Results

Random walk simulation data are available for the multi-scale concrete model for several aggregate size distributions (sieve analyses) and a number of choices of the conductivity contrast between ITZ and matrix [4,15]. In these data, the aggregates always had zero conductivity ( σ_agg =  0). The random walk simulation data are accurate to within a few percent, so they can be used to check the results of the new D-EMT. If the new D-EMT is able to replace these lengthy simulations by achieving an uncertainty of 10 % to 20 %, that would be a successful application. Experimental measurements, which the multi-scale theory hopes to predict, are probably only accurate to within a factor of two [4].

Table 1 shows the values {C_i } of the four different sieve analyses used (cfcc, fffc, ffcc, and cffc, see Ref. [4] for details of these sieve analyses). Figure 4 shows the results of the new D-EMT, plotted against the data of Table 2, taken from Ref. [4]. Good agreement, 10 % or better, is seen for most of the values, with somewhat higher disagreement but still less than 20 % for some data points. It is interesting to note that most of the D-EMT results are systematically lower than the simulation results. This is probably at least partly an artifact of the D-EMT calculation, because even at fairly low contrast, the percolation of the ITZ regions will have some effect. It is also possible that the simulation results are a bit high, which would be the case if the random walkers were not allowed to diffuse for a long enough time. The random walkers start out diffusing at the matrix diffusivity, and only gradually, through colliding with many aggregates, do their effective diffusivity and conductivity come down to the concrete values. Spot checks of some of the random walk data indicate that the random walk values would become about 5 % lower with more random steps being made, which would significantly improve the agreement with the new D-EMT [29].

Table 1: Definition of four different sieve analyses used for the concrete systems of Table 2. The numbers given in the table in the four righthand columns are the volume fraction of total aggregate contained in each sieve (f_j ). Details are given in Ref. [4].

di (mm)	di+1 (mm)	cfcc	fffc	ffcc	cffc
0.075	0.15	0	0.04	0.04	0
0.15	0.30	0.02	0.08	0.08	0.02
0.30	0.60	0.08	0.12	0.12	0.08
0.60	1.18	0.1	0.1	0.1	0.1
1.18	2.36	0.12	0.09	0.06	0.15
2.36	4.75	0.06	0.06	0	0.12
4.75	9.525	0.26	0.33	0.24	0.35
9.525	12.7	0.3	0.18	0.3	0.18
12.7	19.05	0.06	0	0.06	0

Figure 4: Showing the D-EMT results for σ / σ _bulk plotted against the simulation results in Table 2. The dashed line is the line of equality.

Table 2: Table of parameters for different systems, along with simulation and D-EMT results.

Sieve Analysis	c	h(µm)	σITZ/σbulk	$\sigma /\sigma _{bulk}$ / σbulk (simulation)	σ / σbulk (D-EMT)	Error(%)
cffc	0.753	0.01	2.95	0.20	0.18	−10
cffc	0.601	0.03	4.22	0.42	0.42	0.0
fffc	0.754	0.03	2.54	0.28	0.29	3.6
fffc	0.594	0.01	5.0	0.42	0.42	0.0
ffcc	0.602	0.01	2.84	0.36	0.33	−8.3
ffcc	0.752	0.03	3.31	0.34	0.37	8.8
cfcc	0.675	0.01	1.08	0.23	0.19	−17.4
cfcc	0.675	0.01	1.88	0.24	0.21	−12.5
cfcc	0.599	0.03	2.24	0.34	0.34	0.0
cfcc	0.675	0.01	2.32	0.26	0.22	−15.4
cfcc	0.524	0.01	4.06	0.42	0.39	−7.1
cfcc	0.824	0.01	4.14	0.16	0.14	12.5
cfcc	0.757	0.01	4.94	0.23	0.21	−8.7
cfcc	0.675	0.01	7.53	0.33	0.31	−6.1

