Mathematical Model for Establishing the Time-Dependent Behavior of Rocks by the Gradient Method

Abstract

In the underground activity domain, most problems related to mining pressure and mining stability need to be solved by taking into account the time behavior of rocks through an approach of the interaction amidst the rock massif, support system, time through the elastic, viscous and plastic models, namely a rheological approach. In order to choose a rational support system, one needs to know the sustainment solicitation at different time intervals. The change in the sustainment in time is emphasized only in the analytical research in which the massif is studied and characterized in terms of the rheological behavior. The gradient method applied in this regard is based on the evolution of the final deformations at a given time, compared to their previous evolution. The paper is structured into two parts, the experimental and interpretation of the experimental data, showing the author’s methodology to assess the rheological behavior of analyzed andesite, the result of the theoretical and experimental research being carried out on the analyzed rock types. Based on the deformation and the time curve of the horizontal mining work contour, the mathematical function was established, which expresses the law of sought deformation. At the same time, the rheological model capable of describing the behavior under a load and under extremely adverse conditions is proposed.

1. Introduction

Underground constructions are structures whose operational life is long and, consequently, they require optimum technical and economical solutions to provide stability throughout their existence, and therefore to ensure their reliability through an imposed stability [1,2,3,4,5,6,7,8,9,10,11]. In the last decades, the technical literature [12,13,14,15,16,17,18,19,20,21,22,23] mentions that an efficient solution for the stability of the underground structures can only be solved by considering the time parameter, as in taking into account the behavior of rocks in time [24,25,26]. The knowledge of the rheological behavior of rocks must be considered as a fundament for the research into and the design of constructions with a long-term existence, as it determines and creates prerequisites for the future to provide real solutions for the full range of issues that underground constructions are facing in terms of their stability; it provides a logical and numerical rigor frame, in which the principles, methods and calculating procedures offered by rock mechanics may apply and develop. On the other hand, the calculus relations shown in the literature, that do not take into account the time, allow us to determine the size and final nature of the manifestation of pressure, of sustaining, for works such as tunnels, underground mining works or other types of structures requiring sustainment.

In order to choose a rational support system, one needs to know the sustainment solicitation at different time intervals. A changing in the sustainment in time is emphasized only in the analytical research in which the massif is studied and characterized in terms of its rheological behavior. In such a context, we must underline that the rheological characterization of rocks did not exist until this research was realized in the analyzed perimeter, and often the solutions for ensuring the stability of constructions/underground structures were designed and put into practice based only on knowing the geomechanical characteristics (geological, physical, strength and elastic) and some theories, without considering the time factor. The consequence of such solutions has resulted in the high consumption of materials and frequent manifestations, respectively periodical ones of instability, which required a repeated reshaping of these underground constructions. By analyzing the data from the literature, we note that, currently, solving mining geomechanics issues that contribute to achieving the stability and reliability of underground constructions also involves the knowledge of the rheological behavior of rocks. Some of these problems are: elucidating the natural state of stress; the characterization of the deformation behavior of a rock massif; secondary state stress and the deformation of rocks around their underground workings; the characterization of geomechanical conditions of stability in which the work will be performed; the study of pressure and its calculus in the context of the interaction mechanism between the rock massif, the support system and time; and the rheological characterization of consolidated rocks, etc.

The geo-mining conditions for the location of the main horizontal mining works are the natural factors on which the stability of underground mining works depends. However, there are situations where underground mining works are located under difficult and complex geological conditions. The analyzed rock massif is characterized by such complex geological conditions: rocks altered with a low strength, water affluence, cracks and advanced tectonic degrees (micro-tectonic) [26,27,28,29,30,31,32,33,34,35]. The andesitic rock types have different degrees of alteration and therefore different percentages of clay minerals. The higher the degree of alteration, the less resistant the rock will be, and this will create special problems in terms of ensuring the stability of underground works developed in such rocks [24,25,26,27]. For the underground works to be stable, it is necessary not only to know the phenomena and assumptions that may represent the basis of designing and carrying out these works, but also to be familiarized with the probabilistic concept of ensuring these constructions, thus replacing the anachronistic deterministic concept of computation, namely a design without a calculus, based on experience, the empirical rules and an intuitive application of the laws of mechanics, as there are many situations encountered in practice.

The mine workings are the constructions which required and still require to perform the largest investments. For this reason, they must be resistant, durable and economical, namely, to present stability and reliability. In order to achieve this and to achieve the optimal results from a technical and economic point of view, it is required to have knowledge of the actual causes and results of the complex natural geological, geomechanical, technical, mining and production factors which determine whether or not the loss of stability from the main horizontal underground workings from any mining perimeter.

The analysis of the stability of mine workings is based on its evaluation criteria that may guide the geomechanical and specific technical mining conditions to the design of the natural stability or required stability of the analyzed mining workings. Finally, the inclusion of such criteria in mining construction design theory creates the prerequisites for discovering objective connections between the forms and manifestation sizes of the pressure regime and the working conditions of mining supports [36,37,38,39,40,41,42,43,44,45,46,47,48]. Moreover, the stability criteria of the rock massif and mining work and the parameters of the support system for the situation of stability being imposed in the context of the interaction mechanism can form the basis of the alternative choice of the model for the calculus of prediction regime pressure, of establishing the logical connections between the classes and the forms of a manifestation of its regimes, and of the fundamental principles of such a calculus method of the support systems in order to solve the stability and reliability of mine workings throughout its duration of activity.

The stability notion is cumulative, having as an object the knowledge as real as possible, first of the geomechanical characteristics of rocks, natural stress state, the secondary stress-strain state and rock massif pressure and, secondly, the geotectonic processes, the presence of underground water and even the production activity, referring to the construction of underground excavations [9,14,15,24,25,27,28,29,30,31,32,36,38]. Due to the complexity of approaching stability, its general theory has not been so far developed. As a result, the assessment of the stability of mining works has been attempted in several directions over time: analytical, experimental laboratory, in situ observations and measurements. However, it should be noted that the development of these directions of research was done in parallel. Thus, the development of experimental research (laboratory tests and in situ) is not achieved at the expense of the analytical direction and vice versa, because where the developed mathematical models, no matter how general they should be, cannot be satisfactory, then the experimental research methods are welcomed. However, where the existence of an elementary, stable, constant or evolving in time probabilistic structure is noted in a well-defined manner, analytical research reveals its complete value and economic effectiveness [10,11,24,25,26,30,31,32,34].

