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Models for elastic properties and microstructure

The numerical algorithm utilized in the model to determine elastic moduli and shrinkage has previously been described [7]. It uses a linear finite element code to operate on a two- dimensional digital image of a random, multi-phase composite. Each pixel is treated as a simple bi-linear finite element, with displacements defined at the pixel corners. The moduli of each pixel are defined by its phase label. The finite element mesh is then simply defined by the digital image. Digital images of arbitrary microstructures can be analyzed, such as those of portland cement paste [8].

Two different computations can be performed. First, the effective moduli of a composite system can be computed by applying an overall strain to the system. The algorithm uses a modified conjugate gradient solver to determine the set of nodal displacements which minimizes the overall energy of the system, subject to the constraints of the applied strain. The elastic stresses and strains in each pixel can be computed from the displacements.

In order to compute the shrinkage, the size of the system, in addition to the nodal displacements, is allowed to vary. Each phase has a certain unrestrained shrinkage which competes with the rigidity of the rest of the system to produce an overall shrinkage. When the energy is minimized, the average stress is zero, but the individual stresses in each pixel are generally non-zero. The new size of the sample then gives the composite, or overall, shrinkage value.

Because the algorithm models two-dimensional space, it is necessary to decide how to map 3-D elastic properties into a 2-D medium by choosing either a plane stress or a plane strain criterion. Plane stress is appropriate [9], assuming that the two-dimensional digital images represent thin sheets of 3-D material. In such a situation, the Young's modulus, Poisson's ratio, and shear modulus are unchanged between three- and two- dimensions. The 2-D bulk modulus, K₂, is calculated as:

	(1)

where K₃ and G₃ are the bulk and shear moduli in 3-D. The assumption of plane strain or stress does not actually make a significant qualitative difference in the overall computed behavior.

Another important assumption is employed with regard to the width of the interfacial transition zone in the model. The ITZ is usually described as a region 30-50 micrometers in thickness [10] in which the cement paste composition differs significantly from that of the bulk cement paste. This region typically displays much higher porosity and CH volume fractions, and lower C-S-H gel and unhydrated cement volume fractions [11] than in the bulk cement paste. This is attributed, in part, to reduced efficiency in particle packing near the surface of the aggregate particle [10,11].

In the computations described in this paper, the ITZ is assumed to have a width of approximately h=20 micrometers [12]. The properties and composition of the zone are considered to be constant across this 20 micrometer width and represent the average properties of the entire zone region. This simplification is necessary because of the difficulties in simulating the effective properties of a region when properties and composition change rapidly over small length scales. To simulate the overall arrangement of sand grains and interfacial zone areas, the hard core/soft shell model, previously used for 3-D simulation of mortar microstructures, is used [12]. Composite grains, consisting of a circular hard core (sand), surrounded by a concentric circular soft shell (interfacial zone cement paste), are deposited randomly in a matrix (bulk cement paste). The hard cores may not be overlapped, but the soft shells may overlap each other or the bulk cement paste phase. Figure 1 shows a typical 2-D mortar microstructure generated with this algorithm, using the size distribution of sand grains given in Table 1. No attempt has been made to determine the 2-D sand size representations by taking slices of a real 3-D sand size distribution, as has been done in other works [13]. The distributions are only two-dimensional, but are suitable for qualitative insight.

Figure 1: Digital image of mortar using sand distribution in Table 1. (White=sand, Grey=bulk cement paste, Black=interfacial transition zone cement paste)

There are obviously other differences between 2-D and 3-D. In 3-D, there is a much higher volume percentage of interfacial transition zone per particle than in two dimensions, as shown in Table 2 for isolated particles. This is a geometrical fact, and is easily illustrated. If the interfacial transition thickness h << r, the particle radius, then in 2-D this ratio is 2h/r, and in 3-D it is 3h/r. The ratio of these two ratios would then be 3/2. In table 2, it can be seen that this ratio of ratios is above 3/2, but is decreasing continuously towards this value as r increases at a fixed value of h. It has been found that the actual volume of interfacial zone cement paste in realistic 3-D models [12] is within 5% of that calculated by simply multiplying the sand surface area by h, the interfacial transition zone thickness. This implies that the ratio of ITZ volume to particle volume is nearly the same as for a single particle. This result has been seen to be valid for sand volume fractions as high as 50%. The same effect occurs in 2-D, as will be discussed in the next section. This results in a higher percentage of interfacial transition zone cement paste in the 3-D mortar than in the 2-D mortar with the same sand content. However, in 2-D there is usually even more overlap of the ITZ's, which results in an effectively lower are of ITZ at a given sand area fraction.

Percolation of the ITZ's is another important factor. The 2-D models in this paper mostly did not have percolated ITZ's, inconsistent with 3-D [12]. However, overall shrinkage should be relatively unchanged, as the elastic properties of the ITZ and bulk cement paste are within a factor of 2 or 3. This is not a large difference in terms of percolation effects [14].

In order to test the accuracy of the elastic algorithm, a series of computations was performed on a set of mortar images. The properties of the shells (interfacial zone cement paste) were taken to be identical to the matrix (bulk cement paste), in order to reduce the number of elastically different phases from three to two. Each microstructure was created using monosized circular sand grains of diameters 110, 210, 310, 410, and 510 micrometers in a 5.12 by 5.12 mm image. For each particle size, a range of sand area fractions was used, from 10% to 55%. The Young's modulus of the sand grains was set to three times the bulk cement paste matrix. The Poisson's ratios of the bulk cement paste and sand grains were set to 0.3 and 0.2, respectively. These values were derived from elastic moduli values [15] and are reasonable choices for mortars, as will be discussed later. For shrinkage analysis, the sand phase was assumed to be nonshrinking, while the bulk cement paste was given an arbitrary shrinkage. Exact values for the bulk cement paste Young's modulus and unrestrained shrinkage are not important in these 2-D composites since all results are normalized by these values. All values of the effective elastic moduli and shrinkage strains agreed well with previous numerical and analytical results [16,17,18]. In particular, the Hashin-Rosen [19] exact prediction for shrinkage in terms of the effective bulk modulus for a two-phase system agreed very closely with the numerical tests. For the two-phase test, the effective elastic moduli were independent of the sand size used, which was expected for monosize circular inclusions.