versión On-line ISSN 0718-2813
Obras y Proyectos no.11 Concepción 2012
Obras y Proyectos 11, 48-60
ON THE MILD AND WILD MECHANICAL ROUGHNESS OF THE LITHOSPHERE
Sobre la suave y salvaje rugosidad mecánica de la litósfera
Institute of Soil Mechanics and Rock Mechanics, Karlsruhe Institute of Technology, Karlsruhe, Germany, email@example.com
The spatio-temporal mechanical fractality of the lithosphere can be represented by means of Lévy-statistics and fractional calculus. This novel concept is outlined with geometrical and energetic arguments. The fractality originates from the pore system and appears in displacements, stresses and seismicity. It is shown that the roughness may be neglected for the stable range, but not for critical phenomena at the verge of energetic convexity. The outline leads from rough coastlines via sand samples to tectonic chain reactions. Equations and theorems are physically interpreted without algebra and proofs.
Keywords: fractality, lithosphere, roughness.
La fractalidad mecánica espacio-temporal de la litósfera puede ser representada por medio de la estadística de Lévy y cαlculo fraccional. Este nuevo concepto es planteado con argumentos geométricos y energéticos. La fractalidad se origina del sistema poroso y aparece en desplazamientos, tensiones y sismicidad. Se muestra que la rugosidad podría no ser considerada en el rango estable, pero esto no es posible en fenómenos críticos al borde de convexidad energética. Este planteamiento cubre bordes costeros rugosos, muestras de arena y reacciones tectónicas en cadena. Ecuaciones y teoremas son interpretados físicamente sin la necesidad de αlgebra ni pruebas.
Palabras clave: fractalidad, litósfera, rugosidad.
The external fractality of the lithosphere was discovered by Mandelbrot (1982). Rough coastlines can be reproduced in hatched magnifications of sections. The log-log plot of the minimum number N of squares needed to pave a coastline (Figure 1a) with squares versus their size d is a straight line as shown in Figure 1b. This means N = Nr (dr /d)γ with reference values Nr , dr and a fractal dimension 1 < γ < 2. The self-similarity holds within bounds or cutoffs dmin and dmax. The coastline length is thus l ~ Nr dr (dr /d)γ-1. The rough surface of a landscape can exhibit a similar fractality (Figure 2a). The minimum number N of cubes of size d needed to enclose the surface is N = Nr (dr /d)2Υ if plane cross sections exhibit α as above. The area is thus A ~ Nr dr(dr /d)2(γ-1). Length and area are no more extensive as without roughness, i.e. for γ = 1.
A rough line or surface is called monofractal if its α is the same everywhere, otherwise multifractal. This concept works likewise for time-dependent evolutions observed at the surface of the lithosphere, e.g. seismograms of points or hydrograms of rivers. Mandelbrot (1999) uses the rather poetic expression mild and wild roughness in case of mono or multifractality, respectively. These notions may be taken over to spatially and temporally fractal distributions of displacements and forces, i.e. to mechanics with roughness. This requires an objective definition of gradients and time rates although rough distributions versus site xi and time t have no unique tangents. Corresponding objective integrals over Xi and t are needed for balances of classically extensive quantities, viz. mass, energy and momentum or parts of them.
Mandelbrot knew of course that landscapes are shaped by tectonics, erosion and sedimentation, but his algorithms produce at best look-alikes with roughness and do not represent mechanisms. He attributed multifractals to 'multiplicative cascades', but this again is mere mathematics and not yet physics. Humid sand surfaces, e.g. in a box or at the beach (Figure 2b), can exhibit fractality, and may substitute theoretical physical models with roughness. Scaling up model test results to bigger sizes and longer times requires a dimensional backbone from physical theories. This is not supplied by the classical rules (Gudehus 2011). of similarity in case of fractality.
The internal fractality of lithosphere sections - or of small-scale substitutes - is not as thoroughly visible as the external one, boreholes and sensors give at best fragmentary hints. Natural pore systems (πopoζ = passage) appear to be fractal. Their monofractality can be explained by means of rather fictitious data (see in the next section). Systems of tectonic faults, which are more porous than their vicinity, exhibit self-similarity in hatched sections. This kind of fractality is likewise observed in model tests and hypoplastic simulations as shown in Figure 3 (Gudehus 2011). The non-uniformity of groundwater flow points also to fractal pore systems (the permeability is proportional to the square of the pore size). It appears that the complete lithosphere, i.e. from its free surface down to the Moho discontinuity, has a fractal pore system, of course with lower pore fractions for higher pressures and temperatures.
