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Rock damage evolution as failure process of hierarchical structure
Journal Pre-proof Rock damage evolution as failure process of hierarchical structure
S.N. Fedotov PII:
S0031-9201(19)30168-2
DOI:
https://doi.org/10.1016/j.pepi.2020.106449
Reference:
PEPI 106449
To appear in:
Physics of the Earth and Planetary Interiors
Received date:
18 June 2019
Revised date:
14 September 2019
Accepted date:
14 February 2020
Please cite this article as: S.N. Fedotov, Rock damage evolution as failure process of hierarchical structure, Physics of the Earth and Planetary Interiors(2018), https://doi.org/ 10.1016/j.pepi.2020.106449
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© 2018 Published by Elsevier.
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Rock damage evolution as failure process of hierarchical structure S. N. Fedotov* National Research Nuclear University (MEPhI), Moscow, 115409 Russia
Sechenov First Moscow State Medical University, Moscow 119991, Russia *e-mail: [email protected] Abstract
In present study the model is proposed to describe the dynamics of the rock fracture process. Rock formation is assumed
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in the form of a nested self-similar hierarchical structure. The hierarchical structure scheme is chosen in accordance with the experimentally verified criterion of fracture. The failure probability of each structure level is calculated as a function of
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time, temperature, material parameters and loading conditions. Knowing the failure probabilities allows one to calculate the stress dynamics in a rock material, failure volumes for each hierarchy level, energy release, etc. The influence of
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temperature, type of the hierarchical structure, rock material, stress rate on the fracture dynamics is considered. The results
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of calculations of the rock fracture dynamics are compared with the results of experiments.
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1. INTRODUCTION
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Key words: acoustic emission, fracture dynamics, fractals, hierarchical structure.
The analysis of numerous data accumulated over a long period of observations of earthquakes indicates the spatial and temporal stability of the Gutenberg-Richter law (Gutenberg and Richter, 1954). In this case, the distribution of earthquake
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magnitude obeys an apparently universal scaling law despite the different types of movements, seismic activity levels, despite the different deformation rates, chemical and physical characteristics of the medium. In addition, the distribution of aftershocks does not differ from that of ordinary background earthquakes. In the opinion of many researchers, this stability of the earthquake repeatability law is due to the general fundamental laws of the structure of the medium (Makarov, 2010; Weiss, 2003; Yukalov et al. 2004) and to a single mechanism of fracture. The most common regularity in the fracture of solids is the concentration criterion established in the course of experimental studies (Kuksenko et al. 1996). According to this criterion, the concentration parameter K = R / L (L is the length of cracks formed, R is the distance between cracks of this length) at the instant of failure of solids takes the critical value Kcr = 2.7-3 in a wide range of crack linear sizes, regardless of the type of materials and load regimes. This behavior of solids indicates that materials can be considered as self-similar hierarchical structures. This conclusion is based both on the results of long-term studies of the rocks fracture and on the results of earthquake studies. There are some papers that considered the rock material in the form of self-similar hierarchical structures (Lei et al. 2005; Chmel et al. 2007). However, the models described in these papers are not based on experimentally verified criteria for the material destruction. This paper proposes a model for describing the rock destruction dynamics. This model is based on the kinetic approach. The materials themselves are considered in the form of self-similar hierarchical structures. The model uses the regularities
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2. MODEL AND CALCULATIONS
For simplicity the structure in the form of a cube is considered as the simplest three-dimensional nested hierarchical selfsimilar structure. The next level of the hierarchy of such a structure occurs when the length of each of the cube ribs of the previous hierarchy level is doubled (in Figure 1 three levels of the hierarchy of such a structure are depicted). For failure of the cube consisting of 8 cubes of the previous hierarchy level (a crack must pass through the entire volume of the cube connecting opposite faces), it is necessary that at least two cubes of eight are destroyed. This rule is true for any type of crystal lattice (we get 8 identical polyhedral similar to the original polyhedron (8 identical volumes) by doubling the ribs of
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any type of polyhedron).
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Fig. 1. Three hierarchy levels of self-similar structure.
Let the self-similar structure contains N levels of the hierarchy. Then according to the concentration criterion at the instant
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of failure the equality Ri/Li = Кcr ( i = 0, 1, ... N -1) will be valid for each i-th level of the hierarchy. Considering Ri3 as the material volume per crack of length Li and a Li3 as the volume fractured by this crack, the ratio Li3/ Ri3 can be considered as a relative fractured part of volume Ri3. Also the ratio Li3/ Ri3 can be treated as the failure probability Pi for a volume of the i-th hierarchy level. In this case, the concentration criterion takes the following form: at the instant of failure the failure probabilities Pi are equal to the same value Рcr = Li3/ Ri3 = Кcr-3 for all hierarchy levels. Let the failure probability of each of the eight cubes for the lowest level of the hierarchy be independent and equal to the same value P. Then the failure probability F(P) of the cube (the next level of the hierarchy) will be described by the binomial distribution as the failure probability for two or more cubes of eight F(Р) = 1 - (1 - Р)8 – 8P(1 - Р)7. For a self-similar structure hierarchies of all levels behave in a similar manner. Therefore at time t the failure probability Pi(t) of the i-th level of the hierarchy will be completely determined by the failure probability Рi-1(t) of the previous (i-1) -th level of the hierarchy. So the following expression Рi(t) = F( Рi-1(t)), will be valid for all levels of the hierarchy, where F(P) is the function describing the hierarchical scheme of the structure. In this case at the instant of failure the equation Pcr = F(Pcr) will be valid. The root of this equation is the value Рcr = Кcr-3. Thus within the framework of this model, the fracture process is a branching random process and the value of P = Pcr is the critical (degenerate) probability of this process (Karlin, 1966). The similar approach was used in papers devoted to the rock failure during earthquakes (Allegre et al. 1982; Allegre and Mouel, 1994; Allegre e. al. 1995).
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Journal Pre-proof However the models described in these papers are not based on experimentally verified criteria for the material destruction. For a given hierarchical structure the critical probability (the root of the equation F(P) = P) will be Pcr = 0.0423 with the concentration criterion taking the value К = (Рcr)-1/3 = 2.87. This value is in good agreement with the results of many experiments (Kuksenko et al. 1996). In the two-dimensional case, the structure will take the form of embedded squares (the face of the cube in Figure 1). When a similar two-dimensional structure (a film, a thin sheet material) is the hierarchical scheme of the structure will be described by the expression F(Р) = 1 - (1 - Р)4 – 4Р(1 - Р)3. In addition the critical probability will be Pcr = 0.232 and the value of the concentration criterion will be К = (Рcr)-1/2 = 2.08 (in case of the stretching F(Р) = 1 - (1 - Р)4 – 4Р(1 - Р)3 - 2Р2(1 - Р)2, Pcr = 0.382 and K = 1.62). Such values of K agree well with experiment (Shamina and Strizhkov, 1975). Thus the value of the concentration criterion depends only on the dimensionality of the space and to a much lesser extent
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on the loading regime. The three-dimensional structures (Pcr ≈ 0.05) are the least durable, and the most stable ones are one-dimensional (Pcr = 1). So the higher the dimensionality of space, the lower the value of Pcr because the more
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possibilities (variants) for cracking.
