3. Individual phase properties of cement paste

The elastic modulus tensor of an isotropic elastic material has only two independent values, which are usually given as any two of the set: the Young's modulus, E, the Poisson's ratio, ν, the bulk modulus, K, and the shear modulus, G. The following equations show the relationship between these parameters [23, 24]:

(1)

These equations are given because different sources for the elastic moduli of individual phases are given in terms of different combinations of these parameters.

Some individual cement and cement paste phase elastic properties can be found in the geological literature, as they are crystalline minerals found in nature whose elastic properties have been measured. In general, the elastic moduli of a single crystal will give an anisotropic elastic moduli tensor reflecting the symmetry of the crystal lattice. An isotropic average can be performed on this tensor, which will result in an isotropic average elastic modulus tensor (two independent elastic moduli). Two ways in which this can be done, called the Voigt and Reuss averages [25, 26], respectively, result in an upper bound and a lower bound. The average of these bounds for various cement and cement paste phases are reported in Table 1. This isotropic tensor should be more representative of what appears in real cement paste, where the crystalline phases are polycrystalline and randomly dispersed. For the sake of completeness, all four of the elastic moduli are reported in Table 1. A tighter set of bounds for the averaged moduli can be generated, the Hashin bounds [25, 26], but the average of these compares quite closely with the average of the Voigt-Reuss bounds. The Voigt-Reuss bounds are much simpler mathematically, so these are used in Table 1. References [27-31] are general compilations of the elastic properties of various minerals, usually given in terms of averages of the Voigt-Reuss bounds, not the full anisotropic elastic tensor.

The elastic moduli of some cement/cement paste phases, phases not usually found in nature, have been reported in journal articles. The full anisotropic elastic tensor has been measured for calcium hydroxide (CH) via Brillouin scattering [32], and the Hashin bounds have been computed [33]. The average of the Hashin bounds is what is listed in Table 1. The polycrystalline isotropic elastic moduli of fully dense hot-pressed powder compacts of C₃S have been measured [34], and are listed in Table 1. The Young's moduli of the other clinker phases, C₂S, C₃A, and C₄AF, have been measured via nano-indentation [35], which have shown that the Young's modulus of these phases, including C₃S, are all about the same, certainly within 15 %. For simplicity, and since the moduli of C₃S have been previously measured carefully, these values are used for all of the clinker phases in the model reported in this paper.

Because of their volumetric abundance, the three phases that play the largest role in determining cement paste elastic moduli are clinker, CH, and C-S-H. The Young's modulus of late-age C-S-H can be estimated from well-hydrated systems [36] to be about 25 GPa. The Poisson's ratio is estimated to be about 0.25. Nano-indentation has recently been used to measure the Young's modulus of C-S-H [5]. A bi-modal distribution of Young's modulus values was found, which accords with the hypothesized two kinds of C-S-H present in cement paste [37]. When averaged together, a value of E = 23.8 GPa was found [5]. A value of 0.24 for Poisson's ratio was assumed. These numbers accord well with the estimates made above, and with other nanoindentation results [38]. In this paper, the elastic moduli values taken for C-S-H were: E = 22.4 GPa and ν = 0.25 (see Table 1). The values for Young's modulus for all three estimates agree within experimental uncertainty. No measurement of the Poisson's ratio of C-S-H has been made at the present time. Measuring the Poisson's ratio with nano-indentation is not an easy task [39], hence the estimate here.

