- Open Access

# Modeling of Mechanical Properties of Concrete Mixed with Expansive Additive

- Hyeonggil Choi
^{1}and - Takafumi Noguchi
^{2}Email author

**9**:113

https://doi.org/10.1007/s40069-015-0113-8

© The Author(s) 2015

**Received:**8 December 2014**Accepted:**23 August 2015**Published:**16 September 2015

## Abstract

This study modeled the compressive strength and elastic modulus of hardened cement that had been treated with an expansive additive to reduce shrinkage, in order to determine the mechanical properties of the material. In hardened cement paste with an expansive additive, hydrates are generated as a result of the hydration between the cement and expansive additive. These hydrates then fill up the pores in the hardened cement. Consequently, a dense, compact structure is formed through the contact between the particles of the expansive additive and the cement, which leads to the manifestation of the strength and elastic modulus. Hence, in this study, the compressive strength and elastic modulus were modeled based on the concept of the mutual contact area of the particles, taking into consideration the extent of the cohesion between particles and the structure formation by the particles. The compressive strength of the material was modeled by considering the relationship between the porosity and the distributional probability of the weakest points, i.e., points that could lead to fracture, in the continuum. The approach used for modeling the elastic modulus considered the pore structure between the particles, which are responsible for transmitting the tensile force, along with the state of compaction of the hydration products, as described by the coefficient of the effective radius. The results of an experimental verification of the model showed that the values predicted by the model correlated closely with the experimental values.

## Keywords

- expansive additive
- compressive strength
- elastic modulus
- pore volume
- modeling

## 1 Introduction

The application of an expansive additive is known to be an effective means of reducing shrinkage and increasing crack resistance. Thus, such an application is gradually becoming more common in construction projects (Choi et al. 2012a, b). In this study, we attempted to theoretically model the compressive strength and elastic modulus of hardened cement that had been treated with an expansive additive to reduce the shrinkage. The compressive strength and elastic modulus of hardened cement paste mixed with an expansive additive are closely related to the structure formation of the hardened cement paste by the hydration reaction of the cement and expansive additive (Al-Rawi 1976; Woods 1933; Taplin 1959). In other words, hydrates are generated as a result of the hydration between the cement and expansive additive; these hydrates then fill up the pores in the hardened cement. Consequently, a dense, compact structure is formed through the contact between the particles of the expansive additive and cement, which leads to the manifestation of the strength and elastic modulus. Hence, it is important to determine the change in the organizational structure of a hardened cement paste by the progression of the hydration to estimate the strength and elastic modulus of a hardened cement paste mixed with an expansive additive.

## 2 Modeling of Compressive Strength

### 2.1 Equation of Strength Development

The suggested \( B \) value of Ryshkewitch’s equation is 5.43 for a low pore volume of less than 0.3 based on the concept of the contact area in the C-CBM model (Maruyama 2003). In addition, for the high-porosity range, values of 64.5 and 0.523 are suggested for \( C \) and \( P_{cr} \) of Schiller’s equation, respectively, based on the least squares method, with the data showing the relationship between the porosity and strength from Schiller’s equation.

### 2.2 Modeling of Micro-pore Structure

The total porosity of cement paste mixed with an expansive additive was acquired by adding the porosities of the cement and expansive additive parts based on a hydration reaction. This study utilized an existing space formation model (Maruyama 2003; Park and Lee 2005; Park 2004) based on the hydration of the cement, which was used to acquire the porosities of the cement and expansive additive.

^{3}cube. The density of water is denoted as \( \rho_{w} \), the density of the expansive additives is denoted as \( \rho_{e} \), and the water-to-binder ratio is denoted as \( x \). Given these conditions, the volume of the expansive additive in the cube can be expressed as follows:

^{3}cube is obtained as

Here, the hydration reaction rate \( a \) can be used to define the amounts of cement and expansive additive that have reacted from their corresponding total amounts. The calories generated during hydration were measured using a multi-microcalorimeter (MMC-511 SV), and this value was used to determine the hydration reaction rate.

