Computational modelling of fibre-reinforced cementitious composites: An analysis of discrete and mesh-independent techniques

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Computational modelling of

fibre-reinforced cementitious composites:

An analysis of discrete and

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Computational modelling of

fibre-reinforced cementitious composites:

An analysis of discrete and

mesh-independent techniques

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus Prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op woensdag 12 december 2012 om 12.30 uur

door

Frank Kurt Friedrich RADTKE Diplom-Ingenieur, Universität Hannover

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Copromotor: Dr. A. Simone

Samenstelling promotiecommissie:

Rector Magnificus Voorzitter

Prof. dr. ir. L. J. Sluys Technische Universiteit Delft, promotor Dr. A. Simone Technische Universiteit Delft, copromotor Prof. dr. ir. E. Schlangen Technische Universiteit Delft

Prof. dr. ir. J. C. Walraven Technische Universiteit Delft Prof. dr. habil. S. Lomov Katholieke Universiteit Leuven Prof. dr. P. D´ıez Universitat Polit`echnica de Catalunya Prof. dr. M. di Prisco Politecnico di Milano

Prof. dr. ir. K. van Breugel Technische Universiteit Delft, reservelid

This research has been supported by the Netherlands Science Foundation STW (under grant 06623) and the Ministry of Public Works and Water Management.

Keywords: fibre-reinforced concrete, finite element method, partition of unity, failure analysis

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Aknowledgements

The research reported in this thesis has been carried out in the Computational Mechanics Group of the Civil Engineering Faculty of Delft University of Technology.

I would like to gratefully acknowledge the guidance and the support of my supervisors Prof. L.J. Sluys and Dr. A. Simone. Being allowed to work in such a fruitful, international atmosphere surrounded by the beautiful city of Delft was a privilege. I would like to thank my colleagues Peter Moonen, Frans van der Meer, Ronnie Pedersen, Oriol Lloberas Vals, Ilse Vegt, Xuming Shan, Kristian Ølgaard, Mehdi Nikbakht, Nguyen Tien Dung, Huan He and Zahid Shabir for all the discussions and the good time we had. Special thanks are directed to Vinh Phu Nguyen, whose Jem/Jive knowledge and implementations helped me a lot and build the basis for my own Jem/Jive coding, and to Frank Everdij for his support with all the countless computer problems. Furthermore, I am very grateful for the trust, support and encouragement of Prof. K. Thoma, Dr. Chr. Mayrhofer and all my colleagues at Fraunhofer EMI.

The financial support by the Netherlands Science Foundation STW and the Ministry of Public Works and Water Management, that made this work possible, is gratefully ac-knowledged.

Frank Radtke December 2012

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Chapter 1 Introduction

1.1 Aims and scope

The objective of this thesis is the development of a finite element based approach to model fibre-reinforced cementitious composites by including relatively large numbers of discrete short, thin fibres into a continuus matrix without meshing the fibres. The ba-sic idea is shown in Figure 1.1: fibres, treated as discrete entities, are superimposed on a background mesh describing the continuous matrix material —fibres and background mesh do not coincide. The proposed numerical techniques are intended as computational tools for numerical analysis of materials on a meso-scale rather than modelling of struc-tural members or complete structures. Apart from fibre-reinforced cementitious compos-ites (FRCCs) such as fibre-reinforced concrete (FRC) other fibre-reinforced composcompos-ites could be analysed using the proposed techniques. The idea of modelling each individual fibre is based on the assumption that this is the most direct way of representing the influ-ence of the fibre orientation and distribution in a specimen on its mechanical properties including its failure behaviour.

FRCCs consist of a cement matrix, aggregates and reinforcement as shown in the picture of a steel fibre-reinforced concrete sample in Figure 1.2. Reinforcement refers

discrete fibres matrix background mesh fibre-reinforced composite fibres

fibres do not coincide with mesh

Figure 1.1: Modeling of discrete fibres embedded in a continuous matrix material by superimposing the fibres on a background mesh discribing the matrix material.

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to different types of fibres but it might also include steel bars or textiles. Aggregates are mostly of small diameter, if they are at all present and they are therefore neglected in this work. Important parameters in the material design are fibre type, fibre-matrix in-terface [62], and fibre orientation and distribution in a specimen. Mechanical properties and failure behaviour are influenced by the design parameters [12, 68, 106]. Another important aspect is the flaw distribution in the cementitious matrix [117]. In this work, it is assumed that flaws mainly serve as damage initiation points, which are then accounted for in a simplified way by using arbitrary matrix strength distributions. A detailed de-scription of the different constituents and their influence on the mechanical behaviour of the composite material can be found e.g. in [12, 112].

The mechanical behaviour of cementitious composites such as unreinforced concrete is characterised by its brittleness combined with a rather high compressive and low ten-sile strength. It is possible to enhance the tenten-sile strength, the ductility, and its ability to dissipate energy by adding fibres. As depicted in Figure 1.2, the response of FRCCs under tensile loading can range from strain softening to strain hardening accompanied by localised or distributed cracking depending on the material design. An example of FR-CCs are engineered cementitious composites (EFR-CCs) [61] which show strain hardening instead of softening behaviour and distributed crack patterns instead of localised crack-ing. Taking advantage of their superior material behaviour such as their ductility and high tensile strength, FRCCs are often used [12, 59] in industrial floors (slabs on grade), in tunnels (fibre shotcrete), in precast concrete and in earthquake resisting buildings. In thin walled structures, where there is limited room for the use of normal reinforcement, fibre reinforcement might be advantageous for e.g. crack-width control.

The proposed approaches are currently limited to two dimensional, quasi-static analy-sis and small deformations. Furthermore, the simplest possible constitutive relations that allow to represent the basic behaviour of FRCCs are used. The approaches are applied to model FRCCs and it is shown that they can represent the influence of fibres on character-istic mechanical properties of FRCCs such as distributed cracking and strain hardening within the specified limitations.

1.2 Literature overview

An overview of different modelling approaches applicable to FRCCs, placing the pro-posed new approaches in their context, is given. The focus lies on the possibility of the models to handle large numbers of discrete fibres or rebars in an efficient way and to represent the typical mechanical behaviour of FRCCs as described in Section 1.1.

Different continuum models have been developed to model fibre-reinforced concrete. Constitutive models such as damage models and microplane models are able to represent the overall mechanical behaviour of fibre-reinforced concrete [11, 34, 41, 79]. These models are computationally efficient and can be integrated in existing approaches to model structures. But they can account for the structure of the material, as for exam-ple the fibre distribution, only in a phenomenological way. These continuum models are

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fibre

flaw

concrete matrix load

distributed crack pattern

localised crack pattern displacement concrete fibre−reinforced concrete load strain hardening strain softening high performance fibre−reinforced concrete

Figure 1.2: Tensile test of a fibre-reinforced concrete sample and the corresponding qual-itative load-displacement curves and crack patterns of concrete, fibre-reinforced concrete and high performance fibre-reinforced concrete.

