COMPUTATIONAL MODELING OF FIBER REINFORCED CONCRETE WITH APPLICATION TO PROJECTILE PENETRATION

Transcription

1 ALMA MATER STUDIORUM UNIVERSITA DEGLI STUDI DI BOLOGNA FACOLTA DI INGEGNERIA Corso di laurea in Ingegneria civile D.I.S.T.A.R.T. Dipartimento di Ingegneria delle Strutture, dei Trasporti, delle Acque,del Rilevamento e del Territorio Tesi di laurea in Progetto di strutture COMPUTATIONAL MODELING OF FIBER REINFORCED CONCRETE WITH APPLICATION TO PROJECTILE PENETRATION Candidato: Pietro Marangi Relatore: Pro. Ing. Marco Savoia Correlatore: Pro. Ing. Gianluca Cusatis Anno accademico Sessione III

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3 Ritorna con il tuo scudo o sopra di esso - Regina Gorgo -

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5 Abstract Concrete is a material widely used in Civil Engineering due to its good resistance capacity and durability. Even thought engineers have designed the reinorcement o concrete structures or centuries, concrete material is still not well understood. This statement is particularly true outside the elastic range when racture and other inelastic phenomena occur. The main reason or this hard behavior is due to the extreme complexity o concrete internal structure that is highly heterogeneous. To overcome this problem, in the past, several model have been ormulated using dierent approaches. In this work, the goal is to study at irst the Lattice Discrete Particle Model (LDPM) and then to develop one method that allows to study the FRC. The LDPM will be extended to include the eect o dispersed ibers with the objective o simulating the behavior o iber reinorced concrete or armoring system applications. The developed model, named LDPM-F, is validated by carrying out numerical simulations o three-point bending tests on iber reinorced concrete mixed characterized by various iber volume ractions. Finally, LDPM-F is applied to the analysis o the penetration resistance o iber reinorced slabs. In Chapter 1 the ormulation o the LDPM is presented and explained, showing the geometry o the model. In Chapter 2 the constitutive law or interaction o iber-concrete is described and explained. In Chapter 3 an experimental campaign o uniaxial compression tests and three point bending tests on plain concrete specimens are simulated. In Chapter 4 an experimental campaign o three point bending tests on FRC specimens is simulated and the calibration and validation phases are described, in order to clariy the LDPM-F. In Chapter 5 the results ound in the previous chapters will be used or armoring system applications, in order to predict the FRC response.

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7 Contents The lattice Discrete Particle Model (LDPM)...1 Introduction...1 Features o the model...2 Geometrical Characterization o Concrete Mesostructure...4 First step: Number and size o coarse aggregate pieces...4 Second step: Particle position...7 Third step: Inter-particle connection...7 Fourth step: acets generation...8 Discrete Compatibility and Equilibrium Equations...12 LDPM Constitutive Law...17 Elastic behavior...17 Inelastic behavior...18 Fracturing and cohesive behavior...18 Poor collapse and Material Compaction...21 Frictional behavior...23 Numerical Implementation and Stability Analysis...25 Constitutive law or the concrete-iber interaction...28 Introduction...28 Modeling o the single iber behavior...3 Matrix micro-spalling...37 Cook-Gordon eect...38 Two way pullout...4 Analysis stages...4 Stage 1: neither embedment ends completely debonded...4 Stage 2:short embedment end complete debonded...41 Stage 3: both embedment ends completely debonded...42 Parameters analysis o the model...44 Presentation o iber parameters...44 Pullout hardening behavior...45 Two way pullout...46 Debonding racture energy...47 Bond strength...48 Snubbing eect...49 Spalling eect...5 Numerical analysis plain concrete subject to tension and compression...56 Introduction...56 Presentation o the parameters...57 Model sensibility...57

8 Phases o the calibration...58 Specimens geometry...59 Three point bending test (3PBT)...59 Uniaxial unconined compression test (UC)...6 Aggregate generation...61 Parametric analysis...63 Three point bending test (3PBT)...63 Uniaxial unconined compression test (UC)...67 Calibration stage...71 Fracture energy control...75 Numerical analysis iber reinorced concrete (FRC) subject to tension...78 Introduction...78 Phases o pre-calibration...79 Fiber generation...79 Presentation o parameters...81 Model sensibility...81 Calibration stage...84 Validation stage...87 Numerical analysis o projectile penetration or FRC slabs...92 Introduction...92 Geometry o the test...93 Penetration test o concrete slab...94 Penetration test o FRC slab...97 Design o Armoring System...1 Conclusions...14

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10 Chapter 1: The Lattice Discrete Particle Model (LDPM) Chapter 1 The lattice Discrete Particle Model (LDPM) Introduction Concrete is an heterogeneous material, constituted by three phases, with dierent property: aggregate, matrix and interacial boundary. In order to investigate these aspects, a lot o models have been proposed in recent years, but although these allowed us to achieve good results, the concrete eature is not still completely understood. The principal reason o this, is the complexity o the internal structure, that is linked with the length scales o observation used or the model. By this point o view, changing the observation level, is possible to show, dierent aspects that are directly matched with the heterogeneity o the material. The scale length range rom the atomistic scale (1-15 m), characterized by the behavior o crystalline particles o hydrated Portland, to the macroscopic scale (1 1 m), at which concrete has been traditionally considered homogeneous. In the last twenty years, several authors, have done materials models that use miniscale or mesoscale, to study this kind o material, in special way or the geological problems. Miniscale models in which concrete is treated as three-phase composites have been proposed by Wittmann (1988), and Carol (21). They used inite element technique to model with dierent constitutive laws coarse aggregate pieces, mortar matrix, and inclusion-matrix interace. Another remarkable study is done by Van Mier and coworkers (1992), they proposed a model realized with inite elements but without continuum hypothesis. Concrete was modeled through a discrete system o beams. Important is also the experience by Bolander (1999), which realized a discrete miniscale model based on the interaction between rigid particles obtained though the Voronoi tessellation o the domain. The main advantage o miniscale models is they are able to reproduce realistic simulation o cracking, coalescence o multiple distribution cracks into localized cracks, and racture propagation. The only problem that aect these kind o model is that they tend to be computationally intensive especially or 3D modeling. Mesoscale models appear because they are computationally less demanding than the miniscale models. For mesoscale model, concrete is modeled by the whole - 1 -

11 Chapter 1: The Lattice Discrete Particle Model (LDPM) aggregate pieces and the layer o mortar matrix between them. The technique used is inite element. Some examples are Cusatis et al (23a,b), Cusatis et al (26), Cusatis & Cedolin (26). Summarizing every thing, the principal dierence among these two approaches is that, miniscale model describe the concrete like three-phase material: cement paste, aggregate and interacial transitional zone, and the length scale is o 1-4 m or less. The mesoscale, is undamentally dierent, because this show only the mortar and the coarse aggregate, and the scale order is 1-3 m. Both approaches have the problem, which are very numerically expensive, but or the concrete has been seen that using the mesoscale model, that is relative expensive, and is also possible to achieve airly good results like the miniscale. The old models, implemented or macroscale, allowed to obtain good result, but they are not able to simulate material heterogeneity and its eect on damage evolution and racture. For solving these problems now, one possible manner is to use models, that are able to simulate the concrete at the level o the mesostructure, with discrete approach. In this way, it is possible to replicate damage and racture, and to capture the phenomena related to the randomness o the material. Features o the model The model present in this work, use an analysis at the mesoscale level and is called Lattice Discrete Particle Model (LDPM). It is the result o the union by two dierent models: Discrete Particle Model (DPM) and Shear Lattice Model (CSL). The principals eatures o the model are: a) Concrete mesostructure is simulated by a lattice that match a system o particles that are in interaction into their through triangular acets. This lattice is obtained by a Delaunay triangulation o the aggregate centers. b) The specimen is created by a randomly distribution o particles, that is computed taking in account the basic properties and the granulometric o the aggregate. c) The geometrical interaction between the particles is obtained by threedimensional domain tessellation deining a set o polyhedral cells each including one piece o the aggregate. d) Two adjacent pieces o aggregates are connected by the generic lattice element, that transmits shear and normal stresses. These are assumed unctions o normal and shear strains. Allotting the displacements the model works, and permit to ind the stress ield

12 Chapter 1: The Lattice Discrete Particle Model (LDPM) e) The stresses act on contact areas which are deined by constructing a barycentric dual complex o the Delaunay triangulation. ) Starting rom the strains is possible, using the constitutive law to deine the stresses, taking into account the stress-strain boundaries. g) The constitutive law give the ollowing behavior: sotening or pure tension, shear-tension and shear with low compression and it is hardening or pure compression and shear with high compression. h) Friction and cohesion are shown into the shear response. LDPM has inherited rom DPM the MARS computational environment that includes long range contact capabilities typical o the classical ormulation o Discrete Element Methods (DEM). This eature is particularly important to simulate pervasive ailure and ragmentation. LDPM allows to capture a number o new eatures that greatly improve its modeling and predictive capabilities. These new aspects can be explained as ollows: 1) The particles interaction is ormulated by the assemblage o our aggregate pieces whose center are the vertex o the Delaunay tetrahedralization. This model geometry allows to include into the constitutive law the volumetric eects, that the other models are not able to capture. 2) Each single aggregate is contained into one polyhedral cells that is made by dierent triangle acet. Stresses and strains are deined at each single acet. This coniguration allows a better stress resolution in the mesostructure, which, in turn, lead to a better representation o mesoscale racture and damage. 3) The constitutive law simulates the most relevant physical phenomena governing concrete damage and ailure under tension as well as compression. This law compared with the previously existed provides better modeling and predictive capabilities especially or the macroscopic behavior in compression with coninement eects. The LDPM is able to simulate all aspect o concrete response under quasistatic loading, including tensile racturing, cohesive racture and also size eect, compression-shear behavior with sotening zero or mild coninement, and high conined compression, and strength increase under biaxial loading. The ollowing sections will explain beore the geometrical characterization o the model and then equilibrium equation with constitutive law

13 Chapter 1: The Lattice Discrete Particle Model (LDPM) Geometrical Characterization o Concrete Mesostructure The geometrical structure o the concrete mesostructure is obtained with our divers steps. In these phases the objectives is deine: 1) Number and size o coarse aggregate pieces; 2) Particle position; 3) Inter-particle connections; 4) The creation o a surace, into the two adjacent particles that permit to exchange the orces. First step: Number and size o coarse aggregate pieces The irst step is to deine the particles diameters and the respective number, that will be used or reill the specimen volume, and generate the its suraces. In this sense, there are dierent practice to make this in the LDPM, irst o all it will compute the particles and then the zero node point, that also will be explain in the ollowing section. However, the irst hypothesis is to consider the particles with sphere shape, under this assumption, the concrete granulometric distribution can be represented by particle size distribution unction (Psd), proposed by Stroeven: ( d ) = q q+ 1 [ 1 ( d d ) ] d where d a is the maximum aggregate size and d represent the minimum particle size. The precedent Psd, can be interpreted as probability density unction (Pd), this will allow to ind the percentage associated with a certain diameter, using the cumulative distribution unction (Cd), that is expressed as: P qd q d q 1 ( ) ( ) ( d d ) d = d δ d = 1 ( d d ) d It may be shown that the Psd is associated with a sieve curve in this way: a a q - 4 -

14 Chapter 1: The Lattice Discrete Particle Model (LDPM) F ( d ) = where n = 3 q. For q = 2. 5, the relation represent the classical Fuller curve extensively used or the concrete. With this curve is possible to check the concrete granulometric, analyzing the percentage o passing in unction o the diameter, in this way the concrete quality is guaranteed i this curve is contained into the Fuller range. This technique allows to have the best specimen reill and to avoid the empty space, as showed in the Cusatis (21). The specimens used in the simulations have particles created with the 3% o the total curve granulometric (Fuller curve), starting rom the coarse aggregate; this allows us to ind good results, because the computational cost is lower and at the same time there are not big dierences between results ound using another percentage o generation. For simulate the specimen in terms o number and diameters o particles, it is important to know: c = cement content, w/c = water to cement ratio, V = specimen volume, d a = maximum aggregate size, d = minimum particle size. The ollowing procedure is used to ind and then place the particles inside the specimen volume: a) Calculate the volume aggregate raction as va = 1 c ρc w ρw vair, where w = ( w c)c is the water mass content per unit volume o concrete, 3 ρ c = 315Kg m is the mass density o cement, d d a n 3 ρ w = 1 Kg m is the mass density o the water, and v air is the volume raction o entrapped or entrained air, ( usually 3-4%); b) Compute the volume raction o simulated aggregate using the ollowing n v 1 F d v = 1 d d v ; [ a ] a relation: [ ( )] ( ) ao = o a c) Compute the total volume o simulated aggregate as Vao = vaov ; d) Calculate the particle diameter by sampling the Cd by a random numeric q q generator: d d [ 1 P ( 1 d d )] i = i / a, where P i is a sequence o random numbers between and 1. e) The total number o the particles is obtained by checking, or each new generated diameter in the sequence, that the total volume o the generated ~ 3 particles V = Σ( πd 6) does not exceed V a.when or the irst time, a i 1 q - 5 -

15 Chapter 1: The Lattice Discrete Particle Model (LDPM) ~ V a > V a occurs, the current generated particle is discarded and the particle generation is arrested, Cusatis and coworkers (29). The graphical representation o the procedure explained beore is shown in the Fig. 1.1, whereas in the Fig. 1.2 is represent the computational sieve curve obtained during the generation o a cube specimen characterized by c = 3 kg/m 3, w/c =.5, d = 4 mm and d a = 12 mm. Figure 1.1: Cumulative distribution curve The diameters and number particles, ound in this way, will be used to ill the specimen, now the procedure will be completed creating the zero-diameters particles (nodes), that will be placed over the external suraces. That practice, is important, or the third step, when the tetrahedral will be generated. Figure 1.2: Computational sieve curve - 6 -

