Numerical simulation of fracture in plain and fibre reinforced concrete
1 INTRODUCTION Chapter INTRODUCTION 1.1 Background The rapid development of fracture mechanics of quasibrittle materials in the last three decades was essentially dictated by the realisation that its application can lead to a satisfactory simulation and prediction of the local damage phenomena and the effect of structural size to fracture (Baˇant and Planas, 1998) Moreover, it z offers a logical approach to structural analysis and design based on sound mathematical and mechanics concepts Furthermore, the advent of new materials such as high-strength concrete, fibre-reinforced concrete and polymer composites necessitates the use of fracture mechanics to effectively exploit their material properties for reasons of safety and economy At present we are entering a period in which the introduction of fracture mechanics into concrete design is becoming possible (Mindess, 2002) This will help achieve more uniform safety margins, especially for structures of different sizes This, in turn, will improve economy as well as structural reliability It will make it possible to introduce new designs and utilise new concrete materials Applications of fracture mechanics are most urgent for structures such as concrete dams, long span bridges, and nuclear reactor vessels or containments, for which the safety concerns are particularly high and the consequences of potential disaster enormous The applicability of fracture mechanics to real engineering problems depends on the availability of fracture models that can simulate satisfactorily the behaviour of quasibrittle fracture One such model is the cohesive crack model whose early development can be attributed to the independent works of Dugdale (1960) and INTRODUCTION Barenblatt (1962) The cohesive crack models were developed to simulate the nonlinear material behaviour near the crack tip In these models, the crack is assumed to extend and to open while still transferring stress from one face to the other The cohesive model proposed by Barenblatt (1959, 1962) aimed to relate the macroscopic crack growth resistance to the atomic binding energy, while relieving the stress singularity Barenblatt postulated that the cohesive forces were operative on only a small region near the crack tip, and assumed that the shape of the crack profile in this zone was independent of the body size and shape Dugdale (1960), in an investigation of yielding in steel sheets containing slits, formulated a model of a line crack with a cohesive zone having constant yield stress Although formally close to Barenblatt’s, this model was intended to represent a completely different physical situation: macroscopic plasticity rather than microscopic atomic interactions Both models share a convenient picture in which the stress singularity is removed Despite being very simplified, Dugdale’s approach to plasticity gave a good description of ductile fracture for small plastic zone sizes However, it was not intended to describe fracture itself and, in Dugdale’s formulation, the plastic zone extended forever without any actual crack extension The cohesive crack model came to the forefront in the mid 1970s with the work of Hillerborg and co-workers (Hillerborg et al., 1976) The cohesive crack model served as a suitable nonlinear model for mode I fracture Their research acted as a catalyst in rousing the interest of studying quasibrittle materials in fracture mechanics perspective Since then, a number of fracture models have been introduced and used to predict and investigate fracture behaviour of concrete-like materials In general, all the foregoing fracture mechanics theories require a pre-existing crack to analyse the failure of a structure or component This is not so with Hillerborg’s fictitious crack model It is a cohesive crack in the classical sense described above, but it is more than that because it includes crack initiation rules for any situation This means that it can be applied to initially uncracked concrete structures and describe all the fracture processes from no crack at all to complete structural breakage It provides a continuous link between the INTRODUCTION classical strength-based analysis of structures and the energy-based classical fracture mechanics: cohesive cracks start to open as dictated by a strength criterion that naturally and smoothly evolves towards an energetic criterion for large cracks Other nonlinear models such as the two-parameter model by Jenq and Shah (1985) and effective crack model by Nallathambi and Karihaloo (1986) have also been proposed All these models use simplifying assumptions to reduce the computational complexities inherent in fracture analysis The cohesive crack model defines a relationship between normal crack opening and normal cohesive stresses, and assumes that there are neither sliding displacements nor shear stresses along the process zone This assumption is only partially valid for concrete materials Based on experimental observations, it is indeed correct that a crack is usually