· este exemplar foi revisado e corrigido em relação à versão original, sob responsabilidade...

LUÍS ANTÔNIO GUIMARÃES BITENCOURT JÚNIOR

Numerical modeling of failure processes in steel fiber reinforcedcementitious materials

São Paulo2015



Thesis submitted to the Polytechnic School atthe University of São Paulo for award the Doc-tor of Science Degree

São Paulo2015



Thesis submitted to the Polytechnic School atthe University of São Paulo for award the Doc-tor of Science Degree

Area of Concentration: Structural Engineering

Advisor: Prof. Dr. Túlio Nogueira BittencourtCo-Advisor: Prof. Dr. Osvaldo Luís Manzoli

São Paulo2015

Este exemplar foi revisado e corrigido em relação à versão original, sob responsabilidade única do autor e com a anuência de seu orientador. São Paulo, 07 de janeiro de 2015. Assinatura do orientador _______________________ Assinatura do autor ____________________________

Catalogação-na-publicação

Bitencourt Júnior, Luís Antônio Guimarães

Numerical modeling of failure processes in steel fi ber reinforced cementitious materials / L.A.G. Bitencou rt Júnior. -- versão corr. -- São Paulo, 2015.

184 p.

Tese (Doutorado) - Escola Politécnica da Universida de de São Paulo. Departamento de Engenharia de Estruturas e Geotécnica.

1.Steel Fiber Reinforced Cementitious Materials 2.N on-matching Meshes 3.Coupling Finite Elements 4.Damage Constitutive Models 5.Mesh Fragmentation Technique 6.IMPL-EX Integration Method I.Universidade de São Paulo.

Escola Politécnica. Departamento de Engenharia de E struturas e Geotécnica II.t.

This thesis is lovingly dedicated to my family.

Acknowledgments

This research was sponsored by the São Paulo Research Foundation (FAPESP)through the process 2009/07451-2 (Doctoral Scholarship) and the process 2012/05430-0 (BEPE - Research Internship Abroad). The foundation’s support is gratefullyacknowledged.

I would like to thank my research advisor, Prof. Túlio N. Bittencourt, for hissupport, encouragement and advice during this research. I also wish to thank himfor all the opportunities he gave me to work on different research projects and incollaboration with several research groups in Brazil and abroad.

I also wish to express my deep gratitude to my co-advisor, Prof. Osvaldo LuísManzoli, for his interest in my research, guidance and assistance. I am sure thatall the discussions about different topics in computational mechanics will be veryvaluable in my career. Without his support, this work would not have been possible.

I would like to express my sincere thanks and appreciation to Prof. Frank JohnVecchio for giving me the opportunity to work at the University of Toronto (UofT),Canada for one year. My experience at UofT has been very rewarding and unfor-gettable. I am also grateful to my colleagues, Rizwan, Alessandro and Dr. Lee, whoshared the office with me during this period.

I am grateful to all the professors and office staff of the Department of Structuraland Geotechnical Engineering of the Polytechnic School at the University of SãoPaulo (PEF/ EPUSP) for their help and support during this research.

To my colleagues and members of the Structural Concrete Modelling Group (GMEC)and Laboratory of Computational Mechanics (LMC), I wish to express my sincereappreciation for creating a friendly environment, which helped me complete thisresearch.

I am indebted to my friends Fabiano and Jisa, Cayo, Batá, Judson, Jose and Mar-quinhos, and Guil and Roshid for their support and friendship during my research

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internship in Toronto. Without their friendship, my time in Toronto would not havebeen so enjoyable.

I am especially grateful to my friends in São Paulo, Diogo, Bruno, Leandro andRejane, Társis and Darlene, Henrique and Mônica, Plínio and Márcia, Júlio andCreuza, Papito, Júnior and Renata, Marly, and many others.

I wish to express my love and respect to all members of my family, especially myparents Maria and Luis Bitencourt and my sisters Mariane and Manuela Bitencourtfor their continuous support and love. I also would like to thank my parents-in-law,Helia and Jorge Lobato, for all their words of encouragement. A special thanksgoes to my aunt Edileusa Batista, who left us this year. She will always be in mymemories.

I owe the greatest thanks to my wife, Gleicy Bitencourt. She gave the support andlove that I needed over these past years.

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Abstract

This work presents a numerical strategy developed using the Finite Element Method(FEM) to simulate the failure process of Steel Fiber Reinforced Cementitious Com-posites (SFRCCs). The material is described as a composite made up by threephases: a cementitious matrix (paste, mortar or concrete), discrete discontinuousfibers, and a fiber-matrix interface.

A novel coupling scheme for non-matching finite element meshes has been developedto couple the independent generated meshes of the bulk cementitious matrix anda cloud of discrete discontinuous fibers based on the use of special finite elementsdeveloped, termed Coupling Finite Elements (CFEs). Using this approach, a non-rigid coupling procedure is proposed for modeling the complex nonlinear behaviorof the fiber-matrix interface by adopting an appropriate constitutive damage modelto describe the relation between the shear stress (adherence stress) and the relativesliding between the matrix and each fiber individually. This scheme has also beenadopted to account for the presence of regular reinforcing bars in the analysis ofreinforced concrete structural elements.

The steel fibers are modeled using two-node finite elements (truss elements) witha one-dimensional elastoplastic constitutive model. They are positioned using anisotropic uniform random distribution, considering the wall effect of the mold.

Continuous and discontinuous approaches are developed to model the brittle be-havior of the bulk cementitious matrix. For the former, an isotropic damage modelincluding two independent scalar damage variables for describing the composite be-havior under tension and compression is considered. The discontinuous approachis based on a mesh fragmentation technique that employs degenerated solid finiteelements in between all regular (bulk) elements. In this case, a tensile damageconstitutive model, compatible with the Continuum Strong Discontinuity Approach(CSDA), is proposed to predict crack propagation. To increase the computabil-ity and robustness of the continuum damage models used to simulate the failure

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processes in both of the strategies, an implicit-explicit integration scheme is used.

Numerical analyses are performed throughout the presentation of the work. Ini-tially, numerical examples with a single reinforcement are presented to validate thetechnique and to investigate the influence of the fiber’s geometrical properties andits position relative to the crack surface. Then, more complex examples involving acloud of steel fibers are considered. In these cases, special attention is given to theanalysis of the influence of the fiber distribution on the composite behavior relativeto the cracking process. Comparisons with experimental results demonstrate thatthe application of the numerical tool for modeling the behavior of SFRCCs is verypromising and may constitute an important tool for better understanding the effectsof the different aspects involved in the failure process of this material.

Keywords: Steel Fiber Reinforced Cementitious Materials; Non-matching Meshes;Coupling Finite Elements; Damage Constitutive Models; Mesh Fragmentation Tech-nique; IMPL-EX Integration Method.

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ResumoEste trabalho apresenta uma estratégia numérica desenvolvida usando o métododos elementos finitos para simular o processo de falha de compósitos cimentíciosreforçados com fibras de aço. O material é descrito como um compósito compostopor três fases: matriz cimentícia (pasta, argamassa ou concreto), fibras descontínuasdiscretas, e interface fibra-matriz.

Um novo esquema de acoplamento para malhas de elementos finitos não-conformesfoi desenvolvido para acoplar as malhas geradas independentes, da matriz cimen-tícia e de uma nuvem de fibras de aço, baseado na utilização de novos elementosfinitos desenvolvidos, denominados elementos finitos de acoplamento. Utilizandoeste esquema de acoplamento, um procedimento não-rígido é proposto para a mode-lagem do complexo comportamento não linear da interface fibra-matriz, utilizandoum modelo constitutivo de dano apropriado para descrever a relação entre a tensãode cisalhamento (tensão de aderência) e deslizamento relativo entre a matriz e cadafibra de aço individualmente. Este esquema também foi adotado para considerar apresença de barras de aço para as análises de estruturas de concreto armado.

As fibras de aço são modeladas usando elementos finitos lineares com dois nós (ele-mentos de treliça) com modelo material elastoplástico. As fibras são posicionadasusando uma distribuição randômica uniforme isotrópica, considerando o efeito pa-rede.

Uma abordagem contínua e outra descontínua são investigadas para a modelagem docomportamento frágil da matriz cimentícia. Para a primeira, é utilizado um modelode dano isotrópico com duas variáveis de dano para descrever o comportamento dedano à tração e à compressão. A segunda emprega uma técnica de fragmentaçãode malha que utiliza elementos finitos degenerados, posicionados entre todos oselementos finitos que formam a matriz cimentícia. Para esta técnica é propostoum modelo constitutivo à tração, compatível com a abordagem descontínua fortecontínua, para prever a propagação de fissura. Para acelerar o cálculo e aumentara robustez dos modelos de dano contínuos para simular o processamento de falhas,um esquema de integração implícito-explícito é utilizado.

