LUÍS ANTÔNIO GUIMARÃES BITENCOURT JÚNIOR
Numerical modeling of failure processes in steel fiber reinforced
cementitious materials
Numerical modeling of failure processes in steel fiber reinforced
cementitious materials
Thesis submitted to the Polytechnic School at the University of São Paulo for award the Doc-tor of Science Degree
Area of Concentration: Structural Engineering
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 fiber reinforced cementitious materials / L.A.G. Bitencourt Júnior. -- versão corr. -- São Paulo, 2015.
184 p.
Tese (Doutorado) - Escola Politécnica da Universidade de São Paulo. Departamento de Engenharia de Estruturas e Geotécnica.
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 gratefully acknowledged.
I would like to thank my research advisor, Prof. Túlio N. Bittencourt, for his support, encouragement and advice during this research. I also wish to thank him for all the opportunities he gave me to work on different research projects and in collaboration with several research groups in Brazil and abroad.
I also wish to express my deep gratitude to my co-advisor, Prof. Osvaldo Luís Manzoli, for his interest in my research, guidance and assistance. I am sure that all the discussions about different topics in computational mechanics will be very valuable 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 John Vecchio 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, who shared the office with me during this period.
I am grateful to all the professors and office staff of the Department of Structural and Geotechnical Engineering of the Polytechnic School at the University of São Paulo (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 sincere appreciation for creating a friendly environment, which helped me complete this research.
I am especially grateful to my friends in São Paulo, Diogo, Bruno, Leandro and Rejane, Társis and Darlene, Henrique and Mônica, Plínio and Márcia, Júlio and Creuza, Papito, Júnior and Renata, Marly, and many others.
I wish to express my love and respect to all members of my family, especially my parents Maria and Luis Bitencourt and my sisters Mariane and Manuela Bitencourt for 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 thanks goes to my aunt Edileusa Batista, who left us this year. She will always be in my memories.
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 three phases: a cementitious matrix (paste, mortar or concrete), discrete discontinuous fibers, and a fiber-matrix interface.
A novel coupling scheme for non-matching finite element meshes has been developed to couple the independent generated meshes of the bulk cementitious matrix and a cloud of discrete discontinuous fibers based on the use of special finite elements developed, termed Coupling Finite Elements (CFEs). Using this approach, a non-rigid coupling procedure is proposed for modeling the complex nonlinear behavior of the fiber-matrix interface by adopting an appropriate constitutive damage model to describe the relation between the shear stress (adherence stress) and the relative sliding between the matrix and each fiber individually. This scheme has also been adopted to account for the presence of regular reinforcing bars in the analysis of reinforced concrete structural elements.
tially, numerical examples with a single reinforcement are presented to validate the technique and to investigate the influence of the fiber’s geometrical properties and its position relative to the crack surface. Then, more complex examples involving a cloud of steel fibers are considered. In these cases, special attention is given to the analysis of the influence of the fiber distribution on the composite behavior relative to the cracking process. Comparisons with experimental results demonstrate that the application of the numerical tool for modeling the behavior of SFRCCs is very promising and may constitute an important tool for better understanding the effects of the different aspects involved in the failure process of this material.
Keywords: Steel Fiber Reinforced Cementitious Materials; Non-matching Meshes;
Resumo
Este trabalho apresenta uma estratégia numérica desenvolvida usando o método dos elementos finitos para simular o processo de falha de compósitos cimentícios reforçados com fibras de aço. O material é descrito como um compósito composto por três fases: matriz cimentícia (pasta, argamassa ou concreto), fibras descontínuas discretas, e interface fibra-matriz.
Um novo esquema de acoplamento para malhas de elementos finitos não-conformes foi desenvolvido para acoplar as malhas geradas independentes, da matriz cimen-tícia e de uma nuvem de fibras de aço, baseado na utilização de novos elementos finitos desenvolvidos, denominados elementos finitos de acoplamento. Utilizando este esquema de acoplamento, um procedimento não-rígido é proposto para a mode-lagem do complexo comportamento não linear da interface fibra-matriz, utilizando um modelo constitutivo de dano apropriado para descrever a relação entre a tensão de cisalhamento (tensão de aderência) e deslizamento relativo entre a matriz e cada fibra de aço individualmente. Este esquema também foi adotado para considerar a presença 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 posicionadas usando uma distribuição randômica uniforme isotrópica, considerando o efeito pa-rede.
Uma abordagem contínua e outra descontínua são investigadas para a modelagem do comportamento frágil da matriz cimentícia. Para a primeira, é utilizado um modelo de dano isotrópico com duas variáveis de dano para descrever o comportamento de dano à tração e à compressão. A segunda emprega uma técnica de fragmentação de malha que utiliza elementos finitos degenerados, posicionados entre todos os elementos finitos que formam a matriz cimentícia. Para esta técnica é proposto um modelo constitutivo à tração, compatível com a abordagem descontínua forte contínua, para prever a propagação de fissura. Para acelerar o cálculo e aumentar a robustez dos modelos de dano contínuos para simular o processamento de falhas, um esquema de integração implícito-explícito é utilizado.
atenção especial é dada à influência da distribuição das fibras no comportamento do compósito relacionado ao processo de fissuração. Comparações com resultados experimentais demonstram que a aplicação da ferramenta numérica para modelar o comportamento de compósitos cimentícios reforçados com fibras de aço é muito pro-missora e pode ser utilizada como uma importante ferramenta para melhor entender os efeitos dos diferentes aspectos envolvidos no processo de falha deste material.
