Simulation of the Flexural Response of Ultrahigh Performance Fiber-Reinforced Concrete with Lattice Fracture Model

The ﬂexural response of ultrahigh performance ﬁber-reinforced concrete (UHPFRC) was simulated based on the lattice fracture model. Fiber was modelled as separated beam that was connected to the matrix with interface beams. The simulated results were compared with the experimental results. Deviations occurred at the late stage of the strain-softening period. But both the strain-hardening behavior and multicracking phenomenon were observed in the simulation. The eﬀects of ﬁber orientation and ﬁber content were studied with the lattice fracture model. The ﬂexural strength and toughness of UHPFRC improved as the ﬁbers were aligned distributed or the ﬁber content increased. The proposed model has the potential to help with the materials design of UHPFRC, and the limitations of the model were also discussed in the paper.


Introduction
Ultrahigh performance fiber-reinforced concrete (UHPFRC) was initially invented in 1980s in France [1,2]. In the revised recommendations on UHPFRC published by AFGC (Association Francaise de Génie Civil) [3], UHPFRCs are defined as materials with a cement matrix and a characteristic compressive strength of 150 MPa-250 MPa. e most common methodology to prepare UHPFRC is cement + silica fume + very low water to binder (w/b) ratio + fine aggregate + superplasticizer + fiber [4]. Fibers are added to improve the ductility of UHPFRC.
anks to its extremely excellent mechanical properties and durability, UHPFRC has been considered as the potential construction material for the next generation infrastructures [5]. e applications of UHPFRC are growing all over the world, especially in Europe, North America, Japan, Korea, and Australia. It has been widely applied for bridges, buildings, coastal structures, structural repairing, military structures, and so on [4]. However, the UHPFRC material design guides or codes are not fully developed right now, which inhibits the wider application of UHPFRC in infrastructure construction [4,6].
With the increasing applications of UHPFRC, there is a clear need in developing material design methods for UHPFRC. Conventionally, UHPFRC is designed in the laboratory, with series of experimental tests. However, experiments will consume time, money, natural resources, energy, and so on. erefore, numerical simulation has become popular to predict the properties of materials and could assist in the material design [7]. As for UHPFRC, its fracture process under the flexural load is very important when UHPFRC is applied for structures as large-span bridges and thin-wall shields. Many models have been proposed to simulate the fracture process of concrete under loads, such as discrete crack model [8,9], smeared crack model [10,11], damage model [12], and lattice fracture model [13]. Among them, the lattice fracture model is more suitable for the simulation of the flexural response of UHPFRC. e lattice fracture model was rstly developed by Schlangen and van Mier in 1990s [14]. Over the last few decades, it has been successfully used for the simulation of the fracture process in concrete. Based on the lattice fracture model, Arslan et al. simulated the strain-softening behavior of concrete under tension [15]; Vervuurt studied the interface fracture behavior of concrete [16]; Van Vliet investigated the size e ect in tensile fracture of concrete and rock [17]; Man revealed the e ect of aggregate shape on the compressive strength of concrete [18]; Cadu and van Mier analyzed the compressive fracture behavior of normal concrete, high-performance concrete, and foam concrete [19]; Grassl and Davies simulated the corrosion-induced cracking in reinforced concrete [20]; Montero and Schlangen modelled the fracture process of engineered cementitious composites (ECC) under uniaxial tension [21]; and Qian et al. simulated the fracture process of concrete considering the structural information of concrete at di erent scales [13]. Nevertheless, the application of the lattice fracture model for simulating the fracture process of ber-reinforced concrete, especially UHPFRC, under the exural load is rarely seen. In order to aid the material design, in this study, the exural response of UHPFRC was simulated with the lattice fracture model; moreover, the e ects of ber content and orientation were studied and discussed.