Exemplos numéricos são apresentados ao longo do desenvolvimento desta tese. Ini-cialmente, exemplos numéricos com um único reforço são apresentados para validara técnica desenvolvida e para investigar à influência das propriedades geométricas

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das fibras e sua posição em relação à superfície de falha. Posteriormente, exemplosmais complexos são considerados envolvendo uma nuvem de fibras. Nestes casos,atenção especial é dada à influência da distribuição das fibras no comportamentodo compósito relacionado ao processo de fissuração. Comparações com resultadosexperimentais demonstram que a aplicação da ferramenta numérica para modelar ocomportamento de compósitos cimentícios reforçados com fibras de aço é muito pro-missora e pode ser utilizada como uma importante ferramenta para melhor entenderos efeitos dos diferentes aspectos envolvidos no processo de falha deste material.

Palavras-chave: Materiais Cimentícios Reforçados com Fibras de Aço; MalhasNão-conformes; Elementos Finitos de Acoplamento; Modelos Constitutivos de Dano;Técnica de Fragmentação de Malha, Método de Integração IMPL-EX.

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List of Figures

1.1 Steel fibers bridging cracks (, viewed Octo-ber, 2014). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2 Influence of the steel fibers on the stress (σ) x crack opening displace-ment (w) curve [122]. . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3 Typical shapes of steel fibers commonly used in concrete [87]. . . . . 23

2.1 Coupling procedure for non-matching finite element meshes: (a) def-inition of the problem; (b) process of identification of the nodes thatwill compose the CFEs; (c) creation and insertion of the CFEs; (d)detail of coupling in overlapping meshes; and (e) detail of coupling innon-overlapping meshes. . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2 2D and 3D coupling finite elements with linear interpolation functionsof displacements: (a) 3-node triangle + Cnode, (a) 4-node quadrilateral+ Cnode, (a) 4-node tetrahedral + Cnode, and (d) 8-node cube + Cnode. 41

2.3 2D basic tests. (a) Setup of the compression test, material and ge-ometrical properties. (b) Setup of the shear test and non-matchingmeshes employed. (c) Matching reference mesh. . . . . . . . . . . . . 45

2.4 Convergence of horizontal (a) and vertical (b) elongation in the 2Dcompression test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.5 Convergence of energy for the 2D shear test. . . . . . . . . . . . . . 472.6 Results obtained in the 2D tests. (a) Vertical displacement field for

the compression test. (b) Horizontal displacement field for the sheartest (with scaling factor of 5). . . . . . . . . . . . . . . . . . . . . . . 48

2.7 3D finite element mesh employed for the basic tests: (a) before ofthe coupling procedure and (b) complete mesh (after the couplingprocedure). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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2.8 Displacement field over 3D deformed FE mesh in x-direction for: (a)compression and (b) shear tests, and for the tension test in (c) x-direction and (d) y-direction (scaling factor of 5). . . . . . . . . . . . 49

2.9 Cylinder with curved reinforcing layers: (a) problem analyzed; and(b) numerical model for the case with one curved reinforcing layer. . 50

2.10 Steel stress. A quarter of the cylinder with one curved reinforcinglayer with rigid coupling (without bond-slip). . . . . . . . . . . . . . 51

2.11 A quarter of the cylinder with two curved reinforcing layers: (a) nu-merical model; and (b) total strain field. . . . . . . . . . . . . . . . . 52

2.12 Steel stress. A quarter of the cylinder with two curved reinforcinglayers with rigid coupling (without bond-slip). . . . . . . . . . . . . . 52

2.13 Three-point bending beam simulated numerically using: (a) a mesoscalemodel, and (b) a concurrent multiscale model (dimensions in mm). . 54

2.14 Non-matching meshes: (a) coupling procedure; (b) detail of the cou-pling finite elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.15 Total displacement contour field (mm): (a) mesoscale model; (b) mul-tiscale model (with scaling factor of 5). . . . . . . . . . . . . . . . . . 57

2.16 Horizontal normal stress in concrete (in MPa): (a) mesoscale model;(b) multiscale model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1 3D truss finite element: (a) D.O.F. of the element in global coordinatesystem, and (b) D.O.F. in local coordinate system, which influencethe local stiffness matrix and internal force vector. . . . . . . . . . . . 60

3.2 Typical stress-strain curve for one-dimensional elastoplastic materialmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 Scheme adopted for coupling a linear reinforcement element to itscorresponding 2D matrix elements. . . . . . . . . . . . . . . . . . . . 70

3.4 Reinforcement allowing slip. . . . . . . . . . . . . . . . . . . . . . . 713.5 Influence length of the coupling node. . . . . . . . . . . . . . . . . . . 733.6 Interface stress bond-slip relationship (monotonic loading) proposed

by CEB-FIP model code 90 [1]. . . . . . . . . . . . . . . . . . . . . . 753.7 Setup of the pullout tests. . . . . . . . . . . . . . . . . . . . . . . . . 773.8 Finite element mesh of the pullout test with lb = 75mm: (a) complete

mesh, and (b) detail of the coupling finite elements. . . . . . . . . . 783.9 Evolution of average bond stress with respect to slip at the ends (A)

and (B). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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3.10 Deformed matrix (concrete) mesh at: (a) peak load and (b) end ofanalysis (with scaling factor of 5000). . . . . . . . . . . . . . . . . . 80

3.11 Normal stress in concrete (in MPa) at: (a) peak load and (b) end ofanalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.12 Evolution of average bond stress with respect to slip at loaded end(B) for different mesh refinements. . . . . . . . . . . . . . . . . . . . 81

3.13 Axial stress along steel bar refined with 5, 10 and 20 finite elements. . 823.14 Pullout test with lb = 600mm: (a) finite element mesh, (b) deformed

mesh at the end of analysis (with scaling factor of 5000), and (c)normal (axial) stress in concrete (in MPa). . . . . . . . . . . . . . . 83

3.15 Evolution of normal stress in steel with respect to displacement atthe loaded end (lb = 600mm). . . . . . . . . . . . . . . . . . . . . . . 83

3.16 2D numerical model of the bond bending test M.16.16.R of Bigaj [12]. 843.17 3D numerical model of the bond bending test M.16.16.R of Bigaj: (a)

FE mesh before the coupling procedure and (b) deformed FE mesh(after the coupling procedure) at yield load (with a scaling factor of20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.18 Strain distribution along the steel bar at yield load. . . . . . . . . . . 863.19 Fiber-matrix interface model employed. . . . . . . . . . . . . . . . . . 883.20 3D numerical model of the pullout test of single straight fiber embed-

ded on one side: (a) setup of the pullout tests, and (b) detail of thecoupling procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.21 Fiber stress at crack with la = 0.5lf for straight fiber. . . . . . . . . . 893.22 Variation of the slip along the fiber when end slip is 0.1mm. . . . . . 893.23 3D numerical model of the pullout test of single straight fiber em-

bedded on both sides: (a) setup of the pullout tests, (b) detail of thecoupling procedure, and (c) deformed FE mesh (with a scaling factorof 10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.24 Detail of the numerical models with different fiber embedment lengths:(a) la = 0.1lf , (b) la = 0.2lf , (c) la = 0.3lf , and (d) la = 0.4lf . . . . . 91

3.25 Fiber stress at crack against crack width response. . . . . . . . . . . . 923.26 Crack width at maximum pullout stress. . . . . . . . . . . . . . . . . 933.27 Numerical models for pullout tests with different fiber inclination an-

gles: (a) 15º, (b) 30º, (c) 45º, and (d) 60º. . . . . . . . . . . . . . . . 943.28 Fiber stress at crack against fiber inclination angle. . . . . . . . . . . 95

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3.29 Slip at frictional pullout strength against fiber inclination angle. . . . 95

4.1 Specimen’s geometries and their respective restrictions to fiber po-sitioning. (a) Steel fiber. (b) Cylindrical specimen. (c) Prismaticspecimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2 Pseudo code of the algorithm developed for generating steel fibers. . . 103

5.1 Mesh fragmentation process: (a) standard (bulk) finite element meshand identification of the region to be fragmented; (b) separation ofthe finite elements (with an exaggerated scale factor for clarity); (c)insertion of interface finite elements (depicted in green); (d) detailof interface elements in FE mesh; and e) detail of a couple interfacefinite elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2 Interface solid finite elements: (a) three-node triangular element, and(b) four-node tetrahedron element. . . . . . . . . . . . . . . . . . . . 115

5.3 Uniaxial test setup: (a) tension and (b) compression load. . . . . . . 1265.4 Loading history considered in the uniaxial test. . . . . . . . . . . . . 1265.5 Horizontal stress x imposed displacement. . . . . . . . . . . . . . . . 1275.6 Interface finite elements under uniaxial loading: (a) geometry, bound-

ary conditions and finite element mesh, (b) tension load and (c) com-pression load (results with scaling factor of 250). . . . . . . . . . . . . 128

5.7 Horizontal stress x imposed displacement. . . . . . . . . . . . . . . . 1295.8 Horizontal stress x crack width. . . . . . . . . . . . . . . . . . . . . . 130

6.1 Bond-slip relation adopted to described the fiber-matrix interaction. . 1326.2 Direct tension tests: (a) typical specimen tested and (b) tensile test-

ing machine [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.3 2D numerical model: (a) geometrical properties (dimensions in mm),

(b) boundary conditions and loading, and (c) finite element mesh ofthe concrete sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.4 Fiber distribution in the direct tension test specimens for : (a) Vf =0.5%, (b) Vf = 1.0% and (c) Vf = 1.5%. . . . . . . . . . . . . . . . . 135

