Hindawi Publishing Corporation Journal of Earthquakes Volume 2016, Article ID 6302546, 21 pages http://dx.doi.org/10.1155/2016/6302546

Research Article Initiation of Triangle Zones by Delamination, Shear, and Compaction at the Front of Fold-and-Thrust Belts

Chang Liu1,2 and Yaolin Shi1

1 Key Laboratory of Computational Geodynamics, Chinese Academy of Sciences, Beijing, China 2LaboratoiredeGeologie,´ CNRS-UMR 8538, Ecole´ Normale Superieure,´ Paris, France

Correspondence should be addressed to Chang Liu; [email protected]

Received 21 July 2015; Accepted 16 December 2015

Academic Editor: Youshun Sun

Copyright © 2016 C. Liu and Y. Shi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The interest of this paper is to investigate the initiation of triangle zones at the front of fold-and-thrust belts by analyzing the virtual velocity fields in triangle wedges. It allows achieving five collapse mechanisms by delamination, shear, and compaction of competing for the formation of triangle zones as follows. The first mechanism is the classical Coulomb shear thrust. The second is delamination at the frontal part of the decollement´ with straight back thrust, while the third is delamination with curvy back thrust. The fourth is the combination of ramp with Coulomb shear and shear-enhanced compact fault, while the fifth is the combination of the exchanging motion on the ramp and thrust. The dominating mechanism in the formation of triangle zones relies onthe competition of the least upper bound of each mechanism when subjected to tectonic force. The controlling factors of the competition are discovered as follows: (1) the frictional characters and cohesion of horizontal decollements´ and thrust, (2) the slope of the topography of accretion wedge, and (3) the thickness and rock density of the front toe of accretion wedge.

1. Introduction velocities discontinuities which are highlighted in the same figure. The proposition for this prototype comes from the The style of deformation at the front of fold-and-thrust following review of the literatures on triangle zones. belts is often characterized by triangle zones with forward- Charlesworth and Gagnon [6] studied the Rocky moun- dipping ramps (back thrust) (e.g., Southeastern Canadian tains foothills of Central Alberta. They noticed the stacking Cordillera [1] and the foothill of the Longmen Shan [2–5] of horses resulting in a duplex within the internal part of which is the eastern margin of the Tibetan Plateau). The the triangle zone, also called the tectonic wedge, the region objective of this paper is to understand the mechanics of three features of the triangle zones which are summarized in the bounded by the lower decollement´ and our curvy back prototype presented in Figure 1. This prototype is a triangle thrust. We shall not consider the formation of this series wedge with an accretion layer at its front. The first of the of horses and dissipation is accounted for solely along the three features of interest is the change in decollement´ from shear plane rooting on the lower decollement´ tip. These thebasalplaneofourprototypetoanupperdecollement,´ authors suggested that the development of the duplex is thanks to the activation of a short ramp connecting the two accompanied by the initiation of the back-thrusting of the planar surfaces. Second, there is delamination of the upper upper sedimentary layer. That argument seems difficult to decollement´ starting at the point of intersection with the justify from a mechanical point of view if Coulomb materials shortrampandalongasegmentofunknownlength.Theright areconsidered.Itisbelievedthatthevergenceofthethrusting end of this segment is the centre of rotation responsible for the system on the upper decollement´ should be towards the delamination.Thethirdfeatureisthecurvatureofthefault foreland and not hinterland. One of the objectives of this emanating from the upper decollement.´ These deformation paperistoappealtorockrheologytojustifytheinterpretation mechanisms required either shear, opening, or compaction of of Charlesworth and Gagnon [6]. 2 Journal of Earthquakes

the mechanical reasons remain unclear. He used finite- element, Drucker-Prager, and small hardening and needed this pinning of the upper decollement´ to obtain the stress Fault concentration which could lead to back-thrusting. Note that mesh sensitivity could prevent the strain to localize and thus Décollement for this back thrust to be observed. Couzens and Wiltschko [9] noticed that the mechanical Décollement Ramp strength of the three units (duplex, roof decollement,´ and Figure 1: This prototype summarizes the three features of the back thrust) of triangle zones is different. The duplex has the triangle zones which are studied: the change in depth of the same relatively strong strength with the roof decollement,´ decollement,´ the delamination of the upper decollement,´ and the while the overlying cover sequence is weaker. By recon- curving of the fault (back thrust). These deformation mechanisms structing the frontal stratigraphy of the Wyoming thrust require a combination of slip, opening, and compaction on the belt they suggested that triangle zones may form in the late various velocity discontinuities. stages of thrust belt evolution, when significant synorogenic deposits accumulated at the deformation front. It is suggested that the back thrust may be located within the synorogenic Price [1] provided several examples from the Canadian deposits, instead of at their base, because the synorogenic Rockies where the tectonic wedge results from the activation deposits often provide the weak shale-rich rocks for the of thrust splaying from the lower decollement´ and resulting in cover sequence to produce back thrust. They appealed to thethickeningofthetectonicwedgeandthepryingupwardof the strength of the cover stratigraphy to determine triangle thehinterlandvergingupperthrustinglayer.Thrustfoldsin zones to form in the thrust belt, where the cover sequence is the tectonic wedge are also mentioned as well as the absence consistently weaker and shakier. Although material strength of change in depth of the decollement.´ Price mentioned is used to explain the back thrust preferred to the weak young the concept of delamination of the allochthonous frontal sediment, the strength of the duplex and the roof decollement´ sediments along weak interface suggesting our proposal areassumedtothesame.Inthisresearchwewilljustifyhow to account for weak tensile strength in our mechanical the weak decollement´ influences the formation of triangle prototype. The hinterland vergence of the upper thrusting zone. system is justified by Price as the selection of a conjugate shear Jamison [10] used finite-element models to investigate failure plane in a compression test done in the laboratory. This certain mechanical and deformational characteristics of an argument, already present in the analysis of Charlesworth active triangle zone system by employing converging and andGagnon[6],iscertainlycentraltoouranalysis.Notethat nonconverging upper and lower detachment. His results Price [1] proposed that the delamination process could also suggestedthatthefrictionalcharacteristicsoftheupperand occur at the much larger scale of the lithosphere, the lower lower detachments play a significant role in the distribution decollement´ being within, or at the base of the crust. This of deformation in an evolving triangle zone. kind of deep crust lower decollement´ is detected to be existing Varsek [11] described that deformed tectonic wedges at the bottom of the upper crust beneath the eastern Tibetan delaminate the autochthon or upper plate at the flanks of margin to form the lower decollementofthetrianglezone[2].´ the orogenic prism by interpretation of seismic reflection Jamison [7] defined that a triangle zone is the area data from Rocky Mountain fold-and-thrust belt and on the bounded by a back thrust at the foreland margin of the thrust west of the Cascadia subduction zone. Varsek suggested that belt, a floor thrust that terminates up dip at the back thrust, the tectonic wedges occur at various crustal levels including and the most proximal thrust (on the hinterland side of within the upper crust (e.g., the eastern flank of the Rocky the reference back thrust) that reaches the erosional surface. Mountain and coast belts) and at the crust-mantle boundary Jamison is interested in the longevity of the back thrust which (e.g., in the interior of the Cascadia subduction zone). This could be terminated by a foreland thrusting through the observation indicated that tectonic wedging and delamina- upper sedimentary layer or by the initiation of a new back tion are a fundamental feature of crustal deformation. thrust more frontal: the triangle zone is abandoned for a new Couzens-Schultz et al. [12] demonstrated that the strength one. Jamison invoked the minimum dissipation argument of the decollements´ of the intervening and overlying rock to select the best scenarios but does not have the means to layers is the key parameter controlling on the development of compute the various dissipation sources involved. Because passive-roof duplexes (triangle zones) by presenting a series the triangle zone is much smaller than the complete fold- of physical experiments. and-thrust belt, the cohesion effects are potentially significant Adam et al. [13] studied the mechanics of landward through the entire hanging wall of the triangle zone. He convergent thrusts and its response to rapid sedimentation used an approximate solution which extends the critical taper in the frontal subduction zone of the Cascadia convergent theory by accounting for cohesion to decide on the geometry margin using Coulomb frictional wedge analysis, which does of the back thrust wedge. Our methodology is more rigorous not need to invoke very low-basal friction. In this research mathematically and will not require any assumptions on the the taper of the back thrust wedge was defined by the surface internal state of stress. slope and the bounding back thrust. It is noticed that all Erickson [8] invoked the pinning of the upper decol-´ faultsegmentsoftheboundingbackthrustareactivebecause lement to be responsible for the back-thrusting although of the tectonic stresses exceeding the fault strength, since Journal of Earthquakes 3

̂J

t

Décollement |𝜏|

3 T 4 2 𝜏 CD Bulk 1 |𝜏| 2 n 𝜙 𝜙D 𝜎n 0 ∗ C 2 𝜎 5 4 𝜙 1 n 6 ∗ 0 −P 𝜎n TD (a) (b) (c)

Figure 2: The bulk material is frictional, cohesive, and compactant. Its compressive strength is set by the maximum compressive strength ∗ ∗ 𝑃 and the compaction angle 𝜙 (a). The decollement´ is frictional and dilatant with a maximum tensile strength 𝑇𝐷 (b). The two strength domains are presented in the half plane (𝜎𝑛,|𝜏|), these two stresses being defined in (c).

∘ dip variations of individual ramp segments between 25 and on the selection of the virtual velocities to explore the ∘ 45 are observable. The back thrust wedge is mechanically various regions of the strength domain of interest. These stable and can be passively upward delaminated. Adam et al. consequences are important to understand the velocity fields proposed that the accretion mechanism is controlled by the considered in this contribution. The first applications of the contrasting mechanical stratigraphy of the thick incoming maximum strength theorem in Sections 3 and 4 are presented sediment succession entering the subduction zone rather for a triangle wedge, which is the most appropriate prototype than by atypical mechanically boundary conditions in the for a comparison with the critical taper theory of Dahlen [17]. frontal accretionary prism. It is also suggested to explain Six velocity fields, describing each collapse mechanism, are themechanicsoftrianglezoneandbackthrustwedge presented and put in competition. Section 5 is concerned with at mountain fronts of continental fold-and-thrust belts by the change in depth of the active decollement´ assuming fric- Adam et al. tional properties only. The details of the collapse mechanisms Montanari et al. [14] conducted analogue models and attheveryfrontofthewedgearefurtheranalyzedinSection6 evidencedthatthemajorroleintheformationoftriangle where four collapse mechanisms are competing. zone at Vena del Gesso Basin (Romagna Apennines, Italy) (1) is syntectonic erosion that promoted the development 2. The Maximum Strength Theorem of passive-roof duplex style and (2) theroleofdecollement´ level pinch-out that determined an oblique progression of The name of maximum strength theorem was proposed deformation. by Maillot and Leroy [18] to emphasize that the strength Tanner et al. [15] pointed out that triangle zones can even propertiesoftherockareatthecoreoftheapproach.This occur on a very small as tens of meters scale on the condition limited rock strength is illustrated for the bulk material and that this is supported by the mechanical stratigraphy by for the material composing the decollement´ in Figure 2. The studying the siliciclastic Carboniferous strata of the Harz stress space is spanned by the normal stress 𝜎𝑛 =𝑇⋅𝑛 Mountains in northern Germany. This triangle zone exists and the tangential stress 𝜏=𝑇⋅𝑡in which 𝑛 and 𝑡 are within a thin high-strain zone within weakly deformed the normal and tangent vector orienting the facet on which strata. It is suggested that the main controlling factor for the the stress vector 𝑇 is acting (Figure 2(c)). The continuum evolution of a triangle zone was the mechanical stratigraphy, convention sign is adopted: compressive normal stress is which is similar to the proposition by Couzens-Schultz et al. negative. The two strength domains share a common feature [12] by examining the controlling factors on the large scale which is the boundary set by the Coulomb line defined by triangle zone in fold-and-thrust belt. the friction angle 𝜙 or 𝜙𝐷 and the cohesion 𝐶 or 𝐶𝐷.The The contents of this paper are as follows. Section 2 is bulk material is porous and could compact under sufficiently devoted to the maximum strength theorem which is classi- large compressive stresses. This limit in compression is ∗ cally referred to as the kinematic approach of limit analysis defined by the maximum compressive strength 𝑃 .This by Salencon [16]. The central idea of this approach is the compactionisenhancedbytheshearonthefacetandasimple dualization of the forces acting on the faults and decollement´ linear relation is proposed to close the strength domain in ∗ in the sense of power so that the basic unknowns are virtual compression, characterized by the compaction angle 𝜙 .The velocity discontinuities. This dualization has consequences material within the decollement´ does not have this limit in 4 Journal of Earthquakes

