Advances in Materials Science and Engineering

Volume 2017, Article ID 7989346, 14 pages

https://doi.org/10.1155/2017/7989346

## Effects of Voids on Concrete Tensile Fracturing: A Mesoscale Study

College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China

Correspondence should be addressed to Lei Xu; nc.ude.uhh@uxiel

Received 21 December 2016; Accepted 30 March 2017; Published 24 April 2017

Academic Editor: Francesco Caputo

Copyright © 2017 Lei Xu and Yefei Huang. 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

A two-dimensional mesoscale modeling framework, which considers concrete as a four-phase material including voids, is developed for studying the effects of voids on concrete tensile fracturing under the plane stress condition. Aggregate is assumed to behave elastically, while a continuum damaged plasticity model is employed to describe the mechanical behaviors of mortar and ITZ. The effects of voids on the fracture mechanism of concrete under uniaxial tension are first detailed, followed by an extensive investigation of the effects of void volume fraction on concrete tensile fracturing. It is found that both the prepeak and postpeak mesoscale cracking in concrete are highly affected by voids, and there is not a straightforward relation between void volume fraction and the postpeak behavior due to the randomness of void distribution. The fracture pattern of concrete specimen with voids is controlled by both the aggregate arrangement and the distribution of voids, and two types of failure modes are identified for concrete specimens under uniaxial tension. It is suggested that voids should be explicitly modeled for the accurate fracturing simulation of concrete on the mesoscale.

#### 1. Introduction

Concrete is widely used as a construction material and is traditionally treated as a homogeneous continuum on the structural scale (macroscale). This homogenization assumption can hold well as long as the mechanical response of concrete remains in the elastic regime [1, 2]. However, when fracturing occurs, the macroscale mechanical behavior of concrete is greatly controlled by its components and their interactions taking place on a finer scale (mesoscale) [3, 4], which means accurate modeling of concrete fracturing calls for the consideration of its mesostructure.

Up to date, several mesoscale models have been developed to provide tools for a better understanding of concrete fracturing. From the simulation strategy point of view, most of the existing concrete mesoscale models can be broadly grouped into two types: the continuum model and the lattice model. In the continuum model, concrete is usually characterized by a continuum composite material with each component discretized by finite elements, while, for the lattice model, a discrete system composed of lattice elements is used to represent concrete. Moreover, the discrete element method (DEM) has been recently used to perform the mesoscale simulation of concrete [5], and it is shown that the discrete model requires a huge numerical effort that is necessary for this approach to obtain a reasonable representation of concrete mesostructure.

Several researchers studied the concrete fracturing by employing the continuum modeling strategy, and representative contributions can be found in [6–12]. The most recent investigations following this strategy were carried out by Du et al. [13] who studied the dynamic tensile fracturing of concrete by assuming concrete to be composed of aggregate and mortar matrix, by Huang et al. [14] who performed a 3D mesoscale fracturing simulation based on the actual concrete mesostructure, and by Wang et al. [15, 16] who developed a computational technology using the interface element with a cohesive law to perform Monte Carlo simulations of concrete fracturing and to study the 3D mesostructure effects on concrete damage and failure. Overall, the principal merit of the continuum model lies in the detailed representation of concrete mesostructure, which ensures the ability to realize reasonable simulations of cracking initiation on the mesoscale and coalescence of multiple distributed cracks into localized macroscale cracks and fracture propagation. However, it tends to be computationally intensive even for laboratory-scale specimens, especially for three-dimensional cases.

With respect to the lattice modeling strategy, representative studies were carried out in [17–20], and the most recent improvements are performed by Cusatis et al. [21, 22] who proposed a novel model named the lattice discrete particle model (LDPM) by exploiting the merits of both the lattice model and the discrete particle model. In contrast to the continuum model, the lattice model is considered computationally less demanding as concrete mesostructure is roughly represented by a discrete system with relatively less degrees of freedom and meanwhile can still possess the ability to capture the most important aspects of concrete fracturing. However, it is hard to investigate the interactions of concrete components in a real sense since the actual concrete mesostructure is not fully taken into account in the lattice model.

Voids (or pores) with different sizes always exist in concrete and typically take up 2–6% of the total volume, and the use of entrained air void system is a common approach in concrete technology to resist cyclic freezing and thawing degradation [23]. However, the effects of voids on concrete fracturing on the mesoscale are still not well understood. Wang et al. [15] built numerical concrete samples with pores using interface elements and studied the effects of porosity on concrete loading-carrying capability under uniaxial tension, but the fracturing mechanism on the mesoscale was not detailed. Huang et al. [14] reported the distribution of voids greatly influences the tensile strength and crack patterns based on the simulation results of a single 3D specimen. On the whole, it has been recognized that the existence of voids affects the concrete mechanical behavior to a large extent, but further research is needed to reveal the effects of voids on concrete fracturing.

With this in mind, a 2D finite element (FE) mesoscale modeling framework for concrete is proposed in this study in which concrete is considered as a four-phase material composed of aggregate, mortar, interfacial transitional zone (ITZ), and void, and the effects of voids on concrete tensile fracturing under the plane stress condition are detailed by performing several simulations. The rest of this paper is organized as follows: Section 2 presents the generation procedures of concrete mesostructure; the FE modeling methodology including mesh discretization, insertion of ITZ elements, and constitutive modeling of mortar and ITZ is described in Section 3; in Section 4, the effects of voids on concrete tensile fracturing are discussed in detail based on the simulation results of several concrete specimens with different mesostructures; finally, the study is summarized with conclusions in Section 5.

