stability of accretionary wedges based on the maximum strength

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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 60 (2012) 643–664

0022-50

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jmps

Stability of accretionary wedges based on the maximum strengththeorem for fluid-saturated porous media

A. Pons, Y.M. Leroy n

Laboratoire de Geologie, CNRS, Ecole Normale Superieure, 24 rue Lhomond, 75005 Paris, France

a r t i c l e i n f o

Article history:

Received 22 June 2011

Received in revised form

18 October 2011

Accepted 19 December 2011Available online 27 December 2011

Keywords:

Optimization

Geological material

Ideally plastic material

Porous material

Limit analysis

96/$ - see front matter & 2011 Elsevier Ltd. A

016/j.jmps.2011.12.011

esponding author.

ail address: [email protected] (Y.M. Lero

a b s t r a c t

Accretionary wedges are idealized as triangular-shaped regions, the top surface

corresponding to the topography, the bottom surface to the weak, frictional contact

(decollement) with the subducting plate and the last side, the back-wall, to the contact

with the continent. New critical stability conditions are obtained and defined by the

sets of geometrical and material parameters for which the deformation occurs

concurrently in two regions of the wedge e.g., the front and the rear. They extend the

classical critical stability conditions, restricted to triangular regions of infinite extent

and composed of frictional materials, by accounting for arbitrary topography, finite

geometry and cohesive materials. They are obtained by the application of the kinematic

approach of limit analysis, referred to as the maximum strength theorem, which is first

extended to fluid-saturated porous media. The basic failure mode used to defined these

stability conditions consists of two reverse faults intersecting the decollement at a

common point, the root. The decollement is activated from the back wall to the root.

The root position is indeterminate for our extended critical stability conditions. The

proposed theory predicts the classical stability conditions as a special case. A first

application to the Barbados prism is proposed accounting for a non-linear gradient in

fluid pressure within the decollement. The details of the topography is responsible for

several sets of critical stability conditions. Our pressure predictions necessary to explain

the activation of the frontal thrust are significantly less than the estimates done during

the drilling on site. Finally, the theory is applied to a wedge with the decollement

partitioned into two regions, the deepest corresponding to the seismogenic zone. The

associated failure modes include one composed of two blocks sliding at different

velocities over the decollement and separated by a single discontinuity, dipping either

towards the front or towards the rear of the wedge. This special mode dominance is

facilitated by large pressures and small friction coefficients in the shallowest region of

the decollement.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The objective of this work is to propose a simple method to analyze the deformation occurring in accretionary wedgesextending the stability conditions presented by Dahlen (1984) and Lehner (1986) for a cohesionless wedge of infinite

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Fig. 1. An illustration of the geodynamics context: the oceanic plate is subducting below the continental plate and the sediments which are either

accreted or coming from the continent form the wedge (a). The idealized geometry of the wedge with a back-wall perpendicular to the straight

decollement dipping at b (b). The topography, if assumed straight (dashed line), has the slope a.

A. Pons, Y.M. Leroy / J. Mech. Phys. Solids 60 (2012) 643–664644

extent. The kinematic approach of limit analysis (Salenc-on, 1974, 2002), as presented by Maillot and Leroy (2006), isextended to fluid-saturated porous media for that purpose.

The geodynamics context of interest here is subduction zones. Typically, an oceanic plate is subducting below acontinental plate of larger thickness, Fig. 1a. The main interest for these subduction zones is the earthquake hazard and theassociated tsunamis. The triangular-shaped region above the first line of contact between the two plates is theaccretionary wedge. It is bounded above and laterally by the topography and the contact with the continent. The lowerside is the contact with the subducting plate, a frictional interface referred to as the decollement. The wedge is composedof oceanic sediments, which were not dragged with the subducting plate but accreted to the wedge, and sediments comingfrom the continent. There are four external forces acting on the wedge. The first is the tectonic force from the continentopposing the subduction of the wedge. The second is due to gravity. The third is the fluid pressure acting on the submergedtopography and the fourth force is the frictional reaction of the decollement. The deformation within the wedge is typicallymarked by a series of ramps dipping towards the continent (solid lines in Fig. 1a). To first-order, the wedge geometry isidealized as a triangular-shaped region with a back-wall perpendicular to the inclined decollement considered straight(dip b), Fig. 1b. In some cases, the topography is assumed straight and its slope is the angle a.

The first point to construct the classical critical stability conditions is the passive Rankine stress state for wedges ofinfinite extent and with a straight topography. The stress state at a given point within the wedge is function of the distanceto this topographic surface. The equations of equilibrium in 2D are integrated with the boundary conditions on thetopography to obtain the stress state up to a single component. This last component is determined by imposing that thematerial is failing according to the Coulomb strength criterion. Stress is proportional to the lithostatic stress everywherewithin the wedge and all points fail concurrently if the material is cohesionless. The second point to determine the criticalstability conditions is the activation of the decollement. There is a critical value of the topographic slope ac for given valuesof the decollement dip and of the material friction angles for which failure occurs concurrently in the bulk and by slip onthe decollement. This angle ac , function of material properties and geometry, defines the critical stability conditions of theaccretionary wedge determined analytically by Dahlen (1984) and with a Mohr construction by Lehner (1986). They areoften associated to the name of the critical-taper theory.

These critical stability conditions are complemented by the following interpretation. If the average topographic slope ais less than ac , the wedge is said to be sub-critical and deforms by thrusting at the back (close to the back-wall) and slip onpart of the decollement only. The development of the thrusting sequence from the back towards the wedge toe increasesthe average slope until ac is reached and the whole decollement becomes then activated. To complete the stabilityconditions, if the slope a is larger than the critical value, the wedge is said to be super-critical and glides on thedecollement without any internal deformation. The best illustration of this long-term development is seen with analogmaterials such as sand tested in the so-called sand-boxes in the laboratory (e.g., Davis et al., 1983).

The objective of this contribution is to extend the critical-taper theory and to obtain new stability conditions applicableto wedges of finite length, having an arbitrary topography and composed of cohesive, frictional sediments. This objectivewas certainly at the base of the work of Cubas et al. (2008) who were interested by thrusting sequences in fold-and-thrustbelts. The new ingredient of this contribution is that the rocks are now considered as fluid-saturated porous materials.Each material point is occupied by a fluid and a solid phase and failure is controlled by the solid phase stress. This newaspect is fundamental to accretionary wedges where the young sediments are fully saturated and compacting (e.g., Pichonet al., 1993). Pore-pressure evolves in time and affects the deformation style (e.g., Saffer, 2003). The development of

A. Pons, Y.M. Leroy / J. Mech. Phys. Solids 60 (2012) 643–664 645

elevated pore pressure is inferred from wave velocity measurements (e.g., Cochrane et al., 1996), from a few directmeasurements (e.g., Screaton et al., 2002) or from laboratory consolidation tests (e.g., Moore and Tobin, 1997). These datacan be compared to theoretical predictions, as it is proposed here for the case of the Barbados prism.

The contents of this contribution are as follows. The maximum strength theorem is first extended to porous mediacontaining faults or interfaces with specific material properties and pressure conditions. The central failure modeconsidered in Section 3 consists of two reverse faults intersecting at the same point on the decollement, the root. It isshown that the position of this root is indeterminate for the critical-taper conditions, a consequence of the bulk failureoccurring at any point within the wedge. The proposed theory is then applied in Section 4 to the Barbados wedge toillustrate its potential in dealing with arbitrary topography and non-trivial fluid pressure distribution within thedecollement. Section 5 is a preliminary to practical applications of the proposed theory which will inevitably considerdiscontinuous properties of the decollement because of the finite extent of the seismogenic zone. The simplest prototype(Kimura et al., 2007) has two regions of constant frictional properties. The associated dominant failure mode considered byCubas et al. (in preparation) for super-critical conditions, consists of two blocks sliding on the decollement and separatedby a single discontinuity rooting at the point of material discontinuity on the decollement and dipping either towards thefront or the rear of the wedge. The associated fundamental questions, examined in Section 5, are the details of theoptimum velocity fields for this class of problems as well as the equivalence between the proposed theory and one basedon the critical-taper theory and making use of equivalent friction coefficients.

2. The maximum strength theorem in terms of effective stress

The maximum strength theorem is now extended to fluid-saturated media. The formulation is close to the onepresented by Corfdir (2004) with, however, an emphasis on inherited faults having specific mechanical properties. Thistheorem corresponds to the kinematic approach of limit analysis, as presented by Salenc-on (1974, 2002), and is not linkedto a plasticity theory. The symmetric gradient of the virtual velocity is not interpreted as a plastic strain rate, avoiding theeverlasting discussion on the existence of bounds for non-associated plasticity theory.