Spatial distributions of internal pressures and stresses exhibit likewise fractality. This force-roughness plays already a role for soil samples, when one tries to capture it by hidden variables like back stress or intergranular strain Such approaches are not explicitly fractal, but tacitly scale-independent within reasonable bounds. The force-roughness is known from model tests and building sites with sand, it causes an indeterminacy of structural forces in or at the ground. It is more marked for rocks so that stresses at samples or structures in situ can hardly be determined.
The world stress map of the lithosphere exhibits spatial fluctuations with wavelenghts from ca. 1 km to 1000 km (Heidbach et al. 2010). Stresses in the lithosphere are estimated by means of borehole measurements and plate tectonics. Several decades of fluctuation lengths suggest fractality, certainly multifractality in connection with which convection cycles of the asthenosphere impose some order to the lithosphere.
The force-roughness can be frozen for a while in stable sections of the lithosphere (what that means will be explained later on). Changes at boundaries - from above, below or aside - lead to changes of position and state of lithosphere sections or its small substitutes. Apart from smooth and rather slow changes due to thermal activation, such changes are jerky and accompanied by seismicity. The sound-roughness is audible as crackling of soil and rock samples, and more dramatically with breaking structures at or in the ground. Earthquakes exhibit similar emissions with lower frequencies due to bigger extensions. The spectra are often simplified as 1fnoise, this notion is also used by Mandelbrot (1999) if the slope in a log-log spectrum diverges from -1 and indicates fractality. One may understand force-roughness as latent sound-roughness, and would like to know when and how the latter arises.
The dynamics of the lithosphere is evidently multifractal and so complex that at best sections or substitutes of them can be captured by mechanical models. Initial states are inevitably partly subjective, and an additional indeterminacy arises from arbitrary boundaries separating near-fields from far-fields (Gudehus 2011). The interplay of thermal and seismic activations depends on space and time scales and it should not be separated from the fractality. Gain and loss of stability is evidently a key issue, but the treatment of it with fractality poses open questions. Critical phenomena with pattern formation and chaos for geo-matter were first considered with minute sand avalanches (Bak et al. 1987). Similar attempts were made with fractals in structural geology (Turcotte 1997) and rock mechanics (Xie 1993).
Recently, Gudehus and Touplikiotis (2012) proposed a more comprehensive approach by means of the fractional calculus, and a more mathematical paper on fractional stability is underway (Gorenflo et al. 2012). The present paper is less mathematical, however, it employs further physical arguments. First, the transition from fractal and Lévy-type distributions to fractional balances is outlined with the focus on geo-matter. Second, attenuation and subordination are introduced for fractionally uniform zones and indicated beyond. Third, the quasi-local loss of stability and then with chain reactions of increasing size are presented. Fourth, the interplay of seismic and thermal interactions is touched. Finally, conclusions with summary and outlook complete this more qualitative than quantitative paper.
From fractals to fractional balances
Consider the fractal pore system of an idealized layer as shown in Figure 4. Straight lines cut solid and void fractions in such a way that hatched magnifications exhibit a similar rough sequence (Figure 4a). The solid mass m in a cube of size d yields a straight log-log plot (Figure 4b) with reference values mr,dr and a fractal dimension ca.
A log-log histogram of numbers Np of pores with size dp yields a straight line section with slope -3â, Figure 4c. This corresponds to the major part of a skew stable Lévy-distribution (Sato 1999). More precisely speaking, such a probability distribution comes close to a power-law in the falling branch. It is called stable since any sum of independent random values with such a distribution has the same kind of distribution. This uncommon central limit theorem holds strictly without cutoffs, but then mean values and higher moments diverge. There is no divergence with cutoffs, and random sums tend again to such distributions except for extremely big numbers (Mantegna and Stanley 1994). Such distributions can arise from successive fragmentation or agglomeration, e.g. sizes of grown-up animals from microbes to giants in habitats can exhibit them.
Vertical pressures by weight in fractally porous layers can be calculated by adding up the solid volumes of cubes, multiplying by specific gravity gps and dividing by the base area. Figure 4d shows that a prism of height h and cross section b2 has thus
More generally speaking, the Riemann integral over a fractal distribution is equivalent to a fractional integral (Ren et al. 2003). The latter is a generalization of Cauchy's formula for multiple integrals to repeated integration of fractional degree a (Gorenflo and Mainardi 2000). It requires the Gamma function α and dimensionless variables, e.g. by means of a reference length dr. With fractal uniformity (1)
and (2) indicate that the weight is no more extensive as without roughness (cf. length and area in the Introduction). We call quantities fractionally extensive which would be extensive (in Rankine's sense) without spatial roughness, but they have to be expressed by fractional integrals with it. Cutoffs can be applied using expression (2) as well as (1) by keeping b within dmin and dmax < h.
