# Simulations of Fractures of Heterogeneous Orthotropic Fiber-Reinforced Concrete with Pre-Existing Flaws Using an Improved Peridynamic Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Classical BB-PD Theory

_{x}around it through the vector value function

**f**, which is called the pairwise force function, defined as follows:

**u**(

**x**,t) and

**u**(

**x**′,t) are the displacement vectors of material points

**x**and

**x**′, respectively.

**x**at any time t is [21,22]:

**ü**is the acceleration vector of

**x**,

**b**is the applied physical density vector, dV

_{x}

_{′}is the infinitesimal volume linked to point

**x**′, and H

_{x}is the horizon of

**x**. The concept of the horizon is as follows:

_{x}.

**ξ**and

**η**are the relative position vector and relative displacement vector of the points

**x**and

**x**′, respectively.

**s**to represent the bond deformation between material points, which is defined as follows [22]:

_{0}is the bond’s critical stretch; when the stretch of the bond exceeds s

_{0}, the bond will be broken irretrievably.

**f**is defined as follows [22]:

**f**can be obtained from the derivative of micropotential energy ω:

_{PD}at a material point

**x**can be expressed as follows:

_{0}is determined on the basis of the breaking energy G

_{0}, which can be expressed as follows:

**φ**at material point

**x**is defined as follows [22]:

## 3. Improved Orthotropic PD Model

**x**and any other material point in the horizon range of

**x**—that is, the size effect of the nonlocal long-range force—increases.

_{1}in the fiber bond direction, elastic modulus E

_{2}in the matrix bond direction, Poisson’s ratio ν

_{12}, and shear modulus G

_{12}. The micromodulus c can be written as follows [23]:

_{m}and c

_{a}are the micromodulus of the matrix bond and fiber bond, respectively, and φ is the angle between any bond and the positive X-axis.

_{PD}in Equation (10) can be rewritten as follows [23]:

_{m}and s

_{a}represent the stretch of the matrix bond and the fiber bond, respectively, V

_{a}is the volume occupied by another material point in the horizon range of the material point x, and the expression is as follows [23]:

_{ij}is the stiffness matrix.

_{m}

_{0}and s

_{a}

_{0}represent the critical stretch of the matrix bond and the fiber bond, respectively, while T

_{m}and T

_{a}represent the uniaxial tensile strength in the direction of the matrix bond and the fiber bond, respectively.

**f**

_{S}can be expressed as follows:

## 4. Numerical Solution Method

_{i}interacts with the point x

_{j}within the horizon; thus, Equation (17) can be replaced using Riemann sums, as follows:

_{k}is the volume of point x

_{k}. The material points on the horizon boundary are reduced proportionally on the basis of the relationship between the material points on the horizon boundary and the horizon radius. The point volume can be expressed as follows:

## 5. Model Validation

#### 5.1. Elastic Deformation of a Cantilever Beam

^{3}. The PD numerical model was discretized to 90,000 material points, with a node spacing Δx = 0.002 m, and the horizon δ = 3Δx.

_{y}(L) of the midpoint on the right side of the elastic cantilever beam can be expressed as follows:

#### 5.2. Tensile Failure of a Fiber-Reinforced Orthotropic Plate with Pre-Existing Flaws

_{1}= 106 Gpa, the elastic modulus in the matrix bond direction E

_{2}= 8.5 Gpa, Poisson’s ratio ν

_{12}= 1/3, and density ρ = 1801 kg/m

^{3}. The critical stretch s

_{m}

_{0}= 0.02 and s

_{a0}= 0.03. The specimen was discretized into 11,200 material points, with the node spacing Δx = 0.5 mm, horizon δ = 3.015Δx, and time step Δt = 1 × 10

^{−8}s. The tensile displacement load rate was 0.03 mm/s.

## 6. Case Study

_{0}is the scale parameter representing the average value of the parameter p, and m is the shape parameter that determines the basic shape of the probability density function and reflects the homogeneity of the material structure.

_{0}. Due to the inconsistent size of s

_{0}of each particle, the mechanical properties of each point are affected. To ensure that the interaction force between two material points is equal, the critical stretch s

_{0}was taken as the average value of the interacting material points, and its fracture judgment criterion can be expressed as s ≥ (s

_{0}(i) + s

_{0}(j))/2. The fractures of specimens with m values of 10, 20, and 30 were considered. Based on the characteristics of the Weibull distribution function, the lower the m value, the more discrete the distribution of the critical stretch of the concrete materials, and the greater the m value, the closer the critical stretch of the concrete materials to the mean value.

^{−8}s. The upper and lower (UAL) ends of the specimen were subjected to a tensile displacement load with a rate of 0.03 mm/s. Four cases were considered: homogeneous isotropy (without fiber), heterogeneous isotropy (without fiber), homogeneous anisotropy (with fiber), and heterogeneous anisotropy (with fiber).

#### 6.1. Tensile Failure of an Isotropic Concrete Plate (without Fiber)

^{3}. The homogenization and critical stretch s

_{0}were considered to obey the Weibull distribution, with a mean value of 0.02, and shape parameters (m values) of 10, 20, and 30.

