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Shear Band Thermodynamic Model of Fracturing
Chapter 10
Shear Band Thermodynamic Model of Fracturing
Roman Teisseyre
10.1 INTRODUCTION In this chapter we present a new model of the seismic source zone based on the thermodynamics of Une defects. As has been already discussed (Chapter 9), in the thermodynamics of line defects, the concept of the dislocational superlattice plays an essential role. The line vacancies (vacant dislocations) enable us to construct a superlattice consisting of dislocations and vacant dislocations. The depth distribution of the superlattice constant A in the Earth's mantle may follow from the consistency condition between the dislocation and thermodynamic models of the seismic source. The model introduced fits plastic deformations. Because of stress load, we may expect some changes of dislocation density related to the superlattice model. Such changes can be related to a change in the number of vacant dislocations or to a change in the superlattice parameter A. An increase in the number of dislocations in this case corresponds to a hardening process and is related to the space structure of the superlattice. As assumed in our model, the plastic deformations are realized by the formation of shear bands. A dislocation number increases intensively along the shear planes and exceeds the number prescribed by the A structure of the superlattice. However, this A structure will be preserved in the direction perpendicular to the shear band planes. Depending on the conditions, we may enter into a softening regime in which the dislocation arrays are formed in parallel band planes. According to this shear band model, the earthquake process would not be expected unless we assume an existence of some rigid subregions not subjected to plastic flow. We assume that in such subregions there is a very low density of dislocations that is below the number prescribed by the A structure. In other words, such subregions are related to sites where the vacant dislocations have formed clusters. Such clusters would stop a plastic deformation and form a barrier that can be broken by a dynamical coalescence of those opposite (in sign) dislocations that thus annihilate a cluster consisting of vacant dislocations. Interaction between the positive and negative dislocations concentrated at the opposite sides of such cluster results in the Earthquake Thermodynamics and Phase
Transformations in the Earth's Interior
279
Copyright © 2001 by Academic Press
AH rights of reproduction in any form reserved.
280
Roman Teisseyre HAXD/A
A7A?1 r
Figure 10.1 The shear band model with stratification according to the A structure.
formation of dislocation arrays and, consequently, may lead to the occurrence of a seismic event. Figure 10.1 shows a general scheme of the shear band model while Fig. 10.2 describes a possible pattern of a vacant dislocation cluster and the surrounding arrays in this model. The shear band thermodynamic model is combined with the method of estimating the top stress values before fracturing for sequences of events in a confined seismic region, e.g., aftershocks (Teisseyre and Wiejacz, 1993, 1995; Teisseyre, 1995). The obtained relations permit us to estimate from independent data for events from the same region, namely, from a set consisting of seismic moment, source radius, and radiated energy, the other independent set of parameters: the source thickness and the seismic efficiency, the total released energy, and the source damage parameter (defined later), as well as the entropy increase due to earthquake events and the full stress diagrams. Applications to earthquake data (Teisseyre, 1996) are presented. With some modifications, we applied this model to data related to mine events (Teisseyre, 1997) and icequakes (Gorski, 1997). For such cases, it is necessary
Figure 10.2 The shear band model with a rigid inclusion (barrier).
Chapter 10 Shear Band Thermodynamic Model
281
to modify the thermodynamic model to take into account a nonshear component in a fracturing process. The thermodynamic earthquake model is developed on the basis of the thermodynamics of the line defects considered in the previous chapter. Under a high stress load, the density of dislocation increases and, because of dislocation repulsive interactions, a kind of dislocation network is formed; this network is called the A superlattice. The notion of a superlattice is equivalent to the definition of vacant dislocation in a medium with a high density of dislocations (high enough to take into account the repulsive interaction between the dislocations). The vacant dislocations are introduced in such a way as to obtain, together with the existing dislocations, the A network. Such a procedure can be realized by a suitable numerical program (Kuklinska, 1996) for a given distribution of real dislocations. The order of magnitude of a superlattice constant A is discussed later; it is several orders higher than that of the crystal lattice A. The equilibrium number of the vacant dislocations is given by [Eq. (9.19), Chapter 9]
"e, = ^ e x p ( - - | ; j ,
(10.1)
where g^ is the Gibbs formation energy for the vacant dislocation and ft^^ is the equilibrium number of vacant dislocations (self-energy of vacancy and an irreversible part of work done—related to entropy increase), and the superlattice constant A is related to the total number of line defects (dislocations and vacant dislocations N = n -\- ft) hy the relation N = A^/A^, where As is a surface element. 10.2 JOGS AND KINKS When considering the dislocation superlattice, we include also the influence of some distortions on the dislocation lines, such as kinks and jogs. A kink is a step on a dislocation line when the step lies along the glide plane (the case for a common glide plane of the crossing dislocations); a jog is a step on a dislocation line when the step lies perpendicular to the glide plane (the case for perpendicular glide planes of the crossing dislocations) (Fig. 10.3). The jogs hamper the motion of dislocation, influencing effective resistance stress. The kinks and jogs lying on the same dislocation line may form pairs. We follow some formula after Kocks et al. (1975), writing approximately for microscopic dislocations b ^ X: A glide motion of dislocation can be realized by kink motion: i;disl =
A'
^kink
QQ 2)
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Roman Teisseyre
Kink
Figure 10.3 Kink and jog on a dislocation line.
A kink average equilibrium spacing A' (kink pair spacing): A' = Aexp
16kT
(10.3)
If this last formula might be extended to the jog spacing, we can relate A' to the superlattice constant A. 10.3 SHEAR BAND MODEL When the real number of dislocation increases by An, we can expect both changes of A^ and An. However, when An decreases by coalescence process (mutual annihilation of dislocation pair), then Ah increases while N remains constant. The basic assumption of the thermodynamical shear band model is that the slip planes in a seismic source follow a superlattice structure and the distance between them A determines the total number of the gauges per unit volume A^ = 3F/A^ (the factor 3 relates to the three main possible orientations of the gauge dislocations). However, along the slip planes, the number of dislocations greatly exceeds that of the gauge dislocations related to the A structure; these dislocations become squeezed under the action of external field 5*0 and form linear arrays, exerting stress concentration at the array tip.
