# Microstructure-Based Prediction of the Elastic Behaviour of Hydrating Cement Pastes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Mixtures

^{−bDn}, where D is the diameter of the cement particle, and b and n are coefficients. By employing a curve fitting analysis, the values of 0.062 and 1.00 were calculated for b and n, respectively. The fitted curve is also plotted in Figure 1.

## 3. Microstructure of the Cement Paste

## 4. Mechanical Model

^{3}—is initially discretised into small 1.0 × 1.0 × 1.0 μm

^{3}cubic voxels. Two mechanical approaches, described in the following sections, are adopted to predict numerically the effective elastic modulus of the simulated REV microstructure of the cement paste: (i) the lattice and (ii) the FEM models.

#### 4.1. Lattice Model

#### 4.1.1. Lattice Construction

#### 4.1.2. Boundary Conditions and Effective Elastic Modulus

^{−4}μm to the top surface nodes (plane $Z=n\cdot l$). The effective elastic modulus is derived from the following equation:

^{−4}μm). It should be noted that the differences between the results when loading along direction Z or along directions X or Y were negligible (less than 1%), meaning that the effective elastic modulus is not influenced by the loading direction in the REV.

#### 4.1.3. Beam Elements

#### 4.1.4. Definition of Beam Properties

#### 4.1.5. Definition of Beam Geometry

_{1}for Beam 1 and l

_{2}for Beam 2 can be directly calculated from the coordinates at their nodal extremities. In the proposed regular model, these values are constant and given by:

^{3}, discretised with voxels of 1 × 1 × 1 μm

^{3}. Each voxel represents a single material phase, with a local Young’s modulus ${E}_{l}$. A 3D lattice model constructed with this REV-based unit cell is reproduced in Figure 5a, where ${L}_{eff}=20$ and $l=1.0$. The uniaxial tensile boundary conditions described in Section 4.1.2 are assigned to this lattice model, as shown in Figure 8a. For ${d}_{1}/{l}_{1}$ and ${d}_{2}/{l}_{2}$ ranging from 0.2 to 2.0, the results of the lattice model in terms of ${E}_{eff}/{E}_{l}$ are reproduced in Figure 9a as a surface. An example of the deformation field obtained from such direct tensile test is illustrated in Figure 8a. By increasing both ${d}_{1}/{l}_{1}$ and ${d}_{2}/{l}_{2}$, the value of ${E}_{eff}/{E}_{l}$ increases significantly. Since the model is homogenous (a single phase REV is considered), theoretically the following condition has to be satisfied:

#### 4.2. FEM Model

## 5. EMM-ARM Testing Method

## 6. Results and Discussion

#### 6.1. Lattice vs. FEM Results

#### 6.2. Experimental vs. Numerical Results

^{1}H-NMR testing technique [17] for the materials under testing. It is also criticised that

^{1}H-NMR does not measure all the volume of capillary pores since only pores filled by capillary water are measured for a sealed cement paste sample. Further analysis is thus still required for validation of mechanical properties on the same paste evaluated at different w/c values, and tested by SEM, MIP and possibly

^{1}H-NMR. Regardless of the deficiency that was observed for the cement paste with w/c = 0.48, the overall predictive performance of both the lattice and FEM models can be acceptable.

#### 6.3. Parametric Study

#### 6.3.1. Influence of PSD

#### 6.3.2. Influence of Voxel Size

^{3}. In fact, the voxel size of 0.67 × 0.67 × 0.67 μm

^{3}can be an optimised voxel size for discretisation of the REV. This can be explained by the fact that the artificial solid phase connection becomes a minimum when the voxel size decreases. When the artificial solid phase connection decreases, it reduces the total solid part connectivity in REV, leading to a decrease of the effective Young’s modulus. This resolution is important to capture properly the effect due to the solid percolation threshold of the cement paste at very early ages. As shown in Figure 15, at the age of ~5 h, and decreasing the voxel size from 1.25 μm to 0.57 μm, the accuracy of the model to estimate the experimental value increases.