2.1. The governing equations

Consider the fully saturated porous medium which occupies the domain Ot of arbitrary geometry, Fig. 2. Each materialpoint of this domain is labeled by the vector x and is characterized by its density r, the fluid pressure p and the porosity f.Each material point is composed of two phases, the solid and the fluid phase. The material density is related to theporosity, the material density of the solid phase rs and of the fluid rf by

r¼frf þð1�fÞrs: ð1Þ

In many geological problems, the fluid pressure is given in terms of the Hubbert and Rubey (1959) pressure ratio

lðxÞ ¼�pðxÞ�rf gDðxÞ

szðxÞþrf gDðxÞwith szðxÞ ¼ rgðzþDðxÞÞ�rf gDðxÞ, ð2Þ

in which sz is the negative stress corresponding to the pressure resulting from the weight of the column above the point x ofinterest. The z-axis is vertical and directed upwards, Fig. 2, so that the gravity vector is g ¼�gez. In this contribution, one and two

Fig. 2. A fluid-saturated medium occupies the domain Ot of arbitrary geometry. The distance DðxÞ defines the depth of the submerged domain over any

material point located by the vector x . The surface Ss is a potential discontinuity in the stress vector. The faults, such as SU , are velocity and pressure

discontinuities. The zoom within the fault illustrates the equilibrium conditions between the stress vectors applied on the positive and the negative side

of the discontinuity of thickness e, disregarded in comparison to its characteristic length Lc.


bars underlining variables identify vectors and tensors, respectively. The positive scalar DðxÞ in (2) is the height of the free fluidcolumn above the material point of interest at x, Fig. 2. The scalar l is thus the ratio between the change in fluid pressure and thechange in lithostatic pressure as one moves from point x 0 to point x. The ratio l can vary between rf =r and 1. For these twoextreme values, the fluid pressure is equal to the hydrostatic and to the lithostatic pressure, respectively. It is convenient for therest of this contribution to express the fluid pressure in terms of the pressure ratio l and (2) is replaced by

p¼ g½�lrzþðrf�rlÞD�: ð3Þ

The local equations of mechanical equilibrium for the body composed by the saturated porous medium are

divðs Þþrg ¼ 0 8x 2 Ot , ð4aÞ

1sU � n ¼ 0 8x 2 Ss, ð4bÞ

T ¼ T d8x 2 @OT

t , ð4cÞ

in which s , T ¼ s � n and n denotes the stress tensor, the stress vector and the unit normal vector n of the potential discontinuitysurface Ss, respectively. The sign convention for stresses is such that a uniaxial tension results in a positive stress. The doublebrackets in Eq. (4b) denote the difference between the quantity in argument on the two sides of any stress discontinuity. Eq. (4c)stipulates that the stress vector should be equal to the force density T d which is prescribed on the part @OT

t of the boundary.

2.2. The effective stress and the material strength domain

The Cauchy stress s introduced in the governing equations is the total stress acting on both the solid phase and thefluid phase. The strength of the porous medium depends on the strength of the solid skeleton which sustains only part ofthe stress referred to as the effective stress and denoted s 0. The most commonly used effective stress is due to Terzaghiand corresponds to the difference between the total stress and the stress acting on the fluid phase �pd ,

s 0 ¼ sþpd , ð5Þ

in which the tensor d is the second-order identity tensor. The strength of the solid phase requires that this effective stressis within or at the boundary of the strength domain

G¼ fs 09f ðs 0Þr0g, ð6Þ

in which f ðs 0Þ represents the constraints in the stress space.In rock or soil mechanics, most of the failures can be described by frictional rupture and by compaction (e.g., Paterson

and Wong, 2005). Frictional and cohesive materials rupture in shear and this failure mode is well described for most rocksby a strength domain bounded by the Coulomb criterion. For a pre-defined fault of normal n, this criterion reads

f ðs 0Þ ¼ 9t09þtan ðjÞs0n�C, ð7Þ

in which j and C are the friction angle and the cohesion, respectively. The scalars t0 and s0n are the tangential and thenormal component of the effective stress vector

T 0 ¼ s 0 � n ¼ s0nnþt0t ð8Þ

decomposed in the right-handed basis fn,tg in which t is the tangent vector to the fault in our 2D problems. The stressspace of interest is two-dimensional and the Coulomb constraints in (7) are illustrated by the two semi-infinite lines inFig. 3a. The combination of the two constraints in (6) defines the convex, although open strength domain in the stress

Fig. 3. The strength domain for a Coulomb material (cohesion C and friction angle j) (a). The velocity jump 1V U is oriented in the same basis as the

stress vector by the angle Z and the different cases, in terms of this angle and introduced with the support function (16), are also defined (b).


space. The convexity property is essential for the maximum strength theorem which is found after the presentation of thetheorem of effective, virtual powers which follows.

2.3. The theorem of effective, virtual powers

This theorem is constructed starting from the classical theorem of virtual power, applied to the saturated porousmedium, which is derived from the local equations of equilibrium (4). This theorem states that the internal and theexternal powers are equal

PextðV Þ ¼PintðV Þ 8 V KA ð9Þ

for any kinematically admissible (KA) velocity field. Such velocity field is piecewise continuous and should be consistentwith the displacement prescribed on the part of the boundary @OU

t . The internal power

PintðV Þ ¼

ZOt

s : d dVþ

ZSU

T � 1V U dS ð10Þ

has two contributions, the first due to the power of the stress on the rate of deformation d based on the virtual velocity field. Thedouble dot product in (10) and between any two tensors A and B is defined by A : B ¼ trðA � B Þ. The second contribution in (10)is the power of the stress vector on the jump in the velocity field across any velocity discontinuity SU within or on the boundaryof the domain. This discontinuity could be a pre-existing fault and constitute a pressure discontinuity. The external power

PextðV Þ ¼

ZOt

rg � V dVþ

Z@OT

t

T d� V dS ð11Þ

corresponds to the power of the body forces, only gravity in our case, and of the prescribed forces on the boundary. The objectiveis now to express (9) in terms of the effective stress to obtain the theorem of effective, virtual powers.

The stress tensor in the first term in the right-hand side of (10) is replaced by the effective stress according to (5) so thatthe diffuse internal power becomes the addition of s 0 : d and �p divðV Þ. The second term in (10) needs some precautionssince faults are viewed as surfaces with specific fluid pressure which could differ from the pressure in the host rock. Themechanisms responsible for the build-up, the preservation or the evolution in time of these pressure discontinuities willnot be discussed here. The faults are seen as two parallel discontinuities separated by the distance e which is smallcompared to the fault characteristic length Lc, see inset of Fig. 2. The continuity of the total traction vector for equilibriumsake across the two adjacent discontinuities means that T þ�TS

¼ TSþT� ¼ 0 as explained after (4b) and with TS

representing the stress vector on the mid-surface, within the fault, oriented with the normal n. The continuity of the stressvector does not imply the continuity of the effective stress vector because of the two pressure discontinuities between theinternal part of the fault and the positive and negative side of the domain Ot . The continuity condition of the total stressvector, in terms of effective stress vectors, is then

T 0þ�pþn ¼�T 0��p�n ¼ T 0S�pSn, ð12Þ

in which pS is the pressure within the fault. It is the third expression in (12) which is proposed in replacement of the stressvector in the second term of the right-hand side of (10). These substitutions in the expressions for the diffuse and localizedpower in favor of the effective stress and effective stress vector lead to the following definition of the effective, internal power

P0intðV Þ ¼

ZOt

s 0 : d dVþ

ZSU

T 0S � 1V U dS, ð13Þ

where the velocity discontinuity is seen as a pre-existing fault.The effective external power equal to this effective internal power is the virtual power in (11) minus the contributions

of the fluid phase missing now from (13)

P0extðV Þ ¼

ZOt

rg � V dVþ

Z@OT

t

T d� V dSþ

ZOt

p divðV Þ dVþ

ZSU

pSn � 1V U dS, ð14Þ

so that the following theorem, referred to as the effective, virtual powers theorem is proposed

P0extðV Þ ¼P0intðV Þ 8 V KA: ð15Þ

Note that the application of this theorem relies on the pressure distribution and discontinuities to be known, anassumption made through out this contribution.

2.4. The support function and the maximum strength theorem

The maximum strength theorem provides an upper bound to the internal power for any KA velocity field and,consequently, an upper bound to the tectonic force applied at the boundary of the structure. The name maximum strengththeorem was used by Maillot and Leroy (2006) to emphasize that only the concept of strength is required. It correspondsmore classically to the kinematic approach of limit analysis (Salenc-on, 1974). Central to this theorem is the proof that, forstrength domains which are convex in the appropriate stress space, the maximum internal power is the support function p


(Salenc-on, 2002). This function depends on the geometry of the strength domain and is function of either the velocity jump1V U (magnitude and orientation) or the rate of deformation d for localized or diffuse deformation, respectively. Consider afault that sustains a virtual velocity jump of magnitude J and oriented by the angle Z from the normal vector n, Fig. 3b. Forthe strength domain bounded by the Coulomb criterion (7), the support function for this fault reads

case 1 : 0rZop=2�j,

case 10 : �p=2þjoZo0,

(pð1V UÞ ¼ JC cotanðjÞ cosðZÞ,

cases 2 & 20 : Z¼ 7 ðp=2�jÞ, pð1V UÞ ¼ JC cos j,

case 3 : p=2�joZrp,

case 30 : �poZr�p=2þj,

(pð1V UÞ ¼ þ1: ð16Þ

The function varies smoothly from a maximum to the minimum reached in case 2 and 20 as the angle Z orienting thevelocity jump from the normal varies from zero to 7 ðp=2�jÞ. Cases 3 and 30, defined for an angle Z larger in absolutevalue than p=2�j correspond to an infinite support function and thus provide an infinite upper bound to the dissipationand to the tectonic force which is of no use. Avoiding these cases 3 and 30 has the following severe consequence on theselection of the virtual velocity fields: one should only consider velocity fields whose jumps over the faults are in a cone ofaxis the normal n and internal angle p=2�j. Velocity fields having this property will be said to be pertinent. Theycorrespond to cases 1, 10, 2 and 20 in (16) and are presented in Fig. 3b.