Chapter 10 Shear Band Thermodynamic Model
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The superlattice structure defines the sHp planes in a seismic source, becoming separated by a distance equal to A; in Fig. 10.1 we present the model in which a number of slip planes is related to the A structure, while such structure is destroyed along the slip plane. The total Burgers vector results from a shear band structure Z)/A (where D is the source thickness) and from the value of dislocation concentration in arrays ^^A along the slip planes. The rip^ dislocations of an array exert multiple stress concentration at the array tip; we get for the local stress value S = HJ^SQ. However, a number of slip planes remains related to the A structure. We can now write the relation between the crystal lattice constant A, the Burgers vector along slip planes rij^X, and the total Burgers vector b resulting from the shear band structure as expressed by D/A (where D is the source thickness and D/A is the number of slip planes): D b=n^X—. (10.4) A The leading dislocations of an array may form a crack (microfracture) when the stress concentration exceeds the material strength. Opposite arrays, formed by dislocations of opposite signs, may mutually coalesce, releasing internal energy; An will denote the portion of actually executed coalescence processes from all possible coalescences in a given earthquake source. Thus, we assume that only a part of the source is active, and that the A structure of slip planes (fracturing) determines the demage structure of source; this A structure might correspond to the ultramylonite zones observed in focal structures. For the sake of simplicity, we assume a flat cylindrical model of an earthquake source with radius R and thickness D (Fig. 10.1). From the input data {E^^^^, MQ, R), we can calculate the surface densities of radiated energy and moment, 7rad
^0 = ^ ^ ,
Mo m^ = ixb = ^ ^ ,
(10.5)
and then we compute the stress drop using the formula after Randall (1972, 1973): 7 Mr.
ITT
nin
10.4 ENERGY RELEASE AND STRESSES The total internal energy release as expressed by stress and stress drop, assuming that total stored energy, including plastic energy (energies of
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Roman Teisseyre
defects, etc.), may be expressed by an equation similar to that of internal elastic energy, would amount to E'""' = —^SASiTTR^D),
(10.7)
where, however, constant /x* may differ from the rigidity modulus /x because of the complexity of the emerging fracture zone (material hardening due to stress interactions or material softening due to the existence of an older fracture plane) and because this expression refers to the total energy stored (not only elastic energy). 10.4.1 Seismic Events Teisseyre and Wiejacz (1993; see also Teisseyre, 1995) presented a method of estimating the top stress values before fracturing for sequences of closely interrelated events, such as aftershocks. The method is based on solving the following consecutive inequalities for consecutive events (stress drop smaller than the top stress level, and the bottom stress level smaller than the top stress level related to the next event): 0 <5,. - A5,- <5,.+ i.
(10.8)
Here, index / marks the consecutive events. Assuming small differences in the thicknesses of the neighboring sources (D^ ~ ^i+i), we could solve this system by replacing it with an approximative system (Teisseyre and Wiejacz, 1993), Tj^'ASfTTRf
rj^'AS^AS,^ .TTRU i
S.J
^.
7]^'ASf7rRf
(10.9)
with the average seismic efficiency r/^^. Moreover, one supplementary equation is added: e.g., for the last value of source thickness D^ we may assume the value equal to the average of the values D^ for / from 1 to AZ. We can take the average between the left and right values of the inequalities as the search solutions for D^. The discussed system of inequalities has been applied to some series of seismic events from the several confined regions. After Teisseyre (1996), in Fig. 10.4 we present the obtained stress values and subsequent stress drops for the series of events described by Fletcher and Boatwright (1991). Then, we compare this stress evolution diagram to that in which the top stresses are simply taken as proportional to the subsequent stress drops; the very high correlation of these diagrams justified the use of a simple formula for stresses: S = S^AS, 5o = 1.2. (10.10)
Chapter 10 2.0
2.51
P Waves
S Waves
2.o|
1.5
285
Shear Band Thermodynamic Model
1.5
1.0
1.0
0.5
0.51 10
20
30
40 50 Days
60
70
10
30 40 Days
20
50
60
70
Figure 10.4 Stress evolution diagrams after Teisseyre (1996) for the series of events described by Fletcher and Boatwright (1991): top stress value before event and stress value after a stress drop; upper curve according to present model [Eq. (10.10)] and lower curve according to method of inequalities [Eq. (10.8)]; S given in the scale of 10^; the vertical shift of the first curve, introduced for joint presentation, is related to the unit of scale on the figure.
10.4.2
Source Damage Parameter
Having estimated the S and A 5 values, we can compute the value of the total released energy E^^^ as it is expressed by Eq. (10.7). The obtained values of released energies can be treated rather like the relative ones. The shear modulus /x* along a fault may differ from the bulk value. The source damage parameter A', denoting the active part of the source (atomic bonds broken), can be obtained when comparing E^^^ with the thermodynamical formula for total energy release in the source. kT
^
kT
^'-^lTR'D,
\3 '
A
(10.11)
where k is the Boltzmann constant. Energy release, according to the earthquake dislocation theory (Teisseyre, 1961; Teisseyre et al., 1995), is related to a coalescence of dislocations. Dislocations concentrate in arrays and coalesce with the opposite dislocation arrays. Such a model is very simplified; however, it may be helpful to estimate the orders of the magnitudes of the discussed quantities. We notice that in an earthquake preparation zone, the dislocations of opposite sign—a and j8—group on opposite sides of the barrier zone. This is related to different signs of stress gradients (Teisseyre et al., 1995). In a simple ID case, we have the relations for dislocations and stress gradients: 1 f^ dx
while j8 =
1 ^
dx
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Roman Teisseyre
Hence, during an earthquake process, the numbers of dislocations and of vacant dislocations undergo essential changes. Considering the distribution of dislocations (including their arrays) along the slip planes of the A structure, we assume that in some subregions of the source zone there may be a very low density of dislocations; in other words, such more rigid subregions are related to sites where the vacant dislocations have formed the clusters. Such clusters would stop plastic deformation and form a barrier to be broken by a dynamic coalescence of those dislocations coming from the opposite sides through a barrier (cluster of vacant dislocations). We remember that the dislocations of the opposite signs, a and j8, group on the opposite sides of such clusters because of different signs of stress gradients (Czechowski et al., 1994; Teisseyre et al., 1995). Figure 10.2 may thus visualize the preseismic pattern in which dislocation arrays formed along the slip planes stop before a more rigid subregion. Breaking this subregion, the opposite dislocations coalesce and seismic energy is released. Of course, a seismic source zone contains many such rigid subregions; one is shown in Fig. 10.2. Thus, from thermodynamical considerations (Teisseyre and Majewski, 1995), it follows that total released energy can be directly related to the number of coalescences, and we can identify it with the source damage parameter A'. We obtain for the crystallographic cubic system A^ kT
kT
A^
A^
while for the shear band model we can redefine the A' using the cuboid system AA^, kT
kT
E'""' = A—^TTR^D,
e'""' = A—-^,
(10.12)
AA^ AA^ where A' is the source damage parameter related to cube A^ and A is the factor defined for cuboid AA^ (assuming constant ratio A/A); A marks an active part of the source (atomic bonds broken). Factor A is directly related to the number of coalescences, is exactly proportional to the density of the total energy release, and can be determined from (10.12) and (10.7) for the given stresses. Teisseyre (1996), compared the total entropy increase ^^ = —r T
=
V
— T
(10.13)
with the source damage parameter A (for a given temperature in a given depth interval); a high correlation was observed.