A second set of simulation data has recently become available [16], for models with volume fractions of aggregate of 0.62 and 0.70, and a range of conductivity values for the ITZ region, with σ_ITZ / σ _bulk both less than and greater than unity. The sieve analysis for these systems is shown in Table 3. Table 4 shows the simulation and D-EMT data for the different concrete mixtures and parameter choices. Good agreement with simulation results is shown for the D-EMT results for all parameter values, with the differences well below 10 % for most of the data, and only a few differences as high as 13 %. It is somewhat curious to note that the agreement between the D-EMT and simulation actually appears to be better at the higher values of σ_ITZ/σ _bulk. This is the regime where, as was stated above, the ITZ percolation plays more of a role in determining overall properties. Since the D-EMT does not include ITZ percolation, the D-EMT formula might be expected to do worse at these values. This phenomenon can probably be explained by the fact that the simulation results are probably about 5 % too high, as was already pointed out. If all the simulation results would be reduced by this amount, the disagreement between D-EMT and simulation in Table 2 would be roughly constant at about 5 % to 6 %.

Table 3: Sieve analysis used in the systems of Table 4.

di (mm)	di+1 (mm)	Vol. Frac. of Agg.
0.075	0.15	0.02
0.15	0.30	0.05
0.30	0.60	0.10
0.60	1.18	0.10
1.18	2.36	0.105
2.36	4.75	0.06
4.75	9.525	0.295
9.525	12.7	0.240
12.7	19.05	0.03

Table 4: Table of parameters for different systems, along with simulation and D-EMT results. (see Figs. 5 and 6 )

c	σITZ / bulk	$\sigma /\sigma _{bulk}$ / bulk	D-EMT	% Error
0.70	0.5	0.168	0.148	-11.8
0.70	0.75	0.184	0.163	-11.4
0.70	1.0	0.198	0.176	-11.1
0.70	1.25	0.214	0.189	-11.7
0.70	1.5	0.218	0.201	-7.8
0.70	2.0	0.237	0.224	-5.5
0.70	2.5	0.257	0.245	-4.7
0.70	3.0	0.278	0.264	-5.0
0.70	4.0	0.305	0.301	-1.3
0.70	7.0	0.393	0.397	1.0
0.70	10.0	0.486	0.480	-1.2
0.70	12.0	0.531	0.531	0.0
0.70	17.5	0.643	0.660	2.6
0.70	21.0	0.744	0.735	-1.2
0.62	0.5	0.243	0.216	-11.1
0.62	0.75	0.258	0.231	-10.5
0.62	1.0	0.275	0.244	-11.3
0.62	1.25	0.279	0.257	-7.9
0.62	1.5	0.290	0.269	-7.2
0.62	2.0	0.305	0.292	-4.3
0.62	2.5	0.337	0.312	-7.4
0.62	3.0	0.346	0.332	-4.0
0.62	4.0	0.386	0.368	-4.7
0.62	7.0	0.450	0.460	2.2
0.62	10.0	0.541	0.538	-0.6
0.62	12.0	0.591	0.586	-0.8
0.62	15.0	0.664	0.651	-2.0
0.62	21.0	0.773	0.769	-0.5

Figure 5 shows the D-EMT data plotted against the simulation data from Table 4. The dashed line is the line of equality. The data points are seen to follow the dashed line quite well. The D-EMT predictions are again mostly seen to err on the small side, being slightly under the real values.

Figure 5: Showing, for the concrete data given in Table 4, the D-EMT results vs. the simulation results for $\sigma /\sigma _{bulk}$. The dashed line is the line of equality.

Figure 6 shows the same data as in Table 4, but now plotted as a function of σ_ITZ /σ_bulk, separately for the 0.62 and 0.70 aggregate volume fraction concrete systems. The D-EMT correctly captures the shape of these curves [16,26].

Figure 6: Showing, for the concrete data of Table 4, the overall effective conductivity vs. the the ratio of ITZ to bulk conductivity, for V _agg equal 0.62 and 0.70.