6.5 Force x displacement curves. Comparison between numerical (con-tinuum damage model) and experimental responses for Vf = 0.5%. . . 136


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6.8 Numerical analyses using damage model. Plain concrete and steelfiber reinforced concrete with steel fiber volume fractions of 0.5, 1.0and 1.5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.9 Failure pattern (damage distribution) at the end of the analyses for:(a) plain concrete, (b) Vf = 0.5%, (c) Vf = 1.0% and (d) Vf = 1.5%. . 139

6.10 Horizontal displacement contour (mm): (a) plain concrete; (b) Vf =0.5%, (c) Vf = 1.0% and (d) Vf = 1.5%. . . . . . . . . . . . . . . . . 140

6.11 Force x displacement curves for Vf = 1% using the damage model for4 generated fiber structures. . . . . . . . . . . . . . . . . . . . . . . . 141

6.12 Horizontal displacement contour (mm) for 4 generated fiber structureswith Vf = 1.0%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.13 Force x displacement curves. Comparison between numerical (meshfragmentation technique) and experimental responses for Vf = 0.5%. . 143



6.16 Failure pattern obtained using mesh fragmentation technique for Vf =0.5%: (a) deformed mesh at final load, and detail of the failure processfor horizontal displacements (b) δ = 0.24mm, (c) δ = 0.9mm and (d)δ = 3.3mm (figures with scaling factor of 3). . . . . . . . . . . . . . . 145

6.17 Comparison between the numerical approaches for steel fiber volumefractions of 0.5, 1.0, and 1.5%. . . . . . . . . . . . . . . . . . . . . . . 146

6.18 3D finite element mesh of the direct tension test: (a) concrete phaseand (b) fiber phase for Vf = 0.5%. . . . . . . . . . . . . . . . . . . . 147

6.19 3D numerical analyses of plain concrete and steel fiber concrete withfiber volume fraction of Vf = 0.5%. . . . . . . . . . . . . . . . . . . . 147

6.20 Failure pattern obtained using damage model with Vf = 0.5%: (a)damage and (b) horizontal displacement contour field. . . . . . . . . . 148

6.21 Failure pattern obtained using mesh fragmentation technique for Vf =0.5%: (a) specimen tested and (b) detail of the failure (with a scalingfactor of 25). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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6.22 Comparison between force x displacement curves obtained for 2D and3D numerical analyses using damage model of steel fiber concrete withfiber volume fraction of Vf = 0.5%. . . . . . . . . . . . . . . . . . . . 150

6.23 Comparison between force x displacement curves obtained for 2D and3D numerical analyses using damage model for plain concrete. . . . . 150

6.24 Geometrical properties and finite element meshes for the three-pointbending beam with: (a) centered notch and (b) eccentric notch (di-mensions in mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.25 Steel fiber distributions for beams with centered notch: (a) Vf =0.25% and (b) Vf = 0.50%. . . . . . . . . . . . . . . . . . . . . . . . 153

6.26 Load x deflection curves for beams of plain concrete with centerednotch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.27 Load x deflection curves for beams of SFRC (Vf = 0.25%) with cen-tered notch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.28 Load x deflection curves for beams of SFRC (Vf = 0.50%) with cen-tered notch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.29 Failure pattern for beams with centered notch: (a) plain concrete andSFRC with (b) Vf = 0.25% and (b) Vf = 0.50% (with a scaling factorof 5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.30 Numerical responses for beams with centered notch of plain concreteand SFRC with 0.25 and 0.5% steel fiber contents. . . . . . . . . . . . 157

6.31 Steel fiber distributions for beams with eccentric notch: (a) Vf =0.25% and (b) Vf = 0.50%. . . . . . . . . . . . . . . . . . . . . . . . . 158

6.32 Load x deflection curves for beams of plain concrete and eccentricnotch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.33 Load x deflection curves for beams of SFRC (Vf = 0.25%) and eccen-tric notch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.34 Load x deflection curves for beams of SFRC (Vf = 0.50%) and eccen-tric notch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.35 Failure pattern for beams with eccentric notch: (a) plain concreteand SFRC with (b) Vf = 0.25% and (b) Vf = 0.50% (with a scalingfactor of 5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.36 Numerical responses for beams with eccentric notch. Plain concreteand SFRC with 0.25% and 0.5% steel fiber contents. . . . . . . . . . 161

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8.1 IMPL-EX integration scheme: (a) extrapolation of the strain-like in-ternal variable; and (b) prediction-correction stages of the IMPL-EXscheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

8.2 3D truss finite element in global coordinate system. . . . . . . . . . 171

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List of Tables

2.1 Material parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.2 Comparison of the data of the mesoscale and multiscale models. . . . 57

3.1 Classification of the elastoplastic material behavior. . . . . . . . . . . 653.3 Ingredients of the one-dimensional elastoplastic material model. . . . 693.4 IMPL-EX integration scheme for the continuum damage model to

describe bond-slip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.1 Initialization of the variables for generating a cloud of steel fibers. . . 100

5.1 Modified IMPL-EX integration scheme for the continuum damagemodel with distinct tensile and compressive responses. . . . . . . . . 111

5.2 IMPL-EX integration scheme for the tensile damage model. . . . . . . 1225.3 Components of the discrete relation of the IFE for 2D and 3D cases. . 1235.4 Continuum and discrete constitutive equations for the tensile damage

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.1 Effective material parameters for the macroscale subdomains. . . . . 153

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Contents

1 Introduction 201.1 General aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3.1 Numerical strategies for modeling the failure behavior of SFR-CCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3.2 Coupling methods for non-matching meshes . . . . . . . . . . 291.3.3 Robustness issues in numerical modeling of material failure . . 32

1.4 Overview and limitations of the thesis . . . . . . . . . . . . . . . . . . 321.5 Main contributions of the thesis . . . . . . . . . . . . . . . . . . . . . 331.6 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2 Multiscale approach for modeling SFRCC 362.1 Overview of the coupling technique for non-matching FE meshes . . . 372.2 Coupling finite element . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2.1 CFE formulation . . . . . . . . . . . . . . . . . . . . . . . . . 402.2.1.1 CFE internal force vector . . . . . . . . . . . . . . . 422.2.1.2 CFE stiffness matrix . . . . . . . . . . . . . . . . . . 42

2.2.2 Linear elastic model . . . . . . . . . . . . . . . . . . . . . . . 432.3 Rigid coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.4 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4.1 Case study 01: Basic tests . . . . . . . . . . . . . . . . . . . . 442.4.2 Case study 02: Cylinder with curved reinforcing layers . . . . 502.4.3 Case study 03: Non-matching meshes of a concurrent multi-

scale model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 58

i

3 Reinforcement and reinforcement-matrix interaction 603.1 Two-node truss finite element . . . . . . . . . . . . . . . . . . . . . . 61

3.1.1 Truss internal force vector . . . . . . . . . . . . . . . . . . . . 623.1.2 Truss stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 One-dimensional elastoplastic model . . . . . . . . . . . . . . . . . . 633.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 643.2.2 Loading/unloading conditions . . . . . . . . . . . . . . . . . . 663.2.3 Tangent constitutive operator . . . . . . . . . . . . . . . . . . 67

3.3 Non-rigid reinforcement-matrix coupling . . . . . . . . . . . . . . . . 693.3.1 Elastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.3.2 Bond-slip model . . . . . . . . . . . . . . . . . . . . . . . . . 723.3.3 A Continuum damage model to describe bond slip . . . . . . 74

3.3.3.1 Implicit-explicit integration scheme for the contin-uum damage model to describe bond-slip . . . . . . . 75

3.4 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.4.1 Case study 01: 2D pullout tests of reinforcing bars . . . . . . 77

3.4.1.1 Short embedment length (lb = 75mm) . . . . . . . . 783.4.1.2 Long embedment length (lb = 600mm) . . . . . . . . 82

3.4.2 Case study 02: Bending bond test . . . . . . . . . . . . . . . . 843.4.3 Case study 03: Pullout behavior of steel fibers . . . . . . . . . 86

3.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 96

4 Fiber distribution and orientation in cementitious matrix 974.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 Program for generation and orientation of steel fibers . . . . . . . . . 984.3 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 104

5 Cementitious material modeling 1055.1 Continuous approach based on isotropic damage model . . . . . . . . 105

5.1.1 A continuum isotropic damage model with distinct tensile andcompressive responses . . . . . . . . . . . . . . . . . . . . . . 1055.1.1.1 Modified implicit-explicit integration scheme for the

continuum damage model with distinct tensile andcompressive responses . . . . . . . . . . . . . . . . . 109

5.2 Discontinuous approach . . . . . . . . . . . . . . . . . . . . . . . . . 1125.2.1 Overview of the mesh fragmentation technique . . . . . . . . . 113

ii

5.2.2 Interface solid finite elements . . . . . . . . . . . . . . . . . . 1155.2.3 Tension damage model . . . . . . . . . . . . . . . . . . . . . . 120

