Bulletin of the Seismological Society of America, Vol. 97, No. 3, pp. 915–925, June 2007, doi: 10.1785/0120060107
A Constitutive Criterion for the Fault: Modified Velocity-Weakening Law by Letı´cia Fleck Fadel Miguel and Jorge Daniel Riera
Abstract Expanding on a previous article, constitutive laws for solid friction are examined jointly with available experimental results. The models are evaluated by means of numerical dynamic analysis of two sliding blocks simulating adjacent fault sections. Effective constitutive laws are determined as relations between mean values of the relevant variables on a selected area of the sliding surfaces. The material is initially assumed elastic and homogeneous and the influence of nonhomogeneity evaluated next by modeling mass density, Young’s modulus, and friction coefficient as correlated random fields. The effect of fractures in rock close to the fault is also numerically assessed. Finally, the influence of rupture of protrusions (microasperities) between the sliding surfaces is analyzed. The influence of size of the averaging interface area on the parameters of the effective constitutive law is then obtained by means of Monte Carlo simulation. When the rock regions adjacent to the fault are assumed to be linearly elastic and homogeneous or nonhomogeneous no size effect is observed. On the other hand, when the friction coefficient is characterized by a random field, a size effect is detected. Fracture occurrence in the region surrounding the fault does not cause significant alteration of the macroconstitutive law, producing only minor perturbations of the mean law determined without fracture, but it should also introduce high-frequency slave vibrations. Finally, a macroconstitutive law that takes into account the shear rupture of microasperities on the sliding surfaces is suggested. The proposed modified velocity-weakening law, constitutes a more general and flexible constitutive law. Introduction acterize the motion on the fault surface, even in the idealized situation described previously. Therefore, a constitutive law for the fault that correctly defines the relation between the normal and tangential stresses and stress rates and the socalled kinetic variables (displacements, velocities, and accelerations) on the interface is needed for a reliable prediction of the seismic motion in the epicentral region. With such objective in mind, we expand in this article results recently advanced by Miguel et al. (2006a), beginning with a review of the constitutive models that have been most widely used in the dynamic analysis of the fault region. Next, experimental evidence on friction between solids, obtained in laboratory tests without fracture or significant wear, is also reviewed. A three-dimensional numerical model based on the discrete element method (DEM) is next used to simulate seismic pseudorupture, in which each local grid point on the fault is assumed to follow a micro (or local) friction law. In this article the performance of the slipweakening law, not previously considered, is examined in detail. It is understood that the so-called microfriction law should be apt to describe the phenomenon at the scale of a basic element of the numerical model. Macro (or global) fault constitutive laws are obtained by fitting appropriate
The dynamics of rupture propagation during earthquakes is one of the most complex and important issues in seismology. Recent studies of the rupture process along the fault (e.g., Fukuyama and Madariaga, 1998; Madariaga et al., 1998) led to a better understanding of various aspects of the rupture phenomena, highlighting the fundamental role of sliding with friction. However, the development of analyticalnumerical models for the study of rupture propagation continues being a difficult problem, in which many important issues, such as the initial stress field within the adjacent rock masses, the spatial distribution of its mechanical properties, and, in particular, the laws that govern sliding with friction are still insufficiently known. The laws that govern sliding with friction should control the initiation of motion, that is, the pseudorupture by shear failure on the fault, the development and the end of sliding, if no fracture of protrusions or microasperities occur in the process. (Sliding along an existing discontinuity surface, such as a fault, should be distinguished from a true rupture, in which new cracks or fractures are formed.) Coulomb’s model for friction between solids is clearly insufficient to quantify the relation between the resultant shear stress in the direction of rupture propagation and the variables that char915
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curves to the numerically evaluated response curves. These laws relate mean variables evaluated over a predefined area, which determines the scale of the macrolaw. Both homogeneous and nonhomogenous material properties are considered. Finally, the occurrence of fracture near or on the surfaces in contact, as well as the randomness of the fractured areas, is taken into account, to our knowledge, for the first time in the technical literature. Note that Dalguer et al. (2003) assess the effect of fractures occurring in the surrounding region, not adjacent to the fault. On such basis, a more general constitutive law for sliding with friction is proposed. This resulting model is basically a modified velocityweakening law.
