A twodimensional mesoscale modeling framework, which considers concrete as a fourphase 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.
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 [
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 [
Several researchers studied the concrete fracturing by employing the continuum modeling strategy, and representative contributions can be found in [
With respect to the lattice modeling strategy, representative studies were carried out in [
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 [
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 fourphase 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
In this study, concrete is treated as a fourphase 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.
The aggregate size distribution of concrete is described by Talbot’s equation as
For a concrete specimen with total volume
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 [
In order to build numerical concrete specimens automatically, a mesostructure generator for concrete (MGC) is developed using MATLAB based on the takeandplace method [
In the takeprocess, aggregates and voids, which will be placed into the specimen volume in the placeprocess, are generated separately. For the aggregate generation, the aggregate volume for each grading segment is first calculated according to (
Generate a random number representing the aggregate size
For gravel, a circle with radius of
Repeat the previous two steps until the remaining volume left is less than
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 placeprocess).
In the placeprocess, the generated aggregates and voids are first sorted according to their volume, respectively. Then, for the convenience of mesh discretization discussed in Section
Define the shape of concrete specimen using
Generate random numbers to define the position (and orientation if polygon is used to represent the crushed aggregate) of the aggregate using
Perform the aggregate placing. The placement is considered to be successful if the following four conditions are satisfied: (
Repeat Steps 23 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 4segment Fuller curve is used to describe the aggregate grading for all specimens. For aggregates,
Figures
Numerical concrete specimens. (a) Circular aggregate (
Once the concrete mesostructure is obtained, a corresponding FE model is required for performing numerical simulations. The details of the FE modeling methodology developed in this study are presented as follows.
In order to automatically carry out the mesh discretization of the concrete specimen with complex mesostructure, a mesh generator is developed using MATLAB by exploiting the powerful preprocessing modules provided by the commercial finite element software ABAQUS. For a numerical concrete specimen with pregenerated mesostructure, a twopart python file, which defines the boundary of the specimen together with the locations and shapes of aggregates and voids using the first part of the file and specifies the mesh discretization parameters using the second part of the file, is first generated using the mesh generator by taking concrete mesostructure as input data. Then, a FE mesh composed of linear triangular elements can be obtained by the mesh generator through calling ABAQUS/CAE kernel to execute the generated python file. An example of the FE mesh discretization with aggregate elements highlighted is shown in Figure
FE mesh discretization (polygonal aggregate).
As stated in Section
To this end, a fourstep procedure is proposed. Firstly, the original nodes on the boundaries of amplified aggregates are identified, followed by the definition of new nodes based on the coordinates of original nodes on the boundaries and the given thickness of ITZ. Then, the connectivities of the aggregate elements associated with these nodes are redefined by replacing the number of the original nodes by the number of the corresponding new nodes. Subsequently, 4noded ITZ elements are formulated one by one using the original nodes and the corresponding new nodes. Finally, an updated input file for ABAQUS, which contains final mesh data, is generated. An inhouse MATLAB program, which follows the above procedure, is developed, and part of the final FE mesh discretization corresponding to Figure
FE mesh discretization after inserting and adjusting.
Without considering voids, it is well recognized that ITZ is weaker than aggregate and mortar, and consequently mesoscale cracking in concrete under loading is commonly considered to first appear in ITZs. After that, mesoscale cracks propagate into mortar and additional cracks initiate within mortar with the further increase of loading [
In CDP model, two independent hardening variables, that is, equivalent compressive and tensile plastic strains
Then, the damaged elastic modulus
Based on the concept of damage mechanics, the effective stress
The yield function of CDP model is given in the effective stress space as
Yield surface in plane stress.
In order to describe the dilatancy reasonably, nonassociated flow rule is employed in CDP model, and the flow potential takes the form as
As presented earlier, the material softening under tension is defined by the relationship between the uniaxial tensile effective strength and equivalent tensile plastic strain (see (
Due to the highly nonlinear and softening behavior of concrete in the process of fracturing, the ABAQUS/Explicit solver is employed in all simulations with the aim of capturing the entire fracturing process of concrete.