Palavras-chave: Materiais Cimentícios Reforçados com Fibras de Aço; Malhas
List of Figures
1.1 Steel fibers bridging cracks (<http://www.liv.ac.uk/>, 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 that will compose the CFEs; (c) creation and insertion of the CFEs; (d) detail of coupling in overlapping meshes; and (e) detail of coupling in non-overlapping meshes. . . 38 2.2 2D and 3D coupling finite elements with linear interpolation functions
of 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-matching meshes employed. (c) Matching reference mesh. . . 45 2.4 Convergence of horizontal (a) and vertical (b) elongation in the 2D
compression test. . . 46 2.5 Convergence of energy for the 2D shear test. . . 47 2.6 Results obtained in the 2D tests. (a) Vertical displacement field for
the compression test. (b) Horizontal displacement field for the shear test (with scaling factor of 5). . . 48 2.7 3D finite element mesh employed for the basic tests: (a) before of
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 reinforcing
layer 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 reinforcing
layers with rigid coupling (without bond-slip). . . 52 2.13 Three-point bending beam simulated numerically using: (a) a mesoscale
model, 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 coordinate system, and (b) D.O.F. in local coordinate system, which influence the local stiffness matrix and internal force vector. . . 60 3.2 Typical stress-strain curve for one-dimensional elastoplastic material
model. . . 68 3.3 Scheme adopted for coupling a linear reinforcement element to its
corresponding 2D matrix elements. . . 70 3.4 Reinforcement allowing slip. . . 71 3.5 Influence length of the coupling node. . . 73 3.6 Interface stress bond-slip relationship (monotonic loading) proposed
by CEB-FIP model code 90 [1]. . . 75 3.7 Setup of the pullout tests. . . 77 3.8 Finite element mesh of the pullout test withlb = 75mm: (a) complete
mesh, and (b) detail of the coupling finite elements. . . 78 3.9 Evolution of average bond stress with respect to slip at the ends (A)
3.10 Deformed matrix (concrete) mesh at: (a) peak load and (b) end of analysis (with scaling factor of 5000). . . 80 3.11 Normal stress in concrete (in MPa) at: (a) peak load and (b) end of
analysis. . . 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. . 82 3.14 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 at
the loaded end (lb = 600mm). . . 83
3.16 2D numerical model of the bond bending test M.16.16.R of Bigaj [12]. 84 3.17 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 of 20). . . 85 3.18 Strain distribution along the steel bar at yield load. . . 86 3.19 Fiber-matrix interface model employed. . . 88 3.20 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 the coupling procedure. . . 88 3.21 Fiber stress at crack with la = 0.5lf for straight fiber. . . 89
3.22 Variation of the slip along the fiber when end slip is 0.1mm. . . 89 3.23 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 the coupling procedure, and (c) deformed FE mesh (with a scaling factor of 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. . . 92 3.26 Crack width at maximum pullout stress. . . 93 3.27 Numerical models for pullout tests with different fiber inclination
4.1 Specimen’s geometries and their respective restrictions to fiber po-sitioning. (a) Steel fiber. (b) Cylindrical specimen. (c) Prismatic
specimen. . . 99
4.2 Pseudo code of the algorithm developed for generating steel fibers. . . 103
5.1 Mesh fragmentation process: (a) standard (bulk) finite element mesh and identification of the region to be fragmented; (b) separation of the finite elements (with an exaggerated scale factor for clarity); (c) insertion of interface finite elements (depicted in green); (d) detail of interface elements in FE mesh; and e) detail of a couple interface finite 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. . . 126
5.4 Loading history considered in the uniaxial test. . . 126
5.5 Horizontal stress x imposed displacement. . . 127
5.6 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. . . 129
5.8 Horizontal stress x crack width. . . 130
6.1 Bond-slip relation adopted to described the fiber-matrix interaction. . 132
6.2 Direct tension tests: (a) typical specimen tested and (b) tensile test-ing machine [6]. . . 132
6.3 2D numerical model: (a) geometrical properties (dimensions in mm), (b) boundary conditions and loading, and (c) finite element mesh of the 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
6.7 Force x displacement curves. Comparison between numerical (con-tinuum damage model) and experimental responses for Vf = 1.5%. . . 137
6.8 Numerical analyses using damage model. Plain concrete and steel fiber reinforced concrete with steel fiber volume fractions of 0.5, 1.0 and 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 forVf = 1% using the damage model for
4 generated fiber structures. . . 141 6.12 Horizontal displacement contour (mm) for 4 generated fiber structures
with Vf = 1.0%. . . 142
6.13 Force x displacement curves. Comparison between numerical (mesh fragmentation technique) and experimental responses forVf = 0.5%. . 143
6.14 Force x displacement curves. Comparison between numerical (mesh fragmentation technique) and experimental responses forVf = 1.0%. . 143
6.15 Force x displacement curves. Comparison between numerical (mesh fragmentation technique) and experimental responses forVf = 1.5%. . 144
6.16 Failure pattern obtained using mesh fragmentation technique forVf =
0.5%: (a) deformed mesh at final load, and detail of the failure process for 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 volume
