### Stress–pressure curves

The stress–pressure curves of the triaxial loading and unloading assessments are proven in Fig. 4, and the check outcomes are proven in Desk 1. It may be seen that the rocks beneath totally different stress paths present apparent elastoplastic traits, and the curves modifications from a linear relationship to a nonlinear relationship earlier than reaching the height power. In contrast with the loading situation, the pressure softening tendency is weakened and the brittleness traits are extra apparent of the rock beneath the unloading situation, and the height power and peak pressure of the pattern are decrease.

### Failure traits

Determine 5 present the failure traits of argillaceous sandstone beneath totally different stress paths, respectively, and each present shear failure traits (blue curve is the shear failure floor). The fracture floor throughout loading is slim and the diploma of settlement is excessive, indicating that the event and penetration of compression-shear fractures prompted argillaceous sandstone to endure compression-shear failure. The fracture floor throughout unloading is vast, and powder and fragments are dropped. Plenty of secondary tensile cracks lengthen close to the primary shear aircraft, indicating that the rock deforms within the unloading course and the rock pattern undergoes tensile and shear failure. When the confining strain is relieved, the excessive axial stress degree causes the speedy growth and propagation of microcracks. The excessive confining strain will decelerate the event of secondary cracks reminiscent of tensile cracks, ensuing within the discount of secondary cracks because the unloading confining strain will increase.

### Shear power parameters evaluation

The Mohr–Coulomb power criterion believes that the sum of the cohesive pressure of the rock and the friction pressure generated by the traditional stress on the shear floor is the shear power of the rock. The cohesive pressure *c* and the inner friction angle *φ* could be obtained by way of the stress state when the rock undergoes shear failure. The expression is^{29}:

$$ sigma_{1} { = }Asigma_{3} + B $$

(1)

the place *σ*_{1} and *σ*_{3} are the utmost and minimal principal stresses when the rock failure, respectively, (A = frac{1 + sin varphi }{{1 – sin varphi }}), (B = frac{2ccos varphi }{{1 – sin varphi }}).

Utilizing Eq. (1) to carry out regression evaluation on the information in Desk 1, the parameters *A* and *B* could be obtained, and *c* and *φ* could be decided by:

$$ varphi = arcsin frac{A – 1}{{A + 1}} $$

(2)

$$ c = frac{{Bleft( {1 – sin varphi } proper)}}{2cos varphi } $$

(3)

The Mohr–Coulomb power criterion doesn’t take into account the affect of the intermediate principal stress. By numerous rock triaxial compression assessments, Mogi^{30} believes that the affect of intermediate principal stress on rock power can’t be ignored, and proposed the octahedral power criterion:

$$ tau_{{{textual content{oct}}}} = fleft( {sigma_{m,2} } proper) $$

(4)

the place *τ*_{oct} and *σ*_{m,2} are the octahedral shear stress and the efficient intermediate stress, respectively, the expression is:

$$ tau_{oct} = frac{1}{3}sqrt {left( {sigma_{1} – sigma_{2} } proper)^{2} + left( {sigma_{2} – sigma_{3} } proper)^{2} + left( {sigma_{3} – sigma_{1} } proper)^{2} } $$

(5)

$$ sigma_{m,2} = {{left( {sigma_{1} + sigma_{3} } proper)} mathord{left/ {vphantom {{left( {sigma_{1} + sigma_{3} } proper)} 2}} proper. kern-nulldelimiterspace} 2} $$

(6)

Since *σ*_{2} = *σ*_{3} within the typical triaxial check, so:

$$ tau_{oct} = frac{sqrt 2 }{3}left( {sigma_{1} – sigma_{3} } proper) $$

(7)

Al-Ajmi and Zimmerman^{31,32} discovered that there’s a linear relationship between *τ*_{oct} and *σ*_{m,2}, and proposed the Mogi–Coulomb power criterion:

$$ tau_{{{textual content{oct}}}} = a + bsigma_{m,2} $$

(8)

the place *a* and *b* are check parameters.

