Interface Plastic Constitutive Law With End-Cap 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 rigid-plastic 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 far-field displacement boundary conditions evolving time are used, different modes of evolution emerge, and transitions happen, from quasi-static 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 quasi-static 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 quasi-static models (Fitzenz & Miller, 2004), quasi-dynamic models (Dieterich, 1995; Robinson et al., 2011; Zoeller et al., 2004), and full elastodynamic models (Ben-Zion & 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 poro-rigid-plastic 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 Cam-Clay (MCC) law. We denote in brief the obtained ICL as generic end-cap (GEC-ICL) for “generic end-cap” 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 far-field 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).

Figure 1

Main features of the fault model (P & I) considered in Maury et al. (2019) (not to scale). The displacements (U(y,t), V(y,t)) stand in the plane (x,y) with plane strain assumption in the third direction z. They only depend upon the spatial variable y and are antisymmetric with regards to the x axis. In red: fault zone domain, modeled from the geometrical point of view as a zero thickness interface (I) with end-cap poroplastic behavior, permitting shear and normal displacement discontinuities. From the physical point of view, (I) is considered to have a finite thickness 2H°(t), evolving indirectly with time as a function of strain localization transversally to the fault. Above and below (I): elastic pads (P) with thickness h ≫ H°(t). The whole model is denoted (P & I).

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 quasi-static 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 (GEC-ICL)

In the following we denote by GEC-ICL 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 GEC-ICL

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 GEC-ICL 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 2-D or 3-D standard numerical tools (e.g., finite or boundary element methods) for solving problems with known geometry interfaces (curves in 2-D, surfaces in 3-D). 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 GEC-ICL 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 quasi-flat 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 GEC-ICL 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 GEC-ICL is that it does not refer explicitly to the physical time, unlike the “R&S” models. For this we restrict the formulation of the GEC-ICL 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 quasi-static 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 sum up, the criteria retained for the formulation of the GEC-ICL are as follows:

1. to be without thickness from the geometrical point of view
2. 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
3. to derive from the (standard) plasticity framework
4. 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 GEC-ICL from theoretical considerations and assessing its relevance a posteriori, by comparing its effects in models with on-site 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 GEC-ICL includes two steps: first, we select a 3-D (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 3-D constitutive law into an ICL.

2.2 Choice of the MCC Constitutive Law as a Starting Point

The tensorial (standard) plastic laws with end-cap 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 Mao-Hong 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 end-cap 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.

Several variants of the MCC model can be found. Here we use the set of following equations:

(1)

(2)

(3)

where

= (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 (rigid-plastic 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.

Figure 2

3-D elliptic modified Cam-Clay criterion in the (p′, q) plane for three different values of p′_c (recall that p′ is positive for compressive effective states of stress). p′_cⁱ is the on-site initial value of p′_c depending on the past history of the soil. For example, for normally consolidated soil, p′_cⁱ is equal to h ρ_overburden g(1+2K_h)/3 with ρ_overburden = average specific density of the overburden, K_h= horizontal stress factor.

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 Cam-Clay model, this role is devolved to the (plastic) void index). In keeping with Cam-Clay 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 GEC-ICL. 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 rigid-plastic 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).

Figure 3

Two close modes of shear band modeling. (top) Zero thickness interface, with (shear) displacement jumps. (bottom) 3-D medium with finite thickness and continuous (shear) displacements.

One is to consider (dI) as a zero thickness interface as referred to in the rest of the article. Then the shear and normal displacements and stresses, characterizing the mechanical conditions applying to the interface are ⟦U⟧,⟦V⟧ and σ_xy,σ_yy with

1. ⟦U⟧= difference of shear displacement between the two lips of (dI)
2. ⟦V⟧= difference of the normal displacement, also called dilatance for ⟦V⟧ > 0 or contractance for ⟦V⟧ < 0.

