Mesoscopic simulations of a fracture process in reinforced concrete beam in bending using a 2D coupled DEM/ micro-CT approach

In this study, a complex fracture process in a short rectangular concrete (RC) beam reinforced by one longitudinal bar (without vertical reinforcement) and subjected to quasi-static three-point bending was numerically explored in 2D conditions. A critical diagonal shear crack in the beam caused it to fail during the experiment. The numerical simulations were conducted with a classical particle discrete element method (DEM). A three-phase concrete description (aggregate, mortar, and interfacial transitional zones (ITZs) around aggregates) accounted for the concrete heterogeneity. In mesoscopic DEM calculations based on a 2D X-ray CT scan, the actual shape and placement of aggregate particles in concrete were taken for granted. In the study, the steel bar with ribs was replicated. ITZ was also assumed between the bar and mortar. Without imposing any bond-slip law, a geometrical bar/concrete interface condition was explicitly considered. The focus was on the force – deflection diagram, fracture process, contact forces, and stresses along the bar. A good level of agreement about the evolution of the vertical force versus the deflection and failure mechanism was attained between DEM analyses and in-house laboratory tests despite simplified 2D conditions. A strong effect of concrete mesostructure on the crack pattern was found.


Introduction
Concrete is the most often used composite building material in the world in terms of volume because it is simple to make and has the lowest cost-to-strength ratio when compared to other building materials.It has a high compressive strength despite possessing low tensile strength and ductility.Low tensile strength makes concrete materials susceptible to cracks under static and dynamic loads.Cracks are a fundamental phenomenon in concrete [1][2][3][4].A good understanding of crack formation and propagation is crucial for the assessment of structural optimization and concrete safety.The heterogeneous concrete structure, which can range in length from a few nanometers (hydrated cement) to millimeters (aggregate particles), greatly influences a fracture process.The aggregate size, shape, roughness, distribution, and volume, as well as mortar porosity, are the main determining elements [3][4][5][6][7].Concrete can be divided into mortar, aggregate, interfacial transition zones (ITZs), and macro-voids when analyzed at the mesoscale [5].Porous ITZs around aggregates at a specific distance of 0.015-0.100mm [8][9][10] are substantially weaker than aggregate and mortar and have higher permeability, diffusivity, and conductivity [11][12][13].Thus, they operate as attractors for the growth of microcracks and should therefore be modeled at the mesoscale to accurately represent a fracture process [14][15][16].Mesoscale modeling has the advantage that it E-mail addresses: micnitka@pg.edu.pl(M.Nitka), tejchmk@pg.edu.pl(J.Tejchman).

