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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.

Fiber reinforced concrete is a multi-phase composite material, which includes coarse and fine aggregates, mortar, fibers, interfaces and porosity, etc. [

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.

The coarse aggregates with a continuous gradation of 5 - 20 mm were used, and the mix proportions had been given in [

P c ( D < D 0 ) = P k ( 1.065 D 0 0.5 D m i x − 0.5 − 0.053 D 0 4 D m i x − 4 − 0.012 D 0 6 D m i x − 6 − 0.0045 D 0 8 D m i x − 8 + 0.0025 D 0 10 D m i x − 10 ) (1)

where P c ( D < D 0 ) 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% [_{max} is the maximum size of aggregates.

Hooked-end steel (HES) fiber was adopted in this study.

Aggregate grain diameter (mm) | 5 - 10 | 10 - 15 | 15 - 20 |
---|---|---|---|

Aggregate ratio (%) | 19.2% | 15.3 | 5.1% |

N = r o u n d ( 4 × W × H × V f π × d 2 × l ) (2)

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.

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

The damage initiation criterion followed the quadratic nominal stress criterion:

( 〈 t n 〉 t n 0 ) 2 + ( t s t s 0 ) 2 = 1 (3)

where t n and t s denote the tractions; t n 0 and t s 0 are the critical traction; the 〈 〉 is the Macaulay bracket 〈 x 〉 = { x , x ≥ 0 0 , x < 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:

D = δ m f ( δ m max − δ m 0 ) δ m max ( δ m f − δ m 0 ) (4)

where δ m is the effective relative displacement, δ m = 〈 δ n 〉 2 + δ s 2 ; δ m max is the maximum value of the effective relative displacement attained during the loading history. δ m 0 is the effective relative displacement at the damage initiation; δ m f is the effective relative displacement at failure.

The tractions are affected by damage according to the following equations:

t n = { ( 1 − D ) t ¯ n , t ¯ n ≥ 0 t ¯ n , t ¯ n < 0 t s = ( 1 − D ) t ¯ s (5)

where t ¯ n and t ¯ s are the stress components predicted by the elastic traction-separation behavior for the current strain without damages.

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.

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 [

The initial stiffness of cohesive elements, i.e., k n 0 and k s 0 , 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 ( k n 0 / k s 0 ) and the relationship between macro-elastic modulus and initial normal stiffness was proposed by [

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 [

Fiber type | Young’s modulus E (GPa) | Poisson’s ratio μ | Density ρ (10^{−9} ton/mm^{3}) | Initial stiffness k n 0 (MPa/mm) | critical traction t n 0 (MPa) | Fracture energy G_{f}_{ }(N/mm) |
---|---|---|---|---|---|---|

Aggregate | 55 | 0.2 | 2.6 | - | - | - |

HES fiber | 200 | 0.3 | 7.8 | - | - | - |

Mortar | 28 | 0.2 | 2.2 | - | - | - |

ITZ_AGG | - | - | 2.2 | 30,000 | 3.0 | 0.1 |

ITZ_HES | - | - | 2.2 | 30,000 | 6.0 | 0.2 |

MII | - | - | 2.2 | 30,000 | 6.0 | 0.5 |

appear in the intermediate portion of the specimen, and a major crack was formed gradually upward and downward; finally, the major crack penetrated the surface, failing specimen. For normal concrete, as shown in

As shown in

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 significantly flatter, and the residual stress was higher than specimens without fibers.

The authors gratefully acknowledge the support of the Fundamental Research Funds for the Central Universities of Chang’an University [grant number 300203211121], the Science and Technology Planning Project in Henan Province of China [grant number 182102311091], the Natural Science Research Project of Zhengzhou Institute of Science and Technology [grant number 2017-XYZK-002].

The authors declare no conflicts of interest regarding the publication of this paper.

Feng, J.J., Yin, G.S., Liu, Z., Liang, J.H., Zhang, Y.J. and Wen, C.G. (2021) Mesoscale Modeling of Hooked-End Steel Fiber Reinforced Concrete under Uniaxial Compression Using Cohesive Elements. Journal of Applied Mathematics and Physics, 9, 2909-2917. https://doi.org/10.4236/jamp.2021.911184