The other possible way of modeling (dI) is to consider it as a shear band with a small but finite thickness 2H (simplifying here H°(t) into H) with displacements U(x,y),V(x,y) being quasi-linear with y and with variation in x being small compared to those with y. The strain tensor components can then be approximated as ε_xx ≈ 0, , , that is,

(4)

This analogy makes it possible to convert, among other things, the 3-D plastic constitutive laws (criterion + flow rule) into ICLs. For this, two possible methods coming back to the same are presented in Appendix A and applied to the MCC criterion (equation (1)). Doing so, we get the following interface plastic criterion equation (that is slightly simplified in the following sections):

(5)

Besides, it is shown in Appendix A that the 3-D standard flow rule turns into a “standard interface flow rule,” that is,

(6)

2.4 ICL Model Used in the Following

Here since our aim is mostly to deal with a generic ICL, we ultimately use in the following the criterion function F obtained by omitting the coefficient into equation (5), that is,

(7)

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 3-D 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).

In equation (7) still plays the role of a yield parameter, but related to the yield variable ⟦V⟧ instead of the volumetric strain ε_v as in equation (2); precisely, referring to this equation, we write

(8)

with α being a no nil positive coefficient with unit inverse to that of a length. α is considered as constant in the following.

About the flow rule, applying equation (6) to F, gives

(9)

(10)

with , f ≤ 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 .

In this way the model accounts for the probably general phenomenology that can take place in many materials or media, with the formation of plastic shear bands (and subsequent changes in strain localization) behaving

1. as dilatant and softening under relatively small (effective) normal stresses,
2. or vice versa, as contractant and hardening (macro) under relatively high normal (effective) stresses

with a transition stress area in the vicinity of the CSL for which only small or zero volumetric changes happen.

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 end-cap ICL with a shape close to the one here derived was also proposed by De Gennaro and Franck (2005) to account for experiments on soil-pile 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 Quasi-static Regime

3.1 Equations for the (P & I) Model in the Static Domain

In the quasi-static domain (slow evolutions) the equations for the upper part (y ≥ 0) of the antisymmetric fault model (P & I) of Figure 1 are written as follows:

(19)

with

h,λ,μ

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

σ_yy,σ_xy

= normal and shear stresses in the structure

V₁,U₁

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

V₂,U₂

= uniform far-field 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 quasi-static 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 U₁ = U₂ = V₁ = V₂ = 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 GEC-ICL 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 far-field displacements U₂,V₂ (or ) and possibly to the fluid pressure u_ext. The first case is that of rigid-elastic evolutions. The case of plastic evolutions is then discussed.

3.2 Rigid-Elastic Evolutions

We call “rigid-elastic” evolutions those for which the stress path is strictly below the current interface criterion (f < 0). Indeed, in that case the interface behaves as rigid, that is or still due to antisymmetry of the (P&I) model. Only the pads are subject to elastic strain evolution due to the far-field conditions The total strains evolve according to equations (11) and (12). The equation for the pore pressure reduces to the following:

which only accounts for the potential drainage between the interface and a pressure reservoir. In particular, u is constant during rigid-elastic evolutions in the undrained case (R = 0).

Actually as discussed in Appendix F and CP1 it would be possible to add another term on the right-hand 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.

Thus, as shown in Appendix F (equation (34)), (19) could be given the more general shape:

(20)

that reduces to

and then from (11),

in the case of rigid-plastic evolution.

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).

Let t be a given value of time, during a plastic evolution, characterized by the two equations f(t) = 0 and . We first compute the time derivatives of the problem variables (displacements, pore pressure, stresses) as functions of the following:

• the far-field 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 , .

By combination between (19) and (16), we get for

(21)

From the time derivative of (11),

and by combination with (16), it becomes

By summation of this equation with the previous one for , we get

Here just note that the use of equation (20) instead of (19) would lead to

which does not change much the shape of the previous equation and therefore the following calculations, for ς < 1.

From the time derivative of (12) and by combination with (17)

And finally, by derivation of (15) and combination with (16)

These equations show that the whole set of the time derivative problem variables is known at time t, if the value of is known.