fractions of 0.5, 1.0, and 1.5%. . . 146 6.18 3D finite element mesh of the direct tension test: (a) concrete phase
and (b) fiber phase forVf = 0.5%. . . 147
6.19 3D numerical analyses of plain concrete and steel fiber concrete with fiber volume fraction ofVf = 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 forVf =
fiber volume fraction ofVf = 0.5%. . . 150
6.23 Comparison between force x displacement curves obtained for 2D and 3D numerical analyses using damage model for plain concrete. . . 150 6.24 Geometrical properties and finite element meshes for the three-point
bending 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 centered notch. . . 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 and
SFRC with (b)Vf = 0.25% and (b)Vf = 0.50% (with a scaling factor
of 5). . . 156 6.30 Numerical responses for beams with centered notch of plain concrete
and 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 eccentric notch. . . 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 concrete
and SFRC with (b) Vf = 0.25% and (b) Vf = 0.50% (with a scaling
factor of 5). . . 160 6.36 Numerical responses for beams with eccentric notch. Plain concrete
2.1 Material parameters. . . 54 2.2 Comparison of the data of the mesoscale and multiscale models. . . . 57
3.1 Classification of the elastoplastic material behavior. . . 65 3.3 Ingredients of the one-dimensional elastoplastic material model. . . . 69 3.4 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 damage model with distinct tensile and compressive responses. . . 111 5.2 IMPL-EX integration scheme for the tensile damage model. . . 122 5.3 Components of the discrete relation of the IFE for 2D and 3D cases. . 123 5.4 Continuum and discrete constitutive equations for the tensile damage
Contents
1 Introduction 20
1.1 General aspects . . . 20
1.2 Motivation . . . 24
1.3 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 . . . 29
1.3.3 Robustness issues in numerical modeling of material failure . . 32
1.4 Overview and limitations of the thesis . . . 32
1.5 Main contributions of the thesis . . . 33
1.6 Structure of the thesis . . . 34
2 Multiscale approach for modeling SFRCC 36 2.1 Overview of the coupling technique for non-matching FE meshes . . . 37
2.2 Coupling finite element . . . 40
2.2.1 CFE formulation . . . 40
2.2.1.1 CFE internal force vector . . . 42
2.2.1.2 CFE stiffness matrix . . . 42
2.2.2 Linear elastic model . . . 43
2.3 Rigid coupling . . . 43
2.4 Case studies . . . 44
2.4.1 Case study 01: Basic tests . . . 44
2.4.2 Case study 02: Cylinder with curved reinforcing layers . . . . 50
2.4.3 Case study 03: Non-matching meshes of a concurrent multi-scale model . . . 53
3.1.1 Truss internal force vector . . . 62
3.1.2 Truss stiffness matrix . . . 63
3.2 One-dimensional elastoplastic model . . . 63
3.2.1 Basic definitions . . . 64
3.2.2 Loading/unloading conditions . . . 66
3.2.3 Tangent constitutive operator . . . 67
3.3 Non-rigid reinforcement-matrix coupling . . . 69
3.3.1 Elastic model . . . 72
3.3.2 Bond-slip model . . . 72
3.3.3 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 . . . 77
3.4.1 Case study 01: 2D pullout tests of reinforcing bars . . . 77
3.4.1.1 Short embedment length (lb = 75mm) . . . 78
3.4.1.2 Long embedment length (lb = 600mm) . . . 82
3.4.2 Case study 02: Bending bond test . . . 84
3.4.3 Case study 03: Pullout behavior of steel fibers . . . 86
3.5 Summary and conclusions . . . 96
4 Fiber distribution and orientation in cementitious matrix 97 4.1 Introduction . . . 97
4.2 Program for generation and orientation of steel fibers . . . 98
4.3 Summary and conclusions . . . 104
5 Cementitious material modeling 105 5.1 Continuous approach based on isotropic damage model . . . 105
5.1.1 A continuum isotropic damage model with distinct tensile and compressive responses . . . 105
5.1.1.1 Modified implicit-explicit integration scheme for the continuum damage model with distinct tensile and compressive responses . . . 109
5.2 Discontinuous approach . . . 112
5.2.2 Interface solid finite elements . . . 115 5.2.3 Tension damage model . . . 120
5.2.3.1 Implicit-explicit integration scheme for the tensile damage model . . . 121 5.2.4 Discrete relation of the interface finite element . . . 123 5.3 Case studies . . . 125 5.3.1 Case study 01: Continuum damage model under uniaxial loading125 5.3.2 Case study 02: Interface elements under uniaxial loading . . . 127 5.4 Summary and conclusions . . . 130
6 Applications 131
6.1 Example 01: Direct tension tests . . . 132 6.2 Example 02: Three-point bending beams . . . 151 6.3 Summary and conclusions . . . 161
7 General conclusions and recommendations for future research 163
7.1 General conclusions . . . 163 7.2 Recommendations for future research . . . 166
8 Appendix 168
8.1 IMPL-EX integration scheme . . . 168 8.2 Transformation matrix for the two-node linear finite element . . . 171
1.1 General aspects
Cementitious materials, such as mortar and concrete, are among the most widely used construction materials. These materials are classified as quasi-brittle materials, i.e., materials with low tensile strength and strain capacities. To overcome these main drawbacks, discontinuous fibers have been added to cementitious matrices, resulting in so-called Fiber Reinforced Cementitious Composites (FRCCs). Despite the wide use of FRCCs in recent years, the addition of fibers for improving material properties comes from ancient times. Egyptians used straw to reinforce sunbaked bricks [73], and there is also evidence that asbestos fiber was already being employed to reinforce clay posts approximately 5000 years ago [81].