5.2.3.1 Implicit-explicit integration scheme for the tensiledamage model . . . . . . . . . . . . . . . . . . . . . 121

5.2.4 Discrete relation of the interface finite element . . . . . . . . . 1235.3 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.3.1 Case study 01: Continuum damage model under uniaxial loading1255.3.2 Case study 02: Interface elements under uniaxial loading . . . 127

5.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 130

6 Applications 1316.1 Example 01: Direct tension tests . . . . . . . . . . . . . . . . . . . . 1326.2 Example 02: Three-point bending beams . . . . . . . . . . . . . . . . 1516.3 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 161

7 General conclusions and recommendations for future research 1637.1 General conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.2 Recommendations for future research . . . . . . . . . . . . . . . . . . 166

8 Appendix 1688.1 IMPL-EX integration scheme . . . . . . . . . . . . . . . . . . . . . . 1688.2 Transformation matrix for the two-node linear finite element . . . . . 171

Bibliography 173

iii

1 Introduction

1.1 General aspects

Cementitious materials, such as mortar and concrete, are among the most widelyused construction materials. These materials are classified as quasi-brittle materials,i.e., materials with low tensile strength and strain capacities. To overcome thesemain drawbacks, discontinuous fibers have been added to cementitious matrices,resulting in so-called Fiber Reinforced Cementitious Composites (FRCCs). Despitethe wide use of FRCCs in recent years, the addition of fibers for improving materialproperties comes from ancient times. Egyptians used straw to reinforce sunbakedbricks [73], and there is also evidence that asbestos fiber was already being employedto reinforce clay posts approximately 5000 years ago [81].

The first investigations concerning the use of fibers were performed in the early 60sand are reported in the papers published by [104] and [105]. The authors investigatedthe ability of fibers to improve the tensile ductility of concrete. In the decades thatfollowed, a large number of experimental studies were developed to verify the effectsof fibers on the properties of cementitious matrices, such as tension, compression,bending and shear.

Several types of discontinuous fibers have been used to enhance the properties ofcementitious matrices. They are typically classified based on their material composi-tion, such as steel, glass, synthetic, and natural; their physical/chemical properties;and their mechanical properties, such as elastic modulus, tensile strength, and duc-tility. Glass and natural fibers exhibit susceptibility to environmental conditions,leaving steel and synthetic fibers as the most viable concrete reinforcement options[2]. The use of synthetic fibers is usually related to improving the resistance tocracking caused by drying shrinkage (secondary reinforcement). However, due totheir high elastic modulus and their strong bond with the surround cementitiousmatrix, steel fibers are employed as a primary reinforcement.

20

1.1 General aspects

Currently, it is well known that the addition of a small volume of steel fibers mayincrease the ductility and toughness of cementitious matrices [11]. The role playedby fibers is most obvious after matrix cracking has occurred, as fibers offer resistanceto crack propagation, which is illustrated in Fig. 1.1.

Figure 1.1: Steel fibers bridging cracks (, viewed Octo-ber, 2014).

Therefore, the main benefits of the addition of steel fibers in cementitious matricesare directly related to their ability to transfer stresses across cracks. Fig. 1.2 showsthe influence of steel fibers in hardened concrete cementitious matrices for a typicalstress versus crack width curve. As noted in this figure, before the addition of steelfibers and after matrix cracking, the tensile stress immediately decreases (see thecurve for the matrix). However, after the addition of a certain volume of fibers andafter matrix cracking, the fibers are able to maintain a certain load bearing capacity,avoiding an abrupt failure of the composite (see curve matrix + fibers). In addition,the crack widths are less than those of plain concrete [31].

According to [11] this process of stress transfer depends on the internal structure ofthe composite, and the main factors that influence the composite’s behavior are (i)the structure of the bulk cementitious matrix, (ii) the shape and distribution of thefibers, and (iii) the structure of the fiber-matrix interface.

Concerning the cementitious matrices, concrete is the most widely used. This ma-terial is composed of cement, coarse aggregate, fine aggregate and water. Its me-chanical behavior depends on the properties of each ingredient and on the pro-portions used in the mixture. The addition of steel fibers to overcome concrete’sbrittle behavior usually requires superplasticizers to maintain its workability. For

21

1.1 General aspects

Figure 1.2: Influence of the steel fibers on the stress (σ) x crack opening displace-ment (w) curve [122].

conventional steel fiber reinforced concrete, the volume of fiber to be added to thecementitious matrix does not exceed 2%. However, in the last decade, large volumesof fibers have been added to develop so-called High Performance Fiber ReinforcedCementitious Composites (HPFRCCs). HPFRCCs exhibit multiple cracking withstrain-hardening under direct tension. A general classification for fiber reinforcedcomposites based on their tensile response (strain- hardening or strain-softening)can be found in [86].

Today, there are a number of discontinuous steel fibers on the market, with differentshapes, lengths, diameters or equivalent diameters, and surface deformation [87].New fibers with different characteristics are continuously being proposed to improvethe compatibility with the cementitious matrix. Fig. 1.3 illustrates the typical shapesof steel fibers commonly used in concrete.

The distribution and orientation of the steel fibers influence the behavior of thefiber reinforced composite in numerous ways. While fibers aligned perpendicularto the failure plane provide a maximum efficiency, fibers parallel to this plane havealmost no influence. The main properties influencing the orientation of steel fibersinclude the fresh-state properties [41, 80], formwork geometry [112, 34], flow of the

22

1.1 General aspects

Figure 1.3: Typical shapes of steel fibers commonly used in concrete [87].

fresh composite [42], vibration [44] and casting procedure [7, 119, 123]. Amongthese properties, the wall effects introduced by the formwork and the fresh-stateproperties have the greatest effects [79]. Usually, a parameter called the orientationnumber (η) is used to quantify the influence of one of the aforementioned factors onthe fiber distribution [112].

In the last years, a set of techniques has been developed to determine the distributionof steel fibers in hardened cementitious matrices. These techniques are categorized asdestructive and non-destructive methods based on direct and indirect measurements[59]. These techniques are often applied to quantify the wall effects on idealizedisotropic SFRCCs.

The fiber-matrix interaction plays one of the most important roles in the behavior ofthe cementitious composite. According to [85], the transmission forces between thefibers and matrix are produced through the interfacial bond defined as the shearingstress at the interface between the fiber and the surrounding matrix. The descrip-tion of this process is very complex because many factors, such as the mechanicalcomponents of the bond (as in, for example, crimped and hooked fibers), the physi-cal and chemical adhesion between the fibers and the matrix, and the fiber-to-fiberinterlock, may influence its behavior [85]. Thus, steel fibers with different shapesexhibit different slip characteristics and pullout energies, even if the same maximumshear stress is obtained.

Based on the experimental tests, analytical models have been developed to describe

23

1.2 Motivation

the behavior of concrete subject to tension [78, 43, 122, 63, 64]. Among thesemodels, the analysis model called the Diverse Embedment Model (DEM) [63, 64] isthe most realistic model available in the literature. This model considers the effectsof each fiber on the behavior of the composite, including the fibers’ geometry, theirdistribution and orientation, characteristics of the single pullout response and theinfluence of the member dimension.

1.2 Motivation

As described previously, by adding a small volume of steel fibers to cementitious ma-trices, a material with greater ductility and energy absorption is obtained. Althoughthe application of SFRCCs has increased in the last years, being very attractive inmany structures today, such as tunnel linings, bridges, pavements, and pipes, thereremains a lack of numerical models for simulating the behavior (including the fail-ure process) of SFRCCs that consider the contribution of each component (fibers,matrix and fiber-matrix interface) in a fully independent way.

This type of approach is very appealing because the mechanical response of thismaterial is highly dependent on both the distribution of the steel fibers and theinteraction of each fiber with the cementitious matrix. Hence, a numerical modelwith a discrete treatment of the fibers seems to be a natural way to simulate thefailure behavior of this material.

In numerical models based on the finite element method with a discrete treatmentof the fibers, non-conformal meshes are often considered between the cloud of steelfibers and the cementitious matrix. A coupling procedure is then applied to couplethese independent meshes. A rigid coupling is usually applied, and the fiber-matrixinteraction is included in the constitutive model adopted to describe the behaviorof the fibers. In addition, these methods usually present problems of convergence,and analyses are limited to 2D problems.

Numerical models with a discrete treatment of the fibers may be very useful forsimulating a range of cases with different fiber distributions and orientations, fibergeometries and fiber-matrix interactions and for considering other factors that in-fluence the composition, such as the casting procedure, vibrations and wall effectsintroduced by the formwork. Covering all these cases using only experimental in-vestigations would be very expensive.

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1.3 Literature review

Currently, several experimental tests, such as 3-point bending tests, 4-point bendingtests, splitting tensile tests, round determinate tensile tests, uniaxial tensile tests,and Barcelona tests, have been used to characterize the behavior of SFRCCs undertensile load. The major problem found in these tests is the variability of the obtainedresults. Moreover, due to the small size of the specimens compared to the size ofconventional structural members, it is very difficult to use the results obtained inexperimental tests to predict the behavior of structural members. In this case,only different boundary conditions and casting methods will produce a compositewith different mechanical properties. Therefore, a numerical model that considersthe influence of these factors could be very useful in developing a link betweenexperimental tests and practical applications.