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popular discrete approach is the classical particle discrete element method (DEM) [27,29,30,[32][33][34][35].The other discrete models are interface element models with constitutive laws based on non-linear fracture mechanics [26], lattice methods [25,28,37], rigid-bodyspring models [38-40], and lattice discrete particle approaches [41][42][43].One of its drawbacks is that mesoscopic models are difficult to apply to industrial-scale engineering problems due to the long computation times resulting from a large number of particles.As a result, mesoscopic models cannot take the place of macroscopic continuum models in numerical analyses and should not be directly applied to calculations involving large concrete and RC elements.The significant results of those mesoscale simulations may help to enhance macroscale continuum models since they are more realistic in fracture process simulations.
The mechanical behavior of RC structures is significantly influenced by the bond strength between reinforcement and concrete.The stress transfer between those two materials is caused mainly by adhesion, mechanical interaction, and friction.The steel reinforcement roughness, aggregate size, concrete cover, concrete's compressive strength, bar yield stress, bar diameter, and lateral confinement have the most effects on RC elements' bond strength [3,[44][45][46].In the literature, several bond-slip rules between concrete and reinforcement were suggested, based on laboratory experiments (e.g.[47][48][49][50]).In general, they are reliant on the boundary conditions of the entire system.Thus, there are no universal slip-bond laws for RC elements.In addition, it is impossible to investigate the mesomechanical failure mechanism of the bond between steel bars and concrete with the use of slip-bond laws.
The research work being discussed is numerically oriented and concerns a simply supported short reinforced concrete (RC) beam with one steel ribbed bar under 3-point bending that was earlier tested at the laboratory scale [51].The beam was designed to fail due to shear because it was over-reinforced, had a low depth-to-shear span ratio, and lacked vertical shear reinforcement.The crack pattern included one critical diagonal shear crack with some secondary cracks.The force-deflection curve exhibited both softening and rehardening.The numerical simulations were carried out at the mesoscale using the classical particle discrete element method (DEM).The beam material was modeled as an assembly of individual particles that could move independently within the domain and interact.We chose DEM over other discrete models because it has the potential to accurately depict a concrete fracture process [5][6][7]33,34].Additionally, it accurately represents material mesostructure and contact forces by using straightforward dynamic equations.Particularly, the DEM model offers incredibly accurate fracture findings when the mesostructure is taken into account in 3D simulations based on micro-CT images [5,7,34].Three different phases were distinguished in concrete in the DEM analyses that covered aggregate, mortar, and interfacial transitional zones (ITZs) adjacent to aggregate particles.Based on X-ray CT scans, the actual shape and placement of aggregate particles in concrete were taken for granted.To simulate aggregates, a clumped-particle geometry was employed.ITZs were assumed to be 0-thick.The use of such a simple representation of ITZ is usually sufficient in calculations [5,7] although the assumption of ITZs of a defined width provides more accurate outcomes [6].In simulations, the bar/concrete interface was explicitly reproduced.The steel bar was modeled by taking directly its ribs into account.Moreover, ITZ was added between the mortar and steel [52].Thus, DEM simulations for the RC beam used a 5-phase material.In the computations, the force-deflection curve, fracture process, contact forces, and stresses along the steel bar were carefully studied.The preceding experiment [51] and the 2D numerical results (load-deflection diagram and fracture formation) were directly compared.Furthermore, the DEM outcomes utilizing the real mesostructure were juxtaposed with those utilizing the artificial one.
The motivation for the current numerical mesoscopic simulations were two interesting phenomena occurring during the laboratory test [51]: 1) the occurrence of the rehardening phase during deformation and 2) a complex pattern of secondary cracks.The first phenomenon cannot be reproduced by macroscopic continuum models [53] without the artificial assumption of material rehardening.The second one requires the consideration of internal imperfections by the assumption of the actual concrete mesostructure and the shape of the steel bar.
In the current study, the novel research aspect was an in-depth numerical investigation of a complex fracture process at the aggregate level in a short reinforced concrete beam under quasi-static 3-point bending by taking the actual mesostructure of concrete and steel bar shape into account, based on a coupled DEM/micro-CT approach.This coupled approach was not used for RC elements.In the simulations, a geometric interface between the concrete and steel bar was assumed to prevent the reliance of bond-slip rules on the boundary conditions of the entire system.A bond-slip relationship was not thus imposed in DEM simulations; it was the outcome of them.In the first step, 2D simulations were performed.There exist already numerical mesoscopic simulations within continuum mechanics wherein ribs on the surface of rebars were explicitly taken into account [54,55].However, in numerical modelling of the interface between concrete and reinforcement, the bond-slip laws are usually assumed.This DEM study has also some limitations, M. Nitka and J. Tejchman Downloaded from mostwiedzy.pl including 2D simulations (the spatial interference of cracks surrounding the bar was thus ignored), simplified 0-wide ITZs around both aggregates and a steel bar, disregarded macro-pores, and the absence of aggregate fragmentation in the concrete mesostructure.The 2D simulations were carried out for a beam cross-section with a steel bar.DEM simulations were performed using the YADE opensource software [56,57].
There are many DEM solutions for concrete.For RC elements, on the other hand, DEM is far less common.The existing studies (e.g.[58][59][60][61]) confirm that DEM may be also reliably used to examine a fracture process in RC elements

