Interface Plastic Constitutive Law With EndCap and Structural Model Applied to Geological Faults Behavior
Abstract
This paper proposes a framework for the mesoscale modeling of potentially complex fault cores. Such faults cannot be modeled as simple frictional interfaces because they may be thicker and made of several slip surfaces embedded in a material that can contain both grains and broken blocks. Two main aspects are considered here. One, which is the main innovation in the paper, deals with the derivation for the specific rigidplastic interface constitutive law which was developed to account for the hardening/softening behavior of highly stressed rocks like in geological faults. The other one deals with the theoretical and numerical study of the response of the structural model in which this constitutive law was implemented. This first structural model was chosen to represent the behavior of faults surrounded by elastic rock masses. The results obtained from this model already illustrate the high potentiality of the proposed interface constitutive law in the context of seismology. When farfield displacement boundary conditions evolving time are used, different modes of evolution emerge, and transitions happen, from quasistatic to dynamic or vice versa with timescales extending from tens of years to seconds. The instability domain and the dynamic solution are detailed together with the related energy considerations. An extensive analysis of specific case studies is available in Maury et al., companion paper, in particular as regards the possibility for slow slip events to transition into seismic slip (or vice versa). The implementation of the interface model into more realistic and advanced numerical fault models should offer many perspectives.
Key Points
 We derive a poroplastic interface constitutive law (ICL) with end cap and hardening/softening properties to model complex fault cores
 We derive the equations for the ICL problem sandwiched between two elastic pads and describe the numerical code to compute the solution
 Emerging behaviors include slow stable slip as well as dynamic events; the case study analyses and classification are in the companion paper (Maury et al., 2020)
Plain Language Summary
When one thinks of a fault as a frictional interface, what comes to mind is two rough rock walls being pressed past each other until their frictional strength is exceeded and the two walls start sliding past each other. This slip can be seismic or slow. When fault cores are thicker and can contain both grains and broken blocks, it is useful to take a step back and find a modeling strategy that can account for the effective behavior of the whole. This is what this paper is concentrating on: the derivation of a purely plastic interface constitutive law for faults with complex fault cores. Our interface model accounts for two main modes of evolution that rocks can undergo within a fault zone, depending upon the fault maturity and the magnitude of normal stresses. One mode of deformation is considered to reflect a multicracking process with loss of resistance and void formation in the rock shear band. The other mode of deformation is seen as the crushing of already cracked rocks, resulting in an increase of fault resistance and tightening. Once integrated within an elastic medium, the model predicts different modes of evolution, from quasistatic to dynamic or vice versa.
1 Introduction
It is common to model simple, mature fault zones using frictional models and/or a Coulomb failure criterion. This is done in quasistatic models (Fitzenz & Miller, 2004), quasidynamic models (Dieterich, 1995; Robinson et al., 2011; Zoeller et al., 2004), and full elastodynamic models (BenZion & Rice, 1997; Lapusta et al., 2000). Thicker or more complex fault zones have been studied in particular to model the behavior of deformation bands (Fossen et al., 2007) and their influence on permeability structures around reservoirs of interest. In this article (referenced as CP2), we propose a generic framework to model potentially complex and thick fault cores, both in their hardening and their softening regimes, both during their stable deformation and their transitions to dynamic rupture.
The paper is divided into six main sections.
Section 2 deals with the derivation of the specific pororigidplastic interface constitutive law (ICL) used to account for fault behavior. With the aim to obtain a constitutive law likely to be either contractant and strain hardening or rather dilatant and strain softening, this one is derived from the more usual framework of tensorial plastic constitutive laws with end cap that are already known to possess these properties. We more specifically use as a starting point the formalism of the modified CamClay (MCC) law. We denote in brief the obtained ICL as generic endcap (GECICL) for “generic endcap” ICL. Theoretical background of the derivation is given in Appendix A.
The following sections focus on the behavior of the structural model (also known as P & I), obtained by inserting the previous ICL (I) between two elastic pads (P), submitted to some given farfield displacements (Figure 1). Incidentally, this is the structural model used in Maury et al. (2020, companion paper, referred to as CP1). It was inspired from field observations reported in seismology (Fitzenz & Miller, 2001, 2003; Tanaka et al., 2001).
Thus, section 3 lists the equations of the problem (P&I) and develops the solution in terms of time derivatives, in the case of plastic quasistatic evolution. The equations of the problem are reduced to the single one governing the evolution of the “plastic multiplier” and from there discussed in terms of the existence and stability of solution, as a function of the actual state of stress. Since the model comprises a poromechanical component, we also show the possibility of a metastable regime for the solution in the drained case.
The developments in section 4 are similar to those of section 3, except that the evolution of the problem (P & I) is now supposed “dynamic” and linked to the generation and propagation of elastic waves into the pads. The “plastic multiplier” equation is reestablished and shown to insure in that case the existence of a solution, whereas the static problem had none. Specific issues related to dynamic evolutions as well as energetic aspects are discussed.
Section 5 and especially its related Appendix E describe the algorithms used in static and dynamic regimes and implemented into a Visual Basic (VB) code, to compute the evolution of the whole solution with time.
We illustrate in section 6 some typical results (stress path, evolutions of displacements, velocities and accelerations) that can be obtained from the model and the previous code for a given example of data set. However, the interested reader is invited to find more scenarios in the companion paper denoted as CP1, more deeply discussed in the context of geological faults. The purpose of CP1 is to study the roles of the in situ state of stress and fluids in the subsequent occurrence or not of slow slip events and earthquakes.
Some perspectives for further use of the ICL derived in this paper in relation with more advanced models of geological faults behavior are raised in section 7 which concludes the main text.
Note that throughout the paper we use the sign convention of continuum mechanics (e.g., compressive stress <0, contraction strain <0). However, we consider as positive in compression the quantities with the name “pressure” in their denomination, that is, the pore fluid pressure u, the mean stress pressure p and the mean Terzaghi's effective stress pressure p′. Hence, with this convention, , , , where quantities underlined twice designate tensors. A list of the main abbreviations and symbols used in the text is provided following the appendix.
2 Derivation of a Generic Poroplastic Interface Constitutive Equation With End Cap (GECICL)
In the following we denote by GECICL the plastic ICL that we want to develop to account for faults behavior. G stands for generic and EC for end cap, meaning a closed criterion in the stress plane related to the interface. Of course, the constitutive law proposed in this article is only one example among all the possibilities to formulate ICL with end cap.
2.1 Objectives for the Construction of the GECICL
Figure 1 shows the simplified geological fault model used in CP1, consisting of an interface (I) embedded between two elastic pads (P). The ICL chosen for (I) is one of the main elements used for obtaining the results presented in CP1. Hence, more details are given here about the main features sought for the GECICL and the steps used for its formulation.
From the mathematical and geometrical point of view, we consider the interface (I) with no thickness allowing discontinuous shear and normal displacements ⟦U⟧, ⟦V⟧ between both its sides. This modeling technique is widely used in seismology, especially in combination with the “Rate and State” (R&S) constitutive laws and is commonly introduced in the 2D or 3D standard numerical tools (e.g., finite or boundary element methods) for solving problems with known geometry interfaces (curves in 2D, surfaces in 3D). It is an efficient mean to avoid the difficult mathematical issues raised by strain localization in continuous media.
However, from the physical point of view, we aim through the GECICL to deal not only with the behavior of thin faults but also with the more varied behavior of faults possibly having metric or decametric thicknesses. This is a significant difference with the “R&S” models, which in literature, rather address the behavior of (very) thin faults, mostly characterized by the frictional mechanism and friction ratio μ_{R & S} between two quasiflat rock surfaces with small asperities, sliding one against the other (Dieterich, 1972, 1978); Ruina, 1983; Scholz, 1998; Segall & Rice, 1995).
Here we are looking for a mesoscopic ICL able to describe not only the case of (matured) frictional faults with already highly localized shear strains, but also the case of faults still in the process of damaging, breaking and moving rocks in their vicinity and with shear strains still spread over a relatively large interval in transverse direction. We are especially interested by the changes of volume (contractance/dilatance) that may accompany fault sliding, which in turn can strongly impact on fault behavior from several aspects: mechanical resistance, fluid pressure, effective stress, and rock and fluid temperatures.
Thus, despite its zero geometrical thickness, we assimilate the interface (I) to an area with a finite width 2H°(t) (being given the symmetry with regards to the x axis) which can possibly vary with time between plurimetric and centimetric sizes. Indeed, (I) is supposed to represent in our model the y interval inside which the shear and normal strains “concentrate at most” at time t. Hence, the same fault might be viewed with large H° at small ages, before important deformations occur (e.g., fault still in the process of formation) but then with decreasing values H°(t), as the (shear) displacements grow and the strains localize. In particular, rapid evolutions of H° could happen during seismic events with large sliding displacements. The bands comprised between the initial and the latter values of H° might constitute the damaged zones surrounding the fault core, which are sometimes considered in literature. Although it is important to keep in mind this physical meaning of the GECICL we are looking for, we will see, however, that H°(t) does not appear in the equations and that consequently its value does need to be specified.
The other key point that we impose to the GECICL is that it does not refer explicitly to the physical time, unlike the “R&S” models. For this we restrict the formulation of the GECICL to the framework of plasticity. Doing so, we do not pretend that time has no physical impact on faults behavior (the presence of geophysical fluids with healing effects may show the contrary for instance). But here one of our goals is to check whether a purely plastic ICL can naturally account for evolutions switching from quasistatic to dynamic regimes and vice versa, when such a law is incorporated into an elastic structure like the (P & I) one. Of course, reintroducing some part of time dependency in such ICL remains feasible in future models, if necessary to obtain more realistic descriptions of fault behavior.
 to be without thickness from the geometrical point of view
 to account at mesoscopic scale not only for the frictional behavior of faults but also for their more general behavior (e.g., dilatant/softening), related to strain localization still relatively weakly pronounced
 to derive from the (standard) plasticity framework
 to account for the possible presence of a compressible fault fluid, through Terzaghi's effective principle.
With these objectives in mind, there is no much other choice than deriving the GECICL from theoretical considerations and assessing its relevance a posteriori, by comparing its effects in models with onsite observations. Here it is also an important difference with the “Rate & State” models, for which laboratory experiments on small rock samples are possible and sometimes used to determine evolution laws of μ_{R & S} with sliding shear rate and shear displacement.
At best in our case, as often in physics, can we use analogy with other contexts believed to involve phenomena “similar in some way” to those we want to report (at smaller scale for instance). In particular, it is the principle sometimes used by geologists when performing experiments on “small physical soil models” to simulate mechanisms at the geological scale. In our case the transposition process may consist to adapt plastic constitutive laws already proven in other domains, but involving similar constraints as the ones listed above.
Thus, our methodology used to build the GECICL includes two steps: first, we select a 3D (tensorial) plastic constitutive law already possessing part of the properties listed above—this one is borrowed from soil mechanics; then we transform and adapt this 3D constitutive law into an ICL.
2.2 Choice of the MCC Constitutive Law as a Starting Point
The tensorial (standard) plastic laws with endcap criterion and yield evolution are known in general to account for both the plastic contract/hardening or dilatant/softening behaviors that can be experimentally found for a wide diversity of materials like soils, cement concretes, or rocks when subjected to different magnitudes and the ratio between deviatoric and compressive stresses (see for instance MaoHong Yu, 2011, for a general discussion on materials and related plastic models; see also Schulz & Siddhartan, 2005, for the application of Cam cap models to deformation bends in porous granular rocks).
In some way, it shows that the similar formalism can apply to different contexts and scales, provided that the values of the parameters are adapted to the situation at stake.
Among the numerous existing tensorial endcap models, here we make the choice to start from the MCC constitutive law developed in soil mechanics (Roscoe & Burland, 1968; Schofield & Wroth, 1968). Indeed, this one accounts for both the proven contract/hardening and dilatant/softening behavior of soft soils, depending upon the actual state of compaction (density) and the magnitude of (Terzaghi's effective) mean stress pressure. Among other advantages, the MCC only depends on a relatively few number of parameters and is easy to handle for analytical calculations.