16 Chapter 1: The Lattice Discrete Particle Model (LDPM) The method used to search the number o surace nodes is the ollowing, assuming that the external surace o the specimen volume can be described with a set o polyhedral ace, and that is L e is the generic length o the edge specimen, and A p the generic surace area, the number o the nodes will be given by these 2 ratio Le hs and A p h s, where h s is the average surace mesh size, chosen such that the discretization resolution on the resolution on the surace is comparable to the one inside the specimen. This is done using this relation hs = ξsd, by numerical experiment which showed that putting ξ s = 1. 5 leads to obtain good results. Clearly or each vertex on the external surace will be placed one node. Second step: Particle position The particle computed beore, will be now placed in the volume specimen, using a random distribution, on vertexes, edges, suraces, aces and interior volume. The ollowing idea or the ranked, is irst o all, set the vertex nodes, and ater in the edges and surace created among these vertex nodes, place other particles, by allowing a minimum distance o δ sd to minimize the geometrical error o the discretization. The typical value used or δ s = 1. 1, permit to achieve good mesh. Using a procedure introduced by Bažant, it is possible to generate a statistically isotropic random mesostructure, where the center o particles are placed in the volume o the specimen one by one, rom the largest to the smallest. When the procedure, or generate the particles position, is inished, one control is done, or avoid possible overlaps o this particle with the previously placed particles and with surace nodes. In this phase, at the surace nodes are assigned a ictitious diameter o δ sd and a minimum distance o di 2 + d j 2 + ξd among the center o the particles with diameters d i and d j is enorced. The value o ζ, is very important, because it determines the concrete behavior, or ζ= or very small values, the particles distribution is not statically isotropic, and present areas where there are low particle density and others zones with high particle density. In the same manner using very large values o ζ, the specimen volume is saturated quickly and not all particles can be positioned. Ater several numerical experiments, the best value ound is ζ =.2, using this, it is possible to avoid the volume saturation while leading uniorm particle distribution. Third step: Inter-particle connection The third step lets to deine the topology o the interaction among the particles. This is perormed using a Delaunay tetrahedralization, that utilizing the nodal coordinates o the particles center allows to build three-dimensional tetrahedral, which does not overlap, ill all the volume o the specimen. In each vertex o the - 7 -

17 Chapter 1: The Lattice Discrete Particle Model (LDPM) tetrahedrals is placed a particle, generated in the irst step, and permit to ill the total volume specimen. In this study, the Delaunay tetrahedralization is carried out by using the TenGen code. Fourth step: acets generation The last step, the orth has the aim to deine the surace through which the orced are exchanged rom one particle to another. There are dierent ideas proposed or describing this aspect o the problem, in previous works, the interaction between the particles were realized by the edge that was created as a strut. It connected the two adjacent particles, and the eective area o this strut was deined by perorming a tessellation o the domain anchored to Delaunay tetrahedralization. This approach gives a good result in terms o racture behavior and also or the concrete ailure under unconined compression. These methods have the lacuna that does not say something about the volumetric eects that are essential or description o the concrete behavior under high conining pressure. In the LDPM model, the principal thing is the elementary cell, that is the tetrahedral. There be possible, to see this like the union o our sub-domains, where each sub-domain contains one particle, see Fig The single acet o these sub-domains will be the surace, where will be applied the relation to allow the orce exchange. The last phase will be realized or this local geometry or every elementary cell and or each particles. Starting rom the tetrahedral, the tessellation will be construction o the ollowing procedure: a) One point on each edge (edge-point) is deined at midway o the counterpart o the edge not belonging to the associated particles ( point E ij in the Fig. 1.3 ); b) One point on each triangular ace (ace point) o the tetrahedron is deined as ollows. First points located on the straight lines connecting each ace node to the edge point located on the edge opposite to the node under consideration are considered. Similarly to the edge points, these points are located at midway o the line counterpart not belonging to the associated particles. See point F * kl in Fig Later, the acet point is selected as the centroid o the F * kl point. See point F l, in Fig

18 Chapter 1: The Lattice Discrete Particle Model (LDPM) c d j h ij P j E ij P i h ij d i d Figure 1.3: Construction or edge-point k h ij k h ij P j P k k E ij * F lk E ij P i Figure 1.4: Construction or ace-point - 9 -

19 Chapter 1: The Lattice Discrete Particle Model (LDPM) c) One point in the interior o the tetrahedron ( tet-point) is deined by the centroid ( T l in Fig. 1.5 ) o the points identiied, similarly to what is done or the ace points in item 2, on the straight lines connecting each node o the tetrahedron with the ace points on the ace opposite to the node under consideration ( points T l * in Fig. 1.5 ). Again, these point are located at midway o the line counterpart not belonging to the associated particles. e P l P k * F l * T l l h ijk l h ijk P j F l P i Figure 1.5: Construction or Tet-point d) Finally, the tessellation o the tetrahedron is obtained by the set o triangles by connecting the tet-point with the one o the edge-points and one o the ace-points. Each tetrahedron result tessellated with twelve triangular ace ( Fig. 1.6). This is only one type o tessellation possible, basically dierent procedure could be realized but numerical experiments show that this is the best or minimize the intersection between the tessellating surace and the particles. Isolating one particle and its acets, a polyhedral cell is obtained. This is the sub domain that has irregular and also random shape, this characteristic is an important property that ensures a realistic representation o the kinematics o the concrete mesostructure, and it especially allows avoiding the excessive rotation

20 Chapter 1: The Lattice Discrete Particle Model (LDPM) P l E jl E kl T F i P k F j E il F k P j E ik F l E ij P i Figure 1.6: Tessellation o the tetrahedron. Figure 1.7: Particle sub-domain

21 Chapter 1: The Lattice Discrete Particle Model (LDPM) Discrete Compatibility and Equilibrium Equations The tetrahedron ormed by the basic our-particles, as showed in the Fig.1.7, is the primary element used to derive the governing equations o the model. Every particle, which orms this tetrahedron, is included in a sub-domains V i ( i =1,.,4), and all together create the original element. Each sub-domain, has a portion o the three tetrahedron edges attached to the node and also, six triangular tessellation acet joint to the three edges, Fig One o the most important things is to describe, the displacement ield in every domain, this is done by rigid body kinematics. a C k θ i3 V i P i u i3 θ i1 u i1 u i2 θ i2 x 3 x 2 x 1 Figure 1.8: Sub-domain geometry. In order to know the movement o the mid-point, situated over every tetrahedron edge, is important know the displacement and also the coordinates o the center T or each particles. Describing x = [ x1; x2; x3] Vi, the vector that contains the node coordinates, one can write the ollowing expression: u ( x) = ui + θi ( x xi ) = Ai ( x) Qi

22 Chapter 1: The Lattice Discrete Particle Model (LDPM) where: A i ( x) = x x 3i 2 x x 3 2i x x 3 1i x 3i x 1 x 2i x 1 x x 2 1i In this equation, the vector x i contain the position coordinate o node i, the T T T Q = u θ which is realized with the displacement ield is described by [ ] T i i i T T translation u = [ u u ], and the rotation θ = [ θ θ ] i u1 i 2 i 3 i i 1 i 2 i θ3 i, that are the degrees o reedom o node i. The displacement jump at the centroid C, o each ace, is deined as: [ uck ] = ucj uci where i and j, are the nodes adjacent to acet k, and: u u Cj Ci = = u u + ( xck ) ( x ) Ck represent the displacement ield at the acet centroid C k or see Fig x Ck Vi and xck V j, 1 In order to ind the acet strain vector, deine as l [ u ] are divided by l e, where: e Ck, the displacement jump le = x j x i = 1 T [( x x ) ( x x )] 2 j i j i that represents the length o the edge e. To show the real behavior not symmetrical tension compression o concrete, and be able to ormulate an 1 appropriate constitutive law, the strain vector le [ u Ck ] is decomposed in normal and shear components, (Fig. 1.1 ). This is done taking in the count the projection o the tessellation acets, into the planes that are orthogonal to the edge. The projected acets, are used or the deinition o LDPM strain components to avoid non-symmetric behavior under pure shear. It is possible to understand this idea, seeing the Fig. 1.1, where the relative nodal displacement orthogonal to an edge, produce only shear on the projected ace at the place o original acet where there also are tension or compression

23 Chapter 1: The Lattice Discrete Particle Model (LDPM) Proceeding in this way, the strain components are deined as: ε Nk = n T k l u e Ck = B jk N Q j B ik N Q i ε Lk = l T k u l e Ck = B jk L Q j B ik L Q i ε Mk = m T k l u e Ck = B jk M Q j B ik M Q i n = x x l, m k and l k are two direction that are mutually and where k ( j i ) e pk T orthogonal in the plane o the projected acets, and BN ( 1 le ) nk Ap ( xck ) pk T pk T B = ( 1 ) l A ( ) and B = ( 1 ) m A ( ), p=i,j. L le k p xck M le k p xck =, b P j n n k x N x L P i x M Figure 1.9: Projection o the acets, into planes orthogonal to the edge. The equations shown here are the discrete compatibility equations o the LDPM ormulation

24 Chapter 1: The Lattice Discrete Particle Model (LDPM) The normal and shear stresses or each acet is calculate by mesoscale constitutive law, usually is written σ k = F( ε k, ξk ) where σ k, ε k and ξ k are vectors collecting acet stresses, strain and internal variable, respectively. c Tension P j P j n k Shear n Pure Shear P i P i Figure 1.1: Tension and shear components. Employing the Principle o Virtual Work (PVW), the governing equations can be completed, with the imposing o equilibrium. Using the PVW, in the aspect o virtual displacement, in this case the internal work match with a generic acet can be expressed like: k e k T k k e k ( σ δε + σ δε σ δε ) δ W = l A σ δε = l A + Nk where A k is the area o the projected acet. Substituting the previously equations into the virtual work relation is obtained: Nk Lk Lk Mk Mk where: F T ik F T jk = l = l e δ W = F δq + F e A A k k T ik i T jk δq ik ik ik k ( σ Nk BN + σ Mk BM + σ Lk BL ) kj kj kj ( σ B + σ B + σ B ) Nk N Mk M j Lk L

25 Chapter 1: The Lattice Discrete Particle Model (LDPM) This relation express the nodal orce at the nodes i and j, associated with the acet k. Then putting all together the contributions o the several acets and comparing the total internal work with the total external work one can obtain the discrete equilibrium equations o the LDPM ormulation. It is also possible, to prove that the equilibrium equations obtained through the PVW corresponds exactly to the translational and rotational equilibrium o each LDPM cell

26 Chapter 1: The Lattice Discrete Particle Model (LDPM) LDPM Constitutive Law The constitutive law, used or LDPM, allows us to put in match the strain vector with the stress vector at the acet level. For realizing the study over concrete behavior, it is better to divide it into: elastic and inelastic behaviors. The stress domain view two dierent regions that are the or tensile and compressive behavior, Fig This partition is very important because in the tensile ield shear and compression are directly matched, but this relation changes completely, in the compression stage, where only the shear in connected with the compression. Figure 1.11: Total LDPM domain. Elastic behavior In the elastic behavior, the normal and shear stresses are assumed proportional to the corresponding strains: σ N = ENε N σ M = ETε M σ L = ETε L where E N = E, E T = αe, E = eective normal modulus, and α = shear normal coupling parameter. These parameters now presented, are the mesostructure parameters, and are assumed to be material properties. This can be demonstrated, looking at the experimental text result obtained in elastic regime

27 Chapter 1: The Lattice Discrete Particle Model (LDPM) As mentioned in the introduction, the concrete behavior changes with the observation scale. At the macroscopic level it is consider statistically homogeneous and isotropic material, and thereore the concrete elastic behavior is modeled in the literature by the classic theory o elasticity. The irst objective now, is to ind one relationship among mesoscale LDPM parameters (α and E ), and the macroscopic Young s modulus ( E ), and Poisson ratio. This may be obtained by considering the ew case in which an ininite number o acets surrounds one aggregate piece. In the LDPM ormulation, there is the kinematically constrained like the microplane model, that are without deviatoric/volumetric split o the normal strain component. Under these guesses one can write: and: E α = E E = 1 2ν 4 + α o E o 1 4ν 1 α α = ν = 1+ ν 4 + α These relations can be obtained by the kinematically constrained homogenization o a random assemblage o the rigid spherical particles o various sizes interacting through elastic contacts. The eqs. can be used or estimate the LDPM elastic parameters in correct way, starting rom macroscopic experimental measuring o Young s modulus and Poisson s ratio. Inelastic behavior In this section the aim is to deine the ormulation o the non linear and inelastic part o the constitutive law, that is characterized by three dierent physical mechanisms o mesoscale behavior: a) Fracturing and cohesive behavior. b) Pore collapse and material compaction under high compressive stress. c) Frictional behavior. Fracturing and cohesive behavior The racturing behavior is present or tensile normal strain when it is N ε >. As has just done in other papers publishing, is better to do, one kind o ormulation