Chapter 10
10.5
Shear Band Thermodynamic Model
287
SOURCE THICKNESS AND SEISMIC EFFICIENCY
Our model based on the A-structure consists of the D / A sHp planes forming a shear band (Fig. 10.1). From the surface density of moment (10.5) and with help of (10.4) we get mr
(10.14)
MAD| — | .
Seismic moment is proportional to the arm of moment D, represented here by the source thickness. We can find its relative values assuming that certain parameters such as ix, A, and n^^/K are constant in a given depth interval and for a given region. From the surface density of radiated energy we find the seismic efficiency
e'^^ = J]
(10.15)
TTR^D
After Teisseyre (1996) we present for the considered series of earthquakes (Fletcher and Boatwright, 1991) a comparison between the seismic efficiency J] and the inverse of source thickness 1/D (Fig. 10.5). Seismic efficiency evidently depends on fault thickness. As can be seen in this figure, the efficiency appears to be inversely proportional to the source thickness. Seismic radiation probably comes mainly from the upper and lower surfaces of a fault, and contributions from the inner surfaces are negligible because of interferences, while the deformation work is appreciably greater for a thick source. We have mentioned that for closely related events, such as aftershocks, we can estimate the top stress values S^ by solving a sequence of inequalities (Teisseyre and Wiejacz, 1993). However, from the total released energy
P Waves
II
S Waves
i 30 40 Days
10
20
30 40 Days
50
60
70
Figure 10.5 Inverse of source thickness and seismic efficiency after Teisseyre (1996): upper and lower curves present respectively (17 + 0.2), in scale 1 0 ~ \ {^) in scale 10^.
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Roman Teisseyre
multiplied by the average value of seismic efficiency 7/ = 77^^, a value of radiated energy can be obtained: E
=V
(10.16)
TTTR^D.
We may compare this value with the observed radiated energy, ultimately obtaining from this formula the source thickness D^ for consecutive events. Teisseyre (1996) found a remarkably good correlation between the independently obtained values of D/R, confirming the presented model. 10.6 SHEAR AND TENSILE BAND MODEL: MINING SHOCKS AND ICEQUAKES Teisseyre (1997; data after Gibowicz et al., 1990) has presented stress diagrams, entropy releases, source damage parameters, and seismic efficiencies compared with inverse values of source thicknesses for a series of mine events in Ruhr basin. Another set of mine events that occurred in a South African mine in April 1993 was analyzed in a similar way. Figure 10.6 presents the respective efficiencies 17 and the 1/D values (Teisseyre, 1997). Solving the inequalities (9.9) for the consecutive events, it has been found that the approximation (10.10) can also be used for mining events with same value ^0 = 1.2.
1/D
hm/^ 2
4 (40)
6
Days
8 (43) 10
12
14
Figure 10.6 Seismic efficiency and inverse of source thickness after Teisseyre (1997) for a series of events in a South African mine: upper curve (17 + 0.5), and lower curve - {^), in scale 2 X 10^; the corrected values marked by the arrows are estimated as due to part of the nonshear mechanism. Reprinted from Rockbursts and Seismicity in Mines—Proceedings of the 4th International Symposium, Krakow, Poland, 11-14 August 1997, Gibowicz, S. J. & S. Lasocki (eds.). 90 5410 890 8, 1997, 30 cm, 450 pp. EUR137.50/US$162.00/GBP97.00. Please order from: Kubicz Wydawnictwa Importowane, P.O. Box 9, 50-940 Wroclaw 2 [tel.: (071) 372-01-41; fax: (071) 322-14-17].
Chapter 10 Shear Band Thermodynamic Model
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The application of the outlined theory to the data related to mine events should take into account the possible compression/opening (nonshear component) in the source mechanism. Assuming that the seismic moment tensor contains two components, for example, M^2 ^^^ M^, responsible for shear and tensile modes, we can write for the scalar seismic moment (Silver and Jordan, 1982) 1
\i/'
M=\-LM'
/
=\M^,
1
+ -MU
^^/'
.
(10.17)
The possible opening/compression might be proportional to the number n\ of dislocations entering a crack; we may write D
^open/compr _ ^0 ^ _ ^
A
n^
^open/compr _ ^shear_A^ AZA
n^
^ = DX—,
A
(10.18)
where c is the total value of crack opening/compression proportional to the source thickness, and for the ratio of total opening/compression and shear displacements we put open/compr ^shear
„ = 7 '
-^ = J
(10.19)
Thus, according to (10.17), instead of formula (10.14) we might write HK ^A I
1/2
1
mo = /xA-^ 1 + - y 2
D.