Experiments
Laboratory tests were extensively detailed in [51].The length of the rectangular RC beam with one longitudinal steel bar was L = 80 mm, and its height and width were H = B = 40 mm (Fig. 1).The effective height was D = 30 mm, and the effective depth-to-shear  The beam failed in shear due to three factors: a) a high reinforcement ratio of ρ r = 1.8 %, b) a low effective depth-to-shear span ratio of a/D = 1.0, and c) a lack of vertical reinforcement [62] Based on images of both beam sides (Fig. 3), the diagonal critical shear crack between the beam support and loading point caused the beam to collapse in a quick brittle manner.Before the beam failure, a few Fig. 4. Sketch of final crack pattern for short RC beam on all 4 sides (critical shear crack is marked as thick red line, and reinforcement is marked in green) [51].M. Nitka and J. Tejchman Downloaded from mostwiedzy.pl secondary cracks could also be observed (Fig. 4).
The 2D micro-CT image in the vertical beam mid-plane crossing the bar is shown in Fig. 5 (chosen for DEM simulations).The crack pattern was extremely intricate.In addition to the critical inclined shear crack on the left beam side (under ≅ 45 • to the horizontal), two secondary cracks were also apparent: the first was a parallel inclined shear crack above the critical shear crack, and the second was an inclined shear crack on the right beam side.The cracks were attracted by weak ITZs around aggregates (Fig. 5).No cracks were noticed in the concrete cover.

DEM for cohesive-frictional materials
The 3D open-source DEM software YADE [56,57], which permits overlap between two contacted bodies (soft-particle model), was used to run DEM simulations.Particles in a DEM interact with one another during translational and rotational motions using Newton's second law of motion and an explicit time-stepping approach [63].Because DEM takes into account inertial forces, it is a dynamic technique.The model predicts a cohesive bond at the grain contact with a brittle failure below the critical normal tensile force.Shear cohesion failure under typical compression results in contact stick and slip, which are governed by the Coulomb friction law.Regularization is not required for DEM that does not use partial differential equations, so using this approach yields a discrete equation system that is well-posed.Fig. 6 shows the mechanical response of DEM for particles.The following is a list of DEM formulas: M. Nitka and J. Tejchman Downloaded from mostwiedzy.pl F s max = CR 2 and F n min = TR 2 (6) where F → − the normal contact force, U − the overlap between discrete elements, N → − the unit normal vector at the contact point, F → s − the tangential contact force, F → s,prev − the tangential contact force in the previous iteration, X The following material constants are required for DEM simulations: E c , υ c , μ c , C, and T. Non-viscous damping was assumed [64] (Eq.7) in simulations to speed up convergence.Additionally necessary are the parameters R, ρ (mass density), and α d .To accurately replicate the distribution of shear and tensile cracks, the relationship between the uniaxial compressive and tensile strength, and the failure mode of specimens (brittle or quasi-brittle, shear or tensile), the particle contact ratio C/T must be carefully considered [65].As the C/T ratio grows, a more brittle and tensile failure mode dominates, and more tensile cracks appear.
Typically, the material constants are determined by running several DEM simulations and comparing the outcomes to experimental data from simple tests, such as uniaxial compression [5,29], triaxial compression, and simple shear.In simulations, we assumed material constants based on a simple laboratory test (as uniaxial compression) on concrete specimens due to a lack of detailed information concerning the strength properties of mortars with different initial porosity.The damping factor was set to α d = 0.08 consistently.The loading velocity v for this value has no impact on the outcomes [29].Damage is assumed if a cohesive joint (Eq.6) between two spheres disappears after passing a critical threshold.The cohesion does not exist if any connections between spheres are re-established after failure (Eq.5).Material softening is not taken into account by the DEM model.Due to the potential for particle overlap in DEM, any micro-porosity may be reached.The model was successfully used by the authors to represent the behavior of numerous engineering materials having a particulate structure, such as granulates [66][67][68][69], concretes [5][6][7][32][33][34], and rocks [65,70,71].This was accomplished by taking into account shear localization and fracture.Grain fragmentation is not taken into consideration by the current DEM model.This issue can be taken into account in DEM simulations although there will be a significant increase in computing time.Due to calculating time constraints, DEM cannot be recommended as a numerical tool for large concrete or reinforced concrete elements.There are, some up-scaling techniques used with DEMs, such as mass/density/gravity scaling or grain up-scaling (precise scaling of grains and coarse-graining).However, when strain localization occurs in concrete, those up-scaling techniques fail.Since the size of the localized strain cannot be scaled because it depends on the initial and boundary conditions of the entire system, scaling is not feasible.It must be noted that enhanced continuum models enhanced by a characteristic length of mesostructure (related to the aggregate size) require very fine meshes to obtain mesh-independent results in the case of strain localization.For example, when using non-local models, the element size should not be greater than the double characteristic length ( [3,72,73]).