compressive strength and the strength domain remains open where 𝐿𝐴𝐵 and 𝑆𝐴𝐵𝐶 stand for the total length of the in compression. The decollement´ material is weak compared decollement´ and the surface of the wedge, respectively. In to the bulk material and its friction angle 𝜙𝐷 will be assumed what follows, subscripts are sets of two points or one or two smaller than 𝜙. This material has also a finite tensile strength letters which identify the lengths, the surfaces, or the material 𝑇𝐷 which could be small favoring the delamination of the properties. Note that the upper bound is not a function of any upper decollement,´ a characteristic of the triangle zone. free variable which, typically, would be optimized to obtain The maximum strength theorem relies on a dualization theleastupperbound.Inthissense,(1)isalsotheleastupper of the stress problem and the conjugate quantities, in the 𝑄(1). bound lu sense of mechanical power, are velocities. Velocity fields are Collapsemechanismnumber2isathrustfoldwhich proposed which characterizes the initiation of the structure partitions the wedge into three regions (Figure 3(b)). The collapseandmorepreciselywhichpartofthestrength region 𝐺𝐸𝐹 is the hanging wall (HW) which is moving domain boundary could be attained. Consider, for example, forward over the fault 𝐺𝐸.Materialintheregion𝐴𝐺𝐹𝐶, ̂ the velocity jump 𝐽 over the facet in Figure 2(c): its orientation called the back stop (BS), is gliding over the decollement´ with respect to the normal to the fact determines which activated between points 𝐴 and 𝐺.Thetoeregion𝐺𝐵𝐸 is part of the strength domain boundary, numbered from 0 to at rest with respect to the observer. Note that the vergence 6, could be probed. For example, if the velocity jump is a of the hanging wall is not identified. The two faults 𝐺𝐸 and pure opening mode (oriented along 𝑛), the relevant region is 𝐺𝐹, oriented by the angles 𝜃𝐺𝐸 and 𝜃𝐺𝐹,respectively,play numbered 0. If this orientation is −𝑛, the stress space probed identicalroleinthistheory.Theycanacteitherasaramp is in pure compaction, region 6. Region 2 corresponds exactly and shear plane (foreland vergence) or as a shear plane and a to the angle 𝜋/2 − 𝜙 between the velocity jump and the ramp (hinterland vergence) or finally as two ramps (pop-up normal and defines slip on the fault according to Coulomb. structure).Theapplicationofthemaximumstrengththeorem Shear-enhanced compaction, region 4, corresponds to the provides ∗ angle 𝜋/2 + 𝜙 . There are two additional regions 1 and 3 𝑄(2) 𝜙 =𝜌𝑔(𝑆 (𝜙 +𝛽) corresponding to the simultaneous activation of opening u cos 𝐷 BS sin 𝐷 and shear and of shear-enhanced compaction and Coulomb + 𝑈̂ 𝑆 (𝜙 +𝜃 +𝛽))+𝐿 𝐶 𝜙 shear. Note that the decollement´ can only sustain jumps HW HW sin 𝐺𝐸 𝐺𝐸 𝐴𝐺 𝐷 cos 𝐷 (2) which is oriented according to cases 0 to 2. A formal +𝐶 𝐿 𝐽̂ 𝜙 +𝐶 𝐿 𝑈̂ 𝜙 . introduction of the rationale behind this theory is found 𝐺𝐹 𝐺𝐹 𝐺𝐹 cos 𝐺𝐹 𝐺𝐸 𝐺𝐸 HW cos 𝐺𝐸 in Salencon [16], and has been presented in Maillot and 𝑈̂ Leroy[18]andalsoinageologicalcontextbyCubasetal. The norm of the velocity HW of the hanging wall and of the ̂ [19]. The details of the conditions relating the velocity and jump 𝐽𝐺𝐹 over the fault 𝐺𝐹 is related by the regions on the strength domain boundary are presented 𝑈̂ 𝐽̂ again in Appendices for sake of completeness. It contains HW = 𝐺𝐹 also all the calculations leading to the least upper bounds sin (𝜙𝐷 +𝜙𝐺𝐹 +𝜃𝐺𝐹) sin (𝜙𝐺𝐸 +𝜃𝐺𝐸 −𝜙𝐷) (3) presented in this contribution. We shall concentrate solely on 1 the presentation of the velocity field and, more precisely, on = . the potential collapse mechanisms defined by these velocity sin (𝜃𝐺𝐹 +𝜃𝐺𝐸 +𝜙𝐺𝐹 +𝜙𝐺𝐸) fields and at the origin of the instantaneous deformation of The upper bound in (2) is a function of three parameters, the wedge. the two faults angles 𝜃𝐺𝐸 and 𝜃𝐺𝐹 as well as the distance 𝐿𝐴𝐺. It is the minimum value of (2) which is of interest and 3. Triangle Wedge: Six Collapse the associated optimum values of these three parameters. Mechanisms in Competition 𝑄(2) The minimum load is then the least upper bound lu .For The motivation for proposing each mechanism and the that minimization, we follow Cubas et al. [19] and discretize 𝑖 the topography and the decollement´ and select the optimum associated upper bound 𝑄 areprovidedinthissection.The u position of the three points 𝐸, 𝐹,and𝐺 (see Figure 3(b) for details of the calculations are postponed to Appendices. The definition). maximum strength theorem ensures that the smallest of the This second mechanism is proposed for comparison sake six upper bounds is associated with the dominant mode of with the critical taper theory for a purely frictional material. collapse. This comparison is presented in the next section. The distance 𝐿𝐴𝐺 should be small and thus the thrust fold at the back for subcritical conditions (𝛼<𝛼𝑐)andequaltothe 3.1. Mechanism Numbers 1 and 2: For Comparison with the 𝐿 𝛼>𝛼 Critical Taper Theory. Collapse mechanism number 1 is the distance 𝐴𝐵, for super-critical conditions ( 𝑐)sincethe simplest and should be dominant for super-critical slopes deformation is then to the front. In the latter case the first and if only friction prevails. It consists of the gliding of the second collapse mechanisms are identical. wholewedgeonthedecollement´ (Figure 3(a)). The maximum strength theorem provides the upper bound to the applied 3.2. Mechanism Number 3: Delamination on the Frontal Part of force the Decollement.´ Collapse mechanism number 3 is the first to beproposedforexploringthepossibilityforthefrontalpartof 𝑄(1) 𝜙 =𝐶 𝜙 𝐿 +𝜌𝑔 (𝜙 +𝛽)𝑆 , u cos 𝐷 𝐷 cos 𝐷 𝐴𝐵 sin 𝐷 𝐴𝐵𝐶 (1) the decollement´ to delaminate, a basic mechanism observed Journal of Earthquakes 5

C C F g 2 1 𝛽 BS 2 HW E 2 𝜃GF 2 𝛼+𝛽 A 222 TOE A B 2 2 2 2 2 B G 𝜃GE (a) (b) C C s F F 1 4 3 2 −𝜔ê 1 −𝜔ê 3 3 BS 1 𝜃 (s) HW GF 𝜃GF 1 2 22 2 0 00 A 2 2 2 0 000 0 B A G 0 B G (c) (d) C C F e2 F 5 e 6 1 4 2 e E 3 𝜃 4 E 𝜃GF GF 2 4 2 4 𝜃GE 2 B A 2 2 2 B A 2 2 2 G G 𝜃GE (e) (f)

Figure 3: The six collapse mechanisms in competition: the complete gliding on thedecollement´ in (a), the classical frictional thrust fold for comparison with the critical taper theory (b), the delamination mechanism with a straight fault (c) and a curvy fault (d), and finally the compacting thrust fold to promote hinterland (e) and foreland vergence (f). The numbers 0, 1, 2, and 4 identify the regions of the strength domain activated and are defined in Figure 2.

6𝐼 =(𝐿3 −[𝐿 −𝑥 (𝛼+ 𝛽)] 3) in most triangle zones (Figure 3(c)). The wedge is partitioned 1HW 𝐴𝐵 𝐴𝐵 2𝐹 cotan into two regions, the first region being the back stop 𝐴𝐺𝐹𝐶 𝐴𝐺 3 3 gliding on the section of the decollement.´ The second ⋅ tan (𝛼+ 𝛽) − (𝐿 𝐴𝐺 −[𝐿𝐴𝐺 −𝑥2𝐹 cotan 𝜃𝐺𝐹] ) region is the hanging wall 𝐺𝐵𝐹 which sustains a rigid rotation 𝜔̂ around the point 𝐵 as illustrated in Figure 3(c). ⋅ tan 𝜃𝐺𝐹, The decollement´ between points 𝐴 and 𝐺 is sheared 6𝐼 =𝑥3 [ (𝛼+ 𝛽) − 𝜙 ], whereas points between 𝐺 and 𝐵 have a velocity normal to 2HW 2𝐹 cotan cotan 𝐺𝐹 the decollement´ and proportional to the distance to point (4) 𝐵, corresponding to the delamination process. The angular 𝜔̂ 𝐺 velocity is chosen such that the jump in velocity at point in which 𝑥2𝐹, 𝐼1 ,and𝐼2 are the second coordinate of ̂ HW HW between the fault 𝐺𝐹 and the back stop (norm 𝐽𝐺)isprobing point 𝐹 and the moment of the hanging wall with respect the strength domain in region 2. The other points on the to the first and second axis, respectively. The norm ofthe fault 𝐺𝐹 areinregion1,meaningthatthefault𝐺𝐹 sustains velocity jump at point 𝐺 and the angular velocity is related amixedmodeofshearingandopening.Theupperboundto by the tectonic force is ̂ 𝜔𝐿̂ 𝐴𝐵 1 𝐽𝐺 𝑄(3) 𝜙 =𝜌𝑔(𝑆 (𝜙 +𝛽) = = . (5) u cos 𝐷 BS sin 𝐷 sin (𝜙𝐷 +𝜙𝐺𝐹 +𝜃𝐺𝐹) cos (𝜙𝐺𝐹 +𝜃𝐺𝐹) cos 𝜙𝐷

+ 𝜔[̂ cos 𝛽(𝐿𝐴𝐵𝑆 −𝐼1 )+sin 𝛽𝐼2 ]) HW HW HW Thisupperboundisthefunctionofthelength𝐿𝐴𝐵 and 𝜃 1 the angle 𝐺𝐹. These two parameters are varied making use +𝐶 𝐿 𝜙 + 𝜔𝑇̂ 𝐿2 of the same spatial discretization of the topography and 𝐷 𝐴𝐺 cos 𝐷 𝐷 2 𝐺𝐵 the decollement´ as for mechanism number 2. There is an 𝐺 𝐹 𝑄(3) 1 optimum set of points ( , )forwhich u in (4) is minimum ̂ (3) +𝐶𝐺𝐹𝐿𝐺𝐹 (𝐽𝐺 cos 𝜙𝐺𝐹 + 𝐿𝐺𝐵𝜔̂ cotan 𝜙𝐺𝐹), 𝑄 2 and this value is denoted as lu . 6 Journal of Earthquakes

