A novel three-dimensional nonlinear unified failure criterion for rock materials

Abstract

To study the failure (strength) regularities of rocks under complex stress states, we propose a novel three-parameter deviatoric function that includes the shape of the failure envelope on the Mohr–Coulomb (MC), Ottosen, Bigoni–Piccolroaz, and Wu–Zhang deviatoric planes. This function realizes the deviatoric plane shape of numerous classic criteria, including the Rankine, Tresca, von Mises, generalized Tresca, MC, Drucker–Prager (DP), Matsuoka–Nakai (MN), Lade–Duncan (LD), and Ottosen. On this basis, taking the nonlinear characteristics of the power function, we establish a novel 3D NUF criterion, namely MCNUF and LDNUF criteria, where the novel parameter γ is based on the basic parameters of two classical criteria, MC and LD, respectively. Based on the triaxial test data of different rocks, the proposed MCNUF and LDNUF criteria are compared with the previous criteria (the generalized nonlinear failure criterion (MNGNF), the DP and MN unified (DPMNu) criterion). Results show that: (1) the internal friction angles predicted by each criterion are equal under the same meridian plane parameters. (2) For four kinds of rock materials, the prediction performance of the proposed criteria is generally better than that of the MNGNF and DPMNu criteria. (3) The proposed criteria describe the hydrostatic stress and IPS and their coupled effect on various rock materials. (4) The 3D graphical visualization of the proposed criteria is carried out systematically, which plays a positive guiding role in enriching and perfecting the strength theory.

Appendices

Appendix A: Proof of smoothness of the novel three-parameter deviatoric function

The first and second derivatives of Eq. (15) are expressed as:

$$g^{\prime}\left( {\theta_{\sigma } } \right) = \gamma \sin 3\theta_{\sigma } \frac{{\sin \left( {\frac{\pi }{6}\alpha { + }\beta { + }\frac{1}{3}\cos^{ - 1} \gamma } \right)\cos \left[ {\frac{\pi }{6}\alpha + \frac{1}{3}\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right)} \right]}}{{\sin^{2} \left[ {\frac{\pi }{6}\alpha + \frac{1}{3}\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right)} \right]\sqrt {1 - \gamma^{2} \cos^{2} 3\theta_{\sigma } } }}$$

(32)

$$\begin{aligned} g^{\prime \prime } \left( {\theta_{\sigma } } \right) = & \left\{ {\frac{{3\gamma \cos 3\theta_{\sigma } \left( {1 - \gamma^{2} \cos^{2} 3\theta_{\sigma } - \gamma \sin^{2} 3\theta_{\sigma } } \right)\cos \left[ {\frac{\pi }{6}\alpha + \frac{1}{3}\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right)} \right]}}{{\sin^{2} \left[ {\frac{\pi }{6}\alpha + \frac{1}{3}\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right)} \right]\left( {1 - \gamma^{2} \cos^{2} 3\theta_{\sigma } } \right)^{\frac{3}{2}} }}} \right. \\ & - \left. {\gamma^{2} \sin^{2} 3\theta_{\sigma } \frac{{1{ + }\cos^{2} \left[ {\frac{\pi }{6}\alpha + \frac{1}{3}\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right)} \right]}}{{\sin^{3} \left[ {\frac{\pi }{6}\alpha + \frac{1}{3}\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right)} \right]\left( {1 - \gamma^{2} \cos^{2} 3\theta_{\sigma } } \right)}}} \right\}\sin \left( {\frac{\pi }{6}\alpha { + }\beta { + }\frac{1}{3}\cos^{ - 1} \gamma } \right) \\ \end{aligned}$$

(33)

Equation (32) shows that sin3θ_σ = 0 when γ ≠ 1, θ_σ = 0°, and θ_σ = 60°. g′(0°) = 0 and g′(60°) = 0. Research shows that the novel three-parameter deviatoric plane function satisfies the smoothness requirement at the corner point.