Constitutive Laws Several models have been used in the dynamic analysis of the fault region to predict features of the rupture process that give rise to seismic excitation. The simplest and understandably more often employed laws are the slip-weakening laws and the so-called velocity- or rate-weakening laws. By way of introduction to the ensuing proposals, a succinct overview of previous developments is included next. The slip-weakening model, originally proposed by Ida (1972) and Palmer and Rice (1973) and later formalized by Andrews (1976), has been used extensively for dynamic simulations of fault-rupture processes. This model, in which the frictional strength is a function of slip, can explain some features of the dynamic rupture process of earthquakes (e.g., Day, 1982; Ohnaka and Kuwahara, 1990; Olsen et al., 1997; Fukuyama and Madariaga, 1998). The velocity-weakening law, proposed by Burridge and Knopoff (1967) and then modified by Madariaga and Cochard (1994), is used to study the healing process. In this model, the frictional strength jumps at the beginning of slip reaching its lowest values for large slip rates, then as slip velocity decreases, the frictional strength tends to increase again. Carlson and Langer (1989), Huang and Turcotte (1990), and Heaton (1990) also furnish evidence on the dependence of friction forces on the slip rate. Fukuyama and Madariaga (1998) discuss both the slipand velocity-weakening models, noting that according to the slip-weakening law, friction stresses decrease as the velocity increases at rupture initiation. Thus, the process is initially quite different from the theoretical motion predicted using Coulomb’s friction model, but the closing mechanisms are identical. If friction is assumed velocity dependent, the changes in the process are more pronounced and its end is controlled by the increase of friction as the velocity decreases. Madariaga et al. (1998) propose a combination of both models in a slip- and velocity-weakening law, but admit that its use should be restricted to specific cases. A more complex model, the rate- and state-dependent friction law, proposed by Dieterich (1979) and Ruina (1983), considers that the frictional strength is a function of slip rate and one or more
additional state variables. This model appears to correctly simulate the nucleation process and earthquake cycles (e.g., Ben-Zion and Rice, 1995). The models used in seismology cited earlier are not based directly on experimental evidence. Actual testing of rock samples in laboratory constitutes an alternative or complementary approach to developing constitutive criteria. Ohnaka et al. (1997) and Ohnaka and Shen (1999), for instance, presented valuable contributions along this line. However, the scaling of frictional models from laboratory to full-scale earthquakes is still a challenge because of the complexity of real shear faults. Faults are characterized by zones in which inelastic deformations as well as fracture occurs, the material is nonhomogeneous, and the contact surfaces are rough. In the following sections we attempt to assess the performance of friction models, which should lead to macro (or global) constitutive laws in which all those complex features of real earthquakes are somehow accounted for. Note that the scale of the macroconstitutive law is related to the size of the elements in the numerical model of the fault region for which a friction law must be specified.
Laboratory Experiments on Friction between Solids Relevant contributions on the physics of friction between solids and experimental data for small-scale tests were published by Barton and Choubey (1977), Bandis et al. (1981), and Grasselli and Egger (2003). Typical results obtained by the authors in laboratory tests (Miguel, 2002) are presented in Figure 1, which shows friction force versus velocity curves for several slip amplitudes, clearly suggesting the applicability to metallic surfaces of a velocity-weakening law. The experiments consisted of cyclic displacementcontrolled tests of metallic plates in contact under predefined normal pressure. The tests were conducted in a servo hydraulic universal test machine MTS 810 with 100 kN loading capacity. Test frequencies and amplitudes ranged from 0.1 to 3 Hz and 0.1 to 8 mm, respectively, resulting in a range of sliding velocities from 0.06 to 150 mm/sec. The results
Figure 1. Relation between friction force and velocity measured in laboratory tests.