This collapse mechanism has the merits of its geometrical 3.4.MechanismNumbers5and6:CompactionandThrusting. simplicity with the drawback that the fault 𝐺𝐹 is in a slip The last two collapse mechanisms are proposed to break the mode only at point 𝐺 andisinamixedmodeofslipand symmetric role of the two faults 𝐺𝐸 and 𝐺𝐹 in mechanism opening along 𝐺𝐹 by construction. This mode of deformation number2.Thissymmetrybreakingisdonebyassuming is certainly not very realistic and the next collapse mechanism that one of the faults is a ramp and deforming by Coulomb is hoped to improve on this deficiency. shear (region 2) whereas the material crossing the other fault is shear-enhanced compacted (region 4) (Figures 3(e) and 3.3. Mechanism Number 4: Delamination with a Curved Fault. 3(f)). The ramp is shearing repeatedly the same material and Mechanism number 4, the second to account for delamina- compaction cannot be the relevant deformation mechanism. tion, proposes a curved fault which separates the wedge into On the contrary, the conjugate plane is crossed by new two velocity regions. The fault is tangent to the decollement´ material constantly and this material could compact during at point 𝐺,anditscurvatureissuchthatCoulombfrictionis its shearing. activated all along (Figure 3(d)). The hanging wall is rotating These collapse mechanisms number 5 and number 6 are 𝑃∗ around point 𝐵 such that the section 𝐺𝐵 of the decollement´ explored assuming that the compaction strength in the is delaminated, as for collapse mechanism number 3. This bulk is not homogeneous but sustains a gradient typical of fault geometry is computed with a second-order asymptotic thecompactioninasedimentarybasin.Thenecessityfor development in the arc length 𝑠, parameter of the fault from this proposition is documented in the next section. This 𝐺 𝐹 gradientisassumedtobeaconstantvectornormaltothe point to . The application of the maximum strength ∗ ∗ topography ∇𝑃 =𝐺𝑁,where𝑁 is the normal to the theorem then provides the following upper bound: ∗ topography pointing externally and 𝐺 is the gradient norm. 𝑄(4) 𝜙 =𝜌𝑔(𝑆 (𝜙 +𝛽) The compaction strength at any point within the wedge u cos 𝐷 BS sin 𝐷 ∗0 ∗ pointed by the vector 𝑥 is then 𝑃 +∇𝑃 ⋅(𝑥𝐶 −𝑥) taking + 𝜔[̂ 𝛽(𝐿 𝑆 −𝐼 )+ 𝛽𝐼 ]) 𝐶 𝑃∗0 cos 𝐴𝐵 HW 1HW sin 2HW the point as a reference point and standing for the surface value of the compaction strength. The application 1 2 +𝐶𝐷𝐿𝐴𝐺 cos 𝜙𝐷 + 𝜔𝑇̂ 𝐷 𝐿𝐺𝐵 +𝐶𝐺𝐹 cos 𝜙𝐺𝐹 of the maximum strength theorem provides the two upper 2 bounds: 𝑠𝐹 (5) ̂ 󸀠 󸀠 𝑄 cos 𝜙𝐷 =𝜌𝑔(𝑆𝐵𝐶 sin (𝜙𝐷 +𝛽) ⋅ ∫ 𝐽𝐺𝐹 (𝑠 )𝑑𝑠 , u 0 (6) + 𝑈̂ 𝑆 (𝜃 −𝜙∗ +𝛽))+𝐿 𝐶 𝜙 HW HW sin 𝐺𝐸 𝐺𝐸 𝐴𝐺 𝐷 cos 𝐷 𝐼 = ∫ 𝑥 𝑑𝑠, ∗ 1HW 1 ̂ ̂ +𝐶𝐺𝐹𝐿𝐺𝐹𝐽𝐺𝐹 cos 𝜙𝐺𝐹 + 𝑈 sin 𝜙 HW HW 𝐺𝐸 (8) ∗0 ∗ 1 2 ∗ 𝐼 = ∫ 𝑥 𝑑𝑠, ⋅(𝐿𝐺𝐸 [𝑃𝐺𝐸 +∇𝑃𝐺𝐸 ⋅(𝑥𝐶 −𝑥𝐺)] − 𝐿𝐺𝐸∇𝑃𝐺𝐸 2HW 2 2 HW (𝜙 +𝜙 ) ⋅𝑡 ), 𝜔𝐿̂ = sin 𝐷 𝐺𝐹 , 𝐺𝐸 𝐴𝐵 𝜙 cos 𝐺𝐹 (6) 𝑄 cos 𝜙𝐷 =𝜌𝑔(𝑆𝐵𝐶 sin (𝜙𝐷 +𝛽) ̂ u in which 𝐽𝐺𝐹 and 𝑠𝐹 are the norm of the velocity jump + 𝑈̂ 𝑆 (𝜃 +𝜙 +𝛽))+𝐿 𝐶 𝜙 over 𝐺𝐹 and the arc length 𝑠 measured at point 𝐹.The HW HW sin 𝐺𝐸 𝐺𝐸 𝐴𝐺 𝐷 cos 𝐷 moments 𝐼 ,𝐼 , the surface 𝑆 ,andanyintegralare 1HW 2HW HW +𝐶 𝐿 𝑈̂ 𝜙 + 𝐽̂ 𝜙∗ estimated by numerical means. The angle 𝜃𝐺𝐹(𝑠), defining the 𝐺𝐸 𝐺𝐸 HW cos 𝐺𝐹 𝐺𝐹 sin 𝐺𝐹 𝑠 (9) orientation of the tangent to the fault at any (see illustration 1 ⋅(𝐿 [𝑃∗0 +∇𝑃∗ ⋅(𝑥 −𝑥 )] − 𝐿2 ∇𝑃∗ in Figure 3(d)), is obtained from the condition that the fault 𝐺𝐹 𝐺𝐹 𝐺𝐹 𝐶 𝐺 2 𝐺𝐹 𝐺𝐹 is always in case 2

̂ ⋅𝑡𝐺𝐹), 𝐽𝐺𝐹2 tan (𝜃𝐺𝐹 (𝑠) +𝜙𝐺𝐹)=− 𝐽̂ 𝐺𝐹1 where 𝑡𝐺𝐹 and 𝑡𝐺𝐸 are the unit vectors tangent to the fault indicatedinsubscriptandorientedfrom𝐺 towards the free ̂𝐽 (𝑥)=−( 𝜙 − 𝜔𝑥̂ )𝑒 (7) with 𝐺𝐹 cos 𝐷 2 1 surface. The velocity norms in (8) and (9) are defined by ̂ ̂ +[𝜔(𝐿𝐴𝐵 −𝑥1)−sin 𝜙𝐷]𝑒2 , 𝐽𝐺𝐹 no 5: ∗ sin (𝜃𝐺𝐸 −𝜙𝐺𝐸 −𝜙𝐷) in which (𝑥1,𝑥2) are the coordinates of the point on the fault 𝐺𝐹 𝑠 1 at the arc length . = ∗ The minimization of the upper bound in (6) is con- sin (𝜃𝐺𝐹 +𝜙𝐺𝐹 +𝜃𝐺𝐸 −𝜙𝐺𝐸) ducted by computing for every point 𝐺 on the descritized 𝐹 𝑈̂ decollement´ and point on the descritized decollement.´ The = HW , 𝑄(4) (𝜃 +𝜙 +𝜙 ) associated least upper bound is denoted as lu . sin 𝐺𝐹 𝐺𝐹 𝐷 Journal of Earthquakes 7

1 1

=0

D 0.8 ̃ LAB = 1000 0.9 C=C 300 0.6 =0 (1) lu

D AB 100 /L /Q 50 (2) lu AG

C=C Q L 0.4 0.8 1000 =

AB 300 100 50 ̃ L 0.2

0.7 0 0312 456 7 0312 456 7 Topographic angle 𝛼 (deg) Topographic slope 𝛼 (deg) (a) (b)

Figure 4: Influence of the length 𝐿𝐴𝐵 on the cohesive wedge stability and comparison with the cohesionless case (dashed curve). The ratio of the least upper bounds (mechanisms 2 and 1) is presented in (a) and the extent of the activated decollement´ in (b). The jump for the cohesionless wedge at the critical taper slope 𝛼𝑐 is replaced by a smooth transition of the thrust fold root from the back to the front if cohesion is accounted for.

̂ 𝐽𝐺𝐹 Table1:Geometricalandmaterialpropertiesforthetrianglewedge no 6: sin (𝜃𝐺𝐸 +𝜙𝐺𝐸 −𝜙𝐷) and the six collapse mechanisms presented in Figure 3 unless they are varied (var.) in the analysis. 1 = ∗ Symbol Definition Value Unit sin (𝜃𝐺𝐹 −𝜙𝐺𝐹 +𝜃𝐺𝐸 +𝜙𝐺𝐸) 𝛽 Decollement´ dip 2deg 𝑈̂ 𝜌𝑔 = HW . Material unit weight 2500 Pa/m ∗ 𝜙 sin (𝜃𝐺𝐹 −𝜙𝐺𝐹 +𝜙𝐷) Bulk friction angle 30 deg 𝜙∗ (10) Bulk compaction angle 40 deg 𝐶 Bulk cohesion 10 MPa The minimization of (8) and (9) is done as for mechanism ∗0 𝑃 Var. MPa number 2: the load is computed for every set of points (𝐸, 𝐹, Bulk compaction strength at surface 𝐺∗ and 𝐺) and the selected set is the one leading to the minimum Bulk compaction strength gradient Var. MPa (5) (6) 𝑄 𝑄 𝜙𝐷 Decollement´ friction angle 10 deg bounds lu and lu . 𝐶𝐷 Decollement´ cohesion Var. MPa 𝑇 4. Triangle Wedge: Stability Predictions 𝐷 Decollement´ tensile strength 0MPa The following dimensional analysis is proposed. The refer- ence stress 𝜎𝑅 is taken equal to the bulk cohesion 𝐶 set to is small compared to 𝐿𝐴𝐵,whichsignalsthattheoptimum 𝜌𝑔 25 ∗ 104 10 MPa. The material weight is Pa/m so that thrust folds for each 𝛼 is rooting at the back of the wedge, the ratio 𝜌𝑔/𝜎𝑅 provides the characteristic length 𝐿𝑅 = as expected for subcritical slope conditions (𝛼<𝛼𝑐). The 400 m. Material and geometrical parameters are summarized force ratio is equal to one exactly for the topographic slope in Table 1. Dimensionless quantities will be noted with a 𝛼𝑐 = 2.106 predicted by the critical taper theory. The length 𝐿̃ =𝐿 /𝐿 superposed tilde; for example, 𝐴𝐵 𝐴𝐵 𝑅. 𝐿𝐴𝐺 for 𝛼𝑐 is indeterminate and jumps from a small value to avaluecloseto𝐿𝐴𝐵.Thepositionofthethrustfoldcannotbe 4.1. Comparison with the Critical Taper Theory: Mechanism anywhere within the wedge. For larger 𝛼,theforceratioisone Numbers 1 and 2. The first objective is to compare the meaning that the whole wedge is gliding on the decollement´ predictionsofthecollapsemechanismnumbers1and2with and the deformation is indeed found at the front, limited the critical taper theory. The ratio of least upper bounds by the numerical discretization. The topographic conditions (2) (1) 𝑄 /𝑄 are thus super-critical (𝛼> 𝛼 𝑐).Mechanismnumbers1and lu lu is presented as a function of the topographic angle 𝛼 in Figure 4(a). The distance from the back wall to the root 2 are thus in agreement with the classical taper theory for of the thrust fold 𝐿𝐴𝐺 is seen in Figure 4(b). The dashed cohesionless wedges. curves correspond to the cohesionless wedge that is first The stability prediction accounting for cohesion is pre- ̃ ̃ analyzed. The ratio is increasing approximately linearly for sented in the same two figures with 𝐶𝐷 = 𝐶=1.The small values of the topographic angle, and the distance 𝐿𝐴𝐺 wedge length 𝐿𝐴𝐵 is varied between 50 and 1000 times the 8 Journal of Earthquakes

1 0 1 0 0.8

0.9 =C=0

(1) 0.6 lu AB D /L /Q 0 0.5 1 C (2) lu AG Q L 0.5 =0 0.4 1 0.5 0.5 ̃ 0 C̃ =1 D C =1 D 0.8 D

=C

C 0.2

0.7 0 0 123 456 7 0 123 456 7 Topographic angle 𝛼 (deg) Topographic slope 𝛼 (deg) ̃ ̃ LAB =50 LAB =50 ̃ ̃ LAB = 300 LAB = 300 (a) (b) ̃ Figure 5: Influence of the decollement´ cohesion on the cohesive wedge stability for two lengths 𝐿𝐴𝐵 =50and 300 in dotted-dashed and solid curves, respectively. The results for the cohesionless wedges are the dashed curves.

reference length 𝐿𝑅 oftheproblem.Thefirstobservation CM 4 is that the force ratio is not anymore a bilinear function of 1 the topographic slope. There is a gradual transition from the approximately linear relation for small 𝛼 to the horizontal ) (with softening) asymptote at one for large values of 𝛼. This transition is 4 CM 4 ̃ ̃ or 𝐿 𝐿 3 0.9 smoother for the smaller values of 𝐴𝐵.Inthelimitof 𝐴𝐵 , CM 3

tending to infinity, the solution converges towards the results =2 i obtained in the absence of cohesion. The smooth transition ( (1) from subcritical to super-critical conditions is due to the lu

