The influence of microstructure geometry on the scale effect in mechanical behaviour of heterogeneous materials

2.1 Tested materials

The composites studied in this work are made of ready-to-use mixes, commonly available on the market. They are prepared by simply adding a specific, provided by the manufacturer, amount of water to the mix containing Portland cement with sand or “pure” gypsum plaster, respectively. The mortar powder CEMAX (Kreisel, Poznan, Poland) used to make the sample is the type of mortar often applied to build concrete screeds. In the European Standard EN 13813-2003 [23], the properties of this mortar are designated as CT-C16-F4. The gypsum plaster (Dolina Nidy, Leszcze 15, Poland) is a typical building material fulfilling the requirements of the European Standard EN 13279-1-A1 [24]. However, the properties of each of the mixes have been investigated in order to present their detailed composition, for this research to become comparable with other studies concerning cement- or gypsum-based composites. The proportions of the constituents composing each of the materials under investigation are listed in Table 1. In addition, the sieve analysis was performed in order to gather information about the grain sizes of sand composing the mortar mix. The results of the test are presented in Figure 1.

Figure 1:

Results of sieve analysis of sand.

PCV forms of different sizes are used for samples preparation (Figure 2). Vaseline is applied to grease the inner part of the tubes before placing the material in them to facilitate its extraction. All specimens are removed from the moulds after 24 h and wet cured in a curing container for 14 days. A large number of cylindrical samples with length/diameter (L/D) ratio equal to 2 are prepared. For both materials, four different sample diameters are considered, namely, D=28, 36, 45 or 72 mm. To perform mechanical tests on sufficient representative specimens, it was decided to have the ratio between the specimen diameter D and maximum heterogeneity size d_max equal to, at least, D/d_max=9. This corresponds to the worst case, i.e. the smallest sample diameter D=28 mm and the maximum sand particle size d_max=3 mm. Therefore, it can be assumed that the specimens are representative volume elements (e.g. [21, 25]). In Figure 2 the exemplary samples of 28 mm diameter are shown just after removing from the moulds (before the treating procedure).

Figure 2:

Twenty-eight millimetre samples of gypsum plaster and mortar after removing from cylinders (before the treating procedure) and PCV forms used for preparation of the composites.

2.2 X-ray micro-computed tomography

X-ray microCT is a non-destructive test method which gives a possibility to examine and analyse the microstructure of the material. This technique has been known for several decades already; however, it has been commonly used mainly in medicine and biological sciences to visualise the bones and soft tissues. Recently, it has become a useful method to investigate materials, and it plays an important role in engineering, material science, electronics etc. (e.g. [26–30]).

The principle of the microCT technique is that from the 2D projections of the scanned object one can acquire the 2D cross-sections of the material (the so-called “slices”) and, as a result, a 3D reconstruction of the investigated sample. The construction of the typical scanner is presented in Figure 3. It consists of the X-ray source built of two electrodes: anode and cathode, between which a high voltage is applied in order to produce radiation. The X-ray beam is transmitted through the sample, rotating on a stage within the pre-set value of unit angle. The projections, for each angle, are recorded on the detector covered by a scintillator – the material that transforms the ionising radiation (absorbed energy) in the form of light. The scintillator is connected to a charge-coupled device camera that changes the light into a digital signal [31]. The scanning requires adjusting a large number of settings such as unit angle, use of filter, voltage, power of the X-ray source etc. The analysis of each material needs a different set of parameters due to its unique properties. As a result of a scan, a collection of object projections is acquired. In order to obtain the cross sections of the sample, the mathematical reconstruction process is necessary. Finally, the set of slices on the height of the object investigated is gathered, which allows for further image processing including binarisation etc. Based on the binary images, a variety of microstructure measures can be calculated, e.g. porosity, shape factor, specific surface, correlation functions etc. In addition, a 3D reconstruction of the object’s volume can be obtained and analysed [32].

Figure 3:

The construction of the microCT system.

The total attenuation of an X-ray passing through the object can be expressed as a “line integral” – the sum of the absorptions along the path of the beam. The intensity I of the transmitted X-ray beam, according to Beer’s law, verifies the relation [31]

where I represents the intensity of the beam that has already passed through the sample, I_o is the initial intensity of the beam (at x=0) and μ is the material’s linear attenuation coefficient equal to

where i denotes an atomic element, ρ_i is a density of the material, f_i is the atomic weight and μ_i represents the mass attenuation coefficient of the beam energy [31].