So the material fracture process (hierarchical structure levels destroying) depends on the hierarchical scheme of the
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structure only. Physical properties of material are defined on the lowest hierarchical level of structure. A formula for the
( )
(
(
)(
(
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destruction probability P0 (t) of the lowest level was obtained in (Fedotov, 2017) using the kinetic approach:
)
(
)))
,
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where τ is the average time the element of the structure is in the non-destroyed state, and z is the average time the element is in the destroyed state. In addition τ = (28/ Pcr)0.5 · τ0 ∙ exp ((U - γσ) / kT) , where U is the activation energy of the
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destruction process, k is the Boltzmann constant, γ is a parameter characterizing the actual mechanical properties of the material, T is the temperature (K °), σ is the mechanical stress. The coefficient τ0 coincides in order of magnitude with the thermal oscillations period of atoms in a solid. Knowledge of the hierarchical scheme and the failure probability of the
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zero level of the hierarchy make it possible to calculate the failure probability for each level of the hierarchy. Consider the simplest case which, if necessary, can be added and complicated. Suppose that the material with volume V and Young's modulus S is under the mechanical stress σ. In addition, the mechanical energy
E
2V 2S
(1)
will be accumulated in the volume V. In this case the energy gain ΔE in a time t will cause the stress change Δσ and the volume failure ΔV :
E
S
V
2 2S
V
(2)
Let us determine j as j = (tj) for discrete times tj = t0 + jt ( j = 0, 1, 2, …). The change in the failure probability of the i-th hierarchy level at the j-th step (on a time interval t ) will be
Pi(t j , j , T) = Pi(t j+1 , j+1 ,T) - Pi(t j , j ,T). In this connection the change in the volume of the i-th hierarchy level Vi over the time interval t will be described by the following formula
(Vi ) j 1 (Vi ) j (1 Pi (t j , j , T ))
(3)
Knowing of the failure probability distribution for all levels of the hierarchy makes it possible to calculate the average failure volume Vav over the time interval t:
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1 N Vi 8 N i Pi (t j , j , T ) , N 1 i 0
(4)
where N is the number of the highest level of the hierarchy, Vi is the current volume of the i-th level of the hierarchy (initial volume Vi =8i). In this case, the average intact volume at the j-th step Vavj equals to
Vav j Assuming j = j+1
1 N (Vi ) j 8 N i . N 1 i 0
(5)
- j and using (2), the stress is changed at the j-th step as
j 1 j
SE j j
2
2 Vav j
(6)
jVav j
When partially failed, the structure resists fracturing and strives for the stress relief. The release of elastic energy Wj on a
j S
Vav j j
(
)
( )
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)
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The average failure volume Dj
∑(
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Wj
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time interval t is
The obtained dependences make it possible to calculate the stress (t), the average failure volume
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D(t) and the elastic energy release W(t) both for an arbitrary dependence E(t) and for a specified external load as functions of time.
In this study the calculation of the granite fracture dynamics was carried out according to the described method. It is
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supposed that before loading rock material is considered as an ideal intact material with no any defects (ideal structure). Lattice defects occur due to thermal-fluctuation mechanism. The results of calculations for fracture of granite at a
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temperature of 20°C are shown on Figures 2, 3 and 4 in case of an axially compressed specimen. Calculations were performed for the structure containing N = 20 levels of the hierarchy. For granite under compressed load the following parameters were used U = 152 KJ/mol and γ = 0.453 KJ/(mol·MPa). It was supposed that granite specimen had the following sizes 100×100×200 mm3.
Figure 2. The time dependence for the stress (t) the granite specimen (T = 20°C) .
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Figure 3. The top part of the time dependence for the stress (t) that depicted in Figure2.
Figure 4. The time dependences for the elastic energy release W(t) - (1), the average failure volume of specimen D(t) - (2), the creep strain L(t) (nonlinear deformation (mm) ) – (3),
The average failure volume of specimen D(t) is presented in relative units. Here it was assumed that the energy gain is a constant dE/dt = const. Prior to the onset of fracture, until critical parameters are exceeded the specimen behaves like an elastic medium
(t ) const E (t ) . Further the stress does not increase and the energy gain is consumed in fracturing
the volume. Figures 2 and 3 demonstrate that for a self-similar hierarchical structure the fracture process is similar to a random process, though random numbers are not used in the calculations. The time interval from the onset of fracture to the maximum elastic energy release (creep mode) depends both on the rate of energy gain (or a stress rate) as well as on the temperature and characteristics of the specimen material. Up to the maximum elastic energy release there are observed a number of small releases called foreshocks in seismology (Figure 4 and Figure 5). After the maximum energy release a significant part of the volume storing the elastic energy is fractured and subsequent releases (aftershocks) become substantially smaller in magnitude (Figure 4). The creep strain L(t) (nonlinear deformation in millimeters) depicted in Figure 4 (curve 3) was calculated proportionally cubic root from average failure volume of specimen D(t). From Figure 5
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Journal Pre-proof it can be seen the time dependence of elastic energy release W(t) at the initial fracture stage corresponding to the data in Figure 4 on an enlarged scale. Evidently the sequence of the elastic energy releases is similar to the acoustic emission
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signals.
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Figure 5. The time dependence of elastic energy release W(t) on an enlarged scale. The fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels is shown in Figure 6. The curve
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number corresponds to i level of hierarchy. At relatively low levels of hierarchy there is slow failure of relative intact volumes. These levels are destroyed first. The fracture of the volume for the highest hierarchy level observed in the experiment fails instantaneously without any visible preparation. When the volume of the highest hierarchy level breaks,
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the crack passes through the entire specimen.
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Journal Pre-proof Figure 6. The fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels. The curve number i corresponds to the hierarchy level.
The behavior of the different hierarchy levels in the fracture process is of interest. The state of the relative failure volumes di(t) = 1 - Vi(t)/8 i is shown in Figure 7 for each level of the hierarchy i at different fracture stages (at different
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moment of time). All dependencies are in accordance with the data in Figure 6.
Figure 7. Relative failure volumes di(t) for each hierarchy level i at different fracture stages.
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Dependence 1 corresponds to the beginning of the fracture process (t = 10000 s), dependence 2 - failure of low hierarchy levels (t = 20000 s). Dependence 3 corresponds to the time before the failure of the highest hierarchy level (t = 50000 s). Dependence 4 corresponds to the time immediately after the failure of the highest hierarchy level (t = 55000 s).
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In fact this dependence determines the dispersed composition of fragments after the rock failure. At the beginning the fracture process the i level of hierarchy contains 8N-i intact volumes expressed in the number of elements V(0)i = 8i . In the fracture process the ratio Vi(t)/8 i can be considered as a relative part of intact volumes for the i level of hierarchy. So the number of intact volumes for the i level of hierarchy equals to Vi(t) 8N-2i . The linear size of the element for the i-th hierarchy level equals to xi = 2i-N 200 mm. It is not difficult to calculate the probability density function for fragment size G(xi) = Vi(55000) 8N-2iC, where C is normalizing constant. Figure 8 depicts the probability density function for fragment size G(x) defined as log-normal approximation of discrete values G( 2i-N 200).
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Figure 8.The probability density function for fragment size G(x) defined as log-normal approximation
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of discrete values G( 2i-N 200).
As a rule, the probability density function for crack sizes is postulated ( usually the Weibull distribution) in papers devoted
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to materials fracture processes ( X.P. Zhou, H.Q. Yang, 2004; X.P. Zhou, H.Q. Yang, , 2018). Knowledge of the behavior of the failure volumes in time di(t) make it possible to calculate the probability density function
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for crack size P(t, x). It is assumed that the crack size of the i-th hierarchy level is proportional to the cube root of the failure volume of this hierarchy. The probability density function for crack size P(t, x) is shown on Figure 9 for three
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moments of time t = 20000 s, t = 35000 s and t = 50000 s.
Figure 9. The probability density function P(t, x) for crack size x (mm) for different moments of time t. 1 – P(20000, x), 2 – P(35000, x), 3 - P(50000, x).
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Journal Pre-proof Also it is not difficult to calculate the a length of main crack in time as L(t) =
( )
The length of main crack
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in time L(t) is depicted in Figure 10.
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Figure 10. The length of main crack in time L(t).