/Q 0.8 (i) lu CM 2 progressive migration of the dominant thrust fold from the CM 3 rear to the front of the wedge with increasing values of 𝛼.Note Q (with softening) that this transition is not activated at 𝛼𝑐 but with a definite ̃ 0.7 delay which is increasing for the smaller values of 𝐿𝐴𝐵. 123 456 7 The cohesion of the decollement´ is now varied with Topographic slope 𝛼 (deg) respect to bulk material cohesion for the two values of the ̃ Figure 6: Comparison of the bounds for collapse mechanisms length 𝐿𝐴𝐵 of50and300.Theresultsarepresentedin ̃ for thrusting (CM 2), solid line, for delamination with a straight Figure 5. Consider first the results for 𝐿𝐴𝐵 =50,thedotted- fault (CM 3), dashed curves, and delamination with a curved ramp 𝐶̃ =1 ̃ dashed curves presented for 𝐷 ,0.5,and0.Reducing (CM 4). The wedge length is 𝐿𝐴𝐵 =10and the softening for the cohesion by half has shifted the curve of the bound ratio the delamination mechanisms is such that the fault 𝐺𝐹 has the towards the critical taper solutions. There is still a delay in properties of the decollement´ instead of the bulk material. the beginning of the smooth transition of the root 𝐺 from thereartothefrontofthewedgebutthisdelayisreduced by half (Figure 5(b)). The next value of the decollement´ is transition or the cohesionless decollement´ occurs for an angle 𝐶 =0 ∘ 𝐷 , and in that instance the least upper bound for approximately 0.5 less than for the critical taper theory. thesecondmechanismisalwaysgreaterthanone.Thislimit of one is exceeded because of the numerical discretization 4.2. Delamination: Mechanism Numbers 3 and 4. The first but the important result is that the cohesive wedge on the observation is that the delamination mechanisms are always ̃ cohesionless decollement´ with 𝐿𝐴𝐵 =50is always super- tothefrontofthetrianglewedgeandcanonlycompete critical. This conclusion is confirmed by the length of the with thrusting mechanism number 2 for wedge sufficiently ̃ activated decollement,´ which is found to be equal to the total small. Results are presented for 𝐿𝐴𝐵 =10in Figure 6. The length of the wedge (Figure 5(b)). Complementary results are first results were obtained assuming the straight or curvy ̃ presented or the length 𝐿𝐴𝐵 = 300,solidcurves.Thenew fault 𝐺𝐹 had the properties of the bulk material. The ratio ̃ information is that the bound ratios for 𝐶𝐷 = 0.5 and 0 of bounds for CM 3 and 4 was always larger than the ratio are around the critical taper solutions. Note that the stability for the classical thrusting CM 2. Moreover, the CM 3 which Journal of Earthquakes 9

1 1

) 0.8 0.8 6 ∗ CM 2 ̃ ∗

or G =3 G̃ =3 5 ,

AB 0.6

=2 0.6 /L i ( AG (1) lu CM 2 L 0.4 /Q ∗ 0.4 G̃ =0 (i) lu

Q CM 6 CM 5 0.2 CM 6 ∗ 0.2 G̃ =0 CM 5 0 1 24573 6 1 2453 6 7 Topographic slope 𝛼 (deg) Topographic slope 𝛼 (deg) (a) (b)

Figure 7: Comparison of collapse mechanism numbers 5, 6, and 2, to question the thrusting vergence, in terms of the bound ratio, (a) and ̃ thepositionoftheroot𝐺 on the decollement,´ (b). Results obtained for 𝐿𝐴𝐵 = 100.

is supposed to improve on CM 2 was always super-critical, activation which is often observed at the front of fold and positionedtotheveryfrontofthetrianglewedgeandwitha thrust belts. Only frictional properties are considered here bound ratio equal to one. A second set of stability predictions (region 2 of the strength domain) and the contributions of were obtained by introducing some softening on the fault 𝐺𝐹. delamination, opening, and compaction are examined in the It was assigned, for sake of simplicity, the same properties as next section with a simplified prototype. the decollement.Theresultsaremoresatisfactoryinthesense´ The front of our wedge is presented in Figure 8(a) where that CM 3 dominates CM 2 for 𝛼 greater than approximately the lower decollement´ is at the base on the accretion layer ∘ 5 ,approximately.Theroot𝐺 is at the dimensionless distance of thickness 𝐻 with properties (𝜙𝐷,𝐶𝐷). The length of the of 8 and 5, respectively, for this specific angle. Mechanism wedge from the back wall to the point 𝐼 in the topography number 4 is already super-critical and found at the tip of the is denoted as 𝐿𝐼.Theseconddecollement´ (dashed line) is wedge. parallel to the firstecollement d´ at the distance ℎ from the 󸀠 󸀠 accretion topography and has the properties (𝜙𝐷,𝐶𝐷). Two 4.3. Compaction and Thrusting: Mechanism Numbers 5 and collapsemechanismsofthetypeCM2definedinthesection above could be proposed for an accretion layer thickness 6. ThecomparisonbetweenCM5,6,and2isproposedto 𝐻 ℎ question the vergence of the thrusting. Results are presented and and correspond solely to the activation of either 𝐿̃ = 100 lower or upper decollement.´ They both terminate at the front in Figure 7 for 𝐴𝐵 . Two sets of results are presented, 𝐹 𝐺 𝐸 ̃∗ by thrusting to the free surface: points l, l,and l and thefirstintheabsenceofcompactiongradient𝐺 = 𝐹 𝐺 𝐸 0 points u, u,and u in Figure 8(a), the subscripts l and u and the second with a gradient of 3. The first set of referring to the lower and upper decollement,´ respectively. results are interesting since they show that the two thrusting The upper decollement´ activation mechanism is nevertheless mechanisms with compaction dominate with a preference for not possible for an accretion layer of thickness 𝐻,sincethe mechanismnumber5proposedforahinterlandvergence. 𝐺 whole wedge is compressed and the lower decollement´ has Nevertheless, the position of the root (point )ofthis to be activated at least over a minimum section to ensure mechanism is at the back of the wedge for all topographic thetransferofactivitytotheupperdecollement.´ This new 𝐺̃∗ angles. Introducing a gradient has the definite advantage mechanism is presented in Figure 8(b) in the case where that there is a stability transition which is rather close to the transfer occurs close to the wedge front. The fault 𝐺𝐹 is the one recorded for classical mechanism number 2 with ∘ againashearplaneseparatingthebackstop(BS)fromthe a delay in transition not exceeding 1 .Theratioofbound lower hanging wall region (HWl). The ramp of this lower is still in favor of the two compacting mechanisms with a thrusting is however not cutting through the entire structure 󸀠󸀠 slight advantage for mechanism number 5 which promotes but stops as it reached the upper decollement´ at point 𝐺 . 󸀠󸀠 󸀠 hinterland vergence. The sliding wall (SW) above the segment 𝐺 𝐺 is gliding rigidlyovertheupperdecollement.´ It is limited at its rear 󸀠󸀠 󸀠󸀠 by the normal fault 𝐺 𝐹 which is also a shear plane. The 5. Two Décollements and an Accretion Front 󸀠 󸀠 󸀠 󸀠 󸀠 point 𝐺 is the root of the two thrust faults 𝐺 𝐹 and 𝐺 𝐸 Theobjectiveofthenexttwosectionsistoaccountfor bounding the upper hanging wall (HWu) which defines the thepresenceofanaccretionfronttoourtrianglewedge. mostfrontalpartofthethrustingmechanism.Theapplication Thissectionfocusesonthepotentialchangeindecollement´ of the maximum strength theorem is rather systematic and 10 Journal of Earthquakes provides the following expression for the upper bound to the Table 2: Geometrical and material properties for the study of the force necessary to activate this new collapse mechanism wedge with an accretion front. The missing information is found in Table 1. The reference stress and length are still 𝜎𝑅 =10MPa and (7) 𝑄 𝜙 =𝜌𝑔(𝑆 (𝜙 +𝛽) 𝐿𝑅 = 400 m. u cos 𝐷 𝐵𝐶 sin 𝐷 Symbol Definition Value Unit + 𝑈̂ 𝑆 (𝜃 󸀠󸀠 +𝜙 +𝛽))+𝑈̂ 𝑆 HW HWl sin 𝐺𝐺 𝐺𝐸 SW SW ̃ 𝐿𝐼 Length of triangle region 100 m 󸀠 𝐻̃ ⋅ (𝜙 +𝛽)+𝑈̂ 𝑆 (𝜃 󸀠 󸀠 +𝜙 +𝛽) Accretion layer thickness 2.5 m sin 𝐷 HWu HWu sin 𝐺 𝐸 𝐺𝐸 ̃ (11) ℎ Depth of upper decollement´ 1.25 m 󸀠 󸀠 𝜙 +𝐿𝐴𝐺𝐶𝐷 cos 𝜙𝐷 +𝐿𝐺󸀠𝐺󸀠󸀠 𝐶𝐷 cos 𝜙𝐷 +𝐶𝐺𝐸 cos 𝜙𝐺𝐸 𝐷 Lower decollement´ friction angle 20 deg ̃ 𝐶𝐷 Lower decollement´ cohesion 0.5 MPa 󸀠 ⋅(𝐿 󸀠󸀠 𝑈̂ +𝐿 󸀠 󸀠 𝑈̂ )+𝐶 𝜙 𝐺𝐺 HWl 𝐺 𝐸 HWu 𝐺𝐹 cos 𝐺𝐹 𝜙𝐷 Upper decollement´ friction angle 10 deg 𝐶̃󸀠 ̂ ̂ ̂ 𝐷 Upper decollement´ cohesion 0 MPa ⋅(𝐿𝐺𝐹𝐽𝐺𝐹 +𝐿𝐺󸀠󸀠𝐹󸀠󸀠 𝐽𝐺󸀠󸀠𝐹󸀠󸀠 +𝐿𝐺󸀠𝐹󸀠 𝐽𝐺󸀠𝐹󸀠 ), which is numbered seventh in our series. The two first 󸀠󸀠 󸀠 󸀠 󸀠󸀠 lines correspond to the work of gravity, the third one to unknown positions of the seven points 𝐹, 𝐹 , 𝐹 , 𝐸 , 𝐺, 𝐺 , 󸀠 the resisting power of the two activated sections of the and 𝐺 . The computation time required for optimization is of decollements,´ and the last two lines to the resisting power of the order of 𝑀𝑁,where𝑀 isthenumberofpointsonthe the ramps and the shear planes. Note that all shear planes discretized surfaces and 𝑁=7is the number of unknown 𝜙 𝐶 and ramps are assigned the friction properties ( 𝐺𝐹, 𝐺𝐹) points. Such calculation could be done advantageously on a 𝜙 𝐶 and ( 𝐺𝐸, 𝐺𝐸), respectively, corresponding to the properties GPU architecture or using linear procedures. A simplified introduced for CM 2. The velocity norms in (11) are defined approach is proposed in this contribution in the first attempt by to gain some insight on this decollement´ transition. It is proposed that the selection of the frontal part of the thrusting 𝐽̂ 󸀠 󸀠 󸀠 𝐺𝐹 mechanism (points 𝐸 , 𝐺 ,and𝐹 ) is independent of the four sin (𝜃𝐺𝐺󸀠󸀠 +𝜙𝐺𝐸 −𝜙𝐷) others and determined directly from the optimization of CM 2 proposed in Figure 8(a) for the upper decollement.´ The 1 4 3 = computation cost is then reduced to 𝑀 +𝑀 and is not (𝜃 +𝜙 +𝜃 󸀠󸀠 +𝜙 ) sin 𝐺𝐹 𝐺𝐹 𝐺𝐺 𝐺𝐸 an issue anymore. This proposition is not steered only by 𝑈̂ these technical difficulties only but by the observation that = HWl , the upper thrusting part of CM 7 is identical to CM 2 with (𝜃 +𝜙 +𝜙 ) sin 𝐺𝐹 𝐺𝐹 𝐷 the only difference that the back stop velocity of CM 2 is now the velocity of the sliding wall. It is thus legitimate to assume 𝐽̂ 󸀠󸀠 󸀠󸀠 𝐺 𝐹 𝐸󸀠 𝐺󸀠 𝐹󸀠 󸀠 that the optimum positions of points , ,and could be (𝜃 󸀠󸀠 +𝜙 −𝜙 ) sin 𝐺𝐺 𝐺𝐸 𝐷 obtained by isolating the frontal part of CM 7. The positions 󸀠 󸀠 󸀠 of 𝐸 , 𝐺 ,and𝐹 arethustakenasthoseof𝐸u, 𝐺u,and𝐹u 𝑈̂ = SW (12) defined in Figure 8(a). sin (𝜃𝐺󸀠󸀠𝐹󸀠󸀠 +𝜙𝐺𝐹 +𝜃𝐺𝐺󸀠󸀠 +𝜙𝐺𝐸) The first results constitute a comparison between CM 7 and CM 2 for the geometrical and material parameters pro- 𝑈̂ = HWl , videdinTable2.Theupperboundforthetwomechanisms 󸀠 sin (𝜃𝐺󸀠󸀠𝐹󸀠󸀠 −𝜙𝐺𝐹 +𝜙𝐷) is provided in Figure 9(a) as a function of the topographic 𝛼 ̂ slope of the triangle part of the wedge. The dotted-dashed 𝐽𝐺󸀠𝐹󸀠 curveforCM2isbelowtheCM7curveformostoftherange (𝜃𝐺󸀠𝐸󸀠 +𝜙𝐺𝐸 −𝜙𝐷) in values of 𝛼 considered. There is however a window in the sin ∘ ∘ range between 6 and 7.5 forwhichthechangeindecollement´ 𝑈̂ SW can indeed occur. The inset within Figure 9 provides the = ∘ sin (𝜃𝐺󸀠𝐹󸀠 +𝜙𝐺𝐹 +𝜃𝐺󸀠𝐸󸀠 +𝜙𝐺𝐸) position of the two collapse mechanisms for the slope of 6.5 within this small interval. The transition from the lower to 𝑈̂ = HWu , the upper decollementoccursattherearofthewedgeand´ 󸀠 sin (𝜃𝐺󸀠𝐹󸀠 +𝜙𝐺󸀠𝐹󸀠 +𝜙𝐷) the collapse mechanism (CM) 7 requires the main part of the upper decollement´ to be activated up to the thrust at the very 󸀠 with the same notation as before; any angle 𝜃𝑋𝑌 defines the frontofthewedge.TherampsofCM2andthepoint𝐹 of slope of the generic fault 𝑋𝑌. CM 7 are exactly at the breaking slope point 𝐼. The technical difficulty with this mechanism concerns the These results could be explained with the critical taper ∘ 󸀠 ∘ minimization of the upper bound in (11). There are three theory noting that 𝛼𝑐 = 6.36 and 𝛼𝑐 = 2.10 for the lower shear planes dips, two ramp dips, and the two lengths of the and the upper decollement,´ respectively. For that purpose, 󸀠 󸀠󸀠 activated sections of the decollement´ which are unknown. the 𝑥1-coordinate of points 𝐺 , 𝐺 ,and𝐺l, normalized by These seven unknowns are equivalently replaced by the the length 𝐿𝐼, have been plotted in Figure 9(b) as a function Journal of Earthquakes 11