Appendix B: Proof of convexity of the novel three-parameter deviatoric function

2.1 Appendix B.1: Determination of the upper limit of parameter α

By substituting Eqs. (15), (32), and (33) into Eq. (5), we obtain the following:

$$\sin^{2} \left[ {\frac{\pi }{6}\alpha + \beta + \frac{1}{3}\cos^{ - 1} \gamma } \right]\left( {1 - \gamma^{2} } \right)\left\{ {\frac{{\sin \left[ {\frac{\pi }{6}\alpha + \frac{1}{3}\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right)} \right]\left( {1 - \gamma^{2} \cos^{2} 3\theta_{\sigma } } \right)^{\frac{3}{2}} }}{{\sin^{3} \left[ {\frac{\pi }{6}\alpha + \frac{1}{3}\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right)} \right]\left( {1 - \gamma^{2} \cos^{2} 3\theta_{\sigma } } \right)^{\frac{5}{2}} }} - } \right.$$

$$\left. {\frac{{\left( {3\gamma^{3} \cos^{3} 3\theta_{\sigma } - 3\gamma \cos 3\theta_{\sigma } } \right)\cos \left[ {\frac{\pi }{6}\alpha + \frac{1}{3}\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right)} \right]}}{{\sin^{3} \left[ {\frac{\pi }{6}\alpha + \frac{1}{3}\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right)} \right]\left( {1 - \gamma^{2} \cos^{2} 3\theta_{\sigma } } \right)^{\frac{5}{2}} }}} \right\} \ge 0$$

(34)

When γ ∈ [0,1), there is (1-γ²) > 0 and $\left( {1 - \gamma^{2} \cos^{2} 3\theta_{\sigma } } \right) > 0$. According to the inequality's basic properties, both sides multiply (or divide) the same number greater than zero. The unequal direction remains unchanged. Therefore, both ends of Eq. (34) are divided by (1-γ²) and then multiplied by $\left( {1 - \gamma^{2} \cos^{2} 3\theta_{\sigma } } \right)$ for simplification to obtain:

$$1 - \frac{{3\gamma \cos 3\theta_{\sigma } \cos \left[ {\frac{\pi }{6}\alpha + \frac{1}{3}\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right)} \right]}}{{\left( {1 - \gamma^{2} \cos^{2} 3\theta_{\sigma } } \right)^{\frac{1}{2}} \sin \left[ {\frac{\pi }{6}\alpha + \frac{1}{3}\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right)} \right]}} \ge 0$$

(35)

Let $\cos^{ - 1} \left( {\gamma \cos 3\theta_{\sigma } } \right) = \lambda$ and γ ∈ [0,1]; Eq. (35) is further rewritten as:

$$\sin \lambda \sin \left( {\frac{\pi }{6}\alpha + \frac{1}{3}\lambda } \right) - 3\cos \lambda \cos \left( {\frac{\pi }{6}\alpha + \frac{1}{3}\lambda } \right) \ge 0$$

(36)

According to the trigonometric function, Eq. (36) is simplified as:

$$\cos \left( {\frac{\pi }{6}\alpha - \frac{2}{3}\lambda } \right) + 2\cos \left( {\frac{\pi }{6}\alpha + \frac{4}{3}\lambda } \right) \le 0$$

(37)

where $\lambda \in \left[ {\cos^{ - 1} \gamma ,\pi - \cos^{ - 1} \gamma } \right]$.

According to the trigonometric function and assuming that k = 2/3δ, Eq. (37) is transformed into:

$$\frac{\sin k - 2\sin 2k}{{\cos k + 2\cos 2k}}\sin \frac{\pi }{6}\alpha + \cos \frac{\pi }{6}\alpha \ge 0$$

(38)

where $k \in \left[ {\frac{2}{3}\cos^{ - 1} \gamma ,\frac{2}{3}\left( {\pi - \cos^{ - 1} \gamma } \right)} \right]$.