Currently, a wealth of theoretical and practical experience in the forecasting of the stability of a mine’s workings has been accumulated. However, there is still an essential discordance between theoretical and experimental studies, expressed mainly by the fact that the theoretical approaches and the conclusions are not verified in the experimental activities [49,50]. Such a deficiency can be assumed to be unknowledgeable in terms of the time-deformation behavior of rocks and, consequently, of the impossibility of using the calculation models based on the rheological parameters of rocks. On the other hand, the researchers who develop the experimental methods often do not use rock mechanics theory and the achievements of underground construction mechanics. A first step towards researching the stability conditions of the main horizontal mine workings is to determine and know the geomechanical properties because these data give the required parameters of the design calculus of these works. On the issue of the efficient development of the mining production activity, of particular importance is the knowledge of those geomechanical properties that give the possibility of elucidating the deformation behavior of the rocks. For this purpose, there are several properties without which the stability issue cannot be approached. Such a conclusion is based on the certainty that the manner of the deformation of rock and establishing the optimal solutions from a technical and economic viewpoint requires a quantitative–qualitative assessment of the three basic properties, the elasticity, plasticity and viscosity, which are found in different ratios according to the rock varieties of each massif. Even the stability criteria reported in the literature confirms the necessity to take into account the rheological parameters.

The knowledge of the rheological behavior of rocks must be considered as a basis in the research and design of excavations with a long-term existence because it determines and creates the future basis in order to provide the real solutions with respect to the full range of issues faced by the underground constructions viewpoint of stability; it provides the logical framework and numerical rigor that may apply and develop the principles, methods and calculus procedures offered by rock mechanics. On the other hand, the calculus relationships offered by the literature enables us to determine the size and final character of the manifestation of pressure and of load support. For the choice of a rational support’s character of service, the load on the support at different time ranges must be known. The change in the load over time is highlighted only in the analytical research where the massif is studied and characterized in terms of the rheological behavior.

The implications of understanding the rheological behavior of rocks on the stability of the main horizontal mining works are also a result of the wealth of information that the specialized literature has recently provided. Thus, we mention the works and treaties published in the domain by several researchers: Glusko (1973–1975), Erjanov (1964), Duduskina (1970), Usacenko (1964), Bulâcev (1982), Baklaşov (1988), Salustowicz, Filcek, Kwasniewski and many others. From the featured ones, the importance of the knowledge of the rheological behavior of rocks and where the mine workings with a high activity duration will be carried out is clear. Mining pressure largely depends on the mechanical characteristics of the surrounding rock and implicitly on their rheological behavior, which influences the convergence and deformation contour in a nonlinear manner.

Regarding the geomining conditions, it was found that a large volume of the main horizontal mining works that constitute the opening network of the Suior deposit are made in altered rocks with a reduced resistance, water influx, cracks and an advanced tectonic’ degree. Andesitic rocks show different degrees of weathering and different percentages of clay materials. Most of the horizontal mining works are located in pyroclastic andesitic rocks (breccia) and andesitic lavas with hypersthene, kaolinized, sericitized and strong cracks as well as in pyroxene andesite and pyroclastic intensely hydrothermally metamorphosed. Considering this distribution of mining works, the problem of their stability becomes essential, both from a technical and economic point of view. The following conclusions resulted from the observations made:

-

The unsatisfactory state of the stability of the mining works at Suior is the result of the work profile’s shape not being correlated with the geomechanical and geomining conditions of their location;

-

The mining works with straight walls, vaulted ceiling and an unsupported floor were mostly made in rocks with pronounced tendencies towards an alteration and the swelling of the floors. This situation was often encountered at the 900 m, 850 m and 800 m horizons;

-

Profiles with straight walls, a vaulted ceiling and an unsupported floor located in Suior conditions are only applicable in directional galleries made on the vein and in areas where the vein has a high resistance;

-

In conditions where the rocks comprise kaolinized andesite intercalations and observations shows a tendency for the floors to swell, and circular and horseshoe-shaped profiles with a supported floor were required.

Observations regarding the excavation support technology have highlighted deficiencies that have influenced the stability of the mining works (the incomplete filling of the voids behind the support; the pronounced cracking of the rocks on the contour and in depth cracking due to exceeding the consumption of the explosives used in digging; and the uncontrolled tightening of the metal supporting elements, which favored either their premature sliding or their complete stiffening). The convergence of the entire contour recorded values between 400 mm and 600 mm in the mining works from the 850 m horizon. The maintenance expenses for the mining works on the 850 m horizon exceeded 35% of the total expenses. As a result of this situation, it was considered necessary to know the phenomena and the laws that are the basis for the design and achievement of underground resistance constructions which are safe, durable and economical.

2. General Context of the Research

Starting from the considerations presented above, it is imperative to observe the behavior in time, based on which an analysis of the stability of the main horizontal mining works can be made, depending on the factors that compete for the stability of these underground constructions. An important factor that influences the stability of any underground work is the time by the rheological behavior of the rocks’ creep and relaxation [2,3,4,5,6,7,10,11,12,13,22,23,24,25,26,27,28,29,38].

It seems that to explain all aspects of the rocks’ deformation, knowing that their deformation rate is finite, namely, it occurs over time, it is imperative to study the deformation curve as a function of time. The allure of such a rheological curve at the creep (Figure 1) in its generalized form defines a total deformation (ε_t) which is constituted according to the principle of the partition of deformations, from:

$${\mathsf{\epsilon }}_{\mathrm{t}}={\mathsf{\epsilon }}_{\mathrm{0}}+{\mathsf{\epsilon }}_{\mathrm{p}}+{\mathsf{\epsilon }}_{{\mathrm{p}}^{\prime }}+{\mathsf{\epsilon }}_{{\mathrm{p}}^{″}}$$

(1)

where ε₀ is an instantaneous deformation created by the stress under which the test is performed; ε_p is a primary plastic deformation that represents the recorded deformation over time and which characterizes the area of the unstabilized creep; ε_p′ is a secondary plastic deformation represented by the recorded deformation in the area in which the curve follows an oblique or horizontal asymptote and which characterizes the stabilized creep area; and ε_p″ is a tertiary deformation, measured in the area of the quickly increasing area of deformation until the breaking and characterizes the breaking zone.