Description of the Lattice Fracture
Model for UHPFRC e main idea of the lattice fracture model is to replace the composite by a network of beam elements [21]. e di erent phases are represented by beams with di erent materials properties. e procedures to perform lattice fracture simulation are shown in Figure 1. e rst step is to construct the lattice network. UHPFRC is considered to be composed of two phases, that is, the matrix and bers [22]. Discrete bers are embedded in the matrix. Correspondingly, in the lattice fracture model, bers are implemented as discrete beam elements and are connected to the matrix elements with interface beams [21,23], as shown in Figure 2. e network constructed by the matrix beams and nodes represents the matrix of UHPFRC. Since the matrix of UHPFRC is very homogenous, the lattice nodes are generated to at the center of the square grids so that a uniform network can be constructed. With a given volume fraction, ber beams with a given diameter and length are generated and randomly distributed within the matrix. Along with the ber beam, extra nodes were generated at the location where the ber beam intersects the square grid. In order to represent the bermatrix bond, interface beams were generated between the extra nodes and lattice nodes. More details on the lattice network construction for ber-reinforced concrete can be found in [23].
After the lattice network construction, the mechanical properties of all the beams have to be assigned. According to the previous studies [21], the matrix and ber beams are set to be brittle and will fail under tension at their corresponding tensile strength, and the interface beams are set to be ductile and can fail either in tension or compression. e material properties of the matrix and ber beam can be obtained directly with experiments, and the properties of interface beam normally can be determined by tting the experimental results of the single ber pullout test [21,24]. e lattice fracture model could simulate a series of mechanical tests in the lab, such as compressive test [6], tensile test [13], and exural test. With di erent boundary condition settings on the lattice network, di erent mechanical tests can be simulated. e fracture process simulation is described in detail in [13,23,25]. In summary, the fracture process is simulated step by step. At every step, a prescribed force or displacement is applied on the lattice network, and the stresses in the beams can be calculated. e beam with the highest stress/strength ratio is removed from the lattice network, representing the crack growth process. After the simulation, the stress-strain response diagram, the crack pattern, and the crack propagation process can be obtained.

Proportion and Properties of UHPFRC.
e exural response of a typical UHPFRC with 2% steel bers was simulated in the paper. e proportion of the UHPFRC is   shown in Table 1. Cement, y ash, and silica fume were used as binders. River sand had a maximum particle size of 2.36 mm. e superplasticizer was a type of liquid agent with a solid content of 28%. e length, diameter, and tensile strength of steel ber were 13 mm, 0.2 mm, and 1800 MPa, respectively. e compressive strength and four-point exural strength of UHPFRC after 90 d standard curing were 156.1 MPa and 34.4 MPa, respectively.

Lattice Network Construction.
Due to the limitation of computing e ciency, four-point exural tests were simulated on a 10 mm × 10 mm × 40 mm prism. e bers were considered to be randomly distributed in the prism. e ber beams were generated following the method described in [21]. e simulated ber distribution and corresponding lattice network are shown in Figures 3 and 4. e lattice network was constructed with the method mentioned in Section 2, and the mesh size was 1 mm. e matrix beam and ber beam are shown in blue and red, respectively, in Figure 4.

Local Mechanical Properties Assignment.
From a general point of view, the exural response of UHPFRC depends on the matrix parameters, ber parameters, and ber-matrix interface parameters. ese parameters are also the inputs for the lattice fracture model. e input parameters for this study are shown in Table 2. e parameters for the matrix and steel ber were obtained with experiments, and the parameters for the interface beams were tted based on the experimental data from [26]. e matrix and bers were considered as linear elastic ( Figure 5(a)), while a seven segment ductile stress-strain response [21] (Figure 5(b)) was applied for interface elements in order to obtain more realistic results.

