# Microscale Testing and Modelling of Cement Paste as Basis for Multi-Scale Modelling

^{*}

## Abstract

**:**

^{3}) and beams with a square cross section of 400 × 400 µm

^{2}. By loading the cubes to failure with a Berkovich indenter, the global mechanical properties of cement paste were obtained with the aid of a nano-indenter. Simultaneously the 3D images of cement paste with a resolution of 2 µm

^{3}/voxel were generated by applying X-ray microcomputed tomography to a micro beam. After image segmentation, a cubic volume with the same size as the experimental tested specimen was extracted from the segmented images and used as input in the lattice model to simulate the fracture process of this heterogeneous microstructure under indenter loading. The input parameters for lattice elements are local mechanical properties of different phases. These properties were calibrated from experimental measured load displacement diagrams and failure modes in which the same boundary condition as in simulation were applied. Finally, the modified lattice model was applied to predict the global performance of this microcube under uniaxial tension. The simulated Young’s modulus agrees well with the experimental data. With the method presented in this paper the framework for fitting and validation of the modelling at microscale was created, which forms a basis for multi-scale analysis of concrete.

## 1. Introduction

^{3}) using nano-indenter, which provides an unprecedented opportunity for validation of mechanical simulation results at the microscale [32]. The new method uses nano-indentation equipment to assess the fracture properties of small specimens, unlike regular nano-indentation testing that is used to assess elastic modulus and hardness. This method is further developed and presented in this paper. The method for experimental testing and numerical simulation of specimens on the same size under the same boundary conditions is addressed. The adopted mechanical properties of local phases are evaluated by comparing the simulated damage evolution and load displacement diagram with the experimental observation. In addition, the calibrated model is applied to predict the global mechanical properties of micro cement cube under uniaxial tension. The predicted results are compared with the results of the previous works.

## 2. Experimental

#### 2.1. Sample Preparation

^{3}) for global mechanical performance test and microbeams with a cross section of 400 × 400 µm

^{2}for CT scan. Standard grade OPC CEM I 42.5 N cement paste with w/c ratio 0.40 was cast in a PVC cylinder (diameter, 24 mm, height 39 mm) in sealed condition. After 24 h rotation and curing 28 days at room temperature (20 °C), specimens were demoulded and two discs with the thickness of 2 mm were cut from the middle part. One of the pieces was used to create the microcubes, while the other was used to create the microbeams. The hydration was arrested by solvent exchange method using isopropanol [33]. In order to enable faster water-solvent exchange, samples were immerged five times and taken out for a period of one minute. Afterwards, they were placed for 3 days in isopropanol and subsequently taken out, and solvent was removed by evaporation at ambient conditions.

^{3}and beams are with a cross-section of 400 × 400 µm

^{2}as shown in Figure 2.

#### 2.2. Global Micro-Mechanical Performance Using Microcube Indentation

^{−1}. More detail about the loading procedure is discussed in [34] in which carbon nano tube bundles are loaded to failure with a flat nano-indenter. The load-displacement response up until failure of the microcube is shown in Figure 4. In total, 8 load-displacements are measured in the experiments for cement paste with a w/c ratio 0.4. Multiple measurements on different cubes show a high degree of repeatability. Two regimes can be distinguished from the paragraph. In regime (I), the load on sample increases monotonically for increasing indenting until reaches the peak load. The maximum load that can be applied before the microcube collapses is between 350 mN and 450 mN at a critical displacement between 10 µm and 15 µm. Once this load is exceeded, the system transitions from a stable regime (I) towards an unstable regime (II) with rapid displacement bursts. The horizontal line in regime (II) indicates structural collapse of the microcube, which results in an overshoot of the nano-indentation tip towards the substrate. Since displacement control of the nano-indenter is not fast enough, it is not possible at present to capture the post-peak behaviour of the specimen. Therefore, the calibration of the numerical model was carried out only in regime (I). It is observed that the test results still show variability which is induced by the inherent heterogeneity of the material.

