A generalized nonlinear failure criterion for frictional materials

Abstract

A generalized nonlinear failure criterion formulated in terms of stress invariants is proposed for describing the failure characteristics of different frictional materials. This failure criterion combines a power function and a versatile function in the meridian and deviatoric plane, respectively, which is a generalization of several classic criteria, including the Tresca, Drucker–Prager, Mohr–Coulomb, Lade–Duncan and Matsuoka–Nakai failure criterion. The procedure for determination of the strength parameters was demonstrated in detail. Comparisons between the failure criterion and experimental results were presented for uncemented/cemented Monterey sand, normally consolidated Fujinomori clay, rockfill, concrete, Mu-San sandstone and granite, which reveal that the proposed failure criterion captures experimental trend quite well.

Appendix : Proof of the requirements for smoothness and convexity of the new two-parameter deviatoric function

The first and second derivative of Eq. (14) can be expressed as follows:

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

(30)

$$g^{\prime\prime}(\theta_{\sigma } ) = \sin \left( {\alpha \frac{\pi }{3} - \frac{1}{3}\sin^{ - 1} A} \right)\left[ \begin{aligned} \frac{{3A\sin 3\theta_{\sigma } (1 - A^{2} )}}{{(1 - A^{2} \sin^{2} 3\theta_{\sigma } )^{3/2} }}\frac{{\cos \left[ {\alpha \frac{\pi }{3} + \frac{1}{3}\sin^{ - 1} (A\sin 3\theta_{\sigma } )} \right]}}{{\sin^{2} \left[ {\alpha \frac{\pi }{3} + \frac{1}{3}\sin^{ - 1} (A\sin 3\theta_{\sigma } )} \right]}} \hfill \\ + \frac{{A^{2} \cos^{2} 3\theta_{\sigma } }}{{1 - A^{2} \sin^{2} 3\theta_{\sigma } }}\frac{{1 + \cos^{2} \left[ {\alpha \frac{\pi }{3} + \frac{1}{3}\sin^{ - 1} (A\sin 3\theta_{\sigma } )} \right]}}{{\sin^{3} \left[ {\alpha \frac{\pi }{3} + \frac{1}{3}\sin^{ - 1} (A\sin 3\theta_{\sigma } )} \right]}} \hfill \\ \end{aligned} \right]$$

(31)

Apparently, when $\theta_{\sigma } = \pm 30^{ \circ }$,$g^{\prime}( \pm 30^{ \circ } ) = 0$ and $A \ne 1$, which means the new two-parameter deviatoric function is smooth at the corner.

On the other hand, through substitution of Eqs. (14), (30), (21) into Eq. (5), thus yielding

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

(32)

where $\theta_{\sigma } \in \left[ { - 30^{ \circ } ,30^{ \circ } } \right]$, and let $x = \sin^{ - 1} (A\sin 3\theta_{\sigma } )$, then Eq. (32) can be transformed into

$$2\sin \left( {\alpha \frac{\pi }{3} - \frac{2}{3}x} \right) - \sin \left( {\alpha \frac{\pi }{3} + \frac{4}{3}x} \right) \ge 0$$

(33)

where $x \in \left[ { - \sin^{ - 1} A,\sin^{ - 1} A} \right]$ and then can be expressed as:

$$\frac{{ - 2\cos^{2} k + 2\cos k + 1}}{2\sin k(1 + \cos k)}\sin \left( {\alpha \frac{\pi }{3}} \right) - \cos \left( {\alpha \frac{\pi }{3}} \right) \ge 0$$

(34)

with $k = \frac{2}{3}x \in \left[ { - \frac{2}{3}\sin^{ - 1} A,\frac{2}{3}\sin^{ - 1} A} \right]$, and an inequality can be obtained as follows:

$$\alpha \ge \frac{3}{\pi }\tan^{ - 1} \left[ {\frac{2\sin k(1 + \cos k)}{{ - 2\cos^{2} k + 2\cos k + 1}}} \right]$$

(35)

where the right function of inequality sign in Eq. (35) is monotonically increasing with $k$ and varies within the interval $\left[ { - 1,1} \right]$, thus yielding

$$\alpha \ge 1$$

(36)

which means that the new two-parameter deviatoric function is convex as long as Eq. (36) is satisfied and $A \in \left[ {0,1} \right]$.

As for the aspect ratio $K$, which varies within the interval $\left[ {0.5,1} \right]$, the upper limit value of $\alpha$ can be obtained when $A \in \left[ {0,1} \right]$

$$\alpha \le 1.5$$

(37)

Finally, it is necessary to emphasize that within the interval of $\alpha \in \left[ {1,1.5} \right]$, the new two-parameter deviatoric function is convex independently of the values assumed by $A$. Besides, one can achieve the same effect as the Bigoni–Piccolroaz dependence simply by widening the interval of $\alpha$ to $\left[ {1,2} \right]$.