The experimental campaign started with performing the trial scanning procedure. This trial scanning revealed some difficulties in distinguishing the microstructure of mortar from that of the gypsum. Even though the presence of sand grains in the case of mortar samples was expected, no heterogeneities were visible. It was presumed that the problem might stem from the fact that the constituents of the investigated materials have similar densities or average atomic numbers. This prediction is apparent due to the fact that the attenuation coefficient μ of the material depends on three major factors: the thickness of the specimen, its density and the atomic number of the substance (see Equation (2)). Therefore, if there is a noticeable difference in densities or atomic numbers of components of the investigated material, then its internal structure can be successfully visualised on a radiograph. On the other hand, if the components of the scanned material have similar densities or average atomic numbers, then it is difficult to distinguish them on the microCT image.

The artificial composite materials considered in the work consist of hydrated cement/gypsum and aggregate. Differences in the densities of their components are almost negligible, e.g. hydrated cement=1800–3300 kg/m³ and sand=2600 kg/m³. The above is the reason why the individual components could not be extracted from the whole microstructure. In order to enhance the contrast, different settings and parameters of the scanning device were chosen but with no satisfactory effect. Finally, it was decided to stain the samples with a contrast agent. Iodine solution (3%) in ethanol was used. Iodine is a chemical element with a high atomic number, and its compounds are commonly used in medicine as contrast agents, due to the fact that they are non-toxic. The specimens were immersed in the contrast agent and left for approximately 48 h until they dred out (the ethanol evaporates). The iodine was absorbed by the capillary pores network in the hydrated matrix.

Figure 4 shows the influence of the staining substance on the imaging of the cement mortar specimen. As mentioned before, after trial scanning of the whole sample, no aggregate was visible. Additional scanning, with a higher resolution of 13.5 μm, did not give satisfactory results – almost no heterogeneities can be distinguished (see Figure 4, top right). Application of contrast agent, without changing the resolution of scanning, demonstrates the significant differences in the image (see Figure 4, middle right). The outline of the sand grains is visible, as the cement matrix starts to absorb more radiation due to the contrast agent present. Nevertheless, the quality of the images is not sufficient for the quantitative analysis of the grain size distribution. After final scanning, with a resolution of 4.8 μm and an angle step of 0.05°, the detailed microstructure of mortar specimen is noticeable (see Figure 4, bottom right).

Figure 4:

The effect of the iodine staining on the microCT imaging of mortar.

It should be mentioned that all specimens were scanned with the use of Skyscan 1172 (Bruker MicroCT, Kontich, Belgium) X-ray microtomography with different resolutions and angle steps. The X-ray tube voltage was set to 100 kV, and the tube’s power was constant and equal to 10 W. All the scans were performed using built-in Al+Cu filter (Al 1 mm and Cu 0.05 mm). In addition, for each scan, the flat field correction was applied. The reconstruction was performed using NRecon (Bruker MicroCT, Kontich, Belgium) based on Feldkamp algorithm [33]. For the 3D visualisation of the images, CTVox (Bruker MicroCT, Kontich, Belgium) and CTVol (Bruker MicroCT, Kontich, Belgium) software were utilised.

2.3 Mechanical tests

Uniaxial compression tests for both gypsum and mortar specimens were performed in accordance with the guidelines of ASTM [34]. The bottom and top surfaces of each cylindrical specimen were treated in order to ensure their parallelism and planarity and that they are perpendicular to the peripheral surface. The tests were performed using a compression tester PROETI (Proeti S.A., Algete-Madrid, Spain) with a 3000 kN capacity. The rate of displacement was set to 0.003 mm/s. During each test, the compressive load on the specimen P as well as the change in measured axial length of specimen ∆L was recorded. The latter was obtained by making use of the High-Accuracy Digital Contact Sensor GT2-H12K (Keyence, Osaka, Japan). The measuring range of the sensor is 12 mm, while its resolution is on the level of 1 μm. A change in the measured axial length ∆L was then used to compute the axial strain ε, i.e. ε=∆L/L. The compressive stress in the test specimen σ was calculated as the quotient of the compressive load and the initial cross-sectional area of the specimen A; the UCS is therefore calculated using the maximum (peak) compressive load P_max on the specimen, i.e. UCS=P_max/A. In the case of the Young modulus, three different methods of its evaluation are proposed in [34], namely, (1) tangent modulus E_t, (2) average modulus E_av and (3) secant modulus E_s. In the current paper, the consideration is limited to one type of the Young modulus, i.e. the average one. The modulus was evaluated for the average slopes of the more-or-less straight line portion of the axial stress (σ)-axial strain (ε) curve (see Figure 5).

Figure 5:

Method for calculating Young modulus from axial stress-axial strain curve; average modulus of linear portion of stress-strain curve.

