The three-dimensional microstructure of fiber-reinforced concrete is represented within the computer using the hard core/soft shell model. This model has recently been described in detail , and software and documentation for the version of the model based entirely on spherical particles are available for downloading from the /ftp/pub/bfrl/bentz/HCSSMODEL subdirectory at ftp.nist.gov (18.104.22.168) or by accessing an electronic monograph at http://ciks.cbt.nist.gov/garboczi. The model has been used in the past to study the chloride diffusivity of concrete as a function of mixture proportions [35,36], leading to the development of an equation for estimating chloride diffusivity from mixture proportions and the degree of hydration of the cement .
The model simulates the microstructure of a cubic volume of concrete, typically 27000 mm3 in volume for this study. Thus, the sample is 30 mm on a side, the same scale at which spalling failures are commonly observed . With a sample of this size, up to one million individual particles may be required depending on the specific aggregate gradation and volume fraction. The user specifies the particle size distribution of the aggregates and the number of particles to place within the 3-D volume, and the program creates a random microstructure, ensuring that no two aggregate particles overlap within the 3-D cubic volume. Typically, the PSD is specified via the measured sieve size classification of the aggregates. Within each sieve classification, the particle sizes are distributed uniformly by volume. For this study, the basic computer program was modified to include ellipsoidal fibers in the microstructure. Assessing the overlap of two general ellipsoids is more complex than the simple distance check which can be employed to determine if two spheres overlap, but computer codes for this purpose have been developed in the past and used to study the influence of aggregate shape on the percolation of their surrounding ITZ regions . The rigid 3-D ellipsoids serve as a convenient computational abstraction for the polypropylene fibers, which are, in reality, deformable cylinders and can ``bend'' around aggregates, etc. in the concrete.
The computer program is divided into three modules for: 1) the random placement of particles, 2) the assessment of the percolation characteristics of the surrounding ITZs, and 3) systematic point sampling to estimate the volumes of all phases (aggregates, fibers, ITZs, and bulk cement paste) in the concrete microstructure. Generally, the particles are placed from largest to smallest in size. For fibers, placement size is characterized by the largest of the three principal axis directions. For increased computational efficiency, the 3-D microstructure is subdivided into a set of cubic bins. This reduces the computational time required when assessing overlaps, as each particle's position need only be compared with the other particles sharing a common bin (as opposed to every other particle in the system). For the simulations presented in this paper, there were either 20 bins or 30 bins per dimension (8000 bins or 27000 bins in the 3-D volume). Percolation is assessed by determining if there exists a pathway across the 3-D microstructure (in one principal direction) composed of the overlapping ITZ regions. All particles which are a part of this pathway are assigned a special label, so that their volume fraction may be conveniently determined during the systematic point sampling. Additionally, during this sampling, a 3-D digital image of the microstructure (typically 150 voxels x 150 voxels x 150 voxels in size) is created in which each point is labelled as aggregate, fiber, ITZ, or bulk paste, along with its percolation state.
To examine the influence of aggregate volume fraction and gradation and fiber content on the percolation of the ITZ regions, a variety of parameters were varied in a systematic fashion. The aggregate gradations were chosen based on those designated in ASTM C 33 , and presented in Figure 2. The coarse aggregate followed a nominal size range of 12.5 mm to 4.75 mm with a maximum aggregate size of 19.0 mm. The coarse to fine aggregate ratio was fixed at a value of 1.5:1, a value typically employed in concrete mixture proportions , although the influence of this ratio will be discussed in the "Application to Mixture Proportioning" section to follow. Aggregate volume fractions studied included 0.6, 0.65, 0.7, and 0.75 to span the range typically encountered in construction concrete. Because of the necessity of including only integer numbers of particles in the simulated concrete volume, these values varied slightly with the specific aggregate gradation being employed. Although air voids present in the concrete will behave similarly to aggregates (in that each will be surrounded by an ITZ region), their influence was not examined in this study based on the assumption that most of the HPC used in high rise buildings and tunnels will not incorporate air entrainment agents. However, the computational techniques presented herein are equally applicable to air-entrained concretes, as has been demonstrated previously . Recently, some evidence that air-entrained concretes may provide improved spalling resistance has been presented .
For each concrete mixture proportion, the ITZ thickness was varied between 5 µm and 30 µm, to span the projected difference between HPCs and conventional concretes. In each case, systems with no fibers were first evaluated, and then systems with various fiber contents and geometries. Typically, fiber lengths of 10 mm and 20 mm were investigated and in both cases, a value of 0.25 mm was used for the fiber diameter, resulting in fiber aspect ratios of 40:1 and 80:1, respectively. For a few limited simulations, the fiber diameter was reduced to 0.1 mm to examine its effect on the percolation properties of the ITZ regions. For fiber diameters smaller than this, the diameter of the fibers would be similar to that of the original unhydrated cement particles and there would not be a well defined ITZ region surrounding each fiber. Reducing the fiber diameter may thus reduce the volume of fibers required to achieve percolation, but there is a lower limit based on the size of the cement particles in the concrete.
All fibers were randomly oriented in the 3-D microstructure by generating a set of Euler angles corresponding to a random point on the surface of a unit sphere. Thus, it is being assumed that the fibers are not preferentially oriented by the concrete mixing process. Fiber contents were varied by adding different numbers (50, 75, 100, 200, etc.) of fibers to the microstructure volume to estimate the critical volume fraction needed to achieve percolation of the ITZs in those systems whose ITZ regions were originally discontinuous across the microstructure. Finally, simulations were conducted for systems containing only fibers and their ITZs, to determine their approximate percolation threshold. This value could be of particular relevance in lightweight aggregate concrete where the ITZs surrounding aggregates may be effectively eliminated, as an ITZ denser than the bulk paste is formed . Indeed, Bilodeau et al.  have observed that a fiber content sufficient to diminish the spalling of a normal weight high strength concrete had little influence on the spalling behavior of a comparable lightweight high strength concrete.