To compute this coefficient, let us use the equation , that is,

and let us plug in this one, the above expressions obtained for , and .

It becomes

that is,

(22)

with

(23)

(24)

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 far-field 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 rigid-elastic 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.

To simplify the notations and ease the geometrical representation of the 2 conditions above, we set temporarily

Then the equations f = 0 and D = 0 write respectively

(25)

(26)

In the (X,Y > 0) half-plane, 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.

Figure 4

Illustration of the two conditions f = 0,D = 0 in the (σ′_yy, σ_xy) plane for two values S₁, S₂ of S with S₁ < S₂ (see section 3.5) (all other data being the same). Case 1: existence of two intersection points A, B for D₁ = D(S₁) = 0. Case 2: absence of intersection for D₂ = D(S₂) = 0.

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 half-plane (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.

Figure 5

Typical response of the (P&I) structure for a dilatant stress path. Rigid-elastic phase QP (green), followed by a stable plastic phase PA (blue, duration ~ 10 years, ∆U₁~ 20 cm, ~ 2 cm/yr) and an unstable plastic phase ABB′ (red brown, duration ~ 20 s, ∆U₁~ 2 m). C = final state of stress located in the actual rigid-elastic domain of the (P&I) structure (smallest ellipse), after dissipation of elastic waves. Curvilinear triangle in red = zone of potential instability. White point next to B = predicted passage of the stress path, from the static approximation of section 4.6. Nota: the stress path was arbitrarily stopped at point C, but it continues to evolve beyond due to tectonic loading and/or fluid pressure evolution.

The condition for the existence of solutions (X,Y) defining the boundary ∂Ω_Nosp of the domain Ω_Nosp and their computation can be easily obtained by eliminating Y (for instance) between equations (25) and (26), which leads to the following second-order polynomial equation:

The existence of X is given by the condition

that is,

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 quasi-static rigid-elastic 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, S₁ < S₂ ⇒Ω_Nosp(S₂) ⊂ Ω_Nosp(S₁).

Indeed, let (σ′_yy, σ_xy) be a stress point in Ω_Nosp(S₂) corresponding to some given value of p′_c such as . For that point is negative. But as shown by its expression (24), D is a linear increasing function of S. Therefore, for S₁ < S₂

showing that (σ′_yy, σ_xy) also belongs to Ω_Nosp(S₁).

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.

In drained regime, the situation is different. If comes to be nil, then

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:

By deferring this equation into the one governing the fluid pressure 19, we get

That is by denoting Δu = u − u_ext and taking u_ext as constant,

or still since

(27)

with

(28)

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 .

Since from equation (21)

(29)

the comment above also stands for and then for all variables of the problem (P & I).

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 far-field 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 quasi-static 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 quasi-static 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).

Thus, let us assume that during the dynamic stress path AB, inertia forces do not affect too importantly the elastic calculation of stresses given by equations (11) and (12). Then from these two equations and (19), (13), and (15) we get

(30)

(by neglecting the “diffusion term” R(u_ext − u) during AB path and assuming S constant)

(31)

(32)

where the subscripts A,B denote the quantities at these points.

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

Equations (13) to (19) already introduced remain valid here. Only equations (11) and (12) relating the far-field and interface displacements to stresses have to be revised by considering the dynamical equilibrium of the pads. The displacements fields in the upper pad ∆V(t,y),∆U(t,y) now depend not only upon time, but also on the y coordinate and comply with the (uncoupled) wave equations:

(33)

(34)

as well as with the initial and boundary conditions, given by

(35)

(36)

(37)

(38)

(39)

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 far-field displacements are considered as negligible due to the very short duration of the dynamic phases (as found latter on in the numerical applications).

Then taking into account the successive reflections of the waves on the boundaries y = 0,y = h, the solutions to these equations are written as

In particular, these equations make it possible to connect stresses and displacements at the interface y = 0. By dropping the prefix ∆ in front of the derivatives , , we get

(40)

(41)

where , stand for the “past history” of , , that is,

(42)

(43)

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 far-field 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.

