Mesoscale Modeling of Hooked-End Steel Fiber Reinforced Concrete under Uniaxial Compression Using Cohesive Elements

Based on the cohesive zone model, the 2D mesostructures were developed for numerical studies of multi-phase hooked-end steel fiber reinforced concrete under uniaxial compression. The zero-thickness cohesive interface elements were inserted within the mortar, on interfaces of mortar and aggregates and interfaces of mortar and fibers to simulate the failure process of fiber reinforced concrete. The results showed that the numerical results matched well the experimental results in both failure modes and stress-strain behavior. Hooked-end steel fiber reinforced concrete exhibited ductile failure and maintained integrity during a whole failure process. Compared with normal concrete, HES fiber reinforced concrete was greater stiffness and compressive strength; the descending branch of the stress-strain curve was significantly flatter; the residual stress was higher.


Introduction
Fiber reinforced concrete is a multi-phase composite material, which includes coarse and fine aggregates, mortar, fibers, interfaces and porosity, etc. [1]. The micro-cracks begin to appear in a hardening part of the stress-strain curve, and macro-cracks form in a softening branch under the compression. The experiments and numerical simulations show that the failure process is very complex and depends on mesostructure [2]. Meso-mechanical characteristics of concrete reflect its macro-mechanic response. Specific mesoscale structure, material and internal property may predict the macro-mechanical performance of concrete [2]. The cohesive zone model and extended finite element method, which are low mesh sensitivity and effective convergence, can simulate discrete cracks. Moreover, the cohesive zone model has greater superiority in multi-cracks propagation. There are rarely studies on the mesoscale numerical simulation of fiber reinforced concrete failure process.
The objective of this study is to generate 2D mesostructures with randomly packed aggregates and steel fibers with a volume fraction of 1.5% and investigate the application of the cohesive zone model in fiber reinforced concrete under uniaxial compression.

Aggregates and Fibers
The coarse aggregates with a continuous gradation of 5 -20 mm were used, and the mix proportions had been given in [3]. For normal concrete, coarse aggregates represent 40% -50% of the concrete volume. In this study, the concrete model with 40% aggregate content was adopted. The shape of aggregates is assumed as the circle in the numerical model, which has been accepted and unitized. To reduce computation, the aggregates with diameters ranged from 5 to 20 mm were modeled, and the fine aggregates less than 5 mm merged with cement were treated as mortar. The 2D mesostructure model with dimensions was adopted. The aggregates decomposed into two dimensions follow the classic Walraven function [4]: where ( ) 0 c P D D < presents the area proportion of aggregates with a particle diameter smaller than D 0 ; P k denotes the percentage of the total area of aggregates to the quadrilateral area (about 75% [4]); D max is the maximum size of aggregates. Table 1 lists the aggregate size distribution in mesoscale models. Hooked-end steel (HES) fiber was adopted in this study. Figure 1 shows the characteristics of HES fiber. The volume fraction of HES fibers is 1.5%. The number (N) of HES fibers in mesoscale models is calculated as follows: where W and H are the width and height of specimen, respectively; V f , d and l are the volume fraction, diameter and length of HES fiber, respectively.

Cohesive Zone Model
The constitutive behavior of zero-thickness cohesive interface element controlled by traction-separation laws was described by the damage initiation criterion and damage evolution law. A bilinear traction-separation law was adopted in this study, illustrated in Figure 2.
The damage initiation criterion followed the quadratic nominal stress criterion: where n t and s t denote the tractions; 0 n t and 0 s t are the critical traction; the is the Macaulay bracket , 0 0, 0 The damage evolution is characterized by a scalar parameter, D. Once the damage criterion is met, all physical mechanisms cause the overall extension of the crack across the elements and the damage parameter D monotonically evolves from 0 (no damage) to 1 (complete damage). The damage parameter D can be formulated by: where n t and s t are the stress components predicted by the elastic traction-separation behavior for the current strain without damages.