(10.20)
Comparing this formula with Eq. (10.14), we obtain for the corrected value of source thickness D' =
D
772-
(1 + h')'
(10.21)
The values of y remain small and become important only for some events (Teisseyre, 1997). For the data from South Africa considered here, we were obliged to introduce such a correction only for two separate events (numbers 40 and 43 in the series) for which the 1/D values strongly deviate from proportionality to the corresponding efficiencies J]. When correcting the value of source thickness, the recalculated new efficiency values, the new total energies, and the new values of source fulfillment factors practically do not change. These corrections are shown on the 1/D plot in Fig. 10.6 (lower curve), where two new values \/D' are introduced (dotted line and arrows) with the very small y 3.15 X 10"^ and 4.3 X 10"^, respectively, which give
290
Roman Teisseyre S.OOi
J
3.00-
^
1.00-
< g-1.00-
J/
_l
-3.00-5.00-8.00
20 40 Event No.
w
/ — '
-6.00
1
-4.00 LOG A
'
1
-2.00
Figure 10.7 Correlation between the logarithms of entropy increments and damage parameters for icequakes (Gorski, 1997).
ratios of the tensile and shear components respectively equal to 7.8 X 10~^ and 4.3 X 10"^ Gorski (1997) applied the same shear band model for series of icequakes. The approximation (9.9) can also be used in this case, but with a different value for 5*0, namely 5o = 3. Figures 10.7 and 10.8 present some of Gorski's results.
0.00 n
o o
-1.00 H
-2.00 H
— I —
-1.00
-1
1—
0.00 LOG 1/D
1.00
Figure 10.8 Correlation between the logarithms of inverse of source thickness and seismic efficiencies for icequakes (Gorski, 1997).
Chapter 10
Shear Band Thermodynamic Model
291
10.7 RESULTS FOR EARTHQUAKES, MINE SHOCKS, AND ICEQUAKES The thermodynamical source model permits us to estimate the relative values of several source parameters. We have demonstrated a fairly good correlation between seismic efficiency and the inverse values of source thickness, whereas total released energy (and hence also entropy increase) correlates with the source damage parameter (the part of active volume of a source—atomic bonds broken). The obtained results also show that the seismic efficiency is inversely proportional to the total energy release: rj a 1/E^^\ Some deviations from our model may be observed for data related to stronger seismic events that might be better described by the crack propagation mode rather than by defect coalescence processes. Also, for some deeper earthquakes, mine events, and icequakes, we observe some deviations that might, however, be explained by the part of a nonshear component in the source processes. The presented model permits us to include a nonshear component that would lead to correction of the value for source thickness. Such a correction is reasonable only for separate events.
10.8
DISCUSSION
As we have mentioned, our results show that the seismic efficiency is inversely proportional to the thickness of an earthquake source. We may thus write Vi = -^
(10.22)
where D^ is constant. A possible interpretation of this finding is that seismic radiation comes mainly from the outer layers of a source, while energy release at inner layers is spent mainly on fracturing work. We may assume this result as the postulate for the shear band model, and in this way we can find the top stress values before seismic events. Thus, instead of using the relation (10.10), we get from (10.16) and (10.22) a simple estimate for the top stress value valid for the separate events:
^SlTR^DQ
(10.23)
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Roman Teisseyre
We may note that this expression is close to the effective and apparent stress values: S^ = VS,,,= - ^ ^ .
(10.24)
REFERENCES Fletcher, J. B., and Boatwright, J. (1991). Source parameters of Loma aftershocks along the San Francisco Peninsula from a joint inversion of digital seismograms. Bull. Seism. Soc. Am. 81, 1783-1812. Gibowicz, S. J., Harjes, H. P., and Schaefer, M. (1990). Source parameters of seismic events at Heinrich Robert mine, Ruhr Basin, Federal Republic of Germany: Evidence for nondoublecouple events. Bull. Seism. Soc. Am. 80, 88-109. Gorski, M. (1997). Seismicity of the Hornsund region, Spitsbergen: Icequakes and earthquakes. Pubis. Inst. Geophys., Pol. Ac. Sci. B-20, 77. Kocks, U. F., Argon, A. S., and Ashby, M. F. (1975). Thermodynamics and Kinetics of Slip. Pergamon Press, Oxford, New York, Toronto, Sydney, Braunschweig, 288 p. Kuklinska, M. (1996). Thermodynamics of line defects: construction of a dislocation superlattice. Acta Geophys. Polon. 44, 237-249. Randall, M. J. (1972). Stress drop and ratio of seismic energy to moment. / . Geophys. Res. 77, 969-970. Randall, M. J. (1973). The spectral theory of seismic sources. Bull. Seism. Soc. Am. 66, 1157-1182. Silver, P. G., and Jordan, T. H. (1982). Optimal estimation of scalar seismic moment. Geophys. J. Roy. Astr. Soc. 70, 755-787. Teisseyre, R. (1961). Dynamic and time relations of the dislocation theory of earthquakes. Acta Geophys. Polon. 9, 3-58. Teisseyre, R. (1995). Earthquake series: Evaluation of stresses. In: Theory of Earthquake Premonitory and Fracture Processes (R. Teisseyre, ed.), pp. 436-446. PWN, Warszawa. Teisseyre, R. (1996). Shear band thermodynamical earthquake model. Acta Geophys. Polon. 43, 219-236. Teisseyre, R. (1997). Shear band thermodynamical model of fracturing with a compressional component. In Rockburst and Seismicity in Mines (S. Gibowicz and S. Lasocki, eds.), pp. 17-21. A. A. Balkema, Rotterdam-Brookfield. Teisseyre, R., and Wiejacz, P. (1993). Earthquake sequences: Stress diagrams. Acta Geophys. Polon. 41, 85-100. Teisseyre, R., and Wiejacz, P. (1995). Estimation of stresses for the mining-induced events. Pubis. Inst. Geophys., Pol. Ac. Sci. M-19(281), 5-14. Teisseyre, R., Yamashita, T., and Czechowski, Z. (1995). Earthquake premonitory and fracture rebound theory: a synopsis and a simulation model in time and space. In Theory of Earthquake Premonitory and Fracture Processes. (R. Teisseyre, ed.), pp. 405-435. PWN, Warszawa.