As mentioned earlier, the ratio between the length and diameter of the specimen was set to 2:1. This choice is supported by the results and conclusions obtained by different researchers (e.g. Pan et al. [12]). Moreover, the problem of the appropriate L/D ratio was also studied in the authors’ previous work [35] where a series of numerical simulations was performed. The conclusion derived from those considerations is as follows: the minimum L/D ratio, for the results to be stable, is 2. For more details we refer the reader to [35].

3.1 Imaging of specimen microstructure

After scanning of the samples with the final settings (described in the previous section), the reconstruction procedure was applied. As a result, sets of 2D images were obtained. In Figure 6 the comparison of the mortar (Figure 6A) and gypsum (Figure 6B) microstructures is presented. The colours of the images are inverted – higher grey levels (white colour) indicate particles of lower density, whereas lower grey levels are assigned to the denser constituents. Observing Figure 6, one can notice demonstrable differences in the microstructure geometry of the two artificial materials under consideration. In the mortar microstructure large distinguishable sand grains are present, which are obviously not present in that of the gypsum. In addition, it can be observed that the pore sizes differ significantly as well. For the proper comparison of the two microstructures, it is evident, however, that the visual observation has to be supplemented by the quantitative analyses. For that purpose, different microstructure measures – attenuation, porosity, pore and particle size distribution – were chosen and calculated separately for the two investigated materials.

Figure 6:

Microstructure of the investigated materials and 3D reconstruction; results for (A) mortar and (B) gypsum (inverted colours).

Before calculation, the reconstructed 2D attenuation maps (images) are processed. The image processing methodology is based on a trial and error process. Due to the high noise, denoising procedures are applied, i.e. Gaussian and Kuwahara filters. For the purpose of size and shape calculation for each of the material components, namely, porosity and particles properties, the appropriate segmentation procedures have to be implemented. Here, the Otsu’s segmentation method [36] is applied with the use of Bruker CTAn software [37]. Due to the partial volume effect, additional morphological procedures on the binarised images have to be employed to support the segmentation procedure, including erosion, opening, dilation and closing.

Firstly, the attenuation profiles of the materials were compared. The attenuation of the material is expressed in grayscale units. Due to the fact that the attenuation coefficient μ is proportional to the density of the material (Equation (2)), very dense elements are presented with white colour in the image, whereas objects of low density are distinguishable as dark grey or even black (in the case of air). For the purpose of this article, the colours were inversed (Figure 6). In Figure 7 the attenuation profiles are plotted versus the distance given in millimetres. The mortar diagram (Figure 7, top) exhibits the presence of the larger heterogeneities (sand grains) with lower frequency of peaks compared to the attenuation profile obtained for gypsum (Figure 7, bottom). Furthermore, the amplitude of the mortar’s profile is higher than in the case of that of the gypsum. It proves, therefore, that the microstructure of mortar is more heterogeneous – the differences in attenuation for each pixel are higher than in the gypsum images.

Figure 7:

Attenuation profiles for the mortar (top) and gypsum (bottom) specimens.

Another measure used for the comparison procedure was porosity. In order to obtain the specific values for each material, firstly, the sets of images were binarised to allow the segmentation of pores from the microstructure. Finally, based on the binarised images, the porosity understood as the number of black pixels divided by the total number of pixels of the chosen region of interest for each 2D slice (cross-section) was calculated. In Figure 8 the changes of porosity versus the height of the sample are graphically presented. It can be noticed that, in general, the porosity of the mortar is higher when compared to that of the gypsum.

Figure 8:

Porosity versus height of the sample.

Furthermore, the variation of porosity, when moving from point to point, is also greater for the mortar. In the case of gypsum, one can observe a rather constant value of porosity with the distance (height of the sample). In addition, the 3D visualisation of the pores was also performed to show their location within the cylindrical sample (Figure 9). The pore distribution suggests that there are differences in the pore size between the mortar and gypsum specimens. Hence, in order to verify the above statement, calculation of a pore size distribution was done. The results of this analysis are graphically presented in Figures 10 and 11.

Figure 9:

Three-dimensional visualisation of the pores in (A) mortar and (B) gypsum.

Figure 10:

Pore size distribution.

Figure 11:

Three-dimensional surface models of the pore size distribution for (A) mortar and (B) gypsum.