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random triangular lattice fibre beam lattice beam lattice node fibre node bond beam

Figure 1.3: Example of a random lattice to model fibre-reinforced material.

well suited to model fibre-reinforced concrete on the macro-scale, thus on the scale of structural members. They are not well suited to study the structure of the material itself.

Rigid particle models or lattice models are used to model the structure of materials [15, 21, 55, 95–97, 103]. In [13, 14, 63, 112, 120] numerous examples studying the behaviour of fibre-reinforced concrete using lattice models are described. They are natu-rally suited for the representation of fibres as depicted in Figure 1.3, but they also depend on the representation of the matrix by lattice systems. Consequently, the solution depends on the lattice configuration. Elastic homogeneity cannot be reproduced locally [89] and some material parameters like the Poisson’s ratio are linked to the properties of the lat-tice [114]. Furthermore, these approaches may suffer from discretisation dependency.

Analytical models have been developed to describe fibre-reinforced concrete espe-cially for material development [12, 61, 64, 74]. They are largely based on a micro-mechanical representation of fibre pull-out behaviour (refer to Figure 1.4). Some of these models have been applied successfully in the development of FRCCs such as engineered cementitious composites (ECCs) [61, 64]. But mostly they do not directly account for the influence of fibre distributions and orientations on mechanical properties of the composite material.

A similar approach —but using numerical techniques to solve the problem— is to re-solve each fibre and the micro-structure of the matrix material around the fibre using con-tinuum mechanics, the finite element method and conforming meshes [39, 72, 82, 119]. Although it allows to represent the micro-structure of the material in detail, computations become extremely expensive. Hence, it only allows the study of systems with limited numbers of fibres or rebars. This prevents the analysis of the influence of fibre distribu-tions on the mechanical properties of FRCCs.

Instead of modelling the influence of fibres or rebars on the overall mechanical be-haviour of a specimen, the assumption that fibres or rebars are active only during cracking can also be used as a starting point. In this case, fracture mechanics approaches might be of advantage. For example, cohesive zone models have been developed to model cracks in concrete [46]. They have also been applied to fibre-reinforced concrete by adapting the

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pull−out distance pull−out force matrix debonding fibre pull−out force pull−out distance deformation of the hooked fibre end

Figure 1.4: Analytical modelling of the fibre pull-out problem in order to optimise the material design of FRCCs [53, 64].

cohesive forces representing fibres

bridging a crack representing a single fibre or rebar crack bridging forces

crack crack

crack−bridging forces cohesive forces

rebar

Figure 1.5: Schematic drawing of a cohesive crack model (left) and a bridged crack model (right).

cohesive law [52, 65, 107, 119] such that it accounts for the influence of the matrix mate-rial as well as for the influence of the reinforcing phase. However, since cohesive forces are mostly applied continuously along the crack surface (refer to Figure 1.5) neither the discrete distribution of fibres nor their load distributing effect when bridging cracks can be represented. In contrast to cohesive zone models, which consider the material as homo-geneous composite, bridged crack models treat reinforced concrete as a bi-phase material in the sense that the reinforcement bars or individual fibres are represented by discrete forces [19, 20] (refer to Figure 1.5). They have been applied successfully to model re-inforced concrete [18]. Cracks passing a reinforcement bar lead to the typical drop in the force displacement curve that can also be found in experimental results (for example ECCs [60]). Since the forces are applied to the crack surfaces, therefore neglecting the influence of fibre length and load distributing effects, it seems to be difficult to simulate typical properties of fibre-reinforced concrete as for example the formation of distributed

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fluid

forces representing the action of the solid on the fluid

solid moving through the fluid immersed boundary method

discretization

modelling of pre−stressed concrete

concrete tendon

ancre plate

forces acting from tendon to matrix

Figure 1.6: Schematic drawing of a membrane moving through a fluid modelled using the immersed boundary method [80, 81] (left) and the representation of tendons using their reaction forces [57, 67] (right).

cracking using bridged crack models.

In [92] a cohesive crack model has been combined with discrete forces representing reinforcement as it is done in bridged crack models. Instead of applying them to the crack surface, the forces are applied in the centre of the shear force distribution along the reinforcement bar that bridges the crack. The smeared crack tip approach is used to model cohesive cracks [8] and this approach seems to be difficult to extend to compli-cated geometries with multiple interacting cracks. In [77] this approach is applied in the context of the boundary element method to model lightly reinforced concrete. In this case the forces representing the reinforcement bars are again applied to the crack faces which neglects the load distributing effect of the reinforcing phase. This idea of representing the action of the reinforcement within the composite material by the use of the reaction forces from reinforcement to matrix has been applied earlier in a different fashion for modelling and design of reinforced [57, 67] and especially prestressed concrete [90] (re-fer to Figure 1.6). In the field of fluid structure interaction the immersed boundary method exploits this approach [80, 81]. Boundaries of a solid which is immersed in a fluid are represented by forces applied to the nodes of the discretisation as shown in Figure 1.6. In this way, even complicated geometries can be modelled using rather coarse discreti-sations, which can also be advantageous for the modelling of FRCCs as intended in this work and specified in Section 1.1.

In the first approach proposed in this thesis (Chapter 2) [85], discrete fibres are added to a continuous matrix by applying the reaction forces between fibre and matrix material to the matrix as shown in Figure 1.7 —thus, discrete fibres are treated without meshing them or increasing the number of degrees of freedom of the system. The interface be-tween fibre and matrix is described by incorporating the fibre pull-out behaviour into the simulation. This data can either be directly measured in fibre pull-out experiments or it can be modelled on a lower scale for example using micro mechanical models [64]. In this way, the approach provides an interface to include lower-scale information into

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finite element mesh cement matrix

fibre

crack fibre−forces

representing the fibre 00 00 11 11 0 0 1 1 00 00 11 11 0 0 1 1 00 11 01 0 0 1 1 00 00 11 11 00 00 11 11 00 11 0 0 1 1 0 1 0 1 0 0 1 1 0 0 1 1

Figure 1.7: Fibre-force approach: a fibre bridging a crack in a sample under tensile load; the fibre is represented by the reaction forces from the fibre to the matrix (fibre-forces) applied to the finite element mesh.

the computation in a very efficient and simple way. The approach allows to model sam-ples with large numbers of fibres treating each fibre individually with limited computa-tional effort. The crack bridging effect of fibres and the development of distributed crack patterns can be represented. Depending on the fibre type, including the fibre-matrix in-terface, the amount of fibres and their distribution in a sample in combination with the properties of the cement matrix the range between strain softening and strain hardening can be captured. Recently, another approach based on the immersed boundary method for the modelling of fibre-reinforced concrete has been published [83].

