment P(υ) Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of 1.5 mm an an embee length of 3. mm (combination 8 in Table 3.2); Experiment an simulation.

47 38 3 Application an valiation of the cohesive interface moel Pullout force P in N Pullout force P in N 3 Simulation P(ω) Experiment P(ω) 3 Simulation P(υ) Experiment P(υ) Displacement ω in mm Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of 1.5 mm an an embee length of 27.1 mm (combination 9 in Table 3.2); Experiment an simulation. Pullout force P in N 3 2 Simulation P(ω) Experiment P(ω) Pullout force P in N 3 2 Simulation P(υ) Experiment P(υ) Displacement ω in mm Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of 1.2 mm an an embee length of 27.1 mm (combination 1 in Table 3.2); Experiment an simulation. Pullout force P in N Pullout force P in N 2 Simulation P(ω) Experiment P(ω) 2 Simulation P(υ) Experiment P(υ) Displacement ω in mm Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of 1. mm an an embee length of 3. mm (combination 11 in Table 3.2); Experiment an simulation.

48 3 Application an valiation of the cohesive interface moel 39 Pullout force P in N 2 15 Simulation P(ω) Experiment P(ω) Pullout force P in N 2 Simulation P(υ) Experiment P(υ) Displacement ω in mm Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of 1. mm an an embee length of 22.1 mm (combination 12 in Table 3.2); Experiment an simulation. Pullout force P in N 15 Simulation ω Experiment ω Displacement ω in mm Pullout force P in N 15 Simulation υ Experiment υ Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of.8 mm an an embee length of 22.1 mm (combination 13 in Table 3.2); Experiment an simulation. Pullout force P in N 15 1 Simulation P(ω) Experiment P(ω) Pullout force P in N 15 1 Simulation P(υ) Experiment P(υ) Displacement ω in mm Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of.6 mm an an embee length of 3. mm (combination 14 in Table 3.2); Experiment an simulation.

49 4 3 Application an valiation of the cohesive interface moel Pullout force P in N 1 75 Simulation P(ω) Experiment P(ω) Pullout force P in N 1 75 Simulation P(υ) Experiment P(υ) Displacement ω in mm Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of.6 mm an an embee length of 17.1 mm (combination 15 in Table 3.2); Experiment an simulation. Pullout force P in N 75 5 Simulation P(ω) Experiment P(ω) Pullout force P in N 75 5 Simulation P(υ) Experiment P(υ) Displacement ω in mm Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of.3 mm an an embee length of 3. mm (combination 16 in Table 3.2); Experiment an simulation. Pullout force P in N 3 Simulation P(ω) Experiment P(ω) 2 1 Pullout force P in N Simulation P(υ) Experiment P(υ) Displacement ω in mm Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of.3 mm an an embee length of 13.6 mm (combination 17 in Table 3.2); Experiment an simulation.

50 3 Application an valiation of the cohesive interface moel 41 Pullout force P in N 3 2 Simulation P(ω) Experiment P(ω) Pullout force P in N 3 2 Simulation P(υ) Experiment P(υ) Displacement ω in mm Displacement υ in mm Fig Loa versus active fiber en isplacement iagram P(ω) an loa versus passive fiber en isplacement iagram P(υ), for a steel fiber iameter of.3 mm an an embee length of 12.1 mm (combination 18 in Table 3.2); Experiment an simulation. All of the liste results an graphs show the typical an expecte behavior foun in pull-out tests; an increasing part with positive graient governe by elastic bon, followe by a ecreasing softening stage (negative graient) an finally by a flattening stage controlle by friction where the pull-out force comes close to constant. As the fiber is pulle through the surrouning matrix (see chapter 3.2), the frictional bon resistance stays constant although it is possible that some minor changes can be etecte ue to abrasion effects or wege aggregates. The variation of the loa eflection curves is epenent on the fiber iameter an embee length use in the test. With increasing fiber iameter an/or embee length the variation ecreases. 3.5 Application an valiation of the moel The erivation of the above escribe moel (chapter 2.4) is base on the test configuration A1 of a pull-push test, i.e. a fiber is pulle out of the matrix against a restraint, in contrast to the experimental tests presente, which are pull-pull tests (the matrix is restraine at the base). However, as has been shown in [Zho92] an [Braa] the influence of a ifferent loaing conition on the expecte results can be neglecte, if the ratio between the stiffness of the fiber an that of the matrix is greater than 1, i.e. E MA M E FA F 1. In the present case the extensional stiffness of the fiber is foun to be e.g. for a fiber iameter of 2 mm E F A F = 21, N / mm² π2 4 2 mm² 659kN The Young s moulus of the fine-graine concrete is etermine to be 35, N/mm², but the loa carrying area of the matrix is not known. From pure expert opinion it is state that the whole cross-sectional area of 5 5mm² will not contribute to the loa transmission. Nevertheless, own experimental tests carrie out uner the use of fiber optic strain sensors which are embee in the matrix at ifferent raial istances from the fiber showe that the loa carrying area is, uner all circumstances, large enough to guarantee, that an extensional

51 42 3 Application an valiation of the cohesive interface moel stiffness ratio greater than 1 is maintaine [Geu3]. Hence the propose solution routine is not only vali for a pull-push test but also for a pull-pull test. Applying this routine for the evaluation of the loa isplacement curves P(ω) presente in Fig. 3.3 to Fig. 3.2, the unerlying BSR τ(s) can be evaluate for each test carrie out. For example the erive τ(s) curve for one pull-out test of a.8 mm iameter steel fiber embee over 22.1 mm in fine-graine concrete (combination 13 in Table 3.2) is liste in Fig. 3.21; note the oscillations an refer to Fig Bon stress τ in N/mm² 4 Inverse Determintaion of τ(s) 3 Smoothing: Ajacent averaging Slip s in mm Fig Evaluate bon stress versus slip relation τ(s) for one pull-out test on a steel / finegraine concrete (PZ-899-1) system;.8 mm steel fiber iameter an an embee length of 22.1 mm (combination 13 in Table 3.2). If an ajacent averaging technique is aopte to smooth the progression of the BSR (black squares in Fig. 3.21) an the presente routine is use for all 5 test results of combination 13 in Table 3.2, the BSR liste in Fig can be obtaine. The bon laws of the investigate test series show variation corresponing to those of the P(ω) relations, see Fig The black squares in Fig refer to the calculate average BSR of combination 13. Bon stress τ in N/mm² 4 3 Iniviual BSR for combination 13 Average BSR for combination Slip s in mm Fig Evaluate bon stress versus slip relations τ(s) for all tests of combination 13 in Table 3.2 (.8 mm fiber iameter an an embee length of 22.1 mm, steel fiber / fine-graine concrete (PZ-899-1) system).

52 3 Application an valiation of the cohesive interface moel 43 If the unerlying average bon laws are erive using an averaging technique for all the combinations liste in Table 3.2 an all replications of tests within a series, an overall average BSR τ(s) can be evaluate as presente in Fig Base on this evaluate τ(s) relationship, the irect bounary value problem (chapter 2.3) is use to simulate the expecte pull-out response for the ifferent fiber iameters an embee lengths teste. The results of these simulations are shown on the left han sie of Fig. 3.3 to Fig. 3.2 together with the experimental pull-out test results. In general it may be state that a goo consistency with the experimental tests is achieve. Bon stress τ in N/mm² Slip s in mm Fig Evaluate bon stress versus slip relation τ(s) for the material combination steel / finegraine concrete (PZ-899-1, see [Bro1]). The graph on the left of Fig. 3.9 which shows the P(ω) relationships measure an simulate for the combination with a 1.5 mm fiber iameter an an embee length of 8. mm, inicates that the increasing part of the graph as well as its peak are overestimate by the simulation. This may be ue to two reasons: Firstly, the specimens experience a mechanical interference when compacte, leaing to possible segregation an seimentation effects in the transition zone between fiber an bulk matrix. Seconly, the relative isplacement between the two markings, one on the steel fiber itself an one on a reference plate fixe to the bench, is measure an taken as ω. However if the embee length increases, the absolute eformation of the matrix also increases an can no longer be ignore. Unfortunately this effect has not been consiere uring this stuy but the fact that the simulation of the loa isplacement curves measure on the free fiber en - i.e. P(υ) - agrees with the measure results of the same combination suggests that the erive bon law is also vali for this combination; see the right han sie graph of Fig Note that these simulate P(υ) relations (on the right sie of Fig. 3.3 to Fig. 3.2) are only use for the valiation of the propose moel, an they show, ue to their goo agreement with the measure relationships, that the evaluate τ(s) relationship can be seen as a material parameter which explicitly characterizes the bon between a steel fiber an the fine-graine concrete. Aitionally, these results show that the propose cohesive interface moel is capable of simulating the pull-out processes of fibers embee in a cementitious matrix.

53 44 3 Application an valiation of the cohesive interface moel Hence this moel may be use for further applications in this thesis, i.e. to etermine the bon characteristics between AR-glass an fine-graine concrete. 3.6 Stochastic size effects It is well unerstoo that concrete is an inhomogeneous material, i.e. pores, imperfections an vois are foun in the bulk matrix an in the transition zone, which influence the mechanical an chemical behavior an certainly the bon characteristics of the material. From this point of view an objective simulation cannot be carrie out if these imperfections are not incorporate in the moel. However if the basic geometric parameters are consiere, which are use in the cohesive interface moel escribe in chapter 2, an applie as before, the assumption is mae that the fiber (or in the following case the filament) is embee over the full embee length an the full circumference in the matrix. However, as the SEM micrographs show, vois an pores influence the contact area an hence the bon quality consierably, especially as the fiber iameter ecreases, because the pore size istribution of the matrix remains constant an only the fiber iameter changes. In literature this effect is calle a stochastic size effect. Examples of impressions of steel fibers of 2.5 mm an.3 mm iameter in the fine-graine concrete PZ are given in Fig to clarify this effect. 1 µ m 1 µ m A) Fig A) Interphase (transition zone) between steel fiber iameter 2. mm an fine-graine concrete (Mag = 4 X, black area is the impression of the fiber). B) Interphase (transition zone) between steel fiber iameter.3 mm an fine-graine concrete (Mag = 4 X, black area is the impression of the fiber). B) In the present stuy these stochastic size effects are also observe, as Fig. 3.2 shows, which relates to the pull-out tests on a.3 mm iameter steel fiber embee over 12.1 mm in finegraine concrete (combination 18 in Table 3.2). It can be seen in Fig. 3.2 that the simulate P(ω) istribution only fits the results of one of the measure pull-out tests, with the remaining 4 experimental relationships falling well below this simulate graph. This might be ue to ajacent pores or aggregates in the interface which isturb the ahesive bon between cement

54 3 Application an valiation of the cohesive interface moel 45 paste an steel fiber. Note the maximum grain size is.6 mm which is, in this case, twice larger than the fiber iameter. Hence it may be state that these imperfection concentrations cause the variations in the experimentally etermine pull-out curves, an this variability also causes the BSR τ(s) to vary. Certain approaches may be feasible to investigate these variations in the interfacial structure an later on such stochastic size effects are taken into account in the evaluate bon law τ(s). One possibility is to use a eterministic an stochastic analysis; e.g. the well known Monte Carlo metho, in which ifferent realizations of the ranom variables are generate accoring to their probability istribution. In the present case one of these variables coul be the pore size istribution an hence the influence circumference an embee length. However, in orer to use this metho, etaile information must be available with regar to the pore, voi, an imperfection istribution along the interface of a fiber matrix system for many specimens. Only if this information is given, can a probability istribution be evaluate an use in this analysis. Using the so-calle finite element reliability metho, [Gut99] stuie such a size-effect phenomenon for the pull-out problem of steel anchors. The theoretical results [Gut99] obtaine are in agreement with the experimental observations of this stuy, in the sense that a small specimen size, particularly the fiber iameter, is quite sensitive to stochastic imperfections an exhibits a large scatter in the pull-out response. In aition to the require information on the pore, voi an imperfection istribution, a probability ensity function must also be known or assume for this analysis. A thir possibility to stuy the interfacial contact area is by optical microscopy. However, this is a estructive test metho an therefore the results can only be obtaine either from specimens, which have alreay experience a pull-out loaing an thus an alteration of the interfacial zone, or from specimens which have been prepare only for this test. Thus, it is not possible to relate the results of the interfacial pore istribution irectly to the pull-out test carrie out. A further metho to etermine an quantify the global interfacial structure coul be the measurement of the electrical resistivity of the interface [Fu95, Fu98]. If a correlation between the contact resistivity an the effect of the voi, imperfection an pore istribution at the interface on the pull-out response is foun, a test metho is given, which provies an inication of the variability of the interfacial structure. It is to be unerstoo that only conuctive fibers can be use. [Fu98] showe for steel fibers that the contact resistivity ecrease linearly with the maximum bon strength τ cr. Hence it may be assume, that the correlation between the contact area an the contact resistivity is also linear. Unfortunately, none of the testing proceures or analysis methos escribe above are use in this stuy to account for stochastic size effects uring the evaluation of the particular bon laws τ(s) because of the complexity of the testing proceures or analytical implementations in the cohesive interface moel. If it is assume that the highest pull-out force correspons to the

55 46 3 Application an valiation of the cohesive interface moel largest contact area between fiber / filament an matrix, an therefore is closest to the iealize contact area efine by the embee length L an circumference π in the cohesive interface moel (see [Fu98]) then the stochastic size effect can be inirectly consiere uring the etermination of the BSR, which preominates the bon between either steel fibers an fine-graine concrete or later on glass filaments an fine-graine concrete. 3.7 Summary The cohesive interface moel erive in chapter 2 is applie an valiate by using the results of pull-out tests which have been carrie out on smooth an straight steel fiber embee in a fine-graine concrete. A goo agreement between simulation an experimental test results is obtaine, an hence an appropriate analytical tool is foun to etermine an evaluate the principal boning characteristics an behaviors between an AR-glass filament an a surrouning cement base matrix (sub-problem I, chapter 1). It is further conclue that the etermine bon stress versus slip relation τ(s) is a material parameter, an hence not epenent on geometric factors of the fiber an matrix such as fiber iameter an embee length. However for small fiber iameters ( < maximum grain size) so-calle stochastic size effects are observe, but these are so far not accounte for in the cohesive interface moel. The stochastic size effect of the fine-graine concrete implies a ranom voi, pore an imperfection istribution in the interface, i.e. inhomogeneity, which influences the bon quality consierably, if the size of, e.g., the pores is of the same magnitue as the iameter of the fiber. Nevertheless, with the presente moel the bon characteristics of ifferent fiber / matrix systems can be compare in much more etail, because the complete bon stress versus slip relationship τ(s) can be evaluate. Furthermore, if τ(s) is known, a tailoring of the interface properties by surface preparation with regar to an optimize mechanical performance of the composite material with enhance characteristics is possible. Future work on this moel coul inclue, for example, the implementation of effects ue to lateral pressure on the fiber, time epenent effects, an repeate loa cycles.