3. COMPARISON WITH EXPERIMENT
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To verify the general mechanism of rock destruction the calculated data were compared with the results of experiments on the granite specimens failure described in (Ponomarev et al. 1997). In this study the acoustic emission signals were recorded during the experiments.
As a result of signal processing in two experiments, the fractal dimensionality df of the material to be failed was determined both at the stage preceding the fracture and at the moment of fracture. At the stage preceding the fracture the fractal dimension was determined as df1 = 2.86 ± 0.07 and df2 = 2.80 ± 0.02. At the moment of fracture, the following values for the fractal dimension were obtained the following values df1 = 2.4 ± 0.4 and df2 = 2.23 ± 0.05. In addition, in this study the coefficient b was estimated in the formula describing the Gutenberg-Richter's law lg N = - bM + const . The value of M was determined as the energy class of the earthquake M = lg E + const, where E is the earthquake energy. For the coefficient b were obtained the estimates b1 = 0.95 and b2 = 0.91 . In the present paper a self-similar cubic structure is considered as a regular fractal (Feder, 1987). For this fractal the element of the next hierarchy level consists of 8 elements of the previous hierarchy level and the similarity coefficient is 0.5. In this case, the fractal dimension is defined as df = lnV / ln2, where V is the volume expressed in the number of elements obtained when the linear dimension of the structure is doubled. In the case of the cubic self-similar structure the volume V can be considered as V = 23Vr, where Vr is the average relative intact volume. For a intact material the fractal dimension is df = 3, since the relative intact volume is Vr = 1.
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Journal Pre-proof In the present study the average relative intact volume Vr(t) was calculated as the volume averaged over all hierarchy levels.
Vr (t )
1 N Vi (t ) 8 i N 1 i 0
At the stage preceding the failure (t = 20000 s), the average relative intact volume is equal to Vr = 0.847. At the moment of failure (t = 49000 s) the average relative intact volume is Vr = 0.493. For these instants of time, the fractal dimension df takes values of 2.76 and 1.98 respectively. These values are in satisfactory agreement with the experiment. In this paper the estimation for the coefficient b of the earthquake repeatability law was carried out using the energy release sequence W (t) shown in Figure 5. For this purpose the energy release range was divided into equal intervals. After that the number of W values corresponding to each i-th interval were counted as mi. Further the repeatability dependence was calculated as
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()
( )
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∑
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The dependence lg N as function of M calculated in this way, where M = lg W is shown in Figure 11.
Figure 11. The dependence lg N as function of M, where M = lg W.
The straight line lg N = - bM + const is plotted using the value b = 0.9.
The straight line lg N = - bM + const in Figure 11 is plotted using the value b = 0.9, which indicates a good agreement between the calculated data and the experimental results. Satisfactory coincidence of the results of calculations and experimental data allows concluding the fracture process can be considered as a process of the hierarchical structure failure. On the other hand the proposed model can be used for description of damage evolution for different rock materials as different the hierarchical structures .
4. EFFECT OF TEMPERATURE (BRITTLENESS) ON THE RUPTURE DYNAMICS Of interest is the behavior of damage evolution for the hierarchical structures with different the structure types at different temperatures. Also in present study the damage evolution calculations were made for granite at temperature of T = 500°C.
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Journal Pre-proof The dependencies for the stress (t) for this case is depicted in Figure 12. Figure 13 depicts the fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels (T = 500°C). The curve number i corresponds to the hierarchy level. The dependence lg N as function of M (curve 1) calculated for T = 500°C is shown in Figure 14. The straight line
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(curve 3) in Figure 14 is plotted using the value b = 0.9 as in Figure 11.
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Figure 12. The time dependence for the stress (t) the granite specimen (T = 500°C) .
Figure 13. The fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels (T = 500°C). The curve number i corresponds to the hierarchy level.
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Figure 14. The dependences lg N as function of M, where M = lg W. 1 – at T = 500°C, 2 – at T = 20°C, 3, 4 - straight lines lg N = - bM + const are plotted using the value b = 0.9,
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5 - straight line is plotted using the value b = 0.65 in range of M about from -7 to -6.
Figure 15.The probability density function for fragment size G(x) at T = 500° defined as log-normal approximation of discrete values G( 2i-N 200).
In Figure 12 it can be seen that at temperature T = 500°C the creep time (the time interval from the onset of fracture to the maximum elastic energy release) and maximum stress in the specimen were significantly reduced in compare with calculations at temperature T = 20°C (Figure 2). This fact is associated with behavior change of seven initial hierarchy levels (Figure 13). The behavior of senior levels that determine the rupture nature does not change. Therefore the form of dependences lg N of M in Figure 14 (curves 1 and 2) are practically the same (straight lines correspond to b = 0.9). As the temperature rises, the materials become less brittle. The effect of reducing brittleness with increasing temperature is manifested in an increase in the average size of fragments of the destroyed specimen. Figure depicts 15 the probability
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Journal Pre-proof density function for fragment size G(x) corresponding to T = 500 C°. The average fragment size at a temperature T = 20° was 0.51 mm and at a temperature T = 500° it was 1.03 mm. The rupture dynamics for rocks with different brittleness is compared in the paper (Nejati and Ghazvinian 2014). As a result of the experiments it was shown by authors of this paper that more brittle rocks produce a greater number of acoustic signals during fracture (a greater number of cracks). Also in this paper, the ambiguous behavior of the slope coefficient b was obtained depending on the brittleness of the rock. This behavior of the slope is explained by the presence of a threshold in measuring the amplitudes of the acoustic emission signals. The figure 14 shows that for the range of M about from -7 to -6, the two curves have different values of b. In this case a smaller value of b corresponds to a more brittle material. If we consider an even less fragile material then the curve for it will pass below curve 1. Then a sharply falling part of the curve corresponding to the least fragile material will fall into this range of M value. In this case, the largest value of slope B will correspond to the least brittle material.
5. EFFECT OF HIERARCHICAL STRUCTURE TYPE
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The fact that at the instant of failure, the concentration parameter K takes on the critical value KCR = 2.7 – 3.0 being the same almost for all materials whether with or with no lattice is likely because crystal structure disturbances (defects,
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impurities) are accumulated with increasing the number of hierarchy levels such that material loses its “memory” about the structure at the lower levels. However the structure of material at low hierarchy levels influences its characteristics.
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Figures 15, 16 and 17 present calculation data for hypothetical material which structure has quasi-flat lattice ( 4 elements) of hierarchy levels (0 – 4) and cubic lattice at the rest levels ( 8 elements). For instance mica has such quasi-flat
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structure. So that is beginning with 5-th level, the structure loses its ordering. It is typical situation for polycrystalline
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materials.
Figure 16. The top part of the time dependence for the stress (t) (hypothetical material). In the calculations the parameters U and γ remained the same at all hierarchy levels as in previous case for the granite specimen at T = 20°C. It is seen from Figure 16 that 5 first hierarchy levels influence on the strength properties of material (maximal value of (t) and creep time slightly increased). The maximal value of (t) corresponds to the beginning of the fracture process and to the destruction of more durable lower structure levels (each such structure level has 4 elements and its destruction probability equals to Pcr = 0.232). Figure 17 depicts the more intensive destruction of low hierarchy levels (0 – 4) in compare with Figure 6. This fact is well illustrated by Figure 18. The time dependence of elastic energy release W(t) on an enlarged scale for hypothetical material . The release of elastic energy W(t) on an enlarged scale for a hypothetical material is more at the beginning of the destruction process than in Figure 5. However the behavior of senior
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Figure 11. The straight line in Figure 19 corresponds to b = 0.9.
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Figure 17. The fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels (hypothetical material).
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The curve number i corresponds to the hierarchy level.
Figure 18. The time dependence of elastic energy release W(t) on an enlarged scale for hypothetical material.
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Figure 19. The dependence lg N as function of M, where M = lg W.
