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, Crack in a fiber-reinforced concrete specimen at the end of a three-point bending test, with fibers bridging the notch
, softening (red): (a) stress-strain diagram for the bulk material (elasticity + hardening phase + elastic unloading); (b) tractionseparation cohesive law at the discontinuity (softening phase); (c) microcracks (blue) and macro-crack (red) in a specimen, Crack development in concrete -three phases of material behavior: elasticity (black), hardening (blue)
, 12 2.3 Standard shape functions and incompatible shape function with its derivative for a truss bar
, 13 2.5 Example of the incompatible shape function M for for a CST element when the discontinuity passes through the middle of the element: (a) 3D view; (b) front view
, 21 2.7 Traction at the discontinuity vs. crack opening (displacement jump): (a) normal direction; (b) tangential direction, Fracture energy G f represented by the grey area under the softening part of the response
, Six specimens containing a fiber: three with 1/4 embedded length, and three with 1/2 embedded length
, 31 2.10 Different embedded lengths for the fiber: l fe 7.5 mm (one quarter of the fiber length), and l fe 15 mm (one half of the fiber length), Single-fiber pull-out tests: (a) specimen with embedded fiber; (b) experimental setup, p.31
, Results for the single-fiber pull-out tests: (a) specimens with 1/4 embedded length; (b) specimens with 1/2 embedded length
, Results of single-fiber pull-out tests with different embedded lengths, p.32
, Fiber at the end of the analysis, where the straightening of the hook is visible on the pull-out side
, 33 2.15 Linear bond-slip law for the complete pull-out of the fiber, p.36
, Exponential bond-slip law for the complete pull-out of the fiber, p.36
, Non-conforming mesh: (a) a randomly oriented fiber in the domain; (b) degrees of freedom of a fully enriched element
, 42 3.3 Two phases of the composite behavior: (a) uncracked state; (b) cracked state. 42 3.4 Comparison of total computational time for a monolithic and partitioned approach
, Algorithm flow-chart for the multi-scale framework
, Geometry of the reinforced concrete specimen in 2D, p.52
, Finite element mesh with enriched elements shown in grey, p.53
, Force-displacement diagram for the tension test on a reinforced concrete specimen
, 54 3.10 Bond stress: (a) distribution of ? bs along the reinforcement bar (values at Gauss points); (b) evolution of ? bs in time for all enriched elements, Distribution of slip ? bs along the reinforcement bar at the end of the analysis (nodal values), p.54
, Plastic slip: (a) distribution of ? bs,p along the reinforcement bar at the end of the analysis (values at Gauss points); (b) evolution of ? bs,p in time for the enriched elements that have entered the plastic phase, p.55
, Crack opening in concrete: (a) distribution of ? c along the reinforcement bar at the end of the analysis (values at Gauss points); (b) evolution of ? c in time for the cracked element in the middle of the specimen, p.56
, Comparison of the evolution in time of ? c and ? bs : the crack opening in concrete in the middle of the element is nearly equal to the sum of the absolute values of slip in the left and right node of the same element, p.56
, Linear elastic analysis for the domain with a medium fiber (l f 2 mm): (a) Finite element mesh with enriched elements shown in black ; (b) Contour plot of displacements in x-direction at the end of the analysis, p.58
, Force-displacement diagram for the linear elastic behavior of the matrix material, for different fiber lengths
, 59 3.18 Damage analysis for the domain with a medium fiber (l f 2 mm): (a) finite element mesh with enriched elements shown in black and weakened elements shown in grey; (b) contour plot of displacements in x-direction at the end of the analysis, where the cracks are shown with red lines, p.60
, Force-displacement curves for the damage case for different fiber lengths, p.60
, Slip distribution along the fiber for different fiber lengths, p.61
, Force-displacement curve with and without bond-slip for the medium fiber, p.61
, Stress distribution in concrete in x-direction: (a) with bond-slip; (b) without bond-slip
, Influence of the choice of the fixed node: (a) on the global level (forcedisplacement diagram); (b) on the local level of the fiber (slip distribution), p.62
, Bond stress vs. bond strain for different values of the parameter ? bs, p.63
, Bond stress along the fiber for different values of the tangent modulus ? bs, vol.63
, Bond stress and fiber stress along the fiber at the end of the analysis for the example with medium fiber and ? bs 200
, Evolution of bond stress in time for three elements on the left, p.64
, Position of the weak zone in the specimen (a) completely on the left -element 1; (b) in the middle-left -element 3; (c) in the middle-right -element 6; (d) completely on the right -element 8
, Force-displacement diagrams for different crack locations, p.65
, 66 3.31 Mesh refinement study for three different meshes (128, 512 and 2048 elements): (a) force-displacement diagrams; (b) slip distribution along the fiber; (c) bond stress along the fiber; (d) convergence: bond stress at the left end of the fiber plotted against the number of elements in the mesh, Slip along the fiber for different crack locations, p.66
, The fiber bridging the notch
, Three-point bending test [41]: (a) specimen; (b) experimental setup, p.68
, Results of the three point bending test [41]: load-displacement curve for three specimens
, 69 3.36 Mesh and boundary conditions for the modeled specimen, where the notch is represented with grey elements, and the enriched elements containing fiber are shown in black
, Results of the numerical simulation for the three-point bending test, p.70
, 71 3.39 Numerical simulation of the three-point bending test -distribution along the fiber at the end of the analysis: (a) slip; (b) bond stress, Three point bending test with a fiber bridging the notch, p.71
, A truss with a damaged bar (shown in green)
, Displacements of the truss bar, with measurement points shown with red dots, p.77
, Beam with measurement points shown in red
, Upper diagonals (yellow) and lower diagonals (green) in a truss structure, p.80
, Error minimization for different combinations of diagonal stiffness, p.81
, Poisson distribution with mean ¯ x 3
, Distribution of n 1000 fibers inside a square domain: (a) fiber midpoints; (b) whole fibers
, Square domain with fibers: (a) subdivision of square in zones; (b) number of points in each zone
, Sample with n 10000 points, and 2500 zones
, Plot of variance vs. mean in logarithmic scale for different zone sizes, p.85
, Plot of variance of number of points vs. zone size
, 86 4.16 Distribution of fibers in a concrete specimen of dimensions 40x10x10 cm: (a) fiber midpoints; (b) whole fibers, Distribution of fibers in 3D: (a) fiber midpoints; (b) whole fibers, vol.87
, X-ray image of a fiber-reinforced concrete specimen [68], p.88
, Sorted fiber lengths and their subdivision in bins
, Comparison of the Gauss distribution (blue) and a histogram obtained from order statistics (red), for 50 bins
, Length based model: (a) individual constitutive behavior for three fibers; (b) force-displacement diagram obtained from the contribution of all the fibers' responses
,
, Stiffness based model: (a) individual constitutive behavior for three fibers; (b) force-displacement diagram obtained from the contribution of all the fibers' responses
, Results of the parameter identification procedure, for different values of the damage threshold
, (a) different stiffness variance; (b) different force damage threshold; (c) different stiffness mean