Once the solution 22 has been obtained, the dislocation distribution can be computed from 14 ; the slip amplitude and the dislocation stress can be easily computed from the following see also Bonafede et al. Bonufede and M. From 24 it appears explicitly that s, is generally infinite at crack tips. As noted in Appendix B, two features are shared between the truncated and the exact generally unknown solution: the initial and the final configurations are exactly the same, i.
Outside this region the plastic threshold is infinite so that slip is confined within -u, a. The eigenvalues kj of the operator d M as functions of the truncation order M. Since a and L' are fixed parameters. The way in which the eigenvalues of the truncated operator r Mdepend upon the truncation order M is shown by Fig. The slip function for three values of T. Truncation order need be larger only for short times. SCIS 3 2 1 Figure 6.
The stress history inside and outside the crack. Note the stress singularity just outside the crack tip, and the complete release of stress just inside it. As shown by Fig. This crack model of course presents stress singularities at crack tips: the stress history inside and zyxwvut outside the crack is shown in Fig. This is not possible if the positions of crack tips are fixed as it was the case in the previous section , since in such a case the stress contribution produced by crack slip is, in general, not bounded equation If however the active stress s, -- r is non-uniform, Bonafede et al.
Equation 33 seems more suitable to solve than 3 2 ; however, it must be born in mind that 3 2 provides a bounded stress field even for the truncated solution while 33 will not, in general.
Borrafede and M. Furthermore, the characteristic time T in equation 9 is proportional t o a, so that the adimensional time 7 equation 8 is n o longer proportional to the real time t. Bonafede et al. It is shown in the Appendix C that. Consistency is hence achieved between crack models of intense-shear deformation zones and constitutive laws of plastic media e. Fung S : creep takes place if and only if the local value of the plastic threshold is exceeded. Hence, in the present case the crack equations are given by 16 and 32 t o be solved simultaneously.
These values will then be inserted into 16 - 19 to obtain the approximate solutions 23 - Apart from the initial and final limit configurations, which are exact for the truncated solution as noted at the end of the Section 3. However, this value approxi- mates to the plastic threshold Y at the crack tip only when the truncation order is high enough. Accordingly, the discrepancy between s and rhs of 2 , as computed from the truncated solution for a!
M , gives an indication regarding the speed of convergence of the truncated solution itself. If, however, the stress field inside the crack is computed from the lhs of 1 S , it coincides with Y at the crack tips but does not match exactly the value of zyxwv the stress outside.
This mismatch is generally quite small and is mostly localized in a small neighbourhood of the crack tip, as will be shown by the following examples see Fig. Crack half-length u as a function of time for parabolically varying yield threshold and constant effective viscosity thin lines or parabolically varying viscosity thick line. Again the truncation order need be large only for short time values.
Dragoni zyxwvutsrq 4. Accordingly G,, is independent of crack parameters and the same holds for the eigenvalues Xi. Slip history for the non-singular crack model described in the text.
The stress within the crack tips evolves from zero at time zero, t o Y x when time goes t o infinity. It is com- puted in two ways: as the left and right side of 15 from the truncated solution for a!
Outside the slipping section, stress is zyx computed from 24 : it is finite at crack tips, where it merges continuously with the internal zyxwvut stress and decreases sharply outward. Stress field produced by crack slip. Parabolic yield threshold and constant viscosity profiles. Bonafkde and M. The solution for a as a function of time is obtained from 32 and is shown as a thick line in Fig. Considerations very similar to those made in the previous case apply to the present one.
Furthermore, the higher value of the viscosity at the crack tips makes both Au and a vary slower than in the previous case.
Comparing Fig. Dragoni P 1. Parabolic profiles of yield threshold and effective viscosity. Solutions for a t. A u x , t , s, x, t are shown in Figs 13, 14, 1 5 and 16, respectively. These models are consistent with constitutive laws of plasticity since the classical singularity of the stress field outside the crack tips is removed, provided that the crack tips are not fixed or imposed a prion but are solved through 32 or Crack half-length Q as a function of time when tectonic stress increases linearly in time.
The parabolic profile 46 is employed for the yield threshold. The thin line is a plot of the crack half length computed after neglecting the viscous stress contribution in 3 rate independent plasticity. The bottom time-scale applies t o curve a , the top scale to b and c. Both scales apply to the thin and dashed curves depending on the value assumed for the stress rate. Convergence of the truncated solution has been shown to be fairly rapid and the behaviour of the truncated solution is exact at time zero and infinity.
In the following we wish to discuss briefly how such models may be pertinent to geo- physical situations. Sibson , recently proposed a crustal model which allows a better understanding of fault-zone mechanics: the vertical profile of strength in the crust is basically pressure-dependent at shallow depth, where the frictional regime dominates, and mainly temperature-dependent at greater depths, where plastic deformation is enhanced from the mylonitization of quartz-bearing rock.