56 47 CHAPTER 4 BOND BETWEEN GLASS FILAMENTS AND A CEMENT BASED MATRIX In the previous chapters the cohesive interface moel which allows the etaile evaluation of the bon properties of a single fiber / matrix system was introuce (chapter 2) an verifie on the basis of pull-out tests on steel fibers (chapter 3). As the aim of this thesis is to characterize the bon of a stran / matrix system which is in turn preominate by the bon between the iniviual AR-glass filaments an the surrouning or penetrate fine-graine concrete (Fig. 1.1), the propose moel is now systematically aopte in this chapter to evaluate the BSR τ(s) between an AR-glass filament an a cement base matrix (subproblem I, chapter 1). As shown before, the principal input values for this moel are the loa versus isplacement relationship P(ω), the corresponing embee length L as well as the extensional stiffnesses of the filament an the matrix respectively (see Eq. (2.16) an Eq. (2.18)). Hence, in the following, the experimental testing techniques are introuce which are use to etermine these parameters; e.g. a single filament tensile test to etermine E F, optical microscopy to etermine A F, an a single filament pull-out test which allows the experimental ientification of the loa versus isplacement relation P(ω). Subsequently the cohesive interface moel is applie to erive the unerlying BSR τ(s) of the teste material combinations. 4.1 Introuction In 1966 scientists at the Builing Research Establishment (BRE) in the UK starte to evelop glass strans containing Zirconia (ZrO 2 ) which show a high resistance to glass corrosion in an alkaline meium, i.e. the attack by hyroxyl ions from the hyrating cement matrix; this prouct is now known as alkali resistant (AR) glass. Since that time, a variety of chemical compositions of mineral glasses have been use to prouce AR glass strans. Most of them are base on silica (SiO 2 ) with aitions of oxies an other constituents; see Table 4.1. Glass strans can be prouce either in continuous filament or staple form. The continuous glass filaments are generate from molten glass (about 155 C) by being rawn through small orifices (platinum-rhoium bushings). The resulting filament iameter is controlle by the orifice size, the rawing spee (approaching 1 to 2 m/min) an the viscosity of the molten glass. Immeiately after rawing, the filaments are rapily coole own an size.

57 48 4 Bon between glass filaments an a cement base matrix Table 4.1. Typical AR glass composition 1. Constituent Content % SiO Al 2 O 3-5 ZrO CaO -1 Na 2 O 1-15 TiO accoring to Vetrotech Saint-Gobain safety sheet ( The size layer aroun the stran which consists of organic proucts isperse in water, is esigne to give the glass stran certain characteristics necessary for final processing. Each size is specially esigne for a moling or compouning process an for a ifferent matrix type (e.g. resin or concrete) an usually contains a silane type chemical "coupling agent" which contributes to enhance the mechanical properties of composites an in particular their resistance to aging. After sizing, the filaments are loosely bone into strans. These strans usually contain 24 filaments with an approximate iameter of between 1 an 3 µm assemble in the form of a flattene bunle. If two or more strans are brought together, a socalle roving is prouce. Such a roving may contain over 2 iniviual filaments. Only a few meters after rawing an sizing the stran or roving respectively is woun up onto a spool an is then reay for further processing. To simplify matters a stran an a roving will not explicitly be istinguishe any more but will only be referre to as strans from here on. Due to technical reasons it is, so far, not possible to prouce a single filament or withraw a single filament irectly uring the prouction process escribe above. Hence, to perform tensile tests on filaments or pull-out tests on filament / matrix systems, single filaments have to be manually extracte from a stran for further specimen preparation. Such extracte filaments were use in recent years by many researchers to carry out tensile tests or pull-out tests to evaluate the tensile strength as well as the Young s moulus an bon characteristics between ifferent types of resin an filaments respectively. A review on these stuies is foun in [Shi97]. Some of the few stuies which pertain to pull-out tests on filament embee in a cementitious matrix are [Vek68, Hil85, Kat95a, Kat95b] but, no etaile information on the specimen preparation an the experimental methos coul be foun, an hence suitable testing proceures ha to be evelope uring this project from scratch. Information with regar to the preparation of samples of a polymeric matrix was consiere but coul not easily be transferre to the present problem of a filament embee in a cementitious matrix, ue to, for example, ifferent harening times of the resin, the general matrix composition, etc.

58 4 Bon between glass filaments an a cement base matrix Materials composition an specimen preparation AR-glass filaments The following tensile an pull-out tests are performe on filaments extracte from strans use within the collaborative research center SFB 532. See Table 4.2 for a listing. A variety of 5 ifferent types of AR glass strans from two ifferent manufacturers with ifferent filament iameters are use. The nomenclature use within the collaborative research center is aopte. For a etaile reference see [Gri2]. From here on, the stran types will be referre to by the names in column 1 of Table 4.2. Table 4.2. Stran types. Stran type Tensile Strength Tensile Moulus Failure strain N/mm² N/mm² % 1-NEG NEG-RO-ARG , NEG NEG-RO-ARG , NEG NEG-RO-ARG ,4 5 74, VET VET-RO-ARG o 2, 3 3, 4 73, VET VET-RO-ARG , 4 73, Nippon Electric Glass; 31, 11, an 24 tex alkali resistant glass roving respectively; prouction ate 2. Vetrotech Saint-Gobain; 32, an 24 tex alkali resistant glass roving respectively; prouction ate 21. without sizing accoring to the manufacturer ( accoring to the manufacturer ( The 3-NEG an 5-VET strans are 24 tex rovings, i.e. one km of this stran weighs 2.4 kg (linear weight). The ensity of the AR glass is, accoring to the manufacturers specifications, approximately 2.69 kg / m³, an hence the cross-sectional area of these strans can be calculate to be.889 mm²; likewise 11 tex (2-NEG) correspons to.49 mm², 32 tex (3-VET) to.119 mm², an 31 tex (1-NEG) to.115 mm² as cross-sectional areas of the ARglass strans. No etaile information on the composition of the size is given by the manufacturers. However, investigations within the SFB 532 showe that the main component of the size applie on the NEG strans is polyethylene glycol whereas the main component of the size applie on the VET strans is a silane [Hof1]. Aitional information gaine in extraction tests inicate that the quantity of size applie on the 5-VET strans is more than three times higher in comparison to all other strans [Hof1], which correspons to a layer thickness of aroun.25.5 µm. Note that no size at all has been applie on the 4-VET roving Fine-graine concrete As for the pull-out tests on steel fibers the filaments, which are extracte from all the above liste types of strans, are teste in combination with the previously introuce PZ fine-graine concrete (see also Table 4.3). To evaluate whether a ifferent matrix composition

59 5 4 Bon between glass filaments an a cement base matrix has any influence on the pull-out response, a secon composition of fine-graine concrete (FA-12-1) is introuce where, in comparison with the PZ matrix, a cement fraction of 28 kg/m³ has been replace by fly ash. Compare with the cement the fly ash features a larger particle size an ifferent harening characteristics. Both compositions are presente in Table 4.3. Table 4.3. Fine-graine concrete composition PZ an FA-12-1 accoring to [Bro1]. PZ FA-12-1 Constituent Content Content kg/m³ kg/m³ Biner CEM I 52,5 N Fly ash Silica fume Amixtures Superplasticizer 14 6 Aggregates Siliceous fine san San Accoring to [Bra3] both concrete mixtures show an extremely ense structure, but, the FA mixture offers a significantly coarser pore size istribution with pore raii >.1 µm in comparison with the PZ mixture. The Young s moulus has been etermine in compressive tests to be 35, N/mm² for the PZ an 24,8 N/mm² for the FA mixture respectively [Bra2] Specimen preparation tensile test Due to the fragility an brittleness of the filaments the specimen preparation requires a very careful hanling. A single filament is extracte from a stran an place in the mile between the upper an lower strip of a carboar winow. See Fig. 4.1 for illustration. Filament Cut Use for the evaluation of the filament iameter by optical microscopy. 1 mm Glue Carboar Cut Fig Pre-arrangement for tensile an pull-out tests on filaments.

60 4 Bon between glass filaments an a cement base matrix 51 The filament is fixe with ahesive an the carboar winow is fole in the center such that the filament rests in the fol itself (Fig. 4.1). The carboar allows an easy hanling an uring testing the easy clamping of the filaments in the testing machine (Fig. 4.4). The extening ens of the filament are cut off an use for a subsequent evaluation of the filament iameter by optical microscopy Specimen preparation pull-out test For the preparation of the pull-out test specimens the same filament carboar arrangement as escribe above is place in a mol consisting of a bottom layer of soft foam plastic, with a slit in the center to allow for a central positioning (see Fig. 4.2). Sie view Front view Fine graine concrete Formwork Foam plastic L Slotte hole Fig Specimen preparation for a filament pull-out test. Using a slotte hole, the outer formwork of the mol can be ajuste such, that ifferent embee lengths L can be caste. For the presente test series embee lengths between 1 mm L 2 mm are prepare. However, ue to the viscous consistency of the concrete the embee length to be reache is uner- or overfille an hence the actual L neee for the erivation of the BSR τ(s) with the cohesive interface moel has to be etermine for each specimen separately immeiately after testing. The specimens are compacte after casting for 6 sec. on a V-B table with a vibration amplitue of.5 mm an cure in the mol for 24 hours. The specimens are store at 2 C an 95% RH until testing at an age of 3 an 28 ays respectively. Compare with the storage conitions for the steel fiber pull-out test specimens, a higher humiity is chosen to prevent a rapi rying of the specimens, which woul certainly occur ue to their small imensions (thickness of a few mm) in a ryer climate an hence alter the boning characteristics Experiment sequences In Table 4.4 the material combinations teste with regar to the filament type, matrix composition, an storage conition are liste. All filaments extracte from the stran types which are liste in Table 4.2 are teste in combination with the PZ composition at an age of 3 ays after 1 ay curing in the mol an 2 further ays storage at 2 C an 95 % RH (combinations C1 to C5). Aitionally, filaments extracte from the 5-VET stran are teste at a total specimen age of 28 ays (1 ay in the mol an 27 ays at 2 C an 95 % RH) in combination with the PZ an FA-12-1 mixture (combination C6 an C7

61 52 4 Bon between glass filaments an a cement base matrix respectively). The values in brackets in Table 4.4 refer to the number of replications carrie out for each test series. Table 4.4. Experimental scheme for filament pull-out tests. PZ Stran type Storage conitions 1 ay mol + 2 ays at 2 C an 95 % RH 1-NEG C1 (6) 2-NEG C2 (8) 3-NEG C3 (8) 4-VET C4 (8) 1 ay mol + 27 ays at 2 C an 95 % RH FA ay mol + 27 ays at 2 C an 95 % RH 5-VET C5 (15) C6 (6) C7 (6) 4.3 Experimental methos Tensile test The tensile test is carrie out on specimens which are prepare in accorance to chapter Using a high-resolution close-loop DC-Mike by Physik Instrumente inc. featuring a precision of.5 µm, the test is isplacement controlle at a strain rate of.1 mm/mm/min at 2 C. Note that the testing length correspons to approximately 1 mm; see Fig Alligator crimps known from electrical engineering are use to clamp both ens of the specimen, see Fig However, the applie pressure by this clamp is not suppose to amage the filaments, an so lea to an early failure before the actual tensile strength is reache. Thus the pressure on the filament itself must be minimize an as a result the specimen might slip between the grips. To avoi such problems resulting unknowingly, a vieo extensometer is introuce to gain information on the Young s moulus E F an the elastic strain ε F (similar to chapter 3.3). Ientification winow Markings Fig Markings of the filament for ientification by the vieo extensometer for the tensile test.