On the other hand, for the cement volume that contributes to the reaction in the hydration reaction model, the volume increase rate *V* of the cement is obtained by adding approximately 75 % of the water volume in the total volume of gel with the volume of gel produced through the hydration reaction by chemically combining cement and water that constitute approximately 25 % of the weight of cement (Tashiro 1993). Therefore, the volume changes calculated using the cement density indicate that the appropriate volume increase rate during the complete hydration of the cement is approximately 1.59. However, the volume increase rate is expected to increase if the adsorbed water present in the hydration product or the water in the gel pores that does not contribute to the hydration is considered. For this reason, in previous studies (Powers and Brownyard 1947; van Breugel 1997), the volume increase rate of cement was defined as being in the range of 1.9–2.2. Using this range as a reference, in this study, we set the volume increase rate to 2.0 by considering the time that it converged to a certain value with regard to the hydration reaction rate, along with the adsorbed water present in the hydration product or the water in the gel pores that did not contribute to the hydration reaction.

_{3}A in the cement. In other words, in a case where only the reaction of the expansive additive is considered, it is necessary to consider the ettringite hydration product on the surface of the C

_{3}A in the cement. Therefore, in this study, the volume of the hydration product layer produced by the hydration of only the expansive additive was acquired by considering the volume ratio of each of the assumed hydration products (calcium hydroxide and ettringite) that were expanded as a result of the expansive additive. Figure 4 shows the volume of the ettringite hydration product created on the surface of the C

_{3}A in the cement by the hydration of the expansive additive. This is the volume of the hydration product with the porosity created by the hydration reaction of the expansive additive. Considering the volume change of the hydrates, as previously discussed, the volume increase rate \( V \) of the expansive additive is defined as being 3.34, which is a relatively constant value.

Area of surface, and volume and porosity of particles by outermost radius \( R_{t} \) (Maruyama 2003).

Limitless part: \( R_{t} < 0.5 \) | |

Area of a surface | \( S_{1} = 4\pi R_{t}^{2} \) |

Volume | \( V_{1} = \frac{4}{3}\pi R_{t}^{3} \) |

Porosity | \( P_{1} = (1 - V_{1} )\cdot p_{EX} \) |

First period of contact: \( 0.5 \le R_{t} < \sqrt 2 /2 \) | |

Area of a surface | \( S_{2} = 4\pi R_{t}^{2} - 12\pi \left( {1 - \frac{0.5}{{R_{t} }}} \right) \) |

Volume | \( V_{2} = \frac{4}{3}\pi R_{t}^{3} - 6\pi \left( {\frac{2}{3}R_{t}^{3} - \frac{1}{2}R_{t}^{2} + \frac{1}{24}} \right) \) |

Porosity | \( P_{2} = (1 - V_{2} )\cdot p_{EX} \) |

Later period of contact: \( \sqrt 2 /2 \le R_{t} < \sqrt 3 /2 \) | |

Area of a surface | \( S_{3} = \mathop \int \limits_{{\sqrt {R_{t}^{2} - 1/2} }}^{1/2} \mathop \int \limits_{{(R_{t}^{2} - 1/2)/(4 - x^{2} )}}^{1/2} \frac{{R_{t} }}{{\sqrt {R_{t}^{2} - x^{2} - y^{2} } }}dxdy \) |

Volume | \( \begin{aligned} V_{3} = \,&2\sqrt {R_{t} ^{2} - 1/2} + 16\int\limits_{{\sqrt {R_{t} ^{2} - 1/2} }}^{{1/2}} \left( {\frac{1}{2}\cdot 0.5 \cdot \sqrt {R_{t} ^{2} - x^{2} - 1/4} } \right)\\ &+\, \frac{{R_t ^2}-{x^2}}{2} \times \left[ \frac{\pi}{4}-Arc\,cos \left (\frac{0.5}{\sqrt{{R_t ^2}-{x^2}}} \right )\right] {\rm d}x \end{aligned}\) |

Porosity | \( P_{3} = (1 - V_{3} )\cdot p_{EX} \) |

The previous discussion showed that the compressive strength of cement paste mixed with an expansive additive can be acquired using Eqs. (4) and (5), which show the relations of the pore volume and strength at each range of porosity using the total pore volume acquired by Eq. (11).