of secants for different section lengths, or as an objective gradient of a fractal distribution by means of the Gamma function and a reference length for normalization. This gradient is no more local in the classical continuum sense, but refers to a point in the fractional image of roughness. In the case considered here the fractal distribution represents the internal force-roughness by weight (Figure 4e). More
Turning to temporal fractality, we consider now a fractally uniform cubical range ofjammed rock fractions. The latter may be grains or more densely packed pieces of rock. They constitute a perfect sample without deviations from fractal uniformity along the boundaries. We consider evolutions of spatial averages with time t, keeping in mind that spatial distributions are fractal so that classically extensive quantities are fractionally extensive as outlined above for weight. Imagine non-disturbing sensors which indicate seismicity, stress and shape of our cube. Figure 5 shows idealized plots of microseismic bursts and simultaneous sudden changes of stress and shape. This may be concluded from triaxial tests with dry hard-grained samples, confined by a membrane under constant pressure (Gudehus and Touplikiotis 2012). Slow axial loading causes audible microseismic bursts, and almost simultaneous changes of overall state and shape as long as the travel times of microseismic waves within our cube are much shorter than the intervals between seismic bursts.
Cube-averaged extensive properties Xi, in particular seismic energy, boundary forces (expressed by stress components) and lengths (changing by deformation) exhibit the same temporal fractality. Although the temporal fluctuations have different intensities box-counting as in Figure 1, would thus yield the same fractal dimension. This justifies fractional balance equations
(3) works only for increasing t, this may be attributed to causality for dissipative evolutions.
Balance equations like (3) are coupled by means of Xi-dependent Ki and μi. In an economic sense they express that the capital of market partners varies fractally with time by capital-dependent loss, and by input or output among each other and with the surroundings. For our idealized cube of jammed rock fractions coupled balances refer to shape and energies (elastic, seismic and thermal). This will be partly outlined later on and balances of extensive quantities will already be treated in the sequel. Coupled balances like (3) can be transformed to classical, i.e. non-fractional ones by means of fractional time stretching. This means that t is transformed so that fractional integrals change into ordinary ones, and it is justified for the stable range with ca. 0.9 < α < 1.
Spatio-temporal balances of fractionally extensive quantities Xi can in simple cases be written as
with reference quantities Xir, tr, dr, and fractalities α, â as outlined further above and summation over j for fractional gradients. Expression (4) represents a balance for a section of space-time with fractal distribution. The loss term with Ki does not need to be proportional to the fractional gradient of Xi as written for simplicity in (4). In addition to shape and energies as for fractally uniform cubes, balanced quantities are mass and momentum, both are conserved for sums of constituents.
We assume isofractality, i.e. α = β. This is justified by the same narrow range of α and β and by the spatio-temporal coupling via microseismic elastic waves. The latter suggests dr /tr = cs with the velocity of elastic shear waves cs for isotropic states. With higher void ratios and lower pressures α decreases, which depends also on mineral and temperature. Apart from this moderate multifractality (see more in subsequent sections) one can transform (4) into non-fractional balances by means of a fractional space-time stretching (Gudehus and Touplikiotis 2012). This is justified for the stable regime with the narrow range of α.
Attenuation and subordination
As proposed by Gudehus (2010) for sand, we assume the specific elastic energy
with normalization as further above and summation in j. It expresses the conservation of momentum with fractality as by (4) without input as small deviations do not change the weight. Insertion of (6), (8) and (9) into (10) yields the fractional wave equation
The vectorial fractional differential equation (11) is not given in the literature. Only scalar special versions, which can be derived by confining to isotropic states (Gudehus and Touplikiotis 2012), have been solved for certain initial and boundary conditions (Mainardi and Tomoirotti 1997). They reveal a hysteretic damping for the hypoelastic range which would be missed by spatio-temporal stretching so that (11) is simplified with α = 1 to the version of Mayer and Liu (2010). A novel solution of (11) was obtained with simultaneous Laplace and Fourier transformations and Green functions (Gorenflo et al. 2012). Other than without roughness, whereby a classical ansatz leads to an eigenvalue problem as mentioned above, a fractional eigenvalue problem is only obtained with integral transformations. For the stable range all eigenvalues are real so that propagations can occur. However, due to the fractality elastic waves are no more sinusoidal and dissipation occurs since any wave generates smaller irregular ones. Extended exponential functions, which were proposed by Mittag-Leffler 100 years ago, provide an orthogonal base.