62 4 Bon between glass filaments an a cement base matrix 53 As mentione before, this vieo-extensometer operates irectly as a non-contact isplacement measurement evice by etermining the change in istance between two targets. In the present case of the tensile test, two small roplets of white paint about 1/1 mm in iameter an approximately 2 to 3 mm apart are cautiously place on the filaments an use as targets; see Fig The fiel of view uring measuring is chosen to be 5 x 5 mm², an hence in the present case the precision of measurements is etermine as.25 µm. The effective istance between the two targets is evaluate also with the help of the vieo extensometer, which is calibrate before the tests by means of a calibration boy. This is possible, if the focal point of the optical system an hence the istance between the camera an the specimen stays the same for the calibration an the measurement, an hence a new, unknown istance can be calculate by rules of proportion. Immeiately before testing, the carboar is cut (Fig. 4.4) an a pre-loaing force is applie up to about.5 N to straighten the filament an hence reuce the possibility of the vieo extensometer losing its target because of an uncontrolle movement of the filament. Again the relative isplacement between the two targets is transferre into an analog voltage an recore simultaneously with the loae force, measure by a miniature loa cell (Sensotec, measuring range.5 to 2.5 N) at a rate of one reaing per secon until filament failure. Thus a force versus isplacement relation P(ω) for a tensile test is etermine. F F Alligator crimp Filament Cut Cut Cut L Free length Steel plate Fig F Loaing of the specimens uring tensile an pull-out tests Pull-out test In orer to examine the eboning an pull-out process of the filament, the pull-out specimen escribe in chapter is glue (X6) on its rear sie to a steel block, such that the extening part of the filament is place through a slot in the steel plate (Fig. 4.4). The steel block itself is fixe to the bench of the testing machine an the top en of the fiber is clampe with an alligator crimp, leaving a free length of a couple of millimeters. The pull-out test is again isplacement controlle with a high-resolution close-loop DC-Mike at a strain rate of.1 mm/mm/min at 2 C. Two LVDTs are use to aitionally measure the isplacement of the cross-hea of the testing machine (Fig. 4.5). If this cross-hea isplacement is use to gain information on the filament slip ω of the filament at the uppermost ege of the matrix at

63 54 4 Bon between glass filaments an a cement base matrix x = L which is neee for the evaluation of the BSR τ(s) with the cohesive interface moel (compare Fig. 2.1), the elastic strain contribution over the free length of the filament between the clamping an the point of intersection with the matrix has to be subtracte. However, unwante isplacements - e.g. ue to a slipping in the clamping - may again be mistakenly taken as introuce loa, an falsify the results. Thus, the relative isplacement of the filament at the point of intersection with the matrix is measure with the vieo extensometer, between a marking on the filament in form of a small roplet of white paint about a tenth of a millimeter above the matrix an a secon one on a reference plate which is fixe to the top of the concrete specimen. Again a isplacement winow of 5 x 5 mm² is chosen for the non contact measurement an the ata is recore at one reaing per secon resulting in a force versus isplacement relation P(ω) for a pull-out test. The complete test set-up is illustrate in Fig Note that the carboar an the filament at the rear (Fig. 4.4) are cut before the actual testing, an the filament is pre-loae, in a similar way to the tensile test with a loa of about.5 N. The actual embee length L (Fig. 4.4) is etermine by the use of a igital sliing calliper after the pull-out test at a metering precision of.5 mm right next to the embee filament. LVDT Loa Cell DC-Mike Drive Vieo - extensiometer Filament Matrix Restraint Reference plate P ω Fig Set-up for pull-out tests on filaments embee in a cement base matrix Optical microscopy A further important geometric parameter neee for the calculation of the Young s moulus E F an tensile strength f t of the filaments as well as for the analytical evaluation of the BSR τ(s) is the cross-sectional area A F of the filament efine by the filament iameter. As the prouction process of an AR-glass roving - see chapter oes not guarantee a uniformity of the filament iameters even within a single stran, has to be etermine for every single filament teste either in a tensile or in a pull-out test. Note that uring the preparation of the tensile as well as of the pull-out specimens, the en sections of the filaments which exten beyon the carboar (see Fig. 4.1) are use for the evaluation of to assure consistency between the iameter use for the analytical etermination of τ(s) an the filament iameter teste in the pull-out test. In this project optical microscopy in combination

64 4 Bon between glass filaments an a cement base matrix 55 with a computer aie image analyzing system is chosen to gain information on ; see Fig µ m Fig Determination of the filament iameter with optical microscopy. However, this metho carries a certain error potential. As shown in Fig. 4.6 the image of the filament visualize is surroune by an aurora an hence in many cases the iameter is not exactly efine. This error sensitivity increases when the iameter of the filaments gets relatively small, e.g. in the case of filaments extracte from the 1-NEG stran featuring a iameter of about 14 µm. Since an accuracy of only.5 µm can be achieve this correspons to a possible error of 3.4 % for the 1-NEG filament. 4.4 Test results Tensile tests on filaments As an illustration, the evaluation proceure to etermine the Young s moulus E F an the tensile strength f t of the filaments is emonstrate for one filament extracte from a 1-NEG stran. The output from the tensile test is a loa versus isplacement relation P(ω) as shown in the left han sie graph of Fig Loa P in N Displacement ω in mm Stress σ in N/mm² Strain ε in mm/m Fig Loa versus isplacement relation an stress versus strain relation for a 1-NEG filament uner tension. Note, that P(ω) starts at a loa level of.5 N because the filament is pre-loae for reasons liste in chapter The corresponing iameter is etermine by optical microscopy as

65 56 4 Bon between glass filaments an a cement base matrix = 14.7 µm an the test length with the vieo extensometer as L = 3.3 mm. Using this information a stress versus strain relation σ(ε) can be calculate as presente in the right han sie graph of Fig As has been expecte, this relation is linear until failure. Hence with a simple least squares metho the slope an therefore the Young s moulus is foun. For the given experiment E F is evaluate as about 65, N/mm² an the tensile strength as approximately f t = 2,62 N/mm². However, these test results inclue a certain error accumulation. In this example the maximum loa occurs at a isplacement of about 8 µm an the non contact isplacement measurement by the vieo extensometer has an accuracy of about.25 µm, which therefore correspons to an error of.32 % (see chapter 4.3.1). The inaccuracy of the measure testing length may be assume with.8 % as negligible, in contrast to a 3.4 % inaccuracy cause by the etermination of the filament iameter (see chapter 4.3.3). Thus, by rule of error accumulation a total testing imprecision of 3.7 % for the presente filament type might be assume. Each test series of a certain filament type contains a number of 2 tests. Only those tests which neither have faile at the clamping nor at the paint markings are taken for a further evaluation. For all following parameters a Gaussian istribution is assume an checke with the Kolmogorov-Smirnov Test. In all cases a high enough significance level is foun not to reject the hypothesis, which implies that the mean average values an the relative stanar eviations may be liste to represent the experimental ata. Table 4.5 shows the average of the etermine tensile strengths an Young s mouli for each filament type of Table 4.2 with the corresponing relative stanar eviation. Aitionally, the average of the measure filament iameters with the corresponing relative stanar eviation as well as the particular testing imprecision are liste. Table 4.5. Experimental results of the tensile tests on filaments (see Table 4.2). Number of Tensile strength Young s moulus Filament iameter specimens analyse f t E F a Average r.. 1 Average r.. 1 Average r.. 1 Testing imprecision at f t - N/mm² % N/mm² % µm % % 1-NEG , NEG , NEG , VET , VET , r.. = relative stanar eviation A comparison shows that only filament type 1-NEG shows a stanar erivation variability for the tensile strength of smaller than 1%. Tests on E-glass filaments showe [Zin99], that sample sizes of about 35 4 are necessary to achieve constant values of average tensile

66 4 Bon between glass filaments an a cement base matrix 57 strengths an corresponing relative stanar eviations. However, these stanar eviations are of the same magnitue as evaluate in the present stuy. In principle the finings liste in Table 4.5 are in fair agreement with the results given by the manufacturer (Table 4.2). The observe iscrepancies between the material strength of the 4- VET an 5-VET strans which have been specifie both by Vetrotex Saint-Gobain an evaluate uring this thesis, perhaps are cause by the ifferent implie loaing rates uring the test. Whereas Vetrotex carries out tensile tests at a rate of aroun.1 to 4. mm/mm/min the chosen loaing rate in this stuy is.1 mm/mm/min to ensure conformance between tensile an pull-out tests. However, one exception is the filament extracte from the 5-VET stran. The Young s moulus foun in the experiment an subsequent analysis is far below the expecte value of about 68, 73, N/mm² for AR glass. This may have a simple reason: accoring to Vetrotex Saint-Gobain the filament iameter of the filaments are suppose to be in a range of 26 ± 1.5 µm. But as Table 4.5 column 5 shows that the values etermine in this stuy are about 28.8 µm. This may be cause by the size applie uring the prouction process. This aitional layer certainly oes not contribute to the loaing an is taken wrongly as extra filament area; compare chapter On the filaments extracte from all other strans only one thir of that quantity of size has been foun [Hof1]. Hence similar effects but in a much smaller scale may exist for these filament types as well. Nevertheless, the Young s mouli are aopte as etermine an presente in Table Pull-out tests on filaments To maintain clarity in the following iagrams, which specify the results of the pull-out tests carrie out accoring to the experimental scheme liste in Table 4.4, only 3 representative loa versus isplacement relations are given. Note that the corresponing embee length L an iameter for each performe test are also presente in the iagram. Note further that ue to these ifferences in embee length an filament iameter the liste loa versus isplacement curves cannot be compare irectly. As explaine before, the irregularities in the embee length are a consequence of the casting process an the varieties in the filament iameter are cause by the manufacturing process of the stran. Aitionally the average maximum tensile loas an the corresponing stanar eviations which have been calculate on the basis of the average tensile strengths (Table 4.5, columns 3, 4) an the average iameters (Table 4.5, columns 7, 8) of the teste types of filaments are presente in the iagrams as a failure range (gray shae area). Similar to the previous chapter 3.4 the simulate P(ω) relations are plotte as symbols in each iagram as well. As will be explaine in chapter 4.5, these simulate loa versus isplacement curves are base on a BSR τ(s) evaluate with the help of the previously propose cohesive interface moel. Fig. 4.8 shows three loa versus isplacement relations P(ω) etermine for combination C1 (1-NEG filament / PZ matrix, 3 ays age at testing). Analogous to the steel fiber pull-out test results in chapter 3.4 it may be state that all of the presente P(ω) curves show

67 58 4 Bon between glass filaments an a cement base matrix the tren generally foun in pull-out tests: An increasing although nonlinear part, followe by a ecreasing softening branch an finally by a fae out controlle by friction. During this fae out the filaments are pulle through the matrix an hence the frictional bon resistance stays about constant although some minor changes can be etecte, possibly on account of abrasion affects. Pull-out loa P in N Failure range of tensile tests = 16.7 µm, L = 1.4 mm Fig = 13.7 µm, L = 1.5 mm Slip s in mm = 13. µm, L = 1.5 mm Simulation: = 16.7 µm, L = 1.4 mm = 13. µm, L =1.5 mm = 13.7 µm, L = 1.5 mm Loa versus active fiber en isplacement iagram P(ω) for 1-NEG filaments of ifferent filament iameters an embee lengths (see legen), combination C1 Table 4.4; Experiment (lines) an simulation (ots) base on BSR τ(s) state in Fig Testing age is 3 ays. Note that the embee lengths L of the tests are 1.4 to 1.5 mm which is only about three times the maximum grain size of the fine-graine concrete. However, as can be seen in Fig. 4.9, the bon between a filament of type 2-NEG an the PZ matrix (concrete age 3 ays) is alreay goo enough to transfer a force (Table 4.5) within the given embee length into the surrouning concrete which correspons to its tensile strength. Consequently, this iagram shows only the increasing part of the pull-out process, until the tensile strength of the filament is reache an tensile failure occurs. For the liste pull-out tests the failure loas of the 2-NEG filaments are in the lower part of the failure range calculate on the basis of the results of the previously liste tensile tests. A possible explanation might be the lateral pressure on the filaments by the surrouning fine-graine concrete which aitionally stresses the material an thus leas to a reuce strength. Furthermore, possible notches on the outsie of the filaments in combination with intrue hyration proucts of the cement paste may lea to a reuce tensile strength. Two of the presente 3-NEG filaments (Fig. 4.1) an all liste 4-VET filaments (Fig. 4.11) break because their tensile strength is reache uring the pull-out test at a specimen age of three ays, an hence no shear bon failure can occur. Only one 3-NEG filament is actually pulle out of the matrix.

68 4 Bon between glass filaments an a cement base matrix 59 Pull-out loa P in N.6.4 = 2. µm, L = 1.8 mm = 16.5 µm, L = 1.8 mm Failure range of tensile tests.2. = 18.5 µm, L = 1.5 mm Slip s in mm Simulation: = 2. µm, L = 1.8 mm = 16.5 µm, L = 1.8 mm = 18.5 µm, L = 1.5 mm Fig Loa versus active fiber en isplacement iagram P(ω) for 2-NEG filaments of ifferent filament iameters an embee lengths (see legen), combination C2 Table 4.4; Experiment (lines) an simulation (ots) base on BSR τ(s) state in Fig Testing age is 3 ays. Pull-out loa P in N = 22.7 µm, L = 1.8 mm Failure range of tensile tests.4.2. = 21.9 µm, L = 1.8 mm Slip s in mm = 26. µm, L = 1.7 mm Simulation: = 22.7 µm, L = 1.8 mm = 26. µm, L = 1.7 mm = 21.9 µm, L = 1.8 mm Fig Loa versus active fiber en isplacement iagram P(ω) for 3-NEG filaments of ifferent filament iameters an embee lengths (see legen), combination C3 Table 4.4; Experiment (lines) an simulation (ots) base on BSR τ(s) state in Fig Testing age is 3 ays. A totally ifferent response is observe, if a 5-VET filament is pulle out of the PZ matrix after 3 ays; see Fig In contrast to all other test results which have been presente so far, no ahesional bon seems to evelop between the filament an the surrouning matrix, an hence the pull-out behavior is preominate only by frictional bon which results in the presente steay-going an constant P(ω) istribution. One aspect not consiere in the results presente so far is that the boning characteristics may change over time because of e.g. the ongoing formation of hyration proucts in the interphase between filament an matrix. For steel fiber matrix systems this has been observe up to an age of 28 ays; hereafter the BSR is foun to be time inepenent.