28 Chapter 1: The Lattice Discrete Particle Model (LDPM) that put in relationship the eective strain and eective stress, that are expressed by the ollowing relations. This is convenient or ormulate the racture and damage evolutions. The principal relations are: ε = ε + α ( ) ( σ M + σ ) ε + ε σ = σ + 2 L N M L N α 2 The relation used as eective strain, is similar to the strain measured used in the interace element model o Camacho and Ortiz (1996), and supply a total measure o the material straining. The normal and shear strain can be calculated rom eective and shear strain using the ollowing relation, in a way similar to simple damage models: ε N σ N = σ ε αε M σ M = σ ε αε L σ L = σ ε The eective stress σ, is incrementally elastic, & σ = E & ε and have to respect the inequality σ σ bt ( ε, ω), where the E is the eective strain that is one o LDPM parameters. In according with the Cusatis (28) the strain dependent boundary σ bt ( ε, ω), may be written as: σ bt exp ( ε, ω) = σ ( ω) H ( ω) ε max σ ε ( ω) ( ω) In which the brackets are used in Macaulay sense: x = max{ x,}. These relations, are used or describe the elastic-sotening concrete domain when it is over tension stress, looking the relation is possible understand how when the ε >, the strength drop or the sotening eect, see Fig max ε Now every behavior is summarized into the total domain o LDPM model, that is characterized by ω, the internal variable. This is written as ollowing: tan ε N ω = = αε T σ N α σ T

29 Chapter 1: The Lattice Discrete Particle Model (LDPM) that is characterized by the ratio between normal strain ε N and the total shear 2 2 strain ε T, obtained in this way ε T = ε M + ε L, or equivalently using the ratio into 2 2 the normal stress σ and the total stress, so deined asσ = σ + σ. N T M L Figure 1.12: Tensile behavior, changing the internal parameter ω The inelastic behavior σ bt is commanded with the exponentially relation, that also represent the boundary in the total domain, as a unction o the maximum eective strain, which is a history dependent variable deined as: where: ε = ε N + αε max 2,max 2 T,max ε N εt,max,max ( t) = max[ ε N ( τ )] τ < t () t = max[ ε ( τ )] are the value o maximum normal and total shear strain, that are present during the loading history. It is worth noting that in absence o unloading ε max ε. The point over the boundary in the total domain, are decrypted by the unction σ ω that is the strength limit or the eective stress and is deined as: ( ) σ ( ω) sin = σ t τ < t 2 2 ( ω) + sin ( ω) + 4α cos ( ω) 2 2α cos ( ω) / rst T / r st - 2 -

30 Chapter 1: The Lattice Discrete Particle Model (LDPM) where: rs = σ σ represent the ratio among the shear strength σ s, and the tensile s t strength σ t. In the total domain, or stress space σ N σ T, the equation previously present describe a parabola with its axis coincident with the σ N axis. The transition between the two ields (elastic and inelastic domain) occurs, when the maximum eective strain achieves its elastic limit ( ω) σ ( ω) E ε =, in this circumstances, the boundary σ bt start to decay in exponential way. The speed o this decaying is governed by the post peck slope (sotening modulus), that is considered to be a power unction o the internal variable ω: H ( ω) = H t 2ω π Using this expression it is possible to have a smooth transition, rom the sotening behavior subject to pure tensile stress ( ω = π 2, H ( ω) = Ht ), to perectly plastic eature under pure shear ( ω =, H ( ω) = ). For avoiding problems with the dissipation energy during the mesoscale damage localization, this expression is used or the sotening modulus in pure tension: Ht = 2E /( lcr / l 1), and where 2 G t represent the mesoscale racture energy, lcr = 2E G t / σ t, and l is the length o the tetrahedron edge associated with the current ace. Poor collapse and Material Compaction Another typically inelastic behavior is present when the concrete is under high compressive hydrostatic deormations, in this set is possible to see one strainhardening plasticity. This plastic ield really can be divide into two dierent phases, the irst is connected with the pores collapse under load and a second phase, when the pores are closed, and this give to the concrete structure one major density. Computing these two eect, in terms o stress-strain response, is showed or the irst problem, one sudden decrease o the stiness, that is in the second stage regained with a rehardening behavior. Experiments show that ater the densiication phases both the tangent plastic stiness and the unloading elastic stiness, can be even higher than the initial elastic stiness, in the rehardening phase. However, in the case o present o a signiicant deviatoric deormations, the rehardening phase result limited or sometimes also negligible, this really produce a horizontal plateau in the measured stress versus strain curve, Fig The LDPM constitutive law, simulates this eature using a strain-dependent normal boundary σ bc( ε D, εv ), that is assumed unction o volumetric strain ε V and the deviatoric strain ε D. These kind o deormation are computed respectively ε = V V V, where V is the current volume and at the level o tetrahedron as V ( ) nt

31 Chapter 1: The Lattice Discrete Particle Model (LDPM) V is the initial volume o the basic element. For each tetrahedron there are twelve acets, or these is guessed that its are subjected at the same volumetric strain. Although, there is one think that change or every acets, this is the deviatoric strain characterized by several value. It computes as ε D = ε N εv, where ε N is the normal strain. The deinition o the volumetric and deviatoric strains are equivalent to the same quantities deined at each microplane in the microplane ormulation. The compressive boundary is expressed as unction o rdv = ε D εv (deviatoric strain to volumetric strain ratio), that when assume constant value give at σ bc( rdv, εv ) an initial linear evolution ( modeling the pore collapse) ollowed by an exponential evolution. It is possible to write: σ ( ε, ε ) bc D V = σ c + εv ε c Hc( rdv ) σ ( r ) exp[ ( ε ε ) H ( r ) σ ( r )] c1 DV V c1 c DV c1 DV or otherwise ε V ε c1 where: σ c = yielding compressive stress, ε c = σ c /E = volumetric strain at the onset o pore collapse, H c ( r DV ) = initial hardening modulus, ε c1 = k c ε c = volumetric strain at which hardening begins, k c = material parameter governing σ r σ + ε + ε H r c1 DV = c c1 co c DV. the onset o hardening, ( ) ( ) ( ) When there is a increment o r DV, the slope o the initial hardening modulus needs to go to zero, in order to the simulate the observed horizontal plateau eatured by typical experimental data. One way or obtaining this is the ollowing: For compressive loading, the normal stress is calculated imposing the inequality σ bc( ε D, εv ) σ N. Inside the domain, descript with the previously equation, the behavior is assumed to be incrementally elastic & σ N = E & Ncε N. In agree to the model the increased stiness during unloading, the loading-unloading stiness E is deined: Nc H c ( r ) DV = H c 1+ kc2 rdv kc 1 E Nc = E or σ N < σ c Ed otherwise where E d is the densiied normal modulus. The Fig shows the typically hardening behavior or compression

32 Chapter 1: The Lattice Discrete Particle Model (LDPM) Figure 1.13: Elastic-hardening behavior under compression stress. Frictional behavior The concrete present when in compression stress stage one increasing o shear stress due the rictional eects, is possible to compute this eect using the classical incremental plasticity. The relation used or calculate the shear stress is the ollowing: & σ M & σ L = = E E T L p (& ε & M ε M ) p (& ε & ε ) M L Where the plastic strain increments are assumed to obey the normality rule: p & σ M p & σ L = & λ ϕ σ M = & λ ϕ σ L 2 2 The plastic potential is written as ϕ = σ + σ σ ( σ ) strength σ i ormulated with a nonlinear rictional law: bs M L bs N in which shear σ bs ( σ ) = σ + ( µ + µ ) σ µ σ ( µ µ ) σ ( σ σ ) N s N N N exp N N Where σ s = cohesion, µ and µ are the initial and inal internal riction coeicients, respectively, and σ N = the normal stress at which the internal riction coeicient transitions rom µ to µ

33 Chapter 1: The Lattice Discrete Particle Model (LDPM) Finally, equations governing the shear stress evolution must be completed by the loading-unloading conditions ϕ & λ and & λ

34 Chapter 1: The Lattice Discrete Particle Model (LDPM) Numerical Implementation and Stability Analysis The LDPM model, is implemented into one sotware called MARS ( Modeling and Analysis o the Response o Structures), which is a multi-purpose structural analysis program based on a modern object-oriented architecture. The MARS uses one kind o the explicit dynamic algorithm or solving the analysis o structural problems. The dynamic algorithm take or the our problems is the central dierence method, that is developed rom central dierence ormulas or the velocity and acceleration. This is particularly important or the research presented in this model, because it is characterized by several thousands o degrees o reedom, and this permits to solve the problem using a standard computer, because it does not need more computational space. In any way the explicit scheme is not aected by the convergence problems that implicit schemes have in handling sotening behavior. In act, the explicit algorithms are not always stable, and these oten need an accurate analysis on the numerical stability o the numerical simulation. In order to have the stability, it is possible to make this kind o approach. In the elastic regime, the stability condition may be expressed, with the respect o the ollowing relation: t < 2 ωmax. In this relation ω max represents the highest natural requency o the computational system. The irst step, or ensure the stability is to ind this value, is possible demonstrate that ωmax < max( ω I ), where the ω I is the natural requencies o the individual unrestrained elements composing the mesh used in the simulation. Basically, the objective is to study every element and to ind its natural requency, then to take a higher value which will represent one limited or the our system. In this way is estimated ω max, which will be contained in the range that see ω I like last value. The second step will be to compute the time step, using this relation: t < 2 ω I, this procedure permits to avoid every instable convergence problem. 2 Knowing the correct time step, the ollowing problem det( K ω M ) = is computed, where K is the stiness matrix and M the mass-matrix. The elastic energy associated with the generic acet k is: U K = 1 2 l e A K k k k k ( ENε Nk + ETε Lk + ETε Mk ) = ( K ij + K jj ) Q j + ( Kii + K ji ) Q i where: pk T qk pk T qk pk T qk ( B ) B + E ( B ) B + E ( B ) ( p, q i, j. ) k K pq = EN N N T M M T L BL =

35 Chapter 1: The Lattice Discrete Particle Model (LDPM) The total tetrahedron stiness matrix is obtained by assembling the strain energy contributions o all twelve acet: K = 12 K k = 12 k = 1 k = 1 K K k ii k ji K K k ij k jj where the symbol Σ is used to identiy the assembly operation. For a generic acet k, the kinetic energy associated is subdivided into two terms relative to the two nodes, that belonged at the acet. This may be showed in this way: I = I + I. Each individual term read as: k ki kj I ki 1 T T T T T ρ u &( x) u& ( x) dv = Q ρap ( x) Ap ( x) dv Q p = Q p M pqp = p 2 2 V kp Vkp 1 & & 1 2 & & & where: M k p Vkp = ρ z Skp y Skp V kp S S z kp x kp V S kp y kp z kp S I S y kp S + I I I z kp y kp z kp xy kp xz kp I S I x kp S z kp + I I x kp xy kp z kp yz kp I S I I x kp S y kp x kp xz kp yz kp y kp + I The elements that appear in this matrix are: ρ = material density, V kp = volume identiied by the acet k and the node p. The others terms present in the matrix are: S = the volume irst order, and I = the second order moments o the volume V kp about the axes o the Cartesian system o the reerence with origin at the node p. In the same way, the overall mass matrix is computed using the assemblage o the contribution or each acets: M = M k K i M = k = 1 k = 1 k M j

36 Chapter 1: The Lattice Discrete Particle Model (LDPM)

37 Chapter 2: Constitutive law or the concrete-iber interaction Chapter 2 Constitutive law or the concrete-iber interaction Introduction Fibers, whose use can be traced back to the Roman Empire, are important to restrict and, in some sense, protect against the coalescence o microcracks, microvoids into wide cracks. Nowadays there are many types o materials with this structure among which Fiber-Reinorced Concrete (FRC) and the Engineered Cementitious Composites (ECC), whose unction is exactly what I just pointed out. The main unction o ibers, besides giving a moderate increase to the tensile and compressive strength, is to ameliorate the energy transmission and absorption capability that is undamental in resisting impacts and dierent kinds o shocks. More speciically ibers limit crack width and permeability and have a substantial impact in reducing corrosion risk, Begnini, Bažant, Zhou, Gouirand and Caner (27). Previous literature mainly ocuses on uniaxial tension, Li et al. (1998), Nataraja et al. (1999), Ramesh et al. (23). More recently, multiaxial loading models and experiments have been reported by Kullaa (1994), Nataraja et al.(1999), Grimaldi and Luciano (2), Peng and Meyer (2), Li and Li (21), Kholmyansky (22), Kwak et al. (22), Cho and Kim (23), Ramesh et al. (23), and Kabele (24), Yin et al. (1989), Traina and Mansour (1991), Chern et al. (1992), Pantazopoulou and Zanganeh (21), Mirsayah and Banthia (22). The objective is to create one constitutive law that permits to describe the relationship between the bridging stress σ transerred across a crack and the opening o this crack w and apply it to the LDPM. In this way it include the eects o randomly dispersed ibers in order to simulate the behavior o ibers. During the pre-processing phase, each individual iber is inserted into the specimen volume. Fibers positions and orientations are randomly generated, and the intersections between ibers and LDPM acets are detected

38 Chapter 2: Constitutive law or the concrete-iber interaction The stress on each LDPM acet can be computed as: σ = σ c + σ = σ c + 1 A c Ac P ( w, n, d L, L ) s l This relation assume a parallel coupling between the ibers and the concrete σ = σ σ σ, matrix. The elements present in this relation are: [ ] T N M L σ = [ σ σ σ ] T, = [ P P P ] T A = acet area, c Nc Mc Lc opening, embedment length, and d = iber diameter, P N M L, c L s = short embedment length, w = acet crack L l = long n = iber orientation with respect to the crack (acet) plane. The terms relative to the concrete stress are computed in according to the LDPM constitutive law. Basically, this construction starts rom modeling a single pull-out iber behavior against the surrounding matrix. The relationship σ = σ ( w) can be then obtained by averaging the contributions rom ibers with dierent embedment length and orientation across the crack plane. In the theory that is proposed in this study the ollowing aspects are considered: two-way iber debonding-pullout due to a sliphardening interacial bond, matrix micro spalling and also the Cook-Gordon eect. This constitutive law will be used so to describe the concrete-ibers interaction behavior, into the LDPM. The idea used to achieve this aim takes the ormulation proposed or E.H. Yang, S. Wang, Y. Yang and V.C. Li (28), as a base, and develops this idea or the mesoscale approach. In the ollowing sections, will be shown the analysis that starts rom the single iber pull-out and later takes into account the total eects that are present in the iber eature. In the next chapter the constitutive load will be veriied using the results that are available rom the experimental tests