3. Deformation Characteristics in Time of Altered Andesite

3.1. Deformation Characteristics in Time of Altered Andesite Established in Laboratory

For the study of the creep characteristics and the mathematical establishment of the law of deformation in time of the andesite rocks type, the rheological tests were performed on intensively hydrothermal metamorphosed pyroxene andesite; these being the most unfavorable in terms of their stability. The creep tests were performed on uniaxial compression, with different stresses corresponding to the load degree of the andesite samples, according to the data in

Table 1. Data concerning creep tests.

Rock Type	Samples Dimensions (mm)		Load Stress, σ0, (MPa)	Tests Duration, t		Mean Breaking Stress, σmed, (MPa)	Load Degree, σ0/σmed
	d	h		Hours	Days
Altered pyroxene andesite 1	42	42	11.55	1198	50	28.00	0.41
			14.44	1198	50		0.51
			21.60	168	7		0.70
			28.00	12	0,5		1.00

¹ Rock samples taken from the experimental sections from Suior mine, Romania.

. The obtained results are shown in

Table 2. Obtained results from creep tests.

Rock Type	Load Stress, σ0, (MPa)	Stabilization Time, t (Hours)	Stable Area, σ0/σcritic	Relative Stability Area, σ0/σcritic	Unstable Area, σ0/σcritic
Altered pyroxene andesite 1	11.55	165	0.41	0.41–0.71	0.71
	14.44
	21.60
	28.00

¹ Rock samples taken from the experimental sections from Suior mine, Romania.

and Figure 2 and Figure 3.

Through analyzing the experimental creep curve obtained by the tests, based on the data provided by the specialized literature [3,4,5,10,15,19,24,25,26,30,32,33,40,41,42], it is found that the Poynting–Thomson model is the model with which we could assimilate the deformation in time of strongly hydrothermal metamorphosed pyroxene andesite rock types. Because the assumption implies a confirmation in this regard, we took the model with the creep equation and analyzed it based on the obtained data from the tests through an analytical processing. The functional equation of the rock deformation in time can be expressed in the following form:

$$\mathsf{\epsilon }=\mathrm{f}\left(\mathrm{t}\right)$$

(2)

The explanation of this function according to the Poynting–Thomson model for a uniaxial compression test becomes:

$$\mathsf{\epsilon }=\frac{{\mathsf{\sigma }}_0}{\mathrm{E}}-\left(\frac{{\mathsf{\sigma }}_0}{\mathrm{E}}-{\mathsf{\epsilon }}_0\right){\mathrm{e}}^{-\frac{\mathrm{E}}{\mathsf{\eta }}\mathrm{t}}$$

(3)

where σ₀ is the load stress, MPa; E is the modulus of elasticity, MPa; ε₀ is the instantaneous elastic deformation; $\mathsf{\eta }=\mathrm{T}\cdot \mathrm{E}$ is the coefficient of viscosity, Ns/m²; and t is the time.

Or, Equation (3) can be written in the form:

$$\mathsf{\epsilon }=\mathrm{a}-\mathrm{b}{\mathrm{e}}^{-\mathrm{c}\mathrm{t}}$$

(4)

in which the following substitutions were made:

$$\mathrm{a}=\frac{{\mathsf{\sigma }}_0}{\mathrm{E}} ; \mathrm{b}=\frac{{\mathsf{\sigma }}_0}{\mathrm{E}}-{\mathsf{\epsilon }}_0 ; \mathrm{c}=\frac{\mathrm{E}}{\mathsf{\eta }}=\frac{\mathrm{E}}{\mathrm{T}\mathrm{E}}=\frac{1}{\mathrm{T}}$$

(5)

Based on the fact that from the experimental data we can choose two arbitrary points with the abscissa t₁ and t₂, and having the obtained values from the creep test for a stress equal to σ₀ = 11.55 MPa, then:

$${\mathrm{t}}_3=\frac{{\mathrm{t}}_1+{\mathrm{t}}_2}{2} ; \mathrm{a}=\frac{{\mathsf{\epsilon }}_1{\mathsf{\epsilon }}_2-{\mathsf{\epsilon }}_3^2}{{\mathsf{\epsilon }}_1+{\mathsf{\epsilon }}_2-2{\mathsf{\epsilon }}_3}$$

(6)

which become:

$${\mathrm{t}}_3=\frac{7.135+51.5}{2} \Rightarrow {\mathrm{t}}_3=29.3125 \mathrm{days}$$

(7)

and:

$$\mathrm{a}=\frac{3637\times {10}^{-6}\times 5291\times {10}^{-6}-{\left(5180\times {10}^{-6}\right)}^2}{3637\times {10}^{-6}+5291\times {10}^{-6}-2\times 5180\times {10}^{-6}} \Rightarrow \mathrm{a}=5299.50\times {10}^{-6}$$

(8)

where the value 0.005180 represents ε₃, which corresponds to t₃ and which was determined by interpolation.

Knowing that:

$$\mathrm{b}=\frac{{\mathsf{\sigma }}_0}{\mathrm{E}}-{\mathsf{\epsilon }}_0=\mathrm{a}-{\mathsf{\epsilon }}_0$$

(9)

then:

$$\mathrm{b}=5299.50\times {10}^{-6}-2125\times {10}^{-6} \Rightarrow \mathrm{b}=3174.50\times {10}^{-6}$$

(10)

Equation (3) can also be written in the form:

$$\mathsf{\epsilon }-\mathrm{a}=-\mathrm{b}{\mathrm{e}}^{-\mathrm{c}\mathrm{t}} \mathrm{o}\mathrm{r} \mathrm{a}-\mathsf{\epsilon }=\mathrm{b}{\mathrm{e}}^{-\mathrm{c}\mathrm{t}}$$

(11)

making the logarithm of Equation (11), it results in:

$$\lg \left(\mathrm{a}-\mathsf{\epsilon }\right)=\lg \mathrm{b}-\mathrm{c}\mathrm{t}\lg \mathrm{e}$$

(12)

noting:

$$\mathrm{p}=\lg \left(\mathrm{a}-\mathsf{\epsilon }\right) ; \mathrm{B}=\lg \mathrm{b} ; \mathrm{C}=\mathrm{c}\lg \mathrm{e}$$