69 6 4 Bon between glass filaments an a cement base matrix Pull-out loa P in N.4 = 14.9 µm, L = 1.4 mm.3.2 = 15.5 µm, L = 1.65 mm Failure range of tensile tests.1. = 13.1 µm, L = 1.4 mm Slip s in mm Simulation: = 14.9 µm, L = 1.4 mm = 15.5 µm, L = 1.65 mm = 13.1 µm, L = 1.4 mm Fig Loa versus active fiber en isplacement iagram P(ω) for 4-VET filaments of ifferent filament iameters an embee lengths (see legen), combination C4 Table 4.4; Experiment (lines) an simulation (ots) base on BSR τ(s) state in Fig Testing age is 3 ays. Pull-out loa P in N Failure range of tensile tests = 28.8 µm, L = 1.7 mm.4.2 = 26. µm, L = 1.7 mm. = 26. µm, L = 1.6 mm Slip s in mm Simulation: = 28.8 µm, L = 1.6 mm = 26. µm, L = 1.7 mm = 26. µm, L = 1.6 mm Fig Loa versus active fiber en isplacement iagram P(ω) for 5-VET filaments of ifferent filament iameters an embee lengths (see legen), combination C5 Table 4.4; Experiment (lines) an simulation (ots) base on BSR τ(s) state in Fig Testing age is 3 ays. To investigate possible changes in the BSR τ(s) as a result of a changing microstructure in the transition zone, the same filament type 5-VET is also examine in combination with the PZ mixture at a specimen age of 28 ays; 1 ay curing in the mol an 27 ays storage at 2 C an 95 % RH. The results are presente in the familiar form in Fig That which is later confirme by the etermination of the unerlying BSR τ(s) can alreay be anticipate by a comparison of the tests results presente in Fig an Fig. 4.13, which refer to the P(ω) relationships of 5-VET filaments pulle out of a PZ matrix at 3 an 28 ays respectively: The bon characteristics for this combination o not change within this time perio an presumably not thereafter.

70 4 Bon between glass filaments an a cement base matrix 61 Pull-out loa P in N Failure range of tensile tests = 27.8 µm, L = 1.5 mm.4.2. = 26.9 µm, L = 1.4 mm = 26.1 µm, L = 1.7 mm Slip s in mm Simulation: = 27.8 µm, L = 1.5 mm = 26.9 µm, L = 1.4 mm = 26.1 µm, L = 1.7 mm Fig Loa versus active fiber en isplacement iagram P(ω) for 5-VET filaments of ifferent filament iameters an embee lengths (see legen), combination C6 Table 4.4; Experiment (lines) an simulation (ots) base on BSR τ(s) state in Fig Testing age is 28 ays. Fig shows the output of 3 pull-out tests on a 5-VET filament / FA-12-1 matrix system after 28 ays of storage (combination C7 in Table 4.4). The etermine steay-going an constant P(ω) istributions as well as a comparison with the results of combination C6 (Fig. 4.13) inicate that the pull-out behavior is preominate by a frictional bon, an that the general boning characteristics of these two combinations are probably very similar. Pull-out loa P in N Failure range of tensile tests = 26.6 µm, L = 1.6 mm = 29.6 µm, L = 1.7 mm = 27. µm, L = 1.6 mm Slip s in mm Simulation: = 27. µm, L = 1.6 mm = 26.6 µm, L = 1.6 mm = 29.6 µm, L = 1.7 mm Fig Loa versus active fiber en isplacement iagram P(ω) for 5-VET filaments of ifferent filament iameters an embee lengths (see legen), combination C7 Table 4.4; Experiment (lines) an simulation (ots) base on BSR τ(s) state in Fig Testing age is 28 ays.

71 62 4 Bon between glass filaments an a cement base matrix 4.5 Bon stress versus slip relations τ(s) To analyze the afore liste pull-out test results an thus allow a comparison of the ifferent responses, the unerlying BSR τ(s) has to be etermine for each evaluate P(ω) relation, an a best fit for all experimental results for a given filament / matrix combination has to be foun. The proceure which is use to erive this bon stress versus slip istribution is ientical to that which has been escribe before for the steel fiber pull-out tests; hence chapter 3.5 shoul be referre to for etails. The BSR τ(s) evaluate for a 1-NEG filament / PZ concrete system is shown in Fig The corresponing pull-out tests can be foun in Fig Note that the average Young s moulus of 66,881 N/mm² (Table 4.5) an the iniviually evaluate filament iameters of the teste filaments as well as the measure embee lengths are use uring the erivation. Aopting this BSR the pull-out response of the 1-NEG filaments for the given filament iameters an embee lengths L can be simulate. These simulations are presente in Fig. 4.8 an are inicate by ifferent shape symbols in the iagram. The progressions of the P(ω) relations of the combinations = 16.7 / L = 1.4 an = 13. / L = 1.5 are well simulate. However, the response for the combination = 13.7 / L = 1.5 is overestimate. This may be ue to the fact that pores or vois influence the contact area between the filament an the surrouning matrix. Similar stochastic size effects have been observe for the steel fiber / cement base matrix system for small fiber iameters (chapter 3.6). Bon stress τ in N/mm² 8 6 NEG-RO-ARG Slip s in mm 1-NEG / PZ / 3 ays i s τ - mm N/mm² Fig Evaluate bon stress versus slip relation τ(s) for the material combination 1-NEG filament / fine-graine concrete (PZ-899-1); combination C1 Table 4.4. Bon conitions after 3 ays. In a similar proceure the unerlying BSR τ(s) is evaluate for the 2-NEG / PZ combination an presente in Fig The associate pull-out tests are presente in Fig Due to the fact that the pull-out process coul only be recore in the increasing part of the P(ω) relationship until the filaments reache their tensile strength, the corresponing an

72 4 Bon between glass filaments an a cement base matrix 63 unerlying BSR τ(s) coul only be irectly evaluate as far as i = 4 (see table in Fig. 4.16, row 4). The remaining progression is aapte from the BSR evaluate for combination C1. Bon stress τ in N/mm² 8 6 NEG-RO-ARG Slip s in mm 2-NEG / PZ / 3 ays i s τ - mm N/mm² Fig Evaluate bon stress versus slip relation τ(s) for the material combination 2-NEG filament / fine-graine concrete (PZ-899-1); combination C2 Table 4.4. Bon conitions after 3 ays. The simulate pull-out responses for the ifferent liste fiber iameters an embee lengths L are base on the BSR erive earlier. They are inicate again by ifferent shape symbols an liste together with the corresponing experimental P(ω) relations (see Fig. 4.9). In the presente cases the simulations fit the experiments quite well although only the increasing parts of the P(ω) relations are experimentally recore. Evaluating the unerlying BSR for combination C3 (3-NEG / PZ-899-1) reveals the same relationship as that euce for the 2-NEG filament / PZ system (Fig. 4.17). The simulations base on this τ(s) relation are again introuce as ifferently shape symbols in the experimental P(ω) iagram presente in Fig In general a goo consistency with the experimental results is foun. Bon stress τ in N/mm² 8 6 NEG-RO-ARG Slip s in mm 3-NEG / PZ / 3 ays i s τ - mm N/mm² Fig Evaluate bon stress versus slip relation τ(s) for the material combination 3-NEG filament / fine-graine concrete (PZ-899-1); combination C3 Table 4.4. Bon conitions after 3 ays.

73 64 4 Bon between glass filaments an a cement base matrix So far all evaluate BSR τ(s) refer to filaments which were size right after the rawing process. Hence, the evaluate bon characteristics are possibly influence by this aitional film on the glass surface. On the 4-VET filament however no size has been applie an thus the erive bon stress versus slip relation pertains to a pure AR-glass / cement base matrix bon. Fig shows the erive τ(s) relation for combination C4 (4-VET filament an PZ matrix). Unfortunately, only the ascening branch of the P(ω) relations coul be experimentally recore again because the tensile strength of the filaments was reache before a bon failure coul occur. Thus the BSR state in Fig coul only be irectly evaluate as far as i = 4. Nevertheless, the maximum bon strength τ cr (i = 3) coul be etermine. The resiual part of the BSR is again aapte from the τ(s) relationships which have been evaluate for the afore state combinations which might be an amissible assumption, in this case, ue to the similarity of both etermine initial parts of τ(s) (compare Fig an Fig. 4.18). The simulations base on this BSR can be foun in Fig which generally again show a goo consistency with the experimental results. Bon stress τ in N/mm² 8 6 VET-RO-ARG o Slip s in mm 4-VET / PZ / 3 ays i s τ - mm N/mm² Fig Evaluate bon stress versus slip relation τ(s) for the material combination 4-VET filament / fine-graine concrete (PZ-899-1); combination C4 Table 4.4. Bon conitions after 3 ays. The assumption of the bon between a 5-VET filament an a fine-graine concrete PZ lacking ahesion after 3 ays of harening is true, if the unerlying BSR τ(s) is evaluate with the help of the cohesive interface moel; see Fig The resulting bon stress τ is almost inepenent of the slip s. This inicates the formation of a constant frictional bon between the filament an the matrix. The simulate P(ω) response base on this BSR is again presente by ifferent shape symbols in Fig The simulations show a goo agreement with the experimental results for all 3 of the teste filament iameter an embee length combinations.

74 4 Bon between glass filaments an a cement base matrix 65 Bon stress τ in N/mm² 8 6 VET-RO-ARG Slip s in mm 5-VET / PZ / 3 & 28 ays i s τ - mm N/mm² Fig Evaluate bon stress versus slip relation τ(s) for the material combination 5-VET filament / fine-graine concrete (PZ-899-1); combination C5 an C6 Table 4.4. Bon conitions etermine at 3 an 28 ays respectively. An ientical BSR τ(s) is foun for the material combination C6 (Fig. 4.19). This emonstrates the time inepenent behavior of the boning characteristics at least for the investigate perio an material combination. The simulations base on the propose BSR an those presente in Fig show a goo consistency with the experimental results. The principal boning characteristics of the two material combinations C6 an C7, for both, the 5-VET filament / PZ concrete system an the 5-VET filament / FA-12-1 concrete system, are also very similar at a testing age of 28 ays (compare Fig an Fig. 4.2). Bon stress τ in N/mm² 8 6 VET-RO-ARG Slip s in mm 5-VET / FA-12-1 / 28 ays i s τ - mm N/mm² Fig Evaluate bon stress versus slip relation τ(s) for the material combination 5-VET filament / fine-graine concrete (FA-12-1); combination C7 Table 4.4. Bon conitions after 28 ays. These two graphs (Fig an Fig. 4.2) show that no ahesional bon is eveloping an thus the interfacial properties are preominate by friction. The frictional bon between 5- VET filament an FA-12-1 matrix is about 15 % lower compare with the frictional bon between 5-VET filament an PZ matrix.

75 66 4 Bon between glass filaments an a cement base matrix 4.6 Discussion The pull-out tests on filaments presente above which have been extracte from glass strans an embee in a fine-graine concrete, as well as the subsequent analysis with the cohesive interface moel to evaluate the unerlying BSR τ(s), are carrie out in this stuy for the first time. However, the experimental results an the corresponing boning characteristics inicate that the propose algorithm is suitable for etermining the interfacial properties of ifferent combinations of filaments an cement base matrices. The scatter in the pull-out test results presumably reflects the ifficulty in preparing reproucible specimens because, for example, the hanling of fragile filaments with small iameters is quite complicate an concrete is an inhomogeneous material, rather than the inaccuracy of the test an evaluation methos themselves. In contrast to the outputs of the pull-out tests carrie out on steel fiber / fine-graine concrete combinations (chapter 3.4) where only geometric parameters such as fiber iameters an embee lengths were change an an average BSR τ(s) was foun vali for all combinations, the teste glass filaments, which are prouce by 2 ifferent manufactures, iffer aitionally in the type an amount of the size. See also chapter As mentione before, the composition of the size is not publishe as it is a carefully kept secret of each manufacturer. However, the main component of the 3-NEG stran size was foun to be polyethylenglycol [Hof1] which is water soluble an hence probably issolves uring the casting process of a single extracte filament in a cement base matrix. This is backe up by the fact that an ientical BSR to that for the 3-NEG filament is ientifie for the 4-VET filament, which has been prepare without any size; although only the increasing part an the first point of the softening branch of this BSR coul be ientifie for certain, since the filaments reache their tensile strength before a pull-out an thus bon failure coul occur. Aitionally the unerlying BSR for the combination 2-NEG filaments an PZ matrix was also foun to be alike to those for the above mentione 2 filament types even though it is known that the type of size is subject to change between prouction charges an ifferent stran types of the same year. The τ(s) relationship referring to the bon between a 1-NEG filament an the PZ matrix is similar although not ientical to those corresponing to the 3 filament types mentione above. The main ifference is the maximum bon strength τ cr reache, which is significantly lower for the filament of the lower tex stran. Whether this is cause by the smaller iameter an resulting statistical size effects or by a ifferent size use uring the prouction process of the stran coul not be verifie within the scope of this stuy. An entirely ifferent pull-out behavior is foun for the 5-VET filaments embee in a PZ matrix an store uner ientical conitions (combination C5). This is likely to be ue to an application of a silane type chemical coupling agent in the prouction process of the strans which is not water-soluble. This aitional relatively thick layer (see

76 4 Bon between glass filaments an a cement base matrix 67 chapter 4.2.1) between the glass surface an the cement base matrix an the fact that the applie size swells if place in contact with water might prevent the formation of an ahesion bon. Aitional investigations on the time epenent behavior of the interfacial properties show that the etermine BSR oes not change between 3 an 28 ays. This is in contrast to finings for steel fiber / cementitious matrix combinations, where the boning characteristics are influence by the formation of aitional hyration proucts up to an age of 28 ays an only thereafter are foun to be time inepenent. Whether this is ue to the applie size whose material properties presumably o not change over time an which causes the composite to fail in this aitional layer, or ue to other effects coul not be investigate within the framework of this stuy. However, it may be state here, that the filaments pulle out i not show any resiual cement fractions on their surface. They appeare to have faile not actually in the interphase or transition zone of the matrix as propose by many authors but actually in the interface, or the size layer. A comparison between the BSR for ifferent combinations of fiber an matrix, e.g. steel fiber / PZ (chapter 3.5) an 4-VET / PZ-899-1, shows, that the ahesion bon between steel an concrete is weaker than between AR-glass an concrete. In the present case the maximum bon strength between steel an concrete is foun to be τ cr = 2.4 N/mm² whereas between AR-glass an concrete it is τ cr = 5.8 N/mm². Uner the assumption of a constant shear stress istribution over the embee length L, [Vek68] evaluate a maximum bon strength of τ cr = 6.38 N/mm² for an unsize E-glass filament of 5 µm iameter embee in a Portlan cement paste with a water / cement ratio of.3. Although the matrix an filament composition is ifferent, the magnitue of the bon strength is very similar. Bon stress τ in N/mm² 8 Steel fiber - PZ (28) 4-VET - PZ (3) 6 5-VET - PZ (3 & 28) Slip s in mm Fig BSR τ(s) etermine for a steel fiber / fine-graine concrete system, a 4-VET / PZ system an a 5- VET / PZ system.

