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Chapter 8. Elasto-plastic microplane formulation for FRCC0.3 and σ N ,C AP0 = −1.0 MPa; the residual strengths χ mi cr= 0.0 MPa, c mi cr= 0.0 MPa,tan(φ mi cr ) = 0.1 and σ N ,C APr = −0.1 MPa; and all the shape coefficients α pi = 0.0.The expressions, defining the rate of plastic work ẇ cr , are given by means of thefollowing expressionsẇ cr = σ N · ˙ε p N · l cs I + σ T,k · ˙ε p T,k · l cs I I if σ N < σ N ,C AP or σ N ≥ 0()ẇ cr = σ T,k · ˙ε p T,k · l csI I 1 − |σ N |t an(φ mi c )‖σ T ‖if σ N ,C AP ≤ σ N < 0(8.23)in which the strain components in each microplane, ε mi c = [ ε N ,ε T1 ,ε T2] t , can be easilytransformed in terms of crack-opening displacement through the following expressionsε N = ul I csε T1 = v 1l I I acsε T2 = v 2l I I acs(8.24)where u, v 1 and v 2 are the relative crack displacements, while lcs I and l csI I a are homogenizationcharacteristic lengths to be calibrated for the considered failure modes. Theymainly represent the crack spacings in direct tension, under shear stress states withvery high confinement, or the damaged zone under compressive compactions.In principle, the adopted inelastic model, its evolution laws and the flow rule representationsare similar to that proposed in the previous chapters for discrete crack analysesat material and mesoscale observation level.8.5 Crack-bridging effect of fibersFiber actions deriving from fracture opening processes in concrete, and schematicallyproposed in Figure 8.7, take relevant bridging effects on the FRCC constitutive response.On the one hand the bond mechanism between fibers and concrete matrix is takeninto account as an axial (tensile) stress acting at the microplanes σ f . It can be derivedby equilibrium conditions of shear stresses reacting on the lateral contact surfaces offibers embedded within the concrete matrix as outlined in Figure 8.7. On the otherhand the dowel action, V d of Figure 8.7 as result of the shear transfer mechanism,represents another important component on the overall bridging effect of steel fibersin fracture processes of FRCC. A simple analytical model for reproducing the dowelaction of fibers crossing cracks is developed. It is based upon the definition of bothstiffness and strength of a generic fiber embedded in the concrete matrix and subjectedto a transverse force vs. displacement behavior.164
8.6. Numerical analyses about the material failureaxial strainnε Nsingle fiberaxial actiondowel strengthx=0dowel strainτ aσ f + dσ fx=0 Tτ al f σ fτ adxV dFigure 8.7: Microplane strain activating the fiber bridging effects.For the sake of brevity, many details of the model derivations are not provided in thisChapter. The complete formulations for both the bond-slip and dowel action hascompletely been detailed in Chapters 3 and 4.8.6 Numerical analyses about the material failureThis section presents the main features and capabilities of the proposed formulationscomparing some numerical results against experimental data available in literature.In order to evaluate the numerical stability of the aforementioned formulation, thefollowing simulation cases are analyzed in this section:• uniaxial tension, and• simple shear.Particularly, the schematized test cases reported in Figure 8.8, considering a 1.0cmedgecubic sample under plane stress condition, are considered.8.6.1 Tensile testsComparison with experimental dataNumerical predictions carried out in both plain and Steel Fiber Reinforced Concrete(SFRC), by adopting the proposed microplane formulation for analyzing tensile testcases, are presented in this section. The comparisons between model predictions165
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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
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Chapter 1. Introduction• "c" if 0
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Chapter 1. Introductionsumed due to
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Chapter 1. Introductionaleatoric na
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Chapter 1. IntroductionFigure 1.13:
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Chapter 1. IntroductionElement Meth
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Chapter 2. Experimental characteriz
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Chapter 3. Zero-thickness interface
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4 Bond behavior of fibers in cement
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4.2. Bond behavior of fibers in con
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4.3. Elasto-plastic joint/interface
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4.4. Fracture energy-based interfac
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4.4. Fracture energy-based interfac
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4.5. Numerical results and experime
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4.6. Closing remarksMPa3.02.5Pi N2.
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5 Model performance and numericalpr
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5.1. Numerical analysesones mention
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5.1. Numerical analysesshown in Fig
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5.1. Numerical analysesconsidered,
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5.1. Numerical analysestic behavior
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5.1. Numerical analysesσ MPa121086
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5.2. Cracking analysis of the propo
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4 03 02 01 001 0 09 08 07 06 05 04
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Page 206 and 207: BibliographyN. Banthia and N. Nanda
Page 208 and 209: BibliographyM. Butler, V. Mechtcher
Page 210 and 211: BibliographyCUR. Bepaling van de Bu
Page 212 and 213: BibliographyEN-12390-3. Testing of
Page 214 and 215: BibliographyT. Guttema. Ein Beitrag
Page 216 and 217: BibliographyE. Kuhl, P. Steinmann,
Page 218 and 219: BibliographyO. Manzoli, J. Oliver,
Page 220 and 221: BibliographyK. Park, G. Paulino, an
Page 222 and 223: BibliographyI. Singh, B. Mishra, an
Page 224: BibliographyG. Wells and L. Sluys.
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