39 Chapter 2: Constitutive law or the concrete-iber interaction Modeling o the single iber behavior A well-known technique to study iber-matrix interacial behavior is single iber pull-out, Katz. A., Li V.C. (1996). In Fig. 2.1 is shown the typical pull-out curve. Three stages are identiied: initial elastic stretching o the iber ree length, ollowed by debonding stage and at the end the pullout phase Elastic stage P [KN] Debonding stage Pullout stage.5.1 v.15 d v [mm] Figure 2.1: Single pullout curve o steel iber. The iber in pullout irst is subject to debonding beore the pulling out phase takes place. Basically, this can be described as a crack propagation rom the surrounding matrix crack to the embedded end. This stage lasts until there is the load drop, that represents the transition to the pull-out stage, when we are let just with rictional bond that can actually increase, whereas the chemical bond, that is present during this irst stage, is lost. Stress analysis and energy balance process are used to model the debonding and the pullout o a single iber. The ollowing are the main assumptions o this model by Z. Lin, T. Kanda and V.C. Li (1999): a) Fibers are high aspect ratios (>1), so that the inal eect on the total debonding load is negligible; - 3 -

40 Chapter 2: Constitutive law or the concrete-iber interaction b) During the debonding stage, the slip-dependent eect is negligible since relative slippage between the iber and the matrix in the debonded portion is small. Hence, the rictional stress within the debonding zone remains at a constant τ ; c) Poisson s eect is negligible. For lexible iber-cement systems, Poisson s eect is usually diminished due to inevitable slight misalignment and surace roughness o the iber; d) Elastic stretch o the iber ater complete debonding is negligible, compared with slip magnitude. The relation between the load P and the displacement w is written. This is divided in two dierent equations, matched with the dierent stages present in the single iber behavior. At each stage, the iber extending across the matrix microcrack is always in equilibrium. It is possible to say that the orce exerted on the short embedment side and the long embedment side are equal: P = P = s ( v ) P ( v ) s According to the previous statement, it is also true to declare that in absence o the dierent eects, the crack opening can be written as: w = v + v s l l l this is the case when the crack opening is done only by the ibers-matrix slippage. The crack opening, implemented into the code, is composed by three dierent contributions, given that the model allows to study 3d problems: [ w + w w ] 2 w = + N In this way, is identiied the vector identiying the crack opening created only by the ibers slippage. This crack opening vector is the result by sum o the slippages values, later when the other eects will appear there will be a new crack opening vector, called w '. This will be dierent than w or direction and modulus. Considering a crack crossed by straight iber (Fig. 2.2), characterized by the embedment lengths L s and L l. I one neglects the iber bending stiness and the elastic deormation o the crack-bridging segment, the length o such a segment (distance between points A and B in Fig. 2.2) can be computed as: ' M w = w + 2s n L

41 Chapter 2: Constitutive law or the concrete-iber interaction ' where w is the crack opening vector and s the reduction o embedment lengths due to micro-spalling. a) L l w N w T w w B φ n s A L s Figure 2.2: Schematic o inclined bridging with matrix spalling. In additional, the iber orce con be assumed to be coaxial with the crack-bridging segment and expressed as: P = P n ' where: n = w w and ' ' ' w = w + w + w. ' '2 N '2 M '2 L The Fig. 2.3 shows the best situation possible, characterized by iber orthogonal to the crack plane. In this particular coniguration the spalling and the snubbing, that will be later explained, are not present. In the next section, will be explained how the crack opening vector changes according to the other eect that the ibers issue shows. The new eect that will be considered is Cook-Gordon eect. According to the change just show beore, will be explained also the micro-spalling. The dierent stages or a single iber will be described using dierent relations that permit to ind the link between P v and also σ v. In this irst step, the relation between orce and crack opening will be analyzed, assuming that the pullout orce is always applied orthogonal to the crack plane. Later on, the same analysis will be perormed but looking at most common cases

42 Chapter 2: Constitutive law or the concrete-iber interaction In the second step, the strength relations will be introduced. In this way the constitutive law is done. b) P, v L Figure 2.3: Fiber pullout with orce applied orthogonal to the crack plane. Following the detailed derivation by the Z. Lin, T. Kanda and V.C. Li (1999), the equation or the debonding stage can be written as: P () v π τ E d ( 1+ η) π G d = v + 2 E 2 d v v where vd is the crack opening that corresponds to the displacement at which ulldebonding is completed, and it is expressed as: d v d 2 ( 1+ η) 8G L ( 1 η) 2 2τ L d + = + E d E d The elements that appear here are: η = V E VmEm, this is a parameter expressing the ratio o the eective (accounting or the volume raction) iber stiness to eective matrix stiness, rictional stress τ and debonding racture energy G d (also reerred to as chemical bond). When the debonding phase is inished, it is possible to see a sudden drop due to unstable extension o the tunnel crack. Subsequently, the iber is held back into the matrix only by rictional bonding

43 Chapter 2: Constitutive law or the concrete-iber interaction The magnitude o the drop can be used to calibrate the chemical bond G d. During the debonding and pullout stage, the iber may rupture i the load P exceeds the iber tensile strength. According to this eect, the study is developed under the assumption that the iber rupture never happen. The relation that describe the orce in the pullout stage is: P () v = πd τ 1+ β ( v v ) d ( L v + v ) v v L d The dierent elements present in this equation, are shown in Fig It is also possible to write the equation or the debonding stage in terms o the debonding length. In this case the debonding load P can be also expressed as: at the ull debonding P 2 3 ( L) = π d τ L + π G E d 2 d L = Le. The maximum debonding load is given by: d d P a = P + b 2 π G d E d 3 2 where Pb = πd τ Le is the initial riction pullout load. This equation allows calibrating the chemical bond strength G d and the rictional bond strength τ, rom the maximum debonding strength load P a and the initial rictional pullout load P b. The slip-hardening coeicient β, is obtained rom the relation that is used or the pullout stage. c) P, v L (v v d ) Figure 2.4: Slippage eect and variable o the problem

44 Chapter 2: Constitutive law or the concrete-iber interaction β is the parameter that controls the behavior in the tall o the curve P v. It can be hardening or sotening according to the kind o iber used in the specimen. These simple relations are the building blocks to construct the constitutive law or composite material like ibers-concrete. The next step will be the writing o the relations in terms o strength and displacement. The single iber-bringing stress σ debonding versus the iber displacement (at exit point) relative to the matrix crack surace v is given by: σ debonding = ( τ v + G ) 2 d 2E ( 1+ η) d where E and d are iber Young s modulus and diameter, respectively o iber. Ater the debonding is completed, the iber is in the pull-out stage. In this stage, there is one load drop that represents the absence o chemical bonding. Here the strength is guaranteed by its rictional bond only. For some types o ibers, particularly PVA ibers, signiicant slip-hardening response was observed during long-range pullout. Assuming the linearity o the rictional stress with respect to the slip distance with the coeicient β ( reerred as slip-hardening coeicient, Lin, Z. and Li, V.C. (1997), the iber stress σ pullout during the pullout stage can be expressed as: τ ( ) β σ pullout = Le v vd 1+ d d ( v v ) In the short iber composite system, most ibers are oriented to an arbitrary angle ϕ relative to the crack plane, which takes values in the range o π 2, with ϕ = or ibers perpendicular to the plane and ϕ = π 2 or ibers parallel to the plane. Furthermore, the interaction with the matrix, when this last one exits the iber puts the iber under urther stress. Speciically or the polymeric iber as Morton and Groves (1979) and Z. Lin, T. Kanda and V.C. Li (1999) suggest the ollowing holds: d P = P s ' ksnϕ ( v ) e = P ( v ) s l l e ' snϕ k Where ksn > is reerred to the snubbing coeicient. This relation accounts the increase o bridging orce P due to an inclination angle ϕ by making analogy to an Euler riction pulley at the iber exit point. As shown in Fig. 2.5, at the exit point, i.e. where the iber begins to protrude beyond the crack ace, there is a kink

45 Chapter 2: Constitutive law or the concrete-iber interaction in the iber. At this point, the matrix acts as a rictional pulley. This phenomenon is typically called snubbing eect. d) e) P P φ φ φ s P s or P l Figure 2.5: Snubbing eect geometry. Moreover dierent ibers reacts dierently to bending and lateral stress. For instance, in order to study this eect, or the PVA iber, Kanda and Li (1998) introduced a strength reduction coeicient: σ u ( ϕ) = σ ( ) u k ϕ rup e where σ u is the ultimate tensile strength o the ibers associated with the theoretical situation where the iber s bridging segment and its embedded segment are collinear, and k rup is a material parameter. Basically the previously relation can be written also in term o orce. The previously equations are valid only i the iber orce does not cause iber rupture. This can be checked by making sure that the iber orce magnitude is less than the rupture orce: P < P u =.25πd σ e 2 u ' k rup ϕ Until this point I just tries to illustrate and explain the ibers behavior. Later on in my work I will ocus on three important aspects: two-way iber pullout, spalling eect and the Cook Gordon eect, according with E.H. Yang, S. Wang, Y. Yang and V.C. Li (28)

46 Chapter 2: Constitutive law or the concrete-iber interaction Matrix micro-spalling An issue that should be taken into account is the misalignment o the pullout orce with the orientation o the embedment segments at the iber exit (Fig. 2.2). This is called, matrix micro-spalling, common in the case o random iber reinorced brittle matrix composites, as PVA-ECC, Kanda and Li (1998). For the deinition o the spalling length, s, various models have been proposed in the literature, including Cailleux et al (25) and Leung & Li (1992). Herein, the ormula proposed by Yang et al. (28) is adopted. The spalling length, expressed as a unction o external load on the iber, matrix strength, matrix stiness, and inclination angle inluences the size o the spalled matrix piece, that is estimated through the ollowing equation: s = k sp P ϕ N sen 2 2 d σ cos ϕ t 2 P N is the normal component o the external orce acting on the iber, ϕ is where the iber inclination angle, σ t is the matrix tensile strength, k sp is a dimensionless constant related to the iber geometry and matrix stiness, calibrated through experimental observation and called spalling parameter. This speciication, typical o LDPM, allows to consider the inclination crack, that is not possible by simply ollowing Yang, S. Wang, Y. Yang and V.C. Li (28) that only consider crack opening normal to the crack plane. In the above speciication, the spalling size is considered as proportional to the external load on the iber exit and inversely proportional to the matrix tensile strength and iber diameter. For typical orces in PVA, iber in ECC, the predicted spall size (or k sp = 3), is in the range o the 28µ m, consistent with those observed experimentally. Matrix micro-spalling has an impact on stress concentration on strengthen bridging ibers and on the iber inclination angle ϕ to a smaller ϕ '. In a PVA-ECC iber, the spalling size s ranges rom several micrometers to iber diameter. Moreover, starting rom a steady-state crack width o 6 µ m, the maximum elastic stretch o the iber segments reed rom spalling will not exceed 2 µm at the peak load, which is negligible compared without signiicant inluence on the modeling o single iber-bridging behavior with consideration o both two

47 Chapter 2: Constitutive law or the concrete-iber interaction way pullout and matrix spalling can be reduced to the problem o two-way pullout with a modiied w, as is just shown beore. ' Cook-Gordon eect The Cook-Gordon eect is linked to the tensile stress exerted on a blunt matrix describes a premature iber/matrix interace debonding normal to the iber axis caused by a tensile stress located ahead o blunt matrix crack propagating towards a iber under remote tensile load as depicted in Fig.5. In the Fig. let is represented the induces iber-matrix separation due to the tensile stress in the horizontal direction associated with the elastic crack tip ield o the approaching matrix crack. Whereas the Fig. on the right leads to an additional crack opening v cg due to elastic stretching o the iber segment. v = v+v cg k cg Figure 2.6: Cook Gordon eect. In other words, we have the stretching o a ree iber segment k cg (Cook-Gordon parameter), and additional crack opening v cg, as shown in Fig. 2.6, due to debonding ahead i the matrix crack. Following Li at al: (1993), v cg can be directly computed rom the bridging orce P or all ibers without considering the orientation. Thereore, or a single iber, the ormula or the additional displacement is the ollowing: 4k cg vcg = 2 πd P E

48 Chapter 2: Constitutive law or the concrete-iber interaction When P is independent o ϕ and z, the integration o next relation gives the number o ibers bridging across the crack per unit area times P. where B A σ, ( z φ) dzd PNB = P p V φ N, the number o iber-bridging across the crack per unit area, is: and N B A =, V p A ( z φ ) dzdφ = ηb V η B = p, ( z φ) dzdφ this is the eiciency o bridging in terms o the amount o iber bridging across a crack with respect to orientation eect. The P v relationship or the single iber, can be written also in this way: cg v cg kcgσ = V E η B In this way, the iber-bridging stress σ is computed as a unction o the numerical procedure or calculating the bridging stress vs. crack opening displacement relationship