77 68 4 Bon between glass filaments an a cement base matrix However, if a non water-soluble silane base size is applie on the filaments investigate (5- VET), the ahesion bon is broken but the frictional bon is foun stronger in comparison with the steel fiber / cementitious matrix system. A further aspect which has to be consiere if the bon between AR glass filaments an a fine-graine concrete matrix is concerne might be the stochastic size effects which were introuce an explaine in etail for a steel fibers / cement base matrix systems in chapter 3.6. The variations of the pull-out test results which have been observe in the previous chapters are presumably cause by the ranomly istribute voi, pore an grain istribution in the filament pull-out specimens. Remember that the average embee length of 1.5 mm in the presente test series is only about 2.5 times the maximum grain size of the matrix. As a result any ajacent pores an grains in the transition zone between filament an matrix may have a lasting effect on the boning characteristics etermine uring the pull-out tests an the subsequent analytical erivation of the BSR τ(s) because the reuce contact area is not consiere in the evaluation. To account for this effect in the moel, the ranom istribution of these vois, imperfections, grains an pores in the matrix an explicitly in the transition zone has to be ascertaine for each pull-out test iniviually, which is a complex an expensive task. See chapter Summary By using an enhance single filament pull-out test, which allows the straightforwar evaluation of P(ω) relations, an the application of the cohesive interface moel in form of the inverse bounary value problem, average bon stress versus slip relationships τ(s) are erive (Table 4.4) for ifferent filament / fine-graine concrete combinations which efine the bon of the composite at an age of 3 an 28 ays. The etermine BSR s are in turn use to simulate the experimental response of filaments, an a goo agreement for the majority of the teste combinations is foun. Hence experimental tests an analytical as well as numerical tools to etermine an evaluate the principal boning characteristics an mechanisms (chapters 2 an 3) between a filament an a surrouning cement base matrix have been elaborate (sub-problem I, chapter 1). It is confirme from this stuy that the size applie on the stran uring the prouction process consierably influences the quality of the bon. This size can lea to a significant egraation of the interfacial properties, if it prevents the formation of an ahesional bon between the glass surface of the filament an the surrouning cementitious matrix. One of the questions which has been not consiere in this work is whether or not, for example, the interfacial properties are unique to an iniviual fine-graine concrete type, an hence also epenent on for eample the grain-size istribution, the water / biner ratio, the cement type, the curing conitions an so on. This is a possible fiel for further investigation an stuy.

78 69 CHAPTER 5 BOND BETWEEN GLASS STRANDS AND A CEMENT BASED MATRIX In the previous chapter 4 the principal boning characteristics between iniviual AR-glass filaments an ifferent cementitious matrices uner varying storage conitions were evaluate an expresse by bon versus slip relations. Although the bon of a stran in a cementitious matrix is certainly preominate by the bon properties between filaments an matrix, more etaile information is neee to evaluate the failure mechanisms of such a complex system uner a pull-out loa an hence to allow an analytical an numerical simulation of this composite. Thus, in this chapter ifferent innovative test methos are evelope an use to ientify the eboning process of a stran as a result of the pull-out process an ascertain the contact faces between the iniviual filaments an the matrix (sub-problem II, chapter 1). Aitionally, numerical proceures are propose which use the finings to allow, for the first time, a irect relationship between the loa history of a pull-out test, to the failure process of an AR-glass stran, by means of a mathematical function (sub-problem III, chapter 1); the socalle active filament versus isplacement relation N F (Ω). Together with the loa versus isplacement relationship P(Ω) also erive uring the pull-out test an the results of chapter 4, an analytical characterization an simulation of the bon between an AR-glass stran an a cement base matrix will be achieve in chapter Introuction Various mechanical tests exist toay to etermine the parameters of interfacial action an thus the principal boning mechanisms between an AR-glass stran an a cementitious matrix. Nevertheless, no stanar metho has as yet been recommene or set for the etermination of these parameters, although all these tests can generally be classifie in two major groups. The first group (I) involves one sie pull-out tests (e.g. [Bar82]) where the embee stran is pulle out off the specimen against a restraint while recoring the introuce isplacement an loa P(Ω) an the secon group concerns (II) ouble sie pull-out tests (e.g. [Maj74]) where an artificial crack brige by one or more strans is prouce in the specimen by iviers an the crack extension w is recore together with the loa P applie at the faces of the cast concrete blocks. In recent years both tests have been use, moifie an enhance by ifferent researchers to stuy the pull-out response of strans. The ouble sie an the one sie test set-up show

79 7 5 Bon between glass strans an a cement base matrix avantages an isavantages with regar to the quality of the information gaine from the experimental results, i.e. either the loa versus crack extension relation P(w) or the loa versus cross hea isplacement relation P(Ω). If the outputs of the ouble sie pull-out test are evaluate, it has to be unerstoo that uring the eboning stage an in the case of a high frictional bon or mechanical anchorage, the obtaine isplacement is a result of the extensions on both sies of the specimen. In such a case the recore pull-out curve woul epen on isplacements that are mixe from each sie of the specimen uring the initial part of the curve an influence by the failure of iniviual filaments on both sies. Certainly this test set up is closer to the processes occurring in a real structure where a crack is brige by a reinforcing element, but for research purposes one nees a efine failure process. Therefore it may be not reasonable. The one sie pull-out test on the other han implies some uncertainties as well. Many of the one sie pull-out tests carrie out so far (among others [Bar82, Pel]) inclue a free length between the point of embement an the loa introuction. Hence, elastic strain energy is store uring the pull-out test in the system, which results in a moifie failure process. Aitionally it has to be ensure that at the point of loa introuction every single filament is clampe an no slip occurs. Uner consieration of the above state aspects the one sie pull-out test is chosen an moifie in this stuy in orer to gain information on the pull-out response P(Ω) which is neee as one main input parameter for the analytical moeling of the composite which will subsequently be carrie out. However, the etermine P(Ω) relationship is not sufficient to characterize the actual failure process of the stran / cement base matrix system an therefore other experimental techniques are introuce within this stuy to ientify the onset an subsequent stages of failure of the reinforcing filaments. Unfortunately many of the methos which have been propose in recent stuies to investigate the failure of composite materials are not applicable in this stuy for numerous reasons. On the one han most of the introuce techniques o not offer the ability to visualize the exact location of a filament failure or eboning process because of an inaequate resolution, for example acoustic emission analysis [Gro94], the Impact-Echo technique, or the air-couple ultrasoun metho [Pre2]. On the other han other methos are not practical for an online testing, for example lock-in-thermography, vibration- an moal analysis [Ebe99, Pre2] or laser scanning microscopy (LSM). Thus no functional relationship between the loa history an the failure process of the composite can be establishe. However, the iea of the latter technique is exploite in this stuy to etermine the final filament slippage at the rear of the specimen an thus gain information on the plastic strain of the filaments at the en of the pullout test. In 21 [Tof1] an [Pra1] came up with the iea to monitor the electrical continuity an resistance of electrically conuctive coate glass filaments online uring tensional tests. The results showe that a straight forwar relationship between the increasing resistance an the

80 5 Bon between glass strans an a cement base matrix 71 number of filaments failing uner tension exists. Unfortunately this test can only be performe on fibers which are electroconuctive or have been coate with an electrically conuctive layer. However, such a coating oes change the boning properties an consequently the pull-out response of the composite, an thus this technique is not applie in this stuy. Although electrical conuctivity has been eliminate there exists another property of glass filaments which may be use for online measuring: that is they transfer light. This feature has recently been use to buil fiber optic strain sensors [Hab]. During this project a new technique is introuce, which uses these light transmission properties to set up a test scheme, to allow an online measurement of the failure mechanisms occurring uring a stran pull-out. Because this test investigates the failure mechanisms of a stran by light transmission, it is calle the FILT test (Failure Investigation using Light Transmission properties). On the basis of these test results an active filament versus isplacement relationship N F (Ω) can be erive, which allows a straightforwar relation between the loa history of a pull-out test P(Ω) an the failure process of an AR-glass stran to be establishe. To gain some aitional information on the structural features of an AR-glass stran / cement base matrix system, scanning electron microscopy (SEM) is applie in this stuy. However, these investigations are only use to back up certain assumptions or complete the finings from the FILT test an LSM analysis. A more etaile quantitative escription of the microstructure of the composite is introuce later on in chapter Materials composition an specimen preparation AR-glass strans The pull-out tests liste in the following are performe with Vetrotech Saint-Gobain 24 tex alkali resistant glass rovings prouce in 21 (5-VET, see chapter 4.2.1). The strans teste are taken straight from the spool an caution is pai not to contaminate the filaments with ust or later on with formwork release oil. The cross-sectional area of the strans can be calculate to be.889 mm². Using the average iameter etermine in chapter an liste in Table 4.5 the number of filaments per stran N F,m can be calculate to be approximately Fine-graine concrete All pullout tests are carrie out using the previously introuce PZ matrix (see chapter an Table 4.3) Specimen preparation pull-out test For specimen preparation the glass stran is firstly place in a polymer plastic mol (Rhoorsil RTV V-34) an cast in epoxy resin over a length of 3 mm an a cross-sectional 2 area of 1 1 mm centric. This block offers high protection for the fragile filaments against an early an uncontrolle failure cause by clamping later on in the pull-out test. Care must

81 72 5 Bon between glass strans an a cement base matrix be taken that, ue to capillary effects, the resin oes not penetrate the stran over a greater length. This epoxy resin block is use in the following pull-out tests for loa introuction an guarantees the same introuce isplacement on all iniviual filaments of the stran. In SEM investigations the uniform penetration of the epoxy resin into the interior of the stran was verifie. Stran embee in epoxy resin block 1 mm 3 mm 5 mm Fine-graine concrete L 5 mm Fig Specimen for the one sie stran pull-out test. The stran is then cast in a fine-graine concrete matrix. The imensions of the specimens are L mm with L being the embee length of the stran. L is chosen for the presente tests to be 3 mm (Fig. 5.1). The stran is situate horizontally in the mol an extens on the rear sie about by 3 mm through a hole which is seale with silicon paste before casting. A slight pre-stress is applie to align the stran. On the front sie the concrete is caste straight against the epoxy block which itself is place in a cut-out of the mol. Hence a zero free length of the stran is accomplishe. After casting, the specimens are compacte for 6 secons on a V-B table with a vibration amplitue of.5 mm an cure in the mol for 24 hours. After emoling the extening filaments on the rear sie of the concrete block are cut off. The specimens are store at 2 C an 95% RH until testing at a total age of 28 ays Experimental sequences Only one material combination is closely investigate with regar to the pull-out response an failure mechanisms because the testing proceure introuce below is rather complex. A 5- VET stran / PZ fine-graine concrete system is chosen as this combination buils the so-calle backbone in the collaborative research center SFB 532 (series 12); that is most of the chemical investigations, structural experiments an numerical analyses are carrie out on this combination in the relevant projects. All tests are carrie out on one series with three replications; specimens A to D Testing proceure In orer to examine the complex failure mechanism of a stran / cement base matrix composite, the pull-out an FILT tests as well as the investigations by LSM an SEM are utilize as outline in the following testing proceure (Table 5.1).

82 5 Bon between glass strans an a cement base matrix 73 Table 5.1. Testing Proceure. Step Time Specification Results 1 Sample preparation 2 1 Demoling 3 27 LSM Profile contour I 4 28 Pull-out test in combination with FILT test Loa versus isplacement curve, failure mechanism of stran 5 29 LSM Profile contour II slip of filaments 6 4 SEM Information on penetrate concrete 5.3 Experimental methos The pull-out test The pull-out tests are carrie out using a universal testing machine (Instron 5566) at a isplacement rate of.1 mm/min at 2 C until a maximum isplacement of about 1.7 mm is reache. The specimen is mounte in the machine such that the epoxy resin block is place through a cut-out of a steel plate an then clampe in a mechanical grip fixe to the cross hea of the machine. The steel plate itself is fixe by a frame to the bench of the testing machine. All of this creates a set up of the pull-push test, i.e. the stran is pulle out of the matrix against a restraint. For illustration see Fig The pull-out loas an the cross hea isplacements are recore every 2 N change in force resulting in a loa versus isplacement relationship P(Ω). A capital Ω is chosen to refer to the stran en isplacements in contrast to the small letter ω for the single fiber or filament en isplacements. Loa cell Light source P, Ω 123 Steel plate Matrix Stran Digital camera Epoxy resin P ω Fig Test set-up for a pull-out test on a stran.