Without fractality hypoelastic oscillations would last forever and could be decomposed into harmonics or eigenmodes. There are no harmonics with fractality, but with stability the wave equation dictates a characteristic autogeneous dwindling after an initial disturbance. It can be proven (Gorenflo et al. 2012) that the power-law spectrum of an initial seismicity experiences a downward and rightward shift in the log-log plot (Figure 6). The amplitudes fade with t by power-laws as all wave fractions induce erratic oscillations with higher frequencies due to the fractal pore system. This kind of attenuation works as an autogeneous attractor for different initial seismicities and is characteristic of stable states with fractality. This works already for the hypoelastic range and without gradients. With them the attenuation is enhanced by radiation. With dissipative dislocations the attenuation after an initial small disturbance is faster than hypoelastic and leaves back a small change of position and state, whereas it would not cause any trace without roughness.
A cubical fractally uniform zone of jammed grains may now be deformed monotonously from a stable state of rest. After a hypoelastic onset microseismicity arises with an intensity Ts which may be named seismocrasy (Gudehus 2010). This name corresponds to the Greek 'thermocrasy' for temperature and is more suitable than 'granular
strain is augmented by overall stretching and reduced by seismically activated relaxation.
Better than with the classical calculus, the balance equations can be written as in (3) for temporal fractality (Gudehus and Touplikiotis 2012). They are coupled via Ts and state that
- the seismic energy rises by average relative velocities of grains and dwindles by transition into heat,
- the elastic energy rises by changes of shape and dwindles by seismic relaxation.
Monotonous deformations with fractional rate D lead to subordination, which is characterized by a monotonous exogeneos attractor (Gudehus 2010), works in the stable range. It can be reduced for non-fractional hypoplastic relations by fractional time-stretching (Gudehus and Touplikiotis 2012). Subordination implies rate-independence.
One may argue that the additional effort of using the fractional calculus is not worth it if it turns out that it is not needed. Well, not really, because the mechanical roughness is real and should not just be ignored. With the above argument the success of non-fractional models is further justified in spite of fractality for the stable range (but not otherwise, see next section). The argument works also for evolutions with pulsating seismicity in the stable range, i.e. for subordinations which imply pulsations of Ts. The behavior is then intermittently hypoelastic and hypoplastic with likewise rate-independent transitions. This kind of subordination is characterized by cyclic exogeneous attractors (Gudehus 2011). Fractional time-stretching leads to non-fractional equations which produce realistic attractors (Gudehus et al. 2010). This means that hidden variables like back stress or intergranular strain for the internal force-roughness are not needed in energy-based models in spite of the actual fractality. This helps in particular for cumulative effects (Gudehus 2010) and for rather fractal driven variations of Ts.
A kind of quantum seismodynamics is desirable for explaining the microseismicity during subordinations of cubical zones of jammed grains. The acoustic emission indicates seismons, i.e. seismic monads (Gudehus 2010), with a truncated power-law spectrum. These can be attributed to successive bursts of fractally distributed groups of grains which are jammed up to the stability limit. In the stable range seismons have randomly distributed polarizations, but other than with phonons for thermal oscillations the seismic energy is not conserved and not Gauss-like partitioned. The fractional Schrodinger equation
m and potential V. Like its precursor for particles (Laskin 2000), expression (12) yields truncated Lévy-spectra by means of Laplace and Fourier transformations. This suits to the stability of Lévy-distributions and the truncated fractality of microseismic bursts. The original Schrodinger equation, i.e. (12) with α= 1, yields much richer discrete spectra. The matter becomes more intricate with critical phenomena (next section).
Subordination in the large can occur in lithosphere sections, or in small substitutes at beaches or building sites or in model setups. It requires stability and excludes thermal activation (see last section). Coupled fractional balance equations such as (4) may then be reduced to non-fractional ones by fractional space-time stretching (as in previous section). This can suffice for the mild subcritical roughness with ca. 0.9 < α < 1 for suitably chosen regions. Relics of former critical phenomena (next section) cause an inevitable indeterminacy which is linked with rather vague notions such as fabric and inherent anisotropy. The indeterminacy can be reduced sometimes by means of attractors in the large (Gudehus 2011), which constitute a subordination in suitable soil regions.
The issue is more complex with rock and with bigger sections of the lithosphere. The mechanical roughness is apparently wilder for building sites and mines, this can be attributed to spatially variable cutoffs from previous critical phenomena and implies more indeterminacy than with soil. Fault patterns in the lithosphere are evidently multifractal and can lead to further critical phenomena (next section). Cases of coarse-graining may still correspond to subordination of the earth crust by magma convection below, but even in those cases seismic features of interest can at best be crudely estimated.
Loss of stability and chain reactions
Consider a fault system with a fractionally uniform zone at the verge of energetic convexity as shown in Figure 7a. This may represent a fractal fault zone with negligible fractional gradients at the verge of stability. One may imagine a zone in a fault system which is critical in a range with much smaller size than its depth. This specification is inevitably imprecise, an interpretation will be offered further below. Fractional uniformity means in particular that the stress tensor has the same alignment throughout our zone. Criticality with the simplest reasonable specific energy we by (5) means that we has a chair-point for the Coulomb condition r2p2 = 2b/(v - 2) (see Figure 7b). Then an increasing part of the elastic energy becomes kinetic for a one-sided section of elastic strain deviations.