Further examples of support function are found in Salenc-on (2002) including support functions for diffuse deformation.In that instance, the function depends on the virtual rate of deformation d and pðd ÞZ d : s 0 for any stress s 0 within thestrength domain. Also, the open strength domain considered here could be amended to account for compactionmechanisms, as proposed by Maillot and Leroy (2006). This modification will be essential to capture the deformation orcompaction bands found at the toe of the Nankai prism, South-East Japan, which could either be due to the lithostaticpressure or to the tectonic forces (e.g., Karig and Lundberg, 1990).

The integration of the support function over all the discontinuities SU found within or on the boundary of Ot defines themaximum resisting power. This power is larger than the effective, internal power

P0intðV Þ r P0mrðV Þ ¼

ZOt

pðd Þ dVþ

ZSU

pð1V UÞ dS ð17Þ

for any KA velocity field. The combination of (17) with the effective, virtual power theorem (15) leads to

P0extðV ÞrP0mrðV Þ 8 V KA, ð18Þ

an inequality which constitutes the maximum strength theorem. The presence of the tectonic force Q in the externalpower will become clear in the various examples considered in what follows, as in the triangular wedge considered next.The inequality (18) leads to an upper bound Qu on the tectonic force which is finite for any pertinent, KA velocity field.

3. Stability of a triangular wedge

The objective is first to check the consistency of our theory with the critical-taper theory by comparing their respectivepredictions. The latter is restricted to a triangular submarine wedge of infinite extent having the constant topographicslope a and resting on a decollement dipping at the angle b. The prototype proposed in Fig. 4a has the same two slopes butis of finite extent since it is terminated at the rear by the back-wall which is perpendicular to the decollement. Theobjective is to find the upper bound to the force Q applied to the back-wall and to capture the critical stability conditions interms of geometry, material properties and pressure distribution for which the whole wedge is failing by faulting and,concomitantly, by slipping along the whole decollement. The second objective is to study the influence of the friction onthe back-wall on the critical stability conditions.

The water depth DðxÞ needed in the definition of the pressure ratio in (2) and of the pressure in (3) is D0 at the toe of thewedge, point C, and decreases linearly as �tan ðaÞx with the first coordinate. The decollement presents a different value ofthe pressure ratio denoted lD. The pressure distribution with depth at an arbitrary x-coordinate is represented in Fig. 4a.The pressure is hydrostatic between the fluid top surface and the topography of the wedge. Within the wedge, up to thedecollement contact, the pressure gradient is constant, equal to lrg, comprised between the hydrostatic and the lithostaticgradient. At the decollement, there is a discontinuity in the pressure field from the bulk pressure, p(x), to the pressurewithin the decollement denoted pD(x).

Upper bounds to the force Q are now computed for the three different failure mechanisms presented in Fig. 4b–d. The firstmechanism is typical of a super-critical wedge sliding on its decollement as a rigid block with a constant virtual velocity. Thesecond mechanism, Fig. 4c, concerns a sub-critical wedge and the internal deformation is close to the back-wall. Thisdeformation is tentatively captured by a single ramp rooting at the base of the back-wall and separating the hanging-wall,which has a uniform virtual velocity, from the foot-wall which is at rest. The third mechanism is valid for both sub-critical andsuper-critical wedges depending on the length d of the activated decollement measured from the back-wall to the root

Fig. 4. The triangular wedge and the distribution of the fluid pressure with depth (a). Three failure mechanisms are studied corresponding to the full

activation of the decollement in (b), the activation of a ramp rooting at the back-wall in (c) and the partial activation of the decollement, of a ramp (GE)

and of a back-thrust (GF) in (d). The inset defines the virtual velocity field of the third failure mechanism, which is piecewise uniform.


(point G) of the two reverse faults composing the failure mechanism. The two regions, the back-stop and the hanging-wall,have uniform velocities. The length d should be small or equal to the total length of the decollement L for sub-critical or super-critical wedges, respectively. In the latter case, mechanism (3) is identical to mechanism (1). The maximum strength theoremdoes provide this optimum length d and thus predict the wedge stability conditions. The comparison of mechanisms (2) and(3), for sub-critical conditions, provides us with some insight on the role of the back-stop, not accounted for in the critical-tapertheory and explored by numerical means by Souloumiac et al. (2010) for dry materials.

A virtual velocity field which is KA and pertinent is constructed for each failure mechanism. The three mechanisms share thesame velocity of the back-wall V BW , which is parallel to the decollement and of norm denoted V BW and set to one. The interfacesbetween the back-wall and the bulk and between the decollement and the bulk are assumed to be cohesive and frictional. Theirfriction angles and cohesions are identified by the subscripts BW and D, respectively. The ramp (segment AE or GE) and the back-thrust (segment GF) are crossing the bulk, frictional and cohesive material. They could be assigned different properties, for sake ofgenerality, and their cohesions and friction angles are identified by the subscripts R and BT, respectively. Table 1 summarizes thenotations and the numerical values of the geometrical and material parameters.

3.1. Mechanism (1): decollement fully activated

The first failure mechanism corresponds to the rigid translation of the wedge on the fully activated decollement at theuniform velocity of norm V . This velocity is also the jump across the decollement J

D, a vector oriented by the angle ZD from

Table 1Geometrical and material parameters for the critical wedge analysis. The third and fourth columns correspond to the values which are kept constant and

to the ranges of values which are explored.

Notation Definition Value Range Unit

a Topographic slope Variable 0–8 deg.

b Decollement angle 3 – deg.

L Decollement length 50 – km

D0 Depth of the toe 6 – km

jR , jBT Ramp and back-thrust friction angle 25 – deg.

jD Decollement friction angle 15 – deg.

jBW Back-wall friction angle 25 0–25 deg.

Ca Cohesions (a¼ R,BT ,BW or D) 0 – Pa

rf Fluid phase density 1000 – kg/m3

rs Solid phase density 2600 – kg/m3

f Porosity 30 – %

r Saturated rock density 2120 – kg/m3

lhydro Hydrostatic pressure ratio 0.47 – –

l Bulk pressure ratio 0.6 0.47–1 –

lD Decollement pressure ratio 0.6 0.47–1 –

g Gravity acceleration 9.81 – m/s2

Fig. 5. Hodogram for the velocity jump across the back-wall of mechanism (1) (a), and the constraints on the angles nBW and ZD (b). The optimum

velocity orientations correspond to the point marked by the solid square.


the normal (see Fig. 4b). The velocity jump at the back-wall is the difference V�V BW , a vector of norm J BW and oriented bythe angle ZBW from the normal. This angle is negative and it is preferred in the following sections to introduce nBW ¼�ZBW .The hodogram of the velocity jump at the back-wall is presented in Fig. 5a and the law of sines provides

J BW

cos ðZDÞ¼

V

sin ðnBW Þ¼

�V BW

cos ðZDþnBW Þ: ð19Þ

This hodogram is valid as long as the internal angles of the triangle are positive. Moreover, the velocities have to bepertinent, corresponding to cases 1, 10, 2 and 20 defined with the support function in (16). The cone of pertinent velocities isbounded by the shaded areas found in Fig. 4b for each discontinuity. These various constraints are summarized by

ZDþnBW 4p2

,p2�jDZZDZ0,

p2�jBW ZnBW 40: ð20Þ

The velocity is uniform over the triangular wedge and the associated rate of deformation tensor is zero. Consequently,the only sources of dissipation are the velocity jumps over the decollement and the interface to the back-wall. Themaximum resisting power defined in (17) combined with the definition of the support function in (16) provides

P0rmðV Þ ¼ CDL cotan ðjDÞ cosðZDÞV þCBW H cotan ðjBW Þ cosðnBW ÞJ BW : ð21Þ

The effective, external power defined in (14) is composed of four terms. The first term is the power of the gravity field. Thesecond contribution in our problem corresponds to the power of the fluid pressure on the surface CB and of the tectonicforce of magnitude Q. The third term is zero since the velocity field is homogeneous over the wedge thus of zerodivergence. Finally the last term corresponds to the power of the pressure in all the interfaces which are the interface at


the back-wall and the decollement for this mechanism. After integration, the effective, external power reads

P0extðV Þ ¼ �1

2rgLH cosðZD�bÞV þQ�

pðCÞþpðBÞ

2

L

cosðaþbÞcosðZD�a�bÞV

þpðCÞþpDðAÞ

2L cosðZDÞV þ

pðAÞþpðBÞ

2H cosðnBW ÞJ BW : ð22Þ

The maximum strength theorem in (18) combined with the results in (21) and (22) leads to the upper bound to thetectonic force applied to the back-wall for this first mechanism

Qu1 ¼ Aþcos ðZDÞ

cos ðZDþnBW ÞðB sinðnBW ÞþC cosðnBW ÞÞ ð23aÞ

with

A¼H

2ðrgL sinðbÞþpðCÞþpðBÞÞ, ð23bÞ

B¼L

2ðpDðAÞ�pðBÞ�2CD cotan ðjDÞ�rgH cosðbÞÞ, ð23cÞ

C¼H

2ðpðAÞ�pðCÞ�2CBW cotan ðjBW Þ�rgL sinðbÞÞ: ð23dÞ

The three scalars A, B and C are only functions of the geometry, the material properties of the wedge and the fluidpressures but not of the virtual velocity field.