83 74 5 Bon between glass strans an a cement base matrix The FILT test During the actual pull-out test the specimen is aitionally expose by an artificial light source from the front via the epoxy resin block as sketche in Fig If a small chargecouple evice camera (CCD-camera) an a zoom lens is use, the stran can be istinguishe on the rear of the specimen from the surrouning matrix ue to its exposure an therefore bright appearance (on the left han sie of Fig. 5.3 an Fig. 5.4). After a tensile failure a filament is no longer capable of transferring light, an therefore the bright appearance vanishes for the next loa step in the next image. A) B) Fig A) Digital image of 5-VET stran. B) Binarize image of the etecte filaments using a numerical image analyzing routine. If a numerical image analyzing routine evelope by ECM Datensysteme Lt accoring to the nees is applie uring this stuy, the optical image of the CCD-camera presente in Fig. 5.3 (A) can be analyze an converte in a binarize image (black an white pixels) as shown in Fig. 5.3 (B). Possible changes of lighting conitions can be eliminate if a manual fine ajustment of a threshol value is carrie out. Depening on the chosen resolution an the zoom lens use it is possible to visualize every single filament. For illustration see Fig. 5.4 (A) an (B). However, care must be taken that the complete stran is recore uring the test. A) B) Fig Enlargement of Fig. 5.3 (A) an (B): A) Digital image of the stran. B) Binarize image of the etecte filaments using a numerical image analyzing routine. In the same way as the recoring of the loas an cross hea isplacements, the images of the etecte filaments are save every 2 N change of force which means the FILT test provies binarize images of the etecte an thus intact filaments at the same point of time at which the force an the isplacement of the pull-out test are recore. In combination with the

84 5 Bon between glass strans an a cement base matrix 75 recore force isplacement curves it is therefore possible to etermine the principal failure mechanism of a stran / matrix system uner a pull-out loa. A numerical computer aie proceure has been evelope, which allows the evaluation of the binarize FILT images with regar to the number of white pixels the image contains [Kob3]. By using this computer routine it is possible to plot the number of pixels recore versus the corresponing loas, an isplacements from the pull-out test respectively. Since only an optical image of the CCD-camera visualizing the filaments is analyze an converte in a binarize image outlining the filaments as white pixels (Fig. 5.3), the number of these ientifie pixels can simply be irectly relate to the cross-sectional area of the stran an thus to the number of filaments in a stran. Therefore the number of filaments N F which have not faile can be plotte versus the corresponing loas or isplacements respectively applie uring the pull-out test. To o so, the number of pixels N P, m recore at the beginning of the test are correlate to the total cross-sectional area of the teste stran A S by a proportional factor C 2 as given in the following: C 2 N P, m = A S TEX = 1 ρ G 3 (5.1) TEX correspons to the linear weight or tex count in g/km of the stran an ρ G is the ensity of the AR glass in g/cm³. Note that the proportional factor C² is the only unknown parameter an thus can be etermine from Eq. (5.1). As the average filament iameter a is known (Table 4.5), the correlation between recore pixels N P at each loa step an filaments which are still active N F is given by N F 4 A ( Ω) = π S 2 a N P N ( Ω) P, m (5.2) Thus a straightforwar relationship is given between the loa history of a pull-out test P(Ω) an the failure process of an AR-glass stran N F (Ω) which can be use as a main input parameter in the subsequent analytical moel to simulate the pull-out process of a stran out off a cementitious matrix. As a supplementary evaluation process, the binarize image of each loa step is analyze with regar to the number of white pixels N P, C in contact with surrouning black pixels, an hence the contact perimeter U C of the loa carrying filaments with the surrouning layer can be etermine. Similar to the above escribe proceure the with an height of a pixel can be correlate to a real length by using the proportional parameter C which is evaluate beforehan. This contact perimeter U C can be plotte against the applie isplacement, resulting in an active contact perimeter versus isplacement relation U C (Ω). However, since the computer routine only ifferentiates between white an black pixels, but cannot istinguish whether the black pixel refers to the matrix, vois or alreay faile filaments, the contact perimeter U C can be seen only as a qualitative an not as a quantitative parameter.

85 76 5 Bon between glass strans an a cement base matrix Laser scanning microscopy (LSM) LSM is mainly use in the area of biology an meicine for the measurement of cells or for DNA analysis. Nowaays LSM also has an increasing application for the evaluation of technical surfaces. Base on the principle of confocal microscopy [Lic94] iniviual, very sharp, bright an high-contrast profile contour images can be taken from samples by LSM. Similar to tomography iniviual sections are reconstructe to a true scale an threeimensional moel of the sample surface. With the help of a high peak algorithm an image can be create, in which the shaings correspon to the topography of the surface of the specimen. However, this testing technique allows no online measurements an is hence aopte in this test series only for a comparison of the unteste an teste state, i.e. a contour profile image of the filaments is taken on the rear sie of the specimen before an after the pull-out test. A marking on the specimen an a fixe restraint guarantees that neither a twist nor a horizontal shift of the specimen takes place uring the test an therefore the position of the specimen is ientical uring the taking of the two contour profiles. However, very small iscrepancies in the positions, especially rotations, may cause ifficulties when both images are superimpose for the numerical subtraction. From these contour images, the height of each filament before an after the pull-out test is etermine an by a numerical subtraction of the ata the final slip of the filaments at the rear of the specimen ue to the pull-out loaing can be calculate. Thus information is gaine on the resiual strain of the filaments at the en of the pull-out test; similar to the LVDT measurement uring the steel fiber pull-out tests in chapter 3. The applie numerical algorithm was programme accoring to the nees as a stuent research project at the Laboratory for Machine Tools an Prouction Engineering (WZL) uring this stuy [Mue1]. The final slip of the filaments can be observe in images by using ifferent shaings whereby each represents a particular isplacement. 64 shaings are use an istribute uniformly over the measure levels. In each image one shaing represents the so-calle zero level i.e. no slip occurre at these places. Since the maximum slip of the teste samples eviates from each other, varying colors are assigne to the images as zero levels Scanning electron microscopy (SEM) Finally the specimens are examine by means of SEM. Because imperfections (air), cement paste an glass iffer in their elementary composition, a goo classification of these components is possible. Again the rear sie of the specimen is examine after the pull-out test to analyze the microstructural morphology an the amount of stran penetrate by concrete, which obviously changes within the embee length of the stran. Nevertheless, it is believe that a general statement relating the type of the failure mechanism an the amount of concrete penetrate into the core of the stran may be given. For a possible numerical quantification of the microstructural morphology base on the taken SEM images, refer to chapter 6.

86 5.4 Tests results 5 Bon between glass strans an a cement base matrix The pull-out test Fig. 5.5 shows the recore pull-out loa versus isplacement iagrams P(Ω). Some etails are shown in aitional iagrams on the left an right of Fig. 5.5 for a etaile analysis. The one on the left shows the origin of the pull-out up to a isplacement of.4 mm an emphasizes the ifferent starting graients of the P(Ω) relation. The P(Ω) iagram in the center of Fig. 5.5 shows the complete loa versus isplacement istribution. Finally the iagram on the right refers to the frictional stage of the pull-out process. In general, the principal tren of a pull-out relation can be observe. An almost linearly increasing part with a subsequent non-linear region until the maximum pull-out loa is reache, followe by a ecreasing softening branch. Although specimen B starts up with the highest slope followe by specimens A an C an then specimen D, the maximum pull-out loa at approximately 5 N is almost ientical for specimens B an C. Specimen D carries a maximum pull-out loa of 433 N which is about 14 % lower an specimen A has a maximum pull-out loa of 41 N corresponing to a eficit of 18 %. This variety is quite acceptable for pull-out tests on strans embee in a cement base matrix. [Maj74] an [Law86] experience more than 4 % scatter in their test results, which is cause by the uncontrollable an ranom penetration of the matrix into the core of the stran. However, maybe ue to the careful an reproucible preparation of the specimens an the low viscosity of the matrix, the scatter of this stuy s test results coul be significantly reuce. Pull-out loa P in N 6 4 Pull-out loa P in N 6 4 Specimen A Specimen B Specimen C Specimen D Pull-out loa P in N Displacement Ω in mm Displacement Ω in mm Displacement Ω in mm Fig Pull-out responses of specimens A to D. A noticeable an consierable ifference between specimens B an C, which reache the same maximum pull-out loa of 5 N, is that they experience a ifferent failure behavior in the softening branch of the P(Ω) relationship. Whereas the progression for specimen B rops rapily to a lower loa level of about 28 N at a pull-out isplacement of.37 mm, the pullout response of specimen C takes a smoother path an reaches the same loa level at a

87 78 5 Bon between glass strans an a cement base matrix consierably higher isplacement of.78 mm. The P(Ω) istributions of the remaining two specimens stay somewhere in between. The P(Ω) progressions of all 4 specimens en within a range of 6 N between the uppermost an lowermost final loas, with specimen B at the lower en with 5 N an specimen A with about 1 N at the higher en. Note that ue to the fact that the stran is cut off at the rear of the specimen uring specimen preparation, the core filaments are pulle out an not pulle through. Hence no constant frictional loa will be achieve at the en of the tests The FILT test As explaine in chapter the images of the FILT test are taken, binarize an recore simultaneously with the loas an isplacements of the pull-out test. The loa versus isplacement iagram P(Ω) recore for specimen A is presente in the secon column of Table 5.2 for ifferent stages of the pull-out process to clarify this proceure: Ω at maximum loa an at 3 subsequent loa steps. Table 5.2. Observe failure process of a 5-VET stran embee in PZ matrix uner a pull-out loa (FILT tests); specimen A. L-S 1 Loa versus isplacement iagram P(Ω) FILT test image 1 Pull-out loa P in N Displacement Ω in mm 2 Pull-out loa P in N Displacement Ω in mm 3 Pull-out loa P in N Displacement Ω in mm

88 5 Bon between glass strans an a cement base matrix 79 4 Pull-out loa P in N Displacement Ω in mm 1 L-S = loa step For these selecte loa steps the corresponing binarize FILT images are presente in the thir column of Table 5.2. Remaining intact filaments are visualize by white pixels. The loa steps are labele in the first column of Table 5.2. A B Fig C FILT images of specimen A to D respectively at the maximum introuce isplacement of about 1.7 mm. Thin white line outlines the perimeter of the stran before the pull-out test. D It is obviously visible that with an increasing pull-out isplacement the amount of intact filaments ecreases. However, this failure process is not consistent over the cross-section of the stran but ifferent groups of filaments form together, which are finally pulle out. In Fig. 5.6 the FILT images of all 4 specimens are given at the final stran-en isplacement of about 1.7 mm. A comparison of these FILT images shows the variety of results for these 3 replications, i.e. that ifferent amounts of filaments at ifferent locations are left intact at the en of the pull-out tests. To verify whether the outer filaments break own first an then a successive failure occurs layer by layer from the sleeve filaments to the core filaments, the amounts of filaments failing

89 8 5 Bon between glass strans an a cement base matrix between two successive loa levels of Table 5.2 are ascertaine an presente in Table 5.3. In contrast to the results presente earlier, the white pixels now represent the broken filaments. Again the secon column shows the loa versus isplacement iagram, but now the segment between two ajacent loa steps (first column) is highlighte. Table 5.3. Observe telescopic failure of a stran (specimen A). L-S 1 Loa history Telescopic failure image FF Pull-out loa P in N Displacement Ω in mm 2-3 Pull-out loa P in N Displacement Ω in mm 3-4 Pull-out loa P in N Displacement Ω in mm 1 2 L-S = loa segment Number of faile filaments within the loa segment. Total number of filaments is 14. Between loa step one an two the stran experiences a ecrease of loa of approximately 127 N an a change in isplacement of.22 mm. The thir column of Table 5.3 shows the location an the amount of faile filaments which cause that loa ecay. It is obviously visible that an outer ring of filaments breaks own but the core filaments still contribute to the loa transmission. A similar phenomenon is foun for the following segment where a ecrease of loa of 16 N within a isplacement change of.71 mm is observe. Again the failure of a ring of filaments causes this ecay in loa. The last segment shows only a ecrease of 1 N between the corresponing loa steps within a isplacement increase of.23 mm. In turn filaments surrouning the core fail in tension. Fig shows all the

90 5 Bon between glass strans an a cement base matrix 81 filaments which have faile uring the complete pull-out process, i.e. at a loa of 113 N an an introuce isplacement of 1.78 mm. Although no complete an continuous core remains, it is obvious that only core filaments stay intact. This image sequence an the corresponing loa versus isplacement iagrams are a valuable source of information for the analytical moeling of the pull-out response of a stran aresse later in chapter 7. In a further step the binarize images which are recore uring the FILT test (Table 5.2 column 3) can be evaluate with regar to the number of white pixels N P, by using the numerical proceure propose in chapter If N P is plotte against the corresponing isplacement Ω a pixel versus isplacement iagram is obtaine. For the four specimens this result is presente in the graph on the left of Fig Pixels recore N P 25 Specimen A 2 Specimen B Specimen C 15 Specimen D Displacement Ω in mm Active filaments N F Displacement Ω in mm Fig A) Pixel versus isplacement iagram for specimen A-D. B) Active filaments versus isplacement iagram for specimens A-D. On the basis of Eq. (5.2) the recore pixel versus isplacement relationship N P (Ω) can be transferre into an active filament versus isplacement relationship N F (Ω) referring to the number of filaments which are left intact uring the pull-out test. N F (Ω) is presente for the four specimens on the right of Fig In principle all progressions are characterize by a short almost horizontal line between Ω.12 mm followe by an exponential ecay, i.e. almost no filaments fail in tension before the maximum pull-out loa is reache. A more etaile comparison of the presente active filament versus isplacement iagrams reveals the varying breakown of the filaments of the 4 strans teste. Whereas the filaments of specimen B fail rapily so that at a isplacement of.5 mm only 23 % of the filaments are left intact, specimens A an D are left with 61 % of intact filaments at the same isplacement. Specimen C retains about 4 % lying somewhere in between. The graient of the N F (Ω) relationship for specimen B ecreases noticeably at the en of the pull-out test finally leaving about 6 % of filaments which have not faile in tension at a maximum isplacement of approximately 1.7 mm. Specimen A is left with the highest percentage (2 %) of unbroken filaments at the same introuce isplacement.