Laplace and Fourier transformation with Green functions (Gorenflo et al. 2012) then produces a shift of an initial truncated power-law spectrum to the left and upwards as can be observed in Figure 7c. In other words, at a critical point an initial seismicity grows by the same factors for wave-length and -height throughout the wave package.
As for attenuations (previous section) the elastic strain components change with such a ratio that the loss of weis maximal. With two invariants and weby (5) this means that the spontaneous deviation from a chair point follows the steepest descent. This can be interpreted as maximal rate of entropy production due to the positive feedback of seismons (Gudehus 2010). The Lévy-spectrum exhibits a slight rightwards rotation of the log-log plot, but this kind of seismogenesis is still a stable Lévy-process. If the initial weak seismicity is not Lévy-like it is there with the named magnification so that one can speak of an autogeneous strange attractor (Gudehus 2011; Gorenflo et al. 2012).
The same spectral shift is also obtained with further degrees of freedom. The simplest model has only two invariants and two different stress components, one can incorporate a third invariant and a less symmetric stress state. One could extend the approach with polar quantities which arise from shear banding, this implies gradients which will be discussed below. One could also incorporate the extension of cracks by means of surface integrals (Budiansky and Rice 1973), and could include sorption energies for cracks (Rice 1978). The path-independent integrals enable energy balances for fractal crack systems, including critical points in the energetic sense of Griffith (1921). Fractional critical phenomena engulf thus the verge of energetic convexity and fractal distributions. This was addressed already in an overview paper on the mechanics of natural solids (Gudehus 2009), therein convexity was called concavity by considering an energy landscape from above, not from below as in mathematics and theoretical physics.
This approach is apparently insufficient as critical zones cannot exist with any extension due to their instability. One should instead imagine seismic reaction fronts which propagate to a certain extent. The front is fractal as with a forest fire or with irregular dominoes standing on a table. The reaction (e.g. burning or overturning) occurs in the front with a rather fuzzy extension, it leaves back a stable dead zone and propagates as long as a reaction in the zone ahead is triggered by the propagation. A chain reaction in a fractal fault zone propagates as long as the part ahead of the front is nearly critical so that the stress redistribution due to the collapse in the front leads to critical points immediately ahead.
One may try to extend Hadamard' s concept of propagating discontinuities to fractional models of fractal evolutions with reaction fronts, but this will be mathematically difficult. Instead and less strictly, one can estimate the rate of propagation via a gradient-delayed collapse. This means that any critical zone has gradients of state towards its non-critical vicinity which reduce the rate of extension of seismic waves due to sidewards radiation (Gorenflo et al. 2012). The rate of extension can be reduced to zero, then the propagation is stuck. The upper cutoff bounds the rate of extension, this is an estimate of the propagation velocity of the seismic reaction front. This estimate is crude as it stems from a fractional image of a fractal reality, since gradients are fractional and can at best be estimated, as well as fractional balances for a reaction front are partly represented.
Consider real earthquakes for illustration. The 9.5 moment magnitude earthquake in Valdivia, Chile of 1960, is still the biggest ever recorded, involved roughly 20 m dislocation in a subduction zone of ca. 1000 km long and roughly 100 km deep within ca. 1000 s. This means that the propagation of a seismic reaction front with up to ca. 1 km/s horizontally and 100 m/s vertically which came to an end farther off. In our fractional image a seismic chain reaction started somewhere near the surface and propagated with fractional speeds (in the sense of (8)) of the indicated order of magnitude. This represents a fractal reality, i.e. the extension of the critical zone and its propagation are rather fuzzy. The mechanism may be called a seismic soliton, in Mandelbrot's (1999) terms it means a transition from mild to wild roughness. The propagation could principally be recorded by suitably placed seismometers, and simulated by means of coupled fractional balances. Opaqueness and multifractality of the lithosphere delimit validations, but at least observed Lévy-spectra support our approach.
Data from other earthquakes could enable a sort of calibration, but this would be premature as our model should first be extended for further degrees of freedom (indicated above) and for thermal activation (next section). The Gutenberg-Richter rule confirms the Lévy-stability for any kind of superposition. The usual localization of active faults by means of P-waves (Hyndman and Hyndman 2009) could be improved by means of the actual fractality. Other than Burridge-Knopoff models (Cartwright et al. 1997), which are tectonically and tribologically over-simplified, our source mechanism is energetically sound and engulfs the real fractality. We concede that the real multifractality, which prevails due to earlier critical phenomena and arises again at critical points, is not yet properly captured. The narrow range ca. 0.9 < α < 1, however, would require only a minor correction of balances. What counts more is the strong variation of cutoffs and minerals; both could principally be taken into account, but the lack of data will always require hypothetic completions.