In this section, a brief literature review of the numerical models available for mod-eling the failure behavior of SFRCCs is performed. Attention will be given to themodels that consider the individual influence of each fiber on the failure behavior ofthe composite. Other important subjects in this thesis, such as coupling methods fornon-matching meshes and robustness issues in the numerical modeling of materialfailure, will also be addressed.

1.3.1 Numerical strategies for modeling the failure behavior ofSFRCCs

[114] describes the failure process of cementitious materials as follows: “micro-cracksfirst arise which change gradually into dominant macroscopic discrete cracks up torupture”. In the literature, many numerical models have been proposed for modelingthis failure process. In general, these models can be classified as continuous ordiscontinuous. An overview can be found in [114] and [49].

Among the numerous continuous models available in the literature, elastoplastic,damage and smeared crack models are widely used. In the smeared crack models,a fracture zone is considered to be distributed in a certain region of the solid [14].To overcome problems of mesh size dependence presented by the models of thisclass, several regularization procedures have been proposed [106]. Other continuous

25


models used to solve this problem are based on fracture mechanics concepts that leadto fracture energy release regularization [9, 19]. However, the loss of objective of thedeformation pattern is also presented. More recently, the mesh dependence problemhas been addressed using integral-type, non-local and second-gradient approaches[114].

Discontinuous models are characterized by the introduction of displacement or straindiscontinuities into standard finite elements to represent cracks. The models avail-able in the literature using this approach include cohesive crack models [57, 45], Em-bedded Strong Discontinuities (E-FEM)[91, 95, 76, 92], lattice models [54, 55, 107],zero-thickness interface models [70, 71, 18], Element-free Galerkin [10, 113], eX-tended Finite Element (X-FEM) [77, 5] and particle models [50]. These methodsare referenced by some authors as discrete crack models.

Several approaches have also been proposed for modeling the failure process of SFR-CCs. Some continuum models have been developed based on the results of experi-mental tests of structural elements, such as 3- and 4-point bending beams and slabs[109]. These models are very limited because they are only able to reproduce thesame conditions applied in the laboratory test. Moreover, this type of model is veryexpensive due to the large number of tests required to calibrate the model.

Stress-strain relations obtained from the inverse analysis of the laboratory test re-sults are also available [38, 37, 117]. Moment-curve or load-displacement relationsare used as input data. As the previous models, the relations that describe thebehavior of the material are developed for specific structural elements.

Various analytical models [62, 63] used to describe the tension behavior of SFRCCshave been proposed and implemented in computational programs. An interestingmodel called Diverse Embedment Model (DEM) [63] has been recently developed.The DEM considers the effects of the fiber geometry, the fibers’ distribution andorientation, characteristics of the single pullout response and the influence of thestructural member dimensions.

The models described above are usually implemented in numerical tools using acontinuous approach, such as damage, elastoplastic and smeared crack models.

Models based on mixture theory have also been developed [74, 84]. In these models,the composite stresses are obtained by summing the stresses of each constituent,which are weighted according to their corresponding volumetric participation. Thus,these models do not allow to take into account the distribution of the fibers. [74]

26


and [84] use a continuum strong discontinuous approach (CSDA) for modeling thefailure behavior of the composite.

Recently, various studies have focused on the development of models that include adiscrete treatment of fibers. An explicit representation is adopted in some models,whereas in other models, only interaction forces are considered to account for thepresence of the fibers. This type of approach was adopted in this thesis with thefibers being explicitly represented. Therefore, the main references used in this thesisare described next.

[101] developed an approach for modeling SFRCCs in which the meshes of the con-crete bulk and fiber cloud are generated in a completely independent way. Theinteractions between the independent meshes are formed by the application of con-cepts of the Immersed Boundary Methods (IBMs1). Hence, in the developed model,the individual fibers immersed in the concrete bulk are accounted for at their actuallocation and with their orientation. The interactions between fibers and concreteare considered in the stress-strain relation adopted to describe the fiber behavior.In turn, this relation is obtained through the expressions proposed by [59] whichare deduced from pullout tests [60, 61]. Adopting this strategy, the model is able tocapture each fiber shape (straight or hooked), the inclination angle and the fiber em-bedment length. The concrete matrix is modeled using a nonlinear damage model.In [98] a 3D extension of this formulation is also presented. In addition, the failureof concrete can also be represented using a heuristic crack model with joints. Inthis material model, the whole specimen is described by an elastic material, and thecracking pattern is described using joint elements. The numerical analyses demon-strated that the approach captures the main features of the composite behavior. Forall the numerical examples performed, the failure pattern exhibits only one crack.Moreover, to define the constitutive model for each fiber, the angle between the fiberand the failure pattern must be known beforehand.

The numerical approach proposed by [26] for modeling the behavior of SFRCCsconsiders the composite as a heterogeneous material made up of two phases: a ho-mogeneous matrix (aggregate and paste) and another phase composed of a cloud ofsteel fibers. A fixed smeared crack model is employed to describe the failure processof the cementitious matrix. The steel fibers are explicitly represented using linearelements with an embedded approach, modeled using a perfectly bonded formula-

1A general description of the IBMs can be found in [83].

27


tion. Hence, the fiber-matrix interaction is considered in an indirect fashion in theconstitutive model adopted for the fibers. This procedure consists of obtaining astress-strain relation from the load-slip relation of a pullout test. In this model, thefibers are distributed using the Monte Carlo method. The 3D numerical simula-tions of both uniaxial tensile tests and 3-point bending tests revealed a very goodagreement with the experimental results. Only experiments with a previously de-fined fracture plane have been analyzed. More details about the formulation of thismodel can also be found in [27, 28].

[103] developed an approach to describe the failure process in FRCCs that consid-ers the features of the matrix, fibers and fiber-matrix interaction. The fibers areconsidered as discrete entities, but they are not explicitly discretized to avoid highcomputational costs. Instead, only reaction forces from the fiber on the matrix areconsidered, and they are activated only when they bridge a crack. The reactionforces are assumed to be equal to the fiber pullout forces, which can be modeledeither analytically or numerically. A damage model (regularized or non-local) isemployed for modeling the matrix. The 2D numerical examples demonstrated thatthe model is able to represent the influence of fibers, their distribution and theirinteraction with the matrix. Moreover, several features of FRCCs, such as theirductile behavior, strain hardening and multiple cracking, were obtained.

In the model proposed by [40], the failure behavior of SFRCCs is evaluated at boththe macro and mesoscale levels of observation. This model employs a discrete crackformulation based on the use of zero-thickness interface elements proposed by [17]for plain concrete. The well-known “Mixture Theory” from [120] is considered for in-terface elements to account for the fiber-to-concrete/mortar interactions, where theeffects of both the bond-slip and dowel mechanisms are considered. A failure crite-rion defined in terms of the normal shear stress components acting on the joint planeis adopted. For the pre-peak regime, a linear elastic model is considered, while thepost-peak is formulated in terms of the fracture energy release under failure modesI and/or II. The fibers are not represented explicitly, and their distribution is de-fined through statistical procedures. Analytical expressions proposed by [112] areemployed to define the number of crossing fibers per interface and the respective ori-entation factor. The 2D numerical analyses performed by the authors demonstratedthat the nonlinear interface formulation can capture the main features of the failureprocess of SFRCCs.

28


A discrete irregular lattice model was proposed by [56]. The meso-scale model iscomposed of the cement matrix, aggregate inclusions, steel fibers explicitly modeledwith a realistic stochastic distribution within the concrete, matrix-inclusion interfaceand matrix-fiber interface. The aggregates are randomly positioned in the domainaccording to a granulometric distribution with a circular or spherical shape for a2D or 3D model, respectively. The model employs rigid rod elements, creating alattice that breaks according to a simple rule. Moreover, each element is associatedwith a particular material phase or interface. Two and three-dimensional quasi-static simulations were performed, and the results were qualitatively compared withexperiments. The effects of the fiber length, fiber orientation distributions, interfacestrength, fiber volume, and specimen size were captured by the proposed model.

In the model proposed by [102], the material constitutive behavior of each con-stituent of the composite, matrix, fiber, and fiber-matrix interaction is defined in anindependent way. The presence of fibers is defined using the Partition of Unity Fi-nite Element Method (PUFEM)2. Thus, the presence of discrete fibers is considered,employing the partition of unity property of finite element shape functions, withoutexplicitly meshing them to ensure numerical efficiency. A linear elastic behavior isused to describe the fibers, whereas the matrix behavior is described by an isotropicdamage model with an exponential softening law. This damage model is equippedwith a simple regularized fracture energy model (details can be found in [9]) andwith the gradient-enhanced damage model proposed by [99] to avoid the mesh de-pendence. The nonlinear behavior of the fiber-matrix interaction is described bythe model proposed by [47]. The strategy presented is general, and any materialmodel can be applied to represent each constituent. Several 2D numerical exampleshave been used to measure the influence of the fiber distribution, fiber length, in-terface model and mesh refinement dependency. According to various authors, thecomputational costs can be dramatically reduced compared with the standard finiteelement method.