Boundary Condition Setting.
e boundary condition was set following that happened in the experiments. A fourpoint bending test was set for the simulation, and an illustration is shown in Figure 6. In the lattice network, the arrow pointed nodes at the bottom were xed, and a prescribed displacement was imposed on the arrow pointed nodes at the top for every step. e beams connected to these nodes were shown in red in Figure 6. Figure 7 shows the simulated loaddeformation curve of UHPFRC. It can be seen from Figure 7(a) that the lattice fracture model could give a reasonable exural response of UHPFRC. Both the strain-hardening behavior and multiple cracking could be simulated. Nevertheless, it is unrealistic that the curve begins to go up when the deformation exceeds 2.5 mm. At this stage, a large portion of the matrix beam have been broken, and the ber beams and interface beams, which have higher mechanical properties, were connected to each other and might be bearing most of the loads. Hence, the load went up instead at the late stage. In Figure 7(b), the load-deformation curve before deformation reaches 0.2 mm is emphasized. Before step 100, UHPFRC showed an elastic behavior. When the stress in the matrix exceeded the matrix cracking strength, the rst crack of the UHPC matrix occurred, and the crack began to propagate until bers took over the load at step 500. en, the second crack occurred at step 1000. e crack bridging e ect of the bers led to a strainhardening behavior of UHPFRC. During this process, the number of the cracks increased and individual cracks widened. At step 5000, UHPC began to fail. Fibers started to pull out from the matrix, and cracks began to localize. Figure 8 shows the crack patterns in UHPFRC at different steps, and the broken beams are shown in blue. e white beams are broken beams at the end of the simulation. It can be seen from Figure 8 that, at step 100, UHPFRC started to crack. Only some beams at the bottom were    Advances in Civil Engineering broken. After that, the crack developed, and one straight crack formed at step 500. en, this crack was bridged by bers, and cracks at other location began to occur at step 1000. At step 5000, several cracks can be seen and UHPFRC started to fail.

Experimental Validation.
In order verify the lattice fracture model, four-point bending tests were performed on UHPFRC specimens with a dimension of 40 mm × 40 mm × 160 mm. e simulated and measured loaddeformation curves are compared in Figure 9. Although different specimen sizes were adopted in the simulation and experiments, it still can be seen that the load-deformation curves obtained by simulation and experiments were in a similar shape. For the simulated curve, the load increased very fast during the elastic and strain-hardening periods. e slow development of the load for the measured curve may be due to that the clamp and the specimen were not fully contacted at the beginning of the test. Flexural strengths of UHPFRC obtained from the simulation result and experimental result were 31.3 MPa and 34.1 MPa, respectively. e deviation is very small. e most obvious di erence of the two curves occurred at the late stage of the strain-softening period. e load of the simulated curve increased at ending of the simulation, which was not in accordance with the reality. In the simulation, the interface beams were connected with the ber beam and matrix nodes.
ere is also a chance that one matrix node was connected to two or more interface beams so that the interface beams and ber beams may be connected to each other and hold the load together, when most of the matrix beams were broken. e input of the tensile strength for interface and ber beams was much higher than the matrix beam; hence, the load increased at the late stage. One of the solutions to this problem is to reduce the grid size to avoid the direct connection between interface beams. But, under this circumstance, the computation will be more expensive.
In general, the presented lattice fracture model could reliably simulate the exural response of UHPFRC before failure, and improvements have to be made to eliminate the deviation of the simulated load-deformation curve at the late stage.

Effect of Fiber Orientation and Fiber
Content on the Flexural Response of UHPFRC

Fiber Orientation.
It is well acknowledged that ber orientation will de nitely in uence the mechanical properties of UHPFRC, especially the exural and tensile strength. But, it is very di cult to control the ber orientation in the experiments, while di erent ber orientation could be implemented easily in the simulation.
In the lattice fracture model, theoretically, ber can be distributed in any desired form. To demonstrate the potential application of the lattice fracture model, the e ect of aligned and randomly distributed bers on the exural behavior of UHPFRC was studied. 2% steel bers were added in UHPFRC, and the inputs and settings were the same as those in Section 3. Aligned distributed bers are shown in Figure 10, and randomly distributed bers are presented in Figure 3.
e simulated load-deformation curves of UHPFRC with aligned and randomly distributed bers are shown in Figure 11. It can be seen that UHPFRC with aligned distributed bers has a higher exural strength than UHPFRC with randomly distributed bers. Moreover, the area under the load-deformation curve of UHPFRC with aligned distributed bers was also bigger than UHPFRC with randomly    Advances in Civil Engineering distributed bers, which implied UHPFRC with aligned distributed bers had a higher toughness. In general, the aligned distributed bers showed a better reinforcing and toughening e ect on UHPFRC. e results are as expected. Figure 12 shows the crack patterns of UHPFRC with aligned and randomly distributed bers. Compared with UHPFRC with randomly distributed bers, more cracks occurred in UHPFRC with aligned distributed bers. e aligned distributed bers did better in inhibiting the crack propagation than randomly distributed bers. Hence, when preparing UHPFRC materials, a particular method could be applied to make bers tend to be aligned distributed and be perpendicular to the crack, so that the exural property of UHPFRC could be improved.

