Evaluating Size Effects for a Porous, Weak, Homogeneous Limestone

Abstract

In rock engineering, size effects have been a topic of extensive research since the early 1960s, and despite many advances over the years, our understanding of size effect remains incomplete, especially for weak, porous, homogeneous rocks. Indeed, the vast majority of studies related to size effect have specifically considered low porosity rocks (generally crystalline). To bridge this gap in knowledge, we conducted unconfined compression tests on cubic limestone blocks ranging in size from 0.1 to 0.9 m. Texas Cream Limestone, which is a porous, homogeneous, weak rock, was chosen for this study. As this rock has not previously been studied in the literature, conventional compression tests and indirect tensile strength tests on cylindrical specimens were completed prior to testing the cube specimens. For the largest specimens, 3D digital image correlation (3D-DIC) was employed to track the surficial displacements as a function of the applied load. The tests revealed a lack of size effect for the entire range of block sizes considered. To evaluate size effects more broadly, data from prior studies on sedimentary rocks were compiled, and a tendency for the magnitude of the size effect on strength to decline with increasing porosity was noted. Some hypotheses regarding this trend are presented and evaluated based on strain-field heterogeneity metrics obtained from the 3D-DIC analysis.

Highlights

• Unconfined compression tests were conducted on limestone blocks ranging in size from 0.1 m to 0.9 m side length.

• Negligible size effect on strength was observed in this weak, porous, homogeneous limestone.

• 3D-Digital Image Correlation analysis was performed to obtain strain fields as a function of applied load for the two largest specimens.

• Contrary to low-porosity rocks, more heterogeneity in strain field was noted in the axial direction in comparison to the lateral direction

• A compilation of data from the literature indicates that increased porosity may dampen size effects.

Appendices

Appendix A: Geometrical Details of Test spEcimens

The mean and standard deviation of the heights of the 0.2–0.9 m specimens are listed in Table

Table A1 Mean and standard deviation of the height of each specimen

A1, while the mean dimensions in three perpendicular directions are listed in Table

Table A2 Mean dimension of each test specimen

A2. The 0.1 m specimens were only measured once in the three perpendicular directions. While congruence in height measured along different vertical faces does not imply parallelism for the entirety of the loading surfaces (i.e. central portion versus edges), we believe it is reasonable to assume the surfaces to be planar when cut using a saw. Accordingly, the dimensions measured along the edges are considered to be representative of the entire loading surface.

The mean dimensions along the x, y, and z directions are most dissimilar for the 0.1 m specimens (Table 3). The reason is that the 0.1 m specimens were saw-cut from one 0.2 m block and then ground until two opposite faces were parallel. This grinding process and minor errors in sub-dividing the 0.2 m block led to the variation in the edge lengths. In any case, the width to height ratio of all 0.1 m specimens was within the interval [0.98, 1.06], and this change in aspect ratio is unlikely to affect the strengths in any meaningful way (Du et al. 2019).

The ASTM suggested method covering the preparation of UCS specimens (ASTM 4543-85, 2001) requires ends to be flat to ± 0.025 mm and not depart from perpendicularity to the longitudinal axis by more than 0.25 degrees (or 0.22 mm in 50 mm). It is noted here that the ASTM specifications were developed for smaller cylindrical specimens and possibly represent a lower bound (in terms of stringency) of specimen requirement (Cvitanović et al. 2015). The flatness of the 0.2–0.9 m specimens were not measured explicitly, but if the variability in specimen size (Table 2) is normalized with respect to the edge length, then all values are lower than the corresponding recommended value in Cvitanović et al. (2015). In terms of parallelism of the loading surfaces, if 2*variability/mean edge length is computed for all specimens in Table 2, then the value ranges from [0.0005, 0.0027]; the corresponding ASTM requirement is 0.0044. All specimens, therefore, satisfy the parallelism requirement of ASTM, but some of the specimens exceed the stricter ISRM stipulations (0.001 radians or 0.05 mm in 50 mm; Fairhurst and Hudson 1999). In any case, the fact that highly consistent stress–strain curves and fracture patterns were obtained (as reported in Sect. 4) provides confidence in the test results.

Appendix B: Adjustment in Strain Calculation due to Lighting Issues

During the calculation of strain heterogeneity, significant noise was observed in the strain field generated by GOM Correlate along the right edge of the 0.9 m specimen. The raw images captured by 3D-DIC were revisited, and some issues with lighting were identified. In particular, about ~ 0.35 m of the specimen right side was found to be darker than the rest of the image (Fig. 14a). This issue was not present in the 0.7 m specimen images, and the lighting was consistent across the entire surface. To filter out the noise in the DIC data, the mean and standard deviation of ${\varepsilon }_{yy}$ was computed within rectangular bins that were 0.75 mm wide (3 pixels) and spanned the entire specimen height. The results at 20%, 40%, 60%, and 80% peak strength are plotted against the X-coordinate of the bin midpoints in Fig. 14b–e. Since the standard deviation of the ${\varepsilon }_{yy}$ strain started to fluctuate dramatically in the rightmost ~ 350 mm region (Fig. 14b–e), that portion was omitted while preparing Fig. 9.

Fig. 14

a Raw image from one of the 3D-DIC camera at a load level corresponding to 20% peak strength. Mean and standard deviation of ${\varepsilon }_{yy}$ strain field computed within rectangular strips 0.75 mm wide across the entire width of the specimen at b 20% peak strength, c 40% peak strength, d 60% peak strength, and e 80% peak strength. X-coordinate is the center point of the moving rectangular window

Appendix C: Discussion on the Two Outliers: Gambier Limestone and Bathstone

The data used to develop Fig. 12 are presented in Table

Table C1 Data (and references) used for plotting Fig. 12

4. With respect to β, there are two notable outliers—Gambier Limestone and Bathstone. The exact reason why β < − 0.4 for these two rocks is not obvious. Gambier Limestone is reported to have ~ 44–50% porosity and is composed of 98% calcite (Khishvand et al. 2016; Zhai et al. 2020; Armstrong et al. 2021). It is hypothesized that such a homogeneous, high porosity rock can exhibit size effect if the pore size distribution itself is playing a role in the failure process (i.e. the volumetric proportion of larger pores is not attaining a constant value over the range of specimen sizes considered). This is somewhat evident in Fig. 1, where Gambier Limestone attains its maximum UCS at 119 mm size that is significantly larger than what is typical for most rock types (40–60 mm; Kong et al. 2021). Pores can act as crack arrestors or stress concentrators; in this case, it is possible that the pores interacted with the surface to lead to premature failure in the < 119 mm diameter specimens and continued to play a role in the failure process through internal mechanisms (e.g., stress concentrating around larger pores, crack initiation) in the larger specimens. The fact that larger pores can lower material strength has been illustrated by Biao et al. (2017) and Li et al. (2020) using composites. Khishvand et al. (2016) also recently reported pore diameters as large as 0.8 mm for Gambier Limestone. The same reasoning may or may not be applicable to Bathstone that has a porosity of 23–26% (Marker 2015), and there could also be an interaction between the pores and its oolitic structure (Elliot and Brown 1985). It is interesting, however, that the UCS of Bathstone drops by 25.7% between 54 and 74 mm diameter and only by 15% between 74 and 150 mm diameter.