DEM input data
The experiment's 40 mm × 80 mm geometry was replicated in the 2D numerical beam with a reinforcing steel ribbed bar (Fig. 7).To deform the beam assembly, a constant vertical velocity of v = 5 mm/s was applied at the position of the vertical force F using a steel loading cylinder.This vertical velocity was small enough to consider the test as quasi-static.The mean inertial number I which quantifies the significance of dynamic effects was kept below 10 -4 which always corresponds to a quasi-static regime [74].Concrete was modeled as a three-phase material made up of mortar, aggregate, and ITZs.Macro-pores were not taken into account.The  M. Nitka and J. Tejchman Downloaded from mostwiedzy.pl aggregate range (with ITZs) was between 1 mm and 16 mm.Clusters of 0.4 mm diameter spheres were used as the aggregate particles in order to faithfully reproduce their shape.Based on micro-CT scans, the dimensions, configuration, and placement of each aggregate were directly retrieved from the actual specimen (Fig. 5).Spheres with diameters ranging from 0.35 mm to 1.0 mm were used to The geometry of the steel ribbed bar was replicated in 2D simulations.Densely packed spherical components with diameters varying between 0.15 mm and 0.35 mm were used to construct the steel bar with rib heights of 0.70 mm.The bar height was equal to its diameter.The normal bilinear contact model in DEM for particles imitating a steel bar is shown in Fig. 8.It was calibrated to match the global elastic modulus of steel of E = 200 GPa and steel yielding of f y = 650 MPa during uniaxial tension (Section 2).The bar pullout failure mode may occur in the calculations since the mortar particles were not fixed to the bar.ITZs around aggregates and between steel bar and mortar particles were modeled as weaker contacts with no physical width explicitly defined [5,[34][35][36].The ratio E c,ITZ /E c, cm = 0.70 was chosen based on the nanoindentation experiments [6,76].The remaining two ratios, C ITZ /C cm and T ITZ /T cm , were also assumed to be equal to 0.70 as in earlier DEM simulations [5,34,35] due to a lack of experimental data for mortars of different initial porosity.Based on triaxial compression tests with clumped granulates, the inter-particle friction angle of μ c = 18 • was postulated [77].
The same ratios of E c,bond /E c,cm = 0.70, C bond /C cm = 0.70, and T bond /T cm = 0.70 were also assumed in ITZ between the ribbed bar and mortar again due to a lack of experimental pull-out data for with mortars of different initial porosity.A total of 32'000 DEM elements were used, with 14'000 elements modeling steel bar, 12,500 elements modeling mortar, and 5'500 elements modeling aggregate (Fig. 8).The width of the beam cross-section was equal to the grains' diameter.The aggregates in amount 4 in the concrete cover of a thickness of 7 mm were disregarded since there were no visible cracks beneath the bar (Fig. 5).About five days were needed to finish the calculations on a computer with a 3.30 GHz CPU.The computational cost of the simulation was relatively high because the existing DEM model was parallelized on threads only but not in a distributed mode (on cluster computer nodes).In Table 1, the material parameters chosen in DEM computations are listed (with the C/T ratio equal to 1).The parameter F n lim (Fig. 7) was taken as |F N lim |= 18.15 × 10 6 kN.
The DEM material parameters for concrete (Tab.1) were validated with the aid of 2D uniaxial compression tests on rectangular concrete specimens measuring 10 × 10 cm 2 (Fig. 9) without performing micro-CT images.The calculated compressive strength, modulus of elasticity, and Poisson's ratio were compared with the experimental values.The tensile fracture energy was not calibrated since the DEM model does not include softening.The concrete mesostructure in the specimen under compression was similar to that in the beam (the segment of the beam mesostructure was a few times replicated within the concrete specimen).The initial porosity was also the same p = 2.7 %.The same calibration procedure was proved to be sufficient in earlier 2D DEM simulations for concrete by assuming the artificial mesostructure [5][6][7]34,36].The predicted compressive strength of concrete was approximately f c = 46 MPa (Fig. 9a), slightly lower than the experimental one of f c = 49.75MPa (Section 2).The predicted elastic modulus was E = 32 GPa (Fig. 9a), also slightly lower than the experimental value of E = 34.8GPa (Section 2).The predictive Poisson's ratio was ν = 0.21 as in the laboratory test (Section 2).Inside the specimen three evident macro-cracks were numerically obtained: two inclined and one almost vertical (Fig. 9b).