#### 6.2. Tensile Failure of an Anisotropic FRC Plate (θ = 0°)

_{1}in the direction of the fiber bond was 17.2 GPa, and the critical stretch s

_{a}

_{0}= 0.03. The properties in the other directions were the same as those of the isotropic specimens. In addition to the homogeneous condition, it was considered that the critical stretch of the matrix bond s

_{m}

_{0}and the critical stretch of the fiber bond s

_{a}

_{0}obeyed the Weibull distribution, with mean values of 0.02 and 0.03, respectively, and shape parameter m values of 10, 20, and 30.

#### 6.3. Tensile Failure of an Anisotropic FRC Plate (θ = 45°)

#### 6.4. Damage Degree Analysis

## 7. Conclusions

- (1)
- Comparing the analytical solution to the cantilever beam deformation and the experimental results of a fiber-reinforced orthotropic plate with flaws, it was verified that the improved PD model could effectively simulate the failure process of anisotropic FRC materials with flaws.
- (2)
- Under isotropic conditions, the stronger the specimen’s heterogeneity, the more severe the failure degree, but the failure mode remained unchanged as tensile failure.
- (3)
- Under orthotropic conditions, the specimen changed from tensile failure to shear failure when the fiber bond was 0°. When the fiber bond was 45°, the cracks propagated in strict accordance with the direction of the fiber bond, eventually forming sections that ran through the entire specimen, and leading to its failure.
- (4)
- The stronger the concrete’s heterogeneity, the greater the increase in the damage degree. In the range of 0°–90°, the greater the angle between the fiber bond and the tensile load, the lower the damage degree of the specimen.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of the variables in the BB-PD model. (

**a**) Horizon of x and pairwise force function; (

**b**) Deformation diagram.

**Figure 8.**Geometric models of orthotropic thin plates with pre-existing flaws under tensile loads. (

**a**) Flaw is a fracture; (

**b**) Flaw is a hole.

**Figure 9.**Experimental and PD simulation results of the tensile failure of a plate with fractures (θ = 0°). (

**a**) 5.5 μs; (

**b**) 8 μs; (

**c**) 14 μs; (

**d**) Experimental results [38].

**Figure 10.**Experimental and PD simulation results of the tensile failure of a plate with fractures (θ = 45°). (

**a**) 11 μs; (

**b**) 12.5 μs; (

**c**) 15.3 μs; (

**d**) Experimental results [38].

**Figure 11.**Experimental and PD simulation results of the tensile failure of a plate with holes (θ = 0°). (

**a**) 5.5 μs; (

**b**) 8 μs; (

**c**) 15 μs; (

**d**) Experimental results [38].

**Figure 12.**Experimental and PD simulation results of the tensile failure of a plate with holes (θ = 45°). (

**a**) 12 μs; (

**b**) 13.5 μs; (

**c**) 15.3 μs; (

**d**) Experimental results [38].

**Figure 14.**Crack propagation process of a homogeneous isotropic plate with flaws. (

**a**) 8.8 μs; (

**b**) 9.1 μs; (

**c**) 9.5 μs; (

**d**) 10.6 μs; (

**e**) 11.3 μs.

**Figure 15.**Failure modes of heterogeneous isotropic plates with flaws. (

**a**) m = 10, 10.1 μs; (

**b**) m = 20, 11.3 μs; (

**c**) m = 30, 11.3 μs.

**Figure 16.**Crack propagation process of a homogeneous plate with flaws when the fiber bond was 0°. (

**a**) 7.5 μs; (

**b**) 13.5 μs; (

**c**) 19.9 μs.

**Figure 17.**Failure modes of the heterogeneous plate with flaws when the fiber bond was 0°. (

**a**) m = 10, 14.8 μs; (

**b**) m = 20, 18.8 μs; (

**c**) m = 30, 19.6 μs.

**Figure 18.**Crack propagation process of the homogeneous plate with flaws when the fiber bond was 45°. (

**a**) 7.5 μs; (

**b**) 11.0 μs; (

**c**) 12.8 μs.

**Figure 19.**Failure modes of a heterogeneous plate with flaws when the fiber bond was 45°. (

**a**) m = 10, 12.2 μs; (

**b**) m = 20, 12.8 μs; (

**c**) m = 30, 12.8 μs.

**Figure 20.**Curve of the change in damage degree. (

**a**) Isotropy; (

**b**) 0° fiber bond; (

**c**) 45° fiber bond.

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**MDPI and ACS Style**

Zhou, L.; Zhu, S.; Zhu, Z.; Xie, X.
Simulations of Fractures of Heterogeneous Orthotropic Fiber-Reinforced Concrete with Pre-Existing Flaws Using an Improved Peridynamic Model. *Materials* **2022**, *15*, 3977.
https://doi.org/10.3390/ma15113977

**AMA Style**

Zhou L, Zhu S, Zhu Z, Xie X.
Simulations of Fractures of Heterogeneous Orthotropic Fiber-Reinforced Concrete with Pre-Existing Flaws Using an Improved Peridynamic Model. *Materials*. 2022; 15(11):3977.
https://doi.org/10.3390/ma15113977

**Chicago/Turabian Style**

Zhou, Luming, Shu Zhu, Zhende Zhu, and Xinghua Xie.
2022. "Simulations of Fractures of Heterogeneous Orthotropic Fiber-Reinforced Concrete with Pre-Existing Flaws Using an Improved Peridynamic Model" *Materials* 15, no. 11: 3977.
https://doi.org/10.3390/ma15113977