 = (total) stress tensor (considered with the usual convention of continuum mechanics)


 = 3 × 3 unit tensor

 u

 = pore pressure (positive)


 = effective Terzaghi stress tensor =

 p′

 = mean Terzaghi effective pressure = (positive for compressive stress states)


 = deviatoric stress tensor =

 q

 =deviatoric stress = with summation on repeated indexes 1 ≤ i ≤ 3,1 ≤ j ≤ 3)

 ε_{v}

 = (plastic) volumetric strain =


 = (plastic) deviatoric strain tensor =

 f

 = MCC plastic criterion


 = actual value of the “material consolidation pressure” (yield parameter) depending upon the algebraic value of ε_{v} (yield variable)


 = initial consolidation pressure of the material for ε_{v} = 0

 M

 = slope of the critical state line

 a

 = no nil positive coefficient


 = plastic multiplier, positive or nil real number.
Here it can be first noted that the total strains are assumed equal to the plastic strains, meaning that we neglect for (I) the elastic strains with respect to the plastic ones (rigidplastic model). This nonessential hypothesis makes it possible to simplify the calculations hereafter while retaining the main aspects we want to report.
Equation (1) is typical of the plastic criterion used in the MCC model (note that in the bibliography this one is sometimes found with the shape where . This equation is indeed equal to (1) by setting .
For any given value of the equation f = 0 is that of a half ellipse in the (p′, q) plane (Figure 2), passing through the origin (p′ = 0, q = 0), having one axis confounded with the horizontal line q = 0 and with its two other summits located at and . Due to its closed shape, limiting p′ values to , this criterion well belongs to the family of plastic models with end cap. Besides, f is obviously also a convex function of p′and q.
As shown by equation (2) the consolidation pressure plays as a yield parameter, linked to the volumetric strain ε_{v} having the role of a yield variable (in the original CamClay model, this role is devolved to the (plastic) void index). In keeping with CamClay models, the relationship between and ε_{v} is given the shape of a decreasing exponential.
Thus, negative values of ε_{v} induce an increase of and of the material criterion, related to an increase of mechanical performance (strain hardening) as observed on soft soils. Conversely, positive values of ε_{v} (dilatance) induce a decrease of and of the material failure properties, as also observed on soft soils.
When varies the ellipses f = 0 have their upper summits on the straight line with equation q = Mp′ called the critical state line (CSL).
Equation (3) is the flow rule, connecting the rates of volumetric and deviatoric strains with the plastic multiplier and the position of the stress tensor along the criterion (note that in the derivation of , s_{ij} and s_{ji} are considered as two independent variables for i ≠ j; of course they are taken as equal afterward). Here the choice is done of the standard flow rule expressing the collinearity of the vector ( , ) with the outer normal to the criterion at the actual stress point for . In particular, in the case of the MCC criterion, it implies plastic dilatant strains on the right side of the ellipse, contractant ones on the left side and nil volumetric strains on the CSL. The choice of a standard flow rule looks well adapted to the construction of a generic constitutive law as intended for the GECICL. Indeed, for convex criteria (which is the case here), it automatically insures the requirement of the second Thermodynamics principle for the positivity of the plastic energy dissipation rate ( . Moreover standard flow rules are generally known to reflect reasonably well the real behavior of plastic media from the physical point of view (as an example, Pouya et al., 1998, found it was the case for fine grained sediments tested in laboratory).
From these equations, let us now infer the formulation of a plastic constitutive law for an interface going through the material, using the following principles. Let us start from a general standard convex criterion f(σ) ≤ 0.
2.3 Derivation of the ICL From the Tensorial MCC Law
Let us consider a segment with small length (dI) representing a shear band through a rigidplastic material. We assume (dI) to be locally confounded with the axis x and for the sake of simplicity the problem is supposed to be in strain plane conditions in the z direction. For the time being, we also ignore the porous and effective stress aspects, which do not change the approach followed here.
Two geometrical and mechanical ways of modeling the presence of (dI) are possible (Figure 3).
 ⟦U⟧= difference of shear displacement between the two lips of (dI)
 ⟦V⟧= difference of the normal displacement, also called dilatance for ⟦V⟧ > 0 or contractance for ⟦V⟧ < 0.
2.4 ICL Model Used in the Following
It can be checked that the previous simplification, which is not necessary but just convenient for hand calculations, does not change the main physical meaning and properties of the ICL defined in section 2.1.
Actually, equation (7) has a similar shape to the MCC 3D criterion, except for being expressed in the (σ′_{yy} < 0, σ_{xy}) interface stress plane, instead of the (p′, q) plane (just note that contrary to q, σ_{xy} can be either positive or negative, but in the following we only consider situations for which σ_{xy} > 0).
With σ′_{yy} < 0 and assuming σ_{xy} ≥ 0, the equation F = 0 defines half ellipses in the (σ′_{yy}, σ_{xy}) stress plane, passing through the origin (σ′_{yy}= 0, σ_{xy} = 0) and the two other summits (σ′_{yy} = −p′_{c}/2, ) and (σ′_{yy} = −p′_{c}, σ_{xy} = 0). A CSL, going through the upper summits can still be defined with equation . On the left side of that one, the plastic behavior of the interface is contractant with growing size of ; at the opposite, on the right side of the CSL, the interface is dilatant with decreasing size of .
 as dilatant and softening under relatively small (effective) normal stresses,
 or vice versa, as contractant and hardening (macro) under relatively high normal (effective) stresses
The magnitude of p′_{c} and of the other model coefficients makes it possible to scale the ICL to different contexts of stresses and interface thickness at play, such as landslides in soft soils and geological faults in deep rocks. By the way, it is interesting to note that the thickness H°(t) introduced to help the physical interpretation and construction of the model does not appear in the final equations. From this aspect, it avoids the need to specify numerical values for this parameter possibly evolving with time.
It is also worth to note that an endcap ICL with a shape close to the one here derived was also proposed by De Gennaro and Franck (2005) to account for experiments on soilpile interaction.
Besides, within a given context, one can think that introducing variations of p′_{c} on a discontinuity area through initial functions (along a curve with r = curvilinear abscissa) or (for a surface) can be an easy mean to model spatial variations of interface maturity or of initial stress (with depth in particular) and study the evolution of heterogeneous structural situations. However, this analysis is beyond the scope of this paper.
3 The “Pads+Interface” (P & I) Structural Problem: Definition and Study of Its Time Derivative Response in Quasistatic Regime
3.1 Equations for the (P & I) Model in the Static Domain

 h,λ,μ

 = thickness along y and Lame's coefficients of the pads

 σ_{yy},σ_{xy}

 = normal and shear stresses in the structure

 V_{1},U_{1}

 = normal (dilatance) and shear displacements in y and x of the upper fault edge

 V_{2},U_{2}

 = uniform farfield imposed displacements in y and x directions of the upper pad boundary (y = h)