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are typical of dry friction in solids, in situations in which there is negligible wear of the sliding surfaces or fracture in the adjacent material; in addition, stick-and-slip does not occur. These conditions do not hold when there is fracture of rock protrusions on the fault surface, herein denoted microasperities, as will be discussed later. The velocity-weakening law shown in Figure 2 was fitted to the experimental data. Despite the rather overwhelming laboratory evidence in favor of a velocity-weakening (micro) law, we also examined the applicability of the slip-weakening law (Miguel and Riera, 2003; Riera et al., 2005; and Miguel, 2005). In fact, a slip-weakening law was employed in the numerical determination of dynamic effects caused by the seismic source during the Chi-Chi Earthquake (Taiwan 1999) (Dalguer et al., 2001). Miguel and Riera (2003) conclude that, if the appropriate parameters are used, both laws should lead to similar results during roughly the first two-thirds of the predicted motions, but differ at the end, with evidence pending in favor of the velocity-weakening model. In the final part of the motion, the slip-weakening law underestimates the friction forces. It then becomes necessary to introduce arbitrary assumptions, as for example, the existence of an arresting mechanism. In this context, it is germane to point out that in the analysis of Dalguer et al. (2001), the correlation between predicted and observed displacement and velocity ground motion at all seismological stations is perceptibly better in the first half of the records in relation to the second half. Hartzell et al. (2005) also concluded that a velocityweakening law is more consistent for the 1994 Northridge Earthquake than a slip-weakening law is. However, according to recent investigations of real earthquakes (e.g., Guatteri and Spudich, 2000; Peyrat et al., 2004) the solution of the dynamic problem is not unique. Consequently, it may not be possible to assess the friction properties of the fault by using rupture modeling with the current bandwidth limitations. This implies that due to the upper-frequency limit in the ground motion that numerical models are able to simulate, many dynamic rupture solutions may fit the observed ground motion. However, the inclusion of higher-frequency components from high-resolution near-field records of large earthquakes may enable their separation in the future by increasing the resolution of the models.
Description of the Numerical Model Herein a three-dimensional numerical model based on the discrete element method (DEM) is used to represent the rock layers adjacent to the fault. The model consists of a three-dimensional periodic trusslike structure with lumped masses at the nodes, interconnected by unidimensional viscoelastic elements, which has been extensively used in linear and nonlinear dynamic analyses (e.g., Dalguer et al., 2001, 2003). The model employed in the present article for the numerical simulation of a seismic fault zone is shown in Figure 3. The dimensions of the model are the following:
Figure 2.
Velocity-weakening law fitted to exper-
imental data.
Figure 3.
DEM model employed in seismic-fault
simulation.
• • • •
Length of longitudinal elements 10 m. Number of elements in x direction 25. Number of elements in y direction 5. Number of elements in z direction 10 (5 elements in each block of rock).
So, each block measures 250 50 50 m, and mass density, Young’s modulus, and Poisson’s ratio of the rock in the homogeneous model are equal to q 2700 kg/m3, E 7.5E10 N/m2, and v 0.25, respectively. In the nonhomogeneous models the values indicated previously represent the mean value of the corresponding random fields. Normal and tangential forces are applied at the top surface of the upper block and at the bottom surface of the lower
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block, as shown schematically in Figure 4. Vertical and horizontal forces are also applied at nodes on the end faces of the blocks to account for the effect of the adjacent (infinite) rock fields. At each integration step, if slip occurs, friction forces are applied at all sliding nodes on the interface, according to the specified friction law. Relevant properties, such as mass density, Young’s modulus, and friction coefficient, are assumed to be correlated Gaussian random fields of the spatial coordinates, whereas the specific fracture energy is modeled as an uncorrelated Weibull field. Note that the normal and tangential stresses at the interface, as well as the horizontal and vertical displacements, or corresponding velocities, at the interface and upper and lower boundaries of the model, are viewed as two-dimensional random processes. Freezing time, twodimensional random fields result. The resulting macro (or global) constitutive law for the fault should relate the basic parameters of those fields. The idea is to assume that a friction law governs the (microscale) sliding at individual contact points or elements of the model. The macrofriction law can then be determined from the average, over a region of the fault, of all relevant variables, such as slip, slip rate, or shear stress. The influence of inelastic material behavior, fracture, and nonhomogeneous material properties may then be taken into account.