Then as previously, we have the equations

by neglecting in this last one the “slow” effect of drainage on fluid pressure in comparison with the instantaneous impact of dilatance.

By derivation with time of equations (40) and (42) and with , we get

and from the time derivation of equation (16):

that is,

(44)

In the same way, we get

that is,

(45)

Then by setting (44) and (45) into the equation , we obtain the following consistency equation for :

(46)

with

(47)

(48)

4.3 Behavior of Plastic Dynamic Solutions for Small Time Values

The first-order 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 .

At the beginning of the dynamic phase 0 ≤ t ≤ 2h/c_p for which the right-hand side term in the previous equation is nil and is small, the evolution law for can be approximated by

with

Then considering A and B as quasi-constant in the beginning of the dynamic phase, is found to behave as an exponential function of time:

meaning that the condition is satisfied, at least for a while, just by taking a strictly positive value for .

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 ).

From the expression (24) of denominator D, A also writes

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.

Since

τ appears to be between the durations and needed for the waves to travel across the pads. As expected it confirms that solutions in the dynamic regime initiate in an “unstable” way, with rapid increase of the accelerations, velocities and displacements at the interface and within the pads.

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 quasi-static 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.

Indeed, let us consider the solutions of the linear differential equation for the two different initial conditions and (assuming for instance ). Those write and and are then related by the relationship

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 right-hand 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 far-field 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 “plastic-elastic + rigid-elastic” 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

= steady-state stress point, reached after dissipation of kinetic energy

Then using these points as subscripts, let us denote

1. ( the researched static state of stress
2. , = the stress field and far-field displacements at the beginning of the dynamic plastic phase (or at the end of the quasi-static phase);
3. = the interface displacements and fluid pressure at the end of the dynamic plastic phase.

From the elastic behavior of the pads (equations (11), (12)), we get

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.

Let us first apply the FLT to the (upper) pad only, neglecting any external thermal sources and temperature variations within this one. Let us also consider a dynamic phase during which the change of far-field displacements can be neglected ( . Then for the domain Ω_pad(x,H° < y < h,z) with unit area ∆x = 1,∆z = 1, we have

with

1. U_pad = internal energy inside Ω_pad= stored elastic energy
2. K_pad = kinetic energy inside Ω_pad
3. = power of external forces acting on Ω_pad boundary = forces × displacements velocities at the pad/interface contact (y = H°) = negative quantity since in practice

Rewriting this equation as

(49)

then shows that a dynamic phase is initiated when the strain energy release rate due to the decrease of the interface resistance (criterion decrease) exceeds the “plastic” energy dissipation rate . The difference between both quantities is then transformed into kinetic energy.

On the other hand, the FLT applied to the (upper-half) interface area denoted (I/2) with ∆x = 1,∆z = 1 (and physical thickness 0 < y < H°) writes

with

1. U_(I) = internal energy within the interface area (I/2), including rocks and fluid
2. = power of external forces acting on (I) = opposite quantity to the one considered for Ω_pad.

With this equation we neglect any kinetic energy in (I) and also assume that no heat exchange occurs with the external medium, that is, the upper pad. Furthermore, we can consider U_(I) as essentially related to temperature, neglecting free as well as blocked elastic energy, except the one due to the fluid compressibility combined with volumetric changes. Here we also neglect for the sake of simplicity any feedback loop possibly induced by fluid temperatures changes (thermal dilation, vaporization, etc.), that could be considered in future works. Then making use of the linearization

the FLT for (I/2) writes

(50)

where

1. = internal energy for Kelvin temperature Tⁱ (J/m²)

2. 

3. C=average volumetric calorific heat (including rocks and fluid) for the domain (I/2) (J/m³/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.