initiated in pure mode I (i.e opening mode) in concrete, even for mixed mode loading (Saouma, 2000) However, during crack propagation, the crack may curve due to stress redistribution or non-proportional loading, and significant sliding displacements develop along the crack Therefore, it is desirable to incorporate these shear effects Interface elements were first proposed by Goodman et al (1968) to model nonlinear behavior of rock joints Since then, numerous interface constitutive models have been proposed for a wide range of applications such as rock mechanics (Goodman et al., 1968), masonry structures (Lotfi, 1992) and concrete fracture (Stankowski, 1990; Feenstra et al., 1991; Carol et al., 1992; ˇ Cervenka, 1994) These models are basically the extension of Hillerborg’s cohesive crack model for shear effects, and as such it can be also used to model interface cracks All fracture models are governed by a constitutive law The cohesive crack model, for instance, requires a tension-softening relation (softening law) to characterise the fracture behaviour of cementitious materials In the practical application of the cohesive crack model, the shape of the softening law is simplified and is assumed INTRODUCTION σ σy ft Crack opening displacement σy GF Real crack Process zone w wc Figure 1.1: Linear softening law and the cohesive crack model to be known a priori Among the simplest softening relationships developed is the linear softening law that was used by Hillerborg and co-workers (1976) to illustrate the applicability of their proposed fracture model As shown in Figure 1.1, only two parameters need to be specified to sufficiently characterise the model One can use any of the combinations of the tensile strength ft and fracture energy GF or tensile strength ft and the critical crack width wc Petersson (1981) proposed the two-branch law that is generally acknowledged to provide a better approximation of the fracture behaviour of concrete The two-branch law, in general, is fully characterised by specifying four parameters, except if the breakpoint is known Details and the application of these models will be discussed further in the next section τ ϕ c σ Figure 1.2: Mohr-Coulomb criterion and shear band predicted in principal stress space (De Borst, 1986) Up to now, the most practical failure models that incorporate shear have been INTRODUCTION the Mohr-Coulomb type models, which limit and control the shear stress at a plane as a function of the normal stress on that plane (Figure 1.2) Though they are formulated in principal stress space, they actually limit the shear stress on certain planes Figure 1.2 shows a shear band in a specimen loaded in compression as predicted by the use of a Mohr-Coulomb continuum model When a shear plane is known, it is possible to use a Mohr-Coulomb type of model for the description of interface behaviour Simple interface models of this type have been used by Roelfstra and Sadouki (1986); Roelfstra (1989); Lorig and Cundall (1989); Vonk (1992) In these models a tension cut-off criterion is added to the shear failure criterion A more complex model for the combination of tensile and shear loading including softening has been proposed by Stankowski (1990) The normality rule and/or the association of the flow laws with the yield function in classical plasticity refer to the following circumstance: in the space of the stress and strain components superposed, the plastic strain rate vector is normal to the activated yield surface at the stress point Nonassociated constitutive law refers to circumstances otherwise (Koiter, 1960; Maier, 1969) The safety and durability of concrete structures are significantly influenced by the fracture behaviour of the concrete There are many fracture formulations which assume concrete as a homogeneous material or as a two-phase material composed of aggregate particles dispersed in a cement paste matrix and provide reasonable simulations However, such models not include the effects of the transition zone between coarse aggregate and cement paste It is well known that this zone has a significant effect on the elastic properties, but little is known on how it affects the softening process Therefore, it is necessary to access the adequacy of fracture models considering the heterogeneous nature of concrete with three distinctive phases (Vonk, 1992; van Mier, 1997; Leite et al., 2004) INTRODUCTION 1.2 Aim and motivation of the research The broad aim of the project is to develop novel methods apt to simulate fracture behaviour and softening processes in plain and fibre-reinforced concrete as a quasibrittle material Specifically, this study deals with identifying different modes of failure, i.e tension, shear, and compression with several questions about the interaction between shear and tension A mathematical programming based discrete interface formulation is employed to achieve this goal Several benchmark problems are tackled including the compressive softening of a concrete cube and crack interaction in a beam 1.3 Objective scope of study A composite model is