The first investigations concerning the use of fibers were performed in the early 60s and are reported in the papers published by [104] and [105]. The authors investigated the ability of fibers to improve the tensile ductility of concrete. In the decades that followed, a large number of experimental studies were developed to verify the effects of fibers on the properties of cementitious matrices, such as tension, compression, bending and shear.
1.1 General aspects
Currently, it is well known that the addition of a small volume of steel fibers may increase the ductility and toughness of cementitious matrices [11]. The role played by fibers is most obvious after matrix cracking has occurred, as fibers offer resistance to crack propagation, which is illustrated in Fig. 1.1.
Figure 1.1: Steel fibers bridging cracks (<http://www.liv.ac.uk/>, viewed
Octo-ber, 2014).
Therefore, the main benefits of the addition of steel fibers in cementitious matrices are directly related to their ability to transfer stresses across cracks. Fig. 1.2 shows the influence of steel fibers in hardened concrete cementitious matrices for a typical stress versus crack width curve. As noted in this figure, before the addition of steel fibers and after matrix cracking, the tensile stress immediately decreases (see the curve for the matrix). However, after the addition of a certain volume of fibers and after 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 of the 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 the fibers, and (iii) the structure of the fiber-matrix interface.
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 the cementitious matrix does not exceed 2%. However, in the last decade, large volumes of fibers have been added to develop so-called High Performance Fiber Reinforced Cementitious Composites (HPFRCCs). HPFRCCs exhibit multiple cracking with strain-hardening under direct tension. A general classification for fiber reinforced composites 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 different shapes, lengths, diameters or equivalent diameters, and surface deformation [87]. New fibers with different characteristics are continuously being proposed to improve the compatibility with the cementitious matrix. Fig. 1.3 illustrates the typical shapes of steel fibers commonly used in concrete.
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]. Among these properties, the wall effects introduced by the formwork and the fresh-state properties have the greatest effects [79]. Usually, a parameter called the orientation number (η) is used to quantify the influence of one of the aforementioned factors on the fiber distribution [112].
In the last years, a set of techniques has been developed to determine the distribution of steel fibers in hardened cementitious matrices. These techniques are categorized as destructive and non-destructive methods based on direct and indirect measurements [59]. These techniques are often applied to quantify the wall effects on idealized isotropic SFRCCs.
The fiber-matrix interaction plays one of the most important roles in the behavior of the cementitious composite. According to [85], the transmission forces between the fibers and matrix are produced through the interfacial bond defined as the shearing stress 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 mechanical components of the bond (as in, for example, crimped and hooked fibers), the physi-cal and chemiphysi-cal adhesion between the fibers and the matrix, and the fiber-to-fiber interlock, may influence its behavior [85]. Thus, steel fibers with different shapes exhibit different slip characteristics and pullout energies, even if the same maximum shear stress is obtained.
the behavior of concrete subject to tension [78, 43, 122, 63, 64]. Among these models, the analysis model called the Diverse Embedment Model (DEM) [63, 64] is the most realistic model available in the literature. This model considers the effects of each fiber on the behavior of the composite, including the fibers’ geometry, their distribution and orientation, characteristics of the single pullout response and the influence 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. Although the application of SFRCCs has increased in the last years, being very attractive in many structures today, such as tunnel linings, bridges, pavements, and pipes, there remains 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 this material is highly dependent on both the distribution of the steel fibers and the interaction of each fiber with the cementitious matrix. Hence, a numerical model with a discrete treatment of the fibers seems to be a natural way to simulate the failure behavior of this material.
In numerical models based on the finite element method with a discrete treatment of the fibers, non-conformal meshes are often considered between the cloud of steel fibers and the cementitious matrix. A coupling procedure is then applied to couple these independent meshes. A rigid coupling is usually applied, and the fiber-matrix interaction is included in the constitutive model adopted to describe the behavior of the fibers. In addition, these methods usually present problems of convergence, and analyses are limited to 2D problems.
1.3 Literature review
Currently, several experimental tests, such as 3-point bending tests, 4-point bending tests, splitting tensile tests, round determinate tensile tests, uniaxial tensile tests, and Barcelona tests, have been used to characterize the behavior of SFRCCs under tensile load. The major problem found in these tests is the variability of the obtained results. Moreover, due to the small size of the specimens compared to the size of conventional structural members, it is very difficult to use the results obtained in experimental tests to predict the behavior of structural members. In this case, only different boundary conditions and casting methods will produce a composite with different mechanical properties. Therefore, a numerical model that considers the influence of these factors could be very useful in developing a link between experimental tests and practical applications.