Shear Band Thermodynamic Model of Fracturing
Roman Teisseyre
10.1 INTRODUCTION In this chapter we present a new model of the seismic source zone based on the thermodynamics of Une defects. As has been already discussed (Chapter 9), in the thermodynamics of line defects, the concept of the dislocational superlattice plays an essential role. The line vacancies (vacant dislocations) enable us to construct a superlattice consisting of dislocations and vacant dislocations. The depth distribution of the superlattice constant A in the Earth's mantle may follow from the consistency condition between the dislocation and thermodynamic models of the seismic source. The model introduced fits plastic deformations. Because of stress load, we may expect some changes of dislocation density related to the superlattice model. Such changes can be related to a change in the number of vacant dislocations or to a change in the superlattice parameter A. An increase in the number of dislocations in this case corresponds to a hardening process and is related to the space structure of the superlattice. As assumed in our model, the plastic deformations are realized by the formation of shear bands. A dislocation number increases intensively along the shear planes and exceeds the number prescribed by the A structure of the superlattice. However, this A structure will be preserved in the direction perpendicular to the shear band planes. Depending on the conditions, we may enter into a softening regime in which the dislocation arrays are formed in parallel band planes. According to this shear band model, the earthquake process would not be expected unless we assume an existence of some rigid subregions not subjected to plastic flow. We assume that in such subregions there is a very low density of dislocations that is below the number prescribed by the A structure. In other words, such subregions are related to sites where the vacant dislocations have formed clusters. Such clusters would stop a plastic deformation and form a barrier that can be broken by a dynamical coalescence of those opposite (in sign) dislocations that thus annihilate a cluster consisting of vacant dislocations. Interaction between the positive and negative dislocations concentrated at the opposite sides of such cluster results in the Earthquake Thermodynamics and Phase
Transformations in the Earth's Interior
279
Copyright © 2001 by Academic Press
AH rights of reproduction in any form reserved.
280
Roman Teisseyre HAXD/A
A7A?1 r
Figure 10.1 The shear band model with stratification according to the A structure.
formation of dislocation arrays and, consequently, may lead to the occurrence of a seismic event. Figure 10.1 shows a general scheme of the shear band model while Fig. 10.2 describes a possible pattern of a vacant dislocation cluster and the surrounding arrays in this model. The shear band thermodynamic model is combined with the method of estimating the top stress values before fracturing for sequences of events in a confined seismic region, e.g., aftershocks (Teisseyre and Wiejacz, 1993, 1995; Teisseyre, 1995). The obtained relations permit us to estimate from independent data for events from the same region, namely, from a set consisting of seismic moment, source radius, and radiated energy, the other independent set of parameters: the source thickness and the seismic efficiency, the total released energy, and the source damage parameter (defined later), as well as the entropy increase due to earthquake events and the full stress diagrams. Applications to earthquake data (Teisseyre, 1996) are presented. With some modifications, we applied this model to data related to mine events (Teisseyre, 1997) and icequakes (Gorski, 1997). For such cases, it is necessary
Figure 10.2 The shear band model with a rigid inclusion (barrier).
Chapter 10 Shear Band Thermodynamic Model
281
to modify the thermodynamic model to take into account a nonshear component in a fracturing process. The thermodynamic earthquake model is developed on the basis of the thermodynamics of the line defects considered in the previous chapter. Under a high stress load, the density of dislocation increases and, because of dislocation repulsive interactions, a kind of dislocation network is formed; this network is called the A superlattice. The notion of a superlattice is equivalent to the definition of vacant dislocation in a medium with a high density of dislocations (high enough to take into account the repulsive interaction between the dislocations). The vacant dislocations are introduced in such a way as to obtain, together with the existing dislocations, the A network. Such a procedure can be realized by a suitable numerical program (Kuklinska, 1996) for a given distribution of real dislocations. The order of magnitude of a superlattice constant A is discussed later; it is several orders higher than that of the crystal lattice A. The equilibrium number of the vacant dislocations is given by [Eq. (9.19), Chapter 9]
"e, = ^ e x p ( - - | ; j ,
(10.1)
where g^ is the Gibbs formation energy for the vacant dislocation and ft^^ is the equilibrium number of vacant dislocations (self-energy of vacancy and an irreversible part of work done—related to entropy increase), and the superlattice constant A is related to the total number of line defects (dislocations and vacant dislocations N = n -\- ft) hy the relation N = A^/A^, where As is a surface element. 10.2 JOGS AND KINKS When considering the dislocation superlattice, we include also the influence of some distortions on the dislocation lines, such as kinks and jogs. A kink is a step on a dislocation line when the step lies along the glide plane (the case for a common glide plane of the crossing dislocations); a jog is a step on a dislocation line when the step lies perpendicular to the glide plane (the case for perpendicular glide planes of the crossing dislocations) (Fig. 10.3). The jogs hamper the motion of dislocation, influencing effective resistance stress. The kinks and jogs lying on the same dislocation line may form pairs. We follow some formula after Kocks et al. (1975), writing approximately for microscopic dislocations b ^ X: A glide motion of dislocation can be realized by kink motion: i;disl =
A'
^kink
QQ 2)
282
Roman Teisseyre
Kink
Figure 10.3 Kink and jog on a dislocation line.
A kink average equilibrium spacing A' (kink pair spacing): A' = Aexp
16kT
(10.3)
If this last formula might be extended to the jog spacing, we can relate A' to the superlattice constant A. 10.3 SHEAR BAND MODEL When the real number of dislocation increases by An, we can expect both changes of A^ and An. However, when An decreases by coalescence process (mutual annihilation of dislocation pair), then Ah increases while N remains constant. The basic assumption of the thermodynamical shear band model is that the slip planes in a seismic source follow a superlattice structure and the distance between them A determines the total number of the gauges per unit volume A^ = 3F/A^ (the factor 3 relates to the three main possible orientations of the gauge dislocations). However, along the slip planes, the number of dislocations greatly exceeds that of the gauge dislocations related to the A structure; these dislocations become squeezed under the action of external field 5*0 and form linear arrays, exerting stress concentration at the array tip.
