Size effects on the strength and cracking behavior of flawed rocks under uniaxial compression: from laboratory scale to field scale

Abstract

The failure behavior of field-scale rock masses has long been studied indirectly through laboratory compression tests on rock specimens with preexisting flaws. However, little to no attention has been paid to size effects on these cracking processes that may be governed by the relative size between the fracture process zone and the rock structure. Here, we investigate such size effects on the compressive strength and cracking behavior of flawed rocks through high-fidelity simulations of mixed-mode fracture in quasi-brittle materials. We perform a series of numerical uniaxial compression tests on geometrically similar gypsum specimens with single and double flaws, across a wide range of sizes from 0.25 times a standard laboratory specimen size to 16 times. The results suggest strong size effects on both the uniaxial compressive strength and cracking patterns. The size effect on the compressive strength appears qualitatively similar to Bažant’s size effect law derived for the tensile strength of notched structures. However, the quantitative changes in the strength deviate from the existing size effect law which does not account for mixed-mode fracture. As for the cracking behavior, three types of cracking patterns per flaw configuration are identified as the specimen size changes. Remarkably, the cracking patterns that emerge at the field scale, where the size of the fracture process zone is negligible, are analogous to those observed from laboratory experiments on highly brittle materials. The findings of this work provide important insights into how to bridge observations on rock fracturing processes across scales.

Appendices

Appendix 1: Validation of the phase-field model for investigating energetic size effects

This appendix validates that under pure tensile failure, the phase-field model used in this study produces nominal strengths that follow Bažant’s size effect law. To this end, we use the phase-field model to simulate a series of three-point bending tests, which is a standard problem used for examining size effects on tensile strengths. Figure 12 depicts the setup of the three-point bending tests conducted herein. To investigate size effects, the tests are repeated for specimens of ten different size factors: $D=0.01,0.0625,0.125,0.25,0.5,1,2,4,8$, and 16. The numerical tests use the same material parameters and the discretization scheme as those in the uniaxial compression tests on the flawed specimens.

Fig. 12

Setup of the numerical three-point bending tests: geometry and boundary conditions. D denotes the size factor

Figure 13 presents the nominal strengths of the notched beam specimens with varied D. Also shown is Bažant’s size effect law, Eq. (1), fitted to the nominal strengths. One can see that the nominal strengths show size dependence that is well-described by the existing size effect law—an analytical expression derived for notched structures made of quasi-brittle materials. As such, it is confirmed that the phase-field model is a suitable means for investigating energetic (FPZ-related) size effects.

Fig. 13

Nominal strengths in the three-point bending tests with varied D, and Bažant’s size effect law fitted to the nominal strengths ($\sigma _0=4.2$ MPa and $D_0=0.257$ for Eq. 1)

Appendix 2: Selection of the phase-field length parameter

This appendix describes how the value of the phase-field length parameter, L, is selected for the numerical experiments in this study. We note that although the double-phase-field model is built on formulations where stress–strain responses are insensitive to L at the material point level, it can still show some degree of sensitivity to L at the structural level, see, e.g., [29, 32]. For our purpose, a good value of L should be (i) small enough to produce physically meaningful results and (ii) large enough to be computationally affordable. To find such a value, we test three different values of L for conducting a series of numerical uniaxial compression tests on the 45-single specimens with varied D.

Figure 14 shows how the UCS and their size dependence are affected by the value of L. It can be seen that as L becomes smaller, the UCS slightly increase particularly when the structure size is larger. This trend may be explained by that as the structure becomes larger, it fails in a more brittle manner, and hence its strength becomes more dependent on the width of the phase-field regularization. Note, however, that our interest in this study is not the values of UCS per se. Instead, we are interested in the decay rate (the slope in the bi-logarithmic plot) of UCS, which quantifies the energetic size effects. From the figure, one can find that the decay rate shows a convergent value of 1/1.34 from $L=0.15D$ mm. So it can be concluded that the choice of $L=0.15D$ mm is physically meaningful, while involving less computational cost than a smaller value of L. Also, while not presented for brevity, the cracking patterns obtained with the three values of L are virtually identical. Therefore, we select $L=0.15D$ mm for the numerical experiments in this study.

Fig. 14

Uniaxial compressive strengths in the 45-single specimens with $D=0.25,0.5,1,2,4,8$, and 16, obtained with three different values of L