Finally, by summing the FLT equations (49) for the (upper) pad and (51) for the (upper-half) interface, the term disappears as now being the power of internal forces for the global system (P & I). Then we get

still,

(51)

showing that during a dynamic phase the elastic energy dissipation rate (pad + possibly fault fluid) is (mostly) transformed into kinetic energy within the pads and heat within the interface.

It is also interesting to evaluate the difference of energy stored in the model between the condition at the end of the dynamic phase (point B′ along the stress path) and the one reached when assuming the pads have returned to static state (point C, cf. section 4.5). Indeed, this difference can be compared with the radiated energy E_R considered in seismology (e.g., Kanamori & Rivera, 2006). For the upper part of the model, is given by the equation

with

= 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.

These quantities can be calculated using the stress state at the beginning of the dynamic phase and the work of the stress vector exerted by the interface on the upper pad all along the dynamic phase, that is,

Indeed, from integration with time of equation (49), we get

where from

using the notation ∆ for changes of quantities since the beginning of the dynamic phase (point A).

But between the static state in A and any other static state,

that is with , , ,

Hence, since the displacements in the static state are the same as the ones at the end of the dynamic phase, can be computed as

(52)

For illustration, Figure 7 in section 4 shows the time evolution of the three following quantities computed in the dynamic regime for the data set example, given in Table 1:

Table 1. Data Set Used in the Present Example

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
Cfl	8. 10^−3 MPa^−1
R	0 (undrained)
S estimated as 1/(H ° ϕ ° Cfl) with H °  = 10 m, ϕ °  = 10%	125 MPa/m
Initial conditions (for a dilatant stress path)
σyy°	−48 MPa
σxy°	12 MPa
u°(initial fault zone pore pressure)	40 MPa
Loading conditions
	10 cm/yr
Far field compressive regime	−2 cm/yr

The first one W_int/pad is the work of the stress vector exerted by the interface upon the pad since the beginning of the dynamic phase. The second one is an approximation of W_int/pad when neglecting the normal component with regards to the shear one. The last one W_eff computed with the normal effective stress corresponds to the right term of equation (50). Actually it can be observed from the curves of Figure 7 that these integrals are closed to each other proving the predominance of the shear term in these calculations. In particular, equation (50) can be approximated by

We also compute in section 6 the numerical value of which for the chosen data set is found to be small compared to the relaxation of elastic energy in the pads during the dynamic phase. That is,

5 Implementation of the Solution Into a Numerical Code

A numerical code (VB) was developed to compute the solution with time of the (P & I) problem for any given initial conditions and functions of far-field displacements U₂(t), V₂(t) and parameters u_ext(t),R(t),S(t) defining the pore pressure evolution. The code comprises two main routines:

1. one dealing with quasi-static evolutions;
2. 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 far-field regime is taken as compressive ( instead of extensive ( .

Figure 5 shows the computed stress path as well as the evolution of the interface criterion, here characterized by a monotonous decrease of p′_c. The choice of the initial stress and pore pressure conditions illustrated by point Q locates most of the scenario at the right of the CSL. Four consecutive regimes are observed, as expected from the theoretical study:

1. an initial rigid-elastic phase (QP) (i.e., rigid for the fault zone, elastic for pads)
2. a stable plastic phase with dilatant deformation (PA)
3. a dynamic plastic phase (ABB′)
4. 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 far-field 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 quasi-static 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 far-field loading and/or the evolution of the pore pressure (not represented here).

Figure 6

Examples of time curves for a certain number of variables. Left column = static evolution; right column = dynamic evolution. Row (a),(a′): V₁(t),U₁(t) (same scale); row (b),(b′): u(t); row (c),(c′): ; row (d),(d′): − left scale, − right scale on (d′) only.

The rows of Figure 6 illustrate successively the time curves for the following:

1. the interface displacements U₁,V₁: Figure 6 (a),(a′)
2. the pore pressure u: Figure 6 (b),(b′)
3. the interface stresses σ_xy,σ′_yy: Figure 6 (c),(c′)
4. the interface horizontal velocity : Figure 6 (d),(d′)
5. the interface horizontal acceleration in the dynamic regime: Figure 6d (′).