According to the double-angle formula of a trigonometric function, Eq. (38) is further simplified as:

$$\frac{{\sin k\left( {1 - 4\cos k} \right)}}{{4\cos^{2} k + \cos k - 2}}\sin \frac{\pi }{6}\alpha + \cos \frac{\pi }{6}\alpha \ge 0$$

(39)

The parameter α in Eq. (39) is solved to obtain:

$$\alpha \ge - \frac{6}{\pi }\tan^{ - 1} \left[ {\frac{{4\cos^{2} k + \cos k - 2}}{{\sin k\left( {1 - 4\cos k} \right)}}} \right]$$

(40)

The domain of the function y = arctan(x) is (-∞, + ∞), the range is $\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)$, and it is a monotone increasing function in the domain.

When the value range of k is $k \in \left[ {\frac{2}{3}\cos^{ - 1} \gamma ,\frac{2}{3}\left( {\pi - \cos^{ - 1} \gamma } \right)} \right]$, the value range at the right end of Eq. (40) can be calculated as [-3,1]. The lower bound value range of α is:

$$\alpha \ge 1$$

(41)

The research shows that when α ≥ 1, the novel three-parameter deviatoric plane function (Eq. (15)) satisfies the convexity requirement.

2.2 Appendix B.2: Determination of the lower limit of parameter α

Herein, the upper limit of α is solved by using the aspect ratio K ∈ [0.5,1], β = π and γ ∈ [0,1].

• (1) When K = 0.5, γ = 0, Eq. (15) is simplified as:

  $$0.5\sin \left( {\frac{\pi }{6}\alpha { + }\frac{\pi }{6}} \right) = \sin \left( {\frac{\pi }{6}\alpha + \frac{\pi }{6}} \right)$$

  (42)

That is

$$\frac{\pi }{6}\alpha { + }\frac{\pi }{6}{ = }z\pi ,z = 1,2, \ldots ,n$$

(43)

Solving Eq. (43), we obtain:

$$\alpha = 5,\;11,\; \ldots ,\;\left( {6n - 1} \right)$$

(44)

• (2) When K = 0.5, γ = 1, Eq. (15) can be simplified as:

  $$0.5\sin \left( {\frac{\pi }{6}\alpha } \right) = \sin \left( {\frac{\pi }{6}\alpha + \frac{\pi }{3}} \right)$$

  (45)

Based on the trigonometric function, Eq. (45) can be simplified as:

$$\frac{\sqrt 3 }{2}\cos \frac{\pi }{6}\alpha = 0$$

(46)

That is

$$\frac{\pi }{6}\alpha { = }z\frac{\pi }{2},z = 1,2, \ldots ,n$$

(47)

Solving Eq. (47), we obtain the following:

$$\alpha = 3,\;6,\; \ldots ,\;3n$$

(48)

• (3) When K = 1.0, γ = 0, Eq. (15) can be simplified as:

  $$\sin \left( {\frac{\pi }{6}\alpha { + }\frac{\pi }{6}} \right) = \sin \left( {\frac{\pi }{6}\alpha + \frac{\pi }{6}} \right)$$

  (49)

Solving Eq. (49), we obtain the following:

$$\alpha = 2,\;3,\; \ldots ,\;n$$

(50)

When K = 1.0, γ = 1.0, Eq. (15) can be simplified as:

$$\sin \left( {\frac{\pi }{6}\alpha } \right) = \sin \left( {\frac{\pi }{6}\alpha + \frac{\pi }{3}} \right)$$

(51)

Based on the trigonometric function, Eq. (51) can be simplified as follows:

$$\sin \left( {\frac{\pi }{6}\alpha - \frac{\pi }{3}} \right) = 0$$

(52)

That is

$$\frac{\pi }{6}\alpha - \frac{\pi }{3} = z\pi ,z = 0,1, \ldots ,n$$

(53)