#### 2.3. Microstructure Characterization Using of Micro-CT

^{3}. Reconstructed slices were carried out with Phoenix Datos software. For saving the calculation time of mechanical model, the original resolution of reconstructed slices was reduced to 2 µm

^{3}/voxel. Image segmentation was performed using a so-called global threshold approach [7,9,13,35]. In this method, phases were isolated from the original grey-scale map by choosing the corresponding threshold step by step as shown in Figure 6. Firstly, two threshold grey values are defined on the basis of the grey-level histogram as shown in Figure 7: T

_{1}, pore/solid phase threshold, is assumed as the grey value at the inflection point in the cumulative fraction curve of the histogram; T

_{2}, hydration products/anhydrous cement phase threshold, is a critical point at which the tangent slope of the histogram changes suddenly. Three phases can be isolated: pore, anhydrous cement and hydration product. More details about this approach can be found in previous work [13,35]. However, it is well known that at least three types of hydration product with different mechanical properties [14,15,16,17] exist: inner product C-S-H

_{LD}, outer product C-S-H

_{HD}and C-H. In order to simplify procedure, C–H was not considered as a separate phase, and therefore was not explicitly modelled. This simplification is considered not to significantly affect the results of mechanical properties simulation [30]. However, in further work, C-H as a separated phase should be considered.

_{LD}and C-S-H

_{HD}. As shown in Figure 8, the input for this model are w/c ratio and degree of hydration which can be estimated on the basis of volume fraction of anhydrous cement V

_{anhydrous}and hydration products V

_{hydrated}according to equation:

_{3}can be determined from the cumulative volume fraction cure of grey-histogram as shown in Figure 7. The voxels with a grey value lower than T

_{3}are regarded as outer hydration product, while the ones with higher values are inner hydration product. A cubical region of interest with 100 µm in length was extracted from the segmented microstructure for lattice fracture analysis as shown in Figure 9a. Microstructure characterization of cement paste with w/c ratio 0.3 and 0.5 at the same curing age of 28 days were carried out using the same procedure and shown in Figure 9c.

## 3. Modelling

#### 3.1. Modelling Approach

^{3}) is divided into a cubic grid with a cell size of 2 µm

^{3}. Then, a sub-cell was defined within each cell in which a node is randomly placed. The ratio between the length of sub-cell and cell is defined as randomness. As shown in preview study [38], the choice of randomness affects the simulated the fracture behaviour of materials, because the simulated crack shape is affected by the orientation of meshes. In order to avoid big variations in length of elements and introduce geometry disorder of material texture, a randomness of 0.5 is adopted. Then, Delaunay triangulation is performed to connect the four nodes that are closest to each other with lattice elements. Afterwards, the cross-section of lattice element is determined by altering this parameter in a homogenous lattice model until the simulated global Young’s modulus matches the assumed local value.

_{i}, E

_{j}and E

_{ij}, are the Young’s modulus or shear modulus for phase i, phase j and element which connects phase i and phase j, respectively. The compressive strength and tensile strength take the lower value of the connected two phases, which can be expressed in [30]:

_{ij}= min(f

_{i}, f

_{j}),

_{i}, f

_{j}and f

_{ij}, are the compressive strength or tensile strength for phase i, phase j and element which connects phase i and phase j, respectively. The mechanical properties of these pure phases was preferred to be measured in laboratory test, but in the case of a lack of experimental data, properties of these phases are commonly derived from the nano-indentation measurements [30,31]. However, no data is available defining the relationship between the model parameters (tensile and compressive strength, Young’s modulus) and nano-indentation results (indentation hardness and modulus of elasticity). Herein, in order to work out this relationship, the simulated fracture performance is compared with the experimental results to find out the best simulation, and these values are further used in Section 3.3 to predict the mechanical and fracture properties of hydrated cement pastes with different w/c ratio. Ratios of tensile strength (and compressive strength) among each phase are assumed to be equal to the ratios of measured hardness among these phases in this calibration. For individual phases, values reported in [14] (Table 2) were used in this work. Since scanning electron microscope was adopted to reflect phases at the tested location for the statistics analysis, this method gives more reliable results.