1.3 Literature review
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 the models that consider the individual influence of each fiber on the failure behavior of the composite. Other important subjects in this thesis, such as coupling methods for non-matching meshes and robustness issues in the numerical modeling of material failure, will also be addressed.
1.3.1 Numerical strategies for modeling the failure behavior of
SFRCCs
[114] describes the failure process of cementitious materials as follows: “micro-cracks first arise which change gradually into dominant macroscopic discrete cracks up to rupture”. In the literature, many numerical models have been proposed for modeling this failure process. In general, these models can be classified as continuous or discontinuous. An overview can be found in [114] and [49].
models used to solve this problem are based on fracture mechanics concepts that lead to fracture energy release regularization [9, 19]. However, the loss of objective of the deformation pattern is also presented. More recently, the mesh dependence problem has been addressed using integral-type, non-local and second-gradient approaches [114].
Discontinuous models are characterized by the introduction of displacement or strain discontinuities 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 methods are 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 the same conditions applied in the laboratory test. Moreover, this type of model is very expensive 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 relations are used as input data. As the previous models, the relations that describe the behavior of the material are developed for specific structural elements.
Various analytical models [62, 63] used to describe the tension behavior of SFRCCs have been proposed and implemented in computational programs. An interesting model called Diverse Embedment Model (DEM) [63] has been recently developed. The DEM considers the effects of the fiber geometry, the fibers’ distribution and orientation, characteristics of the single pullout response and the influence of the structural member dimensions.
1.3 Literature review
and [84] use a continuum strong discontinuous approach (CSDA) for modeling the failure behavior of the composite.
Recently, various studies have focused on the development of models that include a discrete treatment of fibers. An explicit representation is adopted in some models, whereas in other models, only interaction forces are considered to account for the presence of the fibers. This type of approach was adopted in this thesis with the fibers being explicitly represented. Therefore, the main references used in this thesis are 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. The interactions 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 actual location and with their orientation. The interactions between fibers and concrete are considered in the stress-strain relation adopted to describe the fiber behavior. In turn, this relation is obtained through the expressions proposed by [59] which are deduced from pullout tests [60, 61]. Adopting this strategy, the model is able to capture 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 failure of concrete can also be represented using a heuristic crack model with joints. In this material model, the whole specimen is described by an elastic material, and the cracking pattern is described using joint elements. The numerical analyses demon-strated that the approach captures the main features of the composite behavior. For all the numerical examples performed, the failure pattern exhibits only one crack. Moreover, to define the constitutive model for each fiber, the angle between the fiber and the failure pattern must be known beforehand.
The numerical approach proposed by [26] for modeling the behavior of SFRCCs considers 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 of steel fibers. A fixed smeared crack model is employed to describe the failure process of the cementitious matrix. The steel fibers are explicitly represented using linear elements with an embedded approach, modeled using a perfectly bonded
tion. Hence, the fiber-matrix interaction is considered in an indirect fashion in the constitutive model adopted for the fibers. This procedure consists of obtaining a stress-strain relation from the load-slip relation of a pullout test. In this model, the fibers 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 good agreement with the experimental results. Only experiments with a previously de-fined fracture plane have been analyzed. More details about the formulation of this model 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, fibconsid-ers and fiber-matrix interaction. The fibconsid-ers are considered as discrete entities, but they are not explicitly discretized to avoid high computational costs. Instead, only reaction forces from the fiber on the matrix are considered, and they are activated only when they bridge a crack. The reaction forces are assumed to be equal to the fiber pullout forces, which can be modeled either analytically or numerically. A damage model (regularized or non-local) is employed for modeling the matrix. The 2D numerical examples demonstrated that the model is able to represent the influence of fibers, their distribution and their interaction with the matrix. Moreover, several features of FRCCs, such as their ductile behavior, strain hardening and multiple cracking, were obtained.
1.3 Literature review
A discrete irregular lattice model was proposed by [56]. The meso-scale model is composed of the cement matrix, aggregate inclusions, steel fibers explicitly modeled with a realistic stochastic distribution within the concrete, matrix-inclusion interface and matrix-fiber interface. The aggregates are randomly positioned in the domain according to a granulometric distribution with a circular or spherical shape for a 2D or 3D model, respectively. The model employs rigid rod elements, creating a lattice that breaks according to a simple rule. Moreover, each element is associated with a particular material phase or interface. Two and three-dimensional quasi-static simulations were performed, and the results were qualitatively compared with experiments. The effects of the fiber length, fiber orientation distributions, interface strength, 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 an independent 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, without explicitly meshing them to ensure numerical efficiency. A linear elastic behavior is used to describe the fibers, whereas the matrix behavior is described by an isotropic damage model with an exponential softening law. This damage model is equipped with a simple regularized fracture energy model (details can be found in [9]) and with the gradient-enhanced damage model proposed by [99] to avoid the mesh de-pendence. The nonlinear behavior of the fiber-matrix interaction is described by the model proposed by [47]. The strategy presented is general, and any material model can be applied to represent each constituent. Several 2D numerical examples have been used to measure the influence of the fiber distribution, fiber length, in-terface model and mesh refinement dependency. According to various authors, the computational costs can be dramatically reduced compared with the standard finite element 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
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. As a consequence, another problem may arise in the transition between coarse and fine meshes, since the presence of distorted elements can invalidate the solution in the transition regions [3]. Another strategy widely used today consists of discretizing the regions of the problem (subdomains), in a totally independent way, according to the interest of the analyst, and then use a coupling technique to connect their non-matching interfaces. This strategy has been applied extensively to problems with adaptive mesh refinement [125, 69, 67], multiscale problems [121, 68, 116] or multiphysics 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 interaction effects between the subdomains, initially subdivided to be computed by different processors [72, 118].