used to represent the heterogeneity of plain concrete consisting of coarse aggregates, mortar matrix and the mortar-aggregate interface The composite elements of plain concrete are modelled using triangular finite element units which have six interface nodes along their sides Fracture is captured through a constitutive single branch softening-fracture law at the interface nodes, which bounds the elastic domain inside each triangular unit The inelastic displacement at an interface node represent the crack opening and/or sliding displacement and is conjugate to the internodal forces The path-dependent softening behaviour is developed within a quasi-prescribed displacement control formulation The crack profile is restricted to the interface boundaries of the defined mesh No re-meshing is carried out Solutions to the rate formulation are obtained using a mathematical programming procedure in the form of a linear complementary problem Fibre particles are modelled by introducing additional linear elements interconnecting distant interface nodes in the matrix media after the generation of matrix-aggregate structure The allocation of fibres is associated with the mesh structure by choosing all possible combinations of distant nodes in the matrix which have a designated length INTRODUCTION range and not cross any present aggregate particles Limited experiments have been undertaken on plain and fiber-reinforced concrete specimens which are used to verify the analytical model developed 1.4 Organisation of the research This dissertation deals with the numerical simulation of fracture in plain concrete and fibre reinforced concrete and is organised into nine chapters and three appendices Each chapter starts with an introduction and ends with a summary The introduction provides an overview, and if necessary a brief review, of the topics contained therein The summary highlights the important points discussed in the chapter Moreover, it also provides a smooth transition to the next chapter The contents of each chapter are briefly described in the following The first chapter naturally constitutes the introduction to the thesis, aims, motivation of the research and objective scope of the work This chapter also contains several assumptions and common notations employed throughout the thesis Chapter comprises the literature survey of topics related to this work, i.e fracture mechanics in plain and fibre reinforced concrete and the cohesive crack model Topics directly related to this thesis requiring more detailed discussion or derivation are separately covered in the subsequent chapters The literature survey provides a brief historical overview of the early development of fracture mechanics and introduces the different fracture models developed over the years starting with linear elastic fracture mechanics (LEFM) The fundamental ideas underlying the concept of the cohesive crack model are explained and the simplifying assumptions adopted are discussed The tension-softening relationship required of the model is described and the fracture parameters characterising this softening behaviour, and their significance to fracture mechanics, are also discussed INTRODUCTION Chapter deals with the formulation of the state problem expressed as a linear complementarity problem (LCP) It covers the mathematical descriptions of basic equations for elastic-plastic relations in structural mechanics The concepts and formulation of a structure into a finite number of six-node interface triangular units each consisting of nine constant strain triangle are then presented The implementation of a piecewise linear inelastic failure surface and softening constitutive law is described The single branch softening laws in tension and shear are formulated in a complementarity format The structural relations are cast into a nonholonomic (irreversible) rate formulation Also introduced in this chapter is a review of the linear complementarity problem and its applications in engineering mechanics as well as some of the computational algorithms employed in the thesis, such as Lemke (Lemke, 1965) and the industry standard solver PATH (Dirkse and Ferris, 1995) Chapter discusses the methods and algorithms used in the automated mesh generation and the composite model to include the heterogeneous nature of concrete (modelling at meso-level) Concrete is modelled as a three-phase material with coarse aggregates, a mortar matrix and the mortar-aggregate interfaces Properties of each constituent available in the literature is likewise mentioned Chapter analyses several verification examples using actual experimental data One of them is the interacting crack problem The formulation developed in Chapter is employed to report the investigation of multiple interacting cracks in the four-point bending test of a simple plain concrete beam Material properties are assigned in a homogeneous manner The solution algorithm concentrates on the analysis of various fracture modes in a plain concrete beam under four point bending with several notches and examines the interacting