A recent work by Silva Bustos (2008) may serve for illustration. A cross section with seismic sources indicates that a subduction is a fractal fault system so that its extension cannot be precisely determined (Figure 8). The depicted sources were localized from arrivals of waves in several earthquakes, but many minor sources cannot yet be localized. Thus the fractality of the subduction zone is but roughly represented. It appears that seismic solitons were stuck after running from the outcrop ca. 200 km downwards. It will be difficult to localize moving seismic sources because of the fractal indeterminacy.
A log-log-plot of number N versus magnitude Ms of seismic events shown in Figure 9, confirms the Gutenberg-Richter rule. It suits to the reproduction of Lévy-spectra by weighted additions as the seismic energy Es follows logEs ~ Ms+4.8 empirically (Hyndman and Hyndman 2009). In other words, Gutenberg-Richter plots are a consequence of balaces with fractality and energetics at critical points (Gorenflo et al. 2012). The different slopes indicate a slight variation of fractality in our sense. This is not strictly valid as the usual determination of Ms and Es is metrically imprecise with fractality, but could be improved by means of the fractional calculus. As indicated further above this argument holds also true with new ruptures which require a Griffith-type and not a Coulomb-type verge of convexity.
The lithosphere dynamics implies further critical phenomena than just seismic chain reactions. One is the original freezing and cracking, a second one is presumably related with the Moho discontinuity. Slow tectonics with formation and modification of faults is more complex if it is accompanied by earthquakes, and is multifractal anyway. Outbursts of pore fluid and gas mean cold volcanism, A.v. Humboldt noticed already its relation with faults. Bigger magmatic spots of colliding plates, causing volcanoes and craters. Erosion and sedimentation imply critical phenomena with fluids. Man-made earthquakes occur if injections produce propagating critical zones which are delayed by diffusion, and similar effects occur after strong earthquakes (next section). A delicate issue is of course how recurrence times of natural earthquakes can thus increase.
The energetics at critical points means new territory in a physical and mathematical sense. The verge of convexity of the free energy is a general feature ofthe loss of stability, this holds from groups of molecules via conservative structures up to galaxies. An ensemble of molecules with Brownian motion exhibits then second-order phase transitions and arising fractality, this includes quantum effects (Binney 1992). Granular zones or formations of fissured rock are already fractal and become multifractal at critical points. The spontaneous growth of previously inactive degrees of freedom, typically related to shear banding or cracking, requires extensions of the fractional approach (indicated above). Balances of fractionally extensive quantities require fractionally extended differentials, this implies generalized seismic and thermal entropy productions. The quantum seismodynamics at critical points cannot be captured with (12) as for stability. It appears that two coupled fractional Schrodinger equations are needed as in quantum electrodynamics (Gudehus and Touplikiotis 2012).
Seismic and thermal activations
Deforming slowly, i.e. below ca. 10-3/s, a cube ofjammed rock fractions implies microseismic bursts with heatflashes. Thermography of dry sand samples revealed T jumps at grain contacts beyond 1300°K within milliseconds (Luong 1982). Except for rapid deformation the heat is rapidly diffused after seismic flashes so that an increase of T can hardly be observed. This holds also true for rock fractions in laboratory tests except for so high pressures p that the diffusion of heat takes more time than its generation. The molecules at seismogeneus particle contacts are dislocated when past each other with higher rates by simultaneously higher T. The seismic activation is thus partly thermal due to T-rises within very short times (compared with encounter times) at minute mass fractions of the solid skeleton.
Seismic bursts in fractal fault zones can produce less localized T-jumps with overall deformation rates of around 10-3 %/s as thermal diffusion times can exceed heat production times. T-rises within active fault zones can thus engulf more than a minute fraction of all molecules. This can lead to softening and melting, but hardly to runaway instability (Vardoulakis 2002) and reactions (Hatzo et al. 2009). This type of co-seismic thermal activation is confined to temporarily seismogeneous zones, which are fractally distributed and exist only for short times in passing seismic reaction fronts (as seen in the previous section). More confined seismic bursts during subordinations cannot produce T-rises as encounter times of rock fractions exceed by far thermal diffusion times. Thus heat flashes due to seismic bursts characterize also lithosphere sections in a coarse-grained view.