Force-deflection curve
For the vertical beam mid-plane passing the ribbed bar (Fig. 5), DEM computations were performed.Fig. 10 contrasts the experimental evolution with the expected evolution F = f(u).Despite some differences, there is reasonable agreement regarding the M. Nitka and J. Tejchman Downloaded from mostwiedzy.pl force-deflection curve's shape.The experimental data (F max = 10.5 kN and u = 0.30 mm) and the calculated maximum vertical force (F max = 10.5 kN) and deflection corresponding to F max (u = 0.277 mm) were comparable.Initially, there was also a similar stiffness.The highest value of the vertical force was followed by a decline in the maximum vertical force.In DEM simulations, this drop, however, was too strong and too fast.The simulation then went through the same re-hardening phase as the experiment did.In DEM analyses, the beam failure occurred too quickly (deflection u = 0.58 mm versus u = 0.74 mm in the experiment).The 3D simulations with a larger amount of discrete elements in the mortar should be used to decrease the beam brittleness after reaching the maximum force F max , based on our earlier DEM calculations for a concrete beam [5] which indicated the concrete brittleness growth in 3D Fig. 11.Deformed A) and non-deformed beam B) with broken contacts in three-point bending during deflection u (Fig. 10): a) u = 0.277 mm (before load peak, point '1' in Fig. 10), b) u = 0.302 mm (before re-hardening phase, point '2' in Fig. 10), c) u = 0.538 mm (after re-hardening phase, point '3' in Fig. 10) and d) u = 0.570 mm (failure phase, point '4' in Fig. 10) (colors denote in broken contacts: magentamortar, yellow -ITZs around aggregates, and blue -ITZ along steel bar in Fig. 11A and redbeam in Fig. 11B).simulations compared to 2D ones.