Numerical Simulations of the Uniaxial Compression Test
Uniaxial compression tests rectangle specimens were modeled in this study. The area proportion of aggregates and number of HES fibers are met by generating random aggregates and fibers in Digimat software. The solid elements for fibers, aggregates and mortar were assumed to behave linear elastically. Triangle elements provide the most numerous interfaces with the same nodes in the plane and mesh the 2D mesostructures. A 3-node linear triangle plane stress element (CPS3) in ABAQUS is applied to model the fibers, aggregates and mortar. The 4-node two-dimensional cohesive elements (COH2D4) are inserted within the mortar (MII), on interfaces of mortar and aggregates (ITZ_AGG), and on interfaces of mortar and fibers (ITZ_HES) to simulate the stress-strain behavior and failure modes of fiber reinforced concrete. Figure 3 shows the bulk and cohesive elements creating finite element mesh. All models were fixed at the bottom boundary and subjected to a uniformly distributed displacement at the top boundary, i.e., the displacement-controlled loading scheme was adopted. All analyses were ended at a displacement of 2 mm. Xiong and Xiao [2] indicated that the compressive performance is more sensitive to mesh precision. The model with an element length of 1 mm and loading time of 0.01 s used in the ABAQUS/Explicit solver was validated [5].
Hence, the mesh size of 1 mm was chosen in this study due to its good simulation accuracy and acceptable time consumption.

Model Parameter
The initial stiffness of cohesive elements, i.e., 0 n k and 0 s k , are defined for opening and sliding, but the choice of appropriate values is not straightforward. It needs to be high enough to prevent excessive deformations in the elastic regime, but a too high value results in ill-conditioning of the global stiffness matrix. As a guideline for the initial stiffness selection, the relationship between the macro-Poisson ratio and initial stiffness ratio ( 0 0 n s k k ) and the relationship between macro-elastic modulus and initial normal stiffness was proposed by [6].
The trial-and-error method for determining the acceptable initial stiffness was adopted. The ratio of normal to shear properties varies between 2 to 10. The critical traction and fracture energy for normal mode range from 2 to 3 MPa and between 0.01 to 0.1 N/mm, respectively [7]. It is necessary to define the interaction characteristics to simulate the collision and fraction during compression. The friction behavior mainly influences the descending stage of the stress-strain curve and compressive strength. As the suggestion by [6], 0.5 was adopted as an appropriate value for friction coefficient μ. Table 2 summarizes the material properties. Figure 4 shows the failure modes of the numerical and experimental specimen under the uniaxial compression. The specimens were subjected to the following failure process: after reaching the peak stress, several micro-cracks began to

Stress-Strain Curve
As shown in Figure 5, the numerical results matched well with experimental curves by the trial-and-error method. Afterward, the cohesive parameters were determined in the mesostructure model, illustrated in Table 2. The stress-strain curve of fiber reinforced concrete had similar shapes to normal concrete. Compared with the normal concrete specimen, the slope of ascending branch for specimen with HES fibers was steeper, and the peak stress was greater. It indicated that HES fiber reinforced concrete was greater stiffness and compressive strength. However, the descending branch was significantly flatter than that of mixes without fibers. Due to the addition of fibers, the stress decreased slower after the peak stress, and the residual stress was higher. The result meant that the incorporation of fibers improved the deformability and energy absorption capacity after peak stress. This find is that the concrete matrix has a dominant effect in the ascending branch, while the fibers play their roles in the descending branch and increase the crack resistance [10]. Journal of Applied Mathematics and Physics

Conclusions
The paper has developed the 2D mesostructures of normal concrete and HES fiber reinforced concrete and investigated uniaxial compressive behavior. The following conclusions can be summarized from the numerical and experimental results.  The failure modes of the numerical specimen are similar to experimental results. Without fibers, the specimen failed in a brittle manner, accompanying peeling off and dislocation. With HES fibers, the specimen developed a group of the wide shear band, exhibited ductile failure, and maintained integrity during a whole failure process.  The stress-strain curves of the mesostructure model under uniaxial compression matched well with experimental results. Compared with normal concrete, HES fiber reinforced concrete was greater stiffness and compressive strength. The descending branch of specimens with HES fibers was signifi-Journal of Applied Mathematics and Physics cantly flatter, and the residual stress was higher than specimens without fibers.