In rock mass engineering, the criterion of rock mass stability has complex nonlinear characteristics, so the process of instability for local rock mass system cannot be quantified by the traditional ways of displacement criterion and the criterion of development of plastic zones, which are strongly empirical. Based on the research about the criterion of rock mass stability, criterion of improved strain energy catastrophe is put forward by virtue of catastrophe theory in this paper. After regularizing potential function, the stability of the system can be determined by catastrophe characteristic values. Take a certain slope for example; the results show that the criterion can quantitatively reflect the behavioral process of instability for rock mass system, which is consistent with the engineering practice and possesses a certain engineering reference value.
The stability of surrounding rock is an important criterion to determine the reasonable design and operation safety of the underground rock engineering, for example, the tunnels. Due to the extremely complex rock behavior and occurrence condition and numerous factors, which may influence the stability of rock mass and have features of diversity, variability, and uncertainty, criterion of rock mass stability has not yet reached a mature stage; that is, a series of basic problems have not formed a definite system, from the definition of stability and the criterion of quantization to the theory, the criterion, and the method of analysis [
Instability of surrounding rock which is a fairly complex process, often accompanied by heterogeneity, discontinuity, and large displacement of deformation, is a highly nonlinear scientific problem and nonlinearity is the essential characteristics of mechanical behavior of surrounding rock. Therefore, to forecast and control its mechanical behavior, modern nonlinear science is applied. Since the 1970s, developed nonlinear theory, such as fractal, bifurcation, mutation, chaos, and dissipation theory, has become a powerful tool to solve the problem of complicated nonlinear system and to provide theory and method to the nonlinear problem of rock mass. Among them, the application of catastrophe theory established by Thom is used more frequently. Since founded, catastrophe theory has been widely used. It can not only be applied to the “hard science” such as physics, but can also be in the “soft science” like sociology. There exists “a gray area” between soft disciplines and hard disciplines; most of rock mechanics and geoscience belong to this area. Catastrophe theory belongs to qualitative application in rock mechanics. Shan [
It has been proven by Thom that when the number of state variable is no more than two and control variable is not greater than four, there will be seven different basic catastrophe types, including fold catastrophe, cusp catastrophe, swallowtail catastrophe, butterfly catastrophe, hyperbolic umbilical catastrophe, elliptic umbilical catastrophe, and parabolic umbilical catastrophe. The first two are simpler and more widely used. In this paper, cusp catastrophe is improved to be applied.
There are two control variables and one state variable in the cusp catastrophe model. Its potential function is
Equilibrium surface and sets of crossing points in cusp catastrophe.
A cubic polynomial is used to fit equilibrium surfaces of system. Based on it, transpose the polynomial to the integral to obtain the potential function of the system. Then regularize it and determine the stability of the system by catastrophe characteristic values.
Take strain energy catastrophe model for example; according to actual excavation step, increment series of strain energy of the system are obtained by numerical calculation, The variables and the independent variables of the fitting polynomial change. The increment of strain energy is an independent variable and the excavation parameter is a variable. Adopt a cubic polynomial to fit.
The fitting polynomial is
Consider the fitting formula as the manifolds of potential function of system (equilibrium surface).
Transposition and integration are as follows:
Choose the following as a potential function of the system:
It must be pointed out that the quartic polynomial fitting about the excavation parameter
A slope is composed of three rock-soil layers. Its geometric size and boundary conditions are shown in Figure
Geometric model and boundary conditions of a slope.
According to strength reduction method, the direct relationship between the maximum displacement of the slope and the strength reduction factor is gotten by continuously reducing material parameters and conducting conventional elastic-plastic finite element calculation, as is shown in Figure
Relation curve graph of maximum displacement and reduction factor.
A total displacement sequence can be gotten:
Displacement sequence under different reduction factors is the subset of the total displacement sequence. Analyze displacement sequence under the reduction factor at all levels with least square method to fit it into quartic polynomial (at least from level 5) and then regularize the polynomial to get standard form of catastrophe model.
The fitting relationship between the maximum displacement and reduction factor of slope under reduction factor at levels 5 to 10 and the fitting formula of quartic polynomial with the square value of the correlation coefficient are given in Figure
Control variables
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1.2 | 1.25 | 1.3 | 1.31 | 1.32 | 1.33 |
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0.07375 |
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0.138 |
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0.517397 |
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State | No mutation | No mutation | No mutation | No mutation | No mutation | Mutation |
Fitting curves of displacement reduction factor under reduction factor at levels 5 to 10.