A discrete treatment of rebars or fibres is also employed in models originally devel-oped for “normal” reinforced concrete, (see for example [66, 67, 88]). One of the most straightforward approaches is to explicitly consider fibres or reinforcement bars in finite element approaches as shown in Figure 1.8. In the simplest case, the discretisations of bars and matrix material have to be conforming. This makes the mesh generation pro-cess more difficult and tedious when considering large numbers of fibres. In [115] fibres are modelled as beams using non-conforming meshes by mapping the degrees of free-dom of beam elements to solid elements. Distributed crack patterns in bending could be reproduced and good agreement with experimental results was achieved. But neither the interface between fibre and matrix nor the strain-softening behaviour of the concrete matrix was taken into account, yet. In most cases beam or truss elements, representing reinforcement bars, are coupled to solid elements, representing matrix material, using in-terface elements [84, 92, 93, 102]. Other approaches [4, 25, 48] employ slip degrees of freedom to represent the influence of the reinforcement on the mechanical behaviour of the composite material. A number of different non conforming approaches have already been implemented into commercial codes as for example shown in [116].

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conform-fibre element matrix element

matrix node

connecting element

fibre node

Figure 1.8: Modelling of fibres or rebars in a concrete matrix using an embedded formu-lation.

ing meshes or coupling fibres via interface elements to the matrix, partition of unity finite element (PUFEM) like methods [3, 28, 70, 75] can be employed. They allow to represent complicated geometries without using conforming meshes by enhancing the function space used to approximate the solution. This is applied for example to model cracks [73, 118] or inclusions in a continuum matrix [72, 111]. Approaches using parti-tion of unity finite element methods have been successfully applied to model reinforced materials like reinforced concrete [23, 44, 45].

The second approach presented in this thesis (Chapters 3, 4 and 5) [86, 87] is based on the PUFEM [3, 28, 70, 75]. As described above, it allows to model material boundaries including the interfaces in a mesh independent way [45, 109] by enhancing the function space used to approximate the solution. In the proposed approach an enrichment function is developed that represents a discrete fibre embedded in a continuous cement matrix as shown in Figure 1.9. Avoiding the necessity to create conforming meshes including dis-crete fibres helps to keep the approach computationally efficient enough to model FRCC samples considering large numbers of fibres. However, extra degrees of freedom are needed to describe slip between fibre and matrix again leading to higher computational costs. Slip between fibre and matrix and stress and strain field around the fibre are re-solved in detail. The theoretical framework of the PUFEM [70, 73, 109] allows to easily extend this approach to model thick fibres without the need of conforming meshes, or to use additional or different functions as enrichment to improve its convergence behaviour.

1.3 Outline of the thesis and introduction to the

devel-oped approaches to model fibre-reinforced concrete

In Chapter 2 the first proposed approach [85] is presented to model fibre-reinforced con-crete representing the fibres by their reaction forces as shown in Figure 1.7. Basically, the fibre-force approach solves the problem of modelling fibre-reinforced concrete, rep-resenting each fibre individually. As shown in Chapter 2, the fibre-force approach is capable of modelling the mechanical behaviour of FRCCs as stated in Section 1.1. It

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x y

enrichment

specimen with embedded fibre enrichment representing a fibre

cement matrix

finite element mesh fibre x y 0 0 1 1 00 00 11 11 00 00 11 11 0 0 1 1 0 1 01 0011 01 0 0 1 1 0 0 1 1 0 1 01

Figure 1.9: PUFEM based approach: the fibre is represented by an enrichment function.

is computationally efficient since fibres are not discretized meaning that the number of degrees of freedom of the system does not increase with an increasing number of fibres. No conforming meshes have to be created. Furthermore, the approach allows to use the available information describing the fibre-matrix interface, e.g. from fibre pull-out ex-periments, directly in the numerical model. However, it is inherent to the approach that neither the fibre nor the interface between fibre and matrix is modelled individually. In-formation on their constitutive behaviour needs to be integrated into the description of the fibre pull-out behaviour. Furthermore, detailed information on quantities like strain or stress along the fibre in the fibre-matrix interface is not available since neither fibre nor interface is actually modelled.

To circumvent these shortcomings, a second novel approach based on the partition of unity finite element method is presented in Chapter 3. Compared to the approach presented in Chapter 2, fibre and fibre-matrix interface are now directly modelled by using enrichment functions representing the fibres (refer to Figure 1.9). Extra degrees of freedom are needed to describe slip between fibre and matrix, which leads to higher computational costs. However, slip between fibre and matrix and stress and strain field around the fibre are resolved in detail. The use of the enrichment function to describe short thin fibres avoids the meshing of fibres.

In Chapter 4 the PUFEM approach developed in Chapter 3 is coupled to damage mechanics [87] to represent cracking in the concrete matrix. Damage models are compu-tationally efficient and provide reasonable capabilities to model concrete. Furthermore, a wide range of different formulations is available. To resolve the issue of mesh dependency a gradient-enhanced damage model is employed. In this way, both the representation of

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discrete fibres, including slip between matrix and fibre, and damage of the matrix can be modelled independently of the mesh. This increases the quality of the resulting crack patterns when considering specimens with complex geometries, arbitrary distributions of fibres and flaws, distributed matrix cracking, fibre debonding and tunnelling cracks around fibres.

In Chapter 5 the approach developed in Chapters 3 and 4 is applied to model rein-forced concrete. The results are compared to experiments published in literature. It is shown that this approach can capture the mechanical behaviour of fibre-reinforced con-crete in a reasonable way. Furthermore, it can be easily applied to model “normal” re-inforced concrete as well as concrete rere-inforced with combinations of fibres and rebars. Apart from providing some assessment of the proposed approach it is shown how the numerical approach can be used to analyse material behaviour.

The thesis is organised in and based on journal papers containing the main results of the work performed during this PhD study. The notation is explained within each paper. Due to the organisation of the thesis in journal papers, the notation might not be completely consistent.

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Chapter 2 Fibre-force approach

∗

2.1 Introduction

This chapter has been published in [85]. In this paper we present a novel approach to model fibre-reinforced cementitious composites (FRCCs). We propose a computational framework to treat discrete fibres embedded in a quasi brittle matrix such as concrete. This enables us to analyse the impact of a discrete distribution of fibres on the mechanical properties of the material. Nevertheless, the fibres are not explicitly discretized to ensure computational efficiency of the model.

Many approaches have been developed to study the behaviour of fibre-reinforced con-crete at different scales. On the macro-scale, damage, crack and microplane models are well suited to represent the behaviour of FRCCs [11, 34, 41, 79], while rigid particle or lattice models are used on meso- and micro-scale [14, 63, 97]. A sequential multi-scale framework for FRCCs has been presented by Kabele [50]. His approach links analytical and computational models covering scales from micro- to macro-level. With respect to the development of FRCCs, analytical models have been applied successfully [61, 64, 74].

A discrete treatment of fibres can be achieved by employing models originally devel-oped for reinforced concrete (see for an overview [67]). One way is to explicitly consider fibres or reinforcement bars. In finite element approaches beam or truss elements, repre-senting reinforcement bars, are coupled to solid elements, reprerepre-senting matrix material, using interface elements [84, 93, 102]. In the simplest case, the discretization of the bars and the matrix material have to be conforming. This makes the mesh generation process more difficult and tedious when considering large numbers of fibres. A number of differ-ent non conforming approaches have already been implemdiffer-ented into commercial codes as for example Atena [116]. Vanalli et al. [115] have included fibres modelled as beams using non-conforming meshes by mapping the degrees of freedom of the beam elements to the solid elements.

