# Mesoscale Fracture Analysis of Multiphase Cementitious Composites Using Peridynamics

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## Abstract

**:**

## 1. Introduction

#### 1.1. Mesoscale Modeling

#### 1.2. Analysis Methods Available for Fracture Modeling

#### 1.3. Summary of Work Presented in the Remaining Sections

#### 1.4. Novel Aspects of the Research

## 2. Proposed Mesoscale Concrete Model

#### 2.1. Mesoscopic Constituent Materials

#### 2.2. Distribution of Constituent Materials into Multi-Phases

- Determine the total area of the coarse aggregate, ${A}_{agg}$, with the aggregate size between ${d}_{i}$ and ${d}_{i+1}$, using Equation (1).
- Generate a random number, which defines the aggregate diameter, d, within the segment $[{d}_{i},{d}_{i+1}]$. The aggregate diameter, d, is obtained by $d={d}_{i}+\mathsf{\eta}({d}_{i+1}-{d}_{i})$, where $\mathsf{\eta}$ is a variable selected from a uniform distribution of numbers between zero and one by using the “$RANDOM\_NUMBER$” command in FORTRAN.
- Generate two sets of random numbers to define the location of current aggregate. Two numbers are selected from a uniform distribution, with equal probability for all values, of the random variables between zero and one.
- Check the placement of aggregate: Two conditions must be met to position the aggregate. First, the aggregate must be located within the analytical specimen boundary with a minimum clearance distance from the specimen boundary, ${\mathsf{\gamma}}_{1}$. There must be no overlapping area between current aggregate and previously-placed aggregate, if any. A minimum distance of ${\mathsf{\gamma}}_{2}$ between the two aggregates must be considered. These two conditions assure that the current aggregate is reasonably surrounded by the cementitious matrix. For the coarse aggregate distribution, $0.1d$ and $0.1(d+{d}^{\prime})/2$ are used for ${\mathsf{\gamma}}_{1}$ and ${\mathsf{\gamma}}_{2}$, respectively. d is the diameter of the current aggregate being positioned, and ${d}^{\prime}$ is the diameter of previously-positioned aggregate.
- Repeat the random placement in Steps 2–3 until the conditions in Step 4 are satisfied.
- Determine the total area of generated aggregates, ${A}_{agg}^{\prime}$, in this segment, and find the remaining aggregate area by subtracting ${A}_{agg}^{\prime}$ from the total aggregate area, ${A}_{agg}$, determined in Step 1.
- Repeat Steps 2–6 until the remaining area is no longer available to generate additional aggregate in the grading segment.

#### 2.3. Formulation of the Proposed Analysis Framework

#### 2.4. Determination of Statistically-Significant Sample Size

## 3. Analysis of a Concrete Specimen in Tension

#### 3.1. Description of a Numerical Model

#### 3.2. Material Models

## 4. Analysis Results and Discussion of the Results

#### 4.1. Effect of Particle Spacing and Convergence

#### 4.2. Effect of Loading Increment and Dynamic Relaxation Threshold

#### 4.3. Comparison with Available Finite Element Analysis Results

## 5. Sensitivity Study

#### 5.1. Effect of Aggregate Volume Fraction

#### 5.2. Effect of Porosity

#### 5.3. Comparison with the FEM Results and Damage Patterns

## 6. Discussion

- Studying the influence of particle shape and surface texture of the coarse aggregate.
- Refinement in particle spacing and/or horizon size in ITZs to reduce the computational time while providing accuracy.
- Enhancing material models and damaged parameters for complex loading conditions, such as shear.
- Extending the proposed analysis approach to three-dimensional (3D) models in which, for instance, non-planar (or 3D) fracture surfaces are characterized.
- Advanced computational techniques to optimize computational expense.