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 U₁ 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/m² 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, , ⇒ ).

Figure 7

Time evolution of energy functions W_int/pad(t),W_eff(t), W_approx(t) introduced in section 4.6.

Using equation (50) and the typical value C = 5 MJ/m³/ ° C for the volumetric calorific heat of the fault, we then get the following estimation of the temperature rise ∆θ (°C) expected within the shear band during the dynamic phase, that is,

if considering H° as constant and expressed in meters. It shows the strong impact of the shear band thickness on ∆θ, around 5°C for H° = 1 m but reaching 500 °C for H° = 1 cm. (For recall the shear displacement ∆U₁ during that phase is about 1,6 m).

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/m² 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).

Figure 8

Scenario on the contractant side of the criterion. Undrained and far-field extensive displacement conditions. QP: elastic static regime, PDD′: plastic static regime.

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 3-D MCC constitutive law, which has the advantage of relying on a well-established 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 2-D or 3-D 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 2-D or 3-D 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

GEC-ICL

generic end-cap interface constitutive law

(I)

interface

ICL

interface constitutive law

MMC

modified Cam-Clay 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 = rigid-elastic 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 (U₁, V₁) = (U₂, V₂) = (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

U₁ = ⟦U⟧/2 = U(t,0)

displacement along x of the upper lip of the interface

U₂

far-field 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

V₂

far-field 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

Let be the stress tensor related to the strain tensor of equation (4) in section 2.3:

with its components being slowly variable within the shear band.

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, .

For a plastic evolution of (dI) with , we have the two following equations according to the tensorial flow rule (3), when neglecting elasticity:

(A1)

But since 0 and this last equation implies

(A2)

meaning that is an implicit function of (the two vectors having the same size).

Let us denote this relationship as

Then by deferring this equation into f, we get the function

(A3)

which actually defines the plastic interface criterion F we are looking for, this one only depending upon the stress vector .

Besides, it can also be observed that the interface “strain” rate derives from F through the standard flow rule. Indeed, by computing it becomes

since for .

Therefore, for a plastic evolution with , from A1, or still

which after multiplication by the positive factor 2H can finally be rewritten in the shape of the standard flow rule for the interface (dI) (keeping with the notation for the plastic multiplier instead of ):

(A4)

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 , .

Indeed, referring to Salençon (1983), let us note π this quantity, that is,

Then by introducing the Lagrangian

with f being a (smooth) convex criterion, π is obtained by looking for the stationarity of with regards to its parameters, that is,

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).

From the derivatives, and , we have and then from the 3-D flow rule A3

That is with

(A5)

(A6)

The two last equations allow to express σ_xx and σ_zz as functions of σ_yy, that is,

(A7)

and then

Finally, setting these expressions into f(p,q), we deduce the plastic interface criterion

(A8)

As stated in the general case in section 2.3, let us check that the standard flow rule (A4) is verified.

Here we have

(A9)

and

(A10)

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 .

Consequently, we get , which after multiplication by the positive factor 2H can be rewritten indeed as standard flow rule for the interface (I), that is (still noting the plastic multiplier , rather than ),

(A11)

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 V₁(t) for the (P & I) problem). For the sake of simplicity, the mineral phase of the rock phase is considered as uncompressible.

At time t, the fluid volume V_fl inside (I) is equal to the difference between the volume of (I) itself and the volume of the rock phase, that is,

Then

assuming for the sake of simplicity ⟦V⟧ ≪ 2H °∅ °.

Therefore, introducing the fluid compressibility defined as

it becomes

with

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.

By combination of equations (30), (31), and (32) we get

(A14)

Besides, by elimination of the term 3σ_xy²/M² between the equations f = 0 and D = 0, we have

(A15)

Then with the equation

(A16)

we get a reduced system with three equations and three unknowns ( .