Simulation Results It is initially assumed that fracture does not occur during sliding. (New fractures in the region surrounding the fault surface may occur and should not be confused with the fracture of microasperities on the fault surface. The effect of the latter should be included in the constitutive law for the fault.) Thus, in accordance with the usual models in seismology, the material is assumed to be linearly elastic and isotropic, but mass density, Young’s modulus, and the static friction coefficient may vary with the spatial coordinates, because q, E, and ls were defined as random fields. Both the velocityweakening and the slip-weakening laws are considered at the microscale, from which a macro (or global) law is obtained relating the mean value of the shear strength with the mean value of normal stress at the interface and the mean tangential displacement and velocity. The applied normal and tangential stresses are shown in Figure 5. Preliminary results were obtained with the micro (or local) velocity-weakening law shown in Figure 6 and with a coefficient of variation equal to zero for all properties (Miguel et al., 2006a). Figure 7 illustrates the evolution with time of the mean tangential stress at the fault surface. Unless specifically indicated, this mean is evaluated over the entire interface between the blocks. It may be seen that for a friction coefficient higher than 2.0, there is no sliding and the reaction simply presents high-frequency elastic oscillations around the pseudostatic sinusoidal reaction pulse, similar to the applied stress pulse. Figure 8 presents the evolution with time of the mean shear strength at the fault surface for a
Figure 4.
Applied stresses in the fault region.
Figure 5.
Evolution of the normal and tangential stresses on model boundaries (from Miguel et al., 2006a).
Figure 6.
Micro (or local) velocity-weakening law adopted in this study (from Miguel et al., 2006a).
range of values of the static friction coefficient. A bathtub shape can be observed due to the increase of the friction force when the velocity decreases. Figures 9 and 10 show the mean slip and mean slip rate on the fault surface, where the extreme cases may be easily identified: without friction the block slides freely, but for a friction coefficient higher than 2.0 it remains glued to the lower block. The resulting
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A Constitutive Criterion for the Fault: Modified Velocity-Weakening Law
Figure 7. Mean tangential stress at fault surface versus time (from Miguel et al., 2006a).
Figure 8. Mean shear strength at fault surface versus time (from Miguel et al., 2006a).
Figure 9.
Mean slip on fault surface versus time (from Miguel et al., 2006a).
Figure 10.
Mean slip rate on fault surface versus
time.
macroslip law, presenting a bathtub shape, may be seen in Figure 11. The macro velocity-weakening law, presented in Figure 12, is very similar to the microlaw (Fig. 6), confirming that no size effect exists, as can be seen in Tables 1 and 2. The second micro (or local) constitutive relation employed in the dynamic analysis of the fault region is the slipweakening law shown in Figure 13. The evolution with time of the mean tangential stress, mean shear strength, mean slip, and mean slip rate on the fault surface are similar to those determined for a velocity-weakening microlaw. The resulting macrorelations between the mean shear strength, mean slip, and mean slip rate on the fault surface are presented in Figures 14 and 15, respectively. It may be seen that the macrolaw expressing the dependence of friction on the mean slip is single valued and similar in shape to the microlaw (Fig. 13), whereas the relation between friction and velocity is double valued. Again, the macro (Fig. 14) and the micro (Fig. 13) laws are very similar, confirming that no size effect exists, as can be seen in Tables 3 and 4.
Figure 11.
Macro slip-weakening laws, assuming a micro velocity-weakening law and homogeneous material properties.
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mal and tangential stresses are those indicated in Figure 5. Figure 16 shows the resulting macroconstitutive laws, which are a relation between mean shear strength and mean slip rate on the fault surface, for the micro velocity-weakening law shown in Figure 6. Figure 17, on the other hand, presents the resulting macroconstitutive laws, which are a relation between mean shear strength and mean slip on the fault surface, corresponding to the assumption that the microconstitutive law is the slip weakening shown in Figure 13. From Figures 16 and 17 and Tables 1–4, it may be concluded that the macrofriction model is still similar to the microlaw, when the mass density and/or Young’s modulus are assumed to be random fields, but the friction strength suffers a small reduction when the friction coefficient is considered a random field. In summary, randomness of the mass density and Young’s modulus does not introduce a significant size effect and may be disregarded to reduce computational costs, but the random distribution of the friction coefficient does cause a size effect (Miguel et al., 2006a).
Figure 12. Macro velocity-weakening laws, assuming a micro velocity-weakening law and homogeneous material properties.