crack itineraries by identifying the various equilibrium solutions available Next, two of the most cited problems in identifying parameters of cohesive crack model in concrete, i.e the Brazilian test INTRODUCTION and the three-point bending test, are numerically simulated using the same formulation in conjunction with the composite model prepared in Chapter The boundary condition and factors that affect the outcome of these tests are examined Chapter deals with an articulated particle/interface model of concrete and the introduction of a compression cap to the Mohr-Coulomb failure surface to further track compressive failure As an example, results on the fracture process in a cube of concrete under compression are studied All major factors that affect the softening behaviour in uniaxial compression - e.g the influence of size, the boundary condition, etc - are alike discussed Chapter presents experimental results on fracture in plain and fibre-reinforced concrete Material tests, shear tests and three-point bending tests are in turn presented Basically, all parameters in the particle/interface model are derived Different fibre dosage is used to verify how fibre content affects the fracture energy and critical crack opening displacement of shear and beam specimens Chapter is the further development of the presented model to include fibres Simulation of several tests in the literature are performed and compared with experimental results These consist of the three-point bending test and the push-off shear test Lastly, the experimental results obtained in this study are simulated using the proposed model Chapter concludes the thesis with key summaries and recommendations for future research INTRODUCTION 1.5 10 Assumptions and notations Where applicable, assumptions are stated immediately following the derivation and formulation of mathematical expressions used in the thesis The following are assumed throughout: The formulation is applied to quasibrittle materials Structural modes of failure are opening, shear and/or compression for concrete constituents; tension and/or pullout failure for steel fibres Linear softening laws are employed for all modes of fracture Displacements are assumed to be small The loading path is piecewise linearised (i.e., any given nonlinear load path is divided into a finite number of proportional loading stages) The following conventions are used for general description throughout the thesis while specific ones are indicated where appropriate Vectors and matrices are indicated by bold type symbols Column vectors are assumed throughout A scalar quantity is denoted in italics A real vector a of size n is indicated by a ∈ Rn and a real m × n matrix A by A ∈ Rm×n denotes a null vector Transpose of a vector or a matrix is indicated by the superscript T; the inverse of a matrix by the superscript -1 The complementarity relationship between two nonnegative vectors f and z is written as fT z = which implies the componentwise condition of fk zk = for all k Vector inequalities apply componentwise A Shear test results 298 30 Mix (HSC-F1) - Specimen 12 Initial confinement: 0.81 MPa Shear bond strength: 24.32 MPa Res bond strength: 10.80 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.25: Shear stress-displacement curves for Mix (HSC-F1) - Specimen 12 30 Mix (HSC-F1) - Specimen 14 Initial confinement: 0.80 MPa Shear bond strength: 24.30 MPa Res bond strength: 10.08 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.26: Shear stress-displacement curves for Mix (HSC-F1) - Specimen 14 A Shear test results 299 30 Mix (HSC-F1) - Specimen 15 Initial confinement: 0.64 MPa Shear bond strength: 23.51 MPa Res bond strength: 9.24 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.27: Shear stress-displacement curves for Mix (HSC-F1) - Specimen 15 30 Mix (HSC-F1) - Specimen 10 Initial confinement: 0.63 MPa Shear bond strength: 23.47 MPa Res bond strength: 9.21 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.28: Shear stress-displacement curves for Mix (HSC-F1) - Specimen 10 A Shear test results 300 30 Mix (HSC-F1) - Specimen 11 Initial confinement: 0.59 MPa Shear bond strength: 23.76 MPa Res bond strength: 9.13 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.29: Shear stress-displacement curves for Mix (HSC-F1) - Specimen 11 30 Mix (HSC-F1) - Specimen Initial confinement: 0.42 MPa Shear bond strength: 22.31 MPa Res bond strength: 7.77 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.30: Shear stress-displacement curves for Mix (HSC-F1) - Specimen A Shear test results 301 30 Mix (HSC-F1) - Specimen Initial confinement: 0.38 MPa Shear bond strength: 21.64 MPa Res bond strength: 7.10 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.31: Shear stress-displacement curves for Mix (HSC-F1) - Specimen 30 Mix (HSC-F1) - Specimen Initial confinement: 0.37 MPa Shear bond strength: 21.50 MPa Res bond strength: 6.98 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.32: Shear stress-displacement curves for Mix (HSC-F1) - Specimen A Shear test results 302 30 Mix (NSC-F2) - Specimen