The next step consists in minimizing the bound Qu1 in (23a) in terms of the two angles nBW and ZD which characterizethe velocity field. The search is conducted within the dark grey triangle found in Fig. 5b in view of the constraints (20). Thedomain of search includes the vertical and horizontal sides but not the hypotenuse. As one approaches this hypotenusefrom within the domain of search, the trigonometric function in the denominator in the right-hand side of (23a) tends to0� leading to an upper bound equal to �1, unless the term delimited by the large parentheses is negative

B sinðnBW ÞþC cosðnBW Þo0 8 nBW 2�jD;p=2�jBW �: ð24Þ

This condition warrants the upper bound Qu1 to be positive and thus meaningful.Condition (24) being met, the optimum velocity field is found by inspection of the partial derivatives of the right-hand

side of (23a) with respect to nBW and ZD. These partial derivatives are both negative in the domain of search and theminimum is thus attained at the point the further away from the hypotenuse, the point of coordinateðZD ¼ ðp=2Þ�jD,nBW ¼ ðp=2Þ�jBW Þ illustrated by the square in Fig. 5b. The upper bound in (23), for this specific set ofangles, is the least upper bound to the tectonic force for failure by full activation of the decollement

Qlu1 ¼ A�sinðjDÞ

cosðjDþjBW ÞðB cosðjBW ÞþC sinðjBW ÞÞ: ð25Þ

3.2. Mechanism (2): a thrust rooting at the back-wall

The second failure mechanism does not make use of the decollement and consists of a thrust rooting at the base of theback-wall and dipping at g (Fig. 4c). This mode of failure is certainly difficult to justify from field observations alone and ismore pertinent for experimental and numerical investigations where finite-size wedges are considered and properboundary conditions with the back-stop are necessary. The virtual velocity field is such that the velocity of the foot-wall iszero corresponding to the absence of activity of the decollement. The hanging-wall has the uniform velocity field of normV HW . It corresponds also to the jump on the ramp J

Rwhich is oriented by the angle ZR from the normal to the ramp, Fig. 4c.

The jump in velocity across the back-wall interface is JBW¼ V HW�V BW and is oriented with the angle ZBW from the normal

to the back-wall (Fig. 4c). The positive angle nBW ¼�ZBW is introduced again for the sake of simplicity.The rest of the derivation of the upper bound and of the minimization in terms of the velocity orientations follows the

same reasoning as for mechanism (1) and is postponed to Appendix A. It is shown that the optimum angles are

ZR ¼p2�jR, nBW ¼

p2�jBW , ð26Þ

so that the upper bound for the second mechanism is

Qu2 ¼ A0�sinðjRþg�bÞðB

0 cosðjBW ÞþC0 sinðjBW ÞÞ

cosðjRþjBWþg�bÞ, ð27aÞ

with

A0 ¼ �CR cotan ðjRÞLEA sinðg�bÞþrgSAEB sinðbÞþpðEÞþpðAÞ

2LEA sinðg�bÞþ pðEÞþpðBÞ

2LEB sinðaþbÞ, ð27bÞ


B0 ¼ �CR cotan ðjRÞLEA cosðg�bÞ�rgSAEB cosðbÞþpðEÞþpðAÞ

2LEA cosðg�bÞ�pðEÞþpðBÞ

2LEB cosðaþbÞ, ð27cÞ

C0 ¼ CR cotan ðjRÞLEA sinðg�bÞ�CBW cotan ðjBW ÞH�pðEÞþpðAÞ

2LEA sinðg�bÞþ pðAÞþpðBÞ

2H

�pðEÞþpðBÞ

2LEB sinðaþbÞ�rgSAEB sinðbÞ: ð27dÞ

The least upper bound Qlu2 should now be obtained by minimization of (27) with respect to the only left unknown, theramp dip g. This angle is contained within the interval ½b; minðp=2�jR�jBW ,p=2þbÞ� for the hodogram construction inAppendix A to be valid and for the hanging-wall velocity to be pointing towards the toe. This minimization is rathertedious to conduct analytically and the result would not be an explicit relation for the optimum g and would require anumerical approach. Consequently, and for sake of simplicity, it is proposed to search directly the minimum Qlu2 from(27a) by exploring the range of possible g by numerical means.

3.3. Mechanism (3): a ramp and a back-thrust rooting on the decollement

The third failure mechanism consists of the ramp GE dipping at g and the back-thrust GF dipping at y, Fig. 4d. They bothroot at point G on the decollement which is activated from point A to G. These two points are separated by the distance d,an unknown of the problem. The geometry of this third failure mechanism is thus described with three parameters: g, yand d. Note that this mechanism is identical to mechanism (1) for d¼L but it never collapses to mechanism (2) since theback-thrust and the interface to the back-wall have different material properties.

The velocities of the back-stop and the hanging-wall are uniform and denoted V BS and V HW , respectively (see inset ofFig. 4d). They induce the jumps J

Rand J

Dover the ramp and the decollement, two vectors oriented by the angles ZR and ZD

from the corresponding normal. There are two additional velocity discontinuities corresponding to the back-thrust(J

BT¼ V HW�V BS) and to the interface with the back-wall (J

BW¼ V BS�V HW Þ. These two jumps are oriented by the angles ZBT

and ZBW from the normal to the back-thrust and to the back-wall, respectively. The angles ZBT and ZBW are negative and thepositive angles nBT ¼�ZBT and nBW ¼�ZBW are preferred in what follows.

Appendix B shows how the application of the maximum strength theorem provides the expression for the upper boundto the tectonic force associated to this third failure mechanism

Qu3 ¼ rgSHW cosðZR�gÞV HWþrgSBS cosðZD�bÞV BSþLEFpðEÞþpðFÞ

2cosðZR�g�aÞV HW

þLFBpðFÞþpðBÞ

2cosðZD�b�aÞV BSþ CD cotðjDÞ�

pDðGÞþpDðAÞ

2

� �cosðZDÞV BS

þ CR cotðjRÞ�pðGÞþpðEÞ

2

� �LGE cosðZRÞV HWþ CBT cotðjBT Þ�

pðGÞþpðFÞ

2

� �LFG cosðnBT ÞJ BT

þ CBW cotðjBW Þ�pðAÞþpðBÞ

2

� �H cosðnBW ÞJ BW , ð28Þ

where SHW and SBS are the surfaces of the hanging-wall and the back-stop.The least upper bound Qlu3 is obtained by minimization of (28) with respect to the three geometrical parameters g, y

and d as well as with respect to the four angles describing the velocity jumps. This optimization is done also by numericalmeans. One of the results of this exercise is that the least upper bound Qlu3 is found for the optimum angles

ZD ¼p2�jD, ZR ¼

p2�jR, nBW ¼

p2�jBW and nBT ¼

p2�jBT ð29Þ

for all triplets of geometrical parameters ðd,g,yÞ. They correspond to case 2 or 20 of the support function in (16).

3.4. Comparison with the critical-taper theory

The objective is now to compare the three least upper bounds found in or derived from (25), (27) and (28) for thematerial properties summarized in the third column of Table 1. The least of the three least upper bounds is the closest tothe exact, unknown tectonic force and the corresponding mechanism is considered to be the dominant failure mechanism.If the first mechanism is dominant, or equivalently, if the third is dominant with a fully activated decollement d¼L, thewedge is super-critical. Sub-critical conditions are associated with the dominance of the second or the third failuremechanism with a partial activation of the decollement. These predictions are compared with the critical-taper theory forwhich the topographic slope ac takes the value of 4.361 for the selected material properties.

The results are represented in Fig. 6a where the least upper bounds, normalized by rgHL, for the three failuremechanisms are presented as functions of the topographic slope a. The graph in Fig. 6b presents the optimum value of d

also as a function of a. For sub-critical slopes (aoac), the least upper bound Qlu3 is the smallest and thus the third failuremechanism is dominant. The value of d is small and the deformation occurs at the back of the wedge. The dominant failuremode requires a partial activation of the decollement instead of the onset of a ramp rooting at the bottom of the back-wall.The role of the back-wall interface friction angle on that property will be further investigated below. Failure mechanisms

Fig. 6. Comparison of the least upper bounds for the three failure mechanisms, functions of the topographic slope a (a). The evolution of the optimum

normalized length d=L, of the activated decollement, with a (b). This optimum ratio jumps from a small value typical of a deformation at the back of the

wedge to the value of 1, exactly as the slope a reaches the critical-taper angle, ac .

Fig. 7. Influence of the pressure ratio in the decollement on the critical slope ac according to the maximum strength theorem (circular and triangular

symbols) and to the critical-taper theory (dashed curves) (a). In (b), the evolution of the normalized length, d=L for a constant topographic slope a¼ 31

with the pressure ratio lD . The bulk pressure ratio l is either kept constant to 0.6 (triangular, red symbols) or set equal to lD (circular, blue symbols). (For

interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


(1) and (3) have the same least upper bound exactly for the critical slope ac: the distance d can take any value between asmall value (dictated by the need for the back-thrust to outcrop at the back-wall contact) and L. The distance d is said to beindeterminate. For super-critical slopes (a4ac), the mechanisms (1) and (3) are identical (d¼L) and the wholedecollement is activated. The maximum strength theorem has thus captured exactly the critical-taper theory.