(13)

then, we get the relationship:

$$\mathrm{p}=\mathrm{B}-\mathrm{C}\mathrm{t}$$

(14)

the value of B is determined according to the relation (13), resulting:

$$\mathrm{B}=\lg \left(3174.5\times {10}^{-6}\right)=3.5016753+\lg \left({10}^{-6}\right)$$

(15)

knowing the value of B, from Equation (13), we can obtain the value of C:

$${\mathrm{p}}_{\mathrm{i}}=\mathrm{n}\mathrm{B}-\stackrel{-}{\mathrm{C}}{\mathrm{t}}_{\mathrm{i}}$$

(16)

from where:

$${\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{p}}_{\mathrm{i}}}=\mathrm{n}\mathrm{B}-\stackrel{-}{\mathrm{C}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{t}}_{\mathrm{i}}}$$

(17)

therefore, the value of C will be given by the relation:

$$\stackrel{-}{\mathrm{C}}=\frac{\mathrm{n}\mathrm{B}-{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{p}}_{\mathrm{i}}}}{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{t}}_{\mathrm{i}}}}$$

(18)

where ${\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{p}}_{\mathrm{i}}}; {\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{t}}_{\mathrm{i}}}$ were determined based on resulted data from the creep test and n = 25 is the number of obtained data.

Namely:

$$\stackrel{-}{\mathrm{C}}=\frac{25\times \left(3.5016753+\lg {10}^{-6}\right)-\left(53.64918405+\lg {10}^{-6}\right)}{673.665} \Rightarrow \stackrel{-}{\mathrm{C}}=0.0503$$

(19)

whence:

$$\stackrel{-}{\mathrm{c}}=\frac{\stackrel{-}{\mathrm{C}}}{\lg \mathrm{e}}=\frac{0.0503}{0.434} \Rightarrow \stackrel{-}{\mathrm{c}}=0.115898617 \Rightarrow \stackrel{-}{\mathrm{c}}=0.116$$

(20)

it is known that:

$${\mathsf{\epsilon }}_{\mathrm{i}}={\stackrel{-}{\mathsf{\epsilon }}}_{\mathrm{i}}+{\left(\frac{\partial \stackrel{-}{\mathsf{\epsilon }}}{\partial \stackrel{-}{\mathrm{c}}}\right)}_{\mathrm{i}}\mathrm{s} ; \mathrm{i}=1,2,\dots ,\mathrm{n}$$

(21)

and:

$$\mathrm{s}=\frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathsf{\epsilon }-\stackrel{-}{\mathsf{\epsilon }}\right)}_{\mathrm{i}}}}{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\left(\frac{\partial \stackrel{-}{\mathsf{\epsilon }}}{\partial \stackrel{-}{\mathrm{c}}}\right)}_{\mathrm{i}}}}$$

(22)

once the value of $\stackrel{-}{\mathrm{c}}$ (the first approximation) is established, Equation (3) becomes:

$$\stackrel{-}{\mathsf{\epsilon }}=\mathrm{a}-\mathrm{b}{\mathrm{e}}^{-\stackrel{-}{\mathrm{c}}\mathrm{t}}$$

(23)

which the derivative with respect to $\stackrel{-}{\mathrm{c}}$ will be:

$$\frac{\partial \stackrel{-}{\mathsf{\epsilon }}}{\partial \stackrel{-}{\mathrm{c}}}=\mathrm{b}{\mathrm{t}}_{\mathrm{i}}{\mathrm{e}}^{-\stackrel{-}{\mathrm{c}}{\mathrm{t}}_{\mathrm{i}}}$$

(24)

from the processing of the obtained experimental data, it can easily determine the value of (s), that is:

$$\mathrm{s}=\frac{-190.78}{84,512.8} \Rightarrow \mathrm{s}=-0.0225$$

(25)

in this case, it turns out:

$$\stackrel{=}{\mathrm{c}}=\stackrel{-}{\mathrm{c}}+\mathrm{s} \Rightarrow \stackrel{=}{\mathrm{c}}=0.0935$$

(26)

where $\stackrel{=}{\mathrm{c}}$ represents the second approximation.

On the basis of $\stackrel{=}{\mathrm{c}}$, the accuracy increases from 0.520404 to 0.097772. Equation (3), respectively (23), becomes:

$$\stackrel{=}{\mathsf{\epsilon }}=\mathrm{a}-\mathrm{b}{\mathrm{e}}^{-\stackrel{=}{\mathrm{c}}\mathrm{t}}$$

(27)

returning to the substitutions made, the relation (5), we can now determine the search parameters of the creep curve. So:

$$\begin{array}{l}\mathrm{E}=\frac{{\mathsf{\sigma }}_0}{\mathrm{a}}\\ \mathrm{E}=\frac{11.55}{0.0052995} \Rightarrow \mathrm{E}=2179.4 \mathrm{MPa}\end{array}$$

(28)

this value confirms the good results of the elasticity module determined in the laboratory.

$$\mathrm{T}=\frac{1}{\stackrel{=}{\mathrm{c}}} \iff \mathrm{T}=\frac{1}{0.0935} \Rightarrow \mathrm{T}=10.7 \mathrm{days}$$

(29)

and:

$$\mathsf{\eta }=\mathrm{T}\times \mathrm{E} \iff \mathsf{\eta }=2179.4\times 10.7 \Rightarrow \mathsf{\eta }=23319.6 \mathrm{MPa}\times \mathrm{day}$$

(30)

therefore, Equation (6) becomes:

$$\mathsf{\epsilon }=0.00529816-0.00317316{\mathrm{e}}^{-0.0935\mathrm{t}}$$

(31)

Equation (31) represents the analytical equation established for the strongly metamorphosed andesite that has been analyzed and which confirms that this rock can be assimilated as a manner of behavior in time with the model given by the Poynting–Thomson model (see Figure 4).

3.2. Deformation Characteristics in Time of Altered Andesite Established In Situ

In parallel with the tests performed in the laboratory, the in situ measurements have also been made regarding the deformation in time of the rocks (convergence) around the mining works that intercept the rock types of the highly hydrothermal metamorphosed andesite (Appendix A). The measurements were made in cross sections in the direction of the galleries in the following underground works: a trapezoidal cross gallery (Figure 5a); a directional gallery with an arched profile with straight walls (Figure 5b); and a directional gallery with a circular profile (Figure 5c).