In this context, a number of coupling methods have been developed to capture interface effects accurately [46]. Ref. [52] defines the continuity and compatibil-ity conditions at non-matching interfaces between subdomains as the fundamental requirements, and, also assert that the strain fields must be transferred correctly through the non-matching interfaces. Constraint equations have been used to make a 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 loose coupling boundary nodes of the local subdomain (fine mesh) using the displacement interpolation of the adjacent finite element of the global subdomain (coarse mesh) [121].
1.3 Literature review
On the side of the primal methods one could cite the penalty methods [96, 97]. Also the discontinuous Galerkin (DG) methods (a good review of DG methods is given in [4]) and Nitsche methods (originally described in [88]) are among the most widely used.
In these methods, the interface is represented by its displacement field and no dual variables are introduced [46]. For this reason, and in contrast with the dual ap-proaches, primal methods are not subject to theinf-sup or Ladyzhenskaya-Babuˇska (LBB) restrictions. However, a stabilization parameter is needed. Ref. [46] proposed a primal interface formulation that uses local enrichment of the interface elements to enable an unbiased enforcement of geometric compatibility at all interface nodes without 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 problems of fluid-structure interaction.
1.3.3 Robustness issues in numerical modeling of material failure
When a numerical model for modeling material failure is proposed, an important issue is presented: robustness. This is because when a crack propagates within the strain softening regime, the algorithmic tangent operator may become singular, and as a consequence, the solution of the resulting systems of nonlinear equations using a fully implicit discretization methodology cannot be obtained. According to [93], this problem may also be presented in the numerical modeling of material failure even 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 the integration of the constitutive models was proposed by Oliver [93, 94]. Among the main beneficial aspects of the use of the IMPL-EX algorithm are the guaranteed convergence, regardless of the length of the load step, and the low computational cost compared with standard implicit schemes.
The guaranteed convergence and robustness of the IMPL-EX algorithm result from the positive definite algorithmic tangent operator, which, in turn, is due to the internal 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 only one iteration per load step is sufficient to establish equilibrium, which is given by the difference between external and internal forces. This is because the algorithmic tangent operator is constant (when an infinitesimal strain format is adopted) during a load step.
In sec. 8.1, the idea behind the IMPL-EX integration scheme and its application to continuum damage models with scalar damage variables is presented. A complete description and details on the use of this integration technique with other families of constitutive models, such as elastoplastic models, can be found in [93, 94].
1.4 Overview and limitations of the thesis
per-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 factors that influence the behavior of the composite, such as the fiber geometry and the distribution and orientation of the fibers; adopted bond-slip model; and properties of 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 discrete treatment is applied to the fibers, and their discretization is independent of the finite element mesh of the bulk cementitious matrix. Here, the focus is on the development of the coupling scheme to couple non-matching meshes, which allows a 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 the specimen, such as the casting direction, vibrations, and flow of fresh composite, are not considered.
In spite of the mesoscale nature of the proposed numerical model, coarse aggregates and the mortar are homogenized, thus requiring the adoption of effective properties to characterize the specimen’s behavior.
Although the model can be applied to any volume content of fibers, this study is limited to 2% for reasons of computational costs. Moreover, the interaction between fibers 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 integrate the constitutive models of both approaches is investigated.
1.5 Main contributions of the thesis
• A new coupling technique for coupling non-matching finite element meshes. • A strategy for modeling the reinforcement elements (reinforcing bars and
fibers) 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 cementitious matrices based on a mesh fragmentation technique and on finite elements with high aspect ratio.
• An implicit-explicit integration scheme to increase the robustness of the con-stitutive models and to accelerate the nonlinear convergence.
1.6 Structure of the thesis
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 Chapter 2. The focus herein is on the formulation of the new finite elements, termed Coupling Finite Elements (CFEs), which are responsible for coupling the non-matching finite element meshes generated during the discretization of the different scale materials. In this chapter, the formulation for the case of a rigid-coupling procedure is assessed and validated.
1.6 Structure of the thesis
of pullout tests of reinforcing bars and steel fibers. In addition, the results obtained from the pullout tests for steel fibers are compared with the analytical expressions from the basic assumptions of the Diverse Embedment Model (DEM) proposed by [63], in which important variables, such as inclination angle and embedment lengths are considered.
Chapter 4 addresses the steel fiber distribution and orientation in cementitious ma-trices. An algorithm implemented in MATLAB© has been developed for generating
clouds of steel fibers, given the fiber content and geometrical properties of both the steel fiber and concrete specimen. The fibers are randomly generated using an isotropic 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 approach based on a continuum damage model with two independent scalar damage variables used for capturing different responses under tension and compression is employed. For the discontinuous approach, a mesh fragmentation technique that introduces degenerate solid finite elements in between all regular (bulk) elements is developed. In this case, a tensile damage constitutive model, compatible with the Continuum Strong 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 and robustness.