In the field of fracture mechanics, bridged [18, 19] and cohesive crack [46, 107, 119] ∗_{This chapter is based on [85]}

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models can be used to explain the mechanical behaviour of reinforced and fibre-reinforced concrete. Ruiz [92] has combined a cohesive crack with discrete forces; these forces represent the effect of reinforcement bridging a crack as depicted in Figure 2.1. When modelling one dominant crack and a reinforcement layer, good agreement with experimental results is achieved.

The concept of representing the action of reinforcement within a composite material by using reaction forces from reinforcement to matrix has been applied earlier in the modelling and design of reinforced and prestressed concrete [67]. In the field of fluid-structure interaction the Immersed Boundary Method exploits the same idea [80, 81]. Boundaries of a solid, immersed in a fluid, are represented by forces applied to the nodes of the discretization. In this way even complicated geometries can be modelled using rather coarse grids.

In our approach, we split the composite material into its two major components: ma-trix and fibres. As shown in Figure 2.2, the mama-trix material is represented by a background mesh while fibres are treated as discrete entities which are not related to the underlying discretization of the background mesh. We continue to follow the idea of representing the fibre action and the behaviour of the fibre-matrix interface by using the reaction forces from the fibre to the matrix as illustrated in Figure 2.1. To represent the matrix material we employ a regularized damage model. Our approach is computationally efficient, can be easily adapted to existing simulation codes, and can represent the interaction between the structure of FRCCs and their mechanical properties.

The new aspects of our approach are the introduction of the concept discussed for example by Ruiz [92] and, in a different context, by Peskin [80, 81] into damage me-chanics using the finite element method, and the direct inclusion of information from the micro-scale regarding the fibre matrix interface, by using fibre pull-out relations, to model fibre-reinforced concrete on the meso-scale.

2.2 Approach

2.2.1 Problem statement

Cementitious materials are characterised by a quasi-brittle response under tensile loading. The inclusion of fibres improves mechanical properties such as peak strength, ductility and residual load carrying capacity as shown in Figure 2.3.

While plain concrete (C) and fibre-reinforced concrete (FRC) show strain softening behaviour accompanied by localized cracking, high performance fibre-reinforced con-crete (HPFRC) develops strain hardening behaviour and distributed cracking.

A model intended for the analysis and design of fibre-reinforced cementitious com-posites has to represent these phenomena in a way that clearly links them to the underly-ing physical processes.

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reinforcement bridging a crack reinforcement crack physical problem fibres bridging cracks fibre crack damage zone

cohesive crack model crack model bridged crack force bridging cohesive crack

with bridging forces

force bridging

fibre position

damage model with bridging forces representing fibres position of reinforcement proposed model existing models

Figure 2.1: Different approaches to model reinforcement bridging a crack (top left) and fibres bridging cracks (top right): bridged crack model [18] (bottom left), cohesive crack model with bridging forces [92] (bottom middle) and proposed approach (bottom right).

matrix fibres fibre reinforced

concrete

background mesh

discrete fibres

fibres do not coincide with mesh

Figure 2.2: Matrix material and discrete fibres are merged to obtain a numerical fibre-reinforced concrete.

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macro cracks stress micro tensile cracks displacement HPFRC FRC C

Figure 2.3: Characteristic force displacement plots and crack patterns of concrete (C), fibre-reinforced concrete (FRC) and high-performance fibre-reinforced concrete (HPFRC).

2.2.2 Model description

In order to develop a model that can represent the features described in Section 2.2.1, a simplified description of the physical processes governing the material behaviour is needed. Consider a plain concrete sample loaded in tension as shown in Figure 2.3. At first, the response of the material is linear elastic. With increasing load, micro cracks start growing, initiating from weak spots like pores, defects or existing cracks. They coalesce and form macro cracks. Finally, the material loses its load carrying capacity. In fibre-reinforced cementitious composites with a fibre volume up to 5 or 6% the main action of fibres is crack bridging. When a crack reaches a fibre, it is either arrested or it passes the fibre. In the latter case the fibre bridges the crack. Depending on the fibre type and on the interface between fibre and matrix the following three characteristic failure modes can develop: the crack opens and the fibre is pulled out of the matrix, fibre rupture occurs, or the crack grows without opening. Considering the latter case, the crack can pass through the whole sample without any drop in the load carrying capacity since fibres transfer the forces across the crack. With a further load increase other cracks will form. Their location is determined by the occurrence of weak spots in the matrix and concentrations of the forces redistributed from the fibres to the matrix. The process of crack formation will continue until the material is fully saturated with cracks. Finally, one or more cracks become dominant and the material fails due to fibre pull-out or fibre rupture.

Three aspects governing the material behaviour can be identified using this simplified description of the composite: the cracking process in the matrix, which is linked to the distribution, number and size of weak spots in the material; the fibre type and especially the interface between fibre and matrix, which have a major influence on the properties of the composite material; and the amount of fibres, their distribution and orientation. Re-garding the distribution of weak spots and fibres in a sample, there exists not yet enough experimental data to include this information into numerical simulations (for some exper-iments on fibre distributions in FRC specimen see [105]). These aspects can be treated in

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a stochastic way for example using Monte Carlo simulations. In this contribution we do not focus on the stochastic aspects at the micro mechanical level but only on the mechan-ical description of fibre-reinforced concrete on the meso-scale.

The major steps required to build a mechanical model based on this material descrip-tion are shown in Figure 2.4. Within our approach we consider two components, namely fibres and matrix. The matrix material is described as a continuum, where pores and initial cracks are represented by weak areas or notches. In order to be able to deal with a large number of fibres at low computational costs, fibres are represented by means of their reaction forces. Thus, we neglect the fibres and keep only the matrix and the reac-tion forces from the fibre to the matrix, which we call for convenience fibre-forces. This assumption holds for fibre-reinforced cementitious composites with a limited amount of fibre volume. Furthermore, we initially assume that fibres contribute to the properties of the material only when they bridge a crack. This implies that fibres are neglected during the elastic phase. The effect of this assumption is shown in Section 2.4.3. To calculate the fibre-forces, when a fibre bridges a crack, we assume that they are equal to fibre pull-out forces. Fibre pull-out forces are measured during a fibre pull-out experiment as a function of the pull-out distance. We assume that the pull-out distance is equal to the opening of the crack, which is bridged by the fibre, at the intersection between fibre and crack. Apart from their experimental quantification, pull-out forces can be modelled either analytically [64] or numerically [82]. The introduction of the fibre pull-out behaviour provides a way to include information from lower scales–we might think for instance of the properties of the interface between matrix and fibre. A number of assumptions concerning the force distribution have been made in literature [64, 92]. For convenience and because of the rather short length of the fibres we assume that the fibre-forces can be lumped around the fibre ends. In reality however, the fibre-forces are distributed along the fibre.