Solving Eq. (53), we obtain the following:

$$\alpha = 2,\;8,\; \ldots ,\;\left( {6n + 2} \right)$$

(54)

In summary, the upper limit value of the parameter α is:

$$\alpha \le 2$$

(55)

Appendix C: Influence of different meridian plane functions on the failure criterion

3.1 Appendix C.1: Influence of the meridian plane function on failure criterion for the Yamaguchi marble true triaxial test

3.1.1 Appendix C.1.1 Meridian plane of the linear function passing through the I ₁ negative semi-axis

Based on Table 2, the test data at θ_σ = 0° are used for fitting, and d = 184.4 MPa is obtained. From Eq. (22), δ_t = 138.3 MPa and ξ_t = 79.8 MPa. Based on Sect. 5.1, the aspect ratio K of the Yamaguchi marble material is 0.8252. The meridian plane parameters of the MCNUF and LDNUF criteria are N_f = 0.2416, δ_t = 138.3 MPa, and n = 1 (see Fig. 

Fig. 25

Predicted results of the MCNUF and LDNUF criteria under the I₁ negative axis linear meridian plane and the Yamaguchi marble true triaxial test data: a best-fit envelope of the meridian plane, b deviatoric plane failure envelope of the MCNUF criterion, c deviatoric plane failure envelope of the LDNUF criterion, d 3D space failure envelope surface of the MCNUF criterion and d 3D space failure envelope surface of the LDNUF criterion

25a).

The deviatoric plane parameters of the MCNUF and the LDNUF criteria are determined according to the meridian plane parameters and the aspect ratio. The deviatoric plane parameters of the MCNUF criterion are α = 1.6007, β = π or − π, γ_MC = 0.9557, and φ = 31.3°. The deviatoric plane parameters of the LDNUF criterion are α = 1.2304, β = π or − π, γ_LD = 0.6155, and φ = 31.3°.

Similarly, the strength parameters of MNGNF criterion are ω = 0.2416, β = 138.3 MPa, n = 1, α = 1.2046, A = 0.7666, and φ = 31.3°. The strength parameters of the DPMNu criterion are A = 0.5918, ξ_t = 79.8 MPa, n = 1, ω = 0.7666, α = 0.4075, and φ = 31.3°. The prediction results of the MCNUF and LDNUF criteria (when the linear meridian plane passes through the I₁ negative semi-axis and the Yamaguchi marble true triaxial test data) are shown in Fig. 25, and the prediction results of the MNGNF and DPMNu criteria are shown in Fig. 

Fig. 26

Predicted results of the MNGNF and DPMNu criteria under the I₁ negative axis linear meridian plane and the Yamaguchi marble true triaxial test data: a deviatoric plane failure envelope of the MNGNF criterion and b deviatoric plane failure envelope of the DPMNu criterion

26.

3.1.2 Appendix C.1.2 Meridian plane of the power function passing through the origin

On the meridian plane (see Fig. 

Fig. 27

Prediction results of the MCNUF and LDNUF criteria under the power function meridian plane passing through the origin point and the Yamaguchi marble true triaxial test data: a best-fit envelope of the meridian plane, b deviatoric plane of the MCNUF criterion, c deviatoric plane of the LDNUF criterion, d 3D space failure envelope surface of the MCNUF criterion and d 3D space failure envelope surface of the LDNUF criterion