Thermally activated dislocations with constant T occur with rates which are proportional to exp(-Ea/kBT) where kB is the Boltzmann constant. Dislocation units in aggregates of jammed rock fractions have activation energies Eafrom about 0.5 eV for soft to 5 eV for hard minerals at T ~ 270°K with 40kBT ~ 1 eV. Their thermal activation causes moderate or low rate-dependence or relaxation and/ or moderate or very slow creep. Thus lithosphere sections are more slowly jammed by tectonic deformations, without these jammed zones relax and pore systems close gradually. Critical points are therefore later or not attained with sufficiently slow or interrupted tectonic base shift. This thermal effect is evidently more marked with higher T and lower Ea.
Pore fluids have Ea < kBT, adsorbates of them have Ea<10kBT. After squeezing or injecting pore fluid this diffuses towards equilibrium, before the solid partial pressure ps= p-pf is temporarily lower due to a higher fluid pressure pf (principle of effective stress). As the solid shear stress ô is thus not changed the solid skeleton can attain critical states so that the seismicity rises. Man-made earthquakes could thus increase the recurrence times of natural ones. Calculations require additional fractional balance equations for the pore fluid, coupled with the solid balances and with isofractality due to the fractal pore system and the seismicity. Again the roughness may be eliminated in the stable range by fractional stretching, but no more at critical points.
The interplay of thermal and seismic activations could approximately be followed up by separating mildly rough and rather aseismic slow periods from short seismic episodes with wild roughness. The latter are hardly influenced by thermal activation with uniform T as seismic solitons pass by within short times. There is no diffusion of pore fluid during such episodes, but certainly in rest periods thereafter. Fractally distributed seismic bursts produce flash-like T -jumps, rapid dislocations can heat mylonite bands in faults so that reactions are enhanced. Other than assumed by some authors (Vardoulakis 2002; Hatzo et al. 2009), however, such reactions occur in jerks and can at best enhance the seismogenesis like fluid injections. The spontaneous generation of kinetic energy occurs again at critical points of the free energy, now including chemical potentials with capillary effects. The Lévy-type seismicity can only arise from fractality. This holds also true for the seismogenesis deeply below the lithosphere with spontaneous phase transitions in shear bands.
The solid-fluid interaction is the most important type of interplay of seismic and thermal activations. Except for methane cushions the fractal pore system is filled with almost incompressible water and dissolved minerals. This fluid experiences fractional diffusion (Gudehus and Touplikiotis 2012) with seepage and dwindling porosity in aseismic periods, then solubles can also condensate. Seismic episodes in critical zones are far shorter so that simultaneous diffusion and condensation are negligible. Then the pore pressure pw is higher or lower than the previous hydrostatic one as the porosity tends to decrease or increase, respectively (Gudehus 2011). The tendency for contractancy or dilatancy depends on the porosity, and on the ratio of mean solid pressure ps and hardness hs. Similar changes ofpw andps are produced by injection or depletion of pore water.
The diffusion of pore water from a zone with seismogenesis by pumping in water causes a spreading of the zone with higher than hydrostatic pw. As the total pressure p=ps+pw and the deviatoric stress ô remain rather constant critical zones can arise in the neighbourhood if pw rises. This leads to delayed natural aftershocks or man-made earthquakes. It may suffice to estimate diffusion and related stress redistribution in the stable range without roughness, as this is mild and can be formally removed by a fractional stretching of spacetime, but this is no more legitimate at critical points. There is no way around the fractional calculus in case of seismogenesis, particularly with solid-fluid coupling.
We conclude in particular that lithosphere sections have a fractal system of pores and internal forces. Balances of classically extensive properties require fractional integrals or fractional differential equations. Attenuation and subordination in the stable range imply Lévy-type seismicity can be captured with fractional calculus, but balances without this mild roughness may often suffice.
can only be captured with the fractional calculus and wild roughness arises alongside from multifractality. Seismic bursts imply heat flashes which can cause local softening and reactions in short episodes, whereas thermal activation causes slow relaxation and creep so that tectonic jamming is reduced. The water pressure in the pores rises in seismic episodes, this rise spreads by diffusion and can cause aftershocks, this can occur similarly after pumping in water.
The agenda for further research comprehends in particular, inherent and changing multifractality, fault systems with gradients and multifractality, Griffith-Rice fracture with fractality, coupling of solid and fluid with seismic and thermal activation, analytical bounds and numerical simulations and monitoring and control in situ.
Andrzej Niemunis, Mauro Poblete and Asterios Touplikiotis improved this paper by valuable hints and technical support, this is great fully acknowledged.
