Linear elastic data were collected via a resonant frequency method (ASTM C215-97) for early age cement paste elastic moduli for the L cement. It was found that the computed elastic moduli, using real particle shapes [9], were much too high, compared to experiment, when a 1003 microstructure was used. Since the early age cement paste microstructure is very tenuous − small amounts of hydration products holding together much stiffer cement particles − it was very possible that higher resolution finite element meshes could be necessary to properly represent this tenuous microstructure. Higher resolution microstructures were made in two ways. First, larger cement particle systems were made, and hydration was carried out at the higher resolution (2003 − 0.5 µm/voxel and 4003 − 0.25 µ m/voxel), which involved some modifications to the CEMHYD3D code [44]. This process resulted in higher resolution systems that were made by hydrating higher resolution model cement particles. A second way of preparing higher resolution systems was to take the original 1003 microstructure and subdivide each voxel. In this way, a better elastic solution was obtained but on the same microstructure. For example, a narrow one-voxel neck would now be represented by several voxels, thus allowing more elastic freedom in that neck. The elastic moduli code was then run on each system. For systems larger than 1003, a parallel version of the code was used, and run on parallel computer clusters at NIST [45].
Figure 7 shows the values of E and G for a w/c = 0.6, α = 0.219 early age cement paste, plotted vs. the system size N, which represents the number of voxel lengths in the 100 µm physical length of the side of the computational unit cell. The resolution scaling here was of the first kind, actually changing the resolution at which the cement particles were made and altering the CEMHYD3D model so as to accurately scale with system resolution [44]. The computed elastic moduli values decrease rapidly with N, and one might guess from the graph that with increased resolution, the values are asymptotically approaching the experimental values. When the resolution was also altered the second way mentioned above, by simply sub-dividing each voxel in the original 1003 microstructure and keeping the same properties from "parent" to "daughter" voxel, similar results were obtained (not shown), implying that scaling the hydration to different resolutions was done correctly and did not introduce any extra unphysical artifacts. Previous work on scaling the resolution may have introduced some unphysical artifacts [8].
Figure 7: Elastic moduli of cement paste system (w/c = 0.60, α= 0.219) for the L cement, plotted against the number of voxels per 100 µm in the model. The straight lines are the experimental values, and the points are from numerical computations.
A more detailed comparison to experiment is illuminating for the above case. For this w/c = 0.6, α = 0.219 cement paste, the experimental elastic moduli values are E = 1.7 GPa and G = 0.66 GPa (both with uncertainties of at least ±5 %, since it can be difficult to make early material age resonance measurements). Note that actual measurements were made at degrees of hydration before and after this value of the degree of hydration, so that the experimental results quoted is only a linear interpolation. Computing the corresponding value of K, the bulk modulus, and ν, the Poisson's ratio, one obtains ν = 0.29. Choosing the value of E to be 5 % higher and the value G to be 5 % lower, one obtains ν = 0.423, so that the value of Poisson's ratio obtained at this early age is quite sensitive to the values of E and G used, since they are so small compared to later ages. There is some early age white cement paste data in the published results and thesis of Boumiz [34]. At a w/c ratio of 0.6 and approximating the degree of hydration from the thesis graph, we obtain E ≈ 1.0 GPa , G ≈ 0.35 GPa, and ν ≈ 0.47. The best simulation values, for the 4003 system, are E = 2.76 GPa and G = 0.98 GPa, with ν = 0.408. Our experimental value of ν = 0.29 then seems much too low for such an early age material and so it is probably true that the real result has been masked by the uncertainty and the interpolation. After all, before set, the value of ν starts from about 0.5, the value for a liquid, and decreases from that point as hydration proceeds.