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The straight line lg N = - bM + const is plotted using the value b = 0.9 (hypothetical material).
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6. EFFECT OF MATERIAL TYPE
Also in this study calculations of the fracture dynamics were carried out for another rock material. Marble was chosen as
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such a rock ( U = 164 KJ/mol, γ = 1.16 KJ/(mol·MPa) ). The results of calculations as time dependences for the stress (t) and relative intact volumes for fracture of marble at a temperature of 20°C are shown on Figures 20 and 21
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correspondently. From the figures it is clear that these dependences have their own characteristics in comparison with granite. It should be noted that the shape of the curve lgN as function of M remains unchanged in case of marble too (curve 1 in Figure 22).
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Figure 20. The top part of the time dependence for the stress (t) (marble).
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Figure 21. The fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels (marble). The curve
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number i corresponds to the hierarchy level.
Figure 22. The dependence lg N as function of M, where M = lg W (marble). 1 – stress rate 10-3 MPa·s-1, 2 – stress rate 10-2 MPa·s-1. The straight line lg N = - bM + const is plotted using the value b = 0.9.
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Of interest is the behavior of the rock material at different stress rates. In all previous calculations the stress rate was -3
10 MPa·s-1. In this study calculations were made of the rapture dynamics for marble at temperature T = 20 C° for various stress rates. Figures 23 and 24 show the time dependences of stress (t) and elastic energy release W(t) respectively for
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stress rate of 10-2 MPa·s-1. These dependences demonstrate a series of aftershocks in case of higher stress rate.
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Figure 23. The top part of the time dependence for the stress (t) (marble, stress rate of 10-2 MPa·s-1).
Figure 24. The time dependence of elastic energy release W(t) (marble, stress rate of 10-2 MPa·s-1).
The fracture dynamics of relative intact volumes for different hierarchy levels (Figure 25) in case of stress rate of 10-2 MPa·s-1 is significantly different from the fracture dynamics of relative intact volumes at stress rate of 10-3 MPa·s-1 (Figure 21).
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Figure 25. The fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels
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(marble, stress rate of 10-2 MPa·s-1). The curve number i corresponds to the hierarchy level.
It is seen that the hierarchical structure does not have time to rebuild under the action of a rapidly increasing load (the
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kinetic process is rather inert) and the failure of the structure levels continues after the moment of the main energy release. Thus a relatively short creep time and the presence of a larger number of aftershocks are characteristic of higher stress
(curve 2 in Figure 22).
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rates. As for the dependence lg N as function of M in case of stress rate of 10-2 MPa·s-1 the curve keeps the same form
In order to compare the behavior of the fractal dimension in the process of rock destruction, in each case the creep time
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(the time interval from the onset of fracture to the maximum elastic energy release) was divided into 4 segments. Calculations of the fractal dimension were made at the points corresponding to the first, second and third ends of the segments or to the relative parts of creep time tc. Figure 26 depicts the results of calculations of the fractal dimension for different tress rates as function of the relative part of the creep time tc. It should be noted that the results of calculations of the fractal dimension for all the considered cases at stress rate of 10-2 MPa·s-1 correspond to the same curve 1 regardless of temperature, type of the hierarchical structure and type of rock.
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Figure 26. The fractal dimension for different stress rates as function of the relative part of the creep time tc :
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1 - 10-3 MPa·s-1, 2 - 3·10-3 MPa·s-1, 3 - 10-2 MPa·s-1, 4 - 10-1 MPa·s-1.
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CONCLUSION
In the present study the model is proposed for describing the rock fracture dynamics. The results of calculations of the fracture dynamics were compared with experimental data of the granite specimen failure. Satisfactory coincidence of
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the results of calculations and experimental data allows concluding the rock fracture process can be considered as a failure process of the hierarchical structure. The calculation results of the dynamics of the crack size distribution and the size
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distribution of fragments as a result of the specimen fracture are presented. The influence of temperature, the type of hierarchical structure, the type of rock and the stress rate on the fracture dynamics is considered. It is shown that the dynamics of the fractal dimension in the process of sample destruction significantly depends only on the stress rate and does not depend on temperature change and type of material and type of hierarchical structure. It is also revealed that the shape of the curve lg N as function of M is not affected by all the above factors including the stress rate (coefficient b ≈ 9 in all cases) . Perhaps this fact explains the spatial and temporal stability of the Gutenberg-Richter law.
Conflict of Interest: Author S. N. Fedotov declares that he has no conflict of interest. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
REFERENCES
Allegre, C.J., Mouel, J.L., Provos,t A., 1982. Scaling rules in rock fracture and possible implications for earthquake prediction. Nature. 297, 47- 49. https://doi.org/10.1038/297047a0. Allegre, C.J., Mouel, J.L., 1994. Introduction of scaling techniques in brittle fracture of rocks. Physics of Earth and Planetary Interiors. 87, 85- 93. https://doi.org/10.1016/0031-9201(94)90023-X.
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Journal Pre-proof Allegre, C.J., Mouel, J.L. , Chau, H., Narteau, C., 1995. Scaling organization of fracture tectonics (SOFT) fracture mechanism. Physics of Earth and Planetary Interiors. 92, 215-233. https://doi.org/10.1016/0031-9201(95)03033-0. Chmel ,A., Kuksenko, V.S., Smirnov, V.S., Tomilin, N.G., 2007. Anomalies of critical state in fracturing geophysical objects. Nonlin. Processes Geophys. 14, 103–108. Feder, J., 1987. Fractals. Plenum Press, London. Fedotov, S.N., 2017. Quasi-brittle fracture as failure of hierarchical structure. Physical mesomechanic. 20 (2), 222 – 228. https://doi.org/10.1134/S1029959917020126 Gutenberg, B., Richter C., 1954. Seismicity of the earth and associated phenomena. Princeton Univ. Press, Princeton. Karlin, S., 1966. A first course in stochastic processes. Academic Press, New York. Kuksenko, V., Tomilin, N., Damaskinskaya, E., Lockner, D., 1996. A two-stage model of fracture of rocks..Pure Appl. Geophys. 146 (2), 253 - 258. https://doi.org/10.1007/BF00876492.
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Lei, X., Nishizawa, O., Moura, A., Saton, T., 2005. Hierarchical fracture process in brittle rocks by means of hight-speed monitoring of AE hipocentre. Journal of AE. 23, 102-108.
mesomechanics. 13 (5-6), 292-305.
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Makarov, P.V., 2010. Self-organized criticality of deformation and prospects for fracture prediction. Physical https://doi.org/10.1016/j.physme.2010.11.010.
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Nejati, H.R., Ghazvinian, A., 2014. Brittleness Effect on Rock Fatigue Damage Evolution. Rock Mech Rock Eng. 47, 1839–1848. https://doi: 10.1007/s00603-013-0486-4
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Ponomarev, A.V., Zavyalov, A.D., Smirnov, V.B., Lockner, D.A., 1997. Physical modeling of the formation and evolution of seismically active fault zones. Tectonophysics .277. 57- 81. https://doi.org/10.1016/S0040-1951(97)00078-4.
17-27.
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Shamina, O.G., Strizhko,,S.A., 1975. Fracture interaction in samples at pressure, Izv. Akad. Nauk. SSSR. Fiz. Zemli. 9,
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Weiss, J,, 2003. Scaling of fracture and faulting of ice on Earth. Survey in Geophysics. 24,185-227. https://doi: 10.1023/A:1023293117309 Yukalov, V.I., Moura, A., Nechad, H., 2004. Self-similar law of energy release before materials fracture. Journal of the
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Mechanics and Physics of Solids. 52( 2), 453-465. https://doi: 10.1016/S0022-5096(03)00088-7. Zhou, X.P., Yang, H.Q., 2004. Analysis of the localization of deformation and the complete stress-strain relation for mesoscopic heterogeneous brittle rocks under dynamic uniaxial tensile loading. Int. Journal of the Solids and Structures, 41, 1725-1738. https://doi: 10.1016/j.ijsolstr.2003.07.007. Zhou, X.P., Yang, H.Q., 2018. Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach. Theoretical and Applied Fracture Mechanics, 97, 289-302. https://doi.org/10.1016/j.tafmec.2017.05.006. Zurkov, S.N., 1965. Kinetic Concept of the Strength of Solids. Int. J. Fracture Mech. 1, 311–323.