91 82 5 Bon between glass strans an a cement base matrix The left graph of Fig. 5.8 shows the numerically evaluate number of white pixels N P, C in contact with surrouning black pixels. By using the proportional factor C which is evaluate on the basis of Eq. (5.1), this relation can be transferre into a contact perimeter versus isplacement istribution U C (Ω) presente on the right of Fig Number of contact pixels N P,C Displacement Ω in mm Contact perimeter U C in mm Specimen A 16 Specimen B Specimen C 12 Specimen D Displacement Ω in mm Fig A) Contact pixels versus isplacement iagram for specimen A-D. B) Contact perimeter versus isplacement iagram for specimen A-D. For all of the specimens presente, the contact perimeter U C steaily increases until a isplacement of about.25 mm is reache. A subsequent ecreasing stage follows. Specimen B starts out with the highest contact perimeter of 12.6 mm, whereas U C for specimen A an D is noticeably lower at about 8.2 mm. The increase in percentage of the progression is about the same for all four specimens. However, the following almost linear rop from a contact perimeter of 15 mm to 8.2 mm within a isplacement range of.18 mm is unique for specimen B although specimen C follows a similar but more graual progression. In contrast the results of specimen A an D show a more steay ecrease. At the final pull-out isplacement of 1.7 mm, specimen B is left with a contact perimeter of less than 4 mm whereas specimen C an D en up with a U C of about 5 mm, although they starte with a consierably ifferent initial value at the origin. Specimen A shows the highest value for U C at Ω = 1.7 mm with 8 mm Laser-scanning microscopy (LSM) As mentione in chapter 5.3.3, an image of the contour profile of the filaments on the rear sie of the specimen is taken before an after the pull-out test. This information is shown in Fig. 5.9 for specimen A by using ifferent shaings, whereby each represents a particular height of the filament. The white lines in Fig. 5.9 outline the perimeter of the stran before the pull-out test has been carrie out. If these profiles are numerically subtracte from each other, then the slip experience by each filament at x = until the en of the pull-out test can be evaluate (resiual strain). Note that the black color in Fig. 5.9 right represents the so-calle zero level, i.e. no slip occurre in these areas.

92 5 Bon between glass strans an a cement base matrix 83 LSM image before test LSM image after test Numerical subtraction of 1 an 2 Fig Results of the LSM analysis (specimen A). Usually the numerical subtraction reveals the amount of slip the filaments have experience by means of color. Regrettably, ue to the black an white print of this thesis a great eal of information is lost. Examples for colore figures are given in [ A B Fig C Results of the LSM analysis, i.e. numerical subtraction of LSM images taken before an after the pull-out test, for specimen A to D. Thin white line outlines the perimeter of the stran before the pull-out test. D In Fig. 5.1 the results of the LSM analysis of all 4 specimens are presente, i.e. the numerical subtractions of the LSM images before an after the pull-out test. Similar to the results of the FILT tests which were previously presente in Fig. 5.6 the LSM results show that the locations an the amount of slip of the filaments of the 4 strans vary consierably within the test series. A visual comparison of the LSM results illustrate in Fig. 5.1 shows that contrary to specimens B an C more filaments of specimens A an D which are more uniformly istribute, experience a pull-out. Aitionally it can be state that almost all of the pulle-out filaments of specimens B an C experience the same slip, whereas the results of

93 84 5 Bon between glass strans an a cement base matrix specimens A an D show that the filaments nearer to the core of the stran have slippe by a consierably higher amount than the ones which are locate nearer the sleeve Scanning electron microscopy (SEM) SEM analysis is applie to analyze the microstructural morphology an the amount of stran penetrate by the concrete. As an example, a micrograph is presente in Fig taken from the rear sie of specimen A after the pull-out test. The areas where the core filaments have been pulle out can be istinctly ifferentiate (black areas in the center of the figure). 2. mm Zoom Zoom Fig SEM image of specimen A after the pull-out test (rear sie). Fig presents an enlargement of the center an of the upper right corner of the stran (marke areas in Fig. 5.11). Note the ifferences in the pull-out behavior. Whereas the filaments in the center of the stran have been completely pulle out together as a bunle, the upper right area of the stran shows that only iniviual filaments experience a pull-out (Fig. 5.12). In general it may be state from these observations that the sleeve filaments which are still visible in the SEM-micrographs are completely embee in cement paste, where hyration proucts have penetrate into the gaps between these filaments..1 mm.2 mm A) Fig B) Enlargements of the SEM micrograph in Fig. 5.11: A) center section B) upper right corner.

94 5 Bon between glass strans an a cement base matrix 85 A SEM-micrograph of a longituinal section of specimen A aitionally shows the matrix strips which have penetrate between the iniviual filaments. For a more etaile investigation of the microstructure refer to chapter 6..2 mm Fig SEM micrograph of a longituinal section of specimen A. 5.5 Discussion In this sub-section, the results of the iniviual test techniques presente above are assemble an analyze in context to gain qualitative an quantitative information on the complex failure mechanisms of a stran embee in a cementitious matrix uner a pull-out loa. A first visual comparison between the results of the pull-out tests - i.e. the P(Ω) relationships, an the N F (Ω) relationships from the FILT tests - reveals that specimen B which ene up with the fewest number of still active an intact filaments (Fig. 5.7) an the smallest final contact perimeter U C (Fig. 5.8), also carries the lowest pull-out loa at the en of the test. Specimen A with 3 times more active filaments at a maximum isplacement of 1.7 mm carries the highest loa an also features the highest contact perimeter at the en of the pullout test. Specimens C an D follow a similar sequence. Such a correlation between the progressions of the pull-out loa versus isplacement relationships P(Ω) an the active filaments versus isplacement istribution N F (Ω) is not only foun in this last stage but is also vali for the complete pull-out process. Fig shows the P(Ω) relationship an the N F (Ω) istribution for specimen A, B, C an D. Note that the active filaments N F axis is on the primary y-axis an the pull-out loas P on the seconary y-axis. Comparing the relationships which are state in these iagrams outlines one funamental ifference: Whereas the progressions N F (Ω) an P(Ω) of specimen B run more or less parallel to each other from the origin point where the pull-out force is almost at a constant loa level of 32 N after maximum loa, the N F (Ω) an P(Ω) relations of specimens A, C an D only run parallel for a short run at the beginning up to a loa level of 5 N, 7 N an 5 N, respectively, after maximum loa an quickly converge with further progression.

95 86 5 Bon between glass strans an a cement base matrix Active filaments N F 15 1 Pull-out loa P in N 75 N F (Ω) P(Ω) 5 Active filaments N F 15 1 Pull-out loa P in N 75 N F (Ω) P(Ω) 5 5 Specimen A 25 5 Specimen B Ω Displacement Ω in mm Displacement Ω in mm NFL Ω NFL Active filaments N F 15 1 Pull-out loa P in N 75 N F (Ω) P(Ω) 5 Active filaments N F 15 1 Pull-out loa P in N 75 N F (Ω) P(Ω) 5 5 Specimen C 25 5 Specimen D Displacement Ω in mm Ω NFL Displacement Ω in mm Ω NFL Fig Active filament versus isplacement iagram N F (Ω) an pull-out loa versus isplacement P(Ω) for specimens A to D. For specimen B it may be state that the tensile failure of filaments after maximum loa, i.e. ecrease of number of active filaments N F, goes along with an almost proportional ecrease in the pull-out loa (i.e. parallel portion of graphs) until a isplacement Ω NFL of.33 mm is reache. Ω NFL efines the isplacement at the en of the proportionality between P(Ω) an N F (Ω). In comparison, this approximate proportionality is true for specimens A to values of Ω NFL =.38 mm, for specimen C to Ω NFL =.31 mm, an for specimen D to values of Ω NFL =.42 mm respectively. A possible explanation for these varying pull-out responses is that a slightly ifferent failure mechanism exists between specimens A an D on the one han an specimens B an C on the other han which can be ientifie if the results of the LSM an SEM investigations are inclue in the consierations. A comparison of the FILT images (Fig. 5.6) an the LSM images (Fig. 5.1) reveals that there are more filaments which actually experience a plastic strain or filament en isplacement accoring to the LSM investigations than there are filaments which, accoring to the FILT images, remain unbroken an thus pulle out. These filaments which fail in tension after experiencing a pull-out can be ientifie by means of an image-superposition of both tests results, where those filaments still foun in the FILT test image (Fig. 5.6) are blackene in the

96 5 Bon between glass strans an a cement base matrix 87 LSM image (Fig. 5.1). Fig shows, as an example, this superposition for specimens A an C of series 12. Filament en isplacement:.4 mm.55 mm A Fig C.7 mm Superposition of FILT test an LSM images of specimens A an C. Thin white line outlines the perimeter of the stran before the pull-out test. Many of the former sleeve filaments of specimen A experience a slip before they fail in tension (see on the left of Fig. 5.15). The group of filaments in the upper right half of the stran experience a pull-out up to a filament en isplacement of about.7 mm. Groups of filaments in the center of the stran an in the lower left part unergo a similar process. They experience a filament en isplacement between.4 an 1. mm. The image-superposition of the results of specimen D (not presente) shows a similar outcome, in contrast to the superposition of the results specimen B (not presente) an C (see on the right of Fig. 5.15) where only a very limite number of filaments experience a pull-out before they fail in tension. In this case an overall filament en isplacement of aroun.6 mm is foun. A comparison of Fig. 5.6 an Fig. 5.1 also shows that some filaments of the strans which have not faile accoring to the information gaine from the FILT tests, o not experience any slip if the results from the LSM analysis are viewe. These iscrepancies foun are probably cause uring the preparation of the specimens for the secon LSM analysis. On the basis of the above presente results a first conclusion may be rawn on how the N F (Ω) an P(Ω) istributions are linke to the failure mechanisms occurring uring the pull-out process of the stran. For specimens B an C the tensile failure of a filament N F (Ω) which is observe with the FILT test goes along with an almost proportional ecrease in the P(Ω) relationship. Aitionally, the superposition of the FILT an LSM results shows that only a very limite number of filaments faile in tension after they experience a pull-out. In contrast, specimens A an D show only a short parallel run between N(Ω) an P(Ω) at the beginning of the pull-out test (see Fig. 5.14) which then isintegrates with further progression implying a pull-out of filaments which then fail in tension before reaching the en of the pullout test. Note that the tensile strength of a filament is likely to ecrease if its surface is amage by abrasion when slipping through the matrix. In general, it may be state that the outer filaments are strongly bone to the surrouning matrix an hence can not be pulle out but instea fail in tension. This is verifie by the SEM micrographs presente in Fig an Fig The sleeve filaments which are completely

97 88 5 Bon between glass strans an a cement base matrix surroune with matrix are well efine. Aitional evience for this occurrence is given by the image-superposition of the FILT test image, showing all the filaments which have faile uring the pull-out test, with the corresponing SEM micrograph. This is shown for specimen A in Fig In general a goo agreement is foun; i.e. most of the filaments which are ientifie in the FILT test results to have faile in tension are foun to be fully embee in fine-graine concrete by SEM analysis. 2. mm 2. mm Fig A) A) Filaments which have faile uring the pull-out tests (Specimen A). B) Superposition of a) with SEM micrograph (Specimen A). Unfortunately no information regaring the failure location in the thir imension can be gaine from the presente test sequence, i.e. it is not known at what istance x from the loa introuction point the filaments fail. A possible testing metho to etermine this last unknown coorinate of the failure location of the single filaments is by acoustic emission analysis [Rei2]. Although this technique oes not have the resolution to etermine the exact spacial coorinates of a filament breakage, it provies enough accuracy to etermine the x-location within the embee length where a failure occurs. Combining this information with the results from the FILT test woul allow a more precise picture of the preominating failure mechanisms. B) 1) 2) Fig Visualize staggere failure of the stran. Unfortunately, a high technical expertise an a subsequent elaborate analyzing proceure are necessary to apply this technique an hence it cannot be use in the framework of this stuy. However pictures taken uring the testing in this stuy show a staggere breakown of the

98 5 Bon between glass strans an a cement base matrix 89 stran (Fig. 5.17); i.e. the filaments break at ifferent x-locations within the embee length. But unfortunately, no quantitative information can be gaine from these images. 5.6 Summary The results presente above support those of [Maj74], [Bar87] an [Cur3] in that the pullout behavior is controlle by a strong boning of the external filaments in the stran an a slip of the inner filaments, again influence by the ranom an therefore unpreictable penetration of matrix into the core. In principle this is not a surprise, although cement grains which measure approximately 1 µm can harly penetrate into the spaces (approximately 3 µm wie) between the filaments if the filaments are assemble in a compact form. However, in many cases the original compact flattene bunle is loosene uring the placing an manufacturing, hence the matrix may penetrate up to a certain egree into the core of the stran. Nevertheless, the formation of hyration proucts within the stran is initially limite. This uncontrolle penetration leas to a ifferent formation of the inner an outer bon characteristics, an hence the failure mechanism after exceeing the maximum pull-out loa is escribe as a so-calle telescopic failure, i.e. a successive break own layer by layer from the sleeve to the core filaments, see Table 5.3. After the tensile failure of the outer filament layers, a core of inner filaments is pulle out of the stran. Note that although some of the filaments were foun to have faile in tension, they also experience a certain egree of pull-out; i.e. uring the actual pull-out process they faile at some stage, presumably ue to the exceeance of the tensile strength which was reuce by abrasion an introuce notches in the filament surface. As a result the real, higher tensile strength of the stran is not activate. Note that the highest theoretical tensile strength of a stran correspons to the strength of a single filament, for example, a filament of the 5-VET stran has a tensile strength of 1,473 N/mm² (Table 4.5) corresponing to a force of about 1,32 N which the stran is suppose to carry, in contrast to the 4 N it actually takes in the pull-out test. Base on these evaluate results, the complex failure process of a pull-out test on a stran / cement base matrix system may be sketche as presente in Fig (Subproblem II, chapter 1). Core filaments Sleeve filaments Bulk matrix Type of failure: Pull-out + tensile failure Pull-out Tensile failure L P( Ω) Fig Penetrate matrix Tensile failure Failure mechanism of a stran embee in a cement base matrix uner a pull-out loa.