C F C F g l F󳰀󳰀

󳰀 E F 󳰀 l I E Fu HW I Eu l HWu BS SW h

󳰀󳰀 Gu G G󳰀 H G A l A G (a) (b)

Figure 8: Two decollements´ are activated and could terminate with a thrusting to the free surface, (a). Mixed collapse mechanism number 7 includes a transition from the lower to the upper decollement´ and terminates with thrusting at the front (b).

400 1 G󳰀

300 C, F

CM 7 Gl 0.5 G󳰀󳰀 200 CM 2

G󳰀󳰀

󳰀 thrust roots of position Normalized Normalized position of thrust roots of position Normalized G A G Gl 100 0 3456789 3456789 Topographic slope 𝛼 (deg) Topographic slope 𝛼 (deg) (a) (b)

Figure 9: The bound for the change in decollement´ (CM 7) is compared to the bound for classical thrusting (CM 2), the two forces being a function of the topographic slope 𝛼 (a). Note the small interval for dominance of CM 7. Note also in the inset, the transition occurs at the 󸀠 󸀠󸀠 rear of the wedge in that interval. The positions of the three points 𝐺, 𝐺 ,and𝐺 are presented in (b) as a function of 𝛼 to discuss stability conditions.

󸀠 of the slope 𝛼.Thecoordinate𝐺 of the upper thrusting Our prototype is certainly too simple to explain the partofCM7isalwaysgreaterthanonesinceitisfound frontal position of CM 7 and its dominance over CM 2. on the accretion layer. This upper thrust is always super- Several reasons could be evoked to explain this result includ- 󸀠 critical since the values of 𝛼 are exceeding 𝛼𝑐.Thechoice ing a gradient in frictional properties. One could imagine of a zero cohesion is certainly responsible for the activation a negative gradient in the frictional properties towards the of the upper decollement´ beyond the point 𝐼.Point𝐺l is front which would penalize the transition from the lower to close to the back of the wedge for 𝛼<𝛼𝑐 and migrates the upper decollementattherearofthewedge.Onecouldalso´ 󸀠󸀠 abruptly to the front as soon as 𝛼 exceeds 𝛼𝑐, as expected. imagine that the transition ramp 𝐺𝐺 is not a new fault but It is around this change in position that CM 7 appears to an inherited structure reactivated in compression. Another 󸀠󸀠 be dominant. However, the position of point 𝐺 signaling explanation is that the shape of the wedge at the front is more the transition to the upper decollement´ is at the back of complex and then idealized in our prototype on a length scale 󸀠󸀠 󸀠 the structure. One has to consider larger topographic slope corresponding to the activated section 𝐺 𝐺 of the upper ∘ 󸀠󸀠 around 𝛼=9 for this point 𝐺 to be found at the front of decollement.´ To investigate this possibility, a trilinear wedge 󸀠󸀠 󸀠 the accretion wedge. Note however that the length 𝐺 𝐺 of has been considered, as seen in Figure 10. The slope of the ∘ the activated upper decollementsectionisthensmall.Itisfor´ segment 𝐾𝐼 is kept constant at 5 to make sure that the upper reason that the change in decollement´ is not so beneficial and thrusting part of CM 7 is at the very front. The slope of the rear that the dominant mechanisms remain CM 2 for these large part of the topography 𝐶𝐾 is changed over a small interval ∘ topographic slopes. around the 9 which was found necessary to obtain a point 12 Journal of Earthquakes

8 8

∘ 8.9 10∘ 6 6 F󳰀󳰀 -coordinate

K -coordinate

2 K 2 󳰀󳰀 x 4 5∘ x 4 F 5∘ I I 2 2 G󳰀󳰀 Dimensionless Dimensionless 󳰀 󳰀󳰀 G G G󳰀 G G G Gl 0 l 0 50 60 70 80 90 50 60 70 80 90

Dimensionless x1-coordinate Dimensionless x1-coordinate (a) (b)

Figure 10: A bilinear topography prior to the accretion front is proposed to control the extent of the activated upper decollement´ for CM 7. ∘ ∘ The rear section dips at 8.9 and 10 in (a) and (b).

󸀠󸀠 𝐺 at the front in our first set of results. The shapes of CM7 4 are also considered. The fourth mechanism is CM 5 which ∘ ∘ and CM 2 are presented for 𝛼= 8.9 and 𝛼=10 in solid and couldexplainthehinterlandvergenceofthetrianglewedge ∘ long dashed lines, respectively. For the slope of 8.9 ,points (in Section 4) by introducing compaction. Note that CM 6 󸀠󸀠 𝐺 and 𝐺 are at the coordinates 62 and 68 on the lower and corresponding to the frontal vergence was not found of any the upper decollement,´ respectively. The section of the upper interest and is not considered further here. 󸀠󸀠 󸀠 decollement´ is activated in 𝐺 𝐺 below the 𝐾𝐼 segmentofthe The velocity fields for these four collapse mechanisms are triangle region. The V-shaped thrust of CM 2 is also presented not different from the one discussed in Section 4. The bound 𝐺 for CM 2 is still given by (2) and (3), the definitions for the inthatfigureforthesametopographyangle.Thepoint l is ahead of 𝐺 atthecoordinate67sothatCM2requiresalonger back stop and the hanging wall remaining the same. activation of the lower decollement´ than CM 7. The results The velocity fields for CM 3 and CM 4 rely on a rotation ∘ 𝑅 ´ for 𝛼=10 differ by the relative position of these two points: around the point on the decollement and its position is a new unknown. The bounds to the tectonic force of CM 𝐺 is now very close to 𝐺l and has shifted by a distance of 6 ∘ 3 and CM 4 in the accretion wedge are similar to those in despitethesmallangularincreaseof1.1.Suchlargevariation thetrianglewedge,andthemaindifferenceistheaccount is typical at the critical stability conditions. of compaction along the ligament 𝑅𝐸. The application of the These preliminary results should be completed by a maximum strength theorem provides the two upper bounds thorough parametric study but two results seem already to presented as emerge. First, the change in decollement´ could dominate 𝑄(3) 𝜙 =𝜌𝑔(𝑆 (𝜙 +𝛽) close to the criticality conditions for the wedge composed u cos 𝐷 BS sin 𝐷 of the lower decollement´ (𝛼𝑐). The transition from the + 𝜔[̂ 𝛽(𝐿 𝑆 −𝐼 )+ 𝛽𝐼 ]) cos 𝐴𝐵 HW 1HW sin 2HW lower to the upper decollement´ occurs at the back of the 1 2 wedge at these critical conditions and to the front of the +𝐶𝐷𝐿𝐴𝐺 cos 𝜙𝐷 + 𝜔𝑇̂ 𝐷 𝐿𝐺𝐵 1.5𝛼 2 (13) wedge for topographic slopes much larger (close to 𝑐 1 for our parameters). Second, a curvature or a change in +𝐶 𝐿 (𝐽̂ 𝜙 + 𝐿 𝜔̂ 𝜙 ) 𝐺𝐹 𝐺𝐹 𝐺 cos 𝐺𝐹 2 𝐺𝐹 cotan 𝐺𝐹 the topographic slope in the frontal region can provide the 1 1 characteristic length controlling the extent of the activated ∗0 2 ∗ 3 + 𝜔(̂ 𝑃𝑅𝐸𝐻 − 𝐺 𝐻 ), section of the upper decollement.´ 2 6 𝐼 𝐼 where 1HW and 2HW follow (4) 𝑄(4) 𝜙 =𝜌𝑔(𝑆 (𝜙 +𝛽) 6. Delamination, Shear, and Compaction at u cos 𝐷 BS sin 𝐷 Accretion Front + 𝜔[̂ 𝛽(𝐿 𝑆 −𝐼 )+ 𝛽𝐼 ]) cos 𝐴𝐵 HW 1HW sin 2HW This section concentrates on the search for the frontal 1 2 +𝐶𝐷𝐿𝐴𝐺 cos 𝜙𝐷 + 𝜔𝑇̂ 𝐷 𝐿𝐺𝐵 mechanism setting aside the change in decollement´ depth 2 (14) 𝑠 just discussed. The prototype seen in Figure 11 has a sin- 𝐹 1 ̂ 󸀠 󸀠 ̂ ∗0 2 gle decollementatitsbase,andtheobjectiveistostudy´ +𝐶𝐺𝐹𝐿𝐺𝐹 ∫ 𝐽𝐺𝐹 (𝑠 )𝑑𝑠 + 𝜔( 𝑃𝑅𝐸𝐻 0 2 the deformation style at the accretion front. Four collapse 1 mechanisms are considered and compared and the basic − 𝐺∗𝐻3), mechanismofCM2forwhichonlyfrictionalpropertiesare 6 𝐼 𝐼 required. The two delamination mechanisms CM 3 and CM where 1HW and 2HW follow (6). Journal of Earthquakes 13

C C F F 1 I E I E 2 2 2 g 3 1 6 2 𝛽 2 1 6 A 2 2 2 0 0 A 2 2 2 G R C G (a) (b)

C F 2 C F I E e I E 4 2 2 2 4 6 e 5 2 1 6 e3 2 A 4 2 2 0 0 A G R 2 2 2 G (c) (d)

Figure 11: The four collapse mechanisms in competition at the front of accretion wedge.