 σ_{yy} ° ,σ_{xy}°

 = uniform initial stresses

 u

 = fault pore pressure

 σ′_{yy}

 = effective normal stress

 S

 = coefficient relating fault pore pressure and dilatance variations

 R

 = coefficient relating fault pore pressure variation to the external fluid reservoir in drained regime.
In quasistatic regime, the quantities σ_{yy},σ_{xy},σ′_{yy} are uniform within the whole structure, being dependent upon time t but not on x.
Equations (11) and (12) reflect the elastic nonporous static behavior of the pads. They also account for the initial stress field (σ_{yy}°, σ_{xy}°) prevailing in the structure for U_{1} = U_{2} = V_{1} = V_{2} = 0 at time t = 0.
Equation (13) is the definition of Terzaghi effective stress within the interface, u being the interstitial pressure.
Equations (14)–(18) correspond to the GECICL introduced above.
Equation (19) is the one used to model the pore pressure evolution. The first term on the right accounts for the combined effect between the (algebraic) dilatance and the fluid compressibility C_{fl}. In Appendix B the coefficient S is shown to be inversely proportional to the value of C_{fl}. The second term at the right of equation (19) accounts in a simplified way for a possible link between the fluid in the interface and a “fluid reservoir” at pressure u_{ext} (considered as a given constant or function of time). The difference of pressure u_{ext} − u and the coefficient R can be viewed as average gradient and permeability terms, relating the fluid reservoir to the interface.
These equations will now allow us to examine the response of the (P & I) structure subjected to the farfield displacements U_{2},V_{2} (or ) and possibly to the fluid pressure u_{ext}. The first case is that of rigidelastic evolutions. The case of plastic evolutions is then discussed.
3.2 RigidElastic Evolutions
Actually as discussed in Appendix F and CP1 it would be possible to add another term on the righthand side of equation (19) to take into consideration the impact of interface elastic deformations on pore pressure evolution. Indeed, even if it may be legitimate to neglect elastic strains in front of plastic ones as we do in this article in the case of plastic regime, it may, however, be necessary to consider the impact of elastic strains on the fault fluid pressure during elastic evolution, which might have a significant importance on the real effective stress in that case.
The coefficient ς has the role of a Skempton coefficient, linking the pore pressure variation to the total normal stress, actually reflecting the effect of the elastic deformation of the interface on the fluid pressure.
However, for the sake of simplicity and to avoid the introduction of a supplementary coefficient in the modeling, ς is set equal to 0 thereafter.
3.3 Derivation of the Consistency Equation for the Plastic Multiplier in the Plastic Regime
An important step for solving plasticity problems is their reformulation in terms of time derivatives. In particular, it makes it possible to exhibit the scalar equation for the plastic multiplier (consistency equation) which plays a central role in the evolution of the solution. This approach is used below to discuss the sign of and the existence of a solution to the problem (P & I) in static regime.
Let us build the equations for the time derivative problem, denoted as (P & I)_{,t}, related to the problem (P & I).
 the farfield displacement rates ( considered as given data
 the plastic multiplier
 the values at time t of all (P & I) variables themselves, supposed to be known.
Equations (16) and (17) have already the good shape, since they are expressed in terms of the time rates , .
These equations show that the whole set of the time derivative problem variables is known at time t, if the value of is known.
We thus obtain the scalar consistency equation governing the evolution of the plastic multiplier and therefore that of all variables of the (P & I) model.
3.4 Discussion About the Condition
Above all equation (24) makes it possible to discuss the inequality which has to be verified to insure the existence of a solution.
Let us assume the numerator N as being positive, by an appropriate choice of . Typically for a problem for which the shear component of the farfield displacement is considered as the dominant driving factor, this condition supposes to choose with the same sign as that of σ_{xy} (or σ_{xy}°) insuring an increase of shear stresses in the pads in the case of a rigidelastic behavior. The opposite condition would lead to unload the pads and to the obvious case of a rigid behavior for the interface, which is of no interest here.
Then for N ≥ 0, the condition implies for D to be strictly positive and leads to examine the possibility for getting both conditions f = 0 (plastic evolution) and D < 0.
Let us first look in the (σ′_{yy}, σ_{xy}) stress plane, the boundary between the authorized and unauthorized areas, defined by the equations f = 0 and D = 0.
In the (X,Y > 0) halfplane, the first equation is that of a circle centered on the origin and with radius p′_{c}.
The second equation is that of an ellipse with axes parallel to X and Y, passing through the 3 summits (0,0), ,
Since , two cases may arise.
Either the ellipse is entirely below the circle or it is “high” enough to intersect it. In the first case, the system f = 0, D = 0 has no solution; in the second case, it admits two solutions in general which can be taken as parametric functions of p′_{c}; we denote them as and (points A and B) with the convention . These two situations are shown in Figure 4, plotted in the original (σ′_{yy}, σ_{xy}) plane.
The limit between these two cases corresponds to the situation for which the circle and the ellipse are tangent. In that case the system f = 0, D = 0 admits a single solution for which the value of p′_{c} is minimal and noted p′_{c lim}.
The curve composed of the two branches of points A and B splits the stress halfplane (σ′_{yy}, σ_{xy}) into two areas, when p′_{c} is varied from p′_{c lim} to +∞. As can be seen in Figure 5, this boundary has the typical shape of an open turned up curvilinear triangle. The denominator D is negative inside the triangle (domain denoted as Ω_{Nosp}), positive outside. Then plastic static evolutions can happen only for stress states outside Ω_{Nosp}. Instead, as shown in section 4, the domain Ω_{Nosp} (Nosp = No static plastic evolution) is the one for which plastic “unstable” dynamic evolutions are likely to initiate and develop.
Thus, p′_{clim} is the smallest value of the consolidation pressure for which the system f = 0, D = 0 is likely to exhibit “dynamic plastic unstable” evolutions (lower tip of Ω_{Nosp}).
This expression shows in some way the role of the different model parameters; in particular, we note that the lower S or the higher α and the higher h, then the lower value of p′_{clim} and the larger the unstable domain Ω_{Nosp}.
However, it is important to remember that the discussion above deals with plastic evolution only, but that the stress domain Ω_{Nosp} can nevertheless be penetrated and traveled during quasistatic rigidelastic evolutions. Moreover such a sequence can suddenly switch to an unstable evolution (as shown further), if the stress path comes into contact with the actual criterion boundary (f = 0) when being inside the domain Ω_{Nosp}. In CP1 this situation is believed to be one possibility for the occurrence of sudden seismic events in the quasi absence of precursor events.
3.5 Incidence of the Fluid Compressibility on the Domain Ω_{Nosp}
Let us examine more precisely the incidence of the parameter S on the domain Ω_{Nosp} which we now note as the function Ω_{Nosp}(S). This one can be easily shown decreasing with S, that is, S_{1} < S_{2} ⇒Ω_{Nosp}(S_{2}) ⊂ Ω_{Nosp}(S_{1}).
In other words, since S is inversely proportional to the fluid compressibility, the higher the fluid compressibility, the larger the domain of unstable plastic evolutions.
In particular, the largest domain for Ω_{Nosp} is Ω_{Nosp}(0) corresponding to infinite fluid compressibility. Consequences of these results in relation with the stability of geological faults are detailed in Maury et al. (2011, 2013) and CP1, in which the presence of a gaseous fluid instead of liquid is shown as an important factor for the occurrence of mainshocks. Based on the same model, the possible effect on fault stability of phase change of sea water due to pressure variations under some specific range of temperature is also discussed in Géli et al. (2016).
3.6 Drained Regime and Metastability of the Solution
The above discussion about the existence of a static solution to the time derivative problem can be put one step further by considering whether the evolution outside the domain Ω_{Nosp}(S) will evolve or not if the driving terms came to be nil.