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 five chapters for modeling experimental tests described in the literature to characterize the tensile failure response of the cementitious materials.
Finally, general conclusions and directions for future research are discussed in Chap-ter 7.
SFRCC
This chapter presents a multiscale approach developed for modeling Steel Fiber Reinforced Cementitious Composites (SFRCCs). The novelty here is the new tech-nique for coupling non-matching finite element meshes, based on the use of special finite elements termed coupling finite elements (CFEs), which shares nodes with non-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 of interfaces similar to cohesive models;
3. Non-matching meshes of any dimension and any type of finite elements can be coupled;
4. Overlapping and non-overlapping meshes can be considered.
The feature 4 above allows the use of the same strategy to deal two problems of non-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 fiber developed can be applied only in that region of interest, increasing the performance in terms of computation time consuming.
2.1 Overview of the coupling technique for non-matching FE meshes
approached. Later, the formulation will be extended for non-rigid coupling cases in chapter 3.
2.1 Overview of the coupling technique for
non-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, the coupling technique is employed to couple both, the interface of the subdomains, which were refined in distinct scales, and the constituents of the meso-model, which were explicitly modeled.
Therefore, let us consider a problem of domain Ω and boundary Γ, in which the domain Ω is subdivided into subdomains Ω1 (mesoscale) and Ω2 (macroscale), as
illustrated in Fig. 2.1(a). After the mesh discretization (see Fig. 2.1(b)), the common boundary interface Γ1,2 = Γ1∩Γ2 of these subdomains and the FE meshes of the
cloud 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 described above can be summarized in the following steps:
1. Identification, at the common boundary interface of the subdomains and in the region where a mesoscale approach was adopted, the loose nodes (here represented by the red nodes in Fig. 2.1(b));
2. Generation of the CFEs based on step 1 (Fig. 2.1(c));
Ω1 Γ1
Γ2 Γ1,2
Ω2
(a) (b)
n m
p
c3
(c)
c1
c2
i j
k
l
(d)
(e) mesoscale
model
macroscale model
Figure 2.1: Coupling procedure for non-matching finite element meshes: (a)
defini-tion of the problem; (b) process of identificadefini-tion of the nodes that will compose the CFEs; (c) creation and insertion of the CFEs; (d) detail of coupling in overlapping meshes; and (e) detail of coupling in non-overlapping meshes.
2.1 Overview of the coupling technique for non-matching FE meshes
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 two coupling finite elements CF E1 = {i, j, k, c1} and CF E2 = {j, l, k, c2} were used,
whose nodes c1, and c2, respectively are their coupling nodes. At the common
boundary 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 triangles defined by two nodes) along at the common boundary interface. An example is shown in Fig. 2.1(d), where the coupling finite element CF E3 = {m, n, p, c3} is
introduced, whosec3, is the coupling node.
These elements that share nodes with both non-matching meshes can then be used to ensure the compatibility of displacements and to transfer interaction forces between non-matching meshes. The interaction forces between the non-matching meshes may also be described by an appropriate constitutive model applied in the CFEs. This is one of the major advantages of the technique, since a rigid (full compatibility of displacements) 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 modeled independently.
Fig. 2.1(c) illustrates the final configuration of the mesh, with all the CFEs. After the application of the coupling procedure, the global internal force vector and the stiffness matrix can be written as:
Fint =AnelΩ1
e=1 (Finte )Ω1 +A
nelΩ2
e=1 (Finte )Ω2 +Aenel=1C(Finte )C (2.1)
K=AnelΩ1
e=1 (Ke)Ω1 +A
nelΩ2
e=1 (Ke)Ω2 +Aenel=1C(Ke)C (2.2)
whereA stands for the finite element assembly operator, the first and second terms of 2.1 and 2.2 are related to the subdomains Ω1 and Ω2, respectively, and the third
The formulation of the CFE and its role are described next.
2.2 Coupling finite element
2.2.1 CFE formulation
Consider a standard isoparametric finite element of domain Ωe, with number of
nodes equal tonn, and shape functionsNi(X) (i= 1, nn), which are defined for the
material points X ∈ Ωe, such that the displacement U at any point in its domain
can be approximated in terms of its nodal displacements Di (i= 1, nn), as follows:
U(X) =
nn X
i=1
Ni(X)Di. (2.3)
The CFE is a finite element which has the above described nodes of the standard isoparametric finite element as well as an additional node, nn+ 1, called coupling node (Cnode), situated at the material point Xc ∈ Ωe, as illustrated in Fig. 2.2.
2.2 Coupling finite element
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 functions
of 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 displacement of the Cnode and the displacement of the material point Xc, can be evaluated using
the shape functions of the underlying finite element,Ni(Xc) (i= 1, nn), as follows:
[[U]] = Dnn+1−U(Xc) = Dnn+1−
nn X
i=1
Ni(Xc)Di =BeDe, (2.4)
where the matrix Be, is given by
Be = [−N1(Xc) −N2(Xc) ... −Nnn(Xc) I], (2.5)
and Ni = NiI, while I is the identity matrix of order 2 or 3, for 2D and 3D
is given as:
De =
D1 D2 ...