The convergence measurements were extended over a period of 3 years and the obtained results were used in order to establish the laws of variation in the time of the rock’s movement on the outline of the mining works, that is, the laws of relaxation and creep.

3.2.1. Relaxation Law

Following the graphical and analytical processing of the convergence values (Figure 6) taken as the displacements, it turns out that, in time, they are determined by a law of variation whose analytical form could be established with the help of mathematical statistics.

A time variation law of the movements on the contour of the gallery resulted in the forming of the equation:

$$\mathrm{y}=\frac{8\mathrm{m}}{\mathrm{x}}+\mathrm{b}$$

(32)

where y is the values of the displacements according to the direction of measurement; x is the time at which these displacements were measured; and m and b are the numerical coefficients of the equation, whose minimum values are shown in

Table 3. The convergence function and the parameter values of this function.

Function Type	Coefficients	Coefficients Values 1
$\mathrm{y}=\frac{8\mathrm{m}}{\mathrm{x}}+\mathrm{b}$	m	4800
	b	1170

¹ according to the direction of measurement.

.

3.2.2. Creep Law

In order to establish such a law, two following deformation methods of the contour of the underground work have been developed in time [10,11,45,46], namely, the threshold method and the gradient method. This second method, chosen for the interpretation of the measurements results, is based on the general considerations regarding the evolution of the deformation phenomenon; that is, when a cause (in our case, an underground excavation) comes to disturb the rock massif equilibrium, it will tend to evolve to a new state of equilibrium, which concretizes in deformations that should be amortized when the disturbing cause ceases.

However, in reality, these deformations increase continuously, and therefore the deformation phenomenon presents an unamortized evolution until the destruction of the mining work (if the support is inadequately designed according to the real working conditions). The problem that arises in such situations is to detect accelerations in the production of deformations, which will in fact indicate the occurrence of the failure phenomenon of the underground work and its destruction.

However, the laboratory tests allow us to specify that a linear evolution of the deformation measurements can be approximated by a damped curve, in which case there is an ambiguity regarding the stability of the underground work and, therefore, the observations cannot be stopped but must be continued (Figure 7).

Only if the evolution of the deformations indicates an acceleration of them, then we can predict with certainty the beginning of the phenomenon of the destruction of the underground work because, from a rheological point of view, the breaking of the rocks is always preceded by a short acceleration of the deformations (Figure 8).

Therefore, the chosen method is not based on the final value of the deformation reached at any given moment but on the analysis of the evolution of these deformations compared to their history, that is to say, with their previous evolution.

Such a research method requires complex observations, capable of memorizing the history of in situ deformations; by this, the fluctuation range due to the vulnerability can be reproduced, the general sense of the evolution and the abnormal variation, and the index of the start of the destruction process of the rock’s support system. Such a result can be achieved by considering from the beginning the making of successive and punctual measurements, corresponding to a range, so that in the end we can obtain the evolution of the deformations as true and real as possible.

4. The Gradient Method

The gradient method is based on general considerations regarding the evolution of the deformation phenomenon (Appendix D). Based on the measurements made in each range of time, it is found that the deformations in this range (defined by at least three points) are defined by a straight line. With the help of the smallest squares, the characteristics of the line were established:

$$\mathrm{D}={\mathrm{a}}^{\mathrm{i}}\mathrm{t}+{\mathrm{b}}^{\mathrm{i}}$$

(33)

where aⁱ is the slope of the straight; bⁱ is the y-coordinate to the origin; and t is the time starting with the first day of the interval measurement.

Taking into account the variability of daily measurements, based on the given data, it was possible to decide whether the observed slope (aⁱ) is significant either for an increase in the deformations, for a decrease in the deformations, or on the contrary, if the deformations were constant during the range of measurement. This decision can be materialized by the concepts of convergence, divergence and constancy.

Therefore, the gradient method actually starts with a comparison between the observed central tendency and the tendency that characterized the past. This comparison is actually stored in the form of the straight line given by Equation (33) in the measurement interval (i) and a certain dispersion (σ). The dispersion will indicate an increase in the amplitude of the jumps in the deformation process (Figure 9), and the comparison of the slopes establishes the concavity of the mean deformation curve (Figure 10). Thus, by comparing the dispersions σ′ and σ″, we can decide, for example, whether σ″ is significantly much larger than σ′, either as a deterioration of the measuring device or an increase in the amplitude of the jumps in the deformation process (Figure 9). If the slope difference (aⁱ⁺¹ – aⁱ) is large, then it means either an acceleration or stabilization according to the sign of this difference.

If the difference (aⁱ⁺¹−aⁱ) is large, taking into account the variability of the daily measurements, it will be concluded by the difference that it is either an acceleration or a stabilization. If the two straight lines are identical, the data for week (i) will be added to the previous data for defining the straight, which will be taken as the reference straight for the week (i + 1):

$$\mathrm{D}={\mathrm{a}}_0^{\mathrm{i}+1}\mathrm{t}+{\mathrm{b}}_0^{\mathrm{i}+1}$$

(34)

and the straight line of the week (i) will be: D = aⁱt + bⁱ as the reference straight line for the week (i + 1).

Observations can be made on a daily basis, comparing the obtained measurement with the reference straight line, taking into account the dispersion (σ) that characterizes the variability of the measurements. This allows you to set a first alarm level, which involves intensifying the observations or, for example, re-editing the observation. If the next measurement is still higher than the previously observed variation range, the second alarm level can be triggered (Figure 11).

Based on this process, deformations were analyzed in the case of mining works located in andesite. The range of measurement was 10 days, in which 3–4 measurements were made (Appendix B). The graphical representation of the evolution of the deformations in time and their analysis by the gradient method is shown in Figure 12.

The graphical form of the function that shows the law of deformation in time of the contour of horizontal mining works, obtained from the analysis by the gradient method (Figure 10), is in fact identical to the shape of the creep curve.