Chapter 10 Shear Band Thermodynamic Model
283
The superlattice structure defines the sHp planes in a seismic source, becoming separated by a distance equal to A; in Fig. 10.1 we present the model in which a number of slip planes is related to the A structure, while such structure is destroyed along the slip plane. The total Burgers vector results from a shear band structure Z)/A (where D is the source thickness) and from the value of dislocation concentration in arrays ^^A along the slip planes. The rip^ dislocations of an array exert multiple stress concentration at the array tip; we get for the local stress value S = HJ^SQ. However, a number of slip planes remains related to the A structure. We can now write the relation between the crystal lattice constant A, the Burgers vector along slip planes rij^X, and the total Burgers vector b resulting from the shear band structure as expressed by D/A (where D is the source thickness and D/A is the number of slip planes): D b=n^X—. (10.4) A The leading dislocations of an array may form a crack (microfracture) when the stress concentration exceeds the material strength. Opposite arrays, formed by dislocations of opposite signs, may mutually coalesce, releasing internal energy; An will denote the portion of actually executed coalescence processes from all possible coalescences in a given earthquake source. Thus, we assume that only a part of the source is active, and that the A structure of slip planes (fracturing) determines the demage structure of source; this A structure might correspond to the ultramylonite zones observed in focal structures. For the sake of simplicity, we assume a flat cylindrical model of an earthquake source with radius R and thickness D (Fig. 10.1). From the input data {E^^^^, MQ, R), we can calculate the surface densities of radiated energy and moment, 7rad
^0 = ^ ^ ,
Mo m^ = ixb = ^ ^ ,
(10.5)
and then we compute the stress drop using the formula after Randall (1972, 1973): 7 Mr.
ITT
nin
10.4 ENERGY RELEASE AND STRESSES The total internal energy release as expressed by stress and stress drop, assuming that total stored energy, including plastic energy (energies of
284
Roman Teisseyre
defects, etc.), may be expressed by an equation similar to that of internal elastic energy, would amount to E'""' = —^SASiTTR^D),
(10.7)
where, however, constant /x* may differ from the rigidity modulus /x because of the complexity of the emerging fracture zone (material hardening due to stress interactions or material softening due to the existence of an older fracture plane) and because this expression refers to the total energy stored (not only elastic energy). 10.4.1 Seismic Events Teisseyre and Wiejacz (1993; see also Teisseyre, 1995) presented a method of estimating the top stress values before fracturing for sequences of closely interrelated events, such as aftershocks. The method is based on solving the following consecutive inequalities for consecutive events (stress drop smaller than the top stress level, and the bottom stress level smaller than the top stress level related to the next event): 0 <5,. - A5,- <5,.+ i.
(10.8)
Here, index / marks the consecutive events. Assuming small differences in the thicknesses of the neighboring sources (D^ ~ ^i+i), we could solve this system by replacing it with an approximative system (Teisseyre and Wiejacz, 1993), Tj^'ASfTTRf
rj^'AS^AS,^ .TTRU i
S.J
^.
7]^'ASf7rRf
(10.9)
with the average seismic efficiency r/^^. Moreover, one supplementary equation is added: e.g., for the last value of source thickness D^ we may assume the value equal to the average of the values D^ for / from 1 to AZ. We can take the average between the left and right values of the inequalities as the search solutions for D^. The discussed system of inequalities has been applied to some series of seismic events from the several confined regions. After Teisseyre (1996), in Fig. 10.4 we present the obtained stress values and subsequent stress drops for the series of events described by Fletcher and Boatwright (1991). Then, we compare this stress evolution diagram to that in which the top stresses are simply taken as proportional to the subsequent stress drops; the very high correlation of these diagrams justified the use of a simple formula for stresses: S = S^AS, 5o = 1.2. (10.10)
Chapter 10 2.0
2.51
P Waves
S Waves
2.o|
1.5
285
Shear Band Thermodynamic Model
1.5
1.0
1.0
0.5
0.51 10
20
30
40 50 Days
60
70
10
30 40 Days
20
50
60
70
Figure 10.4 Stress evolution diagrams after Teisseyre (1996) for the series of events described by Fletcher and Boatwright (1991): top stress value before event and stress value after a stress drop; upper curve according to present model [Eq. (10.10)] and lower curve according to method of inequalities [Eq. (10.8)]; S given in the scale of 10^; the vertical shift of the first curve, introduced for joint presentation, is related to the unit of scale on the figure.
10.4.2
Source Damage Parameter
Having estimated the S and A 5 values, we can compute the value of the total released energy E^^^ as it is expressed by Eq. (10.7). The obtained values of released energies can be treated rather like the relative ones. The shear modulus /x* along a fault may differ from the bulk value. The source damage parameter A', denoting the active part of the source (atomic bonds broken), can be obtained when comparing E^^^ with the thermodynamical formula for total energy release in the source. kT
^
kT
^'-^lTR'D,
\3 '
A
(10.11)
where k is the Boltzmann constant. Energy release, according to the earthquake dislocation theory (Teisseyre, 1961; Teisseyre et al., 1995), is related to a coalescence of dislocations. Dislocations concentrate in arrays and coalesce with the opposite dislocation arrays. Such a model is very simplified; however, it may be helpful to estimate the orders of the magnitudes of the discussed quantities. We notice that in an earthquake preparation zone, the dislocations of opposite sign—a and j8—group on opposite sides of the barrier zone. This is related to different signs of stress gradients (Teisseyre et al., 1995). In a simple ID case, we have the relations for dislocations and stress gradients: 1 f^ dx
while j8 =
1 ^
dx
286
Roman Teisseyre
Hence, during an earthquake process, the numbers of dislocations and of vacant dislocations undergo essential changes. Considering the distribution of dislocations (including their arrays) along the slip planes of the A structure, we assume that in some subregions of the source zone there may be a very low density of dislocations; in other words, such more rigid subregions are related to sites where the vacant dislocations have formed the clusters. Such clusters would stop plastic deformation and form a barrier to be broken by a dynamic coalescence of those dislocations coming from the opposite sides through a barrier (cluster of vacant dislocations). We remember that the dislocations of the opposite signs, a and j8, group on the opposite sides of such clusters because of different signs of stress gradients (Czechowski et al., 1994; Teisseyre et al., 1995). Figure 10.2 may thus visualize the preseismic pattern in which dislocation arrays formed along the slip planes stop before a more rigid subregion. Breaking this subregion, the opposite dislocations coalesce and seismic energy is released. Of course, a seismic source zone contains many such rigid subregions; one is shown in Fig. 10.2. Thus, from thermodynamical considerations (Teisseyre and Majewski, 1995), it follows that total released energy can be directly related to the number of coalescences, and we can identify it with the source damage parameter A'. We obtain for the crystallographic cubic system A^ kT
kT
A^
A^
while for the shear band model we can redefine the A' using the cuboid system AA^, kT
kT
E'""' = A—^TTR^D,
e'""' = A—-^,
(10.12)
AA^ AA^ where A' is the source damage parameter related to cube A^ and A is the factor defined for cuboid AA^ (assuming constant ratio A/A); A marks an active part of the source (atomic bonds broken). Factor A is directly related to the number of coalescences, is exactly proportional to the density of the total energy release, and can be determined from (10.12) and (10.7) for the given stresses. Teisseyre (1996), compared the total entropy increase ^^ = —r T
=
V
— T
(10.13)
with the source damage parameter A (for a given temperature in a given depth interval); a high correlation was observed.