Fracture process
The crack progression in the deformed and non-deformed beam based on computed displacements and broken connections is shown Fig. 11.Initially, many microcracks emerged in ITZs surrounding aggregates and along the ribbed bar up to the maximum vertical force for u = 0.28 mm (point '1' in Fig. 10), a few bending cracks appeared at the bottom of the beam, and a few cracks appeared below the vertical force and at the supports owing to compression (Fig. 11a).Between the vertical force and left support, a branching inclined shear crack (similar to that in the experiment (Fig. 3) also appeared (Fig. 11a).The branched shear crack broke into a primary and secondary shear crack at the end of the softening range (point '2' in Fig. 10), as seen in Fig. 11b.Later on (points '3' and '4' in Fig. 10), the primary shear crack widened into a critical one (Figs.11c and 11d).The calculated mean crack inclination to the horizontal (about 45 • ) was similar to the experiment.The numerical crack propagated under two aggregate particles in the beam midpart and above two aggregate particles at the beam bottom in contrast to the experiment.Precise placement of aggregates in calculations had a crucial effect on the crack geometry due to the presence of weak ITZs that attracted the diagonal shear crack.The estimated failure mechanism was similar to the experimental one (failure in shear due to a diagonal shear crack) (Fig. 5).However, any secondary shear cracks of Fig. 5 were numerically obtained.
Fig. 12 displays the displacement vector maps of individual discrete elements during beam deflection.The vectors were initially multiplied by a factor of 20 and progressively decreased to a factor of 3 for the final steps.The plot's colors represent computed displacement, with the red color designating the biggest displacement (about 0.02 mm).At first, displacements of vertical elements took place below the vertical force (Fig. 12a).Later, the right beam component experienced horizontal movement of elements, whereas the left beam part experienced both vertical (upward) and horizontal movements as a result of the development of a primary inclined shear crack (Figs.12b and 12c).The left beam part finally separated from the reinforcement bar (Fig. 12d).

Contact forces
Figures 13 and 14 show the development of normal and tangential contact forces between discrete elements.The red and blue colors indicate forces higher than their mean compressive/tensile value.The region under the vertical force and two supports initially experienced strong inclined compressive normal forces (Fig. 13a).At both beam sides, they were supported by four bar ribs.Strong tensile forces were also seen in the bending steel bar at the same time.Then, the horizontal compressive normal contact forces began to concentrate along the bar rib and the inclined compressive normal forces started to concentrate in the left beam part (supported by only two bar ribs) (Fig. 13b).Thus, significant redistribution of the contact forces was due to the creation of the inclined shear crack.During the re-hardening phase, all compressive contact forces strengthened (Fig. 13Ac).The compressive strut in the left beam region temporarily wedged between the bar ribs, which led to the global re-hardening (the shear crack was arrested).Later, the contact forces softened (Fig. 13Ad).High tangential contact forces (higher than the mean value) mainly occurred in the critical diagonal shear crack and at the beam supports (Fig. 14).
In Fig. 15, the development of broken contacts during bending is seen.The total number of broken contacts at the end of the numerical test (curve 'a' in Fig. 15) was 2'200 (around 2.8 % of all initial contacts in the RC beam).Up at the maximum vertical force (u = 0-0.28mm) and in the softening region (u = 0.28-0.30mm), the highest number of cracks was created.Long before the peak (curve 'd'  M. Nitka and J. Tejchman Downloaded from mostwiedzy.pl in Fig. 15), the first indications of micro-cracking were seen in ITZ along the steel bar for u = 0.018 mm.Micro-cracks then developed in the mortar and ITZs surrounding the aggregates (see curves 'b' and 'c' in Fig. 15).The mortar and ITZ along the bar had the greatest microcracking just before the peak force for u = 0.28 mm.A few contacts in the mortar were broken throughout the re-hardening phase and the beam damage phase.The mortar, ITZs along the bar, and ITZs near aggregates had sequentially the highest contact breakage rate during beam loading.
The development of the width of the inclined critical shear crack at its midpoint is depicted in Fig. 16.The calculated crack widths w were equal to 0.19 mm before reaching the maximum force (F max ), 0.67 mm in the re-hardening phase mid-point, and 1.35 mm in the failure phase for u = 0.55 mm.They were slightly greater than the experimental ones (0.16 mm, 0.50 mm, and 1.25 mm).The beam experienced the largest rise in the crack width increment in both the softening phases (u = 0.28-0.30mm and u = 0.55-0.60mm) (Fig. 16).A re-hardening process (u = 0.30-0.45mm) did not cause the crack's width to increase since the shear crack was arrested in the bar ribs (Fig. 13).