Initial confinement: 0.77 MPa Shear bond strength: 13.17 MPa Res bond strength: 9.15 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.33: Shear stress-displacement curves for Mix (NSC-F2) - Specimen 30 Mix (NSC-F2) - Specimen Initial confinement: 0.64 MPa Shear bond strength: 12.87 MPa Res bond strength: 8.42 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.34: Shear stress-displacement curves for Mix (NSC-F2) - Specimen A Shear test results 303 30 Mix (NSC-F2) - Specimen Initial confinement: 0.59 MPa Shear bond strength: 12.52 MPa Res bond strength: 8.32 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.35: Shear stress-displacement curves for Mix (NSC-F2) - Specimen 30 Mix (NSC-F2) - Specimen Initial confinement: 0.39 MPa Shear bond strength: 11.89 MPa Res bond strength: 7.99 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.36: Shear stress-displacement curves for Mix (NSC-F2) - Specimen A Shear test results 304 30 Mix (NSC-F2) - Specimen Initial confinement: 0.35 MPa Shear bond strength: 10.96 MPa Res bond strength: 6.81 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.37: Shear stress-displacement curves for Mix (NSC-F2) - Specimen A Shear test results 305 30 Mix (HSC-F2) - Specimen Initial confinement: 0.80 MPa Shear bond strength: 26.14 MPa Res bond strength: 8.77 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.38: Shear stress-displacement curves for Mix (HSC-F2) - Specimen 30 Mix (HSC-F2) - Specimen Initial confinement: 0.60 MPa Shear bond strength: 25.33 MPa Res bond strength: 8.62 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.39: Shear stress-displacement curves for Mix (HSC-F2) - Specimen A Shear test results 306 30 Mix (HSC-F2) - Specimen Initial confinement: 0.55 MPa Shear bond strength: 24.90 MPa Res bond strength: 8.26 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.40: Shear stress-displacement curves for Mix (HSC-F2) - Specimen 30 Mix (HSC-F2) - Specimen Initial confinement: 0.41 MPa Shear bond strength: 24.36 MPa Res bond strength: 7.87 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.41: Shear stress-displacement curves for Mix (HSC-F2) - Specimen A Shear test results 307 30 Mix (HSC-F2) - Specimen Initial confinement: 0.39 MPa Shear bond strength: 23.96 MPa Res bond strength: 7.72 MPa Shear stress (MPa) 25 20 15 10 0 Shear displacement (mm) 10 Figure A.42: Shear stress-displacement curves for Mix (HSC-F2) - Specimen Appendix B THREE-POINT BENDING TEST RESULTS 308 B Three-point bending test results 309 12000 NSC-F1-3 NSC-F1-4 HSC-F1-2 HSC-F1-3 10000 Load (N) 8000 6000 4000 2000 0 10 15 20 25 Displacement (mm) 30 35 40 Figure B.1: Load vs displacement curve for Mix (NSC-F1) and Mix (HSC-F1) 16000 NSC-F2 14000 HSC-F2 Load (N) 12000 10000 8000 6000 4000 2000 0 10 15 20 25 Displacement (mm) 30 35 40 Figure B.2: Load vs displacement curve for Mix (NSC-F2) and Mix (HSC-F2) Appendix C BASIC MATERIAL TESTS 310 C Basic material tests 12 12 Axial Axial Circumferential NSC-F0-2 Stress (MPa) Stress (MPa) 311 Circumferential NSC-F0-4 Strain Strain 0 0.002 0.004 0.006 0.008 0.01 0.002 0.004 0.006 0.008 0.01 Figure C.1: Compressive cylinder test: stress-strain curves for Mix (NSC-F0) 30 Axial Stress (MPa) Circumferential 20 Axial Circumferential Stress (MPa) 30 20 HSC-F0-6 10 HSC-F0-3 10 Strain Strain 0 0.002 0.004 0.006 0.008 0.002 0.004 0.006 0.008 Figure C.2: Compressive cylinder test: stress-strain curves for Mix (HSC-F0) 12 Axial Circumferential NSC-F1-1 Stress (MPa) Stress (MPa) 12 Axial Circumferential NSC-F1-3 Strain Strain 0 0.002 0.004 0.006 0.008 0.01 0.002 0.004 0.006 0.008 0.01 Figure C.3: Compressive cylinder test: stress-strain curves for Mix (NSC-F1) C Basic material tests 30 Stress (MPa) Axial Circumferential 20 Axial Circumferential Stress (MPa) 30 312 20 HSC-F1-3 10 HSC-F1-6 10 Strain Strain 0 0.002 0.004 0.006 0.008 0.01 0.002 0.004 0.006 0.008 0.01 Figure C.4: Compressive cylinder test: stress-strain curves for Mix (HSC-F1) 12 Axial Circumferential NSC-F2-1 Stress (MPa) Stress (MPa) 12 Axial Circumferential NSC-F2-2 Strain Strain 0 0.002 0.004 0.006 0.008 0.01 0.002 0.004 0.006 0.008 0.01 Figure C.5: Compressive cylinder test: stress-strain curves for Mix (NSC-F2) 30 Stress (MPa) Axial Circumferential 20 Axial Circumferential Stress (MPa) 30 20 HSC-F2-1 10 HSC-F2-2 10 Strain Strain 0 0.002 0.004 0.006 0.008 0.01 0.002 0.004 0.006 0.008 0.01 Figure C.6: Compressive cylinder test: stress-strain curves for Mix (HSC-F2) ... simulation of fracture in plain concrete and fibre reinforced concrete and is organised into nine chapters and three appendices Each chapter starts with an introduction and ends with a summary The introduction... the history of fiber reinforced concrete (FRC) and the modelling of its behaviour in light of fracture mechanics 2.2 A review of fracture mechanics and quasibrittle models The advent of fracture. .. interest of studying quasibrittle materials in fracture mechanics perspective Since then, a number of fracture models have been introduced and used to predict and investigate fracture behaviour of concrete- like