2.2.3 Mathematical formulation

The mechanical system representing the material structure of FRCCs is shown in Fig-ure 2.5. We consider a cracked domain Ω loaded in tension, which is bridged by a fibre represented by fibre-forces. The virtual work equation describing the system reads as follows

δWint | {z }

internal virtual work

= δWext

| {z }

external virtual work

+ δWf ib | {z } fibre virtual work

, (2.1)

where the internal virtual workδWint describes the matrix material,δWf ibrepresents the action of the fibre on the matrix andδWext follows from the applied external loads.

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0 0 0 0 1 1 1 1 background mesh P P P ∆ ∆ P 000000 111111 000000111111 0 101 0 1 0 1 0 1010 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

fibre bridging a crack

fibre pull−out relation

analytical or experimental fibre pull−out experiment

= pull−out distance

proposed model

load

damage bridged by a fibre

fibre−forces load

forces acting on the matrix forces acting on the fibre 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 000 000 000 000 000 111 111 111 111 111

Figure 2.4: Development of a simplified mechanical model for fibre-reinforced cementi-tious composites.

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y x fibcr Ω− _Ω+ fibcr Ωf Ω+ Ω− p+ _p− F fibre surface Ω F fibre−forces domain loaded domain

Figure 2.5: Mechanical representation of a cracked domain Ω (Ω=Ω+_{∪ Ω}−_{) bridged} by a fibre represented by fibre-forces.

Written in more detail Equation (2.1) reads Z Ω σσσ :δεεεdΩ | {z } δWint = Z Ω_f fff_ext·δuuu dΩf | {z } δWext +X nf ib Z Ω+ f ibcr ppp+(∆₎_·δ_u_{uu dΩ}+ f ibcr+ Z Ω− f ibcr ppp−(∆₎_·δ_u_{uu dΩ}− f ibcr | {z } δWf ib , (2.2)

whereσσσ is the Cauchy stress tensor,εεε is the linear strain tensor, fff_ext is the external load vector applied to the domain Ωf, uuu is the displacement vector (primary unknowns), nf ib is the number of fibres in the system, ppp+ and ppp−are the fibre-force vectors on each side of the crack, which are distributed over the fibre surface Ω+_{f ibcr} and Ω−_{f ibcr}, and ∆ is the fibre pull-out distance. The effect of fibres is added to the system by adding the fibre virtual work as stated in Equation (2.1). Practically, this means that the reaction forces from the fibres to the matrix are added to the external force vector as can be seen in Equation (2.2). In the following derivations we will elaborate only on the fibre terms on the Ω+_{f ibcr} side of the crack since the derivation of the fibre terms on the negative side of the crack is similar.

2.2.4 Constitutive model for the concrete matrix

The material behaviour of the matrix is assumed to be linear elastic followed by strain softening when damage starts. We use an isotropic damage model with exponential soft-ening to describe the matrix behaviour. Its implementation and use are simple and the model is computationally efficient. The constitutive equation reads

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where ω is the damage variable ranging from zero to one and DDD is the elastic material stiffness tensor.

The damage evolution law

ω = f(κ) (2.4)

is described by an exponential softening law [49] in which

ω =    0 κ<ε₀ 1−ε0 κe −κ−ε0 εf −ε0 κ_≥ε 0, (2.5)

where ε0= ft/E, ft is the tensile strength, E is the Young’s modulus, εf is a parame-ter affecting the slope of the softening branch, and κ is a history variable equal to the largest converged equivalent strain. In some cases a residual stress level is desirable. The exponential softening law is modified by

ω = (1−σresidual

Eκ ) if κ >ε0−(εf−ε0)ln(

σresidual ε0E

), (2.6)

in whichσresidual is the residual stress level. Allowing for a residual stress level implies

that the matrix can transfer stresses even if it is almost fully damaged and the damage variableω is close to one. Thus, the introduction of a residual stress level has no physical meaning in this case, but it is intended to prevent the stiffness matrix from becoming ill-posed when a large number of elements is close to failure (this implies that entries in the stiffness matrix related to "fully" damaged elements are different from zero).

Two expressions of the equivalent strain have been employed: the modified von Mises and the Mazars definitions. The modified von Mises definition for the equivalent strain ˜ε [26] reads ˜ ε= k− 1 2k(1− 2ν)I1+ 1 2k s (k− 1)2 (1− 2ν)2I 2 1+ 6k (1+ν)2J2, (2.7) with I1=tr(εεε), J2=tr(εεε·εεε)− 1 3tr 2₍_εεε₎_, _(2.8)

where k represents the ratio of the compressive strength fc, and the tensile strength ft, ν denotes the Poisson’s ratio and I1 and J2 are invariants of the strain tensor. Mazars

criterion is implemented as given in [49] ˜

ε=phεεεi : hεεεi, (2.9)

whereh i denote McAuley brackets. Regularization of the governing equations is needed to avoid mesh sensitivity of the solution. In this contribution three regularization tech-niques are employed: fracture energy regularization and non-local damage models of the integral and the gradient type.

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The simplest technique is fracture energy regularization [16]. Within this concept the damage evolution law is adapted depending on the element size such that the dissi-pated energy is constant upon mesh refinement. In the damage evolution law given in Equation (2.5) the softening parameterεf is modified as follows:

εf = λ h(εf− ε0 2) + ε0 2, (2.10)

whereλ denotes the width of the damage zone and h is an equivalent element size. Using triangular elements the equivalent element size h is chosen to be the radius of the in-circle of the triangle. Since the model stays local the width of a damage zone is dependent on the element size and the crack path follows the pattern of discretization.

Non-local damage models [10] can overcome these disadvantages. We use the integral type as described in [49]. The equivalent strain

ε = Z

V

αε˜dV , (2.11)

is chosen as non-local quantity. In this expression, α is a normalised version of a non-local weight function α0 and V is its support. As non-local weight function α0, the

piecewise polynomial bell-shaped function

α0(r) = (

(1−_Rr22)2 if |r| ≤ R

0 if |r| > R, (2.12)

is employed, in which r is the distance between a point and the point under consideration. R is called the interaction radius and determines the maximum distance between two points affecting each other. By using this approach the damage path and width of the damage zone become independent of the discretization. The parameter R refers to the length scale of the material.

As a third regularization technique, the implicit gradient damage model is employed as described in [78]. The equivalent strain is used as non-local quantity and is determined by solving

ε−1 2l

2_∇2_ε ₌_ε_˜ _(2.13)

as an additional equation, where l is the length scale of the material.

