Advances in Materials Science and Engineering

Volume 2018, Article ID 2352864, 11 pages

https://doi.org/10.1155/2018/2352864

## Mesoscopic Finite Element Method of the Effective Thermal Conductivity of Concrete with Arbitrary Gradation

^{1}School of Highway, Changan University, Xian 710064, China^{2}School of Architecture and Civil Engineering, Xian University of Science and Technology, Xian 710064, China^{3}School of Materials Science and Engineering, Changan University, Xian 710064, China

Correspondence should be addressed to Yamin Sun; nc.ude.dhc@nimaynus

Received 4 June 2018; Revised 20 November 2018; Accepted 5 December 2018; Published 27 December 2018

Academic Editor: Paolo Andrea Carraro

Copyright © 2018 Pingming Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The effective thermal conductivity (ETC) of concrete is the most important parameter in determining the temperature field and thermal stresses. A 2D random polygonal aggregate model and its modified model considering porosity were established in this paper in order to partially replace the experiment for parametric analysis on the ETC of concrete and to save the experiment cost. A mesoscopic finite element method for the ETC of concrete with arbitrary gradation was also proposed. In addition, the influence factors (thermal conductivity of coarse aggregate, cement mortar, and volume fraction of coarse aggregate) of the effective thermal conductivity of concrete were analyzed. The results show that the 2D gradation curve of coarse aggregates is proved to exist, and there is a corresponding relationship between the 2D and 3D gradation curves of coarse aggregates. The effective thermal conductivity of concrete has a positive exponential relationship with the volume fraction of coarse aggregates, a positive logarithm relationship with the thermal conductivity of coarse aggregates, and a positive linear correlation with the thermal conductivity of cement mortar. The most practical way to improve the effective thermal conductivity of concrete is to increase the ETC of the cement mortar, but the most effective way is to replace the aggregate with a material with a high thermal conductivity.

#### 1. Introduction

Estimating the heat transfer properties of concrete is of great significance in various engineering fields, such as building energy conservation [1, 2], hydraulic structures, nuclear reactor structures [3], bridge structures, and other structures exposed under the thermal loading [4]. The effective thermal conductivity of concrete is the most important parameter in denoting the thermal transmission capacity of concrete and in determining the temperature field in concrete structure. Large temperature gradient due to the low ETC of concrete would induce considerable thermal stresses, which will probably lead to cracking in concrete structures. Hence, estimating the ETC of concrete accurately is important to prevent cracking and ensure the safety of concrete structures.

Previous studies on the ETC of concrete can be divided into three categories: experimental investigation, theoretical model, and mesoscopic finite element model. The experimental investigation indicated that moisture content, water-cement ratio (W/C), age, volume fractions of fine and coarse aggregates, and temperature had significant influence on the ETC of concrete [5–7]. Khan [8] measured the thermal conductivity of concrete at various moisture contents, and developed a relationship between the ETC of concrete and the thermal conductivity of aggregate at different saturations. Kim et al. [9] and Zhang et al. [10] investigated quantitatively the influence factors on the ETC of concrete and proposed different prediction equations for the ETC of concrete. Experimental investigation on the ETC of concrete is the most direct and effective method, but it can be time-consuming, costly, and labor intensive for investigating plentiful concrete specimens. Concrete was regarded as a two-phase composite material consisting of cement mortar (continuous phase) and coarse aggregate (dispersed phase) in the mesoscale [8], and several theoretical models for the ETC of concrete were developed based on the two-phase composite material theory. Campbell-Allen and Thorne [11] considered concrete as a set of cubes of aggregate with uniform size arranged systematically in the matrix of cement mortar, and developed a simplified theory model (namely, Campbell-Thorne model) to calculate the ETC of concrete. Another theoretical model for the ETC of concrete was established on the basis of general model of porous materials, and the representative model was Harmathy model [12]. In addition, some semitheoretical models were developed based on the experimental results. The Hamilton–Crosser model included the effect of aggregate shapes [13]. The Bruggeman model considered the effect of aggregate volume fraction [14]. The Hasselaman-Johnson model took the effect of interfacial thermal resistance into account [15]. The ETC of concrete can be roughly estimated using these theoretical or semitheoretical models; however, they were derived based on many assumptions. Mesoscopic finite element method has been widely employed to study the mechanical behavior of concrete by simulating the anisotropic property of concrete. And the numerical results obtained from mesoscopic finite element method were in good agreement with the experimental results. And the numerical results obtained from the mesoscopic finite element method were in good agreement with the experimental results [16–19]. Thus, this mesoscopic finite element method was also used to simulate the thermal behavior of concrete [20, 21]. A two-dimensional numerical model was developed to calculate the ETC by Tang et al. [22], and the results revealed that the ETC of concrete depended on the degree of heterogeneity strongly, and the size and shape of coarse aggregates had negligible influence on the ETC of concrete. Carson et al. [23] established a two-dimensional mesoscopic finite element model to examine the influences of porosity on the ETC of theoretical porous materials, which simulated a steady-state thermal conductivity measurement device. Shen et al. [3] proposed a two-dimensional mesoscopic numerical method that evaluated the effect of cracking behavior on the ETC of concrete quantitatively. The results displayed that the effective thermal conductivity decreased greatly when some microcracks initiated, and the concrete became anisotropic as the microcracks propagated further. The mesoscopic finite element model can simulate the concrete material more accurately compared to the existing theoretical or semitheoretical models, and the numerical results match well with the experimental values. Theoretically, the mesoscopic finite element models can replace the experimental tests for parametric analysis on the ETC of concrete. However, the existing mesoscopic numerical model of concrete cannot consider the arbitrary gradation of coarse aggregates except for the fuller gradation [24], and the gradation of concrete has a significant effect on the degree of heterogeneity, which further affects the ETC of concrete. Besides, the mesoscopic finite element model has not been verified by experiments. Therefore, the mesoscopic numerical method in estimating the ETC of concrete still has great large application limitations.

This paper mainly consists of four steps. Firstly, a method generating two-dimensional random polygonal aggregate model of concrete with arbitrary gradation was proposed. Secondly, a mesoscopic numerical analysis method for the ETC of concrete was presented. Thirdly, the existing experiment results about the ETC of concrete were used to verify the validity of the proposed method. Finally, the influence of volume fraction and thermal conductivity of coarse aggregates and cement mortar on the ETC of concrete were investigated by the mesoscopic finite element method.

#### 2. Mesoscopic Numerical Method

##### 2.1. 2D Random Polygonal Aggregate Model for Concrete with Arbitrary Gradation

Generally, the aggregate volume fraction of bridge concrete is between 30% and 50%, and the artificial broken stones are used as coarse aggregates. A 2D random polygonal aggregate model can represent the section from an actual concrete structure accurately, as shown in Figure 1. However, the key of this model is to determine the 2D cumulative distribution function (CDF) of the coarse aggregates of concrete with arbitrary gradation. But the CDF of coarse aggregates is a 3D concept, and there is no 2D CDF concept currently. In this research, the 2D CDF is defined as the distribution of different aggregate sizes in the concrete section, which includes the cumulative distribution function of mass, area, and particle number (PNCDF). The 2D PNCDF of coarse aggregates can be determined as follows:(i)A 3D random spherical aggregate model of concrete is generated according to the arbitrary gradation of coarse aggregates determined by the seiving test. The generation process was described by Ruan and Pan [25].(ii)A random cross section is obtained from the 3D model.(iii)A 2D PNCDF of coarse aggregates can be acquired by counting the particle number with different particle sizes in the cross section.