27a), the parameters of the MCNUF and LDNUF criteria are N_f = 6.0236, δ_t = 0 MPa, and n = 1, respectively. The aspect ratio is K = 0.8252. The deviatoric plane parameters of the MCNUF and LDNUF criteria are determined according to the meridian plane parameters and the aspect ratio, where the deviatoric plane parameters of the MCNUF criterion are as follows: when I₁ = 470.1 MPa, α = 1.6155, β = π or − π, γ_MC = 0.9678 and φ = 26.2°. When I₁ = 427.0 MPa, α = 1.6131, β = π or − π, γ_MC = 0.9659 and φ = 27.1°. When I₁ = 356.6 MPa, α = 1.6083, β = π or − π, γ_MC = 0.9621 and φ = 28.7°. When I₁ = 243.4 MPa, α = 1.5973, β = π or − π, γ_MC = 0.9527 and φ = 32.5°. The deviatoric plane parameters of the LDNUF criterion are: when I₁ = 470.1 MPa, α = 1.0934, β = π or − π, γ_LD = 0.5250 and φ = 26.2°. When I₁ = 427.0 MPa, α = 1.1187, β = π or − π, γ_LD = 0.5403 and φ = 27.1°. When I₁ = 356.6 MPa, α = 1.1649, β = π or − π, γ_LD = 0.5698 and φ = 28.7°. When I₁ = 243.4 MPa, α = 1.2573, β = π or − π, γ_LD = 0.6357 and φ = 32.5°.

Similarly, the meridian plane parameters of the MNGNF criterion are ω = 6.0236, β = 0 MPa, and n = 0.6478. The deviatoric plane parameters of the MNGNF criterion are as follows: when I₁ = 470.1 MPa, α = 1.1571, A = 0.6813, and φ = 26.2°. When I₁ = 427.0 MPa, α = 1.1661, A = 0.6965 and φ = 27.1°. When I₁ = 356.6 MPa, α = 1.1822, A = 0.7249 and φ = 28.7°. When I₁ = 243.4 MPa, α = 1.2137, A = 0.7842 and φ = 32.5°. The meridian plane parameters of the DPMNu criterion are ξ_t = 0 MPa, A = 12.1594, and n = 0.6478. The deviatoric plane parameters of the DPMNu criterion are as follows: when ξ = 271.4 MPa, ω = 0.6813, α = 0.3193 and φ = 21.6°. When ξ = 246.5 MPa, ω = 0.6965, α = 0.3362 and φ = 27.1°. When ξ = 205.9 MPa, ω = 0.7249, α = 0.3662 and φ = 28.7°. When ξ = 140.5 MPa, ω = 0.7842, α = 0.4240 and φ = 32.5°.

The prediction results of the MCNUF and LDNUF criteria under the power function meridian plane passing through the origin point and the Yamaguchi marble true triaxial test data are shown in Fig. 27. The prediction results of the MNGNF and DPMNu criteria are shown in Fig. 

Fig. 28

Prediction results of the MNGNF and DPMNu criteria under the power function meridian plane passing through the origin point and the Yamaguchi marble true triaxial test data: a deviatoric plane of the MNGNF criterion and b deviatoric plane of the DPMNu criterion

28.

3.2 Appendix C.2: Influence of the meridian plane function on failure criterion under the Laxiwa granite true triaxial test

3.2.1 Appendix C.2.1 Meridian plane of the power function passing through the I ₁ negative semi-axis

Section 5.2 shows that d = 143.11 MPa (mean value), K = 0.7587 (maximum value), δ_t = 107.33 MPa and ξ_t = 61.97 MPa. The meridian plane parameters of the MCNUF criterion and LDNUF criteria (see Fig. 

Fig. 29

Prediction results of the MCNUF and LDNUF criteria under the power function meridian plane passing through the I₁ negative semi-axis and the Laxiwa granite true triaxial test data: a best-fit envelope of the meridian plane, b deviatoric plane of the MCNUF criterion, c deviatoric plane of the LDNUF criterion, d 3D space failure envelope surface of the MCNUF criterion and d 3D space failure envelope surface of the LDNUF criterion

29a) are N_f = 1.2407, δ_t = 0 MPa, and n = 0.8352, respectively.