99 9 5 Bon between glass strans an a cement base matrix Using the numerical evaluation proceure the ientifie failure mechanisms coul not only be visualize but also quantifie. By introucing an active filament versus isplacement relationship N F (Ω), the pull-out response P(Ω) coul be irectly relate to the number of filaments failing in tension uring the process. Although only 4 specimens were teste in this first step, it may be assume that the combine failure mechanism of filaments (i.e. pull-out then tensile failure) may be ientifie by a comparison of the graients of the two relationships. If both istributions run more or less parallel from the origin to the en, the preominating failure is a tensile break own of the filaments without any prior pull-out. As soon as this parallel progression isintegrates, a pull-out of filaments occurs, followe by a subsequent failure in tension. Furthermore, not only coul the number of filaments, which actually contribute to the loa carrying behavior at each loa step, be quantifie but also a numerical evaluation proceure coul be presente allowing the etermination of the contact perimeter U C of the active filaments with the surrouning layer at the corresponing loa step. Hence the funamentals are given to buil up an analytical moel capable of simulating the complex failure mechanism of a stran embee in a cement base matrix uner a pull-out loa (chapter 7). Thus, sub-problem III is also solve. As the main parameter influencing the failure process an hence the loa carrying behavior of a stran in fine-graine concrete is the amount of matrix penetrating its core, which affects the exten to which the full tensile strength of a stran is reache, future research shoul concentrate among other things on the material an preparation parameters influencing this penetration. An optimize an reproucible loa carrying behavior of this composite, an hence controllable an ajustable boning characteristics between the stran an the surrouning cement base matrix coul be achieve by ifferent ways an means. Either the penetration of matrix into the stran is prevente altogether by ifferent manufacturing techniques of the stran, for example twiste or agglutinate strans, or the cement base matrix is moifie such that fine particles for example silica fume, cement grains or polymers - fully penetrate into the stran. In orer to permit such a complete penetration the size of the strans has to be moifie as well, for example by a plasma treatment or a moification of the chemical composition. Such techniques were suggeste as early as 1988 by [Ben88]. The embee length of the stran uring the pull-out test has - as is shown in [Bra4] - only an inferior effect on the loa carrying mechanisms of a stran/matrix system uner a pull-out loa in a pull-out test (slight change of the post failure behaviour ue to a higher conatct area in the frictional stage) as the embee length neee for a single filament to reach its tensile strength an the pentration characteristics of the matrix into the stran are far more important (see also section 7). Thus the classical statistical size effects observe in tenisle tests on strans, play no sigificant role uring pull-out tests (an may therfore be ignore within this stuy). Besies of these classical size effects, the influence of the irregularity in the yarn

100 5 Bon between glass strans an a cement base matrix 91 structure on the performance of yarns with short effective lengths (e.g. crack brige) has been investigate in [Chu4a] an [Vor4]. Hence future research shoul inclue investigations on other effects influencing the bon between a stran an the surrouning matrix. For example the capillary absorption of water by the stran an hence the storage of excess water within the stran at the time of setting may lea to an increase in the porosity an a ehyration of the matrix near the sleeve filaments [Ben88, Sha91a]. Also the influence of the size on the boning characteristics, in particular the solubility an its effect on the surrouning matrix shoul be examine as well as a possible interference ue to swelling.

101 92 5 Bon between glass strans an a cement base matrix

102 93 CHAPTER 6 THREE DIMENSIONAL ARRANGEMENT OF A STRAND IN A CEMENT BASED MATRIX In the previous chapter the principal failure mechanisms of a stran embee in a cement base matrix uner a pull-out loa were establishe an it was foun that they mainly epene on the ranom penetration of fine-graine concrete into the interior of the stran. So far no quantitative information on the efinite locations where matrix has penetrate the stran is known an thus an aequate characterization of the microstructure, which is a further critical point in the formulation of an analytical moel, is not possible. Therefore in this chapter experimental techniques are presente which allow a spatial illustration of a stran matrix system. However, the overall information gaine on the three-imensional assembly of a stran in a cement base matrix with regar to the arrangement of the filaments, the locations an amounts of matrix penetrate into the stran, as well as vois, pores, an imperfections influencing the interfacial characteristics is restricte ue to technical an financial limitations. 6.1 Introuction Whilst the microstructure can be viewe quite easily in two imensions at a variety of resolutions, e.g. optical or scanning electron microscopy as shown in Fig. 1.1, this is not as useful as a 3-D image since it is the three-imensional array of cement particles an hyration proucts in the interior of the stran which has the greatest influence on the pull-out response of a stran. However, a etaile investigation of this complex 3-D system is still a ifficult task because the imensions of a single filament within the system are quite small compare to the overall size of the pull-out specimen (spaces between filaments =.3 mm, iameter of filament =.2 mm, overall specimen length = 3 mm). In general computer tomography (CT) is an applicable technique to gain information on this 3-D arrangement of a stran / matrix system. Using ifferent visualization toolkits, crosssectional images or "slices" of the investigate structure are either connecte into a stack of planar polygons an then into a 3D triangulate surface moel, or converte into voxels (3D or volumetric pixels) as one by VGStuio as basic elements to represent not only the surface but also the entire interior of the object. If aspects of resolution are consiere, there are two main possibilities, presente in the following, to gain the neee input ata, i.e. series of cross-sectional images.

103 94 6 Three imensional arrangement of a stran in a cement base matrix Recently the so-calle X-ray microtomography (micro XRT) has been use to obtain slice images of the microstructure of cement pastes with a resolution better than one micrometer per voxel, although this only applie to probes with maximum imensions of mm³. The contrast in XRT images is base on the ifference in absorption of X-rays by the constituents of the sample (e.g. silica an air). However, these micro XRT investigations can be carrie out only at two or three facilities worlwie, one of which inclues the European Synchrotron Raiation Facility (ESRF) in Grenoble, France 1. Thanks to a close cooperation between Grenoble an the epartment of physics at RWTH Aachen University, a small specimen of a VET-5 stran embee in PZ matrix coul be investigate at this facility [Len2]. A secon possibility of creating the aforementione image slices has alreay been successfully applie in The Visible Human Project at the National Library of Meicine (NLM) in the USA. CT ata consisting of axial SEM images of the entire human boy taken at about 1 mm intervals were achieve by planing sections from the specimen with a jointer ajuste to minimum thickness. This latter technique is hence aapte an moifie to the given conitions uring this current stuy to gain insight into the microstructure of a stran embee in a fine-graine concrete matrix. In a subsequent analysis, the information gaine is evaluate by means of a computer aie image analyzing system to try to quantify an statistically evaluate the penetration istribution of cement paste an also locate the voi an imperfection istribution within the stran. 6.2 Materials composition an specimen preparation AR-glass strans / fine-graine concrete All of the following microstructural investigations are performe with the material combination 5-VET stran (see chapter 4.2.1) an PZ matrix (see chapter 4.2.2) Specimen preparation To guarantee that the microstructure of the specimens teste in the pull-out tests an the samples use uring the CT analyses are as similar to each other as possible, the specimen preparation is one as escribe in chapter 5.2. However, the stran is not embee in epoxy resin on either sie an aitionally three small steel fibers of.1 mm iameter are place parallel to the stran an perpenicular to the cross-section in a triangular arrangement in the specimen. These steel fibers allow the exact placing an alignment of the sectional images in the CT analysis later on. All specimens are cure after casting for one ay in the mol an store for two further ays in a 2 C an 95 % RH climate. Afterwars they are impregnate with clear resin an the top an rear sies of the specimen are polishe. After preparation each specimen is sectione 1

104 6 Three imensional arrangement of a stran in a cement base matrix 95 along its x-axis, whilst using water to cool the cutting saw. An axial prism, approximately 1 1 mm² in cross-section an 3 mm in length, is slice leaving the stran an the steel fibers near two perpenicular sies. X-ray microtomography The specimen use for the X-ray microtomography is further cut own to a cubic form of mm³ which is the maximum size allowe for these investigations. As a result the steel fibers an a larger part of the cross-sectional area of the stran are remove. Environmental scanning electron microscopy (ESEM) In the case of the ESEM investigations, the front an rear sies of the mm³ axial prism are groun on a flat surface with a wheel griner an polishe with 6# silicon carbie. Further polishing is performe with 1-, 5-, an 1-micron aluminum power on a glass plate. After polishing, the specimens are immerse in acetone an place in an ultrasonic machine in orer to remove the resiual film on their surfaces Experiment sequences One mm³ 5-VET stran / PZ matrix system is investigate by means of the micro XRT an two mm³ specimens are analyze with the ESEM. 6.3 Experimental methos X-ray microtomography (XRT) The mm³ specimen is image on the 3-D microtomography unit at the European Synchrotron Raiation Facility (ESRF) in France using parabolic refractive lenses mae of aluminum to generate har X-ray microbeams at several beamlines. The specimen to be image is mounte on a translation / rotation stage allowing a precise alignment in the beam. A set of 25 X-ray micrographs is recore as a function of sample rotation at equiistant steps in an angular interval from to 18 an with a x1.6 magnification. Filtere backprojection is aopte to reconstruct the cross-sectional images. For more information refer to [Len2]. Using the visualization program VGStuio these slices are finally converte into voxels an reassemble into a three-imensional image of the composite Environmental scanning electron microscopy (ESEM) An environmental SEM is use in this stuy to examine the non-conuctive samples without coating it with a conuctive material, which simplifies the preparation proceure by a consierable egree. The specimen is mounte on a specimen stub an place on the stage allowing a precise alignment in the beam. Electron backscatter iffraction analysis is use to etermine the make-up of the sample. The resulting images show each element in the sample in a ifferent shae, from almost white to black (phase ientification). These images are 124 pixels by 124 pixels where each pixel is mae up of 12 bits of gray tone.

105 96 6 Three imensional arrangement of a stran in a cement base matrix After taking an axial image of the composite, the specimen is remove from the ESEM unit an place again on the wheel griner to remove about 1 to 2 mm of material. The subsequent polishing an the finishing proceure is as escribe above. The specimen is then place in the ESEM again an a further axial image is taken. This proceure is repeate about 1 times per specimen in 1 2 mm intervals. Attention is pai, that each image also contains the three cross-sections of the steel fibers. All sectional images are converte again into voxels (VGStuio ) an reassemble into a three-imensional structure of the composite using the steel fiber bench marks to align the images. 6.4 Results X-ray microtomography (XRT) A section of a 2-D (slice) image is provie on the left of Fig Clearly visible are the circular filaments an the vois between them fille with cement paste to a certain egree (rough appearing areas). The principle of micro XRT is base on the ifference in absorption of X-rays, an since the atomic weight of AR-glass filaments an cement particles are very similar, the contrast in the resulting images is moerate. The generally high X-ray absorption of concrete provies aitional problems to visualize high contrast patterns, i.e. images with a high spectrum of gray levels. 25 µ m 25 µ m 115 µ m 15 µ m Fig Section of a 2-D (slice) image an a subvolume of 3-D image of a 5-VET stran in PZ matrix etermine by means of micro XRT. As the initial spectrum of graylevels is so low, it is not possible to apply a multi-level thresholing operation by means of a computer aie image analyzing system to prouce segmente (phase) images of the specimen. Thus, this metho of investigation unfortunately oes not offer the potential for further evaluation with regar to the locations an amounts of penetrating matrix, vois, pores, or imperfections within the stran.

106 6 Three imensional arrangement of a stran in a cement base matrix Environmental scanning electron microscopy (ESEM) Fig. 6.2 shows, as an example, a 2-D (slice) micrograph of the 5-VET stran / PZ matrix system (section 2, specimen 1). The white line in the image borers the sleeve filaments of the stran an highlights the overall contorte formation an the sprea of the filaments. For most parts AR-glass filaments (white), cement paste (gray) an vois (black) can be ifferentiate. In some cases, however, SiO 2 aggregates which possess an atomic weight almost ientical to that of the glass filaments are also isplaye as white areas. Outline of stran Location 2 Representative sector Location 1.2 mm Location 3 Fig D (slice) image of a 5-VET stran / PZ matrix system (section 2, specimen 1). To allow a more etaile evaluation of the mircostructure of specimen 1, three 2-D (slice) images of a representative sector (see Fig. 6.2) together with the resulting 3-D subvolume are presente in Fig µ m Section 2 Subvolume Section 3 Section 4 2 mm (not in scale) Fig Three 2-D (slice) images an a subvolume of a 3-D image of a 5-VET stran in PZ matrix (specimen 1).

107 98 6 Three imensional arrangement of a stran in a cement base matrix Obviously the microstructure of the system rastically changes within only a couple of millimeters; compare the three 2-D slices on the left of Fig San grains isplace the filaments, cement paste penetrates at ifferent locations an vois form in a ranom manner. Also clearly visible is that filaments which are in close contact with cement paste (upper 2-D slice (section 2), center of slice) at a certain location lose this contact within just a few millimeters away, i.e. from center of section 2 slice to center of section 3 slice. Naturally this microstructure also iffers for the two specimens investigate. Fig Segmente (phase) images of one 3-D subvolume: Black areas represent vois or flaws, gray areas represent cement paste an white areas represent filaments. The visualizing software VGStuio allows to clip a volumetric structure along arbitrary chosen axes or planes so that ifferent sections may be evaluate with regar to contact zones between the three phases, e.g. between filaments an cement paste. Fig. 6.4 represents such a 3-D illustration of a section of the subvolume presente in Fig Computer aie image analyzing system Base on the results presente above, fair quality segmente (phase) images of the 2-D slices an thus also of the 3-D subvolumes can be prouce by means of a multi-level thresholing operation. Fig. 6.5 shows such binarize phase images for the representative sector of Fig µ m 2 µ m 2 µ m ROI A) Path B) C) Fig Segmente (phase) images of one 2-D slice: A) Black areas correspon to filaments an parts of some aggregates (SiO2). B) Black areas correspon to cement paste. C) Black areas correspon to vois or imperfections. The black areas in Fig. 6.5 A represent filaments, in Fig. 6.5 B cement paste an those areas in Fig. 6.5 C vois or imperfections. Note that again ue to the almost ientical atomic