On the stability of geotechnical systems and its fractal progressive loss

Abstract

Geotechnical systems, consisting of soil and embedded solid structures, are practically stable if inevitable actions cause at most harmless redistributions. This kind of robustness can often be achieved with limit state design, i.e. by assuming representative snapshots of worst cases. Changes in configuration and state due to changing boundary conditions can be better judged with quasi-static numerical simulations using validated constitutive relations. The ever-present fractality of the ground may be neglected as long as the system is stable, whereas it gets dominant during a progressive loss of stability with jerky critical phenomena which elude mathematical treatment until present. In this sense geotechnical systems can be or get sensitive, i.e. further actions can trigger detrimental chain reactions with seismogeneous collapse of the soil fabric, pore pressure increase up to liquefaction, erosion, cracking of ground and structural parts and/or tilting. The geotechnical risk can be better mitigated by taking into account chain reactions with wild randomness. It can be further reduced by monitoring the seismic emission in addition to mass flows, structural deformations and pore pressures. The paper is to clarify notions and concepts.

Appendix: Fractality

Stressed fabrics of mineral particles exhibit self-similar roughness already at and near stable equilibria. Stresses are transmitted in force chains as no two particles are equal, and observed stress distributions fluctuate in all scales. The solid volume fraction $n_\mathrm{s}$ (often replaced by the void ratio $e=(1-n_\mathrm{s})/n_\mathrm{s}$) fluctuates with wavelengths from a few particle diameters to several hundred metres. This can be expressed by the mass fractal $m_\mathrm{s}=m_\mathrm{sr}(d/d_\mathrm{r})^{3\alpha }$ for the expected value $m_\mathrm{s}$ of the solid mass in a control cube of size d with a reference value $m_\mathrm{sr}$ for $d=d_\mathrm{r}$, a fractal dimension $\alpha$ just below 1 ($\alpha =1$ without fractality), and lower and upper d-cut-offs. This leads to the solid volume fraction $n_\mathrm{s}\equiv n_\mathrm{sr}(d_\mathrm{r}/d)^{3(1-\alpha )}$ with $n_\mathrm{s}= m_\mathrm{s}/\rho _\mathrm{s} d^3, n_\mathrm{sr}=m_\mathrm{sr}/\rho _\mathrm{s} d_\mathrm{r}^3$ and the particle density $\rho _\mathrm{s}$. Equivalently the solid fraction along straight lines is $n_\mathrm{s}=n_\mathrm{sr}(d_\mathrm{r}/d)^{1-\alpha }$ for control sections of length d. Analogously the elastic energy is $E_\mathrm{e}=E_\mathrm{er}(d/d_\mathrm{r})^{3\alpha }$ so that its amount per unit of volume—called specific elastic energy—is $w_\mathrm{e}\equiv E_\mathrm{e}/d^3=w_\mathrm{er}(d_3/d)^{3(1-\alpha )}$, i.e. $n_\mathrm{s}$ and $w_\mathrm{e}$ dwindle likewise with an increase of d. This idealization cannot directly be validated and quantified, but there are indirect confirmations.

Waves are propagated along force chains; this leads to a diffusion of energy already in the elastic range [12]. Deviations from equilibrium exhibit a similar roughness versus time at a point as along straight lines at a time; therefore, we postulate spatio-temporal iso-fractality. Gradients and time rates can be expressed by fractional derivatives which represent expected values of classical derivatives at the level of particles. With them, and with a stiffness matrix by $w_\mathrm{e}$ relating deviations of elastic strain $\epsilon _{ij}^{e}$ and of stress $\sigma _{ij}=\partial w_\mathrm{e}/\partial \epsilon _{ij}^{e}$ from equilibrium values, the balance of linear momentum leads to an iso-fractional wave equation. Its solutions indicate that wave crests are polarized and propagate as fast as without fractality, and yield features of damping as observed. Power spectra tend to $v^2\sim (f_p/f)^{2\alpha }$ right of a peak frequency $f_p$ given by the extent of excitation. Spectra in the far field of liquefied sand deposits indicate $\alpha$ just below 1, while their randomness prevents a more precise calibration.

A stressed grain fabric is fractally homogeneous if expected values do not depend on the position of control cubes; then, $n_\mathrm{s}$ and $\sigma _{ij}$ are spatially constant. A thought uniform slow deformation causes a successive buckling of force chains so that a kind of micro-seismicity arises, the intensity of which can be expressed by a granular temperature $T_g$. A new equilibrium is reached after the driving deformation stops as long as $w_\mathrm{e}$ is in the convex range, i.e. in the case of average stability in spite of confined collapses of force chains. Micro-seismically activated dislocations are captured by granular solid hydrodynamics (GSH [18]) which implies $T_g$. This theory supports hypoplastic relations for monotonous deformations and elasto-hypoplastic ones for alternating deformations [13], and to a minor extent elasto-plastic and high-cycle cumulative relations. The substitution of fractional rates by classical ones is an acceptable approximation as long as $\alpha$ is just below 1 [11].