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Journal Pre-proof Highlights
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The rock fracture process can be considered as a failure process of the hierarchical structure. The dynamics of the fractal dimension in the process of sample destruction significantly depends only on the stress rate and does not depend on temperature change and type of material and type of hierarchical structure. The fractal dimension in the process of sample destruction significantly depends only on the stress rate and does not depend on temperature change and type of material and type of hierarchical structure.
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S.N. Fedotov PII:
S0031-9201(19)30168-2
DOI:
https://doi.org/10.1016/j.pepi.2020.106449
Reference:
PEPI 106449
To appear in:
Physics of the Earth and Planetary Interiors
Received date:
18 June 2019
Revised date:
14 September 2019
Accepted date:
14 February 2020
Please cite this article as: S.N. Fedotov, Rock damage evolution as failure process of hierarchical structure, Physics of the Earth and Planetary Interiors(2018), https://doi.org/ 10.1016/j.pepi.2020.106449
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© 2018 Published by Elsevier.
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Rock damage evolution as failure process of hierarchical structure S. N. Fedotov* National Research Nuclear University (MEPhI), Moscow, 115409 Russia
Sechenov First Moscow State Medical University, Moscow 119991, Russia *e-mail: [email protected] Abstract
In present study the model is proposed to describe the dynamics of the rock fracture process. Rock formation is assumed
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in the form of a nested self-similar hierarchical structure. The hierarchical structure scheme is chosen in accordance with the experimentally verified criterion of fracture. The failure probability of each structure level is calculated as a function of
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time, temperature, material parameters and loading conditions. Knowing the failure probabilities allows one to calculate the stress dynamics in a rock material, failure volumes for each hierarchy level, energy release, etc. The influence of
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temperature, type of the hierarchical structure, rock material, stress rate on the fracture dynamics is considered. The results
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of calculations of the rock fracture dynamics are compared with the results of experiments.
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1. INTRODUCTION
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Key words: acoustic emission, fracture dynamics, fractals, hierarchical structure.
The analysis of numerous data accumulated over a long period of observations of earthquakes indicates the spatial and temporal stability of the Gutenberg-Richter law (Gutenberg and Richter, 1954). In this case, the distribution of earthquake
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magnitude obeys an apparently universal scaling law despite the different types of movements, seismic activity levels, despite the different deformation rates, chemical and physical characteristics of the medium. In addition, the distribution of aftershocks does not differ from that of ordinary background earthquakes. In the opinion of many researchers, this stability of the earthquake repeatability law is due to the general fundamental laws of the structure of the medium (Makarov, 2010; Weiss, 2003; Yukalov et al. 2004) and to a single mechanism of fracture. The most common regularity in the fracture of solids is the concentration criterion established in the course of experimental studies (Kuksenko et al. 1996). According to this criterion, the concentration parameter K = R / L (L is the length of cracks formed, R is the distance between cracks of this length) at the instant of failure of solids takes the critical value Kcr = 2.7-3 in a wide range of crack linear sizes, regardless of the type of materials and load regimes. This behavior of solids indicates that materials can be considered as self-similar hierarchical structures. This conclusion is based both on the results of long-term studies of the rocks fracture and on the results of earthquake studies. There are some papers that considered the rock material in the form of self-similar hierarchical structures (Lei et al. 2005; Chmel et al. 2007). However, the models described in these papers are not based on experimentally verified criteria for the material destruction. This paper proposes a model for describing the rock destruction dynamics. This model is based on the kinetic approach. The materials themselves are considered in the form of self-similar hierarchical structures. The model uses the regularities
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2. MODEL AND CALCULATIONS
For simplicity the structure in the form of a cube is considered as the simplest three-dimensional nested hierarchical selfsimilar structure. The next level of the hierarchy of such a structure occurs when the length of each of the cube ribs of the previous hierarchy level is doubled (in Figure 1 three levels of the hierarchy of such a structure are depicted). For failure of the cube consisting of 8 cubes of the previous hierarchy level (a crack must pass through the entire volume of the cube connecting opposite faces), it is necessary that at least two cubes of eight are destroyed. This rule is true for any type of crystal lattice (we get 8 identical polyhedral similar to the original polyhedron (8 identical volumes) by doubling the ribs of
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any type of polyhedron).
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Fig. 1. Three hierarchy levels of self-similar structure.
Let the self-similar structure contains N levels of the hierarchy. Then according to the concentration criterion at the instant
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of failure the equality Ri/Li = Кcr ( i = 0, 1, ... N -1) will be valid for each i-th level of the hierarchy. Considering Ri3 as the material volume per crack of length Li and a Li3 as the volume fractured by this crack, the ratio Li3/ Ri3 can be considered as a relative fractured part of volume Ri3. Also the ratio Li3/ Ri3 can be treated as the failure probability Pi for a volume of the i-th hierarchy level. In this case, the concentration criterion takes the following form: at the instant of failure the failure probabilities Pi are equal to the same value Рcr = Li3/ Ri3 = Кcr-3 for all hierarchy levels. Let the failure probability of each of the eight cubes for the lowest level of the hierarchy be independent and equal to the same value P. Then the failure probability F(P) of the cube (the next level of the hierarchy) will be described by the binomial distribution as the failure probability for two or more cubes of eight F(Р) = 1 - (1 - Р)8 – 8P(1 - Р)7. For a self-similar structure hierarchies of all levels behave in a similar manner. Therefore at time t the failure probability Pi(t) of the i-th level of the hierarchy will be completely determined by the failure probability Рi-1(t) of the previous (i-1) -th level of the hierarchy. So the following expression Рi(t) = F( Рi-1(t)), will be valid for all levels of the hierarchy, where F(P) is the function describing the hierarchical scheme of the structure. In this case at the instant of failure the equation Pcr = F(Pcr) will be valid. The root of this equation is the value Рcr = Кcr-3. Thus within the framework of this model, the fracture process is a branching random process and the value of P = Pcr is the critical (degenerate) probability of this process (Karlin, 1966). The similar approach was used in papers devoted to the rock failure during earthquakes (Allegre et al. 1982; Allegre and Mouel, 1994; Allegre e. al. 1995).
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Journal Pre-proof However the models described in these papers are not based on experimentally verified criteria for the material destruction. For a given hierarchical structure the critical probability (the root of the equation F(P) = P) will be Pcr = 0.0423 with the concentration criterion taking the value К = (Рcr)-1/3 = 2.87. This value is in good agreement with the results of many experiments (Kuksenko et al. 1996). In the two-dimensional case, the structure will take the form of embedded squares (the face of the cube in Figure 1). When a similar two-dimensional structure (a film, a thin sheet material) is the hierarchical scheme of the structure will be described by the expression F(Р) = 1 - (1 - Р)4 – 4Р(1 - Р)3. In addition the critical probability will be Pcr = 0.232 and the value of the concentration criterion will be К = (Рcr)-1/2 = 2.08 (in case of the stretching F(Р) = 1 - (1 - Р)4 – 4Р(1 - Р)3 - 2Р2(1 - Р)2, Pcr = 0.382 and K = 1.62). Such values of K agree well with experiment (Shamina and Strizhkov, 1975). Thus the value of the concentration criterion depends only on the dimensionality of the space and to a much lesser extent
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on the loading regime. The three-dimensional structures (Pcr ≈ 0.05) are the least durable, and the most stable ones are one-dimensional (Pcr = 1). So the higher the dimensionality of space, the lower the value of Pcr because the more
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possibilities (variants) for cracking.