2.2.5 Constitutive model for the fibre and the fibre-matrix interface

Fibres and fibre-matrix interfaces are represented by the reaction forces from the fibres to the matrix, which we call for convenience fibre-forces (see Figure 2.4). We assume that these forces can be represented by fibre pull-out forces. While the pull-out forces are measured at the end of the fibre that is pulled out of the matrix in a certain direction, the

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fib n ) x f( p _P x

Figure 2.6: Decomposition of reaction force vector from fibre to matrix ppp (fibre-forces) into scalar fibre pull-out force P, direction vector of the fibre nnnf iband distribution function of the fibre-forces f(xxx).

fibre-forces are distributed over the surface of the fibre. The system, shown in Figure 2.6, is described by the equation

Z

Ω+

f ibcr p

pp+dΩ+_{f ibcr}=PPP+, (2.14)

where ppp+represents the tractions acting on the matrix along the fibre surface Ω+_{f ibcr}, and P

PP+is the fibre pull-out force vector. Splitting the fibre-forces into a distribution function f+(xxx), a direction vector nnn+_{f ib}(xxx)and the scalar fibre pull-out force P leads to

Z Ω+ f ibcr ppp+dΩ=P Z Ω+ f ibcr f+(xxx)nnn+_{f ib}(xxx)dΩ+_{f ibcr} (2.15) with Z Ω+ f ibcr f+(xxx)dΩ+_{f ibcr}=1. (2.16)

Both the distribution function and the direction vector are a function of the position. While the fibre pull-out force can still be measured and/or modelled reasonably well, measuring the distribution function and the direction vector depending on the position is difficult. Since the fibres are relatively short at the scale of interest, we assume that the fibre-forces are concentrated at the fibre end. Furthermore, we assume that the fibre direction vector nnn+_{f ib}(xxx)is constant. We can therefore rewrite Equation (2.15) as

Z

Ω+

f ibcr

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where P is the pull-out force and n nn+_{f ib}= nf x nf y =− cosαf ib sinαf ib , (2.18)

is the direction vector at each fibre end point, with αf ib denoting the angle between the fibre at a fibre end point and the positive x-axis. Assuming that

n n

n+_{f ib}=−nnn−_{f ib}, (2.19)

the fibre direction vector can be written as

nnnf ib=

xxxf ib1− xxxf ib2 ||xxxf ib1− xxxf ib2||

, (2.20)

where xxxf ib1 and xxxf ib2 are the position vector of the first and the second fibre end point, respectively.

The only unknown that still has to be determined is the fibre pull-out force. It is assumed that the fibre pull-out force P is a function of the pull-out distance ∆. P is either described by the direct use of experimental data or by the employment of a micro-mechanically based model, like the one proposed by Lin et al. [64].

For the direct use of experimental data, it is assumed that the pull-out force is available in pairs of data (P and ∆), between which the force is linearly interpolated. In this way not only experimental data but all possible shapes of a fibre pull-out curve can be included into the model.

Another possibility is the use of the micro-mechanical based model developed by [64]. They presented a model for the pull-out of fibres from a cementitious matrix. Their model consists of one part for the debonding range and one part for the pull-out range. The debonding stage is formulated for 0≤ ∆ ≤ ∆0(with ∆0 defined in Equation (2.23))

as P(∆_{) =} s π2τ 0Ef ibd3_{f ib}(1+η) 2 ∆+ π2_G dEf ibd3_{f ib} 2 , (2.21)

withη=Ef ibVf ib/(EVm), E the matrix elastic modulus, Vm the matrix volume fraction, Vf ibthe fibre volume fraction, Ef ibthe fibre elastic modulus, P the fibre pull-out force, ∆ the pull-out distance, df ibthe fibre diameter, Gd the fibre matrix chemical bond strength (defined as energy per area [64]), andτ0the frictional stress on the debonded interface.

The pull-out stage defined in the interval ∆0≤ ∆ ≤ lemis described by

P(∆_{) =}π_d_{f ib}τ₀_[1₊β∆− ∆0

df ib

](lem− ∆+∆0), (2.22)

whereβ is the fibre-matrix interface slip-hardening parameter, lemis the fibre embedded length and ∆0is the pull-out distance at which debonding is completed defined as

∆₀₌ 2τ0l 2 em(1+η) E_{f ib}d_{f ib} + s 8Gdlem2 (1+η) E_{f ib}d_{f ib} . (2.23)

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Effects like snubbing and spalling are neglected within this contribution for the sake of simplicity but can be added rather easily within the given framework by extending the fibre pull-out relation, as shown for example in [64], and then by applying snubbing forces at the intersection between fibre and crack. The major difficulty regarding the former is that a crack direction has to be extracted from the damage field to compute the angle between fibre and crack, although in this case it is maybe more convenient to use models able to represent discrete cracks directly.

Furthermore, an unloading function has to be added since, depending on the crack pattern, fibres will unload during the loading process of the whole sample. In this contri-bution we have employed simple linear unloading. This unloading function, intended for local unloading during the global loading process, reads:

P(∆_{) =}P∆max

∆_max∆, (2.24)

where P∆_max is the pull-out force at the maximum reached pull-out distance. This

for-mulation implies that no irreversible deformation occurs. Taking into account that fibres are pulled out of the matrix during the loading process and that they are most probably not completely pushed back into the matrix when unloading takes place, the validity of this assumption can be doubted. On the other hand thin short fibres often behave like very slender trusses that cannot resist compression very well when being pulled out of the matrix. This means that the deformation of the sample remaining after unloading will be comparetively small. In conclusion, the employed formulation for unloading is valid for short thin fibres whose compressive load bearing capacity is negligible after having been pulled out of the matrix.

2.2.6 Coupling between fibre and matrix models

The coupling between the constitutive equations for the concrete matrix, the fibre and the fibre-matrix interface is established by the fibre pull-out distance. As already mentioned the fibre pull-out distance is assumed to be equal to the opening of the crack bridged by a fibre at the fibre crack intersection. Thus it directly depends on the displacement field. The pull-out force in turn is a function of the pull-out distance. Hence, to compute the pull-out distance and in turn the pull-out forces, a crack opening has to be extracted from the damage model used for the matrix description. Dufour et al. [31] have proposed an approach based on the equivalence between the strain field computed using a regularized damage model and a strong discontinuity approach. We use a fibre which bridges a crack as measuring basis to estimate the crack opening as shown in Figure 2.7 assuming that the fibre is approximately perpendicular to the crack. If we consider a fibre which bridges a damage zone representing a crack, we can assume in the simplest case that the crack opening and thus the fibre pull-out distance ∆ is equal to the elongation of the fibre ˜u_{f ib}:

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fc

ω

damage at the intersection between fibre and damage zone

is the cracked sample original sample elongation fibre F F damage zone damaged sample

Figure 2.7: Sample including a single fibre loaded in tension: comparison between frac-ture and damage mechanics approach in order to extract a crack opening from the damage model.