The velocity field of CM 5 relies on compaction along Table 3: Geometrical and material properties for the wedge with its plane 𝐺𝐸 to motivate the hinterland vergence. Moreover, an accretion layer and the four collapse mechanisms presented in a gradient in the compaction strength was considered in Figure 11. Information unchanged compared to Section 4 is provided 𝜎 =10 Section 4 to justify the position of this collapse mechanism. in Table 1. The reference stress and length are still 𝑅 MPa and 𝐿 = 400 This gradient was proposed to be normal to the triangle 𝑅 m. wedge topography. It is found more convenient for this Symbol Definition Value Unit prototype which has an accretion front to assume that the ̃ 𝐿𝐼 Length of triangle region 0 m gradient is perpendicular to the decollement.´ The expression 𝐻̃ Accretion layer thickness 2.5 m for the bound to the tectonic force of CM 5 is still given by 𝜙 (8). Bulk friction angle 30 deg 𝜙∗ Geometrical and material parameters are presented in Bulk compaction angle 40 deg ̃ Table3.TheboundsforthefourmechanismsofFigure11 𝐶 Bulk cohesion 1 MPa ̃∗0 are presented in Figure 12(a) as a function of the topographic 𝑃 Bulk compaction strength at surface 5 MPa ∗ slope 𝛼. All bounds, except CM 3, are approximately bilinear. 𝐺̃ Bulk compaction strength gradient 2.5 The bends in the curve signal the transition of the collapse 𝜙𝐷 Decollement´ friction angle 10 deg ̃ mechanism from the rear towards the wedge front. Collapse 𝐶𝐷 Decollement´ cohesion 0.5 MPa 𝐺𝐹 ̃ mechanism 3, the delamination with a straight fault ,is 𝑇𝐷 Decollement´ tensile strength 0 MPa alwaysatthefront.Thefirstmechanismtomigratetothe front with increasing 𝛼 is CM 4, the delamination with the curvy ramp. The next mechanism to migrate to the front is CM 2, the classical frictional faulting, and the last is the CM 5. It is this last mechanism which dominates for the decollement,´ however, does not change the dominance of CM set of parameters proposed here. Note that the difference 5 for thicker accretion layers. betweenCM2,4,and5israthersmalloncethesethree The positions of the collapse mechanisms are sensitive to mechanisms are at the front. This difference is less than 2.5% the topographic slope 𝛼 and they are presented in Figure 12(b) ̃ ∘ and it decreases with decreasing 𝐻.Moreover,itwasfound for the specific slope 𝛼= 4.5 . CM 2 (solid lines) and CM that CM 2 became dominant once 𝐻̃ is set to less than 1. 5 (dotted-dashed) are in the same region with faults 𝐺𝐹 In view of the small differences between the various subparallel. The fault 𝐺𝐸 of the compacting mechanism is bounds,imperfectionscouldthusplayanessentialrolein however steeper and, consequently, the surface of the hanging selecting the dominant mechanisms for thin accretion layers. wall is smaller than the one needed for CM 2, explaining For example, assuming the curvy fault 𝐺𝐹 of CM 4 to certainly the dominance of CM 5. The two delamination ∘ ∘ have a reduced frictional angle of 20 instead of 30 is mechanisms are more to the front with a longer delamination enough to reduce the corresponding bound and thus for this segment for CM 3 compared to CM 4. The consequence is CM 4 to become dominant. Changing the cohesion of the again due to gravity forces: the surface of the accretion layer 14 Journal of Earthquakes

5 450 4 400

CM 3 3 350 CM 4 CM 2 CM 5

-coordinate 2 2

300 ̃ x

Dimensionless bounds Dimensionless 250 1

200 0 2 3 4 5 140 150 160 170

Topographic slope 𝛼 (deg) x̃1-coordinate

CM 2 CM 4 CM 2 CM 4 CM 3 CM 5 CM 3 CM 5 (a) (b)

Figure 12: The bounds for the four CM defined in Figure 11 as a function of the topographic slope 𝛼 (a). The positions of these collapse ∘ mechanism are presented in (b) for 𝛼= 4.5 .

lifted by delamination is larger for CM 3 than for CM 4, than that in mechanism number 2. This result suggests that explaining the advantage of the latter mechanism. delamination prefers to occur in shallow sediment where there is lower density than in deep crust. This finding is sup- ported by the observation from Bossort [20] that the density 7. Discussion of shallow section above the triangle zone is significantly lower than that of the underlying thrust-faulted sequence in By analyzing their virtual velocity fields and comparing their southern Alberta. If mechanism number 3 dominates in a least upper bounds, five collapse mechanisms are investigated shallow and lower density sediment, it supports that upward in the competition for the initiation of triangle zones in decease of density across the back thrust favors uplifting triangle and accretion wedges. the overlying section rather than up thrusting the denser By comparing the predictions of collapse mechanism underlying sheets to the surface. The denser thrust sheet numbers 1 and 2 with the critical taper theory, a good could keep flat at the decollement´ level to wedge and uplift agreement is achieved between the limit analysis (supporting the overlying lower density material upward with rotation. this research) and the classical taper theory for cohesionless This investigation is also supported by the observations that wedges. In a triangle wedge the position of the thrust fault triangle zones commonly develop at mountain fronts where cannot be anywhere by following the rule of either subcritical it prefers to occurr at shallow levels in young sediments or super-critical condition predicted by the classical taper [21]. Couzens and Wiltschko [9] suggested that triangle theory. Results from a series of tests indicate that the stability zonesmayforminthelatestagesofthrustbeltevolution, of the cohesive triangle wedges is controlled by three factors: when significant synorogenic deposits accumulated at the thelengthofthebasaldecollement,´ the cohesion of the deformation front by reconstructing the frontal stratigraphy bulk,andthedecollement.´ This finding is consistent with of the Wyoming thrust belt. The triangle zone observed in the previous findings [10, 12]. Couzens-Schultz et al. [12] the Longmen Shan fold-and-thrust best emerges at the depth suggested that the strength of the decollements´ of the rock less than 5 km in the Trias sediment of the Longmen Shan layers plays a significant role in controlling the style of foothill and the adjacent Sichuan Basin [2–4]. In order to duplex. The frictional characteristics of the upper and lower discover the details of how the density and thickness of the detachments have influence not only on the initiation of frontal sediment are playing the role in the competition of triangle zone, but also on the distribution of deformation in different mechanism, thorough parameterization tests should an evolving triangle zone [10]. beconductedinthefuture. Delamination mechanism numbers 3 and 4 always However, a weak (softening) back thrust introduced in emerge at the front of the triangle wedge by following super- mechanism numbers 3 and 4 improves the situation of delam- critical rule, so that mechanism number 2 dominates in the ination mechanisms in the competition. Mechanism number competition with mechanisms 3 and 4. The reason could 3 begins to dominate by delaminating along the horizontal be attributed to the gravity force of the hanging wall, since decollement´ and shear along the thrust in the triangle wedge. the external power consumed in the rotation of the whole The reason could be attributed to the decreasing of the inner hanging wall in mechanism numbers 3 and 4 is much larger power consumed on the activation along the back thrust. Journal of Earthquakes 15

In a triangle wedge mechanism numbers 5 and 6 of the first mechanism is the classical Coulomb shear thrust. The combination of shear-enhanced compaction and Coulomb second is delamination at the frontal part of the decollement´ shear consume less power than mechanism number 2 of with straight back thrust, while the third is delamination with Coulomb shear. Moreover, the hinterland thrust (mechanism curvy back thrust. The forth is the combination of ramp with number 5) is preferred to the foreland thrust (mechanism Coulomb shear and shear-enhanced compact fault, while number 6) in competition of the two compaction mecha- the fifth is the combination of the exchanging motion on nisms. However the advantage is slight in the compaction the ramp and thrust. The dominating mechanism in the by mechanism number 5 over riding mechanism numbers 6 formation of triangle zones relies on the competition of the and 2. Mechanism number 5 also dominates in the accretion least upper bound of each mechanism when subjected to wedge with a thick toe accretion wedge, while mechanism tectonic force. The controlling factors of the competition number 2 dominates in a thin toe accretion wedge. are discovered and summarized as follows: (1) the frictional In an accretion wedge mechanism number 5 loses the characters and cohesion of horizontal decollements´ and advantage and mechanism number 4 becomes dominant with thrust, (2) the slope of the topography of accretion wedge, decreasing the frictional angle of back thrust. This finding and (3) the thickness and the rock density of the front toe of agrees with the previous proposal [7] that the frictional accretion wedge. characteristics (friction angle) of the thrust have influence on the distribution of deformation in an evolving triangle Appendices zone. Collapse mechanism number 5 of Coulomb shear on the back thrust and shear-enhanced compaction on the ramp This section presents the details of the derivation of the upper is a new introduction for a possible collapse mechanism in bounds presented in the main text. The structure of those the accretion wedge. appendices follows exactly the one of the main text except The collapse mechanism in the two decollement´ accretion for the first section which is devoted to the definition ofthe wedges is also investigated. The results indicate the change support function. of the decollementbyaramp(forethrust)fromthelower´ flatecollement d´ to the upper decollement´ happens. However, A. Material Strength and Support Function this changing is depending on the wedge conditions. It occurs either at the rear on the critical condition of the Central to the maximum strength theorem used throughout lower decollement´ or at the front of the wedge with the this contribution is the bounding of the power done on any large topography. The topography of the fold-and-thrust discontinuity. It is the support function which defines this belts of mountain front is always much more complicated maximum power and it is directly related to the material than our proposed bilinear and trilinear prototypes in this strength,asitisdiscussedinthissection. research. The real topography with changing slops plays a Consider a material which is cohesive, frictional, com- significant role in the activation of the length of the lower pactant, and dilatant. An arbitrary plane is crossing this and upper decollement´ and the position of the ramp which material and is oriented by the normal vector 𝑛,pointingto connects the two decollements.´ High topography activates the positive side + and the tangent vector 𝑡 in this 2D setting the changing decollement´ near the front of the wedge by (Figure 13(a)). The stress vector acting on this plane is 𝑇 and is mechanism number 4 with curvy fault. This suggests that decomposed in a normal 𝜎𝑛 and a tangential component 𝜏 in erosional process which shapes the morphology of mountain the right-handed basis {𝑛,𝑡}. Failure occurs if the stress vector front is important in the formation of triangle zones. By reaches the boundary of the strength domain presented in the conducting analogue models Montanari et al. [14] proposed same figure. This boundary is characterized by three scalars ∗ that the syntectonic erosion that promotes the development having dimension of stress, 𝑃 , 𝐶,and𝑇 corresponding to the of passive-roof duplex style is one of the major roles in the maximum compressive strength (positive), the cohesion, and formation of triangle zone at Vena del Gesso Basin (Romagna the tensile strength (not to be confused with the stress vector), Apennines, Italy). The Longmen Shan range locates in the respectively. Two angles are also necessary, the classical ∗ south Asian monsoon zones indicating high incision rate Coulomb frictional angle 𝜙 and the compaction angle 𝜙 , in this region. During the topography building process the orienting the two segments closing the strength domain morphology of the Longmen Shan has been shaping by heavy in compression. Note that this general strength domain is erosion together with the large scale land slide induced by convex in the (𝜎𝑛,𝜏)space. repeating large earthquake like the 2008 Mw7.9 Wenchuan The failure plane considered accommodates a velocity earthquake [22]. The formation of the triangle zone at the discontinuity ̂𝐽 defined as the difference between the velocity termination of the Longmen Shan fold-and-thrust belt may + − (̂𝐽 = 𝑈̂ − 𝑈̂ ) be influenced by the changing slope during the mountain vectors on the positive and the negative side . building process. The superposed hat on the velocities is to remind the reader that any velocity, consistent with boundary conditions, could be considered and not just the exact, unknown velocity. Any 8. Conclusions such velocity field is said to be kinematically admissible (KA). Five collapse mechanisms by delamination, shear, and com- Thepoweratanypointontheplaneisthescalarproduct 𝑇⋅̂𝐽 paction are introduced to the competition in the formation betweentheunknownstressvectorandthevelocityjump of triangle zone front of fold-and-thrust belts as follows. The ̂𝐽. The maximum power is defined by the support function, 16 Journal of Earthquakes

𝜏 t Case 3 𝜏 T 𝜏 𝜏 t Case 2 ̂J 𝜙∗ Case 1 C Case 4 − + −P∗ T ̂− 𝜎 𝜎n 𝜎n n U −+ 𝜂 Case 0 Case 𝜙 5 Case 𝜎n n + −𝜏 Û (a) (b)