Except the diffusion of kinetic energy fractional gradients may be replaced by classical ones for boundary value problems as long as the system is stable, i.e. as long as $w_\mathrm{e}$ is convex with respect to $\epsilon _{ij}^{e}$. This enables in particular to model quasi-static responses to slowly changing boundary conditions as no kinetic energy arises except the one with $T_g$. For such cases, numerical simulations can yield realistic displacements and cruder approximations of internal forces: fractal distributions of displacements can be smoothed approximately, and to a lower extent the ones of stresses. This holds also true with soft soil particles, enhancing thermally activated creep and relaxation, and with diffusion of pore water as long as the system remains stable. The success can be attributed to driven and autogeneous attractors, i.e. the trend to new stable equilibria with minimal redistributions [8]. Attractors work with few degrees of freedom (for soil elements) and likewise with many of them (for finite element meshes) as long as no new degrees of freedom arise by bifurcations. As the latter occurs in a capricious manner during a progressive loss of stability in chain reactions, this can no more be captured with regular attractors.

Quasi-static simulations with polar quantities yield shear band patterns of sand nearly as observed [25]. They require a rather arbitrary initial fluctuation and tend to a loss of well-posedness in spite of the regularization by polar terms. Accelerations should also be taken into account as they arise by collapses at critical points of the specific elastic energy. Within zones of equally aligned average stress, neighboured critical points are attained by negative pressure waves so that shear waves of increasing size are generated by a feedback like with dominoes on a table. Depending on initial and boundary conditions, this chain reaction is blocked rapidly (e.g. in Darwin’s experiments) or leads to an overall collapse (e.g. by toppling or with an avalanche). Such critical phenomena [8] elude mathematical treatment until present already with dry sand.

The wave equation degenerates at critical points of $w_\mathrm{e}$; then, divergent deformations follow the energetic steepest descent like a sphere upon the wrist of your hand, and more intergranular dislocations are activated so that the entropy grows faster than in the stable range. Evolution equations are not yet at hand for such cases, and their probabilistic interpretation will be difficult as expected values should refer to ensembles of chaotic events which cannot easily be established. One may speak of a strange attractor as the chaos arising with successive bifurcations is deterministic in each realization, but not like with conservative systems or with turbulence. So what has this to do with critical state soil mechanics (CSSM, [30]) and self-organized criticality (SOC, [1])?

In CSSM, an elasto-plastic constitutive relation with density hardening is proposed so that uniform critical states in Casagrande’s sense work as driven asymptote or attractor. Actually soil samples lose their uniformity in triaxial, biaxial and simple shear devices in a rather chaotic way when approaching a limit state. The assumed smoothness of critical states gets similarly lost in a shear apparatus with stationary torsional drive, near the outlet of a silo with steady feeding, with horizontal driving of a solid under a free surface, or when pulling a solid horizontally through the ground. Instead of a stable flow equilibrium, the output of mass and energy due to a stationary input can at best get fractally stationary, so one may speak of a driven strange attractor. Bak et al. [1] produced minute avalanches by putting grains on top of a sandpile upon a scales, and found that the number N of avalanches with size s or more depends on s by the power law $N\sim s^{-\tau }$, with $\tau$ ranging from about 0.5–1.5 and with cut-offs. They derived the same relation by means of a rather arbitrary algorithm for a lattice named cellular automaton, and called the obtained attractor ‘SOC with 1 / f-noise’.

Their paper triggered a euphoria in various sciences, while further sand tests did not confirm the SOC concept. It suits apparently to the empirical Gutenberg–Richter relationship for earthquakes, but authors such as Turcotte [32] criticize that there is no generally accepted definition of SOC. Dhar [6] shows with mathematical rigour that the exponent $\tau$ depends on dimension and shape of the chosen lattice, and admits that such algorithms have little in common with sand avalanches. Hergarten [16] proposes SOC algorithms for earthquakes, landslides, forest fires and drainage networks and shows that the exponent $\tau$ depends on chosen algorithms and assumed input data including cut-offs. He states that failure scenarios have to be simulated for lack of field data, and that SOC models should be further developed for this purpose.