Since the fibre is not explicitly modelled, ˜uf ibis expressed as

˜

uf ib=||uuuf ib1− uuuf ib2||, (2.26)

with uuuf ib1and uuuf ib2being the displacement of the matrix at the locations of the fibre end points. Since the fibre does not coincide with the background mesh describing the matrix, uuuf ib1and uuuf ib2have to be interpolated from the background mesh as described in further detail in Section 2.3.3. This formulation implies that the fibre is active as soon as it is elongated, which is in contrast with the assumptions made in Section 2.2.2. If we require that a fibre is only active when it bridges a crack, we have to scale the elongation of the fibre with the bridged damage, so that the pull-out distance is only non-zero if a damaged area is bridged. This leads to

∆₌ω_{f c}u˜f ib, (2.27)

in which ωf c is the damage value at the intersection between fibre and damage zone. ωf c is computed by searching for the largest damage value, averaged over each element, within the set of elements bridged by the fibre. If the part of the domain which is bridged by the fibre is not damaged,ωf c=0, no crack opening is predicted. If the domain is fully damaged,ωf c=1, the whole displacement between the fibre end points is assumed to be localized in the crack–thus the crack opening is equal to the fibre elongation.

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2.3 Numerical aspects and implementational issues

The system of equations describing matrix and fibres is solved using the finite element method. Since the equations are nonlinear, a Newton-Raphson scheme is employed. Next, we focus on linearization and discretization of the governing equations paying par-ticular attention to the fibre terms and skipping the standard terms.

2.3.1 Linearization of the fibre terms within the governing equations

The linearization of the fibre terms using a fracture energy regularized local damage model is elaborated in detail in Appendix A.1. The derivations for the non-local damage model of integral and gradient type are carried out in a similar way. Since they yield a rather lengthy formulations we present here the linearization of the terms which originate from the addition of fibre-forces to the fracture regularized local continuum damage me-chanics equations. For the fibre pull-out description we use Equation (2.27). This leads to the following result:

D∆uuu[δW_{f ib}+] =∂ P ∂∆nnn + f ib ˜ uf ib ∂ff c ∂κf c ∂κf c ∂ε˜_{f c} ∂ε˜_{f c} ∂εεεf c :∇s_∆uuuf c | {z } Term I ·δuuu + ∂P ∂∆nnn + f ib ωf c ˜

u_{f ib}(uuuf ib1− uuuf ib2)·(∆uuuf ib1− ∆uuuf ib2)

| {z }

Term II

·δuuu, (2.28)

whereδW_{f ib}+ represents the fibre virtual work on the positive side of the crack Ω+ and D∆uuudenotes the directional derivative. The index f c stands for quantities at the intersec-tion of fibre and crack or damage zone, respectively. ff cis the damage evolution law,κf c is an internal variable describing the largest equivalent strain reached, ˜εf cis the equivalent strain,εεεf c denotes the linear strain tensor and ∆uuuf cis the displacement increment at the intersection between fibre and crack. ∆uuuf ib1and ∆uuuf ib2are the corresponding displace-ment incredisplace-ments at the particular fibre end points. Term I in Equation (2.28) originates from the linearization of the damage variableωf c in Equation (2.27) and Term II from the linearization of the elongation of the fibre ˜uf ib as given in Equation (2.26). Since the effect of the energy regularization is limited to the damage evolution law, it does not explicitly appear in Equation (2.28). Using a more sophisticated model, like e.g. the gra-dient enhanced damage model [78], yields extra terms related to the linearization of the nonlocal equivalent strain. The fibre virtual work on the negative side of the crackδW_{f ib}− can be written in the same way.

2.3.2 Discretization of the fibre terms within the governing equations

Here we formulate the discretization of the fibre terms within the governing equations using the fracture energy regularized local damage model. For the non-local models the derivation can be done in a similar fashion.

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The discretized system of equations can be written in matrix notation as KKKT∆uuu= fffextj+1+fff j f ib− fff j int, (2.29)

where KKKT is the tangential stiffness matrix, originating from the linearized Equa-tion (2.28), ∆uuu is the displacement increment, fff_ext describes the external forces, fff_{f ib} denotes the forces originating from the fibres and fff_int is the internal force vector. The load step is denoted by j. In the next derivations this index is dropped for convenience. The fibre-forces can be expressed as

fff_{f ib}=[

nf ib

(Pnnn+_{f ib}+Pnnn−_{f ib}), (2.30)

whereSdenotes the assembly procedure and nf ibstands for the number of fibres. Apart from the standard matrix terms KKKmatrix, the tangential stiffness matrix KKKT contains addi-tional contributions originating from the linearized fibre terms KKK_{f ib}+ and KKK_{f ib}− and can be expressed as KKKT = [ nelem KKKmatrix−[ nf ib KKK_{f ib}+− [ nf ib KKK_{f ib}−, (2.31)

in which nelemis the number of elements. In Equation (2.31) the linearised fibre terms are subtracted from the stiffness matrix since the fibre-forces are added to the right-hand side of Equation (2.29). Their sign is determined by the fibre direction vector n_{f ib} as can be seen in Equations 2.33 and 2.35. We will elaborate only on KKK_{f ib}+ since KKK_{f ib}− is derived in the same way. The fibre stiffness matrix KKK_{f ib}+ can be split into two contributions according to the two terms of Equation (2.28):

K K

K_{f ib}+ =KKK_{f ib}+_{,Term I}+KKK_{f ib}+_{,Term II}. (2.32) The discretization of Term I in Equation (2.28) reads

KKK_{f ib}+_{,Term I} = ∂P ∂∆WWWP+nnn + f ibu˜f ib ∂ff c ∂κf c ∂κf c ∂ε˜_{f c}( ∂ε˜_{f c} ∂εεεf c )T_B_B_B f c∆uuuf c, (2.33) in which the matrix BBBf c is the strain-displacement matrix at the fibre crack intersection point. Since the fibres do not coincide with the discretization, all quantities at the fibre end points have to be averaged from the surrounding nodes or have to be distributed to them, as the fibre-forces P have to be distributed to the nodes surrounding the fibre end points. The matrix WWW_P+ contains weighting factors representing the influence of the nodes on the positive side of the crack Ω+ where the fibre-forces are distributed to. It is defined as WWW_P+ =          w_f+_1x 0 .. . ... w_f+_nx 0 0 w_f+_1y .. . ... 0 w_f+_ny          , (2.34)

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in which w_f+_1xrefers to the weight, attributed to the x-component of the fibre-force at the first node, to which the fibre-forces on the positive side of the crack Ω+ are distributed and n denotes the number of nodes the fibre-forces are distributed to. In Section 2.3.3 more details are given regarding the computations of the weighting factors.