Ettringite is another phase that can appear in some quantity in some cement pastes, but probably only plays a minor elastic role. A recent paper by Zohdi et al. [40] takes measurements made on packed powder compacts and extrapolates them to zero porosity to estimate that the isotropically averaged, polycrystalline elastic moduli of ettringite are E = 52 GPa and G = 20 GPa. This extrapolation was complicated mathematically, and gave a very high degree of curvature of the moduli vs. porosity plots at low porosities. The minimum porosity that was actually measured was 23.8 %. We have made similar measurements, but on more porosities and on a minimum porosity of about 11 %. Both sets of data are shown in Fig. 1. Comparing to the extrapolation made in Ref. [40], our 11 % porosity measurement falls well below their extrapolation, which calls into question their extrapolation procedure. At higher porosities, where comparison can be made, the two sets of data agree fairly well, probably within experimental error when taking into account possible differences in compaction. The experimental uncertainty on our measurements was less than 1 % for the resonance measurement and about 3 % to 5 % for the measurement of the porosity of the ettringite powder compact. The resonance measurements are actually quite precise, but differences in compaction tend to negate this precision. In Fig. 1 are also shown simple linear fits to all our data, giving a zero porosity value of E = 25 GPa and G = 10 GPa. The data of Ref. [40] falls fairly well on these lines. These values for fully dense, polycrystalline ettringite accord well with our values for C-S-H, which is another material that incorporates a large amount of water like ettringite. That is why in Table 1, and in the simulations, the elastic moduli of ettringite were taken to be the same as for C-S-H.

Figure 1: Two sets of experimental measurements on powder compacts of ettringite. HG = Haecker-Garboczi (this paper), ZML = Zohdi, Monteiro, and Lamour [40].

A few other phases in Table 1 are listed as having elastic moduli "Same as" another phase. The elastic moduli of these minor phases are not known independently. Hence, the elastic moduli were taken, as a "best guess," to have elastic moduli equal to that of another phase, which phase they most resembled structurally. Also, for hemihydrate gypsum, the E and υ values for this phase, which could not be found in the literature, are just the average of the literature values of dihydrate gypsum and anhydrite gypsum. Other schemes could be proposed, based on the relative water content, but in lieu of data, a simple averaging scheme is just as plausible as a more sophisticated scheme. If a pore space voxel is encountered that has been emptied of water due to self-desiccation, its elastic moduli are both set equal to zero.

The uncertainty in the measured elastic moduli in Table 1 is probably less than 10 %, taking into account both measurement uncertainty, the uncertainty in using averaged values instead of full tensor values, as described above, and the difficulties in preparing or finding pure phases. The "best-guess" values for the minor phases must have greater uncertainty, probably in the range of 50 % to 100 %.

Cement chemistry Notation	Mineral Name	K (GPa)	G (GPa)	E (GPa)		υ	Ref.
H	Water	2.2	0.0	−−		−−	[27]
C3S	Tricalcium silicate	105.2	44.8	117.6		0.314	[34]
C2S	Dicalcium silicate	Same as C3S					[35]
C3A	Tricalcium aluminate	Same as C3S					[35]
C4AF	Tetracalcium aluminoferrite	Same as C3S					[35]
C·H2 (gypsum)	Dihydrate	42.5	15.7	45.7	0.33		[28, 29]
C·H1/2	Hemihydrate	52.4	24.2	62.9	0.30
C	Anhydrite	54.9	29.3	80.0	0.275		[28, 29]
K	Potassium sulfate (arcanite)	31.9	17.4	44.2	0.269		[28, 29]
N	Sodium sulfate (thenardite)	43.4	22.3	57.1	0.281		[28, 29]
SiO2	Silica fume	36.5	31.2	72.8	0.167		[31]
CH	Portlandite	40.0	16.0	42.3	0.324		[32, 33]
C1.7SH4	C-S-H	14.9	9.0	22.4	0.25		[5, 38]
CaCO3	Limestone	69.8	30.4	79.6	0.31		[28, 29]
C3AH6	Hydrogarnet	Same as C-S-H
C6A3H32	Ettringite	Same as C-S-H
C4AH12(Afm)	Monosulfate	Same as CH
FH3	Iron hydroxide	Same as C-S-H
CaCl2	Calcium chloride	Same as CH
C3A(CaCl2)H10	Friedel salt	Same as ettringite
C2ASH8	Stratlingite	Same as C-S-H
C3A(CaCO3)H11 (Afmc)	Monocarbonate	Same as Afm

Table 1: Elastic moduli of individual cement and cement paste phases, taken from several sources in the literature.