Shear banding as a dissipative structure from a thermodynamic viewpoint

Pre‐Eruptive Damage, Weakening and Magma‐Edifice Coupling at Piton De La Fournaise Volcano

“…Reaching complete failure with weakly interacting cracks suggests subcritical crack growth, occurring at low stress threshold (the long-term strength) and low stress transfer with a few finite-size effect in weak, eventually fractured volumes, the fracturing process as a whole being a long-term process. This case is similar to the low-m case in Figures 3a-3e of Tang andKaiser (1998) or in Figures 4.10-4.11 of Tang andHudson (2011), to the compliant layer case in Gudmundsson (2005), to the loose-sand case in geomechanics (see, e.g., Nicot et al, 2023), or to the stable mode of Lyakhovsky and Shalev (2021). The end-member case where there is no crack interaction, that is p′ = 0, corresponds to the failure at constant displacement or strain described, for example, in Turcotte et al (2003) and Gudmundsson (2012), which is possible in already fractured media; therefore n 0 is the event rate without crack interaction.…”

“…When p ′ tends to 1, n ( t ) and the interactions between cracks increase, which requires sufficient rock strength; the characteristic length of the stress field perturbation reaches the size of the body to be ruptured. This may be compared to the high‐m case in Figures 3a’–3e’ of Tang and Kaiser (1998) or in Figures 4.10–4.11 of Tang and Hudson (2011), to the stiff layer case in Gudmundsson (2005), to the dense‐sand case in geomechanics (see, e.g., Nicot et al., 2023), or to the runaway mode of Lyakhovsky and Shalev (2021). In the strong crack interaction case, the continuity is expressed by Equations 15 or 25, and Equation 32 writes (see, e.g., Turcotte et al., 2003 for a similar reasoning): $$$\frac{dN(t)}{dt}={n}_{0}{e}^{\alpha \delta N(t)}$$$ which is a first‐order linear differential equation, the solution of which is: $$$N(t)=-{n}_{0}{t}_{c}\,\mathrm{ln}\left(1-\frac{t}{{t}_{c}}\right)$$$ if N ( t = 0) = 0 and $${t}_{c}=\frac{1}{{n}_{0}\alpha \delta }$$.…”

“…10.1029/2023JB027595 Nicot et al, 2023), or to the runaway mode of Lyakhovsky and Shalev (2021). In the strong crack interaction case, the continuity is expressed by Equations 15 or 25, and Equation 32 writes (see, e.g., Turcotte et al, 2003 for a similar reasoning):…”

Configurational mechanics in granular media

“…Furthermore, for a sufficiently densely packed specimen, a phase transition is observed after a certain level of shearing coinciding with the formation of a strain localization pattern with one or multiple shear bands developing (Desrues and Andò, 2015). It was recently shown that this phase transition is consistent with the extremal entropy production theorems (Nicot et al, 2023), leading to an optimal dissipative structure (namely, the shear band pattern) able to dissipate the most part of the external energy put into the system, and making the system able to sustain external loading without collapsing (Wang et al, 2023).…”

“…This assumption is reasonable in most situations but can fail when dealing with very large-scale problems such as in earthquakes where the mobilized frictional energy under very high tectonic pressures induces fast and significant temperature rises. As shown in Nicot et al (2023), the internal entropy production int S & is proportional to the plastic dissipation power…”