49 Chapter 2: Constitutive law or the concrete-iber interaction Two way pullout Once the crack opening takes place, the short embedment side o the iber debonds. It is the pullout o this side that leads to an increase o the crack opening itsel. Inact, at this point, the long embedment side is characterized only by an elastic stretch whose inluence on the crack opening is minimal. O course, the eects is dierent according to the type o iber. Speciically or the PVA one, Redon and Li (21) suggests that slip-hardening o PVA iber in cement matrix is due to abrasion and to the jamming eect during the slip stage. The embedment length is also important: when it is small the iber is completely pulled out rom the matrix, whereas when the length is large there is a rupture o the iber. Moreover, it should be pointed out, that the signiicant slip-hardening behavior causes the pullout load to be larger at ull debonding rather than at the completion o the debonding. Looking at PVA ibers, we can witness a two-way iber pull-out process, given that, when the short embedment sides goes through bridging, it is the long embedment side that starts the pull-out stage. This eature makes necessary, to have a complete and not biased picture, to include the eect that the slip displacement coming rom both sides o the iber has on the total crack opening ' w. Wang et al (1988) were the irst to suggest this two-way iber pullout, shown in ig. Later, I will describe the dierent phases in more details. Analysis stages The aim is to divide the dierent stages and to analyze every stages so to have a complete picture o the problem. For each stages, the iber is assumed to be no orthogonal to the crack opening, the snubbing and spalling eects or this reason will be always present. In this study, nothing will be said about these eects, the attention will be ocus only on the two way pullout behavior. Stage 1: neither embedment ends completely debonded In this stage, the crack opening v is started and the iber begins to work. Here the magnitude o the crack opening is really and the relation that allows to write the compatibility equations is v = vl + vs = 2vs, where v l, vs are respectively the slippage in the long and in the short embedment iber. In this stage, v = v, because both sides are in the debonding stage. The next subsection shows how the s l - 4 -

50 Chapter 2: Constitutive law or the concrete-iber interaction passage to the short embedment length in the pullout stage will change the above relation. It will also show that the slippage will signiicantly increase in this stage. For the debonding stage, the equation used to express the relation P v is the ollowing: P ( 1+ η) π τ E d π Gd E d = v + v 2 2 v d applied to both sides until there is complete debonding in the short side (Stage 2): v d 2 ( 1+ η) 8G L ( 1 η) 2 2τ Ls d s + = + E d E d where L s is the length or the short iber. This is an assumption already seen in previous literature. It states that the pullout stage is achieved irst in the short side and later in the other side. However, the iber is always in equilibrium: the orce in the iber is independent o the embedment length, Yang at al. (28). For a given crack opening v, the contribution v L and v S rom the long embedment segment (length L l ) and the short embedment segment (length L s ), respectively, can be calculated rom the iber load balance. The upper bound on v or this stage can be deined as: v < 2v d where vd is the crack opening relative to the completely debonding or the short side Stage 2:short embedment end complete debonded In the new stage the short embedment length is at ull debonding and the orce suddenly weakens. Thereore, the bonding embedment is lost. However, the riction eect is still present and it also allows increasing the orce in this stage. It should be noted that, i the rictional pullout is a hardening process, the P will increase until it reaches the value needed to debond the long side. Mathematically, this stage can be depicted using the ollowing equations: v l < v dl 2 ( 1+ η) 8G L ( 1 η) 2 2τ Ll d l + = + E d E d

51 Chapter 2: Constitutive law or the concrete-iber interaction v s > v ds 2 ( 1+ η) 8G L ( 1 η) 2 2τ Ls d s + = + E d E d and the ollowing relations are used to describe the problem or the orce: P P s l = 2 π τ E d 2 = πd τ 1 β 3 ( 1+ η) 2 π Gd E d + 2 ( + ( v v ) d )( L v v ) s where v sd corresponds to the displacement at which ull-debonding is completed and it is given by: sd s 3 s sd v sd 2 ( 1+ η) 8G L ( 1 η) 2 2τ Ls d s + = + E d E d The equilibrium in the iber is assured in this way: ld ( v ) = P ( v L ) P, l sp s s Stage 3: both embedment ends completely debonded In this last stage, the debonding is achieved or both sides o the iber that is in pullout stage. The relations that describe this stage are: v v l s > v > v dl ds 2 ( 1+ η) 8G L ( 1 η) 2 2τ Ll d l + = + E d E d 2 ( 1+ η) 8G L ( 1 η) 2 2τ Ls d s + = + E d E d In this phase the relation or the crack opening changes, and the new orm is: v + v + 2 s = v l s e

52 Chapter 2: Constitutive law or the concrete-iber interaction Where s is the spall size that is included in the rictional pullout equation. It reduces the length o cylinder, where the iber is in rictional contact with the tunnel. The term that includes β, which provides the magnitude o the rictional stress and which is dependent upon the occurrence o the relative slip between the iber and the tunnel, does not change. P ps P pl = πd τ 1 β = πd τ 1 β ( + ( v v ) d )( L v v ) s sd ( + ( v v ) d )( L v v ) l ld s l s l sd ld

53 Chapter 2: Constitutive law or the concrete-iber interaction Parameters analysis o the model The constitutive law implemented into the LDPM will be studied to check and understand how the dierent parameters allow to modiy the curve orm or iber pullout. The single pullout iber will be evaluated trying to change the values o the main parameters, that have a crucial impact on the way the ibers react. With this in mind, the several cases studied are: pullout orce applied in the same iber direction and not (present o spalling eect), dierent values or β (pullout hardening behavior), a geometrical study about the two dierent embedment lengths or the iber, that bring the phenomenon o two way pullout and several more. In additional, the geometry o the problem is investigated. In this way taking ixed the parameters values, is studied how the iber position aect the response. In brie, looking also at the indings o previous literature, or each o these cases just described, I will develop and test several hypotheses on the reaction behavior o the ibers. Presentation o iber parameters The LDPM-F model or the ibers is governed or the 11 parameters. These ones are: E = Elastic Modulus; k sp = Spalling Parameter; σ = Fiber Strength; k rup = Fiber Strength Decay; k sn = Snubbing Parameter; τ = Bond Strength; η = Volume Stiness Ratio; G d = Debonding Fracture Energy; β = Pull Out Hardening; k p = Plastic Parameter; =Cook-Gordon Parameter; k cg The next paragraphs will be a discussion on the most important parameters, and the main eects o this constitutive law. The idea is to hypothesize the eect

54 Chapter 2: Constitutive law or the concrete-iber interaction connect to the dierent parameters, and then to check i the hypothesizes done are corrects. In this way the law is checked over. Pullout hardening behavior The iber behavior ater complete debonding is characterized by the presence o a sudden drop o the load caused by the lost o the chemical bond. In the next stage, called pullout phase, the iber response can change signiicantly according to the kind o iber used in the specimen. Several previous works showed that these parameters can be ranged into 1 < β < 1. The idea is to try now to change each parameter and to check that the reply the one expected. The behavior can be reerred as sotening when β < or hardening β >. This kind o test is not aected by the iber position respect to the crack plane. According to this aspect, the test is realized using one iber positioned orthogonal to the crack plane, thereore the pullout orce is applied coaxial to the iber direction. This problem coniguration is shown in the Fig It is deined as linear when β =. This is shown in the ollowing image: Pullout hardening [β] β =. β =.2 β = P [KN] v [mm] Figure 2.7: Inluence o the pullout parameter on single iber pullout curve

55 Chapter 2: Constitutive law or the concrete-iber interaction Clearly this kind o parametric analysis is done, ixing the other parameters equal to zero, in this way the results are not aected by mixed o dierent phenomenon. Basically the pullout eect can be divided into three dierent groups. On o this, is shown a linear behavior, the remaining two are a no-linear trend. Two way pullout This eect is not matched with the parameters changing. The purpose is to understand how the two way pullout change the response o the iber. As already said in this chapter, the iber is always in equilibrium or both the debonding and pullout stages. The only dierence is that in the short embedment length the ull debonding crack opening is achieved beore the long side, this shows that the slippage in the short time is bigger than the long side. Now in order to check that the model allows or a good description o this aspect, the test done will change the two embedment length. The tests will have embedment length very dierent in the irst stage ( v * ), and almost equal in the second test v. ibers slippage [v l - v s ] * v s [Ll ~L s ] * v l [Ll ~L s ] v s [L l >L s ] v l [L l >L s ] Two way pullout crack opening [v ] Figure 2. 8: Two way pullout behavior due to the dierent embedment lengths. The result expected is that in the test v the slippage or the two sides is the same until the ull debonding crack opening is achieved, ater the test v *. When this

56 Chapter 2: Constitutive law or the concrete-iber interaction limit or the both tests is achieved, it is possible to see how to change the slippage magnitude in the short and length embedment sides. Debonding racture energy In this analysis, the aim is to investigate the eect o the debonding racture energy ( G d ). How is already said in this chapter the iber can be in two dierent stage: debonding and pullout. When the iber pass rom debonding to the pullout stage is possible to see a sudden drop in the orce. The magnitude o the drop is directly proportional to the value o G d. The approached used is to try several values o G d and will check i the results expected will be captured rom the model. The results obtained prove how increasing the value o G d the drop increase. It is also shown one increasing in the orce peak. This is reasonable because with a bigger value o G d in the iber is so increasing the orce that is brought Debonding racture energy [G d ] G d = 4. N/m G d = 6. N/m G d = 12. N/m.12 P [KN] v [mm] Figure 2. 9: Inluence o the Debonding racture energy on single iber pullout curve. This parameter is also not aected by iber position. The angle realized between iber direction and pullout orce is not important to the analysis o the problem

57 Chapter 2: Constitutive law or the concrete-iber interaction The parameters that are important or considering this angle are: spalling parameter and snubbing eect. They are shown later. Bond strength The bond strength τ is a crucial parameter and it is matched with the rictional orce, that it is assumed to be constant in the debonding stage. This parameter appears in both the relations or P. It not only aects the elastic slope but also the peak orce. Notably, its eect is more clear ater the post peak, where the iber is held in the concrete matrix only by the rictional strength. Starting rom this idea, the value o the bonding strength will be increased step by step and later the results will be checked. As the graph shows the results ound are in line with the expectations. Bond strength [τ ].25.2 τ = 2. MPa τ = 4. MPa τ = 6. MPa P [KN] v [mm] Figure 2. 1: The curves show how the single iber pullout is aected by bond strength. This parameter is not aected by the iber inclination respect to the pullout orce direction. According to this, the single iber is positioned orthogonal to the crack opening plane

58 Chapter 2: Constitutive law or the concrete-iber interaction Snubbing eect The snubbing eect is present when the angle φ between the iber and orce direction is dierent rom zero. This phenomenon shows how the orce brought by the iber increases in this new coniguration. We expect that the P value, particularly at the peak, increases with k sn ixed, and changes only the φ angle. Using this approach, the orce increases when the angle becomes larger. In this study, three values o angle will be used. The graph shows that again indings are in line with expectations. Snubbing eect [k sn ].25.2 φ =. o φ = 45 o φ = 6 o P [KN] v [mm] Figure 2.11: Dierent response or snubbing eect caused by iber orientation. According to this igure, the snubbing parameter has the same trend ixing the angle and increasing the value o the parameter. This parameter will be very important in the next chapter, when the model will be calibrated. It allows to it the peak

59 Chapter 2: Constitutive law or the concrete-iber interaction Spalling eect The spalling eect is present when the angle φ between the iber and orce direction is dierent rom zero (Fig.4). This phenomenon shows how the embedment length o the iber is subject to one reduction caused by the rupture o the concrete matrix close to the iber exit. The behavior will be the same in terms o orce sustained, but I expect that the ull debonding crack opening will be achieved earlier than in the case without spalling. Thus, three dierent values o the angle will be used. We expect that the value, in particularly at the peak, increases with the ixed, and a change in the angle. Using this approach, the orce increases when the angle is larger. In this study, three values o the angle will be used. As expected, when the angle value increases, the ull debonding crack opening is achieved earlier. Spalling parameter [k sp ].25.2 φ= φ= 45 φ= 6 P [KN] v [mm] Figure 2. 12: Dierent response or spalling eect caused by iber orientation. The spalling parameter is too important in the processing phase because it determines the spalling length. According to this actor, the crack opening vector changes, and or this reason also the pullout orce that the iber can bring

60 Chapter 2: Constitutive law or the concrete-iber interaction The inverse relation between spalling length and spalling parameter is important. In act, the increasing o spalling parameter produces one reduction in the spalling length

61 Chapter 3: Numerical analysis plain concrete subject to tension and compression Chapter 3 Numerical analysis plain concrete subject to tension and compression Introduction In this chapter will be explained how to calibrate the LDPM in order to it the experimental results by Laps (Bologna), Buratti at al (28). Following this idea, initially will be studied the 3PBT (three point bending test) and UC (uniaxial unconined compression test) just ocusing on the concrete. Ater that, in the next chapter, when the concrete is already calibrated, the ibers will be introduced in the specimens. The irst calibration is done using the common mesostructure values o parameters, shown in previous works. Starting rom here, will be run one parameter study to understand as the dierent parameters aect the model. The experience sais that usually there is no linear relationship among the dierent parameters. In order to obtain the best possible calibration, the ollowing tests should be run: uniaxial unconined compression, three point bending test, triaxial shear test and hydrostatic compression test. In such a way it will be possible to capture each kind o concrete behaviour: compressive and shear-tensile response. Is a common procedure not to run each one o these tests. It is enough to run two tests related to the study o the tensile and compression behaviour. The approach ollowed is aimed at calibrating how the model reacts to a change in the parameters or each type o available test while using or the rest o parameters the common values ound or other kind o concretes. The structure will be as ollow. I illustrate the calibration procedure irst and then I will analyze the dierent topics matched with the LDPM model