Actually in the undrained case (R = 0), the response is obvious. Indeed, for , N (equation (23)) is also nil and so is also the case for all the time derivatives of the (P & I)_{,t} problem. Thus, the solution stops evolving as soon as the driving terms disappear.
Let us assume that at the same moment and u are such that and
u < u_{ext} (high fluid pressure reservoir), giving positive values to N.
(Typically the condition u < u_{ext} is the one obtained during the first moments of a plastic regime on the dilatant side of the CSL, if one considers u_{ext} as constant with time and u = u_{ext} as initial condition for u)
Therefore:
But from section 4.4, equation (28) indicates that for an evolution inside the complementary domain Ω_{Nosp}(0) − Ω_{Nosp}(S) the coefficient r is negative, meaning that the tangent evolution with time of Δu is similar to that of an increasing exponential of time with a positive exponent (Δu(t+dt)~∆u(t) e^{rR dt} with r > 0, ∆u < 0) and will accelerate (in absolute value) rather than stop, despite the conditions .
For this reason, for R > 0 (drained case) and f = 0 (plastic evolution) the domain Ω_{Nosp}(0) − Ω_{Nosp}(S) can be considered as “metastable,” since being governed by the interface drainage properties rather than the farfield displacement boundary conditions.
(By the way, it can be checked that equation (29) implies as expected in plastic regime, since , D ° < 0, D − D° > 0, )
3.7 Static Approximation of the Plastic Stress Path Within the Domain Ω_{Nosp}
As seen before, there can be no plastic quasistatic solution crossing the domain Ω_{Nosp}. To recover a solution, one possible mean which is especially relevant in the context of seismology, is to introduce inertia forces in the equations of section 3. This is the aim of section 4. However, for a given quasistatic stress path, arriving at point A on the right branch of the boundary ∂Ω_{Nosp}(S), it remains possible to define an approximate location of point B, by which the dynamic stress path will quit the domain Ω_{Nosp}(S) without the need for computing the inner path between A and B. The calculation principle is to replace the flow rule equations (16) and (17) by the two conditions f = 0,D = 0 that define ∂Ω_{Nosp}(S).
Then considering the 2 additional equations and , we get a nonlinear system with seven unknowns ( and seven equations.
This system admits 2 solutions, one already known (point A) on the right branch of ∂Ω_{Nosp}(S), the other one on its left branch, corresponding to the searched point B.
In practice, the system above can be easily solved numerically, by the fixed point algorithm for instance (see Appendix C). This computation is implemented in the VB code mentioned in section 5, to compare the true dynamic stress path with this simplified “static calculation”. The example of section 6 shows the good agreement between the two methods, the dynamic stress path passing very close to the point B determined as above.
4 Study of the Time Derivative Problem (P & I)_{,t} in the Plastic Dynamic Regime
Let us now examine the case of solutions to the problem (P & I) with rapid evolution, for which inertial forces generated in the pads cannot be neglected any more. Dynamic forces in the interface itself are neglected, given its relatively small volume and mass. In this section, we denote t the time value since the beginning of the dynamic phase (generally following a static phase with another time origin) and ∆ the prefix used to indicate the variations of the mechanical quantities from that time.
and are the velocities of the elastic compressive and shear waves in the pads supposed to have an homogeneous specific density ρ.
4.1 Relationships Between Stresses and Displacements in Dynamic Regime
In these “pad equations,” as in the static case, the fault thickness is neglected, with the lower bound of the upper pad supposed at y = 0.
In equations (36) and (37) the variations of the farfield displacements are considered as negligible due to the very short duration of the dynamic phases (as found latter on in the numerical applications).
Equations (40) and (41) show that interface stresses at time t are linked for one part to the velocity of the displacements of the interface at the same moment, but are also related to the past values of and at times , previously computed. Note that , are equal to 0 at the beginning of the dynamic phase for 0 ≤ t ≤ 2h/c_{p}, (respectively for 0 ≤ t ≤ 2h/c_{s}) until P waves (respectively S waves) reflect on the farfield boundary y = h and come back to the interface.
Finally, equations (40) and (41) with the ones from (13) to (19) make the new set of equations governing the dynamic regime. However, to insure the existence of a solution to this problem in the plastic regime (f = 0), we still have to show that the value of is positive.
Then let us now derive the new consistency equation for the evolution of .
4.2 Consistency Equation for the Plastic Multiplier in the Dynamic Regime
To simplify the writing, we omit in the following the qualification y = 0 in the expression of σ_{yy}, σ_{xy} at the interface.
4.3 Behavior of Plastic Dynamic Solutions for Small Time Values
The firstorder differential equation with time (46) governs the evolution of during the unstable plastic dynamic phases. By adding to this one, those related with the time derivatives of all other unknowns of the problem (P & I), we obtain a set of differential equations, whose integration with time and use of initial values, makes it possible to compute the dynamic solution at any moment.
However, for (46) to be valid, the solution must check the condition . To discuss this point, let us examine the behavior of at the beginning of the dynamic phase and the choice of the initial value .
Before coming back to the question of the choice for , let us precise the sign of A to know whether the previous exponential is of decaying (A > 0 ) or increasing type (A < 0 ).
But in the vicinity of the boundary ∂Ω_{Nosp}(S), where the dynamic regime is expected to initiate, D is nil or almost nil. Then A is clearly negative, indicating that for small values of time and initial values , is positive and close to the exponentially increasing function , with being a characteristic time.
In particular, within the frame of geological active faults modeling, the value of τ will be found to be with the magnitude of seconds, to be compared with the much longer characteristic time of tectonic loading governing stable phases.
(By the way, since A becomes negative before D reaches negative values, it shows that the consideration of inertia forces makes appear the unstable stress domain slightly larger than the domain Ω_{Nosp}(S) found in the quasistatic regime. This difference is neglected here).
Another consequence of the equation found for for small time values is that the choice of the initial value has not much importance on the evolution of the solutions, as long as is chosen small enough.
It means that a change in the initial condition does not impact the shape of the solution; it has only a time translation effect with the order of τ, which is unimportant for our purpose.
Thus, in the numerical code related to the model (see section 5), is chosen with an arbitrary small positive value, irrespective of whether the unstable dynamic regime originates from a plastic static phase or directly from a rigid static phase.
4.4 Behavior of the Solution at the End of the Plastic Dynamic Phase
Equation (47) shows that A^{*} tends to 0 as the stress path crosses the CSL ( and passes from the dilatant to the contractant side of the interface criterion. Then A^{*} becomes positive, inducing now a strong decrease with time in as well as of all other variables. Due to the presence of the term on the righthand side of equation (46), even becomes nil at a certain time t = t_{end}, which ends the plastic dynamic phase.
For t ≥ t_{end} the interface behaves as rigid, with nil displacement and velocity, inducing a discontinuity of the accelerations which jump from no nil values at to nil ones at . In the absence of dissipation source of energy considered in the pads in the present model, the elastic waves generated during the plastic dynamic phase continue for all t > t_{end} to propagate back and forth between the interface and the farfield boundaries of the model.
4.5 Steady Stress State Reached at the End of the Dynamic Phase After Wave Dissipation
Nevertheless, by comparison with field case for which the kinetic energy of seismic waves always ends up dissipating, especially because of geometrical and material viscous damping, it is worth considering for our model the situation reached at the end of a “plasticelastic + rigidelastic” dynamic event, if the P and S waves generated in the pads finally came to cancel.
We note here Q,P,A,B,B′,C the main points shown further in Figure 5 met along a typical stress path starting from the dilatant side of the criterion. They are defined as the beginning and the end of the following segments:

 QP

 = initial elastic phase

 PA

 = stable plastic evolution

 AB

 = dynamic plastic evolution within the domain Ω_{Nosp}(S)

 BB′

 = end of the dynamic plastic evolution

 C

 = steadystate stress point, reached after dissipation of kinetic energy
 ( the researched static state of stress
 , = the stress field and farfield displacements at the beginning of the dynamic plastic phase (or at the end of the quasistatic phase);
 = the interface displacements and fluid pressure at the end of the dynamic plastic phase.
As shown in Appendix D, this state of stress is strictly located within the current elastic domain resulting from the end of the plastic dynamic phase. In the presence of a “pressure reservoir,” this property is considered in CP1 as being a possible source of earthquake aftershocks by successive repressurizations of the fault.
The case of infinite pads in the y direction is shortly discussed in Appendix G. The dynamic phase AB is shown to end in the plastic regime on the CSL, with the theoretical possibility for the interface to slide forever due to the infinite and homogeneous extension of the (P & I) model in the x direction.
4.6 Energy Aspects
The first law of thermodynamics (FLT) gives an easy macroscopic complementary interpretation of the (P & I) model response in the form of energy budget.
 U_{pad} = internal energy inside Ω_{pad}= stored elastic energy
 K_{pad} = kinetic energy inside Ω_{pad}
 = power of external forces acting on Ω_{pad} boundary = forces × displacements velocities at the pad/interface contact (y = H°) = negative quantity since in practice
 U_{(I)} = internal energy within the interface area (I/2), including rocks and fluid
 = power of external forces acting on (I) = opposite quantity to the one considered for Ω_{pad}.
 = internal energy for Kelvin temperature T^{i} (J/m^{2})

 C=average volumetric calorific heat (including rocks and fluid) for the domain (I/2) (J/m^{3}/K) with physical thickness H°.
This equation shows that the plastic dissipation energy during a short dynamic phase is mostly transformed into temperature rise within the interface; the thinner the interface, the higher the increase of temperature.
= elastic + kinetic energy stored in the pad at the end of the dynamic phase
= elastic energy stored in the pad, after return to static state.
Properties  Values 