5. Results and Discussions

We set out to find the mathematical function that will express the law of the search deformation based on the already known form. From a mathematical point of view, the problem is reduced to determining a functional dependence of the variable y (in our case, this variable is the deformation) against another variable x (namely, the time), which can be written as Equation (35) with the obvious identification of all the parameters that come into the definition of a functional dependence:

$$\mathrm{y}=\mathrm{f}\left(\mathrm{x}; {\mathrm{a}}_0; {\mathrm{a}}_1; {\mathrm{a}}_2;\dots ;{\mathrm{a}}_{\mathrm{n}}\right)$$

(35)

the method that allows us to obtain accurate and consistent estimates of the parameters is: a₀, a₁, a₂, …, a_n and it is the least-squares method using Cebisev’s polynomials. We chose this method because our functional has a form:

$$\mathrm{y}={\mathrm{b}}_0{\mathrm{p}}_0(\mathrm{x})+{\mathrm{b}}_1{\mathrm{p}}_1(\mathrm{x})+{\mathrm{b}}_2{\mathrm{p}}_2(\mathrm{x})+\dots +{\mathrm{b}}_{\mathrm{n}}{\mathrm{p}}_{\mathrm{n}}(\mathrm{x})$$

(36)

where p₀(x); p₁(x); p₂(x); …; p_n(x) is the the orthogonal polynomials of Cebisev on the set of the points x₁, x₂, …, x_n with the weights w(x).

The expression parameters (36) are calculated with the relation:

$${\mathrm{b}}_{\mathrm{j}}=\frac{{\mathrm{y}}_{\mathrm{k}}{\mathrm{p}}_{\mathrm{j}}\left({\mathrm{x}}_{\mathrm{k}}\right){\mathrm{w}}_{\mathrm{k}}}{{\mathrm{p}}_{\mathrm{j}}^2\left({\mathrm{x}}_{\mathrm{k}}\right){\mathrm{w}}_{\mathrm{k}}} , \mathrm{j}=0, 1, 2, \dots , \mathrm{n}$$

(37)

which do not depend on the degree n of the searched polynomial.

If the values of the argument are equidistant:

$${\mathrm{x}}_{\mathrm{k}+1}={\mathrm{x}}_{\mathrm{k}}=\mathrm{h} , \mathrm{k}=1, 2, \dots , \mathrm{N}-1$$

(38)

the calculation of the orthogonal polynomials Cebisev is greatly simplified, if the variable is changed:

$${\mathrm{u}}_{\mathrm{k}}=\frac{{\mathrm{x}}_{\mathrm{k}}-\mathrm{x}}{\mathrm{h}}$$

(39)

where:

$$\mathrm{x}=\frac{{\mathrm{x}}_1+{\mathrm{x}}_{\mathrm{N}}}{2}$$

(40)

taking this into account, the searched polynomial will be:

$$\mathrm{y}={\mathrm{c}}_0{\mathrm{p}}_0(\mathrm{u})+{\mathrm{c}}_1{\mathrm{p}}_1(\mathrm{u})+{\mathrm{c}}_2{\mathrm{p}}_2(\mathrm{u})+\dots +{\mathrm{c}}_{\mathrm{n}}{\mathrm{p}}_{\mathrm{n}}(\mathrm{u})$$

(41)

and the parameters of the polynomial are estimated using the relations:

$${\mathrm{c}}_{\mathrm{j}}=\frac{1}{{\mathrm{H}}_{\mathrm{j}}}{\mathrm{y}}_{\mathrm{k}}{\mathrm{p}}_{\mathrm{j}}\left({\mathrm{u}}_{\mathrm{k}}\right) , \mathrm{j}=0, 1, 2, \dots , \mathrm{n}; \mathrm{n}=\mathrm{N}$$

(42)

where:

$${\mathrm{H}}_{\mathrm{j}}={\mathrm{p}}_{\mathrm{j}}^2\left({\mathrm{u}}_{\mathrm{k}}\right)$$

(43)

and ${\mathrm{u}}_{\mathrm{k}}=\frac{{\mathrm{x}}_{\mathrm{k}}-\mathrm{x}}{\mathrm{h}}$ takes only the whole values: 0; ±1; ±2; …; ±(N − 1)/2 if N is uneven, respectively, and the values ±1/2; ±3/2; …; ± (N − 1)/2 when N is an even number.

Using the special formulas for an even number (N = 2L) and an uneven number (N = 2L + 1) of the obtained observations from the convergence measurements, but also the relations of the calculation of the parameters p₁, p₂, …, p₅, corresponding to the number of measurements within a range of 10 days of the measurement, it was passed to determine the searched functional.

The total number of the measurements varies between 9 and 12 data. For this N, the corresponding relations and parameters were taken (see

Table 4. Example of calculation for 10 measurements data.

Computing Relations for N = 10 = H0 1	u	${\mathrm{p}}_1^{*}$	${\mathrm{p}}_2^{*}$	${\mathrm{p}}_3^{*}$	${\mathrm{p}}_4^{*}$	${\mathrm{p}}_5^{*}$
$\begin{array}{l}{\mathrm{p}}_1=\mathrm{u}=\frac{1}{2}{\mathrm{p}}_1^{*}\\ {\mathrm{p}}_2={\mathrm{u}}^2-\frac{33}{4}=2{\mathrm{p}}_2^{*}\\ {\mathrm{p}}_3={\mathrm{u}}^3-\frac{293}{20}\mathrm{u}=\frac{3}{5}{\mathrm{p}}_3^{*}\\ {\mathrm{p}}_4={\mathrm{u}}^4-\frac{41}{2}{\mathrm{u}}^2+\frac{3861}{80}=\frac{12}{5}{\mathrm{p}}_4^{*}\\ {\mathrm{p}}_5={\mathrm{u}}^5-\frac{155}{6}{\mathrm{u}}^3+\frac{6067}{48}\mathrm{u}=10{\mathrm{p}}_5^{*}\end{array}$	0	1	−4	−12	18	6
	1	3	−3	−31	3	11
	2	5	−1	−35	−17	1
	3	7	2	−14	−22	−14
	4	9	6	42	18	6
	γ	165	264	5.148	6.864	7.800
	H	$\frac{165}{2}$	528	$\frac{15.444}{5}$	$\frac{82.368}{5}$	78.000
	$\frac{{\mathrm{H}}_{\mathrm{j}}}{{\mathrm{H}}_{\mathrm{j}}-1}$	$8\frac{1}{4}$	$6\frac{2}{5}$	$5\frac{17}{20}$	$5\frac{1}{3}$	$4\frac{97}{132}$

¹ computing relations for N = 9; N = 11; and N = 12 are shown in Appendix C.