The discretized version of the second term Equation (2.28) reads K KK_{f ib}+_{,Term II}= ∂ P ∂∆WWWP+nnn + f ib ωf c ˜ uf ib u uuT_{f 12}WWWTLLLT LLLWWW ∆uuuf 12, (2.35)

where LLLWWW ∆uuuf 12 is the discretization of the term (∆uuuf ib1− ∆uuuf ib2) in Equation (2.28) written in matrix notation. Since the fibre end points do not coincide with the discretiza-tion, ∆uuuf ib1and ∆uuuf ib2, which denote the displacement increments at the first and second fibre end point, cannot be directly computed and have to be interpolated from the sur-rounding nodes. More details regarding this approach are given in Section 2.3.3. The in-terpolation is done by multiplying the incremental displacement vector of the surrounding nodes

∆uuuf 12=

∆uf+_1x . . . _∆u_f+_nx ∆u_f−_1x . . . _∆u_f−_nx ∆u_f+_1y . . . _∆u_f+_ny ∆u_f−_1y . . . _∆u_f−_nyT (2.36)

with a weighting matrix

W WW =     w_f+_1x . . . w f+nx 0 0 0 0 0 0 0 0 0 0 0 0 wf−_1x . . . _w_f−_nx 0 0 0 0 0 0 0 0 0 0 0 0 w_f+_1y . . . w_f₊_ny 0 0 0 0 0 0 0 0 0 0 0 0 w_f−_1y . . . w_f−_ny    . (2.37) The component ∆u_f+_1xof the vector ∆uuu_{f 12}in Equation (2.36) denotes the displacement increment in x-direction of the first node from which the displacement at the fibre end on the positive side of the crack Ω+ is interpolated. The corresponding weight, which represents the influence of the particular node, is given by the component w_f+_1x of the matrix WWW (see Equation (2.37)) and n denotes the number of nodes the fibre-forces are distributed to and the displacement is interpolated from. The matrix

L LL= 1 −1 0 0 0 0 1 −1 (2.38) is used to express the difference between ∆uuuf ib1and ∆uuuf ib2. We discretize(uuuf ib1− uuuf ib2) in Equation (2.28) similarly. The discretization of the derivative of the fibre-forces is done in the same way as in Equation (2.33). For convenience, the fibre-force P is distributed over the same nodes that are used to approximate the displacements at the fibre end points. Thus, the same weighting factors are used in WWW_P+ and WWW .

With these definitions the contributions of the fibres to the tangential stiffness matrix can be computed. Two fibre terms have to be computed for each fibre end (see Equa-tions (2.31) and (2.32)). They are added to the global stiffness matrix according to the corresponding degrees of freedom.

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2.3.3 Distribution of the fibre-forces to the background mesh and

interpolation of the displacements at the fibre end points from

the background mesh

For the computation of the fibre pull-out force P, the fibre pull-out distance ∆, which is equal to the slip of the fibre, must be computed. In reality, the slip is distributed along the fibre. But since we do not calculate any slip, we have to evaluate it from the displacement field. The displacement at the fibre endpoints, which is used to compute the fibre pull-out distance (see Section 2.2.6), should not be based on the local displacement at the exact position at the fibre endpoint only, but on the averaged displacement around the fibre end. This is to avoid the unrealistic influence of local phenomena such as stress concentrations on the computation of the slip. The simplest way to account for that is to interpolate the displacement from the nodes surrounding the fibre endpoint.

Furthermore, we have assumed that the reaction forces acting from the fibre to the matrix are transfered at the end point of the fibre, thus at a single point. Within the framework of the finite element method these forces could be applied for example at the node which is the closest to the fibre end point. Apart from being physically unrealistic, a refinement of the discretization around the fibre end point would lead to increasing stresses and thus to mesh dependent results. To avoid this, the fibre-forces are distributed over a certain area. In this case, half circles around the fibre end points are used as shown in Figure 2.8. The shape of a half circle has been chosen because otherwise points behind the fibre end points are influenced and are influencing the fibre-forces.

The forces are distributed according to a constant distribution function or to a nor-malised inverse distance function. The latter one reads

wi= 1 Pnon l=1 Pnon k=1ldist,k ldist,l Pnon m=1ldist,m ldist,i , (2.39)

where wiis the weight of a particular node i within the area of influence of the fibre end point and non denotes the number of nodes situated in this area. ldist,k is the distance (> 0) between a particular node k and the fibre end point. The impact of the radius of the half-circle and the type of the distribution function is investigated in Sections 2.4.4.

Following this procedure, we can use the same nodes and weights that are employed to distribute the forces to the background mesh for the displacement interpolation.

2.3.4 Numerical requirements for the fibre pull-out relations

A general requirement for fibre pull-out relations is smoothness. In our simulations we have noticed that smoothing the initially rigid part of the pull-out relation described in Section 2.2.5 improves the numerical performance of our approach. This is accomplished by pre-multiplying the first part of the pull-out relation (Equation (2.21)) with

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influence radius of influence area of fibre crack P P

Figure 2.8: Schematic drawing of the area of influence with the radius of influence.

where the parameters k and a prescribe the slope of the function and its shift along the x-axis. In Figure 2.9 the influence of the smoothing on the pull-out relation is shown. The following set of parameters is used: Young’s modulus of the matrix E=20000 N/mm2, fibre Young’s modulus Ef ib=22000 N/mm2, chemical bond strength Gd=0.06 J/mm2, frictional stressτ0=2.5 N/mm2, slip hardening parameterβ =1, fibre diameter df ib= 0.04 mm and fibre length lf ib=2 mm. The embedded length is assumed to be half of the fibre length. The sample is 10 mm long and 2 mm high. In the top part of Figure 2.9 one can see that the overall influence of the smoothing is small. In the detail below the smoothing in the initial part of the curve becomes visible. How fast the original curve is reached depends on the choice of the smoothing parameters: k influences the slope of the curve and a determines its offset.

2.3.5 Solution algorithm

The set of equations describing matrix and fibres can either be solved in a monolithic or a staggered way. In the monolithic approach the fibre-forces are updated within the Newton-Raphson iteration. In the staggered approach the equations describing the matrix material are solved first and based on this solution the fibre-forces are computed. This is repeated until the equations for the matrix and the fibres are converged. Both approaches yield the same results. Based on the authors’ experience, the staggered approach provides a more robust convergence behaviour. Convergence problems may arise if a highly brittle matrix is combined with strong fibres and a large number of cracks has already formed. In normal cases the monolithic approach shows generally a faster convergence. Both schemes are used in the following examples.

Computational modelling of fibre-reinforced cementitious composites: An analysis of discrete and mesh-independent techniques

Computational modelling of

fibre-reinforced cementitious composites:

An analysis of discrete and

Computational modelling of

fibre-reinforced cementitious composites:

An analysis of discrete and

mesh-independent techniques

Aknowledgements

Contents

Chapter 1

Introduction

1.1

Aims and scope

1.2

Literature overview

1.3

Outline of the thesis and introduction to the

devel-oped approaches to model fibre-reinforced concrete

Chapter 2

Fibre-force approach

∗

2.1

Introduction

2.2

Approach

2.2.1

Problem statement

2.2.2

Model description

2.2.3

Mathematical formulation

2.2.4

Constitutive model for the concrete matrix

2.2.5

Constitutive model for the fibre and the fibre-matrix interface

2.2.6

Coupling between fibre and matrix models

2.3

Numerical aspects and implementational issues

2.3.1

Linearization of the fibre terms within the governing equations

2.3.2

Discretization of the fibre terms within the governing equations

2.3.3

Distribution of the fibre-forces to the background mesh and

interpolation of the displacements at the fibre end points from

the background mesh

2.3.4

Numerical requirements for the fibre pull-out relations

2.3.5

Solution algorithm