1.3.2 Coupling methods for non-matching meshes

The accuracy of results in numerical analysis by the finite element method is re-lated directly to the adequacy of the discretization. However, the finer the mesh the

2Details of the PUFEM formulation can be found in [82] .

29


greater the computational effort required to solve the problem. Thus, a common so-lution for large scale problems is to use a fine mesh only in the region of interest. Asa consequence, another problem may arise in the transition between coarse and finemeshes, since the presence of distorted elements can invalidate the solution in thetransition regions [3]. Another strategy widely used today consists of discretizingthe regions of the problem (subdomains), in a totally independent way, accordingto the interest of the analyst, and then use a coupling technique to connect theirnon-matching interfaces. This strategy has been applied extensively to problemswith adaptive mesh refinement [125, 69, 67], multiscale problems [121, 68, 116] ormultiphysics analysis [35, 48, 8, 100]. In addition, with the advent of parallel com-puting, this kind of approach has been extensively used to deal with the interactioneffects between the subdomains, initially subdivided to be computed by differentprocessors [72, 118].

In this context, a number of coupling methods have been developed to captureinterface effects accurately [46]. Ref. [52] defines the continuity and compatibil-ity conditions at non-matching interfaces between subdomains as the fundamentalrequirements, and, also assert that the strain fields must be transferred correctlythrough the non-matching interfaces. Constraint equations have been used to makea strong coupling between the subdomains. This method is usually known as multi-point constraint (MPC). The main idea is to evaluate the displacement of the loosecoupling boundary nodes of the local subdomain (fine mesh) using the displacementinterpolation of the adjacent finite element of the global subdomain (coarse mesh)[121].

Other existing classes of methods for coupling non-matching meshes are based ona weak coupling approach. In this case, the displacement compatibility is onlysatisfied in an average sense. Consequently, displacement compatibility between thesubdomains may not be satisfied, which may result in small gaps or overlaps [121].

Another general classification for the methods to couple non-matching meshes is todivide them as dual or primal methods. In the dual approaches [46] the methodsare based on the use of a field of Lagrange multipliers to impose the boundaryconditions, as for instance, in the mortar method [124, 58] and the Arlequin method[32, 33]. Unlike the former, the Arlequin method presents a region in which thesubdomains overlap. The main drawback of these methods is the introduction ofextra unknowns (additional degrees of freedom) to the system of equations.

30


On the side of the primal methods one could cite the penalty methods [96, 97]. Alsothe discontinuous Galerkin (DG) methods (a good review of DG methods is givenin [4]) and Nitsche methods (originally described in [88]) are among the most widelyused.

In these methods, the interface is represented by its displacement field and no dualvariables are introduced [46]. For this reason, and in contrast with the dual ap-proaches, primal methods are not subject to the inf-sup or Ladyzhenskaya-Babuška(LBB) restrictions. However, a stabilization parameter is needed. Ref. [46] proposeda primal interface formulation that uses local enrichment of the interface elementsto enable an unbiased enforcement of geometric compatibility at all interface nodeswithout inducing over-constraint and additional variables. In [29] some primal cou-pling methods such as the nearest neighbor interpolation [115], projection method[21] and methods based on spline interpolation [115, 110] are compared for problemsof fluid-structure interaction.

Another class of coupling strategies are the methods that convert standard finiteelements on non-matching interfaces into special types of elements. Methods basedon the interface element method (IEM) [51, 52] may be included in this class. Inthese methods, interface elements are defined on the finite elements bordering thenon-matching interfaces, and the moving least square (MLS) approximations areused to construct the shape functions of these interface elements [52]. This approachintroduces no additional degrees of freedom, and hence the system matrix remainspositive-definite. However, the algorithm for the construction of interface elementsis not negligible and is computationally expensive. Moreover, it is necessary to storeinformation about the shape functions at all the integration points in the interfaceregion, according to the conventional way to calculate shape functions using MLS[25]. Based on the IEM, [53] developed three-dimensional interface elements forcoupling non-matching hexahedral meshes. To overcome the main disadvantages ofthe IEM, [25] proposed a new improved interface element called MLS-based variable-node element. Other applications using this strategy can be found in [23, 24, 65,66]. In the literature some authors call the interface elements transition elements[69, 111].

31

1.4 Overview and limitations of the thesis

1.3.3 Robustness issues in numerical modeling of material failure

When a numerical model for modeling material failure is proposed, an importantissue is presented: robustness. This is because when a crack propagates within thestrain softening regime, the algorithmic tangent operator may become singular, andas a consequence, the solution of the resulting systems of nonlinear equations usinga fully implicit discretization methodology cannot be obtained. According to [93],this problem may also be presented in the numerical modeling of material failureeven when powerful continuation methods to pass structural points are used, i.e.,arc length methods to transverse limit and turning points.

To address this problem, an implicit-explicit (IMPL-EX) integration scheme for theintegration of the constitutive models was proposed by Oliver [93, 94]. Among themain beneficial aspects of the use of the IMPL-EX algorithm are the guaranteedconvergence, regardless of the length of the load step, and the low computationalcost compared with standard implicit schemes.

The guaranteed convergence and robustness of the IMPL-EX algorithm result fromthe positive definite algorithmic tangent operator, which, in turn, is due to theinternal variable update procedure adopted in the IMPL-EX scheme. As a side effect,there is an error associated with the use of this alternative integration methodology,which can be reduced (or controlled) by decreasing the load step length.

Regarding the computational cost, an advantage of the IMPL-EX scheme is that onlyone iteration per load step is sufficient to establish equilibrium, which is given bythe difference between external and internal forces. This is because the algorithmictangent operator is constant (when an infinitesimal strain format is adopted) duringa load step.

In sec. 8.1, the idea behind the IMPL-EX integration scheme and its application tocontinuum damage models with scalar damage variables is presented. A completedescription and details on the use of this integration technique with other familiesof constitutive models, such as elastoplastic models, can be found in [93, 94].

1.4 Overview and limitations of the thesis

This thesis focuses on the development of a numerical approach for modeling thefailure process of steel fiber reinforced cementitious composites. Therefore, the per-

32

1.5 Main contributions of the thesis

formed analyses attempt to assess various important aspects, such as mesh depen-dency; problems of convergence; capacity of the model to capture important factorsthat influence the behavior of the composite, such as the fiber geometry and thedistribution and orientation of the fibers; adopted bond-slip model; and propertiesof the cementitious matrix.

Numerical analyses are performed in 2D and 3D. However, the 3D analyses are lim-ited to only few case studies because of the computational effort required. Moreover,the analyses are restricted to quasi-static loading.

To study the effect of each fiber on the failure process of the composite, a discretetreatment is applied to the fibers, and their discretization is independent of thefinite element mesh of the bulk cementitious matrix. Here, the focus is on thedevelopment of the coupling scheme to couple non-matching meshes, which allowsa non-rigid coupling, to account for the fiber-matrix interaction.

The fibers are randomly generated using an isotropic uniform random distribution.Moreover, it is worth mentioning that factors that influence the behavior of thespecimen, such as the casting direction, vibrations, and flow of fresh composite, arenot considered.

In spite of the mesoscale nature of the proposed numerical model, coarse aggregatesand the mortar are homogenized, thus requiring the adoption of effective propertiesto characterize the specimen’s behavior.

Although the model can be applied to any volume content of fibers, this study islimited to 2% for reasons of computational costs. Moreover, the interaction betweenfibers is disregarded. Thus, overlaps between fibers are allowed.

Because the focus of this study is on the failure process, continuous and discontin-uous approaches are developed. The use of an implicit-explicit scheme to integratethe constitutive models of both approaches is investigated.

1.5 Main contributions of the thesis

The main contribution of this thesis consists of the development and implementationof a new numerical model for modeling the failure process in steel fiber reinforcedcementitious composites with a discrete treatment of the fibers. The components ofthis model can be summarized as follows:

33

1.6 Structure of the thesis

• A new coupling technique for coupling non-matching finite element meshes.

• A strategy for modeling the reinforcement elements (reinforcing bars andfibers) with a discrete treatment.

• A procedure to account for the fiber-matrix interaction.

• A computational program to generate fiber clouds.

• A continuous approach for modeling the failure process of cementitious matri-ces based on continuum damage mechanics theory.

• A discontinuous approach for modeling the failure process of cementitiousmatrices based on a mesh fragmentation technique and on finite elements withhigh aspect ratio.

• An implicit-explicit integration scheme to increase the robustness of the con-stitutive models and to accelerate the nonlinear convergence.


This thesis is organized into eight chapters. Chapter 1 contains the introduction.The strategy for modeling the behavior of steel fiber reinforced cementitious materi-als using a multiscale approach (discrete treatment of fibers) is presented in Chapter2. The focus herein is on the formulation of the new finite elements, termed CouplingFinite Elements (CFEs), which are responsible for coupling the non-matching finiteelement meshes generated during the discretization of the different scale materials.In this chapter, the formulation for the case of a rigid-coupling procedure is assessedand validated.