In Figure 10 the frequency of the pores, which is the total volume of pores of specified diameter (horizontal axis) divided by the volume of the sample, is plotted against the pore diameter (mm). The pore size diagram exhibits large dissimilarities between the two types of materials under study. It is apparent that the functions of frequency are entirely different. For the gypsum samples, the pore sizes are small, and they almost do not vary within the sample. On the other hand, for mortar there is no dominant pore size – they are distributed evenly in the material, and there is a great variety of sizes. In Figure 11 the pore size distribution is presented, however, as a 3D model of pores for each type of material investigated. Particular pore size intervals are shown with miscellaneous colours to illustrate their location within the composite. It is noteworthy that the pores in mortar (Figure 11A) are big and relatively uniformly distributed, whereas in gypsum the pores are smaller. In addition, in the gypsum plaster, some dense particles were observed. The results of the analysis of those constituents are presented in Figure 12. It is interesting that the maximum size of the particles, 0.12 mm, is greater than the size of the pores; thus, these particles can be treated as gypsum’s heterogeneities. Comparing this value to the maximum size of heterogeneity in sand, which is, based on the sieve analysis, approximately 3 mm, it is clearly visible that the materials are significantly different. The right panel of Figure 12 represents the particle shape factor (sphericity); as this value converges towards 1, the shape of the particle is close to a sphere.

Figure 12:

Gypsum particle analysis.

It was shown in the previous section that the materials (gypsum and mortar) exhibit significant differences in the microstructure geometry. Based on microCT, and as a consequence on microstructural measures (attenuation, porosity and pore size distribution), it was demonstrated that mortar has a more heterogeneous microstructure when compared to the one of gypsum. Now, it is verified whether these materials behave also in a different manner at uniaxial compression test.

3.2 Uniaxial compressive strength and Young modulus

As mentioned, UCS as well as the Young modulus E_av were determined for different diameters D. It has to be, however, strongly emphasised that for each value of D a series of tests was performed, and the final result, for a given D, is calculated as the mean value averaged over a specified number of tests. Furthermore, it is obvious that as the specimen size (diameter D) increases, the variance of both the uniaxial compressive strength and the Young modulus should decrease. Therefore, according to the central limit theorem (e.g. [38]) the number of tests, required for a given value of D, should decrease with the increase of specimen size. In other words, comparing the samples with D=28 mm and D=72 mm, it is evident that the required number of tests is significantly greater in the case of the specimen with smaller diameter, i.e. D=28 mm.

After each uniaxial compression test i (for a given diameter D) the mean values of UCS and the Young modulus E_av as well as the variances of these quantities were evaluated. The mean values were calculated in the following way:

Having the mean values (Equations (3) and (4)) then, the variances were estimated using the following expressions:

In the equations above, Eav,D(i) and UCSD(i) are the Young modulus and the uniaxial compressive strength obtained from the ith test performed on the specimen of diameter D, respectively, and n stands for the number of tests. As mentioned above, the number of tests n that could be recognised as sufficient is affected by the size of the specimen – diameter D. The number of tests n that was assumed to be appropriate was obtained by observing the influence of successive results on the change in both mean values (Equations (3) and (4)) and the variances (Equations (5) and (6)). In other words, if successive results did not affect a significant change of mean values (Equations (3) and (4)) and the variances (Equations (5) and (6)), the number of performed tests was recognised as sufficient. In what follows, some chosen plots are presented, namely, in Figures 13 and 14, the results obtained for mortar and gypsum specimens of diameter D= 28 mm.

Figure 13:

The results obtained for mortar specimens of diameter D=28 mm.

(A) the mean value of Young modulus, (B) the mean value of UCS, (C) variance of Young modulus and (D) variance of UCS, as a function of the number of performed tests n.

Figure 14:

The results obtained for gypsum specimens of diameter D=28 mm.

(A) the mean value of Young modulus, (B) the mean value of UCS, (C) variance of Young modulus and (D) variance of UCS, as a function of the number of performed tests n.

Analysing the results, two main features can be observed. For both mortar and gypsum samples the mean value of UCS exhibits faster convergence than the mean value of the Young modulus. Therefore, it is the Young modulus that determines the number of tests n. Furthermore, the convergence of results for gypsum specimens is faster when compared to the results for mortar samples. This is due to the smaller variation of successive results around the mean value for gypsum specimens. This can be observed in Figures 15 and 16 where the variation of Young modulus and UCS for successive tests is presented for mortar and gypsum specimens, respectively. One can simply observe the greater scatter in the results for mortar samples (for both Young modulus and UCS) compared to the ones obtained for gypsum specimens; in both figures the continuous line is the mean value (Equations (3) and (4)), while the dashed line describes the mean value ±σ¯, namely, the standard deviation (square root of variance given by Equations (5) and (6)). Due to fact that the same conclusions were drawn when the results for other diameters were analysed and compared, it can be stated that the appropriate number of tests n for gypsum specimens is smaller than that for the mortar samples.

Figure 15:

The results of successive uniaxial compression tests for mortar samples (D=28 mm).

(A) variation of Young modulus and (B) variation of UCS.

Figure 16:

The results of successive uniaxial compression tests for gypsum samples (D=28 mm).

(A) variation of Young modulus and (B) variation of UCS.