So the material fracture process (hierarchical structure levels destroying) depends on the hierarchical scheme of the
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structure only. Physical properties of material are defined on the lowest hierarchical level of structure. A formula for the
( )
(
(
)(
(
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destruction probability P0 (t) of the lowest level was obtained in (Fedotov, 2017) using the kinetic approach:
)
(
)))
,
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where τ is the average time the element of the structure is in the non-destroyed state, and z is the average time the element is in the destroyed state. In addition τ = (28/ Pcr)0.5 · τ0 ∙ exp ((U - γσ) / kT) , where U is the activation energy of the
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destruction process, k is the Boltzmann constant, γ is a parameter characterizing the actual mechanical properties of the material, T is the temperature (K °), σ is the mechanical stress. The coefficient τ0 coincides in order of magnitude with the thermal oscillations period of atoms in a solid. Knowledge of the hierarchical scheme and the failure probability of the
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zero level of the hierarchy make it possible to calculate the failure probability for each level of the hierarchy. Consider the simplest case which, if necessary, can be added and complicated. Suppose that the material with volume V and Young's modulus S is under the mechanical stress σ. In addition, the mechanical energy
E
2V 2S
(1)
will be accumulated in the volume V. In this case the energy gain ΔE in a time t will cause the stress change Δσ and the volume failure ΔV :
E
S
V
2 2S
V
(2)
Let us determine j as j = (tj) for discrete times tj = t0 + jt ( j = 0, 1, 2, …). The change in the failure probability of the i-th hierarchy level at the j-th step (on a time interval t ) will be
Pi(t j , j , T) = Pi(t j+1 , j+1 ,T) - Pi(t j , j ,T). In this connection the change in the volume of the i-th hierarchy level Vi over the time interval t will be described by the following formula
(Vi ) j 1 (Vi ) j (1 Pi (t j , j , T ))
(3)
Knowing of the failure probability distribution for all levels of the hierarchy makes it possible to calculate the average failure volume Vav over the time interval t:
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1 N Vi 8 N i Pi (t j , j , T ) , N 1 i 0
(4)
where N is the number of the highest level of the hierarchy, Vi is the current volume of the i-th level of the hierarchy (initial volume Vi =8i). In this case, the average intact volume at the j-th step Vavj equals to
Vav j Assuming j = j+1
1 N (Vi ) j 8 N i . N 1 i 0
(5)
- j and using (2), the stress is changed at the j-th step as
j 1 j
SE j j
2
2 Vav j
(6)
jVav j
When partially failed, the structure resists fracturing and strives for the stress relief. The release of elastic energy Wj on a
j S
Vav j j
(
)
( )
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)
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The average failure volume Dj
∑(
(7)
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Wj
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time interval t is
The obtained dependences make it possible to calculate the stress (t), the average failure volume
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D(t) and the elastic energy release W(t) both for an arbitrary dependence E(t) and for a specified external load as functions of time.
In this study the calculation of the granite fracture dynamics was carried out according to the described method. It is
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supposed that before loading rock material is considered as an ideal intact material with no any defects (ideal structure). Lattice defects occur due to thermal-fluctuation mechanism. The results of calculations for fracture of granite at a
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temperature of 20°C are shown on Figures 2, 3 and 4 in case of an axially compressed specimen. Calculations were performed for the structure containing N = 20 levels of the hierarchy. For granite under compressed load the following parameters were used U = 152 KJ/mol and γ = 0.453 KJ/(mol·MPa). It was supposed that granite specimen had the following sizes 100×100×200 mm3.
Figure 2. The time dependence for the stress (t) the granite specimen (T = 20°C) .
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Figure 3. The top part of the time dependence for the stress (t) that depicted in Figure2.
Figure 4. The time dependences for the elastic energy release W(t) - (1), the average failure volume of specimen D(t) - (2), the creep strain L(t) (nonlinear deformation (mm) ) – (3),
The average failure volume of specimen D(t) is presented in relative units. Here it was assumed that the energy gain is a constant dE/dt = const. Prior to the onset of fracture, until critical parameters are exceeded the specimen behaves like an elastic medium
(t ) const E (t ) . Further the stress does not increase and the energy gain is consumed in fracturing
the volume. Figures 2 and 3 demonstrate that for a self-similar hierarchical structure the fracture process is similar to a random process, though random numbers are not used in the calculations. The time interval from the onset of fracture to the maximum elastic energy release (creep mode) depends both on the rate of energy gain (or a stress rate) as well as on the temperature and characteristics of the specimen material. Up to the maximum elastic energy release there are observed a number of small releases called foreshocks in seismology (Figure 4 and Figure 5). After the maximum energy release a significant part of the volume storing the elastic energy is fractured and subsequent releases (aftershocks) become substantially smaller in magnitude (Figure 4). The creep strain L(t) (nonlinear deformation in millimeters) depicted in Figure 4 (curve 3) was calculated proportionally cubic root from average failure volume of specimen D(t). From Figure 5
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Journal Pre-proof it can be seen the time dependence of elastic energy release W(t) at the initial fracture stage corresponding to the data in Figure 4 on an enlarged scale. Evidently the sequence of the elastic energy releases is similar to the acoustic emission
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signals.
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Figure 5. The time dependence of elastic energy release W(t) on an enlarged scale. The fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels is shown in Figure 6. The curve
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number corresponds to i level of hierarchy. At relatively low levels of hierarchy there is slow failure of relative intact volumes. These levels are destroyed first. The fracture of the volume for the highest hierarchy level observed in the experiment fails instantaneously without any visible preparation. When the volume of the highest hierarchy level breaks,
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the crack passes through the entire specimen.
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Journal Pre-proof Figure 6. The fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels. The curve number i corresponds to the hierarchy level.
The behavior of the different hierarchy levels in the fracture process is of interest. The state of the relative failure volumes di(t) = 1 - Vi(t)/8 i is shown in Figure 7 for each level of the hierarchy i at different fracture stages (at different
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moment of time). All dependencies are in accordance with the data in Figure 6.
Figure 7. Relative failure volumes di(t) for each hierarchy level i at different fracture stages.
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Dependence 1 corresponds to the beginning of the fracture process (t = 10000 s), dependence 2 - failure of low hierarchy levels (t = 20000 s). Dependence 3 corresponds to the time before the failure of the highest hierarchy level (t = 50000 s). Dependence 4 corresponds to the time immediately after the failure of the highest hierarchy level (t = 55000 s).
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In fact this dependence determines the dispersed composition of fragments after the rock failure. At the beginning the fracture process the i level of hierarchy contains 8N-i intact volumes expressed in the number of elements V(0)i = 8i . In the fracture process the ratio Vi(t)/8 i can be considered as a relative part of intact volumes for the i level of hierarchy. So the number of intact volumes for the i level of hierarchy equals to Vi(t) 8N-2i . The linear size of the element for the i-th hierarchy level equals to xi = 2i-N 200 mm. It is not difficult to calculate the probability density function for fragment size G(xi) = Vi(55000) 8N-2iC, where C is normalizing constant. Figure 8 depicts the probability density function for fragment size G(x) defined as log-normal approximation of discrete values G( 2i-N 200).
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Figure 8.The probability density function for fragment size G(x) defined as log-normal approximation
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of discrete values G( 2i-N 200).