62 Chapter 3: Numerical analysis plain concrete subject to tension and compression Presentation o the parameters The Lattice Discrete Particle Model (LDPM) is governed by the 16 mesostructural parameters, that allow to describe the concrete behaviour. In the next chapter will be introduce additional parameters, speciically used or the ibers calibration in addition to the previous one. The parameters used or the ibers calibration that will be add to these. For the concrete calibration the parameters are: E = Normal Modulus; E d E = Densiication Ratio; α = Shear-normal coupling parameter; σ t = Tensile Strength; σ c = Compressive Strength σ s σ t = Shear Strength Ratio; l cr = Tensile Characteristic Length; n t = Sotening Exponent; H c = Initial Hardening Modulus Ratio; ε c1 ε c = Transitional Strain Ratio; µ = Initial Friction; µ = Asymptotic Friction; σ N = Transitional Stress; β = Volumetric Deviatoric Coupling; k 1= Deviatoric Strain Threshold Ratio k 2 = Deviatoric Damage Parameter. Model sensibility The LDPM allows to reproduce the concrete behaviours, showing tension, compression behaviour and also the shear eect. It will involve tests replicating both behaviours. Through these experimental tests, will be study ew parameters. For calibrate the model all kind o test or the same concrete need. In this way will be able to it each parameter. Using this approach, will be shown the parameters that aect the test behaviour or 3PBT and UC. However, I will reer to the precious works done in this ield, in order to have a clear idea about the dierent values ranges in which the parameter luctuate and or understand what kind o

63 Chapter 3: Numerical analysis plain concrete subject to tension and compression parameter aect my test. In the recent years, the ollowing links test parameters were identiied: Tensile test: the elastic part is linked with the E and α. The peak is moved by σ t and the inelastic part is a unction o l cr that allows to change the post-peak slope and σ t, important in order to move the curve tail; Uniaxial unconined compression test: E and α represent the principal parameters needed to it the elastic part. The peak is governed σ t, σ s σ t and µ. The post-peak by σ t and l cr ; Hydrostatic compression test: the elastic part is unction o E and α. The limit pressure value o the elastic behaviour by σ c, the initial slope o the hardening part by H c. Looking at the previous literature according to Mencarelli (27): For tests like tensile and the uniaxial unconined compression the curve is scaled proportionally to σ t out o the elastic part (peak and post-peak); l cr does not allow to scale in proportional way the curves out o the elastic part or the tensile and uniaxial unconined compressive tests; The ratio between the value o the peak o the Uniaxial Unconined Compression Test and the value o the peak o the tensile test is governed by the σ σ ; s t The σ c is the parameter that allows to scale the curve out o the elastic part or the hydrostatic test; Phases o the calibration Two types o tests are available in order to calibrate the LDPM only or the concrete. The tests used are 3PBT and UC. The calibration will be done, starting rom the 3PBT Aterwards, will be checked i the results ound in this irst phase are in line with the experimental data or the UC test

64 Chapter 3: Numerical analysis plain concrete subject to tension and compression The diiculty o this calibration is the non-linearity o relationship among parameters. The inal aim, then, will be that to ind a ir compromise among successive calibrations, so to account or and understand the nonlinearity o the problem. The ollowing section is divided into dierent subsections according to the dierent steps needed to perorm the calibration. Specimens geometry The specimens used in these tests are the beam and the cube respectively or 3PBT and UC. In general the approach is always characterize by these three phases, as I pointed out already in the irst chapter o this work. In this subchapter the irst stage will be explained. Basically, the Mars requires the only dimension o the specimen that can be directly assigned, when it has a common shape, as it is done or the UC test. Three point bending test (3PBT) The approach used or the 3PBT specimen is completely dierent rom the one usually done. There are two main reasons: the beam is realized with the notch in the middle o the span, making the specimen geometry a no-easy shape. F a b a $ F/2 F/2 Figure 3.1: Three point bending test specimen geometry

65 Chapter 3: Numerical analysis plain concrete subject to tension and compression The Mars allows to overcome this problem, by reerring to the nodes coordinate o the solid that will be used later on in the second stage. The second problem that aect this test is the specimen dimension. The beam used has one span o 5 mm and the sectional area is 15x125 mm, in the middle section where the notch is present. In order to reduce the computational cost and to take into account that all cracks are in the central area, where the notch is done, the beam is divided into three zones. The two areas near the supports, indicated with a in the Fig. 3.1, will be realized with inite element. The central part, called b is the LDPM volume. The dimensions o these regions are: a=15mm and b=2mm. These dimensions are ound according to the possibility cracks opening distribution, helping to get better clear scenario. In this way it is possible to realize a good numerical test without spending a lot o time, because the number o the particles has been reduced. The Fig. 3.1 shows the beam used in my test. Uniaxial unconined compression test (UC) Cube, cylinder and prism are the common shapes o specimen used in this test. These specimens are positioned between a top and a bottom plate, where the vertically compression is applied by the top plate with no lateral coninement. F D D Figure 3.2: Uniaxial unconined compression specimen. In my test, the specimen used is a cube o dimension 15x15x15mm. The characteristic used or the contact plate-specimen is high riction; in the F - 6 -

66 Chapter 3: Numerical analysis plain concrete subject to tension and compression experimental analysis sometimes this eect is reduced putting Telon or an oil layer between the plate and the specimen. The Fig. 3.2 shows the specimen used. This volume will be later illed with the particles deined in the ollowing subchapter. The image shows a triangle construction, that is realized in the last stage, when the particles and the zero-nodes are positioned into the volume, leading to the Delaunay triangulations is done. Aggregate generation The composition o the concrete, used into the LDPM model, is simulated starting rom the parameters o the mix design. This is very important in order to reproduce the mesostructure o the concrete. The tables present in the Buratti N. at al (28) show the concrete mix used in the Laps tests, this will be an obvious starting point. According to these tables, it is possible to rebuild the Fuller curve, that is expressed by this equation: F = d D max n where n is the Fuller coeicient, that will be estimated. This unction allows me to describe the granulometric distribution o the particles. By perorming a study on the principles parameters, it is possible to ind this distribution. The principles parameters are: the cement content c (mass/cubic length), the water to cement ratio w/c, the aggregate cement ratio a/c. At this stage, the main aim is to deine the concrete characteristic and the diameters range used to generate the particles that the LDPM will use to ill the specimen volume. Basically, only the coarse aggregate are reproduced, these are deined as the particles with characteristic size greater that 5 mm. It should be noted that, in order to obtain a reasonable computational time, the whole sieve curve cannot be simulated and, thereore, very small size aggregates must be discarded. According to this idea, the range 1-22 mm is used in my simulations. This choice allows to respect the two criteria undamentally empirically. First o all, later dierent years o simulation is been ound that covering the 3% o the total Fuller curve. I am able to deine one mixed design that allows to obtain good results, that are not aected by the choice itsel. Dierent percentage coverages are considered, but given some slight dierence in the results, the 3% is deined as the best value in terms o quality o reaction and computational time cost. The characteristics used in my test are: Cement Content = 357 kg/m3 Water To Cement Ratio =.49; Aggregate To Cement Ratio = 5.2;

67 Chapter 3: Numerical analysis plain concrete subject to tension and compression Min Aggregate = 1 mm; Max Aggregate = 22 mm; Fuller Coeicient =.5131; The ollowing image shows the estimated Fuller curve, where the green part represent the percentage simulated 1 Fuller curve.9.8 Mass percentage [%] Diameter [mm] Figure 3.3: Computed sieve curve. Experimental Fuller curve n=.51 Fuller curve silmulated by Mars The generation phase ends by positioning the particles into the specimens. This is done by using a random algorithm that replaces the particles according to the command seed. The seed is assigned into the mars input ile, and allows to simulate the concrete randomness structure as well. This will be very interesting in the calibration stage, where the results will be aected by seed. this is in line with experimental date with the experimental data, where a scatter is always present or the same materials specimens

68 Chapter 3: Numerical analysis plain concrete subject to tension and compression At the end o this study, the specimens appear in this way: F F 2 mm 15 mm Figure 3.4: Specimens ill with the generate particles. Parametric analysis The LDPM, as pointed out beore, is governed by 16 parameters. These parameters are needed in analyzing o the concrete and in capturing the compressive and shear-tensile behaviour. The parameters will be calibrated according to the experimental tests available. The experience say that the principal parameters that aect these two tests are: tensile strength σ t, shear-strength ratio σ s σ t, tensile characteristic length l cr and initial riction µ. The reaming parameters aect strongly the others kind o tests, whereas, in my tests, they imply just a small change in the results. The main objective is to take averaged values o each o the parameters and then start studying them. In this way, I will be able to it one concrete that can represent a good response also to other types o behaviour, or which the analysis cannot be perormed. Basically, it is important understand how these parameters change the responses o the model. According to this goal, a parameter study is done. This is the reason why sometimes it is possible to run one test, or instance 3PBT, but lose the good calibration speciic or the UC. In general, to have a good it, is needed is to achieve a balance among the dierent eects. Three point bending test (3PBT) The 3PBT is a test dominated by tension where a load is applied vertically in the middle o a simple support beam specimen o span l with or without notch in the central section. The aim is to obtain the orce-crack Mouth Opening

69 Chapter 3: Numerical analysis plain concrete subject to tension and compression Displacement (CMOD) curve, where the two magnitudes are measured respectively in the point where the load is applied and at the extremes o the CMOD, when the notch is done. In other situations, the displacement considered is vertically measured in the same point where the orce is applied. The rupture starts vertically rom the center o the specimen, where the notch is realized or in this central area when is without the notch. In this case, the area under the orce-displacement curve divided by the area o the ligament is deined like the Fracture Energy G. The 3PBT can be calibrated, by ocusing the attention on two main parameters, that are know to aect more this kind o test. The parameters are: σ t (tensile strength) and l cr (tensile characteristic length). In order to this kind o approach, this parametric analysis is done: 15 Tensile characteristic length [l cr ] l cr = 3 mm l cr = 6 mm l cr = 7 mm 1 F F[KN] 5 2 mm CMOD[mm] Figure 3.5: Tensile characteristic length analysis or 3PBT on plain concrete specimen. In this irst image, is shown how the tensile characteristic length cr l aects the peak and the slope or the plastic stage and, then, the position o tail curve. According to these results, this parameter will be used to estimate the post-peak slope and then when this portion o the curve become ixed o curve is ixed the peak will be ound. The peak is moved using other parameters. The image allows to understand how only one parameter aect dierent parts o the curve and also the behaviour. It is important to keep in mind this, because oten changing a

70 Chapter 3: Numerical analysis plain concrete subject to tension and compression parameter to it one part o the curve, would imply a change in another portion that was ixed beore. This can be very expensive in terms o computational time. The tensile strength σ t is another important actor that is used or the 3PBT. By varying this, according to its three values realized, the results obtained are: Tensile strength [σ t ] 25 2 σ t = 2.7 MPA σ t = 3.2 MPA σ t = 4. MPA F[KN] 15 1 F 2 mm CMOD[mm] Figure 3.6: Tensile strength analysis or 3PBT on plain concrete specimen. In this case, the tensile strength appears like a parameter that aect only the peak, and no-eect are shown or the post-peak slope. It allows to scale the curve out o the elastic part. As expected, by increasing the tensile, the strength increased the peak and at the same time also the curve tail went up. The method used or the calibration has been the ollowing: starting rom the 3PBT and then I tried or dierent values o these two parameters. Later on in the study, when a good it is achieved, the same parameters or the UC behaviour will be checked. How was clear to aspect the results ound in this stage, was not good also or the UC. With this in mind, I re-start my analysis by checking or what values there is a good it in the UC test, which has to be done by using shear strength ratio and initial riction. These changes aect also 3PBT. Thus, in order to explain this type o behaviour the ollowing analyses are run:

71 Chapter 3: Numerical analysis plain concrete subject to tension and compression 25 2 Initial riction [µ ] µ =.3 µ = 1.5 µ = F F[KN] 1 2 mm CMOD[mm] Figure 3.7: Initial riction analysis Shear strength ratio [σ s /σ t ] σ s /σ t =.1 σ s /σ t =.3 σ s /σ t =.5 12 F F [KN] mm CMOD[mm] Figure 3.8: Shear strength ratio analysis

72 Chapter 3: Numerical analysis plain concrete subject to tension and compression The initial riction µ does not aect the 3PBT and will be used later on to it the pick in the UC test. Notable, this goal is achieved also by using the shear strength ratio σ s /σ t. This completely changes the curve, as it is shown in the next igure. Moreover this parameter is important or the triaxial test. For this reason it is preerred not to change this parameter value. Thereore, the common value shown by the experience is used. Notable, in order to solve the calibration problem and to cover UC and 3PBT experimental results, the initial riction will be used or the UC test. The next subsection show how these parameters aect the UC test. Uniaxial unconined compression test (UC) It is a test where a prismatic, cubic or cylindrical specimen positioned between a top and bottom plate is compressed vertically by the top plate with no lateral coninement. For the prismatic and cylindrical specimen, the ratio between the dimension o the side L, o the square section or the diameter D o the circular section and the height H is usually 1:2 or 1:3. In both cases, the height o the specimens helps to decrease the coninement because the eect o the boundary plate does not inluence the behaviour in the central part o the specimen or the De Saint Venant eect. Cube specimens are used in my tests. The goal is to obtain the stress- strain curve, that is ound to divide the orce and the displacement, as measured on the plate or the area o the section and or the height o the specimen. The rupture response is matched with the level o coninement and can be explosive in the case o high riction. Basically, the problem linked with the calibration stage, was to take the pick. This can be achieved by using with shear strength ratio σ s /σ t and the initial riction µ. The elastic slope is already knew by the Bažant relations, that are the only relations that create a link into macro-parameters and meso-parameters. The results ound changing dierent values or these parameters are the ollowing. The irst parameter used to it the peak was the shear strength ratio σ s /σ t, rom the parametric study is possible to see how, by increasing it, the value the peak goes down. The experience also says that there is a linear match into the value o this parameter and the compressive strength. According to this, it is possible to it the curve, making a linear study. This method will create a problem into the 3PBT curve, because this kind o test is strongly inluenced by the shear strength ratio. The solution used is to work with the initial riction that does not change the results ound in the 3PBT. The sensibility o this parameter is shown in the Fig