By evaluating Eqs. (6) to (8) with the Mohr–Coulomb expression, the connection between the check parameters *a*, *b* and the Mohr–Coulomb parameters *c*, *φ* could be obtained:

$$ a = frac{2sqrt 2 }{3}ccos varphi $$

(9)

$$ b = frac{2sqrt 2 }{3}sin varphi $$

(10)

The Drucker–Prager power criterion not solely considers the affect of the intermediate principal stress, but in addition considers the impact of hydrostatic strain, the expression is:

$$ sqrt {J_{2} } = alpha I_{1} + {textual content{okay}} $$

(11)

the place *α* and *okay* are experimental constants, *I*_{1} is the primary invariant of stress, and *J*_{2} is the second invariant of stress deviator. The expressions of *I*_{1} and *J*_{2} are:

$$ I_{1} = sigma_{1} + sigma_{2} + sigma_{3} $$

(12)

$$ J_{2} = frac{1}{6}left[ {left( {sigma_{1} – sigma_{2} } right)^{2} + left( {sigma_{2} – sigma_{3} } right)^{2} + left( {sigma_{3} – sigma_{1} } right)^{2} } right] $$

(13)

The experimental constants within the D–P criterion could be transformed to the Mohr–Coulomb parameters in 4 methods ^{33,34}: (1) the M–C outer nook circumscribed circle criterion (DP1), (2) the M–C internal nook circumscribed circle criterion (DP2), (3) the M–C inscribed circle criterion (DP3), (4) M–C equal-area circle criterion (DP4), the precise expressions are proven in Desk 2. The right utility of the D–P collection of standards depends upon the totally different stress states of the rock mass. For the stress state that satisfies the situations of *σ*_{1} > *σ*_{2} = *σ*_{3}, reminiscent of unidirectional compression and standard triaxial compression, DP1 matches the M-C criterion, so this paper selects the DP1 criterion for evaluation.

Based on the check knowledge in Desk 1, regression evaluation was carried out utilizing Eqs. (1), (8) and (11), as proven in Fig. 6. The *c* and *φ* are obtained by way of calculation strategies beneath totally different standards, as proven in Desk 3. Equally, the above technique is used to resolve the residual shear power parameters *c*_{r} and *φ*_{r}, as proven in Desk 4.

It may be seen from Fig. 6 that the Mogi–Coulomb and Drucker–Prager power criterion beneath totally different stress paths have a greater regression impact than the Mohr–Coulomb. The intermediate principal stress is taken into account within the first two power standards, indicating that it has a sure affect on rock failure. In an effort to decide which parameter is extra correct, one technique is to make preliminary judgments primarily based on the engineering subject expertise, and the opposite technique is to make use of mathematical strategies to calculate the sum of absolutely the values of the becoming deviations for optimization^{35}.Subsequently, the common power deviation *f*_{a} is launched:

$$ {textual content{f}}_{a} { = }sum {frac{{ABSleft( {{textual content{f}}_{c} – {textual content{f}}_{t} } proper)}}{N}} $$

(14)

the place *f*_{c} and *f*_{t} are the calculated and experimental values of *σ*_{1}, *τ*_{oct} and (sqrt {J_{2} }) within the three standards, and *N* is the variety of teams of check knowledge.

The ultimate outcomes of the 2 strategies each present that *c,* *φ,* *c*_{r} and *φ*_{r} obtained by the Mogi–Coulomb are extra acceptable. The Mogi-Coulomb can higher replicate the loading and unloading failure power and residual power traits of argillaceous sandstone. It may be calculated from Tables 3 and 4 that *c* decreased by 30.87%, *c*_{r} decreased by 49.31%, *φ* elevated by 30.87%, and *φ*_{r} elevated by 68.15% within the unloaded state in contrast with the loaded state. Earlier than the rock reaches its peak power throughout unloading, the tensile cracks produced will destroy the rock particles and cements, and ultimately trigger the rock *c* to lower. The tensile-shear fracture floor is usually rougher than the compression-shear fracture floor, so *φ* will improve when the rock is unloaded.

https://www.nature.com/articles/s41598-022-20433-y