Note that (A15) can be either rewritten in the form of a second-order polynomial in , that is,

(A17)

or in the form of a second-order polynomial in σ^′B_yy, that is,

(A18)

Also, equations (11) and (15) can be rewritten

(A19)

(A20)

Now if considering V₁^B as “primary unknown,” equations (A14), (A15), and (A16) define a nonlinear system in V₁^B which can be written as V₁^B = f(V₁^B). One can then attempt to solve it by the usual fixed point algorithm V₁^B(i+1) = f(V₁^B(i)), with V₁^B(i) being the value of V₁^Bat 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(V₁^B)/dV₁^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 V₁^B(i+1) = f(V₁^B(i)), the second one, based on the reverse order of previous operations, computes V₁^B(i+1) as V₁^B(i+1) = f⁻¹(V₁^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 V₁^B(i) obtained at iteration n° i

1. Compute σ^′B_yy(i+1) from V₁^B(i) using equation (A14)

2. 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 V₁^B with the second algorithm

3. Compute

4. If |V₁^B(i+1) − V₁^B(i)|< ε (small value) then

   Do V₁^B = V₁^B(i+1) and compute σ^′B_yy(V₁^B),σ^B_xy(V₁^B); stop

5. If the V₁^B(i) cycle appears to be divergent, restart from algorithm 2

If the V₁^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 V₁^B(i) obtained at iteration n° i

1. Compute from (A3.3):
2. 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)

3. Compute V₁^B(i+1) from and 19

4. If |V₁^B(i+1) − V₁^B(i)|< ε (small value) then

   Do V₁^B = V₁^B(i+1) and compute σ^′B_yy(V₁^B),σ^B_xy(V₁^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 V₁, undrained regime) and therefore .

Then it can be observed that for t >^B′, are both periodic functions of time with respective periods equal to and . Indeed, let us consider for instance the function ∆σ_xy(t). From equation (41), it is written as

that is since for t > t^B′

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.

Then let us show that and are the mean values of and σ_xy(t) over their respective periods. Considering ∆σ_xy(t) for example and the integration time slot t^B′ ≤ t ≤ t^B′+T_s, we have

since ∆U₁(t^B′ − nT_s) is nil from some given value of n.

But from equation (12), we also have

and then

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 rigid-elastic 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 Quasi-static Evolutions

The time step for such a regime is chosen in connection with the evolution of far-field conditions (e.g., ∆t such as or possibly with the order of magnitude of 1/R (e.g., in the case of drained regime.

Let us consider the time integration of equations (11)–(19) between times t and t+∆t in plastic regime. We denote

1. q^t = the values of the different quantities q at time step t (those are all supposed already computed and known)
2. q^t+∆t= the values of the quantities q at time step t+∆t (those are the unknowns to be computed)
3. ∆q = q^t+∆t − q^t = increment of the different quantities q from t to t+∆t

By considering the Euler fully implicit algorithm with the approximation applied to equations (11)–(19), considered below in a different order, we have to solve the following set of equations:

(if the stress state at time t+∆t is found “plastic,” with ∆κ > 0

This set of eight equations (ignoring provisory the inequality ∆κ > 0) defines a nonlinear system with eight unknowns .

Actually, similarly to what has been done for the derivation of the plastic multiplier equation (section 3.3), this system can be reduced to a single scalar equation in terms of ∆κ only, by expressing all the unknowns as a function of ∆κ and by looking for the value of ∆κ that satisfies the condition

To do so, this equation can be solved using by instance Newton's iterative algorithm that is,

where

i = iteration index

Δκⁱ = value of Δκ already computed (and then known) at iteration i

Δκⁱ⁺¹= new value of Δκ at iteration i+1.

In this equation, the derivative of f with κ can be approximated by the finite difference

with ε being a “small” positive real value.