According to the meridian plane parameters and the aspect ratio, the deviatoric plane parameters of the MCNUF criterion can be calculated as follows: when I₁ = 479.95 MPa, α = 1.4332, β = π or − π, γ_MC = 0.9537 and φ = 32.1°. When I₁ = 390.06 MPa, α = 1.4299, β = π or − π, γ_MC = 0.9516 and φ = 32.9°. When I₁ = 299.99 MPa, α = 1.4258, β = π or − π, γ_MC = 0.9489 and φ = 33.9°. When I₁ = 224.99 MPa, α = 1.4215, β = π or − π, γ_MC = 0.9461 and φ = 35.0°. The meridian plane parameters of the LDNUF criterion are as follows: when I₁ = 479.95 MPa, α = 0.9761 (1 in this paper), β = π or − π, γ_LD = 0.6295 and φ = 32.1°. When I₁ = 299.99 MPa, α = 0.9986 (1 in this paper), β = π or − π, γ_LD = 0.6433 and φ = 32.9°. When I₁ = 299.99 MPa, α = 1.0253, β = π or − π, γ_LD = 0.6602 and φ = 33.9°. When I₁ = 224.99 MPa, α = 1.10521, β = π or − π, γ_LD = 0.6776 and φ = 35.0°.

Similarly, the meridian plane parameters of the MNGNF criterion are ω = 1.2407, β = 107.33 MPa, and n = 0.8352, respectively. The deviatoric plane parameters of the MNGNF criterion are as follows: when I₁ = 479.95 MPa, α = 1.0983, A = 0.7788 and φ = 32.1°. When I₁ = 390.06 MPa, α = 1.1063, A = 0.7907 and φ = 32.9°. When I₁ = 299.99 MPa, α = 1.2256, A = 0.8047 and φ = 33.9°. When I₁ = 224.99 MPa, α = 1.1249, A = 0.8188 and φ = 35.0°. The meridian plane parameters of the DPMNu criterion are ξ_t = 61.97 MPa, A = 2.7761, n = 0.8352. The deviatoric plane parameters of the DPMNu criterion are as follows: when ξ = 277.10 MPa, ω = 0.7447, α = 0.1980, and φ = 32.1°. When ξ = 225.20 MPa, ω = 0.7907, α = 0.2132 and φ = 32.9°. When ξ = 173.20 MPa, ω = 0.8047, α = 0.2308 and φ = 33.9°. When ξ = 224.99 MPa, ω = 0.8188, α = 0.2482 and φ = 35.0°.

The prediction results of the MCNUF and LDNUF criteria for the power function meridian plane passing through the I₁ negative semi-axis and the Laxiwa granite true triaxial test data are shown in Fig. 29, and the prediction results of the MNGNF and DPMNu criteria are shown in Fig. 

Fig. 30

Prediction results of the MNGNF and DPMNu criteria for the power function meridian plane passing through the I₁ negative semi-axis and the Laxiwa granite true triaxial test data: a deviatoric plane of the MNGNF criterion and b deviatoric plane of the DPMNu criterion

30.

3.2.2 Appendix C.2.2 Meridian plane of the power function passing through the origin

This paper considers the aspect ratio as K = 0.7587 (maximum value). The meridian plane parameters of the MCNUF criterion (see Fig. 

Fig. 31

Prediction results of the MCNUF criterion under the power function meridian plane passing through the origin point and Laxiwa granite true triaxial test data: a best-fit envelope of the meridian plane, b deviatoric plane of the MCNUF criterion and c 3D space failure envelope surface of the MCNUF criterion

31a) are N_f = 8.1028, δ_t = 0 MPa, and n = 0.6331. According to the meridian plane parameters and the aspect ratio, the deviatoric plane parameters of the MCNUF criterion can be calculated: when I₁ = 479.95 MPa, α = 1.4421, β = π or − π, γ_MC = 0.9592, and φ = 29.9°. When I₁ = 390.06 MPa, α = 1.4334, β = π or − π, γ_MC = 0.9538 and φ = 32.0°. When I₁ = 299.99 MPa, α = 1.4214, β = π or − π, γ_MC = 0.9460 and φ = 35.0°. When I₁ = 224.99 MPa, α = 1.4069, β = π or − π, γ_MC = 0.9363 and φ = 38.6°. Among them, the deviatoric plane of the LDNUF criterion does not satisfy the convexity requirement, and it is not analyzed in this paper.