Figure 13: The general strength domain for a frictional, cohesive, compactant, and dilatant material is characterized by the friction angle 𝜙, ∗ ∗ the compaction angle 𝜙 , the cohesion 𝐶,thetensilestrength𝑇, and the maximum compressive strength 𝑃 (a). The support function 𝜋(̂𝐽) is defined in terms of the orientation of the velocity jump vector ̂𝐽,theangle𝜂 measured anticlockwise from the normal to the interface. The values of 𝜋(̂𝐽) are defined in (13) for the six orientation cases presented in(b). constructedbyagraphicalmethod[16],anditreadsforthe and 2𝜋 is also partitioned in five cases, completely symmetric general strength domain to the one illustrated here. This general strength domain summarizes all the proper- 𝜂=0, case 0: ties which could be of interest in this contribution but they are 𝜋(̂𝐽)=𝐽𝑇,̂ not all assigned to a given material. In fact, the bulk material is considered to be frictional, cohesive, and compactant so 𝜋 that its tensile strength is set by the Coulomb intersection 0<𝜂< −𝜙, 𝜎 𝐶 𝜙 case 1: 2 with the 𝑛 axis to cotan .Thedecollement´ is cohesive, frictional, and compactant but its compressive strength is cos (𝜂 + 𝜙) infinite leading to an unbounded strength domain. Cases 3 𝜋(̂𝐽)=𝐽[̂ 𝑇+𝐶sin (𝜙)] , cos (𝜙) to 5 in (A.1) are associated with an infinite support function. 𝜋 These three cases are of no interest and their angular ranges case 2: 𝜂= −𝜙, will be avoided. These differences between the bulk and the 2 decollement´ properties are illustrated in Figure 14. 𝜋(̂𝐽)=𝐽𝐶̂ cos (𝜙) , (A.1) 𝜋 𝜋 −𝜙<𝜂< +𝜙, B. Triangle Wedge case 3: 2 2 B.1. Mechanism Numbers 1 and 2. Collapse mechanism num- 𝜋(̂𝐽)=𝐽𝐶̂ cos (𝜂) 𝛿+sin (𝜂) 𝜏, ber 1 is the simplest and should be dominant for super-critical slopes if only friction prevails. It consists of the gliding of 𝜋 ∗ case 4: 𝜂= +𝜙 , the whole wedge on the decollement.´ This mechanism is now 2 used to illustrate the application of the maximum strength ∗ ∗ 𝜋(̂𝐽)=𝐽𝑃̂ sin (𝜙 ), theorem. Thevirtualvelocityofthewedgeisofunitnormand 𝜋 𝜙 +𝜙∗ <𝜂<𝜋, oriented at 𝐷 from the decollement´ in the orthonormal case 5: 2 basis illustrated in Figure 15. This vector orientation corre- ̂ ̂ ∗ sponds to case 2 of the support function associated with the 𝜋(𝐽)=𝐽𝑃 cos (𝜂) , decollement.´ The external forces include the gravity force (−𝜌𝑔𝑒2),where𝜌 and 𝑔 are the material density and the in which the two scalars introduced for case 3 are acceleration, respectively, as well as the tectonic force of 𝐶−𝑃∗ 𝜙∗ magnitude 𝑄 appliedonthebackwall𝐴𝐶,inadirection 𝛿= tan , tan 𝜙+tan 𝜙∗ parallel to the decollement.´ The external power is the power of these forces over the velocity field and reads (−𝜌𝑔 sin(𝜙𝐷 + ∗ ∗ (A.2) (𝐶 + 𝑃 𝜙) 𝜙 𝛽)𝑆𝐴𝐵𝐶 +𝑄cos(𝜙𝐷)),where𝑆𝐴𝐵𝐶 is the surface of the wedge. 𝜏= tan tan . tan 𝜙+tan 𝜙∗ This power will be compared to the maximum resisting power which is the sum of support function over the dissipative They correspond to the intersection of the Coulomb line and discontinuities, here the decollement.´ This power in case the compaction truncation line presented in Figure 13(a). The 2isthus𝐶𝐷 cos 𝜙𝐷𝐿𝐴𝐵. The maximum strength theorem support function in (A.1) is defined in five cases, correspond- stipulates that the external power is always smaller than the ing to different values or ranges of the orientation angle 𝜂,and maximum resisting power providing an upper bound to the presentedinFigure13(b).Notethattherangeof𝜂 between 𝜋 applied force presented in (1) in the main text. Journal of Earthquakes 17

|𝜏| |𝜏|

Bulk Décollement

𝜎n 𝜎n

(a) (b)

Figure 14: The bulk material is frictional, cohesive, and compactant. Its tensile strength is set by the Coulomb line (a). Thedecollement´ is frictional and dilatant. Its compressive strength is infinite (b).

C ̂ Û −UBS HW g F ̂J 𝛽 GF ̂ ̂ 𝜙 UHW UBS e D − + 2 𝜙GF E 𝜙GE e e 1 𝜃GF 𝜃 B 3 A GE 𝜃GF 𝜃 G GE (a) (b)

Figure 15: Collapse mechanism number 2 consists of a thrust fold with the partial activation of the decollement´ from the backwall to the common root of the two faults on the decollement.´

The technical difficulties start with mechanism number 2 The external work associated with this velocity field reads for which collapse occurs by a thrust fold which decomposes (𝐺𝐵𝐸) ̂ the wedge into three regions, the toe ,thehangingwall 𝑃ext (𝑈)=𝑄cos 𝜙𝐷 −𝜌𝑔(𝑆𝐵𝐶 sin (𝜙𝐷 +𝛽) (𝐺𝐹𝐸), and the back stop (𝐴𝐶𝐹𝐺), respectively (Figure 15(a)). (B.3) + 𝑈̂ 𝑆 (𝜙 +𝜃 +𝛽)) This mechanism has been studied by Cubas et al. [19] and HW HW sin 𝐺𝐸 𝐺𝐸 thederivationoftheupperboundisrepeatedhereforsake 𝑆 𝑆 of completeness. in which BS and HW arethesurfaceofthebackstopandof Thevirtualvelocityfieldisasfollows.Thebackstopregion the hanging wall, respectively. The maximum resisting power 𝑈̂ has the velocity BS which is a unit vector oriented at the correspondstothesumofthesupportfunctionoverevery angle 𝜙𝐷 from the decollement,´ as for mechanism number 1. discontinuity. The support function is always in case 2 (A.1), 𝑈̂ 𝜙 Thehangingwallvelocityis HW and oriented at 𝐺𝐸 from the basic assumption considered to build the virtual velocity the fault 𝐺𝐸.Thejumpoverthefault𝐺𝐹 is the difference field. This power reads ̂ ̂ ̂ 𝐽 = 𝑈 − 𝑈 , a vector which is oriented at 𝜙𝐺𝐹 with 𝐺𝐹 HW BS 𝑃 (𝑈̂)=𝐿 𝐶 𝜙 +𝐶 𝐿 𝐽̂ 𝜙 respect to the fault 𝐺𝐹. The norm of this jump and the norm mr 𝐴𝐺 𝐷 cos 𝐷 𝐺𝐹 𝐺𝐹 𝐺𝐹 cos 𝐺𝐹 of the hanging wall velocity are obtained by application of the (B.4) +𝐶 𝐿 𝑈̂ 𝜙 . law of sine to the hodogram presented in Figure 15(b): 𝐺𝐸 𝐺𝐸 HW cos 𝐺𝐸 The application of the maximum strength theorem for this 𝑃 ≤𝑃 𝑈̂ 𝐽̂ second mechanism which stipulates that ext mr for any HW = 𝐺𝐹 KA velocity field results in the upper bound in the applied (𝜙 +𝜙 +𝜃 ) (𝜙 +𝜃 −𝜙 ) sin 𝐷 𝐺𝐹 𝐺𝐹 sin 𝐺𝐸 𝐺𝐸 𝐷 forcewhichisprovidedinthemaintextas(2). (B.1) 1 = . sin (𝜃𝐺𝐹 +𝜃𝐺𝐸 +𝜙𝐺𝐹 +𝜙𝐺𝐸) B.2. Mechanism Number 3: Delamination of the Frontal Part of the Decollement.´ The wedge is partitioned into two regions for mechanism number 3. The frontal part 𝐺𝐹𝐵 is called the The validity of this construction relies on the following hanging wall and the rest is the back stop (Figure 16(a)). The constraints over the friction angles and dips: velocity field of the back stop is the same as for the two previous mechanisms. The hanging wall sustains a rotation −𝜔𝑒̂ 3 around point 𝐵 such that the velocity at any point 𝑥 −𝜔𝑒̂ ∧(𝑥−𝑥 ) 𝜙 +𝜙 +𝜃 <𝜋, positioned by the vector is 3 𝐵 .Thevelocity 𝐷 𝐺𝐹 𝐺𝐹 ̂ ̂ jump at point 𝐺 on the decollement´ is 𝐽𝐺 = 𝜔𝐿̂ 𝐺𝐵𝑒2 − 𝑈𝐺𝐹. 𝜙𝐺𝐸 +𝜃𝐺𝐸 −𝜙𝐷 <𝜋, (B.2) The hodogram of this velocity difference is presented in Figure 16(b) in which the jump is oriented according to case 𝜃 +𝜃 +𝜙 +𝜙 <𝜋. 𝐺𝐹 𝐺𝐸 𝐺𝐹 𝐺𝐸 2ofthesupportfunction.Forthevelocityofthebackstop 18 Journal of Earthquakes

̂ −UBS The maximum resisting power is the sum of the support 𝜉 nGF

̂J 2 function over the dissipative surfaces, the decollement,´ and e e Û 2 G the fault 𝐺𝐹. The dissipation over the section 𝐴𝐺 of the F HW GB + e1 𝐶 𝜙 𝐿

̂ ̂ U 𝜔L ´ 𝐷 cos 𝐷 𝐴𝐺 BS − e3 decollement is, as for the other mechanisms, = and the opening of section 𝐺𝐵 provides 𝜔𝑇̂ 𝐷1/2𝐿𝐺𝐵,sincethe 𝜃 𝜙GF + G ̂ GF B U velocity jump is oriented according to case 0 of the support 𝜃GF G function. The dissipation over the fault 𝐺𝐹 is the sum of the support function defined by the product (a) (b) 𝐶 𝜙 𝑛 ⋅(̂𝐽 +𝜉𝜔𝑛̂ ), Figure 16: The virtual velocity field for collapse mechanism number 𝐺𝐹 cotan 𝐺𝐹 𝐺 𝐺𝐹 (B.12) 3 and the hodogram of the velocity jump at point 𝐺. at any point along the 𝜉-axis. This sum is calculated noting that the velocity jump at 𝐺 is oriented according to case 2 being fully determined, the virtual angular velocity is found (𝑛 ⋅ ̂𝐽 = 𝜙 𝐽̂ ) 𝐺 cos 𝐺𝐹 𝐺 ,andthetotal,maximumresistingpower by application of the law of sines to this hodogram reads ̂ 𝜔𝐿̂ 𝐺𝐵 1 𝐽𝐺 𝑃 (𝑈̂) = = , (B.5) mr sin (𝜙𝐷 +𝜙𝐺𝐹 +𝜃𝐺𝐹) cos (𝜙𝐺𝐹 +𝜃𝐺𝐹) cos 𝜙𝐷 1 =𝐿 𝐶 𝜙 + 𝜔𝑇̂ 𝐿2 which provides also the magnitude of the velocity jump at 𝐴𝐺 𝐷 cos 𝐷 𝐷 2 𝐺𝐵 (B.13) point 𝐺.Thevelocityalongthefault𝐺𝐹 on the hanging wall + 1 𝑈̂ +𝜉𝑛 𝜉 +𝐶 ( 𝜙 𝐽̂ 𝐿 + 𝜔𝐿̂ 2 𝜙 ). side is expressed as 𝐺 𝐺𝐹 where the -axis, parallel to 𝐺𝐹 cos 𝐺𝐹 𝐺 𝐺𝐹 𝐺𝐹 2 cotan 𝐺𝐹 the fault, originates at point 𝐺 and 𝑛𝐺𝐹 is the normal vector ̂𝐽 ̂𝐽 + (Figure 16(a)). The jump over the fault 𝐺𝐹 is thus the sum 𝐺 Application of the maximum strength theorem then results in 𝜉𝑛𝐺𝐹, a vector which is oriented in the cone corresponding to the expression for the upper bound found in (4) in the main case 1 of the support function, regardless of the orientation text. 𝜃𝐺𝐹 of the fault. The power of the external work has three contributions, 𝑄 𝜙 B.3.MechanismNumber4:DelaminationwithaCurvedFault. the first being due to the tectonic force cos 𝐷 and the The main difficulty with this mechanism is the determination second to the work of gravity on the back stop sin(𝜙𝐷 + 𝐺𝐹 𝛽)𝜌𝑔𝑆 of the locus of the curved fault parameterized by the arc BS.Thethirdcontributionisthesumoftheproduct length 𝑠 with origin at point 𝐺 (Figure 17(a)). The boundary condition at point 𝐺 is that the curve is tangent to the −𝜌𝑔 (sin 𝛽𝑒1 + cos 𝛽𝑒2)⋅(−𝜔𝑒̂ 3)∧(𝑥−𝑥𝐵) (B.6) decollement so 𝜃𝐺𝐹(𝑠 = 0) =, 0 𝑡(𝑠 = 0) = 1−𝑒 ,and𝑛(𝑠) =2 𝑒 , over the hanging wall. This product could be rewritten in these quantities being defined at arbitrary 𝑠 in Figure 17(a). terms of the classical triple product The curve 𝐺𝐹 is constructed with the following second-order asymptotic development in increment Δ𝑠: 𝜌𝑔𝜔̂ {sin 𝛽 [𝑒1,𝑒3,𝑥−𝑥𝐵] + cos 𝛽 [𝑒2,𝑒3,𝑥−𝑥𝐵]} , (B.7) Δ𝑠2 𝑥 (𝑠+Δ𝑠) =𝑥(𝑠) +Δ𝑠𝑡(𝑠) + 𝜅 (𝑠) 𝑛 (𝑠) , (B.14) which is simplified to 2 𝜌𝑔𝜔{̂ 𝛽[𝑒 ,𝑒 ,𝑥 𝑒 ] sin 1 3 2 2 𝜅 (B.8) having introduced the Serret-Frenet relations and defining the curvature found from + cos 𝛽[𝑒2,𝑒3,(𝑥1 −𝑥1𝐵)𝑒1]} , 𝑑 𝑑 because of the trilinearity and the orthogonality properties. 𝜅 (𝑠) =𝑛(𝑠) + 𝑡 (𝑠) = 𝜃𝐺𝐹 (𝑠) . (B.15) The resulting scalar is 𝑑𝑠 𝑑𝑠 𝑀 𝐺𝐹 𝜌𝑔𝜔{−̂ sin 𝛽𝑥2 + cos 𝛽(𝑥1 −𝑥1𝐵)} (B.9) Consider now an arbitrary point on the fault of coordinates (𝑥1,𝑥2) and arc length 𝑠.Thevelocityontheplus + which has to be integrated over the hanging wall. The end side of point 𝑀, 𝑈̂ (𝑠) is perpendicular to the segment 𝑀𝐵 result is and the velocity jump reads 𝜌𝑔𝜔̂ [ 𝛽 (𝐿 𝑆 −𝐼 ) + 𝛽𝐼 ] , cos 𝐴𝐵 HW 1HW sin 2HW (B.10) ̂𝐽 (𝑠) =(𝜔𝑥̂ − 𝜙 )𝑒 𝐺𝐹 2 cos 𝐷 1 𝐼 𝐼 (B.16) where the moments 1HW and 2HW are defined in the main +(𝐿 −𝑥 − 𝜙 )𝑒 . text with (4). In summary, the total external power reads 𝐴𝐵 1 sin 𝐷 2 𝑃 (𝑈̂)=𝑄 𝜙 −𝜌𝑔(𝑆 (𝜙 +𝛽) ̂𝐽 ext cos 𝐷 𝐵𝐶 sin 𝐷 At the origin, this jump is the vector 𝐺 which is the ̂ (B.11) difference 𝜔𝐿̂ 𝐺𝐵𝑒 − 𝑈 . The hodogram of this velocity jump + 𝜔[̂ 𝛽(𝐿 𝑆 −𝐼 )+ 𝛽𝐼 ]) . 2 BS cos 𝐴𝐵 HW 1HW sin 1HW is presented in Figure 17(b) and the virtual rotation rate 𝜔̂ is Journal of Earthquakes 19