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

∆q^t+∆t,i+1,q^t+∆t,i+1= values of quantities ∆q or q to be computed at iteration i+1and for a given value of ∆κ, let C(∆κ) be the following sequence of consecutive calculations ending with the calculation of f

Then to compute the solution from time t to t+Δt, the algorithm comprised the following operations:

1. Initialization for time step t → t+∆t

• ○

  

2. Check if the solution at time t+∆t is elastic

• ○

  Operate the sequence C(∆κ⁰) = 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”

3. Iterative “Newton plastic loop” i

• ○

  Operate the sequence C(∆κⁱ) leading to f(∆κⁱ)

• ○

  Operate C(Δκ^i,ε) with Δκ^i,ε = Δκⁱ+ε, leading to f(Δκ^i,ε)

• ○

  Compute

• Test for the sign of Δκⁱ⁺¹
• If Δκⁱ⁺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 Δκⁱ⁺¹ > 0, continue as follows:
• Test for convergence of Δκⁱ⁺¹
• If |Δκⁱ⁺¹ − Δκⁱ| > 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)

At time t and at iteration (t+Δt, i), we consider as being known

1. all quantities q^t,Δq^t
2. all quantities q^t+Δt,i,Δq^t+Δt,i
3. 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

We introduce the sequence of computation , ending by the calculation of , defined as follows:

Then using Newton's iterative algorithm to solve for , the whole algorithm between times t and t+Δt writes as follows:

1. Initialization for time step t → t+∆t (if t+∆t < T = final computation time)

   • ○

     i = 0, ,

2. 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 (Quasi-static Regime Case)

Adding an elastic component to the ICL considered in section 3 of the article comes to modify equations (11)–(19) in the following way:

(A29)

where the subscripts “e” and “p” respectively stand for the elastic and plastic components of the interface displacements, which now split into

(A30)

(A31)

Besides, this set of equations must be completed by the relationships between the elastic interface displacements and stresses. Assuming for example independency between (effective) normal and shear stresses, one can write

(A32)

(A33)

with l,m being two positive coefficients, with the dimension of an elastic modulus divided by a length (e.g., interface physical thickness). Neglecting elasticity comes to consider l → +∞, m → +∞.

Then by elimination of U₁^e between (A32) and (A22), one gets

which is similar to (12), if considering U₁ = U₁^p and replacing μ by .

In the same way, elimination of V₁^e between (A33), (A29), (A23), and (A21) comes to the equation

which is close to the equation derived from (13) and (11):

if considering V₁ = V₁^p and replacing λ+2μ by and σ_yy° by

It only introduces a corrective factor in front of the pore pressure variable.

Actually the main change in the equations of the problem comes from the pore fluid rate evolution equation, which now writes

or still

(A34)

which is the shape of equation (20) of section 3.2 if changing notations ( . But it can be easily shown that the additional term , compared to equation (19), does not alter much the derivation of the plastic multiplier equation in the plastic regime (see note in section 3.3).

Finally, it can be concluded that when written in terms of V₁^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 = +∞ (semi-infinite pads)

G1. Static Regime

Let us first consider the static regime. In that case equations (11) and (12) can be rewritten as

where and are considered as two given (linear) functions of time, reflecting the far-field (tectonic) boundary conditions and replacing the driving factors and .

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 → +∞.

For instance the numerator and denominator equations (23) and (24) of equation (22) turn into

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 V₁^B is fixed with zero dilatance ( ) but the shear displacement U₁^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.

The terms , in equations (40) and (41) are nil in the absence of wave reflection; the stress change reduces to the classical relationships for semi-infinite elastic media:

or still, since ∆u =  − S∆V₁ for rapid changes

where = . is nil since the stress point in B is on the CSL.

But we also have the equations at point B

where from we deduce as the implicit positive solution ( > 0) of the equation

It follows the whole set of variables in B, among which the constant velocity , given by

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 GEC-ICL. 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

A package of supplementary materials has been prepared to support this paper as well as CP1. These ones supply the following:

1. the “VBA executable file” inserted within XCEL framework, used to generate most of the results and figures contained in the two papers
2. some short instructions to run this file open to any users data (within some ranges of parameter values)
3. 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