Effect of loading rates and stress paths on rock strengths: a novel approach based on experimental evidence

Abstract

Tests of rock strength under different loading rates and stress paths often lead to ambiguous results, as both factors are inter-playing each other. As a result, there is surprisingly little research into the study of unifying both factors when comparing the rock strengths from various triaxial compression tests. In this technical note, we propose a simple but rigorous mathematics expression to assess and compare the rock strengths under different loading rates and stress paths. To make the proposed method as accessible as possible to a broad application, we have derived the explicit forms of solutions for three popular rock failure criteria. We then examine and validate the proposed method using the results of triaxial tests on marbles and sandstones, based on the important Mohr-Coulomb criterion.

Appendix. An example of FAR calculation

In this section, an example is given to illustrate how to calculate FAR and what is the relationship between the Mohr-Coulomb failure criterion and FAR. For a better illustration purpose, let us look at the loading condition of a conventional compression test (CCT) of marble 1 in Table 3. There are mainly three steps involved.

• Step 1: triaxial compression tests of rock specimens

The marble specimens were axially loaded at a rate of 0.5 MPa/s, while the corresponding confining pressure for each specimen remains unchanged, such that the loading condition can be expressed using the loading vector ${v}_{\sigma }=\left({v}_{\sigma_1},{v}_{\sigma_2},{v}_{\sigma_3}\right)$ = (0.5, 0, 0). The strength of each specimen at failure was measured.

• Step 2: use of Mohr-Coulomb failure criterion

Plotted the measured strengths on σ₁-σ₃ plane, given that the principal stresses follow the condition of σ₁ ≥ σ₂ ≥ σ₃; fitted the strength data with Mohr-Coulomb failure criterion to estimate the corresponding internal friction angle, φ, e.g., 37.4° as shown in Table 3.

• Step 3: FAR calculation

Substituting φ into Eq. (2) or $\left(1,0,-\frac{1+\sin \varphi }{1-\sin \varphi}\right)$ in Table 1 to calculate the direction n = (1, 0, − 4.09), and the norm ∣n∣ = 4.21; then, calculating the value of FAR as per Eq. (3): $FAR={v}_{\sigma}\cdot \frac{n}{\mid n\mid }=\left(0.5,0,0\right)\cdot \frac{\left(1,0,-4.09\right)}{4.21}\approx 0.12$.

The rest FARs for the other tests in Table 3 can be conveniently calculated by following the above procedure.