73 Chapter 3: Numerical analysis plain concrete subject to tension and compression Shear strength ratio [σ s /σ t ] σ s /σ t =.5 σ s /σ t =.3 σ s /σ t =.1 Rcm [MPa] F 15 mm ε [-] Figure 3.9: Shear strength ratio analysis or UC test. Initial riction [µ ] x µ =.5 µ =.4 µ =.3 Rcm [MPa] F 15 mm ε [-] Figure 3.1: Initial riction analysis or UC test. x

74 Chapter 3: Numerical analysis plain concrete subject to tension and compression The remain two parameters that are important into the 3PBT calibration test are these types o responses in the UC. Rcm [MPa] F 15 mm Tensile characteristic length [l cr ] l cr = 3 mm l cr = 6 mm l cr = 7 mm ε [-] x 1-3 Figure 3.11: Tensile characteristic length or UC test. Understand how the UC react when these two parameters are changed is very important in the last calibration phases, when both test are almost it and small changes are needed to complete the model. In this way, the sensibility analysis permit to have just one idea also or this kind o study. Concluding this will be the order ollowed or it the result: a) The 3PBT will be calibrated using σ t and l cr ; b) The UC will be checked with the calibration used or the 3PBT; c) I it is not correct this will be changed using σ s σ t and µ ; d) The new parameters values set will be checked or the 3PBT; e) I the curves ound do not allow to reproduce both the behaviours, the loop restarts rom a point, until to achieve a good calibration

75 Chapter 3: Numerical analysis plain concrete subject to tension and compression 12 1 σ t = 2.7 MPa σ t = 3.2 MPa σ t = 4. MPa Tensile strength [σ t ] 8 F Rcm [MPa] mm ε [-] Figure 3.12: Tensile strength or UC test. x

76 Chapter 3: Numerical analysis plain concrete subject to tension and compression Calibration stage For the calibration o the model the 3PBT and UC, previously described, are used. The tests perormed are. the uniaxial unconined compression test allows to it the elastic part E and the peak σ c or the compressive behaviour; the three point bending test is the principal test used or to ind the tensile reply o the concrete. The experimental data available are: two test or the 3PBT and six or the UC or the specimen made only by concrete. The approach used is divided in two dierent phases. The irst step is the analysis o the experimental curves or both tests. One average curve will be generated to it at the start the calibration. According with the heterogeneity o the concrete, the experimental tests present a certain scatter and this approach allows to it the numerical curve on the average. Starting rom this, the calibration begins by using a generic seed or the Mars ile. It is a tool that is used or positioning the particle into the volume specimen. Changing this value, the randomness algorithm change the position and the heterogeneity o concrete can be reproduced also in the numerical model. The calibration starts using the common parameters values that are: Normal Modulus MPa, Densiication Ratio 1, Alpha.25, Tensile Strength 4.3 MPa, Compressive Strength 15 MPa, Shear Strength Ratio 2.7, Tensile Characteristic Length 12 mm,sotening Exponent.2, Initial Hardening Modulus Ratio.4, Transitional Strain Ratio 2, Initial Friction.2, Asymptotic Friction, Transitional Stress 6 MPa, Volumetric Deviatoric Coupling, Deviatoric Strain Threshold Ratio 1, Deviatoric Damage Parameter 5. The idea is to initially look at how the model response is ound and ater that start changing the parameters that the previous study has identiied has the most important. When a good calibration is ound, the second phase start. In this second step, the idea is to realize one general study being able to show the heterogenic structure o the concrete. With this in mind I lock the parameters values o the previously step, and six tests are run with dierent seed values. The tables 3.1 and 3.2 summarize the particle and tetrahedrons number present or the dierent specimens, or both UC and 3PBT. The curves ound are averaged and then compared with the experimental average curve. The magenta curve are the curve obtained by the model using dierent seeds. I the two curve are itted the calibration phase is over. Otherwise, it should continue, ollowing the same approach

77 Chapter 3: Numerical analysis plain concrete subject to tension and compression The results ound or 3PBT are: Concrete calibration: 3PBT test Experimental curves Numerical curves Experimental average curve Numerical average curve 1 F F [KN] mm CMOD[mm] Figure 3.13: Comparison between numerical and experimental results, or 3PBT. In this igure is possible to see the complete study. The blue curves are the experimental test results and the cyan blue curve is the their average. This is compared with the red curve that is the model average response. As it is shown, the model allows to have a very good it o the experimental data. Specimen Particles Tetrahedrons A B C D E F Table 3. 1: UC specimens. The objective or the UC test is dierent. In this case only numerical values are available, or it the pick and the elastic part. These values are picked using

78 Chapter 3: Numerical analysis plain concrete subject to tension and compression dierent seeds and then the average curve is used to check the inal results. The calibration ound or this test is shown here. Specimen Particles Tetrahedrons A B C D E F Table 3.2: 3PBT single notch specimens. The images shown how the LDPM allows to it every part o the curve with an high quality Concrete calibration: UC test Numerical curves Numerical average curve Elastic slope Num. Rcm= 41 Mpa Exp. Rcm= 43 Mpa Rcm [MPa] 3 2 F 1 15 mm ε [-] x 1-3 Figure 3.14: Comparison between numerical and experimental results or UC test. The specimens or UC and 3PBT appear later the test respectively as show in the Fig and In this images is possible to see the cracks patterns and also the ragmentation due to the compressive load in the UC specimen

79 Chapter 3: Numerical analysis plain concrete subject to tension and compression Figure 3.15: UC specimen damage. Cracks evolution. Figure 3.16: 3PBT specimen damage. Cracks propagation. The parameters value that allow this calibration are: E = Normal Modulus 5655 MPa E d /E o = Densiication Ratio α = Alpha.25 σ t = Tensile Strength 2.21 MPa σ c = Compressive Strength 195 MPa σ s /σ t = Shear Strength Ratio 2.5 l cr = Tensile Characteristic Length 9 mm n t = Sotening Exponent.2 H c = Initial Hardening Modulus Ratio.2 ε c1 /ε c = Transitional Strain Ratio

80 Chapter 3: Numerical analysis plain concrete subject to tension and compression µ = Initial Friction.25 µ = Asymptotic Friction σ N = Transitional Stress 6 MPa β = Volumetric Deviatoric Coupling k 1 = Deviatoric Strain Threshold Ratio.5 k 2 = Deviatoric Damage Parameter 5 These value are close to the common range shown by the experience. This allows to say that a good calibration is done. The only strange value is or the tensile characteristic length l cr = 9 mm. Normally, this value range into 12 mm. To check i this is a result acceptable, in the next sub-section one energy analysis is done. Fracture energy control The racture energy control will be done to justiy the large value ound or the tensile characteristic length, because it is the irst time that such large values are ound. In line with this, a new test will be done. This test is a dog bone test, that is a test that allows to study the tensile behaviour. No experimental test are available, but the idea is to reproduce one specimen with my calibrate parameters value ound and to calculate the Fracture energy. This test is necessary because generally the area under the curve orce-displacement divided by the ligament represent the Fracture energy value. In my test, is not possible to done this, because orce and CMOD are not two comparable magnitude. to solve this problem, dog bone test is done. Tensile test are the best or evidence the racture energy, or this reason this is used. The dog bone specimen is generated with the same material o my calibration and later in the post processing phase the racture energy is calculate. The Fig shown the specimen. The value o the racture energy ound calculating the area under the curve, divided by the ligament is: * s 24Nmm G = = = N d A 4x6mm 81 2 lig m The objective now, is to compare this value with the value o G d that can be analytically ound using the relation that match racture energy and characteristic length, written in the irst chapter. It is:

81 Chapter 3: Numerical analysis plain concrete subject to tension and compression 2 2 N 2 9x2.28 l 3 t cr G mm d = σ = = 63 N 2E N 2x mm m This control shows that the two Fracture energy ound are comparable as value. The reason o this large change with a higher probability is due to the dimension o aggregate. In my test, the particles are arranged in the range o 1-22 mm. These are values bigger than the common values that the experience show. This will be the principle reason. F H H W Figure 3.17: Dog bone specimen geometry. In the next chapter, using this concrete calibration ound, the ibbers will be introduced

82 Chapter 3: Numerical analysis plain concrete subject to tension and compression

83 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension Chapter 4 Numerical analysis iber reinorced concrete (FRC) subject to tension Introduction The goal o this chapter is to introduce the iber into the LDPM and calibrate this new model. The constitutive law that is applied or the relations among concreteiber is explained in the second chapter o the thesis. In brevity, the methods used to achieve the objective o this study will be discussed herein. The developed model is called Lattice Discrete Particle Model or Fiber Reinorced Concrete (LDPM-F). Starting rom the results ound or the plain concrete, using 3PBT and UC, the ibers are introduced into the specimen, this can be considered the irst step. The second step is to calibrate the typical parameters o the constitutive law, ollowed by validating the results o the analysis. Figure 4.1: Steel ibers used in the experimental specimens. The two steps, calibration and validation, are done utilizing experimental data rom Buratti, Mazzotti, Savoia and Thoot (28). The tests done are 3PBT. The specimen is o the same geometry as that shown in the previous chapter, with the same plain concrete plus random distribution o iber. There are two percentages o iber distribution or the test. The specimen called SF1B_2 have.26 % o

84 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension iber and the group SF1B_35 is characterized by.45% o ibers randomly distributed into the specimen. This parameter is identiied as V. These two groups will be very important or the study, in act the irst is used or the calibration and then the second group allows the validation o the result. In Fig. 4.1 are shown the ibers used in the experimental tests. Phases o pre-calibration The pre-calibration phases are the same as those just described in the previous chapter, however the only dierence is that there is one more step. According to the order just shown, the new step is introduced ater the aggregate generation. This stage is called iber generation. Fiber generation At the global level, the distribution o iber-acet intersections is accomplished by generating a uniorm distribution o ibers throughout a simulated specimen. From a geometry point o view, physical iber can be described using a ew parameters: iber density, length, diameter, and tortuosity. The latter parameter is used to characterize the iber shape: a straight iber has no tortuosity while a iber with many kinks is very tortuous. These geometric parameters are used in MARS or generating random ibers inside a control volume, which can either be the volume o the entire specimen or a larger volume that contains the specimen. Each iber is modelled using a sequence o one or more segments linked together. Single segments are suicient or generating straight ibers. Multiple segments are necessary or generating tortuous ibers. Fiber location and orientation are assumed to be randomly distributed. All ibers are completely contained in the control volume. For cast FRC parts, the control volume should be equal to the volume o the simulated specimen. For machined parts, the control volume should be larger than the simulated specimen. The ibers that intersect the external surace o the part are treated as cut. The portion o a cut iber which lays inside the part is shorter than the length o original iber and this aects the mechanical characteristics o the iber-acet interaction near the external suraces. In the spirit o the discrete, multi-scale physical character o LDPM, the occurrences o iberacet intersection are determined by computing the actual locations where ibers cross inter-cell acets. This computation can require signiicant resources or large models in terms o both time and memory. For this reason, an eicient bin-sorting algorithm was developed or computing these intersections. For each intersection, the iber length on each side o the acet and the orientation at which the iber intersects the acet (crack plane) are computed. These parameters are saved in the acet data structure and used during the simulation or calculating the iber

85 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension contribution to the LDPM acet behaviour. It is worth noting that the uniorm distribution o ibers throughout a volume is sometimes not realistic due to the intrinsic technical diiculties o dispersing the ibers evenly in an actual concrete specimen. This issue needs to be taken into account careully when calibrating and validating the model, G.Cusatis, and co-worker (29). In these tests used steel ibers or calibrating the model. The values o the principle characteristics implemented in the code to represent and describe the ibers characteristics are: Fiber Length=5 cm Element Size=5.1 cm Fiber Section Area=.785 mm 2 Tortuosity=. Volume Fraction=.26 Elastic Modulus=21 GPa Fiber Strength=11 MPa According to these properties the specimens can be generated. The geometry is the same as that used or the plain concrete test, in other words a prismatic notched specimen with span o 5 mm. Fig. 4.2 shows the specimen, where it is possible to see the ibers presence. They are represented in blue colour and the particles are plotted in transparent way to show a better view o the ibers. F 2 mm Figure 4.2: 3PBT specimen with ibers. In this Fig. 4.2 it is possible to see that the only central part, where damage is expected to occur due to the presence o notch, is modelled thought LDPM-F, while the two lateral parts are modelled with standard elastic inite elements. All o the specimens have an out o plane thickness o 15 mm. The second step is to deine the parameters that will be calibrated to make the model usable. The tests done or calibrating the model were 3PBT

86 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension Presentation o parameters The Lattice Discrete Particle Model or Fiber Reinorced Concrete (LDPM-F) is governed by 11 parameters that allow the description o the concrete-ibers behaviour. These parameters that are now shown will be added to the other 16 meso-structure parameters that are calibrated or plain concrete. The strain and strength that are measured on the generic triangular ace o model are the combination o concrete and iber contribution. The parameters are: E = Elastic Modulus k sp = Spalling Parameter σ = Fiber Strength k rup = Fiber Strength Decay k sn = Snubbing Parameter τ = Bond Strength η = Volume Stiness Ratio G d = Debonding Fracture Energy β = Pullout Hardening k p = Plastic Parameter =Cook Gordon Parameter k cg Model sensibility The LDPM-F allows reproducing the concrete-iber behaviour. The ibers are designed to work when the cracks initiate, or this reason their eects on tests characterized by tension are more important compared to tests characterized by compression and shear. Accordingly, only this kind o tests are reproduced. The calibration phase or concrete-iber specimens starts by conducting a sensibility analysis o the model. This is the irst work done with this model, since there are no available results or examples rom the literature. In the sensibility analysis a speciic study over the parameters was done. For each parameter three numerical tests were done by changing the number o the parameters analyzed. Clearly the other parameters excluding Debonding Fracture Energy G d, Bond Strength τ, Elastic Modulus E and Fiber Strength σ, are put equal to zero. The results ound are presented in the next series o igures