The influence of the pressure ratio l and lD on the critical slope is now discussed. Results are presented in Fig. 7a wherethe critical slope ac is computed at a series of values of lD marked by circles (if the pressure ratio in the bulk is identical to

Fig. 8. Influence of the back-wall friction angle on the selection of the dominant mode of failure for sub-critical conditions (a) and for super-critical

conditions (b). Mechanism (2) is the dominant failure mode for small enough friction angles under sub-critical conditions. This second mechanism

increases the exact value of the critical topographic slope for small friction angles as seen in (b).


lD) or inverted triangles (if l is set constant to 0.6) and compared to the critical-taper theory (dashed curves). The firstobservation is, again, the complete agreement between the predictions of the maximum strength theorem and the critical-taper theory. The increase of the pressure ratio in the decollement, whether or not the bulk pressure ratio is kept constant,leads to a decrease of the critical slope ac. Consider for example the case of a¼ 31 and l¼ lD (circles and upper dashedcurve), the wedge is then sub-critical for lDoln

D2 ¼ 0:79 and is super-critical otherwise. If the pressure ratio lD increases,the deformation would be transferred from the back to the front of the wedge once this value ln

D2 was overcome. Thiscritical pressure ratio is less (ln

D1 ¼ 0:68) if the pressure ratio in the bulk is kept constant to 0.6. This difference in thecritical value of lD for a topographic slope a¼ 31 is further explored in Fig. 7b where the value of d=L, defining the relativelength of the activated decollement for the third failure mechanism, is presented as a function of the pressure ratio lD forboth l¼ lD (circular, blue symbols) and l¼ 0:6 (triangular, red symbols). The discontinuities in d at ln

D1 and ln

D2

correspond to the critical-taper conditions. The values of d for sub-critical conditions are influenced only in a minor way bysetting l to lD or keeping it constant at 0.6. However, the critical value of lD is sensitive to the overpressure in the bulk.Indeed, the decollement is, relatively to the bulk, weaker for l set to 0.6 than for l equals to lD in the sense that ln

D1oln

D2.This result suggests that over-pressured wedges require greater decollement overpressure to slip super-critically.

3.5. Influence of the back-wall on the dominant failure mode

The comparison of the least upper bounds for the three failure modes is now continued for sub-critical slopesconditions a¼ 31oac ¼ 4:361. The results are presented in Fig. 8a where the three forces are presented as functions of theback-wall interface friction angle. For small values of this friction angle the second mechanism is dominant (Qlu2oQlu1

and Qlu3): failure is by the onset of a ramp rooting at the base of the back-wall. There is no back-stop and the hanging-wallis sliding up the ramp and also along the interface with the back-wall. As the friction angle of this interface is increased,the least upper bound for this failure mode is increasing more rapidly than the bound for the third mechanism requiring apartial activation of the decollement. The two forces are identical for the critical value jn

BW ¼ 7:881. For friction angleslarger than this critical value, the smallest upper bound is associated to mechanism (3). The bound corresponding to thedominant failure is thus approximately a bi-linear function of the back-wall friction angle with a slope change at jn

BW .Such bi-linear variation of the load was found numerically by Souloumiac et al. (2010) for dry frictional materials.

The back-wall friction angle does not enter the critical-taper conditions which are based on an infinite wedge. Thepresent approach does not suffer from the same limitation and the back-wall has a definite, although small influence onthe proposed stability conditions. To illustrate this point, a comparison is made between the three least upper bounds forsuper-critical conditions according to the critical-taper theory (a¼ 4:414ac ¼ 4:361). Results are presented in Fig. 8bwhere the bounds are functions of the back-wall friction coefficient. The least upper bound for the mechanisms (1) and (3)are identical as it was found earlier for super-critical conditions. However, for small enough friction angle on the back-wall, the second mechanism provides a smaller bound than the bounds for the two other mechanisms. It is thus predictedthat the conditions are sub-critical and not super-critical as thought with the critical-taper theory. This difference vanishesas soon as the friction angle on the back-wall becomes larger than the critical value jn

BW ¼ 6:681.

4. A first application to Barbados

The Barbados accretionary complex marks the subduction front of the Atlantic oceanic crust below the Caribbeancrystalline crust which takes place at a rate estimated to be between 2 and 4 cm/yr (Bangs et al., 1990). The Ocean DrillingProgram (Legs 156 and 171) has reached the decollement in the Northern Barbados accretionary prism and provided

Fig. 9. Interpretation of the seismic section oriented West to East at the toe of the Northern Barbados wedge, modified from Saffer (2003) (a). The

optimized failure mode is composed of the ramp (GE), the back-thrust (GF) and the activated decollement from A to G (b). The pressure ratio within the

decollement is assumed to vary linearly from the internal to the external part of the section and to be constant in the proto-decollement (c). The position

along the decollement of point G is presented as a function of the internal pressure ratio (d). For the case l¼ lE ¼ 0:5, there are three possible positions

for the optimum failure mode numbered 1–3, defined in (d) and positioned in (a). (For interpretation of the references to color in this figure legend, the

reader is referred to the web version of this article.)


valuable information on the physical properties of the material in this region (Moore et al., 1998). The seismicinterpretation of the frontal part of this prism found in Fig. 8a of Saffer (2003) is simplified and presented in Fig. 9a.Point B on the left-hand side of the topography is 4945 m below sea level and is 610 m above the decollement which isassumed to be straight, dipping at 2.31 on average and of a total length of 5500 m. The decollement is partitioned in tworegions. The first region, corresponding to the segment AC (length 3800 m) and presented as a solid line, has accumulatedslip during the development of the thrusting sequence above. The second region, towards the toe or the external part of thewedge and presented with a dashed line, is referred to as the proto-decollement and is yet undisturbed or with littledeformation since there is no thrusting on top. Since the proto-decollement is yet undisturbed, it is assigned the sameproperties and pressure ratio as the bulk material for simplicity sake. The boundary between the two regions of thedecollement (point C) is marked by a frontal thrust (ramp in blue). Saffer (2003) concluded that the pressure ratio waslikely to vary along the activated decollement and the profile presented in Fig. 9c is proposed to describe this potentialvariation. It is assumed to decrease linearly from lI to lE from the internal to the external part of the wedge and to remainconstant in the undisturbed, proto-decollement (lE ¼ l). The change in slope of this function is exactly at the root of thefrontal thrust. The objective of this section is to find one set of material properties and pressure ratios which could explainthe current position of this frontal thrust. This exercise constitutes a non-trivial application of the theory presented in thiscontribution to a complex topography and pressure distribution.

The failure mechanism proposed to find this optimum thrust is the back-thrust and ramp presented in Fig. 9b with partof the internal decollement activated. It is identical to the mechanism (3) considered in the previous section. The commonroot of these two reverse faults (point G) as well as their dips (y and g) are the basic geometrical unknowns. The virtualvelocity field is constant over the back-stop (BS) and the hanging-wall (HW), with the velocity jumps over the back-thrust(BT) and the ramp (R) oriented so that the support function for the frictional material considered here is in case 2 or 20 (16)as found for all mechanisms in Section 3. The norm of the velocity vector of the back-stop is set to one.


The back-wall and the region from this boundary to the line AB are not considered since their contributions remainconstant for all failure mechanisms and thus have no influence on the position of the critical mechanism searched for. Theexpression for the upper bound to the tectonic force is

Qu cosðjDÞ ¼ rgðSBS sinðbþjDÞþSHW sinðgþjRÞV HW ÞþLAGCD cosðjDÞþLGECR cosðjRÞV HWþLGFCBT cosðjBT ÞJ BT

�

Z F

BpðxÞn � ðcosðbþjDÞexþsinðbþjDÞezÞ dS�V HW

Z E

FpðxÞn � ðcosðgþjRÞexþsinðgþjRÞezÞ dS

�V HW

Z E

GpðxÞ dS sinðjRÞ�J BT

Z F

GpðxÞ dS sinðjBT Þ�

Z G

ApDðxÞ dS sinðjDÞ, ð30Þ

with the same notation as in the previous section. The various velocities and jumps are related by

J BS

sinðjRþg�b�jDÞ¼

V HW

sinðjBþyþbþjDÞ¼

1

sinðjBTþyþgþjRÞ: ð31Þ

The first two lines in the right-hand side of (30) are the contributions of the power of the velocity on the gravity force andof the support functions along all the velocity discontinuities. The next two lines correspond to the power of the pressureon the topography of normal n. The last two lines result from the internal power of the fluid pressure on the velocity jumpsacross the ramp, the back-thrust and the decollement, respectively. The numerical procedure to compute this upper load isinspired from the one proposed in Appendix of Cubas et al. (2008). The topography and the decollement are discretised in aseries of points. The force is computed for every possible set of points (G,E,F) defining completely the failure mechanismsketched in Fig. 9b. The quadratures in (30) are computed numerically and require also a discretisation of the ramp and theback-thrust. The whole procedure is available in a fortran code (TectonoErosion, 2011). A convergence analysis has beenconducted to ensure that the results presented next are independent of the various discretisations. All cohesions are set tozero, the bulk friction angle j is set to 301 and equal to the friction angle of the ramp and back-thrust (j¼jR ¼jBT Þ.These equalities would be typical of faults at their onset prior to the development of any damage. The material densitieshave the values indicated in Table 1. Before proceeding to the analysis of the results, note that we assume that theNorthern Barbados prism is super-critical in the sense that the deformation is taking place within the section consideredwhich is the frontal part of the wedge. We do not check this assumption by working at the global scale of the wedge andprefer to conduct a stability analysis at a smaller length scale related to the toe, as shown in Fig. 9a.