#### 3.2. Calibration and Discussion

#### 3.3. Tensile Strength Prediction and Discussion

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 5.**Three stages in the nano-indentation loading process of microcubes observed in ESEM: (

**a**) initial stage of loading; (

**b**) three main cracks running to the sides of the cubes; (

**c**) complete crushing of the sample (adapted from [32]).

**Figure 6.**2D Schematic view of image segmentation process: (

**a**) original grey-scale map; (

**b**) pore (blue) and solid phases (yellow) are isolated from the grey-scale map; (

**c**) anhydrous cement (grey) and hydration product (yellow) are isolated form solid phases; (

**d**) outer product (yellow) and inner product (red) are segmented from hydration product.

**Figure 9.**3D segmented microstructure (100 × 100 × 100 µm

^{3}) of cement paste with w/c ratio (

**a**) 0.3; (

**b**) 0.4; (

**c**) 0.5 (grey-anhydrous cement; red-inner product; yellow-outer product; blue-pore).

**Figure 10.**Schematic view of lattice model generation: (

**a**) lattice network construction (5 × 5 × 5); (

**b**) overlay procedure for 2D lattice mesh (yellow-outer product; red-inner product; grey-anhydrous cement).

**Figure 11.**Comparison between simulated load displacement diagrams and experimental results of cement paste with w/c ratio 0.4.

**Figure 12.**Crack patterns in the final failure state: (

**a**) simulation of S1; (

**b**) simulation of S2; (

**c**) simulation of S3; (

**d**) element types (black-damaged element).

**Figure 14.**Crack patterns in the final failure state of cement paste with w/c ratio (

**a**) 0.3; (

**b**) 0.4; (

**c**) 0.5 (left: whole specimen; right: only damage).

Element Type | Phase 1 | Phase 2 |
---|---|---|

A–A | Anhydrous cement | Anhydrous cement |

I–I | Inner product | Inner product |

O–O | Outer product | Outer product |

A–I | Anhydrous cement | Inner product |

I–O | Inner product | Outer product |

A–O | Anhydrous cement | Outer product |

**Table 2.**Measured mechanical properties of individual local phases from [14].

Phases | Modulus of Elasticity (GPa) | Hardness (GPa) |
---|---|---|

Anhydrous cement | 99.2 | 8.24 |

Inner product | 31.6 | 1.14 |

Outer product | 25.2 | 0.75 |

Set | Anhydrous Cement | Inner Product | Outer Product | ||||||
---|---|---|---|---|---|---|---|---|---|

E (GPa) | f_{t} (GPa) | f_{c} (GPa) | E (GPa) | f_{t} (GPa) | f_{c} (GPa) | E (GPa) | f_{t} (GPa) | f_{c} (GPa) | |

S1 | 99.2 | 0.683 | 6.830 | 31.6 | 0.092 | 0.92 | 25.2 | 0.0583 | 0.58 |

S2 | 99.2 | 0.683 | 68.3 | 31.6 | 0.092 | 9.2 | 25.2 | 0.0583 | 5.8 |

S3 | 99.2 | 0.683 | ∞ | 31.6 | 0.092 | ∞ | 25.2 | 0.058 | ∞ |

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**MDPI and ACS Style**

Zhang, H.; Šavija, B.; Chaves Figueiredo, S.; Lukovic, M.; Schlangen, E. Microscale Testing and Modelling of Cement Paste as Basis for Multi-Scale Modelling. *Materials* **2016**, *9*, 907.
https://doi.org/10.3390/ma9110907

**AMA Style**

Zhang H, Šavija B, Chaves Figueiredo S, Lukovic M, Schlangen E. Microscale Testing and Modelling of Cement Paste as Basis for Multi-Scale Modelling. *Materials*. 2016; 9(11):907.
https://doi.org/10.3390/ma9110907

**Chicago/Turabian Style**

Zhang, Hongzhi, Branko Šavija, Stefan Chaves Figueiredo, Mladena Lukovic, and Erik Schlangen. 2016. "Microscale Testing and Modelling of Cement Paste as Basis for Multi-Scale Modelling" *Materials* 9, no. 11: 907.
https://doi.org/10.3390/ma9110907