), with which the function of the searched polynomial was written.

Because the methodology is identical for each group of measurements, we summarize here the description of a single case, namely when N = 11 (11 measurement data):

$${\mathrm{p}}_0=1 ; {\mathrm{p}}_1=\mathrm{u} ; {\mathrm{p}}_2={\mathrm{u}}^2-10 ; {\mathrm{p}}_3={\mathrm{u}}^3-\frac{89}{5}\mathrm{u}$$

(44)

and then:

$$\mathrm{y}={\mathrm{C}}_0{\mathrm{p}}_0+{\mathrm{C}}_1{\mathrm{p}}_1+{\mathrm{C}}_2{\mathrm{p}}_2+{\mathrm{C}}_3{\mathrm{p}}_3$$

(45)

or:

$$\mathrm{y}={\mathrm{C}}_0+{\mathrm{C}}_1\mathrm{u}+{\mathrm{C}}_2\left({\mathrm{u}}^2-10\right)+{\mathrm{C}}_3\left({\mathrm{u}}^3-17.8\mathrm{u}\right)$$

(46)

from which the following equation results:

$$\mathrm{y}=\left({\mathrm{C}}_0+10{\mathrm{C}}_2\right)+\left({\mathrm{C}}_1-17.8{\mathrm{C}}_3\right)\mathrm{u}+{\mathrm{C}}_2{\mathrm{u}}^2+{\mathrm{C}}_3{\mathrm{u}}^3$$

(47)

but:

$$\begin{array}{l}\mathrm{y}=\left(+10{\mathrm{C}}_2\right)+\left(-17.8{\mathrm{C}}_3\right)\mathrm{u}+{\mathrm{C}}_2{\mathrm{u}}^2+{\mathrm{C}}_3{\mathrm{u}}^3\\ {\mathrm{C}}_0=\frac{1}{\mathrm{N}}{\displaystyle \sum {\mathrm{y}}_{\mathrm{k}}}\\ {\mathrm{C}}_1=\frac{1}{110}\left({\mathrm{y}}_1-{\mathrm{y}}_{-1}\right)+2\left({\mathrm{y}}_2-{\mathrm{y}}_{-2}\right)+3\left({\mathrm{y}}_3-{\mathrm{y}}_{-3}\right)+4\left({\mathrm{y}}_4-{\mathrm{y}}_{-4}\right)+5\left({\mathrm{y}}_5-{\mathrm{y}}_{-5}\right)\\ {\mathrm{C}}_2=\frac{1}{858}-10{\mathrm{y}}_0-9\left({\mathrm{y}}_1+{\mathrm{y}}_{-1}\right)-6\left({\mathrm{y}}_2+{\mathrm{y}}_{-2}\right)-\left({\mathrm{y}}_3+{\mathrm{y}}_{-3}\right)+6\left({\mathrm{y}}_4+{\mathrm{y}}_{-4}\right)+15\left({\mathrm{y}}_5+{\mathrm{y}}_{-5}\right)\\ {\mathrm{C}}_3=\frac{1}{5148}-14\left({\mathrm{y}}_1-{\mathrm{y}}_{-1}\right)-23\left({\mathrm{y}}_2-{\mathrm{y}}_{-2}\right)-22\left({\mathrm{y}}_3-{\mathrm{y}}_{-3}\right)-6\left({\mathrm{y}}_4-{\mathrm{y}}_{-4}\right)+30\left({\mathrm{y}}_5-{\mathrm{y}}_{-5}\right)\end{array}$$

(48)

knowing:

$$\begin{array}{l}\mathrm{u}=\frac{{\mathrm{x}}_{\mathrm{k}}-\mathrm{x}}{\mathrm{h}} ; \mathrm{h}=\frac{{\mathrm{x}}_{\mathrm{k}+1}-{\mathrm{x}}_{\mathrm{k}}}{1}={\mathrm{x}}_{\mathrm{k}+1}-{\mathrm{x}}_{\mathrm{k}}\\ \mathrm{x}=\frac{110+10}{2}=60\end{array}$$

(49)

then:

$$\mathrm{u}=\frac{{\mathrm{x}}_{\mathrm{k}}-60}{10} \Rightarrow \mathrm{u}=0.1\mathrm{x}-6$$

(50)

it results:

$$\begin{array}{l}\mathrm{u}=0.1\mathrm{x}-6\\ {\mathrm{u}}^2=0.01{\mathrm{x}}^2-1.2\mathrm{x}+36\\ {\mathrm{u}}^3=0.001{\mathrm{x}}^3-0.18{\mathrm{x}}^2+10.8\mathrm{x}-216\end{array}$$

(51)

based on the calculations, according to

Table 5. Computing scheme of the coefficients.

x	yk 1	Values of yk	Size of (y+i − y−i)	Value	Size of (y+i + y−i)	Value
10	y−5		y+5 − y−5		-	-
20	y−4		y+4 − y−4		-	-
30	y−3		y+3 − y−3		-	-
40	y−2		y+2 − y−2		-	-
50	y−1		y+1 − y−1		-	-
60	y0	-	-		-	-
70	y+1		-	-	y+1 + y−1
80	y+2		-	-	y+2 + y−2
90	y+3		-	-	y+3 + y−3
100	y+4		-	-	y+4 + y−4
110	y+5		-	-	y+5 + y−5

¹ for 11 measurements data.

, the value of the C₀, C₁, C₂ and C₃ coefficient results are obtained.

The equation becomes:

$$\mathrm{y}=461.97846+10.01506\mathrm{u}+0.25279{\mathrm{u}}^2+0.28044{\mathrm{u}}^3$$

(52)

based on the substitutions made, according to the relations (51) and given that the variable x represents the time, it results in:

$$\mathrm{y}=350.41+3.73\mathrm{t}-0.048{\mathrm{t}}^2+0.00028{\mathrm{t}}^3$$

(53)

All the data processing led us to the equations of the form obtained above, so we can specify that the law of the deformation in time of the contour of the main horizontal mining works which were made in altered andesite rock type has the main form, given by the relation (53), and whose graphical representation has the shape shown in Figure 13:

$$\mathsf{\epsilon }\left(\mathrm{t}\right)=\mathrm{A}+\mathrm{B}\mathrm{t}+\mathrm{C}{\mathrm{t}}^2+\mathrm{D}{\mathrm{t}}^3$$

(54)

Based on the rheological tests performed, the characterization of the altered andesite rock type, it was concluded that for these rocks, the rheological behavior can be assimilated as being the type given by the Poynting–Thomson model. Given the fact that in the actual analyzed in situ conditions, where, due to external factors like the water infiltration but also due to the relative humidity, the alteration phenomenon of the rocks around the horizontal mining works intensifies, having as a consequence the decrease in the bearing capacity, the appearance of the phenomenon of swelling, that is, their change in time, leads us to the conclusion that the use of the Poynting–Thomson model becomes uncertain.