The results are presented in Fig. 9d where the x-coordinate of the point G, common root to the two reverse faultscomposing the failure mechanism, is presented as a function of the pressure ratio lI . The red curve is obtained for thepressure ratio in the bulk l and the external pressure ratio lE set to 0.5, a value close to hydrostatic conditions. This valuewill be increased latter on. The friction angle of the decollement is set to 101. This value is typical of laboratoryexperiments, and used in many numerical simulations, and is proposed tentatively on the field length scale. As the internalpressure ratio is increased, the position of point G sustains two jumps and remains approximately constant over threeplateaux which are numbered 1–3. These three characteristic positions of the failure mechanism are presented in red inthe seismic interpretation of Fig. 9a. The existence of discrete positions for the optimum failure mode has similarity withthe results of the global stability analysis presented in the previous section where the position of the root of the ramp andback-thrust goes from the back to the front, if the decollement pressure ratio is increased. In this problem, the topographyis irregular and the stability argument is that there are not two but three different positions for the common root of thefailure mechanism. Note that the third position is very close to the frontal thrust with a root less than 100 m within theproto-decollement.

To explore the role of over-pressures on the stability conditions, it is now proposed to increase the value of l and lE to0.6. Results are also presented in Fig. 9d. There are only two positions for the root of the failure mechanisms which can beselected instead of three for the previous set of results. The first position is close to the position of no. 1 described aboveand the second position is well ahead of the frontal thrust, rooting in the proto-decollement (x-coordinate of 4350 m). It isinteresting to note that the stability conditions for larger pressure ratios are thus closer to the ones obtained for the criticaltaper: either to the front or to the back. The influence of the irregular topography is not sufficient to modify this simpleresult in the presence of large over-pressures.

A third set of calculations was conducted with l¼ lE ¼ 0:7 for which the failure mode is always rooting well aheadwithin the proto-decollement at the same x-coordinate. This solution is not presented in Fig. 9d. To counter-balance thesuper-critical effects of the increase in pressure introduced here, a fourth set of calculations is reported for which the samevalues of the pressure ratios are used but the decollement friction angle is increased to 151. The positions of the failuremechanism root are again grouped in three main regions although the second group sees a scatter in position unseen forthe previous data sets, Fig. 9d. All mechanisms remain rather far away from the current position of the frontal root and thefailure is thus sub-critical on the length scale of the wedge toe.

Our results should now be compared with the pressures reported during the drilling campaign. Drilling through theroot of the frontal thrust (Site 1047) provided an estimate of the pressure ratio between 0.68 and 0.84. At three quarters ofthe distance to the back-wall of our structure in Fig. 9a, another well (Site 948) has provided the pressure ratio of 0.70–0.92 (Saffer, 2003). Our first set of predictions (red curve in Fig. 9d) which was the closest to the actual frontal thrust hasthus too small pressure ratios except in the internal part of the decollement. If pressure ratios close to the field estimates


are considered, our predictions either over-shoot or remain behind the frontal thrust. There are many reasons which couldexplain this discrepancy, including the assumption of a straight decollement and of an active ramp with the same frictionas the surrounding bulk material. Improving our predictions is beyond the objective of this first application and deserve acomplete study by itself with a better prototype. It could be conducted in a rather systematic manner by finding theprobability distributions of the various material properties using the inverse method considered by Cubas et al. (2010).

5. Consequences of the partitioning of the decollement in two mechanical units

The present theory will inevitably be applied to heterogeneous decollement as considered in the previous section. Theobjective of this section is to study the effect of a simple heterogeneity on the decollement on the stability conditions. Thedecollement is partitioned into two regions, the external and the internal parts, as it was proposed by Kimura et al. (2007)for Nankai accretionary wedge, South-East Japan, to differentiate between the upper part of the seismic region and theaseismic external part. The same partition was considered by Miyakawa et al. (2010) for their numerical simulations. Theprototype is presented in Fig. 10a with constant friction angles, jI and jE, and pressure ratios, lI and lE, for the internaland the external regions of the decollement, respectively. Point D is the boundary between the two regions and it is at30 km from the front (point C) for a decollement of total length 50 km. The internal part of the decollement has constantmaterial properties throughout this analysis: jI ¼ 151 and lI ¼ l¼ 0:6. The parameters for the external part of thedecollement are varied. All other material parameters are set to the values used in Section 3 (third column of Table 1).

The maximum strength theorem has been applied to similar prototypes by Cubas et al. (in preparation) and theirfindings, based on the numerical algorithm of Souloumiac et al. (2010), reveals a failure mechanism which differs from thebasic one: the internal and external parts of the wedge are sliding at different velocities and are separated by a singlediscontinuity rooting at point D, without the conjugate fault or back-trust we have considered so far in this contribution. Asimilar mechanism with a single fault was considered by Del Castello and Cooke (2008). The first fundamental question isto check by analytical means the potential of this failure mode compared to the ones considered above. Furthermore, moststability analyses rely on a procedure where the over-pressures in the discontinuities are accounted for with an equivalentfriction coefficient

m0d ¼ tanðjdÞð1�ldÞ=ð1�lÞ, d¼ E,I, ð32Þ

which is based on the critical-taper theory, for a homogeneous decollement. The second fundamental question is toconfirm or not that such simplified approach is correct by comparing the outcome with our predictions. Underlying thisquestion is the optimization of the virtual velocity field which has been found in the previous section to be always suchthat all jumps are on the cone of the pertinent velocities defined by case 2 or 20 for the support function (16). Exploration ofcases 1 and 10, corresponding to the whole cone, is considered here for the decollement.

The first failure mechanism considered here corresponds to the third mechanism studied in Section 3 and is thuscomposed of the gliding of the back-stop on the rear part of the decollement and of a hanging-wall sliding over the ramp.The foot-wall remains at rest. The position of the root of the ramp and the back-thrust, point G, is constrained to bebetween D and A. The external and, possibly part of the internal decollement thus remain inactivated. This mechanismshould be appropriate for sub-critical stability conditions. This first, classical mechanism is compared with the secondmechanism presented in Fig. 10b. It differs from the first mechanism by the non-zero velocity of the foot-wall which isdenoted V E and the root of the two faults attached to point D. The two other regions of the wedge have velocities of normV I and V C . The three subscripts of the velocity vectors E, I and C are referring to the external, the internal and the centralregions, respectively. These three vectors are oriented by the angles ZE and ZI from the normal to the decollement and by

Fig. 10. The decollement of this triangular wedge is divided into the internal and the external parts (a). The virtual velocity field (in blue) and the various

velocity jumps (in green) of the second failure mechanism considered for stability analysis are defined in (b). The root of the two faults bounding the

central region is set at the transition between the two parts of the decollement, point D. (For interpretation of the references to color in this figure legend,

the reader is referred to the web version of this article.)


the angle w from the vertical direction, as presented in Fig. 10b. The angles ZE and ZI are assumed to be in case 1 or 2 of (16)and all values will be explored. The range of values for w is from �b to p=2�b. The orientation of the three velocity jumpsJ CE, J CI and J BW over the discontinuities DE, DF and the back-wall, respectively, is set to their critical value defined by case 2or 20 in (16). This second failure mechanism will collapse to the one found by Cubas et al. (in preparation) if one of thevelocity jumps (CE or CI) has a zero norm. The upper bound for this second mechanism is

QuðV Þ ¼ J BW H cosðnBW Þ CBW cotðjBW Þ�pBW ðAÞþpðBÞ

2

� �þ V I SIrg cosðZI�bÞþcosðZI�a�bÞLFB

pðFÞþpðBÞ

2

� �

þ V I cosðZIÞLDA CI cotðjIÞ�pDIðAÞþpDIðDÞ

2

� �þ V C SCrg cosðwÞþcosðw�aÞLEF

pðFÞþpðEÞ

2

� �

þ V E SErg cosðZE�bÞþcosðZE�a�bÞLCEpðCÞþpðEÞ

2

� �þ V E cosðZEÞLCD CE cotðjEÞ�

pDEðDÞþpðCÞ

2

� �

þ J CI cosðZCIÞLDF C cotðjÞ�pðFÞþpðDÞ

2

� �þ J CE cosðZCEÞLDE C cotðjÞ�pðEÞþpðDÞ

2

� �, ð33Þ

in which the subscripts DE and DI for the pressure denote the external and the internal parts of the decollement. Thevelocity of the back-wall is set to one. The minimization of this upper bound is done by numerical means.