#### 6.3.3. Influence of Elastic Properties of C-S-H

## 7. Summary and Conclusions

- (i).
- While the volume of cement was kept constant in the cement hydration modelling, a variation of the Particle Size Distribution (PSD) of cement grains resulted in different morphologies in the cement paste microstructure. When the number of particle increases (adopting a finer PSD curve), finer pore networks are formed, leading to a higher number of solid phase contacts in the microstructure. The higher number of solid connections increases the effective Young’s modulus of the cement paste. The phenomenon was opposite when a coarser PSD curve was adopted. Therefore, the pore network morphology has a strong influence on the numerical prediction of the effective elastic modulus of the cement paste.
- (ii).
- The voxel size, directly related to the microstructure discretisation, plays an important role when using numerical models for predicting the elastic modulus of cement pastes. To reduce the artificial solid connections in digitalised microstructures, the size of voxel shall be adequately small. Decreasing the voxel size to about 70% of the minimum size of cement grains seems to be a promising strategy to improve the accuracy of the numerical predictions.
- (iii).
- At very early ages, by decreasing Young’s modulus of C-S-H the numerical models capture the experimental value of the effective Young’s modulus of the cement paste. At these early ages, only low dense C-S-H starts to form in the microstructure, and consequently, a division of the C-S-H product into inner and outer phases leads to an overestimation of the effective Young’s modulus of the cement paste.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**(

**a**) Cement hydration products in HYMOSTRUC3D, (

**b**) Overlapping of the product layers in particle contacts.

**Figure 3.**Microstructure formation of the cement paste in a 2D view: (

**a**) after 1 h of hydration; (

**b**) after 170 h of hydration (grey: unreacted cement grain; red: Inner C-S-H; yellow: Outer C-S-H; green: CH grain; blue: water or void).

**Figure 5.**(

**a**) Construction of the lattice unit cell based on truncated octahedral; (

**b**) positioning of the lattice unit cell on the discretised Representative Elementary Volume (REV) voxels.

**Figure 6.**(

**a**) 3D regular lattice model, (

**b**) FE model—grey: unreacted cement, red: C-S-H inner, yellow: C-S-H outer, green: CH, black: all interface beams, blue: pore phases.

**Figure 7.**Effective Young’s modulus of an interface element calculated with two different methods: the serial/parallel and the Hashin–Shtrikman classical bounds theory

**Figure 8.**The configuration of a homogeneous lattice model of a 20 × 20 × 20 μm

^{3}REV and deformation fields: (

**a**) direct tensile model; (

**b**) direct shear model (DtX, DtY, and DtZ are, respectively, deformation in X, Y and Z directions).

**Figure 9.**Homogeneous and isotopic lattice model for a 20 × 20 × 20 μm

^{3}REV for variations of d

_{1}/l and d

_{2}/l regarding (

**a**) E

_{eff}/E

_{l}; (

**b**) Poisson’s ratio.

**Figure 10.**Calibrated geometrical values for d

_{1}/l and d

_{2}/l are assigned to, respectively, Beams 1 and 2 in the proposed lattice model.

**Figure 11.**Experimental setup for the Elasticity Modulus Measurement through Ambient Response Method (EMM-ARM) testing of cement paste and data processing.

**Figure 12.**Elastic results of lattice and FEM models regarding (

**a**) Young’s modulus; (

**b**) Poisson’s ratio.

**Figure 14.**(

**a**) Three adopted PSD curves in the hydration model by HYMOSTRUC3D; (

**b**) Young’s moduli obtained for the three different PSD curves.

**Figure 16.**Effective Young’s modulus of the cement paste in REV: (

**a**) influence of the outer C-S-H Young’s modulus; (

**b**) influence of the average Young’s modulus of C-S-H when it is assumed as a single-phase product with no division to inner and outer.

**Table 1.**Mineral composition of the white cement [12].

Cement clinker component | C3S | C2S | C3A | C4AF | Others | |
---|---|---|---|---|---|---|

Fraction (%) | 66.89 | 20.59 | 3.62 | 1.03 | 7.87 | |

Density (g/cm^{3}) | 3.21 | 3.28 | 3.03 | 2.32 | ~3.0 | Avg. 3.15 |

Cement Clinker | Velez et al. (2001) [18] | Manzano et al. (2009) [19] | Present Study | Clinker | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

E | G | υ | E | G | υ | E | G | υ | E | G | υ | |

(GPa) | (GPa) | - | (GPa) | (GPa) | - | (GPa) | (GPa) | - | (GPa) | (GPa) | - | |

Alite (C_{3}S) | 135 | 51.9 | 0.3 | 138.9 | 54.5 | 0.28 | 137 | 53 | 0.30 | 137 | 53 | 0.3 |