Pad properties  
h  10,000 m 
E  40,000 MPa 
ν  0.3 
ρ  2,500 kg/m^{3} 
Fault Core properties  
80 MPa  
M  1 
α  26 
Fault zone fluid  
C_{fl}  8. 10^{−3} MPa^{−1} 
R  0 (undrained) 
S estimated as 1/(H ° ϕ ° C_{fl}) with H ° = 10 m, ϕ ° = 10%  125 MPa/m 
Initial conditions (for a dilatant stress path)  
σ_{yy}°  −48 MPa 
σ_{xy}°  12 MPa 
u°(initial fault zone pore pressure)  40 MPa 
Loading conditions  
10 cm/yr  
Far field compressive regime  −2 cm/yr 
5 Implementation of the Solution Into a Numerical Code
 one dealing with quasistatic evolutions;
 the other one dealing with dynamic evolutions.
Each of them is governed by its own timescale, and the code enables automatic transitions from one mode of evolution to the other one, as well as between elastic and plastic regimes. Both routines are based on incremental implicit time schemes with constant time steps ∆t (different between slow and rapid evolutions) and iterative algorithms to integrate the set of nonlinear differential equations of the problem. Details about the algorithms are given in Appendix E.
6 Illustration of the Computed Solution in the Dynamic Plastic and Elastic Regimes
Miscellaneous scenarios of evolution of the (P & I) problem are shown and commented in Maury et al. (2011, 2013, CP1) and Géli et al. (2016), in relation to the behavior of active geological faults. Much of the discussion is based on the stress paths obtained in the plane as a function of data set.
The example below just aims to illustrate one of the typical responses of the model, computed with the VB code of section 5, in the case of a dilatant stress path without drainage (R = 0). The data set considered here (Table 1) is similar to the one mainly used in CP1, except that the farfield regime is taken as compressive ( instead of extensive ( .
 an initial rigidelastic phase (QP) (i.e., rigid for the fault zone, elastic for pads)
 a stable plastic phase with dilatant deformation (PA)
 a dynamic plastic phase (ABB′)
 a final elastic phase below B′, ending at point (C), when assuming the dissipation of elastic waves after some while.
It can be observed that the QP segment is directed to the top left due to the contractant farfield condition leading to negative values for and also , since and in that phase.
It can also be noted that point A for which the code automatically switches from stable to unstable regime is precisely located on the boundary of the domain Ω_{Nosp}(curvilinear triangle in red), which was determined directly from the equations of section 4.3.
In the same way, as a mean of verification of the VB code implementation, it can be checked that point B of the dynamic stress path is well predicted by the “static approximation” described at section 4.6 (cf. white point in Figure 5).
Figure 6 shows the corresponding time evolution for a certain number of variables. It is divided into two columns: the left one deals with the quasistatic evolution at the scale of years associated to the stress path QPA of Figure 5, whereas the right one is related to the dynamic evolution at the scale of seconds (≈6, 9 s), associated to the stress path ABB′C. The curves show the corresponding position of the stress path points Q,P,A,B,B′with time. Note that after point C, the stress path would continue due to the farfield loading and/or the evolution of the pore pressure (not represented here).
In particular, these curves allow compare the variations of the different quantities between the static and dynamic regimes, which are generally higher in the second case despite its much smaller duration. For instance it can be noticed from Figure 6a that the variation of displacement U_{1} is around 0.2 m after 16 years (P to A), whereas from Figure 6a′ it is around 0.2 m after 6.9 s (A to B) and 1.6 m after 14 s (A to B′). Figure 6d′ also shows how the dynamic plastic phase for the (P & I) structure ends, with an abrupt stop of the interface sliding, characterized by a final jump of the acceleration to 0 (this last value being due to the perfect antisymmetry of the model). The fault displacements and pore pressure do not change any more after.
However, as expected from the initial assumptions of the model, stress oscillations continue to occur within the pads after the end of the plastic dynamic regime (see Figure 6c′) since no material, nor geometric loss of energy are in play once the interface has stopped sliding. Instead it is more interesting to look at the situation which can be computed at point C, when the kinetic energy is supposed to have totally disappeared after some time (cf. section 4.5). This one shows a “slight” return into the elastic domain which can reflect the behavior of some faults after a seismic event.
Figure 7 presents the time evolution for the dynamic phase of the energy quantities W_{int/pad}(t), W_{eff}(t), W_{approx}(t) introduced in section 4.6. It can be noticed that the 3 curves are relatively close to each other, all leading to the order of magnitude of 25MJ/m^{2} per unit area of interface at the end of the dynamic phase. The functions W_{eff}(t) and W_{approx}(t) are slightly higher than W_{int/pad}(t) due to dilatance, which induces negative work (σ_{yy} < 0, , ⇒ ).
The results obtained at points A,B,B ′,C of Figure 5 and the value of also make it possible to compute the “radiated energy” given by equation (52). For the present data set, this one is found to be equal to 0,30MJ/m^{2} showing that this value is small (about 1%) in comparison with the elastic energy relaxation in the upper pad between points A and C with magnitude .
Finally, Figure 8, borrowed from CP1, illustrates the stress path QPDD′ obtained for an undrained scenario on the contractant side of the criterion. As expected, no dynamic regime appears in this case; the stress path remains on the left of the CSL without intersecting the domain Ω_{Nosp}(S).
7 Conclusions
This article is a companion paper to Maury et al. (2020) (here also referenced as CP1), intended to detail the technical elements used in it. The present paper is focused on two topics. One (developed in section 2) is the construction and justification of the poroplastic ICL chosen for CP1. The constitutive law accounts for the macroscopic behavior of active geological faults assumed to be deformable either in contract/hardening or dilatant/softening mode, depending upon their degree of maturity (p′_{c}) and the magnitude of the effective normal stress. The second topic addressed in sections 3–6, 3–6 deals with the analytical and numerical study of the response of a poroelastoplastic (P&I) structural model including the previous ICL sandwiched between two elastic pads.
By the fact that it does not depend explicitly on physical time, the ICL developed in section 2 is distinct from the “rate and state” model, generally used in seismology. It can then constitute a useful complementary route to the modeling work carried out in this domain. One of its main features is to derive from the 3D MCC constitutive law, which has the advantage of relying on a wellestablished formalism, but also to be known as representative of a proven phenomenology about contract/hardening or dilatant/softening behavior for some geomaterials. Although this formalism was originally developed for porous soft soils in drained or undrained conditions, here it is believed that it can be extended macroscopically to other situations and scales, like the case of fracturing rock faults submitted to high compressive states of stress.
For its part, the (P&I) model shows two interrelated benefits. As already mentioned, one is to provide a complementary modeling approach to the rate and state model, for the interpretation of the drivers of some typical distinct behaviors of active geological faults (e.g., slow slip events, main events, and aftershocks) under different sets of hypotheses. These topics can be addressed through the evolution of stresses and displacements or through energy considerations. The interested reader is invited to go on to CP1. The other merit of the (P & I) model is to provide a simple framework for the semianalytical study and illustration of the potentialities of plastic (standard) ICLs with end caps, when plunged into 2D or 3D structures.
The present work offers several perspectives. A particularly promising one in the domain of seismology would be to extend, possibly in a semianalytical way first, the (P & I) model to heterogeneous situations along the x direction. This would make it possible to account for different degrees of maturity of a fault along x (especially by considering the yield parameter as a function of x and t) and study in that way the evolution and “spatial propagation” of faults for miscellaneous scenarios of initial conditions (e.g., ), tectonic loading, fault fluid drainage, etc.
Another important avenue could be the implementation of (elasto) plastic ICL, such as the one developed in this article, in 2D or 3D finite element codes, in order to widely extend the possibilities of investigation (geometry, boundary and initial conditions, material properties, etc.) of phenomena involving stable/unstable evolutions due to the formation and evolution of shear bands (e.g., seismology, mine roofs stability, and landslides). The algorithms and numerical code developed in relation with this article and CP1 already give some valuable indications on how one should proceed to insure the detection and management of regime changes (stable ↔ unstable) by properly analyzing the tangent evolution problems. CP1 also defines a few monitoring tracks to be carried out on fault sites, at the surface or in depth, to help better specify interface constitutive laws such as the one developed here and the numerical values of their related parameters.
Abbreviations

 CP1

 companion paper number 1

 CSL

 critical state line

 (dI)

 interface with elementary length

 FLT

 first law of thermodynamics

 GECICL

 generic endcap interface constitutive law

 (I)

 interface

 ICL

 interface constitutive law

 MMC

 modified CamClay constitutive law

 (P)

 pads

 (P & I)

 model with the interface sandwiched between the two pads

 Points Q, P, A, B, B′, C

 points along a general stress path with QP = rigidelastic phase, PA = static plastic phase, AB = plastic dynamic phase inside the unstable stress domain, B′ = point at the end of the dynamic plastic regime, C = point after return to static state

 Ω_{Nosp},Ω_{Nosp}(S)

 stress domain without possible static plastic evolution (function of S)

 ∂Ω_{Nosp}

 boundary of domain Ω_{Nosp} composed of two branches: points A on the right side, points B on the left side

 R&S

 rate and state
Symbols Used in the Equations

 a

 no nil positive coefficient in the consolidation pressure yield curve of MMC

 α

 no nil positive coefficient in the consolidation pressure yield curve of the ICL

 c_{p},c_{s}

 velocity of primary and secondary elastic waves in the pads

 C

 average volumetric calorific heat (including rocks and fluid) in the half domain (I/2)

 C_{fl}

 interface fluid compressibility


 denominator of the static plastic multiplier evolution equation


 value of D for S = 0


 elastic energy contained in one pad


 radiated energy in the upper half of the model (P & I)


 strain tensor


 (plastic) deviatoric strain tensor =

 ε_{v}

 (plastic) volumetric strain =


 vector with strain rate components

 f

 plastic criterion (MMC or interface)

 H°(t)

 thickness of the shear band, concentrating most of the shear rate deformation at time t

 H° ϕ°

 fault thickness × porosity, used to estimate S


 3 × 3 unit tensor

 K_{pad}

 kinetic energy contained in one pad


 plastic multiplier (positive or nil)

 λ,μ

 Lamé's coefficients for the pads

 μ_{R & S}

 friction ratio in R & S models

 M

 slope of the critical state line

 N

 numerator of the static plastic multiplier equation

 p′

 mean Terzaghi effective pressure = (positive for compressive σ states)


 actual value of the “material consolidation pressure” (yield parameter) depending upon the algebraic value of ε_{v} (yield variable)


 initial consolidation pressure of the material for ε_{v} = 0


 surface density of maximal plastic dissipation power

 q

 deviatoric stress = with summation on repeated indexes—positive number)

 R

 (macroscopic) drainage coefficient between the interface and the external source of pressure