Table 1 Parameters of Macroconstitutive Laws for Velocity Weakening Condition
Macro (or Global) Constitutive Laws for the Seismic Fault
Parameters
ls ls ls ls
0.2 0.5 1.0 1.5
CV All 0% CV All 0% CV All 0% CV All 0%
a a a a
ls ls ls ls
1.0 1.0 1.0 1.0
CV Density 25% CV Young 25% CV Friction 25% CV All 25%
a 1.000 a 0.999 a 0.876 a 0.867
0.200 0.500 0.999 1.500
b b b b
0.250 0.250 0.250 0.253
b 0.251 b 0.250 b 0.249 b 0.250
c c c c
4.020 3.973 3.959 3.942
c 3.967 c 3.978 c 4.008 c 4.000
Fault constitutive relations can be obtained by fitting appropriate curves to the numerical results. The first micro (or local) constitutive law assessed by us is the velocityweakening law (Fig. 6), described by equation (1): sr arn
CV, coefficient of variation.
The nonhomogeneous character of rock is considered next. Mass density, Young’s modulus, and friction coefficient were assumed to be Gaussian random fields of the spatial coordinates, modeled following Shinozuka and Deodatis (1996). The mean static friction coefficient is set equal to 1.0 for all random fields, and a coefficient of variation of 25% and a correlation length of 25 m are adopted for all fields. These are just plausible values, because we did not find field data in the literature. Again, the boundary nor-
1 b e c˙x , 1 b
(1)
where a ls, b 0.25, and c 4.0. The parameters of the macro (or global) constitutive law fitted to the numerical simulations are listed in Table 1. Table 2 presents the mean values and coefficients of variation of the parameters of the macroconstitutive laws fitted by zones. The zones previously referred are five adjacent square regions with sides equal to 50 m, that is, twice the correlation length. The macroconstitutive laws for the fault relate the mean shear strength (sr) at the interface with the mean normal stress (rn) and the mean slip rate (x˙) at the interface.
Table 2 Parameters of Macroconstitutive Laws, by Zones, for Velocity Weakening Condition
Parameters
ls ls ls ls
0.2 0.5 1.0 1.5
CV All 0% CV All 0% CV All 0% CV All 0%
am am am am
0.200 0.500 1.000 1.499
acv acv acv acv
0.00% 0.11% 0.05% 0.06%
bm 0.250 bm 0.250 bm 0.250 bm 0.253
bcv bcv bcv bcv
0.54% 0.36% 0.54% 1.14%
cm cm cm cm
4.014 4.024 3.988 3.946
ccv 0.53% ccv 1.87% ccv 1.49% ccv 1.51%
ls ls ls ls
1.0 1.0 1.0 1.0
CV Density 25% CV Young 25% CV Friction 25% CV All 25%
am am am am
1.000 1.000 0.870 0.864
acv acv acv acv
0.00% 0.04% 16.3% 17.3%
bm 0.249 bm 0.250 bm 0.250 bm 0.251
bcv bcv bcv bcv
0.44% 0.91% 0.46% 0.98%
cm cm cm cm
4.001 3.998 4.005 4.000
ccv 0.74% ccv 2.07% ccv 1.08% ccv 1.60%
CV, coefficient of variation.
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Figure 15.
Figure 13.
Macro velocity-weakening laws, assuming a micro slip-weakening law and homogeneous material properties.
Micro (or local) slip-weakening law employed in the analysis.
Table 3 Parameters of Macroconstitutive Laws for Slip Weakening Condition
Parameters
ls ls ls ls
0.2 0.5 1.0 1.5
CV All 0% CV All 0% CV All 0% CV All 0%
a a a a
0.200 0.500 1.000 1.500
b b b b
0.501 0.499 0.500 0.501
ls ls ls ls
1.0 1.0 1.0 1.0
CV Density 25% CV Young 25% CV Friction 25% CV All 25%
a a a a
1.000 1.000 0.876 0.866
b b b b
0.501 0.500 0.500 0.501
CV, coefficient of variation.
Figure 14.
Macro slip-weakening laws, assuming a micro slip-weakening law and homogeneous material properties.
As shown in Tables 1 and 2, the parameters of the macrofriction laws remain practically unchanged when compared with the values of the microconstitutive law (equation 1). No significant size effect was found, except for the decrease of the friction coefficient when this parameter is assumed to be a two-dimensional random field. The second microconstitutive law assessed in this study is the slip-weakening law shown in Figure 13 and represented by equation (2): sr arn (arn 0.8arn)
x , b
(2)
where a ls and b 0.50. The parameters of the macroconstitutive laws fitted to the simulated results are given in Table 3. Table 4 presents the mean values and coefficients of variation of the parameters of the macroconstitutive laws evaluated for the same five zones on the blocks interface described earlier for coefficients of variation null and 25%. The macroconstitutive
laws for the fault relate the mean shear strength (sr) at the interface with the mean normal stress (rn) and the mean slip (x) at the interface. Again, as shown in Tables 3 and 4, the parameters of the macrofriction laws remain practically unchanged when compared with the values of the microconstitutive law (equation 2). No significant size effect was detected, except for the decrease of the friction coefficient when this parameter is assumed to be a two-dimensional random field.