Fiber Content.
It is anticipated that the increase of ber content could improve the exural strength and toughness of UHPFRC. is was also studied by comparing simulated  Step 500 Step 100 Step 1000 Step 5000 0 results of UHPFRC with di erent ber contents. e other inputs and settings were also the same as those in Section 3. Four ber volume fractions (V f ) were studied in this paper: 0%, 1%, 2%, and 3%. In Figure 13, the simulated loaddeformation curves of UHPFRC with di erent ber contents are presented. It is observed that the exural strength and toughness increased with the increase in ber content. e results are also in accordance with the experimental results [26]. e crack patterns in UHPFRC with di erent ber contents are shown in Figure 14. When bers were not presented, the material was brittle and one single crack was developed (Figure 14(a)). As the ber volume fraction increased, the main crack was bridged by bers and other cracks were generated. With more bers, more cracks were generated, and in other words, the multicracking phenomenon was more remarkable.

Potential Applications and Limitations
e lattice fracture model has been successfully used for simulating the uniaxial tensile behavior of pain concrete and ber-reinforced concrete [13,21,23]. e results of this study showed that the lattice fracture model was also capable to simulate the exural response of UHPFRC, which is one of typical ber-reinforced concretes. Besides the ber orientation and ber content, the e ect of di erent variables, for example, ber strength, ber sti ness, and ber size, can be explored with the lattice fracture model as well. e e ect of matrix composition on the exural behavior of UHPFRC can also be addressed with the lattice fracture model at di erent length scales. e outcome of the simulation could contribute to the materials design of UHPFRC.
One drawback of the presented simulation is that the demand for computational time is very long, which caused that the sample size used in the simulations was much smaller than that in experiments. According to the simulated results, it is acceptable to simulate the exural response of UHPFRC with a small size, but it is better to perform the simulation on UHPFRC with the same size as that used in experiments, if the computing e ciency could be improved. Hence, perfecting the algorithm of the lattice fracture model is necessary for the further improvements. Another solution of this problem is to develop an approach to determine the representative elementary volume (REV) size of UHPFRC for exural response simulation. REV is the smallest volume, whose properties are representative of the whole material. Simulated exural response of REV of UHPFRC could represent the actual exural behavior of UHPFRC. Under this circumstance, the computational time could be reduced.
At present, the properties of interface beams can be determined only by tting the experimental results. It may cause the problem that the tted data could not be used for other UHPFRCs with con dence. Hence, a deeper understanding on the relationship between the matrix and interface properties is needed, and this can be obtained by simulations at a lower scale (nanoscale or microscale), or by series of experimental tests. More simulations or experiments should be performed to study the bond strength between the bers and the matrix. It is important for the correct setting of the interface beam parameters in the lattice fracture model.

Concluding Remarks
In this paper, the exural response of UHPFRC was simulated with the lattice fracture model. e properties of matrix, ber, and interface were used as inputs. Both the strain-hardening behavior and multicracking phenomenon could be observed in the simulations. e simulated and tested load de ection curves were in a similar shape, except that deviations occurred at the late strain-softening period. e e ects of ber orientation and ber content were studied with simulations. e exural strength and toughness of UHPFRC were improved with aligned distributed bers and with increased ber contents.
In general, the lattice fracture model has proved to be an e cient numerical tool to simulate the exural response of

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UHPFRC. Parameter studies will be conducted to reveal the effects of various parameters and testing configurations, and further studies could be focused on the determination of the interface beam properties of UHPFRC.

Data Availability
e authors declare that all the data supporting the findings of this study are available within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.