Stresses in concrete along bar
Next, along the steel bar, starting from the height just above the ribs, the normal and shear stresses in concrete were calculated.This region's height was chosen to be equal to d 50 = 2 mm.The Cauchy-Weber formula, a standard method for calculating stresses in discrete mechanics, was used [75] where A is the REV area, N c is the number of contacts inside REV, f c i is the contact force between two elements in the i-direction, and l c j is the distance between two centers of elements in contact in the j-direction.Fig. 17 shows the normal and shear stresses along the steel bar length L at various deflections u (u x denotes the horizontal coordinate against the beam's vertical centerline).Because there were ribs, the wall stress distributions were not uniform along the bar.Due to the extremely high wall roughness of the steel bar, the shear stresses were similar to the normal ones (the wall friction angle was roughly 45 • ).Before the fracture process, the stresses along the bar were almost symmetric against the vertical axis (Fig. 17A).Afterward, they slightly increased on the left beam side due to the appearance of the shear crack (Figs.17B and C).As the shear crack developed, the stresses continued to be non-symmetric and higher on the left beam side (Figs.17C-E).The maximum shear stress reached approximately 20 MPa.During damage for u = 0.570 mm (Fig. 17F), the stresses dropped drastically.
The evolution of the wall shear stress τ w versus the horizontal displacement u s in concrete along the steel bar (slip) for 3 points in the left beam region: u x = -20 mm (on the right of the diagonal shear crack), u x = -30 mm (under the shear crack) and u x = -38 mm (on the left of the shear crack) is shown in Fig. 18.The shape of the bond-slip curve τ w = f(u s ) during bending was different in all three points after reaching the maximum value of about 9 MPa in contrast to unique bond-slip laws assumed between the mortar and steel bar.

Effect of material parameters in ITZ
It was investigated how material parameters in ITZ between concrete and a steel bar affected the force-deflection curve and failure mode (Figs.19 and 20).About the mortar characteristics, the basic values of E c , C, and T in ITZ (Tab.1) were reduced by 15 % and 60 %, respectively.
There were no observable variations in the maximum vertical force (Fig. 19).The rate of softening was likewise comparable.However, the ITZ weakening strongly shortened the length of the re-hardening zone whereas the ITZ strengthening enlarged the rehardening zone on the force-deflection diagram (Fig. 19).
The failure modes in Fig. 20 demonstrate how the they were impacted by the material characteristics in ITZ between concrete and a steel bar.When E c , C, and T values in ITZ (Tab.1) were reduced by 60 % (Fig. 20a), there was a steeper critical diagonal shear fracture.

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Values of E c , C, and T in ITZ (Tab.1) less by 15 % resulted in more secondary shear cracks (Fig. 20b).

Effect of concrete mesostructure
Finally, an investigation was conducted into how the various concrete mesostructures affected the crack pattern (Fig. 21).The artificial concrete mesostructure was identical to the original one, but it was shifted horizontally by a different magnitude.The failure modes show that the postulated mesostructure had a significant impact on the crack pattern (Fig. 21).When comparing the artificial mesostructure beam to the actual mesostructure beam, the critical shear crack had a greater inclination towards the horizontal (70 • versus 45 • ).The secondary crack pattern also had a distinct appearance.On the left side of the critical shear crack in the beam of Fig. 21a, there was another secondary shear crack.In the beam of Fig. 21b, a second secondary shear fracture occurred on the opposite side of the critical shear crack.

Summary and conclusions
The classical particle discrete element technique (DEM) was used in this study to predict the behavior of an over-reinforced concrete beam with one steel ribbed bar and without vertical reinforcement under quasi-static 2D conditions.The beam in the experiment failed in shear due to a critical diagonal shear crack.A three-phase concrete description, aggregate, mortar, and interfacial transitional zones (ITZs) around aggregates were used to account for the heterogeneity of the concrete at the mesoscale.In DEM calculations based on a 2D X-ray CT scan, the actual shape and placement of aggregate particles in concrete were taken for granted.In analyses, the shape of the steel ribbed bar was precisely reconstructed.ITZ was also assumed in the mortar along the steel bar.A geometrical interface condition was assumed to exist between the concrete and a steel bar without imposing a bond-slip law (being the outcome of simulations).The experiment and the DEM simulations were compared.The 2D DEM analysis results can lead to the following conclusions:

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 3 .
Fig. 3. 3D micro-CT images of short RC beam reinforced with steel ribbed bar on both sides at failure (pores and cracks are in red, steel bar is in green) [51].

