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Chapter 4. Bond behavior of fibers in cementitious materials: a unifiedformulationFigure 4.1: Considered scheme of fiber under pull-out loading.being ˙σ f the axial stress rate of the fiber, ˙ε f the rate axial strain and the tangent elastoplasticmodulus E ep takes the following two distinct valuesf{EepfE epf= E f el asti c/unloadi ng r eg i me= 0 l oadi ng wi th λ ˙ f > 0(4.2)where ˙λ f is the non-negative plastic multiplier obtained by means of the Kuhn-Tuckerand consistency conditions and E f represents the uniaxial elastic module of the fiber.Finally, the yielding criterion takes the following expressionf f = |σ f | − σ y,f ≤ 0 (4.3)in which σ y,f ≥ 0 is the yield limit of the steel.4.2.2 Interface bond-slip modelsTwo rate-independent contact laws are proposed in this work with the aim to study thefiber-to-concrete debonding:• Elasto-plastic model with linear strain-softening: the same one adopted in thecomposite model outlined in Chapter 3.• Fracture energy-based debonding model: a richer proposal based on fractureenergy-based concepts and conceived within a work-softening plasticity formulation;70
4.3. Elasto-plastic joint/interface model with isotropic linear softeningTable 4.1 describes the key aspects of both models. In particular, f (τ,κ) and g (τ,κ)represent the two yielding criteria based on the interface shear stress τ and the internal(strain-like) variable κ of each considered model; τ y represents the failure shearstrength.Table 4.1: Interface bond-slip models.Fracture energy-based modelStrain-softening elasto-plasticityLoading criterion f (τ,κ) = τ 2 − τ 2 y ≤ 0 g (τ,κ) = |τ| − (τ y,0 +Q) ≤ 0)Stress-like internal variables τ y = τ y,0(1 − w slG˙κ = ẇ f sl = τ · ṡ p˙κ = ˙Q = ˙λ · k HPlastic flowṡ p ∂f= ˙λ∂τ = 2 · ˙λ · τConstitutive equation ˙τ = k E (ṡ − ṡ p )ṡ p ∂g= ˙λ∂τ = ˙λ · si g n[τ]Loading-unloading condition ˙λ ≥ 0, f ≤ 0, ˙λ · f = 0 ˙λ ≥ 0, g ≤ 0, ˙λ · g = 0Constitutive tangent operator⎛k ept an = k E ,2 · ⎝1 −( ) ∂f 2+∆λ ∂f∂τ( ∂f∂τ˙τ = k ept an · ṡ( ∂f ∂κ) ⎞∂τ ∂κ ∂sp · ∂2 f) ∂τ2⎠2+H/kE,2k ept an = k Eel asti c/unloadi ngk ep (t an = k E 1 − k )Ek E +k HBoth bond-slip models can be directly implemented in plasticity-type constitutive lawsfor interface elements. The rate of elastic relative slip, ṡ e , is introduced and related tothe shear stress through the elastic stiffness, k E . In the framework of the incrementalplasticity theory, the following basic equation can be usedṡ = ṡ e + ṡ pṡ e = ˙τk E(4.4)where the inelastic, ṡ p , and the total interface slip ṡ are introduced in rate form.Integrating each constitutive model, the constitutive laws can be defined in terms ofthe tangent elasto-plastic constitutive operator, k ept an, which is specified for loading orunloading/elastic processes in Table 4.1.4.3 Elasto-plastic joint/interface model with isotropic linearsofteningThis section presents a classical one-dimensional plasticity model aimed at simulatingthe bond-slip behavior of fiber-to-concrete interface.71
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Dottorato di Ricercain Ingegneria d
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AcknowledgementsI would like to exp
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Abstractinclusion of fibers and the
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ContentsAcknowledgementsAbstractTab
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Contents8.6.1 Tensile tests . . . .
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List of Figures2.1 Fine and coarse
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List of Figures4.16 Pull-out simula
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List of Figures6.14 Crack paths of
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List of Tables2.1 Mix design per cu
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Chapter 1. Introduction1.1.1 Fiber
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Chapter 1. Introductionthe fiber ad
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Chapter 1. Introductionand mechanic
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Chapter 1. Introduction[Vrech and E
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Chapter 1. IntroductionDespite the
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Chapter 1. Introduction(Figure 1.4)
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Chapter 1. IntroductionFigure 1.8:
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Chapter 1. Introductionfor structur
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Chapter 1. Introductioncracking beh
Page 46 and 47: Chapter 1. Introduction• "c" if 0
Page 48 and 49: Chapter 1. Introductionsumed due to
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Page 52 and 53: Chapter 1. IntroductionFigure 1.13:
Page 54 and 55: Chapter 1. IntroductionElement Meth
Page 56 and 57: Chapter 2. Experimental characteriz
Page 70 and 71: Chapter 3. Zero-thickness interface
Page 93 and 94: 4 Bond behavior of fibers in cement
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Page 99 and 100: 4.3. Elasto-plastic joint/interface
Page 107 and 108: 4.4. Fracture energy-based interfac
Page 109 and 110: 4.4. Fracture energy-based interfac
Page 111 and 112: 4.5. Numerical results and experime
Page 117 and 118: 4.6. Closing remarksMPa3.02.5Pi N2.
Page 119 and 120: 5 Model performance and numericalpr
Page 121 and 122: 5.1. Numerical analysesones mention
Page 123 and 124: 5.1. Numerical analysesshown in Fig
Page 125 and 126: 5.1. Numerical analysesconsidered,
Page 127 and 128: 5.1. Numerical analysestic behavior
Page 129 and 130: 5.1. Numerical analysesσ MPa121086
Page 131 and 132: 5.2. Cracking analysis of the propo
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Page 139: 5.3. Closing remarksconnecting node
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Chapter 6. Structural scale failure
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7 Cracked hinge numerical modelfor
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7.1. Basic assumptionsSection at no
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7.1. Basic assumptions1997, Stankow
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7.2. Bond-slip bridging of fibers o
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7.4. Numerical predictionsvertical
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7.5. Closing remarksthe experimenta
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Chapter 8. Elasto-plastic microplan
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Chapter 9. Conclusionsof first-crac
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Chapter 9. Conclusionsfibers in con
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Chapter 9. Conclusions9.6 Future re
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BibliographyN. Banthia and N. Nanda
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BibliographyM. Butler, V. Mechtcher
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BibliographyCUR. Bepaling van de Bu
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BibliographyEN-12390-3. Testing of
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BibliographyT. Guttema. Ein Beitrag
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BibliographyE. Kuhl, P. Steinmann,
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BibliographyO. Manzoli, J. Oliver,
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BibliographyK. Park, G. Paulino, an
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BibliographyI. Singh, B. Mishra, an
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BibliographyG. Wells and L. Sluys.
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