The influence of the properties of the external part of the decollement on the selection of the dominant failuremechanism is studied by exploring a large range of values for the friction angle jE and for the pressure ratio lE, ½11;251�and ½lhydro;1�, respectively. The topographic angle is first set to a¼ 31, corresponding to sub-critical conditions for theinternal region (cf Fig. 6). The results are represented in Fig. 11a where colored contours of the angle g (in 1) of the frontaldiscontinuity are presented in the space spanned by the friction angle and the pressure ratio of the external part of thedecollement. This angle g is defined in Fig. 10b and would correspond to the dip of the ramp if it is less than p=2. If largerthan this value, it still defines the orientation of the discontinuity but its dip would be p=2 less and oriented towards theexternal part of the wedge. The dark blue region in Fig. 11a corresponds to large values of the friction angle and to small

Fig. 11. The angle g orienting the ramp or the single discontinuity is presented as colored isocontours in the plane spanned by jE and lE for a¼ 31 (a).

Iso-values of the effective coefficient of friction of the decollement external part, m0E , are presented as dashed or solid curves. The closest iso-value to the

boundary between the sub- and super-critical regions, m0E ¼ 0:137, is in white. The failure mode at point nos. 1–4 are presented in (b). The angle g and the

back-thrust dip y, if there is one, are function of jE for l¼ 0:6 in (c). (For interpretation of the references to color in this figure legend, the reader is

referred to the web version of this article.)


values of the pressure ratio and it is associated to the classical failure mechanism with the dip of the two faults at g¼ 26:61and y¼ 44:11. This mechanism is at the back of the wedge which is thus sub-critical as shown by the mechanismnumbered 3 in Fig. 11b. Slip is not activated in the external part of the decollement and the dominant failure mode is thusindependent of jE and lE. The second mechanism is dominant in the rest of the graph where the friction and pressureconditions render the wedge super-critical since the whole decollement is activated, although at two different velocities.Furthermore, a single discontinuity is indeed observed since the velocity jump on one of the two discontinuities is alwaysof a zero norm. The optimum orientation of the velocities V I and V E is always found to be ZI ¼ p=2�jI and ZE ¼ p=2�jE,corresponding to case 2 in (16) for the two parts of the decollement. The orientation of the single discontinuity betweenthe two regions of the super-critical wedge is very sensitive to the friction angle jE and to the pressure ratio lE, varying bymore than 901 in Fig. 11a. This large variation means that the dip is towards the internal part of the wedge at point no. 2whereas it is towards in external part of the wedge at the point no. 1, as illustrated in Fig. 11b.

The three points, numbered 1–3, were chosen for the same value of the pressure ratio (lE ¼ 0:6) so that the orientationof the two or the single discontinuity could be presented as functions of the only varying parameter, the friction angle,Fig. 11c. Note the discontinuity in g (blue curve) at jn

E ¼ 7:91 corresponding to the critical stability conditions. The dip ofthe ramp (g) and of the back-thrust (y, red curve) are independent of jE in the super-critical region. Note that the angle jn

E

is an increasing function of lE.Drastic changes in g are not restricted to the critical stability conditions and appear also in the super-critical regions for

jE ¼jI , Fig. 11a. Crossing this line for increasing values of jE means a sudden change in g: for lE ¼ 0:85, for instance, thesingle discontinuity direction switches from towards the rear (g� 701) to towards the front (g� 1201), as illustrated withthe mechanism of point no. 4 presented in Fig. 11b. For values of lE almost equals to 1, the discontinuity is in the reverseorder, the dip towards the front becomes a dip towards the rear. Concerning the sense of shear on the single discontinuityof the mechanisms at point nos. 1, 2 and 4, we find a normal, a reverse and a reverse sense of shear, respectively. The onlyrule which is observed is that the internal region is moving up with respect to the external region if jI 4jE and is movingdown otherwise.

A final word on the results of Fig. 11 concerns the relevance of the equivalent friction coefficient defined in (32).Isocontours of this equivalent friction coefficient were plotted as solid or dashed lines in Fig. 11a. It is surprising that theisocontour m0E ¼ 0:137 (white solid line) is very close to the boundary of the sub- and super-critical stability regions. It isthus a good approximation to the critical stability conditions despite the material heterogeneity. For super-critical stabilityconditions, this equivalent friction coefficient could be rather misleading to understand the dip of the discontinuity of thedominant failure mode. This point is illustrated by plotting the orientation of the single velocity discontinuity along theisocontour m0E ¼ 0:10. Results are presented in Fig. 12 and show that g is approximately constant at 56.21 (towards the rear)and 125.01 (towards the front) for jE less and greater than jI , respectively. The discontinuity occurs exactly for jE ¼jI

and would not be captured with the equivalent friction coefficient analysis.This stability analysis is continued by considering a larger topographic slope set to a¼ 51 for which the internal part of

the wedge is super-critical according to the critical-taper theory (cf Fig. 6). Results are presented in Fig. 13 with the samestructure as for the results in Fig. 11 for sake of comparison. The domain of dominance of the classical failure mechanics(dark-blue domain at the lower right) has shrunk considerably in size. Furthermore, the ramp and back-thrust of theclassical failure mechanism in that domain are rooting at the material discontinuity, point D on the decollement, andare dipping at 22.41 and 41.91, respectively. This domain, illustrated by the mechanism at point no. 3 in Fig. 13, is neither

Fig. 12. The orientation g of the single velocity discontinuity of the dominant failure mode for super-critical conditions setting m0E ¼ 0:10 and m0I ¼ 0:27.

The friction angle jE and the pressure ratio lE are related by (32). The dip towards the rear for small pressure ratios and small friction angles is becoming

towards the front for larger values of the same parameters. The discontinuity is exactly at jE ¼jI .

Fig. 13. The angle g orienting the ramp or the single discontinuity as colored isocontours in the plan spanned by jE and lE for a¼ 51 (a). Iso-values of the

effective coefficient of friction of the external part of the decollement m0E are also presented as dashed or solid curves. The closest to the boundary

between the sub- and super-critical regions, m0E ¼ 0:285, is the white curve. The failure mode at point nos. 1–4 are presented in (b). The angle g and the

back-thrust dip y, if there is one, are function of jE for l¼ 0:6 in (c). (For interpretation of the references to color in this figure legend, the reader is

referred to the web version of this article.)


sub- nor super-critical and is said to be intermediate. Note that the boundary between the intermediate and the super-critical stability regions is approximately matched by the isocontour m0E ¼ 0:285 (white solid line). Outside theintermediate stability region, the dominant failure mechanism is defined again by two blocks at constant velocitiesgliding over the decollement and separated by a single discontinuity rooting at point D. The dip of this discontinuity is afunction of both jE and the pressure ratio lE and is plotted in Fig. 13c as a function of jE for a constant lE ¼ 0:6. There arenow two discontinuities, the first corresponding to the point where jE ¼jI and the second to the boundary of theintermediate stability region. Between these two discontinuities, the dip of the velocity discontinuity is towards the front.This change in orientation can also be studied by following the isocontour m0E ¼ 0:1 in Fig. 13a. The mechanisms, at pointno. 2 close to this isocontour and at point no. 4, are presented in Fig. 13b and have different dip directions but the samesense of reverse shear.

The final comment of this section concerns the equivalent friction coefficient which approximates best the criticalstability conditions. Its values have been found from a series of plots of the type presented in Fig. 11 obtained for differentlengths of the external decollement. Results are presented in Fig. 14 for a topographic slope a¼ 31 for which the wedgeinternal part is sub-critical. The critical equivalent coefficient is well approximated by the critical-taper theory for anexternal decollement extending at least 80% of the total decollement. For smaller ratios, the critical equivalent coefficientdecreases drastically and is equal to zero for a relative length of 40 %. These results illustrate the impossibility of judgingthe stability of heterogeneous wedges with the critical-taper theory.

6. Conclusion

The proposed extension of the maximum strength theorem to fluid-saturated porous media renders possible thestability analysis of accretionary wedges having complex topography, non-trivial fluid pressures and material propertiesdistributions.

Fig. 14. The critical equivalent friction coefficient, which best approximates the critical stability conditions of the wedge, as a function of the relative

length of the external decollement, for a topographic slope a¼ 31.


The proposed approach does predict the same stability conditions as the critical-taper theory in the special case of aperfectly triangular wedge despite its finite extent. The failure mode considered to establish this connection is composedof a ramp and back-thrust rooting at the same point on the decollement. The common root of the two faults is shown tobecome indeterminate for the critical stability conditions meaning that thrusting could occur anywhere within the wedgeand activating any length of the decollement. The dominant failure mode for sub-critical conditions could be onecomposed of a single ramp rooting at the base of the back-wall for small enough friction angles at its interface with thewedge. The influence on the critical stability conditions of this friction angle is minor and is essentially of interest fornumerical and analog experimentalists. The influence of the back-wall, with a more complex geometry and less trivialorientation, could however be more fruitful to understand doubly verging wedges such as in the Barbados. Note that all thecalculations reported for this special geometry came to the conclusion that the velocity jumps across all discontinuities arealways along the boundary of the pertinent velocity cones. These cones are the consequence of the requirement that thesupport function be finite and of the use of the Coulomb strength domain which is open in compression.