#### 2. Generation of Concrete Mesostructure

In this study, concrete is treated as a four-phase composite material, that is, coarse aggregate, mortar composed of cement matrix and fine aggregate, interfacial transitional zone (ITZ), and void randomly distributed in the mortar. Regarding aggregate generation, gravel is idealized as circle, while crushed aggregate is considered as polygon. Mortar is assumed as a homogenous continuum, and the interface with a specified thickness between coarse aggregate (hereinafter referred to as aggregate) and mortar is used to represent ITZ. Moreover, void is viewed as circle for simplicity.

##### 2.1. Size Distribution of Aggregates and Voids

The aggregate size distribution of concrete is described by Talbot’s equation aswhere is the size of aggregate, is the maximum size of aggregate, represents the ratio of aggregates by weight passing through a sieve of characteristic size equal to , and is the exponent of Talbot’s equation. For , the corresponding curve is known as Fuller’s curve extensively employed in concrete grading design for optimal packing properties.

For a concrete specimen with total volume , the volume of aggregates within a grading segment can be calculated by where is the minimum size of aggregate and represents the aggregate volume fraction.

Currently, the size distribution of voids in concrete has not been detailed. In general, these voids can be broadly grouped into two types according to different formation ways and the resulting different sizes: the (smaller) entrained voids with typical sizes on the order of 0.1 mm and the (larger) entrapped voids with typical sizes commonly more than 1 mm. In this study void size is considered to be uniformly distributed, and the same assumption is also employed by other researchers [15, 16]. Thus, denoting the size range of void by , the void size can be calculated by ( is a uniformly distributed random number between 0 and 1).

##### 2.2. Generation and Placement of Aggregates and Voids

In order to build numerical concrete specimens automatically, a mesostructure generator for concrete (MGC) is developed using MATLAB based on the take-and-place method [24, 25].

In the take-process, aggregates and voids, which will be placed into the specimen volume in the place-process, are generated separately. For the aggregate generation, the aggregate volume for each grading segment is first calculated according to (2). Then, starting with the grading segment with the maximum average size, the aggregates are generated one by one for each grading segment. For a certain grading segment , the generation of aggregates takes the following procedures.

*Step 1. *Generate a random number representing the aggregate size , which is assumed to follow a uniform distribution and therefore can be taken as .

*Step 2. *For gravel, a circle with radius of is defined to represent the aggregate, while, for crushed aggregate, a polygon with the random number of sides ranging from 4 to 10 and with the smallest width equal to is generated to represent the aggregate (see [24] for more details). Then, the volume of the current generated aggregate is calculated.

*Step 3. *Repeat the previous two steps until the remaining volume left is less than , namely, not enough to generate a new aggregate.

*Step 4. *Transfer the remaining volume to the next grading segment.

Following the similar procedures for generating gravel aggregates, the generation of voids can be performed with ease provided by the given void volume fraction and size range, which is followed by the placement of aggregates and voids (the place-process).

In the place-process, the generated aggregates and voids are first sorted according to their volume, respectively. Then, for the convenience of mesh discretization discussed in Section 3, the size of each aggregate is increased by a specified value (the thickness of ITZ, ) to consider the surrounding ITZ, which means the aggregate finally placed consists of two parts (i.e., aggregate piece and the surrounding ITZ with a specified thickness). After the modification of aggregate size, all aggregates are placed into the specified specimen one by one starting with the aggregate with the largest volume, followed by the placement of voids starting from the biggest one. The procedures of the placement of aggregates and voids are detailed as follows.

*Step 1. *Define the shape of concrete specimen using and coordinates of the boundary vertices numbered clockwise or anticlockwise, which will be used in Step 3 to check if an aggregate is inside the concrete specimen.

*Step 2. *Generate random numbers to define the position (and orientation if polygon is used to represent the crushed aggregate) of the aggregate using , , , and , which represent the minimum and maximum value of and coordinates of all boundary vertices, respectively.

*Step 3. *Perform the aggregate placing. The placement is considered to be successful if the following four conditions are satisfied: () the whole aggregate should be within the concrete specimen; () no overlapping/intersection occurs between the aggregate to be placed and any existing aggregate; () a minimum distance (defined by ) should exist between the aggregate and the specimen boundary; and () a minimum gap (defined by ) should exist between any two aggregates. If any of the four conditions is violated, Step 2 is repeated to make a new attempt until the placement of the aggregate is completed.

*Step 4. *Repeat Steps 2-3 until all aggregates are placed inside the specimen.

After the placement of aggregates, voids should also be placed into the specimen, which can be carried out by following the similar steps given above. It is worth noting that voids are considered to be embedded in mortar in this study.

Using MGC, numerical concrete specimens can be built with ease. The specimens shown in this paper are 100 mm squares, and the 4-segment Fuller curve is used to describe the aggregate grading for all specimens. For aggregates, and are set to 10 mm and 4 mm, respectively, while, for voids, = 2 mm and = 1 mm are used. In addition, is approximately set to 100 m according to the experimental observation [1], and both and are taken as 0.1 times of the size of aggregate or void to be placed.

Figures 1(a) and 1(b) sketch two numerical samples using circular aggregates with the same void volume fraction () and the aggregate volume fraction and , respectively, whereas two numerical samples using polygonal aggregates with and and , respectively, are plotted in Figures 1(c) and 1(d).