Belite (C_{2}S) | 130 | 50.0 | 0.3 | 137.9 | 53.1 | 0.3 | 134 | 52 | 0.30 | |||

C_{3}A | 145 | 55.8 | 0.3 | - | - | - | 145 | 56 | 0.30 | |||

C_{4}AF | 125 | 48.1 | 0.3 | - | - | - | 125 | 48 | 0.30 |

Model Hydration Parameters | w/c = 0.32 | w/c = 0.40 | w/c = 0.48 |
---|---|---|---|

k_{0} (reaction rate of cement particle) | 0.035 | 0.055 | 0.04 |

d_{tr} (transition thickness of cement particles) | 2.0 | 2.0 | 2.0 |

β_{1} (calibration parameters) | 1.0 | 1.0 | 1.0 |

β_{2} (calibration parameters) | 1.0 | 1.0 | 1.0 |

Dimensions the REV cube | 100 × 100 × 100 μm^{3} | ||

Reactant | C3S, C2S, C3A, C4AF, and gypsum | ||

Product phases | Inner layer C-S-H; Outer layer C-S-H; CH grain | ||

Minimum and maximum particle size | 1 μm–70 μm | ||

Temperature | 20 °C | ||

PSD parameters: b, n | 0.06, 1.0 (see Figure 1) |

Hydration Product | Elastic Modulus (Average Value) (GPa) | Poisson’s Ratio |
---|---|---|

C-S-H Inner (high-dense) | 26–32 (29) | 0.25 |

C-S-H Outer (low-dense) | 13–26 (19.5) | 0.25 |

CH | 35 | 0.30 |

**Table 5.**Computational details for the lattice and Finite Element Method (FEM) models for w/c = 0.40.

Hydration Time (Hours) | Model | Number of Element | Number of Nodes | Number of Interfaces | Interface Percentage (%) | Solving Time * (sec) | Young’s Modulus (GPa) | Poisson’s Ratio (-) |
---|---|---|---|---|---|---|---|---|

5 | Lattice | 5,199,531 | 1,366,992 | 222,459 | 4.28 | 2641 | 3.60 | 0.248 |

FE | 515,200 | 897,518 | - | 0.0 | 242 | 3.62 | 0.237 | |

15 | Lattice | 6,042,059 | 1,486,326 | 431,944 | 7.15 | 2932 | 6.32 | 0.252 |

FE | 593,551 | 942,238 | - | 0.0 | 242 | 6.32 | 0.239 | |

28 | Lattice | 6,688,144 | 1,564,456 | 543,773 | 8.13 | 3145 | 9.56 | 0.253 |

FE | 651,815 | 964,251 | - | 0.0 | 263 | 9.53 | 0.241 | |

50 | Lattice | 7,278,024 | 1,630,933 | 578,575 | 7.95 | 4086 | 12.31 | 0.255 |

FE | 704,181 | 979,932 | - | 0.0 | 271 | 12.34 | 0.239 | |

84 | Lattice | 7,557,111 | 1,660,055 | 572,667 | 7.58 | 4592 | 13.84 | 0.257 |

FE | 728,502 | 985,321 | - | 0.0 | 274 | 13.86 | 0.241 | |

111 | Lattice | 7,725,450 | 1,677,884 | 574,797 | 7.44 | 5123 | 14.72 | 0.261 |

FE | 743,217 | 988,786 | - | 0.0 | 281 | 14.68 | 0.243 | |

156 | Lattice | 7,904,593 | 1,696,642 | 577,694 | 7.31 | 5637 | 15.32 | 0.263 |

FE | 758,822 | 992,332 | - | 0.0 | 296 | 15.32 | 0.246 |

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**MDPI and ACS Style**

Mazaheripour, H.; Faria, R.; Ye, G.; Schlangen, E.; Granja, J.; Azenha, M.
Microstructure-Based Prediction of the Elastic Behaviour of Hydrating Cement Pastes. *Appl. Sci.* **2018**, *8*, 442.
https://doi.org/10.3390/app8030442

**AMA Style**

Mazaheripour H, Faria R, Ye G, Schlangen E, Granja J, Azenha M.
Microstructure-Based Prediction of the Elastic Behaviour of Hydrating Cement Pastes. *Applied Sciences*. 2018; 8(3):442.
https://doi.org/10.3390/app8030442

**Chicago/Turabian Style**

Mazaheripour, Hadi, Rui Faria, Guang Ye, Erik Schlangen, José Granja, and Miguel Azenha.
2018. "Microstructure-Based Prediction of the Elastic Behaviour of Hydrating Cement Pastes" *Applied Sciences* 8, no. 3: 442.
https://doi.org/10.3390/app8030442