Dnn+1
. (2.6)
2.2.1.1 CFE internal force vector
The internal virtual work of the CFE is given by
δWeint =δ[[U]]TF([[U]]), (2.7)
whereF([[U]]) is the reaction force owing to the relative displacement [[U]] andδ[[U]] is an arbitrary virtual relative displacement, compatible with the boundary conditions of the problem. Using the same approximation for the virtual relative displacement as that used for the relative displacement given by 2.4, i.e., δ[[U]] = BeδDe, the
internal force vector of the coupling finite element can be expressed as follows:
Finte =BTeF([[U]]). (2.8)
2.2.1.2 CFE stiffness matrix
Accordingly, the corresponding tangent stiffness matrix of the CFE can be obtained by the following expression:
Ke=
∂Finte
∂De =B
T
eCtgBe (2.9)
where Ctg = ∂F([[U]])/∂[[U]] is the tangent operator of the constitutive relation between reaction force and the relative displacement.
2.3 Rigid coupling
2.2.2 Linear elastic model
A linear elastic model can be used to describe the relation between the reaction force and the relative displacement:
F = C[[U]] (2.10)
= CBeDe (2.11)
where Cis the matrix of elastic constants.
Consequently, from 2.11, and considering 2.8 and 2.9 above, the internal force vector and stiffness matrix of a CFE become:
Finte =BTeCBeDe (2.12)
and
Ke=BTeCBe. (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 suppresscorrespond-ing the 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 elastic stiffness, such that the matrix of elastic constants is expressed as follows:
C=
˜
C 0 0 0 C˜ 0 0 0 C˜
(2.14)
when the elastic constants tend towards a very high value, the relative displacement components [[U]] must tend to zero.
Note that, for a rigid coupling analysis in 2.14, the third term in 2.2 corresponds to penalty terms that impose constrains between displacements of the conventional finite element approximations and the displacements at coupling nodes. Note also that the rigid coupling is achieved by setting [[U]]≃ 0, assuming a very high value of the elastic stiffness. In that case, the methodology can be seen as imposing linear multipoint constraints (LMPCs) using penalty coefficients (for penalty methods used to 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 examples were performed. For all the cases, rigid coupling procedure was employed. The first example aims to demonstrate the efficiency of the technique to couple 2D and 3D non-overlapping meshes. A case of non-matching overlapping meshes is considered in the second example. In this example, the versatility of the proposed technique is shown through of the coupling of curved reinforcing bars and bulk finite elements. Finally, an example of a concurrent multiscale model is performed, considering in the same example both the cases of overlapping and non-overlapping meshes.
2.4.1 Case study 01: Basic tests
2.4 Case studies
(a)
F
100
2
0
0
2
0
20 20
t=140
rigid base plates E=1.0x103
ν=0.2
t=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.
-0.5 0.0 0.5 1.0 1.5 2.0
0.01 0.1 1 10 100 1000
re la ti v e e rr o r (% )
Cnand Csx106
compression test - horizontal elongation
(a) -0.25 0.00 0.25 0.50 0.75
0.01 0.1 1 10 100 1000
re la ti v e e rr o r (% )
CnandCsx106
compression test - vertical elongation
(b)
Figure 2.4: Convergence of horizontal (a) and vertical (b) elongation in the 2D
compression test.
Compression and shear tests were initially performed to investigate the influence of the 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 ofF = 5×105 was applied in the compression test (Fig. 2.3(a)),
while for the shear test a horizontal displacement ofd= 10 was imposed on the top plate (Fig. 2.3(b)). In both tests, the fixed boundary condition was applied at the bottom of the base plate.
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 the energy 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 obtained with the matching meshes.
0 10 20 30 40 50 60
0.01 0.1 1 10 100 1000
re
la
ti
v
e
e
rr
o
r
(%
)
CnandCsx106
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 for
the elastic components of the coupling procedure ensures a perfect coupling between the non-matching meshes. The vertical displacement field for the compression test and horizontal displacement field for the shear test, both with Cn = Cs = 109, are
(b) (a)
Figure 2.6: Results obtained in the 2D tests. (a) Vertical displacement field for
the compression test. (b) Horizontal displacement field for the shear test (with scaling factor of 5).
To illustrate the use of the coupling procedure for 3D cases, the problem described above 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 used for coupling the non-matching meshes. Fig. 2.7 shows the 3D mesh and the view from 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 the
coupling procedure and (b) complete mesh (after the coupling procedure).
2.4 Case studies
tests, a fixed boundary condition was applied at the bottom of the base plate, and the elastic coupling parameters employed were Cn = Cs =Ct = 109. A prescribed
vertical displacement field ofd = 10 was applied on the top plate in the compression and tension tests (in opposite directions), and a horizontal prescribed displacement of 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 continuity obtained in the displacement field, even under the Poisson’s effect observed in tension and compression tests.
(d) (c)
(a) (b)
Figure 2.8: Displacement field over 3D deformed FE mesh in x-direction for: (a)