The issue is more intricate for geotechnical systems as they exhibit also autogeneous and triggered strange attractors. For example, after a perturbation the pore water pressure increases by contractant collapses of a loose soil fabric; thereafter, it grows in the vicinity by diffusion and can trigger a next seismogeneous chain reaction. This can lead to liquefaction and chaotic sidewards flow up to an avalanche. Earthquakes and storms increase also the pore water pressure, after therefore enhanced intergranular dislocations a system can collapse as a whole, thus, for example, retaining structures failed by spreading of harbour islands 1995 near Kobe/Japan, and offshore platforms got lost. Erosion in the ground and at embedded structures can likewise lead to a chaotic overall loss of stability. Solid structures at or in the ground have an elastic range so that interacting parts of both can snap through (as in Darwin’s experiments), yield plastically (i.e. thermally activated) or crack (at critical points in Griffith’s [7] sense) in chain reactions. A delayed progressive collapse of structures at or in the ground can occur due to relaxation of the ground, such as at Nicoll Highway or after a dilation with diffusion of pore water in slopes with stiff clay.

Such critical phenomena exhibit wild randomness. In Mandelbrot’s [23] sense ‘wild’ self-affinity means that a single extreme event can matter as much for expected values as many minor events. This requires power law tails of probability distributions so that weighted sums of them with the same exponent are again power laws. It is achieved more precisely with stable Lévy distributions for successions of independent random jumps for which moments diverge [29]. The divergence is avoided with cut-offs, but a plethora of sums tend to power laws with other exponents. Mandelbrot speaks of mild randomness if sums tend to Gaussian distributions and variances dwindle by the Law of Large Numbers. We call the randomness of robust systems mild as it is nearly averaged out in the stable range, but no more at its verge. Our fractal dimension $\alpha$ tends mildly towards 1 (without reaching it) by regular attractors in the stable range, and drops wildly with strange attractors otherwise (but not below about 0.9).

We cannot take over SOC for estimating the risk as geotechnical processes are rarely stationary and as SOC algorithms are rather arbitrary. Geotechnical chain reactions are wildly random as initial configurations and states—with traces of former critical phenomena—are not fully known even with the best investigation, and as jerky successions of bifurcation collapses are rather chaotic. We propose

$$\begin{aligned} N & = N_\mathrm{r}(E/E_\mathrm{r})^{-\tau }\hbox { for }E\ge E_\mathrm{r}\hbox { and }\\ N &= N_\mathrm{r}\hbox { for } E<E_\mathrm{r}, N=0\hbox { for }E>E_\mathrm{m},E_\mathrm{m}\gg E_\mathrm{r} \end{aligned}$$

for the expected number of chain reactions with released energy E or more. The upper bound $E_\mathrm{m}$ of E can be estimated for a worst-case chain reaction via the loss of elastic and gravitational energy, it is correlated with the seismic energy radiated into the far field and with the size of damage. We presume $\tau \approx 1$ as the mechanical response of geomatter is nearly rate independent; this suits to the Gutenberg–Richter relationship with b near 1. The reference value $N_\mathrm{r}$, viz. N for $E=E_\mathrm{r}$, is determined by the number of geotechnical operations.

A chain reaction is principally determined by the initial configuration and state, external actions, conservation laws and constitutive relations. Ensembles of them are wildly random as our knowledge of these factors is confined, and as every single chain reaction is chaotic. The power law proposed above is a heuristic approximation for such ensembles. Field records are heterogeneous and scarce so that they cannot suffice for calibration in a statistical sense. This inevitable lack could be partly compensated by simulations although their validation is difficult because of the wild randomness. Differently from thermodynamics and quantum mechanics with Gauss statistics and deduction from strong principles, geotechnical chain reactions require induction by means of case studies, but despite or just because of the wild randomness objective estimates are principally feasible.

The classification of events starts with recognizing the kind of strange attractor—driven, autogeneous or triggered. Simulations of chain reactions will require coupled balance equations and constitutive relations, wherein critical points of potential energy, entropy production and feedback by pressure waves plus pore water diffusion play a key role. The degrees of freedom of numerical grids need not be very numerous because of the self-similarity, and as only the total released energy (i.e. the overall increase in entropy) counts. For a class of events a still harmless, one can be defined by a lower bound reference energy $E_\mathrm{r}$; then, the energy loss $E_\mathrm{m}$ in a worst case has to be estimated. For calculating the expected energy $\bar{E}$ dissipated in chain reactions as a function of $E_\mathrm{r}$ and $E_\mathrm{m}$ influences of geological and technical past, of technical and other actions and of informational entropy—i.e. incomplete data—have to be taken into account. The geotechnical risk, i.e. the expected damage for an operation, can be calculated via $\bar{E}$ by means of the vulnerability with respect to chain reactions.