Chapter 10
10.5
Shear Band Thermodynamic Model
287
SOURCE THICKNESS AND SEISMIC EFFICIENCY
Our model based on the A-structure consists of the D / A sHp planes forming a shear band (Fig. 10.1). From the surface density of moment (10.5) and with help of (10.4) we get mr
(10.14)
MAD| — | .
Seismic moment is proportional to the arm of moment D, represented here by the source thickness. We can find its relative values assuming that certain parameters such as ix, A, and n^^/K are constant in a given depth interval and for a given region. From the surface density of radiated energy we find the seismic efficiency
e'^^ = J]
(10.15)
TTR^D
After Teisseyre (1996) we present for the considered series of earthquakes (Fletcher and Boatwright, 1991) a comparison between the seismic efficiency J] and the inverse of source thickness 1/D (Fig. 10.5). Seismic efficiency evidently depends on fault thickness. As can be seen in this figure, the efficiency appears to be inversely proportional to the source thickness. Seismic radiation probably comes mainly from the upper and lower surfaces of a fault, and contributions from the inner surfaces are negligible because of interferences, while the deformation work is appreciably greater for a thick source. We have mentioned that for closely related events, such as aftershocks, we can estimate the top stress values S^ by solving a sequence of inequalities (Teisseyre and Wiejacz, 1993). However, from the total released energy
P Waves
II
S Waves
i 30 40 Days
10
20
30 40 Days
50
60
70
Figure 10.5 Inverse of source thickness and seismic efficiency after Teisseyre (1996): upper and lower curves present respectively (17 + 0.2), in scale 1 0 ~ \ {^) in scale 10^.
288
Roman Teisseyre
multiplied by the average value of seismic efficiency 7/ = 77^^, a value of radiated energy can be obtained: E
=V
(10.16)
TTTR^D.
We may compare this value with the observed radiated energy, ultimately obtaining from this formula the source thickness D^ for consecutive events. Teisseyre (1996) found a remarkably good correlation between the independently obtained values of D/R, confirming the presented model. 10.6 SHEAR AND TENSILE BAND MODEL: MINING SHOCKS AND ICEQUAKES Teisseyre (1997; data after Gibowicz et al., 1990) has presented stress diagrams, entropy releases, source damage parameters, and seismic efficiencies compared with inverse values of source thicknesses for a series of mine events in Ruhr basin. Another set of mine events that occurred in a South African mine in April 1993 was analyzed in a similar way. Figure 10.6 presents the respective efficiencies 17 and the 1/D values (Teisseyre, 1997). Solving the inequalities (9.9) for the consecutive events, it has been found that the approximation (10.10) can also be used for mining events with same value ^0 = 1.2.
1/D
hm/^ 2
4 (40)
6
Days
8 (43) 10
12
14
Figure 10.6 Seismic efficiency and inverse of source thickness after Teisseyre (1997) for a series of events in a South African mine: upper curve (17 + 0.5), and lower curve - {^), in scale 2 X 10^; the corrected values marked by the arrows are estimated as due to part of the nonshear mechanism. Reprinted from Rockbursts and Seismicity in Mines—Proceedings of the 4th International Symposium, Krakow, Poland, 11-14 August 1997, Gibowicz, S. J. & S. Lasocki (eds.). 90 5410 890 8, 1997, 30 cm, 450 pp. EUR137.50/US$162.00/GBP97.00. Please order from: Kubicz Wydawnictwa Importowane, P.O. Box 9, 50-940 Wroclaw 2 [tel.: (071) 372-01-41; fax: (071) 322-14-17].
Chapter 10 Shear Band Thermodynamic Model
289
The application of the outlined theory to the data related to mine events should take into account the possible compression/opening (nonshear component) in the source mechanism. Assuming that the seismic moment tensor contains two components, for example, M^2 ^^^ M^, responsible for shear and tensile modes, we can write for the scalar seismic moment (Silver and Jordan, 1982) 1
\i/'
M=\-LM'
/
=\M^,
1
+ -MU
^^/'
.
(10.17)
The possible opening/compression might be proportional to the number n\ of dislocations entering a crack; we may write D
^open/compr _ ^0 ^ _ ^
A
n^
^open/compr _ ^shear_A^ AZA
n^
^ = DX—,
A
(10.18)
where c is the total value of crack opening/compression proportional to the source thickness, and for the ratio of total opening/compression and shear displacements we put open/compr ^shear
„ = 7 '
-^ = J
(10.19)
Thus, according to (10.17), instead of formula (10.14) we might write HK ^A I
1/2
1
mo = /xA-^ 1 + - y 2
D.