The first application concerns the toe of the Northern Barbados wedge, searching for the set of frictional angles andpressure ratios which could explain the current position of the active thrust. The stability predictions obtained by varyingthe fluid ratio gradient in the internal part of the decollement reveal that there are two or three main positions for the rootof the predicted ramp because of the complexity of the topography. There are more fluctuations around the three mainpositions for larger values of pressure ratios in the bulk and in the internal part of the decollement. The pressure ratioswhich lead to the best fit between the predicted and the current thrust appear to be small compared to the ones estimatedby the Ocean Drilling Program (Saffer, 2003). Various reasons could explain this discrepancy, including our assumptionthat the decollement is straight. A better prototype and a dedicated inverse analysis based on the statistical approach ofTarantola (2005), as followed by Cubas et al. (2010), could resolve this issue. Also, the numerical algorithm of Souloumiacet al. (2010), once extended to fluid-saturated media, could be applied to a prototype with a complex decollement andshed light on the active deformation mode.

The proposed theorem was applied to a triangular wedge with a decollement partitioned into two regions, the deepestof the two corresponding to the seismogenic zone. The pressure conditions for the dominance of the failure mode,observed by Cubas et al. (in preparation) and consisting of a single discontinuity separating two regions of the wedgegliding at different velocities, are established. The dip of this discontinuity and its sense of shear are a complex function ofboth the friction angles and of the pressure ratios within the decollement. For example, a large friction angle (201) andlarge pressure ratios (0.85) in the aseismic region of the decollement would lead to the landward vergence which isexceptionally observed in Cascadia. Our argument is still based on a Coulomb-type of wedge although the decollement isassumed heterogeneous. The distribution of pressure ratio could thus have a similar effect as the visco-elasticityintroduced by Gustscher et al. (2001). It seems that the activation of the aseismic part of the decollement, either completein our analytical analysis or partial, because of visco-elasticity or of a delamination process (Cooke, 2011), is essential toexplain the anomalous vergence. Note that our predictions were also used to determine the equivalent friction coefficientfor critical stability. It is found that the critical value of this coefficient depends on the relative length of the externaldecollement and cannot be determined form the critical-taper theory directly.

The future development and applications of this contribution could be geared towards determining the evolution intime of accretionary wedges as done for fold-and-thrust belts by Cubas et al. (2010). Sedimentation, erosion and tectonicscould be coupled in a relatively simple computational tool (TectonoErosion, 2011). The account of more realistic pressurefields by solving the pressure transient as a secondary problem would also be of interest. Finally, the strength criterioncould be amended to account for compacting mechanisms to assess the merits of tectonic, compressive stress andlithostatic pressure in generating compaction bands, such as in the specimen extracted from the toe of accretionary prismsof Nankai, South-East Japan (Karig and Lundberg, 1990).


Acknowledgments

The authors would like to thank M. Conin, P. Henry (Cerege, Aix-en-Provence) and S. Lallemant (University of Cergy-Pontoise) for many insightful exchanges on the physics of accretionary prisms and the interpretation of the recent data-acquisition campaigns. N. Cubas (California Institute of Technology) suggested the problem considered in Section 5. Thecomments to two anonymous reviewers contributed significantly to the preparation of the final version of this manuscript.

Appendix A. Mechanism (2) for the triangular wedge

The objective of this first appendix is to derive the upper bound Qu2 for the second mechanism discussed in Section 3and to find the optimum velocity field (angles ZR and nBW ) for a given ramp dip g (definitions in Fig. 4c).

The first step consists in expressing the norms of the hanging-wall velocity V HW and of the velocity jump over the back-wall J BW in terms of the back-wall velocity norm V BW thanks to hodogram presented in Fig. A1a. The law of sine associatedto this triangular construction reads

J BW

cosðZR�gþbÞ¼

V HW

sinðnBW Þ¼

�V BW

cosðZRþnBW�gþbÞ: ðA:1Þ

There are several constraints on the two angles ZR and nBW defining this second failure mode which are summarized by

ZRþnBW 4p2þg�b, ðA:2Þ

p2�jRZZRZg�b, ðA:3Þ

p2�jBW ZnBW Z0: ðA:4Þ

The lower bound on ZR is to warrant that the hanging-wall velocity points towards the wedge toe. The other boundsensure that the velocity field is pertinent (dictated by the support function in (16)) or that the hodogram constructionis valid.

The second step in the derivation of the upper bound consists in estimating the maximum resisting power, defined in(17),

P0rmðV Þ ¼ CRLEA cotðjRÞ cosðZRÞV HWþCBW H cotðjBW Þ cosðnBW ÞJ BW , ðA:5Þ

resulting from the contribution of the ramp and the interface with the back-wall, and then the effective, external power,defined in (15)

P0extðV Þ ¼Q�rgSAEB cosðZR�gÞV HWþpðEÞþpðAÞ

2LEA cosðZRÞV HWþ

pðAÞþpðBÞ

2H cosðnBW ÞJ BW

�pðEÞþpðBÞ

2LEB cosðZR�g�aÞV HW : ðA:6Þ

Next, the application of the maximum strength theorem (18), combined with the law of sine (A.1), leads to the upperbound

Qu2 ¼ A0 þcosðZR�gþbÞðB

0 sinðnBW ÞþC0 cosðnBW ÞÞ

cosðZRþnBW�gþbÞ, ðA:7Þ

in which the three scalar functions A0, B0 and C0 are found in the main text in (27).The structure of this bound in (A.7) is of course reminiscent of the expression for the bound of mechanism (1) found in

(25). Similar to the scalars A, B and C in (25), the three scalars A0, B0 and C0 are functions of the geometry and independent

Fig. A1. Hodogram for the velocity jump across the back-wall of mechanism (2) (a), and the constraints on the angles nBW and ZR (b). The optimum

velocities angles are at the point marked by the solid square.


of the velocity field, although they depend on the dip of the ramp g. The analogy between the two failure modes is that thehanging-wall of the second mechanism is equivalent to the triangular wedge of the first mechanism if the ramp AE is seenas playing the role of the decollement. It is for that reason that the ratio of the two trigonometric functions in (A.7) isindeed identical to the ratio introduced in (25), if the angle ZD is replaced by ZR�gþb. This equivalence is now put in use tofind the optimum velocity field for a given g. The domain of search for the angles ZR and nBW is bounded by the constraints(A.2) and corresponds to the dark grey triangle in Fig. A1b. The hypotenuse is singular and the condition

B0 sinðnBW ÞþC0 cosðnBW Þo0 8 nBW 2 ½jDþg�b;p=2�jBW � ðA:8Þ

must be met for the upper bound to be positive and meaningful. The optimum angles are the coordinates of the solid square inFig. A1b which is the point the further away from the hypotenuse. These coordinates are given in (26) in the main text.

Appendix B. Mechanism (3) for the triangular wedge

The objective of this appendix is to derive the upper bound Qu3 for the third failure mechanism for the triangular wedgeintroduced in Section 3.

The velocity jumps over the back-thrust and at the back-wall are presented in two hodograms in Fig. B1. Theapplication of the law of sines provides

V BS

sinðZRþnBT�g�yÞ¼

�V HW

sinðyþb�nBT�ZDÞ¼

J BT

sinðg�b�ZRþZDÞðB:1Þ

for the back-thrust jump, as well as Eq. (19) replacing V by V BS, for the jump at the back-wall contact. These two sets ofequations ((B.1) and modified (19)) determine the magnitude of all the velocity vectors in terms of the norm of the velocityof the back-wall V BW , used for normalization, and of the four angles nBW , nBT , ZR, and ZD. These four angles are the freeparameters describing the whole velocity field for a given set ðy,gÞ of the fault dips. They are constrained such that thevelocity field and more precisely the four velocity jumps are pertinent according to the support function (16). Also, theangles should be consistent with positive interior angles of the hodograms in Fig. B1 and ensure that the velocity of thehanging-wall and the back-stop are pointing towards the wedge toe. All these constraints are summarized by

ZRþnBT 4gþy,p2�jRZZRZg�b,

p2�jDZZDZ0,

ZDþnBW 4p2

,p2�jBW ZnBW Z0,

p2�jBT ZnBT Z0: ðB:2Þ

The partially activated decollement, the ramp, the back-thrust and the interface at the back-wall are dissipative sourcesand contribute to the maximum resisting power in (17) which reads

P0rmðV Þ ¼ CD cotðjDÞd cosðZDÞV BSþCR cotðjRÞLGE cosðZRÞV HWþCBT cotðjBT ÞLGF cosðnBT ÞJ BT

þCBW cotðjBW ÞH cosðnBW ÞJ BW : ðB:3Þ

The effective, external power (15) is

P0extðV Þ ¼ Q�rgSHW cosðZR�gÞV HW�rgSBS cosðZD�bÞV BSþpDðGÞþpDðAÞ

2d cosðZDÞV BSþ

pðGÞþpðEÞ

2LGE cosðZRÞV HW

þpðGÞþpðFÞ

2LGF cosðnBT ÞJ BTþ

pðAÞþpðBÞ

2H cosðnBW ÞJ BW�

pðFÞþpðEÞ

2LFE cosðZR�g�aÞV HW

�pðFÞþpðBÞ

2LFB cosðZD�b�aÞV BS, ðB:4Þ

where SHW and SBS are the surfaces of the hanging-wall and the back-stop.Finally, to obtain the upper bound to the tectonic force associated to this third failure mechanism, Qu3, the maximum

strength theorem (18) is applied, combining (B.4), (B.3), (B.1) and (19) (with V BS instead of V ). The expression for Qu3 isfound in the main text in (28).

Fig. B1. Hodogram for the velocity jumps across the back-wall and the back-thrust of mechanism (3) for the triangular wedge.


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