Chapter 3 describes the reinforcement and the reinforcement-matrix interaction.The same strategy is used to represent both the traditional reinforcing bars in rein-forcement concrete and the discrete discontinuous steel fibers. The two-node, linearfinite element (truss element) with the traditional (one-dimensional) elastoplasticconstitutive model is adopted. The formulation of the CFE presented in the previ-ous chapter is extended to consider the reinforcement-matrix interaction, for which acontinuum damage model is applied to describe the relation between the shear stress(adherence stress) and the relative sliding between the matrix and the reinforcement.In this chapter, the formulation is validated through the numerical analysis of a set

34


of pullout tests of reinforcing bars and steel fibers. In addition, the results obtainedfrom the pullout tests for steel fibers are compared with the analytical expressionsfrom the basic assumptions of the Diverse Embedment Model (DEM) proposed by[63], in which important variables, such as inclination angle and embedment lengthsare considered.

Chapter 4 addresses the steel fiber distribution and orientation in cementitious ma-trices. An algorithm implemented in MATLAB© has been developed for generatingclouds of steel fibers, given the fiber content and geometrical properties of boththe steel fiber and concrete specimen. The fibers are randomly generated using anisotropic uniform random distribution, considering the wall effect of the mold.

In Chapter 5, the two approaches adopted for modeling the failure process in ce-mentitious matrices are introduced. In the first approach, a continuous approachbased on a continuum damage model with two independent scalar damage variablesused for capturing different responses under tension and compression is employed.For the discontinuous approach, a mesh fragmentation technique that introducesdegenerate solid finite elements in between all regular (bulk) elements is developed.In this case, a tensile damage constitutive model, compatible with the ContinuumStrong Discontinuity Approach (CSDA), is proposed to predict crack propagation.Moreover, for all the constitutive models in both approaches, a special implicit-explicit integration scheme has been developed to increase their computability androbustness.

In Chapter 6, numerical analyses are performed and compared with experimental re-sults. The focus here is the application of the strategy developed in the previous fivechapters for modeling experimental tests described in the literature to characterizethe tensile failure response of the cementitious materials.

Finally, general conclusions and directions for future research are discussed in Chap-ter 7.

It is also important to mention that each chapter of this thesis has been written ina self-contained manner and may be read almost entirely independently.

35

2 Multiscale approach for modelingSFRCC

This chapter presents a multiscale approach developed for modeling Steel FiberReinforced Cementitious Composites (SFRCCs). The novelty here is the new tech-nique for coupling non-matching finite element meshes, based on the use of specialfinite elements termed coupling finite elements (CFEs), which shares nodes withnon-matching meshes. The main features of the proposed technique are:

1. No additional degree of freedom is introduced to the problem;

2. Non-rigid coupling can be considered to describe the nonlinear behavior ofinterfaces similar to cohesive models;

3. Non-matching meshes of any dimension and any type of finite elements can becoupled;

4. Overlapping and non-overlapping meshes can be considered.

The feature 4 above allows the use of the same strategy to deal two problems ofnon-matching meshes addressed in this work. One is regarding the coupling of em-bedded bars into the bulk finite elements (overlapping meshes), and the other cor-responds to the coupling of different subdomains of a concurrent multiscale model(non-overlapping meshes). Thus, for problems where the material failure concen-trates in a specific region, the numerical model with a discrete treatment of fiberdeveloped can be applied only in that region of interest, increasing the performancein terms of computation time consuming.

In the remainder of this chapter, the idea behind the mesh coupling technique andformulation of the coupling finite elements are presented. Afterward the versatilityof the proposed technique is illustrated by a variety of 2D and 3D examples withdifferent non-matching mesh configurations. Here, only the rigid-coupling cases are

36

2.1 Overview of the coupling technique for non-matching FE meshes

approached. Later, the formulation will be extended for non-rigid coupling cases inchapter 3.

2.1 Overview of the coupling technique fornon-matching FE meshes

An overview of the coupling technique is given, considering the most general case,i.e., when a concurrent multiscale model is adopted. For this type of problem, thecoupling technique is employed to couple both, the interface of the subdomains,which were refined in distinct scales, and the constituents of the meso-model, whichwere explicitly modeled.

Therefore, let us consider a problem of domain Ω and boundary Γ, in which thedomain Ω is subdivided into subdomains Ω1 (mesoscale) and Ω2 (macroscale), asillustrated in Fig. 2.1(a). After the mesh discretization (see Fig. 2.1(b)), the commonboundary interface Γ1,2 = Γ1 ∩ Γ2 of these subdomains and the FE meshes of thecloud of fibers and bulk finite elements (in subdomain Ω1) are non-matching meshes.

Thus, the use of the coupling procedure developed to solve the problem describedabove can be summarized in the following steps:

1. Identification, at the common boundary interface of the subdomains and inthe region where a mesoscale approach was adopted, the loose nodes (hererepresented by the red nodes in Fig. 2.1(b));

2. Generation of the CFEs based on step 1 (Fig. 2.1(c));

3. Assembly of the CFEs in the system of equations of the problem, accordingto the law that describes the interaction between the subdomains.

37


Ω1

Γ1

Γ2

Γ1,2

Ω2

(a) (b)

n m

p

c3(c)

c1

c2

ij

k

l

(d)

(e)

mesoscalemodel

macroscalemodel

Figure 2.1: Coupling procedure for non-matching finite element meshes: (a) defini-tion of the problem; (b) process of identification of the nodes that will compose theCFEs; (c) creation and insertion of the CFEs; (d) detail of coupling in overlappingmeshes; and (e) detail of coupling in non-overlapping meshes.

According to Fig. 2.1(c), each CFE has the same nodes of an underlying finite ele-ment of the existing mesh and an extra node, coinciding with the loose node (thisnode is called the coupling node) that belongs to its domain. As a consequence, theCFEs overlap the finite elements of the original mesh around the coupling nodes.

38


Note that for each loose node, one coupling finite element is required.

Fig. 2.1(d) shows an example of coupling between overlapping meshes, where twocoupling finite elements CFE1 = {i, j, k, c1} and CFE2 = {j, l, k, c2} were used,whose nodes c1, and c2, respectively are their coupling nodes. At the commonboundary interface, to each loose node, a coupling finite element is also inserted,using as base an existing finite element, which has one face (for 3-node trianglesdefined by two nodes) along at the common boundary interface. An example isshown in Fig. 2.1(d), where the coupling finite element CFE3 = {m,n, p, c3} isintroduced, whose c3, is the coupling node.

These elements that share nodes with both non-matching meshes can then be used toensure the compatibility of displacements and to transfer interaction forces betweennon-matching meshes. The interaction forces between the non-matching meshes mayalso be described by an appropriate constitutive model applied in the CFEs. Thisis one of the major advantages of the technique, since a rigid (full compatibility ofdisplacements) or non-rigid (degrading interface) coupling can be considered easily.Thus, the use of this technique for modeling reinforced composite is very appeal-ing, since reinforcement, matrix and reinforcement-matrix interface can be modeledindependently.

Fig. 2.1(c) illustrates the final configuration of the mesh, with all the CFEs. Afterthe application of the coupling procedure, the global internal force vector and thestiffness matrix can be written as:

Fint = AnelΩ1e=1 (Finte )Ω1 + AnelΩ2e=1 (Finte )Ω2 + A

nelCe=1 (Finte )C (2.1)

K = AnelΩ1e=1 (Ke)Ω1 + AnelΩ2e=1 (Ke)Ω2 + AnelCe=1 (Ke)C (2.2)

where A stands for the finite element assembly operator, the first and second termsof 2.1 and 2.2 are related to the subdomains Ω1 and Ω2, respectively, and the thirdterm is tied to the introduction of the CFEs. Note that this coupling procedure,with the generation of new coupling elements, can be regarded as a pre-processingstage if the multiscale problem is "static", in the sense that the incompatible meshesare defined a priori and do not change during the analysis.

39

2.2 Coupling finite element

The formulation of the CFE and its role are described next.


2.2.1 CFE formulation

Consider a standard isoparametric finite element of domain Ωe, with number ofnodes equal to nn, and shape functions N i(X) (i = 1, nn), which are defined for thematerial points X ∈ Ωe, such that the displacement U at any point in its domaincan be approximated in terms of its nodal displacements Di (i = 1, nn), as follows:

U(X) =nn∑i=1

N i(X)Di. (2.3)

The CFE is a finite element which has the above described nodes of the standardisoparametric finite element as well as an additional node, nn + 1, called couplingnode (Cnode), situated at the material point Xc ∈ Ωe, as illustrated in Fig. 2.2.The coupling node can be anywhere in the element, including at its boundary (seeFig. 2.2(b)).

40


Ωe

nn=3

Cnode (nn+1)Ωe

nn=4

Cnode (nn+1)

(a) (b)

Ωe

nn=4

Cnode (nn+1)

Ωe

nn=8

Cnode (nn+1)

(c) (d)

Figure 2.2: 2D and 3D coupling finite elements with linear interpolation functionsof displacements: (a) 3-node triangle + Cnode, (a) 4-node quadrilateral + Cnode,(a) 4-node tetrahedral + Cnode, and (d) 8-node cube + Cnode.