As is well known, the dynamic effect inevitably exists in an explicit FE analysis, and its influence on the solution of a quasistatic problem should be small enough to be neglected. In order to minimise the dynamic effect, the loading time should be large enough, while, on the other hand, the computational effort increases proportionally with the increase of loading time. Hence, a balance has to be made between the computational efficiency and simulation accuracy, which can be achieved through comparing the results under different loading time (or loading rates).
Aiming to investigate the effects of voids on the tensile fracturing of concrete with different aggregate volume fractions, three sets of numerical concrete specimens with dimensions of 100 mm × 100 mm using polygonal aggregates are generated, and each set contains four specimens with the same aggregate arrangement and different
For each specimen, uniaxial tensile fracturing is simulated. In FE simulations, the left end of concrete specimen is fixed in the horizontal direction, while the opposite end is subjected to a uniformly distributed horizontal displacement up to 0.06 mm, namely, a displacementcontrolled loading scheme is used. Following the strategy discussed in Section
The same mechanical properties of aggregate, mortar, and ITZ are adopted for all specimens, as listed in Table
Mechanical properties of concrete components.
Material 

Poisson’s ratio (—) 






Aggregate  60  0.2  —  —  —  —  — 
Mortar  20  0.2  1.94  0.1  35  0.12  2.0 
ITZ  15  0.2  1.46  0.1  35  0.12  2.0 
Void  —  —  —  —  —  —  — 
Total stressstrain relation under uniaxial compression and compression hardening curve of mortar.
Total stressstrain relation under uniaxial tension and tension softening curve of mortar.
In order to study the effects of voids on concrete tensile fracturing mechanism, the simulation results of two specimens in Set II, which include one without void and the other one with voids taking up
The macroscopic stress versus displacement curve of the concrete specimen without void (
Stressdisplacement curve (
Mesoscale cracking development at different loading levels of the concrete specimen without void
It can be observed that although concrete specimen initially exhibits elastic response on the macroscale, mesoscale cracking still occurs in ITZs (see Figure
The macroscopic stress versus displacement curve of the concrete specimen (
Stressdisplacement curve (
Mesoscale cracking development at different loading levels of concrete specimen with
Overall, different fracturing mechanisms can be observed for the two concrete specimens by comparing the development processes of mesoscale cracking shown in Figures
In this section, the effects of aggregate volume fraction (
The specimens without void in Sets I, II, and III are simulated, and the macroscopic stress versus displacement curves are depicted in Figure
Stressdisplacement curves of different aggregate volume fractions.
Provided the simulation results of Sets I, II, and III, the effects of void volume fraction (
Stressdisplacement curves for (a)
Fracture patterns of concrete specimens with
Fracture patterns of concrete specimens with
Fracture patterns of concrete specimens with
A finite element modeling strategy of concrete with random mesostructure explicitly taking void into consideration has been proposed in the present work. The tensile fracturing mechanism of concrete with voids is detailed on the mesoscale by comparing the simulation results of two specimens consisting of one without void and the other one with voids with the same aggregate arrangement. Then, several simulations are carried out with prime attention placed on the effects of void volume fraction on concrete tensile fracturing. The main conclusions are as follows:
Different fracturing mechanisms are observed for the two concrete specimens with the same aggregate arrangement including one without void and the other one with voids, and the fracture pattern of concrete specimen with voids is controlled by both the aggregate arrangement and the distribution of voids.
Compared to aggregate volume fraction, void volume fraction has a larger influence on concrete tension strength. The elastic modulus of concrete in the prepeak stage can be considered to be independent of both aggregate volume fraction and void volume fraction.
The relation between concrete postpeak behavior and void volume fraction is not straightforward due to the randomness of void distribution and the resulting fracture pattern.
Two types of failure modes are identified for concrete specimens under uniaxial tension, in which Type I is characterized by a single macroscale crack and Type II by two. Due to the longer crack length, the postpeak stress of Type II drops less quickly than that of Type I.
It is necessary to model void explicitly for the accurate fracturing simulation of concrete on the mesoscale.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by projects of the National Natural Science Foundation of China (Grants nos. 11132003 and 51109067).