s Û e BS F + 2 Û ̂J HW GF − + e1 −Û BS 2 + e ̂ t ̂J UHW n G GB t ̂ ̂ 𝜔L UBS =

+ G M M G ̂ 𝜙GF U 𝜙GF 𝜃GF (s) B G 𝛾(s) 𝜃GF (s)

(a) (b)

Figure 17: The virtual velocity field for collapse mechanism number 4 and the hodograms of the velocity jump at point 𝐺 and any point 𝑀 on the curved fault.

chosen such that the velocity jump is oriented at 𝜙𝐺𝐹 of the betakenin(6)inthemaintext.Themaximumresisting fault, tangent to the decollement at point 𝐺. The law of sines, power is 1 𝑃 (𝑈̂)=𝐿 𝐶 𝜙 + 𝜔𝑇̂ 𝐿2 ̂ mr 𝐴𝐺 𝐷 cos 𝐷 𝐷 2 𝐺𝐵 𝜔𝐿̂ 𝐺𝐵 1 𝐽𝐺 = = , (B.17) (𝜙 +𝜙 ) 𝜙 𝜙 𝑠(𝐹) (B.20) sin 𝐷 𝐺𝐹 cos 𝐺𝐹 cos 𝐷 ̂ 󸀠 󸀠 +𝐶𝐺𝐹 ∫ 𝐽𝐺𝐹 (𝑠 )𝑑𝑠 𝑠(𝐺)=0 provides the value of this rotation rate as well as the magni- thesupportfunctionbeingincase2alongthefault𝐺𝐹 tudeofthevelocityjumpatpoint𝐺. by construction. The quadrature in (B.18) is estimated by The shape of the fault is determined from the same numerical means. The application of the maximum strength condition that the velocity jump is always at 𝜙𝐺𝐹 from its theorem results in the expression for the upper bound found tangent at any 𝑠, as illustrated in the left hodogram of in (6) in the main text. Figure 17(b). This condition reads B.4.MechanismNumbers5and6:CompactionandThrusting. 𝐿 −𝑥 − 𝜙 The velocity of the back stop is the same as for the other (𝜃 (𝑠) +𝜙 )= 𝐴𝐵 1 sin 𝐷 . mechanisms.Thevelocityofthehangingwallisoriented tan 𝐺𝐹 𝐺𝐹 𝜙 − 𝜔𝑥̂ (B.18) ∗ cos 𝐷 2 either at 𝜙𝐺𝐸 from the fault 𝐺𝐸 or at 𝜙𝐺𝐸 for the foreland and the hinterland vergence, mechanisms 5 and 6, respectively. The two corresponding hodograms are presented in Figure 18 Taking the derivative of (B.16) with respect to 𝑠 provides and the application of the law of sines provides the various the searched expression for the curvature which is finally velocity norms proposed in (10) in the main text. presented as One could note that the velocity jump over the com- pacting fault has been set to the orientation of case 4, corresponding to the boundary of the cone for compaction 𝜔̂ 𝜅 (𝑠) = 𝜙 . and shear. This choice is based on the experience that the ̂ cos 𝐺𝐹 (B.19) 𝐽𝐺𝐹 optimum velocity field is always observed on such bounding surfacesandnotwithintheconesinteriorandisnotfurther justified in this contribution. The expression for the external work is still given by (B.11) Theexternalworkassociatedwiththesetwovelocityfields 𝐼 𝐼 although the definition of the moments 1HW and 2HW should reads

𝑃 (𝑈̂)=𝑄 𝜙 −𝜌𝑔(𝑆 (𝜙 +𝛽)+𝑈̂ 𝑆 (𝜙 −𝜃∗ +𝛽)), CM 5: ext cos 𝐷 𝐵𝐶 sin 𝐷 HW HW sin 𝐺𝐸 𝐺𝐸 (B.21) 𝑃 (𝑈̂)=𝑄 𝜙 −𝜌𝑔(𝑆 (𝜙 +𝛽)+𝑈̂ 𝑆 (𝜙 +𝜃 +𝛽)). CM 6: ext cos 𝐷 𝐵𝐶 sin 𝐷 HW HW sin 𝐺𝐸 𝐺𝐸 20 Journal of Earthquakes

𝜙D 𝜙 −Û D BS Û ̂ HW −UBS ̂J GF 𝜙 ̂J GE ̂ GF UHW 𝜙∗ 𝜙GF GF ∗ 𝜙GE e2 𝜃 𝜃 GE 𝜃GF 𝜃GE e1 GF G G (a) (b)

Figure 18: The hodograms of the velocity jumps across the faults 𝐺𝐸 and 𝐺𝐹 for the compaction mechanism numbers 5 and 6 corresponding to hinterland and foreland vergence in (a) and (b), respectively.

The maximum resisting power is

𝑃 (𝑈̂) CM 5: mr ̂ =𝐿𝐴𝐺𝐶𝐷 cos 𝜙𝐷 +𝐶𝐺𝐹 cos 𝜙𝐺𝐹𝐽𝐺𝐹𝐿𝐺𝐹 1 + 𝑈̂ 𝜙∗ (𝐿 [𝑃∗0 +∇𝑃∗ ⋅(𝑥 −𝑥 )] − 𝐿2 ∇𝑃∗ ⋅𝑡 ), HW sin 𝐺𝐸 𝐺𝐸 𝐺𝐸 𝐺𝐸 𝐶 𝐺 2 𝐺𝐸 𝐺𝐸 𝐺𝐸 (B.22) 𝑃 (𝑈̂) CM 6: mr =𝐿 𝐶 𝜙 +𝐶 𝜙 𝑈̂ 𝐿 𝐴𝐺 𝐷 cos 𝐷 𝐺𝐸 cos 𝐺𝐸 HW 𝐺𝐸 1 + 𝐽̂ 𝜙∗ (𝐿 [𝑃∗0 +∇𝑃∗ ⋅(𝑥 −𝑥 )] − 𝐿2 ∇𝑃∗ ⋅𝑡 ), 𝐺𝐹 sin 𝐺𝐹 𝐺𝐹 𝐺𝐹 𝐺𝐹 𝐶 𝐺 2 𝐺𝐹 𝐺𝐹 𝐺𝐹

where 𝑡𝐺𝐹 and 𝑡𝐺𝐸 are the unit vectors tangent to the fault The maximum resisting power of mechanism number indicatedinsubscriptandorientedfrom𝐺 towards the free 3 is the sum of the support function over the dissipative surface. The expressions for the upper bounds are found in surfaces, the decollement,´ the fault 𝐺𝐹 and the ligament 𝑅𝐸. (8) in the main text. The sum of the support function over the dissipative surfaces, the decollement,´ and the fault 𝐺𝐹 has been illustrated in (B.5). C. Accretion Wedge The gradient compaction strength along the ligament reads ∗ 0∗ ∗ The velocity field for mechanism numbers 3 and 4 in the 𝑃 =𝑃 +𝐺 (𝐻 −2 𝑥 ), (C.2) 𝑅 accretion wedge relies on a rotation around the point on 𝐺∗ the decollement´ resulting in compaction along the ligament where standsforthegradientnorm,sothattheresisting 𝑅𝐸, Figure 19, and its position of point 𝑅 is a new unknown. power of the compaction on the ligament reads 𝐻 The velocity fields of other parts (back stop and hanging wall) ∗ ̂ ∗ 1 0∗ 2 1 ∗ 3 𝑃 𝐽𝑅𝐸 = ∫ 𝑃 𝜔𝑥̂ 2𝑑𝑥2 = 𝜔(̂ 𝑃𝑅𝐸𝐻 + 𝐺 𝐻 ). (C.3) in accretion wedge are similar to those in triangle wedge 0 2 6 which has been described in Appendices B.3 and B.4. The The total maximum resisting power is given hanging wall sustains a rotation −𝜔𝑒̂ 3 around point 𝑅 such that the velocity at ligament 𝑅𝐸 positioned by the vector 𝑥 𝑃 (𝑈̂) mr is −𝜔𝑒̂ 3 ∧𝑥2𝑒2. Since the very front of the accretion wedge is stable, the velocity jump at any point 𝐺 at ligament 𝑅𝐸 is 1 2 =𝐿𝐴𝐺𝐶𝐷 cos 𝜙𝐷 + 𝜔𝑇̂ 𝐷 𝐿𝐺𝐵 −𝜔𝑥̂ 2𝑒1,sothatthevelocityjumpatpoint𝐸 on the surface is 2 −𝜔𝐻𝑒̂ 1. 1 (C.4) The total external work of mechanism 3 reads +𝐶 ( 𝜙 𝐽̂ 𝐿 + 𝜔𝐿̂ 2 𝜙 ) 𝐺𝐹 cos 𝐺𝐹 𝐺 𝐺𝐹 𝐺𝐹 2 cotan 𝐺𝐹 ̂ 𝑃 (𝑈)=𝑄cos 𝜙𝐷 −𝜌𝑔(𝑆𝐵𝐶 sin (𝜙𝐷 +𝛽) ext 1 1 (C.1) ̂ ∗0 2 ∗ 3 + 𝜔[̂ 𝛽(𝐿 𝑆 −𝐼 )+ 𝛽𝐼 ]) . + 𝜔( 𝑃𝑅𝐸𝐻 + 𝐺 𝐻 ). cos 𝐴𝐵 HW 1HW sin 2HW 2 6 Journal of Earthquakes 21

of central Alberta,” Bulletin of Canadian Petroleum Geology,vol. Û g C F HW 33, no. 1, pp. 22–30, 1985. 𝛽 I E E [7] W. R. Jamison, “Mechanical stability of the triangle zone: the Û e2 E 0 backthrust wedge,” Journal of Geophysical Research, vol. 98, no. ̂ UBS H 11, pp. 20015–20030, 1993. e1 A R [8] S. G. Erickson, “Mechanics of triangle zones and passive- e G R ̂J = Û =−𝜔He 3 E E 1 roof duplexes: implications of finite-element models,” Tectono- physics,vol.245,no.1-2,pp.1–11,1995. Figure 19: The virtual velocity field for collapse mechanism number [9] B. A. Couzens and D. V. Wiltschko, “The control of mechanical 3 for accretion wedge and the hodogram of the velocity jump at point stratigraphy on the formation of triangle zones,” Bulletin of 𝐸 . Canadian Petroleum Geology,vol.44,no.2,pp.165–179,1996. [10] W. R. 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