## 7. Conclusions

- The results indicate that particle spacing affects the stress convergence, as well as crack patterns. For the tensile loading condition considered herein, the particle spacing of $1/3$ mm provides the most effective discretization with a reasonable computational effort.
- The results of the mesoscale analysis show that the simulation time is sensitive to the displacement loading increment used for dynamic relaxation. Based on the analysis results reported in this paper, the optimal displacement increment and DR threshold is $5\times {10}^{-4}$ mm and ${10}^{-7}$, respectively, for tensile loading.
- By means of interfacial transition zones (ITZs) characterized in the proposed mesoscale model, it is capable of reflecting the effect of varying coarse aggregate volume fractions on the load carrying capacity.
- It is concluded that the strength reduction due to increased air content is reflected in the proposed model, by removing particle points from the areas of voids.
- Finally, in the proposed mesoscale peridynamics analysis, it is possible to identify a statistically-significant sample size for reasonably representing coarse aggregate gradation and distribution and to predict the fracture mechanism in concrete specimens.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 3.**Aggregate gradations achieved by simulation versus the Fuller curve. (

**a**) ${P}_{agg}=20\%$; (

**b**) ${P}_{agg}=30\%$; (

**c**) ${P}_{agg}=40\%$; (

**d**) ${P}_{agg}=50\%$.

**Figure 5.**Numerical specimens with varying particle spacing, h. (

**a**) $h=1/2$ mm; (

**b**) $h=1/3$ mm; (

**c**) $h=1/4$ mm.

**Figure 6.**Tensile stress versus applied displacement for varying particle spacing. (

**a**) $h=1/2$ mm; (

**b**) $h=1/3$ mm; (

**c**) $h=1/4$ mm.

**Figure 7.**Effect of discretization on: (

**a**) mean stress-displacement curves; and (

**b**) sample number with reference to mean peak stress.

**Figure 8.**Effect of particle spacing size on crack pattern. (

**a**) $h=1/2$ mm; (

**b**) $h=1/3$ mm; (

**c**) $h=1/4$ mm.

**Figure 10.**Effect of (

**a**) the displacement loading increment and (

**b**) the dynamic relaxation threshold on the mean stress curve.

**Figure 11.**The effect of (

**a**) the displacement loading increment and (

**b**) the dynamic relaxation threshold on the simulation time.

**Figure 13.**Effect of (

**a**) aggregate volume fraction, ${P}_{agg}$, and (

**b**) porosity, ${P}_{pore}$, on the mean stress curve.

**Figure 14.**Mean peak stress for varying (

**a**) aggregate volume fraction, ${P}_{agg}$, and (

**b**) porosity, ${P}_{pore}$.

**Table 1.**Gradation of the coarse aggregate [5].

Sieve size (mm) | 19.00 | 12.70 | 9.50 | 4.75 | 2.36 |

Total percentage passing (%) | 100 | 97 | 61 | 10 | 1.4 |

**Table 2.**Material properties [5]. ITZ, interfacial transition zone.

Constituent | Young’s Modulus, E (MPa) | Poisson’s Ratio, $\mathsf{\nu}$ | Fracture Energy, ${\mathit{G}}_{\mathit{f}}$ (N/mm) |
---|---|---|---|

Aggregate | 70,000 | 0.2 | - |

Mortar | 25,000 | 0.2 | 0.06 |

ITZ | 25,000 | 0.2 | 0.03 |

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**MDPI and ACS Style**

Yaghoobi, A.; Chorzepa, M.G.; Kim, S.S.; A., S. Mesoscale Fracture Analysis of Multiphase Cementitious Composites Using Peridynamics. *Materials* **2017**, *10*, 162.
https://doi.org/10.3390/ma10020162

**AMA Style**

Yaghoobi A, Chorzepa MG, Kim SS, A. S. Mesoscale Fracture Analysis of Multiphase Cementitious Composites Using Peridynamics. *Materials*. 2017; 10(2):162.
https://doi.org/10.3390/ma10020162

**Chicago/Turabian Style**

Yaghoobi, Amin, Mi G. Chorzepa, S. Sonny Kim, and Stephan A. 2017. "Mesoscale Fracture Analysis of Multiphase Cementitious Composites Using Peridynamics" *Materials* 10, no. 2: 162.
https://doi.org/10.3390/ma10020162