As a rule, the probability density function for crack sizes is postulated ( usually the Weibull distribution) in papers devoted
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to materials fracture processes ( X.P. Zhou, H.Q. Yang, 2004; X.P. Zhou, H.Q. Yang, , 2018). Knowledge of the behavior of the failure volumes in time di(t) make it possible to calculate the probability density function
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for crack size P(t, x). It is assumed that the crack size of the i-th hierarchy level is proportional to the cube root of the failure volume of this hierarchy. The probability density function for crack size P(t, x) is shown on Figure 9 for three
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moments of time t = 20000 s, t = 35000 s and t = 50000 s.
Figure 9. The probability density function P(t, x) for crack size x (mm) for different moments of time t. 1 – P(20000, x), 2 – P(35000, x), 3 - P(50000, x).
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( )
The length of main crack
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in time L(t) is depicted in Figure 10.
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Figure 10. The length of main crack in time L(t).
3. COMPARISON WITH EXPERIMENT
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To verify the general mechanism of rock destruction the calculated data were compared with the results of experiments on the granite specimens failure described in (Ponomarev et al. 1997). In this study the acoustic emission signals were recorded during the experiments.
As a result of signal processing in two experiments, the fractal dimensionality df of the material to be failed was determined both at the stage preceding the fracture and at the moment of fracture. At the stage preceding the fracture the fractal dimension was determined as df1 = 2.86 ± 0.07 and df2 = 2.80 ± 0.02. At the moment of fracture, the following values for the fractal dimension were obtained the following values df1 = 2.4 ± 0.4 and df2 = 2.23 ± 0.05. In addition, in this study the coefficient b was estimated in the formula describing the Gutenberg-Richter's law lg N = - bM + const . The value of M was determined as the energy class of the earthquake M = lg E + const, where E is the earthquake energy. For the coefficient b were obtained the estimates b1 = 0.95 and b2 = 0.91 . In the present paper a self-similar cubic structure is considered as a regular fractal (Feder, 1987). For this fractal the element of the next hierarchy level consists of 8 elements of the previous hierarchy level and the similarity coefficient is 0.5. In this case, the fractal dimension is defined as df = lnV / ln2, where V is the volume expressed in the number of elements obtained when the linear dimension of the structure is doubled. In the case of the cubic self-similar structure the volume V can be considered as V = 23Vr, where Vr is the average relative intact volume. For a intact material the fractal dimension is df = 3, since the relative intact volume is Vr = 1.
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Journal Pre-proof In the present study the average relative intact volume Vr(t) was calculated as the volume averaged over all hierarchy levels.
Vr (t )
1 N Vi (t ) 8 i N 1 i 0
At the stage preceding the failure (t = 20000 s), the average relative intact volume is equal to Vr = 0.847. At the moment of failure (t = 49000 s) the average relative intact volume is Vr = 0.493. For these instants of time, the fractal dimension df takes values of 2.76 and 1.98 respectively. These values are in satisfactory agreement with the experiment. In this paper the estimation for the coefficient b of the earthquake repeatability law was carried out using the energy release sequence W (t) shown in Figure 5. For this purpose the energy release range was divided into equal intervals. After that the number of W values corresponding to each i-th interval were counted as mi. Further the repeatability dependence was calculated as
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()
( )
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∑
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The dependence lg N as function of M calculated in this way, where M = lg W is shown in Figure 11.
Figure 11. The dependence lg N as function of M, where M = lg W.
The straight line lg N = - bM + const is plotted using the value b = 0.9.
The straight line lg N = - bM + const in Figure 11 is plotted using the value b = 0.9, which indicates a good agreement between the calculated data and the experimental results. Satisfactory coincidence of the results of calculations and experimental data allows concluding the fracture process can be considered as a process of the hierarchical structure failure. On the other hand the proposed model can be used for description of damage evolution for different rock materials as different the hierarchical structures .
4. EFFECT OF TEMPERATURE (BRITTLENESS) ON THE RUPTURE DYNAMICS Of interest is the behavior of damage evolution for the hierarchical structures with different the structure types at different temperatures. Also in present study the damage evolution calculations were made for granite at temperature of T = 500°C.
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Journal Pre-proof The dependencies for the stress (t) for this case is depicted in Figure 12. Figure 13 depicts the fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels (T = 500°C). The curve number i corresponds to the hierarchy level. The dependence lg N as function of M (curve 1) calculated for T = 500°C is shown in Figure 14. The straight line
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(curve 3) in Figure 14 is plotted using the value b = 0.9 as in Figure 11.
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Figure 12. The time dependence for the stress (t) the granite specimen (T = 500°C) .
Figure 13. The fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels (T = 500°C). The curve number i corresponds to the hierarchy level.
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Figure 14. The dependences lg N as function of M, where M = lg W. 1 – at T = 500°C, 2 – at T = 20°C, 3, 4 - straight lines lg N = - bM + const are plotted using the value b = 0.9,
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5 - straight line is plotted using the value b = 0.65 in range of M about from -7 to -6.
Figure 15.The probability density function for fragment size G(x) at T = 500° defined as log-normal approximation of discrete values G( 2i-N 200).
In Figure 12 it can be seen that at temperature T = 500°C the creep time (the time interval from the onset of fracture to the maximum elastic energy release) and maximum stress in the specimen were significantly reduced in compare with calculations at temperature T = 20°C (Figure 2). This fact is associated with behavior change of seven initial hierarchy levels (Figure 13). The behavior of senior levels that determine the rupture nature does not change. Therefore the form of dependences lg N of M in Figure 14 (curves 1 and 2) are practically the same (straight lines correspond to b = 0.9). As the temperature rises, the materials become less brittle. The effect of reducing brittleness with increasing temperature is manifested in an increase in the average size of fragments of the destroyed specimen. Figure depicts 15 the probability
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Journal Pre-proof density function for fragment size G(x) corresponding to T = 500 C°. The average fragment size at a temperature T = 20° was 0.51 mm and at a temperature T = 500° it was 1.03 mm. The rupture dynamics for rocks with different brittleness is compared in the paper (Nejati and Ghazvinian 2014). As a result of the experiments it was shown by authors of this paper that more brittle rocks produce a greater number of acoustic signals during fracture (a greater number of cracks). Also in this paper, the ambiguous behavior of the slope coefficient b was obtained depending on the brittleness of the rock. This behavior of the slope is explained by the presence of a threshold in measuring the amplitudes of the acoustic emission signals. The figure 14 shows that for the range of M about from -7 to -6, the two curves have different values of b. In this case a smaller value of b corresponds to a more brittle material. If we consider an even less fragile material then the curve for it will pass below curve 1. Then a sharply falling part of the curve corresponding to the least fragile material will fall into this range of M value. In this case, the largest value of slope B will correspond to the least brittle material.
5. EFFECT OF HIERARCHICAL STRUCTURE TYPE
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The fact that at the instant of failure, the concentration parameter K takes on the critical value KCR = 2.7 – 3.0 being the same almost for all materials whether with or with no lattice is likely because crystal structure disturbances (defects,
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impurities) are accumulated with increasing the number of hierarchy levels such that material loses its “memory” about the structure at the lower levels. However the structure of material at low hierarchy levels influences its characteristics.
at 5 first
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Figures 15, 16 and 17 present calculation data for hypothetical material which structure has quasi-flat lattice ( 4 elements) of hierarchy levels (0 – 4) and cubic lattice at the rest levels ( 8 elements). For instance mica has such quasi-flat
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structure. So that is beginning with 5-th level, the structure loses its ordering. It is typical situation for polycrystalline
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materials.