 ρ

 specific density of pads


 (total) stress tensor (considered with the usual convention of continuum mechanics)


 effective Terzaghi stress tensor =


 vector of stress components (σ_{yy}, σ_{xy})


 vector of stress components (σ_{xx}, σ_{zz})

 σ_{xy} ° ,σ_{yy}°

 initial stress values within the structure for (U_{1}, V_{1}) = (U_{2}, V_{2}) = (0,0)


 deviatoric stress tensor =

 S

 coefficient relating with the interface pore pressure

 u

 interface pore pressure (positive)

 u_{ext}

 pressure of possible « external » source of fluid related with the interface

 U(t,y)

 pad field displacement along x

 ⟦U⟧

 jump of shear displacement at the interface

 = d⟦U⟧/dt

 slip rate at the interface

 U_{1} = ⟦U⟧/2 = U(t,0)

 displacement along x of the upper lip of the interface

 U_{2}

 farfield displacement along x at the top boundary of the upper pad

 U_{(I)}

 internal energy within the interface half domain

 U_{pad}

 internal energy for one pad

 V(t,y)

 pad field displacement along y

 ⟦V⟧

 jump of normal displacement at the interface

 = d⟦V⟧/dt

 slip rate at the interface


 displacement along y of the upper lip of the interface

 V_{2}

 farfield displacement along y at the top boundary of the upper pad

 W_{int/pad}(t)

 work exerted by the interface over the upper pad

 W_{approx}(t)

 work of shear forces only

 W_{eff}(t)

 work of shear forces and effective normal stresses
Acknowledgments
The authors warmly thank the Reviewers and the Editor for their numerous and constructive comments and questions. This notably led us to extend the article to the case of faults with variable thickness over time, as well as to introduce considerations on the energy aspects linked to the model.
Appendix A: Derivation of the Interface From the Tensorial MCC Law
A1. Method of Calculation
To derive the interface plastic criterion from the tensorial one, let us introduce the (line) vectors , , and .
The first two vectors both characterize the “interface” strain and stress states. The stress vector contains the “complementary” stress quantities which only appear in the tensorial approach.
Thus, the tensorial plastic criterion for the material can be considered as a function of and , that is, .
Interestingly it can be noted that the ICL here derived insures the maximum value of the plastic dissipation power than can be dissipated at the interface for given values of the displacement jumps , .
We then obtain the same equations as the ones above (especially (A1) and (A2)) used to transform f into F and to show that derive from F through the standard flow rule. In other words it shows that the kinematic approximations of figure 3 are equivalent to the more mathematical definition of the function π.
A2. Application to the MCC Model
Let us apply these calculations to the MCC criterion (1) (removing temporarily the effective stress “prime” notation).
As stated in the general case in section 2.3, let us check that the standard flow rule (A4) is verified.
Then it can be easily checked that replacing σ_{xx} and σ_{zz} in (A5) by equation (A7) leads to the same equation as (A9), that is .
In the same way, equations (A10) and (A6) are the same to a factor of 2; then .
Appendix B: Equation for the Interface Pore Pressure (Undrained Case)
To roughly relate the coefficient S in equation (19) with the properties of the porous interface model, let us consider the evolution of the pore fluid pressure within the domain (I) in undrained regime (no exchange between the interface and the domain around). Here we suppose (I) to be a segment with unit length in the x and z directions and with constant (half) thickness H° and (saturated) porosity ∅°. These quantities are related to the volume of fluid in “close relation” with the shear band.
We also suppose (I) submitted to the contractance/dilatance ⟦V⟧(t)( = 2 V_{1}(t) for the (P & I) problem). For the sake of simplicity, the mineral phase of the rock phase is considered as uncompressible.
In the ICL model, S is considered as a global “fault coefficient,” which avoids separating the roles of H°,∅°,C_{fl}. Nevertheless, this expression can be used as an initial guess to determine relevant orders of magnitude for S and to predict its evolution in relation with possible changes of fluid compressibility.
Appendix C: Algorithm for the Computation of the “Static Approximation” of Point B
Let us first reduce the number of unknowns from 3 to 7, in the equations of section 3.7.
Now if considering V_{1}^{B} as “primary unknown,” equations (A14), (A15), and (A16) define a nonlinear system in V_{1}^{B} which can be written as V_{1}^{B} = f(V_{1}^{B}). One can then attempt to solve it by the usual fixed point algorithm V_{1}^{B}(i+1) = f(V_{1}^{B}(i)), with V_{1}^{B}(i) being the value of V_{1}^{B}at the ith iteration.
However, the convergence of this algorithm can only be achieved if the derivative of f in the vicinity of the solution is contractant, that is, checks the condition df(V_{1}^{B})/dV_{1}^{B} < 1. Then since this condition depends upon the data sets of (P&I), two “reciprocal” algorithm based either on (A14), (A17), and (A20) or on (A16), (A18), and (A19) are envisaged and implemented (see below). If the first one is considered as solving V_{1}^{B}(i+1) = f(V_{1}^{B}(i)), the second one, based on the reverse order of previous operations, computes V_{1}^{B}(i+1) as V_{1}^{B}(i+1) = f^{−1}(V_{1}^{B}(i)) with f^{−1} being the reciprocal function of f.
But as f′f^{−1}′=1, either we have in practice f′ < 1 or f^{′−1} < 1, implying the convergence of one of the algorithm.
Algorithm n°1 based on equations (A14), (A17) and (A20): iteration i ➔ i + 1
For the value V_{1}^{B}(i) obtained at iteration n° i
 Compute σ^{′B}_{yy}(i+1) from V_{1}^{B}(i) using equation (A14)

Compute if possible as the positive root of (A17).
If the discriminant of the equation is negative, then stop the iteration process and restart the calculation of V_{1}^{B} with the second algorithm
 Compute

If V_{1}^{B}(i+1) − V_{1}^{B}(i)< ε (small value) then
Do V_{1}^{B} = V_{1}^{B}(i+1) and compute σ^{′B}_{yy}(V_{1}^{B}),σ^{B}_{xy}(V_{1}^{B}); stop
 If the V_{1}^{B}(i) cycle appears to be divergent, restart from algorithm 2
If the V_{1}^{B}(i) cycle appears convergent, restart an iteration
Algorithm n°2 based on equations (A16), (A18) and (A19): iteration i ➔ i + 1
For the value V_{1}^{B}(i) obtained at iteration n° i
 Compute from (A3.3):
 Compute σ^{′B}_{yy}(i+1) as the positive root of (A18)
(the discriminant of the equation must be positive, since algorithm n°1 was first tested without success)
 Compute V_{1}^{B}(i+1) from and 19