The relative displacement, [[U]], defined as the difference between the displacementof the Cnode and the displacement of the material point Xc, can be evaluated usingthe shape functions of the underlying finite element, N i(Xc) (i = 1, nn), as follows:

[[U]] = Dnn+1 −U(Xc) = Dnn+1 −nn∑i=1

N i(Xc)Di = BeDe, (2.4)

where the matrix Be, is given by

Be = [−N1(Xc) −N2(Xc) ... −Nnn(Xc) I], (2.5)

and Ni = N iI, while I is the identity matrix of order 2 or 3, for 2D and 3Dproblems, respectively. Note that the matrix Be, in 2.5 is not the well known strain-displacement relationship, B matrix, commonly used in finite element notation. Thevector De which stores the displacement components of the coupling finite element

41


is given as:

De =

D1D2...

Dnn+1

. (2.6)

2.2.1.1 CFE internal force vector

The internal virtual work of the CFE is given by

δW inte = δ[[U]]TF([[U]]), (2.7)

where F([[U]]) is the reaction force owing to the relative displacement [[U]] and δ[[U]] isan arbitrary virtual relative displacement, compatible with the boundary conditionsof the problem. Using the same approximation for the virtual relative displacementas that used for the relative displacement given by 2.4, i.e., δ[[U]] = BeδDe, theinternal force vector of the coupling finite element can be expressed as follows:

Finte = BTe F([[U]]). (2.8)

2.2.1.2 CFE stiffness matrix

Accordingly, the corresponding tangent stiffness matrix of the CFE can be obtainedby the following expression:

Ke =∂Finte∂De

= BTe CtgBe (2.9)

where Ctg = ∂F([[U]])/∂[[U]] is the tangent operator of the constitutive relationbetween reaction force and the relative displacement.

Note that it could be argued that the CFE is not a finite element in a strict sense,since it is used only to construct the constraints for coupling nodes using the shapefunctions of the underlying finite element, or in other words, there is no specificintegration rule for it.

42

2.3 Rigid coupling

2.2.2 Linear elastic model

A linear elastic model can be used to describe the relation between the reactionforce and the relative displacement:

F = C[[U]] (2.10)= CBeDe (2.11)

where C is the matrix of elastic constants.

Consequently, from 2.11, and considering 2.8 and 2.9 above, the internal force vectorand stiffness matrix of a CFE become:

Finte = BTe CBeDe (2.12)

and

Ke = BTe CBe. (2.13)

The next formulations will be developed for 3D problems. However, the correspond-ing matrices and vectors of the 2D formulation can be obtained while suppressingthe third component.

2.3 Rigid coupling

A rigid coupling enforcing displacement compatibility of two non-matching meshes,as depicted in Fig. 2.1, can be imposed by assuming a very high value for the elasticstiffness, such that the matrix of elastic constants is expressed as follows:

C =

C̃ 0 00 C̃ 00 0 C̃

(2.14)

where C̃ stands for a high elastic stiffness value, which plays the role of a penaltyvariable on the relative displacement. It is important to note that, because of theequilibrium conditions, the interaction force F in 2.10 must be bounded. Hence,

43

2.4 Case studies

when the elastic constants tend towards a very high value, the relative displacementcomponents [[U]] must tend to zero.

Note that, for a rigid coupling analysis in 2.14, the third term in 2.2 correspondsto penalty terms that impose constrains between displacements of the conventionalfinite element approximations and the displacements at coupling nodes. Note alsothat the rigid coupling is achieved by setting [[U]] ' 0, assuming a very high valueof the elastic stiffness. In that case, the methodology can be seen as imposing linearmultipoint constraints (LMPCs) using penalty coefficients (for penalty methods usedto join non-matching meshes see, e.g., [16, 96, 13, 97]).

2.4 Case studies

To validate the proposed strategy to couple non-matching meshes, three exampleswere performed. For all the cases, rigid coupling procedure was employed. The firstexample aims to demonstrate the efficiency of the technique to couple 2D and 3Dnon-overlapping meshes. A case of non-matching overlapping meshes is consideredin the second example. In this example, the versatility of the proposed technique isshown through of the coupling of curved reinforcing bars and bulk finite elements.Finally, an example of a concurrent multiscale model is performed, considering inthe same example both the cases of overlapping and non-overlapping meshes.

2.4.1 Case study 01: Basic tests

This first example serves to illustrate the ability of the proposed strategy for couplingnon-matching meshes through the numerical analysis of a set of 2D and 3D basictests. The tests are conducted using a column with rectangular cross-section withtwo rigid parallel plates connected to the base and to the top of the column. Themain challenge here is to couple the non-matching meshes of the column and plates,discretized independently, using two different mesh refinements and types of finiteelements. Hence, this problem has three layers of finite element meshes and two non-matching interfaces. The geometry, boundary conditions and material propertiesadopted for the 2D tests are depicted in Fig. 2.3.

44

2.4 Case studies

(a )

F

100

200

20

20 20

t=140

rigid base

plates

E=1.0x10 3

ν=0.2t=100

(b)

d =10

coupling finite

elements

(c)

Figure 2.3: 2D basic tests. (a) Setup of the compression test, material and geomet-rical properties. (b) Setup of the shear test and non-matching meshes employed.(c) Matching reference mesh.

For the 2D analyses, the column is discretized with a fine mesh using four-nodequadrilateral finite elements, while for the rigid plates, a coarse mesh is employedwith three-node triangular finite elements (Fig. 2.3(b)). Here, the rigid couplingprocedure described in sec. 2.3 is considered using four-node triangular couplingfinite elements (T4). A total of 42 T4 have been used for coupling the two non-matching column-plate interfaces. For comparison, the numerical analyses havebeen also made employing the matching fine mesh shown in Fig. 2.3(c).

45

2.4 Case studies

-0.5

0.0

0.5

1.0

1.5

2.0

0.01 0.1 1 10 100 1000

rela

tive

erro

r (%

)

Cn and Cs x 106

compression test - horizontal elongation

(a)

-0.25

0.00

0.25

0.50

0.75

0.01 0.1 1 10 100 1000

rela

tive

erro

r (%

)

Cn and Cs x 106

compression test - vertical elongation

(b)

Figure 2.4: Convergence of horizontal (a) and vertical (b) elongation in the 2Dcompression test.

Compression and shear tests were initially performed to investigate the influence ofthe values adopted for the elastic components Cn and Cs of the rigid coupling pro-cedure. For both tests, elastic components varying from 104 to 109 were considered.A concentrated load of F = 5×105 was applied in the compression test (Fig. 2.3(a)),while for the shear test a horizontal displacement of d = 10 was imposed on the topplate (Fig. 2.3(b)). In both tests, the fixed boundary condition was applied at thebottom of the base plate.

For the compression test, the vertical elongation and horizontal elongation (at mid-height of the column) were measured and the relative error was calculated basedon the results obtained with the matching mesh (the reference problem). Fig. 2.4

46

2.4 Case studies

illustrates the relative error calculated for horizontal and vertical elongations.

In the shear test, the efficiency of the coupling procedure is measured based on theenergy calculated for each value adopted for the parameters of the coupling scheme(Fig. 2.5). Here, the relative error was also calculated based on the results obtainedwith the matching meshes.

0

10

20

30

40

50

60

0.01 0.1 1 10 100 1000

rela

tive

err

or

(%)

Cn and Cs x 106

shear test - energy

Figure 2.5: Convergence of energy for the 2D shear test.

The results obtained demonstrated that the adoption of values higher than 106 forthe elastic components of the coupling procedure ensures a perfect coupling betweenthe non-matching meshes. The vertical displacement field for the compression testand horizontal displacement field for the shear test, both with Cn = Cs = 109, areshown in Fig. 2.6(a) and (b), respectively.

47

2.4 Case studies

(b)(a )

Figure 2.6: Results obtained in the 2D tests. (a) Vertical displacement field forthe compression test. (b) Horizontal displacement field for the shear test (withscaling factor of 5).

To illustrate the use of the coupling procedure for 3D cases, the problem describedabove is discretized using eight-node hexahedral finite elements and four-node tetra-hedral finite elements for the regions of the column and rigid plates, respectively(Fig. 2.7). A total of 242 five-node tetrahedral finite elements (TETR5) were usedfor coupling the non-matching meshes. Fig. 2.7 shows the 3D mesh and the viewfrom the top of the specimen, before and after the coupling procedure.

(a ) (b)

Figure 2.7: 3D finite element mesh employed for the basic tests: (a) before of thecoupling procedure and (b) complete mesh (after the coupling procedure).

To verify the behavior of the non-matching interfaces in different load situations,the numerical 3D model was subjected to compression, tension and shear. In all the

48

2.4 Case studies

tests, a fixed boundary condition was applied at the bottom of the base plate, andthe elastic coupling parameters employed were Cn = Cs = Ct = 109. A prescribedvertical displacement field of d = 10 was applied on the top plate in the compressionand tension tests (in opposite directions), and a horizontal prescribed displacementof the same value was applied on the top base in the shear test.

Fig. 2.8 illustrates the results obtained for the set of 3D tests. Note, the continuityobtained in the displacement field, even under the Poisson’s effect observed in tensionand compression test

· este exemplar foi revisado e corrigido em relação à versão original, sob responsabilidade...

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