Influence of Fracture Near the Sliding Surfaces In addition to the nonhomogeneity of material properties, other factors may influence the macroconstitutive law of the fault. The effect of the possible occurrence of fracture in the neighborhood of the sliding surfaces has been recently considered by Miguel (2005) and Riera et al. (2005). Macrorelations between the resultant normal and shear stresses at the boundary of the blocks and the mean displacement and/or velocity are sought, which may be used in the form of effective constitutive relations, in conjunction with large DEM elements.
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Table 4 Parameters of Macroconstitutive Laws, by Zones, for Slip Weakening Condition
Parameters
ls ls ls ls
0.2 0.5 1.0 1.5
CV All 0% CV All 0% CV All 0% CV All 0%
am 0.200 am 0.500 am 0.999 am 1.500
acv acv acv acv
0.00% 0.00% 0.05% 0.09%
bm bm bm bm
0.500 0.500 0.500 0.499
bcv bcv bcv bcv
0.37% 0.54% 0.00% 0.41%
ls ls ls ls
1.0 1.0 1.0 1.0
CV Density 25% CV Young 25% CV Friction 25% CV All 25%
am 1.000 am 1.000 am 0.870 am 0.862
acv acv acv acv
0.00% 0.04% 16.3% 17.4%
bm bm bm bm
0.501 0.500 0.500 0.505
bcv bcv bcv bcv
0.33% 0.49% 0.18% 1.70%
CV, coefficient of variation.
Figure 16. Macro velocity-weakening laws, assuming a micro velocity-weakening law and nonhomogeneous material properties (from Miguel et al., 2006a).
Figure 17.
Macro slip-weakening laws, assuming a micro slip-weakening law and nonhomogeneous material properties.
Miguel (2005) employs two methods to take into account the occurrence of fracture in the neighborhood of the seismic fault. The results show that the macroconstitutive laws that take into account the occurrence of fracture, ob-
Figure 18. Macro velocity-weakening law when fracture occurs in the neighborhood of the fault.
tained by both methods, remain similar to the respective macrolaws without fracture, presenting oscillations, however, that do not occur when fracture is not considered, as may be seen in Figure 18. The effect was observed with homogeneous and inhomogeneous material properties and for both the slip- and velocity-weakening friction laws. It suggests that the occurrence of fractures adjacent to or springing from the fault surface, with dimensions of the order of dozens of meters, should induce slave vibrations, but not cause significant changes in the mean macrofriction law. The occurrence of large fissures, when compared with the length of the elements used in the analysis, can be explained by DEM models or other numerical methods. Cracks smaller than the size of the elements in the mesh, in seismology typically in the range between 0.5 and 1 km, must be taken into consideration, when necessary, by indirect means, as suggested in the preceding paragraph. Note, in this context, that large cracks, exceeding the range indicated previously, may be responsible for perceptible differences in the surface accelerations, as shown by Dalguer et al. (2003) and should not be confused with the small-scale cracking discussed in this section.