However, Sibson implicitly assumes in his model that strain rate is constant with depth. Slip history when stress increases linearly in time. Total stress-field during crack slippage for the model described in Fig. The dashed lines show the value of the tectonic stress as time increases. Each curve is labelled by the appropriate value of rI7-y.
The thick line is the parabolic yield threshold This mechanism must however avoid the stress singularity provided by classical crack models, in order to be consistent with the basic principles of the theory of plasticity. The crack model presented above satisfies these criteria. The assumption that the anelastic deformation takes place within a narrow layer of plastic material seems justified, since the downward continuation of fault zones as localized mylonite shear-belts in the mid- t o deep-crust is well documented.
Other causes may add to enhance the rheological properties of a fault zone, such as chemical alteration Stierman , the presence of fault gouge and the deep circulation of water Wang Further- more, Yuen et al. An effective viscosity profile which varies with depth allows rheological variations due to compositional layering, pressure and temperature gradients to be taken into account.
Of course our models employ the idealized constitutive relation of a Bingham fluid, which is the simplest constitutive relation to be employed in the framework of crack mechanics. Although simplified. Not much is known regarding the plastic properties of rocks, but still the concept of a plastic threshold, above which plastic flow is strongly enhanced, is reasonably well- defined at least for quartz bearing rock e. Dragoni zyxwv plastic flow, as studied in the laboratory, is a strongly non-linear phenomenon, SO that our model can be applied only if linearization about Some reference state is introduced.
Apart zyxw from linearization suggestions have sometimes appeared in the literature that the constitutive relation of fault gauge may be linear at very slow strain rates e.
Etheridge er al. An apparent limitation of the previous model is the lack of a free surface. However, a free surface could be easily introduced into the present framework, employing the method outlined by Erdogan et al. The assumption of a yield profile which presents a minimum value at mid-crustal depth was a crucial assumption in the previous models; if the yield threshold was t o be a monotonically decreasing function of depth, non-singularity of the stress field at the bottom tip could not be achieved [in such a case equation 32 would provide an infinite solution for c - a , which would hinder the use of Chebyshev polynomials].
Our assumption is, however, consistent with the compositional layering which is currently thought to provide Bruhn However the theory has been developed in complete generality and there is no difficulty in applying it to given realistic profiles, provided that linear flow laws can be used.
Our ability to work out crack models which are compatible with stress-threshold slip criteria bears important consequences on our understanding of stress-transfer processes between shallow fault-sections, locked by high friction, and deep creeping fault sections.
The present crack model sheds light on the modalities of vertical stress-redistribution following post-seismic rebound or pre-seismic stress build-up. From the case studies presented in this paper, at least qualitative inferences can be drawn that lead us to envisage the following sequence of processes in the earthquake cycle. While tectonic stress ,s builds up, over large-scale areas, stress in excess of the yield threshold relaxes on deep fault sections and, in order to maintain equilibrium, additional stress is thrown on to the shallow fault section.
This stress contribution has a sharp maximum in the proximity of the crack tips, but is not infinite as in classical crack models. The latter mechanism is much more efficient in transferring stress. As time goes on, the aseismic crack tip will eventually transfer the maximum stress t o a point where the total stress equals the brittle rock strength, defined as the static frictional threshold according t o Sibson Before the present model can be used to simulate the deformation observed in real fault regions, however, knowledge is required of realistic yield threshold and viscosity profiles.
Furthermore the presence of the free surface and the viscoelastic coupling with the asthenosphere should also be accounted for. This will possibly be the subject of a zyxwvutsr future paper. Boschi and R. Lupini are gratefully acknowledged. Thanks are due to M. Bacchetti and M. Jannuzzi for technical help. Barenblatt, G. The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses, J. Barker, T. Quasi-static motions near the San Andreas fault zone, Geophys.
Bilby, B. Dislocations and the theory of fracture, in Fracture - an Advanced Treatise vol. I, pp. Liebowitz, H. Blacic, J. Plasticity and hydrolytic weakening of quartz single crystals, J. Bolt, B. A,, Bonafede, M. Implications of stress concentration on a strike-slip fault in an elastic plate subject to basal shear stress, Geophys.
Quasi-static crack models and the frictional stress threshold criterion for slip arrest, Geophys. Focal depths of intracontinental and intraplate earthquakes and their implications for the thermal and mechanical properties of the lithosphere, J.
Dmowska, R. A mechanical model of precursory source processes for some large earth- quakes, Geophys. Dugdale, D. Yielding of steel sheets containing slits, J. Solids, 8, Erdogan, F. Numerical solution of singular integral equations, in Mechanics of Fracture, Vol.
Sih, G. Etheridge, M. High fluid pressures during regional metamorphism and deformation: implications for mass transport and deformation mechanisms, J. R e x , 89, Explore Podcasts All podcasts.
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