Bak, P., Tang, C. and Wiesenfeld, K. (1987). Self-organized criticality: an explanation of 1/f noise. Physical Review Letters 59 (4): 381-384. [ Links ]
Binney, J. (1992). The Theory of Critical Phenomena. Clarendon Press. [ Links ]
Budiansky, B. and Rice, J.R. (1973). Conservation laws and energy release rates. Journal of Applied Mechanics 40, 201-203. [ Links ]
Cartwright, J.H.E., Hernαndez-García, E. and Oreste Piro, O. (1997). Burridge-Knopoff Models as Elastic Excitable Media. Physical Review Letters 79, 527-30. [ Links ]
Gorenflo, R. and Mainardi, F. (2000). Essentials of Fractional Calculus. Printed from MaPhySto Center, free online. [ Links ]
Gorenflo, A., Gudehus, G. and Touplikiotis, A. (2012). On the stability of a fractal fault system. Zeitschrift für Angewandte Mathematik und Mechanik ZAMM (under preparation). [ Links ]
Griffith, A.A. (1921). The Phenomena of Rupture and Flow in Solids. Philosophical Transactions of the Royal Society of London A 221, 163-198. [ Links ]
Gudehus, G. (2009). Mechanics of natural solids - a symposium in Horton, Greece. Obras y Proyectos 6, 4-7. [ Links ]
Gudehus, G. (2010). Psammodynamics - Attractors and Energetics. 9th HSTAMInt. Congr. Mechan., Limassol, online. [ Links ]
Gudehus, G. (2011) Physical Soil Mechanics. Springer, Berlin. [ Links ]
Gudehus, G., Jiang, Y. and Liu, M. (2010). Seismo- and thermodynamics of granular solids. Granular Matter 13 (4), 319-340. [ Links ]
Gudehus, G. and Touplikiotis, A. (2012). Clasmatic seismodynamics - Oxymoron or pleonasm?. Soil Dynamics and Earthquake Engineering 38, 1-14. [ Links ]
Guyon, E. et Troadec, J.-P. (1994). Du sac de billes au tas de sable. Odile Jacob, Paris. [ Links ]
Hatzo, Y., Sulem, J. and Vardoulakis, I. (2009). Meso-scale Shear Physics in Earthquake and Landslide Mechanics. Taylor and Francis. [ Links ]
Heidbach, O.,Tingay, M. and Wenzel, F. (2010). Frontiers in Stress Research: Preface. Tectonophysics 482, 1-4. [ Links ]
Hyndman, D. and Hyndman, D. (2009). Natural Hazards and Disasters. Chapter 3: Earthquakes and their causes. Brooks/ Cole: Cengage Learning, 2nd ed. [ Links ]
Laskin, N. (2000). Fractional quantum mechanics. Physical Review E 62/3, 3135-3145. [ Links ]
Luong, M.P. (1982). Mechanical aspects and thermal effects of cohesionless soils under cyclic and transient loading. IUTAAM Conf. Deform. Fail. Gran. Mat. Delft, 239-246. [ Links ]
Mainardi, F. and Tomoirotti M. (1997). Seismic pulse propagation with constant Q and stable probability distributions. Annali di Geofisica XL 5, 1311-1328. [ Links ]
Mandelbrot, B. (1982). The Fractal Geometry of Nature. W.H. Freeman, New York. [ Links ]
Mandelbrot, B. (1999). Multifractals and 1/f Noise - Wild Self-Affinity in Physics. Springer, New York. [ Links ]
Mantegna, R. and Stanley, H.E. (1994). Stochastic process with ultraslow convergence to Gaussian: The truncated Levy Flight. Physical Review Letters 73, 2946-2994. [ Links ]
Mayer, M. and Liu, M. (2010). Propagation of elastic waves in granular solid hydrodynamics. Physical Review E 82, art. 042301. [ Links ]
Ren, F.-Y., Liang, J.-R., Wang, X.-T. and Qiu, W.-Y. (2003). Integrals and derivatives on net fractals. Chaos, Solitons and Fractals 16, 107-117. [ Links ]
Rice, J.R. (1978). Thermodynamics of the quasi-static growth of Griffith cracks. Journal of the Mechanics and Physics of Solids 26, Issue 3, 61-78. [ Links ]
Sato, K.-I. (1999). Levy Processes and Infinitely Divisible Distributions. Cambridge University Press. [ Links ]
Silva Bustos, N.A. (2008). Caracterización y determinación del peligro sísmico en la región metropolitana. Civil Eng. thesis, Universidad de Chile, online. [ Links ]
Turcotte, D.L. (1997). Fractals and Chaos in Geology and Geophysics. 2nd ed., Cambridge University Press. [ Links ]
Vardoulakis, I. (2002). Steady shear and thermal run-away in clayey gauges. International Journal of Solids and Structures 39, Issues 13-14, 3831-3844. [ Links ]
Xie, H. (1933). Fractals in Rock Mechanics. Balkema, Rotterdam. [ Links ]
Fecha de entrega: 12 de diciembre 2011 Fecha de aceptación: 20 de abril 2012.