Figure 16. The top part of the time dependence for the stress (t) (hypothetical material). In the calculations the parameters U and γ remained the same at all hierarchy levels as in previous case for the granite specimen at T = 20°C. It is seen from Figure 16 that 5 first hierarchy levels influence on the strength properties of material (maximal value of (t) and creep time slightly increased). The maximal value of (t) corresponds to the beginning of the fracture process and to the destruction of more durable lower structure levels (each such structure level has 4 elements and its destruction probability equals to Pcr = 0.232). Figure 17 depicts the more intensive destruction of low hierarchy levels (0 – 4) in compare with Figure 6. This fact is well illustrated by Figure 18. The time dependence of elastic energy release W(t) on an enlarged scale for hypothetical material . The release of elastic energy W(t) on an enlarged scale for a hypothetical material is more at the beginning of the destruction process than in Figure 5. However the behavior of senior
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Figure 11. The straight line in Figure 19 corresponds to b = 0.9.
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Figure 17. The fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels (hypothetical material).
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The curve number i corresponds to the hierarchy level.
Figure 18. The time dependence of elastic energy release W(t) on an enlarged scale for hypothetical material.
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Figure 19. The dependence lg N as function of M, where M = lg W.
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The straight line lg N = - bM + const is plotted using the value b = 0.9 (hypothetical material).
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6. EFFECT OF MATERIAL TYPE
Also in this study calculations of the fracture dynamics were carried out for another rock material. Marble was chosen as
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such a rock ( U = 164 KJ/mol, γ = 1.16 KJ/(mol·MPa) ). The results of calculations as time dependences for the stress (t) and relative intact volumes for fracture of marble at a temperature of 20°C are shown on Figures 20 and 21
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correspondently. From the figures it is clear that these dependences have their own characteristics in comparison with granite. It should be noted that the shape of the curve lgN as function of M remains unchanged in case of marble too (curve 1 in Figure 22).
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Figure 20. The top part of the time dependence for the stress (t) (marble).
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Figure 21. The fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels (marble). The curve
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number i corresponds to the hierarchy level.
Figure 22. The dependence lg N as function of M, where M = lg W (marble). 1 – stress rate 10-3 MPa·s-1, 2 – stress rate 10-2 MPa·s-1. The straight line lg N = - bM + const is plotted using the value b = 0.9.
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Journal Pre-proof 6. EFFECT OF STRESS RATE
Of interest is the behavior of the rock material at different stress rates. In all previous calculations the stress rate was -3
10 MPa·s-1. In this study calculations were made of the rapture dynamics for marble at temperature T = 20 C° for various stress rates. Figures 23 and 24 show the time dependences of stress (t) and elastic energy release W(t) respectively for
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stress rate of 10-2 MPa·s-1. These dependences demonstrate a series of aftershocks in case of higher stress rate.
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Figure 23. The top part of the time dependence for the stress (t) (marble, stress rate of 10-2 MPa·s-1).
Figure 24. The time dependence of elastic energy release W(t) (marble, stress rate of 10-2 MPa·s-1).
The fracture dynamics of relative intact volumes for different hierarchy levels (Figure 25) in case of stress rate of 10-2 MPa·s-1 is significantly different from the fracture dynamics of relative intact volumes at stress rate of 10-3 MPa·s-1 (Figure 21).
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Figure 25. The fracture dynamics of relative intact volumes Vi(t)/8 i for different hierarchy levels
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(marble, stress rate of 10-2 MPa·s-1). The curve number i corresponds to the hierarchy level.
It is seen that the hierarchical structure does not have time to rebuild under the action of a rapidly increasing load (the
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kinetic process is rather inert) and the failure of the structure levels continues after the moment of the main energy release. Thus a relatively short creep time and the presence of a larger number of aftershocks are characteristic of higher stress
(curve 2 in Figure 22).
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rates. As for the dependence lg N as function of M in case of stress rate of 10-2 MPa·s-1 the curve keeps the same form
In order to compare the behavior of the fractal dimension in the process of rock destruction, in each case the creep time
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(the time interval from the onset of fracture to the maximum elastic energy release) was divided into 4 segments. Calculations of the fractal dimension were made at the points corresponding to the first, second and third ends of the segments or to the relative parts of creep time tc. Figure 26 depicts the results of calculations of the fractal dimension for different tress rates as function of the relative part of the creep time tc. It should be noted that the results of calculations of the fractal dimension for all the considered cases at stress rate of 10-2 MPa·s-1 correspond to the same curve 1 regardless of temperature, type of the hierarchical structure and type of rock.
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Figure 26. The fractal dimension for different stress rates as function of the relative part of the creep time tc :
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1 - 10-3 MPa·s-1, 2 - 3·10-3 MPa·s-1, 3 - 10-2 MPa·s-1, 4 - 10-1 MPa·s-1.
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CONCLUSION
In the present study the model is proposed for describing the rock fracture dynamics. The results of calculations of the fracture dynamics were compared with experimental data of the granite specimen failure. Satisfactory coincidence of
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the results of calculations and experimental data allows concluding the rock fracture process can be considered as a failure process of the hierarchical structure. The calculation results of the dynamics of the crack size distribution and the size
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distribution of fragments as a result of the specimen fracture are presented. The influence of temperature, the type of hierarchical structure, the type of rock and the stress rate on the fracture dynamics is considered. It is shown that the dynamics of the fractal dimension in the process of sample destruction significantly depends only on the stress rate and does not depend on temperature change and type of material and type of hierarchical structure. It is also revealed that the shape of the curve lg N as function of M is not affected by all the above factors including the stress rate (coefficient b ≈ 9 in all cases) . Perhaps this fact explains the spatial and temporal stability of the Gutenberg-Richter law.
Conflict of Interest: Author S. N. Fedotov declares that he has no conflict of interest. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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Journal Pre-proof Allegre, C.J., Mouel, J.L. , Chau, H., Narteau, C., 1995. Scaling organization of fracture tectonics (SOFT) fracture mechanism. Physics of Earth and Planetary Interiors. 92, 215-233. https://doi.org/10.1016/0031-9201(95)03033-0. Chmel ,A., Kuksenko, V.S., Smirnov, V.S., Tomilin, N.G., 2007. Anomalies of critical state in fracturing geophysical objects. Nonlin. Processes Geophys. 14, 103–108. Feder, J., 1987. Fractals. Plenum Press, London. Fedotov, S.N., 2017. Quasi-brittle fracture as failure of hierarchical structure. Physical mesomechanic. 20 (2), 222 – 228. https://doi.org/10.1134/S1029959917020126 Gutenberg, B., Richter C., 1954. Seismicity of the earth and associated phenomena. Princeton Univ. Press, Princeton. Karlin, S., 1966. A first course in stochastic processes. Academic Press, New York. Kuksenko, V., Tomilin, N., Damaskinskaya, E., Lockner, D., 1996. A two-stage model of fracture of rocks..Pure Appl. Geophys. 146 (2), 253 - 258. https://doi.org/10.1007/BF00876492.
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Mechanics and Physics of Solids. 52( 2), 453-465. https://doi: 10.1016/S0022-5096(03)00088-7. Zhou, X.P., Yang, H.Q., 2004. Analysis of the localization of deformation and the complete stress-strain relation for mesoscopic heterogeneous brittle rocks under dynamic uniaxial tensile loading. Int. Journal of the Solids and Structures, 41, 1725-1738. https://doi: 10.1016/j.ijsolstr.2003.07.007. Zhou, X.P., Yang, H.Q., 2018. Dynamic damage localization in crack-weakened rock mass: Strain energy density factor approach. Theoretical and Applied Fracture Mechanics, 97, 289-302. https://doi.org/10.1016/j.tafmec.2017.05.006. Zurkov, S.N., 1965. Kinetic Concept of the Strength of Solids. Int. J. Fracture Mech. 1, 311–323.
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The rock fracture process can be considered as a failure process of the hierarchical structure. The dynamics of the fractal dimension in the process of sample destruction significantly depends only on the stress rate and does not depend on temperature change and type of material and type of hierarchical structure. The fractal dimension in the process of sample destruction significantly depends only on the stress rate and does not depend on temperature change and type of material and type of hierarchical structure.
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