Similarly, the meridian plane parameters of the MNGNF criterion are ω = 8.1028, β = 0 MPa, and n = 0.6331, respectively. The deviatoric plane parameters of the MNGNF criterion are as follows: when I₁ = 479.95 MPa, α = 1.0749, A = 0.7447 and φ = 29.9°. When I₁ = 390.06 MPa, α = 1.0978, A = 0.7781 and φ = 32.0°. When I₁ = 299.99 MPa, α = 1.1251, A = 0.8191 and φ = 35.0°. When I₁ = 224.99 MPa, α = 1.1528, A = 0.8615 and φ = 38.6°. The meridian plane parameters of the DPMNu criterion are ξ_t = 0 MPa, A = 16.2249, and n = 0.6331, respectively. The deviatoric plane parameters of the DPMNu criterion are as follows: when ξ = 277.10 MPa, ω = 0.7447, α = 0.1527, and φ = 29.9°. When ξ = 225.20 MPa, ω = 0.7781, α = 0.1971 and φ = 32.0°. When ξ = 173.20 MPa, ω = 0.8191, α = 0.2487 and φ = 35.0°. When ξ = 224.99 MPa, ω = 0.8615, α = 0.2998 and φ = 38.6°.

The prediction results of the MCNUF criterion for the power function meridian plane passing through the origin point and the Laxiwa granite true triaxial test data are shown in Fig. 31, and the prediction results of the MNGNF and DPMNu criteria are shown in Fig. 

Fig. 32

Prediction results of the MNGNF and DPMNu criteria under the power function meridian plane passing through the origin point and the Laxiwa granite true triaxial test data: a deviatoric plane of the MNGNF criterion and b deviatoric plane of the DPMNu criterion

32.

3.3 Appendix C.3: Influence of the meridian plane function on the failure criterion for the Tien–Liao mudstone true triaxial test

Based on Sect. 5.3, this paper takes d = 10.3177 MPa (mean value) and K = 0.6975 (maximum value) as examples for calculation. The meridian plane parameters of the MCNUF criterion (see Fig. 

Fig. 33

Prediction results of the MCNUF criterion for the linear function meridian plane passing through the I₁ negative semi-axis and the Tien–Liao mudstone true triaxial test data: a best-fit envelope of the meridian plane, b deviatoric plane of the MCNUF criterion and c 3D space failure envelope surface of the MCNUF criterion

33a) are N_f = 0.3141, δ_t = 7.7383 MPa, and n = 1. According to the meridian plane parameters and the aspect ratio, the deviatoric plane parameters of the MCNUF criterion can be calculated as α = 1.2396, β = π or − π, γ_MC = 0.9329 and φ = 39.9°. Among them, the deviatoric plane of the LDNUF criterion does not satisfy the convexity requirement, and it is not analyzed in this paper.

The strength parameters of the MNGNF criterion are ω = 0.3141, β = 7.7383 MPa, n = 1, α = 1.0718, A = 0.8744 and φ = 39.9°. The strength parameters of the DPMNu criterion are A = 0.7694, ξ_t = 4.4677 MPa, n = 1, ω = 0.8744, α = 0.1415 and φ = 39.9°.

Figure 33 shows the prediction results of the MCNUF criterion passing through the I₁ negative semi-axis linear meridian plane and the Tien–Liao mudstone true triaxial test data, and the prediction results of the MNGNF and DPMNu criteria are shown in Fig. 

Fig. 34

Prediction results of the MCNUF and DPMNu criteria for the linear function meridian plane passing through the I₁ negative semi-axis and the Tien–Liao mudstone true triaxial test data: a deviatoric plane of the MNGNF criterion and b deviatoric plane of the DPMNu criterion

34.