It can be found in the table that the situation of the system will mutate when the reduction factor is 1.33. Therefore, it can be concluded that the safety coefficient of the slope stability is 1.32 and, according to the cusp catastrophe model, the system characteristic values of mutation are
Spencer method in the traditional limit equilibrium slice method is also used to verify the calculation results in [
In the model above, based on potential function of the maximum displacement of the slope, the cusp catastrophic model is established. From the fitting formula and the regularized equation, it is known that the state variable
When adopting the displacement criterion to determine the stability of slope rock mass, it would certainly bring out displacement catastrophe at the time of slope failure, which is also consistent with the practical engineering experience. Therefore, take the maximum displacement of the slope as state variable
The fitting graphs are shown in Figures
Fitting curves of displacement reduction factor under reduction factor at levels 7 to 12.
The potential function of the system expressed by quartic polynomial is obtained by integral of the fitting curves. Then regularize it into the standard form of cusp catastrophe model. The stability of system can be determined by catastrophe characteristic values.
When the reduction factor reaches 1.32, the maximum slope displacement mutates (Table
Control variables
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1.3 | 1.31 | 1.32 | 1.33 | 1.34 | 1.35 |
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State | No mutation | No mutation | Mutation | Mutation | Mutation | Mutation |
The cusp catastrophe model established in this paper reflects that the slope instability results from the displacement jump, which leads the status of the slope to change and the potential energy of the system also to change. (The specific physical meaning of potential energy is not clear, but the potential function is not the focus of catastrophe research.) Compared with the traditional method using displacement as potential function, cusp catastrophe model has a more definite physical meaning and is consistent with practical engineering, which is established by considering reduction factor as state variable.
When the reduction factor reaches 1.3 (reduction factor at level 7), material strength reduction factors continued being increased, and the catastrophe results of fitting curves on the basis of more than 7 groups of data
Table of control variables
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1.15 | 1.2 | 1.25 |
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State | Mutation | Mutation | Mutation |
Table of control variables
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1.3 | 1.31 | 1.32 | 1.33 | 1.34 | 1.35 |
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State | No mutation | No mutation | No mutation | No mutation | No mutation | No mutation |
Fitting curves of the reduction factor of displacement under reduction factors at levels 4 to 6.
Fitting curves of the reduction factor of displacement under reduction factors at levels 7 to 12.
As is shown in Table
The actual physical meaning of state variables and the sudden jump occurring at the time of catastrophe which results in another state of the system are considered in the catastrophe model in this paper. However, according to the analysis of calculation above, it seems that there is no way to explain the problem why calculation results are still highly accurate when adopting quartic polynomial fitting, using displacement as potential function by conventional catastrophe model and establishing the reduction factors as state variables in the catastrophe model. It is not a particular example for many studies have mentioned it before.
To make a further analysis, not only a conventional method is adopted but also a cubic polynomial fitting is used in this paper. Transpose it into integral to get reduction factors as state variables and cusp catastrophe model including displacement influence. Then determine the stability of the slope based on it.
There is an interesting phenomenon here; that is, at this point catastrophe will not happen again and slope will not lose balance (under the condition of maximum reduction factor at 1.35 which is given in this paper). It cannot give a definite conclusion only by a few examples, “reduction factors cannot be used as state variables to get the critical condition of catastrophe,” but, at least, the conclusion that “when using the reduction factor as state variables a sudden jump of the reduction factor will lead to instability of slope that is not reasonable in its physical meaning” can be pointed out explicitly.
From the perspective of catastrophe mathematical model, a cubic polynomial data fitting is put forward to get a function as the equilibrium surface of the system in this paper. Under an excavation or loading step, the polynomial satisfies the equilibrium conditions approximately (the error is merely the fitting function error at that point). Therefore, this state is on (or near) the equilibrium surface of the potential function. The equilibrium surface obtains a system potential function expressed by a quartic polynomial through integral. After regularization, stability can be determined by the catastrophe characteristic values. This method overcomes the disadvantages of currently commonly used catastrophe criterion types which is not clear in physical meaning.
The authors declare that they have no conflict of interests in this paper. And this research is referenced mutation in mathematics theory to solve the stability problem of geotechnical engineering. It is consistent with the main theme.
Thanks are due to National Science and Technology Support Program (2012BAK03B04/2012BAB03B02) and the Youth Foundation of National Natural Science Fund (51209078).