If V_{1}^{B}(i+1) − V_{1}^{B}(i)< ε (small value) then
Do V_{1}^{B} = V_{1}^{B}(i+1) and compute σ^{′B}_{yy}(V_{1}^{B}),σ^{B}_{xy}(V_{1}^{B}); stop
Otherwise, restart an iteration
The implementation of these two algorithms in the code mentioned in section 6, with the possibility to switch automatically from the first one to the second one, has demonstrated the efficiency of the procedure for all data sets of the (P&I) problem, leading to the computation of point B on the right branch of the boundary ∂Ω_{Nosp}(S).
Appendix D: Location of the Interface Static Stress in the Plane, Following a Dynamic Plastic Phase After Dissipation of Kinetic Energy
Let be the stresses acting at the interface at the end of a dynamic plastic phase and, as defined in section 4.5, let be the static stresses obtained after some while, assuming dissipation of the kinetic energy within the pads. Our aim below is to show that the stress point is strictly within the current elastic (actually rigid) domain of the actual interface criterion.
Let t^{B′} be the time value at the end of the dynamic plastic phase. and let us denote by ∆ the changes of quantities since time t^{A} (e.g., ∆σ_{xy}(t) = σ_{xy}(t) − σ_{xy}(t^{A})).
By definition of t^{B′}, the stress path (here including kinetic energy in the pads) is elastic for t > t^{B′}, that is inside the domain . It is recalled that due to the brevity of elastic dynamic phase(s) and the rigid behavior of the interface, the pore pressure is constant (no change of V_{1}, undrained regime) and therefore .
Therefore,
⟹ periodicity of ∆σ_{xy}(t) with period T_{s}.
Then for t > t^{B′}, the stress path is included within a rectangular domain (with width and height equal to the amplitudes of and σ_{xy}(t)) located below the boundary . By the way the computations show that the “vertical” amplitude of σ_{xy}(t) are much bigger than the “transversal” one.
The same kind of result can be obtained for .
Then, since the mean values + are inside the rectangle , we get the fact that the stress point is strictly inside the current interface rigidelastic domain.
Appendix E: Algorithms for the Numerical Solving of the Plastic Static and Dynamic Evolutions of Problem (P&I)
We only focus in the following on the numerical solving of plastic phases, the case of elastic ones being obvious.
E1. Algorithm for the Numerical Computation of Plastic Quasistatic Evolutions
The time step for such a regime is chosen in connection with the evolution of farfield conditions (e.g., ∆t such as or possibly with the order of magnitude of 1/R (e.g., in the case of drained regime.
 q^{t} = the values of the different quantities q at time step t (those are all supposed already computed and known)
 q^{t+∆t}= the values of the quantities q at time step t+∆t (those are the unknowns to be computed)
 ∆q = q^{t+∆t} − q^{t} = increment of the different quantities q from t to t+∆t
This set of eight equations (ignoring provisory the inequality ∆κ > 0) defines a nonlinear system with eight unknowns .
i = iteration index
Δκ^{i} = value of Δκ already computed (and then known) at iteration i
Δκ^{i+1}= new value of Δκ at iteration i+1.
Once convergence is reached and Δκ is known, all the other quantities of the problem (P & I) can be computed at time t+Δt.
In practice, the implementation in our VB code is as follows.
Let us denote
q^{t+∆t,i}= value of quantity q already computed (and then known) at iteration i
 Initialization for time step t → t+∆t
 ○
 Check if the solution at time t+∆t is elastic
 ○
Operate the sequence C(∆κ^{0}) = C(0)
 ○
If f(0) < 0 (elastic solution at time t+Δt), then
 ▪
For all quantities, store the result q^{t+∆t} = q^{t+∆t,1}
 ▪
Go to the next time step, if t + ∆t < T
 ○
If f(0) > 0, continue with the following “plastic loop”
 Iterative “Newton plastic loop” i
 ○
Operate the sequence C(∆κ^{i}) leading to f(∆κ^{i})
 ○
Operate C(Δκ^{i,ε}) with Δκ^{i,ε} = Δκ^{i}+ε, leading to f(Δκ^{i,ε})
 ○
Compute
 Test for the sign of Δκ^{i+1}
 If Δκ^{i+1} < 0, the solution is no more static; switch to the algorithm for the computation in plastic dynamic regime, starting from the previous time step (having converged)
 If Δκ^{i+1} > 0, continue as follows:
 Test for convergence of Δκ^{i+1}
 If Δκ^{i+1} − Δκ^{i} > η (small positive scalar), then do another iteration with i ⟶ i+1

Otherwise,
 ○
For all the quantities, let q^{t+∆t} = q^{t+∆t,i+1}
 ○

Store q^{t+Δt}
 ○
Go back to the initialization operation to solve for the new time step t+∆t → t+2 ∆t (until reaching the condition t+∆t = T)
 ○
E2. Algorithm for the Numerical Computation of Dynamic Plastic Evolutions
The time step ∆t for this regime is chosen as a small fraction of . The solution from time t to t+∆t is based on the research of the velocities and value of which make it possible for the interface criterion to be nil (for a plastic evolution).
We note:
Δq = variation of the quantity q, since the beginning of the dynamic regime (or since the end of the previous static regime)
δΔq = variation of Δq, between times t and t+Δt
δΔq^{t+∆t,i+1},Δq^{t+∆t,i+1} = values of δΔq or Δq to be computed at iteration (t+∆t,i+1)
 all quantities q^{t},Δq^{t}
 all quantities q^{t+Δt,i},Δq^{t+Δt,i}
 the quantities , since they only require the knowledge of the solution until times t+∆t − 2t_{p} and t+∆t − 2t_{s} which are both smaller than t

Initialization for time step t → t+∆t (if t+∆t < T = final computation time)
 ○
i = 0, ,
 ○
 Iterative “Newton plastic loop” i
 ○
Operate the sequence leading to
 ○
Operate with , leading to
 ○
Compute

Test for convergence of
 If (small positive scalar), then do another iteration with i ⟶ i+1

Otherwise,
 ○
For all the quantities, let q^{t+∆t} = q^{t+∆t,i+1}
 ○

Store q^{t+∆t}
 ○
Test for the sign of
 ○

If (plastic solution at time t+Δt)
 Go back to the initialization operation to solve for the following time step t+∆t → t+2 ∆t

If (elastic solution reached between t and t+Δt)
 Switch to the algorithm for the dynamic elastic regime
 Compute the “static solution” corresponding to the dissipation of kinetic energy within the pads (see section 4.4)
 Switch to the algorithm for static (elastic/plastic) evolutions (if t+Δt < T)
Appendix F: (P&I) Equations When Adding an Elastic Component Into the ICL (Quasistatic Regime Case)
It only introduces a corrective factor in front of the pore pressure variable.
Finally, it can be concluded that when written in terms of V_{1}^{p}, the equations of the problem (P & I) involving an elastic component in the ICL, have globally the same structure as the one considered in section 3.1, with only some minor differences in the coefficients of the equations. Then neglecting interface elasticity has no major qualitative effects on the plastic evolutions of (P&I).
Appendix G: Limit Case of (P&I) Problem for Infinite Thick Pads
The (P&I) model makes it possible to consider the limit case h = +∞ (semiinfinite pads)
G1. Static Regime
Thus, the “static equations” for the case h = +∞ can be easily derived from those of section 3, simply by substituting and for and and by taking the limit of other terms for h → +∞.
Besides, since p′_{clim} → 0 for h → +∞, the left boundary of the domain Ω_{Nosp} (branch of points B) becomes confounded with the entire CSL from the origin O.
G2. Dynamic Regime
In the dynamic regime, let us show that the stress state ( ) at the interface stops at some stationary point (B = B′) located on the CSL, for which the interface slides forever in the plastic regime. The normal displacement V_{1}^{B} is fixed with zero dilatance ( ) but the shear displacement U_{1}^{B} evolves continuously at constant sliding velocity .
Let us note with subscript A the quantities at the time the dynamic regime starts. These quantities are supposed to be known from the computation of the last static phase. During this dynamic phase the stress path will at least reach the left boundary of the domain Ω_{Nosp}, in this case the CSL. Then let us note with subscript B the quantities when the stress state intersects the CSL and let us determine the mechanical state of the interface in B from its state in A.
Once reached, this situation becomes “stationary” for the theoretical problem (P & I) with all the mechanical quantities being set, except for the shear displacement amplitude which grows indefinitely at constant velocity Of course this unrealistic situation is due to the homogeneity and infinite length of the model along the x direction but is not specifically linked to the use of the GECICL. The possibility of infinite displacements for would disappear in the case of infinite models with x and y but with finite extension of the sliding interface (I) along x.
Nevertheless, by continuity, the present study proves that one possible means to increase the amplitude of shear displacements in dynamic phases of (P & I) models, finite in the y direction, is to increase the pad thickness h.
Open Research
Data Availability Statement
 the “VBA executable file” inserted within XCEL framework, used to generate most of the results and figures contained in the two papers
 some short instructions to run this file open to any users data (within some ranges of parameter values)
 some examples of “datasheets” ready to run and especially related to the scenarios described in CP1 and CP2.
These materials can be uploaded from Gustave Eiffel university/IFSTTAR depository address: https://doi.org/10.25578/MK4NQU