(10.20)
Comparing this formula with Eq. (10.14), we obtain for the corrected value of source thickness D' =
D
772-
(1 + h')'
(10.21)
The values of y remain small and become important only for some events (Teisseyre, 1997). For the data from South Africa considered here, we were obliged to introduce such a correction only for two separate events (numbers 40 and 43 in the series) for which the 1/D values strongly deviate from proportionality to the corresponding efficiencies J]. When correcting the value of source thickness, the recalculated new efficiency values, the new total energies, and the new values of source fulfillment factors practically do not change. These corrections are shown on the 1/D plot in Fig. 10.6 (lower curve), where two new values \/D' are introduced (dotted line and arrows) with the very small y 3.15 X 10"^ and 4.3 X 10"^, respectively, which give
290
Roman Teisseyre S.OOi
J
3.00-
^
1.00-
< g-1.00-
J/
_l
-3.00-5.00-8.00
20 40 Event No.
w
/ — '
-6.00
1
-4.00 LOG A
'
1
-2.00
Figure 10.7 Correlation between the logarithms of entropy increments and damage parameters for icequakes (Gorski, 1997).
ratios of the tensile and shear components respectively equal to 7.8 X 10~^ and 4.3 X 10"^ Gorski (1997) applied the same shear band model for series of icequakes. The approximation (9.9) can also be used in this case, but with a different value for 5*0, namely 5o = 3. Figures 10.7 and 10.8 present some of Gorski's results.
0.00 n
o o
-1.00 H
-2.00 H
— I —
-1.00
-1
1—
0.00 LOG 1/D
1.00
Figure 10.8 Correlation between the logarithms of inverse of source thickness and seismic efficiencies for icequakes (Gorski, 1997).
Chapter 10
Shear Band Thermodynamic Model
291
10.7 RESULTS FOR EARTHQUAKES, MINE SHOCKS, AND ICEQUAKES The thermodynamical source model permits us to estimate the relative values of several source parameters. We have demonstrated a fairly good correlation between seismic efficiency and the inverse values of source thickness, whereas total released energy (and hence also entropy increase) correlates with the source damage parameter (the part of active volume of a source—atomic bonds broken). The obtained results also show that the seismic efficiency is inversely proportional to the total energy release: rj a 1/E^^\ Some deviations from our model may be observed for data related to stronger seismic events that might be better described by the crack propagation mode rather than by defect coalescence processes. Also, for some deeper earthquakes, mine events, and icequakes, we observe some deviations that might, however, be explained by the part of a nonshear component in the source processes. The presented model permits us to include a nonshear component that would lead to correction of the value for source thickness. Such a correction is reasonable only for separate events.
10.8
DISCUSSION
As we have mentioned, our results show that the seismic efficiency is inversely proportional to the thickness of an earthquake source. We may thus write Vi = -^
(10.22)
where D^ is constant. A possible interpretation of this finding is that seismic radiation comes mainly from the outer layers of a source, while energy release at inner layers is spent mainly on fracturing work. We may assume this result as the postulate for the shear band model, and in this way we can find the top stress values before seismic events. Thus, instead of using the relation (10.10), we get from (10.16) and (10.22) a simple estimate for the top stress value valid for the separate events:
^SlTR^DQ
(10.23)
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Roman Teisseyre
We may note that this expression is close to the effective and apparent stress values: S^ = VS,,,= - ^ ^ .
(10.24)
REFERENCES Fletcher, J. B., and Boatwright, J. (1991). Source parameters of Loma aftershocks along the San Francisco Peninsula from a joint inversion of digital seismograms. Bull. Seism. Soc. Am. 81, 1783-1812. Gibowicz, S. J., Harjes, H. P., and Schaefer, M. (1990). Source parameters of seismic events at Heinrich Robert mine, Ruhr Basin, Federal Republic of Germany: Evidence for nondoublecouple events. Bull. Seism. Soc. Am. 80, 88-109. Gorski, M. (1997). Seismicity of the Hornsund region, Spitsbergen: Icequakes and earthquakes. Pubis. Inst. Geophys., Pol. Ac. Sci. B-20, 77. Kocks, U. F., Argon, A. S., and Ashby, M. F. (1975). Thermodynamics and Kinetics of Slip. Pergamon Press, Oxford, New York, Toronto, Sydney, Braunschweig, 288 p. Kuklinska, M. (1996). Thermodynamics of line defects: construction of a dislocation superlattice. Acta Geophys. Polon. 44, 237-249. Randall, M. J. (1972). Stress drop and ratio of seismic energy to moment. / . Geophys. Res. 77, 969-970. Randall, M. J. (1973). The spectral theory of seismic sources. Bull. Seism. Soc. Am. 66, 1157-1182. Silver, P. G., and Jordan, T. H. (1982). Optimal estimation of scalar seismic moment. Geophys. J. Roy. Astr. Soc. 70, 755-787. Teisseyre, R. (1961). Dynamic and time relations of the dislocation theory of earthquakes. Acta Geophys. Polon. 9, 3-58. Teisseyre, R. (1995). Earthquake series: Evaluation of stresses. In: Theory of Earthquake Premonitory and Fracture Processes (R. Teisseyre, ed.), pp. 436-446. PWN, Warszawa. Teisseyre, R. (1996). Shear band thermodynamical earthquake model. Acta Geophys. Polon. 43, 219-236. Teisseyre, R. (1997). Shear band thermodynamical model of fracturing with a compressional component. In Rockburst and Seismicity in Mines (S. Gibowicz and S. Lasocki, eds.), pp. 17-21. A. A. Balkema, Rotterdam-Brookfield. Teisseyre, R., and Wiejacz, P. (1993). Earthquake sequences: Stress diagrams. Acta Geophys. Polon. 41, 85-100. Teisseyre, R., and Wiejacz, P. (1995). Estimation of stresses for the mining-induced events. Pubis. Inst. Geophys., Pol. Ac. Sci. M-19(281), 5-14. Teisseyre, R., Yamashita, T., and Czechowski, Z. (1995). Earthquake premonitory and fracture rebound theory: a synopsis and a simulation model in time and space. In Theory of Earthquake Premonitory and Fracture Processes. (R. Teisseyre, ed.), pp. 405-435. PWN, Warszawa.