One might also wonder if early age C-S-H has different elastic properties than what has been determined for later-age C-S-H. The 4003 results of Fig. 7 were re-run using a value for the Young's modulus of the C-S-H phase that was three times lower. The overall elastic moduli then became E = 1.72 GPa and G = 0.6 GPa, so that ν = 0.43. So it is possible that a lower value of Young's modulus for early-age C-S-H is a real effect, since it brings the 4003 simulation results down toward more physical values. However, if early-age C-S-H is similar to the low density form of C-S-H that has been studied [5], its Young's modulus is not much lower than the high density, presumably later-age form of C&345;S-H, 21.7 ± 2.2 GPa vs. 29.4 ± 2.4 GPa [5]. A combination of more nanoindentation experiments and careful early-age resonance data combined with high-resolution modeling can perhaps resolve this question.
The more general question for the simulation results is − why this dependence of elastic moduli on digital resolution? At much higher degrees of hydration, one can check to see if digital resolution matters; it doesn't. At early hydration age, the microstructure, after set, is connected primarily by thin, incomplete, random surface layers of C-S-H and perhaps ettringite linking the cement particles. A simple test case can illustrate this early age cement paste behavior, showing how as an entire microstructure is represented at higher and higher resolution, the elastic moduli are reduced. Take the microstructure consisting of a single sphere, centered in a cubic box, whose diameter D is equal to the side length of the box. With periodic boundary conditions, a true continuum sphere touches six other spheres at point contacts (see Fig. 8). This system will have a finite bulk modulus, since the spheres may push against each other, but a zero shear modulus, since point contacts cannot offer resistance to shear forces. The system is approximated by 3-D digital images, of fixed physical size D. As the number of voxels, N, on a side becomes larger, the system is better resolved with more voxels per unit length (N/D), allowing the digital spheres to look more and more spherical. At low values of N/D, where N is only 3, for example, a diameter = 3 voxel sphere is simply a 3 x 3 x 3 cube with the eight corner voxels removed. The sphere-to-sphere interface will be made up of five voxel faces locked together, so the system will be quite stiff, both in K and in G. As the resolution N/D increases, the contact will become smaller and smaller with respect to the spheres, and so the elastic moduli should go down. Figure 9 shows just this behavior, which is illustrative of the cement paste system. As N/D becomes infinite, the shear modulus should go to zero (neglecting any frictional forces and assuming no simultaneous compression), because of the point contacts, while the bulk modulus should asymptote to a finite value. This microstructure has cubic symmetry, so its three independent moduli, C11, C12, and C44, have been averaged to obtain an isotropic bulk modulus and shear modulus [17, 25, 26].
Figure 8: Illustration of sphere in a periodic box microstructure. The periodic reflections to the top and bottom, left and right, and front and back, are shown to help picture the full 3-D structure.
Figure 9: How stiffness of spherical particle-spherical particle contact can decrease with increasing values of N (N = number of voxels per box length D, N/D = resolution, D is fixed). Image of touching spheres was re-made at each increased resolution.
Unlike the early cement paste model systems studied, the sphere system shown in Fig. 8 will not have significantly smaller moduli when the original microstructure at N = D = 3 is simply sub-divided. Figure 10 shows a slightly different system, a cube periodically linked on all six sides by one-voxel connections to identical cubes. As this microstructure is sub-divided, the single-voxel connections become multiple-voxel connections, 23, 43, and finally 83 voxels in extent, causing the elastic moduli to decrease, since having more voxels in the connection allows for more flexibility in the connection and hence lower values of elastic moduli. The cube was originally 73 voxels in size when only one connecting voxel was used. Figure 11 shows the results for the computed elastic moduli vs. the side length of the microstructure in terms of voxels (N). The cubic moduli have been averaged to obtain isotropic bulk and shear moduli. The drop in modulus is not as dramatic as in Fig. 9, since in the limit of infinite resolution, where the physical length per voxel goes to zero, the moduli will be non-zero, while in the sphere microstructure shown in Fig. 8, the shear modulus goes to zero.
Figure 10: Illustration of part of the microstructure of a cube linked by a single voxel to other cubes in a periodic box.
Figure 11: How the bulk and shear moduli of the Fig. 11 microstructure can decrease with increasing resolution (original microstructure was simply sub-divided).