## 3 Modeling of Elastic Modulus

### 3.1 Elastic Modulus of Cement Paste Without Pore

For the strength development of hardened cement, the strength and porosity seem to be highly correlated because of the stress concentration, which occurs as a result of the strength and pores of the hydration product. However, it is not easy to determine the characteristics of the elastic modulus using only its relationship with the pores because the compressive strength and elastic modulus show the same characteristic in relation to the porosity. However, the relationship between the porosity and compressive strength is expressed as a spatial probability distribution, which is dependent on the existence probability of the weakest point in the particle connections of the hardened cement, whereas the correlation between the porosity and elastic modulus is a parameter that represents the spatial structure that shows how the stress is transferred in the state before reaching the point of destruction (Maruyama 2003). Based on these characteristics, when the paste matrix is thought of as a hardened part of the gel, non-hydrated cement, and non-hydrated expansive additive, and as a two-phase pore material, elastic modulus models for the hardened part with and without pores are needed, both of which use the concept of the contact area. On the other hand, for a hardened cement mixed with the expansive additive, two elastic moduli are needed, one for the cement part and the other for the expansive additive part. In consideration of the mix ratio of the expansive additive, each elastic modulus must have a balancing equation.

Here, \( E_{C,EX} \) and \( E_{gel(C,EX)} \) have values of 50 and 25 GPa, respectively. These values were taken from the results of the study by Maruyama (2003) on the calculation of the elastic modulus of cement. Maruyama determined these values by referring to the values of 40 and 20 GPa suggested by Hua et al. (1997) and 60 and 30 GPa suggested by Lokhorst and van Breugel (1997). In the case of the expansive additive, the cement value was applied because sufficient data could not be found.

### 3.2 Elastic Modulus of Cement Paste with Pore

### 3.3 Elastic Modulus of Concrete

In order to extend the elastic modulus to concrete, the concrete is considered to consist of two materials, the aggregate and paste. It is also assumed that there is no behavior by the aggregate. Hence, the behavior of the paste dominates in the behavior of the concrete. Therefore, the aggregate has a resistor function without affecting the behavior of the paste.

## 4 Verification of Model

### 4.1 Compressive Strength

An experiment was performed to verify the compressive strength model. Water and binder at a ratio of 0.50 was mixed with 0, 5, and 10 % expansive additive (Ettringite-gypsum type, Density 3.05 g/cm^{3}). The increase in the coefficient \( p_{EX} \) of the capillary pores generated by the volume increase with this expansive additive admixture was set by referring to the studies of Yaniamoto and Morioka (Yanimoto et al. 2003; Morioka et al. 1999), in which a mercury porosimeter test measured the change in a specimen’s porosity.

### 4.2 Elastic Modulus

^{3}) was mixed as a binder, where the water/binder ratio was 0.50. The actual values for the paste measured in the experiment and the model’s estimated values are shown in Fig. 14. In addition, this figure shows the estimated elastic modulus of the concrete and Hori’s data for the elastic modulus of the concrete after 3 and 7 days, where the concrete included 30 kg (approximately 10 % of the cement) of the expansive additive, with a water/binder ratio of 0.50 (Hori et al. 2000). Elastic modulus of aggregates were used 60 GPa for the calculation on elastic modulus of concrete. As shown in the figure, we confirmed that the elastic modulus estimates were positively correlated with the experiment values. It was confirmed that the aging of the elastic modulus of the concrete followed the estimate suggested in Hori’s data and experimental data, indicating that the composite model can estimate the elastic modulus of concrete based on the behavior of the paste.

## 5 Conclusions

- (1)
In the modeling of the mechanical properties of the concrete using the expansive additive, in the hardened cement paste with an expansive additive, hydrates are generated as a result of the hydration between the cement and expansive additive; these hydrates then fill up the pores in the hardened cement. Consequently, a dense, compact structure is formed through the contact between the particles of the expansive additive and the cement, leading to the manifestation of the strength and elastic modulus. Hence, modeling was performed to evaluate the compressive strength by assuming that the strength development of the cement paste was closely related to the pore volume.

- (2)
On the other hand, the elastic modulus of the hardened part without pores was modeled by assuming that the paste matrix consisted of the gel, non-hydrated cement, and hardened part of the non-hydrated expansive additive, with two-phase pores in the case where it contained pores. The elastic modulus was modeled using the concept of the effective radius factor and effective contact area.

- (3)
The estimates of the models were positively correlated with experimental values, which verified the compressive strength model and elastic modulus model. In addition, the elastic modulus of the concrete could effectively be estimated based on the composite model of the aggregate and paste.

## Declarations

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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