The uncertainty was created by the creep law because it was presented in the form of a damped curve, which actually indicated a stabilization of the deformations (t ≥ 11 days), but not the moment until which the deformation of the highly metamorphosed andesite still retains a certain bearing capacity, the moment from which the breakage begins. Such a curve was therefore incomplete and always ambiguous, because around the horizontal workings, the deformations increased continuously.

The deformation phenomenon presents a continuous evolution and, from a certain moment, increases more and more. In order to fully characterize the behavior under the load, the in situ measurements led us to complete the Poynting–Thomson model in a way which is much more appropriate to the real conditions. The behavior equation is the following:

$$\mathsf{\epsilon }\left(\mathrm{t}\right)=\frac{{\mathsf{\sigma }}_0}{\mathrm{E}}\left(1+\mathrm{t}\right)+\left(\frac{{\mathsf{\sigma }}_0}{{\mathrm{E}}_{\mathrm{K}}}+\frac{{\mathsf{\sigma }}_{\mathrm{p}}}{{\mathrm{E}}_{\mathrm{C}\mathrm{V}}}\right){\mathrm{e}}^{-\frac{\mathrm{E}}{\mathsf{\eta }}\mathrm{t}}$$

(55)

Thus, we proposed the rheological model presented in Figure 14; in principle, the proposed model is based on the following behavior in time under the load and under extremely adverse conditions (the wet area, swelling tendency, etc.).

The particles of eruptive rock are placed so that they are embedded in cement as a product of alteration. In the behavior of the rock under the load, this characteristic determines the appearance of the two stages of deformation (see

Table 6. Changing the state of the analyzed rocks under the action of the loads and in interaction with the water.

Stage	State	Behavior 1	Flow Criterion	Effects that Appear
1	Initial state	Elastic to elastic–viscous (Poynting–Thomson model)	-	No change in volume
2	Weathering state	Elastic–plastic	$\mathsf{\alpha }{\mathrm{y}}_1+\sqrt{{\mathrm{y}}_2}=\mathrm{K}$	Volume increase
3	Breaking	-	$\mathsf{\alpha }{\mathrm{y}}_1+\sqrt{{\mathrm{y}}_2}>\mathrm{K}$

¹ altered andesite from Suior mine, Romania.

): the viscous-elastic stage with a transition to elastic and the elastic–plastic stage.

When a stress is applied, a disturbance of the particles (which have a minimum volume of voids) is created due to the sliding occurring between them and between which a viscous friction occurs. Then, the particles come into direct contact, making the deformation develop further; if the stress increases or external disturbing factors appear, the deformation phenomenon will continue, creating inelastic deformations, namely, elastic–plastic behavior, keeping its cohesion until the plastic limit is exceeded, and it is at this moment when the breakage can occur. Taking into account such a behavior, it results that, giving the rock the possibility to deform in time (t ≥ T), it passes into the field of an elastic–plastic deformation, an area in which it still presents stability. Such a conclusion becomes extremely important in the rock–support interaction phenomenon. On the other hand, also based on the rheological results and using the principles of hereditary theory [6,7,8,10,13,21,30,35], the nucleus of Abel was determined, obtaining the value Φ_t = 1.49, through which the influence of time can be included in establishing the pressure regime.

6. Conclusions

The necessity to establish the forecast of the stability of the main horizontal mining works in rocks with very different characteristics, involves the assessment of the real value of the properties and firstly of all the anisotropy, heterotrophy and the rheological behavior of the rocks.

The proposed method for studying the evolution of the phenomenon of the deformation in time of rocks around an underground excavation takes into account the fact that the deformation increases continuously, which means that the deformation phenomenon has an undamped evolution until the destruction of the underground work. In such situations, we considered that it is necessary to find accelerations of deformations generation, which will also indicate the occurrence of the delivery phenomenon of the underground work and its failure.

The method chosen is not based on the final value of the deformation reached at any given time but on the analysis of the evolution of these deformations compared to their previous evolution. Based on this process, the deformations in time were analyzed in the case of the mining works located in altered andesite. In relation to the shape of the deformation curve of the horizontal mining works contour, we established the mathematical function that expresses the law of sought deformation. In order to characterize the entire behavior under the load, in situ measurements led us to complete the Poynting–Thomson model much more appropriately to the real conditions. Thus, we have proposed a rheological model that is able to describe the time behavior under the load and in extremely unfavorable conditions (the wet area, swelling tendency, etc.).

The type of the studied rocks is characterized by the fact that they have a low resistance and that, in contact with water, they change their properties. The rocks taken in the study are pyroxenic andesite rocks affected by a medium to high hydrothermal alteration. These are rocks which in their stability area have viscous–elastic–plastic behavior and which can be assimilated by a Poynting–Thomson rheological model when W < W_HM or with the proposed model for W > W_HM, in which case the phenomenon of swelling occurs (W represents the humidity and W_HM is the maximum hygroscopicity). For a period of 56 days, these rocks maintain their stability with the condition that, up to this time limit, the displacements stop. Otherwise, the rocks lose completely their cohesion, the rupture occurs and the pressure on the support will increase, endangering the stability of the underground works.

From this analysis, it follows that a damped evolution alone is not sufficient to predict the stability and finality of in situ observations because when a cause (for example, the execution of an underground excavation) comes to disturb the equilibrium of the rock mass, it will tend to evolve towards a new state of equilibrium, which is characterized by deformations that should be damped when the disturbing cause ceases. In reality, these deformations increase continuously, and therefore the deformation phenomenon shows that an evolution is not damped in time until the moment of the destruction of the mining work (if the support is inadequately designed according to the real working conditions). The problem is to detect accelerations in the generation of deformations that will actually indicate the phenomenon of failure and the destruction of the support.