→
s − the relative tangential displacement increment, K n − the normal contact stiffness, K s − the tangential contact stiffness, E c − the elastic modulus of the particle contact, υ c − the Poisson's ratio of particle contact, R − the particle radius, R A and R B -the contacting particle radii, μ c − the Coulomb inter-particle friction angle, F s max − the critical cohesive contact force, F n min − the minimum tensile force, C − the cohesion at the contact (maximum shear stress at zero pressure), and T − the tensile strength of the contact, F → k damp − the dampened contact force, F → k and v → k p -the k th components of the residual force and translational particle velocity v p , and α d − the positive damping coefficient smaller than 1 (sgn(•) that returns the sign of the k th component of velocity).

Fig. 7 .
Fig. 7. 2D model of RC beam 40 mm × 80 mm: a) entire beam based on Fig.5 and b) part of steel ribbed bar (light grey color denotes mortar, dark grey color denotes aggregate and black color corresponds to steel ribbed).

Fig. 8 .
Fig. 8. Normal contact model in DEM for particles imitating steel bar.

Fig. 13 .
Fig. 13.Evolution of normal contact forces in RC beam (A) and zoom on bar ribs (B) during three-point bending for deflection u: a) u = 0.277 mm (before load peak), b) u = 0.302 mm (before re-hardening phase), c) u = 0.538 mm (after re-hardening phase) and d) u = 0.570 mm (damage phase) (red color denotes large compressive forces, and blue color denotes large tensile forceslarger than the average ones).

Fig. 15 .
Fig. 15.Evolution of broken contacts N during beam deflection u: a) entire beam, b) mortar, c) ITZs around aggregates and d) ITZ along steel bar.

Fig. 16 .
Fig. 16.Evolution of critical crack width w during three-point bending versus deflection u.

Fig. 18 .
Fig. 18.Evolution between wall shear stress τ w and horizontal displacement u s for three points along steel bar: a) u x = -20 mm, b) u x = -30 mm and c) u x = -38 mm (u x denotes horizontal coordinate against vertical centerline).

Fig. 19 .
Fig. 19.Force-deflection curve F = f(u): a) experiment [51] and b-d) DEM results for different parameters of ITZ between steel bar and mortar: b) E c,bond /E c,cm = 0.70, C bond /C cm = 0.70 and T bond /T cm = 0.70, c) E c,bond /E c,cm = 0.85, C bond /C cm = 0.85 and T bond /T cm = 0.85 and d) E c,bond /E c,cm = 0.40, C bond /C cm = 0.40 and T bond /T cm = 0.40.

Fig. 20 .
Fig. 20.Deformed beam with broken contacts in three-point bending with two different parameters of ITZ between steel bar and mortar: a) E c,bond / E c,cm = 0.40, C bond /C cm = 0.40 and T bond /T cm = 0.40 and b) E c,bond /E c,cm = 0.85, C bond /C cm = 0.85 and T bond /T cm = 0.85 (colors denote in broken contacts: magentamortar, yellow -ITZs around aggregates, and blue -ITZ along steel bar).

Fig. 21 .
Fig. 21.Deformed beam with broken contacts in three-point bending with two different initial mesostructures a) and b) (colors denote in broken contacts: magentamortar, yellow -ITZs around aggregates, and blue -ITZ along steel bar).

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. Nitka and J. Tejchman Downloaded from mostwiedzy.pl . The most M. Nitka and J.Tejchman

Table 1
Main DEM parameters for different phases assumed in simulations.