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Influence of Rupture of Protrusions (Microasperities) on the Sliding Surfaces When a shear failure occurs, the blocks of rock slide, one in relation to the other. During this sliding process, rupture of protrusions (microasperities) on the sliding surfaces may occur at isolated points on the fault. Thus, it is important to determine how the rupture by shear and/or by shear and compression of such microasperities on the fault surface may affect the macroconstitutive law of the seismic fault (Miguel, 2005; Miguel et al., 2006a). Because the macroconstitutive law is intended for use in conjunction with mesh sizes of the order of 0.5 km or more, attention is restricted here to small protrusions, with dimensions up to dozens of meters. To simulate the rupture of microasperities by shear and/ or by shear and compression, numerical simulations were performed using cubic samples of rock, which generated data for deriving a constitutive relation for fracture of representative samples of rock protrusions (Miguel et al., 2006b). The response of cubic elements fixed at the base to shear loads on the opposite face and the influence of restraints to vertical motion, which would introduce compression in these dowel elements, was evaluated numerically. Simulations were performed for various element sizes, which led to a mean constitutive law, relating the mean shear stress with the mean displacement on the top face of the cube, both with (i.e., shear and compression) and without restraint to vertical motion (i.e., shear only). Figures 19 and 20 show the mean stress-displacement curves for the 1-m cube, without restraint to vertical motion and with restraints to vertical motion, respectively. Next, a curve was fitted to the descending branch of the constitutive relation, which corresponds to fracture. The laws fitted to the cases without compression and with compression, illustrated in Figures 21 and 22, respectively, are considered as the micro (or local) laws in randomly chosen nodes at the interface of the numerical model of the seismic fault, to represent the fracture of existing protrusions (or microasperities).
Figure 19. Constitutive relation for a 1-m-long cubic rock sample, without restraints to vertical motion.
Figure 20. Constitutive relation for a 1-m-long cubic rock sample, with restraints to vertical motion.
Figure 21. Simulated descending branch of the constitutive relation and proposed approximation, without restraints to vertical motion.
Figure 22. Simulated descending branch of the constitutive relation and proposed approximation, with restraints to vertical motion.
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Figure 23. Macro velocity-weakening law when rupture of microasperities occurs on the fault surface (from Miguel et al., 2006a).
Because rupture by shear and/or by shear and compression of these microasperities on the surfaces in contact must occur mainly at the beginning of sliding, this phenomenon was modeled by the macroconstitutive law shown in Figure 23, called modified velocity-weakening law (Miguel et al., 2006a). The modified velocity-weakening criterion is compatible with numerous studies that suggest that a symmetric velocity-weakening law tends to overestimate the friction forces as the velocity decreases to zero. The law is also compatible with studies that show that the slip-weakening model underestimates the friction forces at the end of the motion. One drawback of the model under discussion is that, if the assumed mechanisms are correct, that is, if the resultant tangential forces at the fault interface result from the addition of friction forces and fracture of microasperities, then the law proposed by Madariaga et al. (1998), that combines both models in a slip- and velocity-weakening law, might reflect more directly the physical phenomenon. Nevertheless, it is still not clear to us how these microasperities should be typically distributed along the slip zone. It seems obvious that at least one such protrusion should fracture before sliding begins. This would be compatible with the empirical observation that the shear stress is higher at the initiation of sliding, and also at the end, because it seems very likely that sliding stops when another microasperity is activated.
Conclusions This article aims to determine a macroconstitutive law for the seismic fault that correctly describes the relation between the relevant static and kinetic variables. Although both velocity-weakening and slip-weakening laws are widely used in seismology, on the basis of experimental results with solids at laboratory scale, we concluded that only a velocity-weakening law would be formally correct when there is no fracture or damage on the sliding surfaces. To assess scale effects, numerical estimates of macro-
constitutive laws were determined, assuming that the microconstitutive law is specified at local discrete points of a numerical model of the fault region, whereas the macrolaw is the average over larger areas on the interface. The blocks that model regions adjacent to the fault were first assumed to be linearly elastic and homogeneous, in which case no size effect was observed, as expected. However, minor changes in the parameters of the macrolaws, in relation to the microconstitutive law, were detected. Next, nonhomogeneous materials were considered, assuming that mass density, Young’s modulus, and the friction coefficient on the fault were correlated Gaussian random fields. Size effects remained marginal, except for the influence of a random distribution of the friction coefficient. Fracture in the region surrounding the fault, which might be an important factor, does not cause significant modifications in the macroconstitutive laws, although it produces some perturbations around the law without fracture, in the form of high-frequency ripples. The numerical simulations suggest that the macroconstitutive law for the fault reproduces the microfriction model, if fracture of microasperities does not occur at the beginning of sliding. Actually, the friction law seems to be better modeled by the modified velocity-weakening law proposed in the present article. This law is compatible with numerous studies that suggest that a traditional velocity-weakening law tends to overestimate the friction forces at the end of the motion.
Acknowledgments We acknowledge the support of CNPq and CAPES, Brazil.
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[email protected] [email protected] Manuscript received 15 May 2006.