87 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension These igures 4.3 and 4.4 show how the curve or 3PBT change in according to the dierent values o the parameters. In the second chapter only one sensibility analysis was made or the only one iber pullout case, but it can not take completely the big test in account, in which the number o ibers is bigger than single pullout and also the orce which is applied in dierent ways. The results ound shows that in some cases, or example or the Pullout Hardening β the model response is almost the same as that which was ound in the single iber analysis. Worth noting is that it is possible to divide these parameters in two groups, according to the relative importance that they have in the calibration process. The irst group is ormed by Debonding Fracture Energy G, Pullout Hardening β, Bond Strength τ, Fiber strength σ and Elastic Modulus sensibility o these parameters is obvious rom the ollowing igures. d E. The Debonding racture energy [G d ] Fiber strength [σ ] F[KN] 15 1 G d =2 N/m G d =4 N/m G d =8 N/m F[KN] 15 1 σ =13 MPa σ =18 MPa σ =23 MPa CMOD[mm] CMOD[mm] Bond strength [τ ] Pullout hardening [β] F[KN] 15 1 τ =2 MPa τ =5 MPa τ =8 MPa F[KN] 15 1 β=.5 β=1.5 β= CMOD[mm] CMOD[mm] Figure 4.3: Parametric analysis o the principle parameters The second step is done by the remaining parameters. The aim is to work over parameters that allow the calibration the model or the 3PBT, and also or other kinds o tests to achieve a calibration that permit to reproduce the behaviour or every kind o steel ibers

88 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension Following this target, the work will be developed using values o parameters, that allow to it also tensile tests. Parameters like Volume Stiness Ratio η and Cook Gordon Parameter k cg are assumed to be equal to zero, because their eect can be considered negligible. Following are shown the remaining parameters which govern the iber-concrete behaviour, Fig Taking in consideration all the results rom this sensibility analysis, the calibration was done. Cook-gordon parameter [k cg ] Spalling parameter [k sp ] 15 k cg =1 15 k sp =1 MPa F[KN] 1 k cg =3 k cg =6 F[KN] 1 k sp =2 MPa k sp =3 MPa CMOD[mm] Fiber strength decay [k rup ] CMOD[mm] Snubbing parameter [k sn ] 15 k rup =.5 15 k sn =.3 F[KN] 1 k rup =1.5 k rup =3. F[KN] 1 k sn =.6 k sn = CMOD[mm] CMOD[mm] Figure 4.4: Parametric analysis o the remaining parameters

89 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension Calibration stage The experimental data presents two dierent group o specimens characterized by dierent percentages o ibers dispersed into the specimens, this parameter is called V. The SF1B_2 is used or the calibration. This group presents V =.26 % o iber and seven experimental tests are done. The table 4.1 shows the number o particles and tetrahedron present in the dierent specimens. Specimen Particles Tetrahedrons A B C D E F G Table 4.1: 3PBT single notch specimens with V =.26% o ibers. According to the approach used in the concrete calibration stage, one average curve will be generated to it the start the calibration. The experimental tests show a certain scatter among the seven curves, which is very big. Two causes can be attributed to this scatter: the heterogeneity o the concrete and the dierent position that the ibers have in the specimen. For example one specimen that present more iber in the central section allows to have more stress than another specimen where the iber are close to the support. The iber distribution is another important random eect that needs to be considered or this material. The model was built taking into account these two eects. For simulating real situations, the MARS is equipped with two dierent seeds. One that is positioned in the Gen.mrs allows the reproduction o the heterogenic structure o the concrete, and the other located in the Run.mrs allows to change the random distribution o the ibers. By changing the seeds the randomness algorithm change the position o the particles and the ibers. The two iles, nominated beore, are the typical input iles used in the numerical analysis. Starting rom the average experimental curve the calibration begins by using a generic seed or the Gen.mrs and Run.mrs iles. The idea is to initially look at how the model responds and ater that start changing the parameters, deined in the previous section, to identiy the most important ones. When a good calibration is ound the second phase starts

90 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension In the second step, the idea is to deine a general study and to be able to show the heterogenic structure o the concrete and also the random distributions o ibers. Accordingly, utilizing the parameters values rom the previous step, seven tests were run with dierent seed values. Clearly both the seeds in Run.mrs and in Gen.mrs were changed. The curves ound are averaged and then compared with the experimental average curve Three Point Bending Test, V =.26% Experimental Experimental Average Numerical Numerical Average 12 F [KN] CMOD[mm] Figure 4.5: Comparison between numerical and experimental results, or 3PBT with V =.26% o ibers. The agreement between the numerical results and the experimental data is very good. The values ound or the parameters are: Elastic Modulus [ E ]=21 GPa; Spalling Parameter[ k sp ]=15 MPa; Fiber Strength[ σ ]=11 MPa; Fiber Strength Decay[ k rup ]=.; Snubbing Parameter[ k sn ]=.5; Bond Strength[ τ ]=6. MPa; Volume Stiness Ratio[η ]=.;

91 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension Debonding Fracture Energy[ G d ]=5.; Pull Out Hardening[ β ]=.; Plastic Parameter[ k p ]=1.; Cook Gordon Parameter[ k cg ]=2.; Figure 4.6: Comparison between o the damage o experimental and numerical beams subject to 3PBT

92 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension Validation stage In this stage the validation is done. The experimental group characterized by.45 % V is used to check i the model works. The idea is to have new numerical test using the same parameters set and changing only the iber percentage distribution to it the experimental curve. Accordingly, seven specimen were run. Specimen Particles Tetrahedrons A B C D E F G Table 4.2: 3PBT single notch specimens with V =.45% o ibers Three Point Bending Test, V =.45% Experimental Experimental Average Numerical Numerical Average 12 F [KN] CMOD[mm] Figure 4.7: Comparison between numerical and experimental results, or 3PBT with V =.45% o ibers

93 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension The results obtained in this case produce a post-peak load carrying capacity higher than the experimental one. For a CMOD o.8 mm the dierence between the average numerical and experimental curves is about 45%. In order to clariy this apparent discrepancy, it is useul to compare the actual number o ibers bridging the crack in the experimental and in the numerical simulations. During the experimental campaign, post-mortem evaluation was perormed and the number o ibers on the crack surace was counted or all specimens o both FRC mixes. For the experimental test, the number o ibers in each third o the ligament length (i.e. the reduced cross-section with associated notch, as shown in Fig. 4.8). The dierent areas identiied in this way are named bottom, middle and top, respectively. Deining them is very important to understand the ibers response. The same inormation was clearly extracted rom the numerical simulations. V =.45% CMOD =.8mm ligament = 125 mm 125/3 125/3 125/3 Figure 4.8: Crack opening isosuraces. To study the CMOD-Force in the ibers case, another approach will be used. The idea is to analyze the increment o load due to the ibers eect and inds a relation among this increment and the position o ibers in terms o bottom, middle or top area

94 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension The new parameter introduced is the load increment, it is expressed by this relation: L i = 1( 1 P P ), where P is the load or the FRC specimens and P is the average load or the plain concrete specimens. Accordingly, plots o the load increment versus the number o ibers considering the bottom, bottom-middle and total regions were done. The results shows that this relation is very good or only the bottom region and also i the bottom-middle area is considered. Following this results the only two-thirds o the ligament length is plotted. This choice is motivated by analyzing the crack pattern in the numerical simulations. Fig. 4.8 shows a gray-scale isosuraces rom a typical mesoscale crack opening. Each surace corresponds to a dierent crack opening value. The minimum value that was plotted (very light gray) corresponds to a crack opening magnitude o 2 v d. This corresponds to a situation at which a straight iber, orthogonal to the crack surace, would experience ull debonding (onset o the rictional pull-out phase). At this level o crack opening, ibers can be considered ully active. As it can be seen rom the Fig. 4.8, the zone characterized by ully active ibers extends to approximately two-thirds o the ligament length above the notch. This indicates that the results are not perect i the total section is considered. To highlight this inding, the relation among Number o Fibers and Load Increment due to ibers is plotted in Fig In this igure the Load Increment Due to Fiber is plotted or CMOD =.8mm. Load Increment Due to Fibers [%] b) Three Point Bending Test CMOD =.8 mm V =.26 % Exp V =.26 % Num V =.45 % Num = V =.45 % Exp Number o Fibers Figure 4.9: Load increment due to iber versus number o active ibers

95 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension This Fig shows clearly that there is a linear relationship between the load and number o iber detected, and that both the experimental data and the numerical results have a similar trend. Looking with depth to the igure, it is possible to note that one inconsistency is present. This is matched with the number o the detected ibers rom the experimental specimens. For the numerical specimens, the range o detected ibers increases rom (22-39) to (47-7) as the reported iber volume raction is nearly doubles rom V =.26% to V =.45%. In the experimental tests, the increase in the number o detected ibers rom (17-39) to (17-5) is not consistent with the near doubling o V. For the higher V the numerically simulated specimens eature more active ibers than the experimental ones. This explains the discrepancy in the curves shown in the Fig. 4.7, while conirming the validity o the iber-matrix interaction strategy adopted in this study. The dierent between the number o active ibers within the experimental and numerical crack ligament is due to the act that in the numerical model a uniorm iber distribution was obtained, while such uniormity apparently was not obtained in the experimental specimens with higher V. Based on the act that iber dispersion can be signiicant in terms o load carrying capacity, LDPM-F should be extended to include non-uniorm iber distribution and, in addition, experimental data should be provided and more inormation about the actual iber distribution obtained during specimen casting should be provided, G.Cusatis and co-worker(29)

96 Chapter 4: Numerical analysis iber reinorced concrete (FRC) subject to tension

97 Chapter 5: Numerical analysis o projectile penetration or FRC slabs Chapter 5 Numerical analysis o projectile penetration or FRC slabs Introduction The goal o this chapter is to investigate the dynamic loads exerted on plain concrete slabs and FRC slabs by impacting objects. Examples o these problems include icebergs impacting loating concrete drilling platorms in the North Atlantic, aircrat crashing into concrete structure, missile impact. For concrete under quasi-static loading conditions, the ailure mechanisms and constitutive relations or concrete under impact and high strain rate loading are not well known yet; partly because o the experimental and analytical diiculties involved, also relatively ew studies o these relationships have been carried out under impact loading (Midness 1993). Considering the experimental tests described in Hanchak (1992), where impact tests where simulated using steel bullet, the numerical analysis in this study was developed. The objective o this study is to investigate the response o FRC slabs under impact loading, and the eect o the slab thickness, the striking velocity o the bullet, and the percentage ibers in the concrete mix on the slab response. The results will be presented in terms o the bullet striking velocity and residual velocity just ater exiting the slab. The approach used herein to simulate these tests is to start by investigating the response o plain concrete slabs and to check i the model yields the same behaviour or experimental tests. The slabs were subjected to dierent shots to understand the controlling actors o the response. FRC slabs were then used and a parametric study was done to investigate the dierent parameters aecting the slab response. Thus, the ballistic limit is the undamental parameter that will be monitored to understand how the slab response change accordingly, or each test

98 Chapter 5: Numerical analysis o projectile penetration or FRC slabs Geometry o the test In order to simulate the impact test on the slab, two elements will be generated utilizing MARS; the slab and bullet elements. In this paragraph the geometry and the techniques used in generating both elements will be discussed. The slab elements are 58 mm squares with two dierent thicknesses, t =5.8 mm and 11.6 mm. The slab is modelled using LDPM-F assigned the material parameters previously described. Fig. 5.1 shows the layout o the slab element t Figure 5.1: Slab geometry realized by LDPM and LDPM-F The bullet is simulated in accordance with the geometry and material properties described in the Hanchak (1992); The bullets were implemented in the numerical model as a rigid body where its steel properties were assigned according to Vascomax t-25 Maranging Steel. The geometry o the problem was chosen such that the bullet was placed at 25.4 mm rom the slab, where the shot is positioned. Basically, this parameter doesn t aect the numerical test results, but it was ound a reasonable value with respect to the computation time. The Fig. 5.2 shows the bullet geometry

99 Chapter 5: Numerical analysis o projectile penetration or FRC slabs Figure 5.2: Bullet geometry, realized with inite element. Starting with this coniguration the simulations were perormed. Penetration test o concrete slab Plain concrete slabs with two dierent thicknesses were subjected to penetration tests under the bullet impact loading. The slab was subjected to impact velocity o up to 45 m/sec. This range o impact velocity was chosen in order to check how the ballistic limit, expressed as the highest strike velocity associated with zero residual velocity, increases with increasing t. The irst step was to check that the Vs Vr curve presents the same trend obtained rom experimental results. To asses this comparison, dierent shots were done using dierent strike velocity V s. For each velocity six numerical analyses were run by changing the seed parameters according to the reproduced heterogenic structure o the concrete. This approach allows to reduce the local eects caused by the internal structure o the concrete. The points used to create the Vs Vr curve were obtained by averaging the values rom the six point, previously discussed. The table 5.1 gives the bullets velocities used in the dierent shots. These lists o velocities undamentally are chosen considering that the curve trend close to the ballistic limit is nonlinear. For this reason, large number o shots are selected in this zone. This region is identiied as: velocity o ballistic limit-v s = 15 m/sec. However, it is important to highlight that this range is valid only or plain concrete slabs. The ollowing table shows how the residual velocity change with respect to the slab thickness, where the ballistic limit is linked to. Based on these data, the ballistic limit was ound to increase with increasing t. For plain concrete slabs with t = 11.6 mm the ballistic limit was ound to be approximately 95% higher than that or slabs with t = 5.8 mm